From 40b0f5377a2996f5831eb308eeeb9ec6380af7b7 Mon Sep 17 00:00:00 2001 From: Juan Ignacio Polanco Date: Mon, 18 Nov 2024 12:03:40 +0000 Subject: [PATCH] build based on 34165d5 --- dev/.documenter-siteinfo.json | 1 + dev/GlobalFFTParams/index.html | 4 +- dev/PencilFFTs/index.html | 10 +- dev/PencilFFTs_timers/index.html | 4 +- dev/Transforms/index.html | 19 +- dev/assets/documenter.js | 927 +- dev/assets/themes/catppuccin-frappe.css | 1 + dev/assets/themes/catppuccin-latte.css | 1 + dev/assets/themes/catppuccin-macchiato.css | 1 + dev/assets/themes/catppuccin-mocha.css | 1 + dev/assets/themes/documenter-dark.css | 7699 +--------------- dev/assets/themes/documenter-light.css | 7733 +--------------- dev/assets/themeswap.js | 82 +- dev/assets/warner.js | 87 +- dev/benchmarks/index.html | 4 +- dev/generated/ffmpeg.err | 48 +- dev/generated/gradient/index.html | 4 +- dev/generated/in-place/index.html | 6 +- dev/generated/navier_stokes/34e86d4d.png | Bin 0 -> 218781 bytes 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v0.15.2/index.html create mode 100644 v0.15.2/objects.inv create mode 100644 v0.15.2/search_index.js create mode 100644 v0.15.2/siteinfo.js create mode 100644 v0.15.2/tutorial/index.html diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json new file mode 100644 index 00000000..78217f87 --- /dev/null +++ b/dev/.documenter-siteinfo.json @@ -0,0 +1 @@ +{"documenter":{"julia_version":"1.11.0","generation_timestamp":"2024-10-11T08:01:51","documenter_version":"1.7.0"}} \ No newline at end of file diff --git a/dev/GlobalFFTParams/index.html b/dev/GlobalFFTParams/index.html index efd6a708..0a3847df 100644 --- a/dev/GlobalFFTParams/index.html +++ b/dev/GlobalFFTParams/index.html @@ -1,9 +1,9 @@ -Global FFT parameters · PencilFFTs.jl
Edit on GitHub

Global FFT parameters

PencilFFTs.GlobalFFTParams — Type
GlobalFFTParams{T, N, inplace}

Specifies the global parameters for an N-dimensional distributed transform. These include the element type T and global data sizes of input and output data, as well as the transform types to be performed along each dimension.

GlobalFFTParams(size_global, transforms, [real_type=Float64])

Define parameters for N-dimensional transform.

transforms must be a tuple of length N specifying the transforms to be applied along each dimension. Each element must be a subtype of Transforms.AbstractTransform. For all the possible transforms, see Transform types.

The element type must be a real type accepted by FFTW, i.e. either Float32 or Float64.

Note that the transforms are applied one dimension at a time, with the leftmost dimension first for forward transforms.

Example

To perform a 3D FFT of real data, first a real-to-complex FFT must be applied along the first dimension, followed by two complex-to-complex FFTs along the other dimensions:

julia> size_global = (64, 32, 128);  # size of real input data
+Global FFT parameters · PencilFFTs.jl
GitHub
Global FFT parameters
PencilFFTs.GlobalFFTParams — Type
GlobalFFTParams{T, N, inplace}
Specifies the global parameters for an N-dimensional distributed transform. These include the element type T and global data sizes of input and output data, as well as the transform types to be performed along each dimension.
GlobalFFTParams(size_global, transforms, [real_type=Float64])
Define parameters for N-dimensional transform.
transforms must be a tuple of length N specifying the transforms to be applied along each dimension. Each element must be a subtype of Transforms.AbstractTransform. For all the possible transforms, see Transform types.
The element type must be a real type accepted by FFTW, i.e. either Float32 or Float64.
Note that the transforms are applied one dimension at a time, with the leftmost dimension first for forward transforms.
Example
To perform a 3D FFT of real data, first a real-to-complex FFT must be applied along the first dimension, followed by two complex-to-complex FFTs along the other dimensions:
julia> size_global = (64, 32, 128);  # size of real input data
 
 julia> transforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT());
 
 julia> fft_params = PencilFFTs.GlobalFFTParams(size_global, transforms)
 Transforms: (RFFT, FFT, FFT)
 Input type: Float64
-Global dimensions: (64, 32, 128) -> (33, 32, 128)
source

+Global dimensions: (64, 32, 128) -> (33, 32, 128)
source
diff --git a/dev/PencilFFTs/index.html b/dev/PencilFFTs/index.html index b812e561..8f0fbc85 100644 --- a/dev/PencilFFTs/index.html +++ b/dev/PencilFFTs/index.html @@ -1,5 +1,5 @@ -Distributed FFT plans · PencilFFTs.jl
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Distributed FFT plans

Distributed FFTs are implemented in the PencilFFTs module, and are built on top of the PencilArrays package.

Creating plans

PencilFFTs.PencilFFTPlan — Type
PencilFFTPlan{T,N} <: AbstractFFTs.Plan{T}

Plan for N-dimensional FFT-based transform on MPI-distributed data, where input data has type T.

PencilFFTPlan(p::Pencil, transforms; kwargs...)

Create a PencilFFTPlan for distributed arrays following a given Pencil configuration. See variant below for details on the specification of transforms and on possible keyword arguments.

PencilFFTPlan(
+Distributed FFT plans · PencilFFTs.jl
GitHub
Distributed FFT plans
Distributed FFTs are implemented in the PencilFFTs module, and are built on top of the PencilArrays package.
Creating plans
PencilFFTs.PencilFFTPlan — Type
PencilFFTPlan{T,N} <: AbstractFFTs.Plan{T}
Plan for N-dimensional FFT-based transform on MPI-distributed data, where input data has type T.
PencilFFTPlan(p::Pencil, transforms; kwargs...)
Create a PencilFFTPlan for distributed arrays following a given Pencil configuration. See variant below for details on the specification of transforms and on possible keyword arguments.
PencilFFTPlan(
     A::PencilArray, transforms;
     fftw_flags = FFTW.ESTIMATE,
     fftw_timelimit = FFTW.NO_TIMELIMIT,
@@ -19,13 +19,13 @@
 proc_dims = (4, 2)  # 2D decomposition
 comm = MPI.COMM_WORLD
 
-plan = PencilFFTPlan(size_global, transforms, proc_dims, comm)
source
Allocating data
PencilFFTs.allocate_input — Function
allocate_input(p::PencilFFTPlan)          -> PencilArray
+plan = PencilFFTPlan(size_global, transforms, proc_dims, comm)
source
Allocating data
PencilFFTs.allocate_input — Function
allocate_input(p::PencilFFTPlan)          -> PencilArray
 allocate_input(p::PencilFFTPlan, dims...) -> Array{PencilArray}
-allocate_input(p::PencilFFTPlan, Val(N))  -> NTuple{N, PencilArray}
Allocate uninitialised PencilArray that can hold input data for the given plan.
The second and third forms respectively allocate an array of PencilArrays of size dims, and a tuple of N PencilArrays.
In-place plans
If p is an in-place plan, a ManyPencilArray is allocated. This type holds PencilArray wrappers for the input and output transforms (as well as for intermediate transforms) which share the same space in memory. The input and output PencilArrays should be respectively accessed by calling first(::ManyPencilArray) and last(::ManyPencilArray).
Example
Suppose p is an in-place PencilFFTPlan. Then,
@assert is_inplace(p)
+allocate_input(p::PencilFFTPlan, Val(N))  -> NTuple{N, PencilArray}
Allocate uninitialised PencilArray that can hold input data for the given plan.
The second and third forms respectively allocate an array of PencilArrays of size dims, and a tuple of N PencilArrays.
In-place plans
If p is an in-place real-to-real or complex-to-complex plan, a ManyPencilArray is allocated. If p is an in-place real-to-complex plan, a ManyPencilArrayRFFT! is allocated. 
These types hold PencilArray wrappers for the input and output transforms (as well as for intermediate transforms) which share the same space in memory. The input and output PencilArrays should be respectively accessed by calling first(::ManyPencilArray) and last(::ManyPencilArray).
Example
Suppose p is an in-place PencilFFTPlan. Then,
@assert is_inplace(p)
 A = allocate_input(p) :: ManyPencilArray
 v_in = first(A)       :: PencilArray  # input data view
 v_out = last(A)       :: PencilArray  # output data view
Also note that in-place plans must be performed directly on the returned ManyPencilArray, and not on the contained PencilArray views:
p * A       # perform forward transform in-place
 p \ A       # perform backward transform in-place
-# p * v_in  # not allowed!!
source
source
PencilFFTs.allocate_output — Function
allocate_output(p::PencilFFTPlan)          -> PencilArray
 allocate_output(p::PencilFFTPlan, dims...) -> Array{PencilArray}
-allocate_output(p::PencilFFTPlan, Val(N))  -> NTuple{N, PencilArray}

Allocate uninitialised PencilArray that can hold output data for the given plan.

If p is an in-place plan, a ManyPencilArray is allocated.

See allocate_input for details.

source

Methods

+allocate_output(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}

Allocate uninitialised PencilArray that can hold output data for the given plan.

If p is an in-place plan, a ManyPencilArray is allocated.

See allocate_input for details.

source

Methods

PencilArrays.Pencils.MPITopologies.get_comm — Method
get_comm(p::PencilFFTPlan)

Get MPI communicator associated to a PencilFFTPlan.

source
PencilFFTs.Transforms.scale_factor — Method
scale_factor(p::PencilFFTPlan)

Get scale factor associated to a PencilFFTPlan.

source
PencilArrays.Pencils.timer — Method
timer(p::PencilFFTPlan)

Get TimerOutput attached to a PencilFFTPlan.

See Measuring performance for details.

source
PencilFFTs.Transforms.is_inplace — Method
Transforms.is_inplace(p::PencilFFTPlan)

Returns true if the given plan operates in-place on the input data, false otherwise.

source

Internals

PencilFFTs.ManyPencilArrayRFFT! — Type
ManyPencilArrayRFFT!{T,N,M} <: AbstractManyPencilArray{N,M}

Container holding M different PencilArray views to the same underlying data buffer. All views share the same dimensionality N. The element type T of the first view is real, that of subsequent views is Complex{T}.

This can be used to perform in-place real-to-complex plan, see alsoTransforms.RFFT!. It is used internally for such transforms by allocate_input and should not be constructed directly.

ManyPencilArrayRFFT!{T}(undef, pencils...; extra_dims=())

Create a ManyPencilArrayRFFT! container that can hold data of type T and Complex{T} associated to all the given Pencils.

The optional extra_dims argument is the same as for PencilArray.

See also ManyPencilArray

source
diff --git a/dev/PencilFFTs_timers/index.html b/dev/PencilFFTs_timers/index.html index 260dfe04..3a0be9a4 100644 --- a/dev/PencilFFTs_timers/index.html +++ b/dev/PencilFFTs_timers/index.html @@ -1,5 +1,5 @@ -Measuring performance · PencilFFTs.jl
Edit on GitHub

Measuring performance

It is possible to measure the time spent in different sections of the distributed transforms using the TimerOutputs package. This has a (very small) performance overhead, so it is disabled by default. To enable time measurements, call TimerOutputs.enable_debug_timings after loading PencilFFTs (see below for an example). For more details see the TimerOutputs docs.

Minimal example:

using MPI
+Measuring performance · PencilFFTs.jl
GitHub
Measuring performance
It is possible to measure the time spent in different sections of the distributed transforms using the TimerOutputs package. This has a (very small) performance overhead, so it is disabled by default. To enable time measurements, call TimerOutputs.enable_debug_timings after loading PencilFFTs (see below for an example). For more details see the TimerOutputs docs.
Minimal example:
using MPI
 using PencilFFTs
 using TimerOutputs
 
@@ -21,4 +21,4 @@
 
 # [do stuff with `plan`...]
 
-print_timer(to)

+print_timer(to)
diff --git a/dev/Transforms/index.html b/dev/Transforms/index.html index 3445cf53..e1cf66cc 100644 --- a/dev/Transforms/index.html +++ b/dev/Transforms/index.html @@ -1,11 +1,12 @@ -Available transforms · PencilFFTs.jl
Edit on GitHub

Available transforms

Transform types

PencilFFTs.Transforms.BFFT — Type
BFFT()

Unnormalised backward complex-to-complex FFT.

Like AbstractFFTs.bfft, this transform is not normalised. To obtain the inverse transform, divide the output by the length of the transformed dimension.

See also AbstractFFTs.bfft.

source
PencilFFTs.Transforms.BRFFT — Type
BRFFT(d::Integer)
-BRFFT((d1, d2, ..., dN))

Unnormalised inverse of RFFT.

To obtain the inverse transform, divide the output by the length of the transformed dimension (of the real output array).

As described in the AbstractFFTs docs, the length of the output cannot be fully inferred from the input length. For this reason, the BRFFT constructor accepts an optional d argument indicating the output length.

For multidimensional datasets, a tuple of dimensions (d1, d2, ..., dN) may also be passed. This is equivalent to passing just dN. In this case, the last dimension (dN) is the one that changes size between the input and output. Note that this is the opposite of FFTW.brfft. The reason is that, in PencilFFTs, the last dimension is the one along which a complex-to-real transform is performed.

See also AbstractFFTs.brfft.

source

Internals

What follows is used internally in PencilFFTs.

Types

PencilFFTs.Transforms.AbstractCustomPlan — Type
AbstractCustomPlan

Abstract type defining a custom plan, to be used as an alternative to FFTW plans (FFTW.FFTWPlan).

The only custom plan defined in this module is IdentityPlan. The user can define other custom plans that are also subtypes of AbstractCustomPlan.

Note that plan returns a subtype of either AbstractFFTs.Plan or AbstractCustomPlan.

source

Functions

PencilFFTs.Transforms.plan — Function
plan(transform::AbstractTransform, A, [dims];
-     flags=FFTW.ESTIMATE, timelimit=Inf)

Create plan to transform array A along dimensions dims.

If dims is not specified, all dimensions of A are transformed.

For FFT plans, this function wraps the AbstractFFTs.jl and FFTW.jl plan creation functions. For more details on the function arguments, see AbstractFFTs.plan_fft.

source
PencilFFTs.Transforms.binv — Function
binv(transform::AbstractTransform, d::Integer)

Returns the backwards transform associated to the given transform.

The second argument must be the length of the first transformed dimension in the forward transform. It is used in particular when transform = RFFT(), to determine the length of the inverse (complex-to-real) transform. See the AbstractFFTs.irfft docs for details.

The backwards transform returned by this function is not normalised. The normalisation factor for a given array can be obtained by calling scale_factor.

Example

julia> binv(Transforms.FFT(), 42)
+Available transforms · PencilFFTs.jl
GitHub
Available transforms
Transform types
PencilFFTs.Transforms.BFFT — Type
BFFT()
Unnormalised backward complex-to-complex FFT.
Like AbstractFFTs.bfft, this transform is not normalised. To obtain the inverse transform, divide the output by the length of the transformed dimension.
See also AbstractFFTs.bfft.
source
PencilFFTs.Transforms.BRFFT — Type
BRFFT(d::Integer)
+BRFFT((d1, d2, ..., dN))
Unnormalised inverse of RFFT.
To obtain the inverse transform, divide the output by the length of the transformed dimension (of the real output array).
As described in the AbstractFFTs docs, the length of the output cannot be fully inferred from the input length. For this reason, the BRFFT constructor accepts an optional d argument indicating the output length.
For multidimensional datasets, a tuple of dimensions (d1, d2, ..., dN) may also be passed. This is equivalent to passing just dN. In this case, the last dimension (dN) is the one that changes size between the input and output. Note that this is the opposite of FFTW.brfft. The reason is that, in PencilFFTs, the last dimension is the one along which a complex-to-real transform is performed.
See also AbstractFFTs.brfft.
source
Internals
What follows is used internally in PencilFFTs.
Types
PencilFFTs.Transforms.AbstractCustomPlan — Type
AbstractCustomPlan
Abstract type defining a custom plan, to be used as an alternative to FFTW plans (FFTW.FFTWPlan).
The only custom plan defined in this module is IdentityPlan. The user can define other custom plans that are also subtypes of AbstractCustomPlan.
Note that plan returns a subtype of either AbstractFFTs.Plan or AbstractCustomPlan.
source
Functions
PencilFFTs.Transforms.plan — Function
plan(transform::AbstractTransform, A, [dims];
+     flags=FFTW.ESTIMATE, timelimit=Inf)
Create plan to transform array A along dimensions dims.
If dims is not specified, all dimensions of A are transformed.
For FFT plans, this function wraps the AbstractFFTs.jl and FFTW.jl plan creation functions. For more details on the function arguments, see AbstractFFTs.plan_fft.
source
PencilFFTs.Transforms.binv — Function
binv(transform::AbstractTransform, d::Integer)
Returns the backwards transform associated to the given transform.
The second argument must be the length of the first transformed dimension in the forward transform. It is used in particular when transform = RFFT(), to determine the length of the inverse (complex-to-real) transform. See the AbstractFFTs.irfft docs for details.
The backwards transform returned by this function is not normalised. The normalisation factor for a given array can be obtained by calling scale_factor.
Example
julia> binv(Transforms.FFT(), 42)
 BFFT
 
 julia> binv(Transforms.BRFFT(9), 42)
-RFFT
source
PencilFFTs.Transforms.scale_factor — Function
scale_factor(transform::AbstractTransform, A, [dims = 1:ndims(A)])
Get factor required to normalise the given array after a transformation along dimensions dims (all dimensions by default).
The array A must have the dimensions of the transform input.
Important: the dimensions dims must be the same that were passed to plan.
Examples
julia> C = zeros(ComplexF32, 3, 4, 5);
+RFFT
source
PencilFFTs.Transforms.scale_factor — Function
scale_factor(transform::AbstractTransform, A, [dims = 1:ndims(A)])
Get factor required to normalise the given array after a transformation along dimensions dims (all dimensions by default).
The array A must have the dimensions of the transform input.
Important: the dimensions dims must be the same that were passed to plan.
Examples
julia> C = zeros(ComplexF32, 3, 4, 5);
 
 julia> scale_factor(Transforms.FFT(), C)
 60
@@ -29,7 +30,7 @@
 
 julia> scale_factor(Transforms.BRFFT(9), C)
 108
This will fail because the input of RFFT is real, and R is a complex array:
julia> scale_factor(Transforms.RFFT(), C, 2:3)
-ERROR: MethodError: no method matching scale_factor(::PencilFFTs.Transforms.RFFT, ::Array{ComplexF32, 3}, ::UnitRange{Int64})
source
PencilFFTs.Transforms.eltype_input — Function
eltype_input(transform::AbstractTransform, real_type<:AbstractFloat)
Determine input data type for a given transform given the floating point precision of the input data.
Some transforms, such as R2R and NoTransform, can take both real and complex data. For those kinds of transforms, nothing is returned.
Example
julia> eltype_input(Transforms.FFT(), Float32)
+ERROR: MethodError: no method matching scale_factor(::PencilFFTs.Transforms.RFFT, ::Array{ComplexF32, 3}, ::UnitRange{Int64})
source
PencilFFTs.Transforms.eltype_input — Function
eltype_input(transform::AbstractTransform, real_type<:AbstractFloat)
Determine input data type for a given transform given the floating point precision of the input data.
Some transforms, such as R2R and NoTransform, can take both real and complex data. For those kinds of transforms, nothing is returned.
Example
julia> eltype_input(Transforms.FFT(), Float32)
 ComplexF32 (alias for Complex{Float32})
 
 julia> eltype_input(Transforms.RFFT(), Float64)
@@ -38,7 +39,7 @@
 julia> eltype_input(Transforms.R2R(FFTW.REDFT01), Float64)  # nothing
 
 julia> eltype_input(Transforms.NoTransform(), Float64)  # nothing
-
source
PencilFFTs.Transforms.eltype_output — Function
eltype_output(transform::AbstractTransform, eltype_input)
Returns the output data type for a given transform given the input type.
Throws ArgumentError if the input data type is incompatible with the transform type.
Example
julia> eltype_output(Transforms.NoTransform(), Float32)
+
source
PencilFFTs.Transforms.eltype_output — Function
eltype_output(transform::AbstractTransform, eltype_input)
Returns the output data type for a given transform given the input type.
Throws ArgumentError if the input data type is incompatible with the transform type.
Example
julia> eltype_output(Transforms.NoTransform(), Float32)
 Float32
 
 julia> eltype_output(Transforms.RFFT(), Float64)
@@ -48,7 +49,7 @@
 Float32
 
 julia> eltype_output(Transforms.FFT(), Float64)
-ERROR: ArgumentError: invalid input data type for PencilFFTs.Transforms.FFT: Float64
source
PencilFFTs.Transforms.expand_dims — Function
expand_dims(transform::AbstractTransform, Val(N))
Expand a single multidimensional transform into one transform per dimension.
Example
# Expand a real-to-complex transform in 3 dimensions.
+ERROR: ArgumentError: invalid input data type for PencilFFTs.Transforms.FFT: Float64
source
PencilFFTs.Transforms.expand_dims — Function
expand_dims(transform::AbstractTransform, Val(N))
Expand a single multidimensional transform into one transform per dimension.
Example
# Expand a real-to-complex transform in 3 dimensions.
 julia> expand_dims(Transforms.RFFT(), Val(3))
 (RFFT, FFT, FFT)
 
@@ -56,7 +57,7 @@
 (BFFT, BFFT, BRFFT{even})
 
 julia> expand_dims(Transforms.NoTransform(), Val(2))
-(NoTransform, NoTransform)
source
PencilFFTs.Transforms.is_inplace — Function
is_inplace(transform::AbstractTransform)         -> Bool
 is_inplace(transforms::Vararg{AbtractTransform}) -> Union{Bool, Nothing}
Check whether a transform or a list of transforms is performed in-place.
If the list of transforms has a combination of in-place and out-of-place transforms, nothing is returned.
Example
julia> is_inplace(Transforms.RFFT())
 false
 
@@ -71,4 +72,4 @@
 
 julia> is_inplace(Transforms.FFT(), Transforms.R2R!(FFTW.REDFT01)) === nothing
 true
-
source
PencilFFTs.Transforms.length_output — Function
length_output(transform::AbstractTransform, length_in::Integer)
Returns the length of the transform output, given the length of its input.
The input and output lengths are specified in terms of the respective input and output datatypes. For instance, for real-to-complex transforms, these are respectively the length of input real data and of output complex data.
source

+
source
PencilFFTs.Transforms.length_output — Function
length_output(transform::AbstractTransform, length_in::Integer)

Returns the length of the transform output, given the length of its input.

The input and output lengths are specified in terms of the respective input and output datatypes. For instance, for real-to-complex transforms, these are respectively the length of input real data and of output complex data.

source
diff --git a/dev/assets/documenter.js b/dev/assets/documenter.js index 6adfbbbf..82252a11 100644 --- a/dev/assets/documenter.js +++ b/dev/assets/documenter.js @@ -1,15 +1,15 @@ // Generated by Documenter.jl requirejs.config({ paths: { - 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/languages/julia.min', + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min', 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', - 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.12.1/jquery-ui.min', - 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/contrib/auto-render.min', - 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.6.0/jquery.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min', 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', - 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min', - 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/highlight.min', - 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/languages/julia-repl.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min', }, shim: { "highlight-julia": { @@ -70,13 +70,96 @@ $(document).ready(function() { hljs.highlightAll(); }) +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let timer = 0; +var isExpanded = true; + +$(document).on( + "click", + ".docstring .docstring-article-toggle-button", + function () { + let articleToggleTitle = "Expand docstring"; + const parent = $(this).parent(); + + debounce(() => { + if (parent.siblings("section").is(":visible")) { + parent + .find("a.docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + } else { + parent + .find("a.docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + articleToggleTitle = "Collapse docstring"; + } + + parent + .children(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + parent.siblings("section").slideToggle(); + }); + } +); + +$(document).on("click", ".docs-article-toggle-button", function (event) { + let articleToggleTitle = "Expand docstring"; + let navArticleToggleTitle = "Expand all docstrings"; + let animationSpeed = event.noToggleAnimation ? 0 : 400; + + debounce(() => { + if (isExpanded) { + $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); + $("a.docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + + isExpanded = false; + + $(".docstring section").slideUp(animationSpeed); + } else { + $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); + $("a.docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + isExpanded = true; + articleToggleTitle = "Collapse docstring"; + navArticleToggleTitle = "Collapse all docstrings"; + + $(".docstring section").slideDown(animationSpeed); + } + + $(this).prop("title", navArticleToggleTitle); + $(".docstring-article-toggle-button").prop("title", articleToggleTitle); + }); +}); + +function debounce(callback, timeout = 300) { + if (Date.now() - timer > timeout) { + callback(); + } + + clearTimeout(timer); + + timer = Date.now(); +} + }) //////////////////////////////////////////////////////////////////////////////// require([], function() { function addCopyButtonCallbacks() { for (const el of document.getElementsByTagName("pre")) { const button = document.createElement("button"); - button.classList.add("copy-button", "fas", "fa-copy"); + button.classList.add("copy-button", "fa-solid", "fa-copy"); + button.setAttribute("aria-label", "Copy this code block"); + button.setAttribute("title", "Copy"); + el.appendChild(button); const success = function () { @@ -85,7 +168,7 @@ function addCopyButtonCallbacks() { }; const failure = function () { - button.classList.add("error", "fa-times"); + button.classList.add("error", "fa-xmark"); button.classList.remove("fa-copy"); }; @@ -94,7 +177,7 @@ function addCopyButtonCallbacks() { setTimeout(function () { button.classList.add("fa-copy"); - button.classList.remove("success", "fa-check", "fa-times"); + button.classList.remove("success", "fa-check", "fa-xmark"); }, 5000); }); } @@ -138,29 +221,584 @@ require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { // Manages the top navigation bar (hides it when the user starts scrolling down on the // mobile). window.Headroom = Headroom; // work around buggy module loading? -$(document).ready(function() { - $('#documenter .docs-navbar').headroom({ - "tolerance": {"up": 10, "down": 10}, +$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let meta = $("div[data-docstringscollapsed]").data(); + + if (meta?.docstringscollapsed) { + $("#documenter-article-toggle-button").trigger({ + type: "click", + noToggleAnimation: true, + }); + } +}); + }) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +/* +To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +PSEUDOCODE: + +Searching happens automatically as the user types or adjusts the selected filters. +To preserve responsiveness, as much as possible of the slow parts of the search are done +in a web worker. Searching and result generation are done in the worker, and filtering and +DOM updates are done in the main thread. The filters are in the main thread as they should +be very quick to apply. This lets filters be changed without re-searching with minisearch +(which is possible even if filtering is on the worker thread) and also lets filters be +changed _while_ the worker is searching and without message passing (neither of which are +possible if filtering is on the worker thread) + +SEARCH WORKER: + +Import minisearch + +Build index + +On message from main thread + run search + find the first 200 unique results from each category, and compute their divs for display + note that this is necessary and sufficient information for the main thread to find the + first 200 unique results from any given filter set + post results to main thread + +MAIN: + +Launch worker + +Declare nonconstant globals (worker_is_running, last_search_text, unfiltered_results) + +On text update + if worker is not running, launch_search() + +launch_search + set worker_is_running to true, set last_search_text to the search text + post the search query to worker + +on message from worker + if last_search_text is not the same as the text in the search field, + the latest search result is not reflective of the latest search query, so update again + launch_search() + otherwise + set worker_is_running to false + + regardless, display the new search results to the user + save the unfiltered_results as a global + update_search() + +on filter click + adjust the filter selection + update_search() + +update_search + apply search filters by looping through the unfiltered_results and finding the first 200 + unique results that match the filters + + Update the DOM +*/ + +/////// SEARCH WORKER /////// + +function worker_function(documenterSearchIndex, documenterBaseURL, filters) { + importScripts( + "https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min.js" + ); + + let data = documenterSearchIndex.map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; + }); + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", + ]); + + let index = new MiniSearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + + word = word.toLowerCase(); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not + // find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + prefix: true, + boost: { title: 100 }, + fuzzy: 2, + }, + }); + + index.addAll(data); + + /** + * Used to map characters to HTML entities. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const htmlEscapes = { + "&": "&", + "<": "<", + ">": ">", + '"': """, + "'": "'", + }; + + /** + * Used to match HTML entities and HTML characters. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const reUnescapedHtml = /[&<>"']/g; + const reHasUnescapedHtml = RegExp(reUnescapedHtml.source); + + /** + * Escape function from lodash + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + function escape(string) { + return string && reHasUnescapedHtml.test(string) + ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr]) + : string || ""; + } + + /** + * RegX escape function from MDN + * Refer: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions#escaping + */ + function escapeRegExp(string) { + return string.replace(/[.*+?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string + } + + /** + * Make the result component given a minisearch result data object and the value + * of the search input as queryString. To view the result object structure, refer: + * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ + function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + searchstring = escapeRegExp(querystring); + let textindex = new RegExp(`${searchstring}`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + text = text.length ? escape(text) : ""; + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`${escape(searchstring)}`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
+
${escape(result.title)}
+
${result.category}
+
+

+ ${display_result} +

+
+ ${display_link} +
+ + ${search_divider} + `; + + return result_div; + } + + self.onmessage = function (e) { + let query = e.data; + let results = index.search(query, { + filter: (result) => { + // Only return relevant results + return result.score >= 1; + }, + combineWith: "AND", + }); + + // Pre-filter to deduplicate and limit to 200 per category to the extent + // possible without knowing what the filters are. + let filtered_results = []; + let counts = {}; + for (let filter of filters) { + counts[filter] = 0; + } + let present = {}; + + for (let result of results) { + cat = result.category; + cnt = counts[cat]; + if (cnt < 200) { + id = cat + "---" + result.location; + if (present[id]) { + continue; + } + present[id] = true; + filtered_results.push({ + location: result.location, + category: cat, + div: make_search_result(result, query), + }); + } + } + + postMessage(filtered_results); + }; +} + +// `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript! +const filters = [ + ...new Set(documenterSearchIndex["docs"].map((x) => x.category)), +]; +const worker_str = + "(" + + worker_function.toString() + + ")(" + + JSON.stringify(documenterSearchIndex["docs"]) + + "," + + JSON.stringify(documenterBaseURL) + + "," + + JSON.stringify(filters) + + ")"; +const worker_blob = new Blob([worker_str], { type: "text/javascript" }); +const worker = new Worker(URL.createObjectURL(worker_blob)); + +/////// SEARCH MAIN /////// + +// Whether the worker is currently handling a search. This is a boolean +// as the worker only ever handles 1 or 0 searches at a time. +var worker_is_running = false; + +// The last search text that was sent to the worker. This is used to determine +// if the worker should be launched again when it reports back results. +var last_search_text = ""; + +// The results of the last search. This, in combination with the state of the filters +// in the DOM, is used compute the results to display on calls to update_search. +var unfiltered_results = []; + +// Which filter is currently selected +var selected_filter = ""; + +$(document).on("input", ".documenter-search-input", function (event) { + if (!worker_is_running) { + launch_search(); + } +}); + +function launch_search() { + worker_is_running = true; + last_search_text = $(".documenter-search-input").val(); + worker.postMessage(last_search_text); +} + +worker.onmessage = function (e) { + if (last_search_text !== $(".documenter-search-input").val()) { + launch_search(); + } else { + worker_is_running = false; + } + + unfiltered_results = e.data; + update_search(); +}; + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + selected_filter = ""; + } else { + selected_filter = $(this).text().toLowerCase(); + } + + // This updates search results and toggles classes for UI: + update_search(); +}); + +/** + * Make/Update the search component + */ +function update_search() { + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + if (selected_filter == "") { + results = unfiltered_results; + } else { + results = unfiltered_results.filter((result) => { + return selected_filter == result.category.toLowerCase(); + }); + } + + let search_result_container = ``; + let modal_filters = make_modal_body_filters(); + let search_divider = `
`; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + for (var i = 0, n = results.length; i < n && count < 200; ++i) { + let result = results[i]; + if (result.location && !links.includes(result.location)) { + search_results += result.div; + count++; + links.push(result.location); + } + } + + if (count == 1) { + count_str = "1 result"; + } else if (count == 200) { + count_str = "200+ results"; + } else { + count_str = count + " results"; + } + let result_count = `
${count_str}
`; + + search_result_container = ` +
+ ${modal_filters} + ${search_divider} + ${result_count} +
+ ${search_results} +
+
+ `; + } else { + search_result_container = ` +
+ ${modal_filters} + ${search_divider} +
0 result(s)
+
+
No result found!
+ `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(` +
Type something to get started!
+ `); + } +} + +/** + * Make the modal filter html + * + * @returns string + */ +function make_modal_body_filters() { + let str = filters + .map((val) => { + if (selected_filter == val.toLowerCase()) { + return `${val}`; + } else { + return `${val}`; + } + }) + .join(""); + + return ` +
+ Filters: + ${str} +
`; +} }) //////////////////////////////////////////////////////////////////////////////// require(['jquery'], function($) { // Modal settings dialog -$(document).ready(function() { - var settings = $('#documenter-settings'); - $('#documenter-settings-button').click(function(){ - settings.toggleClass('is-active'); +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); }); // Close the dialog if X is clicked - $('#documenter-settings button.delete').click(function(){ - settings.removeClass('is-active'); + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); }); // Close dialog if ESC is pressed - $(document).keyup(function(e) { - if (e.keyCode == 27) settings.removeClass('is-active'); + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); }); }); @@ -168,151 +806,246 @@ $(document).ready(function() { //////////////////////////////////////////////////////////////////////////////// require(['jquery'], function($) { +$(document).ready(function () { + let search_modal_header = ` +
+
+

+ + + + +

+
+
+ +
+
+ `; + + let initial_search_body = ` +
Type something to get started!
+ `; + + let search_modal_footer = ` + + `; + + $(document.body).append( + ` + + ` + ); + + document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); + }); + + document + .querySelector(".close-search-modal") + .addEventListener("click", () => { + closeModal(); + }); + + $(document).on("click", ".search-result-link", function () { + closeModal(); + }); + + document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; + }); + + // Functions to open and close a modal + function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); + } + + function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
Type something to get started!
+ `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); + } + + document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + // Manages the showing and hiding of the sidebar. -$(document).ready(function() { +$(document).ready(function () { var sidebar = $("#documenter > .docs-sidebar"); - var sidebar_button = $("#documenter-sidebar-button") - sidebar_button.click(function(ev) { + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { ev.preventDefault(); - sidebar.toggleClass('visible'); - if (sidebar.hasClass('visible')) { + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { // Makes sure that the current menu item is visible in the sidebar. $("#documenter .docs-menu a.is-active").focus(); } }); - $("#documenter > .docs-main").bind('click', function(ev) { + $("#documenter > .docs-main").bind("click", function (ev) { if ($(ev.target).is(sidebar_button)) { return; } - if (sidebar.hasClass('visible')) { - sidebar.removeClass('visible'); + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); } }); -}) +}); // Resizes the package name / sitename in the sidebar if it is too wide. // Inspired by: https://github.com/davatron5000/FitText.js -$(document).ready(function() { +$(document).ready(function () { e = $("#documenter .docs-autofit"); function resize() { - var L = parseInt(e.css('max-width'), 10); + var L = parseInt(e.css("max-width"), 10); var L0 = e.width(); - if(L0 > L) { - var h0 = parseInt(e.css('font-size'), 10); - e.css('font-size', L * h0 / L0); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); // TODO: make sure it survives resizes? } } // call once and then register events resize(); $(window).resize(resize); - $(window).on('orientationchange', resize); + $(window).on("orientationchange", resize); }); // Scroll the navigation bar to the currently selected menu item -$(document).ready(function() { +$(document).ready(function () { var sidebar = $("#documenter .docs-menu").get(0); var active = $("#documenter .docs-menu .is-active").get(0); - if(typeof active !== 'undefined') { + if (typeof active !== "undefined") { sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; } -}) +}); }) //////////////////////////////////////////////////////////////////////////////// require(['jquery'], function($) { -function set_theme(theme) { - var active = null; - var disabled = []; - for (var i = 0; i < document.styleSheets.length; i++) { - var ss = document.styleSheets[i]; - var themename = ss.ownerNode.getAttribute("data-theme-name"); - if(themename === null) continue; // ignore non-theme stylesheets - // Find the active theme - if(themename === theme) active = ss; - else disabled.push(ss); - } - if(active !== null) { - active.disabled = false; - if(active.ownerNode.getAttribute("data-theme-primary") === null) { - document.getElementsByTagName('html')[0].className = "theme--" + theme; - } else { - document.getElementsByTagName('html')[0].className = ""; - } - disabled.forEach(function(ss){ - ss.disabled = true; - }); - } - - // Store the theme in localStorage - if(typeof(window.localStorage) !== "undefined") { - window.localStorage.setItem("documenter-theme", theme); - } else { - console.error("Browser does not support window.localStorage"); - } -} - // Theme picker setup -$(document).ready(function() { +$(document).ready(function () { // onchange callback - $('#documenter-themepicker').change(function themepick_callback(ev){ - var themename = $('#documenter-themepicker option:selected').attr('value'); - set_theme(themename); + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); }); // Make sure that the themepicker displays the correct theme when the theme is retrieved // from localStorage - if(typeof(window.localStorage) !== "undefined") { - var theme = window.localStorage.getItem("documenter-theme"); - if(theme !== null) { - $('#documenter-themepicker option').each(function(i,e) { - e.selected = (e.value === theme); - }) - } else { - $('#documenter-themepicker option').each(function(i,e) { - e.selected = $("html").hasClass(`theme--${e.value}`); - }) + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); } } -}) +}); }) //////////////////////////////////////////////////////////////////////////////// require(['jquery'], function($) { // update the version selector with info from the siteinfo.js and ../versions.js files -$(document).ready(function() { +$(document).ready(function () { // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the // siteinfo.js file, we just return immediately and not display the version selector. - if (typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === 'boolean' && DOCUMENTER_VERSION_SELECTOR_DISABLED) { + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { return; } var version_selector = $("#documenter .docs-version-selector"); var version_selector_select = $("#documenter .docs-version-selector select"); - version_selector_select.change(function(x) { - target_href = version_selector_select.children("option:selected").get(0).value; + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; window.location.href = target_href; }); // add the current version to the selector based on siteinfo.js, but only if the selector is empty - if (typeof DOCUMENTER_CURRENT_VERSION !== 'undefined' && $('#version-selector > option').length == 0) { - var option = $(""); + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); version_selector_select.append(option); } - if (typeof DOC_VERSIONS !== 'undefined') { + if (typeof DOC_VERSIONS !== "undefined") { var existing_versions = version_selector_select.children("option"); - var existing_versions_texts = existing_versions.map(function(i,x){return x.text}); - DOC_VERSIONS.forEach(function(each) { - var version_url = documenterBaseURL + "/../" + each; + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; var existing_id = $.inArray(each, existing_versions_texts); // if not already in the version selector, add it as a new option, // otherwise update the old option with the URL and enable it if (existing_id == -1) { - var option = $(""); + var option = $( + "" + ); version_selector_select.append(option); } else { var option = existing_versions[existing_id]; @@ -326,6 +1059,6 @@ $(document).ready(function() { if (version_selector_select.children("option").length > 0) { version_selector.toggleClass("visible"); } -}) +}); }) diff --git a/dev/assets/themes/catppuccin-frappe.css b/dev/assets/themes/catppuccin-frappe.css new file mode 100644 index 00000000..32e3f008 --- /dev/null +++ b/dev/assets/themes/catppuccin-frappe.css @@ -0,0 +1 @@ +html.theme--catppuccin-frappe .pagination-previous,html.theme--catppuccin-frappe .pagination-next,html.theme--catppuccin-frappe 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html.theme--catppuccin-frappe .content kbd.button.is-outlined{background-color:transparent;border-color:#414559;box-shadow:none;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #414559 #414559 !important}html.theme--catppuccin-frappe 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.button.is-primary.is-inverted.is-outlined.is-focused,html.theme--catppuccin-frappe .docstring>section>a.button.is-inverted.is-outlined.is-focused.docs-sourcelink{background-color:#fff;color:#8caaee}html.theme--catppuccin-frappe .button.is-primary.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .docstring>section>a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:hover::after,html.theme--catppuccin-frappe .button.is-primary.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .docstring>section>a.button.is-inverted.is-outlined.is-loading.is-hovered.docs-sourcelink::after,html.theme--catppuccin-frappe .button.is-primary.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .docstring>section>a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:focus::after,html.theme--catppuccin-frappe .button.is-primary.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-frappe 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.button.is-primary.is-light.is-hovered,html.theme--catppuccin-frappe .docstring>section>a.button.is-light.is-hovered.docs-sourcelink{background-color:#e2eafb;border-color:transparent;color:#153a8e}html.theme--catppuccin-frappe .button.is-primary.is-light:active,html.theme--catppuccin-frappe .docstring>section>a.button.is-light.docs-sourcelink:active,html.theme--catppuccin-frappe .button.is-primary.is-light.is-active,html.theme--catppuccin-frappe .docstring>section>a.button.is-light.is-active.docs-sourcelink{background-color:#d7e1f9;border-color:transparent;color:#153a8e}html.theme--catppuccin-frappe .button.is-link{background-color:#8caaee;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-link:hover,html.theme--catppuccin-frappe .button.is-link.is-hovered{background-color:#81a2ec;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-link:focus,html.theme--catppuccin-frappe 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.button.is-link.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-link.is-outlined{background-color:transparent;border-color:#8caaee;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-outlined:hover,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-link.is-outlined:focus,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-focused{background-color:#8caaee;border-color:#8caaee;color:#fff}html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading::after{border-color:transparent transparent #8caaee #8caaee !important}html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-link.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-link.is-outlined{background-color:transparent;border-color:#8caaee;box-shadow:none;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe 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.button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #81c8be #81c8be !important}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-light{background-color:#f1f9f8;color:#2d675f}html.theme--catppuccin-frappe .button.is-info.is-light:hover,html.theme--catppuccin-frappe .button.is-info.is-light.is-hovered{background-color:#e8f5f3;border-color:transparent;color:#2d675f}html.theme--catppuccin-frappe .button.is-info.is-light:active,html.theme--catppuccin-frappe .button.is-info.is-light.is-active{background-color:#dff1ef;border-color:transparent;color:#2d675f}html.theme--catppuccin-frappe 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.button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6d189 #a6d189 !important}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-light{background-color:#f4f9f0;color:#446a29}html.theme--catppuccin-frappe .button.is-success.is-light:hover,html.theme--catppuccin-frappe .button.is-success.is-light.is-hovered{background-color:#edf6e7;border-color:transparent;color:#446a29}html.theme--catppuccin-frappe .button.is-success.is-light:active,html.theme--catppuccin-frappe .button.is-success.is-light.is-active{background-color:#e6f2de;border-color:transparent;color:#446a29}html.theme--catppuccin-frappe .button.is-warning{background-color:#e5c890;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:hover,html.theme--catppuccin-frappe .button.is-warning.is-hovered{background-color:#e3c386;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:focus,html.theme--catppuccin-frappe .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:focus:not(:active),html.theme--catppuccin-frappe .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(229,200,144,0.25)}html.theme--catppuccin-frappe .button.is-warning:active,html.theme--catppuccin-frappe .button.is-warning.is-active{background-color:#e0be7b;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning{background-color:#e5c890;border-color:#e5c890;box-shadow:none}html.theme--catppuccin-frappe .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted:hover,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined{background-color:transparent;border-color:#e5c890;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-outlined:hover,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-warning.is-outlined:focus,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-focused{background-color:#e5c890;border-color:#e5c890;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #e5c890 #e5c890 !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-outlined{background-color:transparent;border-color:#e5c890;box-shadow:none;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #e5c890 #e5c890 !important}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-light{background-color:#fbf7ee;color:#78591c}html.theme--catppuccin-frappe .button.is-warning.is-light:hover,html.theme--catppuccin-frappe .button.is-warning.is-light.is-hovered{background-color:#f9f2e4;border-color:transparent;color:#78591c}html.theme--catppuccin-frappe .button.is-warning.is-light:active,html.theme--catppuccin-frappe .button.is-warning.is-light.is-active{background-color:#f6edda;border-color:transparent;color:#78591c}html.theme--catppuccin-frappe .button.is-danger{background-color:#e78284;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:hover,html.theme--catppuccin-frappe .button.is-danger.is-hovered{background-color:#e57779;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:focus,html.theme--catppuccin-frappe .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:focus:not(:active),html.theme--catppuccin-frappe .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(231,130,132,0.25)}html.theme--catppuccin-frappe .button.is-danger:active,html.theme--catppuccin-frappe .button.is-danger.is-active{background-color:#e36d6f;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger{background-color:#e78284;border-color:#e78284;box-shadow:none}html.theme--catppuccin-frappe .button.is-danger.is-inverted{background-color:#fff;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted:hover,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-frappe .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined{background-color:transparent;border-color:#e78284;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-outlined:hover,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-danger.is-outlined:focus,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-focused{background-color:#e78284;border-color:#e78284;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #e78284 #e78284 !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-outlined{background-color:transparent;border-color:#e78284;box-shadow:none;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #e78284 #e78284 !important}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-light{background-color:#fceeee;color:#9a1e20}html.theme--catppuccin-frappe .button.is-danger.is-light:hover,html.theme--catppuccin-frappe .button.is-danger.is-light.is-hovered{background-color:#fae3e4;border-color:transparent;color:#9a1e20}html.theme--catppuccin-frappe .button.is-danger.is-light:active,html.theme--catppuccin-frappe .button.is-danger.is-light.is-active{background-color:#f8d8d9;border-color:transparent;color:#9a1e20}html.theme--catppuccin-frappe .button.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-frappe .button.is-small:not(.is-rounded),html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-frappe .button.is-normal{font-size:1rem}html.theme--catppuccin-frappe .button.is-medium{font-size:1.25rem}html.theme--catppuccin-frappe .button.is-large{font-size:1.5rem}html.theme--catppuccin-frappe .button[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button{background-color:#737994;border-color:#626880;box-shadow:none;opacity:.5}html.theme--catppuccin-frappe .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-frappe .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-frappe .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-frappe .button.is-static{background-color:#292c3c;border-color:#626880;color:#838ba7;box-shadow:none;pointer-events:none}html.theme--catppuccin-frappe .button.is-rounded,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-frappe .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-frappe .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-frappe .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-frappe .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-frappe .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-frappe .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-frappe .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-frappe .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-frappe .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-frappe .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-frappe .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-frappe .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-frappe .buttons.has-addons .button:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-frappe .buttons.has-addons .button:focus,html.theme--catppuccin-frappe .buttons.has-addons .button.is-focused,html.theme--catppuccin-frappe .buttons.has-addons .button:active,html.theme--catppuccin-frappe .buttons.has-addons .button.is-active,html.theme--catppuccin-frappe .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-frappe .buttons.has-addons .button:focus:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-frappe .buttons.has-addons .button:active:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-frappe .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-frappe .buttons.is-centered{justify-content:center}html.theme--catppuccin-frappe .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-frappe .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-frappe .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-frappe .button.is-responsive.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-frappe .button.is-responsive,html.theme--catppuccin-frappe .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-frappe .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-frappe .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-frappe .button.is-responsive.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-frappe .button.is-responsive,html.theme--catppuccin-frappe .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-frappe .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-frappe .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-frappe .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-frappe .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-frappe .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-frappe .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-frappe .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-frappe .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-frappe .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-frappe .content li+li{margin-top:0.25em}html.theme--catppuccin-frappe .content p:not(:last-child),html.theme--catppuccin-frappe .content dl:not(:last-child),html.theme--catppuccin-frappe .content ol:not(:last-child),html.theme--catppuccin-frappe .content ul:not(:last-child),html.theme--catppuccin-frappe .content blockquote:not(:last-child),html.theme--catppuccin-frappe .content pre:not(:last-child),html.theme--catppuccin-frappe .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-frappe .content h1,html.theme--catppuccin-frappe .content h2,html.theme--catppuccin-frappe .content h3,html.theme--catppuccin-frappe .content h4,html.theme--catppuccin-frappe .content h5,html.theme--catppuccin-frappe .content h6{color:#c6d0f5;font-weight:600;line-height:1.125}html.theme--catppuccin-frappe .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-frappe .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-frappe .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-frappe .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-frappe .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-frappe .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-frappe .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-frappe .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-frappe .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-frappe .content blockquote{background-color:#292c3c;border-left:5px solid #626880;padding:1.25em 1.5em}html.theme--catppuccin-frappe .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-frappe .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-frappe .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-frappe .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-frappe .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-frappe .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-frappe .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-frappe .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-frappe .content ul ul ul{list-style-type:square}html.theme--catppuccin-frappe .content dd{margin-left:2em}html.theme--catppuccin-frappe .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-frappe .content figure:not(:first-child){margin-top:2em}html.theme--catppuccin-frappe .content figure:not(:last-child){margin-bottom:2em}html.theme--catppuccin-frappe .content figure img{display:inline-block}html.theme--catppuccin-frappe .content figure figcaption{font-style:italic}html.theme--catppuccin-frappe .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--catppuccin-frappe .content sup,html.theme--catppuccin-frappe .content sub{font-size:75%}html.theme--catppuccin-frappe .content table{width:100%}html.theme--catppuccin-frappe .content table td,html.theme--catppuccin-frappe .content table th{border:1px solid #626880;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--catppuccin-frappe .content table th{color:#b0bef1}html.theme--catppuccin-frappe .content table th:not([align]){text-align:inherit}html.theme--catppuccin-frappe .content table thead td,html.theme--catppuccin-frappe .content table thead th{border-width:0 0 2px;color:#b0bef1}html.theme--catppuccin-frappe .content table tfoot td,html.theme--catppuccin-frappe .content table tfoot th{border-width:2px 0 0;color:#b0bef1}html.theme--catppuccin-frappe .content table tbody tr:last-child td,html.theme--catppuccin-frappe .content table tbody tr:last-child th{border-bottom-width:0}html.theme--catppuccin-frappe .content .tabs li+li{margin-top:0}html.theme--catppuccin-frappe .content.is-small,html.theme--catppuccin-frappe #documenter 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select:not([multiple]){padding-right:2.5em}html.theme--catppuccin-frappe .select select[multiple]{height:auto;padding:0}html.theme--catppuccin-frappe .select select[multiple] option{padding:0.5em 1em}html.theme--catppuccin-frappe .select:not(.is-multiple):not(.is-loading):hover::after{border-color:#99d1db}html.theme--catppuccin-frappe .select.is-white:not(:hover)::after{border-color:#fff}html.theme--catppuccin-frappe .select.is-white select{border-color:#fff}html.theme--catppuccin-frappe .select.is-white select:hover,html.theme--catppuccin-frappe .select.is-white select.is-hovered{border-color:#f2f2f2}html.theme--catppuccin-frappe .select.is-white select:focus,html.theme--catppuccin-frappe .select.is-white select.is-focused,html.theme--catppuccin-frappe .select.is-white select:active,html.theme--catppuccin-frappe .select.is-white select.is-active{box-shadow:0 0 0 0.125em rgba(255,255,255,0.25)}html.theme--catppuccin-frappe 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select.is-focused,html.theme--catppuccin-frappe .select.is-dark select:active,html.theme--catppuccin-frappe .content kbd.select select:active,html.theme--catppuccin-frappe .select.is-dark select.is-active,html.theme--catppuccin-frappe .content kbd.select select.is-active{box-shadow:0 0 0 0.125em rgba(65,69,89,0.25)}html.theme--catppuccin-frappe .select.is-primary:not(:hover)::after,html.theme--catppuccin-frappe .docstring>section>a.select.docs-sourcelink:not(:hover)::after{border-color:#8caaee}html.theme--catppuccin-frappe .select.is-primary select,html.theme--catppuccin-frappe .docstring>section>a.select.docs-sourcelink select{border-color:#8caaee}html.theme--catppuccin-frappe .select.is-primary select:hover,html.theme--catppuccin-frappe .docstring>section>a.select.docs-sourcelink select:hover,html.theme--catppuccin-frappe .select.is-primary select.is-hovered,html.theme--catppuccin-frappe .docstring>section>a.select.docs-sourcelink select.is-hovered{border-color:#769aeb}html.theme--catppuccin-frappe .select.is-primary select:focus,html.theme--catppuccin-frappe .docstring>section>a.select.docs-sourcelink select:focus,html.theme--catppuccin-frappe .select.is-primary select.is-focused,html.theme--catppuccin-frappe .docstring>section>a.select.docs-sourcelink select.is-focused,html.theme--catppuccin-frappe .select.is-primary select:active,html.theme--catppuccin-frappe .docstring>section>a.select.docs-sourcelink select:active,html.theme--catppuccin-frappe .select.is-primary select.is-active,html.theme--catppuccin-frappe .docstring>section>a.select.docs-sourcelink select.is-active{box-shadow:0 0 0 0.125em rgba(140,170,238,0.25)}html.theme--catppuccin-frappe .select.is-link:not(:hover)::after{border-color:#8caaee}html.theme--catppuccin-frappe .select.is-link select{border-color:#8caaee}html.theme--catppuccin-frappe .select.is-link select:hover,html.theme--catppuccin-frappe .select.is-link select.is-hovered{border-color:#769aeb}html.theme--catppuccin-frappe .select.is-link select:focus,html.theme--catppuccin-frappe .select.is-link select.is-focused,html.theme--catppuccin-frappe .select.is-link select:active,html.theme--catppuccin-frappe .select.is-link select.is-active{box-shadow:0 0 0 0.125em rgba(140,170,238,0.25)}html.theme--catppuccin-frappe .select.is-info:not(:hover)::after{border-color:#81c8be}html.theme--catppuccin-frappe .select.is-info select{border-color:#81c8be}html.theme--catppuccin-frappe .select.is-info select:hover,html.theme--catppuccin-frappe .select.is-info select.is-hovered{border-color:#6fc0b5}html.theme--catppuccin-frappe .select.is-info select:focus,html.theme--catppuccin-frappe .select.is-info select.is-focused,html.theme--catppuccin-frappe .select.is-info select:active,html.theme--catppuccin-frappe .select.is-info select.is-active{box-shadow:0 0 0 0.125em rgba(129,200,190,0.25)}html.theme--catppuccin-frappe 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a.navbar-item:focus,html.theme--catppuccin-frappe a.navbar-item:focus-within,html.theme--catppuccin-frappe a.navbar-item:hover,html.theme--catppuccin-frappe a.navbar-item.is-active,html.theme--catppuccin-frappe .navbar-link:focus,html.theme--catppuccin-frappe .navbar-link:focus-within,html.theme--catppuccin-frappe .navbar-link:hover,html.theme--catppuccin-frappe .navbar-link.is-active{background-color:rgba(0,0,0,0);color:#8caaee}html.theme--catppuccin-frappe .navbar-item{flex-grow:0;flex-shrink:0}html.theme--catppuccin-frappe .navbar-item img{max-height:1.75rem}html.theme--catppuccin-frappe .navbar-item.has-dropdown{padding:0}html.theme--catppuccin-frappe .navbar-item.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-frappe .navbar-item.is-tab{border-bottom:1px solid transparent;min-height:4rem;padding-bottom:calc(0.5rem - 1px)}html.theme--catppuccin-frappe .navbar-item.is-tab:focus,html.theme--catppuccin-frappe .navbar-item.is-tab:hover{background-color:rgba(0,0,0,0);border-bottom-color:#8caaee}html.theme--catppuccin-frappe .navbar-item.is-tab.is-active{background-color:rgba(0,0,0,0);border-bottom-color:#8caaee;border-bottom-style:solid;border-bottom-width:3px;color:#8caaee;padding-bottom:calc(0.5rem - 3px)}html.theme--catppuccin-frappe .navbar-content{flex-grow:1;flex-shrink:1}html.theme--catppuccin-frappe .navbar-link:not(.is-arrowless){padding-right:2.5em}html.theme--catppuccin-frappe .navbar-link:not(.is-arrowless)::after{border-color:#fff;margin-top:-0.375em;right:1.125em}html.theme--catppuccin-frappe .navbar-dropdown{font-size:0.875rem;padding-bottom:0.5rem;padding-top:0.5rem}html.theme--catppuccin-frappe .navbar-dropdown .navbar-item{padding-left:1.5rem;padding-right:1.5rem}html.theme--catppuccin-frappe .navbar-divider{background-color:rgba(0,0,0,0.2);border:none;display:none;height:2px;margin:0.5rem 0}@media screen and (max-width: 1055px){html.theme--catppuccin-frappe 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a.navbar-item:hover{background-color:rgba(0,0,0,0);color:#838ba7}html.theme--catppuccin-frappe .navbar-dropdown a.navbar-item.is-active{background-color:rgba(0,0,0,0);color:#8caaee}.navbar.is-spaced html.theme--catppuccin-frappe .navbar-dropdown,html.theme--catppuccin-frappe .navbar-dropdown.is-boxed{border-radius:8px;border-top:none;box-shadow:0 8px 8px rgba(10,10,10,0.1), 0 0 0 1px rgba(10,10,10,0.1);display:block;opacity:0;pointer-events:none;top:calc(100% + (-4px));transform:translateY(-5px);transition-duration:86ms;transition-property:opacity, transform}html.theme--catppuccin-frappe .navbar-dropdown.is-right{left:auto;right:0}html.theme--catppuccin-frappe .navbar-divider{display:block}html.theme--catppuccin-frappe .navbar>.container .navbar-brand,html.theme--catppuccin-frappe .container>.navbar .navbar-brand{margin-left:-.75rem}html.theme--catppuccin-frappe .navbar>.container .navbar-menu,html.theme--catppuccin-frappe .container>.navbar 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body.has-spaced-navbar-fixed-bottom{padding-bottom:6rem}html.theme--catppuccin-frappe a.navbar-item.is-active,html.theme--catppuccin-frappe .navbar-link.is-active{color:#8caaee}html.theme--catppuccin-frappe a.navbar-item.is-active:not(:focus):not(:hover),html.theme--catppuccin-frappe .navbar-link.is-active:not(:focus):not(:hover){background-color:rgba(0,0,0,0)}html.theme--catppuccin-frappe .navbar-item.has-dropdown:focus .navbar-link,html.theme--catppuccin-frappe .navbar-item.has-dropdown:hover .navbar-link,html.theme--catppuccin-frappe .navbar-item.has-dropdown.is-active .navbar-link{background-color:rgba(0,0,0,0)}}html.theme--catppuccin-frappe .hero.is-fullheight-with-navbar{min-height:calc(100vh - 4rem)}html.theme--catppuccin-frappe .pagination{font-size:1rem;margin:-.25rem}html.theme--catppuccin-frappe .pagination.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.pagination{font-size:.75rem}html.theme--catppuccin-frappe .pagination.is-medium{font-size:1.25rem}html.theme--catppuccin-frappe .pagination.is-large{font-size:1.5rem}html.theme--catppuccin-frappe .pagination.is-rounded .pagination-previous,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.pagination .pagination-previous,html.theme--catppuccin-frappe .pagination.is-rounded .pagination-next,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.pagination .pagination-next{padding-left:1em;padding-right:1em;border-radius:9999px}html.theme--catppuccin-frappe .pagination.is-rounded .pagination-link,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.pagination .pagination-link{border-radius:9999px}html.theme--catppuccin-frappe .pagination,html.theme--catppuccin-frappe .pagination-list{align-items:center;display:flex;justify-content:center;text-align:center}html.theme--catppuccin-frappe .pagination-previous,html.theme--catppuccin-frappe .pagination-next,html.theme--catppuccin-frappe .pagination-link,html.theme--catppuccin-frappe .pagination-ellipsis{font-size:1em;justify-content:center;margin:.25rem;padding-left:.5em;padding-right:.5em;text-align:center}html.theme--catppuccin-frappe .pagination-previous,html.theme--catppuccin-frappe .pagination-next,html.theme--catppuccin-frappe .pagination-link{border-color:#626880;color:#8caaee;min-width:2.5em}html.theme--catppuccin-frappe .pagination-previous:hover,html.theme--catppuccin-frappe .pagination-next:hover,html.theme--catppuccin-frappe .pagination-link:hover{border-color:#737994;color:#99d1db}html.theme--catppuccin-frappe .pagination-previous:focus,html.theme--catppuccin-frappe .pagination-next:focus,html.theme--catppuccin-frappe .pagination-link:focus{border-color:#737994}html.theme--catppuccin-frappe .pagination-previous:active,html.theme--catppuccin-frappe .pagination-next:active,html.theme--catppuccin-frappe .pagination-link:active{box-shadow:inset 0 1px 2px 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768px){html.theme--catppuccin-frappe .pagination{flex-wrap:wrap}html.theme--catppuccin-frappe .pagination-previous,html.theme--catppuccin-frappe .pagination-next{flex-grow:1;flex-shrink:1}html.theme--catppuccin-frappe .pagination-list li{flex-grow:1;flex-shrink:1}}@media screen and (min-width: 769px),print{html.theme--catppuccin-frappe .pagination-list{flex-grow:1;flex-shrink:1;justify-content:flex-start;order:1}html.theme--catppuccin-frappe .pagination-previous,html.theme--catppuccin-frappe .pagination-next,html.theme--catppuccin-frappe .pagination-link,html.theme--catppuccin-frappe .pagination-ellipsis{margin-bottom:0;margin-top:0}html.theme--catppuccin-frappe .pagination-previous{order:2}html.theme--catppuccin-frappe .pagination-next{order:3}html.theme--catppuccin-frappe .pagination{justify-content:space-between;margin-bottom:0;margin-top:0}html.theme--catppuccin-frappe .pagination.is-centered .pagination-previous{order:1}html.theme--catppuccin-frappe .pagination.is-centered 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.button.is-white.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-white.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black{background-color:#0a0a0a;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-black:hover,html.theme--catppuccin-latte .button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-black:focus,html.theme--catppuccin-latte .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-black:focus:not(:active),html.theme--catppuccin-latte .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--catppuccin-latte .button.is-black:active,html.theme--catppuccin-latte .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-black[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--catppuccin-latte .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-inverted:hover,html.theme--catppuccin-latte .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-outlined:hover,html.theme--catppuccin-latte .button.is-black.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-black.is-outlined:focus,html.theme--catppuccin-latte .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-light{background-color:#f5f5f5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light:hover,html.theme--catppuccin-latte .button.is-light.is-hovered{background-color:#eee;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light:focus,html.theme--catppuccin-latte .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light:focus:not(:active),html.theme--catppuccin-latte .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(245,245,245,0.25)}html.theme--catppuccin-latte .button.is-light:active,html.theme--catppuccin-latte .button.is-light.is-active{background-color:#e8e8e8;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-light{background-color:#f5f5f5;border-color:#f5f5f5;box-shadow:none}html.theme--catppuccin-latte .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-latte .button.is-light.is-inverted:hover,html.theme--catppuccin-latte .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-latte .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-latte .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;color:#f5f5f5}html.theme--catppuccin-latte .button.is-light.is-outlined:hover,html.theme--catppuccin-latte .button.is-light.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-light.is-outlined:focus,html.theme--catppuccin-latte .button.is-light.is-outlined.is-focused{background-color:#f5f5f5;border-color:#f5f5f5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-latte .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark,html.theme--catppuccin-latte .content kbd.button{background-color:#ccd0da;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark:hover,html.theme--catppuccin-latte .content kbd.button:hover,html.theme--catppuccin-latte .button.is-dark.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-hovered{background-color:#c5c9d5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark:focus,html.theme--catppuccin-latte .content kbd.button:focus,html.theme--catppuccin-latte .button.is-dark.is-focused,html.theme--catppuccin-latte .content kbd.button.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark:focus:not(:active),html.theme--catppuccin-latte .content kbd.button:focus:not(:active),html.theme--catppuccin-latte .button.is-dark.is-focused:not(:active),html.theme--catppuccin-latte .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(204,208,218,0.25)}html.theme--catppuccin-latte .button.is-dark:active,html.theme--catppuccin-latte .content kbd.button:active,html.theme--catppuccin-latte .button.is-dark.is-active,html.theme--catppuccin-latte .content kbd.button.is-active{background-color:#bdc2cf;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark[disabled],html.theme--catppuccin-latte .content kbd.button[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-dark,fieldset[disabled] html.theme--catppuccin-latte .content kbd.button{background-color:#ccd0da;border-color:#ccd0da;box-shadow:none}html.theme--catppuccin-latte .button.is-dark.is-inverted,html.theme--catppuccin-latte .content kbd.button.is-inverted{background-color:rgba(0,0,0,0.7);color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-inverted:hover,html.theme--catppuccin-latte .content kbd.button.is-inverted:hover,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark.is-inverted[disabled],html.theme--catppuccin-latte .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-dark.is-inverted,fieldset[disabled] html.theme--catppuccin-latte .content kbd.button.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-loading::after,html.theme--catppuccin-latte .content kbd.button.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-latte .button.is-dark.is-outlined,html.theme--catppuccin-latte .content kbd.button.is-outlined{background-color:transparent;border-color:#ccd0da;color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-outlined:hover,html.theme--catppuccin-latte .content kbd.button.is-outlined:hover,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-dark.is-outlined:focus,html.theme--catppuccin-latte .content kbd.button.is-outlined:focus,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-focused,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-focused{background-color:#ccd0da;border-color:#ccd0da;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #ccd0da #ccd0da !important}html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-latte .button.is-dark.is-outlined[disabled],html.theme--catppuccin-latte .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-dark.is-outlined,fieldset[disabled] html.theme--catppuccin-latte .content kbd.button.is-outlined{background-color:transparent;border-color:#ccd0da;box-shadow:none;color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ccd0da #ccd0da !important}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined[disabled],html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-primary,html.theme--catppuccin-latte .docstring>section>a.button.docs-sourcelink{background-color:#1e66f5;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-primary:hover,html.theme--catppuccin-latte .docstring>section>a.button.docs-sourcelink:hover,html.theme--catppuccin-latte .button.is-primary.is-hovered,html.theme--catppuccin-latte .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#125ef4;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-primary:focus,html.theme--catppuccin-latte .docstring>section>a.button.docs-sourcelink:focus,html.theme--catppuccin-latte .button.is-primary.is-focused,html.theme--catppuccin-latte .docstring>section>a.button.is-focused.docs-sourcelink{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-primary:focus:not(:active),html.theme--catppuccin-latte .docstring>section>a.button.docs-sourcelink:focus:not(:active),html.theme--catppuccin-latte 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.docstring>section>a.button.is-inverted.docs-sourcelink{background-color:#fff;color:#1e66f5}html.theme--catppuccin-latte .button.is-primary.is-inverted:hover,html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.docs-sourcelink:hover,html.theme--catppuccin-latte .button.is-primary.is-inverted.is-hovered,html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.is-hovered.docs-sourcelink{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-primary.is-inverted[disabled],html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.docs-sourcelink[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-primary.is-inverted,fieldset[disabled] html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.docs-sourcelink{background-color:#fff;border-color:transparent;box-shadow:none;color:#1e66f5}html.theme--catppuccin-latte .button.is-primary.is-loading::after,html.theme--catppuccin-latte 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.button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-outlined{background-color:transparent;border-color:#40a02b;box-shadow:none;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-focused{background-color:#fff;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #40a02b #40a02b !important}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-success.is-light{background-color:#f1fbef;color:#40a12b}html.theme--catppuccin-latte .button.is-success.is-light:hover,html.theme--catppuccin-latte .button.is-success.is-light.is-hovered{background-color:#e8f8e5;border-color:transparent;color:#40a12b}html.theme--catppuccin-latte .button.is-success.is-light:active,html.theme--catppuccin-latte .button.is-success.is-light.is-active{background-color:#e0f5db;border-color:transparent;color:#40a12b}html.theme--catppuccin-latte .button.is-warning{background-color:#df8e1d;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:hover,html.theme--catppuccin-latte .button.is-warning.is-hovered{background-color:#d4871c;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:focus,html.theme--catppuccin-latte .button.is-warning.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:focus:not(:active),html.theme--catppuccin-latte .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(223,142,29,0.25)}html.theme--catppuccin-latte .button.is-warning:active,html.theme--catppuccin-latte .button.is-warning.is-active{background-color:#c8801a;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning{background-color:#df8e1d;border-color:#df8e1d;box-shadow:none}html.theme--catppuccin-latte .button.is-warning.is-inverted{background-color:#fff;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted:hover,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-warning.is-outlined{background-color:transparent;border-color:#df8e1d;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-outlined:hover,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-warning.is-outlined:focus,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-focused{background-color:#df8e1d;border-color:#df8e1d;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #df8e1d #df8e1d !important}html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-outlined{background-color:transparent;border-color:#df8e1d;box-shadow:none;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-focused{background-color:#fff;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #df8e1d #df8e1d !important}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-light{background-color:#fdf6ed;color:#9e6515}html.theme--catppuccin-latte .button.is-warning.is-light:hover,html.theme--catppuccin-latte .button.is-warning.is-light.is-hovered{background-color:#fbf1e2;border-color:transparent;color:#9e6515}html.theme--catppuccin-latte .button.is-warning.is-light:active,html.theme--catppuccin-latte .button.is-warning.is-light.is-active{background-color:#faebd6;border-color:transparent;color:#9e6515}html.theme--catppuccin-latte .button.is-danger{background-color:#d20f39;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:hover,html.theme--catppuccin-latte .button.is-danger.is-hovered{background-color:#c60e36;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:focus,html.theme--catppuccin-latte .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:focus:not(:active),html.theme--catppuccin-latte .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(210,15,57,0.25)}html.theme--catppuccin-latte .button.is-danger:active,html.theme--catppuccin-latte .button.is-danger.is-active{background-color:#ba0d33;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger{background-color:#d20f39;border-color:#d20f39;box-shadow:none}html.theme--catppuccin-latte .button.is-danger.is-inverted{background-color:#fff;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted:hover,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-danger.is-outlined{background-color:transparent;border-color:#d20f39;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-outlined:hover,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-danger.is-outlined:focus,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-focused{background-color:#d20f39;border-color:#d20f39;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #d20f39 #d20f39 !important}html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-outlined{background-color:transparent;border-color:#d20f39;box-shadow:none;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #d20f39 #d20f39 !important}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-light{background-color:#feecf0;color:#e9113f}html.theme--catppuccin-latte .button.is-danger.is-light:hover,html.theme--catppuccin-latte .button.is-danger.is-light.is-hovered{background-color:#fde0e6;border-color:transparent;color:#e9113f}html.theme--catppuccin-latte .button.is-danger.is-light:active,html.theme--catppuccin-latte .button.is-danger.is-light.is-active{background-color:#fcd4dd;border-color:transparent;color:#e9113f}html.theme--catppuccin-latte .button.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-latte .button.is-small:not(.is-rounded),html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-latte .button.is-normal{font-size:1rem}html.theme--catppuccin-latte .button.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .button.is-large{font-size:1.5rem}html.theme--catppuccin-latte .button[disabled],fieldset[disabled] html.theme--catppuccin-latte .button{background-color:#9ca0b0;border-color:#acb0be;box-shadow:none;opacity:.5}html.theme--catppuccin-latte .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-latte .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-latte .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-latte .button.is-static{background-color:#e6e9ef;border-color:#acb0be;color:#8c8fa1;box-shadow:none;pointer-events:none}html.theme--catppuccin-latte .button.is-rounded,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-latte .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-latte .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-latte .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-latte .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-latte .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-latte .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-latte .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-latte .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-latte .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-latte .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-latte .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-latte .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-latte .buttons.has-addons .button:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-latte .buttons.has-addons .button:focus,html.theme--catppuccin-latte .buttons.has-addons .button.is-focused,html.theme--catppuccin-latte .buttons.has-addons .button:active,html.theme--catppuccin-latte .buttons.has-addons .button.is-active,html.theme--catppuccin-latte .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-latte .buttons.has-addons .button:focus:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-latte .buttons.has-addons .button:active:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-latte .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .buttons.is-centered{justify-content:center}html.theme--catppuccin-latte .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-latte .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-latte .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-latte .button.is-responsive.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-latte .button.is-responsive,html.theme--catppuccin-latte .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-latte .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-latte .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-latte .button.is-responsive.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-latte .button.is-responsive,html.theme--catppuccin-latte .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-latte .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-latte .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-latte .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-latte .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-latte .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-latte .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-latte .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-latte .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-latte .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-latte .content li+li{margin-top:0.25em}html.theme--catppuccin-latte .content p:not(:last-child),html.theme--catppuccin-latte .content dl:not(:last-child),html.theme--catppuccin-latte .content ol:not(:last-child),html.theme--catppuccin-latte .content ul:not(:last-child),html.theme--catppuccin-latte .content blockquote:not(:last-child),html.theme--catppuccin-latte .content pre:not(:last-child),html.theme--catppuccin-latte .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-latte .content h1,html.theme--catppuccin-latte .content h2,html.theme--catppuccin-latte .content h3,html.theme--catppuccin-latte .content h4,html.theme--catppuccin-latte .content h5,html.theme--catppuccin-latte .content h6{color:#4c4f69;font-weight:600;line-height:1.125}html.theme--catppuccin-latte .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-latte .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-latte .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-latte .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-latte .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-latte .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-latte .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-latte .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-latte .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-latte .content blockquote{background-color:#e6e9ef;border-left:5px solid #acb0be;padding:1.25em 1.5em}html.theme--catppuccin-latte .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-latte .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-latte .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-latte .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-latte .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-latte .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-latte .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-latte .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-latte .content ul ul ul{list-style-type:square}html.theme--catppuccin-latte .content dd{margin-left:2em}html.theme--catppuccin-latte 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th:not([align]){text-align:inherit}html.theme--catppuccin-latte .content table thead td,html.theme--catppuccin-latte .content table thead th{border-width:0 0 2px;color:#41445a}html.theme--catppuccin-latte .content table tfoot td,html.theme--catppuccin-latte .content table tfoot th{border-width:2px 0 0;color:#41445a}html.theme--catppuccin-latte .content table tbody tr:last-child td,html.theme--catppuccin-latte .content table tbody tr:last-child th{border-bottom-width:0}html.theme--catppuccin-latte .content .tabs li+li{margin-top:0}html.theme--catppuccin-latte .content.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--catppuccin-latte .content.is-normal{font-size:1rem}html.theme--catppuccin-latte .content.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .content.is-large{font-size:1.5rem}html.theme--catppuccin-latte 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#documenter .docs-sidebar form.docs-search>input.file{font-size:.75rem}html.theme--catppuccin-latte .file.is-normal{font-size:1rem}html.theme--catppuccin-latte .file.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .file.is-medium .file-icon .fa{font-size:21px}html.theme--catppuccin-latte .file.is-large{font-size:1.5rem}html.theme--catppuccin-latte .file.is-large .file-icon .fa{font-size:28px}html.theme--catppuccin-latte .file.has-name .file-cta{border-bottom-right-radius:0;border-top-right-radius:0}html.theme--catppuccin-latte .file.has-name .file-name{border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-latte .file.has-name.is-empty .file-cta{border-radius:.4em}html.theme--catppuccin-latte .file.has-name.is-empty .file-name{display:none}html.theme--catppuccin-latte .file.is-boxed .file-label{flex-direction:column}html.theme--catppuccin-latte .file.is-boxed .file-cta{flex-direction:column;height:auto;padding:1em 3em}html.theme--catppuccin-latte .file.is-boxed .file-name{border-width:0 1px 1px}html.theme--catppuccin-latte .file.is-boxed .file-icon{height:1.5em;width:1.5em}html.theme--catppuccin-latte .file.is-boxed .file-icon .fa{font-size:21px}html.theme--catppuccin-latte .file.is-boxed.is-small .file-icon .fa,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-boxed .file-icon .fa{font-size:14px}html.theme--catppuccin-latte .file.is-boxed.is-medium .file-icon .fa{font-size:28px}html.theme--catppuccin-latte .file.is-boxed.is-large .file-icon .fa{font-size:35px}html.theme--catppuccin-latte .file.is-boxed.has-name .file-cta{border-radius:.4em .4em 0 0}html.theme--catppuccin-latte .file.is-boxed.has-name .file-name{border-radius:0 0 .4em .4em;border-width:0 1px 1px}html.theme--catppuccin-latte .file.is-centered{justify-content:center}html.theme--catppuccin-latte .file.is-fullwidth .file-label{width:100%}html.theme--catppuccin-latte .file.is-fullwidth .file-name{flex-grow:1;max-width:none}html.theme--catppuccin-latte .file.is-right{justify-content:flex-end}html.theme--catppuccin-latte .file.is-right .file-cta{border-radius:0 .4em .4em 0}html.theme--catppuccin-latte .file.is-right .file-name{border-radius:.4em 0 0 .4em;border-width:1px 0 1px 1px;order:-1}html.theme--catppuccin-latte .file-label{align-items:stretch;display:flex;cursor:pointer;justify-content:flex-start;overflow:hidden;position:relative}html.theme--catppuccin-latte .file-label:hover .file-cta{background-color:#c5c9d5;color:#41445a}html.theme--catppuccin-latte .file-label:hover .file-name{border-color:#a5a9b8}html.theme--catppuccin-latte .file-label:active .file-cta{background-color:#bdc2cf;color:#41445a}html.theme--catppuccin-latte .file-label:active .file-name{border-color:#9ea2b3}html.theme--catppuccin-latte .file-input{height:100%;left:0;opacity:0;outline:none;position:absolute;top:0;width:100%}html.theme--catppuccin-latte .file-cta,html.theme--catppuccin-latte .file-name{border-color:#acb0be;border-radius:.4em;font-size:1em;padding-left:1em;padding-right:1em;white-space:nowrap}html.theme--catppuccin-latte .file-cta{background-color:#ccd0da;color:#4c4f69}html.theme--catppuccin-latte .file-name{border-color:#acb0be;border-style:solid;border-width:1px 1px 1px 0;display:block;max-width:16em;overflow:hidden;text-align:inherit;text-overflow:ellipsis}html.theme--catppuccin-latte .file-icon{align-items:center;display:flex;height:1em;justify-content:center;margin-right:.5em;width:1em}html.theme--catppuccin-latte .file-icon .fa{font-size:14px}html.theme--catppuccin-latte .label{color:#41445a;display:block;font-size:1rem;font-weight:700}html.theme--catppuccin-latte .label:not(:last-child){margin-bottom:0.5em}html.theme--catppuccin-latte .label.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.label{font-size:.75rem}html.theme--catppuccin-latte .label.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .label.is-large{font-size:1.5rem}html.theme--catppuccin-latte .help{display:block;font-size:.75rem;margin-top:0.25rem}html.theme--catppuccin-latte .help.is-white{color:#fff}html.theme--catppuccin-latte .help.is-black{color:#0a0a0a}html.theme--catppuccin-latte .help.is-light{color:#f5f5f5}html.theme--catppuccin-latte .help.is-dark,html.theme--catppuccin-latte .content kbd.help{color:#ccd0da}html.theme--catppuccin-latte .help.is-primary,html.theme--catppuccin-latte .docstring>section>a.help.docs-sourcelink{color:#1e66f5}html.theme--catppuccin-latte .help.is-link{color:#1e66f5}html.theme--catppuccin-latte .help.is-info{color:#179299}html.theme--catppuccin-latte .help.is-success{color:#40a02b}html.theme--catppuccin-latte .help.is-warning{color:#df8e1d}html.theme--catppuccin-latte .help.is-danger{color:#d20f39}html.theme--catppuccin-latte .field:not(:last-child){margin-bottom:0.75rem}html.theme--catppuccin-latte .field.has-addons{display:flex;justify-content:flex-start}html.theme--catppuccin-latte .field.has-addons .control:not(:last-child){margin-right:-1px}html.theme--catppuccin-latte .field.has-addons .control:not(:first-child):not(:last-child) .button,html.theme--catppuccin-latte .field.has-addons .control:not(:first-child):not(:last-child) .input,html.theme--catppuccin-latte .field.has-addons .control:not(:first-child):not(:last-child) #documenter .docs-sidebar form.docs-search>input,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control:not(:first-child):not(:last-child) form.docs-search>input,html.theme--catppuccin-latte .field.has-addons .control:not(:first-child):not(:last-child) .select select{border-radius:0}html.theme--catppuccin-latte .field.has-addons .control:first-child:not(:only-child) .button,html.theme--catppuccin-latte .field.has-addons .control:first-child:not(:only-child) .input,html.theme--catppuccin-latte .field.has-addons .control:first-child:not(:only-child) #documenter .docs-sidebar form.docs-search>input,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control:first-child:not(:only-child) form.docs-search>input,html.theme--catppuccin-latte .field.has-addons .control:first-child:not(:only-child) .select select{border-bottom-right-radius:0;border-top-right-radius:0}html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) .button,html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) .input,html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) #documenter .docs-sidebar form.docs-search>input,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control:last-child:not(:only-child) form.docs-search>input,html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) .select select{border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .button.is-hovered:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .input.is-hovered:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-hovered:not([disabled]),html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-hovered:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .select select.is-hovered:not([disabled]){z-index:2}html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):focus,html.theme--catppuccin-latte .field.has-addons .control .button.is-focused:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):active,html.theme--catppuccin-latte .field.has-addons .control .button.is-active:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):focus,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):focus,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):focus,html.theme--catppuccin-latte .field.has-addons .control .input.is-focused:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-focused:not([disabled]),html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-focused:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):active,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):active,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):active,html.theme--catppuccin-latte .field.has-addons .control .input.is-active:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-active:not([disabled]),html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-active:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):focus,html.theme--catppuccin-latte .field.has-addons .control .select select.is-focused:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):active,html.theme--catppuccin-latte .field.has-addons .control .select select.is-active:not([disabled]){z-index:3}html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):focus:hover,html.theme--catppuccin-latte .field.has-addons .control .button.is-focused:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):active:hover,html.theme--catppuccin-latte .field.has-addons .control .button.is-active:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):focus:hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):focus:hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):focus:hover,html.theme--catppuccin-latte .field.has-addons .control .input.is-focused:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-focused:not([disabled]):hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-focused:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):active:hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):active:hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):active:hover,html.theme--catppuccin-latte .field.has-addons .control .input.is-active:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-active:not([disabled]):hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-active:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):focus:hover,html.theme--catppuccin-latte .field.has-addons .control .select select.is-focused:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):active:hover,html.theme--catppuccin-latte .field.has-addons .control .select select.is-active:not([disabled]):hover{z-index:4}html.theme--catppuccin-latte .field.has-addons .control.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .field.has-addons.has-addons-centered{justify-content:center}html.theme--catppuccin-latte .field.has-addons.has-addons-right{justify-content:flex-end}html.theme--catppuccin-latte .field.has-addons.has-addons-fullwidth .control{flex-grow:1;flex-shrink:0}html.theme--catppuccin-latte .field.is-grouped{display:flex;justify-content:flex-start}html.theme--catppuccin-latte .field.is-grouped>.control{flex-shrink:0}html.theme--catppuccin-latte .field.is-grouped>.control:not(:last-child){margin-bottom:0;margin-right:.75rem}html.theme--catppuccin-latte .field.is-grouped>.control.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .field.is-grouped.is-grouped-centered{justify-content:center}html.theme--catppuccin-latte .field.is-grouped.is-grouped-right{justify-content:flex-end}html.theme--catppuccin-latte .field.is-grouped.is-grouped-multiline{flex-wrap:wrap}html.theme--catppuccin-latte .field.is-grouped.is-grouped-multiline>.control:last-child,html.theme--catppuccin-latte .field.is-grouped.is-grouped-multiline>.control:not(:last-child){margin-bottom:0.75rem}html.theme--catppuccin-latte .field.is-grouped.is-grouped-multiline:last-child{margin-bottom:-0.75rem}html.theme--catppuccin-latte .field.is-grouped.is-grouped-multiline:not(:last-child){margin-bottom:0}@media screen and (min-width: 769px),print{html.theme--catppuccin-latte .field.is-horizontal{display:flex}}html.theme--catppuccin-latte .field-label .label{font-size:inherit}@media screen and (max-width: 768px){html.theme--catppuccin-latte .field-label{margin-bottom:0.5rem}}@media screen and (min-width: 769px),print{html.theme--catppuccin-latte .field-label{flex-basis:0;flex-grow:1;flex-shrink:0;margin-right:1.5rem;text-align:right}html.theme--catppuccin-latte .field-label.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.field-label{font-size:.75rem;padding-top:0.375em}html.theme--catppuccin-latte .field-label.is-normal{padding-top:0.375em}html.theme--catppuccin-latte .field-label.is-medium{font-size:1.25rem;padding-top:0.375em}html.theme--catppuccin-latte .field-label.is-large{font-size:1.5rem;padding-top:0.375em}}html.theme--catppuccin-latte .field-body .field .field{margin-bottom:0}@media screen and (min-width: 769px),print{html.theme--catppuccin-latte .field-body{display:flex;flex-basis:0;flex-grow:5;flex-shrink:1}html.theme--catppuccin-latte .field-body .field{margin-bottom:0}html.theme--catppuccin-latte .field-body>.field{flex-shrink:1}html.theme--catppuccin-latte .field-body>.field:not(.is-narrow){flex-grow:1}html.theme--catppuccin-latte .field-body>.field:not(:last-child){margin-right:.75rem}}html.theme--catppuccin-latte .control{box-sizing:border-box;clear:both;font-size:1rem;position:relative;text-align:inherit}html.theme--catppuccin-latte .control.has-icons-left .input:focus~.icon,html.theme--catppuccin-latte .control.has-icons-left #documenter .docs-sidebar form.docs-search>input:focus~.icon,html.theme--catppuccin-latte #documenter .docs-sidebar .control.has-icons-left form.docs-search>input:focus~.icon,html.theme--catppuccin-latte .control.has-icons-left .select:focus~.icon,html.theme--catppuccin-latte .control.has-icons-right .input:focus~.icon,html.theme--catppuccin-latte .control.has-icons-right #documenter .docs-sidebar form.docs-search>input:focus~.icon,html.theme--catppuccin-latte #documenter .docs-sidebar .control.has-icons-right form.docs-search>input:focus~.icon,html.theme--catppuccin-latte .control.has-icons-right .select:focus~.icon{color:#ccd0da}html.theme--catppuccin-latte .control.has-icons-left .input.is-small~.icon,html.theme--catppuccin-latte .control.has-icons-left #documenter .docs-sidebar form.docs-search>input~.icon,html.theme--catppuccin-latte #documenter .docs-sidebar .control.has-icons-left form.docs-search>input~.icon,html.theme--catppuccin-latte .control.has-icons-left .select.is-small~.icon,html.theme--catppuccin-latte .control.has-icons-right .input.is-small~.icon,html.theme--catppuccin-latte .control.has-icons-right #documenter .docs-sidebar form.docs-search>input~.icon,html.theme--catppuccin-latte #documenter .docs-sidebar .control.has-icons-right form.docs-search>input~.icon,html.theme--catppuccin-latte .control.has-icons-right .select.is-small~.icon{font-size:.75rem}html.theme--catppuccin-latte .control.has-icons-left .input.is-medium~.icon,html.theme--catppuccin-latte .control.has-icons-left #documenter .docs-sidebar form.docs-search>input.is-medium~.icon,html.theme--catppuccin-latte #documenter .docs-sidebar .control.has-icons-left form.docs-search>input.is-medium~.icon,html.theme--catppuccin-latte .control.has-icons-left .select.is-medium~.icon,html.theme--catppuccin-latte .control.has-icons-right .input.is-medium~.icon,html.theme--catppuccin-latte .control.has-icons-right #documenter .docs-sidebar form.docs-search>input.is-medium~.icon,html.theme--catppuccin-latte #documenter .docs-sidebar .control.has-icons-right form.docs-search>input.is-medium~.icon,html.theme--catppuccin-latte .control.has-icons-right .select.is-medium~.icon{font-size:1.25rem}html.theme--catppuccin-latte .control.has-icons-left 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kbd.button.is-outlined{background-color:transparent;border-color:#363a4f;box-shadow:none;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-macchiato .content 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.button.is-info.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-info.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-focused{background-color:#8bd5ca;border-color:#8bd5ca;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #8bd5ca #8bd5ca !important}html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-info.is-outlined{background-color:transparent;border-color:#8bd5ca;box-shadow:none;color:#8bd5ca}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#8bd5ca}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #8bd5ca #8bd5ca !important}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-light{background-color:#f0faf8;color:#276d62}html.theme--catppuccin-macchiato .button.is-info.is-light:hover,html.theme--catppuccin-macchiato .button.is-info.is-light.is-hovered{background-color:#e7f6f4;border-color:transparent;color:#276d62}html.theme--catppuccin-macchiato .button.is-info.is-light:active,html.theme--catppuccin-macchiato .button.is-info.is-light.is-active{background-color:#ddf3f0;border-color:transparent;color:#276d62}html.theme--catppuccin-macchiato .button.is-success{background-color:#a6da95;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:hover,html.theme--catppuccin-macchiato .button.is-success.is-hovered{background-color:#9ed78c;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:focus,html.theme--catppuccin-macchiato .button.is-success.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:focus:not(:active),html.theme--catppuccin-macchiato .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(166,218,149,0.25)}html.theme--catppuccin-macchiato .button.is-success:active,html.theme--catppuccin-macchiato .button.is-success.is-active{background-color:#96d382;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success{background-color:#a6da95;border-color:#a6da95;box-shadow:none}html.theme--catppuccin-macchiato .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined{background-color:transparent;border-color:#a6da95;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-success.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-focused{background-color:#a6da95;border-color:#a6da95;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6da95 #a6da95 !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-outlined{background-color:transparent;border-color:#a6da95;box-shadow:none;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6da95 #a6da95 !important}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-light{background-color:#f2faf0;color:#386e26}html.theme--catppuccin-macchiato .button.is-success.is-light:hover,html.theme--catppuccin-macchiato .button.is-success.is-light.is-hovered{background-color:#eaf6e6;border-color:transparent;color:#386e26}html.theme--catppuccin-macchiato .button.is-success.is-light:active,html.theme--catppuccin-macchiato .button.is-success.is-light.is-active{background-color:#e2f3dd;border-color:transparent;color:#386e26}html.theme--catppuccin-macchiato .button.is-warning{background-color:#eed49f;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:hover,html.theme--catppuccin-macchiato .button.is-warning.is-hovered{background-color:#eccf94;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:focus,html.theme--catppuccin-macchiato .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:focus:not(:active),html.theme--catppuccin-macchiato .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(238,212,159,0.25)}html.theme--catppuccin-macchiato .button.is-warning:active,html.theme--catppuccin-macchiato .button.is-warning.is-active{background-color:#eaca89;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning{background-color:#eed49f;border-color:#eed49f;box-shadow:none}html.theme--catppuccin-macchiato .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined{background-color:transparent;border-color:#eed49f;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-warning.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-focused{background-color:#eed49f;border-color:#eed49f;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #eed49f #eed49f !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-outlined{background-color:transparent;border-color:#eed49f;box-shadow:none;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #eed49f #eed49f !important}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-light{background-color:#fcf7ee;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-warning.is-light:hover,html.theme--catppuccin-macchiato .button.is-warning.is-light.is-hovered{background-color:#faf2e3;border-color:transparent;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-warning.is-light:active,html.theme--catppuccin-macchiato .button.is-warning.is-light.is-active{background-color:#f8eed8;border-color:transparent;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-danger{background-color:#ed8796;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:hover,html.theme--catppuccin-macchiato .button.is-danger.is-hovered{background-color:#eb7c8c;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:focus,html.theme--catppuccin-macchiato .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:focus:not(:active),html.theme--catppuccin-macchiato .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(237,135,150,0.25)}html.theme--catppuccin-macchiato .button.is-danger:active,html.theme--catppuccin-macchiato .button.is-danger.is-active{background-color:#ea7183;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger{background-color:#ed8796;border-color:#ed8796;box-shadow:none}html.theme--catppuccin-macchiato .button.is-danger.is-inverted{background-color:#fff;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-macchiato .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined{background-color:transparent;border-color:#ed8796;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-danger.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-focused{background-color:#ed8796;border-color:#ed8796;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #ed8796 #ed8796 !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-outlined{background-color:transparent;border-color:#ed8796;box-shadow:none;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ed8796 #ed8796 !important}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-light{background-color:#fcedef;color:#971729}html.theme--catppuccin-macchiato .button.is-danger.is-light:hover,html.theme--catppuccin-macchiato .button.is-danger.is-light.is-hovered{background-color:#fbe2e6;border-color:transparent;color:#971729}html.theme--catppuccin-macchiato .button.is-danger.is-light:active,html.theme--catppuccin-macchiato .button.is-danger.is-light.is-active{background-color:#f9d7dc;border-color:transparent;color:#971729}html.theme--catppuccin-macchiato .button.is-small,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-macchiato .button.is-small:not(.is-rounded),html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-macchiato .button.is-normal{font-size:1rem}html.theme--catppuccin-macchiato .button.is-medium{font-size:1.25rem}html.theme--catppuccin-macchiato .button.is-large{font-size:1.5rem}html.theme--catppuccin-macchiato .button[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button{background-color:#6e738d;border-color:#5b6078;box-shadow:none;opacity:.5}html.theme--catppuccin-macchiato .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-macchiato .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-macchiato .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-macchiato .button.is-static{background-color:#1e2030;border-color:#5b6078;color:#8087a2;box-shadow:none;pointer-events:none}html.theme--catppuccin-macchiato .button.is-rounded,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-macchiato .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-macchiato .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-macchiato .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-macchiato .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-macchiato .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-macchiato .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-macchiato .buttons.are-small 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.container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-macchiato .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-macchiato .content li+li{margin-top:0.25em}html.theme--catppuccin-macchiato .content p:not(:last-child),html.theme--catppuccin-macchiato .content dl:not(:last-child),html.theme--catppuccin-macchiato .content ol:not(:last-child),html.theme--catppuccin-macchiato .content ul:not(:last-child),html.theme--catppuccin-macchiato .content blockquote:not(:last-child),html.theme--catppuccin-macchiato .content pre:not(:last-child),html.theme--catppuccin-macchiato .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-macchiato .content h1,html.theme--catppuccin-macchiato .content h2,html.theme--catppuccin-macchiato .content h3,html.theme--catppuccin-macchiato .content h4,html.theme--catppuccin-macchiato .content h5,html.theme--catppuccin-macchiato .content 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ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-macchiato .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-macchiato .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-macchiato .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-macchiato .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-macchiato .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-macchiato .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-macchiato .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-macchiato .content ul ul ul{list-style-type:square}html.theme--catppuccin-macchiato .content dd{margin-left:2em}html.theme--catppuccin-macchiato .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-macchiato 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form.docs-search>input.is-active{box-shadow:0 0 0 0.125em rgba(255,255,255,0.25)}html.theme--catppuccin-macchiato .is-black.textarea,html.theme--catppuccin-macchiato .is-black.input,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-black{border-color:#0a0a0a}html.theme--catppuccin-macchiato .is-black.textarea:focus,html.theme--catppuccin-macchiato .is-black.input:focus,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-black:focus,html.theme--catppuccin-macchiato .is-black.is-focused.textarea,html.theme--catppuccin-macchiato .is-black.is-focused.input,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-focused,html.theme--catppuccin-macchiato .is-black.textarea:active,html.theme--catppuccin-macchiato .is-black.input:active,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-black:active,html.theme--catppuccin-macchiato 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html.theme--catppuccin-mocha .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-link.is-light{background-color:#ebf3fe;color:#063c93}html.theme--catppuccin-mocha .button.is-link.is-light:hover,html.theme--catppuccin-mocha .button.is-link.is-light.is-hovered{background-color:#dfebfe;border-color:transparent;color:#063c93}html.theme--catppuccin-mocha .button.is-link.is-light:active,html.theme--catppuccin-mocha .button.is-link.is-light.is-active{background-color:#d3e3fd;border-color:transparent;color:#063c93}html.theme--catppuccin-mocha .button.is-info{background-color:#94e2d5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:hover,html.theme--catppuccin-mocha .button.is-info.is-hovered{background-color:#8adfd1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:focus,html.theme--catppuccin-mocha .button.is-info.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:focus:not(:active),html.theme--catppuccin-mocha .button.is-info.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(148,226,213,0.25)}html.theme--catppuccin-mocha .button.is-info:active,html.theme--catppuccin-mocha .button.is-info.is-active{background-color:#80ddcd;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info{background-color:#94e2d5;border-color:#94e2d5;box-shadow:none}html.theme--catppuccin-mocha .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted:hover,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-info.is-outlined{background-color:transparent;border-color:#94e2d5;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-outlined:hover,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-info.is-outlined:focus,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-focused{background-color:#94e2d5;border-color:#94e2d5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #94e2d5 #94e2d5 !important}html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info.is-outlined{background-color:transparent;border-color:#94e2d5;box-shadow:none;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #94e2d5 #94e2d5 !important}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-light{background-color:#effbf9;color:#207466}html.theme--catppuccin-mocha .button.is-info.is-light:hover,html.theme--catppuccin-mocha .button.is-info.is-light.is-hovered{background-color:#e5f8f5;border-color:transparent;color:#207466}html.theme--catppuccin-mocha .button.is-info.is-light:active,html.theme--catppuccin-mocha .button.is-info.is-light.is-active{background-color:#dbf5f1;border-color:transparent;color:#207466}html.theme--catppuccin-mocha .button.is-success{background-color:#a6e3a1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:hover,html.theme--catppuccin-mocha .button.is-success.is-hovered{background-color:#9de097;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:focus,html.theme--catppuccin-mocha .button.is-success.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:focus:not(:active),html.theme--catppuccin-mocha .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(166,227,161,0.25)}html.theme--catppuccin-mocha .button.is-success:active,html.theme--catppuccin-mocha .button.is-success.is-active{background-color:#93dd8d;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success{background-color:#a6e3a1;border-color:#a6e3a1;box-shadow:none}html.theme--catppuccin-mocha .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted:hover,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-success.is-outlined{background-color:transparent;border-color:#a6e3a1;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-outlined:hover,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-success.is-outlined:focus,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-focused{background-color:#a6e3a1;border-color:#a6e3a1;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6e3a1 #a6e3a1 !important}html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-outlined{background-color:transparent;border-color:#a6e3a1;box-shadow:none;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6e3a1 #a6e3a1 !important}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-light{background-color:#f0faef;color:#287222}html.theme--catppuccin-mocha .button.is-success.is-light:hover,html.theme--catppuccin-mocha .button.is-success.is-light.is-hovered{background-color:#e7f7e5;border-color:transparent;color:#287222}html.theme--catppuccin-mocha .button.is-success.is-light:active,html.theme--catppuccin-mocha .button.is-success.is-light.is-active{background-color:#def4dc;border-color:transparent;color:#287222}html.theme--catppuccin-mocha .button.is-warning{background-color:#f9e2af;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:hover,html.theme--catppuccin-mocha .button.is-warning.is-hovered{background-color:#f8dea3;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:focus,html.theme--catppuccin-mocha .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:focus:not(:active),html.theme--catppuccin-mocha .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(249,226,175,0.25)}html.theme--catppuccin-mocha .button.is-warning:active,html.theme--catppuccin-mocha .button.is-warning.is-active{background-color:#f7d997;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning{background-color:#f9e2af;border-color:#f9e2af;box-shadow:none}html.theme--catppuccin-mocha .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted:hover,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined{background-color:transparent;border-color:#f9e2af;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-outlined:hover,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-warning.is-outlined:focus,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-focused{background-color:#f9e2af;border-color:#f9e2af;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #f9e2af #f9e2af !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-outlined{background-color:transparent;border-color:#f9e2af;box-shadow:none;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f9e2af #f9e2af !important}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-light{background-color:#fef8ec;color:#8a620a}html.theme--catppuccin-mocha .button.is-warning.is-light:hover,html.theme--catppuccin-mocha .button.is-warning.is-light.is-hovered{background-color:#fdf4e0;border-color:transparent;color:#8a620a}html.theme--catppuccin-mocha .button.is-warning.is-light:active,html.theme--catppuccin-mocha .button.is-warning.is-light.is-active{background-color:#fcf0d4;border-color:transparent;color:#8a620a}html.theme--catppuccin-mocha .button.is-danger{background-color:#f38ba8;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:hover,html.theme--catppuccin-mocha .button.is-danger.is-hovered{background-color:#f27f9f;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:focus,html.theme--catppuccin-mocha .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:focus:not(:active),html.theme--catppuccin-mocha .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(243,139,168,0.25)}html.theme--catppuccin-mocha .button.is-danger:active,html.theme--catppuccin-mocha .button.is-danger.is-active{background-color:#f17497;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger{background-color:#f38ba8;border-color:#f38ba8;box-shadow:none}html.theme--catppuccin-mocha .button.is-danger.is-inverted{background-color:#fff;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted:hover,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-mocha .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined{background-color:transparent;border-color:#f38ba8;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-outlined:hover,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-danger.is-outlined:focus,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-focused{background-color:#f38ba8;border-color:#f38ba8;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #f38ba8 #f38ba8 !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-outlined{background-color:transparent;border-color:#f38ba8;box-shadow:none;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f38ba8 #f38ba8 !important}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-light{background-color:#fdedf1;color:#991036}html.theme--catppuccin-mocha .button.is-danger.is-light:hover,html.theme--catppuccin-mocha .button.is-danger.is-light.is-hovered{background-color:#fce1e8;border-color:transparent;color:#991036}html.theme--catppuccin-mocha .button.is-danger.is-light:active,html.theme--catppuccin-mocha .button.is-danger.is-light.is-active{background-color:#fbd5e0;border-color:transparent;color:#991036}html.theme--catppuccin-mocha .button.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-mocha .button.is-small:not(.is-rounded),html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-mocha .button.is-normal{font-size:1rem}html.theme--catppuccin-mocha .button.is-medium{font-size:1.25rem}html.theme--catppuccin-mocha .button.is-large{font-size:1.5rem}html.theme--catppuccin-mocha .button[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button{background-color:#6c7086;border-color:#585b70;box-shadow:none;opacity:.5}html.theme--catppuccin-mocha .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-mocha .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-mocha .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-mocha .button.is-static{background-color:#181825;border-color:#585b70;color:#7f849c;box-shadow:none;pointer-events:none}html.theme--catppuccin-mocha .button.is-rounded,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-mocha .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-mocha .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-mocha .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-mocha .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-mocha .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-mocha .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-mocha .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-mocha .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-mocha .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-mocha .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-mocha .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-mocha .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-mocha .buttons.has-addons .button:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-mocha .buttons.has-addons .button:focus,html.theme--catppuccin-mocha .buttons.has-addons .button.is-focused,html.theme--catppuccin-mocha .buttons.has-addons .button:active,html.theme--catppuccin-mocha .buttons.has-addons .button.is-active,html.theme--catppuccin-mocha .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-mocha .buttons.has-addons .button:focus:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-mocha .buttons.has-addons .button:active:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-mocha .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-mocha .buttons.is-centered{justify-content:center}html.theme--catppuccin-mocha .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-mocha .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-mocha .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-mocha .button.is-responsive.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-mocha .button.is-responsive,html.theme--catppuccin-mocha .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-mocha .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-mocha .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-mocha .button.is-responsive.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-mocha .button.is-responsive,html.theme--catppuccin-mocha .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-mocha .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-mocha .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-mocha .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-mocha .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-mocha .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-mocha .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-mocha .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-mocha .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-mocha .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-mocha .content li+li{margin-top:0.25em}html.theme--catppuccin-mocha .content p:not(:last-child),html.theme--catppuccin-mocha .content dl:not(:last-child),html.theme--catppuccin-mocha .content ol:not(:last-child),html.theme--catppuccin-mocha .content ul:not(:last-child),html.theme--catppuccin-mocha .content blockquote:not(:last-child),html.theme--catppuccin-mocha .content pre:not(:last-child),html.theme--catppuccin-mocha .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-mocha .content h1,html.theme--catppuccin-mocha .content h2,html.theme--catppuccin-mocha .content h3,html.theme--catppuccin-mocha .content h4,html.theme--catppuccin-mocha .content h5,html.theme--catppuccin-mocha .content h6{color:#cdd6f4;font-weight:600;line-height:1.125}html.theme--catppuccin-mocha .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-mocha .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-mocha .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-mocha .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-mocha .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-mocha .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-mocha .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-mocha .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-mocha .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-mocha .content blockquote{background-color:#181825;border-left:5px solid #585b70;padding:1.25em 1.5em}html.theme--catppuccin-mocha .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-mocha .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-mocha .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-mocha .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-mocha .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-mocha .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-mocha .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-mocha .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-mocha .content ul ul ul{list-style-type:square}html.theme--catppuccin-mocha .content dd{margin-left:2em}html.theme--catppuccin-mocha .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-mocha .content figure:not(:first-child){margin-top:2em}html.theme--catppuccin-mocha .content figure:not(:last-child){margin-bottom:2em}html.theme--catppuccin-mocha .content figure img{display:inline-block}html.theme--catppuccin-mocha .content figure figcaption{font-style:italic}html.theme--catppuccin-mocha .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--catppuccin-mocha .content sup,html.theme--catppuccin-mocha .content sub{font-size:75%}html.theme--catppuccin-mocha .content table{width:100%}html.theme--catppuccin-mocha .content table td,html.theme--catppuccin-mocha .content table th{border:1px solid #585b70;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--catppuccin-mocha .content table th{color:#b8c5ef}html.theme--catppuccin-mocha .content table th:not([align]){text-align:inherit}html.theme--catppuccin-mocha .content table thead td,html.theme--catppuccin-mocha .content table thead th{border-width:0 0 2px;color:#b8c5ef}html.theme--catppuccin-mocha .content table tfoot td,html.theme--catppuccin-mocha .content table tfoot th{border-width:2px 0 0;color:#b8c5ef}html.theme--catppuccin-mocha .content table tbody tr:last-child td,html.theme--catppuccin-mocha .content table tbody tr:last-child th{border-bottom-width:0}html.theme--catppuccin-mocha .content .tabs li+li{margin-top:0}html.theme--catppuccin-mocha .content.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--catppuccin-mocha .content.is-normal{font-size:1rem}html.theme--catppuccin-mocha .content.is-medium{font-size:1.25rem}html.theme--catppuccin-mocha .content.is-large{font-size:1.5rem}html.theme--catppuccin-mocha .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--catppuccin-mocha .icon.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--catppuccin-mocha .icon.is-medium{height:2rem;width:2rem}html.theme--catppuccin-mocha .icon.is-large{height:3rem;width:3rem}html.theme--catppuccin-mocha .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--catppuccin-mocha .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--catppuccin-mocha .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--catppuccin-mocha .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--catppuccin-mocha div.icon-text{display:flex}html.theme--catppuccin-mocha .image,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--catppuccin-mocha .image img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--catppuccin-mocha .image img.is-rounded,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--catppuccin-mocha .image.is-fullwidth,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--catppuccin-mocha .image.is-square img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--catppuccin-mocha .image.is-square .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--catppuccin-mocha .image.is-1by1 img,html.theme--catppuccin-mocha #documenter .docs-sidebar 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.has-ratio,html.theme--catppuccin-mocha .image.is-5by3 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-5by3 img,html.theme--catppuccin-mocha .image.is-5by3 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-5by3 .has-ratio,html.theme--catppuccin-mocha .image.is-16by9 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-16by9 img,html.theme--catppuccin-mocha .image.is-16by9 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-16by9 .has-ratio,html.theme--catppuccin-mocha .image.is-2by1 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-2by1 img,html.theme--catppuccin-mocha .image.is-2by1 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-2by1 .has-ratio,html.theme--catppuccin-mocha .image.is-3by1 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by1 img,html.theme--catppuccin-mocha 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img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by5 img,html.theme--catppuccin-mocha .image.is-3by5 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by5 .has-ratio,html.theme--catppuccin-mocha .image.is-9by16 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-9by16 img,html.theme--catppuccin-mocha .image.is-9by16 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-9by16 .has-ratio,html.theme--catppuccin-mocha .image.is-1by2 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by2 img,html.theme--catppuccin-mocha .image.is-1by2 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by2 .has-ratio,html.theme--catppuccin-mocha .image.is-1by3 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by3 img,html.theme--catppuccin-mocha .image.is-1by3 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by3 .has-ratio{height:100%;width:100%}html.theme--catppuccin-mocha .image.is-square,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-square,html.theme--catppuccin-mocha .image.is-1by1,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by1{padding-top:100%}html.theme--catppuccin-mocha .image.is-5by4,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-5by4{padding-top:80%}html.theme--catppuccin-mocha .image.is-4by3,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-4by3{padding-top:75%}html.theme--catppuccin-mocha .image.is-3by2,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by2{padding-top:66.6666%}html.theme--catppuccin-mocha .image.is-5by3,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-5by3{padding-top:60%}html.theme--catppuccin-mocha .image.is-16by9,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-16by9{padding-top:56.25%}html.theme--catppuccin-mocha .image.is-2by1,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-2by1{padding-top:50%}html.theme--catppuccin-mocha .image.is-3by1,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by1{padding-top:33.3333%}html.theme--catppuccin-mocha .image.is-4by5,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-4by5{padding-top:125%}html.theme--catppuccin-mocha .image.is-3by4,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by4{padding-top:133.3333%}html.theme--catppuccin-mocha .image.is-2by3,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-2by3{padding-top:150%}html.theme--catppuccin-mocha .image.is-3by5,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by5{padding-top:166.6666%}html.theme--catppuccin-mocha .image.is-9by16,html.theme--catppuccin-mocha #documenter 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code.language-julia-repl>span.hljs-meta{color:#a6e3a1;font-weight:bolder}html.theme--catppuccin-mocha code .hljs{color:#cdd6f4;background:#1e1e2e}html.theme--catppuccin-mocha code .hljs-keyword{color:#cba6f7}html.theme--catppuccin-mocha code .hljs-built_in{color:#f38ba8}html.theme--catppuccin-mocha code .hljs-type{color:#f9e2af}html.theme--catppuccin-mocha code .hljs-literal{color:#fab387}html.theme--catppuccin-mocha code .hljs-number{color:#fab387}html.theme--catppuccin-mocha code .hljs-operator{color:#94e2d5}html.theme--catppuccin-mocha code .hljs-punctuation{color:#bac2de}html.theme--catppuccin-mocha code .hljs-property{color:#94e2d5}html.theme--catppuccin-mocha code .hljs-regexp{color:#f5c2e7}html.theme--catppuccin-mocha code .hljs-string{color:#a6e3a1}html.theme--catppuccin-mocha code .hljs-char.escape_{color:#a6e3a1}html.theme--catppuccin-mocha code .hljs-subst{color:#a6adc8}html.theme--catppuccin-mocha code .hljs-symbol{color:#f2cdcd}html.theme--catppuccin-mocha code 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.search-result-title{color:#cdd6f4}html.theme--catppuccin-mocha .search-result-highlight{background-color:#f38ba8;color:#181825}html.theme--catppuccin-mocha .search-divider{border-bottom:1px solid #5e6d6f50}html.theme--catppuccin-mocha .w-100{width:100%}html.theme--catppuccin-mocha .gap-2{gap:0.5rem}html.theme--catppuccin-mocha .gap-4{gap:1rem} diff --git a/dev/assets/themes/documenter-dark.css b/dev/assets/themes/documenter-dark.css index 0bf1af7d..c41c82f2 100644 --- a/dev/assets/themes/documenter-dark.css +++ b/dev/assets/themes/documenter-dark.css @@ -1,7702 +1,7 @@ -@charset "UTF-8"; -/* Font Awesome 5 mixin. Can be included in any rule that should render Font Awesome icons. */ -@keyframes spinAround { - from { - transform: rotate(0deg); } - to { - transform: rotate(359deg); } } - -html.theme--documenter-dark .tabs, html.theme--documenter-dark .pagination-previous, -html.theme--documenter-dark .pagination-next, -html.theme--documenter-dark .pagination-link, -html.theme--documenter-dark .pagination-ellipsis, html.theme--documenter-dark .breadcrumb, html.theme--documenter-dark .file, html.theme--documenter-dark .button, .is-unselectable, html.theme--documenter-dark .modal-close, html.theme--documenter-dark .delete { - -webkit-touch-callout: none; - -webkit-user-select: none; - -moz-user-select: none; - -ms-user-select: none; - user-select: none; } - -html.theme--documenter-dark .navbar-link:not(.is-arrowless)::after, html.theme--documenter-dark .select:not(.is-multiple):not(.is-loading)::after { - border: 3px solid transparent; - border-radius: 2px; - border-right: 0; - border-top: 0; - content: " "; - display: block; - height: 0.625em; - margin-top: -0.4375em; - pointer-events: none; - position: absolute; - top: 50%; - transform: rotate(-45deg); - transform-origin: center; - width: 0.625em; } - -html.theme--documenter-dark .admonition:not(:last-child), html.theme--documenter-dark .tabs:not(:last-child), html.theme--documenter-dark .message:not(:last-child), html.theme--documenter-dark .list:not(:last-child), html.theme--documenter-dark .level:not(:last-child), html.theme--documenter-dark .breadcrumb:not(:last-child), html.theme--documenter-dark .highlight:not(:last-child), html.theme--documenter-dark .block:not(:last-child), html.theme--documenter-dark .title:not(:last-child), -html.theme--documenter-dark .subtitle:not(:last-child), html.theme--documenter-dark .table-container:not(:last-child), html.theme--documenter-dark .table:not(:last-child), html.theme--documenter-dark .progress:not(:last-child), html.theme--documenter-dark .notification:not(:last-child), html.theme--documenter-dark .content:not(:last-child), html.theme--documenter-dark .box:not(:last-child) { - margin-bottom: 1.5rem; } - -html.theme--documenter-dark .modal-close, html.theme--documenter-dark .delete { - -moz-appearance: none; - -webkit-appearance: none; - background-color: rgba(10, 10, 10, 0.2); - border: none; - border-radius: 290486px; - cursor: pointer; - pointer-events: auto; - display: inline-block; - flex-grow: 0; - flex-shrink: 0; - font-size: 0; - height: 20px; - max-height: 20px; - max-width: 20px; - min-height: 20px; - min-width: 20px; - outline: none; - position: relative; - vertical-align: top; - width: 20px; } - html.theme--documenter-dark .modal-close::before, html.theme--documenter-dark .delete::before, html.theme--documenter-dark .modal-close::after, html.theme--documenter-dark .delete::after { - background-color: white; - content: ""; - display: block; - left: 50%; - position: absolute; - top: 50%; - transform: translateX(-50%) translateY(-50%) rotate(45deg); - transform-origin: center center; } - html.theme--documenter-dark .modal-close::before, html.theme--documenter-dark .delete::before { - height: 2px; - width: 50%; } - html.theme--documenter-dark .modal-close::after, html.theme--documenter-dark .delete::after { - height: 50%; - width: 2px; } - html.theme--documenter-dark .modal-close:hover, html.theme--documenter-dark .delete:hover, html.theme--documenter-dark .modal-close:focus, html.theme--documenter-dark .delete:focus { - background-color: rgba(10, 10, 10, 0.3); } - html.theme--documenter-dark .modal-close:active, html.theme--documenter-dark .delete:active { - background-color: rgba(10, 10, 10, 0.4); } - html.theme--documenter-dark .is-small.modal-close, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.modal-close, html.theme--documenter-dark .is-small.delete, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.delete { - height: 16px; - max-height: 16px; - max-width: 16px; - min-height: 16px; - min-width: 16px; - width: 16px; } - html.theme--documenter-dark .is-medium.modal-close, html.theme--documenter-dark .is-medium.delete { - height: 24px; - max-height: 24px; - max-width: 24px; - min-height: 24px; - min-width: 24px; - width: 24px; } - html.theme--documenter-dark .is-large.modal-close, html.theme--documenter-dark .is-large.delete { - height: 32px; - max-height: 32px; - max-width: 32px; - min-height: 32px; - min-width: 32px; - width: 32px; } - -html.theme--documenter-dark .control.is-loading::after, html.theme--documenter-dark .select.is-loading::after, html.theme--documenter-dark .loader, html.theme--documenter-dark .button.is-loading::after { - animation: spinAround 500ms infinite linear; - border: 2px solid #dbdee0; - border-radius: 290486px; - border-right-color: transparent; - border-top-color: transparent; - content: ""; - display: block; - height: 1em; - position: relative; - width: 1em; } - -html.theme--documenter-dark .hero-video, html.theme--documenter-dark .modal-background, html.theme--documenter-dark .modal, html.theme--documenter-dark .image.is-square img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-square img, -html.theme--documenter-dark .image.is-square .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-square .has-ratio, html.theme--documenter-dark .image.is-1by1 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by1 img, -html.theme--documenter-dark .image.is-1by1 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by1 .has-ratio, html.theme--documenter-dark .image.is-5by4 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by4 img, -html.theme--documenter-dark .image.is-5by4 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by4 .has-ratio, html.theme--documenter-dark .image.is-4by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by3 img, -html.theme--documenter-dark .image.is-4by3 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by3 .has-ratio, html.theme--documenter-dark .image.is-3by2 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by2 img, -html.theme--documenter-dark .image.is-3by2 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by2 .has-ratio, html.theme--documenter-dark .image.is-5by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by3 img, -html.theme--documenter-dark .image.is-5by3 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by3 .has-ratio, html.theme--documenter-dark .image.is-16by9 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-16by9 img, -html.theme--documenter-dark .image.is-16by9 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-16by9 .has-ratio, html.theme--documenter-dark .image.is-2by1 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by1 img, -html.theme--documenter-dark .image.is-2by1 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by1 .has-ratio, html.theme--documenter-dark .image.is-3by1 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by1 img, -html.theme--documenter-dark .image.is-3by1 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by1 .has-ratio, html.theme--documenter-dark .image.is-4by5 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by5 img, -html.theme--documenter-dark .image.is-4by5 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by5 .has-ratio, html.theme--documenter-dark .image.is-3by4 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by4 img, -html.theme--documenter-dark .image.is-3by4 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by4 .has-ratio, html.theme--documenter-dark .image.is-2by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by3 img, -html.theme--documenter-dark .image.is-2by3 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by3 .has-ratio, html.theme--documenter-dark .image.is-3by5 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by5 img, -html.theme--documenter-dark .image.is-3by5 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by5 .has-ratio, html.theme--documenter-dark .image.is-9by16 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-9by16 img, -html.theme--documenter-dark .image.is-9by16 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-9by16 .has-ratio, html.theme--documenter-dark .image.is-1by2 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by2 img, -html.theme--documenter-dark .image.is-1by2 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by2 .has-ratio, html.theme--documenter-dark .image.is-1by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by3 img, -html.theme--documenter-dark .image.is-1by3 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by3 .has-ratio, .is-overlay { - bottom: 0; - left: 0; - position: absolute; - right: 0; - top: 0; } - -html.theme--documenter-dark .pagination-previous, -html.theme--documenter-dark .pagination-next, -html.theme--documenter-dark .pagination-link, -html.theme--documenter-dark .pagination-ellipsis, html.theme--documenter-dark .file-cta, -html.theme--documenter-dark .file-name, html.theme--documenter-dark .select select, html.theme--documenter-dark .textarea, html.theme--documenter-dark .input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input, html.theme--documenter-dark .button { - -moz-appearance: none; - -webkit-appearance: none; - align-items: center; - border: 1px solid transparent; - border-radius: 0.4em; - box-shadow: none; - display: inline-flex; - font-size: 15px; - height: 2.25em; - justify-content: flex-start; - line-height: 1.5; - padding-bottom: calc(0.375em - 1px); - padding-left: calc(0.625em - 1px); - padding-right: calc(0.625em - 1px); - padding-top: calc(0.375em - 1px); - position: relative; - vertical-align: top; } - html.theme--documenter-dark .pagination-previous:focus, - html.theme--documenter-dark .pagination-next:focus, - html.theme--documenter-dark .pagination-link:focus, - html.theme--documenter-dark .pagination-ellipsis:focus, html.theme--documenter-dark 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1056px){.is-invisible-desktop{visibility:hidden !important}}@media screen and (min-width: 1056px) and (max-width: 1215px){.is-invisible-desktop-only{visibility:hidden !important}}@media screen and (min-width: 1216px){.is-invisible-widescreen{visibility:hidden !important}}@media screen and (min-width: 1216px) and (max-width: 1407px){.is-invisible-widescreen-only{visibility:hidden !important}}@media screen and (min-width: 1408px){.is-invisible-fullhd{visibility:hidden !important}}html.theme--documenter-dark{/*! Theme: a11y-dark Author: @ericwbailey Maintainer: @ericwbailey Based on the Tomorrow Night Eighties theme: https://github.com/isagalaev/highlight.js/blob/master/src/styles/tomorrow-night-eighties.css -*/ - /* Comment */ - /* Red */ - /* Orange */ - /* Yellow */ - /* Green */ - /* Blue */ - /* Purple */ } - html.theme--documenter-dark html { - background-color: #1f2424; - font-size: 16px; - -moz-osx-font-smoothing: grayscale; - -webkit-font-smoothing: antialiased; - min-width: 300px; - overflow-x: auto; - overflow-y: scroll; - text-rendering: optimizeLegibility; - text-size-adjust: 100%; } - html.theme--documenter-dark article, - html.theme--documenter-dark aside, - html.theme--documenter-dark figure, - html.theme--documenter-dark footer, - html.theme--documenter-dark header, - html.theme--documenter-dark hgroup, - html.theme--documenter-dark section { - display: block; } - html.theme--documenter-dark body, - html.theme--documenter-dark button, - html.theme--documenter-dark input, - html.theme--documenter-dark select, - html.theme--documenter-dark textarea { - font-family: "Lato Medium", -apple-system, BlinkMacSystemFont, "Segoe UI", "Helvetica Neue", "Helvetica", "Arial", sans-serif; } - html.theme--documenter-dark code, - html.theme--documenter-dark pre { - -moz-osx-font-smoothing: auto; - -webkit-font-smoothing: auto; - font-family: "JuliaMono", "SFMono-Regular", "Menlo", "Consolas", "Liberation Mono", "DejaVu Sans Mono", monospace; } - html.theme--documenter-dark body { - color: #fff; - font-size: 1em; - font-weight: 400; - line-height: 1.5; } - html.theme--documenter-dark a { - color: #1abc9c; - cursor: pointer; - text-decoration: none; } - html.theme--documenter-dark a strong { - color: currentColor; } - html.theme--documenter-dark a:hover { - color: #1dd2af; } - html.theme--documenter-dark code { - background-color: rgba(255, 255, 255, 0.05); - color: #ececec; - font-size: 0.875em; - font-weight: normal; - padding: 0.1em; } - html.theme--documenter-dark hr { - background-color: #282f2f; - border: none; - display: block; - height: 2px; - margin: 1.5rem 0; } - html.theme--documenter-dark img { - height: auto; - max-width: 100%; } - html.theme--documenter-dark input[type="checkbox"], - html.theme--documenter-dark input[type="radio"] { - vertical-align: baseline; } - html.theme--documenter-dark small { - font-size: 0.875em; } - html.theme--documenter-dark span { - font-style: inherit; - font-weight: inherit; } - html.theme--documenter-dark strong { - color: #f2f2f2; - font-weight: 700; } - html.theme--documenter-dark fieldset { - border: none; } - html.theme--documenter-dark pre { - -webkit-overflow-scrolling: touch; - background-color: #282f2f; - color: #fff; - font-size: 0.875em; - overflow-x: auto; - padding: 1.25rem 1.5rem; - white-space: pre; - word-wrap: normal; } - html.theme--documenter-dark pre code { - background-color: transparent; - color: currentColor; - font-size: 1em; - padding: 0; } - html.theme--documenter-dark table td, - html.theme--documenter-dark table th { - vertical-align: top; } - html.theme--documenter-dark table td:not([align]), - html.theme--documenter-dark table th:not([align]) { - text-align: left; } - html.theme--documenter-dark table th { - color: #f2f2f2; } - html.theme--documenter-dark .box { - background-color: #343c3d; - border-radius: 8px; - box-shadow: none; - color: #fff; - display: block; - padding: 1.25rem; } - html.theme--documenter-dark a.box:hover, html.theme--documenter-dark a.box:focus { - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px #1abc9c; } - html.theme--documenter-dark a.box:active { - box-shadow: inset 0 1px 2px rgba(10, 10, 10, 0.2), 0 0 0 1px #1abc9c; } - html.theme--documenter-dark .button { - background-color: #282f2f; - border-color: #4c5759; - border-width: 1px; - color: #375a7f; - cursor: pointer; - justify-content: center; - padding-bottom: calc(0.375em - 1px); - padding-left: 0.75em; - padding-right: 0.75em; - padding-top: calc(0.375em - 1px); - text-align: center; - white-space: nowrap; } - html.theme--documenter-dark .button strong { - color: inherit; } - html.theme--documenter-dark .button .icon, html.theme--documenter-dark .button .icon.is-small, html.theme--documenter-dark .button #documenter .docs-sidebar form.docs-search > input.icon, html.theme--documenter-dark #documenter .docs-sidebar .button form.docs-search > input.icon, html.theme--documenter-dark .button .icon.is-medium, html.theme--documenter-dark .button .icon.is-large { - height: 1.5em; - width: 1.5em; } - html.theme--documenter-dark .button .icon:first-child:not(:last-child) { - margin-left: calc(-0.375em - 1px); - margin-right: 0.1875em; } - html.theme--documenter-dark .button .icon:last-child:not(:first-child) { - margin-left: 0.1875em; - margin-right: calc(-0.375em - 1px); } - html.theme--documenter-dark .button .icon:first-child:last-child { - margin-left: calc(-0.375em - 1px); - margin-right: calc(-0.375em - 1px); } - html.theme--documenter-dark 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html.theme--documenter-dark .button.is-text:active, html.theme--documenter-dark .button.is-text.is-active { - background-color: #1d2122; - color: #f2f2f2; } - html.theme--documenter-dark .button.is-text[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-text { - background-color: transparent; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-white { - background-color: white; - border-color: transparent; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-white:hover, html.theme--documenter-dark .button.is-white.is-hovered { - background-color: #f9f9f9; - border-color: transparent; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-white:focus, html.theme--documenter-dark .button.is-white.is-focused { - border-color: transparent; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-white:focus:not(:active), html.theme--documenter-dark .button.is-white.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); } - html.theme--documenter-dark .button.is-white:active, html.theme--documenter-dark .button.is-white.is-active { - background-color: #f2f2f2; - border-color: transparent; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-white[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-white { - background-color: white; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-white.is-inverted { - background-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .button.is-white.is-inverted:hover, html.theme--documenter-dark .button.is-white.is-inverted.is-hovered { - background-color: black; } - html.theme--documenter-dark .button.is-white.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-white.is-inverted { - background-color: #0a0a0a; - border-color: transparent; - box-shadow: none; - color: white; } - html.theme--documenter-dark 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html.theme--documenter-dark .button.is-white.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - html.theme--documenter-dark .button.is-white.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-white.is-outlined { - background-color: transparent; - border-color: white; - box-shadow: none; - color: white; } - html.theme--documenter-dark .button.is-white.is-inverted.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-white.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-focused { - background-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent white white !important; } - html.theme--documenter-dark .button.is-white.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-white.is-inverted.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - box-shadow: none; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-black { - background-color: #0a0a0a; - border-color: transparent; - color: white; } - html.theme--documenter-dark .button.is-black:hover, html.theme--documenter-dark .button.is-black.is-hovered { - background-color: #040404; - border-color: transparent; - color: white; } - html.theme--documenter-dark .button.is-black:focus, html.theme--documenter-dark .button.is-black.is-focused { - border-color: transparent; - color: white; } - html.theme--documenter-dark .button.is-black:focus:not(:active), html.theme--documenter-dark .button.is-black.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); } - html.theme--documenter-dark .button.is-black:active, html.theme--documenter-dark .button.is-black.is-active { - background-color: black; - border-color: transparent; - color: white; } - html.theme--documenter-dark .button.is-black[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-black { - background-color: #0a0a0a; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-black.is-inverted { - background-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-black.is-inverted:hover, html.theme--documenter-dark .button.is-black.is-inverted.is-hovered { - background-color: #f2f2f2; } - html.theme--documenter-dark .button.is-black.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted { - background-color: white; - border-color: transparent; - box-shadow: none; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-black.is-loading::after { - border-color: transparent transparent white white !important; } - html.theme--documenter-dark .button.is-black.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-black.is-outlined:hover, html.theme--documenter-dark .button.is-black.is-outlined.is-hovered, html.theme--documenter-dark .button.is-black.is-outlined:focus, html.theme--documenter-dark .button.is-black.is-outlined.is-focused { - background-color: #0a0a0a; - border-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .button.is-black.is-outlined.is-loading::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - html.theme--documenter-dark .button.is-black.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-black.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent white white !important; } - html.theme--documenter-dark .button.is-black.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-black.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - box-shadow: none; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-black.is-inverted.is-outlined { - background-color: transparent; - border-color: white; - color: white; } - html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-focused { - background-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - html.theme--documenter-dark .button.is-black.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted.is-outlined { - background-color: transparent; - border-color: white; - box-shadow: none; - color: white; } - html.theme--documenter-dark .button.is-light { - background-color: #ecf0f1; - border-color: transparent; - color: #282f2f; } - html.theme--documenter-dark .button.is-light:hover, html.theme--documenter-dark .button.is-light.is-hovered { - background-color: #e5eaec; - border-color: transparent; - color: #282f2f; } - html.theme--documenter-dark .button.is-light:focus, html.theme--documenter-dark .button.is-light.is-focused { - border-color: transparent; - color: #282f2f; } - html.theme--documenter-dark .button.is-light:focus:not(:active), html.theme--documenter-dark .button.is-light.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(236, 240, 241, 0.25); } - html.theme--documenter-dark .button.is-light:active, html.theme--documenter-dark .button.is-light.is-active { - background-color: #dde4e6; - border-color: transparent; - color: #282f2f; } - html.theme--documenter-dark .button.is-light[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-light { - background-color: #ecf0f1; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-light.is-inverted { - background-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-light.is-inverted:hover, html.theme--documenter-dark .button.is-light.is-inverted.is-hovered { - background-color: #1d2122; } - html.theme--documenter-dark .button.is-light.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted { - background-color: #282f2f; - border-color: transparent; - box-shadow: none; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-light.is-loading::after { - border-color: transparent transparent #282f2f #282f2f !important; } - html.theme--documenter-dark .button.is-light.is-outlined { - background-color: transparent; - border-color: #ecf0f1; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-light.is-outlined:hover, html.theme--documenter-dark .button.is-light.is-outlined.is-hovered, html.theme--documenter-dark .button.is-light.is-outlined:focus, html.theme--documenter-dark .button.is-light.is-outlined.is-focused { - background-color: #ecf0f1; - border-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .button.is-light.is-outlined.is-loading::after { - border-color: transparent transparent #ecf0f1 #ecf0f1 !important; } - html.theme--documenter-dark .button.is-light.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-light.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #282f2f #282f2f !important; } - html.theme--documenter-dark .button.is-light.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-light.is-outlined { - background-color: transparent; - border-color: #ecf0f1; - box-shadow: none; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-light.is-inverted.is-outlined { - background-color: transparent; - border-color: #282f2f; - color: #282f2f; } - html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-focused { - background-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #ecf0f1 #ecf0f1 !important; } - html.theme--documenter-dark .button.is-light.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted.is-outlined { - background-color: transparent; - border-color: #282f2f; - box-shadow: none; - color: #282f2f; } - html.theme--documenter-dark .button.is-dark, html.theme--documenter-dark .content kbd.button { - background-color: #282f2f; - border-color: transparent; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-dark:hover, html.theme--documenter-dark .content kbd.button:hover, html.theme--documenter-dark .button.is-dark.is-hovered, html.theme--documenter-dark .content kbd.button.is-hovered { - background-color: #232829; - border-color: transparent; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-dark:focus, html.theme--documenter-dark .content kbd.button:focus, html.theme--documenter-dark .button.is-dark.is-focused, html.theme--documenter-dark .content kbd.button.is-focused { - border-color: transparent; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-dark:focus:not(:active), html.theme--documenter-dark .content kbd.button:focus:not(:active), html.theme--documenter-dark .button.is-dark.is-focused:not(:active), html.theme--documenter-dark .content kbd.button.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(40, 47, 47, 0.25); } - html.theme--documenter-dark .button.is-dark:active, html.theme--documenter-dark .content kbd.button:active, html.theme--documenter-dark .button.is-dark.is-active, html.theme--documenter-dark .content kbd.button.is-active { - background-color: #1d2122; - border-color: transparent; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-dark[disabled], html.theme--documenter-dark .content kbd.button[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-dark, fieldset[disabled] html.theme--documenter-dark .content kbd.button { - background-color: #282f2f; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-dark.is-inverted, html.theme--documenter-dark .content kbd.button.is-inverted { - background-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .button.is-dark.is-inverted:hover, html.theme--documenter-dark .content kbd.button.is-inverted:hover, html.theme--documenter-dark .button.is-dark.is-inverted.is-hovered, html.theme--documenter-dark .content kbd.button.is-inverted.is-hovered { - background-color: #dde4e6; } - html.theme--documenter-dark .button.is-dark.is-inverted[disabled], html.theme--documenter-dark .content kbd.button.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted, fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted { - background-color: #ecf0f1; - border-color: transparent; - box-shadow: none; - color: #282f2f; } - html.theme--documenter-dark .button.is-dark.is-loading::after, html.theme--documenter-dark .content kbd.button.is-loading::after { - border-color: transparent transparent #ecf0f1 #ecf0f1 !important; } - html.theme--documenter-dark .button.is-dark.is-outlined, html.theme--documenter-dark .content kbd.button.is-outlined { - background-color: transparent; - border-color: #282f2f; - color: #282f2f; } - html.theme--documenter-dark .button.is-dark.is-outlined:hover, html.theme--documenter-dark .content kbd.button.is-outlined:hover, html.theme--documenter-dark .button.is-dark.is-outlined.is-hovered, html.theme--documenter-dark .content kbd.button.is-outlined.is-hovered, html.theme--documenter-dark .button.is-dark.is-outlined:focus, html.theme--documenter-dark .content kbd.button.is-outlined:focus, html.theme--documenter-dark .button.is-dark.is-outlined.is-focused, html.theme--documenter-dark .content kbd.button.is-outlined.is-focused { - background-color: #282f2f; - border-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-dark.is-outlined.is-loading::after, html.theme--documenter-dark .content kbd.button.is-outlined.is-loading::after { - border-color: transparent transparent #282f2f #282f2f !important; } - html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:hover::after, html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after, html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after, html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #ecf0f1 #ecf0f1 !important; } - html.theme--documenter-dark .button.is-dark.is-outlined[disabled], html.theme--documenter-dark .content kbd.button.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined, fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined { - background-color: transparent; - border-color: #282f2f; - box-shadow: none; - color: #282f2f; } - html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined { - background-color: transparent; - border-color: #ecf0f1; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-focused { - background-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #282f2f #282f2f !important; } - html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined[disabled], html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined, fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined { - background-color: transparent; - border-color: #ecf0f1; - box-shadow: none; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-primary, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink { - background-color: #375a7f; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-primary:hover, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:hover, html.theme--documenter-dark .button.is-primary.is-hovered, html.theme--documenter-dark .docstring > section > a.button.is-hovered.docs-sourcelink { - background-color: #335476; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-primary:focus, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:focus, html.theme--documenter-dark .button.is-primary.is-focused, html.theme--documenter-dark .docstring > section > a.button.is-focused.docs-sourcelink { - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-primary:focus:not(:active), html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:focus:not(:active), html.theme--documenter-dark .button.is-primary.is-focused:not(:active), html.theme--documenter-dark .docstring > section > a.button.is-focused.docs-sourcelink:not(:active) { - box-shadow: 0 0 0 0.125em rgba(55, 90, 127, 0.25); } - html.theme--documenter-dark .button.is-primary:active, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:active, html.theme--documenter-dark .button.is-primary.is-active, html.theme--documenter-dark .docstring > section > a.button.is-active.docs-sourcelink { - background-color: #2f4d6d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-primary[disabled], html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-primary, fieldset[disabled] html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink { - background-color: #375a7f; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-primary.is-inverted, html.theme--documenter-dark .docstring > section > a.button.is-inverted.docs-sourcelink { - background-color: #fff; - color: #375a7f; } - html.theme--documenter-dark .button.is-primary.is-inverted:hover, html.theme--documenter-dark .docstring > section > a.button.is-inverted.docs-sourcelink:hover, html.theme--documenter-dark .button.is-primary.is-inverted.is-hovered, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-hovered.docs-sourcelink { - background-color: #f2f2f2; } - html.theme--documenter-dark .button.is-primary.is-inverted[disabled], html.theme--documenter-dark .docstring > section > a.button.is-inverted.docs-sourcelink[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-primary.is-inverted, fieldset[disabled] html.theme--documenter-dark .docstring > section > a.button.is-inverted.docs-sourcelink { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #375a7f; } - html.theme--documenter-dark .button.is-primary.is-loading::after, html.theme--documenter-dark .docstring > section > a.button.is-loading.docs-sourcelink::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-primary.is-outlined, html.theme--documenter-dark .docstring > section > a.button.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #375a7f; - color: #375a7f; } - html.theme--documenter-dark .button.is-primary.is-outlined:hover, html.theme--documenter-dark .docstring > section > a.button.is-outlined.docs-sourcelink:hover, html.theme--documenter-dark .button.is-primary.is-outlined.is-hovered, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-hovered.docs-sourcelink, html.theme--documenter-dark .button.is-primary.is-outlined:focus, html.theme--documenter-dark .docstring > section > a.button.is-outlined.docs-sourcelink:focus, html.theme--documenter-dark .button.is-primary.is-outlined.is-focused, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-focused.docs-sourcelink { - background-color: #375a7f; - border-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .button.is-primary.is-outlined.is-loading::after, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink::after { - border-color: transparent transparent #375a7f #375a7f !important; } - html.theme--documenter-dark .button.is-primary.is-outlined.is-loading:hover::after, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink:hover::after, html.theme--documenter-dark .button.is-primary.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-loading.is-hovered.docs-sourcelink::after, html.theme--documenter-dark .button.is-primary.is-outlined.is-loading:focus::after, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink:focus::after, html.theme--documenter-dark .button.is-primary.is-outlined.is-loading.is-focused::after, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-loading.is-focused.docs-sourcelink::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-primary.is-outlined[disabled], html.theme--documenter-dark .docstring > section > a.button.is-outlined.docs-sourcelink[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-primary.is-outlined, fieldset[disabled] html.theme--documenter-dark .docstring > section > a.button.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #375a7f; - box-shadow: none; - color: #375a7f; } - html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #fff; - color: #fff; } - html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined:hover, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink:hover, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-hovered.docs-sourcelink, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined:focus, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink:focus, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-focused, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-focused.docs-sourcelink { - background-color: #fff; - color: #375a7f; } - html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:hover::after, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-loading.is-hovered.docs-sourcelink::after, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:focus::after, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-loading.is-focused::after, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-loading.is-focused.docs-sourcelink::after { - border-color: transparent transparent #375a7f #375a7f !important; } - html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined[disabled], html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined, fieldset[disabled] html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - html.theme--documenter-dark .button.is-link { - background-color: #1abc9c; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-link:hover, html.theme--documenter-dark .button.is-link.is-hovered { - background-color: #18b193; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-link:focus, html.theme--documenter-dark .button.is-link.is-focused { - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-link:focus:not(:active), html.theme--documenter-dark .button.is-link.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(26, 188, 156, 0.25); } - html.theme--documenter-dark .button.is-link:active, html.theme--documenter-dark .button.is-link.is-active { - background-color: #17a689; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-link[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-link { - background-color: #1abc9c; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-link.is-inverted { - background-color: #fff; - color: #1abc9c; } - html.theme--documenter-dark .button.is-link.is-inverted:hover, html.theme--documenter-dark .button.is-link.is-inverted.is-hovered { - background-color: #f2f2f2; } - html.theme--documenter-dark .button.is-link.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-link.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #1abc9c; } - html.theme--documenter-dark .button.is-link.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-link.is-outlined { - background-color: transparent; - border-color: #1abc9c; - color: #1abc9c; } - html.theme--documenter-dark .button.is-link.is-outlined:hover, html.theme--documenter-dark .button.is-link.is-outlined.is-hovered, html.theme--documenter-dark .button.is-link.is-outlined:focus, html.theme--documenter-dark .button.is-link.is-outlined.is-focused { - background-color: #1abc9c; - border-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .button.is-link.is-outlined.is-loading::after { - border-color: transparent transparent #1abc9c #1abc9c !important; } - html.theme--documenter-dark .button.is-link.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-link.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-link.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-link.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-link.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-link.is-outlined { - background-color: transparent; - border-color: #1abc9c; - box-shadow: none; - color: #1abc9c; } - html.theme--documenter-dark .button.is-link.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - html.theme--documenter-dark .button.is-link.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #1abc9c; } - html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #1abc9c #1abc9c !important; } - html.theme--documenter-dark .button.is-link.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-link.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - html.theme--documenter-dark .button.is-info { - background-color: #024c7d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-info:hover, html.theme--documenter-dark .button.is-info.is-hovered { - background-color: #024470; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-info:focus, html.theme--documenter-dark .button.is-info.is-focused { - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-info:focus:not(:active), html.theme--documenter-dark .button.is-info.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(2, 76, 125, 0.25); } - html.theme--documenter-dark .button.is-info:active, html.theme--documenter-dark .button.is-info.is-active { - background-color: #023d64; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-info[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-info { - background-color: #024c7d; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-info.is-inverted { - background-color: #fff; - color: #024c7d; } - html.theme--documenter-dark .button.is-info.is-inverted:hover, html.theme--documenter-dark .button.is-info.is-inverted.is-hovered { - background-color: #f2f2f2; } - html.theme--documenter-dark .button.is-info.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-info.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #024c7d; } - html.theme--documenter-dark .button.is-info.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-info.is-outlined { - background-color: transparent; - border-color: #024c7d; - color: #024c7d; } - html.theme--documenter-dark .button.is-info.is-outlined:hover, html.theme--documenter-dark .button.is-info.is-outlined.is-hovered, html.theme--documenter-dark .button.is-info.is-outlined:focus, html.theme--documenter-dark .button.is-info.is-outlined.is-focused { - background-color: #024c7d; - border-color: #024c7d; - color: #fff; } - html.theme--documenter-dark .button.is-info.is-outlined.is-loading::after { - border-color: transparent transparent #024c7d #024c7d !important; } - html.theme--documenter-dark .button.is-info.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-info.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-info.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-info.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-info.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-info.is-outlined { - background-color: transparent; - border-color: #024c7d; - box-shadow: none; - color: #024c7d; } - html.theme--documenter-dark .button.is-info.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - html.theme--documenter-dark .button.is-info.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #024c7d; } - html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #024c7d #024c7d !important; } - html.theme--documenter-dark .button.is-info.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-info.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - html.theme--documenter-dark .button.is-success { - background-color: #008438; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-success:hover, html.theme--documenter-dark .button.is-success.is-hovered { - background-color: #007733; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-success:focus, html.theme--documenter-dark .button.is-success.is-focused { - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-success:focus:not(:active), html.theme--documenter-dark .button.is-success.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(0, 132, 56, 0.25); } - html.theme--documenter-dark .button.is-success:active, html.theme--documenter-dark .button.is-success.is-active { - background-color: #006b2d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-success[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-success { - background-color: #008438; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-success.is-inverted { - background-color: #fff; - color: #008438; } - html.theme--documenter-dark .button.is-success.is-inverted:hover, html.theme--documenter-dark .button.is-success.is-inverted.is-hovered { - background-color: #f2f2f2; } - html.theme--documenter-dark .button.is-success.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-success.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #008438; } - html.theme--documenter-dark .button.is-success.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-success.is-outlined { - background-color: transparent; - border-color: #008438; - color: #008438; } - html.theme--documenter-dark .button.is-success.is-outlined:hover, html.theme--documenter-dark .button.is-success.is-outlined.is-hovered, html.theme--documenter-dark .button.is-success.is-outlined:focus, html.theme--documenter-dark .button.is-success.is-outlined.is-focused { - background-color: #008438; - border-color: #008438; - color: #fff; } - html.theme--documenter-dark .button.is-success.is-outlined.is-loading::after { - border-color: transparent transparent #008438 #008438 !important; } - html.theme--documenter-dark .button.is-success.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-success.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-success.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-success.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-success.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-success.is-outlined { - background-color: transparent; - border-color: #008438; - box-shadow: none; - color: #008438; } - html.theme--documenter-dark .button.is-success.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - html.theme--documenter-dark .button.is-success.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #008438; } - html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #008438 #008438 !important; } - html.theme--documenter-dark .button.is-success.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-success.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - html.theme--documenter-dark .button.is-warning { - background-color: #ad8100; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-warning:hover, html.theme--documenter-dark .button.is-warning.is-hovered { - background-color: #a07700; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-warning:focus, html.theme--documenter-dark .button.is-warning.is-focused { - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-warning:focus:not(:active), html.theme--documenter-dark .button.is-warning.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(173, 129, 0, 0.25); } - html.theme--documenter-dark .button.is-warning:active, html.theme--documenter-dark .button.is-warning.is-active { - background-color: #946e00; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-warning[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-warning { - background-color: #ad8100; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-warning.is-inverted { - background-color: #fff; - color: #ad8100; } - html.theme--documenter-dark .button.is-warning.is-inverted:hover, html.theme--documenter-dark .button.is-warning.is-inverted.is-hovered { - background-color: #f2f2f2; } - html.theme--documenter-dark .button.is-warning.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #ad8100; } - html.theme--documenter-dark .button.is-warning.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-warning.is-outlined { - background-color: transparent; - border-color: #ad8100; - color: #ad8100; } - html.theme--documenter-dark .button.is-warning.is-outlined:hover, html.theme--documenter-dark .button.is-warning.is-outlined.is-hovered, html.theme--documenter-dark .button.is-warning.is-outlined:focus, html.theme--documenter-dark .button.is-warning.is-outlined.is-focused { - background-color: #ad8100; - border-color: #ad8100; - color: #fff; } - html.theme--documenter-dark .button.is-warning.is-outlined.is-loading::after { - border-color: transparent transparent #ad8100 #ad8100 !important; } - html.theme--documenter-dark .button.is-warning.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-warning.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-warning.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-warning.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-warning.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-outlined { - background-color: transparent; - border-color: #ad8100; - box-shadow: none; - color: #ad8100; } - html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #ad8100; } - html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #ad8100 #ad8100 !important; } - html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - html.theme--documenter-dark .button.is-danger { - background-color: #9e1b0d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-danger:hover, html.theme--documenter-dark .button.is-danger.is-hovered { - background-color: #92190c; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-danger:focus, html.theme--documenter-dark .button.is-danger.is-focused { - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-danger:focus:not(:active), html.theme--documenter-dark .button.is-danger.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(158, 27, 13, 0.25); } - html.theme--documenter-dark .button.is-danger:active, html.theme--documenter-dark .button.is-danger.is-active { - background-color: #86170b; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-danger[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-danger { - background-color: #9e1b0d; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-danger.is-inverted { - background-color: #fff; - color: #9e1b0d; } - html.theme--documenter-dark .button.is-danger.is-inverted:hover, html.theme--documenter-dark .button.is-danger.is-inverted.is-hovered { - background-color: #f2f2f2; } - html.theme--documenter-dark .button.is-danger.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #9e1b0d; } - html.theme--documenter-dark .button.is-danger.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-danger.is-outlined { - background-color: transparent; - border-color: #9e1b0d; - color: #9e1b0d; } - html.theme--documenter-dark .button.is-danger.is-outlined:hover, html.theme--documenter-dark .button.is-danger.is-outlined.is-hovered, html.theme--documenter-dark .button.is-danger.is-outlined:focus, html.theme--documenter-dark .button.is-danger.is-outlined.is-focused { - background-color: #9e1b0d; - border-color: #9e1b0d; - color: #fff; } - html.theme--documenter-dark .button.is-danger.is-outlined.is-loading::after { - border-color: transparent transparent #9e1b0d #9e1b0d !important; } - html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-danger.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-outlined { - background-color: transparent; - border-color: #9e1b0d; - box-shadow: none; - color: #9e1b0d; } - html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #9e1b0d; } - html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #9e1b0d #9e1b0d !important; } - html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - html.theme--documenter-dark .button.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.button { - border-radius: 3px; - font-size: 0.85em; } - html.theme--documenter-dark .button.is-normal { - font-size: 15px; } - html.theme--documenter-dark .button.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .button.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .button[disabled], fieldset[disabled] html.theme--documenter-dark .button { - background-color: #8c9b9d; - border-color: #dbdee0; - box-shadow: none; - opacity: 0.5; } - html.theme--documenter-dark .button.is-fullwidth { - display: flex; - width: 100%; } - html.theme--documenter-dark .button.is-loading { - color: transparent !important; - pointer-events: none; } - html.theme--documenter-dark .button.is-loading::after { - position: absolute; - left: calc(50% - (1em / 2)); - top: calc(50% - (1em / 2)); - position: absolute !important; } - html.theme--documenter-dark .button.is-static { - background-color: #282f2f; - border-color: #5e6d6f; - color: #dbdee0; - box-shadow: none; - pointer-events: none; } - html.theme--documenter-dark .button.is-rounded, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.button { - border-radius: 290486px; - padding-left: 1em; - padding-right: 1em; } - html.theme--documenter-dark .buttons { - align-items: center; - display: flex; - flex-wrap: wrap; - justify-content: flex-start; } - html.theme--documenter-dark .buttons .button { - margin-bottom: 0.5rem; } - html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth) { - margin-right: 0.5rem; } - html.theme--documenter-dark .buttons:last-child { - margin-bottom: -0.5rem; } - html.theme--documenter-dark .buttons:not(:last-child) { - margin-bottom: 1rem; } - html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large) { - border-radius: 3px; - font-size: 0.85em; } - html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large) { - font-size: 1.25rem; } - html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium) { - font-size: 1.5rem; } - html.theme--documenter-dark .buttons.has-addons .button:not(:first-child) { - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - html.theme--documenter-dark .buttons.has-addons .button:not(:last-child) { - border-bottom-right-radius: 0; - border-top-right-radius: 0; - margin-right: -1px; } - html.theme--documenter-dark .buttons.has-addons .button:last-child { - margin-right: 0; } - html.theme--documenter-dark .buttons.has-addons .button:hover, html.theme--documenter-dark .buttons.has-addons .button.is-hovered { - z-index: 2; } - html.theme--documenter-dark .buttons.has-addons .button:focus, html.theme--documenter-dark .buttons.has-addons .button.is-focused, html.theme--documenter-dark .buttons.has-addons .button:active, html.theme--documenter-dark .buttons.has-addons .button.is-active, html.theme--documenter-dark .buttons.has-addons .button.is-selected { - z-index: 3; } - html.theme--documenter-dark .buttons.has-addons .button:focus:hover, html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover, html.theme--documenter-dark .buttons.has-addons .button:active:hover, html.theme--documenter-dark .buttons.has-addons .button.is-active:hover, html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover { - z-index: 4; } - html.theme--documenter-dark .buttons.has-addons .button.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - html.theme--documenter-dark .buttons.is-centered { - justify-content: center; } - html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth) { - margin-left: 0.25rem; - margin-right: 0.25rem; } - html.theme--documenter-dark .buttons.is-right { - justify-content: flex-end; } - html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth) { - margin-left: 0.25rem; - margin-right: 0.25rem; } - html.theme--documenter-dark .container { - flex-grow: 1; - margin: 0 auto; - position: relative; - width: auto; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .container { - max-width: 992px; } - html.theme--documenter-dark .container.is-fluid { - margin-left: 32px; - margin-right: 32px; - max-width: none; } } - @media screen and (max-width: 1215px) { - html.theme--documenter-dark .container.is-widescreen { - max-width: 1152px; } } - @media screen and (max-width: 1407px) { - html.theme--documenter-dark .container.is-fullhd { - max-width: 1344px; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .container { - max-width: 1152px; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .container { - max-width: 1344px; } } - html.theme--documenter-dark .content li + li { - margin-top: 0.25em; } - html.theme--documenter-dark .content p:not(:last-child), - html.theme--documenter-dark .content dl:not(:last-child), - html.theme--documenter-dark .content ol:not(:last-child), - html.theme--documenter-dark .content ul:not(:last-child), - html.theme--documenter-dark .content blockquote:not(:last-child), - html.theme--documenter-dark .content pre:not(:last-child), - html.theme--documenter-dark .content table:not(:last-child) { - margin-bottom: 1em; } - html.theme--documenter-dark .content h1, - html.theme--documenter-dark .content h2, - html.theme--documenter-dark .content h3, - html.theme--documenter-dark .content h4, - html.theme--documenter-dark .content h5, - html.theme--documenter-dark .content h6 { - color: #f2f2f2; - font-weight: 600; - line-height: 1.125; } - html.theme--documenter-dark .content h1 { - font-size: 2em; - margin-bottom: 0.5em; } - html.theme--documenter-dark .content h1:not(:first-child) { - margin-top: 1em; } - html.theme--documenter-dark .content h2 { - font-size: 1.75em; - margin-bottom: 0.5714em; } - html.theme--documenter-dark .content h2:not(:first-child) { - margin-top: 1.1428em; } - html.theme--documenter-dark .content h3 { - font-size: 1.5em; - margin-bottom: 0.6666em; } - html.theme--documenter-dark .content h3:not(:first-child) { - margin-top: 1.3333em; } - html.theme--documenter-dark .content h4 { - font-size: 1.25em; - margin-bottom: 0.8em; } - html.theme--documenter-dark .content h5 { - font-size: 1.125em; - margin-bottom: 0.8888em; } - html.theme--documenter-dark .content h6 { - font-size: 1em; - margin-bottom: 1em; } - html.theme--documenter-dark .content blockquote { - background-color: #282f2f; - border-left: 5px solid #5e6d6f; - padding: 1.25em 1.5em; } - html.theme--documenter-dark .content ol { - list-style-position: outside; - margin-left: 2em; - margin-top: 1em; } - html.theme--documenter-dark .content ol:not([type]) { - list-style-type: decimal; } - html.theme--documenter-dark .content ol.is-lower-alpha:not([type]) { - list-style-type: lower-alpha; } - html.theme--documenter-dark .content ol.is-lower-roman:not([type]) { - list-style-type: lower-roman; } - html.theme--documenter-dark .content ol.is-upper-alpha:not([type]) { - list-style-type: upper-alpha; } - html.theme--documenter-dark .content ol.is-upper-roman:not([type]) { - list-style-type: upper-roman; } - html.theme--documenter-dark .content ul { - list-style: disc outside; - margin-left: 2em; - margin-top: 1em; } - html.theme--documenter-dark .content ul ul { - list-style-type: circle; - margin-top: 0.5em; } - html.theme--documenter-dark .content ul ul ul { - list-style-type: square; } - html.theme--documenter-dark .content dd { - margin-left: 2em; } - html.theme--documenter-dark .content figure { - margin-left: 2em; - margin-right: 2em; - text-align: center; } - html.theme--documenter-dark .content figure:not(:first-child) { - margin-top: 2em; } - html.theme--documenter-dark .content figure:not(:last-child) { - margin-bottom: 2em; } - html.theme--documenter-dark .content figure img { - display: inline-block; } - html.theme--documenter-dark .content figure figcaption { - font-style: italic; } - html.theme--documenter-dark .content pre { - -webkit-overflow-scrolling: touch; - overflow-x: auto; - padding: 0; - white-space: pre; - word-wrap: normal; } - html.theme--documenter-dark .content sup, - html.theme--documenter-dark .content sub { - font-size: 75%; } - html.theme--documenter-dark .content table { - width: 100%; } - html.theme--documenter-dark .content table td, - html.theme--documenter-dark .content table th { - border: 1px solid #5e6d6f; - border-width: 0 0 1px; - padding: 0.5em 0.75em; - vertical-align: top; } - html.theme--documenter-dark .content table th { - color: #f2f2f2; } - html.theme--documenter-dark .content table th:not([align]) { - text-align: left; } - html.theme--documenter-dark .content table thead td, - html.theme--documenter-dark .content table thead th { - border-width: 0 0 2px; - color: #f2f2f2; } - html.theme--documenter-dark .content table tfoot td, - html.theme--documenter-dark .content table tfoot th { - border-width: 2px 0 0; - color: #f2f2f2; } - html.theme--documenter-dark .content table tbody tr:last-child td, - html.theme--documenter-dark .content table tbody tr:last-child th { - border-bottom-width: 0; } - html.theme--documenter-dark .content .tabs li + li { - margin-top: 0; } - html.theme--documenter-dark .content.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.content { - font-size: 0.85em; } - html.theme--documenter-dark .content.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .content.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .icon { - align-items: center; - display: inline-flex; - justify-content: center; - height: 1.5rem; - width: 1.5rem; } - html.theme--documenter-dark .icon.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.icon { - height: 1rem; - width: 1rem; } - html.theme--documenter-dark .icon.is-medium { - height: 2rem; - width: 2rem; } - html.theme--documenter-dark .icon.is-large { - height: 3rem; - width: 3rem; } - html.theme--documenter-dark .image, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img { - display: block; - position: relative; } - html.theme--documenter-dark .image img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img img { - display: block; - height: auto; - width: 100%; } - html.theme--documenter-dark .image img.is-rounded, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img img.is-rounded { - border-radius: 290486px; } - html.theme--documenter-dark .image.is-square img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-square img, - html.theme--documenter-dark .image.is-square .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-square .has-ratio, html.theme--documenter-dark .image.is-1by1 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by1 img, - html.theme--documenter-dark .image.is-1by1 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by1 .has-ratio, html.theme--documenter-dark .image.is-5by4 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by4 img, - html.theme--documenter-dark .image.is-5by4 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by4 .has-ratio, html.theme--documenter-dark .image.is-4by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by3 img, - html.theme--documenter-dark .image.is-4by3 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by3 .has-ratio, html.theme--documenter-dark .image.is-3by2 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by2 img, - html.theme--documenter-dark .image.is-3by2 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by2 .has-ratio, html.theme--documenter-dark .image.is-5by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by3 img, - html.theme--documenter-dark .image.is-5by3 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by3 .has-ratio, html.theme--documenter-dark .image.is-16by9 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-16by9 img, - html.theme--documenter-dark .image.is-16by9 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-16by9 .has-ratio, html.theme--documenter-dark .image.is-2by1 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by1 img, - html.theme--documenter-dark .image.is-2by1 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by1 .has-ratio, html.theme--documenter-dark .image.is-3by1 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by1 img, - html.theme--documenter-dark .image.is-3by1 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by1 .has-ratio, html.theme--documenter-dark .image.is-4by5 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by5 img, - html.theme--documenter-dark .image.is-4by5 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by5 .has-ratio, html.theme--documenter-dark .image.is-3by4 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by4 img, - html.theme--documenter-dark .image.is-3by4 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by4 .has-ratio, html.theme--documenter-dark .image.is-2by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by3 img, - html.theme--documenter-dark .image.is-2by3 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by3 .has-ratio, html.theme--documenter-dark .image.is-3by5 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by5 img, - html.theme--documenter-dark .image.is-3by5 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by5 .has-ratio, html.theme--documenter-dark .image.is-9by16 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-9by16 img, - html.theme--documenter-dark .image.is-9by16 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-9by16 .has-ratio, html.theme--documenter-dark .image.is-1by2 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by2 img, - html.theme--documenter-dark .image.is-1by2 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by2 .has-ratio, html.theme--documenter-dark .image.is-1by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by3 img, - html.theme--documenter-dark .image.is-1by3 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by3 .has-ratio { - height: 100%; - width: 100%; } - html.theme--documenter-dark .image.is-square, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-square, html.theme--documenter-dark .image.is-1by1, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by1 { - padding-top: 100%; } - html.theme--documenter-dark .image.is-5by4, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by4 { - padding-top: 80%; } - html.theme--documenter-dark .image.is-4by3, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by3 { - padding-top: 75%; } - html.theme--documenter-dark .image.is-3by2, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by2 { - padding-top: 66.6666%; } - html.theme--documenter-dark .image.is-5by3, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by3 { - padding-top: 60%; } - html.theme--documenter-dark .image.is-16by9, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-16by9 { - padding-top: 56.25%; } - html.theme--documenter-dark .image.is-2by1, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by1 { - padding-top: 50%; } - html.theme--documenter-dark .image.is-3by1, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by1 { - padding-top: 33.3333%; } - html.theme--documenter-dark .image.is-4by5, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by5 { - padding-top: 125%; } - html.theme--documenter-dark .image.is-3by4, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by4 { - padding-top: 133.3333%; } - html.theme--documenter-dark .image.is-2by3, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by3 { - padding-top: 150%; } - html.theme--documenter-dark .image.is-3by5, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by5 { - padding-top: 166.6666%; } - html.theme--documenter-dark .image.is-9by16, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-9by16 { - padding-top: 177.7777%; } - html.theme--documenter-dark .image.is-1by2, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by2 { - padding-top: 200%; } - html.theme--documenter-dark .image.is-1by3, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by3 { - padding-top: 300%; } - html.theme--documenter-dark .image.is-16x16, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-16x16 { - height: 16px; - width: 16px; } - html.theme--documenter-dark .image.is-24x24, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-24x24 { - height: 24px; - width: 24px; } - html.theme--documenter-dark .image.is-32x32, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-32x32 { - height: 32px; - width: 32px; } - html.theme--documenter-dark .image.is-48x48, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-48x48 { - height: 48px; - width: 48px; } - html.theme--documenter-dark .image.is-64x64, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-64x64 { - height: 64px; - width: 64px; } - html.theme--documenter-dark .image.is-96x96, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-96x96 { - height: 96px; - width: 96px; } - html.theme--documenter-dark .image.is-128x128, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-128x128 { - height: 128px; - width: 128px; } - html.theme--documenter-dark .notification { - background-color: #282f2f; - border-radius: 0.4em; - padding: 1.25rem 2.5rem 1.25rem 1.5rem; - position: relative; } - html.theme--documenter-dark .notification a:not(.button):not(.dropdown-item) { - color: currentColor; - text-decoration: underline; } - html.theme--documenter-dark .notification strong { - color: currentColor; } - html.theme--documenter-dark .notification code, - html.theme--documenter-dark .notification pre { - background: white; } - html.theme--documenter-dark .notification pre code { - background: transparent; } - html.theme--documenter-dark .notification > .delete { - position: absolute; - right: 0.5rem; - top: 0.5rem; } - html.theme--documenter-dark .notification .title, - html.theme--documenter-dark .notification .subtitle, - html.theme--documenter-dark .notification .content { - color: currentColor; } - html.theme--documenter-dark .notification.is-white { - background-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .notification.is-black { - background-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .notification.is-light { - background-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .notification.is-dark, html.theme--documenter-dark .content kbd.notification { - background-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .notification.is-primary, html.theme--documenter-dark .docstring > section > a.notification.docs-sourcelink { - background-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .notification.is-link { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .notification.is-info { - background-color: #024c7d; - color: #fff; } - html.theme--documenter-dark .notification.is-success { - background-color: #008438; - color: #fff; } - html.theme--documenter-dark .notification.is-warning { - background-color: #ad8100; - color: #fff; } - html.theme--documenter-dark .notification.is-danger { - background-color: #9e1b0d; - color: #fff; } - html.theme--documenter-dark .progress { - -moz-appearance: none; - -webkit-appearance: none; - border: none; - border-radius: 290486px; - display: block; - height: 15px; - overflow: hidden; - padding: 0; - width: 100%; } - html.theme--documenter-dark .progress::-webkit-progress-bar { - background-color: #5e6d6f; } - html.theme--documenter-dark .progress::-webkit-progress-value { - background-color: #dbdee0; } - html.theme--documenter-dark .progress::-moz-progress-bar { - background-color: #dbdee0; } - html.theme--documenter-dark .progress::-ms-fill { - background-color: #dbdee0; - border: none; } - html.theme--documenter-dark .progress.is-white::-webkit-progress-value { - background-color: white; } - html.theme--documenter-dark .progress.is-white::-moz-progress-bar { - background-color: white; } - html.theme--documenter-dark .progress.is-white::-ms-fill { - background-color: white; } - html.theme--documenter-dark .progress.is-white:indeterminate { - background-image: linear-gradient(to right, white 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-black::-webkit-progress-value { - background-color: #0a0a0a; } - html.theme--documenter-dark .progress.is-black::-moz-progress-bar { - background-color: #0a0a0a; } - html.theme--documenter-dark .progress.is-black::-ms-fill { - background-color: #0a0a0a; } - html.theme--documenter-dark .progress.is-black:indeterminate { - background-image: linear-gradient(to right, #0a0a0a 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-light::-webkit-progress-value { - background-color: #ecf0f1; } - html.theme--documenter-dark .progress.is-light::-moz-progress-bar { - background-color: #ecf0f1; } - html.theme--documenter-dark .progress.is-light::-ms-fill { - background-color: #ecf0f1; } - html.theme--documenter-dark .progress.is-light:indeterminate { - background-image: linear-gradient(to right, #ecf0f1 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-dark::-webkit-progress-value, html.theme--documenter-dark .content kbd.progress::-webkit-progress-value { - background-color: #282f2f; } - html.theme--documenter-dark .progress.is-dark::-moz-progress-bar, html.theme--documenter-dark .content kbd.progress::-moz-progress-bar { - background-color: #282f2f; } - html.theme--documenter-dark .progress.is-dark::-ms-fill, html.theme--documenter-dark .content kbd.progress::-ms-fill { - background-color: #282f2f; } - html.theme--documenter-dark .progress.is-dark:indeterminate, html.theme--documenter-dark .content kbd.progress:indeterminate { - background-image: linear-gradient(to right, #282f2f 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-primary::-webkit-progress-value, html.theme--documenter-dark .docstring > section > a.progress.docs-sourcelink::-webkit-progress-value { - background-color: #375a7f; } - html.theme--documenter-dark .progress.is-primary::-moz-progress-bar, html.theme--documenter-dark .docstring > section > a.progress.docs-sourcelink::-moz-progress-bar { - background-color: #375a7f; } - html.theme--documenter-dark .progress.is-primary::-ms-fill, html.theme--documenter-dark .docstring > section > a.progress.docs-sourcelink::-ms-fill { - background-color: #375a7f; } - html.theme--documenter-dark .progress.is-primary:indeterminate, html.theme--documenter-dark .docstring > section > a.progress.docs-sourcelink:indeterminate { - background-image: linear-gradient(to right, #375a7f 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-link::-webkit-progress-value { - background-color: #1abc9c; } - html.theme--documenter-dark .progress.is-link::-moz-progress-bar { - background-color: #1abc9c; } - html.theme--documenter-dark .progress.is-link::-ms-fill { - background-color: #1abc9c; } - html.theme--documenter-dark .progress.is-link:indeterminate { - background-image: linear-gradient(to right, #1abc9c 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-info::-webkit-progress-value { - background-color: #024c7d; } - html.theme--documenter-dark .progress.is-info::-moz-progress-bar { - background-color: #024c7d; } - html.theme--documenter-dark .progress.is-info::-ms-fill { - background-color: #024c7d; } - html.theme--documenter-dark .progress.is-info:indeterminate { - background-image: linear-gradient(to right, #024c7d 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-success::-webkit-progress-value { - background-color: #008438; } - html.theme--documenter-dark .progress.is-success::-moz-progress-bar { - background-color: #008438; } - html.theme--documenter-dark .progress.is-success::-ms-fill { - background-color: #008438; } - html.theme--documenter-dark .progress.is-success:indeterminate { - background-image: linear-gradient(to right, #008438 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-warning::-webkit-progress-value { - background-color: #ad8100; } - html.theme--documenter-dark .progress.is-warning::-moz-progress-bar { - background-color: #ad8100; } - html.theme--documenter-dark .progress.is-warning::-ms-fill { - background-color: #ad8100; } - html.theme--documenter-dark .progress.is-warning:indeterminate { - background-image: linear-gradient(to right, #ad8100 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-danger::-webkit-progress-value { - background-color: #9e1b0d; } - html.theme--documenter-dark .progress.is-danger::-moz-progress-bar { - background-color: #9e1b0d; } - html.theme--documenter-dark .progress.is-danger::-ms-fill { - background-color: #9e1b0d; } - html.theme--documenter-dark .progress.is-danger:indeterminate { - background-image: linear-gradient(to right, #9e1b0d 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress:indeterminate { - animation-duration: 1.5s; - animation-iteration-count: infinite; - animation-name: moveIndeterminate; - animation-timing-function: linear; - background-color: #5e6d6f; - background-image: linear-gradient(to right, #fff 30%, #5e6d6f 30%); - background-position: top left; - background-repeat: no-repeat; - background-size: 150% 150%; } - html.theme--documenter-dark .progress:indeterminate::-webkit-progress-bar { - background-color: transparent; } - html.theme--documenter-dark .progress:indeterminate::-moz-progress-bar { - background-color: transparent; } - html.theme--documenter-dark .progress.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.progress { - height: 0.85em; } - html.theme--documenter-dark .progress.is-medium { - height: 1.25rem; } - html.theme--documenter-dark .progress.is-large { - height: 1.5rem; } - -@keyframes moveIndeterminate { - from { - background-position: 200% 0; } - to { - background-position: -200% 0; } } - html.theme--documenter-dark .table { - background-color: #343c3d; - color: #fff; } - html.theme--documenter-dark .table td, - html.theme--documenter-dark .table th { - border: 1px solid #5e6d6f; - border-width: 0 0 1px; - padding: 0.5em 0.75em; - vertical-align: top; } - html.theme--documenter-dark .table td.is-white, - html.theme--documenter-dark .table th.is-white { - background-color: white; - border-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .table td.is-black, - html.theme--documenter-dark .table th.is-black { - background-color: #0a0a0a; - border-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .table td.is-light, - html.theme--documenter-dark .table th.is-light { - background-color: #ecf0f1; - border-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .table td.is-dark, - html.theme--documenter-dark .table th.is-dark { - background-color: #282f2f; - border-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .table td.is-primary, - html.theme--documenter-dark .table th.is-primary { - background-color: #375a7f; - border-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .table td.is-link, - html.theme--documenter-dark .table th.is-link { - background-color: #1abc9c; - border-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .table td.is-info, - html.theme--documenter-dark .table th.is-info { - background-color: #024c7d; - border-color: #024c7d; - color: #fff; } - html.theme--documenter-dark .table td.is-success, - html.theme--documenter-dark .table th.is-success { - background-color: #008438; - border-color: #008438; - color: #fff; } - html.theme--documenter-dark .table td.is-warning, - html.theme--documenter-dark .table th.is-warning { - background-color: #ad8100; - border-color: #ad8100; - color: #fff; } - html.theme--documenter-dark .table td.is-danger, - html.theme--documenter-dark .table th.is-danger { - background-color: #9e1b0d; - border-color: #9e1b0d; - color: #fff; } - html.theme--documenter-dark .table td.is-narrow, - html.theme--documenter-dark .table th.is-narrow { - white-space: nowrap; - width: 1%; } - html.theme--documenter-dark .table td.is-selected, - html.theme--documenter-dark .table th.is-selected { - background-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .table td.is-selected a, - html.theme--documenter-dark .table td.is-selected strong, - html.theme--documenter-dark .table th.is-selected a, - html.theme--documenter-dark .table th.is-selected strong { - color: currentColor; } - html.theme--documenter-dark .table th { - color: #f2f2f2; } - html.theme--documenter-dark .table th:not([align]) { - text-align: left; } - html.theme--documenter-dark .table tr.is-selected { - background-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .table tr.is-selected a, - html.theme--documenter-dark .table tr.is-selected strong { - color: currentColor; } - html.theme--documenter-dark .table tr.is-selected td, - html.theme--documenter-dark .table tr.is-selected th { - border-color: #fff; - color: currentColor; } - html.theme--documenter-dark .table thead { - background-color: transparent; } - html.theme--documenter-dark .table thead td, - html.theme--documenter-dark .table thead th { - border-width: 0 0 2px; - color: #f2f2f2; } - html.theme--documenter-dark .table tfoot { - background-color: transparent; } - html.theme--documenter-dark .table tfoot td, - html.theme--documenter-dark .table tfoot th { - border-width: 2px 0 0; - color: #f2f2f2; } - html.theme--documenter-dark .table tbody { - background-color: transparent; } - html.theme--documenter-dark .table tbody tr:last-child td, - html.theme--documenter-dark .table tbody tr:last-child th { - border-bottom-width: 0; } - html.theme--documenter-dark .table.is-bordered td, - html.theme--documenter-dark .table.is-bordered th { - border-width: 1px; } - html.theme--documenter-dark .table.is-bordered tr:last-child td, - html.theme--documenter-dark .table.is-bordered tr:last-child th { - border-bottom-width: 1px; } - html.theme--documenter-dark .table.is-fullwidth { - width: 100%; } - html.theme--documenter-dark .table.is-hoverable tbody tr:not(.is-selected):hover { - background-color: #282f2f; } - html.theme--documenter-dark .table.is-hoverable.is-striped tbody tr:not(.is-selected):hover { - background-color: #282f2f; } - html.theme--documenter-dark .table.is-hoverable.is-striped tbody tr:not(.is-selected):hover:nth-child(even) { - background-color: #2d3435; } - html.theme--documenter-dark .table.is-narrow td, - html.theme--documenter-dark .table.is-narrow th { - padding: 0.25em 0.5em; } - html.theme--documenter-dark .table.is-striped tbody tr:not(.is-selected):nth-child(even) { - background-color: #282f2f; } - html.theme--documenter-dark .table-container { - -webkit-overflow-scrolling: touch; - overflow: auto; - overflow-y: hidden; - max-width: 100%; } - html.theme--documenter-dark .tags { - align-items: center; - display: flex; - flex-wrap: wrap; - justify-content: flex-start; } - html.theme--documenter-dark .tags .tag, html.theme--documenter-dark .tags .content kbd, html.theme--documenter-dark .content .tags kbd, html.theme--documenter-dark .tags .docstring > section > a.docs-sourcelink { - margin-bottom: 0.5rem; } - html.theme--documenter-dark .tags .tag:not(:last-child), html.theme--documenter-dark .tags .content kbd:not(:last-child), html.theme--documenter-dark .content .tags kbd:not(:last-child), html.theme--documenter-dark .tags .docstring > section > a.docs-sourcelink:not(:last-child) { - margin-right: 0.5rem; } - html.theme--documenter-dark .tags:last-child { - margin-bottom: -0.5rem; } - html.theme--documenter-dark .tags:not(:last-child) { - margin-bottom: 1rem; } - html.theme--documenter-dark .tags.are-medium .tag:not(.is-normal):not(.is-large), html.theme--documenter-dark .tags.are-medium .content kbd:not(.is-normal):not(.is-large), html.theme--documenter-dark .content .tags.are-medium kbd:not(.is-normal):not(.is-large), html.theme--documenter-dark .tags.are-medium .docstring > section > a.docs-sourcelink:not(.is-normal):not(.is-large) { - font-size: 15px; } - html.theme--documenter-dark .tags.are-large .tag:not(.is-normal):not(.is-medium), html.theme--documenter-dark .tags.are-large .content kbd:not(.is-normal):not(.is-medium), html.theme--documenter-dark .content .tags.are-large kbd:not(.is-normal):not(.is-medium), html.theme--documenter-dark .tags.are-large .docstring > section > a.docs-sourcelink:not(.is-normal):not(.is-medium) { - font-size: 1.25rem; } - html.theme--documenter-dark .tags.is-centered { - justify-content: center; } - html.theme--documenter-dark .tags.is-centered .tag, html.theme--documenter-dark .tags.is-centered .content kbd, html.theme--documenter-dark .content .tags.is-centered kbd, html.theme--documenter-dark .tags.is-centered .docstring > section > a.docs-sourcelink { - margin-right: 0.25rem; - margin-left: 0.25rem; } - html.theme--documenter-dark .tags.is-right { - justify-content: flex-end; } - html.theme--documenter-dark .tags.is-right .tag:not(:first-child), html.theme--documenter-dark .tags.is-right .content kbd:not(:first-child), html.theme--documenter-dark .content .tags.is-right kbd:not(:first-child), html.theme--documenter-dark .tags.is-right .docstring > section > a.docs-sourcelink:not(:first-child) { - margin-left: 0.5rem; } - html.theme--documenter-dark .tags.is-right .tag:not(:last-child), html.theme--documenter-dark .tags.is-right .content kbd:not(:last-child), html.theme--documenter-dark .content .tags.is-right kbd:not(:last-child), html.theme--documenter-dark .tags.is-right .docstring > section > a.docs-sourcelink:not(:last-child) { - margin-right: 0; } - html.theme--documenter-dark .tags.has-addons .tag, html.theme--documenter-dark .tags.has-addons .content kbd, html.theme--documenter-dark .content .tags.has-addons kbd, html.theme--documenter-dark .tags.has-addons .docstring > section > a.docs-sourcelink { - margin-right: 0; } - html.theme--documenter-dark .tags.has-addons .tag:not(:first-child), html.theme--documenter-dark .tags.has-addons .content kbd:not(:first-child), html.theme--documenter-dark .content .tags.has-addons kbd:not(:first-child), html.theme--documenter-dark .tags.has-addons .docstring > section > a.docs-sourcelink:not(:first-child) { - margin-left: 0; - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - html.theme--documenter-dark .tags.has-addons .tag:not(:last-child), html.theme--documenter-dark .tags.has-addons .content kbd:not(:last-child), html.theme--documenter-dark .content .tags.has-addons kbd:not(:last-child), html.theme--documenter-dark .tags.has-addons .docstring > section > a.docs-sourcelink:not(:last-child) { - border-bottom-right-radius: 0; - border-top-right-radius: 0; } - html.theme--documenter-dark .tag:not(body), html.theme--documenter-dark .content kbd:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body) { - align-items: center; - background-color: #282f2f; - border-radius: 0.4em; - color: #fff; - display: inline-flex; - font-size: 0.85em; - height: 2em; - justify-content: center; - line-height: 1.5; - padding-left: 0.75em; - padding-right: 0.75em; - white-space: nowrap; } - html.theme--documenter-dark .tag:not(body) .delete, html.theme--documenter-dark .content kbd:not(body) .delete, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body) .delete { - margin-left: 0.25rem; - margin-right: -0.375rem; } - html.theme--documenter-dark .tag.is-white:not(body), html.theme--documenter-dark .content kbd.is-white:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-white:not(body) { - background-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .tag.is-black:not(body), html.theme--documenter-dark .content kbd.is-black:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-black:not(body) { - background-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .tag.is-light:not(body), html.theme--documenter-dark .content kbd.is-light:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-light:not(body) { - background-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .tag.is-dark:not(body), html.theme--documenter-dark .content kbd:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-dark:not(body), html.theme--documenter-dark .content .docstring > section > kbd:not(body) { - background-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .tag.is-primary:not(body), html.theme--documenter-dark .content kbd.is-primary:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body) { - background-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .tag.is-link:not(body), html.theme--documenter-dark .content kbd.is-link:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-link:not(body) { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .tag.is-info:not(body), html.theme--documenter-dark .content kbd.is-info:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-info:not(body) { - background-color: #024c7d; - color: #fff; } - html.theme--documenter-dark .tag.is-success:not(body), html.theme--documenter-dark .content kbd.is-success:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-success:not(body) { - background-color: #008438; - color: #fff; } - html.theme--documenter-dark .tag.is-warning:not(body), html.theme--documenter-dark .content kbd.is-warning:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-warning:not(body) { - background-color: #ad8100; - color: #fff; } - html.theme--documenter-dark .tag.is-danger:not(body), html.theme--documenter-dark .content kbd.is-danger:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-danger:not(body) { - background-color: #9e1b0d; - color: #fff; } - html.theme--documenter-dark .tag.is-normal:not(body), html.theme--documenter-dark .content kbd.is-normal:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-normal:not(body) { - font-size: 0.85em; } - html.theme--documenter-dark .tag.is-medium:not(body), html.theme--documenter-dark .content kbd.is-medium:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-medium:not(body) { - font-size: 15px; } - html.theme--documenter-dark .tag.is-large:not(body), html.theme--documenter-dark .content kbd.is-large:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-large:not(body) { - font-size: 1.25rem; } - html.theme--documenter-dark .tag:not(body) .icon:first-child:not(:last-child), html.theme--documenter-dark .content kbd:not(body) .icon:first-child:not(:last-child), html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body) .icon:first-child:not(:last-child) { - margin-left: -0.375em; - margin-right: 0.1875em; } - html.theme--documenter-dark .tag:not(body) .icon:last-child:not(:first-child), html.theme--documenter-dark .content kbd:not(body) .icon:last-child:not(:first-child), html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body) .icon:last-child:not(:first-child) { - margin-left: 0.1875em; - margin-right: -0.375em; } - html.theme--documenter-dark .tag:not(body) .icon:first-child:last-child, html.theme--documenter-dark .content kbd:not(body) .icon:first-child:last-child, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body) .icon:first-child:last-child { - margin-left: -0.375em; - margin-right: -0.375em; } - html.theme--documenter-dark .tag.is-delete:not(body), html.theme--documenter-dark .content kbd.is-delete:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body) { - margin-left: 1px; - padding: 0; - position: relative; - width: 2em; } - html.theme--documenter-dark .tag.is-delete:not(body)::before, html.theme--documenter-dark .content kbd.is-delete:not(body)::before, html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body)::before, html.theme--documenter-dark .tag.is-delete:not(body)::after, html.theme--documenter-dark .content kbd.is-delete:not(body)::after, html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body)::after { - background-color: currentColor; - content: ""; - display: block; - left: 50%; - position: absolute; - top: 50%; - transform: translateX(-50%) translateY(-50%) rotate(45deg); - transform-origin: center center; } - html.theme--documenter-dark .tag.is-delete:not(body)::before, html.theme--documenter-dark .content kbd.is-delete:not(body)::before, html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body)::before { - height: 1px; - width: 50%; } - html.theme--documenter-dark .tag.is-delete:not(body)::after, html.theme--documenter-dark .content kbd.is-delete:not(body)::after, html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body)::after { - height: 50%; - width: 1px; } - html.theme--documenter-dark .tag.is-delete:not(body):hover, html.theme--documenter-dark .content kbd.is-delete:not(body):hover, html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body):hover, html.theme--documenter-dark .tag.is-delete:not(body):focus, html.theme--documenter-dark .content kbd.is-delete:not(body):focus, html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body):focus { - background-color: #1d2122; } - html.theme--documenter-dark .tag.is-delete:not(body):active, html.theme--documenter-dark .content kbd.is-delete:not(body):active, html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body):active { - background-color: #111414; } - html.theme--documenter-dark .tag.is-rounded:not(body), html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:not(body), html.theme--documenter-dark .content kbd.is-rounded:not(body), html.theme--documenter-dark #documenter .docs-sidebar .content form.docs-search > input:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-rounded:not(body) { - border-radius: 290486px; } - html.theme--documenter-dark a.tag:hover, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:hover { - text-decoration: underline; } - html.theme--documenter-dark .title, - html.theme--documenter-dark .subtitle { - word-break: break-word; } - html.theme--documenter-dark .title em, - html.theme--documenter-dark .title span, - html.theme--documenter-dark .subtitle em, - html.theme--documenter-dark .subtitle span { - font-weight: inherit; } - html.theme--documenter-dark .title sub, - html.theme--documenter-dark .subtitle sub { - font-size: 0.75em; } - html.theme--documenter-dark .title sup, - html.theme--documenter-dark .subtitle sup { - font-size: 0.75em; } - html.theme--documenter-dark .title .tag, html.theme--documenter-dark .title .content kbd, html.theme--documenter-dark .content .title kbd, html.theme--documenter-dark .title .docstring > section > a.docs-sourcelink, - html.theme--documenter-dark .subtitle .tag, - html.theme--documenter-dark .subtitle .content kbd, - html.theme--documenter-dark .content .subtitle kbd, - html.theme--documenter-dark .subtitle .docstring > section > a.docs-sourcelink { - vertical-align: middle; } - html.theme--documenter-dark .title { - color: #fff; - font-size: 2rem; - font-weight: 500; - line-height: 1.125; } - html.theme--documenter-dark .title strong { - color: inherit; - font-weight: inherit; } - html.theme--documenter-dark .title + .highlight { - margin-top: -0.75rem; } - html.theme--documenter-dark .title:not(.is-spaced) + .subtitle { - margin-top: -1.25rem; } - html.theme--documenter-dark .title.is-1 { - font-size: 3rem; } - html.theme--documenter-dark .title.is-2 { - font-size: 2.5rem; } - html.theme--documenter-dark .title.is-3 { - font-size: 2rem; } - html.theme--documenter-dark .title.is-4 { - font-size: 1.5rem; } - html.theme--documenter-dark .title.is-5 { - font-size: 1.25rem; } - html.theme--documenter-dark .title.is-6 { - font-size: 15px; } - html.theme--documenter-dark .title.is-7 { - font-size: 0.85em; } - html.theme--documenter-dark .subtitle { - color: #8c9b9d; - font-size: 1.25rem; - font-weight: 400; - line-height: 1.25; } - html.theme--documenter-dark .subtitle strong { - color: #8c9b9d; - font-weight: 600; } - html.theme--documenter-dark .subtitle:not(.is-spaced) + .title { - margin-top: -1.25rem; } - html.theme--documenter-dark .subtitle.is-1 { - font-size: 3rem; } - html.theme--documenter-dark .subtitle.is-2 { - font-size: 2.5rem; } - html.theme--documenter-dark .subtitle.is-3 { - font-size: 2rem; } - html.theme--documenter-dark .subtitle.is-4 { - font-size: 1.5rem; } - html.theme--documenter-dark .subtitle.is-5 { - font-size: 1.25rem; } - html.theme--documenter-dark .subtitle.is-6 { - font-size: 15px; } - html.theme--documenter-dark .subtitle.is-7 { - font-size: 0.85em; } - html.theme--documenter-dark .heading { - display: block; - font-size: 11px; - letter-spacing: 1px; - margin-bottom: 5px; - text-transform: uppercase; } - html.theme--documenter-dark .highlight { - font-weight: 400; - max-width: 100%; - overflow: hidden; - padding: 0; } - html.theme--documenter-dark .highlight pre { - overflow: auto; - max-width: 100%; } - html.theme--documenter-dark .number { - align-items: center; - background-color: #282f2f; - border-radius: 290486px; - display: inline-flex; - font-size: 1.25rem; - height: 2em; - justify-content: center; - margin-right: 1.5rem; - min-width: 2.5em; - padding: 0.25rem 0.5rem; - text-align: center; - vertical-align: top; } - html.theme--documenter-dark .select select, html.theme--documenter-dark .textarea, html.theme--documenter-dark .input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input { - background-color: #1f2424; - border-color: #5e6d6f; - border-radius: 0.4em; - color: #dbdee0; } - html.theme--documenter-dark .select select::-moz-placeholder, html.theme--documenter-dark .textarea::-moz-placeholder, html.theme--documenter-dark .input::-moz-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input::-moz-placeholder { - color: rgba(219, 222, 224, 0.3); } - html.theme--documenter-dark .select select::-webkit-input-placeholder, html.theme--documenter-dark .textarea::-webkit-input-placeholder, html.theme--documenter-dark .input::-webkit-input-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input::-webkit-input-placeholder { - color: rgba(219, 222, 224, 0.3); } - html.theme--documenter-dark .select select:-moz-placeholder, html.theme--documenter-dark .textarea:-moz-placeholder, html.theme--documenter-dark .input:-moz-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:-moz-placeholder { - color: rgba(219, 222, 224, 0.3); } - html.theme--documenter-dark .select select:-ms-input-placeholder, html.theme--documenter-dark .textarea:-ms-input-placeholder, html.theme--documenter-dark .input:-ms-input-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:-ms-input-placeholder { - color: rgba(219, 222, 224, 0.3); } - html.theme--documenter-dark .select select:hover, html.theme--documenter-dark .textarea:hover, html.theme--documenter-dark .input:hover, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:hover, html.theme--documenter-dark .select select.is-hovered, html.theme--documenter-dark .is-hovered.textarea, html.theme--documenter-dark .is-hovered.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-hovered { - border-color: #8c9b9d; } - html.theme--documenter-dark .select select:focus, html.theme--documenter-dark .textarea:focus, html.theme--documenter-dark .input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:focus, html.theme--documenter-dark .select select.is-focused, html.theme--documenter-dark .is-focused.textarea, html.theme--documenter-dark .is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .select select:active, html.theme--documenter-dark .textarea:active, html.theme--documenter-dark .input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:active, html.theme--documenter-dark .select select.is-active, html.theme--documenter-dark .is-active.textarea, html.theme--documenter-dark .is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - border-color: #1abc9c; - box-shadow: 0 0 0 0.125em rgba(26, 188, 156, 0.25); } - html.theme--documenter-dark .select select[disabled], html.theme--documenter-dark .textarea[disabled], html.theme--documenter-dark .input[disabled], html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[disabled], fieldset[disabled] html.theme--documenter-dark .select select, fieldset[disabled] html.theme--documenter-dark .textarea, fieldset[disabled] html.theme--documenter-dark .input, fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input { - background-color: #8c9b9d; - border-color: #282f2f; - box-shadow: none; - color: white; } - html.theme--documenter-dark .select select[disabled]::-moz-placeholder, html.theme--documenter-dark .textarea[disabled]::-moz-placeholder, html.theme--documenter-dark .input[disabled]::-moz-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[disabled]::-moz-placeholder, fieldset[disabled] html.theme--documenter-dark .select select::-moz-placeholder, fieldset[disabled] html.theme--documenter-dark .textarea::-moz-placeholder, fieldset[disabled] html.theme--documenter-dark .input::-moz-placeholder, fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input::-moz-placeholder { - color: rgba(255, 255, 255, 0.3); } - html.theme--documenter-dark .select select[disabled]::-webkit-input-placeholder, html.theme--documenter-dark .textarea[disabled]::-webkit-input-placeholder, html.theme--documenter-dark .input[disabled]::-webkit-input-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[disabled]::-webkit-input-placeholder, fieldset[disabled] html.theme--documenter-dark .select select::-webkit-input-placeholder, fieldset[disabled] html.theme--documenter-dark .textarea::-webkit-input-placeholder, fieldset[disabled] html.theme--documenter-dark .input::-webkit-input-placeholder, fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input::-webkit-input-placeholder { - color: rgba(255, 255, 255, 0.3); } - html.theme--documenter-dark .select select[disabled]:-moz-placeholder, html.theme--documenter-dark .textarea[disabled]:-moz-placeholder, html.theme--documenter-dark .input[disabled]:-moz-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[disabled]:-moz-placeholder, fieldset[disabled] html.theme--documenter-dark .select select:-moz-placeholder, fieldset[disabled] html.theme--documenter-dark .textarea:-moz-placeholder, fieldset[disabled] html.theme--documenter-dark .input:-moz-placeholder, fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:-moz-placeholder { - color: rgba(255, 255, 255, 0.3); } - html.theme--documenter-dark .select select[disabled]:-ms-input-placeholder, html.theme--documenter-dark .textarea[disabled]:-ms-input-placeholder, html.theme--documenter-dark .input[disabled]:-ms-input-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[disabled]:-ms-input-placeholder, fieldset[disabled] html.theme--documenter-dark .select select:-ms-input-placeholder, fieldset[disabled] html.theme--documenter-dark .textarea:-ms-input-placeholder, fieldset[disabled] html.theme--documenter-dark .input:-ms-input-placeholder, fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:-ms-input-placeholder { - color: rgba(255, 255, 255, 0.3); } - html.theme--documenter-dark .textarea, html.theme--documenter-dark .input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input { - box-shadow: inset 0 1px 2px rgba(10, 10, 10, 0.1); - max-width: 100%; - width: 100%; } - html.theme--documenter-dark .textarea[readonly], html.theme--documenter-dark .input[readonly], html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[readonly] { - box-shadow: none; } - html.theme--documenter-dark .is-white.textarea, html.theme--documenter-dark .is-white.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-white { - border-color: white; } - html.theme--documenter-dark .is-white.textarea:focus, html.theme--documenter-dark .is-white.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-white:focus, html.theme--documenter-dark .is-white.is-focused.textarea, html.theme--documenter-dark .is-white.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-white.textarea:active, html.theme--documenter-dark .is-white.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-white:active, html.theme--documenter-dark .is-white.is-active.textarea, html.theme--documenter-dark .is-white.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); } - html.theme--documenter-dark .is-black.textarea, html.theme--documenter-dark .is-black.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-black { - border-color: #0a0a0a; } - html.theme--documenter-dark .is-black.textarea:focus, html.theme--documenter-dark .is-black.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-black:focus, html.theme--documenter-dark .is-black.is-focused.textarea, html.theme--documenter-dark .is-black.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-black.textarea:active, html.theme--documenter-dark .is-black.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-black:active, html.theme--documenter-dark .is-black.is-active.textarea, html.theme--documenter-dark .is-black.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); } - html.theme--documenter-dark .is-light.textarea, html.theme--documenter-dark .is-light.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-light { - border-color: #ecf0f1; } - html.theme--documenter-dark .is-light.textarea:focus, html.theme--documenter-dark .is-light.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-light:focus, html.theme--documenter-dark .is-light.is-focused.textarea, html.theme--documenter-dark .is-light.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-light.textarea:active, html.theme--documenter-dark .is-light.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-light:active, html.theme--documenter-dark .is-light.is-active.textarea, html.theme--documenter-dark .is-light.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(236, 240, 241, 0.25); } - html.theme--documenter-dark .is-dark.textarea, html.theme--documenter-dark .content kbd.textarea, html.theme--documenter-dark .is-dark.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-dark, html.theme--documenter-dark .content kbd.input { - border-color: #282f2f; } - html.theme--documenter-dark .is-dark.textarea:focus, html.theme--documenter-dark .content kbd.textarea:focus, html.theme--documenter-dark .is-dark.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-dark:focus, html.theme--documenter-dark .content kbd.input:focus, html.theme--documenter-dark .is-dark.is-focused.textarea, html.theme--documenter-dark .content kbd.is-focused.textarea, html.theme--documenter-dark .is-dark.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .content kbd.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar .content form.docs-search > input.is-focused, html.theme--documenter-dark .is-dark.textarea:active, html.theme--documenter-dark .content kbd.textarea:active, html.theme--documenter-dark .is-dark.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-dark:active, html.theme--documenter-dark .content kbd.input:active, html.theme--documenter-dark .is-dark.is-active.textarea, html.theme--documenter-dark .content kbd.is-active.textarea, html.theme--documenter-dark .is-dark.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active, html.theme--documenter-dark .content kbd.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar .content form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(40, 47, 47, 0.25); } - html.theme--documenter-dark .is-primary.textarea, html.theme--documenter-dark .docstring > section > a.textarea.docs-sourcelink, html.theme--documenter-dark .is-primary.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-primary, html.theme--documenter-dark .docstring > section > a.input.docs-sourcelink { - border-color: #375a7f; } - html.theme--documenter-dark .is-primary.textarea:focus, html.theme--documenter-dark .docstring > section > a.textarea.docs-sourcelink:focus, html.theme--documenter-dark .is-primary.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-primary:focus, html.theme--documenter-dark .docstring > section > a.input.docs-sourcelink:focus, html.theme--documenter-dark .is-primary.is-focused.textarea, html.theme--documenter-dark .docstring > section > a.is-focused.textarea.docs-sourcelink, html.theme--documenter-dark .is-primary.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .docstring > section > a.is-focused.input.docs-sourcelink, html.theme--documenter-dark .is-primary.textarea:active, html.theme--documenter-dark .docstring > section > a.textarea.docs-sourcelink:active, html.theme--documenter-dark .is-primary.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-primary:active, html.theme--documenter-dark .docstring > section > a.input.docs-sourcelink:active, html.theme--documenter-dark .is-primary.is-active.textarea, html.theme--documenter-dark .docstring > section > a.is-active.textarea.docs-sourcelink, html.theme--documenter-dark .is-primary.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active, html.theme--documenter-dark .docstring > section > a.is-active.input.docs-sourcelink { - box-shadow: 0 0 0 0.125em rgba(55, 90, 127, 0.25); } - html.theme--documenter-dark .is-link.textarea, html.theme--documenter-dark .is-link.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-link { - border-color: #1abc9c; } - html.theme--documenter-dark .is-link.textarea:focus, html.theme--documenter-dark .is-link.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-link:focus, html.theme--documenter-dark .is-link.is-focused.textarea, html.theme--documenter-dark .is-link.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-link.textarea:active, html.theme--documenter-dark .is-link.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-link:active, html.theme--documenter-dark .is-link.is-active.textarea, html.theme--documenter-dark .is-link.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(26, 188, 156, 0.25); } - html.theme--documenter-dark .is-info.textarea, html.theme--documenter-dark .is-info.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-info { - border-color: #024c7d; } - html.theme--documenter-dark .is-info.textarea:focus, html.theme--documenter-dark .is-info.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-info:focus, html.theme--documenter-dark .is-info.is-focused.textarea, html.theme--documenter-dark .is-info.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-info.textarea:active, html.theme--documenter-dark .is-info.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-info:active, html.theme--documenter-dark .is-info.is-active.textarea, html.theme--documenter-dark .is-info.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(2, 76, 125, 0.25); } - html.theme--documenter-dark .is-success.textarea, html.theme--documenter-dark .is-success.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-success { - border-color: #008438; } - html.theme--documenter-dark .is-success.textarea:focus, html.theme--documenter-dark .is-success.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-success:focus, html.theme--documenter-dark .is-success.is-focused.textarea, html.theme--documenter-dark .is-success.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-success.textarea:active, html.theme--documenter-dark .is-success.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-success:active, html.theme--documenter-dark .is-success.is-active.textarea, html.theme--documenter-dark .is-success.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(0, 132, 56, 0.25); } - html.theme--documenter-dark .is-warning.textarea, html.theme--documenter-dark .is-warning.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-warning { - border-color: #ad8100; } - html.theme--documenter-dark .is-warning.textarea:focus, html.theme--documenter-dark .is-warning.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-warning:focus, html.theme--documenter-dark .is-warning.is-focused.textarea, html.theme--documenter-dark .is-warning.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-warning.textarea:active, html.theme--documenter-dark .is-warning.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-warning:active, html.theme--documenter-dark .is-warning.is-active.textarea, html.theme--documenter-dark .is-warning.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(173, 129, 0, 0.25); } - html.theme--documenter-dark .is-danger.textarea, html.theme--documenter-dark .is-danger.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-danger { - border-color: #9e1b0d; } - html.theme--documenter-dark .is-danger.textarea:focus, html.theme--documenter-dark .is-danger.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-danger:focus, html.theme--documenter-dark .is-danger.is-focused.textarea, html.theme--documenter-dark .is-danger.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-danger.textarea:active, html.theme--documenter-dark .is-danger.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-danger:active, html.theme--documenter-dark .is-danger.is-active.textarea, html.theme--documenter-dark .is-danger.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(158, 27, 13, 0.25); } - html.theme--documenter-dark .is-small.textarea, html.theme--documenter-dark .is-small.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input { - border-radius: 3px; - font-size: 0.85em; } - html.theme--documenter-dark .is-medium.textarea, html.theme--documenter-dark .is-medium.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .is-large.textarea, html.theme--documenter-dark .is-large.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .is-fullwidth.textarea, html.theme--documenter-dark .is-fullwidth.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-fullwidth { - display: block; - width: 100%; } - html.theme--documenter-dark .is-inline.textarea, html.theme--documenter-dark .is-inline.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-inline { - display: inline; - width: auto; } - html.theme--documenter-dark .input.is-rounded, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input { - border-radius: 290486px; - padding-left: 1em; - padding-right: 1em; } - html.theme--documenter-dark .input.is-static, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-static { - background-color: transparent; - border-color: transparent; - box-shadow: none; - padding-left: 0; - padding-right: 0; } - html.theme--documenter-dark .textarea { - display: block; - max-width: 100%; - min-width: 100%; - padding: 0.625em; - resize: vertical; } - html.theme--documenter-dark .textarea:not([rows]) { - max-height: 600px; - min-height: 120px; } - html.theme--documenter-dark .textarea[rows] { - height: initial; } - html.theme--documenter-dark .textarea.has-fixed-size { - resize: none; } - html.theme--documenter-dark .radio, html.theme--documenter-dark .checkbox { - cursor: pointer; - display: inline-block; - line-height: 1.25; - position: relative; } - html.theme--documenter-dark .radio input, html.theme--documenter-dark .checkbox input { - cursor: pointer; } - html.theme--documenter-dark .radio:hover, html.theme--documenter-dark .checkbox:hover { - color: #8c9b9d; } - html.theme--documenter-dark .radio[disabled], html.theme--documenter-dark .checkbox[disabled], fieldset[disabled] html.theme--documenter-dark .radio, fieldset[disabled] html.theme--documenter-dark .checkbox { - color: white; - cursor: not-allowed; } - html.theme--documenter-dark .radio + .radio { - margin-left: 0.5em; } - html.theme--documenter-dark .select { - display: inline-block; - max-width: 100%; - position: relative; - vertical-align: top; } - html.theme--documenter-dark .select:not(.is-multiple) { - height: 2.25em; } - html.theme--documenter-dark .select:not(.is-multiple):not(.is-loading)::after { - border-color: #1abc9c; - right: 1.125em; - z-index: 4; } - html.theme--documenter-dark .select.is-rounded select, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.select select { - border-radius: 290486px; - padding-left: 1em; } - html.theme--documenter-dark .select select { - cursor: pointer; - display: block; - font-size: 1em; - max-width: 100%; - outline: none; } - html.theme--documenter-dark .select select::-ms-expand { - display: none; } - html.theme--documenter-dark .select select[disabled]:hover, fieldset[disabled] html.theme--documenter-dark .select select:hover { - border-color: #282f2f; } - html.theme--documenter-dark .select select:not([multiple]) { - padding-right: 2.5em; } - html.theme--documenter-dark .select select[multiple] { - height: auto; - padding: 0; } - html.theme--documenter-dark .select select[multiple] option { - padding: 0.5em 1em; } - html.theme--documenter-dark .select:not(.is-multiple):not(.is-loading):hover::after { - border-color: #8c9b9d; } - html.theme--documenter-dark .select.is-white:not(:hover)::after { - border-color: white; } - html.theme--documenter-dark .select.is-white select { - border-color: white; } - html.theme--documenter-dark .select.is-white select:hover, html.theme--documenter-dark .select.is-white select.is-hovered { - border-color: #f2f2f2; } - html.theme--documenter-dark .select.is-white select:focus, html.theme--documenter-dark .select.is-white select.is-focused, html.theme--documenter-dark .select.is-white select:active, html.theme--documenter-dark .select.is-white select.is-active { - box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); } - html.theme--documenter-dark .select.is-black:not(:hover)::after { - border-color: #0a0a0a; } - html.theme--documenter-dark .select.is-black select { - border-color: #0a0a0a; } - html.theme--documenter-dark .select.is-black select:hover, html.theme--documenter-dark .select.is-black select.is-hovered { - border-color: black; } - html.theme--documenter-dark .select.is-black select:focus, html.theme--documenter-dark .select.is-black select.is-focused, html.theme--documenter-dark .select.is-black select:active, html.theme--documenter-dark .select.is-black select.is-active { - box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); } - html.theme--documenter-dark .select.is-light:not(:hover)::after { - border-color: #ecf0f1; } - html.theme--documenter-dark .select.is-light select { - border-color: #ecf0f1; } - html.theme--documenter-dark .select.is-light select:hover, html.theme--documenter-dark .select.is-light select.is-hovered { - border-color: #dde4e6; } - html.theme--documenter-dark .select.is-light select:focus, html.theme--documenter-dark .select.is-light select.is-focused, html.theme--documenter-dark .select.is-light select:active, html.theme--documenter-dark .select.is-light select.is-active { - box-shadow: 0 0 0 0.125em rgba(236, 240, 241, 0.25); } - html.theme--documenter-dark .select.is-dark:not(:hover)::after, html.theme--documenter-dark .content kbd.select:not(:hover)::after { - border-color: #282f2f; } - html.theme--documenter-dark .select.is-dark select, html.theme--documenter-dark .content kbd.select select { - border-color: #282f2f; } - html.theme--documenter-dark .select.is-dark select:hover, html.theme--documenter-dark .content kbd.select select:hover, html.theme--documenter-dark .select.is-dark select.is-hovered, html.theme--documenter-dark .content kbd.select select.is-hovered { - border-color: #1d2122; } - html.theme--documenter-dark .select.is-dark select:focus, html.theme--documenter-dark .content kbd.select select:focus, html.theme--documenter-dark .select.is-dark select.is-focused, html.theme--documenter-dark .content kbd.select select.is-focused, html.theme--documenter-dark .select.is-dark select:active, html.theme--documenter-dark .content kbd.select select:active, html.theme--documenter-dark .select.is-dark select.is-active, html.theme--documenter-dark .content kbd.select select.is-active { - box-shadow: 0 0 0 0.125em rgba(40, 47, 47, 0.25); } - html.theme--documenter-dark .select.is-primary:not(:hover)::after, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink:not(:hover)::after { - border-color: #375a7f; } - html.theme--documenter-dark .select.is-primary select, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select { - border-color: #375a7f; } - html.theme--documenter-dark .select.is-primary select:hover, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select:hover, html.theme--documenter-dark .select.is-primary select.is-hovered, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select.is-hovered { - border-color: #2f4d6d; } - html.theme--documenter-dark .select.is-primary select:focus, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select:focus, html.theme--documenter-dark .select.is-primary select.is-focused, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select.is-focused, html.theme--documenter-dark .select.is-primary select:active, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select:active, html.theme--documenter-dark .select.is-primary select.is-active, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select.is-active { - box-shadow: 0 0 0 0.125em rgba(55, 90, 127, 0.25); } - html.theme--documenter-dark .select.is-link:not(:hover)::after { - border-color: #1abc9c; } - html.theme--documenter-dark .select.is-link select { - border-color: #1abc9c; } - html.theme--documenter-dark .select.is-link select:hover, html.theme--documenter-dark .select.is-link select.is-hovered { - border-color: #17a689; } - html.theme--documenter-dark .select.is-link select:focus, html.theme--documenter-dark .select.is-link select.is-focused, html.theme--documenter-dark .select.is-link select:active, html.theme--documenter-dark .select.is-link select.is-active { - box-shadow: 0 0 0 0.125em rgba(26, 188, 156, 0.25); } - html.theme--documenter-dark .select.is-info:not(:hover)::after { - border-color: #024c7d; } - html.theme--documenter-dark .select.is-info select { - border-color: #024c7d; } - html.theme--documenter-dark .select.is-info select:hover, html.theme--documenter-dark .select.is-info select.is-hovered { - border-color: #023d64; } - html.theme--documenter-dark .select.is-info select:focus, html.theme--documenter-dark .select.is-info select.is-focused, html.theme--documenter-dark .select.is-info select:active, html.theme--documenter-dark .select.is-info select.is-active { - box-shadow: 0 0 0 0.125em rgba(2, 76, 125, 0.25); } - html.theme--documenter-dark .select.is-success:not(:hover)::after { - border-color: #008438; } - html.theme--documenter-dark .select.is-success select { - border-color: #008438; } - html.theme--documenter-dark .select.is-success select:hover, html.theme--documenter-dark .select.is-success select.is-hovered { - border-color: #006b2d; } - html.theme--documenter-dark .select.is-success select:focus, html.theme--documenter-dark .select.is-success select.is-focused, html.theme--documenter-dark .select.is-success select:active, html.theme--documenter-dark .select.is-success select.is-active { - box-shadow: 0 0 0 0.125em rgba(0, 132, 56, 0.25); } - html.theme--documenter-dark .select.is-warning:not(:hover)::after { - border-color: #ad8100; } - html.theme--documenter-dark .select.is-warning select { - border-color: #ad8100; } - html.theme--documenter-dark .select.is-warning select:hover, html.theme--documenter-dark .select.is-warning select.is-hovered { - border-color: #946e00; } - html.theme--documenter-dark .select.is-warning select:focus, html.theme--documenter-dark .select.is-warning select.is-focused, html.theme--documenter-dark .select.is-warning select:active, html.theme--documenter-dark .select.is-warning select.is-active { - box-shadow: 0 0 0 0.125em rgba(173, 129, 0, 0.25); } - html.theme--documenter-dark .select.is-danger:not(:hover)::after { - border-color: #9e1b0d; } - html.theme--documenter-dark .select.is-danger select { - border-color: #9e1b0d; } - html.theme--documenter-dark .select.is-danger select:hover, html.theme--documenter-dark .select.is-danger select.is-hovered { - border-color: #86170b; } - html.theme--documenter-dark .select.is-danger select:focus, html.theme--documenter-dark .select.is-danger select.is-focused, html.theme--documenter-dark .select.is-danger select:active, html.theme--documenter-dark .select.is-danger select.is-active { - box-shadow: 0 0 0 0.125em rgba(158, 27, 13, 0.25); } - html.theme--documenter-dark .select.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.select { - border-radius: 3px; - font-size: 0.85em; } - html.theme--documenter-dark .select.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .select.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .select.is-disabled::after { - border-color: white; } - html.theme--documenter-dark .select.is-fullwidth { - width: 100%; } - html.theme--documenter-dark .select.is-fullwidth select { - width: 100%; } - html.theme--documenter-dark .select.is-loading::after { - margin-top: 0; - position: absolute; - right: 0.625em; - top: 0.625em; - transform: none; } - html.theme--documenter-dark .select.is-loading.is-small:after, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-loading:after { - font-size: 0.85em; } - html.theme--documenter-dark .select.is-loading.is-medium:after { - font-size: 1.25rem; } - html.theme--documenter-dark .select.is-loading.is-large:after { - font-size: 1.5rem; } - html.theme--documenter-dark .file { - align-items: stretch; - display: flex; - justify-content: flex-start; - position: relative; } - html.theme--documenter-dark .file.is-white .file-cta { - background-color: white; - border-color: transparent; - color: #0a0a0a; } - html.theme--documenter-dark .file.is-white:hover .file-cta, html.theme--documenter-dark .file.is-white.is-hovered .file-cta { - background-color: #f9f9f9; - border-color: transparent; - color: #0a0a0a; } - html.theme--documenter-dark .file.is-white:focus .file-cta, html.theme--documenter-dark .file.is-white.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(255, 255, 255, 0.25); - color: #0a0a0a; } - html.theme--documenter-dark .file.is-white:active .file-cta, html.theme--documenter-dark .file.is-white.is-active .file-cta { - background-color: #f2f2f2; - border-color: transparent; - color: #0a0a0a; } - html.theme--documenter-dark .file.is-black .file-cta { - background-color: #0a0a0a; - border-color: transparent; - color: white; } - html.theme--documenter-dark .file.is-black:hover .file-cta, html.theme--documenter-dark .file.is-black.is-hovered .file-cta { - background-color: #040404; - border-color: transparent; - color: white; } - html.theme--documenter-dark .file.is-black:focus .file-cta, html.theme--documenter-dark .file.is-black.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(10, 10, 10, 0.25); - color: white; } - html.theme--documenter-dark .file.is-black:active .file-cta, html.theme--documenter-dark .file.is-black.is-active .file-cta { - background-color: black; - border-color: transparent; - color: white; } - html.theme--documenter-dark .file.is-light .file-cta { - background-color: #ecf0f1; - border-color: transparent; - color: #282f2f; } - html.theme--documenter-dark .file.is-light:hover .file-cta, html.theme--documenter-dark .file.is-light.is-hovered .file-cta { - background-color: #e5eaec; - border-color: transparent; - color: #282f2f; } - html.theme--documenter-dark .file.is-light:focus .file-cta, html.theme--documenter-dark .file.is-light.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(236, 240, 241, 0.25); - color: #282f2f; } - html.theme--documenter-dark .file.is-light:active .file-cta, html.theme--documenter-dark .file.is-light.is-active .file-cta { - background-color: #dde4e6; - border-color: transparent; - color: #282f2f; } - html.theme--documenter-dark .file.is-dark .file-cta, html.theme--documenter-dark .content kbd.file .file-cta { - background-color: #282f2f; - border-color: transparent; - color: #ecf0f1; } - html.theme--documenter-dark .file.is-dark:hover .file-cta, html.theme--documenter-dark .content kbd.file:hover .file-cta, html.theme--documenter-dark .file.is-dark.is-hovered .file-cta, html.theme--documenter-dark .content kbd.file.is-hovered .file-cta { - background-color: #232829; - border-color: transparent; - color: #ecf0f1; } - html.theme--documenter-dark .file.is-dark:focus .file-cta, html.theme--documenter-dark .content kbd.file:focus .file-cta, html.theme--documenter-dark .file.is-dark.is-focused .file-cta, html.theme--documenter-dark .content kbd.file.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(40, 47, 47, 0.25); - color: #ecf0f1; } - html.theme--documenter-dark .file.is-dark:active .file-cta, html.theme--documenter-dark .content kbd.file:active .file-cta, html.theme--documenter-dark .file.is-dark.is-active .file-cta, html.theme--documenter-dark .content kbd.file.is-active .file-cta { - background-color: #1d2122; - border-color: transparent; - color: #ecf0f1; } - html.theme--documenter-dark .file.is-primary .file-cta, html.theme--documenter-dark .docstring > section > a.file.docs-sourcelink .file-cta { - background-color: #375a7f; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-primary:hover .file-cta, html.theme--documenter-dark .docstring > section > a.file.docs-sourcelink:hover .file-cta, html.theme--documenter-dark .file.is-primary.is-hovered .file-cta, html.theme--documenter-dark .docstring > section > a.file.is-hovered.docs-sourcelink .file-cta { - background-color: #335476; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-primary:focus .file-cta, html.theme--documenter-dark .docstring > section > a.file.docs-sourcelink:focus .file-cta, html.theme--documenter-dark .file.is-primary.is-focused .file-cta, html.theme--documenter-dark .docstring > section > a.file.is-focused.docs-sourcelink .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(55, 90, 127, 0.25); - color: #fff; } - html.theme--documenter-dark .file.is-primary:active .file-cta, html.theme--documenter-dark .docstring > section > a.file.docs-sourcelink:active .file-cta, html.theme--documenter-dark .file.is-primary.is-active .file-cta, html.theme--documenter-dark .docstring > section > a.file.is-active.docs-sourcelink .file-cta { - background-color: #2f4d6d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-link .file-cta { - background-color: #1abc9c; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-link:hover .file-cta, html.theme--documenter-dark .file.is-link.is-hovered .file-cta { - background-color: #18b193; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-link:focus .file-cta, html.theme--documenter-dark .file.is-link.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(26, 188, 156, 0.25); - color: #fff; } - html.theme--documenter-dark .file.is-link:active .file-cta, html.theme--documenter-dark .file.is-link.is-active .file-cta { - background-color: #17a689; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-info .file-cta { - background-color: #024c7d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-info:hover .file-cta, html.theme--documenter-dark .file.is-info.is-hovered .file-cta { - background-color: #024470; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-info:focus .file-cta, html.theme--documenter-dark .file.is-info.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(2, 76, 125, 0.25); - color: #fff; } - html.theme--documenter-dark .file.is-info:active .file-cta, html.theme--documenter-dark .file.is-info.is-active .file-cta { - background-color: #023d64; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-success .file-cta { - background-color: #008438; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-success:hover .file-cta, html.theme--documenter-dark .file.is-success.is-hovered .file-cta { - background-color: #007733; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-success:focus .file-cta, html.theme--documenter-dark .file.is-success.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(0, 132, 56, 0.25); - color: #fff; } - html.theme--documenter-dark .file.is-success:active .file-cta, html.theme--documenter-dark .file.is-success.is-active .file-cta { - background-color: #006b2d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-warning .file-cta { - background-color: #ad8100; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-warning:hover .file-cta, html.theme--documenter-dark .file.is-warning.is-hovered .file-cta { - background-color: #a07700; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-warning:focus .file-cta, html.theme--documenter-dark .file.is-warning.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(173, 129, 0, 0.25); - color: #fff; } - html.theme--documenter-dark .file.is-warning:active .file-cta, html.theme--documenter-dark .file.is-warning.is-active .file-cta { - background-color: #946e00; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-danger .file-cta { - background-color: #9e1b0d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-danger:hover .file-cta, html.theme--documenter-dark .file.is-danger.is-hovered .file-cta { - background-color: #92190c; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-danger:focus .file-cta, html.theme--documenter-dark .file.is-danger.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(158, 27, 13, 0.25); - color: #fff; } - html.theme--documenter-dark .file.is-danger:active .file-cta, html.theme--documenter-dark .file.is-danger.is-active .file-cta { - background-color: #86170b; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.file { - font-size: 0.85em; } - html.theme--documenter-dark .file.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .file.is-medium .file-icon .fa { - font-size: 21px; } - html.theme--documenter-dark .file.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .file.is-large .file-icon .fa { - font-size: 28px; } - html.theme--documenter-dark .file.has-name .file-cta { - border-bottom-right-radius: 0; - border-top-right-radius: 0; } - html.theme--documenter-dark .file.has-name .file-name { - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - html.theme--documenter-dark .file.has-name.is-empty .file-cta { - border-radius: 0.4em; } - html.theme--documenter-dark .file.has-name.is-empty .file-name { - display: none; } - html.theme--documenter-dark .file.is-boxed .file-label { - flex-direction: column; } - html.theme--documenter-dark .file.is-boxed .file-cta { - flex-direction: column; - height: auto; - padding: 1em 3em; } - html.theme--documenter-dark .file.is-boxed .file-name { - border-width: 0 1px 1px; } - html.theme--documenter-dark .file.is-boxed .file-icon { - height: 1.5em; - width: 1.5em; } - html.theme--documenter-dark .file.is-boxed .file-icon .fa { - font-size: 21px; } - html.theme--documenter-dark .file.is-boxed.is-small .file-icon .fa, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-boxed .file-icon .fa { - font-size: 14px; } - html.theme--documenter-dark .file.is-boxed.is-medium .file-icon .fa { - font-size: 28px; } - html.theme--documenter-dark .file.is-boxed.is-large .file-icon .fa { - font-size: 35px; } - html.theme--documenter-dark .file.is-boxed.has-name .file-cta { - border-radius: 0.4em 0.4em 0 0; } - html.theme--documenter-dark .file.is-boxed.has-name .file-name { - border-radius: 0 0 0.4em 0.4em; - border-width: 0 1px 1px; } - html.theme--documenter-dark .file.is-centered { - justify-content: center; } - html.theme--documenter-dark .file.is-fullwidth .file-label { - width: 100%; } - html.theme--documenter-dark .file.is-fullwidth .file-name { - flex-grow: 1; - max-width: none; } - html.theme--documenter-dark .file.is-right { - justify-content: flex-end; } - html.theme--documenter-dark .file.is-right .file-cta { - border-radius: 0 0.4em 0.4em 0; } - html.theme--documenter-dark .file.is-right .file-name { - border-radius: 0.4em 0 0 0.4em; - border-width: 1px 0 1px 1px; - order: -1; } - html.theme--documenter-dark .file-label { - align-items: stretch; - display: flex; - cursor: pointer; - justify-content: flex-start; - overflow: hidden; - position: relative; } - html.theme--documenter-dark .file-label:hover .file-cta { - background-color: #e5eaec; - color: #282f2f; } - html.theme--documenter-dark .file-label:hover .file-name { - border-color: #596668; } - html.theme--documenter-dark .file-label:active .file-cta { - background-color: #dde4e6; - color: #282f2f; } - html.theme--documenter-dark .file-label:active .file-name { - border-color: #535f61; } - html.theme--documenter-dark .file-input { - height: 100%; - left: 0; - opacity: 0; - outline: none; - position: absolute; - top: 0; - width: 100%; } - html.theme--documenter-dark .file-cta, - html.theme--documenter-dark .file-name { - border-color: #5e6d6f; - border-radius: 0.4em; - font-size: 1em; - padding-left: 1em; - padding-right: 1em; - white-space: nowrap; } - html.theme--documenter-dark .file-cta { - background-color: #ecf0f1; - color: #343c3d; } - html.theme--documenter-dark .file-name { - border-color: #5e6d6f; - border-style: solid; - border-width: 1px 1px 1px 0; - display: block; - max-width: 16em; - overflow: hidden; - text-align: left; - text-overflow: ellipsis; } - html.theme--documenter-dark .file-icon { - align-items: center; - display: flex; - height: 1em; - justify-content: center; - margin-right: 0.5em; - width: 1em; } - html.theme--documenter-dark .file-icon .fa { - font-size: 14px; } - html.theme--documenter-dark .label { - color: #282f2f; - display: block; - font-size: 15px; - font-weight: 700; } - html.theme--documenter-dark .label:not(:last-child) { - margin-bottom: 0.5em; } - html.theme--documenter-dark .label.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.label { - font-size: 0.85em; } - html.theme--documenter-dark .label.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .label.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .help { - display: block; - font-size: 0.85em; - margin-top: 0.25rem; } - html.theme--documenter-dark .help.is-white { - color: white; } - html.theme--documenter-dark .help.is-black { - color: #0a0a0a; } - html.theme--documenter-dark .help.is-light { - color: #ecf0f1; } - html.theme--documenter-dark .help.is-dark, html.theme--documenter-dark .content kbd.help { - color: #282f2f; } - html.theme--documenter-dark .help.is-primary, html.theme--documenter-dark .docstring > section > a.help.docs-sourcelink { - color: #375a7f; } - html.theme--documenter-dark .help.is-link { - color: #1abc9c; } - html.theme--documenter-dark .help.is-info { - color: #024c7d; } - html.theme--documenter-dark .help.is-success { - color: #008438; } - html.theme--documenter-dark .help.is-warning { - color: #ad8100; } - html.theme--documenter-dark .help.is-danger { - color: #9e1b0d; } - html.theme--documenter-dark .field:not(:last-child) { - margin-bottom: 0.75rem; } - html.theme--documenter-dark .field.has-addons { - display: flex; - justify-content: flex-start; } - html.theme--documenter-dark .field.has-addons .control:not(:last-child) { - margin-right: -1px; } - html.theme--documenter-dark .field.has-addons .control:not(:first-child):not(:last-child) .button, - html.theme--documenter-dark .field.has-addons .control:not(:first-child):not(:last-child) .input, - html.theme--documenter-dark .field.has-addons .control:not(:first-child):not(:last-child) #documenter .docs-sidebar form.docs-search > input, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control:not(:first-child):not(:last-child) form.docs-search > input, - html.theme--documenter-dark .field.has-addons .control:not(:first-child):not(:last-child) .select select { - border-radius: 0; } - html.theme--documenter-dark .field.has-addons .control:first-child:not(:only-child) .button, - html.theme--documenter-dark .field.has-addons .control:first-child:not(:only-child) .input, - html.theme--documenter-dark .field.has-addons .control:first-child:not(:only-child) #documenter .docs-sidebar form.docs-search > input, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control:first-child:not(:only-child) form.docs-search > input, - html.theme--documenter-dark .field.has-addons .control:first-child:not(:only-child) .select select { - border-bottom-right-radius: 0; - border-top-right-radius: 0; } - html.theme--documenter-dark .field.has-addons .control:last-child:not(:only-child) .button, - html.theme--documenter-dark .field.has-addons .control:last-child:not(:only-child) .input, - html.theme--documenter-dark .field.has-addons .control:last-child:not(:only-child) #documenter .docs-sidebar form.docs-search > input, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control:last-child:not(:only-child) form.docs-search > input, - html.theme--documenter-dark .field.has-addons .control:last-child:not(:only-child) .select select { - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - html.theme--documenter-dark .field.has-addons .control .button:not([disabled]):hover, html.theme--documenter-dark .field.has-addons .control .button.is-hovered:not([disabled]), - html.theme--documenter-dark .field.has-addons .control .input:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):hover, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control .input.is-hovered:not([disabled]), - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-hovered:not([disabled]), - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-hovered:not([disabled]), - html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control .select select.is-hovered:not([disabled]) { - z-index: 2; } - html.theme--documenter-dark .field.has-addons .control .button:not([disabled]):focus, html.theme--documenter-dark .field.has-addons .control .button.is-focused:not([disabled]), html.theme--documenter-dark .field.has-addons .control .button:not([disabled]):active, html.theme--documenter-dark .field.has-addons .control .button.is-active:not([disabled]), - html.theme--documenter-dark .field.has-addons .control .input:not([disabled]):focus, - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):focus, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):focus, - html.theme--documenter-dark .field.has-addons .control .input.is-focused:not([disabled]), - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-focused:not([disabled]), - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-focused:not([disabled]), - html.theme--documenter-dark .field.has-addons .control .input:not([disabled]):active, - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):active, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):active, - html.theme--documenter-dark .field.has-addons .control .input.is-active:not([disabled]), - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-active:not([disabled]), - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-active:not([disabled]), - html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]):focus, - html.theme--documenter-dark .field.has-addons .control .select select.is-focused:not([disabled]), - html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]):active, - html.theme--documenter-dark .field.has-addons .control .select select.is-active:not([disabled]) { - z-index: 3; } - html.theme--documenter-dark .field.has-addons .control .button:not([disabled]):focus:hover, html.theme--documenter-dark .field.has-addons .control .button.is-focused:not([disabled]):hover, html.theme--documenter-dark .field.has-addons .control .button:not([disabled]):active:hover, html.theme--documenter-dark .field.has-addons .control .button.is-active:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control .input:not([disabled]):focus:hover, - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):focus:hover, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):focus:hover, - html.theme--documenter-dark .field.has-addons .control .input.is-focused:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-focused:not([disabled]):hover, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-focused:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control .input:not([disabled]):active:hover, - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):active:hover, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):active:hover, - html.theme--documenter-dark .field.has-addons .control .input.is-active:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-active:not([disabled]):hover, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-active:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]):focus:hover, - html.theme--documenter-dark .field.has-addons .control .select select.is-focused:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]):active:hover, - html.theme--documenter-dark .field.has-addons .control .select select.is-active:not([disabled]):hover { - z-index: 4; } - html.theme--documenter-dark .field.has-addons .control.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - html.theme--documenter-dark .field.has-addons.has-addons-centered { - justify-content: center; } - html.theme--documenter-dark .field.has-addons.has-addons-right { - justify-content: flex-end; } - html.theme--documenter-dark .field.has-addons.has-addons-fullwidth .control { - flex-grow: 1; - flex-shrink: 0; } - html.theme--documenter-dark .field.is-grouped { - display: flex; - justify-content: flex-start; } - html.theme--documenter-dark .field.is-grouped > .control { - flex-shrink: 0; } - html.theme--documenter-dark .field.is-grouped > .control:not(:last-child) { - margin-bottom: 0; - margin-right: 0.75rem; } - html.theme--documenter-dark .field.is-grouped > .control.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - html.theme--documenter-dark .field.is-grouped.is-grouped-centered { - justify-content: center; } - html.theme--documenter-dark .field.is-grouped.is-grouped-right { - justify-content: flex-end; } - html.theme--documenter-dark .field.is-grouped.is-grouped-multiline { - flex-wrap: wrap; } - html.theme--documenter-dark .field.is-grouped.is-grouped-multiline > .control:last-child, html.theme--documenter-dark .field.is-grouped.is-grouped-multiline > .control:not(:last-child) { - margin-bottom: 0.75rem; } - html.theme--documenter-dark .field.is-grouped.is-grouped-multiline:last-child { - margin-bottom: -0.75rem; } - html.theme--documenter-dark .field.is-grouped.is-grouped-multiline:not(:last-child) { - margin-bottom: 0; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .field.is-horizontal { - display: flex; } } - html.theme--documenter-dark .field-label .label { - font-size: inherit; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .field-label { - margin-bottom: 0.5rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .field-label { - flex-basis: 0; - flex-grow: 1; - flex-shrink: 0; - margin-right: 1.5rem; - text-align: right; } - html.theme--documenter-dark .field-label.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.field-label { - font-size: 0.85em; - padding-top: 0.375em; } - html.theme--documenter-dark .field-label.is-normal { - padding-top: 0.375em; } - html.theme--documenter-dark .field-label.is-medium { - font-size: 1.25rem; - padding-top: 0.375em; } - html.theme--documenter-dark .field-label.is-large { - font-size: 1.5rem; - padding-top: 0.375em; } } - html.theme--documenter-dark .field-body .field .field { - margin-bottom: 0; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .field-body { - display: flex; - flex-basis: 0; - flex-grow: 5; - flex-shrink: 1; } - html.theme--documenter-dark .field-body .field { - margin-bottom: 0; } - html.theme--documenter-dark .field-body > .field { - flex-shrink: 1; } - html.theme--documenter-dark .field-body > .field:not(.is-narrow) { - flex-grow: 1; } - html.theme--documenter-dark .field-body > .field:not(:last-child) { - margin-right: 0.75rem; } } - html.theme--documenter-dark .control { - box-sizing: border-box; - clear: both; - font-size: 15px; - position: relative; - text-align: left; } - html.theme--documenter-dark .control.has-icons-left .input:focus ~ .icon, html.theme--documenter-dark .control.has-icons-left #documenter .docs-sidebar form.docs-search > input:focus ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-left form.docs-search > input:focus ~ .icon, - html.theme--documenter-dark .control.has-icons-left .select:focus ~ .icon, html.theme--documenter-dark .control.has-icons-right .input:focus ~ .icon, html.theme--documenter-dark .control.has-icons-right #documenter .docs-sidebar form.docs-search > input:focus ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-right form.docs-search > input:focus ~ .icon, - html.theme--documenter-dark .control.has-icons-right .select:focus ~ .icon { - color: #5e6d6f; } - html.theme--documenter-dark .control.has-icons-left .input.is-small ~ .icon, html.theme--documenter-dark .control.has-icons-left #documenter .docs-sidebar form.docs-search > input ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-left form.docs-search > input ~ .icon, - html.theme--documenter-dark .control.has-icons-left .select.is-small ~ .icon, html.theme--documenter-dark .control.has-icons-right .input.is-small ~ .icon, html.theme--documenter-dark .control.has-icons-right #documenter .docs-sidebar form.docs-search > input ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-right form.docs-search > input ~ .icon, - html.theme--documenter-dark .control.has-icons-right .select.is-small ~ .icon { - font-size: 0.85em; } - html.theme--documenter-dark .control.has-icons-left .input.is-medium ~ .icon, html.theme--documenter-dark .control.has-icons-left #documenter .docs-sidebar form.docs-search > input.is-medium ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-left form.docs-search > input.is-medium ~ .icon, - html.theme--documenter-dark .control.has-icons-left .select.is-medium ~ .icon, html.theme--documenter-dark .control.has-icons-right .input.is-medium ~ .icon, html.theme--documenter-dark .control.has-icons-right #documenter .docs-sidebar form.docs-search > input.is-medium ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-right form.docs-search > input.is-medium ~ .icon, - html.theme--documenter-dark .control.has-icons-right .select.is-medium ~ .icon { - font-size: 1.25rem; } - html.theme--documenter-dark .control.has-icons-left .input.is-large ~ .icon, html.theme--documenter-dark .control.has-icons-left #documenter .docs-sidebar form.docs-search > input.is-large ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-left form.docs-search > input.is-large ~ .icon, - html.theme--documenter-dark .control.has-icons-left .select.is-large ~ .icon, html.theme--documenter-dark .control.has-icons-right .input.is-large ~ .icon, html.theme--documenter-dark .control.has-icons-right #documenter .docs-sidebar form.docs-search > input.is-large ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-right form.docs-search > input.is-large ~ .icon, - html.theme--documenter-dark .control.has-icons-right .select.is-large ~ .icon { - font-size: 1.5rem; } - html.theme--documenter-dark .control.has-icons-left .icon, html.theme--documenter-dark .control.has-icons-right .icon { - color: #dbdee0; - height: 2.25em; - pointer-events: none; - position: absolute; - top: 0; - width: 2.25em; - z-index: 4; } - html.theme--documenter-dark .control.has-icons-left .input, html.theme--documenter-dark .control.has-icons-left #documenter .docs-sidebar form.docs-search > input, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-left form.docs-search > input, - html.theme--documenter-dark .control.has-icons-left .select select { - padding-left: 2.25em; } - html.theme--documenter-dark .control.has-icons-left .icon.is-left { - left: 0; } - html.theme--documenter-dark .control.has-icons-right .input, html.theme--documenter-dark .control.has-icons-right #documenter .docs-sidebar form.docs-search > input, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-right form.docs-search > input, - html.theme--documenter-dark .control.has-icons-right .select select { - padding-right: 2.25em; } - html.theme--documenter-dark .control.has-icons-right .icon.is-right { - right: 0; } - html.theme--documenter-dark .control.is-loading::after { - position: absolute !important; - right: 0.625em; - top: 0.625em; - z-index: 4; } - html.theme--documenter-dark .control.is-loading.is-small:after, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-loading:after { - font-size: 0.85em; } - html.theme--documenter-dark .control.is-loading.is-medium:after { - font-size: 1.25rem; } - html.theme--documenter-dark .control.is-loading.is-large:after { - font-size: 1.5rem; } - html.theme--documenter-dark .breadcrumb { - font-size: 15px; - white-space: nowrap; } - html.theme--documenter-dark .breadcrumb a { - align-items: center; - color: #1abc9c; - display: flex; - justify-content: center; - padding: 0 0.75em; } - html.theme--documenter-dark .breadcrumb a:hover { - color: #1dd2af; } - html.theme--documenter-dark .breadcrumb li { - align-items: center; - display: flex; } - html.theme--documenter-dark .breadcrumb li:first-child a { - padding-left: 0; } - html.theme--documenter-dark .breadcrumb li.is-active a { - color: #f2f2f2; - cursor: default; - pointer-events: none; } - html.theme--documenter-dark .breadcrumb li + li::before { - color: #8c9b9d; - content: "\0002f"; } - html.theme--documenter-dark .breadcrumb ul, - html.theme--documenter-dark .breadcrumb ol { - align-items: flex-start; - display: flex; - flex-wrap: wrap; - justify-content: flex-start; } - html.theme--documenter-dark .breadcrumb .icon:first-child { - margin-right: 0.5em; } - html.theme--documenter-dark .breadcrumb .icon:last-child { - margin-left: 0.5em; } - html.theme--documenter-dark .breadcrumb.is-centered ol, - html.theme--documenter-dark .breadcrumb.is-centered ul { - justify-content: center; } - html.theme--documenter-dark .breadcrumb.is-right ol, - html.theme--documenter-dark .breadcrumb.is-right ul { - justify-content: flex-end; } - html.theme--documenter-dark .breadcrumb.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.breadcrumb { - font-size: 0.85em; } - html.theme--documenter-dark .breadcrumb.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .breadcrumb.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .breadcrumb.has-arrow-separator li + li::before { - content: "\02192"; } - html.theme--documenter-dark .breadcrumb.has-bullet-separator li + li::before { - content: "\02022"; } - html.theme--documenter-dark .breadcrumb.has-dot-separator li + li::before { - content: "\000b7"; } - html.theme--documenter-dark .breadcrumb.has-succeeds-separator li + li::before { - content: "\0227B"; } - html.theme--documenter-dark .card { - background-color: white; - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); - color: #fff; - max-width: 100%; - position: relative; } - html.theme--documenter-dark .card-header { - background-color: transparent; - align-items: stretch; - box-shadow: 0 1px 2px rgba(10, 10, 10, 0.1); - display: flex; } - html.theme--documenter-dark .card-header-title { - align-items: center; - color: #f2f2f2; - display: flex; - flex-grow: 1; - font-weight: 700; - padding: 0.75rem; } - html.theme--documenter-dark .card-header-title.is-centered { - justify-content: center; } - html.theme--documenter-dark .card-header-icon { - align-items: center; - cursor: pointer; - display: flex; - justify-content: center; - padding: 0.75rem; } - html.theme--documenter-dark .card-image { - display: block; - position: relative; } - html.theme--documenter-dark .card-content { - background-color: transparent; - padding: 1.5rem; } - html.theme--documenter-dark .card-footer { - background-color: transparent; - border-top: 1px solid #5e6d6f; - align-items: stretch; - display: flex; } - html.theme--documenter-dark .card-footer-item { - align-items: center; - display: flex; - flex-basis: 0; - flex-grow: 1; - flex-shrink: 0; - justify-content: center; - padding: 0.75rem; } - html.theme--documenter-dark .card-footer-item:not(:last-child) { - border-right: 1px solid #5e6d6f; } - html.theme--documenter-dark .card .media:not(:last-child) { - margin-bottom: 1.5rem; } - html.theme--documenter-dark .dropdown { - display: inline-flex; - position: relative; - vertical-align: top; } - html.theme--documenter-dark .dropdown.is-active .dropdown-menu, html.theme--documenter-dark .dropdown.is-hoverable:hover .dropdown-menu { - display: block; } - html.theme--documenter-dark .dropdown.is-right .dropdown-menu { - left: auto; - right: 0; } - html.theme--documenter-dark .dropdown.is-up .dropdown-menu { - bottom: 100%; - padding-bottom: 4px; - padding-top: initial; - top: auto; } - html.theme--documenter-dark .dropdown-menu { - display: none; - left: 0; - min-width: 12rem; - padding-top: 4px; - position: absolute; - top: 100%; - z-index: 20; } - html.theme--documenter-dark .dropdown-content { - background-color: #282f2f; - border-radius: 0.4em; - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); - padding-bottom: 0.5rem; - padding-top: 0.5rem; } - html.theme--documenter-dark .dropdown-item { - color: #fff; - display: block; - font-size: 0.875rem; - line-height: 1.5; - padding: 0.375rem 1rem; - position: relative; } - html.theme--documenter-dark a.dropdown-item, - html.theme--documenter-dark button.dropdown-item { - padding-right: 3rem; - text-align: left; - white-space: nowrap; - width: 100%; } - html.theme--documenter-dark a.dropdown-item:hover, - html.theme--documenter-dark button.dropdown-item:hover { - background-color: #282f2f; - color: #0a0a0a; } - html.theme--documenter-dark a.dropdown-item.is-active, - html.theme--documenter-dark button.dropdown-item.is-active { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .dropdown-divider { - background-color: #5e6d6f; - border: none; - display: block; - height: 1px; - margin: 0.5rem 0; } - html.theme--documenter-dark .level { - align-items: center; - justify-content: space-between; } - html.theme--documenter-dark .level code { - border-radius: 0.4em; } - html.theme--documenter-dark .level img { - display: inline-block; - vertical-align: top; } - html.theme--documenter-dark .level.is-mobile { - display: flex; } - html.theme--documenter-dark .level.is-mobile .level-left, - html.theme--documenter-dark .level.is-mobile .level-right { - display: flex; } - html.theme--documenter-dark .level.is-mobile .level-left + .level-right { - margin-top: 0; } - html.theme--documenter-dark .level.is-mobile .level-item:not(:last-child) { - margin-bottom: 0; - margin-right: 0.75rem; } - html.theme--documenter-dark .level.is-mobile .level-item:not(.is-narrow) { - flex-grow: 1; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .level { - display: flex; } - html.theme--documenter-dark .level > .level-item:not(.is-narrow) { - flex-grow: 1; } } - html.theme--documenter-dark .level-item { - align-items: center; - display: flex; - flex-basis: auto; - flex-grow: 0; - flex-shrink: 0; - justify-content: center; } - html.theme--documenter-dark .level-item .title, - html.theme--documenter-dark .level-item .subtitle { - margin-bottom: 0; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .level-item:not(:last-child) { - margin-bottom: 0.75rem; } } - html.theme--documenter-dark .level-left, - html.theme--documenter-dark .level-right { - flex-basis: auto; - flex-grow: 0; - flex-shrink: 0; } - html.theme--documenter-dark .level-left .level-item.is-flexible, - html.theme--documenter-dark .level-right .level-item.is-flexible { - flex-grow: 1; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .level-left .level-item:not(:last-child), - html.theme--documenter-dark .level-right .level-item:not(:last-child) { - margin-right: 0.75rem; } } - html.theme--documenter-dark .level-left { - align-items: center; - justify-content: flex-start; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .level-left + .level-right { - margin-top: 1.5rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .level-left { - display: flex; } } - html.theme--documenter-dark .level-right { - align-items: center; - justify-content: flex-end; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .level-right { - display: flex; } } - html.theme--documenter-dark .list { - background-color: white; - border-radius: 0.4em; - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .list-item { - display: block; - padding: 0.5em 1em; } - html.theme--documenter-dark .list-item:not(a) { - color: #fff; } - html.theme--documenter-dark .list-item:first-child { - border-top-left-radius: 0.4em; - border-top-right-radius: 0.4em; } - html.theme--documenter-dark .list-item:last-child { - border-bottom-left-radius: 0.4em; - border-bottom-right-radius: 0.4em; } - html.theme--documenter-dark .list-item:not(:last-child) { - border-bottom: 1px solid #5e6d6f; } - html.theme--documenter-dark .list-item.is-active { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark a.list-item { - background-color: #282f2f; - cursor: pointer; } - html.theme--documenter-dark .media { - align-items: flex-start; - display: flex; - text-align: left; } - html.theme--documenter-dark .media .content:not(:last-child) { - margin-bottom: 0.75rem; } - html.theme--documenter-dark .media .media { - border-top: 1px solid rgba(94, 109, 111, 0.5); - display: flex; - padding-top: 0.75rem; } - html.theme--documenter-dark .media .media .content:not(:last-child), - html.theme--documenter-dark .media .media .control:not(:last-child) { - margin-bottom: 0.5rem; } - html.theme--documenter-dark .media .media .media { - padding-top: 0.5rem; } - html.theme--documenter-dark .media .media .media + .media { - margin-top: 0.5rem; } - html.theme--documenter-dark .media + .media { - border-top: 1px solid rgba(94, 109, 111, 0.5); - margin-top: 1rem; - padding-top: 1rem; } - html.theme--documenter-dark .media.is-large + .media { - margin-top: 1.5rem; - padding-top: 1.5rem; } - html.theme--documenter-dark .media-left, - html.theme--documenter-dark .media-right { - flex-basis: auto; - flex-grow: 0; - flex-shrink: 0; } - html.theme--documenter-dark .media-left { - margin-right: 1rem; } - html.theme--documenter-dark .media-right { - margin-left: 1rem; } - html.theme--documenter-dark .media-content { - flex-basis: auto; - flex-grow: 1; - flex-shrink: 1; - text-align: left; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .media-content { - overflow-x: auto; } } - html.theme--documenter-dark .menu { - font-size: 15px; } - html.theme--documenter-dark .menu.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.menu { - font-size: 0.85em; } - html.theme--documenter-dark .menu.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .menu.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .menu-list { - line-height: 1.25; } - html.theme--documenter-dark .menu-list a { - border-radius: 3px; - color: #fff; - display: block; - padding: 0.5em 0.75em; } - html.theme--documenter-dark .menu-list a:hover { - background-color: #282f2f; - color: #f2f2f2; } - html.theme--documenter-dark .menu-list a.is-active { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .menu-list li ul { - border-left: 1px solid #5e6d6f; - margin: 0.75em; - padding-left: 0.75em; } - html.theme--documenter-dark .menu-label { - color: white; - font-size: 0.75em; - letter-spacing: 0.1em; - text-transform: uppercase; } - html.theme--documenter-dark .menu-label:not(:first-child) { - margin-top: 1em; } - html.theme--documenter-dark .menu-label:not(:last-child) { - margin-bottom: 1em; } - html.theme--documenter-dark .message { - background-color: #282f2f; - border-radius: 0.4em; - font-size: 15px; } - html.theme--documenter-dark .message strong { - color: currentColor; } - html.theme--documenter-dark .message a:not(.button):not(.tag):not(.dropdown-item) { - color: currentColor; - text-decoration: underline; } - html.theme--documenter-dark .message.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.message { - font-size: 0.85em; } - html.theme--documenter-dark .message.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .message.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .message.is-white { - background-color: white; } - html.theme--documenter-dark .message.is-white .message-header { - background-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .message.is-white .message-body { - border-color: white; - color: #4d4d4d; } - html.theme--documenter-dark .message.is-black { - background-color: #fafafa; } - html.theme--documenter-dark .message.is-black .message-header { - background-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .message.is-black .message-body { - border-color: #0a0a0a; - color: #090909; } - html.theme--documenter-dark .message.is-light { - background-color: #f9fafb; } - html.theme--documenter-dark .message.is-light .message-header { - background-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .message.is-light .message-body { - border-color: #ecf0f1; - color: #505050; } - html.theme--documenter-dark .message.is-dark, html.theme--documenter-dark .content kbd.message { - background-color: #f9fafa; } - html.theme--documenter-dark .message.is-dark .message-header, html.theme--documenter-dark .content kbd.message .message-header { - background-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .message.is-dark .message-body, html.theme--documenter-dark .content kbd.message .message-body { - border-color: #282f2f; - color: #212526; } - html.theme--documenter-dark .message.is-primary, html.theme--documenter-dark .docstring > section > a.message.docs-sourcelink { - background-color: #f8fafc; } - html.theme--documenter-dark .message.is-primary .message-header, html.theme--documenter-dark .docstring > section > a.message.docs-sourcelink .message-header { - background-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .message.is-primary .message-body, html.theme--documenter-dark .docstring > section > a.message.docs-sourcelink .message-body { - border-color: #375a7f; - color: #2b4159; } - html.theme--documenter-dark .message.is-link { - background-color: #f6fefc; } - html.theme--documenter-dark .message.is-link .message-header { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .message.is-link .message-body { - border-color: #1abc9c; - color: #0b2f28; } - html.theme--documenter-dark .message.is-info { - background-color: #f5fbff; } - html.theme--documenter-dark .message.is-info .message-header { - background-color: #024c7d; - color: #fff; } - html.theme--documenter-dark .message.is-info .message-body { - border-color: #024c7d; - color: #033659; } - html.theme--documenter-dark .message.is-success { - background-color: #f5fff9; } - html.theme--documenter-dark .message.is-success .message-header { - background-color: #008438; - color: #fff; } - html.theme--documenter-dark .message.is-success .message-body { - border-color: #008438; - color: #023518; } - html.theme--documenter-dark .message.is-warning { - background-color: #fffcf5; } - html.theme--documenter-dark .message.is-warning .message-header { - background-color: #ad8100; - color: #fff; } - html.theme--documenter-dark .message.is-warning .message-body { - border-color: #ad8100; - color: #3d2e03; } - html.theme--documenter-dark .message.is-danger { - background-color: #fef6f6; } - html.theme--documenter-dark .message.is-danger .message-header { - background-color: #9e1b0d; - color: #fff; } - html.theme--documenter-dark .message.is-danger .message-body { - border-color: #9e1b0d; - color: #7a170c; } - html.theme--documenter-dark .message-header { - align-items: center; - background-color: #fff; - border-radius: 0.4em 0.4em 0 0; - color: rgba(0, 0, 0, 0.7); - display: flex; - font-weight: 700; - justify-content: space-between; - line-height: 1.25; - padding: 0.75em 1em; - position: relative; } - html.theme--documenter-dark .message-header .delete { - flex-grow: 0; - flex-shrink: 0; - margin-left: 0.75em; } - html.theme--documenter-dark .message-header + .message-body { - border-width: 0; - border-top-left-radius: 0; - border-top-right-radius: 0; } - html.theme--documenter-dark .message-body { - border-color: #5e6d6f; - border-radius: 0.4em; - border-style: solid; - border-width: 0 0 0 4px; - color: #fff; - padding: 1.25em 1.5em; } - html.theme--documenter-dark .message-body code, - html.theme--documenter-dark .message-body pre { - background-color: white; } - html.theme--documenter-dark .message-body pre code { - background-color: transparent; } - html.theme--documenter-dark .modal { - align-items: center; - display: none; - flex-direction: column; - justify-content: center; - overflow: hidden; - position: fixed; - z-index: 40; } - html.theme--documenter-dark .modal.is-active { - display: flex; } - html.theme--documenter-dark .modal-background { - background-color: rgba(10, 10, 10, 0.86); } - html.theme--documenter-dark .modal-content, - html.theme--documenter-dark .modal-card { - margin: 0 20px; - max-height: calc(100vh - 160px); - overflow: auto; - position: relative; - width: 100%; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .modal-content, - html.theme--documenter-dark .modal-card { - margin: 0 auto; - max-height: calc(100vh - 40px); - width: 640px; } } - html.theme--documenter-dark .modal-close { - background: none; - height: 40px; - position: fixed; - right: 20px; - top: 20px; - width: 40px; } - html.theme--documenter-dark .modal-card { - display: flex; - flex-direction: column; - max-height: calc(100vh - 40px); - overflow: hidden; - -ms-overflow-y: visible; } - html.theme--documenter-dark .modal-card-head, - html.theme--documenter-dark .modal-card-foot { - align-items: center; - background-color: #282f2f; - display: flex; - flex-shrink: 0; - justify-content: flex-start; - padding: 20px; - position: relative; } - html.theme--documenter-dark .modal-card-head { - border-bottom: 1px solid #5e6d6f; - border-top-left-radius: 8px; - border-top-right-radius: 8px; } - html.theme--documenter-dark .modal-card-title { - color: #f2f2f2; - flex-grow: 1; - flex-shrink: 0; - font-size: 1.5rem; - line-height: 1; } - html.theme--documenter-dark .modal-card-foot { - border-bottom-left-radius: 8px; - border-bottom-right-radius: 8px; - border-top: 1px solid #5e6d6f; } - html.theme--documenter-dark .modal-card-foot .button:not(:last-child) { - margin-right: 0.5em; } - html.theme--documenter-dark .modal-card-body { - -webkit-overflow-scrolling: touch; - background-color: white; - flex-grow: 1; - flex-shrink: 1; - overflow: auto; - padding: 20px; } - html.theme--documenter-dark .navbar { - background-color: #375a7f; - min-height: 4rem; - position: relative; - z-index: 30; } - html.theme--documenter-dark .navbar.is-white { - background-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-white .navbar-brand .navbar-link { - color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-white .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-white .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-white .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-white .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-white .navbar-brand .navbar-link.is-active { - background-color: #f2f2f2; - color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-brand .navbar-link::after { - border-color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-burger { - color: #0a0a0a; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-white .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-white .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-white .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-white .navbar-end .navbar-link { - color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-white .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-white .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-white .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-white .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-white .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-white .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-white .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-white .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-white .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-white .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-white .navbar-end .navbar-link.is-active { - background-color: #f2f2f2; - color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-white .navbar-end .navbar-link::after { - border-color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-white .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-white .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #f2f2f2; - color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-dropdown a.navbar-item.is-active { - background-color: white; - color: #0a0a0a; } } - html.theme--documenter-dark .navbar.is-black { - background-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-black .navbar-brand .navbar-link { - color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-black .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-black .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-black .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-black .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-black .navbar-brand .navbar-link.is-active { - background-color: black; - color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-brand .navbar-link::after { - border-color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-burger { - color: white; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-black .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-black .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-black .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-black .navbar-end .navbar-link { - color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-black .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-black .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-black .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-black .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-black .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-black .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-black .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-black .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-black .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-black .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-black .navbar-end .navbar-link.is-active { - background-color: black; - color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-black .navbar-end .navbar-link::after { - border-color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-black .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-black .navbar-item.has-dropdown.is-active .navbar-link { - background-color: black; - color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-dropdown a.navbar-item.is-active { - background-color: #0a0a0a; - color: white; } } - html.theme--documenter-dark .navbar.is-light { - background-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-light .navbar-brand .navbar-link { - color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-light .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-light .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-light .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-light .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-light .navbar-brand .navbar-link.is-active { - background-color: #dde4e6; - color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-brand .navbar-link::after { - border-color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-burger { - color: #282f2f; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-light .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-light .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-light .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-light .navbar-end .navbar-link { - color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-light .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-light .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-light .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-light .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-light .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-light .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-light .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-light .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-light .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-light .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-light .navbar-end .navbar-link.is-active { - background-color: #dde4e6; - color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-light .navbar-end .navbar-link::after { - border-color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-light .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-light .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #dde4e6; - color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-dropdown a.navbar-item.is-active { - background-color: #ecf0f1; - color: #282f2f; } } - html.theme--documenter-dark .navbar.is-dark, html.theme--documenter-dark .content kbd.navbar { - background-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-brand > .navbar-item, html.theme--documenter-dark .content kbd.navbar .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-dark .navbar-brand .navbar-link, - html.theme--documenter-dark .content kbd.navbar .navbar-brand .navbar-link { - color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .content kbd.navbar .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-dark .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .content kbd.navbar .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-dark .navbar-brand > a.navbar-item.is-active, html.theme--documenter-dark .content kbd.navbar .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-dark .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .content kbd.navbar .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-dark .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .content kbd.navbar .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-dark .navbar-brand .navbar-link.is-active, - html.theme--documenter-dark .content kbd.navbar .navbar-brand .navbar-link.is-active { - background-color: #1d2122; - color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-brand .navbar-link::after, html.theme--documenter-dark .content kbd.navbar .navbar-brand .navbar-link::after { - border-color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-burger, html.theme--documenter-dark .content kbd.navbar .navbar-burger { - color: #ecf0f1; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-dark .navbar-start > .navbar-item, html.theme--documenter-dark .content kbd.navbar .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-dark .navbar-start .navbar-link, - html.theme--documenter-dark .content kbd.navbar .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-dark .navbar-end > .navbar-item, - html.theme--documenter-dark .content kbd.navbar .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-dark .navbar-end .navbar-link, - html.theme--documenter-dark .content kbd.navbar .navbar-end .navbar-link { - color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .content kbd.navbar .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-dark .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .content kbd.navbar .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-dark .navbar-start > a.navbar-item.is-active, html.theme--documenter-dark .content kbd.navbar .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-dark .navbar-start .navbar-link:focus, - html.theme--documenter-dark .content kbd.navbar .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-dark .navbar-start .navbar-link:hover, - html.theme--documenter-dark .content kbd.navbar .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-dark .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .content kbd.navbar .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-dark .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .content kbd.navbar .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-dark .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .content kbd.navbar .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-dark .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .content kbd.navbar .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-dark .navbar-end .navbar-link:focus, - html.theme--documenter-dark .content kbd.navbar .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-dark .navbar-end .navbar-link:hover, - html.theme--documenter-dark .content kbd.navbar .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-dark .navbar-end .navbar-link.is-active, - html.theme--documenter-dark .content kbd.navbar .navbar-end .navbar-link.is-active { - background-color: #1d2122; - color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-start .navbar-link::after, html.theme--documenter-dark .content kbd.navbar .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-dark .navbar-end .navbar-link::after, - html.theme--documenter-dark .content kbd.navbar .navbar-end .navbar-link::after { - border-color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-item.has-dropdown:focus .navbar-link, html.theme--documenter-dark .content kbd.navbar .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-dark .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .content kbd.navbar .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-dark .navbar-item.has-dropdown.is-active .navbar-link, - html.theme--documenter-dark .content kbd.navbar .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #1d2122; - color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-dropdown a.navbar-item.is-active, html.theme--documenter-dark .content kbd.navbar .navbar-dropdown a.navbar-item.is-active { - background-color: #282f2f; - color: #ecf0f1; } } - html.theme--documenter-dark .navbar.is-primary, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink { - background-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-brand > .navbar-item, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-primary .navbar-brand .navbar-link, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-primary .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-primary .navbar-brand > a.navbar-item.is-active, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-primary .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-primary .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-primary .navbar-brand .navbar-link.is-active, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link.is-active { - background-color: #2f4d6d; - color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-brand .navbar-link::after, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-burger, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-primary .navbar-start > .navbar-item, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-primary .navbar-start .navbar-link, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-primary .navbar-end > .navbar-item, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-primary .navbar-end .navbar-link, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-primary .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-primary .navbar-start > a.navbar-item.is-active, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-primary .navbar-start .navbar-link:focus, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-primary .navbar-start .navbar-link:hover, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-primary .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-primary .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-primary .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-primary .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-primary .navbar-end .navbar-link:focus, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-primary .navbar-end .navbar-link:hover, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-primary .navbar-end .navbar-link.is-active, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link.is-active { - background-color: #2f4d6d; - color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-start .navbar-link::after, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-primary .navbar-end .navbar-link::after, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-item.has-dropdown:focus .navbar-link, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-primary .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-primary .navbar-item.has-dropdown.is-active .navbar-link, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #2f4d6d; - color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-dropdown a.navbar-item.is-active, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-dropdown a.navbar-item.is-active { - background-color: #375a7f; - color: #fff; } } - html.theme--documenter-dark .navbar.is-link { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-link .navbar-brand .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-link .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-link .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-link .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-link .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-link .navbar-brand .navbar-link.is-active { - background-color: #17a689; - color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-brand .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-link .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-link .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-link .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-link .navbar-end .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-link .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-link .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-link .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-link .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-link .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-link .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-link .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-link .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-link .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-link .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-link .navbar-end .navbar-link.is-active { - background-color: #17a689; - color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-link .navbar-end .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-link .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-link .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #17a689; - color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-dropdown a.navbar-item.is-active { - background-color: #1abc9c; - color: #fff; } } - html.theme--documenter-dark .navbar.is-info { - background-color: #024c7d; - color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-info .navbar-brand .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-info .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-info .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-info .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-info .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-info .navbar-brand .navbar-link.is-active { - background-color: #023d64; - color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-brand .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-info .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-info .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-info .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-info .navbar-end .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-info .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-info .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-info .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-info .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-info .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-info .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-info .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-info .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-info .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-info .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-info .navbar-end .navbar-link.is-active { - background-color: #023d64; - color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-info .navbar-end .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-info .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-info .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #023d64; - color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-dropdown a.navbar-item.is-active { - background-color: #024c7d; - color: #fff; } } - html.theme--documenter-dark .navbar.is-success { - background-color: #008438; - color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-success .navbar-brand .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-success .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-success .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-success .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-success .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-success .navbar-brand .navbar-link.is-active { - background-color: #006b2d; - color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-brand .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-success .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-success .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-success .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-success .navbar-end .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-success .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-success .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-success .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-success .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-success .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-success .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-success .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-success .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-success .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-success .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-success .navbar-end .navbar-link.is-active { - background-color: #006b2d; - color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-success .navbar-end .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-success .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-success .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #006b2d; - color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-dropdown a.navbar-item.is-active { - background-color: #008438; - color: #fff; } } - html.theme--documenter-dark .navbar.is-warning { - background-color: #ad8100; - color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-warning .navbar-brand .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-warning .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-warning .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-warning .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-warning .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-warning .navbar-brand .navbar-link.is-active { - background-color: #946e00; - color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-brand .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-warning .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-warning .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-warning .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-warning .navbar-end .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-warning .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-warning .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-warning .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-warning .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-warning .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-warning .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-warning .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-warning .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-warning .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-warning .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-warning .navbar-end .navbar-link.is-active { - background-color: #946e00; - color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-warning .navbar-end .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-warning .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-warning .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #946e00; - color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-dropdown a.navbar-item.is-active { - background-color: #ad8100; - color: #fff; } } - html.theme--documenter-dark .navbar.is-danger { - background-color: #9e1b0d; - color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-danger .navbar-brand .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-danger .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-danger .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-danger .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-danger .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-danger .navbar-brand .navbar-link.is-active { - background-color: #86170b; - color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-brand .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-danger .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-danger .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-danger .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-danger .navbar-end .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-danger .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-danger .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-danger .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-danger .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-danger .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-danger .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-danger .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-danger .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-danger .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-danger .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-danger .navbar-end .navbar-link.is-active { - background-color: #86170b; - color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-danger .navbar-end .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-danger .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-danger .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #86170b; - color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-dropdown a.navbar-item.is-active { - background-color: #9e1b0d; - color: #fff; } } - html.theme--documenter-dark .navbar > .container { - align-items: stretch; - display: flex; - min-height: 4rem; - width: 100%; } - html.theme--documenter-dark .navbar.has-shadow { - box-shadow: 0 2px 0 0 #282f2f; } - html.theme--documenter-dark .navbar.is-fixed-bottom, html.theme--documenter-dark .navbar.is-fixed-top { - left: 0; - position: fixed; - right: 0; - z-index: 30; } - html.theme--documenter-dark .navbar.is-fixed-bottom { - bottom: 0; } - html.theme--documenter-dark .navbar.is-fixed-bottom.has-shadow { - box-shadow: 0 -2px 0 0 #282f2f; } - html.theme--documenter-dark .navbar.is-fixed-top { - top: 0; } - html.theme--documenter-dark html.has-navbar-fixed-top, - html.theme--documenter-dark body.has-navbar-fixed-top { - padding-top: 4rem; } - html.theme--documenter-dark html.has-navbar-fixed-bottom, - html.theme--documenter-dark body.has-navbar-fixed-bottom { - padding-bottom: 4rem; } - html.theme--documenter-dark .navbar-brand, - html.theme--documenter-dark .navbar-tabs { - align-items: stretch; - display: flex; - flex-shrink: 0; - min-height: 4rem; } - html.theme--documenter-dark .navbar-brand a.navbar-item:focus, html.theme--documenter-dark .navbar-brand a.navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .navbar-tabs { - -webkit-overflow-scrolling: touch; - max-width: 100vw; - overflow-x: auto; - overflow-y: hidden; } - html.theme--documenter-dark .navbar-burger { - color: #fff; - cursor: pointer; - display: block; - height: 4rem; - position: relative; - width: 4rem; - margin-left: auto; } - html.theme--documenter-dark .navbar-burger span { - background-color: currentColor; - display: block; - height: 1px; - left: calc(50% - 8px); - position: absolute; - transform-origin: center; - transition-duration: 86ms; - transition-property: background-color, opacity, transform; - transition-timing-function: ease-out; - width: 16px; } - html.theme--documenter-dark .navbar-burger span:nth-child(1) { - top: calc(50% - 6px); } - html.theme--documenter-dark .navbar-burger span:nth-child(2) { - top: calc(50% - 1px); } - html.theme--documenter-dark .navbar-burger span:nth-child(3) { - top: calc(50% + 4px); } - html.theme--documenter-dark .navbar-burger:hover { - background-color: rgba(0, 0, 0, 0.05); } - html.theme--documenter-dark .navbar-burger.is-active span:nth-child(1) { - transform: translateY(5px) rotate(45deg); } - html.theme--documenter-dark .navbar-burger.is-active span:nth-child(2) { - opacity: 0; } - html.theme--documenter-dark .navbar-burger.is-active span:nth-child(3) { - transform: translateY(-5px) rotate(-45deg); } - html.theme--documenter-dark .navbar-menu { - display: none; } - html.theme--documenter-dark .navbar-item, - html.theme--documenter-dark .navbar-link { - color: #fff; - display: block; - line-height: 1.5; - padding: 0.5rem 0.75rem; - position: relative; } - html.theme--documenter-dark .navbar-item .icon:only-child, - html.theme--documenter-dark .navbar-link .icon:only-child { - margin-left: -0.25rem; - margin-right: -0.25rem; } - html.theme--documenter-dark a.navbar-item, - html.theme--documenter-dark .navbar-link { - cursor: pointer; } - html.theme--documenter-dark a.navbar-item:focus, html.theme--documenter-dark a.navbar-item:focus-within, html.theme--documenter-dark a.navbar-item:hover, html.theme--documenter-dark a.navbar-item.is-active, - html.theme--documenter-dark .navbar-link:focus, - html.theme--documenter-dark .navbar-link:focus-within, - html.theme--documenter-dark .navbar-link:hover, - html.theme--documenter-dark .navbar-link.is-active { - background-color: transparent; - color: #1abc9c; } - html.theme--documenter-dark .navbar-item { - display: block; - flex-grow: 0; - flex-shrink: 0; } - html.theme--documenter-dark .navbar-item img { - max-height: 1.75rem; } - html.theme--documenter-dark .navbar-item.has-dropdown { - padding: 0; } - html.theme--documenter-dark .navbar-item.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - html.theme--documenter-dark .navbar-item.is-tab { - border-bottom: 1px solid transparent; - min-height: 4rem; - padding-bottom: calc(0.5rem - 1px); } - html.theme--documenter-dark .navbar-item.is-tab:focus, html.theme--documenter-dark .navbar-item.is-tab:hover { - background-color: transparent; - border-bottom-color: #1abc9c; } - html.theme--documenter-dark .navbar-item.is-tab.is-active { - background-color: transparent; - border-bottom-color: #1abc9c; - border-bottom-style: solid; - border-bottom-width: 3px; - color: #1abc9c; - padding-bottom: calc(0.5rem - 3px); } - html.theme--documenter-dark .navbar-content { - flex-grow: 1; - flex-shrink: 1; } - html.theme--documenter-dark .navbar-link:not(.is-arrowless) { - padding-right: 2.5em; } - html.theme--documenter-dark .navbar-link:not(.is-arrowless)::after { - border-color: #fff; - margin-top: -0.375em; - right: 1.125em; } - html.theme--documenter-dark .navbar-dropdown { - font-size: 0.875rem; - padding-bottom: 0.5rem; - padding-top: 0.5rem; } - html.theme--documenter-dark .navbar-dropdown .navbar-item { - padding-left: 1.5rem; - padding-right: 1.5rem; } - html.theme--documenter-dark .navbar-divider { - background-color: rgba(0, 0, 0, 0.2); - border: none; - display: none; - height: 2px; - margin: 0.5rem 0; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .navbar > .container { - display: block; } - html.theme--documenter-dark .navbar-brand .navbar-item, - html.theme--documenter-dark .navbar-tabs .navbar-item { - align-items: center; - display: flex; } - html.theme--documenter-dark .navbar-link::after { - display: none; } - html.theme--documenter-dark .navbar-menu { - background-color: #375a7f; - box-shadow: 0 8px 16px rgba(10, 10, 10, 0.1); - padding: 0.5rem 0; } - html.theme--documenter-dark .navbar-menu.is-active { - display: block; } - html.theme--documenter-dark .navbar.is-fixed-bottom-touch, html.theme--documenter-dark .navbar.is-fixed-top-touch { - left: 0; - position: fixed; - right: 0; - z-index: 30; } - html.theme--documenter-dark .navbar.is-fixed-bottom-touch { - bottom: 0; } - html.theme--documenter-dark .navbar.is-fixed-bottom-touch.has-shadow { - box-shadow: 0 -2px 3px rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .navbar.is-fixed-top-touch { - top: 0; } - html.theme--documenter-dark .navbar.is-fixed-top .navbar-menu, html.theme--documenter-dark .navbar.is-fixed-top-touch .navbar-menu { - -webkit-overflow-scrolling: touch; - max-height: calc(100vh - 4rem); - overflow: auto; } - html.theme--documenter-dark html.has-navbar-fixed-top-touch, - html.theme--documenter-dark body.has-navbar-fixed-top-touch { - padding-top: 4rem; } - html.theme--documenter-dark html.has-navbar-fixed-bottom-touch, - html.theme--documenter-dark body.has-navbar-fixed-bottom-touch { - padding-bottom: 4rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar, - html.theme--documenter-dark .navbar-menu, - html.theme--documenter-dark .navbar-start, - html.theme--documenter-dark .navbar-end { - align-items: stretch; - display: flex; } - html.theme--documenter-dark .navbar { - min-height: 4rem; } - html.theme--documenter-dark .navbar.is-spaced { - padding: 1rem 2rem; } - html.theme--documenter-dark .navbar.is-spaced .navbar-start, - html.theme--documenter-dark .navbar.is-spaced .navbar-end { - align-items: center; } - html.theme--documenter-dark .navbar.is-spaced a.navbar-item, - html.theme--documenter-dark .navbar.is-spaced .navbar-link { - border-radius: 0.4em; } - html.theme--documenter-dark .navbar.is-transparent a.navbar-item:focus, html.theme--documenter-dark .navbar.is-transparent a.navbar-item:hover, html.theme--documenter-dark .navbar.is-transparent a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-transparent .navbar-link:focus, - html.theme--documenter-dark .navbar.is-transparent .navbar-link:hover, - html.theme--documenter-dark .navbar.is-transparent .navbar-link.is-active { - background-color: transparent !important; } - html.theme--documenter-dark .navbar.is-transparent .navbar-item.has-dropdown.is-active .navbar-link, html.theme--documenter-dark .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:focus .navbar-link, html.theme--documenter-dark .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:focus-within .navbar-link, html.theme--documenter-dark .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:hover .navbar-link { - background-color: transparent !important; } - html.theme--documenter-dark .navbar.is-transparent .navbar-dropdown a.navbar-item:focus, html.theme--documenter-dark .navbar.is-transparent .navbar-dropdown a.navbar-item:hover { - background-color: transparent; - color: #dbdee0; } - html.theme--documenter-dark .navbar.is-transparent .navbar-dropdown a.navbar-item.is-active { - background-color: transparent; - color: #1abc9c; } - html.theme--documenter-dark .navbar-burger { - display: none; } - html.theme--documenter-dark .navbar-item, - html.theme--documenter-dark .navbar-link { - align-items: center; - display: flex; } - html.theme--documenter-dark .navbar-item { - display: flex; } - html.theme--documenter-dark .navbar-item.has-dropdown { - align-items: stretch; } - html.theme--documenter-dark .navbar-item.has-dropdown-up .navbar-link::after { - transform: rotate(135deg) translate(0.25em, -0.25em); } - html.theme--documenter-dark .navbar-item.has-dropdown-up .navbar-dropdown { - border-bottom: 1px solid rgba(0, 0, 0, 0.2); - border-radius: 8px 8px 0 0; - border-top: none; - bottom: 100%; - box-shadow: 0 -8px 8px rgba(10, 10, 10, 0.1); - top: auto; } - html.theme--documenter-dark .navbar-item.is-active .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:focus .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:focus-within .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:hover .navbar-dropdown { - display: block; } - .navbar.is-spaced html.theme--documenter-dark .navbar-item.is-active .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-active .navbar-dropdown.is-boxed, .navbar.is-spaced html.theme--documenter-dark .navbar-item.is-hoverable:focus .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:focus .navbar-dropdown.is-boxed, .navbar.is-spaced html.theme--documenter-dark .navbar-item.is-hoverable:focus-within .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:focus-within .navbar-dropdown.is-boxed, .navbar.is-spaced html.theme--documenter-dark .navbar-item.is-hoverable:hover .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:hover .navbar-dropdown.is-boxed { - opacity: 1; - pointer-events: auto; - transform: translateY(0); } - html.theme--documenter-dark .navbar-menu { - flex-grow: 1; - flex-shrink: 0; } - html.theme--documenter-dark .navbar-start { - justify-content: flex-start; - margin-right: auto; } - html.theme--documenter-dark .navbar-end { - justify-content: flex-end; - margin-left: auto; } - html.theme--documenter-dark .navbar-dropdown { - background-color: #375a7f; - border-bottom-left-radius: 8px; - border-bottom-right-radius: 8px; - border-top: 1px solid rgba(0, 0, 0, 0.2); - box-shadow: 0 8px 8px rgba(10, 10, 10, 0.1); - display: none; - font-size: 0.875rem; - left: 0; - min-width: 100%; - position: absolute; - top: 100%; - z-index: 20; } - html.theme--documenter-dark .navbar-dropdown .navbar-item { - padding: 0.375rem 1rem; - white-space: nowrap; } - html.theme--documenter-dark .navbar-dropdown a.navbar-item { - padding-right: 3rem; } - html.theme--documenter-dark .navbar-dropdown a.navbar-item:focus, html.theme--documenter-dark .navbar-dropdown a.navbar-item:hover { - background-color: transparent; - color: #dbdee0; } - html.theme--documenter-dark .navbar-dropdown a.navbar-item.is-active { - background-color: transparent; - color: #1abc9c; } - .navbar.is-spaced html.theme--documenter-dark .navbar-dropdown, html.theme--documenter-dark .navbar-dropdown.is-boxed { - border-radius: 8px; - border-top: none; - box-shadow: 0 8px 8px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); - display: block; - opacity: 0; - pointer-events: none; - top: calc(100% + (-4px)); - transform: translateY(-5px); - transition-duration: 86ms; - transition-property: opacity, transform; } - html.theme--documenter-dark .navbar-dropdown.is-right { - left: auto; - right: 0; } - html.theme--documenter-dark .navbar-divider { - display: block; } - html.theme--documenter-dark .navbar > .container .navbar-brand, - html.theme--documenter-dark .container > .navbar .navbar-brand { - margin-left: -.75rem; } - html.theme--documenter-dark .navbar > .container .navbar-menu, - html.theme--documenter-dark .container > .navbar .navbar-menu { - margin-right: -.75rem; } - html.theme--documenter-dark .navbar.is-fixed-bottom-desktop, html.theme--documenter-dark .navbar.is-fixed-top-desktop { - left: 0; - position: fixed; - right: 0; - z-index: 30; } - html.theme--documenter-dark .navbar.is-fixed-bottom-desktop { - bottom: 0; } - html.theme--documenter-dark .navbar.is-fixed-bottom-desktop.has-shadow { - box-shadow: 0 -2px 3px rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .navbar.is-fixed-top-desktop { - top: 0; } - html.theme--documenter-dark html.has-navbar-fixed-top-desktop, - html.theme--documenter-dark body.has-navbar-fixed-top-desktop { - padding-top: 4rem; } - html.theme--documenter-dark html.has-navbar-fixed-bottom-desktop, - html.theme--documenter-dark body.has-navbar-fixed-bottom-desktop { - padding-bottom: 4rem; } - html.theme--documenter-dark html.has-spaced-navbar-fixed-top, - html.theme--documenter-dark body.has-spaced-navbar-fixed-top { - padding-top: 6rem; } - html.theme--documenter-dark html.has-spaced-navbar-fixed-bottom, - html.theme--documenter-dark body.has-spaced-navbar-fixed-bottom { - padding-bottom: 6rem; } - html.theme--documenter-dark a.navbar-item.is-active, - html.theme--documenter-dark .navbar-link.is-active { - color: #1abc9c; } - html.theme--documenter-dark a.navbar-item.is-active:not(:focus):not(:hover), - html.theme--documenter-dark .navbar-link.is-active:not(:focus):not(:hover) { - background-color: transparent; } - html.theme--documenter-dark .navbar-item.has-dropdown:focus .navbar-link, html.theme--documenter-dark .navbar-item.has-dropdown:hover .navbar-link, html.theme--documenter-dark .navbar-item.has-dropdown.is-active .navbar-link { - background-color: transparent; } } - html.theme--documenter-dark .hero.is-fullheight-with-navbar { - min-height: calc(100vh - 4rem); } - html.theme--documenter-dark .pagination { - font-size: 15px; - margin: -0.25rem; } - html.theme--documenter-dark .pagination.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.pagination { - font-size: 0.85em; } - html.theme--documenter-dark .pagination.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .pagination.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .pagination.is-rounded .pagination-previous, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.pagination .pagination-previous, - html.theme--documenter-dark .pagination.is-rounded .pagination-next, - html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.pagination .pagination-next { - padding-left: 1em; - padding-right: 1em; - border-radius: 290486px; } - html.theme--documenter-dark .pagination.is-rounded .pagination-link, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.pagination .pagination-link { - border-radius: 290486px; } - html.theme--documenter-dark .pagination, - html.theme--documenter-dark .pagination-list { - align-items: center; - display: flex; - justify-content: center; - text-align: center; } - html.theme--documenter-dark .pagination-previous, - html.theme--documenter-dark .pagination-next, - html.theme--documenter-dark .pagination-link, - html.theme--documenter-dark .pagination-ellipsis { - font-size: 1em; - justify-content: center; - margin: 0.25rem; - padding-left: 0.5em; - padding-right: 0.5em; - text-align: center; } - html.theme--documenter-dark .pagination-previous, - html.theme--documenter-dark .pagination-next, - html.theme--documenter-dark .pagination-link { - border-color: #5e6d6f; - color: #1abc9c; - min-width: 2.25em; } - html.theme--documenter-dark .pagination-previous:hover, - html.theme--documenter-dark .pagination-next:hover, - html.theme--documenter-dark .pagination-link:hover { - border-color: #8c9b9d; - color: #1dd2af; } - html.theme--documenter-dark .pagination-previous:focus, - html.theme--documenter-dark .pagination-next:focus, - html.theme--documenter-dark .pagination-link:focus { - border-color: #8c9b9d; } - html.theme--documenter-dark .pagination-previous:active, - html.theme--documenter-dark .pagination-next:active, - html.theme--documenter-dark .pagination-link:active { - box-shadow: inset 0 1px 2px rgba(10, 10, 10, 0.2); } - html.theme--documenter-dark .pagination-previous[disabled], - html.theme--documenter-dark .pagination-next[disabled], - html.theme--documenter-dark .pagination-link[disabled] { - background-color: #dbdee0; - border-color: #dbdee0; - box-shadow: none; - color: #5e6d6f; - opacity: 0.5; } - html.theme--documenter-dark .pagination-previous, - html.theme--documenter-dark .pagination-next { - padding-left: 0.75em; - padding-right: 0.75em; - white-space: nowrap; } - html.theme--documenter-dark .pagination-link.is-current { - background-color: #1abc9c; - border-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .pagination-ellipsis { - color: #8c9b9d; - pointer-events: none; } - html.theme--documenter-dark .pagination-list { - flex-wrap: wrap; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .pagination { - flex-wrap: wrap; } - html.theme--documenter-dark .pagination-previous, - html.theme--documenter-dark .pagination-next { - flex-grow: 1; - flex-shrink: 1; } - html.theme--documenter-dark .pagination-list li { - flex-grow: 1; - flex-shrink: 1; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .pagination-list { - flex-grow: 1; - flex-shrink: 1; - justify-content: flex-start; - order: 1; } - html.theme--documenter-dark .pagination-previous { - order: 2; } - html.theme--documenter-dark .pagination-next { - order: 3; } - html.theme--documenter-dark .pagination { - justify-content: space-between; } - html.theme--documenter-dark .pagination.is-centered .pagination-previous { - order: 1; } - html.theme--documenter-dark .pagination.is-centered .pagination-list { - justify-content: center; - order: 2; } - html.theme--documenter-dark .pagination.is-centered .pagination-next { - order: 3; } - html.theme--documenter-dark .pagination.is-right .pagination-previous { - order: 1; } - html.theme--documenter-dark .pagination.is-right .pagination-next { - order: 2; } - html.theme--documenter-dark .pagination.is-right .pagination-list { - justify-content: flex-end; - order: 3; } } - html.theme--documenter-dark .panel { - font-size: 15px; } - html.theme--documenter-dark .panel:not(:last-child) { - margin-bottom: 1.5rem; } - html.theme--documenter-dark .panel-heading, - html.theme--documenter-dark .panel-tabs, - html.theme--documenter-dark .panel-block { - border-bottom: 1px solid #5e6d6f; - border-left: 1px solid #5e6d6f; - border-right: 1px solid #5e6d6f; } - html.theme--documenter-dark .panel-heading:first-child, - html.theme--documenter-dark .panel-tabs:first-child, - html.theme--documenter-dark .panel-block:first-child { - border-top: 1px solid #5e6d6f; } - html.theme--documenter-dark .panel-heading { - background-color: #282f2f; - border-radius: 0.4em 0.4em 0 0; - color: #f2f2f2; - font-size: 1.25em; - font-weight: 300; - line-height: 1.25; - padding: 0.5em 0.75em; } - html.theme--documenter-dark .panel-tabs { - align-items: flex-end; - display: flex; - font-size: 0.875em; - justify-content: center; } - html.theme--documenter-dark .panel-tabs a { - border-bottom: 1px solid #5e6d6f; - margin-bottom: -1px; - padding: 0.5em; } - html.theme--documenter-dark .panel-tabs a.is-active { - border-bottom-color: #343c3d; - color: #17a689; } - html.theme--documenter-dark .panel-list a { - color: #fff; } - html.theme--documenter-dark .panel-list a:hover { - color: #1abc9c; } - html.theme--documenter-dark .panel-block { - align-items: center; - color: #f2f2f2; - display: flex; - justify-content: flex-start; - padding: 0.5em 0.75em; } - html.theme--documenter-dark .panel-block input[type="checkbox"] { - margin-right: 0.75em; } - html.theme--documenter-dark .panel-block > .control { - flex-grow: 1; - flex-shrink: 1; - width: 100%; } - html.theme--documenter-dark .panel-block.is-wrapped { - flex-wrap: wrap; } - html.theme--documenter-dark .panel-block.is-active { - border-left-color: #1abc9c; - color: #17a689; } - html.theme--documenter-dark .panel-block.is-active .panel-icon { - color: #1abc9c; } - html.theme--documenter-dark a.panel-block, - html.theme--documenter-dark label.panel-block { - cursor: pointer; } - html.theme--documenter-dark a.panel-block:hover, - html.theme--documenter-dark label.panel-block:hover { - background-color: #282f2f; } - html.theme--documenter-dark .panel-icon { - display: inline-block; - font-size: 14px; - height: 1em; - line-height: 1em; - text-align: center; - vertical-align: top; - width: 1em; - color: white; - margin-right: 0.75em; } - html.theme--documenter-dark .panel-icon .fa { - font-size: inherit; - line-height: inherit; } - html.theme--documenter-dark .tabs { - -webkit-overflow-scrolling: touch; - align-items: stretch; - display: flex; - font-size: 15px; - justify-content: space-between; - overflow: hidden; - overflow-x: auto; - white-space: nowrap; } - html.theme--documenter-dark .tabs a { - align-items: center; - border-bottom-color: #5e6d6f; - border-bottom-style: solid; - border-bottom-width: 1px; - color: #fff; - display: flex; - justify-content: center; - margin-bottom: -1px; - padding: 0.5em 1em; - vertical-align: top; } - html.theme--documenter-dark .tabs a:hover { - border-bottom-color: #f2f2f2; - color: #f2f2f2; } - html.theme--documenter-dark .tabs li { - display: block; } - html.theme--documenter-dark .tabs li.is-active a { - border-bottom-color: #1abc9c; - color: #1abc9c; } - html.theme--documenter-dark .tabs ul { - align-items: center; - border-bottom-color: #5e6d6f; - border-bottom-style: solid; - border-bottom-width: 1px; - display: flex; - flex-grow: 1; - flex-shrink: 0; - justify-content: flex-start; } - html.theme--documenter-dark .tabs ul.is-left { - padding-right: 0.75em; } - html.theme--documenter-dark .tabs ul.is-center { - flex: none; - justify-content: center; - padding-left: 0.75em; - padding-right: 0.75em; } - html.theme--documenter-dark .tabs ul.is-right { - justify-content: flex-end; - padding-left: 0.75em; } - html.theme--documenter-dark .tabs .icon:first-child { - margin-right: 0.5em; } - html.theme--documenter-dark .tabs .icon:last-child { - margin-left: 0.5em; } - html.theme--documenter-dark .tabs.is-centered ul { - justify-content: center; } - html.theme--documenter-dark .tabs.is-right ul { - justify-content: flex-end; } - html.theme--documenter-dark .tabs.is-boxed a { - border: 1px solid transparent; - border-radius: 0.4em 0.4em 0 0; } - html.theme--documenter-dark .tabs.is-boxed a:hover { - background-color: #282f2f; - border-bottom-color: #5e6d6f; } - html.theme--documenter-dark .tabs.is-boxed li.is-active a { - background-color: white; - border-color: #5e6d6f; - border-bottom-color: transparent !important; } - html.theme--documenter-dark .tabs.is-fullwidth li { - flex-grow: 1; - flex-shrink: 0; } - html.theme--documenter-dark .tabs.is-toggle a { - border-color: #5e6d6f; - border-style: solid; - border-width: 1px; - margin-bottom: 0; - position: relative; } - html.theme--documenter-dark .tabs.is-toggle a:hover { - background-color: #282f2f; - border-color: #8c9b9d; - z-index: 2; } - html.theme--documenter-dark .tabs.is-toggle li + li { - margin-left: -1px; } - html.theme--documenter-dark .tabs.is-toggle li:first-child a { - border-radius: 0.4em 0 0 0.4em; } - html.theme--documenter-dark .tabs.is-toggle li:last-child a { - border-radius: 0 0.4em 0.4em 0; } - html.theme--documenter-dark .tabs.is-toggle li.is-active a { - background-color: #1abc9c; - border-color: #1abc9c; - color: #fff; - z-index: 1; } - html.theme--documenter-dark .tabs.is-toggle ul { - border-bottom: none; } - html.theme--documenter-dark .tabs.is-toggle.is-toggle-rounded li:first-child a { - border-bottom-left-radius: 290486px; - border-top-left-radius: 290486px; - padding-left: 1.25em; } - html.theme--documenter-dark .tabs.is-toggle.is-toggle-rounded li:last-child a { - border-bottom-right-radius: 290486px; - border-top-right-radius: 290486px; - padding-right: 1.25em; } - html.theme--documenter-dark .tabs.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.tabs { - font-size: 0.85em; } - html.theme--documenter-dark .tabs.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .tabs.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .column { - display: block; - flex-basis: 0; - flex-grow: 1; - flex-shrink: 1; - padding: 0.75rem; } - .columns.is-mobile > html.theme--documenter-dark .column.is-narrow { - flex: none; } - .columns.is-mobile > html.theme--documenter-dark .column.is-full { - flex: none; - width: 100%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-three-quarters { - flex: none; - width: 75%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-two-thirds { - flex: none; - width: 66.6666%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-half { - flex: none; - width: 50%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-one-third { - flex: none; - width: 33.3333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-one-quarter { - flex: none; - width: 25%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-one-fifth { - flex: none; - width: 20%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-two-fifths { - flex: none; - width: 40%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-three-fifths { - flex: none; - width: 60%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-four-fifths { - flex: none; - width: 80%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-three-quarters { - margin-left: 75%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-two-thirds { - margin-left: 66.6666%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-half { - margin-left: 50%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-one-third { - margin-left: 33.3333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-one-quarter { - margin-left: 25%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-one-fifth { - margin-left: 20%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-two-fifths { - margin-left: 40%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-three-fifths { - margin-left: 60%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-four-fifths { - margin-left: 80%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-0 { - flex: none; - width: 0%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-0 { - margin-left: 0%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-1 { - flex: none; - width: 8.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-1 { - margin-left: 8.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-2 { - flex: none; - width: 16.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-2 { - margin-left: 16.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-3 { - flex: none; - width: 25%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-3 { - margin-left: 25%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-4 { - flex: none; - width: 33.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-4 { - margin-left: 33.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-5 { - flex: none; - width: 41.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-5 { - margin-left: 41.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-6 { - flex: none; - width: 50%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-6 { - margin-left: 50%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-7 { - flex: none; - width: 58.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-7 { - margin-left: 58.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-8 { - flex: none; - width: 66.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-8 { - margin-left: 66.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-9 { - flex: none; - width: 75%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-9 { - margin-left: 75%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-10 { - flex: none; - width: 83.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-10 { - margin-left: 83.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-11 { - flex: none; - width: 91.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-11 { - margin-left: 91.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-12 { - flex: none; - width: 100%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-12 { - margin-left: 100%; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .column.is-narrow-mobile { - flex: none; } - html.theme--documenter-dark .column.is-full-mobile { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-three-quarters-mobile { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-two-thirds-mobile { - flex: none; - width: 66.6666%; } - html.theme--documenter-dark .column.is-half-mobile { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-one-third-mobile { - flex: none; - width: 33.3333%; } - html.theme--documenter-dark .column.is-one-quarter-mobile { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-one-fifth-mobile { - flex: none; - width: 20%; } - html.theme--documenter-dark .column.is-two-fifths-mobile { - flex: none; - width: 40%; } - html.theme--documenter-dark .column.is-three-fifths-mobile { - flex: none; - width: 60%; } - html.theme--documenter-dark .column.is-four-fifths-mobile { - flex: none; - width: 80%; } - html.theme--documenter-dark .column.is-offset-three-quarters-mobile { - margin-left: 75%; } - html.theme--documenter-dark .column.is-offset-two-thirds-mobile { - margin-left: 66.6666%; } - html.theme--documenter-dark .column.is-offset-half-mobile { - margin-left: 50%; } - html.theme--documenter-dark .column.is-offset-one-third-mobile { - margin-left: 33.3333%; } - html.theme--documenter-dark .column.is-offset-one-quarter-mobile { - margin-left: 25%; } - html.theme--documenter-dark .column.is-offset-one-fifth-mobile { - margin-left: 20%; } - html.theme--documenter-dark .column.is-offset-two-fifths-mobile { - margin-left: 40%; } - html.theme--documenter-dark .column.is-offset-three-fifths-mobile { - margin-left: 60%; } - html.theme--documenter-dark .column.is-offset-four-fifths-mobile { - margin-left: 80%; } - html.theme--documenter-dark .column.is-0-mobile { - flex: none; - width: 0%; } - html.theme--documenter-dark .column.is-offset-0-mobile { - margin-left: 0%; } - html.theme--documenter-dark .column.is-1-mobile { - flex: none; - width: 8.3333333333%; } - html.theme--documenter-dark .column.is-offset-1-mobile { - margin-left: 8.3333333333%; } - html.theme--documenter-dark .column.is-2-mobile { - flex: none; - width: 16.6666666667%; } - html.theme--documenter-dark .column.is-offset-2-mobile { - margin-left: 16.6666666667%; } - html.theme--documenter-dark .column.is-3-mobile { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-offset-3-mobile { - margin-left: 25%; } - html.theme--documenter-dark .column.is-4-mobile { - flex: none; - width: 33.3333333333%; } - html.theme--documenter-dark .column.is-offset-4-mobile { - margin-left: 33.3333333333%; } - html.theme--documenter-dark .column.is-5-mobile { - flex: none; - width: 41.6666666667%; } - html.theme--documenter-dark .column.is-offset-5-mobile { - margin-left: 41.6666666667%; } - html.theme--documenter-dark .column.is-6-mobile { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-offset-6-mobile { - margin-left: 50%; } - html.theme--documenter-dark .column.is-7-mobile { - flex: none; - width: 58.3333333333%; } - html.theme--documenter-dark .column.is-offset-7-mobile { - margin-left: 58.3333333333%; } - html.theme--documenter-dark .column.is-8-mobile { - flex: none; - width: 66.6666666667%; } - html.theme--documenter-dark .column.is-offset-8-mobile { - margin-left: 66.6666666667%; } - html.theme--documenter-dark .column.is-9-mobile { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-offset-9-mobile { - margin-left: 75%; } - html.theme--documenter-dark .column.is-10-mobile { - flex: none; - width: 83.3333333333%; } - html.theme--documenter-dark .column.is-offset-10-mobile { - margin-left: 83.3333333333%; } - html.theme--documenter-dark .column.is-11-mobile { - flex: none; - width: 91.6666666667%; } - html.theme--documenter-dark .column.is-offset-11-mobile { - margin-left: 91.6666666667%; } - html.theme--documenter-dark .column.is-12-mobile { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-offset-12-mobile { - margin-left: 100%; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .column.is-narrow, html.theme--documenter-dark .column.is-narrow-tablet { - flex: none; } - html.theme--documenter-dark .column.is-full, html.theme--documenter-dark .column.is-full-tablet { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-three-quarters, html.theme--documenter-dark .column.is-three-quarters-tablet { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-two-thirds, html.theme--documenter-dark .column.is-two-thirds-tablet { - flex: none; - width: 66.6666%; } - html.theme--documenter-dark .column.is-half, html.theme--documenter-dark .column.is-half-tablet { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-one-third, html.theme--documenter-dark .column.is-one-third-tablet { - flex: none; - width: 33.3333%; } - html.theme--documenter-dark .column.is-one-quarter, html.theme--documenter-dark .column.is-one-quarter-tablet { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-one-fifth, html.theme--documenter-dark .column.is-one-fifth-tablet { - flex: none; - width: 20%; } - html.theme--documenter-dark .column.is-two-fifths, html.theme--documenter-dark .column.is-two-fifths-tablet { - flex: none; - width: 40%; } - html.theme--documenter-dark .column.is-three-fifths, html.theme--documenter-dark .column.is-three-fifths-tablet { - flex: none; - width: 60%; } - html.theme--documenter-dark .column.is-four-fifths, html.theme--documenter-dark .column.is-four-fifths-tablet { - flex: none; - width: 80%; } - html.theme--documenter-dark .column.is-offset-three-quarters, html.theme--documenter-dark .column.is-offset-three-quarters-tablet { - margin-left: 75%; } - html.theme--documenter-dark .column.is-offset-two-thirds, html.theme--documenter-dark .column.is-offset-two-thirds-tablet { - margin-left: 66.6666%; } - html.theme--documenter-dark .column.is-offset-half, html.theme--documenter-dark .column.is-offset-half-tablet { - margin-left: 50%; } - html.theme--documenter-dark .column.is-offset-one-third, html.theme--documenter-dark .column.is-offset-one-third-tablet { - margin-left: 33.3333%; } - html.theme--documenter-dark .column.is-offset-one-quarter, html.theme--documenter-dark .column.is-offset-one-quarter-tablet { - margin-left: 25%; } - html.theme--documenter-dark .column.is-offset-one-fifth, html.theme--documenter-dark .column.is-offset-one-fifth-tablet { - margin-left: 20%; } - html.theme--documenter-dark .column.is-offset-two-fifths, html.theme--documenter-dark .column.is-offset-two-fifths-tablet { - margin-left: 40%; } - html.theme--documenter-dark .column.is-offset-three-fifths, html.theme--documenter-dark .column.is-offset-three-fifths-tablet { - margin-left: 60%; } - html.theme--documenter-dark .column.is-offset-four-fifths, html.theme--documenter-dark .column.is-offset-four-fifths-tablet { - margin-left: 80%; } - html.theme--documenter-dark .column.is-0, html.theme--documenter-dark .column.is-0-tablet { - flex: none; - width: 0%; } - html.theme--documenter-dark .column.is-offset-0, html.theme--documenter-dark .column.is-offset-0-tablet { - margin-left: 0%; } - html.theme--documenter-dark .column.is-1, html.theme--documenter-dark .column.is-1-tablet { - flex: none; - width: 8.3333333333%; } - html.theme--documenter-dark .column.is-offset-1, html.theme--documenter-dark .column.is-offset-1-tablet { - margin-left: 8.3333333333%; } - html.theme--documenter-dark .column.is-2, html.theme--documenter-dark .column.is-2-tablet { - flex: none; - width: 16.6666666667%; } - html.theme--documenter-dark .column.is-offset-2, html.theme--documenter-dark .column.is-offset-2-tablet { - margin-left: 16.6666666667%; } - html.theme--documenter-dark .column.is-3, html.theme--documenter-dark .column.is-3-tablet { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-offset-3, html.theme--documenter-dark .column.is-offset-3-tablet { - margin-left: 25%; } - html.theme--documenter-dark .column.is-4, html.theme--documenter-dark .column.is-4-tablet { - flex: none; - width: 33.3333333333%; } - html.theme--documenter-dark .column.is-offset-4, html.theme--documenter-dark .column.is-offset-4-tablet { - margin-left: 33.3333333333%; } - html.theme--documenter-dark .column.is-5, html.theme--documenter-dark .column.is-5-tablet { - flex: none; - width: 41.6666666667%; } - html.theme--documenter-dark .column.is-offset-5, html.theme--documenter-dark .column.is-offset-5-tablet { - margin-left: 41.6666666667%; } - html.theme--documenter-dark .column.is-6, html.theme--documenter-dark .column.is-6-tablet { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-offset-6, html.theme--documenter-dark .column.is-offset-6-tablet { - margin-left: 50%; } - html.theme--documenter-dark .column.is-7, html.theme--documenter-dark .column.is-7-tablet { - flex: none; - width: 58.3333333333%; } - html.theme--documenter-dark .column.is-offset-7, html.theme--documenter-dark .column.is-offset-7-tablet { - margin-left: 58.3333333333%; } - html.theme--documenter-dark .column.is-8, html.theme--documenter-dark .column.is-8-tablet { - flex: none; - width: 66.6666666667%; } - html.theme--documenter-dark .column.is-offset-8, html.theme--documenter-dark .column.is-offset-8-tablet { - margin-left: 66.6666666667%; } - html.theme--documenter-dark .column.is-9, html.theme--documenter-dark .column.is-9-tablet { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-offset-9, html.theme--documenter-dark .column.is-offset-9-tablet { - margin-left: 75%; } - html.theme--documenter-dark .column.is-10, html.theme--documenter-dark .column.is-10-tablet { - flex: none; - width: 83.3333333333%; } - html.theme--documenter-dark .column.is-offset-10, html.theme--documenter-dark .column.is-offset-10-tablet { - margin-left: 83.3333333333%; } - html.theme--documenter-dark .column.is-11, html.theme--documenter-dark .column.is-11-tablet { - flex: none; - width: 91.6666666667%; } - html.theme--documenter-dark .column.is-offset-11, html.theme--documenter-dark .column.is-offset-11-tablet { - margin-left: 91.6666666667%; } - html.theme--documenter-dark .column.is-12, html.theme--documenter-dark .column.is-12-tablet { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-offset-12, html.theme--documenter-dark .column.is-offset-12-tablet { - margin-left: 100%; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .column.is-narrow-touch { - flex: none; } - html.theme--documenter-dark .column.is-full-touch { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-three-quarters-touch { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-two-thirds-touch { - flex: none; - width: 66.6666%; } - html.theme--documenter-dark .column.is-half-touch { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-one-third-touch { - flex: none; - width: 33.3333%; } - html.theme--documenter-dark .column.is-one-quarter-touch { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-one-fifth-touch { - flex: none; - width: 20%; } - html.theme--documenter-dark .column.is-two-fifths-touch { - flex: none; - width: 40%; } - html.theme--documenter-dark .column.is-three-fifths-touch { - flex: none; - width: 60%; } - html.theme--documenter-dark .column.is-four-fifths-touch { - flex: none; - width: 80%; } - html.theme--documenter-dark .column.is-offset-three-quarters-touch { - margin-left: 75%; } - html.theme--documenter-dark .column.is-offset-two-thirds-touch { - margin-left: 66.6666%; } - html.theme--documenter-dark .column.is-offset-half-touch { - margin-left: 50%; } - html.theme--documenter-dark .column.is-offset-one-third-touch { - margin-left: 33.3333%; } - html.theme--documenter-dark .column.is-offset-one-quarter-touch { - margin-left: 25%; } - html.theme--documenter-dark .column.is-offset-one-fifth-touch { - margin-left: 20%; } - html.theme--documenter-dark .column.is-offset-two-fifths-touch { - margin-left: 40%; } - html.theme--documenter-dark .column.is-offset-three-fifths-touch { - margin-left: 60%; } - html.theme--documenter-dark .column.is-offset-four-fifths-touch { - margin-left: 80%; } - html.theme--documenter-dark .column.is-0-touch { - flex: none; - width: 0%; } - html.theme--documenter-dark .column.is-offset-0-touch { - margin-left: 0%; } - html.theme--documenter-dark .column.is-1-touch { - flex: none; - width: 8.3333333333%; } - html.theme--documenter-dark .column.is-offset-1-touch { - margin-left: 8.3333333333%; } - html.theme--documenter-dark .column.is-2-touch { - flex: none; - width: 16.6666666667%; } - html.theme--documenter-dark .column.is-offset-2-touch { - margin-left: 16.6666666667%; } - html.theme--documenter-dark .column.is-3-touch { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-offset-3-touch { - margin-left: 25%; } - html.theme--documenter-dark .column.is-4-touch { - flex: none; - width: 33.3333333333%; } - html.theme--documenter-dark .column.is-offset-4-touch { - margin-left: 33.3333333333%; } - html.theme--documenter-dark .column.is-5-touch { - flex: none; - width: 41.6666666667%; } - html.theme--documenter-dark .column.is-offset-5-touch { - margin-left: 41.6666666667%; } - html.theme--documenter-dark .column.is-6-touch { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-offset-6-touch { - margin-left: 50%; } - html.theme--documenter-dark .column.is-7-touch { - flex: none; - width: 58.3333333333%; } - html.theme--documenter-dark .column.is-offset-7-touch { - margin-left: 58.3333333333%; } - html.theme--documenter-dark .column.is-8-touch { - flex: none; - width: 66.6666666667%; } - html.theme--documenter-dark .column.is-offset-8-touch { - margin-left: 66.6666666667%; } - html.theme--documenter-dark .column.is-9-touch { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-offset-9-touch { - margin-left: 75%; } - html.theme--documenter-dark .column.is-10-touch { - flex: none; - width: 83.3333333333%; } - html.theme--documenter-dark .column.is-offset-10-touch { - margin-left: 83.3333333333%; } - html.theme--documenter-dark .column.is-11-touch { - flex: none; - width: 91.6666666667%; } - html.theme--documenter-dark .column.is-offset-11-touch { - margin-left: 91.6666666667%; } - html.theme--documenter-dark .column.is-12-touch { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-offset-12-touch { - margin-left: 100%; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .column.is-narrow-desktop { - flex: none; } - html.theme--documenter-dark .column.is-full-desktop { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-three-quarters-desktop { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-two-thirds-desktop { - flex: none; - width: 66.6666%; } - html.theme--documenter-dark .column.is-half-desktop { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-one-third-desktop { - flex: none; - width: 33.3333%; } - html.theme--documenter-dark .column.is-one-quarter-desktop { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-one-fifth-desktop { - flex: none; - width: 20%; } - html.theme--documenter-dark .column.is-two-fifths-desktop { - flex: none; - width: 40%; } - html.theme--documenter-dark .column.is-three-fifths-desktop { - flex: none; - width: 60%; } - html.theme--documenter-dark .column.is-four-fifths-desktop { - flex: none; - width: 80%; } - html.theme--documenter-dark .column.is-offset-three-quarters-desktop { - margin-left: 75%; } - html.theme--documenter-dark .column.is-offset-two-thirds-desktop { - margin-left: 66.6666%; } - html.theme--documenter-dark .column.is-offset-half-desktop { - margin-left: 50%; } - html.theme--documenter-dark .column.is-offset-one-third-desktop { - margin-left: 33.3333%; } - html.theme--documenter-dark .column.is-offset-one-quarter-desktop { - margin-left: 25%; } - html.theme--documenter-dark .column.is-offset-one-fifth-desktop { - margin-left: 20%; } - html.theme--documenter-dark .column.is-offset-two-fifths-desktop { - margin-left: 40%; } - html.theme--documenter-dark .column.is-offset-three-fifths-desktop { - margin-left: 60%; } - html.theme--documenter-dark .column.is-offset-four-fifths-desktop { - margin-left: 80%; } - html.theme--documenter-dark .column.is-0-desktop { - flex: none; - width: 0%; } - html.theme--documenter-dark .column.is-offset-0-desktop { - margin-left: 0%; } - html.theme--documenter-dark .column.is-1-desktop { - flex: none; - width: 8.3333333333%; } - html.theme--documenter-dark .column.is-offset-1-desktop { - margin-left: 8.3333333333%; } - html.theme--documenter-dark .column.is-2-desktop { - flex: none; - width: 16.6666666667%; } - html.theme--documenter-dark .column.is-offset-2-desktop { - margin-left: 16.6666666667%; } - html.theme--documenter-dark .column.is-3-desktop { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-offset-3-desktop { - margin-left: 25%; } - html.theme--documenter-dark .column.is-4-desktop { - flex: none; - width: 33.3333333333%; } - html.theme--documenter-dark .column.is-offset-4-desktop { - margin-left: 33.3333333333%; } - html.theme--documenter-dark .column.is-5-desktop { - flex: none; - width: 41.6666666667%; } - html.theme--documenter-dark .column.is-offset-5-desktop { - margin-left: 41.6666666667%; } - html.theme--documenter-dark .column.is-6-desktop { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-offset-6-desktop { - margin-left: 50%; } - html.theme--documenter-dark .column.is-7-desktop { - flex: none; - width: 58.3333333333%; } - html.theme--documenter-dark .column.is-offset-7-desktop { - margin-left: 58.3333333333%; } - html.theme--documenter-dark .column.is-8-desktop { - flex: none; - width: 66.6666666667%; } - html.theme--documenter-dark .column.is-offset-8-desktop { - margin-left: 66.6666666667%; } - html.theme--documenter-dark .column.is-9-desktop { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-offset-9-desktop { - margin-left: 75%; } - html.theme--documenter-dark .column.is-10-desktop { - flex: none; - width: 83.3333333333%; } - html.theme--documenter-dark .column.is-offset-10-desktop { - margin-left: 83.3333333333%; } - html.theme--documenter-dark .column.is-11-desktop { - flex: none; - width: 91.6666666667%; } - html.theme--documenter-dark .column.is-offset-11-desktop { - margin-left: 91.6666666667%; } - html.theme--documenter-dark .column.is-12-desktop { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-offset-12-desktop { - margin-left: 100%; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .column.is-narrow-widescreen { - flex: none; } - html.theme--documenter-dark .column.is-full-widescreen { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-three-quarters-widescreen { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-two-thirds-widescreen { - flex: none; - width: 66.6666%; } - html.theme--documenter-dark .column.is-half-widescreen { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-one-third-widescreen { - flex: none; - width: 33.3333%; } - html.theme--documenter-dark .column.is-one-quarter-widescreen { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-one-fifth-widescreen { - flex: none; - width: 20%; } - html.theme--documenter-dark .column.is-two-fifths-widescreen { - flex: none; - width: 40%; } - html.theme--documenter-dark .column.is-three-fifths-widescreen { - flex: none; - width: 60%; } - html.theme--documenter-dark .column.is-four-fifths-widescreen { - flex: none; - width: 80%; } - html.theme--documenter-dark .column.is-offset-three-quarters-widescreen { - margin-left: 75%; } - html.theme--documenter-dark .column.is-offset-two-thirds-widescreen { - margin-left: 66.6666%; } - html.theme--documenter-dark .column.is-offset-half-widescreen { - margin-left: 50%; } - html.theme--documenter-dark .column.is-offset-one-third-widescreen { - margin-left: 33.3333%; } - html.theme--documenter-dark .column.is-offset-one-quarter-widescreen { - margin-left: 25%; } - html.theme--documenter-dark .column.is-offset-one-fifth-widescreen { - margin-left: 20%; } - html.theme--documenter-dark .column.is-offset-two-fifths-widescreen { - margin-left: 40%; } - html.theme--documenter-dark .column.is-offset-three-fifths-widescreen { - margin-left: 60%; } - html.theme--documenter-dark .column.is-offset-four-fifths-widescreen { - margin-left: 80%; } - html.theme--documenter-dark .column.is-0-widescreen { - flex: none; - width: 0%; } - html.theme--documenter-dark .column.is-offset-0-widescreen { - margin-left: 0%; } - html.theme--documenter-dark .column.is-1-widescreen { - flex: none; - width: 8.3333333333%; } - html.theme--documenter-dark .column.is-offset-1-widescreen { - margin-left: 8.3333333333%; } - html.theme--documenter-dark .column.is-2-widescreen { - flex: none; - width: 16.6666666667%; } - html.theme--documenter-dark .column.is-offset-2-widescreen { - margin-left: 16.6666666667%; } - html.theme--documenter-dark .column.is-3-widescreen { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-offset-3-widescreen { - margin-left: 25%; } - html.theme--documenter-dark .column.is-4-widescreen { - flex: none; - width: 33.3333333333%; } - html.theme--documenter-dark .column.is-offset-4-widescreen { - margin-left: 33.3333333333%; } - html.theme--documenter-dark .column.is-5-widescreen { - flex: none; - width: 41.6666666667%; } - html.theme--documenter-dark .column.is-offset-5-widescreen { - margin-left: 41.6666666667%; } - html.theme--documenter-dark .column.is-6-widescreen { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-offset-6-widescreen { - margin-left: 50%; } - html.theme--documenter-dark .column.is-7-widescreen { - flex: none; - width: 58.3333333333%; } - html.theme--documenter-dark .column.is-offset-7-widescreen { - margin-left: 58.3333333333%; } - html.theme--documenter-dark .column.is-8-widescreen { - flex: none; - width: 66.6666666667%; } - html.theme--documenter-dark .column.is-offset-8-widescreen { - margin-left: 66.6666666667%; } - html.theme--documenter-dark .column.is-9-widescreen { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-offset-9-widescreen { - margin-left: 75%; } - html.theme--documenter-dark .column.is-10-widescreen { - flex: none; - width: 83.3333333333%; } - html.theme--documenter-dark .column.is-offset-10-widescreen { - margin-left: 83.3333333333%; } - html.theme--documenter-dark .column.is-11-widescreen { - flex: none; - width: 91.6666666667%; } - html.theme--documenter-dark .column.is-offset-11-widescreen { - margin-left: 91.6666666667%; } - html.theme--documenter-dark .column.is-12-widescreen { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-offset-12-widescreen { - margin-left: 100%; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .column.is-narrow-fullhd { - flex: none; } - html.theme--documenter-dark .column.is-full-fullhd { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-three-quarters-fullhd { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-two-thirds-fullhd { - flex: none; - width: 66.6666%; } - html.theme--documenter-dark .column.is-half-fullhd { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-one-third-fullhd { - flex: none; - width: 33.3333%; } - html.theme--documenter-dark .column.is-one-quarter-fullhd { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-one-fifth-fullhd { - flex: none; - width: 20%; } - html.theme--documenter-dark .column.is-two-fifths-fullhd { - flex: none; - width: 40%; } - html.theme--documenter-dark .column.is-three-fifths-fullhd { - flex: none; - width: 60%; } - html.theme--documenter-dark .column.is-four-fifths-fullhd { - flex: none; - width: 80%; } - html.theme--documenter-dark .column.is-offset-three-quarters-fullhd { - margin-left: 75%; } - html.theme--documenter-dark .column.is-offset-two-thirds-fullhd { - margin-left: 66.6666%; } - html.theme--documenter-dark .column.is-offset-half-fullhd { - margin-left: 50%; } - html.theme--documenter-dark .column.is-offset-one-third-fullhd { - margin-left: 33.3333%; } - html.theme--documenter-dark .column.is-offset-one-quarter-fullhd { - margin-left: 25%; } - html.theme--documenter-dark .column.is-offset-one-fifth-fullhd { - margin-left: 20%; } - html.theme--documenter-dark .column.is-offset-two-fifths-fullhd { - margin-left: 40%; } - html.theme--documenter-dark .column.is-offset-three-fifths-fullhd { - margin-left: 60%; } - html.theme--documenter-dark .column.is-offset-four-fifths-fullhd { - margin-left: 80%; } - html.theme--documenter-dark .column.is-0-fullhd { - flex: none; - width: 0%; } - html.theme--documenter-dark .column.is-offset-0-fullhd { - margin-left: 0%; } - html.theme--documenter-dark .column.is-1-fullhd { - flex: none; - width: 8.3333333333%; } - html.theme--documenter-dark .column.is-offset-1-fullhd { - margin-left: 8.3333333333%; } - html.theme--documenter-dark .column.is-2-fullhd { - flex: none; - width: 16.6666666667%; } - html.theme--documenter-dark .column.is-offset-2-fullhd { - margin-left: 16.6666666667%; } - html.theme--documenter-dark .column.is-3-fullhd { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-offset-3-fullhd { - margin-left: 25%; } - html.theme--documenter-dark .column.is-4-fullhd { - flex: none; - width: 33.3333333333%; } - html.theme--documenter-dark .column.is-offset-4-fullhd { - margin-left: 33.3333333333%; } - html.theme--documenter-dark .column.is-5-fullhd { - flex: none; - width: 41.6666666667%; } - html.theme--documenter-dark .column.is-offset-5-fullhd { - margin-left: 41.6666666667%; } - html.theme--documenter-dark .column.is-6-fullhd { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-offset-6-fullhd { - margin-left: 50%; } - html.theme--documenter-dark .column.is-7-fullhd { - flex: none; - width: 58.3333333333%; } - html.theme--documenter-dark .column.is-offset-7-fullhd { - margin-left: 58.3333333333%; } - html.theme--documenter-dark .column.is-8-fullhd { - flex: none; - width: 66.6666666667%; } - html.theme--documenter-dark .column.is-offset-8-fullhd { - margin-left: 66.6666666667%; } - html.theme--documenter-dark .column.is-9-fullhd { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-offset-9-fullhd { - margin-left: 75%; } - html.theme--documenter-dark .column.is-10-fullhd { - flex: none; - width: 83.3333333333%; } - html.theme--documenter-dark .column.is-offset-10-fullhd { - margin-left: 83.3333333333%; } - html.theme--documenter-dark .column.is-11-fullhd { - flex: none; - width: 91.6666666667%; } - html.theme--documenter-dark .column.is-offset-11-fullhd { - margin-left: 91.6666666667%; } - html.theme--documenter-dark .column.is-12-fullhd { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-offset-12-fullhd { - margin-left: 100%; } } - html.theme--documenter-dark .columns { - margin-left: -0.75rem; - margin-right: -0.75rem; - margin-top: -0.75rem; } - html.theme--documenter-dark .columns:last-child { - margin-bottom: -0.75rem; } - html.theme--documenter-dark .columns:not(:last-child) { - margin-bottom: calc(1.5rem - 0.75rem); } - html.theme--documenter-dark .columns.is-centered { - justify-content: center; } - html.theme--documenter-dark .columns.is-gapless { - margin-left: 0; - margin-right: 0; - margin-top: 0; } - html.theme--documenter-dark .columns.is-gapless > .column { - margin: 0; - padding: 0 !important; } - html.theme--documenter-dark .columns.is-gapless:not(:last-child) { - margin-bottom: 1.5rem; } - html.theme--documenter-dark .columns.is-gapless:last-child { - margin-bottom: 0; } - html.theme--documenter-dark .columns.is-mobile { - display: flex; } - html.theme--documenter-dark .columns.is-multiline { - flex-wrap: wrap; } - html.theme--documenter-dark .columns.is-vcentered { - align-items: center; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns:not(.is-desktop) { - display: flex; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-desktop { - display: flex; } } - html.theme--documenter-dark .columns.is-variable { - --columnGap: 0.75rem; - margin-left: calc(-1 * var(--columnGap)); - margin-right: calc(-1 * var(--columnGap)); } - html.theme--documenter-dark .columns.is-variable .column { - padding-left: var(--columnGap); - padding-right: var(--columnGap); } - html.theme--documenter-dark .columns.is-variable.is-0 { - --columnGap: 0rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-0-mobile { - --columnGap: 0rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-0-tablet { - --columnGap: 0rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-0-tablet-only { - --columnGap: 0rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-0-touch { - --columnGap: 0rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-0-desktop { - --columnGap: 0rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-0-desktop-only { - --columnGap: 0rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-0-widescreen { - --columnGap: 0rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-0-widescreen-only { - --columnGap: 0rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-0-fullhd { - --columnGap: 0rem; } } - html.theme--documenter-dark .columns.is-variable.is-1 { - --columnGap: 0.25rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-1-mobile { - --columnGap: 0.25rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-1-tablet { - --columnGap: 0.25rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-1-tablet-only { - --columnGap: 0.25rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-1-touch { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-1-desktop { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-1-desktop-only { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-1-widescreen { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-1-widescreen-only { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-1-fullhd { - --columnGap: 0.25rem; } } - html.theme--documenter-dark .columns.is-variable.is-2 { - --columnGap: 0.5rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-2-mobile { - --columnGap: 0.5rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-2-tablet { - --columnGap: 0.5rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-2-tablet-only { - --columnGap: 0.5rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-2-touch { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-2-desktop { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-2-desktop-only { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-2-widescreen { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-2-widescreen-only { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-2-fullhd { - --columnGap: 0.5rem; } } - html.theme--documenter-dark .columns.is-variable.is-3 { - --columnGap: 0.75rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-3-mobile { - --columnGap: 0.75rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-3-tablet { - --columnGap: 0.75rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-3-tablet-only { - --columnGap: 0.75rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-3-touch { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-3-desktop { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-3-desktop-only { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-3-widescreen { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-3-widescreen-only { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-3-fullhd { - --columnGap: 0.75rem; } } - html.theme--documenter-dark .columns.is-variable.is-4 { - --columnGap: 1rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-4-mobile { - --columnGap: 1rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-4-tablet { - --columnGap: 1rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-4-tablet-only { - --columnGap: 1rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-4-touch { - --columnGap: 1rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-4-desktop { - --columnGap: 1rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-4-desktop-only { - --columnGap: 1rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-4-widescreen { - --columnGap: 1rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-4-widescreen-only { - --columnGap: 1rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-4-fullhd { - --columnGap: 1rem; } } - html.theme--documenter-dark .columns.is-variable.is-5 { - --columnGap: 1.25rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-5-mobile { - --columnGap: 1.25rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-5-tablet { - --columnGap: 1.25rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-5-tablet-only { - --columnGap: 1.25rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-5-touch { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-5-desktop { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-5-desktop-only { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-5-widescreen { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-5-widescreen-only { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-5-fullhd { - --columnGap: 1.25rem; } } - html.theme--documenter-dark .columns.is-variable.is-6 { - --columnGap: 1.5rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-6-mobile { - --columnGap: 1.5rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-6-tablet { - --columnGap: 1.5rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-6-tablet-only { - --columnGap: 1.5rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-6-touch { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-6-desktop { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-6-desktop-only { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-6-widescreen { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-6-widescreen-only { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-6-fullhd { - --columnGap: 1.5rem; } } - html.theme--documenter-dark .columns.is-variable.is-7 { - --columnGap: 1.75rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-7-mobile { - --columnGap: 1.75rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-7-tablet { - --columnGap: 1.75rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-7-tablet-only { - --columnGap: 1.75rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-7-touch { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-7-desktop { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-7-desktop-only { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-7-widescreen { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-7-widescreen-only { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-7-fullhd { - --columnGap: 1.75rem; } } - html.theme--documenter-dark .columns.is-variable.is-8 { - --columnGap: 2rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-8-mobile { - --columnGap: 2rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-8-tablet { - --columnGap: 2rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-8-tablet-only { - --columnGap: 2rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-8-touch { - --columnGap: 2rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-8-desktop { - --columnGap: 2rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-8-desktop-only { - --columnGap: 2rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-8-widescreen { - --columnGap: 2rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-8-widescreen-only { - --columnGap: 2rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-8-fullhd { - --columnGap: 2rem; } } - html.theme--documenter-dark .tile { - align-items: stretch; - display: block; - flex-basis: 0; - flex-grow: 1; - flex-shrink: 1; - min-height: min-content; } - html.theme--documenter-dark .tile.is-ancestor { - margin-left: -0.75rem; - margin-right: -0.75rem; - margin-top: -0.75rem; } - html.theme--documenter-dark .tile.is-ancestor:last-child { - margin-bottom: -0.75rem; } - html.theme--documenter-dark .tile.is-ancestor:not(:last-child) { - margin-bottom: 0.75rem; } - html.theme--documenter-dark .tile.is-child { - margin: 0 !important; } - html.theme--documenter-dark .tile.is-parent { - padding: 0.75rem; } - html.theme--documenter-dark .tile.is-vertical { - flex-direction: column; } - html.theme--documenter-dark .tile.is-vertical > .tile.is-child:not(:last-child) { - margin-bottom: 1.5rem !important; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .tile:not(.is-child) { - display: flex; } - html.theme--documenter-dark .tile.is-1 { - flex: none; - width: 8.3333333333%; } - html.theme--documenter-dark .tile.is-2 { - flex: none; - width: 16.6666666667%; } - html.theme--documenter-dark .tile.is-3 { - flex: none; - width: 25%; } - html.theme--documenter-dark .tile.is-4 { - flex: none; - width: 33.3333333333%; } - html.theme--documenter-dark .tile.is-5 { - flex: none; - width: 41.6666666667%; } - html.theme--documenter-dark .tile.is-6 { - flex: none; - width: 50%; } - html.theme--documenter-dark .tile.is-7 { - flex: none; - width: 58.3333333333%; } - html.theme--documenter-dark .tile.is-8 { - flex: none; - width: 66.6666666667%; } - html.theme--documenter-dark .tile.is-9 { - flex: none; - width: 75%; } - html.theme--documenter-dark .tile.is-10 { - flex: none; - width: 83.3333333333%; } - html.theme--documenter-dark .tile.is-11 { - flex: none; - width: 91.6666666667%; } - html.theme--documenter-dark .tile.is-12 { - flex: none; - width: 100%; } } - html.theme--documenter-dark .hero { - align-items: stretch; - display: flex; - flex-direction: column; - justify-content: space-between; } - html.theme--documenter-dark .hero .navbar { - background: none; } - html.theme--documenter-dark .hero .tabs ul { - border-bottom: none; } - html.theme--documenter-dark .hero.is-white { - background-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .hero.is-white a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-white strong { - color: inherit; } - html.theme--documenter-dark .hero.is-white .title { - color: #0a0a0a; } - html.theme--documenter-dark .hero.is-white .subtitle { - color: rgba(10, 10, 10, 0.9); } - html.theme--documenter-dark .hero.is-white .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-white .subtitle strong { - color: #0a0a0a; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-white .navbar-menu { - background-color: white; } } - html.theme--documenter-dark .hero.is-white .navbar-item, - html.theme--documenter-dark .hero.is-white .navbar-link { - color: rgba(10, 10, 10, 0.7); } - html.theme--documenter-dark .hero.is-white a.navbar-item:hover, html.theme--documenter-dark .hero.is-white a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-white .navbar-link:hover, - html.theme--documenter-dark .hero.is-white .navbar-link.is-active { - background-color: #f2f2f2; - color: #0a0a0a; } - html.theme--documenter-dark .hero.is-white .tabs a { - color: #0a0a0a; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-white .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-white .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-white .tabs.is-boxed a, html.theme--documenter-dark .hero.is-white .tabs.is-toggle a { - color: #0a0a0a; } - html.theme--documenter-dark .hero.is-white .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-white .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-white .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-white .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-white .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-white .tabs.is-toggle li.is-active a:hover { - background-color: #0a0a0a; - border-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .hero.is-white.is-bold { - background-image: linear-gradient(141deg, #e8e3e4 0%, white 71%, white 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-white.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #e8e3e4 0%, white 71%, white 100%); } } - html.theme--documenter-dark .hero.is-black { - background-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .hero.is-black a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-black strong { - color: inherit; } - html.theme--documenter-dark .hero.is-black .title { - color: white; } - html.theme--documenter-dark .hero.is-black .subtitle { - color: rgba(255, 255, 255, 0.9); } - html.theme--documenter-dark .hero.is-black .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-black .subtitle strong { - color: white; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-black .navbar-menu { - background-color: #0a0a0a; } } - html.theme--documenter-dark .hero.is-black .navbar-item, - html.theme--documenter-dark .hero.is-black .navbar-link { - color: rgba(255, 255, 255, 0.7); } - html.theme--documenter-dark .hero.is-black a.navbar-item:hover, html.theme--documenter-dark .hero.is-black a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-black .navbar-link:hover, - html.theme--documenter-dark .hero.is-black .navbar-link.is-active { - background-color: black; - color: white; } - html.theme--documenter-dark .hero.is-black .tabs a { - color: white; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-black .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-black .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-black .tabs.is-boxed a, html.theme--documenter-dark .hero.is-black .tabs.is-toggle a { - color: white; } - html.theme--documenter-dark .hero.is-black .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-black .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-black .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-black .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-black .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-black .tabs.is-toggle li.is-active a:hover { - background-color: white; - border-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .hero.is-black.is-bold { - background-image: linear-gradient(141deg, black 0%, #0a0a0a 71%, #181616 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-black.is-bold .navbar-menu { - background-image: linear-gradient(141deg, black 0%, #0a0a0a 71%, #181616 100%); } } - html.theme--documenter-dark .hero.is-light { - background-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .hero.is-light a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-light strong { - color: inherit; } - html.theme--documenter-dark .hero.is-light .title { - color: #282f2f; } - html.theme--documenter-dark .hero.is-light .subtitle { - color: rgba(40, 47, 47, 0.9); } - html.theme--documenter-dark .hero.is-light .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-light .subtitle strong { - color: #282f2f; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-light .navbar-menu { - background-color: #ecf0f1; } } - html.theme--documenter-dark .hero.is-light .navbar-item, - html.theme--documenter-dark .hero.is-light .navbar-link { - color: rgba(40, 47, 47, 0.7); } - html.theme--documenter-dark .hero.is-light a.navbar-item:hover, html.theme--documenter-dark .hero.is-light a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-light .navbar-link:hover, - html.theme--documenter-dark .hero.is-light .navbar-link.is-active { - background-color: #dde4e6; - color: #282f2f; } - html.theme--documenter-dark .hero.is-light .tabs a { - color: #282f2f; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-light .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-light .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-light .tabs.is-boxed a, html.theme--documenter-dark .hero.is-light .tabs.is-toggle a { - color: #282f2f; } - html.theme--documenter-dark .hero.is-light .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-light .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-light .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-light .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-light .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-light .tabs.is-toggle li.is-active a:hover { - background-color: #282f2f; - border-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .hero.is-light.is-bold { - background-image: linear-gradient(141deg, #cadfe0 0%, #ecf0f1 71%, #fafbfc 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-light.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #cadfe0 0%, #ecf0f1 71%, #fafbfc 100%); } } - html.theme--documenter-dark .hero.is-dark, html.theme--documenter-dark .content kbd.hero { - background-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .hero.is-dark a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), html.theme--documenter-dark .content kbd.hero a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-dark strong, - html.theme--documenter-dark .content kbd.hero strong { - color: inherit; } - html.theme--documenter-dark .hero.is-dark .title, html.theme--documenter-dark .content kbd.hero .title { - color: #ecf0f1; } - html.theme--documenter-dark .hero.is-dark .subtitle, html.theme--documenter-dark .content kbd.hero .subtitle { - color: rgba(236, 240, 241, 0.9); } - html.theme--documenter-dark .hero.is-dark .subtitle a:not(.button), html.theme--documenter-dark .content kbd.hero .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-dark .subtitle strong, - html.theme--documenter-dark .content kbd.hero .subtitle strong { - color: #ecf0f1; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-dark .navbar-menu, html.theme--documenter-dark .content kbd.hero .navbar-menu { - background-color: #282f2f; } } - html.theme--documenter-dark .hero.is-dark .navbar-item, html.theme--documenter-dark .content kbd.hero .navbar-item, - html.theme--documenter-dark .hero.is-dark .navbar-link, - html.theme--documenter-dark .content kbd.hero .navbar-link { - color: rgba(236, 240, 241, 0.7); } - html.theme--documenter-dark .hero.is-dark a.navbar-item:hover, html.theme--documenter-dark .content kbd.hero a.navbar-item:hover, html.theme--documenter-dark .hero.is-dark a.navbar-item.is-active, html.theme--documenter-dark .content kbd.hero a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-dark .navbar-link:hover, - html.theme--documenter-dark .content kbd.hero .navbar-link:hover, - html.theme--documenter-dark .hero.is-dark .navbar-link.is-active, - html.theme--documenter-dark .content kbd.hero .navbar-link.is-active { - background-color: #1d2122; - color: #ecf0f1; } - html.theme--documenter-dark .hero.is-dark .tabs a, html.theme--documenter-dark .content kbd.hero .tabs a { - color: #ecf0f1; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-dark .tabs a:hover, html.theme--documenter-dark .content kbd.hero .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-dark .tabs li.is-active a, html.theme--documenter-dark .content kbd.hero .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-dark .tabs.is-boxed a, html.theme--documenter-dark .content kbd.hero .tabs.is-boxed a, html.theme--documenter-dark .hero.is-dark .tabs.is-toggle a, html.theme--documenter-dark .content kbd.hero .tabs.is-toggle a { - color: #ecf0f1; } - html.theme--documenter-dark .hero.is-dark .tabs.is-boxed a:hover, html.theme--documenter-dark .content kbd.hero .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-dark .tabs.is-toggle a:hover, html.theme--documenter-dark .content kbd.hero .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-dark .tabs.is-boxed li.is-active a, html.theme--documenter-dark .content kbd.hero .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-dark .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-dark .tabs.is-toggle li.is-active a, html.theme--documenter-dark .content kbd.hero .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-dark .tabs.is-toggle li.is-active a:hover { - background-color: #ecf0f1; - border-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .hero.is-dark.is-bold, html.theme--documenter-dark .content kbd.hero.is-bold { - background-image: linear-gradient(141deg, #0f1615 0%, #282f2f 71%, #313c40 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-dark.is-bold .navbar-menu, html.theme--documenter-dark .content kbd.hero.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #0f1615 0%, #282f2f 71%, #313c40 100%); } } - html.theme--documenter-dark .hero.is-primary, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink { - background-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .hero.is-primary a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-primary strong, - html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink strong { - color: inherit; } - html.theme--documenter-dark .hero.is-primary .title, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .title { - color: #fff; } - html.theme--documenter-dark .hero.is-primary .subtitle, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .subtitle { - color: rgba(255, 255, 255, 0.9); } - html.theme--documenter-dark .hero.is-primary .subtitle a:not(.button), html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-primary .subtitle strong, - html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-primary .navbar-menu, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar-menu { - background-color: #375a7f; } } - html.theme--documenter-dark .hero.is-primary .navbar-item, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar-item, - html.theme--documenter-dark .hero.is-primary .navbar-link, - html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar-link { - color: rgba(255, 255, 255, 0.7); } - html.theme--documenter-dark .hero.is-primary a.navbar-item:hover, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink a.navbar-item:hover, html.theme--documenter-dark .hero.is-primary a.navbar-item.is-active, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-primary .navbar-link:hover, - html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar-link:hover, - html.theme--documenter-dark .hero.is-primary .navbar-link.is-active, - html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar-link.is-active { - background-color: #2f4d6d; - color: #fff; } - html.theme--documenter-dark .hero.is-primary .tabs a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs a { - color: #fff; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-primary .tabs a:hover, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-primary .tabs li.is-active a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-primary .tabs.is-boxed a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed a, html.theme--documenter-dark .hero.is-primary .tabs.is-toggle a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle a { - color: #fff; } - html.theme--documenter-dark .hero.is-primary .tabs.is-boxed a:hover, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-primary .tabs.is-toggle a:hover, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-primary .tabs.is-boxed li.is-active a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-primary .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-primary .tabs.is-toggle li.is-active a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-primary .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #375a7f; } - html.theme--documenter-dark .hero.is-primary.is-bold, html.theme--documenter-dark .docstring > section > a.hero.is-bold.docs-sourcelink { - background-image: linear-gradient(141deg, #214b62 0%, #375a7f 71%, #3a5796 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-primary.is-bold .navbar-menu, html.theme--documenter-dark .docstring > section > a.hero.is-bold.docs-sourcelink .navbar-menu { - background-image: linear-gradient(141deg, #214b62 0%, #375a7f 71%, #3a5796 100%); } } - html.theme--documenter-dark .hero.is-link { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .hero.is-link a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-link strong { - color: inherit; } - html.theme--documenter-dark .hero.is-link .title { - color: #fff; } - html.theme--documenter-dark .hero.is-link .subtitle { - color: rgba(255, 255, 255, 0.9); } - html.theme--documenter-dark .hero.is-link .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-link .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-link .navbar-menu { - background-color: #1abc9c; } } - html.theme--documenter-dark .hero.is-link .navbar-item, - html.theme--documenter-dark .hero.is-link .navbar-link { - color: rgba(255, 255, 255, 0.7); } - html.theme--documenter-dark .hero.is-link a.navbar-item:hover, html.theme--documenter-dark .hero.is-link a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-link .navbar-link:hover, - html.theme--documenter-dark .hero.is-link .navbar-link.is-active { - background-color: #17a689; - color: #fff; } - html.theme--documenter-dark .hero.is-link .tabs a { - color: #fff; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-link .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-link .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-link .tabs.is-boxed a, html.theme--documenter-dark .hero.is-link .tabs.is-toggle a { - color: #fff; } - html.theme--documenter-dark .hero.is-link .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-link .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-link .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-link .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-link .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-link .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #1abc9c; } - html.theme--documenter-dark .hero.is-link.is-bold { - background-image: linear-gradient(141deg, #0c9764 0%, #1abc9c 71%, #17d8d2 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-link.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #0c9764 0%, #1abc9c 71%, #17d8d2 100%); } } - html.theme--documenter-dark .hero.is-info { - background-color: #024c7d; - color: #fff; } - html.theme--documenter-dark .hero.is-info a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-info strong { - color: inherit; } - html.theme--documenter-dark .hero.is-info .title { - color: #fff; } - html.theme--documenter-dark .hero.is-info .subtitle { - color: rgba(255, 255, 255, 0.9); } - html.theme--documenter-dark .hero.is-info .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-info .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-info .navbar-menu { - background-color: #024c7d; } } - html.theme--documenter-dark .hero.is-info .navbar-item, - html.theme--documenter-dark .hero.is-info .navbar-link { - color: rgba(255, 255, 255, 0.7); } - html.theme--documenter-dark .hero.is-info a.navbar-item:hover, html.theme--documenter-dark .hero.is-info a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-info .navbar-link:hover, - html.theme--documenter-dark .hero.is-info .navbar-link.is-active { - background-color: #023d64; - color: #fff; } - html.theme--documenter-dark .hero.is-info .tabs a { - color: #fff; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-info .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-info .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-info .tabs.is-boxed a, html.theme--documenter-dark .hero.is-info .tabs.is-toggle a { - color: #fff; } - html.theme--documenter-dark .hero.is-info .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-info .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-info .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-info .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-info .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-info .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #024c7d; } - html.theme--documenter-dark .hero.is-info.is-bold { - background-image: linear-gradient(141deg, #003a4c 0%, #024c7d 71%, #004299 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-info.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #003a4c 0%, #024c7d 71%, #004299 100%); } } - html.theme--documenter-dark .hero.is-success { - background-color: #008438; - color: #fff; } - html.theme--documenter-dark .hero.is-success a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-success strong { - color: inherit; } - html.theme--documenter-dark .hero.is-success .title { - color: #fff; } - html.theme--documenter-dark .hero.is-success .subtitle { - color: rgba(255, 255, 255, 0.9); } - html.theme--documenter-dark .hero.is-success .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-success .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-success .navbar-menu { - background-color: #008438; } } - html.theme--documenter-dark .hero.is-success .navbar-item, - html.theme--documenter-dark .hero.is-success .navbar-link { - color: rgba(255, 255, 255, 0.7); } - html.theme--documenter-dark .hero.is-success a.navbar-item:hover, html.theme--documenter-dark .hero.is-success a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-success .navbar-link:hover, - html.theme--documenter-dark .hero.is-success .navbar-link.is-active { - background-color: #006b2d; - color: #fff; } - html.theme--documenter-dark .hero.is-success .tabs a { - color: #fff; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-success .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-success .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-success .tabs.is-boxed a, html.theme--documenter-dark .hero.is-success .tabs.is-toggle a { - color: #fff; } - html.theme--documenter-dark .hero.is-success .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-success .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-success .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-success .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-success .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-success .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #008438; } - html.theme--documenter-dark .hero.is-success.is-bold { - background-image: linear-gradient(141deg, #005115 0%, #008438 71%, #009e5d 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-success.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #005115 0%, #008438 71%, #009e5d 100%); } } - html.theme--documenter-dark .hero.is-warning { - background-color: #ad8100; - color: #fff; } - html.theme--documenter-dark .hero.is-warning a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-warning strong { - color: inherit; } - html.theme--documenter-dark .hero.is-warning .title { - color: #fff; } - html.theme--documenter-dark .hero.is-warning .subtitle { - color: rgba(255, 255, 255, 0.9); } - html.theme--documenter-dark .hero.is-warning .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-warning .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-warning .navbar-menu { - background-color: #ad8100; } } - html.theme--documenter-dark .hero.is-warning .navbar-item, - html.theme--documenter-dark .hero.is-warning .navbar-link { - color: rgba(255, 255, 255, 0.7); } - html.theme--documenter-dark .hero.is-warning a.navbar-item:hover, html.theme--documenter-dark .hero.is-warning a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-warning .navbar-link:hover, - html.theme--documenter-dark .hero.is-warning .navbar-link.is-active { - background-color: #946e00; - color: #fff; } - html.theme--documenter-dark .hero.is-warning .tabs a { - color: #fff; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-warning .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-warning .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-warning .tabs.is-boxed a, html.theme--documenter-dark .hero.is-warning .tabs.is-toggle a { - color: #fff; } - html.theme--documenter-dark .hero.is-warning .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-warning .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-warning .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-warning .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-warning .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-warning .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #ad8100; } - html.theme--documenter-dark .hero.is-warning.is-bold { - background-image: linear-gradient(141deg, #7a4700 0%, #ad8100 71%, #c7b500 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-warning.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #7a4700 0%, #ad8100 71%, #c7b500 100%); } } - html.theme--documenter-dark .hero.is-danger { - background-color: #9e1b0d; - color: #fff; } - html.theme--documenter-dark .hero.is-danger a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-danger strong { - color: inherit; } - html.theme--documenter-dark .hero.is-danger .title { - color: #fff; } - html.theme--documenter-dark .hero.is-danger .subtitle { - color: rgba(255, 255, 255, 0.9); } - html.theme--documenter-dark .hero.is-danger .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-danger .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-danger .navbar-menu { - background-color: #9e1b0d; } } - html.theme--documenter-dark .hero.is-danger .navbar-item, - html.theme--documenter-dark .hero.is-danger .navbar-link { - color: rgba(255, 255, 255, 0.7); } - html.theme--documenter-dark .hero.is-danger a.navbar-item:hover, html.theme--documenter-dark .hero.is-danger a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-danger .navbar-link:hover, - html.theme--documenter-dark .hero.is-danger .navbar-link.is-active { - background-color: #86170b; - color: #fff; } - html.theme--documenter-dark .hero.is-danger .tabs a { - color: #fff; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-danger .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-danger .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-danger .tabs.is-boxed a, html.theme--documenter-dark .hero.is-danger .tabs.is-toggle a { - color: #fff; } - html.theme--documenter-dark .hero.is-danger .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-danger .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-danger .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-danger .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-danger .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-danger .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #9e1b0d; } - html.theme--documenter-dark .hero.is-danger.is-bold { - background-image: linear-gradient(141deg, #75030b 0%, #9e1b0d 71%, #ba380a 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-danger.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #75030b 0%, #9e1b0d 71%, #ba380a 100%); } } - html.theme--documenter-dark .hero.is-small .hero-body, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.hero .hero-body { - padding-bottom: 1.5rem; - padding-top: 1.5rem; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .hero.is-medium .hero-body { - padding-bottom: 9rem; - padding-top: 9rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .hero.is-large .hero-body { - padding-bottom: 18rem; - padding-top: 18rem; } } - html.theme--documenter-dark .hero.is-halfheight .hero-body, html.theme--documenter-dark .hero.is-fullheight .hero-body, html.theme--documenter-dark .hero.is-fullheight-with-navbar .hero-body { - align-items: center; - display: flex; } - html.theme--documenter-dark .hero.is-halfheight .hero-body > .container, html.theme--documenter-dark .hero.is-fullheight .hero-body > .container, html.theme--documenter-dark .hero.is-fullheight-with-navbar .hero-body > .container { - flex-grow: 1; - flex-shrink: 1; } - html.theme--documenter-dark .hero.is-halfheight { - min-height: 50vh; } - html.theme--documenter-dark .hero.is-fullheight { - min-height: 100vh; } - html.theme--documenter-dark .hero-video { - overflow: hidden; } - html.theme--documenter-dark .hero-video video { - left: 50%; - min-height: 100%; - min-width: 100%; - position: absolute; - top: 50%; - transform: translate3d(-50%, -50%, 0); } - html.theme--documenter-dark .hero-video.is-transparent { - opacity: 0.3; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero-video { - display: none; } } - html.theme--documenter-dark .hero-buttons { - margin-top: 1.5rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero-buttons .button { - display: flex; } - html.theme--documenter-dark .hero-buttons .button:not(:last-child) { - margin-bottom: 0.75rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .hero-buttons { - display: flex; - justify-content: center; } - html.theme--documenter-dark .hero-buttons .button:not(:last-child) { - margin-right: 1.5rem; } } - html.theme--documenter-dark .hero-head, - html.theme--documenter-dark .hero-foot { - flex-grow: 0; - flex-shrink: 0; } - html.theme--documenter-dark .hero-body { - flex-grow: 1; - flex-shrink: 0; - padding: 3rem 1.5rem; } - html.theme--documenter-dark .section { - padding: 3rem 1.5rem; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .section.is-medium { - padding: 9rem 1.5rem; } - html.theme--documenter-dark .section.is-large { - padding: 18rem 1.5rem; } } - html.theme--documenter-dark .footer { - background-color: #282f2f; - padding: 3rem 1.5rem 6rem; } - html.theme--documenter-dark hr { - height: 1px; } - html.theme--documenter-dark h6 { - text-transform: uppercase; - letter-spacing: 0.5px; } - html.theme--documenter-dark .hero { - background-color: #343c3d; } - html.theme--documenter-dark a { - transition: all 200ms ease; } - html.theme--documenter-dark .button { - transition: all 200ms ease; - border-width: 1px; - color: white; } - html.theme--documenter-dark .button.is-active, html.theme--documenter-dark .button.is-focused, html.theme--documenter-dark .button:active, html.theme--documenter-dark .button:focus { - box-shadow: 0 0 0 2px rgba(140, 155, 157, 0.5); } - html.theme--documenter-dark .button.is-white.is-hovered, html.theme--documenter-dark .button.is-white:hover { - background-color: white; } - html.theme--documenter-dark .button.is-white.is-active, html.theme--documenter-dark .button.is-white.is-focused, html.theme--documenter-dark .button.is-white:active, html.theme--documenter-dark .button.is-white:focus { - border-color: white; - box-shadow: 0 0 0 2px rgba(255, 255, 255, 0.5); } - html.theme--documenter-dark .button.is-black.is-hovered, html.theme--documenter-dark .button.is-black:hover { - background-color: #1d1d1d; } - html.theme--documenter-dark .button.is-black.is-active, html.theme--documenter-dark .button.is-black.is-focused, html.theme--documenter-dark .button.is-black:active, html.theme--documenter-dark .button.is-black:focus { - border-color: #0a0a0a; - box-shadow: 0 0 0 2px rgba(10, 10, 10, 0.5); } - html.theme--documenter-dark .button.is-light.is-hovered, html.theme--documenter-dark .button.is-light:hover { - background-color: white; } - html.theme--documenter-dark .button.is-light.is-active, html.theme--documenter-dark .button.is-light.is-focused, html.theme--documenter-dark .button.is-light:active, html.theme--documenter-dark .button.is-light:focus { - border-color: #ecf0f1; - box-shadow: 0 0 0 2px rgba(236, 240, 241, 0.5); } - html.theme--documenter-dark .button.is-dark.is-hovered, html.theme--documenter-dark .content kbd.button.is-hovered, html.theme--documenter-dark .button.is-dark:hover, html.theme--documenter-dark .content kbd.button:hover { - background-color: #3a4344; } - html.theme--documenter-dark .button.is-dark.is-active, html.theme--documenter-dark .content kbd.button.is-active, html.theme--documenter-dark .button.is-dark.is-focused, html.theme--documenter-dark .content kbd.button.is-focused, html.theme--documenter-dark .button.is-dark:active, html.theme--documenter-dark .content kbd.button:active, html.theme--documenter-dark .button.is-dark:focus, html.theme--documenter-dark .content kbd.button:focus { - border-color: #282f2f; - box-shadow: 0 0 0 2px rgba(40, 47, 47, 0.5); } - html.theme--documenter-dark .button.is-primary.is-hovered, html.theme--documenter-dark .docstring > section > a.button.is-hovered.docs-sourcelink, html.theme--documenter-dark .button.is-primary:hover, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:hover { - background-color: #436d9a; } - html.theme--documenter-dark .button.is-primary.is-active, html.theme--documenter-dark .docstring > section > a.button.is-active.docs-sourcelink, html.theme--documenter-dark .button.is-primary.is-focused, html.theme--documenter-dark .docstring > section > a.button.is-focused.docs-sourcelink, html.theme--documenter-dark .button.is-primary:active, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:active, html.theme--documenter-dark .button.is-primary:focus, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:focus { - border-color: #375a7f; - box-shadow: 0 0 0 2px rgba(55, 90, 127, 0.5); } - html.theme--documenter-dark .button.is-link.is-hovered, html.theme--documenter-dark .button.is-link:hover { - background-color: #1fdeb8; } - html.theme--documenter-dark .button.is-link.is-active, html.theme--documenter-dark .button.is-link.is-focused, html.theme--documenter-dark .button.is-link:active, html.theme--documenter-dark .button.is-link:focus { - border-color: #1abc9c; - box-shadow: 0 0 0 2px rgba(26, 188, 156, 0.5); } - html.theme--documenter-dark .button.is-info.is-hovered, html.theme--documenter-dark .button.is-info:hover { - background-color: #0363a3; } - html.theme--documenter-dark .button.is-info.is-active, html.theme--documenter-dark .button.is-info.is-focused, html.theme--documenter-dark .button.is-info:active, html.theme--documenter-dark .button.is-info:focus { - border-color: #024c7d; - box-shadow: 0 0 0 2px rgba(2, 76, 125, 0.5); } - html.theme--documenter-dark .button.is-success.is-hovered, html.theme--documenter-dark .button.is-success:hover { - background-color: #00aa48; } - html.theme--documenter-dark .button.is-success.is-active, html.theme--documenter-dark .button.is-success.is-focused, html.theme--documenter-dark .button.is-success:active, html.theme--documenter-dark .button.is-success:focus { - border-color: #008438; - box-shadow: 0 0 0 2px rgba(0, 132, 56, 0.5); } - html.theme--documenter-dark .button.is-warning.is-hovered, html.theme--documenter-dark .button.is-warning:hover { - background-color: #d39e00; } - html.theme--documenter-dark .button.is-warning.is-active, html.theme--documenter-dark .button.is-warning.is-focused, html.theme--documenter-dark .button.is-warning:active, html.theme--documenter-dark .button.is-warning:focus { - border-color: #ad8100; - box-shadow: 0 0 0 2px rgba(173, 129, 0, 0.5); } - html.theme--documenter-dark .button.is-danger.is-hovered, html.theme--documenter-dark .button.is-danger:hover { - background-color: #c12110; } - html.theme--documenter-dark .button.is-danger.is-active, html.theme--documenter-dark .button.is-danger.is-focused, html.theme--documenter-dark .button.is-danger:active, html.theme--documenter-dark .button.is-danger:focus { - border-color: #9e1b0d; - box-shadow: 0 0 0 2px rgba(158, 27, 13, 0.5); } - html.theme--documenter-dark .label { - color: #dbdee0; } - html.theme--documenter-dark .button, - html.theme--documenter-dark .control.has-icons-left .icon, - html.theme--documenter-dark .control.has-icons-right .icon, - html.theme--documenter-dark .input, - html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input, - html.theme--documenter-dark .pagination-ellipsis, - html.theme--documenter-dark .pagination-link, - html.theme--documenter-dark .pagination-next, - html.theme--documenter-dark .pagination-previous, - html.theme--documenter-dark .select, - html.theme--documenter-dark .select select, - html.theme--documenter-dark .textarea { - height: 2.5em; } - html.theme--documenter-dark .input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input, - html.theme--documenter-dark .textarea { - transition: all 200ms ease; - box-shadow: none; - border-width: 1px; - padding-left: 1em; - padding-right: 1em; } - html.theme--documenter-dark .select:after, - html.theme--documenter-dark .select select { - border-width: 1px; } - html.theme--documenter-dark .control.has-addons .button, - html.theme--documenter-dark .control.has-addons .input, - html.theme--documenter-dark .control.has-addons #documenter .docs-sidebar form.docs-search > input, - html.theme--documenter-dark #documenter .docs-sidebar .control.has-addons form.docs-search > input, - html.theme--documenter-dark .control.has-addons .select { - margin-right: -1px; } - html.theme--documenter-dark .notification { - background-color: #343c3d; } - html.theme--documenter-dark .card { - box-shadow: none; - border: 1px solid #343c3d; - background-color: #282f2f; - border-radius: 0.4em; } - html.theme--documenter-dark .card .card-image img { - border-radius: 0.4em 0.4em 0 0; } - html.theme--documenter-dark .card .card-header { - box-shadow: none; - background-color: rgba(18, 18, 18, 0.2); - border-radius: 0.4em 0.4em 0 0; } - html.theme--documenter-dark .card .card-footer { - background-color: rgba(18, 18, 18, 0.2); } - html.theme--documenter-dark .card .card-footer, - html.theme--documenter-dark .card .card-footer-item { - border-width: 1px; - border-color: #343c3d; } - html.theme--documenter-dark .notification.is-white a:not(.button) { - color: #0a0a0a; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-black a:not(.button) { - color: white; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-light a:not(.button) { - color: #282f2f; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-dark a:not(.button), html.theme--documenter-dark .content kbd.notification a:not(.button) { - color: #ecf0f1; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-primary a:not(.button), html.theme--documenter-dark .docstring > section > a.notification.docs-sourcelink a:not(.button) { - color: #fff; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-link a:not(.button) { - color: #fff; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-info a:not(.button) { - color: #fff; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-success a:not(.button) { - color: #fff; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-warning a:not(.button) { - color: #fff; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-danger a:not(.button) { - color: #fff; - text-decoration: underline; } - html.theme--documenter-dark .tag, html.theme--documenter-dark .content kbd, html.theme--documenter-dark .docstring > section > a.docs-sourcelink { - border-radius: 0.4em; } - html.theme--documenter-dark .menu-list a { - transition: all 300ms ease; } - html.theme--documenter-dark .modal-card-body { - background-color: #282f2f; } - html.theme--documenter-dark .modal-card-foot, - html.theme--documenter-dark .modal-card-head { - border-color: #343c3d; } - html.theme--documenter-dark .message-header { - font-weight: 700; - background-color: #343c3d; - color: white; } - html.theme--documenter-dark .message-body { - border-width: 1px; - border-color: #343c3d; } - html.theme--documenter-dark .navbar { - border-radius: 0.4em; } - html.theme--documenter-dark .navbar.is-transparent { - background: none; } - html.theme--documenter-dark .navbar.is-primary .navbar-dropdown a.navbar-item.is-active, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-dropdown a.navbar-item.is-active { - background-color: #1abc9c; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .navbar .navbar-menu { - background-color: #375a7f; - border-radius: 0 0 0.4em 0.4em; } } - html.theme--documenter-dark .hero .navbar, - html.theme--documenter-dark body > .navbar { - border-radius: 0; } - html.theme--documenter-dark .pagination-link, - html.theme--documenter-dark .pagination-next, - html.theme--documenter-dark .pagination-previous { - border-width: 1px; } - html.theme--documenter-dark .panel-block, - html.theme--documenter-dark .panel-heading, - html.theme--documenter-dark .panel-tabs { - border-width: 1px; } - html.theme--documenter-dark .panel-block:first-child, - html.theme--documenter-dark .panel-heading:first-child, - html.theme--documenter-dark .panel-tabs:first-child { - border-top-width: 1px; } - html.theme--documenter-dark .panel-heading { - font-weight: 700; } - html.theme--documenter-dark .panel-tabs a { - border-width: 1px; - margin-bottom: -1px; } - html.theme--documenter-dark .panel-tabs a.is-active { - border-bottom-color: #17a689; } - html.theme--documenter-dark .panel-block:hover { - color: #1dd2af; } - html.theme--documenter-dark .panel-block:hover .panel-icon { - color: #1dd2af; } - html.theme--documenter-dark .panel-block.is-active .panel-icon { - color: #17a689; } - html.theme--documenter-dark .tabs a { - border-bottom-width: 1px; - margin-bottom: -1px; } - html.theme--documenter-dark .tabs ul { - border-bottom-width: 1px; } - html.theme--documenter-dark .tabs.is-boxed a { - border-width: 1px; } - html.theme--documenter-dark .tabs.is-boxed li.is-active a { - background-color: #1f2424; } - html.theme--documenter-dark .tabs.is-toggle li a { - border-width: 1px; - margin-bottom: 0; } - html.theme--documenter-dark .tabs.is-toggle li + li { - margin-left: -1px; } - html.theme--documenter-dark .hero.is-white .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-black .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-light .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-dark .navbar .navbar-dropdown .navbar-item:hover, html.theme--documenter-dark .content kbd.hero .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-primary .navbar .navbar-dropdown .navbar-item:hover, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-link .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-info .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-success .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-warning .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-danger .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark h1 .docs-heading-anchor, html.theme--documenter-dark h1 .docs-heading-anchor:hover, html.theme--documenter-dark h1 .docs-heading-anchor:visited, html.theme--documenter-dark h2 .docs-heading-anchor, html.theme--documenter-dark h2 .docs-heading-anchor:hover, html.theme--documenter-dark h2 .docs-heading-anchor:visited, html.theme--documenter-dark h3 .docs-heading-anchor, html.theme--documenter-dark h3 .docs-heading-anchor:hover, html.theme--documenter-dark h3 .docs-heading-anchor:visited, html.theme--documenter-dark h4 .docs-heading-anchor, html.theme--documenter-dark h4 .docs-heading-anchor:hover, html.theme--documenter-dark h4 .docs-heading-anchor:visited, html.theme--documenter-dark h5 .docs-heading-anchor, html.theme--documenter-dark h5 .docs-heading-anchor:hover, html.theme--documenter-dark h5 .docs-heading-anchor:visited, html.theme--documenter-dark h6 .docs-heading-anchor, html.theme--documenter-dark h6 .docs-heading-anchor:hover, html.theme--documenter-dark h6 .docs-heading-anchor:visited { - color: #f2f2f2; } - html.theme--documenter-dark h1 .docs-heading-anchor-permalink, html.theme--documenter-dark h2 .docs-heading-anchor-permalink, html.theme--documenter-dark h3 .docs-heading-anchor-permalink, html.theme--documenter-dark h4 .docs-heading-anchor-permalink, html.theme--documenter-dark h5 .docs-heading-anchor-permalink, html.theme--documenter-dark h6 .docs-heading-anchor-permalink { - visibility: hidden; - vertical-align: middle; - margin-left: 0.5em; - font-size: 0.7rem; } - html.theme--documenter-dark h1 .docs-heading-anchor-permalink::before, html.theme--documenter-dark h2 .docs-heading-anchor-permalink::before, html.theme--documenter-dark h3 .docs-heading-anchor-permalink::before, html.theme--documenter-dark h4 .docs-heading-anchor-permalink::before, html.theme--documenter-dark h5 .docs-heading-anchor-permalink::before, html.theme--documenter-dark h6 .docs-heading-anchor-permalink::before { - font-family: "Font Awesome 5 Free"; - font-weight: 900; - content: "\f0c1"; } - html.theme--documenter-dark h1:hover .docs-heading-anchor-permalink, html.theme--documenter-dark h2:hover .docs-heading-anchor-permalink, html.theme--documenter-dark h3:hover .docs-heading-anchor-permalink, html.theme--documenter-dark h4:hover .docs-heading-anchor-permalink, html.theme--documenter-dark h5:hover .docs-heading-anchor-permalink, html.theme--documenter-dark h6:hover .docs-heading-anchor-permalink { - visibility: visible; } - html.theme--documenter-dark .docs-light-only { - display: none !important; } - html.theme--documenter-dark pre { - position: relative; - overflow: hidden; } - html.theme--documenter-dark pre code, html.theme--documenter-dark pre code.hljs { - padding: 0 0.75rem !important; - overflow: auto; - display: block; } - html.theme--documenter-dark pre code:first-of-type, html.theme--documenter-dark pre code.hljs:first-of-type { - padding-top: 0.5rem !important; } - html.theme--documenter-dark pre code:last-of-type, html.theme--documenter-dark pre code.hljs:last-of-type { - padding-bottom: 0.5rem !important; } - html.theme--documenter-dark pre .copy-button { - opacity: 0.2; - transition: opacity 0.2s; - position: absolute; - right: 0em; - top: 0em; - padding: 0.5em; - width: 2.5em; - height: 2.5em; - background: transparent; - border: none; - font-family: "Font Awesome 5 Free"; - color: #fff; - cursor: pointer; - text-align: center; } - html.theme--documenter-dark pre .copy-button:focus, html.theme--documenter-dark pre .copy-button:hover { - opacity: 1; - background: rgba(255, 255, 255, 0.1); - color: #1abc9c; } - html.theme--documenter-dark pre .copy-button.success { - color: #259a12; - opacity: 1; } - html.theme--documenter-dark pre .copy-button.error { - color: #cb3c33; - opacity: 1; } - html.theme--documenter-dark pre:hover .copy-button { - opacity: 1; } - html.theme--documenter-dark .admonition { - background-color: #282f2f; - border-style: solid; - border-width: 1px; - border-color: #5e6d6f; - border-radius: 0.4em; - font-size: 15px; } - html.theme--documenter-dark .admonition strong { - color: currentColor; } - html.theme--documenter-dark .admonition.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.admonition { - font-size: 0.85em; } - html.theme--documenter-dark .admonition.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .admonition.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .admonition.is-default { - background-color: #282f2f; - border-color: #5e6d6f; } - html.theme--documenter-dark .admonition.is-default > .admonition-header { - background-color: #5e6d6f; } - html.theme--documenter-dark .admonition.is-info { - background-color: #282f2f; - border-color: #024c7d; } - html.theme--documenter-dark .admonition.is-info > .admonition-header { - background-color: #024c7d; } - html.theme--documenter-dark .admonition.is-success { - background-color: #282f2f; - border-color: #008438; } - html.theme--documenter-dark .admonition.is-success > .admonition-header { - background-color: #008438; } - html.theme--documenter-dark .admonition.is-warning { - background-color: #282f2f; - border-color: #ad8100; } - html.theme--documenter-dark .admonition.is-warning > .admonition-header { - background-color: #ad8100; } - html.theme--documenter-dark .admonition.is-danger { - background-color: #282f2f; - border-color: #9e1b0d; } - html.theme--documenter-dark .admonition.is-danger > .admonition-header { - background-color: #9e1b0d; } - html.theme--documenter-dark .admonition.is-compat { - background-color: #282f2f; - border-color: #137886; } - html.theme--documenter-dark .admonition.is-compat > .admonition-header { - background-color: #137886; } - html.theme--documenter-dark .admonition-header { - background-color: #5e6d6f; - align-items: center; - font-weight: 700; - justify-content: space-between; - line-height: 1.25; - padding: 0.5rem 0.75rem; - position: relative; } - html.theme--documenter-dark .admonition-header:before { - font-family: "Font Awesome 5 Free"; - font-weight: 900; - margin-right: 0.75rem; - content: "\f06a"; } - html.theme--documenter-dark .admonition-body { - color: #fff; - padding: 0.5rem 0.75rem; } - html.theme--documenter-dark .admonition-body pre { - background-color: #282f2f; } - html.theme--documenter-dark .admonition-body code { - background-color: rgba(255, 255, 255, 0.05); } - html.theme--documenter-dark .docstring { - margin-bottom: 1em; - background-color: transparent; - border: 1px solid #5e6d6f; - box-shadow: none; - max-width: 100%; } - html.theme--documenter-dark .docstring > header { - display: flex; - flex-grow: 1; - align-items: stretch; - padding: 0.5rem 0.75rem; - background-color: #282f2f; - box-shadow: 0 1px 2px rgba(10, 10, 10, 0.1); - box-shadow: none; - border-bottom: 1px solid #5e6d6f; } - html.theme--documenter-dark .docstring > header code { - background-color: transparent; } - html.theme--documenter-dark .docstring > header .docstring-binding { - margin-right: 0.3em; } - html.theme--documenter-dark .docstring > header .docstring-category { - margin-left: 0.3em; } - html.theme--documenter-dark .docstring > section { - position: relative; - padding: 0.75rem 0.75rem; - border-bottom: 1px solid #5e6d6f; } - html.theme--documenter-dark .docstring > section:last-child { - border-bottom: none; } - html.theme--documenter-dark .docstring > section > a.docs-sourcelink { - transition: opacity 0.3s; - opacity: 0; - position: absolute; - right: 0.375rem; - bottom: 0.375rem; } - html.theme--documenter-dark .docstring > section > a.docs-sourcelink:focus { - opacity: 1 !important; } - html.theme--documenter-dark .docstring:hover > section > a.docs-sourcelink { - opacity: 0.2; } - html.theme--documenter-dark .docstring:focus-within > section > a.docs-sourcelink { - opacity: 0.2; } - html.theme--documenter-dark .docstring > section:hover a.docs-sourcelink { - opacity: 1; } - html.theme--documenter-dark .documenter-example-output { - background-color: #1f2424; } - html.theme--documenter-dark .outdated-warning-overlay { - position: fixed; - top: 0; - left: 0; - right: 0; - box-shadow: 0 0 10px rgba(0, 0, 0, 0.3); - z-index: 999; - background-color: #282f2f; - border-bottom: 3px solid #9e1b0d; - padding: 10px 35px; - text-align: center; - font-size: 15px; } - html.theme--documenter-dark .outdated-warning-overlay .outdated-warning-closer { - position: absolute; - top: calc(50% - 10px); - right: 18px; - cursor: pointer; - width: 12px; } - html.theme--documenter-dark .outdated-warning-overlay a { - color: #1abc9c; } - html.theme--documenter-dark .outdated-warning-overlay a:hover { - color: #1dd2af; } - html.theme--documenter-dark .content pre { - border: 1px solid #5e6d6f; } - html.theme--documenter-dark .content code { - font-weight: inherit; } - html.theme--documenter-dark .content a code { - color: #1abc9c; } - html.theme--documenter-dark .content h1 code, html.theme--documenter-dark .content h2 code, html.theme--documenter-dark .content h3 code, html.theme--documenter-dark .content h4 code, html.theme--documenter-dark .content h5 code, html.theme--documenter-dark .content h6 code { - color: #f2f2f2; } - html.theme--documenter-dark .content table { - display: block; - width: initial; - max-width: 100%; - overflow-x: auto; } - html.theme--documenter-dark .content blockquote > ul:first-child, html.theme--documenter-dark .content blockquote > ol:first-child, html.theme--documenter-dark .content .admonition-body > ul:first-child, html.theme--documenter-dark .content .admonition-body > ol:first-child { - margin-top: 0; } - html.theme--documenter-dark pre, html.theme--documenter-dark code { - font-variant-ligatures: no-contextual; } - html.theme--documenter-dark .breadcrumb a.is-disabled { - cursor: default; - pointer-events: none; } - html.theme--documenter-dark .breadcrumb a.is-disabled, html.theme--documenter-dark .breadcrumb a.is-disabled:hover { - color: #f2f2f2; } - html.theme--documenter-dark .hljs { - background: initial !important; } - html.theme--documenter-dark .katex .katex-mathml { - top: 0; - right: 0; } - html.theme--documenter-dark .katex-display, html.theme--documenter-dark mjx-container, html.theme--documenter-dark .MathJax_Display { - margin: 0.5em 0 !important; } - html.theme--documenter-dark html { - -moz-osx-font-smoothing: auto; - -webkit-font-smoothing: auto; } - html.theme--documenter-dark li.no-marker { - list-style: none; } - html.theme--documenter-dark #documenter .docs-main > article { - overflow-wrap: break-word; } - html.theme--documenter-dark #documenter .docs-main > article .math-container { - overflow-x: auto; - overflow-y: hidden; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark #documenter .docs-main { - max-width: 52rem; - margin-left: 20rem; - padding-right: 1rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark #documenter .docs-main { - width: 100%; } - html.theme--documenter-dark #documenter .docs-main > article { - max-width: 52rem; - margin-left: auto; - margin-right: auto; - margin-bottom: 1rem; - padding: 0 1rem; } - html.theme--documenter-dark #documenter .docs-main > header, html.theme--documenter-dark #documenter .docs-main > nav { - max-width: 100%; - width: 100%; - margin: 0; } } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar { - background-color: #1f2424; - border-bottom: 1px solid #5e6d6f; - z-index: 2; - min-height: 4rem; - margin-bottom: 1rem; - display: flex; } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar .breadcrumb { - flex-grow: 1; } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right { - display: flex; - white-space: nowrap; } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-icon, html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-label, html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-sidebar-button { - display: inline-block; } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-label { - padding: 0; - margin-left: 0.3em; } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-settings-button { - margin: auto 0 auto 1rem; } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-sidebar-button { - font-size: 1.5rem; - margin: auto 0 auto 1rem; } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar > * { - margin: auto 0; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark #documenter .docs-main header.docs-navbar { - position: sticky; - top: 0; - padding: 0 1rem; - /* For Headroom.js */ - transition-property: top, box-shadow; - -webkit-transition-property: top, box-shadow; - /* Safari */ - transition-duration: 0.3s; - -webkit-transition-duration: 0.3s; - /* Safari */ } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar.headroom--not-top { - box-shadow: 0.2rem 0rem 0.4rem #171717; - transition-duration: 0.7s; - -webkit-transition-duration: 0.7s; - /* Safari */ } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar.headroom--unpinned.headroom--not-top.headroom--not-bottom { - top: -4.5rem; - transition-duration: 0.7s; - -webkit-transition-duration: 0.7s; - /* Safari */ } } - html.theme--documenter-dark #documenter .docs-main section.footnotes { - border-top: 1px solid #5e6d6f; } - html.theme--documenter-dark #documenter .docs-main section.footnotes li .tag:first-child, html.theme--documenter-dark #documenter .docs-main section.footnotes li .docstring > section > a.docs-sourcelink:first-child, html.theme--documenter-dark #documenter .docs-main section.footnotes li .content kbd:first-child, html.theme--documenter-dark .content #documenter .docs-main section.footnotes li kbd:first-child { - margin-right: 1em; - margin-bottom: 0.4em; } - html.theme--documenter-dark #documenter .docs-main .docs-footer { - display: flex; - flex-wrap: wrap; - margin-left: 0; - margin-right: 0; - border-top: 1px solid #5e6d6f; - padding-top: 1rem; - padding-bottom: 1rem; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark #documenter .docs-main .docs-footer { - padding-left: 1rem; - padding-right: 1rem; } } - html.theme--documenter-dark #documenter .docs-main .docs-footer .docs-footer-nextpage, html.theme--documenter-dark #documenter .docs-main .docs-footer .docs-footer-prevpage { - flex-grow: 1; } - html.theme--documenter-dark #documenter .docs-main .docs-footer .docs-footer-nextpage { - text-align: right; } - html.theme--documenter-dark #documenter .docs-main .docs-footer .flexbox-break { - flex-basis: 100%; - height: 0; } - html.theme--documenter-dark #documenter .docs-main .docs-footer .footer-message { - font-size: 0.8em; - margin: 0.5em auto 0 auto; - text-align: center; } - html.theme--documenter-dark #documenter .docs-sidebar { - display: flex; - flex-direction: column; - color: #fff; - background-color: #282f2f; - border-right: 1px solid #5e6d6f; - padding: 0; - flex: 0 0 18rem; - z-index: 5; - font-size: 15px; - position: fixed; - left: -18rem; - width: 18rem; - height: 100%; - transition: left 0.3s; - /* Setting up a nicer theme style for the scrollbar */ } - html.theme--documenter-dark #documenter .docs-sidebar.visible { - left: 0; - box-shadow: 0.4rem 0rem 0.8rem #171717; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark #documenter .docs-sidebar.visible { - box-shadow: none; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark #documenter .docs-sidebar { - left: 0; - top: 0; } } - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo { - margin-top: 1rem; - padding: 0 1rem; } - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img { - max-height: 6rem; - margin: auto; } - html.theme--documenter-dark #documenter .docs-sidebar .docs-package-name { - flex-shrink: 0; - font-size: 1.5rem; - font-weight: 700; - text-align: center; - white-space: nowrap; - overflow: hidden; - padding: 0.5rem 0; } - html.theme--documenter-dark #documenter .docs-sidebar .docs-package-name .docs-autofit { - max-width: 16.2rem; } - html.theme--documenter-dark #documenter .docs-sidebar .docs-package-name a, html.theme--documenter-dark #documenter .docs-sidebar .docs-package-name a:hover { - color: #fff; } - html.theme--documenter-dark #documenter .docs-sidebar .docs-version-selector { - border-top: 1px solid #5e6d6f; - display: none; - padding: 0.5rem; } - html.theme--documenter-dark #documenter .docs-sidebar .docs-version-selector.visible { - display: flex; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu { - flex-grow: 1; - user-select: none; - border-top: 1px solid #5e6d6f; - padding-bottom: 1.5rem; - /* Managing collapsible submenus */ } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu > li > .tocitem { - font-weight: bold; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu > li li { - font-size: 14.25px; - margin-left: 1em; - border-left: 1px solid #5e6d6f; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu input.collapse-toggle { - display: none; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.collapsed { - display: none; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu input:checked ~ ul.collapsed { - display: block; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu label.tocitem { - display: flex; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu label.tocitem .docs-label { - flex-grow: 2; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu label.tocitem .docs-chevron { - display: inline-block; - font-style: normal; - font-variant: normal; - text-rendering: auto; - line-height: 1; - font-size: 11.25px; - margin-left: 1rem; - margin-top: auto; - margin-bottom: auto; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu label.tocitem .docs-chevron::before { - font-family: "Font Awesome 5 Free"; - font-weight: 900; - content: "\f054"; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu input:checked ~ label.tocitem .docs-chevron::before { - content: "\f078"; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu .tocitem { - display: block; - padding: 0.5rem 0.5rem; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu .tocitem, html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu .tocitem:hover { - color: #fff; - background: #282f2f; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu a.tocitem:hover, html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu label.tocitem:hover { - color: #fff; - background-color: #32393a; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu li.is-active { - border-top: 1px solid #5e6d6f; - border-bottom: 1px solid #5e6d6f; - background-color: #1f2424; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu li.is-active .tocitem, html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu li.is-active .tocitem:hover { - background-color: #1f2424; - color: #fff; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu li.is-active ul.internal .tocitem:hover { - background-color: #32393a; - color: #fff; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu > li.is-active:first-child { - border-top: none; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.internal { - margin: 0 0.5rem 0.5rem; - border-top: 1px solid #5e6d6f; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.internal li { - font-size: 12.75px; - border-left: none; - margin-left: 0; - margin-top: 0.5rem; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.internal .tocitem { - width: 100%; - padding: 0; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.internal .tocitem::before { - content: "⚬"; - margin-right: 0.4em; } - html.theme--documenter-dark #documenter .docs-sidebar form.docs-search { - margin: auto; - margin-top: 0.5rem; - margin-bottom: 0.5rem; } - html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input { - width: 14.4rem; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu { - overflow-y: auto; - -webkit-overflow-scroll: touch; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu::-webkit-scrollbar { - width: .3rem; - background: none; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu::-webkit-scrollbar-thumb { - border-radius: 5px 0px 0px 5px; - background: #3b4445; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu::-webkit-scrollbar-thumb:hover { - background: #4e5a5c; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark #documenter .docs-sidebar { - overflow-y: auto; - -webkit-overflow-scroll: touch; } - html.theme--documenter-dark #documenter .docs-sidebar::-webkit-scrollbar { - width: .3rem; - background: none; } - html.theme--documenter-dark #documenter .docs-sidebar::-webkit-scrollbar-thumb { - border-radius: 5px 0px 0px 5px; - background: #3b4445; } - html.theme--documenter-dark #documenter .docs-sidebar::-webkit-scrollbar-thumb:hover { - background: #4e5a5c; } } - html.theme--documenter-dark #documenter .docs-main #documenter-search-info { - margin-bottom: 1rem; } - html.theme--documenter-dark #documenter .docs-main #documenter-search-results { - list-style-type: circle; - list-style-position: outside; } - html.theme--documenter-dark #documenter .docs-main #documenter-search-results li { - margin-left: 2rem; } - html.theme--documenter-dark #documenter .docs-main #documenter-search-results .docs-highlight { - background-color: yellow; } - html.theme--documenter-dark { - background-color: #1f2424; - font-size: 16px; - min-width: 300px; - overflow-x: auto; - overflow-y: scroll; - text-rendering: optimizeLegibility; - text-size-adjust: 100%; } - html.theme--documenter-dark .ansi span.sgr1 { - font-weight: bolder; } - html.theme--documenter-dark .ansi span.sgr2 { - font-weight: lighter; } - html.theme--documenter-dark .ansi span.sgr3 { - font-style: italic; } - html.theme--documenter-dark .ansi span.sgr4 { - text-decoration: underline; } - html.theme--documenter-dark .ansi span.sgr7 { - color: #1f2424; - background-color: #fff; } - html.theme--documenter-dark .ansi span.sgr8 { - color: transparent; } - html.theme--documenter-dark .ansi span.sgr8 span { - color: transparent; } - html.theme--documenter-dark .ansi span.sgr9 { - text-decoration: line-through; } - html.theme--documenter-dark .ansi span.sgr30 { - color: #242424; } - html.theme--documenter-dark .ansi span.sgr31 { - color: #f6705f; } - html.theme--documenter-dark .ansi span.sgr32 { - color: #4fb43a; } - html.theme--documenter-dark .ansi span.sgr33 { - color: #f4c72f; } - html.theme--documenter-dark .ansi span.sgr34 { - color: #7587f0; } - html.theme--documenter-dark .ansi span.sgr35 { - color: #bc89d3; } - html.theme--documenter-dark .ansi span.sgr36 { - color: #49b6ca; } - html.theme--documenter-dark .ansi span.sgr37 { - color: #b3bdbe; } - html.theme--documenter-dark .ansi span.sgr40 { - background-color: #242424; } - html.theme--documenter-dark .ansi span.sgr41 { - background-color: #f6705f; } - html.theme--documenter-dark .ansi span.sgr42 { - background-color: #4fb43a; } - html.theme--documenter-dark .ansi span.sgr43 { - background-color: #f4c72f; } - html.theme--documenter-dark .ansi span.sgr44 { - background-color: #7587f0; } - html.theme--documenter-dark .ansi span.sgr45 { - background-color: #bc89d3; } - html.theme--documenter-dark .ansi span.sgr46 { - background-color: #49b6ca; } - html.theme--documenter-dark .ansi span.sgr47 { - background-color: #b3bdbe; } - html.theme--documenter-dark .ansi span.sgr90 { - color: #92a0a2; } - html.theme--documenter-dark .ansi span.sgr91 { - color: #ff8674; } - html.theme--documenter-dark .ansi span.sgr92 { - color: #79d462; } - html.theme--documenter-dark .ansi span.sgr93 { - color: #ffe76b; } - html.theme--documenter-dark .ansi span.sgr94 { - color: #8a98ff; } - html.theme--documenter-dark .ansi span.sgr95 { - color: #d2a4e6; } - html.theme--documenter-dark .ansi span.sgr96 { - color: #6bc8db; } - html.theme--documenter-dark .ansi span.sgr97 { - color: #ecf0f1; } - html.theme--documenter-dark .ansi span.sgr100 { - background-color: #92a0a2; } - html.theme--documenter-dark .ansi span.sgr101 { - background-color: #ff8674; } - html.theme--documenter-dark .ansi span.sgr102 { - background-color: #79d462; } - html.theme--documenter-dark .ansi span.sgr103 { - background-color: #ffe76b; } - html.theme--documenter-dark .ansi span.sgr104 { - background-color: #8a98ff; } - html.theme--documenter-dark .ansi span.sgr105 { - background-color: #d2a4e6; } - html.theme--documenter-dark .ansi span.sgr106 { - background-color: #6bc8db; } - html.theme--documenter-dark .ansi span.sgr107 { - background-color: #ecf0f1; } - html.theme--documenter-dark code.language-julia-repl > span.hljs-meta { - color: #4fb43a; - font-weight: bolder; } - html.theme--documenter-dark .hljs { - background: #2b2b2b; - color: #f8f8f2; } - html.theme--documenter-dark .hljs-comment, - html.theme--documenter-dark .hljs-quote { - color: #d4d0ab; } - html.theme--documenter-dark .hljs-variable, - html.theme--documenter-dark .hljs-template-variable, - html.theme--documenter-dark .hljs-tag, - html.theme--documenter-dark .hljs-name, - html.theme--documenter-dark .hljs-selector-id, - html.theme--documenter-dark .hljs-selector-class, - html.theme--documenter-dark .hljs-regexp, - html.theme--documenter-dark .hljs-deletion { - color: #ffa07a; } - html.theme--documenter-dark .hljs-number, - html.theme--documenter-dark .hljs-built_in, - html.theme--documenter-dark .hljs-literal, - html.theme--documenter-dark .hljs-type, - html.theme--documenter-dark .hljs-params, - html.theme--documenter-dark .hljs-meta, - html.theme--documenter-dark .hljs-link { - color: #f5ab35; } - html.theme--documenter-dark .hljs-attribute { - color: #ffd700; } - html.theme--documenter-dark .hljs-string, - html.theme--documenter-dark .hljs-symbol, - html.theme--documenter-dark .hljs-bullet, - html.theme--documenter-dark .hljs-addition { - color: #abe338; } - html.theme--documenter-dark .hljs-title, - html.theme--documenter-dark .hljs-section { - color: #00e0e0; } - html.theme--documenter-dark .hljs-keyword, - html.theme--documenter-dark .hljs-selector-tag { - color: #dcc6e0; } - html.theme--documenter-dark .hljs-emphasis { - font-style: italic; } - html.theme--documenter-dark .hljs-strong { - font-weight: bold; } - @media screen and (-ms-high-contrast: active) { - html.theme--documenter-dark .hljs-addition, - html.theme--documenter-dark .hljs-attribute, - html.theme--documenter-dark .hljs-built_in, - html.theme--documenter-dark .hljs-bullet, - html.theme--documenter-dark .hljs-comment, - html.theme--documenter-dark .hljs-link, - html.theme--documenter-dark .hljs-literal, - html.theme--documenter-dark .hljs-meta, - html.theme--documenter-dark .hljs-number, - html.theme--documenter-dark .hljs-params, - html.theme--documenter-dark .hljs-string, - html.theme--documenter-dark .hljs-symbol, - html.theme--documenter-dark .hljs-type, - html.theme--documenter-dark .hljs-quote { - color: highlight; } - html.theme--documenter-dark .hljs-keyword, - html.theme--documenter-dark .hljs-selector-tag { - font-weight: bold; } } - html.theme--documenter-dark .hljs-subst { - color: #f8f8f2; } +*/}html.theme--documenter-dark html{background-color:#1f2424;font-size:16px;-moz-osx-font-smoothing:grayscale;-webkit-font-smoothing:antialiased;min-width:300px;overflow-x:auto;overflow-y:scroll;text-rendering:optimizeLegibility;text-size-adjust:100%}html.theme--documenter-dark article,html.theme--documenter-dark aside,html.theme--documenter-dark figure,html.theme--documenter-dark footer,html.theme--documenter-dark header,html.theme--documenter-dark hgroup,html.theme--documenter-dark section{display:block}html.theme--documenter-dark body,html.theme--documenter-dark button,html.theme--documenter-dark input,html.theme--documenter-dark optgroup,html.theme--documenter-dark select,html.theme--documenter-dark textarea{font-family:"Lato Medium",-apple-system,BlinkMacSystemFont,"Segoe UI","Helvetica Neue","Helvetica","Arial",sans-serif}html.theme--documenter-dark code,html.theme--documenter-dark pre{-moz-osx-font-smoothing:auto;-webkit-font-smoothing:auto;font-family:"JuliaMono","SFMono-Regular","Menlo","Consolas","Liberation Mono","DejaVu Sans Mono",monospace}html.theme--documenter-dark body{color:#fff;font-size:1em;font-weight:400;line-height:1.5}html.theme--documenter-dark a{color:#1abc9c;cursor:pointer;text-decoration:none}html.theme--documenter-dark a strong{color:currentColor}html.theme--documenter-dark a:hover{color:#1dd2af}html.theme--documenter-dark code{background-color:rgba(255,255,255,0.05);color:#ececec;font-size:.875em;font-weight:normal;padding:.1em}html.theme--documenter-dark hr{background-color:#282f2f;border:none;display:block;height:2px;margin:1.5rem 0}html.theme--documenter-dark img{height:auto;max-width:100%}html.theme--documenter-dark input[type="checkbox"],html.theme--documenter-dark input[type="radio"]{vertical-align:baseline}html.theme--documenter-dark small{font-size:.875em}html.theme--documenter-dark span{font-style:inherit;font-weight:inherit}html.theme--documenter-dark strong{color:#f2f2f2;font-weight:700}html.theme--documenter-dark fieldset{border:none}html.theme--documenter-dark pre{-webkit-overflow-scrolling:touch;background-color:#282f2f;color:#fff;font-size:.875em;overflow-x:auto;padding:1.25rem 1.5rem;white-space:pre;word-wrap:normal}html.theme--documenter-dark pre code{background-color:transparent;color:currentColor;font-size:1em;padding:0}html.theme--documenter-dark table td,html.theme--documenter-dark table th{vertical-align:top}html.theme--documenter-dark table td:not([align]),html.theme--documenter-dark table th:not([align]){text-align:inherit}html.theme--documenter-dark table th{color:#f2f2f2}html.theme--documenter-dark .box{background-color:#343c3d;border-radius:8px;box-shadow:none;color:#fff;display:block;padding:1.25rem}html.theme--documenter-dark a.box:hover,html.theme--documenter-dark a.box:focus{box-shadow:0 0.5em 1em -0.125em rgba(10,10,10,0.1),0 0 0 1px #1abc9c}html.theme--documenter-dark a.box:active{box-shadow:inset 0 1px 2px rgba(10,10,10,0.2),0 0 0 1px #1abc9c}html.theme--documenter-dark .button{background-color:#282f2f;border-color:#4c5759;border-width:1px;color:#375a7f;cursor:pointer;justify-content:center;padding-bottom:calc(0.5em - 1px);padding-left:1em;padding-right:1em;padding-top:calc(0.5em - 1px);text-align:center;white-space:nowrap}html.theme--documenter-dark .button strong{color:inherit}html.theme--documenter-dark .button .icon,html.theme--documenter-dark .button .icon.is-small,html.theme--documenter-dark .button #documenter .docs-sidebar form.docs-search>input.icon,html.theme--documenter-dark #documenter .docs-sidebar .button form.docs-search>input.icon,html.theme--documenter-dark .button .icon.is-medium,html.theme--documenter-dark .button .icon.is-large{height:1.5em;width:1.5em}html.theme--documenter-dark .button .icon:first-child:not(:last-child){margin-left:calc(-0.5em - 1px);margin-right:.25em}html.theme--documenter-dark .button .icon:last-child:not(:first-child){margin-left:.25em;margin-right:calc(-0.5em - 1px)}html.theme--documenter-dark .button .icon:first-child:last-child{margin-left:calc(-0.5em - 1px);margin-right:calc(-0.5em - 1px)}html.theme--documenter-dark .button:hover,html.theme--documenter-dark .button.is-hovered{border-color:#8c9b9d;color:#f2f2f2}html.theme--documenter-dark .button:focus,html.theme--documenter-dark .button.is-focused{border-color:#8c9b9d;color:#17a689}html.theme--documenter-dark .button:focus:not(:active),html.theme--documenter-dark .button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(26,188,156,0.25)}html.theme--documenter-dark .button:active,html.theme--documenter-dark .button.is-active{border-color:#343c3d;color:#f2f2f2}html.theme--documenter-dark .button.is-text{background-color:transparent;border-color:transparent;color:#fff;text-decoration:underline}html.theme--documenter-dark .button.is-text:hover,html.theme--documenter-dark .button.is-text.is-hovered,html.theme--documenter-dark .button.is-text:focus,html.theme--documenter-dark .button.is-text.is-focused{background-color:#282f2f;color:#f2f2f2}html.theme--documenter-dark .button.is-text:active,html.theme--documenter-dark .button.is-text.is-active{background-color:#1d2122;color:#f2f2f2}html.theme--documenter-dark .button.is-text[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-text{background-color:transparent;border-color:transparent;box-shadow:none}html.theme--documenter-dark .button.is-ghost{background:none;border-color:rgba(0,0,0,0);color:#1abc9c;text-decoration:none}html.theme--documenter-dark .button.is-ghost:hover,html.theme--documenter-dark .button.is-ghost.is-hovered{color:#1abc9c;text-decoration:underline}html.theme--documenter-dark .button.is-white{background-color:#fff;border-color:transparent;color:#0a0a0a}html.theme--documenter-dark .button.is-white:hover,html.theme--documenter-dark .button.is-white.is-hovered{background-color:#f9f9f9;border-color:transparent;color:#0a0a0a}html.theme--documenter-dark .button.is-white:focus,html.theme--documenter-dark .button.is-white.is-focused{border-color:transparent;color:#0a0a0a}html.theme--documenter-dark .button.is-white:focus:not(:active),html.theme--documenter-dark .button.is-white.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(255,255,255,0.25)}html.theme--documenter-dark .button.is-white:active,html.theme--documenter-dark .button.is-white.is-active{background-color:#f2f2f2;border-color:transparent;color:#0a0a0a}html.theme--documenter-dark .button.is-white[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-white{background-color:#fff;border-color:#fff;box-shadow:none}html.theme--documenter-dark .button.is-white.is-inverted{background-color:#0a0a0a;color:#fff}html.theme--documenter-dark .button.is-white.is-inverted:hover,html.theme--documenter-dark .button.is-white.is-inverted.is-hovered{background-color:#000}html.theme--documenter-dark .button.is-white.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-white.is-inverted{background-color:#0a0a0a;border-color:transparent;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-white.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-white.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-white.is-outlined:hover,html.theme--documenter-dark .button.is-white.is-outlined.is-hovered,html.theme--documenter-dark .button.is-white.is-outlined:focus,html.theme--documenter-dark .button.is-white.is-outlined.is-focused{background-color:#fff;border-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-white.is-outlined.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-white.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-white.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-white.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-white.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-white.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-white.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--documenter-dark .button.is-white.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-white.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-focused{background-color:#0a0a0a;color:#fff}html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-white.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black{background-color:#0a0a0a;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:hover,html.theme--documenter-dark .button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:focus,html.theme--documenter-dark .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:focus:not(:active),html.theme--documenter-dark .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--documenter-dark .button.is-black:active,html.theme--documenter-dark .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--documenter-dark .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted:hover,html.theme--documenter-dark .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark 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.content kbd.button:active,html.theme--documenter-dark .button.is-dark.is-active,html.theme--documenter-dark .content kbd.button.is-active{background-color:#1d2122;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark[disabled],html.theme--documenter-dark .content kbd.button[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark,fieldset[disabled] html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:#282f2f;box-shadow:none}html.theme--documenter-dark .button.is-dark.is-inverted,html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted:hover,html.theme--documenter-dark .content kbd.button.is-inverted:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark 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kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark 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html.theme--documenter-dark .docstring>section>a.button.is-inverted.is-outlined.docs-sourcelink{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-primary.is-light,html.theme--documenter-dark .docstring>section>a.button.is-light.docs-sourcelink{background-color:#f1f5f9;color:#4d7eb2}html.theme--documenter-dark .button.is-primary.is-light:hover,html.theme--documenter-dark .docstring>section>a.button.is-light.docs-sourcelink:hover,html.theme--documenter-dark .button.is-primary.is-light.is-hovered,html.theme--documenter-dark .docstring>section>a.button.is-light.is-hovered.docs-sourcelink{background-color:#e8eef5;border-color:transparent;color:#4d7eb2}html.theme--documenter-dark .button.is-primary.is-light:active,html.theme--documenter-dark .docstring>section>a.button.is-light.docs-sourcelink:active,html.theme--documenter-dark .button.is-primary.is-light.is-active,html.theme--documenter-dark 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.button.is-warning{background-color:#f4c72f;border-color:#f4c72f;box-shadow:none}html.theme--documenter-dark .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#f4c72f}html.theme--documenter-dark .button.is-warning.is-inverted:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f4c72f}html.theme--documenter-dark .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-warning.is-outlined{background-color:transparent;border-color:#f4c72f;color:#f4c72f}html.theme--documenter-dark .button.is-warning.is-outlined:hover,html.theme--documenter-dark 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html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fbefef;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#f8e6e5;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#f6dcda;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 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.button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark 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h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table 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Can be included in any rule that should render Font Awesome icons. */ -@keyframes spinAround { - from { - transform: rotate(0deg); } - to { - transform: rotate(359deg); } } - -.tabs, .pagination-previous, -.pagination-next, -.pagination-link, -.pagination-ellipsis, .breadcrumb, .file, .button, .is-unselectable, .modal-close, .delete { - -webkit-touch-callout: none; - -webkit-user-select: none; - -moz-user-select: none; - -ms-user-select: none; - user-select: none; } - -.navbar-link:not(.is-arrowless)::after, .select:not(.is-multiple):not(.is-loading)::after { - border: 3px solid transparent; - border-radius: 2px; - border-right: 0; - border-top: 0; - content: " "; - display: block; - height: 0.625em; - margin-top: -0.4375em; - pointer-events: none; - position: absolute; - top: 50%; - transform: rotate(-45deg); - transform-origin: center; - width: 0.625em; } - -.admonition:not(:last-child), .tabs:not(:last-child), .message:not(:last-child), .list:not(:last-child), .level:not(:last-child), .breadcrumb:not(:last-child), .highlight:not(:last-child), .block:not(:last-child), .title:not(:last-child), -.subtitle:not(:last-child), .table-container:not(:last-child), .table:not(:last-child), .progress:not(:last-child), .notification:not(:last-child), .content:not(:last-child), .box:not(:last-child) { - margin-bottom: 1.5rem; } - -.modal-close, .delete { - -moz-appearance: none; - -webkit-appearance: none; - background-color: rgba(10, 10, 10, 0.2); - border: none; - border-radius: 290486px; - cursor: pointer; - pointer-events: auto; - display: inline-block; - flex-grow: 0; - flex-shrink: 0; - font-size: 0; - height: 20px; - max-height: 20px; - max-width: 20px; - min-height: 20px; - min-width: 20px; - outline: none; - position: relative; - vertical-align: top; - width: 20px; } - .modal-close::before, .delete::before, .modal-close::after, .delete::after { - background-color: white; - content: ""; - display: block; - left: 50%; - position: absolute; - top: 50%; - transform: translateX(-50%) translateY(-50%) rotate(45deg); - transform-origin: center center; } - .modal-close::before, .delete::before { - height: 2px; - width: 50%; } - .modal-close::after, .delete::after { - height: 50%; - width: 2px; } - .modal-close:hover, .delete:hover, .modal-close:focus, .delete:focus { - background-color: rgba(10, 10, 10, 0.3); 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- left: 0; - position: absolute; - right: 0; - top: 0; } - -.pagination-previous, -.pagination-next, -.pagination-link, -.pagination-ellipsis, .file-cta, -.file-name, .select select, .textarea, .input, #documenter .docs-sidebar form.docs-search > input, .button { - -moz-appearance: none; - -webkit-appearance: none; - align-items: center; - border: 1px solid transparent; - border-radius: 4px; - box-shadow: none; - display: inline-flex; - font-size: 1rem; - height: 2.25em; - justify-content: flex-start; - line-height: 1.5; - padding-bottom: calc(0.375em - 1px); - padding-left: calc(0.625em - 1px); - padding-right: calc(0.625em - 1px); - padding-top: calc(0.375em - 1px); - position: relative; - vertical-align: top; } - .pagination-previous:focus, - .pagination-next:focus, - .pagination-link:focus, - .pagination-ellipsis:focus, .file-cta:focus, - .file-name:focus, .select select:focus, .textarea:focus, .input:focus, #documenter .docs-sidebar form.docs-search > input:focus, .button:focus, .is-focused.pagination-previous, - .is-focused.pagination-next, - .is-focused.pagination-link, - .is-focused.pagination-ellipsis, .is-focused.file-cta, - .is-focused.file-name, .select select.is-focused, .is-focused.textarea, .is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-focused.button, .pagination-previous:active, - .pagination-next:active, - .pagination-link:active, - .pagination-ellipsis:active, .file-cta:active, - .file-name:active, .select select:active, .textarea:active, .input:active, #documenter .docs-sidebar form.docs-search > input:active, .button:active, .is-active.pagination-previous, - .is-active.pagination-next, - .is-active.pagination-link, - .is-active.pagination-ellipsis, .is-active.file-cta, - .is-active.file-name, .select select.is-active, .is-active.textarea, .is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active, .is-active.button { - outline: none; } - .pagination-previous[disabled], - .pagination-next[disabled], - .pagination-link[disabled], - .pagination-ellipsis[disabled], .file-cta[disabled], - .file-name[disabled], .select select[disabled], .textarea[disabled], .input[disabled], #documenter .docs-sidebar form.docs-search > input[disabled], .button[disabled], fieldset[disabled] .pagination-previous, - fieldset[disabled] .pagination-next, - fieldset[disabled] .pagination-link, - fieldset[disabled] .pagination-ellipsis, fieldset[disabled] .file-cta, - fieldset[disabled] .file-name, fieldset[disabled] .select select, .select fieldset[disabled] select, fieldset[disabled] .textarea, fieldset[disabled] .input, fieldset[disabled] #documenter .docs-sidebar form.docs-search > input, #documenter .docs-sidebar fieldset[disabled] form.docs-search > input, fieldset[disabled] .button { - cursor: not-allowed; } - -/*! minireset.css v0.0.4 | MIT License | github.com/jgthms/minireset.css */ -html, -body, -p, -ol, -ul, -li, -dl, -dt, -dd, -blockquote, -figure, -fieldset, -legend, -textarea, -pre, -iframe, -hr, -h1, -h2, -h3, -h4, -h5, -h6 { - margin: 0; 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} - -.is-relative { - position: relative !important; } - -.box { - background-color: white; - border-radius: 6px; - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); - color: #222222; - display: block; - padding: 1.25rem; } - -a.box:hover, a.box:focus { - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px #2e63b8; } - -a.box:active { - box-shadow: inset 0 1px 2px rgba(10, 10, 10, 0.2), 0 0 0 1px #2e63b8; } - -.button { - background-color: white; - border-color: #dbdbdb; - border-width: 1px; - color: #363636; - cursor: pointer; - justify-content: center; - padding-bottom: calc(0.375em - 1px); - padding-left: 0.75em; - padding-right: 0.75em; - padding-top: calc(0.375em - 1px); - text-align: center; - white-space: nowrap; } - .button strong { - color: inherit; } - .button .icon, .button .icon.is-small, .button #documenter .docs-sidebar form.docs-search > input.icon, #documenter .docs-sidebar .button form.docs-search > input.icon, .button .icon.is-medium, .button .icon.is-large { - height: 1.5em; - width: 1.5em; } - .button .icon:first-child:not(:last-child) { - margin-left: calc(-0.375em - 1px); - margin-right: 0.1875em; } - .button .icon:last-child:not(:first-child) { - margin-left: 0.1875em; - margin-right: calc(-0.375em - 1px); } - .button .icon:first-child:last-child { - margin-left: calc(-0.375em - 1px); - margin-right: calc(-0.375em - 1px); } - .button:hover, .button.is-hovered { - border-color: #b5b5b5; - color: #363636; } - .button:focus, .button.is-focused { - border-color: #3c5dcd; - color: #363636; } - .button:focus:not(:active), .button.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(46, 99, 184, 0.25); } - .button:active, .button.is-active { - border-color: #4a4a4a; - color: #363636; } - .button.is-text { - background-color: transparent; - border-color: transparent; - color: #222222; - text-decoration: underline; } - .button.is-text:hover, .button.is-text.is-hovered, .button.is-text:focus, .button.is-text.is-focused { - background-color: whitesmoke; - color: #222222; } - .button.is-text:active, .button.is-text.is-active { - background-color: #e8e8e8; - color: #222222; } - .button.is-text[disabled], fieldset[disabled] .button.is-text { - background-color: transparent; - border-color: transparent; - box-shadow: none; } - .button.is-white { - background-color: white; - border-color: transparent; - color: #0a0a0a; } - .button.is-white:hover, .button.is-white.is-hovered { - background-color: #f9f9f9; - border-color: transparent; - color: #0a0a0a; } - .button.is-white:focus, .button.is-white.is-focused { - border-color: transparent; - color: #0a0a0a; } - .button.is-white:focus:not(:active), .button.is-white.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); } - .button.is-white:active, .button.is-white.is-active { - background-color: #f2f2f2; - border-color: transparent; - color: #0a0a0a; } - .button.is-white[disabled], fieldset[disabled] .button.is-white { - background-color: white; - border-color: transparent; - box-shadow: none; } - .button.is-white.is-inverted { - background-color: #0a0a0a; - color: white; } - .button.is-white.is-inverted:hover, .button.is-white.is-inverted.is-hovered { - background-color: black; } - .button.is-white.is-inverted[disabled], fieldset[disabled] .button.is-white.is-inverted { - background-color: #0a0a0a; - border-color: transparent; - box-shadow: none; - color: white; } - .button.is-white.is-loading::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - .button.is-white.is-outlined { - background-color: transparent; - border-color: white; - color: white; } - .button.is-white.is-outlined:hover, .button.is-white.is-outlined.is-hovered, .button.is-white.is-outlined:focus, .button.is-white.is-outlined.is-focused { - background-color: white; - border-color: white; - color: #0a0a0a; } - .button.is-white.is-outlined.is-loading::after { - border-color: transparent transparent white white !important; } - .button.is-white.is-outlined.is-loading:hover::after, .button.is-white.is-outlined.is-loading.is-hovered::after, .button.is-white.is-outlined.is-loading:focus::after, .button.is-white.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - .button.is-white.is-outlined[disabled], fieldset[disabled] .button.is-white.is-outlined { - background-color: transparent; - border-color: white; - box-shadow: none; - color: white; } - .button.is-white.is-inverted.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - color: #0a0a0a; } - .button.is-white.is-inverted.is-outlined:hover, .button.is-white.is-inverted.is-outlined.is-hovered, .button.is-white.is-inverted.is-outlined:focus, .button.is-white.is-inverted.is-outlined.is-focused { - background-color: #0a0a0a; - color: white; } - .button.is-white.is-inverted.is-outlined.is-loading:hover::after, .button.is-white.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-white.is-inverted.is-outlined.is-loading:focus::after, .button.is-white.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent white white !important; } - .button.is-white.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-white.is-inverted.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - box-shadow: none; - color: #0a0a0a; } - .button.is-black { - background-color: #0a0a0a; - border-color: transparent; - color: white; } - .button.is-black:hover, .button.is-black.is-hovered { - background-color: #040404; - border-color: transparent; - color: white; } - .button.is-black:focus, .button.is-black.is-focused { - border-color: transparent; - color: white; } - .button.is-black:focus:not(:active), .button.is-black.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); } - .button.is-black:active, .button.is-black.is-active { - background-color: black; - border-color: transparent; - color: white; } - .button.is-black[disabled], fieldset[disabled] .button.is-black { - background-color: #0a0a0a; - border-color: transparent; - box-shadow: none; } - .button.is-black.is-inverted { - background-color: white; - color: #0a0a0a; } - .button.is-black.is-inverted:hover, .button.is-black.is-inverted.is-hovered { - background-color: #f2f2f2; } - .button.is-black.is-inverted[disabled], fieldset[disabled] .button.is-black.is-inverted { - background-color: white; - border-color: transparent; - box-shadow: none; - color: #0a0a0a; } - .button.is-black.is-loading::after { - border-color: transparent transparent white white !important; } - .button.is-black.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - color: #0a0a0a; } - .button.is-black.is-outlined:hover, .button.is-black.is-outlined.is-hovered, .button.is-black.is-outlined:focus, .button.is-black.is-outlined.is-focused { - background-color: #0a0a0a; - border-color: #0a0a0a; - color: white; } - .button.is-black.is-outlined.is-loading::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - .button.is-black.is-outlined.is-loading:hover::after, .button.is-black.is-outlined.is-loading.is-hovered::after, .button.is-black.is-outlined.is-loading:focus::after, .button.is-black.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent white white !important; } - .button.is-black.is-outlined[disabled], fieldset[disabled] .button.is-black.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - box-shadow: none; - color: #0a0a0a; } - .button.is-black.is-inverted.is-outlined { - background-color: transparent; - border-color: white; - color: white; } - .button.is-black.is-inverted.is-outlined:hover, .button.is-black.is-inverted.is-outlined.is-hovered, .button.is-black.is-inverted.is-outlined:focus, .button.is-black.is-inverted.is-outlined.is-focused { - background-color: white; - color: #0a0a0a; } - .button.is-black.is-inverted.is-outlined.is-loading:hover::after, .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-black.is-inverted.is-outlined.is-loading:focus::after, .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - .button.is-black.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-black.is-inverted.is-outlined { - background-color: transparent; - border-color: white; - box-shadow: none; - color: white; } - .button.is-light { - background-color: whitesmoke; - border-color: transparent; - color: #363636; } - .button.is-light:hover, .button.is-light.is-hovered { - background-color: #eeeeee; - border-color: transparent; - color: #363636; } - .button.is-light:focus, .button.is-light.is-focused { - border-color: transparent; - color: #363636; } - .button.is-light:focus:not(:active), .button.is-light.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(245, 245, 245, 0.25); } - .button.is-light:active, .button.is-light.is-active { - background-color: #e8e8e8; - border-color: transparent; - color: #363636; } - .button.is-light[disabled], fieldset[disabled] .button.is-light { - background-color: whitesmoke; - border-color: transparent; - box-shadow: none; } - .button.is-light.is-inverted { - background-color: #363636; - color: whitesmoke; } - .button.is-light.is-inverted:hover, .button.is-light.is-inverted.is-hovered { - background-color: #292929; } - .button.is-light.is-inverted[disabled], fieldset[disabled] .button.is-light.is-inverted { - background-color: #363636; - border-color: transparent; - box-shadow: none; - color: whitesmoke; } - .button.is-light.is-loading::after { - border-color: transparent transparent #363636 #363636 !important; } - .button.is-light.is-outlined { - background-color: transparent; - border-color: whitesmoke; - color: whitesmoke; } - .button.is-light.is-outlined:hover, .button.is-light.is-outlined.is-hovered, .button.is-light.is-outlined:focus, .button.is-light.is-outlined.is-focused { - background-color: whitesmoke; - border-color: whitesmoke; - color: #363636; } - .button.is-light.is-outlined.is-loading::after { - border-color: transparent transparent whitesmoke whitesmoke !important; } - .button.is-light.is-outlined.is-loading:hover::after, .button.is-light.is-outlined.is-loading.is-hovered::after, .button.is-light.is-outlined.is-loading:focus::after, .button.is-light.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #363636 #363636 !important; } - .button.is-light.is-outlined[disabled], fieldset[disabled] .button.is-light.is-outlined { - background-color: transparent; - border-color: whitesmoke; - box-shadow: none; - color: whitesmoke; } - .button.is-light.is-inverted.is-outlined { - background-color: transparent; - border-color: #363636; - color: #363636; } - .button.is-light.is-inverted.is-outlined:hover, .button.is-light.is-inverted.is-outlined.is-hovered, .button.is-light.is-inverted.is-outlined:focus, .button.is-light.is-inverted.is-outlined.is-focused { - background-color: #363636; - color: whitesmoke; } - .button.is-light.is-inverted.is-outlined.is-loading:hover::after, .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-light.is-inverted.is-outlined.is-loading:focus::after, .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent whitesmoke whitesmoke !important; } - .button.is-light.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-light.is-inverted.is-outlined { - background-color: transparent; - border-color: #363636; - box-shadow: none; - color: #363636; } - .button.is-dark, .content kbd.button { - background-color: #363636; - border-color: transparent; - color: whitesmoke; } - .button.is-dark:hover, .content kbd.button:hover, .button.is-dark.is-hovered, .content kbd.button.is-hovered { - background-color: #2f2f2f; - border-color: transparent; - color: whitesmoke; } - .button.is-dark:focus, .content kbd.button:focus, .button.is-dark.is-focused, .content kbd.button.is-focused { - border-color: transparent; - color: whitesmoke; } - .button.is-dark:focus:not(:active), .content kbd.button:focus:not(:active), .button.is-dark.is-focused:not(:active), .content kbd.button.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(54, 54, 54, 0.25); } - .button.is-dark:active, .content kbd.button:active, .button.is-dark.is-active, .content kbd.button.is-active { - background-color: #292929; - border-color: transparent; - color: whitesmoke; } - .button.is-dark[disabled], .content kbd.button[disabled], fieldset[disabled] .button.is-dark, fieldset[disabled] .content kbd.button, .content fieldset[disabled] kbd.button { - background-color: #363636; - border-color: transparent; - box-shadow: none; } - .button.is-dark.is-inverted, .content kbd.button.is-inverted { - background-color: whitesmoke; - color: #363636; } - .button.is-dark.is-inverted:hover, .content kbd.button.is-inverted:hover, .button.is-dark.is-inverted.is-hovered, .content kbd.button.is-inverted.is-hovered { - background-color: #e8e8e8; } - .button.is-dark.is-inverted[disabled], .content kbd.button.is-inverted[disabled], fieldset[disabled] .button.is-dark.is-inverted, fieldset[disabled] .content kbd.button.is-inverted, .content fieldset[disabled] kbd.button.is-inverted { - background-color: whitesmoke; - border-color: transparent; - box-shadow: none; - color: #363636; } - .button.is-dark.is-loading::after, .content kbd.button.is-loading::after { - border-color: transparent transparent whitesmoke whitesmoke !important; } - .button.is-dark.is-outlined, .content kbd.button.is-outlined { - background-color: transparent; - border-color: #363636; - color: #363636; } - .button.is-dark.is-outlined:hover, .content kbd.button.is-outlined:hover, .button.is-dark.is-outlined.is-hovered, .content kbd.button.is-outlined.is-hovered, .button.is-dark.is-outlined:focus, .content kbd.button.is-outlined:focus, .button.is-dark.is-outlined.is-focused, .content kbd.button.is-outlined.is-focused { - background-color: #363636; - border-color: #363636; - color: whitesmoke; } - .button.is-dark.is-outlined.is-loading::after, .content kbd.button.is-outlined.is-loading::after { - border-color: transparent transparent #363636 #363636 !important; } - .button.is-dark.is-outlined.is-loading:hover::after, .content kbd.button.is-outlined.is-loading:hover::after, .button.is-dark.is-outlined.is-loading.is-hovered::after, .content kbd.button.is-outlined.is-loading.is-hovered::after, .button.is-dark.is-outlined.is-loading:focus::after, .content kbd.button.is-outlined.is-loading:focus::after, .button.is-dark.is-outlined.is-loading.is-focused::after, .content kbd.button.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent whitesmoke whitesmoke !important; } - .button.is-dark.is-outlined[disabled], .content kbd.button.is-outlined[disabled], fieldset[disabled] .button.is-dark.is-outlined, fieldset[disabled] .content kbd.button.is-outlined, .content fieldset[disabled] kbd.button.is-outlined { - background-color: transparent; - border-color: #363636; - box-shadow: none; - color: #363636; } - .button.is-dark.is-inverted.is-outlined, .content kbd.button.is-inverted.is-outlined { - background-color: transparent; - border-color: whitesmoke; - color: whitesmoke; } - .button.is-dark.is-inverted.is-outlined:hover, .content kbd.button.is-inverted.is-outlined:hover, .button.is-dark.is-inverted.is-outlined.is-hovered, .content kbd.button.is-inverted.is-outlined.is-hovered, .button.is-dark.is-inverted.is-outlined:focus, .content kbd.button.is-inverted.is-outlined:focus, .button.is-dark.is-inverted.is-outlined.is-focused, .content kbd.button.is-inverted.is-outlined.is-focused { - background-color: whitesmoke; - color: #363636; } - .button.is-dark.is-inverted.is-outlined.is-loading:hover::after, .content kbd.button.is-inverted.is-outlined.is-loading:hover::after, .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after, .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-dark.is-inverted.is-outlined.is-loading:focus::after, .content kbd.button.is-inverted.is-outlined.is-loading:focus::after, .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after, .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #363636 #363636 !important; } - .button.is-dark.is-inverted.is-outlined[disabled], .content kbd.button.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-dark.is-inverted.is-outlined, fieldset[disabled] .content kbd.button.is-inverted.is-outlined, .content fieldset[disabled] kbd.button.is-inverted.is-outlined { - background-color: transparent; - border-color: whitesmoke; - box-shadow: none; - color: whitesmoke; } - .button.is-primary, .docstring > section > a.button.docs-sourcelink { - background-color: #4eb5de; - border-color: transparent; - color: #fff; } - .button.is-primary:hover, .docstring > section > a.button.docs-sourcelink:hover, .button.is-primary.is-hovered, .docstring > section > a.button.is-hovered.docs-sourcelink { - background-color: #43b1dc; - border-color: transparent; - color: #fff; } - .button.is-primary:focus, .docstring > section > a.button.docs-sourcelink:focus, .button.is-primary.is-focused, .docstring > section > a.button.is-focused.docs-sourcelink { - border-color: transparent; - color: #fff; } - .button.is-primary:focus:not(:active), .docstring > section > a.button.docs-sourcelink:focus:not(:active), .button.is-primary.is-focused:not(:active), .docstring > section > a.button.is-focused.docs-sourcelink:not(:active) { - box-shadow: 0 0 0 0.125em rgba(78, 181, 222, 0.25); } - .button.is-primary:active, .docstring > section > a.button.docs-sourcelink:active, .button.is-primary.is-active, .docstring > section > a.button.is-active.docs-sourcelink { - background-color: #39acda; - border-color: transparent; - color: #fff; } - .button.is-primary[disabled], .docstring > section > a.button.docs-sourcelink[disabled], fieldset[disabled] .button.is-primary, fieldset[disabled] .docstring > section > a.button.docs-sourcelink { - background-color: #4eb5de; - border-color: transparent; - box-shadow: none; } - .button.is-primary.is-inverted, .docstring > section > a.button.is-inverted.docs-sourcelink { - background-color: #fff; - color: #4eb5de; } - .button.is-primary.is-inverted:hover, .docstring > section > a.button.is-inverted.docs-sourcelink:hover, .button.is-primary.is-inverted.is-hovered, .docstring > section > a.button.is-inverted.is-hovered.docs-sourcelink { - background-color: #f2f2f2; } - .button.is-primary.is-inverted[disabled], .docstring > section > a.button.is-inverted.docs-sourcelink[disabled], fieldset[disabled] .button.is-primary.is-inverted, fieldset[disabled] .docstring > section > a.button.is-inverted.docs-sourcelink { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #4eb5de; } - .button.is-primary.is-loading::after, .docstring > section > a.button.is-loading.docs-sourcelink::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-primary.is-outlined, .docstring > section > a.button.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #4eb5de; - color: #4eb5de; } - .button.is-primary.is-outlined:hover, .docstring > section > a.button.is-outlined.docs-sourcelink:hover, .button.is-primary.is-outlined.is-hovered, .docstring > section > a.button.is-outlined.is-hovered.docs-sourcelink, .button.is-primary.is-outlined:focus, .docstring > section > a.button.is-outlined.docs-sourcelink:focus, .button.is-primary.is-outlined.is-focused, .docstring > section > a.button.is-outlined.is-focused.docs-sourcelink { - background-color: #4eb5de; - border-color: #4eb5de; - color: #fff; } - .button.is-primary.is-outlined.is-loading::after, .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink::after { - border-color: transparent transparent #4eb5de #4eb5de !important; } - .button.is-primary.is-outlined.is-loading:hover::after, .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink:hover::after, .button.is-primary.is-outlined.is-loading.is-hovered::after, .docstring > section > a.button.is-outlined.is-loading.is-hovered.docs-sourcelink::after, .button.is-primary.is-outlined.is-loading:focus::after, .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink:focus::after, .button.is-primary.is-outlined.is-loading.is-focused::after, .docstring > section > a.button.is-outlined.is-loading.is-focused.docs-sourcelink::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-primary.is-outlined[disabled], .docstring > section > a.button.is-outlined.docs-sourcelink[disabled], fieldset[disabled] .button.is-primary.is-outlined, fieldset[disabled] .docstring > section > a.button.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #4eb5de; - box-shadow: none; - color: #4eb5de; } - .button.is-primary.is-inverted.is-outlined, .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #fff; - color: #fff; } - .button.is-primary.is-inverted.is-outlined:hover, .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink:hover, .button.is-primary.is-inverted.is-outlined.is-hovered, .docstring > section > a.button.is-inverted.is-outlined.is-hovered.docs-sourcelink, .button.is-primary.is-inverted.is-outlined:focus, .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink:focus, .button.is-primary.is-inverted.is-outlined.is-focused, .docstring > section > a.button.is-inverted.is-outlined.is-focused.docs-sourcelink { - background-color: #fff; - color: #4eb5de; } - .button.is-primary.is-inverted.is-outlined.is-loading:hover::after, .docstring > section > a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:hover::after, .button.is-primary.is-inverted.is-outlined.is-loading.is-hovered::after, .docstring > section > a.button.is-inverted.is-outlined.is-loading.is-hovered.docs-sourcelink::after, .button.is-primary.is-inverted.is-outlined.is-loading:focus::after, .docstring > section > a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:focus::after, .button.is-primary.is-inverted.is-outlined.is-loading.is-focused::after, .docstring > section > a.button.is-inverted.is-outlined.is-loading.is-focused.docs-sourcelink::after { - border-color: transparent transparent #4eb5de #4eb5de !important; } - .button.is-primary.is-inverted.is-outlined[disabled], .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink[disabled], fieldset[disabled] .button.is-primary.is-inverted.is-outlined, fieldset[disabled] .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - .button.is-link { - background-color: #2e63b8; - border-color: transparent; - color: #fff; } - .button.is-link:hover, .button.is-link.is-hovered { - background-color: #2b5eae; - border-color: transparent; - color: #fff; } - .button.is-link:focus, .button.is-link.is-focused { - border-color: transparent; - color: #fff; } - .button.is-link:focus:not(:active), .button.is-link.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(46, 99, 184, 0.25); } - .button.is-link:active, .button.is-link.is-active { - background-color: #2958a4; - border-color: transparent; - color: #fff; } - .button.is-link[disabled], fieldset[disabled] .button.is-link { - background-color: #2e63b8; - border-color: transparent; - box-shadow: none; } - .button.is-link.is-inverted { - background-color: #fff; - color: #2e63b8; } - .button.is-link.is-inverted:hover, .button.is-link.is-inverted.is-hovered { - background-color: #f2f2f2; } - .button.is-link.is-inverted[disabled], fieldset[disabled] .button.is-link.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #2e63b8; } - .button.is-link.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-link.is-outlined { - background-color: transparent; - border-color: #2e63b8; - color: #2e63b8; } - .button.is-link.is-outlined:hover, .button.is-link.is-outlined.is-hovered, .button.is-link.is-outlined:focus, .button.is-link.is-outlined.is-focused { - background-color: #2e63b8; - border-color: #2e63b8; - color: #fff; } - .button.is-link.is-outlined.is-loading::after { - border-color: transparent transparent #2e63b8 #2e63b8 !important; } - .button.is-link.is-outlined.is-loading:hover::after, .button.is-link.is-outlined.is-loading.is-hovered::after, .button.is-link.is-outlined.is-loading:focus::after, .button.is-link.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-link.is-outlined[disabled], fieldset[disabled] .button.is-link.is-outlined { - background-color: transparent; - border-color: #2e63b8; - box-shadow: none; - color: #2e63b8; } - .button.is-link.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - .button.is-link.is-inverted.is-outlined:hover, .button.is-link.is-inverted.is-outlined.is-hovered, .button.is-link.is-inverted.is-outlined:focus, .button.is-link.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #2e63b8; } - .button.is-link.is-inverted.is-outlined.is-loading:hover::after, .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-link.is-inverted.is-outlined.is-loading:focus::after, .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #2e63b8 #2e63b8 !important; } - .button.is-link.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-link.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - .button.is-info { - background-color: #209cee; - border-color: transparent; - color: #fff; } - .button.is-info:hover, .button.is-info.is-hovered { - background-color: #1497ed; - border-color: transparent; - color: #fff; } - .button.is-info:focus, .button.is-info.is-focused { - border-color: transparent; - color: #fff; } - .button.is-info:focus:not(:active), .button.is-info.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(32, 156, 238, 0.25); } - .button.is-info:active, .button.is-info.is-active { - background-color: #1190e3; - border-color: transparent; - color: #fff; } - .button.is-info[disabled], fieldset[disabled] .button.is-info { - background-color: #209cee; - border-color: transparent; - box-shadow: none; } - .button.is-info.is-inverted { - background-color: #fff; - color: #209cee; } - .button.is-info.is-inverted:hover, .button.is-info.is-inverted.is-hovered { - background-color: #f2f2f2; } - .button.is-info.is-inverted[disabled], fieldset[disabled] .button.is-info.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #209cee; } - .button.is-info.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-info.is-outlined { - background-color: transparent; - border-color: #209cee; - color: #209cee; } - .button.is-info.is-outlined:hover, .button.is-info.is-outlined.is-hovered, .button.is-info.is-outlined:focus, .button.is-info.is-outlined.is-focused { - background-color: #209cee; - border-color: #209cee; - color: #fff; } - .button.is-info.is-outlined.is-loading::after { - border-color: transparent transparent #209cee #209cee !important; } - .button.is-info.is-outlined.is-loading:hover::after, .button.is-info.is-outlined.is-loading.is-hovered::after, .button.is-info.is-outlined.is-loading:focus::after, .button.is-info.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-info.is-outlined[disabled], fieldset[disabled] .button.is-info.is-outlined { - background-color: transparent; - border-color: #209cee; - box-shadow: none; - color: #209cee; } - .button.is-info.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - .button.is-info.is-inverted.is-outlined:hover, .button.is-info.is-inverted.is-outlined.is-hovered, .button.is-info.is-inverted.is-outlined:focus, .button.is-info.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #209cee; } - .button.is-info.is-inverted.is-outlined.is-loading:hover::after, .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-info.is-inverted.is-outlined.is-loading:focus::after, .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #209cee #209cee !important; } - .button.is-info.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-info.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - .button.is-success { - background-color: #22c35b; - border-color: transparent; - color: #fff; } - .button.is-success:hover, .button.is-success.is-hovered { - background-color: #20b856; - border-color: transparent; - color: #fff; } - .button.is-success:focus, .button.is-success.is-focused { - border-color: transparent; - color: #fff; } - .button.is-success:focus:not(:active), .button.is-success.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(34, 195, 91, 0.25); } - .button.is-success:active, .button.is-success.is-active { - background-color: #1ead51; - border-color: transparent; - color: #fff; } - .button.is-success[disabled], fieldset[disabled] .button.is-success { - background-color: #22c35b; - border-color: transparent; - box-shadow: none; } - .button.is-success.is-inverted { - background-color: #fff; - color: #22c35b; } - .button.is-success.is-inverted:hover, .button.is-success.is-inverted.is-hovered { - background-color: #f2f2f2; } - .button.is-success.is-inverted[disabled], fieldset[disabled] .button.is-success.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #22c35b; } - .button.is-success.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-success.is-outlined { - background-color: transparent; - border-color: #22c35b; - color: #22c35b; } - .button.is-success.is-outlined:hover, .button.is-success.is-outlined.is-hovered, .button.is-success.is-outlined:focus, .button.is-success.is-outlined.is-focused { - background-color: #22c35b; - border-color: #22c35b; - color: #fff; } - .button.is-success.is-outlined.is-loading::after { - border-color: transparent transparent #22c35b #22c35b !important; } - .button.is-success.is-outlined.is-loading:hover::after, .button.is-success.is-outlined.is-loading.is-hovered::after, .button.is-success.is-outlined.is-loading:focus::after, .button.is-success.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-success.is-outlined[disabled], fieldset[disabled] .button.is-success.is-outlined { - background-color: transparent; - border-color: #22c35b; - box-shadow: none; - color: #22c35b; } - .button.is-success.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - .button.is-success.is-inverted.is-outlined:hover, .button.is-success.is-inverted.is-outlined.is-hovered, .button.is-success.is-inverted.is-outlined:focus, .button.is-success.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #22c35b; } - .button.is-success.is-inverted.is-outlined.is-loading:hover::after, .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-success.is-inverted.is-outlined.is-loading:focus::after, .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #22c35b #22c35b !important; } - .button.is-success.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-success.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - .button.is-warning { - background-color: #ffdd57; - border-color: transparent; - color: rgba(0, 0, 0, 0.7); } - .button.is-warning:hover, .button.is-warning.is-hovered { - background-color: #ffda4a; - border-color: transparent; - color: rgba(0, 0, 0, 0.7); } - .button.is-warning:focus, .button.is-warning.is-focused { - border-color: transparent; - color: rgba(0, 0, 0, 0.7); } - .button.is-warning:focus:not(:active), .button.is-warning.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(255, 221, 87, 0.25); } - .button.is-warning:active, .button.is-warning.is-active { - background-color: #ffd83e; - border-color: transparent; - color: rgba(0, 0, 0, 0.7); } - .button.is-warning[disabled], fieldset[disabled] .button.is-warning { - background-color: #ffdd57; - border-color: transparent; - box-shadow: none; } - .button.is-warning.is-inverted { - background-color: rgba(0, 0, 0, 0.7); - color: #ffdd57; } - .button.is-warning.is-inverted:hover, .button.is-warning.is-inverted.is-hovered { - background-color: rgba(0, 0, 0, 0.7); } - .button.is-warning.is-inverted[disabled], fieldset[disabled] .button.is-warning.is-inverted { - background-color: rgba(0, 0, 0, 0.7); - border-color: transparent; - box-shadow: none; - color: #ffdd57; } - .button.is-warning.is-loading::after { - border-color: transparent transparent rgba(0, 0, 0, 0.7) rgba(0, 0, 0, 0.7) !important; } - .button.is-warning.is-outlined { - background-color: transparent; - border-color: #ffdd57; - color: #ffdd57; } - .button.is-warning.is-outlined:hover, .button.is-warning.is-outlined.is-hovered, .button.is-warning.is-outlined:focus, .button.is-warning.is-outlined.is-focused { - background-color: #ffdd57; - border-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .button.is-warning.is-outlined.is-loading::after { - border-color: transparent transparent #ffdd57 #ffdd57 !important; } - .button.is-warning.is-outlined.is-loading:hover::after, .button.is-warning.is-outlined.is-loading.is-hovered::after, .button.is-warning.is-outlined.is-loading:focus::after, .button.is-warning.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent rgba(0, 0, 0, 0.7) rgba(0, 0, 0, 0.7) !important; } - .button.is-warning.is-outlined[disabled], fieldset[disabled] .button.is-warning.is-outlined { - background-color: transparent; - border-color: #ffdd57; - box-shadow: none; - color: #ffdd57; } - .button.is-warning.is-inverted.is-outlined { - background-color: transparent; - border-color: rgba(0, 0, 0, 0.7); - color: rgba(0, 0, 0, 0.7); } - .button.is-warning.is-inverted.is-outlined:hover, .button.is-warning.is-inverted.is-outlined.is-hovered, .button.is-warning.is-inverted.is-outlined:focus, .button.is-warning.is-inverted.is-outlined.is-focused { - background-color: rgba(0, 0, 0, 0.7); - color: #ffdd57; } - .button.is-warning.is-inverted.is-outlined.is-loading:hover::after, .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-warning.is-inverted.is-outlined.is-loading:focus::after, .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #ffdd57 #ffdd57 !important; } - .button.is-warning.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-warning.is-inverted.is-outlined { - background-color: transparent; - border-color: rgba(0, 0, 0, 0.7); - box-shadow: none; - color: rgba(0, 0, 0, 0.7); } - .button.is-danger { - background-color: #da0b00; - border-color: transparent; - color: #fff; } - .button.is-danger:hover, .button.is-danger.is-hovered { - background-color: #cd0a00; - border-color: transparent; - color: #fff; } - .button.is-danger:focus, .button.is-danger.is-focused { - border-color: transparent; - color: #fff; } - .button.is-danger:focus:not(:active), .button.is-danger.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(218, 11, 0, 0.25); } - .button.is-danger:active, .button.is-danger.is-active { - background-color: #c10a00; - border-color: transparent; - color: #fff; } - .button.is-danger[disabled], fieldset[disabled] .button.is-danger { - background-color: #da0b00; - border-color: transparent; - box-shadow: none; } - .button.is-danger.is-inverted { - background-color: #fff; - color: #da0b00; } - .button.is-danger.is-inverted:hover, .button.is-danger.is-inverted.is-hovered { - background-color: #f2f2f2; } - .button.is-danger.is-inverted[disabled], fieldset[disabled] .button.is-danger.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #da0b00; } - .button.is-danger.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-danger.is-outlined { - background-color: transparent; - border-color: #da0b00; - color: #da0b00; } - .button.is-danger.is-outlined:hover, .button.is-danger.is-outlined.is-hovered, .button.is-danger.is-outlined:focus, .button.is-danger.is-outlined.is-focused { - background-color: #da0b00; - border-color: #da0b00; - color: #fff; } - .button.is-danger.is-outlined.is-loading::after { - border-color: transparent transparent #da0b00 #da0b00 !important; } - .button.is-danger.is-outlined.is-loading:hover::after, .button.is-danger.is-outlined.is-loading.is-hovered::after, .button.is-danger.is-outlined.is-loading:focus::after, .button.is-danger.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-danger.is-outlined[disabled], fieldset[disabled] .button.is-danger.is-outlined { - background-color: transparent; - border-color: #da0b00; - box-shadow: none; - color: #da0b00; } - .button.is-danger.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - .button.is-danger.is-inverted.is-outlined:hover, .button.is-danger.is-inverted.is-outlined.is-hovered, .button.is-danger.is-inverted.is-outlined:focus, .button.is-danger.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #da0b00; } - .button.is-danger.is-inverted.is-outlined.is-loading:hover::after, .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-danger.is-inverted.is-outlined.is-loading:focus::after, .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #da0b00 #da0b00 !important; } - .button.is-danger.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-danger.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - .button.is-small, #documenter .docs-sidebar form.docs-search > input.button { - border-radius: 2px; - font-size: 0.75rem; } - .button.is-normal { - font-size: 1rem; } - .button.is-medium { - font-size: 1.25rem; } - .button.is-large { - font-size: 1.5rem; } - .button[disabled], fieldset[disabled] .button { - background-color: white; - border-color: #dbdbdb; - box-shadow: none; - opacity: 0.5; } - .button.is-fullwidth { - display: flex; - width: 100%; } - .button.is-loading { - color: transparent !important; - pointer-events: none; } - .button.is-loading::after { - position: absolute; - left: calc(50% - (1em / 2)); - top: calc(50% - (1em / 2)); - position: absolute !important; } - .button.is-static { - background-color: whitesmoke; - border-color: #dbdbdb; - color: #6b6b6b; - box-shadow: none; - pointer-events: none; } - .button.is-rounded, #documenter .docs-sidebar form.docs-search > input.button { - border-radius: 290486px; - padding-left: 1em; - padding-right: 1em; } - -.buttons { - align-items: center; - display: flex; - flex-wrap: wrap; - justify-content: flex-start; } - .buttons .button { - margin-bottom: 0.5rem; } - .buttons .button:not(:last-child):not(.is-fullwidth) { - margin-right: 0.5rem; } - .buttons:last-child { - margin-bottom: -0.5rem; } - .buttons:not(:last-child) { - margin-bottom: 1rem; } - .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large) { - border-radius: 2px; - font-size: 0.75rem; } - .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large) { - font-size: 1.25rem; } - .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium) { - font-size: 1.5rem; } - .buttons.has-addons .button:not(:first-child) { - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - .buttons.has-addons .button:not(:last-child) { - border-bottom-right-radius: 0; - border-top-right-radius: 0; - margin-right: -1px; } - .buttons.has-addons .button:last-child { - margin-right: 0; } - .buttons.has-addons .button:hover, .buttons.has-addons .button.is-hovered { - z-index: 2; } - .buttons.has-addons .button:focus, .buttons.has-addons .button.is-focused, .buttons.has-addons .button:active, .buttons.has-addons .button.is-active, .buttons.has-addons .button.is-selected { - z-index: 3; } - .buttons.has-addons .button:focus:hover, .buttons.has-addons .button.is-focused:hover, .buttons.has-addons .button:active:hover, .buttons.has-addons .button.is-active:hover, .buttons.has-addons .button.is-selected:hover { - z-index: 4; } - .buttons.has-addons .button.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - .buttons.is-centered { - justify-content: center; } - .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth) { - margin-left: 0.25rem; - margin-right: 0.25rem; } - .buttons.is-right { - justify-content: flex-end; } - .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth) { - margin-left: 0.25rem; - margin-right: 0.25rem; } - -.container { - flex-grow: 1; - margin: 0 auto; - position: relative; - width: auto; } - @media screen and (min-width: 1056px) { - .container { - max-width: 992px; } - .container.is-fluid { - margin-left: 32px; - margin-right: 32px; - max-width: none; } } - @media screen and (max-width: 1215px) { - .container.is-widescreen { - max-width: 1152px; } } - @media screen and (max-width: 1407px) { - .container.is-fullhd { - max-width: 1344px; } } - @media screen and (min-width: 1216px) { - .container { - max-width: 1152px; } } - @media screen and (min-width: 1408px) { - .container { - max-width: 1344px; } } -.content li + li { - margin-top: 0.25em; } - -.content p:not(:last-child), -.content dl:not(:last-child), -.content ol:not(:last-child), -.content ul:not(:last-child), -.content blockquote:not(:last-child), -.content pre:not(:last-child), -.content table:not(:last-child) { - margin-bottom: 1em; } - -.content h1, -.content h2, -.content h3, -.content h4, -.content h5, -.content h6 { - color: #222222; - font-weight: 600; - line-height: 1.125; } - -.content h1 { - font-size: 2em; - margin-bottom: 0.5em; } - .content h1:not(:first-child) { - margin-top: 1em; } - -.content h2 { - font-size: 1.75em; - margin-bottom: 0.5714em; } - .content h2:not(:first-child) { - margin-top: 1.1428em; } - -.content h3 { - font-size: 1.5em; - margin-bottom: 0.6666em; } - .content h3:not(:first-child) { - margin-top: 1.3333em; } - -.content h4 { - font-size: 1.25em; - margin-bottom: 0.8em; } - -.content h5 { - font-size: 1.125em; - margin-bottom: 0.8888em; } - -.content h6 { - font-size: 1em; - margin-bottom: 1em; } - -.content blockquote { - background-color: whitesmoke; - border-left: 5px solid #dbdbdb; - padding: 1.25em 1.5em; } - -.content ol { - list-style-position: outside; - margin-left: 2em; - margin-top: 1em; } - .content ol:not([type]) { - list-style-type: decimal; } - .content ol.is-lower-alpha:not([type]) { - list-style-type: lower-alpha; } - .content ol.is-lower-roman:not([type]) { - list-style-type: lower-roman; } - .content ol.is-upper-alpha:not([type]) { - list-style-type: upper-alpha; } - .content ol.is-upper-roman:not([type]) { - list-style-type: upper-roman; } - -.content ul { - list-style: disc outside; - margin-left: 2em; - margin-top: 1em; } - .content ul ul { - list-style-type: circle; - margin-top: 0.5em; } - .content ul ul ul { - list-style-type: square; } - -.content dd { - margin-left: 2em; } - -.content figure { - margin-left: 2em; - margin-right: 2em; - text-align: center; } - .content figure:not(:first-child) { - margin-top: 2em; } - .content figure:not(:last-child) { - margin-bottom: 2em; } - .content figure img { - display: inline-block; } - .content figure figcaption { - font-style: italic; } - -.content pre { - -webkit-overflow-scrolling: touch; - overflow-x: auto; - padding: 0; - white-space: pre; - word-wrap: normal; } - -.content sup, -.content sub { - font-size: 75%; } - -.content table { - width: 100%; } - .content table td, - .content table th { - border: 1px solid #dbdbdb; - border-width: 0 0 1px; - padding: 0.5em 0.75em; - vertical-align: top; } - .content table th { - color: #222222; } - .content table th:not([align]) { - text-align: left; } - .content table thead td, - .content table thead th { - border-width: 0 0 2px; - color: #222222; } - .content table tfoot td, - .content table tfoot th { - border-width: 2px 0 0; - color: #222222; } - .content table tbody tr:last-child td, - .content table tbody tr:last-child th { - border-bottom-width: 0; } - -.content .tabs li + li { - margin-top: 0; } - -.content.is-small, #documenter .docs-sidebar form.docs-search > input.content { - font-size: 0.75rem; } - -.content.is-medium { - font-size: 1.25rem; } - -.content.is-large { - font-size: 1.5rem; } - -.icon { - align-items: center; - display: inline-flex; - justify-content: center; - height: 1.5rem; - width: 1.5rem; } - .icon.is-small, #documenter .docs-sidebar form.docs-search > input.icon { - height: 1rem; - width: 1rem; } - .icon.is-medium { - height: 2rem; - width: 2rem; } - .icon.is-large { - height: 3rem; - width: 3rem; } - -.image, #documenter .docs-sidebar .docs-logo > img { - display: block; - position: relative; } - .image img, #documenter .docs-sidebar .docs-logo > img img { - display: block; - height: auto; - width: 100%; } - .image img.is-rounded, #documenter .docs-sidebar .docs-logo > img img.is-rounded { - border-radius: 290486px; } - .image.is-square img, #documenter .docs-sidebar .docs-logo > img.is-square img, - .image.is-square .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-square .has-ratio, .image.is-1by1 img, #documenter .docs-sidebar .docs-logo > img.is-1by1 img, - .image.is-1by1 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-1by1 .has-ratio, .image.is-5by4 img, #documenter .docs-sidebar .docs-logo > img.is-5by4 img, - .image.is-5by4 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-5by4 .has-ratio, .image.is-4by3 img, #documenter .docs-sidebar .docs-logo > img.is-4by3 img, - .image.is-4by3 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-4by3 .has-ratio, .image.is-3by2 img, #documenter .docs-sidebar .docs-logo > img.is-3by2 img, - .image.is-3by2 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-3by2 .has-ratio, .image.is-5by3 img, #documenter .docs-sidebar .docs-logo > img.is-5by3 img, - .image.is-5by3 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-5by3 .has-ratio, .image.is-16by9 img, #documenter .docs-sidebar .docs-logo > img.is-16by9 img, - .image.is-16by9 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-16by9 .has-ratio, .image.is-2by1 img, #documenter .docs-sidebar .docs-logo > img.is-2by1 img, - .image.is-2by1 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-2by1 .has-ratio, .image.is-3by1 img, #documenter .docs-sidebar .docs-logo > img.is-3by1 img, - .image.is-3by1 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-3by1 .has-ratio, .image.is-4by5 img, #documenter .docs-sidebar .docs-logo > img.is-4by5 img, - .image.is-4by5 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-4by5 .has-ratio, .image.is-3by4 img, #documenter .docs-sidebar .docs-logo > img.is-3by4 img, - .image.is-3by4 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-3by4 .has-ratio, .image.is-2by3 img, #documenter .docs-sidebar .docs-logo > img.is-2by3 img, - .image.is-2by3 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-2by3 .has-ratio, .image.is-3by5 img, #documenter .docs-sidebar .docs-logo > img.is-3by5 img, - .image.is-3by5 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-3by5 .has-ratio, .image.is-9by16 img, #documenter .docs-sidebar .docs-logo > img.is-9by16 img, - .image.is-9by16 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-9by16 .has-ratio, .image.is-1by2 img, #documenter .docs-sidebar .docs-logo > img.is-1by2 img, - .image.is-1by2 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-1by2 .has-ratio, .image.is-1by3 img, #documenter .docs-sidebar .docs-logo > img.is-1by3 img, - .image.is-1by3 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-1by3 .has-ratio { - height: 100%; - width: 100%; } - .image.is-square, #documenter .docs-sidebar .docs-logo > img.is-square, .image.is-1by1, #documenter .docs-sidebar .docs-logo > img.is-1by1 { - padding-top: 100%; } - .image.is-5by4, #documenter .docs-sidebar .docs-logo > img.is-5by4 { - padding-top: 80%; } - .image.is-4by3, #documenter .docs-sidebar .docs-logo > img.is-4by3 { - padding-top: 75%; } - .image.is-3by2, #documenter .docs-sidebar .docs-logo > img.is-3by2 { - padding-top: 66.6666%; } - .image.is-5by3, #documenter .docs-sidebar .docs-logo > img.is-5by3 { - padding-top: 60%; } - .image.is-16by9, #documenter .docs-sidebar .docs-logo > img.is-16by9 { - padding-top: 56.25%; } - .image.is-2by1, #documenter .docs-sidebar .docs-logo > img.is-2by1 { - padding-top: 50%; } - .image.is-3by1, #documenter .docs-sidebar .docs-logo > img.is-3by1 { - padding-top: 33.3333%; } - .image.is-4by5, #documenter .docs-sidebar .docs-logo > img.is-4by5 { - padding-top: 125%; } - .image.is-3by4, #documenter .docs-sidebar .docs-logo > img.is-3by4 { - padding-top: 133.3333%; } - .image.is-2by3, #documenter .docs-sidebar .docs-logo > img.is-2by3 { - padding-top: 150%; } - .image.is-3by5, #documenter .docs-sidebar .docs-logo > img.is-3by5 { - padding-top: 166.6666%; } - .image.is-9by16, #documenter .docs-sidebar .docs-logo > img.is-9by16 { - padding-top: 177.7777%; } - .image.is-1by2, #documenter .docs-sidebar .docs-logo > img.is-1by2 { - padding-top: 200%; } - .image.is-1by3, #documenter .docs-sidebar .docs-logo > img.is-1by3 { - padding-top: 300%; } - .image.is-16x16, #documenter .docs-sidebar .docs-logo > img.is-16x16 { - height: 16px; - width: 16px; } - .image.is-24x24, #documenter .docs-sidebar .docs-logo > img.is-24x24 { - height: 24px; - width: 24px; } - .image.is-32x32, #documenter .docs-sidebar .docs-logo > img.is-32x32 { - height: 32px; - width: 32px; } - .image.is-48x48, #documenter .docs-sidebar .docs-logo > img.is-48x48 { - height: 48px; - width: 48px; } - .image.is-64x64, #documenter .docs-sidebar .docs-logo > img.is-64x64 { - height: 64px; - width: 64px; } - .image.is-96x96, #documenter .docs-sidebar .docs-logo > img.is-96x96 { - height: 96px; - width: 96px; } - .image.is-128x128, #documenter .docs-sidebar .docs-logo > img.is-128x128 { - height: 128px; - width: 128px; } - -.notification { - background-color: whitesmoke; - border-radius: 4px; - padding: 1.25rem 2.5rem 1.25rem 1.5rem; - position: relative; } - .notification a:not(.button):not(.dropdown-item) { - color: currentColor; - text-decoration: underline; } - .notification strong { - color: currentColor; } - .notification code, - .notification pre { - background: white; } - .notification pre code { - background: transparent; } - .notification > .delete { - position: absolute; - right: 0.5rem; - top: 0.5rem; } - .notification .title, - .notification .subtitle, - .notification .content { - color: currentColor; } - .notification.is-white { - background-color: white; - color: #0a0a0a; } - .notification.is-black { - background-color: #0a0a0a; - color: white; } - .notification.is-light { - background-color: whitesmoke; - color: #363636; } - .notification.is-dark, .content kbd.notification { - background-color: #363636; - color: whitesmoke; } - .notification.is-primary, .docstring > section > a.notification.docs-sourcelink { - background-color: #4eb5de; - color: #fff; } - .notification.is-link { - background-color: #2e63b8; - color: #fff; } - .notification.is-info { - background-color: #209cee; - color: #fff; } - .notification.is-success { - background-color: #22c35b; - color: #fff; } - .notification.is-warning { - background-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .notification.is-danger { - background-color: #da0b00; - color: #fff; } - -.progress { - -moz-appearance: none; - -webkit-appearance: none; - border: none; - border-radius: 290486px; - display: block; - height: 1rem; - overflow: hidden; - padding: 0; - width: 100%; } - .progress::-webkit-progress-bar { - background-color: #dbdbdb; } - .progress::-webkit-progress-value { - background-color: #222222; } - .progress::-moz-progress-bar { - background-color: #222222; } - .progress::-ms-fill { - background-color: #222222; - border: none; } - .progress.is-white::-webkit-progress-value { - background-color: white; } - .progress.is-white::-moz-progress-bar { - background-color: white; } - .progress.is-white::-ms-fill { - background-color: white; } - .progress.is-white:indeterminate { - background-image: linear-gradient(to right, white 30%, #dbdbdb 30%); } - .progress.is-black::-webkit-progress-value { - background-color: #0a0a0a; } - .progress.is-black::-moz-progress-bar { - background-color: #0a0a0a; } - .progress.is-black::-ms-fill { - background-color: #0a0a0a; } - .progress.is-black:indeterminate { - background-image: linear-gradient(to right, #0a0a0a 30%, #dbdbdb 30%); } - .progress.is-light::-webkit-progress-value { - background-color: whitesmoke; } - .progress.is-light::-moz-progress-bar { - background-color: whitesmoke; } - .progress.is-light::-ms-fill { - background-color: whitesmoke; } - .progress.is-light:indeterminate { - background-image: linear-gradient(to right, whitesmoke 30%, #dbdbdb 30%); } - .progress.is-dark::-webkit-progress-value, .content kbd.progress::-webkit-progress-value { - background-color: #363636; } - .progress.is-dark::-moz-progress-bar, .content kbd.progress::-moz-progress-bar { - background-color: #363636; } - .progress.is-dark::-ms-fill, .content kbd.progress::-ms-fill { - background-color: #363636; } - .progress.is-dark:indeterminate, .content kbd.progress:indeterminate { - background-image: linear-gradient(to right, #363636 30%, #dbdbdb 30%); } - .progress.is-primary::-webkit-progress-value, .docstring > section > a.progress.docs-sourcelink::-webkit-progress-value { - background-color: #4eb5de; } - .progress.is-primary::-moz-progress-bar, .docstring > section > a.progress.docs-sourcelink::-moz-progress-bar { - background-color: #4eb5de; } - .progress.is-primary::-ms-fill, .docstring > section > a.progress.docs-sourcelink::-ms-fill { - background-color: #4eb5de; } - .progress.is-primary:indeterminate, .docstring > section > a.progress.docs-sourcelink:indeterminate { - background-image: linear-gradient(to right, #4eb5de 30%, #dbdbdb 30%); } - .progress.is-link::-webkit-progress-value { - background-color: #2e63b8; } - .progress.is-link::-moz-progress-bar { - background-color: #2e63b8; } - .progress.is-link::-ms-fill { - background-color: #2e63b8; } - .progress.is-link:indeterminate { - background-image: linear-gradient(to right, #2e63b8 30%, #dbdbdb 30%); } - .progress.is-info::-webkit-progress-value { - background-color: #209cee; } - .progress.is-info::-moz-progress-bar { - background-color: #209cee; } - .progress.is-info::-ms-fill { - background-color: #209cee; } - .progress.is-info:indeterminate { - background-image: linear-gradient(to right, #209cee 30%, #dbdbdb 30%); } - .progress.is-success::-webkit-progress-value { - background-color: #22c35b; } - .progress.is-success::-moz-progress-bar { - background-color: #22c35b; } - .progress.is-success::-ms-fill { - background-color: #22c35b; } - .progress.is-success:indeterminate { - background-image: linear-gradient(to right, #22c35b 30%, #dbdbdb 30%); } - .progress.is-warning::-webkit-progress-value { - background-color: #ffdd57; } - .progress.is-warning::-moz-progress-bar { - background-color: #ffdd57; } - .progress.is-warning::-ms-fill { - background-color: #ffdd57; } - .progress.is-warning:indeterminate { - background-image: linear-gradient(to right, #ffdd57 30%, #dbdbdb 30%); } - .progress.is-danger::-webkit-progress-value { - background-color: #da0b00; } - .progress.is-danger::-moz-progress-bar { - background-color: #da0b00; } - .progress.is-danger::-ms-fill { - background-color: #da0b00; } - .progress.is-danger:indeterminate { - background-image: linear-gradient(to right, #da0b00 30%, #dbdbdb 30%); } - .progress:indeterminate { - animation-duration: 1.5s; - animation-iteration-count: infinite; - animation-name: moveIndeterminate; - animation-timing-function: linear; - background-color: #dbdbdb; - background-image: linear-gradient(to right, #222222 30%, #dbdbdb 30%); - background-position: top left; - background-repeat: no-repeat; - background-size: 150% 150%; } - .progress:indeterminate::-webkit-progress-bar { - background-color: transparent; } - .progress:indeterminate::-moz-progress-bar { - background-color: transparent; } - .progress.is-small, #documenter .docs-sidebar form.docs-search > input.progress { - height: 0.75rem; } - .progress.is-medium { - height: 1.25rem; } - .progress.is-large { - height: 1.5rem; } - -@keyframes moveIndeterminate { - from { - background-position: 200% 0; } - to { - background-position: -200% 0; } } - -.table { - background-color: white; - color: #363636; } - .table td, - .table th { - border: 1px solid #dbdbdb; - border-width: 0 0 1px; - padding: 0.5em 0.75em; - vertical-align: top; } - .table td.is-white, - .table th.is-white { - background-color: white; - border-color: white; - color: #0a0a0a; } - .table td.is-black, - .table th.is-black { - background-color: #0a0a0a; - border-color: #0a0a0a; - color: white; } - .table td.is-light, - .table th.is-light { - background-color: whitesmoke; - border-color: whitesmoke; - color: #363636; } - .table td.is-dark, - .table th.is-dark { - background-color: #363636; - border-color: #363636; - color: whitesmoke; } - .table td.is-primary, - .table th.is-primary { - background-color: #4eb5de; - border-color: #4eb5de; - color: #fff; } - .table td.is-link, - .table th.is-link { - background-color: #2e63b8; - border-color: #2e63b8; - color: #fff; } - .table td.is-info, - .table th.is-info { - background-color: #209cee; - border-color: #209cee; - color: #fff; } - .table td.is-success, - .table th.is-success { - background-color: #22c35b; - border-color: #22c35b; - color: #fff; } - .table td.is-warning, - .table th.is-warning { - background-color: #ffdd57; - border-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .table td.is-danger, - .table th.is-danger { - background-color: #da0b00; - border-color: #da0b00; - color: #fff; } - .table td.is-narrow, - .table th.is-narrow { - white-space: nowrap; - width: 1%; } - .table td.is-selected, - .table th.is-selected { - background-color: #4eb5de; - color: #fff; } - .table td.is-selected a, - .table td.is-selected strong, - .table th.is-selected a, - .table th.is-selected strong { - color: currentColor; } - .table th { - color: #222222; } - .table th:not([align]) { - text-align: left; } - .table tr.is-selected { - background-color: #4eb5de; - color: #fff; } - .table tr.is-selected a, - .table tr.is-selected strong { - color: currentColor; } - .table tr.is-selected td, - .table tr.is-selected th { - border-color: #fff; - color: currentColor; } - .table thead { - background-color: transparent; } - .table thead td, - .table thead th { - border-width: 0 0 2px; - color: #222222; } - .table tfoot { - background-color: transparent; } - .table tfoot td, - .table tfoot th { - border-width: 2px 0 0; - color: #222222; } - .table tbody { - background-color: transparent; } - .table tbody tr:last-child td, - .table tbody tr:last-child th { - border-bottom-width: 0; } - .table.is-bordered td, - .table.is-bordered th { - border-width: 1px; } - .table.is-bordered tr:last-child td, - .table.is-bordered tr:last-child th { - border-bottom-width: 1px; } - .table.is-fullwidth { - width: 100%; } - .table.is-hoverable tbody tr:not(.is-selected):hover { - background-color: #fafafa; } - .table.is-hoverable.is-striped tbody tr:not(.is-selected):hover { - background-color: #fafafa; } - .table.is-hoverable.is-striped tbody tr:not(.is-selected):hover:nth-child(even) { - background-color: whitesmoke; } - .table.is-narrow td, - .table.is-narrow th { - padding: 0.25em 0.5em; } - .table.is-striped tbody tr:not(.is-selected):nth-child(even) { - background-color: #fafafa; } - -.table-container { - -webkit-overflow-scrolling: touch; - overflow: auto; - overflow-y: hidden; - max-width: 100%; } - -.tags { - align-items: center; - display: flex; - flex-wrap: wrap; - justify-content: flex-start; } - .tags .tag, .tags .content kbd, .content .tags kbd, .tags .docstring > section > a.docs-sourcelink { - margin-bottom: 0.5rem; } - .tags .tag:not(:last-child), .tags .content kbd:not(:last-child), .content .tags kbd:not(:last-child), .tags .docstring > section > a.docs-sourcelink:not(:last-child) { - margin-right: 0.5rem; } - .tags:last-child { - margin-bottom: -0.5rem; } - .tags:not(:last-child) { - margin-bottom: 1rem; } - .tags.are-medium .tag:not(.is-normal):not(.is-large), .tags.are-medium .content kbd:not(.is-normal):not(.is-large), .content .tags.are-medium kbd:not(.is-normal):not(.is-large), .tags.are-medium .docstring > section > a.docs-sourcelink:not(.is-normal):not(.is-large) { - font-size: 1rem; } - .tags.are-large .tag:not(.is-normal):not(.is-medium), .tags.are-large .content kbd:not(.is-normal):not(.is-medium), .content .tags.are-large kbd:not(.is-normal):not(.is-medium), .tags.are-large .docstring > section > a.docs-sourcelink:not(.is-normal):not(.is-medium) { - font-size: 1.25rem; } - .tags.is-centered { - justify-content: center; } - .tags.is-centered .tag, .tags.is-centered .content kbd, .content .tags.is-centered kbd, .tags.is-centered .docstring > section > a.docs-sourcelink { - margin-right: 0.25rem; - margin-left: 0.25rem; } - .tags.is-right { - justify-content: flex-end; } - .tags.is-right .tag:not(:first-child), .tags.is-right .content kbd:not(:first-child), .content .tags.is-right kbd:not(:first-child), .tags.is-right .docstring > section > a.docs-sourcelink:not(:first-child) { - margin-left: 0.5rem; } - .tags.is-right .tag:not(:last-child), .tags.is-right .content kbd:not(:last-child), .content .tags.is-right kbd:not(:last-child), .tags.is-right .docstring > section > a.docs-sourcelink:not(:last-child) { - margin-right: 0; } - .tags.has-addons .tag, .tags.has-addons .content kbd, .content .tags.has-addons kbd, .tags.has-addons .docstring > section > a.docs-sourcelink { - margin-right: 0; } - .tags.has-addons .tag:not(:first-child), .tags.has-addons .content kbd:not(:first-child), .content .tags.has-addons kbd:not(:first-child), .tags.has-addons .docstring > section > a.docs-sourcelink:not(:first-child) { - margin-left: 0; - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - .tags.has-addons .tag:not(:last-child), .tags.has-addons .content kbd:not(:last-child), .content .tags.has-addons kbd:not(:last-child), .tags.has-addons .docstring > section > a.docs-sourcelink:not(:last-child) { - border-bottom-right-radius: 0; - border-top-right-radius: 0; } - -.tag:not(body), .content kbd:not(body), .docstring > section > a.docs-sourcelink:not(body) { - align-items: center; - background-color: whitesmoke; - border-radius: 4px; - color: #222222; - display: inline-flex; - font-size: 0.75rem; - height: 2em; - justify-content: center; - line-height: 1.5; - padding-left: 0.75em; - padding-right: 0.75em; - white-space: nowrap; } - .tag:not(body) .delete, .content kbd:not(body) .delete, .docstring > section > a.docs-sourcelink:not(body) .delete { - margin-left: 0.25rem; - margin-right: -0.375rem; } - .tag.is-white:not(body), .content kbd.is-white:not(body), .docstring > section > a.docs-sourcelink.is-white:not(body) { - background-color: white; - color: #0a0a0a; } - .tag.is-black:not(body), .content kbd.is-black:not(body), .docstring > section > a.docs-sourcelink.is-black:not(body) { - background-color: #0a0a0a; - color: white; } - .tag.is-light:not(body), .content kbd.is-light:not(body), .docstring > section > a.docs-sourcelink.is-light:not(body) { - background-color: whitesmoke; - color: #363636; } - .tag.is-dark:not(body), .content kbd:not(body), .docstring > section > a.docs-sourcelink.is-dark:not(body), .content .docstring > section > kbd:not(body) { - background-color: #363636; - color: whitesmoke; } - .tag.is-primary:not(body), .content kbd.is-primary:not(body), .docstring > section > a.docs-sourcelink:not(body) { - background-color: #4eb5de; - color: #fff; } - .tag.is-link:not(body), .content kbd.is-link:not(body), .docstring > section > a.docs-sourcelink.is-link:not(body) { - background-color: #2e63b8; - color: #fff; } - .tag.is-info:not(body), .content kbd.is-info:not(body), .docstring > section > a.docs-sourcelink.is-info:not(body) { - background-color: #209cee; - color: #fff; } - .tag.is-success:not(body), .content kbd.is-success:not(body), .docstring > section > a.docs-sourcelink.is-success:not(body) { - background-color: #22c35b; - color: #fff; } - .tag.is-warning:not(body), .content kbd.is-warning:not(body), .docstring > section > a.docs-sourcelink.is-warning:not(body) { - background-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .tag.is-danger:not(body), .content kbd.is-danger:not(body), .docstring > section > a.docs-sourcelink.is-danger:not(body) { - background-color: #da0b00; - color: #fff; } - .tag.is-normal:not(body), .content kbd.is-normal:not(body), .docstring > section > a.docs-sourcelink.is-normal:not(body) { - font-size: 0.75rem; } - .tag.is-medium:not(body), .content kbd.is-medium:not(body), .docstring > section > a.docs-sourcelink.is-medium:not(body) { - font-size: 1rem; } - .tag.is-large:not(body), .content kbd.is-large:not(body), .docstring > section > a.docs-sourcelink.is-large:not(body) { - font-size: 1.25rem; } - .tag:not(body) .icon:first-child:not(:last-child), .content kbd:not(body) .icon:first-child:not(:last-child), .docstring > section > a.docs-sourcelink:not(body) .icon:first-child:not(:last-child) { - margin-left: -0.375em; - margin-right: 0.1875em; } - .tag:not(body) .icon:last-child:not(:first-child), .content kbd:not(body) .icon:last-child:not(:first-child), .docstring > section > a.docs-sourcelink:not(body) .icon:last-child:not(:first-child) { - margin-left: 0.1875em; - margin-right: -0.375em; } - .tag:not(body) .icon:first-child:last-child, .content kbd:not(body) .icon:first-child:last-child, .docstring > section > a.docs-sourcelink:not(body) .icon:first-child:last-child { - margin-left: -0.375em; - margin-right: -0.375em; } - .tag.is-delete:not(body), .content kbd.is-delete:not(body), .docstring > section > a.docs-sourcelink.is-delete:not(body) { - margin-left: 1px; - padding: 0; - position: relative; - width: 2em; } - .tag.is-delete:not(body)::before, .content kbd.is-delete:not(body)::before, .docstring > section > a.docs-sourcelink.is-delete:not(body)::before, .tag.is-delete:not(body)::after, .content kbd.is-delete:not(body)::after, .docstring > section > a.docs-sourcelink.is-delete:not(body)::after { - background-color: currentColor; - content: ""; - display: block; - left: 50%; - position: absolute; - top: 50%; - transform: translateX(-50%) translateY(-50%) rotate(45deg); - transform-origin: center center; } - .tag.is-delete:not(body)::before, .content kbd.is-delete:not(body)::before, .docstring > section > a.docs-sourcelink.is-delete:not(body)::before { - height: 1px; - width: 50%; } - .tag.is-delete:not(body)::after, .content kbd.is-delete:not(body)::after, .docstring > section > a.docs-sourcelink.is-delete:not(body)::after { - height: 50%; - width: 1px; } - .tag.is-delete:not(body):hover, .content kbd.is-delete:not(body):hover, .docstring > section > a.docs-sourcelink.is-delete:not(body):hover, .tag.is-delete:not(body):focus, .content kbd.is-delete:not(body):focus, .docstring > section > a.docs-sourcelink.is-delete:not(body):focus { - background-color: #e8e8e8; } - .tag.is-delete:not(body):active, .content kbd.is-delete:not(body):active, .docstring > section > a.docs-sourcelink.is-delete:not(body):active { - background-color: #dbdbdb; } - .tag.is-rounded:not(body), #documenter .docs-sidebar form.docs-search > input:not(body), .content kbd.is-rounded:not(body), #documenter .docs-sidebar .content form.docs-search > input:not(body), .docstring > section > a.docs-sourcelink.is-rounded:not(body) { - border-radius: 290486px; } - -a.tag:hover, .docstring > section > a.docs-sourcelink:hover { - text-decoration: underline; } - -.title, -.subtitle { - word-break: break-word; } - .title em, - .title span, - .subtitle em, - .subtitle span { - font-weight: inherit; } - .title sub, - .subtitle sub { - font-size: 0.75em; } - .title sup, - .subtitle sup { - font-size: 0.75em; } - .title .tag, .title .content kbd, .content .title kbd, .title .docstring > section > a.docs-sourcelink, - .subtitle .tag, - .subtitle .content kbd, - .content .subtitle kbd, - .subtitle .docstring > section > a.docs-sourcelink { - vertical-align: middle; } - -.title { - color: #363636; - font-size: 2rem; - font-weight: 600; - line-height: 1.125; } - .title strong { - color: inherit; - font-weight: inherit; } - .title + .highlight { - margin-top: -0.75rem; } - .title:not(.is-spaced) + .subtitle { - margin-top: -1.25rem; } - .title.is-1 { - font-size: 3rem; } - .title.is-2 { - font-size: 2.5rem; } - .title.is-3 { - font-size: 2rem; } - .title.is-4 { - font-size: 1.5rem; } - .title.is-5 { - font-size: 1.25rem; } - .title.is-6 { - font-size: 1rem; } - .title.is-7 { - font-size: 0.75rem; } - -.subtitle { - color: #4a4a4a; - font-size: 1.25rem; - font-weight: 400; - line-height: 1.25; } - .subtitle strong { - color: #363636; - font-weight: 600; } - .subtitle:not(.is-spaced) + .title { - margin-top: -1.25rem; } - .subtitle.is-1 { - font-size: 3rem; } - .subtitle.is-2 { - font-size: 2.5rem; } - .subtitle.is-3 { - font-size: 2rem; } - .subtitle.is-4 { - font-size: 1.5rem; } - .subtitle.is-5 { - font-size: 1.25rem; } - .subtitle.is-6 { - font-size: 1rem; } - .subtitle.is-7 { - font-size: 0.75rem; } - -.heading { - display: block; - font-size: 11px; - letter-spacing: 1px; - margin-bottom: 5px; - text-transform: uppercase; } - -.highlight { - font-weight: 400; - max-width: 100%; - overflow: hidden; - padding: 0; } - .highlight pre { - overflow: auto; - max-width: 100%; } - -.number { - align-items: center; - background-color: whitesmoke; - border-radius: 290486px; - display: inline-flex; - font-size: 1.25rem; - height: 2em; - justify-content: center; - margin-right: 1.5rem; - min-width: 2.5em; - padding: 0.25rem 0.5rem; - text-align: center; - vertical-align: top; } - -.select select, .textarea, .input, #documenter .docs-sidebar form.docs-search > input { - background-color: white; - border-color: #dbdbdb; - border-radius: 4px; - color: #363636; } - .select select::-moz-placeholder, .textarea::-moz-placeholder, .input::-moz-placeholder, #documenter .docs-sidebar form.docs-search > input::-moz-placeholder { - color: rgba(54, 54, 54, 0.3); } - .select select::-webkit-input-placeholder, .textarea::-webkit-input-placeholder, .input::-webkit-input-placeholder, #documenter .docs-sidebar form.docs-search > input::-webkit-input-placeholder { - color: rgba(54, 54, 54, 0.3); } - .select select:-moz-placeholder, .textarea:-moz-placeholder, .input:-moz-placeholder, #documenter .docs-sidebar form.docs-search > input:-moz-placeholder { - color: rgba(54, 54, 54, 0.3); } - .select select:-ms-input-placeholder, .textarea:-ms-input-placeholder, .input:-ms-input-placeholder, #documenter .docs-sidebar form.docs-search > input:-ms-input-placeholder { - color: rgba(54, 54, 54, 0.3); } - .select select:hover, .textarea:hover, .input:hover, #documenter .docs-sidebar form.docs-search > input:hover, .select select.is-hovered, .is-hovered.textarea, .is-hovered.input, #documenter .docs-sidebar form.docs-search > input.is-hovered { - border-color: #b5b5b5; } - .select select:focus, .textarea:focus, .input:focus, #documenter .docs-sidebar form.docs-search > input:focus, .select select.is-focused, .is-focused.textarea, .is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .select select:active, .textarea:active, .input:active, #documenter .docs-sidebar form.docs-search > input:active, .select select.is-active, .is-active.textarea, .is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - border-color: #2e63b8; - box-shadow: 0 0 0 0.125em rgba(46, 99, 184, 0.25); } - .select select[disabled], .textarea[disabled], .input[disabled], #documenter .docs-sidebar form.docs-search > input[disabled], fieldset[disabled] .select select, .select fieldset[disabled] select, fieldset[disabled] .textarea, fieldset[disabled] .input, fieldset[disabled] #documenter .docs-sidebar form.docs-search > input, #documenter .docs-sidebar fieldset[disabled] form.docs-search > input { - background-color: whitesmoke; - border-color: whitesmoke; - box-shadow: none; - color: #6b6b6b; } - .select select[disabled]::-moz-placeholder, .textarea[disabled]::-moz-placeholder, .input[disabled]::-moz-placeholder, #documenter .docs-sidebar form.docs-search > input[disabled]::-moz-placeholder, fieldset[disabled] .select select::-moz-placeholder, .select fieldset[disabled] select::-moz-placeholder, fieldset[disabled] .textarea::-moz-placeholder, fieldset[disabled] .input::-moz-placeholder, fieldset[disabled] #documenter .docs-sidebar form.docs-search > input::-moz-placeholder, #documenter .docs-sidebar fieldset[disabled] form.docs-search > input::-moz-placeholder { - color: rgba(107, 107, 107, 0.3); } - .select select[disabled]::-webkit-input-placeholder, .textarea[disabled]::-webkit-input-placeholder, .input[disabled]::-webkit-input-placeholder, #documenter .docs-sidebar form.docs-search > input[disabled]::-webkit-input-placeholder, fieldset[disabled] .select select::-webkit-input-placeholder, .select fieldset[disabled] select::-webkit-input-placeholder, fieldset[disabled] .textarea::-webkit-input-placeholder, fieldset[disabled] .input::-webkit-input-placeholder, fieldset[disabled] #documenter .docs-sidebar form.docs-search > input::-webkit-input-placeholder, #documenter .docs-sidebar fieldset[disabled] form.docs-search > input::-webkit-input-placeholder { - color: rgba(107, 107, 107, 0.3); } - .select select[disabled]:-moz-placeholder, .textarea[disabled]:-moz-placeholder, .input[disabled]:-moz-placeholder, #documenter .docs-sidebar form.docs-search > input[disabled]:-moz-placeholder, fieldset[disabled] .select select:-moz-placeholder, .select fieldset[disabled] select:-moz-placeholder, fieldset[disabled] .textarea:-moz-placeholder, fieldset[disabled] .input:-moz-placeholder, fieldset[disabled] #documenter .docs-sidebar form.docs-search > input:-moz-placeholder, #documenter .docs-sidebar fieldset[disabled] form.docs-search > input:-moz-placeholder { - color: rgba(107, 107, 107, 0.3); } - .select select[disabled]:-ms-input-placeholder, .textarea[disabled]:-ms-input-placeholder, .input[disabled]:-ms-input-placeholder, #documenter .docs-sidebar form.docs-search > input[disabled]:-ms-input-placeholder, fieldset[disabled] .select select:-ms-input-placeholder, .select fieldset[disabled] select:-ms-input-placeholder, fieldset[disabled] .textarea:-ms-input-placeholder, fieldset[disabled] .input:-ms-input-placeholder, fieldset[disabled] #documenter .docs-sidebar form.docs-search > input:-ms-input-placeholder, #documenter .docs-sidebar fieldset[disabled] form.docs-search > input:-ms-input-placeholder { - color: rgba(107, 107, 107, 0.3); } - -.textarea, .input, #documenter .docs-sidebar form.docs-search > input { - box-shadow: inset 0 1px 2px rgba(10, 10, 10, 0.1); - max-width: 100%; - width: 100%; } - .textarea[readonly], .input[readonly], #documenter .docs-sidebar form.docs-search > input[readonly] { - box-shadow: none; } - .is-white.textarea, .is-white.input, #documenter .docs-sidebar form.docs-search > input.is-white { - border-color: white; } - .is-white.textarea:focus, .is-white.input:focus, #documenter .docs-sidebar form.docs-search > input.is-white:focus, .is-white.is-focused.textarea, .is-white.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-white.textarea:active, .is-white.input:active, #documenter .docs-sidebar form.docs-search > input.is-white:active, .is-white.is-active.textarea, .is-white.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); } - .is-black.textarea, .is-black.input, #documenter .docs-sidebar form.docs-search > input.is-black { - border-color: #0a0a0a; } - .is-black.textarea:focus, .is-black.input:focus, #documenter .docs-sidebar form.docs-search > input.is-black:focus, .is-black.is-focused.textarea, .is-black.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-black.textarea:active, .is-black.input:active, #documenter .docs-sidebar form.docs-search > input.is-black:active, .is-black.is-active.textarea, .is-black.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); } - .is-light.textarea, .is-light.input, #documenter .docs-sidebar form.docs-search > input.is-light { - border-color: whitesmoke; } - .is-light.textarea:focus, .is-light.input:focus, #documenter .docs-sidebar form.docs-search > input.is-light:focus, .is-light.is-focused.textarea, .is-light.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-light.textarea:active, .is-light.input:active, #documenter .docs-sidebar form.docs-search > input.is-light:active, .is-light.is-active.textarea, .is-light.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(245, 245, 245, 0.25); } - .is-dark.textarea, .content kbd.textarea, .is-dark.input, #documenter .docs-sidebar form.docs-search > input.is-dark, .content kbd.input { - border-color: #363636; } - .is-dark.textarea:focus, .content kbd.textarea:focus, .is-dark.input:focus, #documenter .docs-sidebar form.docs-search > input.is-dark:focus, .content kbd.input:focus, .is-dark.is-focused.textarea, .content kbd.is-focused.textarea, .is-dark.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .content kbd.is-focused.input, #documenter .docs-sidebar .content form.docs-search > input.is-focused, .is-dark.textarea:active, .content kbd.textarea:active, .is-dark.input:active, #documenter .docs-sidebar form.docs-search > input.is-dark:active, .content kbd.input:active, .is-dark.is-active.textarea, .content kbd.is-active.textarea, .is-dark.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active, .content kbd.is-active.input, #documenter .docs-sidebar .content form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(54, 54, 54, 0.25); } - .is-primary.textarea, .docstring > section > a.textarea.docs-sourcelink, .is-primary.input, #documenter .docs-sidebar form.docs-search > input.is-primary, .docstring > section > a.input.docs-sourcelink { - border-color: #4eb5de; } - .is-primary.textarea:focus, .docstring > section > a.textarea.docs-sourcelink:focus, .is-primary.input:focus, #documenter .docs-sidebar form.docs-search > input.is-primary:focus, .docstring > section > a.input.docs-sourcelink:focus, .is-primary.is-focused.textarea, .docstring > section > a.is-focused.textarea.docs-sourcelink, .is-primary.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .docstring > section > a.is-focused.input.docs-sourcelink, .is-primary.textarea:active, .docstring > section > a.textarea.docs-sourcelink:active, .is-primary.input:active, #documenter .docs-sidebar form.docs-search > input.is-primary:active, .docstring > section > a.input.docs-sourcelink:active, .is-primary.is-active.textarea, .docstring > section > a.is-active.textarea.docs-sourcelink, .is-primary.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active, .docstring > section > a.is-active.input.docs-sourcelink { - box-shadow: 0 0 0 0.125em rgba(78, 181, 222, 0.25); } - .is-link.textarea, .is-link.input, #documenter .docs-sidebar form.docs-search > input.is-link { - border-color: #2e63b8; } - .is-link.textarea:focus, .is-link.input:focus, #documenter .docs-sidebar form.docs-search > input.is-link:focus, .is-link.is-focused.textarea, .is-link.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-link.textarea:active, .is-link.input:active, #documenter .docs-sidebar form.docs-search > input.is-link:active, .is-link.is-active.textarea, .is-link.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(46, 99, 184, 0.25); } - .is-info.textarea, .is-info.input, #documenter .docs-sidebar form.docs-search > input.is-info { - border-color: #209cee; } - .is-info.textarea:focus, .is-info.input:focus, #documenter .docs-sidebar form.docs-search > input.is-info:focus, .is-info.is-focused.textarea, .is-info.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-info.textarea:active, .is-info.input:active, #documenter .docs-sidebar form.docs-search > input.is-info:active, .is-info.is-active.textarea, .is-info.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(32, 156, 238, 0.25); } - .is-success.textarea, .is-success.input, #documenter .docs-sidebar form.docs-search > input.is-success { - border-color: #22c35b; } - .is-success.textarea:focus, .is-success.input:focus, #documenter .docs-sidebar form.docs-search > input.is-success:focus, .is-success.is-focused.textarea, .is-success.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-success.textarea:active, .is-success.input:active, #documenter .docs-sidebar form.docs-search > input.is-success:active, .is-success.is-active.textarea, .is-success.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(34, 195, 91, 0.25); } - .is-warning.textarea, .is-warning.input, #documenter .docs-sidebar form.docs-search > input.is-warning { - border-color: #ffdd57; } - .is-warning.textarea:focus, .is-warning.input:focus, #documenter .docs-sidebar form.docs-search > input.is-warning:focus, .is-warning.is-focused.textarea, .is-warning.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-warning.textarea:active, .is-warning.input:active, #documenter .docs-sidebar form.docs-search > input.is-warning:active, .is-warning.is-active.textarea, .is-warning.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(255, 221, 87, 0.25); } - .is-danger.textarea, .is-danger.input, #documenter .docs-sidebar form.docs-search > input.is-danger { - border-color: #da0b00; } - .is-danger.textarea:focus, .is-danger.input:focus, #documenter .docs-sidebar form.docs-search > input.is-danger:focus, .is-danger.is-focused.textarea, .is-danger.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-danger.textarea:active, .is-danger.input:active, #documenter .docs-sidebar form.docs-search > input.is-danger:active, .is-danger.is-active.textarea, .is-danger.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(218, 11, 0, 0.25); } - .is-small.textarea, .is-small.input, #documenter .docs-sidebar form.docs-search > input { - border-radius: 2px; - font-size: 0.75rem; } - .is-medium.textarea, .is-medium.input, #documenter .docs-sidebar form.docs-search > input.is-medium { - font-size: 1.25rem; } - .is-large.textarea, .is-large.input, #documenter .docs-sidebar form.docs-search > input.is-large { - font-size: 1.5rem; } - .is-fullwidth.textarea, .is-fullwidth.input, #documenter .docs-sidebar form.docs-search > input.is-fullwidth { - display: block; - width: 100%; } - .is-inline.textarea, .is-inline.input, #documenter .docs-sidebar form.docs-search > input.is-inline { - display: inline; - width: auto; } - -.input.is-rounded, #documenter .docs-sidebar form.docs-search > input { - border-radius: 290486px; - padding-left: 1em; - padding-right: 1em; } - -.input.is-static, #documenter .docs-sidebar form.docs-search > input.is-static { - background-color: transparent; - border-color: transparent; - box-shadow: none; - padding-left: 0; - padding-right: 0; } - -.textarea { - display: block; - max-width: 100%; - min-width: 100%; - padding: 0.625em; - resize: vertical; } - .textarea:not([rows]) { - max-height: 600px; - min-height: 120px; } - .textarea[rows] { - height: initial; } - .textarea.has-fixed-size { - resize: none; } - -.radio, .checkbox { - cursor: pointer; - display: inline-block; - line-height: 1.25; - position: relative; } - .radio input, .checkbox input { - cursor: pointer; } - .radio:hover, .checkbox:hover { - color: #363636; } - .radio[disabled], .checkbox[disabled], fieldset[disabled] .radio, fieldset[disabled] .checkbox { - color: #6b6b6b; - cursor: not-allowed; } - -.radio + .radio { - margin-left: 0.5em; } - -.select { - display: inline-block; - max-width: 100%; - position: relative; - vertical-align: top; } - .select:not(.is-multiple) { - height: 2.25em; } - .select:not(.is-multiple):not(.is-loading)::after { - border-color: #2e63b8; - right: 1.125em; - z-index: 4; } - .select.is-rounded select, #documenter .docs-sidebar form.docs-search > input.select select { - border-radius: 290486px; - padding-left: 1em; } - .select select { - cursor: pointer; - display: block; - font-size: 1em; - max-width: 100%; - outline: none; } - .select select::-ms-expand { - display: none; } - .select select[disabled]:hover, fieldset[disabled] .select select:hover { - border-color: whitesmoke; } - .select select:not([multiple]) { - padding-right: 2.5em; } - .select select[multiple] { - height: auto; - padding: 0; } - .select select[multiple] option { - padding: 0.5em 1em; } - .select:not(.is-multiple):not(.is-loading):hover::after { - border-color: #363636; } - .select.is-white:not(:hover)::after { - border-color: white; } - .select.is-white select { - border-color: white; } - .select.is-white select:hover, .select.is-white select.is-hovered { - border-color: #f2f2f2; } - .select.is-white select:focus, .select.is-white select.is-focused, .select.is-white select:active, .select.is-white select.is-active { - box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); } - .select.is-black:not(:hover)::after { - border-color: #0a0a0a; } - .select.is-black select { - border-color: #0a0a0a; } - .select.is-black select:hover, .select.is-black select.is-hovered { - border-color: black; } - .select.is-black select:focus, .select.is-black select.is-focused, .select.is-black select:active, .select.is-black select.is-active { - box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); } - .select.is-light:not(:hover)::after { - border-color: whitesmoke; } - .select.is-light select { - border-color: whitesmoke; } - .select.is-light select:hover, .select.is-light select.is-hovered { - border-color: #e8e8e8; } - .select.is-light select:focus, .select.is-light select.is-focused, .select.is-light select:active, .select.is-light select.is-active { - box-shadow: 0 0 0 0.125em rgba(245, 245, 245, 0.25); } - .select.is-dark:not(:hover)::after, .content kbd.select:not(:hover)::after { - border-color: #363636; } - .select.is-dark select, .content kbd.select select { - border-color: #363636; } - .select.is-dark select:hover, .content kbd.select select:hover, .select.is-dark select.is-hovered, .content kbd.select select.is-hovered { - border-color: #292929; } - .select.is-dark select:focus, .content kbd.select select:focus, .select.is-dark select.is-focused, .content kbd.select select.is-focused, .select.is-dark select:active, .content kbd.select select:active, .select.is-dark select.is-active, .content kbd.select select.is-active { - box-shadow: 0 0 0 0.125em rgba(54, 54, 54, 0.25); } - .select.is-primary:not(:hover)::after, .docstring > section > a.select.docs-sourcelink:not(:hover)::after { - border-color: #4eb5de; } - .select.is-primary select, .docstring > section > a.select.docs-sourcelink select { - border-color: #4eb5de; } - .select.is-primary select:hover, .docstring > section > a.select.docs-sourcelink select:hover, .select.is-primary select.is-hovered, .docstring > section > a.select.docs-sourcelink select.is-hovered { - border-color: #39acda; } - .select.is-primary select:focus, .docstring > section > a.select.docs-sourcelink select:focus, .select.is-primary select.is-focused, .docstring > section > a.select.docs-sourcelink select.is-focused, .select.is-primary select:active, .docstring > section > a.select.docs-sourcelink select:active, .select.is-primary select.is-active, .docstring > section > a.select.docs-sourcelink select.is-active { - box-shadow: 0 0 0 0.125em rgba(78, 181, 222, 0.25); } - .select.is-link:not(:hover)::after { - border-color: #2e63b8; } - .select.is-link select { - border-color: #2e63b8; } - .select.is-link select:hover, .select.is-link select.is-hovered { - border-color: #2958a4; } - .select.is-link select:focus, .select.is-link select.is-focused, .select.is-link select:active, .select.is-link select.is-active { - box-shadow: 0 0 0 0.125em rgba(46, 99, 184, 0.25); } - .select.is-info:not(:hover)::after { - border-color: #209cee; } - .select.is-info select { - border-color: #209cee; } - .select.is-info select:hover, .select.is-info select.is-hovered { - border-color: #1190e3; } - .select.is-info select:focus, .select.is-info select.is-focused, .select.is-info select:active, .select.is-info select.is-active { - box-shadow: 0 0 0 0.125em rgba(32, 156, 238, 0.25); } - .select.is-success:not(:hover)::after { - border-color: #22c35b; } - .select.is-success select { - border-color: #22c35b; } - .select.is-success select:hover, .select.is-success select.is-hovered { - border-color: #1ead51; } - .select.is-success select:focus, .select.is-success select.is-focused, .select.is-success select:active, .select.is-success select.is-active { - box-shadow: 0 0 0 0.125em rgba(34, 195, 91, 0.25); } - .select.is-warning:not(:hover)::after { - border-color: #ffdd57; } - .select.is-warning select { - border-color: #ffdd57; } - .select.is-warning select:hover, .select.is-warning select.is-hovered { - border-color: #ffd83e; } - .select.is-warning select:focus, .select.is-warning select.is-focused, .select.is-warning select:active, .select.is-warning select.is-active { - box-shadow: 0 0 0 0.125em rgba(255, 221, 87, 0.25); } - .select.is-danger:not(:hover)::after { - border-color: #da0b00; } - .select.is-danger select { - border-color: #da0b00; } - .select.is-danger select:hover, .select.is-danger select.is-hovered { - border-color: #c10a00; } - .select.is-danger select:focus, .select.is-danger select.is-focused, .select.is-danger select:active, .select.is-danger select.is-active { - box-shadow: 0 0 0 0.125em rgba(218, 11, 0, 0.25); } - .select.is-small, #documenter .docs-sidebar form.docs-search > input.select { - border-radius: 2px; - font-size: 0.75rem; } - .select.is-medium { - font-size: 1.25rem; } - .select.is-large { - font-size: 1.5rem; } - .select.is-disabled::after { - border-color: #6b6b6b; } - .select.is-fullwidth { - width: 100%; } - .select.is-fullwidth select { - width: 100%; } - .select.is-loading::after { - margin-top: 0; - position: absolute; - right: 0.625em; - top: 0.625em; - transform: none; } - .select.is-loading.is-small:after, #documenter .docs-sidebar form.docs-search > input.is-loading:after { - font-size: 0.75rem; } - .select.is-loading.is-medium:after { - font-size: 1.25rem; } - .select.is-loading.is-large:after { - font-size: 1.5rem; } - -.file { - align-items: stretch; - display: flex; - justify-content: flex-start; - position: relative; } - .file.is-white .file-cta { - background-color: white; - border-color: transparent; - color: #0a0a0a; } - .file.is-white:hover .file-cta, .file.is-white.is-hovered .file-cta { - background-color: #f9f9f9; - border-color: transparent; - color: #0a0a0a; } - .file.is-white:focus .file-cta, .file.is-white.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(255, 255, 255, 0.25); - color: #0a0a0a; } - .file.is-white:active .file-cta, .file.is-white.is-active .file-cta { - background-color: #f2f2f2; - border-color: transparent; - color: #0a0a0a; } - .file.is-black .file-cta { - background-color: #0a0a0a; - border-color: transparent; - color: white; } - .file.is-black:hover .file-cta, .file.is-black.is-hovered .file-cta { - background-color: #040404; - border-color: transparent; - color: white; } - .file.is-black:focus .file-cta, .file.is-black.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(10, 10, 10, 0.25); - color: white; } - .file.is-black:active .file-cta, .file.is-black.is-active .file-cta { - background-color: black; - border-color: transparent; - color: white; } - .file.is-light .file-cta { - background-color: whitesmoke; - border-color: transparent; - color: #363636; } - .file.is-light:hover .file-cta, .file.is-light.is-hovered .file-cta { - background-color: #eeeeee; - border-color: transparent; - color: #363636; } - .file.is-light:focus .file-cta, .file.is-light.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(245, 245, 245, 0.25); - color: #363636; } - .file.is-light:active .file-cta, .file.is-light.is-active .file-cta { - background-color: #e8e8e8; - border-color: transparent; - color: #363636; } - .file.is-dark .file-cta, .content kbd.file .file-cta { - background-color: #363636; - border-color: transparent; - color: whitesmoke; } - .file.is-dark:hover .file-cta, .content kbd.file:hover .file-cta, .file.is-dark.is-hovered .file-cta, .content kbd.file.is-hovered .file-cta { - background-color: #2f2f2f; - border-color: transparent; - color: whitesmoke; } - .file.is-dark:focus .file-cta, .content kbd.file:focus .file-cta, .file.is-dark.is-focused .file-cta, .content kbd.file.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(54, 54, 54, 0.25); - color: whitesmoke; } - .file.is-dark:active .file-cta, .content kbd.file:active .file-cta, .file.is-dark.is-active .file-cta, .content kbd.file.is-active .file-cta { - background-color: #292929; - border-color: transparent; - color: whitesmoke; } - .file.is-primary .file-cta, .docstring > section > a.file.docs-sourcelink .file-cta { - background-color: #4eb5de; - border-color: transparent; - color: #fff; } - .file.is-primary:hover .file-cta, .docstring > section > a.file.docs-sourcelink:hover .file-cta, .file.is-primary.is-hovered .file-cta, .docstring > section > a.file.is-hovered.docs-sourcelink .file-cta { - background-color: #43b1dc; - border-color: transparent; - color: #fff; } - .file.is-primary:focus .file-cta, .docstring > section > a.file.docs-sourcelink:focus .file-cta, .file.is-primary.is-focused .file-cta, .docstring > section > a.file.is-focused.docs-sourcelink .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(78, 181, 222, 0.25); - color: #fff; } - .file.is-primary:active .file-cta, .docstring > section > a.file.docs-sourcelink:active .file-cta, .file.is-primary.is-active .file-cta, .docstring > section > a.file.is-active.docs-sourcelink .file-cta { - background-color: #39acda; - border-color: transparent; - color: #fff; } - .file.is-link .file-cta { - background-color: #2e63b8; - border-color: transparent; - color: #fff; } - .file.is-link:hover .file-cta, .file.is-link.is-hovered .file-cta { - background-color: #2b5eae; - border-color: transparent; - color: #fff; } - .file.is-link:focus .file-cta, .file.is-link.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(46, 99, 184, 0.25); - color: #fff; } - .file.is-link:active .file-cta, .file.is-link.is-active .file-cta { - background-color: #2958a4; - border-color: transparent; - color: #fff; } - .file.is-info .file-cta { - background-color: #209cee; - border-color: transparent; - color: #fff; } - .file.is-info:hover .file-cta, .file.is-info.is-hovered .file-cta { - background-color: #1497ed; - border-color: transparent; - color: #fff; } - .file.is-info:focus .file-cta, .file.is-info.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(32, 156, 238, 0.25); - color: #fff; } - .file.is-info:active .file-cta, .file.is-info.is-active .file-cta { - background-color: #1190e3; - border-color: transparent; - color: #fff; } - .file.is-success .file-cta { - background-color: #22c35b; - border-color: transparent; - color: #fff; } - .file.is-success:hover .file-cta, .file.is-success.is-hovered .file-cta { - background-color: #20b856; - border-color: transparent; - color: #fff; } - .file.is-success:focus .file-cta, .file.is-success.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(34, 195, 91, 0.25); - color: #fff; } - .file.is-success:active .file-cta, .file.is-success.is-active .file-cta { - background-color: #1ead51; - border-color: transparent; - color: #fff; } - .file.is-warning .file-cta { - background-color: #ffdd57; - border-color: transparent; - color: rgba(0, 0, 0, 0.7); } - .file.is-warning:hover .file-cta, .file.is-warning.is-hovered .file-cta { - background-color: #ffda4a; - border-color: transparent; - color: rgba(0, 0, 0, 0.7); } - .file.is-warning:focus .file-cta, .file.is-warning.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(255, 221, 87, 0.25); - color: rgba(0, 0, 0, 0.7); } - .file.is-warning:active .file-cta, .file.is-warning.is-active .file-cta { - background-color: #ffd83e; - border-color: transparent; - color: rgba(0, 0, 0, 0.7); } - .file.is-danger .file-cta { - background-color: #da0b00; - border-color: transparent; - color: #fff; } - .file.is-danger:hover .file-cta, .file.is-danger.is-hovered .file-cta { - background-color: #cd0a00; - border-color: transparent; - color: #fff; } - .file.is-danger:focus .file-cta, .file.is-danger.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(218, 11, 0, 0.25); - color: #fff; } - .file.is-danger:active .file-cta, .file.is-danger.is-active .file-cta { - background-color: #c10a00; - border-color: transparent; - color: #fff; } - .file.is-small, #documenter .docs-sidebar form.docs-search > input.file { - font-size: 0.75rem; } - .file.is-medium { - font-size: 1.25rem; } - .file.is-medium .file-icon .fa { - font-size: 21px; } - .file.is-large { - font-size: 1.5rem; } - .file.is-large .file-icon .fa { - font-size: 28px; } - .file.has-name .file-cta { - border-bottom-right-radius: 0; - border-top-right-radius: 0; } - .file.has-name .file-name { - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - .file.has-name.is-empty .file-cta { - border-radius: 4px; } - .file.has-name.is-empty .file-name { - display: none; } - .file.is-boxed .file-label { - flex-direction: column; } - .file.is-boxed .file-cta { - flex-direction: column; - height: auto; - padding: 1em 3em; } - .file.is-boxed .file-name { - border-width: 0 1px 1px; } - .file.is-boxed .file-icon { - height: 1.5em; - width: 1.5em; } - .file.is-boxed .file-icon .fa { - font-size: 21px; } - .file.is-boxed.is-small .file-icon .fa, #documenter .docs-sidebar form.docs-search > input.is-boxed .file-icon .fa { - font-size: 14px; } - .file.is-boxed.is-medium .file-icon .fa { - font-size: 28px; } - .file.is-boxed.is-large .file-icon .fa { - font-size: 35px; } - .file.is-boxed.has-name .file-cta { - border-radius: 4px 4px 0 0; } - .file.is-boxed.has-name .file-name { - border-radius: 0 0 4px 4px; - border-width: 0 1px 1px; } - .file.is-centered { - justify-content: center; } - .file.is-fullwidth .file-label { - width: 100%; } - .file.is-fullwidth .file-name { - flex-grow: 1; - max-width: none; } - .file.is-right { - justify-content: flex-end; } - .file.is-right .file-cta { - border-radius: 0 4px 4px 0; } - .file.is-right .file-name { - border-radius: 4px 0 0 4px; - border-width: 1px 0 1px 1px; - order: -1; } - -.file-label { - align-items: stretch; - display: flex; - cursor: pointer; - justify-content: flex-start; - overflow: hidden; - position: relative; } - .file-label:hover .file-cta { - background-color: #eeeeee; - color: #363636; } - .file-label:hover .file-name { - border-color: #d5d5d5; } - .file-label:active .file-cta { - background-color: #e8e8e8; - color: #363636; } - .file-label:active .file-name { - border-color: #cfcfcf; } - -.file-input { - height: 100%; - left: 0; - opacity: 0; - outline: none; - position: absolute; - top: 0; - width: 100%; } - -.file-cta, -.file-name { - border-color: #dbdbdb; - border-radius: 4px; - font-size: 1em; - padding-left: 1em; - padding-right: 1em; - white-space: nowrap; } - -.file-cta { - background-color: whitesmoke; - color: #4a4a4a; } - -.file-name { - border-color: #dbdbdb; - border-style: solid; - border-width: 1px 1px 1px 0; - display: block; - max-width: 16em; - overflow: hidden; - text-align: left; - text-overflow: ellipsis; } - -.file-icon { - align-items: center; - display: flex; - height: 1em; - justify-content: center; - margin-right: 0.5em; - width: 1em; } - .file-icon .fa { - font-size: 14px; } - -.label { - color: #363636; - display: block; - font-size: 1rem; - font-weight: 700; } - .label:not(:last-child) { - margin-bottom: 0.5em; } - .label.is-small, #documenter .docs-sidebar form.docs-search > input.label { - font-size: 0.75rem; } - .label.is-medium { - font-size: 1.25rem; } - .label.is-large { - font-size: 1.5rem; } - -.help { - display: block; - font-size: 0.75rem; - margin-top: 0.25rem; } - .help.is-white { - color: white; } - .help.is-black { - color: #0a0a0a; } - .help.is-light { - color: whitesmoke; } - .help.is-dark, .content kbd.help { - color: #363636; } - .help.is-primary, .docstring > section > a.help.docs-sourcelink { - color: #4eb5de; } - .help.is-link { - color: #2e63b8; } - .help.is-info { - color: #209cee; } - .help.is-success { - color: #22c35b; } - .help.is-warning { - color: #ffdd57; } - .help.is-danger { - color: #da0b00; } - -.field:not(:last-child) { - margin-bottom: 0.75rem; } - -.field.has-addons { - display: flex; - justify-content: flex-start; } - .field.has-addons .control:not(:last-child) { - margin-right: -1px; } - .field.has-addons .control:not(:first-child):not(:last-child) .button, - .field.has-addons .control:not(:first-child):not(:last-child) .input, - .field.has-addons .control:not(:first-child):not(:last-child) #documenter .docs-sidebar form.docs-search > input, - #documenter .docs-sidebar .field.has-addons .control:not(:first-child):not(:last-child) form.docs-search > input, - .field.has-addons .control:not(:first-child):not(:last-child) .select select { - border-radius: 0; } - .field.has-addons .control:first-child:not(:only-child) .button, - .field.has-addons .control:first-child:not(:only-child) .input, - .field.has-addons .control:first-child:not(:only-child) #documenter .docs-sidebar form.docs-search > input, - #documenter .docs-sidebar .field.has-addons .control:first-child:not(:only-child) form.docs-search > input, - .field.has-addons .control:first-child:not(:only-child) .select select { - border-bottom-right-radius: 0; - border-top-right-radius: 0; } - .field.has-addons .control:last-child:not(:only-child) .button, - .field.has-addons .control:last-child:not(:only-child) .input, - .field.has-addons .control:last-child:not(:only-child) #documenter .docs-sidebar form.docs-search > input, - #documenter .docs-sidebar .field.has-addons .control:last-child:not(:only-child) form.docs-search > input, - .field.has-addons .control:last-child:not(:only-child) .select select { - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - .field.has-addons .control .button:not([disabled]):hover, .field.has-addons .control .button.is-hovered:not([disabled]), - .field.has-addons .control .input:not([disabled]):hover, - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):hover, - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):hover, - .field.has-addons .control .input.is-hovered:not([disabled]), - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-hovered:not([disabled]), - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-hovered:not([disabled]), - .field.has-addons .control .select select:not([disabled]):hover, - .field.has-addons .control .select select.is-hovered:not([disabled]) { - z-index: 2; } - .field.has-addons .control .button:not([disabled]):focus, .field.has-addons .control .button.is-focused:not([disabled]), .field.has-addons .control .button:not([disabled]):active, .field.has-addons .control .button.is-active:not([disabled]), - .field.has-addons .control .input:not([disabled]):focus, - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):focus, - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):focus, - .field.has-addons .control .input.is-focused:not([disabled]), - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-focused:not([disabled]), - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-focused:not([disabled]), - .field.has-addons .control .input:not([disabled]):active, - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):active, - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):active, - .field.has-addons .control .input.is-active:not([disabled]), - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-active:not([disabled]), - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-active:not([disabled]), - .field.has-addons .control .select select:not([disabled]):focus, - .field.has-addons .control .select select.is-focused:not([disabled]), - .field.has-addons .control .select select:not([disabled]):active, - .field.has-addons .control .select select.is-active:not([disabled]) { - z-index: 3; } - .field.has-addons .control .button:not([disabled]):focus:hover, .field.has-addons .control .button.is-focused:not([disabled]):hover, .field.has-addons .control .button:not([disabled]):active:hover, .field.has-addons .control .button.is-active:not([disabled]):hover, - .field.has-addons .control .input:not([disabled]):focus:hover, - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):focus:hover, - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):focus:hover, - .field.has-addons .control .input.is-focused:not([disabled]):hover, - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-focused:not([disabled]):hover, - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-focused:not([disabled]):hover, - .field.has-addons .control .input:not([disabled]):active:hover, - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):active:hover, - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):active:hover, - .field.has-addons .control .input.is-active:not([disabled]):hover, - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-active:not([disabled]):hover, - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-active:not([disabled]):hover, - .field.has-addons .control .select select:not([disabled]):focus:hover, - .field.has-addons .control .select select.is-focused:not([disabled]):hover, - .field.has-addons .control .select select:not([disabled]):active:hover, - .field.has-addons .control .select select.is-active:not([disabled]):hover { - z-index: 4; } - .field.has-addons .control.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - .field.has-addons.has-addons-centered { - justify-content: center; } - .field.has-addons.has-addons-right { - justify-content: flex-end; } - .field.has-addons.has-addons-fullwidth .control { - flex-grow: 1; - flex-shrink: 0; } - -.field.is-grouped { - display: flex; - justify-content: flex-start; } - .field.is-grouped > .control { - flex-shrink: 0; } - .field.is-grouped > .control:not(:last-child) { - margin-bottom: 0; - margin-right: 0.75rem; } - .field.is-grouped > .control.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - .field.is-grouped.is-grouped-centered { - justify-content: center; } - .field.is-grouped.is-grouped-right { - justify-content: flex-end; } - .field.is-grouped.is-grouped-multiline { - flex-wrap: wrap; } - .field.is-grouped.is-grouped-multiline > .control:last-child, .field.is-grouped.is-grouped-multiline > .control:not(:last-child) { - margin-bottom: 0.75rem; } - .field.is-grouped.is-grouped-multiline:last-child { - margin-bottom: -0.75rem; } - .field.is-grouped.is-grouped-multiline:not(:last-child) { - margin-bottom: 0; } - -@media screen and (min-width: 769px), print { - .field.is-horizontal { - display: flex; } } - -.field-label .label { - font-size: inherit; } - -@media screen and (max-width: 768px) { - .field-label { - margin-bottom: 0.5rem; } } - -@media screen and (min-width: 769px), print { - .field-label { - flex-basis: 0; - flex-grow: 1; - flex-shrink: 0; - margin-right: 1.5rem; - text-align: right; } - .field-label.is-small, #documenter .docs-sidebar form.docs-search > input.field-label { - font-size: 0.75rem; - padding-top: 0.375em; } - .field-label.is-normal { - padding-top: 0.375em; } - .field-label.is-medium { - font-size: 1.25rem; - padding-top: 0.375em; } - .field-label.is-large { - font-size: 1.5rem; - padding-top: 0.375em; } } - -.field-body .field .field { - margin-bottom: 0; } - -@media screen and (min-width: 769px), print { - .field-body { - display: flex; - flex-basis: 0; - flex-grow: 5; - flex-shrink: 1; } - .field-body .field { - margin-bottom: 0; } - .field-body > .field { - flex-shrink: 1; } - .field-body > .field:not(.is-narrow) { - flex-grow: 1; } - .field-body > .field:not(:last-child) { - margin-right: 0.75rem; } } - -.control { - box-sizing: border-box; - clear: both; - font-size: 1rem; - position: relative; - text-align: left; } - .control.has-icons-left .input:focus ~ .icon, .control.has-icons-left #documenter .docs-sidebar form.docs-search > input:focus ~ .icon, #documenter .docs-sidebar .control.has-icons-left form.docs-search > input:focus ~ .icon, - .control.has-icons-left .select:focus ~ .icon, .control.has-icons-right .input:focus ~ .icon, .control.has-icons-right #documenter .docs-sidebar form.docs-search > input:focus ~ .icon, #documenter .docs-sidebar .control.has-icons-right form.docs-search > input:focus ~ .icon, - .control.has-icons-right .select:focus ~ .icon { - color: #6b6b6b; } - .control.has-icons-left .input.is-small ~ .icon, .control.has-icons-left #documenter .docs-sidebar form.docs-search > input ~ .icon, #documenter .docs-sidebar .control.has-icons-left form.docs-search > input ~ .icon, - .control.has-icons-left .select.is-small ~ .icon, .control.has-icons-right .input.is-small ~ .icon, .control.has-icons-right #documenter .docs-sidebar form.docs-search > input ~ .icon, #documenter .docs-sidebar .control.has-icons-right form.docs-search > input ~ .icon, - .control.has-icons-right .select.is-small ~ .icon { - font-size: 0.75rem; } - .control.has-icons-left .input.is-medium ~ .icon, .control.has-icons-left #documenter .docs-sidebar form.docs-search > input.is-medium ~ .icon, #documenter .docs-sidebar .control.has-icons-left form.docs-search > input.is-medium ~ .icon, - .control.has-icons-left .select.is-medium ~ .icon, .control.has-icons-right .input.is-medium ~ .icon, .control.has-icons-right #documenter .docs-sidebar form.docs-search > input.is-medium ~ .icon, #documenter .docs-sidebar .control.has-icons-right form.docs-search > input.is-medium ~ .icon, - .control.has-icons-right .select.is-medium ~ .icon { - font-size: 1.25rem; } - .control.has-icons-left .input.is-large ~ .icon, .control.has-icons-left #documenter .docs-sidebar form.docs-search > input.is-large ~ .icon, #documenter .docs-sidebar .control.has-icons-left form.docs-search > input.is-large ~ .icon, - .control.has-icons-left .select.is-large ~ .icon, .control.has-icons-right .input.is-large ~ .icon, .control.has-icons-right #documenter .docs-sidebar form.docs-search > input.is-large ~ .icon, #documenter .docs-sidebar .control.has-icons-right form.docs-search > input.is-large ~ .icon, - .control.has-icons-right .select.is-large ~ .icon { - font-size: 1.5rem; } - .control.has-icons-left .icon, .control.has-icons-right .icon { - color: #dbdbdb; - height: 2.25em; - pointer-events: none; - position: absolute; - top: 0; - width: 2.25em; - z-index: 4; } - .control.has-icons-left .input, .control.has-icons-left #documenter .docs-sidebar form.docs-search > input, #documenter .docs-sidebar .control.has-icons-left form.docs-search > input, - .control.has-icons-left .select select { - padding-left: 2.25em; } - .control.has-icons-left .icon.is-left { - left: 0; } - .control.has-icons-right .input, .control.has-icons-right #documenter .docs-sidebar form.docs-search > input, #documenter .docs-sidebar .control.has-icons-right form.docs-search > input, - .control.has-icons-right .select select { - padding-right: 2.25em; } - .control.has-icons-right .icon.is-right { - right: 0; } - .control.is-loading::after { - position: absolute !important; - right: 0.625em; - top: 0.625em; - z-index: 4; } - .control.is-loading.is-small:after, #documenter .docs-sidebar form.docs-search > input.is-loading:after { - font-size: 0.75rem; } - .control.is-loading.is-medium:after { - font-size: 1.25rem; } - .control.is-loading.is-large:after { - font-size: 1.5rem; } - -.breadcrumb { - font-size: 1rem; - white-space: nowrap; } - .breadcrumb a { - align-items: center; - color: #2e63b8; - display: flex; - justify-content: center; - padding: 0 0.75em; } - .breadcrumb a:hover { - color: #363636; } - .breadcrumb li { - align-items: center; - display: flex; } - .breadcrumb li:first-child a { - padding-left: 0; } - .breadcrumb li.is-active a { - color: #222222; - cursor: default; - pointer-events: none; } - .breadcrumb li + li::before { - color: #b5b5b5; - content: "\0002f"; } - .breadcrumb ul, - .breadcrumb ol { - align-items: flex-start; - display: flex; - flex-wrap: wrap; - justify-content: flex-start; } - .breadcrumb .icon:first-child { - margin-right: 0.5em; } - .breadcrumb .icon:last-child { - margin-left: 0.5em; } - .breadcrumb.is-centered ol, - .breadcrumb.is-centered ul { - justify-content: center; } - .breadcrumb.is-right ol, - .breadcrumb.is-right ul { - justify-content: flex-end; } - .breadcrumb.is-small, #documenter .docs-sidebar form.docs-search > input.breadcrumb { - font-size: 0.75rem; } - .breadcrumb.is-medium { - font-size: 1.25rem; } - .breadcrumb.is-large { - font-size: 1.5rem; } - .breadcrumb.has-arrow-separator li + li::before { - content: "\02192"; } - .breadcrumb.has-bullet-separator li + li::before { - content: "\02022"; } - .breadcrumb.has-dot-separator li + li::before { - content: "\000b7"; } - .breadcrumb.has-succeeds-separator li + li::before { - content: "\0227B"; } - -.card { - background-color: white; - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); - color: #222222; - max-width: 100%; - position: relative; } - -.card-header { - background-color: transparent; - align-items: stretch; - box-shadow: 0 1px 2px rgba(10, 10, 10, 0.1); - display: flex; } - -.card-header-title { - align-items: center; - color: #222222; - display: flex; - flex-grow: 1; - font-weight: 700; - padding: 0.75rem; } - .card-header-title.is-centered { - justify-content: center; } - -.card-header-icon { - align-items: center; - cursor: pointer; - display: flex; - justify-content: center; - padding: 0.75rem; } - -.card-image { - display: block; - position: relative; } - -.card-content { - background-color: transparent; - padding: 1.5rem; } - -.card-footer { - background-color: transparent; - border-top: 1px solid #dbdbdb; - align-items: stretch; - display: flex; } - -.card-footer-item { - align-items: center; - display: flex; - flex-basis: 0; - flex-grow: 1; - flex-shrink: 0; - justify-content: center; - padding: 0.75rem; } - .card-footer-item:not(:last-child) { - border-right: 1px solid #dbdbdb; } - -.card .media:not(:last-child) { - margin-bottom: 1.5rem; } - -.dropdown { - display: inline-flex; - position: relative; - vertical-align: top; } - .dropdown.is-active .dropdown-menu, .dropdown.is-hoverable:hover .dropdown-menu { - display: block; } - .dropdown.is-right .dropdown-menu { - left: auto; - right: 0; } - .dropdown.is-up .dropdown-menu { - bottom: 100%; - padding-bottom: 4px; - padding-top: initial; - top: auto; } - -.dropdown-menu { - display: none; - left: 0; - min-width: 12rem; - padding-top: 4px; - position: absolute; - top: 100%; - z-index: 20; } - -.dropdown-content { - background-color: white; - border-radius: 4px; - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); - padding-bottom: 0.5rem; - padding-top: 0.5rem; } - -.dropdown-item { - color: #4a4a4a; - display: block; - font-size: 0.875rem; - line-height: 1.5; - padding: 0.375rem 1rem; - position: relative; } - -a.dropdown-item, -button.dropdown-item { - padding-right: 3rem; - text-align: left; - white-space: nowrap; - width: 100%; } - a.dropdown-item:hover, - button.dropdown-item:hover { - background-color: whitesmoke; - color: #0a0a0a; } - a.dropdown-item.is-active, - button.dropdown-item.is-active { - background-color: #2e63b8; - color: #fff; } - -.dropdown-divider { - background-color: #dbdbdb; - border: none; - display: block; - height: 1px; - margin: 0.5rem 0; } - -.level { - align-items: center; - justify-content: space-between; } - .level code { - border-radius: 4px; } - .level img { - display: inline-block; - vertical-align: top; } - .level.is-mobile { - display: flex; } - .level.is-mobile .level-left, - .level.is-mobile .level-right { - display: flex; } - .level.is-mobile .level-left + .level-right { - margin-top: 0; } - .level.is-mobile .level-item:not(:last-child) { - margin-bottom: 0; - margin-right: 0.75rem; } - .level.is-mobile .level-item:not(.is-narrow) { - flex-grow: 1; } - @media screen and (min-width: 769px), print { - .level { - display: flex; } - .level > .level-item:not(.is-narrow) { - flex-grow: 1; } } -.level-item { - align-items: center; - display: flex; - flex-basis: auto; - flex-grow: 0; - flex-shrink: 0; - justify-content: center; } - .level-item .title, - .level-item .subtitle { - margin-bottom: 0; } - @media screen and (max-width: 768px) { - .level-item:not(:last-child) { - margin-bottom: 0.75rem; } } -.level-left, -.level-right { - flex-basis: auto; - flex-grow: 0; - flex-shrink: 0; } - .level-left .level-item.is-flexible, - .level-right .level-item.is-flexible { - flex-grow: 1; } - @media screen and (min-width: 769px), print { - .level-left .level-item:not(:last-child), - .level-right .level-item:not(:last-child) { - margin-right: 0.75rem; } } -.level-left { - align-items: center; - justify-content: flex-start; } - @media screen and (max-width: 768px) { - .level-left + .level-right { - margin-top: 1.5rem; } } - @media screen and (min-width: 769px), print { - .level-left { - display: flex; } } -.level-right { - align-items: center; - justify-content: flex-end; } - @media screen and (min-width: 769px), print { - .level-right { - display: flex; } } -.list { - background-color: white; - border-radius: 4px; - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); } - -.list-item { - display: block; - padding: 0.5em 1em; } - .list-item:not(a) { - color: #222222; } - .list-item:first-child { - border-top-left-radius: 4px; - border-top-right-radius: 4px; } - .list-item:last-child { - border-bottom-left-radius: 4px; - border-bottom-right-radius: 4px; } - .list-item:not(:last-child) { - border-bottom: 1px solid #dbdbdb; } - .list-item.is-active { - background-color: #2e63b8; - color: #fff; } - -a.list-item { - background-color: whitesmoke; - cursor: pointer; } - -.media { - align-items: flex-start; - display: flex; - text-align: left; } - .media .content:not(:last-child) { - margin-bottom: 0.75rem; } - .media .media { - border-top: 1px solid rgba(219, 219, 219, 0.5); - display: flex; - padding-top: 0.75rem; } - .media .media .content:not(:last-child), - .media .media .control:not(:last-child) { - margin-bottom: 0.5rem; } - .media .media .media { - padding-top: 0.5rem; } - .media .media .media + .media { - margin-top: 0.5rem; } - .media + .media { - border-top: 1px solid rgba(219, 219, 219, 0.5); - margin-top: 1rem; - padding-top: 1rem; } - .media.is-large + .media { - margin-top: 1.5rem; - padding-top: 1.5rem; } - -.media-left, -.media-right { - flex-basis: auto; - flex-grow: 0; - flex-shrink: 0; } - -.media-left { - margin-right: 1rem; } - -.media-right { - margin-left: 1rem; } - -.media-content { - flex-basis: auto; - flex-grow: 1; - flex-shrink: 1; - text-align: left; } - -@media screen and (max-width: 768px) { - .media-content { - overflow-x: auto; } } - -.menu { - font-size: 1rem; } - .menu.is-small, #documenter .docs-sidebar form.docs-search > input.menu { - font-size: 0.75rem; } - .menu.is-medium { - font-size: 1.25rem; } - .menu.is-large { - font-size: 1.5rem; } - -.menu-list { - line-height: 1.25; } - .menu-list a { - border-radius: 2px; - color: #222222; - display: block; - padding: 0.5em 0.75em; } - .menu-list a:hover { - background-color: whitesmoke; - color: #222222; } - .menu-list a.is-active { - background-color: #2e63b8; - color: #fff; } - .menu-list li ul { - border-left: 1px solid #dbdbdb; - margin: 0.75em; - padding-left: 0.75em; } - -.menu-label { - color: #6b6b6b; - font-size: 0.75em; - letter-spacing: 0.1em; - text-transform: uppercase; } - .menu-label:not(:first-child) { - margin-top: 1em; } - .menu-label:not(:last-child) { - margin-bottom: 1em; } - -.message { - background-color: whitesmoke; - border-radius: 4px; - font-size: 1rem; } - .message strong { - color: currentColor; } - .message a:not(.button):not(.tag):not(.dropdown-item) { - color: currentColor; - text-decoration: underline; } - .message.is-small, #documenter .docs-sidebar form.docs-search > input.message { - font-size: 0.75rem; } - .message.is-medium { - font-size: 1.25rem; } - .message.is-large { - font-size: 1.5rem; } - .message.is-white { - background-color: white; } - .message.is-white .message-header { - background-color: white; - color: #0a0a0a; } - .message.is-white .message-body { - border-color: white; - color: #4d4d4d; } - .message.is-black { - background-color: #fafafa; } - .message.is-black .message-header { - background-color: #0a0a0a; - color: white; } - .message.is-black .message-body { - border-color: #0a0a0a; - color: #090909; } - .message.is-light { - background-color: #fafafa; } - .message.is-light .message-header { - background-color: whitesmoke; - color: #363636; } - .message.is-light .message-body { - border-color: whitesmoke; - color: #505050; } - .message.is-dark, .content kbd.message { - background-color: #fafafa; } - .message.is-dark .message-header, .content kbd.message .message-header { - background-color: #363636; - color: whitesmoke; } - .message.is-dark .message-body, .content kbd.message .message-body { - border-color: #363636; - color: #2a2a2a; } - .message.is-primary, .docstring > section > a.message.docs-sourcelink { - background-color: #f6fbfd; } - .message.is-primary .message-header, .docstring > section > a.message.docs-sourcelink .message-header { - background-color: #4eb5de; - color: #fff; } - .message.is-primary .message-body, .docstring > section > a.message.docs-sourcelink .message-body { - border-color: #4eb5de; - color: #1f556a; } - .message.is-link { - background-color: #f7f9fd; } - .message.is-link .message-header { - background-color: #2e63b8; - color: #fff; } - .message.is-link .message-body { - border-color: #2e63b8; - color: #264981; } - .message.is-info { - background-color: #f6fbfe; } - .message.is-info .message-header { - background-color: #209cee; - color: #fff; } - .message.is-info .message-body { - border-color: #209cee; - color: #12537d; } - .message.is-success { - background-color: #f6fdf9; } - .message.is-success .message-header { - background-color: #22c35b; - color: #fff; } - .message.is-success .message-body { - border-color: #22c35b; - color: #0f361d; } - .message.is-warning { - background-color: #fffdf5; } - .message.is-warning .message-header { - background-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .message.is-warning .message-body { - border-color: #ffdd57; - color: #3c3108; } - .message.is-danger { - background-color: #fff5f5; } - .message.is-danger .message-header { - background-color: #da0b00; - color: #fff; } - .message.is-danger .message-body { - border-color: #da0b00; - color: #9b0c04; } - -.message-header { - align-items: center; - background-color: #222222; - border-radius: 4px 4px 0 0; - color: #fff; - display: flex; - font-weight: 700; - justify-content: space-between; - line-height: 1.25; - padding: 0.75em 1em; - position: relative; } - .message-header .delete { - flex-grow: 0; - flex-shrink: 0; - margin-left: 0.75em; } - .message-header + .message-body { - border-width: 0; - border-top-left-radius: 0; - border-top-right-radius: 0; } - -.message-body { - border-color: #dbdbdb; - border-radius: 4px; - border-style: solid; - border-width: 0 0 0 4px; - color: #222222; - padding: 1.25em 1.5em; } - .message-body code, - .message-body pre { - background-color: white; } - .message-body pre code { - background-color: transparent; } - -.modal { - align-items: center; - display: none; - flex-direction: column; - justify-content: center; - overflow: hidden; - position: fixed; - z-index: 40; } - .modal.is-active { - display: flex; } - -.modal-background { - background-color: rgba(10, 10, 10, 0.86); } - -.modal-content, -.modal-card { - margin: 0 20px; - max-height: calc(100vh - 160px); - overflow: auto; - position: relative; - width: 100%; } - @media screen and (min-width: 769px), print { - .modal-content, - .modal-card { - margin: 0 auto; - max-height: calc(100vh - 40px); - width: 640px; } } -.modal-close { - background: none; - height: 40px; - position: fixed; - right: 20px; - top: 20px; - width: 40px; } - -.modal-card { - display: flex; - flex-direction: column; - max-height: calc(100vh - 40px); - overflow: hidden; - -ms-overflow-y: visible; } - -.modal-card-head, -.modal-card-foot { - align-items: center; - background-color: whitesmoke; - display: flex; - flex-shrink: 0; - justify-content: flex-start; - padding: 20px; - position: relative; } - -.modal-card-head { - border-bottom: 1px solid #dbdbdb; - border-top-left-radius: 6px; - border-top-right-radius: 6px; } - -.modal-card-title { - color: #222222; - flex-grow: 1; - flex-shrink: 0; - font-size: 1.5rem; - line-height: 1; } - -.modal-card-foot { - border-bottom-left-radius: 6px; - border-bottom-right-radius: 6px; - border-top: 1px solid #dbdbdb; } - .modal-card-foot .button:not(:last-child) { - margin-right: 0.5em; } - -.modal-card-body { - -webkit-overflow-scrolling: touch; - background-color: white; - flex-grow: 1; - flex-shrink: 1; - overflow: auto; - padding: 20px; } - -.navbar { - background-color: white; - min-height: 3.25rem; - position: relative; - z-index: 30; } - .navbar.is-white { - background-color: white; - color: #0a0a0a; } - .navbar.is-white .navbar-brand > .navbar-item, - .navbar.is-white .navbar-brand .navbar-link { - color: #0a0a0a; } - .navbar.is-white .navbar-brand > a.navbar-item:focus, .navbar.is-white .navbar-brand > a.navbar-item:hover, .navbar.is-white .navbar-brand > a.navbar-item.is-active, - .navbar.is-white .navbar-brand .navbar-link:focus, - .navbar.is-white .navbar-brand .navbar-link:hover, - .navbar.is-white .navbar-brand .navbar-link.is-active { - background-color: #f2f2f2; - color: #0a0a0a; } - .navbar.is-white .navbar-brand .navbar-link::after { - border-color: #0a0a0a; } - .navbar.is-white .navbar-burger { - color: #0a0a0a; } - @media screen and (min-width: 1056px) { - .navbar.is-white .navbar-start > .navbar-item, - .navbar.is-white .navbar-start .navbar-link, - .navbar.is-white .navbar-end > .navbar-item, - .navbar.is-white .navbar-end .navbar-link { - color: #0a0a0a; } - .navbar.is-white .navbar-start > a.navbar-item:focus, .navbar.is-white .navbar-start > a.navbar-item:hover, .navbar.is-white .navbar-start > a.navbar-item.is-active, - .navbar.is-white .navbar-start .navbar-link:focus, - .navbar.is-white .navbar-start .navbar-link:hover, - .navbar.is-white .navbar-start .navbar-link.is-active, - .navbar.is-white .navbar-end > a.navbar-item:focus, - .navbar.is-white .navbar-end > a.navbar-item:hover, - .navbar.is-white .navbar-end > a.navbar-item.is-active, - .navbar.is-white .navbar-end .navbar-link:focus, - .navbar.is-white .navbar-end .navbar-link:hover, - .navbar.is-white .navbar-end .navbar-link.is-active { - background-color: #f2f2f2; - color: #0a0a0a; } - .navbar.is-white .navbar-start .navbar-link::after, - .navbar.is-white .navbar-end .navbar-link::after { - border-color: #0a0a0a; } - .navbar.is-white .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-white .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-white .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #f2f2f2; - color: #0a0a0a; } - .navbar.is-white .navbar-dropdown a.navbar-item.is-active { - background-color: white; - color: #0a0a0a; } } - .navbar.is-black { - background-color: #0a0a0a; - color: white; } - .navbar.is-black .navbar-brand > .navbar-item, - .navbar.is-black .navbar-brand .navbar-link { - color: white; } - .navbar.is-black .navbar-brand > a.navbar-item:focus, .navbar.is-black .navbar-brand > a.navbar-item:hover, .navbar.is-black .navbar-brand > a.navbar-item.is-active, - .navbar.is-black .navbar-brand .navbar-link:focus, - .navbar.is-black .navbar-brand .navbar-link:hover, - .navbar.is-black .navbar-brand .navbar-link.is-active { - background-color: black; - color: white; } - .navbar.is-black .navbar-brand .navbar-link::after { - border-color: white; } - .navbar.is-black .navbar-burger { - color: white; } - @media screen and (min-width: 1056px) { - .navbar.is-black .navbar-start > .navbar-item, - .navbar.is-black .navbar-start .navbar-link, - .navbar.is-black .navbar-end > .navbar-item, - .navbar.is-black .navbar-end .navbar-link { - color: white; } - .navbar.is-black .navbar-start > a.navbar-item:focus, .navbar.is-black .navbar-start > a.navbar-item:hover, .navbar.is-black .navbar-start > a.navbar-item.is-active, - .navbar.is-black .navbar-start .navbar-link:focus, - .navbar.is-black .navbar-start .navbar-link:hover, - .navbar.is-black .navbar-start .navbar-link.is-active, - .navbar.is-black .navbar-end > a.navbar-item:focus, - .navbar.is-black .navbar-end > a.navbar-item:hover, - .navbar.is-black .navbar-end > a.navbar-item.is-active, - .navbar.is-black .navbar-end .navbar-link:focus, - .navbar.is-black .navbar-end .navbar-link:hover, - .navbar.is-black .navbar-end .navbar-link.is-active { - background-color: black; - color: white; } - .navbar.is-black .navbar-start .navbar-link::after, - .navbar.is-black .navbar-end .navbar-link::after { - border-color: white; } - .navbar.is-black .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-black .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-black .navbar-item.has-dropdown.is-active .navbar-link { - background-color: black; - color: white; } - .navbar.is-black .navbar-dropdown a.navbar-item.is-active { - background-color: #0a0a0a; - color: white; } } - .navbar.is-light { - background-color: whitesmoke; - color: #363636; } - .navbar.is-light .navbar-brand > .navbar-item, - .navbar.is-light .navbar-brand .navbar-link { - color: #363636; } - .navbar.is-light .navbar-brand > a.navbar-item:focus, .navbar.is-light .navbar-brand > a.navbar-item:hover, .navbar.is-light .navbar-brand > a.navbar-item.is-active, - .navbar.is-light .navbar-brand .navbar-link:focus, - .navbar.is-light .navbar-brand .navbar-link:hover, - .navbar.is-light .navbar-brand .navbar-link.is-active { - background-color: #e8e8e8; - color: #363636; } - .navbar.is-light .navbar-brand .navbar-link::after { - border-color: #363636; } - .navbar.is-light .navbar-burger { - color: #363636; } - @media screen and (min-width: 1056px) { - .navbar.is-light .navbar-start > .navbar-item, - .navbar.is-light .navbar-start .navbar-link, - .navbar.is-light .navbar-end > .navbar-item, - .navbar.is-light .navbar-end .navbar-link { - color: #363636; } - .navbar.is-light .navbar-start > a.navbar-item:focus, .navbar.is-light .navbar-start > a.navbar-item:hover, .navbar.is-light .navbar-start > a.navbar-item.is-active, - .navbar.is-light .navbar-start .navbar-link:focus, - .navbar.is-light .navbar-start .navbar-link:hover, - .navbar.is-light .navbar-start .navbar-link.is-active, - .navbar.is-light .navbar-end > a.navbar-item:focus, - .navbar.is-light .navbar-end > a.navbar-item:hover, - .navbar.is-light .navbar-end > a.navbar-item.is-active, - .navbar.is-light .navbar-end .navbar-link:focus, - .navbar.is-light .navbar-end .navbar-link:hover, - .navbar.is-light .navbar-end .navbar-link.is-active { - background-color: #e8e8e8; - color: #363636; } - .navbar.is-light .navbar-start .navbar-link::after, - .navbar.is-light .navbar-end .navbar-link::after { - border-color: #363636; } - .navbar.is-light .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-light .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-light .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #e8e8e8; - color: #363636; } - .navbar.is-light .navbar-dropdown a.navbar-item.is-active { - background-color: whitesmoke; - color: #363636; } } - .navbar.is-dark, .content kbd.navbar { - background-color: #363636; - color: whitesmoke; } - .navbar.is-dark .navbar-brand > .navbar-item, .content kbd.navbar .navbar-brand > .navbar-item, - .navbar.is-dark .navbar-brand .navbar-link, - .content kbd.navbar .navbar-brand .navbar-link { - color: whitesmoke; } - .navbar.is-dark .navbar-brand > a.navbar-item:focus, .content kbd.navbar .navbar-brand > a.navbar-item:focus, .navbar.is-dark .navbar-brand > a.navbar-item:hover, .content kbd.navbar .navbar-brand > a.navbar-item:hover, .navbar.is-dark .navbar-brand > a.navbar-item.is-active, .content kbd.navbar .navbar-brand > a.navbar-item.is-active, - .navbar.is-dark .navbar-brand .navbar-link:focus, - .content kbd.navbar .navbar-brand .navbar-link:focus, - .navbar.is-dark .navbar-brand .navbar-link:hover, - .content kbd.navbar .navbar-brand .navbar-link:hover, - .navbar.is-dark .navbar-brand .navbar-link.is-active, - .content kbd.navbar .navbar-brand .navbar-link.is-active { - background-color: #292929; - color: whitesmoke; } - .navbar.is-dark .navbar-brand .navbar-link::after, .content kbd.navbar .navbar-brand .navbar-link::after { - border-color: whitesmoke; } - .navbar.is-dark .navbar-burger, .content kbd.navbar .navbar-burger { - color: whitesmoke; } - @media screen and (min-width: 1056px) { - .navbar.is-dark .navbar-start > .navbar-item, .content kbd.navbar .navbar-start > .navbar-item, - .navbar.is-dark .navbar-start .navbar-link, - .content kbd.navbar .navbar-start .navbar-link, - .navbar.is-dark .navbar-end > .navbar-item, - .content kbd.navbar .navbar-end > .navbar-item, - .navbar.is-dark .navbar-end .navbar-link, - .content kbd.navbar .navbar-end .navbar-link { - color: whitesmoke; } - .navbar.is-dark .navbar-start > a.navbar-item:focus, .content kbd.navbar .navbar-start > a.navbar-item:focus, .navbar.is-dark .navbar-start > a.navbar-item:hover, .content kbd.navbar .navbar-start > a.navbar-item:hover, .navbar.is-dark .navbar-start > a.navbar-item.is-active, .content kbd.navbar .navbar-start > a.navbar-item.is-active, - .navbar.is-dark .navbar-start .navbar-link:focus, - .content kbd.navbar .navbar-start .navbar-link:focus, - .navbar.is-dark .navbar-start .navbar-link:hover, - .content kbd.navbar .navbar-start .navbar-link:hover, - .navbar.is-dark .navbar-start .navbar-link.is-active, - .content kbd.navbar .navbar-start .navbar-link.is-active, - .navbar.is-dark .navbar-end > a.navbar-item:focus, - .content kbd.navbar .navbar-end > a.navbar-item:focus, - .navbar.is-dark .navbar-end > a.navbar-item:hover, - .content kbd.navbar .navbar-end > a.navbar-item:hover, - .navbar.is-dark .navbar-end > a.navbar-item.is-active, - .content kbd.navbar .navbar-end > a.navbar-item.is-active, - .navbar.is-dark .navbar-end .navbar-link:focus, - .content kbd.navbar .navbar-end .navbar-link:focus, - .navbar.is-dark .navbar-end .navbar-link:hover, - .content kbd.navbar .navbar-end .navbar-link:hover, - .navbar.is-dark .navbar-end .navbar-link.is-active, - .content kbd.navbar .navbar-end .navbar-link.is-active { - background-color: #292929; - color: whitesmoke; } - .navbar.is-dark .navbar-start .navbar-link::after, .content kbd.navbar .navbar-start .navbar-link::after, - .navbar.is-dark .navbar-end .navbar-link::after, - .content kbd.navbar .navbar-end .navbar-link::after { - border-color: whitesmoke; } - .navbar.is-dark .navbar-item.has-dropdown:focus .navbar-link, .content kbd.navbar .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-dark .navbar-item.has-dropdown:hover .navbar-link, - .content kbd.navbar .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-dark .navbar-item.has-dropdown.is-active .navbar-link, - .content kbd.navbar .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #292929; - color: whitesmoke; } - .navbar.is-dark .navbar-dropdown a.navbar-item.is-active, .content kbd.navbar .navbar-dropdown a.navbar-item.is-active { - background-color: #363636; - color: whitesmoke; } } - .navbar.is-primary, .docstring > section > a.navbar.docs-sourcelink { - background-color: #4eb5de; - color: #fff; } - .navbar.is-primary .navbar-brand > .navbar-item, .docstring > section > a.navbar.docs-sourcelink .navbar-brand > .navbar-item, - .navbar.is-primary .navbar-brand .navbar-link, - .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link { - color: #fff; } - .navbar.is-primary .navbar-brand > a.navbar-item:focus, .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item:focus, .navbar.is-primary .navbar-brand > a.navbar-item:hover, .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item:hover, .navbar.is-primary .navbar-brand > a.navbar-item.is-active, .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item.is-active, - .navbar.is-primary .navbar-brand .navbar-link:focus, - .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link:focus, - .navbar.is-primary .navbar-brand .navbar-link:hover, - .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link:hover, - .navbar.is-primary .navbar-brand .navbar-link.is-active, - .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link.is-active { - background-color: #39acda; - color: #fff; } - .navbar.is-primary .navbar-brand .navbar-link::after, .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link::after { - border-color: #fff; } - .navbar.is-primary .navbar-burger, .docstring > section > a.navbar.docs-sourcelink .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - .navbar.is-primary .navbar-start > .navbar-item, .docstring > section > a.navbar.docs-sourcelink .navbar-start > .navbar-item, - .navbar.is-primary .navbar-start .navbar-link, - .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link, - .navbar.is-primary .navbar-end > .navbar-item, - .docstring > section > a.navbar.docs-sourcelink .navbar-end > .navbar-item, - .navbar.is-primary .navbar-end .navbar-link, - .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link { - color: #fff; } - .navbar.is-primary .navbar-start > a.navbar-item:focus, .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item:focus, .navbar.is-primary .navbar-start > a.navbar-item:hover, .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item:hover, .navbar.is-primary .navbar-start > a.navbar-item.is-active, .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item.is-active, - .navbar.is-primary .navbar-start .navbar-link:focus, - .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link:focus, - .navbar.is-primary .navbar-start .navbar-link:hover, - .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link:hover, - .navbar.is-primary .navbar-start .navbar-link.is-active, - .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link.is-active, - .navbar.is-primary .navbar-end > a.navbar-item:focus, - .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item:focus, - .navbar.is-primary .navbar-end > a.navbar-item:hover, - .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item:hover, - .navbar.is-primary .navbar-end > a.navbar-item.is-active, - .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item.is-active, - .navbar.is-primary .navbar-end .navbar-link:focus, - .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link:focus, - .navbar.is-primary .navbar-end .navbar-link:hover, - .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link:hover, - .navbar.is-primary .navbar-end .navbar-link.is-active, - .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link.is-active { - background-color: #39acda; - color: #fff; } - .navbar.is-primary .navbar-start .navbar-link::after, .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link::after, - .navbar.is-primary .navbar-end .navbar-link::after, - .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link::after { - border-color: #fff; } - .navbar.is-primary .navbar-item.has-dropdown:focus .navbar-link, .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-primary .navbar-item.has-dropdown:hover .navbar-link, - .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-primary .navbar-item.has-dropdown.is-active .navbar-link, - .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #39acda; - color: #fff; } - .navbar.is-primary .navbar-dropdown a.navbar-item.is-active, .docstring > section > a.navbar.docs-sourcelink .navbar-dropdown a.navbar-item.is-active { - background-color: #4eb5de; - color: #fff; } } - .navbar.is-link { - background-color: #2e63b8; - color: #fff; } - .navbar.is-link .navbar-brand > .navbar-item, - .navbar.is-link .navbar-brand .navbar-link { - color: #fff; } - .navbar.is-link .navbar-brand > a.navbar-item:focus, .navbar.is-link .navbar-brand > a.navbar-item:hover, .navbar.is-link .navbar-brand > a.navbar-item.is-active, - .navbar.is-link .navbar-brand .navbar-link:focus, - .navbar.is-link .navbar-brand .navbar-link:hover, - .navbar.is-link .navbar-brand .navbar-link.is-active { - background-color: #2958a4; - color: #fff; } - .navbar.is-link .navbar-brand .navbar-link::after { - border-color: #fff; } - .navbar.is-link .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - .navbar.is-link .navbar-start > .navbar-item, - .navbar.is-link .navbar-start .navbar-link, - .navbar.is-link .navbar-end > .navbar-item, - .navbar.is-link .navbar-end .navbar-link { - color: #fff; } - .navbar.is-link .navbar-start > a.navbar-item:focus, .navbar.is-link .navbar-start > a.navbar-item:hover, .navbar.is-link .navbar-start > a.navbar-item.is-active, - .navbar.is-link .navbar-start .navbar-link:focus, - .navbar.is-link .navbar-start .navbar-link:hover, - .navbar.is-link .navbar-start .navbar-link.is-active, - .navbar.is-link .navbar-end > a.navbar-item:focus, - .navbar.is-link .navbar-end > a.navbar-item:hover, - .navbar.is-link .navbar-end > a.navbar-item.is-active, - .navbar.is-link .navbar-end .navbar-link:focus, - .navbar.is-link .navbar-end .navbar-link:hover, - .navbar.is-link .navbar-end .navbar-link.is-active { - background-color: #2958a4; - color: #fff; } - .navbar.is-link .navbar-start .navbar-link::after, - .navbar.is-link .navbar-end .navbar-link::after { - border-color: #fff; } - .navbar.is-link .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-link .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-link .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #2958a4; - color: #fff; } - .navbar.is-link .navbar-dropdown a.navbar-item.is-active { - background-color: #2e63b8; - color: #fff; } } - .navbar.is-info { - background-color: #209cee; - color: #fff; } - .navbar.is-info .navbar-brand > .navbar-item, - .navbar.is-info .navbar-brand .navbar-link { - color: #fff; } - .navbar.is-info .navbar-brand > a.navbar-item:focus, .navbar.is-info .navbar-brand > a.navbar-item:hover, .navbar.is-info .navbar-brand > a.navbar-item.is-active, - .navbar.is-info .navbar-brand .navbar-link:focus, - .navbar.is-info .navbar-brand .navbar-link:hover, - .navbar.is-info .navbar-brand .navbar-link.is-active { - background-color: #1190e3; - color: #fff; } - .navbar.is-info .navbar-brand .navbar-link::after { - border-color: #fff; } - .navbar.is-info .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - .navbar.is-info .navbar-start > .navbar-item, - .navbar.is-info .navbar-start .navbar-link, - .navbar.is-info .navbar-end > .navbar-item, - .navbar.is-info .navbar-end .navbar-link { - color: #fff; } - .navbar.is-info .navbar-start > a.navbar-item:focus, .navbar.is-info .navbar-start > a.navbar-item:hover, .navbar.is-info .navbar-start > a.navbar-item.is-active, - .navbar.is-info .navbar-start .navbar-link:focus, - .navbar.is-info .navbar-start .navbar-link:hover, - .navbar.is-info .navbar-start .navbar-link.is-active, - .navbar.is-info .navbar-end > a.navbar-item:focus, - .navbar.is-info .navbar-end > a.navbar-item:hover, - .navbar.is-info .navbar-end > a.navbar-item.is-active, - .navbar.is-info .navbar-end .navbar-link:focus, - .navbar.is-info .navbar-end .navbar-link:hover, - .navbar.is-info .navbar-end .navbar-link.is-active { - background-color: #1190e3; - color: #fff; } - .navbar.is-info .navbar-start .navbar-link::after, - .navbar.is-info .navbar-end .navbar-link::after { - border-color: #fff; } - .navbar.is-info .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-info .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-info .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #1190e3; - color: #fff; } - .navbar.is-info .navbar-dropdown a.navbar-item.is-active { - background-color: #209cee; - color: #fff; } } - .navbar.is-success { - background-color: #22c35b; - color: #fff; } - .navbar.is-success .navbar-brand > .navbar-item, - .navbar.is-success .navbar-brand .navbar-link { - color: #fff; } - .navbar.is-success .navbar-brand > a.navbar-item:focus, .navbar.is-success .navbar-brand > a.navbar-item:hover, .navbar.is-success .navbar-brand > a.navbar-item.is-active, - .navbar.is-success .navbar-brand .navbar-link:focus, - .navbar.is-success .navbar-brand .navbar-link:hover, - .navbar.is-success .navbar-brand .navbar-link.is-active { - background-color: #1ead51; - color: #fff; } - .navbar.is-success .navbar-brand .navbar-link::after { - border-color: #fff; } - .navbar.is-success .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - .navbar.is-success .navbar-start > .navbar-item, - .navbar.is-success .navbar-start .navbar-link, - .navbar.is-success .navbar-end > .navbar-item, - .navbar.is-success .navbar-end .navbar-link { - color: #fff; } - .navbar.is-success .navbar-start > a.navbar-item:focus, .navbar.is-success .navbar-start > a.navbar-item:hover, .navbar.is-success .navbar-start > a.navbar-item.is-active, - .navbar.is-success .navbar-start .navbar-link:focus, - .navbar.is-success .navbar-start .navbar-link:hover, - .navbar.is-success .navbar-start .navbar-link.is-active, - .navbar.is-success .navbar-end > a.navbar-item:focus, - .navbar.is-success .navbar-end > a.navbar-item:hover, - .navbar.is-success .navbar-end > a.navbar-item.is-active, - .navbar.is-success .navbar-end .navbar-link:focus, - .navbar.is-success .navbar-end .navbar-link:hover, - .navbar.is-success .navbar-end .navbar-link.is-active { - background-color: #1ead51; - color: #fff; } - .navbar.is-success .navbar-start .navbar-link::after, - .navbar.is-success .navbar-end .navbar-link::after { - border-color: #fff; } - .navbar.is-success .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-success .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-success .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #1ead51; - color: #fff; } - .navbar.is-success .navbar-dropdown a.navbar-item.is-active { - background-color: #22c35b; - color: #fff; } } - .navbar.is-warning { - background-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-brand > .navbar-item, - .navbar.is-warning .navbar-brand .navbar-link { - color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-brand > a.navbar-item:focus, .navbar.is-warning .navbar-brand > a.navbar-item:hover, .navbar.is-warning .navbar-brand > a.navbar-item.is-active, - .navbar.is-warning .navbar-brand .navbar-link:focus, - .navbar.is-warning .navbar-brand .navbar-link:hover, - .navbar.is-warning .navbar-brand .navbar-link.is-active { - background-color: #ffd83e; - color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-brand .navbar-link::after { - border-color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-burger { - color: rgba(0, 0, 0, 0.7); } - @media screen and (min-width: 1056px) { - .navbar.is-warning .navbar-start > .navbar-item, - .navbar.is-warning .navbar-start .navbar-link, - .navbar.is-warning .navbar-end > .navbar-item, - .navbar.is-warning .navbar-end .navbar-link { - color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-start > a.navbar-item:focus, .navbar.is-warning .navbar-start > a.navbar-item:hover, .navbar.is-warning .navbar-start > a.navbar-item.is-active, - .navbar.is-warning .navbar-start .navbar-link:focus, - .navbar.is-warning .navbar-start .navbar-link:hover, - .navbar.is-warning .navbar-start .navbar-link.is-active, - .navbar.is-warning .navbar-end > a.navbar-item:focus, - .navbar.is-warning .navbar-end > a.navbar-item:hover, - .navbar.is-warning .navbar-end > a.navbar-item.is-active, - .navbar.is-warning .navbar-end .navbar-link:focus, - .navbar.is-warning .navbar-end .navbar-link:hover, - .navbar.is-warning .navbar-end .navbar-link.is-active { - background-color: #ffd83e; - color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-start .navbar-link::after, - .navbar.is-warning .navbar-end .navbar-link::after { - border-color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-warning .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-warning .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #ffd83e; - color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-dropdown a.navbar-item.is-active { - background-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } } - .navbar.is-danger { - background-color: #da0b00; - color: #fff; } - .navbar.is-danger .navbar-brand > .navbar-item, - .navbar.is-danger .navbar-brand .navbar-link { - color: #fff; } - .navbar.is-danger .navbar-brand > a.navbar-item:focus, .navbar.is-danger .navbar-brand > a.navbar-item:hover, .navbar.is-danger .navbar-brand > a.navbar-item.is-active, - .navbar.is-danger .navbar-brand .navbar-link:focus, - .navbar.is-danger .navbar-brand .navbar-link:hover, - .navbar.is-danger .navbar-brand .navbar-link.is-active { - background-color: #c10a00; - color: #fff; } - .navbar.is-danger .navbar-brand .navbar-link::after { - border-color: #fff; } - .navbar.is-danger .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - .navbar.is-danger .navbar-start > .navbar-item, - .navbar.is-danger .navbar-start .navbar-link, - .navbar.is-danger .navbar-end > .navbar-item, - .navbar.is-danger .navbar-end .navbar-link { - color: #fff; } - .navbar.is-danger .navbar-start > a.navbar-item:focus, .navbar.is-danger .navbar-start > a.navbar-item:hover, .navbar.is-danger .navbar-start > a.navbar-item.is-active, - .navbar.is-danger .navbar-start .navbar-link:focus, - .navbar.is-danger .navbar-start .navbar-link:hover, - .navbar.is-danger .navbar-start .navbar-link.is-active, - .navbar.is-danger .navbar-end > a.navbar-item:focus, - .navbar.is-danger .navbar-end > a.navbar-item:hover, - .navbar.is-danger .navbar-end > a.navbar-item.is-active, - .navbar.is-danger .navbar-end .navbar-link:focus, - .navbar.is-danger .navbar-end .navbar-link:hover, - .navbar.is-danger .navbar-end .navbar-link.is-active { - background-color: #c10a00; - color: #fff; } - .navbar.is-danger .navbar-start .navbar-link::after, - .navbar.is-danger .navbar-end .navbar-link::after { - border-color: #fff; } - .navbar.is-danger .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-danger .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-danger .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #c10a00; - color: #fff; } - .navbar.is-danger .navbar-dropdown a.navbar-item.is-active { - background-color: #da0b00; - color: #fff; } } - .navbar > .container { - align-items: stretch; - display: flex; - min-height: 3.25rem; - width: 100%; } - .navbar.has-shadow { - box-shadow: 0 2px 0 0 whitesmoke; } - .navbar.is-fixed-bottom, .navbar.is-fixed-top { - left: 0; - position: fixed; - right: 0; - z-index: 30; } - .navbar.is-fixed-bottom { - bottom: 0; } - .navbar.is-fixed-bottom.has-shadow { - box-shadow: 0 -2px 0 0 whitesmoke; } - .navbar.is-fixed-top { - top: 0; } - -html.has-navbar-fixed-top, -body.has-navbar-fixed-top { - padding-top: 3.25rem; } - -html.has-navbar-fixed-bottom, -body.has-navbar-fixed-bottom { - padding-bottom: 3.25rem; } - -.navbar-brand, -.navbar-tabs { - align-items: stretch; - display: flex; - flex-shrink: 0; - min-height: 3.25rem; } - -.navbar-brand a.navbar-item:focus, .navbar-brand a.navbar-item:hover { - background-color: transparent; } - -.navbar-tabs { - -webkit-overflow-scrolling: touch; - max-width: 100vw; - overflow-x: auto; - overflow-y: hidden; } - -.navbar-burger { - color: #4a4a4a; - cursor: pointer; - display: block; - height: 3.25rem; - position: relative; - width: 3.25rem; - margin-left: auto; } - .navbar-burger span { - background-color: currentColor; - display: block; - height: 1px; - left: calc(50% - 8px); - position: absolute; - transform-origin: center; - transition-duration: 86ms; - transition-property: background-color, opacity, transform; - transition-timing-function: ease-out; - width: 16px; } - .navbar-burger span:nth-child(1) { - top: calc(50% - 6px); } - .navbar-burger span:nth-child(2) { - top: calc(50% - 1px); } - .navbar-burger span:nth-child(3) { - top: calc(50% + 4px); } - .navbar-burger:hover { - background-color: rgba(0, 0, 0, 0.05); } - .navbar-burger.is-active span:nth-child(1) { - transform: translateY(5px) rotate(45deg); } - .navbar-burger.is-active span:nth-child(2) { - opacity: 0; } - .navbar-burger.is-active span:nth-child(3) { - transform: translateY(-5px) rotate(-45deg); } - -.navbar-menu { - display: none; } - -.navbar-item, -.navbar-link { - color: #4a4a4a; - display: block; - line-height: 1.5; - padding: 0.5rem 0.75rem; - position: relative; } - .navbar-item .icon:only-child, - .navbar-link .icon:only-child { - margin-left: -0.25rem; - margin-right: -0.25rem; } - -a.navbar-item, -.navbar-link { - cursor: pointer; } - a.navbar-item:focus, a.navbar-item:focus-within, a.navbar-item:hover, a.navbar-item.is-active, - .navbar-link:focus, - .navbar-link:focus-within, - .navbar-link:hover, - .navbar-link.is-active { - background-color: #fafafa; - color: #2e63b8; } - -.navbar-item { - display: block; - flex-grow: 0; - flex-shrink: 0; } - .navbar-item img { - max-height: 1.75rem; } - .navbar-item.has-dropdown { - padding: 0; } - .navbar-item.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - .navbar-item.is-tab { - border-bottom: 1px solid transparent; - min-height: 3.25rem; - padding-bottom: calc(0.5rem - 1px); } - .navbar-item.is-tab:focus, .navbar-item.is-tab:hover { - background-color: transparent; - border-bottom-color: #2e63b8; } - .navbar-item.is-tab.is-active { - background-color: transparent; - border-bottom-color: #2e63b8; - border-bottom-style: solid; - border-bottom-width: 3px; - color: #2e63b8; - padding-bottom: calc(0.5rem - 3px); } - -.navbar-content { - flex-grow: 1; - flex-shrink: 1; } - -.navbar-link:not(.is-arrowless) { - padding-right: 2.5em; } - .navbar-link:not(.is-arrowless)::after { - border-color: #2e63b8; - margin-top: -0.375em; - right: 1.125em; } - -.navbar-dropdown { - font-size: 0.875rem; - padding-bottom: 0.5rem; - padding-top: 0.5rem; } - .navbar-dropdown .navbar-item { - padding-left: 1.5rem; - padding-right: 1.5rem; } - -.navbar-divider { - background-color: whitesmoke; - border: none; - display: none; - height: 2px; - margin: 0.5rem 0; } - -@media screen and (max-width: 1055px) { - .navbar > .container { - display: block; } - .navbar-brand .navbar-item, - .navbar-tabs .navbar-item { - align-items: center; - display: flex; } - .navbar-link::after { - display: none; } - .navbar-menu { - background-color: white; - box-shadow: 0 8px 16px rgba(10, 10, 10, 0.1); - padding: 0.5rem 0; } - .navbar-menu.is-active { - display: block; } - .navbar.is-fixed-bottom-touch, .navbar.is-fixed-top-touch { - left: 0; - position: fixed; - right: 0; - z-index: 30; } - .navbar.is-fixed-bottom-touch { - bottom: 0; } - .navbar.is-fixed-bottom-touch.has-shadow { - box-shadow: 0 -2px 3px rgba(10, 10, 10, 0.1); } - .navbar.is-fixed-top-touch { - top: 0; } - .navbar.is-fixed-top .navbar-menu, .navbar.is-fixed-top-touch .navbar-menu { - -webkit-overflow-scrolling: touch; - max-height: calc(100vh - 3.25rem); - overflow: auto; } - html.has-navbar-fixed-top-touch, - body.has-navbar-fixed-top-touch { - padding-top: 3.25rem; } - html.has-navbar-fixed-bottom-touch, - body.has-navbar-fixed-bottom-touch { - padding-bottom: 3.25rem; } } - -@media screen and (min-width: 1056px) { - .navbar, - .navbar-menu, - .navbar-start, - .navbar-end { - align-items: stretch; - display: flex; } - .navbar { - min-height: 3.25rem; } - .navbar.is-spaced { - padding: 1rem 2rem; } - .navbar.is-spaced .navbar-start, - .navbar.is-spaced .navbar-end { - align-items: center; } - .navbar.is-spaced a.navbar-item, - .navbar.is-spaced .navbar-link { - border-radius: 4px; } - .navbar.is-transparent a.navbar-item:focus, .navbar.is-transparent a.navbar-item:hover, .navbar.is-transparent a.navbar-item.is-active, - .navbar.is-transparent .navbar-link:focus, - .navbar.is-transparent .navbar-link:hover, - .navbar.is-transparent .navbar-link.is-active { - background-color: transparent !important; } - .navbar.is-transparent .navbar-item.has-dropdown.is-active .navbar-link, .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:focus .navbar-link, .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:focus-within .navbar-link, .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:hover .navbar-link { - background-color: transparent !important; } - .navbar.is-transparent .navbar-dropdown a.navbar-item:focus, .navbar.is-transparent .navbar-dropdown a.navbar-item:hover { - background-color: whitesmoke; - color: #0a0a0a; } - .navbar.is-transparent .navbar-dropdown a.navbar-item.is-active { - background-color: whitesmoke; - color: #2e63b8; } - .navbar-burger { - display: none; } - .navbar-item, - .navbar-link { - align-items: center; - display: flex; } - .navbar-item { - display: flex; } - .navbar-item.has-dropdown { - align-items: stretch; } - .navbar-item.has-dropdown-up .navbar-link::after { - transform: rotate(135deg) translate(0.25em, -0.25em); } - .navbar-item.has-dropdown-up .navbar-dropdown { - border-bottom: 2px solid #dbdbdb; - border-radius: 6px 6px 0 0; - border-top: none; - bottom: 100%; - box-shadow: 0 -8px 8px rgba(10, 10, 10, 0.1); - top: auto; } - .navbar-item.is-active .navbar-dropdown, .navbar-item.is-hoverable:focus .navbar-dropdown, .navbar-item.is-hoverable:focus-within .navbar-dropdown, .navbar-item.is-hoverable:hover .navbar-dropdown { - display: block; } - .navbar.is-spaced .navbar-item.is-active .navbar-dropdown, .navbar-item.is-active .navbar-dropdown.is-boxed, .navbar.is-spaced .navbar-item.is-hoverable:focus .navbar-dropdown, .navbar-item.is-hoverable:focus .navbar-dropdown.is-boxed, .navbar.is-spaced .navbar-item.is-hoverable:focus-within .navbar-dropdown, .navbar-item.is-hoverable:focus-within .navbar-dropdown.is-boxed, .navbar.is-spaced .navbar-item.is-hoverable:hover .navbar-dropdown, .navbar-item.is-hoverable:hover .navbar-dropdown.is-boxed { - opacity: 1; - pointer-events: auto; - transform: translateY(0); } - .navbar-menu { - flex-grow: 1; - flex-shrink: 0; } - .navbar-start { - justify-content: flex-start; - margin-right: auto; } - .navbar-end { - justify-content: flex-end; - margin-left: auto; } - .navbar-dropdown { - background-color: white; - border-bottom-left-radius: 6px; - border-bottom-right-radius: 6px; - border-top: 2px solid #dbdbdb; - box-shadow: 0 8px 8px rgba(10, 10, 10, 0.1); - display: none; - font-size: 0.875rem; - left: 0; - min-width: 100%; - position: absolute; - top: 100%; - z-index: 20; } - .navbar-dropdown .navbar-item { - padding: 0.375rem 1rem; - white-space: nowrap; } - .navbar-dropdown a.navbar-item { - padding-right: 3rem; } - .navbar-dropdown a.navbar-item:focus, .navbar-dropdown a.navbar-item:hover { - background-color: whitesmoke; - color: #0a0a0a; } - .navbar-dropdown a.navbar-item.is-active { - background-color: whitesmoke; - color: #2e63b8; } - .navbar.is-spaced .navbar-dropdown, .navbar-dropdown.is-boxed { - border-radius: 6px; - border-top: none; - box-shadow: 0 8px 8px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); - display: block; - opacity: 0; - pointer-events: none; - top: calc(100% + (-4px)); - transform: translateY(-5px); - transition-duration: 86ms; - transition-property: opacity, transform; } - .navbar-dropdown.is-right { - left: auto; - right: 0; } - .navbar-divider { - display: block; } - .navbar > .container .navbar-brand, - .container > .navbar .navbar-brand { - margin-left: -.75rem; } - .navbar > .container .navbar-menu, - .container > .navbar .navbar-menu { - margin-right: -.75rem; } - .navbar.is-fixed-bottom-desktop, .navbar.is-fixed-top-desktop { - left: 0; - position: fixed; - right: 0; - z-index: 30; } - .navbar.is-fixed-bottom-desktop { - bottom: 0; } - .navbar.is-fixed-bottom-desktop.has-shadow { - box-shadow: 0 -2px 3px rgba(10, 10, 10, 0.1); } - .navbar.is-fixed-top-desktop { - top: 0; } - html.has-navbar-fixed-top-desktop, - body.has-navbar-fixed-top-desktop { - padding-top: 3.25rem; } - html.has-navbar-fixed-bottom-desktop, - body.has-navbar-fixed-bottom-desktop { - padding-bottom: 3.25rem; } - html.has-spaced-navbar-fixed-top, - body.has-spaced-navbar-fixed-top { - padding-top: 5.25rem; } - html.has-spaced-navbar-fixed-bottom, - body.has-spaced-navbar-fixed-bottom { - padding-bottom: 5.25rem; } - a.navbar-item.is-active, - .navbar-link.is-active { - color: #0a0a0a; } - a.navbar-item.is-active:not(:focus):not(:hover), - .navbar-link.is-active:not(:focus):not(:hover) { - background-color: transparent; } - .navbar-item.has-dropdown:focus .navbar-link, .navbar-item.has-dropdown:hover .navbar-link, .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #fafafa; } } - -.hero.is-fullheight-with-navbar { - min-height: calc(100vh - 3.25rem); } - -.pagination { - font-size: 1rem; - margin: -0.25rem; } - .pagination.is-small, #documenter .docs-sidebar form.docs-search > input.pagination { - font-size: 0.75rem; } - .pagination.is-medium { - font-size: 1.25rem; } - .pagination.is-large { - font-size: 1.5rem; } - .pagination.is-rounded .pagination-previous, #documenter .docs-sidebar form.docs-search > input.pagination .pagination-previous, - .pagination.is-rounded .pagination-next, - #documenter .docs-sidebar form.docs-search > input.pagination .pagination-next { - padding-left: 1em; - padding-right: 1em; - border-radius: 290486px; } - .pagination.is-rounded .pagination-link, #documenter .docs-sidebar form.docs-search > input.pagination .pagination-link { - border-radius: 290486px; } - -.pagination, -.pagination-list { - align-items: center; - display: flex; - justify-content: center; - text-align: center; } - -.pagination-previous, -.pagination-next, -.pagination-link, -.pagination-ellipsis { - font-size: 1em; - justify-content: center; - margin: 0.25rem; - padding-left: 0.5em; - padding-right: 0.5em; - text-align: center; } - -.pagination-previous, -.pagination-next, -.pagination-link { - border-color: #dbdbdb; - color: #363636; - min-width: 2.25em; } - .pagination-previous:hover, - .pagination-next:hover, - .pagination-link:hover { - border-color: #b5b5b5; - color: #363636; } - .pagination-previous:focus, - .pagination-next:focus, - .pagination-link:focus { - border-color: #3c5dcd; } - .pagination-previous:active, - .pagination-next:active, - .pagination-link:active { - box-shadow: inset 0 1px 2px rgba(10, 10, 10, 0.2); } - .pagination-previous[disabled], - .pagination-next[disabled], - .pagination-link[disabled] { - background-color: #dbdbdb; - border-color: #dbdbdb; - box-shadow: none; - color: #6b6b6b; - opacity: 0.5; } - -.pagination-previous, -.pagination-next { - padding-left: 0.75em; - padding-right: 0.75em; - white-space: nowrap; } - -.pagination-link.is-current { - background-color: #2e63b8; - border-color: #2e63b8; - color: #fff; } - -.pagination-ellipsis { - color: #b5b5b5; - pointer-events: none; } - -.pagination-list { - flex-wrap: wrap; } - -@media screen and (max-width: 768px) { - .pagination { - flex-wrap: wrap; } - .pagination-previous, - .pagination-next { - flex-grow: 1; - flex-shrink: 1; } - .pagination-list li { - flex-grow: 1; - flex-shrink: 1; } } - -@media screen and (min-width: 769px), print { - .pagination-list { - flex-grow: 1; - flex-shrink: 1; - justify-content: flex-start; - order: 1; } - .pagination-previous { - order: 2; } - .pagination-next { - order: 3; } - .pagination { - justify-content: space-between; } - .pagination.is-centered .pagination-previous { - order: 1; } - .pagination.is-centered .pagination-list { - justify-content: center; - order: 2; } - .pagination.is-centered .pagination-next { - order: 3; } - .pagination.is-right .pagination-previous { - order: 1; } - .pagination.is-right .pagination-next { - order: 2; } - .pagination.is-right .pagination-list { - justify-content: flex-end; - order: 3; } } - -.panel { - font-size: 1rem; } - .panel:not(:last-child) { - margin-bottom: 1.5rem; } - -.panel-heading, -.panel-tabs, -.panel-block { - border-bottom: 1px solid #dbdbdb; - border-left: 1px solid #dbdbdb; - border-right: 1px solid #dbdbdb; } - .panel-heading:first-child, - .panel-tabs:first-child, - .panel-block:first-child { - border-top: 1px solid #dbdbdb; } - -.panel-heading { - background-color: whitesmoke; - border-radius: 4px 4px 0 0; - color: #222222; - font-size: 1.25em; - font-weight: 300; - line-height: 1.25; - padding: 0.5em 0.75em; } - -.panel-tabs { - align-items: flex-end; - display: flex; - font-size: 0.875em; - justify-content: center; } - .panel-tabs a { - border-bottom: 1px solid #dbdbdb; - margin-bottom: -1px; - padding: 0.5em; } - .panel-tabs a.is-active { - border-bottom-color: #4a4a4a; - color: #363636; } - -.panel-list a { - color: #222222; } - .panel-list a:hover { - color: #2e63b8; } - -.panel-block { - align-items: center; - color: #222222; - display: flex; - justify-content: flex-start; - padding: 0.5em 0.75em; } - .panel-block input[type="checkbox"] { - margin-right: 0.75em; } - .panel-block > .control { - flex-grow: 1; - flex-shrink: 1; - width: 100%; } - .panel-block.is-wrapped { - flex-wrap: wrap; } - .panel-block.is-active { - border-left-color: #2e63b8; - color: #363636; } - .panel-block.is-active .panel-icon { - color: #2e63b8; } - -a.panel-block, -label.panel-block { - cursor: pointer; } - a.panel-block:hover, - label.panel-block:hover { - background-color: whitesmoke; } - -.panel-icon { - display: inline-block; - font-size: 14px; - height: 1em; - line-height: 1em; - text-align: center; - vertical-align: top; - width: 1em; - color: #6b6b6b; - margin-right: 0.75em; } - .panel-icon .fa { - font-size: inherit; - line-height: inherit; } - -.tabs { - -webkit-overflow-scrolling: touch; - align-items: stretch; - display: flex; - font-size: 1rem; - justify-content: space-between; - overflow: hidden; - overflow-x: auto; - white-space: nowrap; } - .tabs a { - align-items: center; - border-bottom-color: #dbdbdb; - border-bottom-style: solid; - border-bottom-width: 1px; - color: #222222; - display: flex; - justify-content: center; - margin-bottom: -1px; - padding: 0.5em 1em; - vertical-align: top; } - .tabs a:hover { - border-bottom-color: #222222; - color: #222222; } - .tabs li { - display: block; } - .tabs li.is-active a { - border-bottom-color: #2e63b8; - color: #2e63b8; } - .tabs ul { - align-items: center; - border-bottom-color: #dbdbdb; - border-bottom-style: solid; - border-bottom-width: 1px; - display: flex; - flex-grow: 1; - flex-shrink: 0; - justify-content: flex-start; } - .tabs ul.is-left { - padding-right: 0.75em; } - .tabs ul.is-center { - flex: none; - justify-content: center; - padding-left: 0.75em; - padding-right: 0.75em; } - .tabs ul.is-right { - justify-content: flex-end; - padding-left: 0.75em; } - .tabs .icon:first-child { - margin-right: 0.5em; } - .tabs .icon:last-child { - margin-left: 0.5em; } - .tabs.is-centered ul { - justify-content: center; } - .tabs.is-right ul { - justify-content: flex-end; } - .tabs.is-boxed a { - border: 1px solid transparent; - border-radius: 4px 4px 0 0; } - .tabs.is-boxed a:hover { - background-color: whitesmoke; - border-bottom-color: #dbdbdb; } - .tabs.is-boxed li.is-active a { - background-color: white; - border-color: #dbdbdb; - border-bottom-color: transparent !important; } - .tabs.is-fullwidth li { - flex-grow: 1; - flex-shrink: 0; } - .tabs.is-toggle a { - border-color: #dbdbdb; - border-style: solid; - border-width: 1px; - margin-bottom: 0; - position: relative; } - .tabs.is-toggle a:hover { - background-color: whitesmoke; - border-color: #b5b5b5; - z-index: 2; } - .tabs.is-toggle li + li { - margin-left: -1px; } - .tabs.is-toggle li:first-child a { - border-radius: 4px 0 0 4px; } - .tabs.is-toggle li:last-child a { - border-radius: 0 4px 4px 0; } - .tabs.is-toggle li.is-active a { - background-color: #2e63b8; - border-color: #2e63b8; - color: #fff; - z-index: 1; } - .tabs.is-toggle ul { - border-bottom: none; } - .tabs.is-toggle.is-toggle-rounded li:first-child a { - border-bottom-left-radius: 290486px; - border-top-left-radius: 290486px; - padding-left: 1.25em; } - .tabs.is-toggle.is-toggle-rounded li:last-child a { - border-bottom-right-radius: 290486px; - border-top-right-radius: 290486px; - padding-right: 1.25em; } - .tabs.is-small, #documenter .docs-sidebar form.docs-search > input.tabs { - font-size: 0.75rem; } - .tabs.is-medium { - font-size: 1.25rem; } - .tabs.is-large { - font-size: 1.5rem; } - -.column { - display: block; - flex-basis: 0; - flex-grow: 1; - flex-shrink: 1; - padding: 0.75rem; } - .columns.is-mobile > .column.is-narrow { - flex: none; } - .columns.is-mobile > .column.is-full { - flex: none; - width: 100%; } - .columns.is-mobile > .column.is-three-quarters { - flex: none; - width: 75%; } - .columns.is-mobile > .column.is-two-thirds { - flex: none; - width: 66.6666%; } - .columns.is-mobile > .column.is-half { - flex: none; - width: 50%; } - .columns.is-mobile > .column.is-one-third { - flex: none; - width: 33.3333%; } - .columns.is-mobile > .column.is-one-quarter { - flex: none; - width: 25%; } - .columns.is-mobile > .column.is-one-fifth { - flex: none; - width: 20%; } - .columns.is-mobile > .column.is-two-fifths { - flex: none; - width: 40%; } - .columns.is-mobile > .column.is-three-fifths { - flex: none; - width: 60%; } - .columns.is-mobile > .column.is-four-fifths { - flex: none; - width: 80%; } - .columns.is-mobile > .column.is-offset-three-quarters { - margin-left: 75%; } - .columns.is-mobile > .column.is-offset-two-thirds { - margin-left: 66.6666%; } - .columns.is-mobile > .column.is-offset-half { - margin-left: 50%; } - .columns.is-mobile > .column.is-offset-one-third { - margin-left: 33.3333%; } - .columns.is-mobile > .column.is-offset-one-quarter { - margin-left: 25%; } - .columns.is-mobile > .column.is-offset-one-fifth { - margin-left: 20%; } - .columns.is-mobile > .column.is-offset-two-fifths { - margin-left: 40%; } - .columns.is-mobile > .column.is-offset-three-fifths { - margin-left: 60%; } - .columns.is-mobile > .column.is-offset-four-fifths { - margin-left: 80%; } - .columns.is-mobile > .column.is-0 { - flex: none; - width: 0%; } - .columns.is-mobile > .column.is-offset-0 { - margin-left: 0%; } - .columns.is-mobile > .column.is-1 { - flex: none; - width: 8.3333333333%; } - .columns.is-mobile > .column.is-offset-1 { - margin-left: 8.3333333333%; } - .columns.is-mobile > .column.is-2 { - flex: none; - width: 16.6666666667%; } - .columns.is-mobile > .column.is-offset-2 { - margin-left: 16.6666666667%; } - .columns.is-mobile > .column.is-3 { - flex: none; - width: 25%; } - .columns.is-mobile > .column.is-offset-3 { - margin-left: 25%; } - .columns.is-mobile > .column.is-4 { - flex: none; - width: 33.3333333333%; } - .columns.is-mobile > .column.is-offset-4 { - margin-left: 33.3333333333%; } - .columns.is-mobile > .column.is-5 { - flex: none; - width: 41.6666666667%; } - .columns.is-mobile > .column.is-offset-5 { - margin-left: 41.6666666667%; } - .columns.is-mobile > .column.is-6 { - flex: none; - width: 50%; } - .columns.is-mobile > .column.is-offset-6 { - margin-left: 50%; } - .columns.is-mobile > .column.is-7 { - flex: none; - width: 58.3333333333%; } - .columns.is-mobile > .column.is-offset-7 { - margin-left: 58.3333333333%; } - .columns.is-mobile > .column.is-8 { - flex: none; - width: 66.6666666667%; } - .columns.is-mobile > .column.is-offset-8 { - margin-left: 66.6666666667%; } - .columns.is-mobile > .column.is-9 { - flex: none; - width: 75%; } - .columns.is-mobile > .column.is-offset-9 { - margin-left: 75%; } - .columns.is-mobile > .column.is-10 { - flex: none; - width: 83.3333333333%; } - .columns.is-mobile > .column.is-offset-10 { - margin-left: 83.3333333333%; } - .columns.is-mobile > .column.is-11 { - flex: none; - width: 91.6666666667%; } - .columns.is-mobile > .column.is-offset-11 { - margin-left: 91.6666666667%; } - .columns.is-mobile > .column.is-12 { - flex: none; - width: 100%; } - .columns.is-mobile > .column.is-offset-12 { - margin-left: 100%; } - @media screen and (max-width: 768px) { - .column.is-narrow-mobile { - flex: none; } - .column.is-full-mobile { - flex: none; - width: 100%; } - .column.is-three-quarters-mobile { - flex: none; - width: 75%; } - .column.is-two-thirds-mobile { - flex: none; - width: 66.6666%; } - .column.is-half-mobile { - flex: none; - width: 50%; } - .column.is-one-third-mobile { - flex: none; - width: 33.3333%; } - .column.is-one-quarter-mobile { - flex: none; - width: 25%; } - .column.is-one-fifth-mobile { - flex: none; - width: 20%; } - .column.is-two-fifths-mobile { - flex: none; - width: 40%; } - .column.is-three-fifths-mobile { - flex: none; - width: 60%; } - .column.is-four-fifths-mobile { - flex: none; - width: 80%; } - .column.is-offset-three-quarters-mobile { - margin-left: 75%; } - .column.is-offset-two-thirds-mobile { - margin-left: 66.6666%; } - .column.is-offset-half-mobile { - margin-left: 50%; } - .column.is-offset-one-third-mobile { - margin-left: 33.3333%; } - .column.is-offset-one-quarter-mobile { - margin-left: 25%; } - .column.is-offset-one-fifth-mobile { - margin-left: 20%; } - .column.is-offset-two-fifths-mobile { - margin-left: 40%; } - .column.is-offset-three-fifths-mobile { - margin-left: 60%; } - .column.is-offset-four-fifths-mobile { - margin-left: 80%; } - .column.is-0-mobile { - flex: none; - width: 0%; } - .column.is-offset-0-mobile { - margin-left: 0%; } - .column.is-1-mobile { - flex: none; - width: 8.3333333333%; } - .column.is-offset-1-mobile { - margin-left: 8.3333333333%; } - .column.is-2-mobile { - flex: none; - width: 16.6666666667%; } - .column.is-offset-2-mobile { - margin-left: 16.6666666667%; } - .column.is-3-mobile { - flex: none; - width: 25%; } - .column.is-offset-3-mobile { - margin-left: 25%; } - .column.is-4-mobile { - flex: none; - width: 33.3333333333%; } - .column.is-offset-4-mobile { - margin-left: 33.3333333333%; } - .column.is-5-mobile { - flex: none; - width: 41.6666666667%; } - .column.is-offset-5-mobile { - margin-left: 41.6666666667%; } - .column.is-6-mobile { - flex: none; - width: 50%; } - .column.is-offset-6-mobile { - margin-left: 50%; } - .column.is-7-mobile { - flex: none; - width: 58.3333333333%; } - .column.is-offset-7-mobile { - margin-left: 58.3333333333%; } - .column.is-8-mobile { - flex: none; - width: 66.6666666667%; } - .column.is-offset-8-mobile { - margin-left: 66.6666666667%; } - .column.is-9-mobile { - flex: none; - width: 75%; } - .column.is-offset-9-mobile { - margin-left: 75%; } - .column.is-10-mobile { - flex: none; - width: 83.3333333333%; } - .column.is-offset-10-mobile { - margin-left: 83.3333333333%; } - .column.is-11-mobile { - flex: none; - width: 91.6666666667%; } - .column.is-offset-11-mobile { - margin-left: 91.6666666667%; } - .column.is-12-mobile { - flex: none; - width: 100%; } - .column.is-offset-12-mobile { - margin-left: 100%; } } - @media screen and (min-width: 769px), print { - .column.is-narrow, .column.is-narrow-tablet { - flex: none; } - .column.is-full, .column.is-full-tablet { - flex: none; - width: 100%; } - .column.is-three-quarters, .column.is-three-quarters-tablet { - flex: none; - width: 75%; } - .column.is-two-thirds, .column.is-two-thirds-tablet { - flex: none; - width: 66.6666%; } - .column.is-half, .column.is-half-tablet { - flex: none; - width: 50%; } - .column.is-one-third, .column.is-one-third-tablet { - flex: none; - width: 33.3333%; } - .column.is-one-quarter, .column.is-one-quarter-tablet { - flex: none; - width: 25%; } - .column.is-one-fifth, .column.is-one-fifth-tablet { - flex: none; - width: 20%; } - .column.is-two-fifths, .column.is-two-fifths-tablet { - flex: none; - width: 40%; } - .column.is-three-fifths, .column.is-three-fifths-tablet { - flex: none; - width: 60%; } - .column.is-four-fifths, .column.is-four-fifths-tablet { - flex: none; - width: 80%; } - .column.is-offset-three-quarters, .column.is-offset-three-quarters-tablet { - margin-left: 75%; } - .column.is-offset-two-thirds, .column.is-offset-two-thirds-tablet { - margin-left: 66.6666%; } - .column.is-offset-half, .column.is-offset-half-tablet { - margin-left: 50%; } - .column.is-offset-one-third, .column.is-offset-one-third-tablet { - margin-left: 33.3333%; } - .column.is-offset-one-quarter, .column.is-offset-one-quarter-tablet { - margin-left: 25%; } - .column.is-offset-one-fifth, .column.is-offset-one-fifth-tablet { - margin-left: 20%; } - .column.is-offset-two-fifths, .column.is-offset-two-fifths-tablet { - margin-left: 40%; } - .column.is-offset-three-fifths, .column.is-offset-three-fifths-tablet { - margin-left: 60%; } - .column.is-offset-four-fifths, .column.is-offset-four-fifths-tablet { - margin-left: 80%; } - .column.is-0, .column.is-0-tablet { - flex: none; - width: 0%; } - .column.is-offset-0, .column.is-offset-0-tablet { - margin-left: 0%; } - .column.is-1, .column.is-1-tablet { - flex: none; - width: 8.3333333333%; } - .column.is-offset-1, .column.is-offset-1-tablet { - margin-left: 8.3333333333%; } - .column.is-2, .column.is-2-tablet { - flex: none; - width: 16.6666666667%; } - .column.is-offset-2, .column.is-offset-2-tablet { - margin-left: 16.6666666667%; } - .column.is-3, .column.is-3-tablet { - flex: none; - width: 25%; } - .column.is-offset-3, .column.is-offset-3-tablet { - margin-left: 25%; } - .column.is-4, .column.is-4-tablet { - flex: none; - width: 33.3333333333%; } - .column.is-offset-4, .column.is-offset-4-tablet { - margin-left: 33.3333333333%; } - .column.is-5, .column.is-5-tablet { - flex: none; - width: 41.6666666667%; } - .column.is-offset-5, .column.is-offset-5-tablet { - margin-left: 41.6666666667%; } - .column.is-6, .column.is-6-tablet { - flex: none; - width: 50%; } - .column.is-offset-6, .column.is-offset-6-tablet { - margin-left: 50%; } - .column.is-7, .column.is-7-tablet { - flex: none; - width: 58.3333333333%; } - .column.is-offset-7, .column.is-offset-7-tablet { - margin-left: 58.3333333333%; } - .column.is-8, .column.is-8-tablet { - flex: none; - width: 66.6666666667%; } - .column.is-offset-8, .column.is-offset-8-tablet { - margin-left: 66.6666666667%; } - .column.is-9, .column.is-9-tablet { - flex: none; - width: 75%; } - .column.is-offset-9, .column.is-offset-9-tablet { - margin-left: 75%; } - .column.is-10, .column.is-10-tablet { - flex: none; - width: 83.3333333333%; } - .column.is-offset-10, .column.is-offset-10-tablet { - margin-left: 83.3333333333%; } - .column.is-11, .column.is-11-tablet { - flex: none; - width: 91.6666666667%; } - .column.is-offset-11, .column.is-offset-11-tablet { - margin-left: 91.6666666667%; } - .column.is-12, .column.is-12-tablet { - flex: none; - width: 100%; } - .column.is-offset-12, .column.is-offset-12-tablet { - margin-left: 100%; } } - @media screen and (max-width: 1055px) { - .column.is-narrow-touch { - flex: none; } - .column.is-full-touch { - flex: none; - width: 100%; } - .column.is-three-quarters-touch { - flex: none; - width: 75%; } - .column.is-two-thirds-touch { - flex: none; - width: 66.6666%; } - .column.is-half-touch { - flex: none; - width: 50%; } - .column.is-one-third-touch { - flex: none; - width: 33.3333%; } - .column.is-one-quarter-touch { - flex: none; - width: 25%; } - .column.is-one-fifth-touch { - flex: none; - width: 20%; } - .column.is-two-fifths-touch { - flex: none; - width: 40%; } - .column.is-three-fifths-touch { - flex: none; - width: 60%; } - .column.is-four-fifths-touch { - flex: none; - width: 80%; } - .column.is-offset-three-quarters-touch { - margin-left: 75%; } - .column.is-offset-two-thirds-touch { - margin-left: 66.6666%; } - .column.is-offset-half-touch { - margin-left: 50%; } - .column.is-offset-one-third-touch { - margin-left: 33.3333%; } - .column.is-offset-one-quarter-touch { - margin-left: 25%; } - .column.is-offset-one-fifth-touch { - margin-left: 20%; } - .column.is-offset-two-fifths-touch { - margin-left: 40%; } - .column.is-offset-three-fifths-touch { - margin-left: 60%; } - .column.is-offset-four-fifths-touch { - margin-left: 80%; } - .column.is-0-touch { - flex: none; - width: 0%; } - .column.is-offset-0-touch { - margin-left: 0%; } - .column.is-1-touch { - flex: none; - width: 8.3333333333%; } - .column.is-offset-1-touch { - margin-left: 8.3333333333%; } - .column.is-2-touch { - flex: none; - width: 16.6666666667%; } - .column.is-offset-2-touch { - margin-left: 16.6666666667%; } - .column.is-3-touch { - flex: none; - width: 25%; } - .column.is-offset-3-touch { - margin-left: 25%; } - .column.is-4-touch { - flex: none; - width: 33.3333333333%; } - .column.is-offset-4-touch { - margin-left: 33.3333333333%; } - .column.is-5-touch { - flex: none; - width: 41.6666666667%; } - .column.is-offset-5-touch { - margin-left: 41.6666666667%; } - .column.is-6-touch { - flex: none; - width: 50%; } - .column.is-offset-6-touch { - margin-left: 50%; } - .column.is-7-touch { - flex: none; - width: 58.3333333333%; } - .column.is-offset-7-touch { - margin-left: 58.3333333333%; } - .column.is-8-touch { - flex: none; - width: 66.6666666667%; } - .column.is-offset-8-touch { - margin-left: 66.6666666667%; } - .column.is-9-touch { - flex: none; - width: 75%; } - .column.is-offset-9-touch { - margin-left: 75%; } - .column.is-10-touch { - flex: none; - width: 83.3333333333%; } - .column.is-offset-10-touch { - margin-left: 83.3333333333%; } - .column.is-11-touch { - flex: none; - width: 91.6666666667%; } - .column.is-offset-11-touch { - margin-left: 91.6666666667%; } - .column.is-12-touch { - flex: none; - width: 100%; } - .column.is-offset-12-touch { - margin-left: 100%; } } - @media screen and (min-width: 1056px) { - .column.is-narrow-desktop { - flex: none; } - .column.is-full-desktop { - flex: none; - width: 100%; } - .column.is-three-quarters-desktop { - flex: none; - width: 75%; } - .column.is-two-thirds-desktop { - flex: none; - width: 66.6666%; } - .column.is-half-desktop { - flex: none; - width: 50%; } - .column.is-one-third-desktop { - flex: none; - width: 33.3333%; } - .column.is-one-quarter-desktop { - flex: none; - width: 25%; } - .column.is-one-fifth-desktop { - flex: none; - width: 20%; } - .column.is-two-fifths-desktop { - flex: none; - width: 40%; } - .column.is-three-fifths-desktop { - flex: none; - width: 60%; } - .column.is-four-fifths-desktop { - flex: none; - width: 80%; } - .column.is-offset-three-quarters-desktop { - margin-left: 75%; } - .column.is-offset-two-thirds-desktop { - margin-left: 66.6666%; } - .column.is-offset-half-desktop { - margin-left: 50%; } - .column.is-offset-one-third-desktop { - margin-left: 33.3333%; } - .column.is-offset-one-quarter-desktop { - margin-left: 25%; } - .column.is-offset-one-fifth-desktop { - margin-left: 20%; } - .column.is-offset-two-fifths-desktop { - margin-left: 40%; } - .column.is-offset-three-fifths-desktop { - margin-left: 60%; } - .column.is-offset-four-fifths-desktop { - margin-left: 80%; } - .column.is-0-desktop { - flex: none; - width: 0%; } - .column.is-offset-0-desktop { - margin-left: 0%; } - .column.is-1-desktop { - flex: none; - width: 8.3333333333%; } - .column.is-offset-1-desktop { - margin-left: 8.3333333333%; } - .column.is-2-desktop { - flex: none; - width: 16.6666666667%; } - .column.is-offset-2-desktop { - margin-left: 16.6666666667%; } - .column.is-3-desktop { - flex: none; - width: 25%; } - .column.is-offset-3-desktop { - margin-left: 25%; } - .column.is-4-desktop { - flex: none; - width: 33.3333333333%; } - .column.is-offset-4-desktop { - margin-left: 33.3333333333%; } - .column.is-5-desktop { - flex: none; - width: 41.6666666667%; } - .column.is-offset-5-desktop { - margin-left: 41.6666666667%; } - .column.is-6-desktop { - flex: none; - width: 50%; } - .column.is-offset-6-desktop { - margin-left: 50%; } - .column.is-7-desktop { - flex: none; - width: 58.3333333333%; } - .column.is-offset-7-desktop { - margin-left: 58.3333333333%; } - .column.is-8-desktop { - flex: none; - width: 66.6666666667%; } - .column.is-offset-8-desktop { - margin-left: 66.6666666667%; } - .column.is-9-desktop { - flex: none; - width: 75%; } - .column.is-offset-9-desktop { - margin-left: 75%; } - .column.is-10-desktop { - flex: none; - width: 83.3333333333%; } - .column.is-offset-10-desktop { - margin-left: 83.3333333333%; } - .column.is-11-desktop { - flex: none; - width: 91.6666666667%; } - .column.is-offset-11-desktop { - margin-left: 91.6666666667%; } - .column.is-12-desktop { - flex: none; - width: 100%; } - .column.is-offset-12-desktop { - margin-left: 100%; } } - @media screen and (min-width: 1216px) { - .column.is-narrow-widescreen { - flex: none; } - .column.is-full-widescreen { - flex: none; - width: 100%; } - .column.is-three-quarters-widescreen { - flex: none; - width: 75%; } - .column.is-two-thirds-widescreen { - flex: none; - width: 66.6666%; } - .column.is-half-widescreen { - flex: none; - width: 50%; } - .column.is-one-third-widescreen { - flex: none; - width: 33.3333%; } - .column.is-one-quarter-widescreen { - flex: none; - width: 25%; } - .column.is-one-fifth-widescreen { - flex: none; - width: 20%; } - .column.is-two-fifths-widescreen { - flex: none; - width: 40%; } - .column.is-three-fifths-widescreen { - flex: none; - width: 60%; } - .column.is-four-fifths-widescreen { - flex: none; - width: 80%; } - .column.is-offset-three-quarters-widescreen { - margin-left: 75%; } - .column.is-offset-two-thirds-widescreen { - margin-left: 66.6666%; } - .column.is-offset-half-widescreen { - margin-left: 50%; } - .column.is-offset-one-third-widescreen { - margin-left: 33.3333%; } - .column.is-offset-one-quarter-widescreen { - margin-left: 25%; } - .column.is-offset-one-fifth-widescreen { - margin-left: 20%; } - .column.is-offset-two-fifths-widescreen { - margin-left: 40%; } - .column.is-offset-three-fifths-widescreen { - margin-left: 60%; } - .column.is-offset-four-fifths-widescreen { - margin-left: 80%; } - .column.is-0-widescreen { - flex: none; - width: 0%; } - .column.is-offset-0-widescreen { - margin-left: 0%; } - .column.is-1-widescreen { - flex: none; - width: 8.3333333333%; } - .column.is-offset-1-widescreen { - margin-left: 8.3333333333%; } - .column.is-2-widescreen { - flex: none; - width: 16.6666666667%; } - .column.is-offset-2-widescreen { - margin-left: 16.6666666667%; } - .column.is-3-widescreen { - flex: none; - width: 25%; } - .column.is-offset-3-widescreen { - margin-left: 25%; } - .column.is-4-widescreen { - flex: none; - width: 33.3333333333%; } - .column.is-offset-4-widescreen { - margin-left: 33.3333333333%; } - .column.is-5-widescreen { - flex: none; - width: 41.6666666667%; } - .column.is-offset-5-widescreen { - margin-left: 41.6666666667%; } - .column.is-6-widescreen { - flex: none; - width: 50%; } - .column.is-offset-6-widescreen { - margin-left: 50%; } - .column.is-7-widescreen { - flex: none; - width: 58.3333333333%; } - .column.is-offset-7-widescreen { - margin-left: 58.3333333333%; } - .column.is-8-widescreen { - flex: none; - width: 66.6666666667%; } - .column.is-offset-8-widescreen { - margin-left: 66.6666666667%; } - .column.is-9-widescreen { - flex: none; - width: 75%; } - .column.is-offset-9-widescreen { - margin-left: 75%; } - .column.is-10-widescreen { - flex: none; - width: 83.3333333333%; } - .column.is-offset-10-widescreen { - margin-left: 83.3333333333%; } - .column.is-11-widescreen { - flex: none; - width: 91.6666666667%; } - .column.is-offset-11-widescreen { - margin-left: 91.6666666667%; } - .column.is-12-widescreen { - flex: none; - width: 100%; } - .column.is-offset-12-widescreen { - margin-left: 100%; } } - @media screen and (min-width: 1408px) { - .column.is-narrow-fullhd { - flex: none; } - .column.is-full-fullhd { - flex: none; - width: 100%; } - .column.is-three-quarters-fullhd { - flex: none; - width: 75%; } - .column.is-two-thirds-fullhd { - flex: none; - width: 66.6666%; } - .column.is-half-fullhd { - flex: none; - width: 50%; } - .column.is-one-third-fullhd { - flex: none; - width: 33.3333%; } - .column.is-one-quarter-fullhd { - flex: none; - width: 25%; } - .column.is-one-fifth-fullhd { - flex: none; - width: 20%; } - .column.is-two-fifths-fullhd { - flex: none; - width: 40%; } - .column.is-three-fifths-fullhd { - flex: none; - width: 60%; } - .column.is-four-fifths-fullhd { - flex: none; - width: 80%; } - .column.is-offset-three-quarters-fullhd { - margin-left: 75%; } - .column.is-offset-two-thirds-fullhd { - margin-left: 66.6666%; } - .column.is-offset-half-fullhd { - margin-left: 50%; } - .column.is-offset-one-third-fullhd { - margin-left: 33.3333%; } - .column.is-offset-one-quarter-fullhd { - margin-left: 25%; } - .column.is-offset-one-fifth-fullhd { - margin-left: 20%; } - .column.is-offset-two-fifths-fullhd { - margin-left: 40%; } - .column.is-offset-three-fifths-fullhd { - margin-left: 60%; } - .column.is-offset-four-fifths-fullhd { - margin-left: 80%; } - .column.is-0-fullhd { - flex: none; - width: 0%; } - .column.is-offset-0-fullhd { - margin-left: 0%; } - .column.is-1-fullhd { - flex: none; - width: 8.3333333333%; } - .column.is-offset-1-fullhd { - margin-left: 8.3333333333%; } - .column.is-2-fullhd { - flex: none; - width: 16.6666666667%; } - .column.is-offset-2-fullhd { - margin-left: 16.6666666667%; } - .column.is-3-fullhd { - flex: none; - width: 25%; } - .column.is-offset-3-fullhd { - margin-left: 25%; } - .column.is-4-fullhd { - flex: none; - width: 33.3333333333%; } - .column.is-offset-4-fullhd { - margin-left: 33.3333333333%; } - .column.is-5-fullhd { - flex: none; - width: 41.6666666667%; } - .column.is-offset-5-fullhd { - margin-left: 41.6666666667%; } - .column.is-6-fullhd { - flex: none; - width: 50%; } - .column.is-offset-6-fullhd { - margin-left: 50%; } - .column.is-7-fullhd { - flex: none; - width: 58.3333333333%; } - .column.is-offset-7-fullhd { - margin-left: 58.3333333333%; } - .column.is-8-fullhd { - flex: none; - width: 66.6666666667%; } - .column.is-offset-8-fullhd { - margin-left: 66.6666666667%; } - .column.is-9-fullhd { - flex: none; - width: 75%; } - .column.is-offset-9-fullhd { - margin-left: 75%; } - .column.is-10-fullhd { - flex: none; - width: 83.3333333333%; } - .column.is-offset-10-fullhd { - margin-left: 83.3333333333%; } - .column.is-11-fullhd { - flex: none; - width: 91.6666666667%; } - .column.is-offset-11-fullhd { - margin-left: 91.6666666667%; } - .column.is-12-fullhd { - flex: none; - width: 100%; } - .column.is-offset-12-fullhd { - margin-left: 100%; } } -.columns { - margin-left: -0.75rem; - margin-right: -0.75rem; - margin-top: -0.75rem; } - .columns:last-child { - margin-bottom: -0.75rem; } - .columns:not(:last-child) { - margin-bottom: calc(1.5rem - 0.75rem); } - .columns.is-centered { - justify-content: center; } - .columns.is-gapless { - margin-left: 0; - margin-right: 0; - margin-top: 0; } - .columns.is-gapless > .column { - margin: 0; - padding: 0 !important; } - .columns.is-gapless:not(:last-child) { - margin-bottom: 1.5rem; } - .columns.is-gapless:last-child { - margin-bottom: 0; } - .columns.is-mobile { - display: flex; } - .columns.is-multiline { - flex-wrap: wrap; } - .columns.is-vcentered { - align-items: center; } - @media screen and (min-width: 769px), print { - .columns:not(.is-desktop) { - display: flex; } } - @media screen and (min-width: 1056px) { - .columns.is-desktop { - display: flex; } } -.columns.is-variable { - --columnGap: 0.75rem; - margin-left: calc(-1 * var(--columnGap)); - margin-right: calc(-1 * var(--columnGap)); } - .columns.is-variable .column { - padding-left: var(--columnGap); - padding-right: var(--columnGap); } - .columns.is-variable.is-0 { - --columnGap: 0rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-0-mobile { - --columnGap: 0rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-0-tablet { - --columnGap: 0rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-0-tablet-only { - --columnGap: 0rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-0-touch { - --columnGap: 0rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-0-desktop { - --columnGap: 0rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-0-desktop-only { - --columnGap: 0rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-0-widescreen { - --columnGap: 0rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-0-widescreen-only { - --columnGap: 0rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-0-fullhd { - --columnGap: 0rem; } } - .columns.is-variable.is-1 { - --columnGap: 0.25rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-1-mobile { - --columnGap: 0.25rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-1-tablet { - --columnGap: 0.25rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-1-tablet-only { - --columnGap: 0.25rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-1-touch { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-1-desktop { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-1-desktop-only { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-1-widescreen { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-1-widescreen-only { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-1-fullhd { - --columnGap: 0.25rem; } } - .columns.is-variable.is-2 { - --columnGap: 0.5rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-2-mobile { - --columnGap: 0.5rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-2-tablet { - --columnGap: 0.5rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-2-tablet-only { - --columnGap: 0.5rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-2-touch { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-2-desktop { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-2-desktop-only { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-2-widescreen { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-2-widescreen-only { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-2-fullhd { - --columnGap: 0.5rem; } } - .columns.is-variable.is-3 { - --columnGap: 0.75rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-3-mobile { - --columnGap: 0.75rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-3-tablet { - --columnGap: 0.75rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-3-tablet-only { - --columnGap: 0.75rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-3-touch { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-3-desktop { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-3-desktop-only { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-3-widescreen { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-3-widescreen-only { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-3-fullhd { - --columnGap: 0.75rem; } } - .columns.is-variable.is-4 { - --columnGap: 1rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-4-mobile { - --columnGap: 1rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-4-tablet { - --columnGap: 1rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-4-tablet-only { - --columnGap: 1rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-4-touch { - --columnGap: 1rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-4-desktop { - --columnGap: 1rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-4-desktop-only { - --columnGap: 1rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-4-widescreen { - --columnGap: 1rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-4-widescreen-only { - --columnGap: 1rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-4-fullhd { - --columnGap: 1rem; } } - .columns.is-variable.is-5 { - --columnGap: 1.25rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-5-mobile { - --columnGap: 1.25rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-5-tablet { - --columnGap: 1.25rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-5-tablet-only { - --columnGap: 1.25rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-5-touch { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-5-desktop { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-5-desktop-only { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-5-widescreen { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-5-widescreen-only { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-5-fullhd { - --columnGap: 1.25rem; } } - .columns.is-variable.is-6 { - --columnGap: 1.5rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-6-mobile { - --columnGap: 1.5rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-6-tablet { - --columnGap: 1.5rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-6-tablet-only { - --columnGap: 1.5rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-6-touch { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-6-desktop { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-6-desktop-only { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-6-widescreen { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-6-widescreen-only { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-6-fullhd { - --columnGap: 1.5rem; } } - .columns.is-variable.is-7 { - --columnGap: 1.75rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-7-mobile { - --columnGap: 1.75rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-7-tablet { - --columnGap: 1.75rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-7-tablet-only { - --columnGap: 1.75rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-7-touch { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-7-desktop { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-7-desktop-only { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-7-widescreen { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-7-widescreen-only { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-7-fullhd { - --columnGap: 1.75rem; } } - .columns.is-variable.is-8 { - --columnGap: 2rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-8-mobile { - --columnGap: 2rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-8-tablet { - --columnGap: 2rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-8-tablet-only { - --columnGap: 2rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-8-touch { - --columnGap: 2rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-8-desktop { - --columnGap: 2rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-8-desktop-only { - --columnGap: 2rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-8-widescreen { - --columnGap: 2rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-8-widescreen-only { - --columnGap: 2rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-8-fullhd { - --columnGap: 2rem; } } -.tile { - align-items: stretch; - display: block; - flex-basis: 0; - flex-grow: 1; - flex-shrink: 1; - min-height: min-content; } - .tile.is-ancestor { - margin-left: -0.75rem; - margin-right: -0.75rem; - margin-top: -0.75rem; } - .tile.is-ancestor:last-child { - margin-bottom: -0.75rem; } - .tile.is-ancestor:not(:last-child) { - margin-bottom: 0.75rem; } - .tile.is-child { - margin: 0 !important; } - .tile.is-parent { - padding: 0.75rem; } - .tile.is-vertical { - flex-direction: column; } - .tile.is-vertical > .tile.is-child:not(:last-child) { - margin-bottom: 1.5rem !important; } - @media screen and (min-width: 769px), print { - .tile:not(.is-child) { - display: flex; } - .tile.is-1 { - flex: none; - width: 8.3333333333%; } - .tile.is-2 { - flex: none; - width: 16.6666666667%; } - .tile.is-3 { - flex: none; - width: 25%; } - .tile.is-4 { - flex: none; - width: 33.3333333333%; } - .tile.is-5 { - flex: none; - width: 41.6666666667%; } - .tile.is-6 { - flex: none; - width: 50%; } - .tile.is-7 { - flex: none; - width: 58.3333333333%; } - .tile.is-8 { - flex: none; - width: 66.6666666667%; } - .tile.is-9 { - flex: none; - width: 75%; } - .tile.is-10 { - flex: none; - width: 83.3333333333%; } - .tile.is-11 { - flex: none; - width: 91.6666666667%; } - .tile.is-12 { - flex: none; - width: 100%; } } -.hero { - align-items: stretch; - display: flex; - flex-direction: column; - justify-content: space-between; } - .hero .navbar { - background: none; } - .hero .tabs ul { - border-bottom: none; } - .hero.is-white { - background-color: white; - color: #0a0a0a; } - .hero.is-white a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-white strong { - color: inherit; } - .hero.is-white .title { - color: #0a0a0a; } - .hero.is-white .subtitle { - color: rgba(10, 10, 10, 0.9); } - .hero.is-white .subtitle a:not(.button), - .hero.is-white .subtitle strong { - color: #0a0a0a; } - @media screen and (max-width: 1055px) { - .hero.is-white .navbar-menu { - background-color: white; } } - .hero.is-white .navbar-item, - .hero.is-white .navbar-link { - color: rgba(10, 10, 10, 0.7); } - .hero.is-white a.navbar-item:hover, .hero.is-white a.navbar-item.is-active, - .hero.is-white .navbar-link:hover, - .hero.is-white .navbar-link.is-active { - background-color: #f2f2f2; - color: #0a0a0a; } - .hero.is-white .tabs a { - color: #0a0a0a; - opacity: 0.9; } - .hero.is-white .tabs a:hover { - opacity: 1; } - .hero.is-white .tabs li.is-active a { - opacity: 1; } - .hero.is-white .tabs.is-boxed a, .hero.is-white .tabs.is-toggle a { - color: #0a0a0a; } - .hero.is-white .tabs.is-boxed a:hover, .hero.is-white .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-white .tabs.is-boxed li.is-active a, .hero.is-white .tabs.is-boxed li.is-active a:hover, .hero.is-white .tabs.is-toggle li.is-active a, .hero.is-white .tabs.is-toggle li.is-active a:hover { - background-color: #0a0a0a; - border-color: #0a0a0a; - color: white; } - .hero.is-white.is-bold { - background-image: linear-gradient(141deg, #e8e3e4 0%, white 71%, white 100%); } - @media screen and (max-width: 768px) { - .hero.is-white.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #e8e3e4 0%, white 71%, white 100%); } } - .hero.is-black { - background-color: #0a0a0a; - color: white; } - .hero.is-black a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-black strong { - color: inherit; } - .hero.is-black .title { - color: white; } - .hero.is-black .subtitle { - color: rgba(255, 255, 255, 0.9); } - .hero.is-black .subtitle a:not(.button), - .hero.is-black .subtitle strong { - color: white; } - @media screen and (max-width: 1055px) { - .hero.is-black .navbar-menu { - background-color: #0a0a0a; } } - .hero.is-black .navbar-item, - .hero.is-black .navbar-link { - color: rgba(255, 255, 255, 0.7); } - .hero.is-black a.navbar-item:hover, .hero.is-black a.navbar-item.is-active, - .hero.is-black .navbar-link:hover, - .hero.is-black .navbar-link.is-active { - background-color: black; - color: white; } - .hero.is-black .tabs a { - color: white; - opacity: 0.9; } - .hero.is-black .tabs a:hover { - opacity: 1; } - .hero.is-black .tabs li.is-active a { - opacity: 1; } - .hero.is-black .tabs.is-boxed a, .hero.is-black .tabs.is-toggle a { - color: white; } - .hero.is-black .tabs.is-boxed a:hover, .hero.is-black .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-black .tabs.is-boxed li.is-active a, .hero.is-black .tabs.is-boxed li.is-active a:hover, .hero.is-black .tabs.is-toggle li.is-active a, .hero.is-black .tabs.is-toggle li.is-active a:hover { - background-color: white; - border-color: white; - color: #0a0a0a; } - .hero.is-black.is-bold { - background-image: linear-gradient(141deg, black 0%, #0a0a0a 71%, #181616 100%); } - @media screen and (max-width: 768px) { - .hero.is-black.is-bold .navbar-menu { - background-image: linear-gradient(141deg, black 0%, #0a0a0a 71%, #181616 100%); } } - .hero.is-light { - background-color: whitesmoke; - color: #363636; } - .hero.is-light a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-light strong { - color: inherit; } - .hero.is-light .title { - color: #363636; } - .hero.is-light .subtitle { - color: rgba(54, 54, 54, 0.9); } - .hero.is-light .subtitle a:not(.button), - .hero.is-light .subtitle strong { - color: #363636; } - @media screen and (max-width: 1055px) { - .hero.is-light .navbar-menu { - background-color: whitesmoke; } } - .hero.is-light .navbar-item, - .hero.is-light .navbar-link { - color: rgba(54, 54, 54, 0.7); } - .hero.is-light a.navbar-item:hover, .hero.is-light a.navbar-item.is-active, - .hero.is-light .navbar-link:hover, - .hero.is-light .navbar-link.is-active { - background-color: #e8e8e8; - color: #363636; } - .hero.is-light .tabs a { - color: #363636; - opacity: 0.9; } - .hero.is-light .tabs a:hover { - opacity: 1; } - .hero.is-light .tabs li.is-active a { - opacity: 1; } - .hero.is-light .tabs.is-boxed a, .hero.is-light .tabs.is-toggle a { - color: #363636; } - .hero.is-light .tabs.is-boxed a:hover, .hero.is-light .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-light .tabs.is-boxed li.is-active a, .hero.is-light .tabs.is-boxed li.is-active a:hover, .hero.is-light .tabs.is-toggle li.is-active a, .hero.is-light .tabs.is-toggle li.is-active a:hover { - background-color: #363636; - border-color: #363636; - color: whitesmoke; } - .hero.is-light.is-bold { - background-image: linear-gradient(141deg, #dfd8d9 0%, whitesmoke 71%, white 100%); } - @media screen and (max-width: 768px) { - .hero.is-light.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #dfd8d9 0%, whitesmoke 71%, white 100%); } } - .hero.is-dark, .content kbd.hero { - background-color: #363636; - color: whitesmoke; } - .hero.is-dark a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), .content kbd.hero a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-dark strong, - .content kbd.hero strong { - color: inherit; } - .hero.is-dark .title, .content kbd.hero .title { - color: whitesmoke; } - .hero.is-dark .subtitle, .content kbd.hero .subtitle { - color: rgba(245, 245, 245, 0.9); } - .hero.is-dark .subtitle a:not(.button), .content kbd.hero .subtitle a:not(.button), - .hero.is-dark .subtitle strong, - .content kbd.hero .subtitle strong { - color: whitesmoke; } - @media screen and (max-width: 1055px) { - .hero.is-dark .navbar-menu, .content kbd.hero .navbar-menu { - background-color: #363636; } } - .hero.is-dark .navbar-item, .content kbd.hero .navbar-item, - .hero.is-dark .navbar-link, - .content kbd.hero .navbar-link { - color: rgba(245, 245, 245, 0.7); } - .hero.is-dark a.navbar-item:hover, .content kbd.hero a.navbar-item:hover, .hero.is-dark a.navbar-item.is-active, .content kbd.hero a.navbar-item.is-active, - .hero.is-dark .navbar-link:hover, - .content kbd.hero .navbar-link:hover, - .hero.is-dark .navbar-link.is-active, - .content kbd.hero .navbar-link.is-active { - background-color: #292929; - color: whitesmoke; } - .hero.is-dark .tabs a, .content kbd.hero .tabs a { - color: whitesmoke; - opacity: 0.9; } - .hero.is-dark .tabs a:hover, .content kbd.hero .tabs a:hover { - opacity: 1; } - .hero.is-dark .tabs li.is-active a, .content kbd.hero .tabs li.is-active a { - opacity: 1; } - .hero.is-dark .tabs.is-boxed a, .content kbd.hero .tabs.is-boxed a, .hero.is-dark .tabs.is-toggle a, .content kbd.hero .tabs.is-toggle a { - color: whitesmoke; } - .hero.is-dark .tabs.is-boxed a:hover, .content kbd.hero .tabs.is-boxed a:hover, .hero.is-dark .tabs.is-toggle a:hover, .content kbd.hero .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-dark .tabs.is-boxed li.is-active a, .content kbd.hero .tabs.is-boxed li.is-active a, .hero.is-dark .tabs.is-boxed li.is-active a:hover, .hero.is-dark .tabs.is-toggle li.is-active a, .content kbd.hero .tabs.is-toggle li.is-active a, .hero.is-dark .tabs.is-toggle li.is-active a:hover { - background-color: whitesmoke; - border-color: whitesmoke; - color: #363636; } - .hero.is-dark.is-bold, .content kbd.hero.is-bold { - background-image: linear-gradient(141deg, #1f191a 0%, #363636 71%, #46403f 100%); } - @media screen and (max-width: 768px) { - .hero.is-dark.is-bold .navbar-menu, .content kbd.hero.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #1f191a 0%, #363636 71%, #46403f 100%); } } - .hero.is-primary, .docstring > section > a.hero.docs-sourcelink { - background-color: #4eb5de; - color: #fff; } - .hero.is-primary a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), .docstring > section > a.hero.docs-sourcelink a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-primary strong, - .docstring > section > a.hero.docs-sourcelink strong { - color: inherit; } - .hero.is-primary .title, .docstring > section > a.hero.docs-sourcelink .title { - color: #fff; } - .hero.is-primary .subtitle, .docstring > section > a.hero.docs-sourcelink .subtitle { - color: rgba(255, 255, 255, 0.9); } - .hero.is-primary .subtitle a:not(.button), .docstring > section > a.hero.docs-sourcelink .subtitle a:not(.button), - .hero.is-primary .subtitle strong, - .docstring > section > a.hero.docs-sourcelink .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - .hero.is-primary .navbar-menu, .docstring > section > a.hero.docs-sourcelink .navbar-menu { - background-color: #4eb5de; } } - .hero.is-primary .navbar-item, .docstring > section > a.hero.docs-sourcelink .navbar-item, - .hero.is-primary .navbar-link, - .docstring > section > a.hero.docs-sourcelink .navbar-link { - color: rgba(255, 255, 255, 0.7); } - .hero.is-primary a.navbar-item:hover, .docstring > section > a.hero.docs-sourcelink a.navbar-item:hover, .hero.is-primary a.navbar-item.is-active, .docstring > section > a.hero.docs-sourcelink a.navbar-item.is-active, - .hero.is-primary .navbar-link:hover, - .docstring > section > a.hero.docs-sourcelink .navbar-link:hover, - .hero.is-primary .navbar-link.is-active, - .docstring > section > a.hero.docs-sourcelink .navbar-link.is-active { - background-color: #39acda; - color: #fff; } - .hero.is-primary .tabs a, .docstring > section > a.hero.docs-sourcelink .tabs a { - color: #fff; - opacity: 0.9; } - .hero.is-primary .tabs a:hover, .docstring > section > a.hero.docs-sourcelink .tabs a:hover { - opacity: 1; } - .hero.is-primary .tabs li.is-active a, .docstring > section > a.hero.docs-sourcelink .tabs li.is-active a { - opacity: 1; } - .hero.is-primary .tabs.is-boxed a, .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed a, .hero.is-primary .tabs.is-toggle a, .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle a { - color: #fff; } - .hero.is-primary .tabs.is-boxed a:hover, .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed a:hover, .hero.is-primary .tabs.is-toggle a:hover, .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-primary .tabs.is-boxed li.is-active a, .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed li.is-active a, .hero.is-primary .tabs.is-boxed li.is-active a:hover, .hero.is-primary .tabs.is-toggle li.is-active a, .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle li.is-active a, .hero.is-primary .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #4eb5de; } - .hero.is-primary.is-bold, .docstring > section > a.hero.is-bold.docs-sourcelink { - background-image: linear-gradient(141deg, #1bc7de 0%, #4eb5de 71%, #5fa9e7 100%); } - @media screen and (max-width: 768px) { - .hero.is-primary.is-bold .navbar-menu, .docstring > section > a.hero.is-bold.docs-sourcelink .navbar-menu { - background-image: linear-gradient(141deg, #1bc7de 0%, #4eb5de 71%, #5fa9e7 100%); } } - .hero.is-link { - background-color: #2e63b8; - color: #fff; } - .hero.is-link a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-link strong { - color: inherit; } - .hero.is-link .title { - color: #fff; } - .hero.is-link .subtitle { - color: rgba(255, 255, 255, 0.9); } - .hero.is-link .subtitle a:not(.button), - .hero.is-link .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - .hero.is-link .navbar-menu { - background-color: #2e63b8; } } - .hero.is-link .navbar-item, - .hero.is-link .navbar-link { - color: rgba(255, 255, 255, 0.7); } - .hero.is-link a.navbar-item:hover, .hero.is-link a.navbar-item.is-active, - .hero.is-link .navbar-link:hover, - .hero.is-link .navbar-link.is-active { - background-color: #2958a4; - color: #fff; } - .hero.is-link .tabs a { - color: #fff; - opacity: 0.9; } - .hero.is-link .tabs a:hover { - opacity: 1; } - .hero.is-link .tabs li.is-active a { - opacity: 1; } - .hero.is-link .tabs.is-boxed a, .hero.is-link .tabs.is-toggle a { - color: #fff; } - .hero.is-link .tabs.is-boxed a:hover, .hero.is-link .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-link .tabs.is-boxed li.is-active a, .hero.is-link .tabs.is-boxed li.is-active a:hover, .hero.is-link .tabs.is-toggle li.is-active a, .hero.is-link .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #2e63b8; } - .hero.is-link.is-bold { - background-image: linear-gradient(141deg, #1b6098 0%, #2e63b8 71%, #2d51d2 100%); } - @media screen and (max-width: 768px) { - .hero.is-link.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #1b6098 0%, #2e63b8 71%, #2d51d2 100%); } } - .hero.is-info { - background-color: #209cee; - color: #fff; } - .hero.is-info a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-info strong { - color: inherit; } - .hero.is-info .title { - color: #fff; } - .hero.is-info .subtitle { - color: rgba(255, 255, 255, 0.9); } - .hero.is-info .subtitle a:not(.button), - .hero.is-info .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - .hero.is-info .navbar-menu { - background-color: #209cee; } } - .hero.is-info .navbar-item, - .hero.is-info .navbar-link { - color: rgba(255, 255, 255, 0.7); } - .hero.is-info a.navbar-item:hover, .hero.is-info a.navbar-item.is-active, - .hero.is-info .navbar-link:hover, - .hero.is-info .navbar-link.is-active { - background-color: #1190e3; - color: #fff; } - .hero.is-info .tabs a { - color: #fff; - opacity: 0.9; } - .hero.is-info .tabs a:hover { - opacity: 1; } - .hero.is-info .tabs li.is-active a { - opacity: 1; } - .hero.is-info .tabs.is-boxed a, .hero.is-info .tabs.is-toggle a { - color: #fff; } - .hero.is-info .tabs.is-boxed a:hover, .hero.is-info .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-info .tabs.is-boxed li.is-active a, .hero.is-info .tabs.is-boxed li.is-active a:hover, .hero.is-info .tabs.is-toggle li.is-active a, .hero.is-info .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #209cee; } - .hero.is-info.is-bold { - background-image: linear-gradient(141deg, #05a6d6 0%, #209cee 71%, #3287f5 100%); } - @media screen and (max-width: 768px) { - .hero.is-info.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #05a6d6 0%, #209cee 71%, #3287f5 100%); } } - .hero.is-success { - background-color: #22c35b; - color: #fff; } - .hero.is-success a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-success strong { - color: inherit; } - .hero.is-success .title { - color: #fff; } - .hero.is-success .subtitle { - color: rgba(255, 255, 255, 0.9); } - .hero.is-success .subtitle a:not(.button), - .hero.is-success .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - .hero.is-success .navbar-menu { - background-color: #22c35b; } } - .hero.is-success .navbar-item, - .hero.is-success .navbar-link { - color: rgba(255, 255, 255, 0.7); } - .hero.is-success a.navbar-item:hover, .hero.is-success a.navbar-item.is-active, - .hero.is-success .navbar-link:hover, - .hero.is-success .navbar-link.is-active { - background-color: #1ead51; - color: #fff; } - .hero.is-success .tabs a { - color: #fff; - opacity: 0.9; } - .hero.is-success .tabs a:hover { - opacity: 1; } - .hero.is-success .tabs li.is-active a { - opacity: 1; } - .hero.is-success .tabs.is-boxed a, .hero.is-success .tabs.is-toggle a { - color: #fff; } - .hero.is-success .tabs.is-boxed a:hover, .hero.is-success .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-success .tabs.is-boxed li.is-active a, .hero.is-success .tabs.is-boxed li.is-active a:hover, .hero.is-success .tabs.is-toggle li.is-active a, .hero.is-success .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #22c35b; } - .hero.is-success.is-bold { - background-image: linear-gradient(141deg, #12a02c 0%, #22c35b 71%, #1fdf83 100%); } - @media screen and (max-width: 768px) { - .hero.is-success.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #12a02c 0%, #22c35b 71%, #1fdf83 100%); } } - .hero.is-warning { - background-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .hero.is-warning a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-warning strong { - color: inherit; } - .hero.is-warning .title { - color: rgba(0, 0, 0, 0.7); } - .hero.is-warning .subtitle { - color: rgba(0, 0, 0, 0.9); } - .hero.is-warning .subtitle a:not(.button), - .hero.is-warning .subtitle strong { - color: rgba(0, 0, 0, 0.7); } - @media screen and (max-width: 1055px) { - .hero.is-warning .navbar-menu { - background-color: #ffdd57; } } - .hero.is-warning .navbar-item, - .hero.is-warning .navbar-link { - color: rgba(0, 0, 0, 0.7); } - .hero.is-warning a.navbar-item:hover, .hero.is-warning a.navbar-item.is-active, - .hero.is-warning .navbar-link:hover, - .hero.is-warning .navbar-link.is-active { - background-color: #ffd83e; - color: rgba(0, 0, 0, 0.7); } - .hero.is-warning .tabs a { - color: rgba(0, 0, 0, 0.7); - opacity: 0.9; } - .hero.is-warning .tabs a:hover { - opacity: 1; } - .hero.is-warning .tabs li.is-active a { - opacity: 1; } - .hero.is-warning .tabs.is-boxed a, .hero.is-warning .tabs.is-toggle a { - color: rgba(0, 0, 0, 0.7); } - .hero.is-warning .tabs.is-boxed a:hover, .hero.is-warning .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-warning .tabs.is-boxed li.is-active a, .hero.is-warning .tabs.is-boxed li.is-active a:hover, .hero.is-warning .tabs.is-toggle li.is-active a, .hero.is-warning .tabs.is-toggle li.is-active a:hover { - background-color: rgba(0, 0, 0, 0.7); - border-color: rgba(0, 0, 0, 0.7); - color: #ffdd57; } - .hero.is-warning.is-bold { - background-image: linear-gradient(141deg, #ffae24 0%, #ffdd57 71%, #fffa71 100%); } - @media screen and (max-width: 768px) { - .hero.is-warning.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #ffae24 0%, #ffdd57 71%, #fffa71 100%); } } - .hero.is-danger { - background-color: #da0b00; - color: #fff; } - .hero.is-danger a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-danger strong { - color: inherit; } - .hero.is-danger .title { - color: #fff; } - .hero.is-danger .subtitle { - color: rgba(255, 255, 255, 0.9); } - .hero.is-danger .subtitle a:not(.button), - .hero.is-danger .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - .hero.is-danger .navbar-menu { - background-color: #da0b00; } } - .hero.is-danger .navbar-item, - .hero.is-danger .navbar-link { - color: rgba(255, 255, 255, 0.7); } - .hero.is-danger a.navbar-item:hover, .hero.is-danger a.navbar-item.is-active, - .hero.is-danger .navbar-link:hover, - .hero.is-danger .navbar-link.is-active { - background-color: #c10a00; - color: #fff; } - .hero.is-danger .tabs a { - color: #fff; - opacity: 0.9; } - .hero.is-danger .tabs a:hover { - opacity: 1; } - .hero.is-danger .tabs li.is-active a { - opacity: 1; } - .hero.is-danger .tabs.is-boxed a, .hero.is-danger .tabs.is-toggle a { - color: #fff; } - .hero.is-danger .tabs.is-boxed a:hover, .hero.is-danger .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-danger .tabs.is-boxed li.is-active a, .hero.is-danger .tabs.is-boxed li.is-active a:hover, .hero.is-danger .tabs.is-toggle li.is-active a, .hero.is-danger .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #da0b00; } - .hero.is-danger.is-bold { - background-image: linear-gradient(141deg, #a70013 0%, #da0b00 71%, #f43500 100%); } - @media screen and (max-width: 768px) { - .hero.is-danger.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #a70013 0%, #da0b00 71%, #f43500 100%); } } - .hero.is-small .hero-body, #documenter .docs-sidebar form.docs-search > input.hero .hero-body { - padding-bottom: 1.5rem; - padding-top: 1.5rem; } - @media screen and (min-width: 769px), print { - .hero.is-medium .hero-body { - padding-bottom: 9rem; - padding-top: 9rem; } } - @media screen and (min-width: 769px), print { - .hero.is-large .hero-body { - padding-bottom: 18rem; - padding-top: 18rem; } } - .hero.is-halfheight .hero-body, .hero.is-fullheight .hero-body, .hero.is-fullheight-with-navbar .hero-body { - align-items: center; - display: flex; } - .hero.is-halfheight .hero-body > .container, .hero.is-fullheight .hero-body > .container, .hero.is-fullheight-with-navbar .hero-body > .container { - flex-grow: 1; - flex-shrink: 1; } - .hero.is-halfheight { - min-height: 50vh; } - .hero.is-fullheight { - min-height: 100vh; } - -.hero-video { - overflow: hidden; } - .hero-video video { - left: 50%; - min-height: 100%; - min-width: 100%; - position: absolute; - top: 50%; - transform: translate3d(-50%, -50%, 0); } - .hero-video.is-transparent { - opacity: 0.3; } - @media screen and (max-width: 768px) { - .hero-video { - display: none; } } -.hero-buttons { - margin-top: 1.5rem; } - @media screen and (max-width: 768px) { - .hero-buttons .button { - display: flex; } - .hero-buttons .button:not(:last-child) { - margin-bottom: 0.75rem; } } - @media screen and (min-width: 769px), print { - .hero-buttons { - display: flex; - justify-content: center; } - .hero-buttons .button:not(:last-child) { - margin-right: 1.5rem; } } -.hero-head, -.hero-foot { - flex-grow: 0; - flex-shrink: 0; } - -.hero-body { - flex-grow: 1; - flex-shrink: 0; - padding: 3rem 1.5rem; } - -.section { - padding: 3rem 1.5rem; } - @media screen and (min-width: 1056px) { - .section.is-medium { - padding: 9rem 1.5rem; } - .section.is-large { - padding: 18rem 1.5rem; } } -.footer { - background-color: #fafafa; - padding: 3rem 1.5rem 6rem; } - -h1 .docs-heading-anchor, h1 .docs-heading-anchor:hover, h1 .docs-heading-anchor:visited, h2 .docs-heading-anchor, h2 .docs-heading-anchor:hover, h2 .docs-heading-anchor:visited, h3 .docs-heading-anchor, h3 .docs-heading-anchor:hover, h3 .docs-heading-anchor:visited, h4 .docs-heading-anchor, h4 .docs-heading-anchor:hover, h4 .docs-heading-anchor:visited, h5 .docs-heading-anchor, h5 .docs-heading-anchor:hover, h5 .docs-heading-anchor:visited, h6 .docs-heading-anchor, h6 .docs-heading-anchor:hover, h6 .docs-heading-anchor:visited { - color: #222222; } - -h1 .docs-heading-anchor-permalink, h2 .docs-heading-anchor-permalink, h3 .docs-heading-anchor-permalink, h4 .docs-heading-anchor-permalink, h5 .docs-heading-anchor-permalink, h6 .docs-heading-anchor-permalink { - visibility: hidden; - vertical-align: middle; - margin-left: 0.5em; - font-size: 0.7rem; } - h1 .docs-heading-anchor-permalink::before, h2 .docs-heading-anchor-permalink::before, h3 .docs-heading-anchor-permalink::before, h4 .docs-heading-anchor-permalink::before, h5 .docs-heading-anchor-permalink::before, h6 .docs-heading-anchor-permalink::before { - font-family: "Font Awesome 5 Free"; - font-weight: 900; - content: "\f0c1"; } - -h1:hover .docs-heading-anchor-permalink, h2:hover .docs-heading-anchor-permalink, h3:hover .docs-heading-anchor-permalink, h4:hover .docs-heading-anchor-permalink, h5:hover .docs-heading-anchor-permalink, h6:hover .docs-heading-anchor-permalink { - visibility: visible; } - -.docs-dark-only { - display: none !important; } - -pre { - position: relative; - overflow: hidden; } - pre code, pre code.hljs { - padding: 0 0.75rem !important; - overflow: auto; - display: block; } - pre code:first-of-type, pre code.hljs:first-of-type { - padding-top: 0.5rem !important; } - pre code:last-of-type, pre code.hljs:last-of-type { - padding-bottom: 0.5rem !important; } - pre .copy-button { - opacity: 0.2; - transition: opacity 0.2s; - position: absolute; - right: 0em; - top: 0em; - padding: 0.5em; - width: 2.5em; - height: 2.5em; - background: transparent; - border: none; - font-family: "Font Awesome 5 Free"; - color: #222222; - cursor: pointer; - text-align: center; } - pre .copy-button:focus, pre .copy-button:hover { - opacity: 1; - background: rgba(34, 34, 34, 0.1); - color: #2e63b8; } - pre .copy-button.success { - color: #259a12; - opacity: 1; } - pre .copy-button.error { - color: #cb3c33; - opacity: 1; } - pre:hover .copy-button { - opacity: 1; } - -.admonition { - background-color: #b5b5b5; - border-style: solid; - border-width: 1px; - border-color: #363636; - border-radius: 4px; - font-size: 1rem; } - .admonition strong { - color: currentColor; } - .admonition.is-small, #documenter .docs-sidebar form.docs-search > input.admonition { - font-size: 0.75rem; } - .admonition.is-medium { - font-size: 1.25rem; } - .admonition.is-large { - font-size: 1.5rem; } - .admonition.is-default { - background-color: #b5b5b5; - border-color: #363636; } - .admonition.is-default > .admonition-header { - background-color: #363636; - color: #fff; } - .admonition.is-default > .admonition-body { - color: #fff; } - .admonition.is-info { - background-color: #def0fc; - border-color: #209cee; } - .admonition.is-info > .admonition-header { - background-color: #209cee; - color: #fff; } - .admonition.is-info > .admonition-body { - color: rgba(0, 0, 0, 0.7); } - .admonition.is-success { - background-color: #bdf4d1; - border-color: #22c35b; } - .admonition.is-success > .admonition-header { - background-color: #22c35b; - color: #fff; } - .admonition.is-success > .admonition-body { - color: rgba(0, 0, 0, 0.7); } - .admonition.is-warning { - background-color: #fff3c5; - border-color: #ffdd57; } - .admonition.is-warning > .admonition-header { - background-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .admonition.is-warning > .admonition-body { - color: rgba(0, 0, 0, 0.7); } - .admonition.is-danger { - background-color: #ffaba7; - border-color: #da0b00; } - .admonition.is-danger > .admonition-header { - background-color: #da0b00; - color: #fff; } - .admonition.is-danger > .admonition-body { - color: rgba(0, 0, 0, 0.7); } - .admonition.is-compat { - background-color: #bdeff5; - border-color: #1db5c9; } - .admonition.is-compat > .admonition-header { - background-color: #1db5c9; - color: #fff; } - .admonition.is-compat > .admonition-body { - color: rgba(0, 0, 0, 0.7); } - -.admonition-header { - color: #fff; - background-color: #363636; - align-items: center; - font-weight: 700; - justify-content: space-between; - line-height: 1.25; - padding: 0.5rem 0.75rem; - position: relative; } - .admonition-header:before { - font-family: "Font Awesome 5 Free"; - font-weight: 900; - margin-right: 0.75rem; - content: "\f06a"; } - -.admonition-body { - color: #222222; - padding: 0.5rem 0.75rem; } - .admonition-body pre { - background-color: whitesmoke; } - .admonition-body code { - background-color: rgba(0, 0, 0, 0.05); } - -.docstring { - margin-bottom: 1em; - background-color: transparent; - border: 1px solid #dbdbdb; - box-shadow: 2px 2px 3px rgba(10, 10, 10, 0.1); - max-width: 100%; } - .docstring > header { - display: flex; - flex-grow: 1; - align-items: stretch; - padding: 0.5rem 0.75rem; - background-color: whitesmoke; - box-shadow: 0 1px 2px rgba(10, 10, 10, 0.1); - box-shadow: none; - border-bottom: 1px solid #dbdbdb; } - .docstring > header code { - background-color: transparent; } - .docstring > header .docstring-binding { - margin-right: 0.3em; } - .docstring > header .docstring-category { - margin-left: 0.3em; } - .docstring > section { - position: relative; - padding: 0.75rem 0.75rem; - border-bottom: 1px solid #dbdbdb; } - .docstring > section:last-child { - border-bottom: none; } - .docstring > section > a.docs-sourcelink { - transition: opacity 0.3s; - opacity: 0; - position: absolute; - right: 0.375rem; - bottom: 0.375rem; } - .docstring > section > a.docs-sourcelink:focus { - opacity: 1 !important; } - .docstring:hover > section > a.docs-sourcelink { - opacity: 0.2; } - .docstring:focus-within > section > a.docs-sourcelink { - opacity: 0.2; } - .docstring > section:hover a.docs-sourcelink { - opacity: 1; } - -.documenter-example-output { - background-color: white; } - -.outdated-warning-overlay { - position: fixed; - top: 0; - left: 0; - right: 0; - box-shadow: 0 0 10px rgba(0, 0, 0, 0.3); - z-index: 999; - background-color: #ffaba7; - color: rgba(0, 0, 0, 0.7); - border-bottom: 3px solid #da0b00; - padding: 10px 35px; - text-align: center; - font-size: 15px; } - .outdated-warning-overlay .outdated-warning-closer { - position: absolute; - top: calc(50% - 10px); - right: 18px; - cursor: pointer; - width: 12px; } - .outdated-warning-overlay a { - color: #2e63b8; 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Theme: Default Description: Original highlight.js style Author: (c) Ivan Sagalaev @@ -7644,95 +6,4 @@ code.language-julia-repl > span.hljs-meta { Website: https://highlightjs.org/ License: see project LICENSE Touched: 2021 -*/ -/* -This is left on purpose making default.css the single file that can be lifted -as-is from the repository directly without the need for a build step - -Typically this "required" baseline CSS is added by `makestuff.js` during build. -*/ -pre code.hljs { - display: block; - overflow-x: auto; } - -code.hljs { - padding: 3px 5px; } - -/* end baseline CSS */ -.hljs { - background: #F0F0F0; - color: #444; } - -/* Base color: saturation 0; */ -.hljs-subst { - /* default */ } - -/* purposely ignored */ -.hljs-comment { - color: #888888; } - -.hljs-tag, -.hljs-punctuation { - color: #444a; } - -.hljs-tag .hljs-name, -.hljs-tag .hljs-attr { - color: #444; } - -.hljs-keyword, -.hljs-attribute, -.hljs-selector-tag, -.hljs-meta .hljs-keyword, -.hljs-doctag, -.hljs-name { - font-weight: bold; } - -/* User color: hue: 0 */ -.hljs-type, -.hljs-string, -.hljs-number, -.hljs-selector-id, -.hljs-selector-class, -.hljs-quote, -.hljs-template-tag, -.hljs-deletion { - color: #880000; } - -.hljs-title, -.hljs-section { - color: #880000; - font-weight: bold; } - -.hljs-regexp, -.hljs-symbol, -.hljs-variable, -.hljs-template-variable, -.hljs-link, -.hljs-selector-attr, -.hljs-operator, -.hljs-selector-pseudo { - color: #BC6060; } - -/* Language color: hue: 90; */ -.hljs-literal { - color: #78A960; } - -.hljs-built_in, -.hljs-bullet, -.hljs-code, -.hljs-addition { - color: #397300; } - -/* Meta color: hue: 200 */ -.hljs-meta { - color: #1f7199; } - -.hljs-meta .hljs-string { - color: #4d99bf; } - -/* Misc effects */ -.hljs-emphasis { - font-style: italic; } - -.hljs-strong { - font-weight: bold; } +*/pre code.hljs{display:block;overflow-x:auto;padding:1em}code.hljs{padding:3px 5px}.hljs{background:#F3F3F3;color:#444}.hljs-comment{color:#697070}.hljs-tag,.hljs-punctuation{color:#444a}.hljs-tag .hljs-name,.hljs-tag .hljs-attr{color:#444}.hljs-keyword,.hljs-attribute,.hljs-selector-tag,.hljs-meta .hljs-keyword,.hljs-doctag,.hljs-name{font-weight:bold}.hljs-type,.hljs-string,.hljs-number,.hljs-selector-id,.hljs-selector-class,.hljs-quote,.hljs-template-tag,.hljs-deletion{color:#880000}.hljs-title,.hljs-section{color:#880000;font-weight:bold}.hljs-regexp,.hljs-symbol,.hljs-variable,.hljs-template-variable,.hljs-link,.hljs-selector-attr,.hljs-operator,.hljs-selector-pseudo{color:#ab5656}.hljs-literal{color:#695}.hljs-built_in,.hljs-bullet,.hljs-code,.hljs-addition{color:#397300}.hljs-meta{color:#1f7199}.hljs-meta .hljs-string{color:#38a}.hljs-emphasis{font-style:italic}.hljs-strong{font-weight:bold}.gap-4{gap:1rem} diff --git a/dev/assets/themeswap.js b/dev/assets/themeswap.js index c58e993e..9f5eebe6 100644 --- a/dev/assets/themeswap.js +++ b/dev/assets/themeswap.js @@ -1,20 +1,17 @@ // Small function to quickly swap out themes. Gets put into the tag.. function set_theme_from_local_storage() { - // Intialize the theme to null, which means default + // Initialize the theme to null, which means default var theme = null; // If the browser supports the localstorage and is not disabled then try to get the // documenter theme - if(window.localStorage != null) { + if (window.localStorage != null) { // Get the user-picked theme from localStorage. May be `null`, which means the default // theme. - theme = window.localStorage.getItem("documenter-theme"); + theme = window.localStorage.getItem("documenter-theme"); } - // Check if the browser supports user color preference - var darkPreference = false; // Check if the users preference is for dark color scheme - if(window.matchMedia('(prefers-color-scheme: dark)').matches === true) { - darkPreference = true; - } + var darkPreference = + window.matchMedia("(prefers-color-scheme: dark)").matches === true; // Initialize a few variables for the loop: // // - active: will contain the index of the theme that should be active. Note that there @@ -24,43 +21,64 @@ function set_theme_from_local_storage() { // // - disabled: style sheets that should be disabled (i.e. all the theme style sheets // that are not the currently active theme) - var active = null; var disabled = []; var darkTheme = null; + var active = null; + var disabled = []; + var primaryLightTheme = null; + var primaryDarkTheme = null; for (var i = 0; i < document.styleSheets.length; i++) { var ss = document.styleSheets[i]; // The tag of each style sheet is expected to have a data-theme-name attribute // which must contain the name of the theme. The names in localStorage much match this. var themename = ss.ownerNode.getAttribute("data-theme-name"); // attribute not set => non-theme stylesheet => ignore - if(themename === null) continue; + if (themename === null) continue; // To distinguish the default (primary) theme, it needs to have the data-theme-primary // attribute set. - var isprimary = (ss.ownerNode.getAttribute("data-theme-primary") !== null); - // Check if the theme is primary dark theme - var isDarkTheme = (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null); - // If ss is for dark theme then set the value of darkTheme to the name of the theme - if(isDarkTheme) darkTheme = themename; + if (ss.ownerNode.getAttribute("data-theme-primary") !== null) { + primaryLightTheme = themename; + } + // Check if the theme is primary dark theme so that we could store its name in darkTheme + if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) { + primaryDarkTheme = themename; + } // If we find a matching theme (and it's not the default), we'll set active to non-null - if(themename === theme) active = i; + if (themename === theme) active = i; // Store the style sheets of inactive themes so that we could disable them - if(themename !== theme) disabled.push(ss); + if (themename !== theme) disabled.push(ss); } - if(active !== null) { + var activeTheme = null; + if (active !== null) { // If we did find an active theme, we'll (1) add the theme--$(theme) class to - document.getElementsByTagName('html')[0].className = "theme--" + theme; - // and (2) disable all the other theme stylesheets - disabled.forEach(function(ss){ - ss.disabled = true; - }); + document.getElementsByTagName("html")[0].className = "theme--" + theme; + activeTheme = theme; + } else { + // If we did _not_ find an active theme, then we need to fall back to the primary theme + // which can either be dark or light, depending on the user's OS preference. + var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme; + // In case it somehow happens that the relevant primary theme was not found in the + // preceding loop, we abort without doing anything. + if (activeTheme === null) { + console.error("Unable to determine primary theme."); + return; + } + // When switching to the primary light theme, then we must not have a class name + // for the tag. That's only for non-primary or the primary dark theme. + if (darkPreference) { + document.getElementsByTagName("html")[0].className = + "theme--" + activeTheme; + } else { + document.getElementsByTagName("html")[0].className = ""; + } } - else if(darkTheme !== null && darkPreference === true) { - // If we did find an active theme, we'll (1) add the theme--$(theme) class to - document.getElementsByTagName('html')[0].className = "theme--" + darkTheme; - // and (2) disable all the other theme stylesheets - disabled.forEach(function(ss){ - if (ss.ownerNode.getAttribute("data-theme-name") !== darkTheme) { - ss.disabled = true; - } - }); + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // we'll disable all the stylesheets, except for the active one + ss.disabled = !(themename == activeTheme); } } set_theme_from_local_storage(); diff --git a/dev/assets/warner.js b/dev/assets/warner.js index 5531c885..3f6f5d00 100644 --- a/dev/assets/warner.js +++ b/dev/assets/warner.js @@ -1,49 +1,52 @@ -function maybeAddWarning () { - // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE - // in siteinfo.js. - // If either of these are undefined something went horribly wrong, so we abort. - if ( - window.DOCUMENTER_NEWEST === undefined || - window.DOCUMENTER_CURRENT_VERSION === undefined || - window.DOCUMENTER_STABLE === undefined - ) { - return - }; +function maybeAddWarning() { + // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE + // in siteinfo.js. + // If either of these are undefined something went horribly wrong, so we abort. + if ( + window.DOCUMENTER_NEWEST === undefined || + window.DOCUMENTER_CURRENT_VERSION === undefined || + window.DOCUMENTER_STABLE === undefined + ) { + return; + } - // Current version is not a version number, so we can't tell if it's the newest version. Abort. - if (!/v(\d+\.)*\d+/.test(window.DOCUMENTER_CURRENT_VERSION)) { - return - }; + // Current version is not a version number, so we can't tell if it's the newest version. Abort. + if (!/v(\d+\.)*\d+/.test(window.DOCUMENTER_CURRENT_VERSION)) { + return; + } - // Current version is newest version, so no need to add a warning. - if (window.DOCUMENTER_NEWEST === window.DOCUMENTER_CURRENT_VERSION) { - return - }; + // Current version is newest version, so no need to add a warning. + if (window.DOCUMENTER_NEWEST === window.DOCUMENTER_CURRENT_VERSION) { + return; + } - // Add a noindex meta tag (unless one exists) so that search engines don't index this version of the docs. - if (document.body.querySelector('meta[name="robots"]') === null) { - const meta = document.createElement('meta'); - meta.name = 'robots'; - meta.content = 'noindex'; + // Add a noindex meta tag (unless one exists) so that search engines don't index this version of the docs. + if (document.body.querySelector('meta[name="robots"]') === null) { + const meta = document.createElement("meta"); + meta.name = "robots"; + meta.content = "noindex"; - document.getElementsByTagName('head')[0].appendChild(meta); - }; + document.getElementsByTagName("head")[0].appendChild(meta); + } - const div = document.createElement('div'); - div.classList.add('outdated-warning-overlay'); - const closer = document.createElement('button'); - closer.classList.add('outdated-warning-closer', 'delete'); - closer.addEventListener('click', function () { - document.body.removeChild(div); - }); - const href = window.documenterBaseURL + '/../' + window.DOCUMENTER_STABLE; - div.innerHTML = 'This documentation is not for the latest stable release, but for either the development version or an older release.
Click here to go to the documentation for the latest stable release.'; - div.appendChild(closer); - document.body.appendChild(div); -}; + const div = document.createElement("div"); + div.classList.add("outdated-warning-overlay"); + const closer = document.createElement("button"); + closer.classList.add("outdated-warning-closer", "delete"); + closer.addEventListener("click", function () { + document.body.removeChild(div); + }); + const href = window.documenterBaseURL + "/../" + window.DOCUMENTER_STABLE; + div.innerHTML = + 'This documentation is not for the latest stable release, but for either the development version or an older release.
Click here to go to the documentation for the latest stable release.'; + div.appendChild(closer); + document.body.appendChild(div); +} -if (document.readyState === 'loading') { - document.addEventListener('DOMContentLoaded', maybeAddWarning); +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", maybeAddWarning); } else { - maybeAddWarning(); -}; + maybeAddWarning(); +} diff --git a/dev/benchmarks/index.html b/dev/benchmarks/index.html index 497f8728..1abd0374 100644 --- a/dev/benchmarks/index.html +++ b/dev/benchmarks/index.html @@ -1,5 +1,5 @@ -Benchmarks · PencilFFTs.jl
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Benchmarks

The performance of PencilFFTs.jl is comparable to that of other open-source parallel FFT libraries implemented in lower-level languages. Below, we show comparisons with the Fortran implementation of P3DFFT, possibly the most popular of these libraries. The benchmarks were performed on the Jean–Zay cluster of the IDRIS French computing centre (CNRS).

The figure below shows strong scaling benchmarks of 3D real-to-complex FFTs using 2D ("pencil") decomposition. The benchmarks were run for input arrays of dimensions $N_x × N_y × N_z = 512^3$, $1024^3$ and $2048^3$. Each timing is averaged over 100 repetitions.

+Benchmarks · PencilFFTs.jl
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Benchmarks

The performance of PencilFFTs.jl is comparable to that of other open-source parallel FFT libraries implemented in lower-level languages. Below, we show comparisons with the Fortran implementation of P3DFFT, possibly the most popular of these libraries. The benchmarks were performed on the Jean–Zay cluster of the IDRIS French computing centre (CNRS).

The figure below shows strong scaling benchmarks of 3D real-to-complex FFTs using 2D ("pencil") decomposition. The benchmarks were run for input arrays of dimensions $N_x × N_y × N_z = 512^3$, $1024^3$ and $2048^3$. Each timing is averaged over 100 repetitions.

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a/dev/search_index.js +++ b/dev/search_index.js @@ -1,3 +1,3 @@ var documenterSearchIndex = {"docs": -[{"location":"tutorial/#Tutorial","page":"Tutorial","title":"Tutorial","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The following tutorial shows how to perform a 3D FFT of real periodic data defined on a grid of N_x N_y N_z points.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"
\n \n \n
","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"By default, the domain is distributed on a 2D MPI topology of dimensions N_1 N_2. As an example, the above figure shows such a topology with N_1 = 4 and N_2 = 3, for a total of 12 MPI processes.","category":"page"},{"location":"tutorial/#tutorial:creating_plans","page":"Tutorial","title":"Creating plans","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The first thing to do is to create a domain decomposition configuration for the given dataset dimensions N_x N_y N_z. In the framework of PencilArrays, such a configuration is described by a Pencil object. As described in the PencilArrays docs, we can let the Pencil constructor automatically determine such a configuration. For this, only an MPI communicator and the dataset dimensions are needed:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using MPI\nusing PencilFFTs\n\nMPI.Init()\ncomm = MPI.COMM_WORLD\n\n# Input data dimensions (Nx × Ny × Nz)\ndims = (16, 32, 64)\npen = Pencil(dims, comm)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"By default this creates a 2D decomposition (for the case of a 3D dataset), but one can change this as detailed in the PencilArrays documentation linked above.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"We can now create a PencilFFTPlan, which requires information on decomposition configuration (the Pencil object) and on the transforms that will be applied:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# Apply a 3D real-to-complex (r2c) FFT.\ntransform = Transforms.RFFT()\n\n# Note that, for more control, one can instead separately specify the transforms along each dimension:\n# transform = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())\n\n# Create plan\nplan = PencilFFTPlan(pen, transform)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"See the PencilFFTPlan constructor for details on the accepted options, and the Transforms module for the possible transforms. It is also possible to enable fine-grained performance measurements via the TimerOutputs package, as described in Measuring performance.","category":"page"},{"location":"tutorial/#Allocating-data","page":"Tutorial","title":"Allocating data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Next, we want to apply the plan on some data. Transforms may only be applied on PencilArrays, which are array wrappers that include MPI decomposition information (in some sense, analogous to DistributedArrays in Julia's distributed computing approach). The helper function allocate_input can be used to allocate a PencilArray that is compatible with our plan:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# In our example, this returns a 3D PencilArray of real data (Float64).\nu = allocate_input(plan)\n\n# Fill the array with some (random) data\nusing Random\nrandn!(u)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"PencilArrays are a subtype of AbstractArray, and thus they support all common array operations.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Similarly, to preallocate output data, one can use allocate_output:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# In our example, this returns a 3D PencilArray of complex data (Complex{Float64}).\nv = allocate_output(plan)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"This is only required if one wants to apply the plans using a preallocated output (with mul!, see right below).","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The data types returned by allocate_input and allocate_output are slightly different when working with in-place transforms. See the in-place example for details.","category":"page"},{"location":"tutorial/#Applying-plans","page":"Tutorial","title":"Applying plans","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The interface to apply plans is consistent with that of AbstractFFTs. Namely, * and mul! are respectively used for forward transforms without and with preallocated output data. Similarly, \\ and ldiv! are used for backward transforms.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using LinearAlgebra # for mul!, ldiv!\n\n# Apply plan on `u` with `v` as an output\nmul!(v, plan, u)\n\n# Apply backward plan on `v` with `w` as an output\nw = similar(u)\nldiv!(w, plan, v) # now w ≈ u","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Note that, consistently with AbstractFFTs, normalisation is performed at the end of a backward transform, so that the original data is recovered when applying a forward followed by a backward transform.","category":"page"},{"location":"tutorial/#Accessing-and-modifying-data","page":"Tutorial","title":"Accessing and modifying data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For any given MPI process, a PencilArray holds the data associated to its local partition in the global geometry. PencilArrays are accessed using local indices that start at 1, regardless of the location of the local process in the MPI topology. Note that PencilArrays, being based on regular Arrays, support both linear and Cartesian indexing (see the Julia docs for details).","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For convenience, the global_view function can be used to generate an OffsetArray wrapper that takes global indices.","category":"page"},{"location":"tutorial/#tutorial:output_data_layout","page":"Tutorial","title":"Output data layout","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"In memory, the dimensions of the transform output are by default reversed with respect to the input. That is, if the order of indices in the input data is (x, y, z), then the output has order (z, y, x) in memory. This detail is hidden from the user, and output arrays are always accessed in the same order as the input data, regardless of the underlying output dimension permutation. This applies to PencilArrays and to OffsetArrays returned by global_view.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The reasoning behind dimension permutations, is that they allow to always perform FFTs along the fastest array dimension and to avoid a local data transposition, resulting in performance gains. A similar approach is followed by other parallel FFT libraries. FFTW itself, in its distributed-memory routines, includes a flag that enables a similar behaviour. In PencilFFTs, index permutation is the default, but it can be disabled via the permute_dims flag of PencilFFTPlan.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"A great deal of work has been spent in making generic index permutations as efficient as possible, both in intermediate and in the output state of the multidimensional transforms. This has been achieved, in part, by making sure that permutations such as (3, 2, 1) are compile-time constants.","category":"page"},{"location":"tutorial/#Further-reading","page":"Tutorial","title":"Further reading","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For details on working with PencilArrays see the PencilArrays docs.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The examples on the sidebar further illustrate the use of transforms and provide an introduction to working with MPI-distributed data in the form of PencilArrays. In particular, the gradient example illustrates different ways of computing things using Fourier-transformed distributed arrays. Then, the incompressible Navier–Stokes example is a more advanced and complete example of a possible application of the PencilFFTs package.","category":"page"},{"location":"benchmarks/#Benchmarks","page":"Benchmarks","title":"Benchmarks","text":"","category":"section"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The performance of PencilFFTs.jl is comparable to that of other open-source parallel FFT libraries implemented in lower-level languages. Below, we show comparisons with the Fortran implementation of P3DFFT, possibly the most popular of these libraries. The benchmarks were performed on the Jean–Zay cluster of the IDRIS French computing centre (CNRS).","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The figure below shows strong scaling benchmarks of 3D real-to-complex FFTs using 2D (\"pencil\") decomposition. The benchmarks were run for input arrays of dimensions N_x N_y N_z = 512^3, 1024^3 and 2048^3. Each timing is averaged over 100 repetitions.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"
\n \n \n
","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"As seen above, PencilFFTs generally outperforms P3DFFT in its default setting. This is largely explained by the choice of using non-blocking point-to-point MPI communications (via MPI_Isend and MPI_Irecv), while P3DFFT uses collective MPI_Alltoallv calls. This enables PencilFFTs to perform data reordering operations on the partially received data while waiting for the incoming data, leading to better performance. Moreover, in contrast with P3DFFT, the high performance and scalability of PencilFFTs results from a highly generic code, handling decompositions in arbitrary dimensions and a relatively large (and extensible) variety of transformations.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"Note that PencilFFTs can optionally use collective communications (using MPI_Alltoallv) instead of point-to-point communications. For details, see the docs for PencilFFTPlan and for PencilArray transpositions. As seen above, collective communications generally perform worse than point-to-point ones, and runtimes are nearly indistinguishable from those of P3DFFT.","category":"page"},{"location":"benchmarks/#Benchmark-details","page":"Benchmarks","title":"Benchmark details","text":"","category":"section"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The benchmarks were performed using Julia 1.7-beta3 and Intel MPI 2019. We used PencilFFTs v0.12.5 with FFTW.jl v1.4.3 and MPI.jl v0.19.0. We used the Fortran implementation of P3DFFT, version 2.7.6, which was built with Intel 2019 compilers and linked to FFTW 3.3.8. The cluster where the benchmarks were run has Intel Cascade Lake 6248 processors with 2×20 cores per node.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The number of MPI processes along each decomposed dimension, P_1 and P_2, was automatically determined by a call to MPI_Dims_create, which tends to create a balanced decomposition with P_1 P_2. For instance, a total of 1024 processes is divided into P_1 = P_2 = 32. Different results may be obtained with other combinations, but this was not benchmarked.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The source files used to generate this benchmark, as well as the raw benchmark results, are all available in the PencilFFTs repo.","category":"page"},{"location":"GlobalFFTParams/#Global-FFT-parameters","page":"Global FFT parameters","title":"Global FFT parameters","text":"","category":"section"},{"location":"GlobalFFTParams/","page":"Global FFT parameters","title":"Global FFT parameters","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"GlobalFFTParams/","page":"Global FFT parameters","title":"Global FFT parameters","text":"GlobalFFTParams","category":"page"},{"location":"GlobalFFTParams/#PencilFFTs.GlobalFFTParams","page":"Global FFT parameters","title":"PencilFFTs.GlobalFFTParams","text":"GlobalFFTParams{T, N, inplace}\n\nSpecifies the global parameters for an N-dimensional distributed transform. These include the element type T and global data sizes of input and output data, as well as the transform types to be performed along each dimension.\n\n\n\nGlobalFFTParams(size_global, transforms, [real_type=Float64])\n\nDefine parameters for N-dimensional transform.\n\ntransforms must be a tuple of length N specifying the transforms to be applied along each dimension. Each element must be a subtype of Transforms.AbstractTransform. For all the possible transforms, see Transform types.\n\nThe element type must be a real type accepted by FFTW, i.e. either Float32 or Float64.\n\nNote that the transforms are applied one dimension at a time, with the leftmost dimension first for forward transforms.\n\nExample\n\nTo perform a 3D FFT of real data, first a real-to-complex FFT must be applied along the first dimension, followed by two complex-to-complex FFTs along the other dimensions:\n\njulia> size_global = (64, 32, 128); # size of real input data\n\njulia> transforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT());\n\njulia> fft_params = PencilFFTs.GlobalFFTParams(size_global, transforms)\nTransforms: (RFFT, FFT, FFT)\nInput type: Float64\nGlobal dimensions: (64, 32, 128) -> (33, 32, 128)\n\n\n\n\n\n","category":"type"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"EditURL = \"https://github.com/jipolanco/PencilFFTs.jl/blob/master/docs/examples/in-place.jl\"","category":"page"},{"location":"generated/in-place/#In-place-transforms","page":"In-place transforms","title":"In-place transforms","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Complex-to-complex and real-to-real transforms can be performed in-place, enabling important memory savings. The procedure is very similar to that of out-of-place transforms described in the tutorial. The differences are illustrated in the sections below.","category":"page"},{"location":"generated/in-place/#Creating-a-domain-partition","page":"In-place transforms","title":"Creating a domain partition","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"We start by partitioning a domain of dimensions 163264 along all available MPI processes.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"using PencilFFTs\nusing MPI\nMPI.Init()\n\ndims_global = (16, 32, 64) # global dimensions","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Such a partitioning is described by a Pencil object. Here we choose to decompose the domain along the last two dimensions. In this case, the actual number of processes along each of these dimensions is chosen automatically.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"decomp_dims = (2, 3)\ncomm = MPI.COMM_WORLD\npen = Pencil(dims_global, decomp_dims, comm)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"warning: Allowed decompositions\nDistributed transforms using PencilFFTs.jl require that the first dimension is not decomposed. In other words, if one wants to perform transforms, then decomp_dims above must not contain 1.","category":"page"},{"location":"generated/in-place/#Creating-in-place-plans","page":"In-place transforms","title":"Creating in-place plans","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"To create an in-place plan, pass an in-place transform such as Transforms.FFT! or Transforms.R2R! to PencilFFTPlan. For instance:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"# Perform a 3D in-place complex-to-complex FFT.\ntransform = Transforms.FFT!()\n\n# Note that one can also combine different types of in-place transforms.\n# For instance:\n# transform = (\n# Transforms.R2R!(FFTW.REDFT01),\n# Transforms.FFT!(),\n# Transforms.R2R!(FFTW.DHT),\n# )","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"We can now create a distributed plan from the previously-created domain partition and the chosen transform.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan = PencilFFTPlan(pen, transform)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Note that in-place real-to-complex transforms are not currently supported. (In other words, the RFFT! transform type is not defined.)","category":"page"},{"location":"generated/in-place/#Allocating-data","page":"In-place transforms","title":"Allocating data","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"As with out-of-place plans, data should be allocated using allocate_input. The difference is that, for in-place plans, this function returns a ManyPencilArray object, which is a container holding multiple PencilArray views sharing the same memory space.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"# Allocate data for the plan.\n# Since `plan` is in-place, this returns a `ManyPencilArray` container.\nA = allocate_input(plan)\nsummary(A)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Note that allocate_output also works for in-place plans. In this case, it returns exactly the same thing as allocate_input.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"As shown in the next section, in-place plans must be applied on the returned ManyPencilArray. On the other hand, one usually wants to access and modify data, and for this one needs the PencilArray views contained in the ManyPencilArray. The input and output array views can be obtained by calling first(::ManyPencilArray) and last(::ManyPencilArray).","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"For instance, we can initialise the input array with some data before transforming:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"using Random\nu_in = first(A) # input data view\nrandn!(u_in)\nsummary(u_in)","category":"page"},{"location":"generated/in-place/#Applying-plans","page":"In-place transforms","title":"Applying plans","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Like in FFTW.jl, one can perform in-place transforms using the * and \\ operators. As mentioned above, in-place plans must be applied on the ManyPencilArray containers returned by allocate_input.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan * A; # performs in-place forward transform\nnothing #hide","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"After performing an in-place transform, data contained in u_in has been overwritten and has no \"physical\" meaning. In other words, u_in should not be used at this point. To access the transformed data, one should retrieve the output data view using last(A).","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"For instance, to compute the global sum of the transformed data:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"u_out = last(A) # output data view\nsum(u_out) # sum of transformed data (note that `sum` reduces over all processes)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Finally, we can perform a backward transform and do stuff with the input view:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan \\ A; # perform in-place backward transform\nnothing #hide","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"At this point, the data can be once again found in the input view u_in, while u_out should not be accessed.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"This page was generated using Literate.jl.","category":"page"},{"location":"Transforms/#Available-transforms","page":"Available transforms","title":"Available transforms","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"CurrentModule = PencilFFTs.Transforms","category":"page"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"Transforms","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms","page":"Available transforms","title":"PencilFFTs.Transforms","text":"Defines different one-dimensional FFT-based transforms.\n\nThe transforms are all subtypes of an AbstractTransform type.\n\nWhen possible, the names of the transforms are kept consistent with the functions exported by AbstractFFTs.jl and FFTW.jl.\n\n\n\n\n\n","category":"module"},{"location":"Transforms/#Transform-types","page":"Available transforms","title":"Transform types","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"FFT\nFFT!\nBFFT\nBFFT!\n\nRFFT\nBRFFT\n\nR2R\nR2R!\n\nNoTransform\nNoTransform!","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.FFT","page":"Available transforms","title":"PencilFFTs.Transforms.FFT","text":"FFT()\n\nComplex-to-complex FFT.\n\nSee also AbstractFFTs.fft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.FFT!","page":"Available transforms","title":"PencilFFTs.Transforms.FFT!","text":"FFT!()\n\nIn-place version of FFT.\n\nSee also AbstractFFTs.fft!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BFFT","page":"Available transforms","title":"PencilFFTs.Transforms.BFFT","text":"BFFT()\n\nUnnormalised backward complex-to-complex FFT.\n\nLike AbstractFFTs.bfft, this transform is not normalised. To obtain the inverse transform, divide the output by the length of the transformed dimension.\n\nSee also AbstractFFTs.bfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.BFFT!","text":"BFFT()\n\nIn-place version of BFFT.\n\nSee also AbstractFFTs.bfft!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.RFFT","page":"Available transforms","title":"PencilFFTs.Transforms.RFFT","text":"RFFT()\n\nReal-to-complex FFT.\n\nSee also AbstractFFTs.rfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BRFFT","page":"Available transforms","title":"PencilFFTs.Transforms.BRFFT","text":"BRFFT(d::Integer)\nBRFFT((d1, d2, ..., dN))\n\nUnnormalised inverse of RFFT.\n\nTo obtain the inverse transform, divide the output by the length of the transformed dimension (of the real output array).\n\nAs described in the AbstractFFTs docs, the length of the output cannot be fully inferred from the input length. For this reason, the BRFFT constructor accepts an optional d argument indicating the output length.\n\nFor multidimensional datasets, a tuple of dimensions (d1, d2, ..., dN) may also be passed. This is equivalent to passing just dN. In this case, the last dimension (dN) is the one that changes size between the input and output. Note that this is the opposite of FFTW.brfft. The reason is that, in PencilFFTs, the last dimension is the one along which a complex-to-real transform is performed.\n\nSee also AbstractFFTs.brfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.R2R","page":"Available transforms","title":"PencilFFTs.Transforms.R2R","text":"R2R(kind)\n\nReal-to-real transform of type kind.\n\nThe possible values of kind are those described in the FFTW.r2r docs and the FFTW manual:\n\ndiscrete cosine transforms: FFTW.REDFT00, FFTW.REDFT01, FFTW.REDFFT10, FFTW.REDFFT11\ndiscrete sine transforms: FFTW.RODFT00, FFTW.RODFT01, FFTW.RODFFT10, FFTW.RODFFT11\ndiscrete Hartley transform: FFTW.DHT\n\nNote: half-complex format DFTs (FFTW.R2HC, FFTW.HC2R) are not currently supported.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.R2R!","page":"Available transforms","title":"PencilFFTs.Transforms.R2R!","text":"R2R!(kind)\n\nIn-place version of R2R.\n\nSee also FFTW.r2r!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.NoTransform","page":"Available transforms","title":"PencilFFTs.Transforms.NoTransform","text":"NoTransform()\n\nIdentity transform.\n\nSpecifies that no transformation should be applied.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.NoTransform!","page":"Available transforms","title":"PencilFFTs.Transforms.NoTransform!","text":"NoTransform!()\n\nIn-place version of NoTransform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#Internals","page":"Available transforms","title":"Internals","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"What follows is used internally in PencilFFTs.","category":"page"},{"location":"Transforms/#Types","page":"Available transforms","title":"Types","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"AbstractCustomPlan\nAbstractTransform\nIdentityPlan\nIdentityPlan!\nPlan","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.AbstractCustomPlan","page":"Available transforms","title":"PencilFFTs.Transforms.AbstractCustomPlan","text":"AbstractCustomPlan\n\nAbstract type defining a custom plan, to be used as an alternative to FFTW plans (FFTW.FFTWPlan).\n\nThe only custom plan defined in this module is IdentityPlan. The user can define other custom plans that are also subtypes of AbstractCustomPlan.\n\nNote that plan returns a subtype of either AbstractFFTs.Plan or AbstractCustomPlan.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.AbstractTransform","page":"Available transforms","title":"PencilFFTs.Transforms.AbstractTransform","text":"AbstractTransform\n\nSpecifies a one-dimensional FFT-based transform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.IdentityPlan","page":"Available transforms","title":"PencilFFTs.Transforms.IdentityPlan","text":"IdentityPlan\n\nType of plan associated to NoTransform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.IdentityPlan!","page":"Available transforms","title":"PencilFFTs.Transforms.IdentityPlan!","text":"IdentityPlan!\n\nType of plan associated to NoTransform!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.Plan","page":"Available transforms","title":"PencilFFTs.Transforms.Plan","text":"Plan = Union{AbstractFFTs.Plan, AbstractCustomPlan}\n\nUnion type representing any plan returned by plan.\n\nSee also AbstractCustomPlan.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#Functions","page":"Available transforms","title":"Functions","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"plan\n\nbinv\nscale_factor\n\neltype_input\neltype_output\nexpand_dims\nis_inplace\nkind\nlength_output","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.plan","page":"Available transforms","title":"PencilFFTs.Transforms.plan","text":"plan(transform::AbstractTransform, A, [dims];\n flags=FFTW.ESTIMATE, timelimit=Inf)\n\nCreate plan to transform array A along dimensions dims.\n\nIf dims is not specified, all dimensions of A are transformed.\n\nFor FFT plans, this function wraps the AbstractFFTs.jl and FFTW.jl plan creation functions. For more details on the function arguments, see AbstractFFTs.plan_fft.\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.binv","page":"Available transforms","title":"PencilFFTs.Transforms.binv","text":"binv(transform::AbstractTransform, d::Integer)\n\nReturns the backwards transform associated to the given transform.\n\nThe second argument must be the length of the first transformed dimension in the forward transform. It is used in particular when transform = RFFT(), to determine the length of the inverse (complex-to-real) transform. See the AbstractFFTs.irfft docs for details.\n\nThe backwards transform returned by this function is not normalised. The normalisation factor for a given array can be obtained by calling scale_factor.\n\nExample\n\njulia> binv(Transforms.FFT(), 42)\nBFFT\n\njulia> binv(Transforms.BRFFT(9), 42)\nRFFT\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.scale_factor","page":"Available transforms","title":"PencilFFTs.Transforms.scale_factor","text":"scale_factor(transform::AbstractTransform, A, [dims = 1:ndims(A)])\n\nGet factor required to normalise the given array after a transformation along dimensions dims (all dimensions by default).\n\nThe array A must have the dimensions of the transform input.\n\nImportant: the dimensions dims must be the same that were passed to plan.\n\nExamples\n\njulia> C = zeros(ComplexF32, 3, 4, 5);\n\njulia> scale_factor(Transforms.FFT(), C)\n60\n\njulia> scale_factor(Transforms.BFFT(), C)\n60\n\njulia> scale_factor(Transforms.BFFT(), C, 2:3)\n20\n\njulia> R = zeros(Float64, 3, 4, 5);\n\njulia> scale_factor(Transforms.RFFT(), R, 2)\n4\n\njulia> scale_factor(Transforms.RFFT(), R, 2:3)\n20\n\njulia> scale_factor(Transforms.BRFFT(8), C)\n96\n\njulia> scale_factor(Transforms.BRFFT(9), C)\n108\n\nThis will fail because the input of RFFT is real, and R is a complex array:\n\njulia> scale_factor(Transforms.RFFT(), C, 2:3)\nERROR: MethodError: no method matching scale_factor(::PencilFFTs.Transforms.RFFT, ::Array{ComplexF32, 3}, ::UnitRange{Int64})\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.eltype_input","page":"Available transforms","title":"PencilFFTs.Transforms.eltype_input","text":"eltype_input(transform::AbstractTransform, real_type<:AbstractFloat)\n\nDetermine input data type for a given transform given the floating point precision of the input data.\n\nSome transforms, such as R2R and NoTransform, can take both real and complex data. For those kinds of transforms, nothing is returned.\n\nExample\n\njulia> eltype_input(Transforms.FFT(), Float32)\nComplexF32 (alias for Complex{Float32})\n\njulia> eltype_input(Transforms.RFFT(), Float64)\nFloat64\n\njulia> eltype_input(Transforms.R2R(FFTW.REDFT01), Float64) # nothing\n\njulia> eltype_input(Transforms.NoTransform(), Float64) # nothing\n\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.eltype_output","page":"Available transforms","title":"PencilFFTs.Transforms.eltype_output","text":"eltype_output(transform::AbstractTransform, eltype_input)\n\nReturns the output data type for a given transform given the input type.\n\nThrows ArgumentError if the input data type is incompatible with the transform type.\n\nExample\n\njulia> eltype_output(Transforms.NoTransform(), Float32)\nFloat32\n\njulia> eltype_output(Transforms.RFFT(), Float64)\nComplexF64 (alias for Complex{Float64})\n\njulia> eltype_output(Transforms.BRFFT(4), ComplexF32)\nFloat32\n\njulia> eltype_output(Transforms.FFT(), Float64)\nERROR: ArgumentError: invalid input data type for PencilFFTs.Transforms.FFT: Float64\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.expand_dims","page":"Available transforms","title":"PencilFFTs.Transforms.expand_dims","text":"expand_dims(transform::AbstractTransform, Val(N))\n\nExpand a single multidimensional transform into one transform per dimension.\n\nExample\n\n# Expand a real-to-complex transform in 3 dimensions.\njulia> expand_dims(Transforms.RFFT(), Val(3))\n(RFFT, FFT, FFT)\n\njulia> expand_dims(Transforms.BRFFT(4), Val(3))\n(BFFT, BFFT, BRFFT{even})\n\njulia> expand_dims(Transforms.NoTransform(), Val(2))\n(NoTransform, NoTransform)\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.is_inplace","page":"Available transforms","title":"PencilFFTs.Transforms.is_inplace","text":"is_inplace(transform::AbstractTransform) -> Bool\nis_inplace(transforms::Vararg{AbtractTransform}) -> Union{Bool, Nothing}\n\nCheck whether a transform or a list of transforms is performed in-place.\n\nIf the list of transforms has a combination of in-place and out-of-place transforms, nothing is returned.\n\nExample\n\njulia> is_inplace(Transforms.RFFT())\nfalse\n\njulia> is_inplace(Transforms.NoTransform!())\ntrue\n\njulia> is_inplace(Transforms.FFT!(), Transforms.R2R!(FFTW.REDFT01))\ntrue\n\njulia> is_inplace(Transforms.FFT(), Transforms.R2R(FFTW.REDFT01))\nfalse\n\njulia> is_inplace(Transforms.FFT(), Transforms.R2R!(FFTW.REDFT01)) === nothing\ntrue\n\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.kind","page":"Available transforms","title":"PencilFFTs.Transforms.kind","text":"kind(transform::R2R)\n\nGet kind of real-to-real transform.\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.length_output","page":"Available transforms","title":"PencilFFTs.Transforms.length_output","text":"length_output(transform::AbstractTransform, length_in::Integer)\n\nReturns the length of the transform output, given the length of its input.\n\nThe input and output lengths are specified in terms of the respective input and output datatypes. For instance, for real-to-complex transforms, these are respectively the length of input real data and of output complex data.\n\n\n\n\n\n","category":"function"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"EditURL = \"https://github.com/jipolanco/PencilFFTs.jl/blob/master/docs/examples/navier_stokes.jl\"","category":"page"},{"location":"generated/navier_stokes/#Navier–Stokes-equations","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"In this example, we numerically solve the incompressible Navier–Stokes equations","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"_t bmv + (bmv bm) bmv = -frac1ρ bm p + ν ^2 bmv\nquad bm bmv = 0","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where bmv(bmx t) and p(bmx t) are respectively the velocity and pressure fields, ν is the fluid kinematic viscosity and ρ is the fluid density.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.","category":"page"},{"location":"generated/navier_stokes/#First-steps","page":"Navier–Stokes equations","title":"First steps","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We start by loading the required packages, initialising MPI and setting the simulation parameters.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using MPI\nusing PencilFFTs\n\nMPI.Init()\ncomm = MPI.COMM_WORLD\nprocid = MPI.Comm_rank(comm) + 1\n\n# Simulation parameters\nNs = (64, 64, 64) # = (Nx, Ny, Nz)\nLs = (2π, 2π, 2π) # = (Lx, Ly, Lz)\n\n# Collocation points (\"global\" = over all processes).\n# We include the endpoint (length = N + 1) for convenience.\nxs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls) # = (x, y, z)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's check the number of MPI processes over which we're running our simulation:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"MPI.Comm_size(comm)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"pen = Pencil(Ns, comm)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"The subdomain associated to the local MPI process can be obtained using range_local:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"range_local(pen)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We now construct a distributed vector field that follows the decomposition configuration we just created:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v⃗₀ = (\n PencilArray{Float64}(undef, pen), # vx\n PencilArray{Float64}(undef, pen), # vy\n PencilArray{Float64}(undef, pen), # vz\n)\nsummary(v⃗₀[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We still need to fill this array with interesting values that represent a physical velocity field.","category":"page"},{"location":"generated/navier_stokes/#Initial-condition","page":"Navier–Stokes equations","title":"Initial condition","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"beginaligned\nv_x(x y z) = u₀ sin(k₀ x) cos(k₀ y) cos(k₀ z) \nv_y(x y z) = -u₀ cos(k₀ x) sin(k₀ y) cos(k₀ z) \nv_z(x y z) = 0\nendaligned","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where u₀ and k₀ are two parameters setting the amplitude and the period of the velocity field.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"grid = localgrid(pen, xs_global)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can use this to initialise the velocity field:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"u₀ = 1.0\nk₀ = 2π / Ls[1] # should be integer if L = 2π (to preserve periodicity)\n\n@. v⃗₀[1] = u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)\n@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)\n@. v⃗₀[3] = 0\nnothing # hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's plot a 2D slice of the velocity field managed by the local MPI process:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using GLMakie\n\n# Compute the norm of a vector field represented by a tuple of arrays.\nfunction vecnorm(v⃗::NTuple)\n vnorm = similar(v⃗[1])\n for n ∈ eachindex(v⃗[1])\n w = zero(eltype(vnorm))\n for v ∈ v⃗\n w += v[n]^2\n end\n vnorm[n] = sqrt(w)\n end\n vnorm\nend\n\nlet fig = Figure(resolution = (700, 600))\n ax = Axis3(fig[1, 1]; aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\")\n vnorm = vecnorm(v⃗₀)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, vnorm;\n alpha = 0.2, levels = 4,\n colormap = :viridis, colorrange = (0.0, 1.0),\n )\n cb = Colorbar(fig[1, 2], ct; label = \"Velocity magnitude\")\n fig\nend","category":"page"},{"location":"generated/navier_stokes/#Velocity-in-Fourier-space","page":"Navier–Stokes equations","title":"Velocity in Fourier space","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"bmv(bmx t) =\n_bmk hatbmv_bmk(t) e^i bmk bmx","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where bmk = (k_x k_y k_z) are the discrete wave numbers.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for k_x:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using AbstractFFTs: fftfreq, rfftfreq\n\nks_global = (\n rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]), # kx | real-to-complex\n fftfreq(Ns[2], 2π * Ns[2] / Ls[2]), # ky | complex-to-complex\n fftfreq(Ns[3], 2π * Ns[3] / Ls[3]), # kz | complex-to-complex\n)\n\nks_global[1]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ks_global[2]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ks_global[3]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"See PencilFFTPlan for details on creating plans and on optional keyword arguments.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients hatbmv_bmk:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v̂s = plan .* v⃗₀\nsummary(v̂s[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Note that, in Fourier space, the domain decomposition is performed along the directions x and y:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"pencil(v̂s[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"This is because the 3D FFTs are performed one dimension at a time, with the x direction first and the z direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers bmk associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"grid_fourier = localgrid(v̂s[1], ks_global)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, bmω = bm bmv. In Fourier space, this becomes hatbmω = i bmk hatbmv.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using StaticArrays: SVector\nusing LinearAlgebra: ×\n\nfunction curl_fourier!(\n ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,\n ) where {N}\n @inbounds for I ∈ eachindex(grid_fourier)\n # We use StaticArrays for the cross product between small vectors.\n ik⃗ = im * SVector(grid_fourier[I])\n v⃗ = SVector(getindex.(v̂s, Ref(I))) # = (v̂s[1][I], v̂s[2][I], ...)\n ω⃗ = ik⃗ × v⃗\n for n ∈ eachindex(ω⃗)\n ω̂s[n][I] = ω⃗[n]\n end\n end\n ω̂s\nend\n\nω̂s = similar.(v̂s)\ncurl_fourier!(ω̂s, v̂s, grid_fourier);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We finally transform back to physical space and plot the result:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ωs = plan .\\ ω̂s\n\nlet fig = Figure(resolution = (700, 600))\n ax = Axis3(fig[1, 1]; aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\")\n ω_norm = vecnorm(ωs)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, ω_norm;\n alpha = 0.1, levels = 0.8:0.2:2.0,\n colormap = :viridis, colorrange = (0.8, 2.0),\n )\n cb = Colorbar(fig[1, 2], ct; label = \"Vorticity magnitude\")\n fig\nend","category":"page"},{"location":"generated/navier_stokes/#Computing-the-non-linear-term","page":"Navier–Stokes equations","title":"Computing the non-linear term","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"_t hatbmv_bmk =\n- mathcalP_bmk left widehat(bmv bm) bmv right\n- ν bmk^2 hatbmv_bmk\nquad text with quad\nmathcalP_bmk(hatbmF_bmk) = left( I - fracbmk \nbmkbmk^2 right) hatbmF_bmk","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where mathcalP_bmk is a projection operator allowing to preserve the incompressibility condition bm bmv = 0. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Now that we have the wave numbers bmk, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients hatbmv_bmk of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, hatbmF_bmk = left widehat(bmv bm) bmv right_bmk. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Below we implement a function that computes the non-linear term in Fourier space based on its convective form (bmv bm) bmv = bm (bmv bmv). Note that this equivalence uses the incompressibility condition bm bmv = 0.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using LinearAlgebra: mul!, ldiv! # for applying FFT plans in-place\n\n# Compute non-linear term in Fourier space from velocity field in physical\n# space. Optional keyword arguments may be passed to avoid allocations.\nfunction ns_nonlinear!(\n F̂s, vs, plan, grid_fourier;\n vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),\n )\n # Compute F_i = ∂_j (v_i v_j) for each i.\n # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)\n w, ŵ = vbuf, v̂buf\n @inbounds for (i, F̂i) ∈ enumerate(F̂s)\n F̂i .= 0\n vi = vs[i]\n for (j, vj) ∈ enumerate(vs)\n w .= vi .* vj # w = v_i * v_j in physical space\n mul!(ŵ, plan, w) # same in Fourier space\n # Add derivative in Fourier space\n for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n kj = k⃗[j]\n F̂i[I] += im * kj * ŵ[I]\n end\n end\n end\n F̂s\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"As an example, let's use this function on our initial velocity field:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"F̂s = similar.(v̂s)\nns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)\n ks_max = maximum.(abs, ks_global) # maximum stored wave numbers (kx_max, ky_max, kz_max)\n ks_lim = (2 / 3) .* ks_max\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I]\n if any(abs.(k⃗) .> ks_lim)\n for ŵ ∈ ŵs\n ŵ[I] = 0\n end\n end\n end\n ŵs\nend\n\n# We can apply this on the previously computed non-linear term:\ndealias_twothirds!(F̂s, grid_fourier, ks_global);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Finally, we implement the projection associated to the incompressibility condition:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function project_divergence_free!(ûs, grid_fourier)\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I]\n k² = sum(abs2, k⃗)\n iszero(k²) && continue # avoid division by zero\n û = getindex.(ûs, Ref(I)) # (ûs[1][I], ûs[2][I], ...)\n for i ∈ eachindex(û)\n ŵ = û[i]\n for j ∈ eachindex(û)\n ŵ -= k⃗[i] * k⃗[j] * û[j] / k²\n end\n ûs[i][I] = ŵ\n end\n end\n ûs\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)\nv̂s_proj .≈ v̂s # the last one may be false because v_z = 0 initially","category":"page"},{"location":"generated/navier_stokes/#Putting-it-all-together","page":"Navier–Stokes equations","title":"Putting it all together","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function ns_rhs!(\n dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,\n ) where {N}\n # 1. Compute non-linear term and dealias it\n (; plan, cache, ks_global, grid_fourier) = p\n F̂s = cache.F̂s\n ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])\n dealias_twothirds!(F̂s, grid_fourier, ks_global)\n\n # 2. Project onto divergence-free space\n project_divergence_free!(F̂s, grid_fourier)\n\n # 3. Transform velocity to Fourier space\n v̂s = cache.v̂s\n map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)\n\n # 4. Add viscous term (and multiply projected non-linear term by -1)\n ν = p.ν\n for n ∈ eachindex(v̂s)\n v̂ = v̂s[n]\n F̂ = F̂s[n]\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n k² = sum(abs2, k⃗)\n F̂[I] = -F̂[I] - ν * k² * v̂[I]\n end\n end\n\n # 5. Transform RHS back to physical space\n map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)\n\n nothing\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using OrdinaryDiffEq\nusing RecursiveArrayTools: ArrayPartition\n\nns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)\nvs_init_ode = ArrayPartition(v⃗₀)\nsummary(vs_init_ode)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We now define solver parameters and temporary variables, and initialise the problem:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"params = (;\n ν = 5e-3, # kinematic viscosity\n plan, grid_fourier, ks_global,\n cache = (\n v̂s = similar.(v̂s),\n F̂s = similar.(v̂s),\n )\n)\n\ntspan = (0.0, 10.0)\nprob = ODEProblem(ns_rhs!, vs_init_ode, tspan, params)\nintegrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum E(k), to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity ν must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large k), while high enough to allow the small-scale motions to be correctly resolved.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function energy_spectrum!(Ek, ks, v̂s, grid_fourier)\n Nk = length(Ek)\n @assert Nk == length(ks)\n Ek .= 0\n for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n knorm = sqrt(sum(abs2, k⃗))\n i = searchsortedfirst(ks, knorm)\n i > Nk && continue\n v⃗ = getindex.(v̂s, Ref(I)) # = (v̂s[1][I], v̂s[2][I], ...)\n factor = k⃗[1] == 0 ? 1 : 2 # account for Hermitian symmetry and r2c transform\n Ek[i] += factor * sum(abs2, v⃗) / 2\n end\n MPI.Allreduce!(Ek, +, get_comm(v̂s[1])) # sum across all processes\n Ek\nend\n\nks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])\nEk = similar(ks)\nv̂s = plan .* integrator.u.x\nenergy_spectrum!(Ek, ks, v̂s, grid_fourier)\nEk ./= scale_factor(plan)^2 # rescale energy\n\ncurl_fourier!(ω̂s, v̂s, grid_fourier)\nldiv!.(ωs, plan, ω̂s)\nω⃗_plot = Observable(ωs)\nk_plot = @view ks[2:end]\nE_plot = Observable(@view Ek[2:end])\nt_plot = Observable(integrator.t)\n\nfig = let\n fig = Figure(resolution = (1200, 600))\n ax = Axis3(\n fig[1, 1][1, 1]; title = @lift(\"t = $(round($t_plot, digits = 3))\"),\n aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\",\n )\n ω_mag = @lift vecnorm($ω⃗_plot)\n ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, ω_mag_norm;\n alpha = 0.3, levels = 3,\n colormap = :viridis, colorrange = (0.0, 1.0),\n )\n cb = Colorbar(fig[1, 1][1, 2], ct; label = \"Normalised vorticity magnitude\")\n ax_sp = Axis(\n fig[1, 2];\n xlabel = \"k\", ylabel = \"E(k)\", xscale = log2, yscale = log10,\n title = \"Kinetic energy spectrum\",\n )\n ylims!(ax_sp, 1e-8, 1e0)\n scatterlines!(ax_sp, k_plot, E_plot)\n ks_slope = exp.(range(log(2.5), log(25.0), length = 3))\n E_fivethirds = @. 0.3 * ks_slope^(-5/3)\n @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)\n text!(ax_sp, L\"k^{-5/3}\"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))\n fig\nend\n\nusing Printf # hide\nwith_xvfb = ENV[\"DISPLAY\"] == \":99\" # hide\nnstep = 0 # hide\nconst tmpdir = mktempdir() # hide\nfilename_frame(procid, nstep) = joinpath(tmpdir, @sprintf(\"proc%d_%04d.png\", procid, nstep)) # hide\nrecord(fig, \"vorticity_proc$procid.mp4\"; framerate = 10) do io\n with_xvfb && recordframe!(io) # hide\n while integrator.t < 20\n dt = 0.001\n step!(integrator, dt)\n t_plot[] = integrator.t\n mul!.(v̂s, plan, integrator.u.x) # current velocity in Fourier space\n curl_fourier!(ω̂s, v̂s, grid_fourier)\n ldiv!.(ω⃗_plot[], plan, ω̂s)\n ω⃗_plot[] = ω⃗_plot[] # to force updating the plot\n energy_spectrum!(Ek, ks, v̂s, grid_fourier)\n Ek ./= scale_factor(plan)^2 # rescale energy\n E_plot[] = E_plot[]\n global nstep += 1 # hide\n with_xvfb ? # hide\n save(filename_frame(procid, nstep), fig) : # hide\n recordframe!(io)\n end\nend;\n\nif with_xvfb # hide\n run(pipeline(`ffmpeg -y -r 10 -i $tmpdir/proc$(procid)_%04d.png -c:v libx264 -vf \"fps=25,format=yuv420p\" vorticity_proc$procid.mp4`; stdout = \"ffmpeg.out\", stderr = \"ffmpeg.err\")) # hide\nend # hide\nnothing # hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"
\n \n
","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"This page was generated using Literate.jl.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"EditURL = \"https://github.com/jipolanco/PencilFFTs.jl/blob/master/docs/examples/gradient.jl\"","category":"page"},{"location":"generated/gradient/#Gradient-of-a-scalar-field","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This example shows different methods to compute the gradient of a real-valued 3D scalar field θ(bmx) in Fourier space, where bmx = (x y z). It is assumed that the field is periodic with period L = 2π along all dimensions.","category":"page"},{"location":"generated/gradient/#General-procedure","page":"Gradient of a scalar field","title":"General procedure","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The discrete Fourier expansion of θ writes","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"θ(bmx) = _bmk Z^3 hatθ(bmk) e^i bmk bmx","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"where bmk = (k_x k_y k_z) are the Fourier wave numbers and hatθ is the discrete Fourier transform of θ. Then, the spatial derivatives of θ are given by","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"frac θ(bmx) x_i =\n_bmk Z^3 i k_i hatθ(bmk) e^i bmk bmx","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"where the subscript i denotes one of the spatial components x, y or z.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In other words, to compute bm θ = (_x θ _y θ _z θ), one has to:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"transform θ to Fourier space to obtain hatθ,\nmultiply hatθ by i bmk,\ntransform the result back to physical space to obtain bm θ.","category":"page"},{"location":"generated/gradient/#Preparation","page":"Gradient of a scalar field","title":"Preparation","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In this section, we initialise a random real-valued scalar field θ and compute its FFT. For more details see the Tutorial.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"using MPI\nusing PencilFFTs\nusing Random\n\nMPI.Init()\n\n# Input data dimensions (Nx × Ny × Nz)\ndims = (64, 32, 64)\n\n# Apply a 3D real-to-complex (r2c) FFT.\ntransform = Transforms.RFFT()\n\n# Automatically create decomposition configuration\ncomm = MPI.COMM_WORLD\npen = Pencil(dims, comm)\n\n# Create plan\nplan = PencilFFTPlan(pen, transform)\n\n# Allocate data and initialise field\nθ = allocate_input(plan)\nrandn!(θ)\n\n# Perform distributed FFT\nθ_hat = plan * θ\nnothing # hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, we initialise the output that will hold ∇θ in Fourier space. Noting that ∇θ is a vector field, we choose to store it as a tuple of 3 PencilArrays.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"∇θ_hat = allocate_output(plan, Val(3))\n\n# This is equivalent:\n# ∇θ_hat = ntuple(d -> similar(θ_hat), Val(3))\n\nsummary(∇θ_hat)","category":"page"},{"location":"generated/gradient/#Fourier-wave-numbers","page":"Gradient of a scalar field","title":"Fourier wave numbers","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In general, the Fourier wave numbers are of the form k_i = 0 frac2πL_i frac4πL_i frac6πL_i , where L_i is the period along dimension i. When a real-to-complex Fourier transform is applied, roughly half of these wave numbers are redundant due to the Hermitian symmetry of the complex Fourier coefficients. In practice, this means that for the fastest dimension x (along which a real-to-complex transform is performed), the negative wave numbers are dropped, i.e. k_x = 0 frac2πL_x frac4πL_x .","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The AbstractFFTs package provides a convenient way to generate the Fourier wave numbers, using the functions fftfreq and rfftfreq. We can use these functions to initialise a \"grid\" of wave numbers associated to our 3D real-to-complex transform:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"using AbstractFFTs: fftfreq, rfftfreq\n\nbox_size = (2π, 2π, 2π) # Lx, Ly, Lz\nsample_rate = 2π .* dims ./ box_size\n\n# In our case (Lx = 2π and Nx even), this gives kx = [0, 1, 2, ..., Nx/2].\nkx = rfftfreq(dims[1], sample_rate[1])\n\n# In our case (Ly = 2π and Ny even), this gives\n# ky = [0, 1, 2, ..., Ny/2-1, -Ny/2, -Ny/2+1, ..., -1] (and similarly for kz).\nky = fftfreq(dims[2], sample_rate[2])\nkz = fftfreq(dims[3], sample_rate[3])\n\nkvec = (kx, ky, kz)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that kvec now contains the wave numbers associated to the global domain. In the following, we will only need the wave numbers associated to the portion of the domain handled by the local MPI process.","category":"page"},{"location":"generated/gradient/#gradient_method_global","page":"Gradient of a scalar field","title":"Method 1: global views","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"PencilArrays, returned for instance by allocate_input and allocate_output, take indices that start at 1, regardless of the location of the subdomain associated to the local process on the global grid. (In other words, PencilArrays take local indices.) On the other hand, we have defined the wave number vector kvec which, for each MPI process, is defined over the global domain, and as such it takes global indices.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"One straightforward way of making data arrays compatible with wave numbers is to use global views, i.e. arrays that take global indices. These are generated from PencilArrays by calling the global_view function. Note that, in general, global indices do not start at 1 for a given MPI process. A given process will own a range of data given by indices in (i1:i2, j1:j2, k1:k2).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"θ_glob = global_view(θ_hat)\n∇θ_glob = global_view.(∇θ_hat)\nsummary(θ_glob)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Once we have global views, we can combine data and wave numbers using the portion of global indices owned by the local MPI process, as shown below. We can use CartesianIndices to iterate over the global indices associated to the local process.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"for I in CartesianIndices(θ_glob)\n i, j, k = Tuple(I) # unpack indices\n\n # Wave number vector associated to current Cartesian index.\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n # Compute gradient in Fourier space.\n # Note that modifying ∇θ_glob also modifies the original PencilArray ∇θ_hat.\n ∇θ_glob[1][I] = im * kx * θ_glob[I]\n ∇θ_glob[2][I] = im * ky * θ_glob[I]\n ∇θ_glob[3][I] = im * kz * θ_glob[I]\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The above loop can be written in a slightly more efficient manner by precomputing im * θ_glob[I]:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@inbounds for I in CartesianIndices(θ_glob)\n i, j, k = Tuple(I)\n\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n u = im * θ_glob[I]\n\n ∇θ_glob[1][I] = kx * u\n ∇θ_glob[2][I] = ky * u\n ∇θ_glob[3][I] = kz * u\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Also note that the above can be easily written in a more generic way, e.g. for arbitrary dimensions, thanks in part to the use of CartesianIndices. Moreover, in the above there is no notion of the dimension permutations discussed in the tutorial, as it is all hidden behind the implementation of PencilArrays. And as seen later in the benchmarks, these (hidden) permutations have zero cost, as the speed is identical to that of a function that explicitly takes into account these permutations.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, we can perform a backwards transform to obtain bm θ in physical space:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"∇θ = plan \\ ∇θ_hat;\nnothing #hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that the transform is automatically broadcast over the three fields of the ∇θ_hat vector, and the result ∇θ is also a tuple of three PencilArrays.","category":"page"},{"location":"generated/gradient/#gradient_method_global_explicit","page":"Gradient of a scalar field","title":"Method 2: explicit global indexing","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Sometimes, one does not need to write generic code. In our case, one often knows the dimensionality of the problem and the memory layout of the data (i.e. the underlying index permutation).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Below is a reimplementation of the above loop, using explicit indices instead of CartesianIndices, and assuming that the underlying index permutation is (3, 2, 1), that is, data is stored in (z y x) order. As discussed in the tutorial, this is the default for transformed arrays. This example also serves as a more explicit explanation for what is going on in the first method.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"# Get local data range in the global grid.\nrng = axes(θ_glob) # = (i1:i2, j1:j2, k1:k2)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"For the loop below, we're assuming that the permutation is (3, 2, 1). In other words, the fastest index is the last one, and not the first one as it is usually in Julia. If the permutation is not (3, 2, 1), things will still work (well, except for the assertion below!), but the loop order will not be optimal.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@assert permutation(θ_hat) === Permutation(3, 2, 1)\n\n@inbounds for i in rng[1], j in rng[2], k in rng[3]\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n # Note that we still access the arrays in (i, j, k) order.\n # (The permutation happens behind the scenes!)\n u = im * θ_glob[i, j, k]\n\n ∇θ_glob[1][i, j, k] = kx * u\n ∇θ_glob[2][i, j, k] = ky * u\n ∇θ_glob[3][i, j, k] = kz * u\nend","category":"page"},{"location":"generated/gradient/#gradient_method_local","page":"Gradient of a scalar field","title":"Method 3: using local indices","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Alternatively, we can avoid global views and work directly on PencilArrays using local indices that start at 1. In this case, part of the strategy is to construct a \"local\" grid of wave numbers that can also be accessed with local indices. This can be conveniently done using the localgrid function of the PencilArrays.jl package, which accepts a PencilArray (or its associated Pencil) and the global coordinates (here kvec):","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"grid_fourier = localgrid(θ_hat, kvec)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that one can directly iterate on the returned grid object:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@inbounds for I in CartesianIndices(grid_fourier)\n # Wave number vector associated to current Cartesian index.\n local k⃗ # hide\n k⃗ = grid_fourier[I]\n u = im * θ_hat[I]\n ∇θ_hat[1][I] = k⃗[1] * u\n ∇θ_hat[2][I] = k⃗[2] * u\n ∇θ_hat[3][I] = k⃗[3] * u\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This implementation is as efficient as the other examples, while being slightly shorter to write. Moreover, it is quite generic, and can be made independent of the number of dimensions with little effort.","category":"page"},{"location":"generated/gradient/#gradient_method_broadcast","page":"Gradient of a scalar field","title":"Method 4: using broadcasting","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, note that the local grid object returned by localgrid makes it is possible to compute the gradient using broadcasting, thus fully avoiding scalar indexing. This can be quite convenient in some cases, and can also be very useful if one is working on GPUs (where scalar indexing is prohibitively expensive). Using broadcasting, the above examples simply become:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@. ∇θ_hat[1] = im * grid_fourier[1] * θ_hat\n@. ∇θ_hat[2] = im * grid_fourier[2] * θ_hat\n@. ∇θ_hat[3] = im * grid_fourier[3] * θ_hat\nnothing # hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Once again, as shown in the benchmarks further below, this method performs quite similarly to the other ones.","category":"page"},{"location":"generated/gradient/#Summary","page":"Gradient of a scalar field","title":"Summary","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The PencilArrays module provides different alternatives to deal with MPI-distributed data that may be subject to dimension permutations. In particular, one can choose to work with global indices (first two examples), with local indices (third example), or to avoid scalar indexing altogether (fourth example).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"If one wants to stay generic, making sure that the same code will work for arbitrary dimensions and will be efficient regardless of the underlying dimension permutation, methods 1, 3 or 4 should be preferred. These use CartesianIndices and make no assumptions on possible dimension permutations, which are by default enabled in the output of PencilFFTs transforms. In fact, such permutations are completely invisible in the implementations.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The second method uses explicit (i, j, k) indices. It assumes that the underlying permutation is (3, 2, 1) to loop with i as the slowest index and k as the fastest, which is the optimal order in this case given the permutation. As such, the implementation is less generic than the others, and differences in performance are negligible with respect to more generic variants.","category":"page"},{"location":"generated/gradient/#gradient_benchmarks","page":"Gradient of a scalar field","title":"Benchmark results","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The following are the benchmark results obtained from running examples/gradient.jl on a laptop, using 2 MPI processes and Julia 1.7.2, with an input array of global dimensions 64 32 64. The different methods detailed above are marked on the right. The \"lazy\" marks indicate runs where the wave numbers were represented by lazy Frequencies objects (returned by rfftfreq and fftfreq). Otherwise, they were collected into Vectors. For some reason, plain Vectors are faster when working with grids generated by localgrid.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In the script, additional implementations can be found which rely on a more advanced understanding of permutations and on the internals of the PencilArrays package. For instance, gradient_local_parent! directly works with the raw data stored in Julia Arrays, while gradient_local_linear! completely avoids CartesianIndices while staying generic and efficient. Nevertheless, these display roughly the same performance as the above examples.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":" gradient_global_view!... 89.900 μs\n gradient_global_view! (lazy)... 92.060 μs [Method 1]\n gradient_global_view_explicit!... 88.958 μs\n gradient_global_view_explicit! (lazy)... 81.055 μs [Method 2]\n gradient_local!... 92.305 μs\n gradient_grid!... 92.770 μs\n gradient_grid! (lazy)... 101.388 μs [Method 3]\n gradient_grid_broadcast!... 88.606 μs\n gradient_grid_broadcast! (lazy)... 151.020 μs [Method 4]\n gradient_local_parent!... 92.248 μs\n gradient_local_linear!... 91.212 μs\n gradient_local_linear_explicit!... 90.992 μs","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This page was generated using Literate.jl.","category":"page"},{"location":"PencilFFTs_timers/#PencilFFTs.measuring_performance","page":"Measuring performance","title":"Measuring performance","text":"","category":"section"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"It is possible to measure the time spent in different sections of the distributed transforms using the TimerOutputs package. This has a (very small) performance overhead, so it is disabled by default. To enable time measurements, call TimerOutputs.enable_debug_timings after loading PencilFFTs (see below for an example). For more details see the TimerOutputs docs.","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"Minimal example:","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"using MPI\nusing PencilFFTs\nusing TimerOutputs\n\n# Enable timing of `PencilFFTs` functions\nTimerOutputs.enable_debug_timings(PencilFFTs)\nTimerOutputs.enable_debug_timings(PencilArrays)\nTimerOutputs.enable_debug_timings(Transpositions)\n\nMPI.Init()\n\nplan = PencilFFTPlan(#= args... =#)\n\n# [do stuff with `plan`...]\n\n# Retrieve and print timing data associated to `plan`\nto = timer(plan)\nprint_timer(to)","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"By default, each PencilFFTPlan has its own TimerOutput. If you already have a TimerOutput, you can pass it to the PencilFFTPlan constructor:","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"to = TimerOutput()\nplan = PencilFFTPlan(..., timer=to)\n\n# [do stuff with `plan`...]\n\nprint_timer(to)","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"#PencilFFTs","page":"Home","title":"PencilFFTs","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Fast Fourier transforms of MPI-distributed Julia arrays.","category":"page"},{"location":"#Introduction","page":"Home","title":"Introduction","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package provides multidimensional FFTs and related transforms on MPI-distributed Julia arrays via the PencilArrays package.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The name of this package originates from the decomposition of 3D domains along two out of three dimensions, sometimes called pencil decomposition. This is illustrated by the figure below,[1] where each coloured block is managed by a different MPI process. Typically, one wants to compute FFTs on a scalar or vector field along the three spatial dimensions. In the case of a pencil decomposition, 3D FFTs are performed one dimension at a time, along the non-decomposed direction. Transforms must then be interleaved with global data transpositions to switch between pencil configurations. In high-performance computing environments, such data transpositions are generally the most expensive part of a parallel FFT computation, due to the large cost of communications between computing nodes.","category":"page"},{"location":"","page":"Home","title":"Home","text":"
\n \n
","category":"page"},{"location":"","page":"Home","title":"Home","text":"More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension N. The decompositions can be performed along an arbitrary number M N of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.","category":"page"},{"location":"#Features","page":"Home","title":"Features","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;\nFFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;\nin-place and out-of-place transforms;\nhigh scalability up to (at least) tens of thousands of MPI processes.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"PencilFFTs can be installed using the Julia package manager:","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> ] add PencilFFTs","category":"page"},{"location":"#Similar-projects","page":"Home","title":"Similar projects","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.\nPFFT is a very general parallel FFT library written in C.\nP3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.\n2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[1]: Figure adapted from this PhD thesis.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[2]: For the pencil decomposition represented in the figure, N = 3 and M = 2.","category":"page"},{"location":"PencilFFTs/#Distributed-FFT-plans","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"Distributed FFTs are implemented in the PencilFFTs module, and are built on top of the PencilArrays package.","category":"page"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"PencilFFTs/#Creating-plans","page":"Distributed FFT plans","title":"Creating plans","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"PencilFFTPlan","category":"page"},{"location":"PencilFFTs/#PencilFFTs.PencilFFTPlan","page":"Distributed FFT plans","title":"PencilFFTs.PencilFFTPlan","text":"PencilFFTPlan{T,N} <: AbstractFFTs.Plan{T}\n\nPlan for N-dimensional FFT-based transform on MPI-distributed data, where input data has type T.\n\n\n\nPencilFFTPlan(p::Pencil, transforms; kwargs...)\n\nCreate a PencilFFTPlan for distributed arrays following a given Pencil configuration. See variant below for details on the specification of transforms and on possible keyword arguments.\n\n\n\nPencilFFTPlan(\n A::PencilArray, transforms;\n fftw_flags = FFTW.ESTIMATE,\n fftw_timelimit = FFTW.NO_TIMELIMIT,\n permute_dims = Val(true),\n transpose_method = Transpositions.PointToPoint(),\n timer = timer(pencil(A)),\n)\n\nCreate plan for N-dimensional transform on MPI-distributed PencilArrays.\n\nExtended help\n\nThis creates a PencilFFTPlan for arrays sharing the same properties as A (dimensions, MPI decomposition, memory layout, ...), which describe data on an N-dimensional domain.\n\nTransforms\n\nThe transforms to be applied along each dimension are specified by the transforms argument. Possible transforms are defined as subtypes of Transforms.AbstractTransform, and are listed in Transform types. This argument may be either:\n\na tuple of N transforms to be applied along each dimension. For instance, transforms = (Transforms.R2R(FFTW.REDFT01), Transforms.RFFT(), Transforms.FFT());\na single transform to be applied along all dimensions. The input is automatically expanded into N equivalent transforms. For instance, for a three-dimensional array, transforms = Transforms.RFFT() specifies a 3D real-to-complex transform, and is equivalent to passing (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT()).\n\nNote that forward transforms are applied from left to right. In the last example, this means that a real-to-complex transform (RFFT) is first performed along the first dimension. This is followed by complex-to-complex transforms (FFT) along the second and third dimensions.\n\nInput data layout\n\nThe input PencilArray must satisfy the following constraints:\n\narray dimensions must not be permuted. This is the default when constructing PencilArrays.\nfor an M-dimensional domain decomposition (with M < N), the input array must be decomposed along the last M dimensions. For example, for a 2D decomposition of 3D data, the decomposed dimensions must be (2, 3). In particular, the first array dimension must not be distributed among different MPI processes.\nIn the PencilArrays package, the decomposed dimensions are specified at the moment of constructing a Pencil.\nthe element type must be compatible with the specified transform. For instance, real-to-complex transforms (Transforms.RFFT) require the input to be real floating point values. Other transforms, such as Transforms.R2R, accept both real and complex data.\n\nKeyword arguments\n\nThe keyword arguments fftw_flags and fftw_timelimit are passed to the FFTW plan creation functions (see AbstractFFTs docs).\npermute_dims determines whether the indices of the output data should be reversed. For instance, if the input data has global dimensions (Nx, Ny, Nz), then the output of a complex-to-complex FFT would have dimensions (Nz, Ny, Nx). This enables FFTs to always be performed along the first (i.e. fastest) array dimension, which could lead to performance gains. This option is enabled by default. For type inference reasons, it must be a value type (Val(true) or Val(false)).\ntranspose_method allows to select between implementations of the global data transpositions. See PencilArrays docs docs for details.\ntimer should be a TimerOutput object. See Measuring performance for details.\n\n\n\nPencilFFTPlan(\n dims_global::Dims{N}, transforms, proc_dims::Dims{M}, comm::MPI.Comm,\n [real_type = Float64]; extra_dims = (), kws...\n)\n\nCreate plan for N-dimensional transform.\n\nExtended help\n\nInstead of taking a PencilArray or a Pencil, this constructor requires the global dimensions of the input data, passed via the size_global argument.\n\nThe data is distributed over the MPI processes in the comm communicator. The distribution is performed over M dimensions (with M < N) according to the values in proc_dims, which specifies the number of MPI processes to put along each dimension.\n\nPencilArrays that may be transformed with the returned plan can be created using allocate_input.\n\nOptional arguments\n\nThe floating point precision can be selected by setting real_type parameter, which is Float64 by default.\nextra_dims may be used to specify the sizes of one or more extra dimensions that should not be transformed. These dimensions will be added to the rightmost (i.e. slowest) indices of the arrays. See Extra dimensions below for usage hints.\nsee the other constructor for more keyword arguments.\n\nExtra dimensions\n\nOne possible application of extra_dims is for describing the components of a vector or tensor field. However, this means that different PencilFFTPlans would need to be created for each kind of field (scalar, vector, ...). To avoid the creation of multiple plans, a possibly better alternative is to create tuples (or arrays) of PencilArrays using allocate_input and allocate_output.\n\nAnother more legitimate usage of extra_dims is to specify one or more Cartesian dimensions that should not be transformed nor split among MPI processes.\n\nExample\n\nSuppose we want to perform a 3D FFT of real data. The data is to be decomposed along two dimensions, over 8 MPI processes:\n\nsize_global = (64, 32, 128) # size of real input data\n\n# Perform real-to-complex transform along the first dimension, then\n# complex-to-complex transforms along the other dimensions.\ntransforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())\n# transforms = Transforms.RFFT() # this is equivalent to the above line\n\nproc_dims = (4, 2) # 2D decomposition\ncomm = MPI.COMM_WORLD\n\nplan = PencilFFTPlan(size_global, transforms, proc_dims, comm)\n\n\n\n\n\n","category":"type"},{"location":"PencilFFTs/#Allocating-data","page":"Distributed FFT plans","title":"Allocating data","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"allocate_input\nallocate_output","category":"page"},{"location":"PencilFFTs/#PencilFFTs.allocate_input","page":"Distributed FFT plans","title":"PencilFFTs.allocate_input","text":"allocate_input(p::PencilFFTPlan) -> PencilArray\nallocate_input(p::PencilFFTPlan, dims...) -> Array{PencilArray}\nallocate_input(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}\n\nAllocate uninitialised PencilArray that can hold input data for the given plan.\n\nThe second and third forms respectively allocate an array of PencilArrays of size dims, and a tuple of N PencilArrays.\n\nnote: In-place plans\nIf p is an in-place plan, a ManyPencilArray is allocated. This type holds PencilArray wrappers for the input and output transforms (as well as for intermediate transforms) which share the same space in memory. The input and output PencilArrays should be respectively accessed by calling first(::ManyPencilArray) and last(::ManyPencilArray).ExampleSuppose p is an in-place PencilFFTPlan. Then,@assert is_inplace(p)\nA = allocate_input(p) :: ManyPencilArray\nv_in = first(A) :: PencilArray # input data view\nv_out = last(A) :: PencilArray # output data viewAlso note that in-place plans must be performed directly on the returned ManyPencilArray, and not on the contained PencilArray views:p * A # perform forward transform in-place\np \\ A # perform backward transform in-place\n# p * v_in # not allowed!!\n\n\n\n\n\n","category":"function"},{"location":"PencilFFTs/#PencilFFTs.allocate_output","page":"Distributed FFT plans","title":"PencilFFTs.allocate_output","text":"allocate_output(p::PencilFFTPlan) -> PencilArray\nallocate_output(p::PencilFFTPlan, dims...) -> Array{PencilArray}\nallocate_output(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}\n\nAllocate uninitialised PencilArray that can hold output data for the given plan.\n\nIf p is an in-place plan, a ManyPencilArray is allocated.\n\nSee allocate_input for details.\n\n\n\n\n\n","category":"function"},{"location":"PencilFFTs/#Methods","page":"Distributed FFT plans","title":"Methods","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"get_comm(::PencilFFTPlan)\nscale_factor(::PencilFFTPlan)\ntimer(::PencilFFTPlan)\nis_inplace(::PencilFFTPlan)","category":"page"},{"location":"PencilFFTs/#PencilArrays.Pencils.MPITopologies.get_comm-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilArrays.Pencils.MPITopologies.get_comm","text":"get_comm(p::PencilFFTPlan)\n\nGet MPI communicator associated to a PencilFFTPlan.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilFFTs.Transforms.scale_factor-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilFFTs.Transforms.scale_factor","text":"scale_factor(p::PencilFFTPlan)\n\nGet scale factor associated to a PencilFFTPlan.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilArrays.Pencils.timer-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilArrays.Pencils.timer","text":"timer(p::PencilFFTPlan)\n\nGet TimerOutput attached to a PencilFFTPlan.\n\nSee Measuring performance for details.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilFFTs.Transforms.is_inplace-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilFFTs.Transforms.is_inplace","text":"Transforms.is_inplace(p::PencilFFTPlan)\n\nReturns true if the given plan operates in-place on the input data, false otherwise.\n\n\n\n\n\n","category":"method"}] +[{"location":"tutorial/#Tutorial","page":"Tutorial","title":"Tutorial","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The following tutorial shows how to perform a 3D FFT of real periodic data defined on a grid of N_x N_y N_z points.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"
\n \n \n
","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"By default, the domain is distributed on a 2D MPI topology of dimensions N_1 N_2. As an example, the above figure shows such a topology with N_1 = 4 and N_2 = 3, for a total of 12 MPI processes.","category":"page"},{"location":"tutorial/#tutorial:creating_plans","page":"Tutorial","title":"Creating plans","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The first thing to do is to create a domain decomposition configuration for the given dataset dimensions N_x N_y N_z. In the framework of PencilArrays, such a configuration is described by a Pencil object. As described in the PencilArrays docs, we can let the Pencil constructor automatically determine such a configuration. For this, only an MPI communicator and the dataset dimensions are needed:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using MPI\nusing PencilFFTs\n\nMPI.Init()\ncomm = MPI.COMM_WORLD\n\n# Input data dimensions (Nx × Ny × Nz)\ndims = (16, 32, 64)\npen = Pencil(dims, comm)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"By default this creates a 2D decomposition (for the case of a 3D dataset), but one can change this as detailed in the PencilArrays documentation linked above.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"We can now create a PencilFFTPlan, which requires information on decomposition configuration (the Pencil object) and on the transforms that will be applied:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# Apply a 3D real-to-complex (r2c) FFT.\ntransform = Transforms.RFFT()\n\n# Note that, for more control, one can instead separately specify the transforms along each dimension:\n# transform = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())\n\n# Create plan\nplan = PencilFFTPlan(pen, transform)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"See the PencilFFTPlan constructor for details on the accepted options, and the Transforms module for the possible transforms. It is also possible to enable fine-grained performance measurements via the TimerOutputs package, as described in Measuring performance.","category":"page"},{"location":"tutorial/#Allocating-data","page":"Tutorial","title":"Allocating data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Next, we want to apply the plan on some data. Transforms may only be applied on PencilArrays, which are array wrappers that include MPI decomposition information (in some sense, analogous to DistributedArrays in Julia's distributed computing approach). The helper function allocate_input can be used to allocate a PencilArray that is compatible with our plan:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# In our example, this returns a 3D PencilArray of real data (Float64).\nu = allocate_input(plan)\n\n# Fill the array with some (random) data\nusing Random\nrandn!(u)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"PencilArrays are a subtype of AbstractArray, and thus they support all common array operations.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Similarly, to preallocate output data, one can use allocate_output:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# In our example, this returns a 3D PencilArray of complex data (Complex{Float64}).\nv = allocate_output(plan)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"This is only required if one wants to apply the plans using a preallocated output (with mul!, see right below).","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The data types returned by allocate_input and allocate_output are slightly different when working with in-place transforms. See the in-place example for details.","category":"page"},{"location":"tutorial/#Applying-plans","page":"Tutorial","title":"Applying plans","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The interface to apply plans is consistent with that of AbstractFFTs. Namely, * and mul! are respectively used for forward transforms without and with preallocated output data. Similarly, \\ and ldiv! are used for backward transforms.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using LinearAlgebra # for mul!, ldiv!\n\n# Apply plan on `u` with `v` as an output\nmul!(v, plan, u)\n\n# Apply backward plan on `v` with `w` as an output\nw = similar(u)\nldiv!(w, plan, v) # now w ≈ u","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Note that, consistently with AbstractFFTs, normalisation is performed at the end of a backward transform, so that the original data is recovered when applying a forward followed by a backward transform.","category":"page"},{"location":"tutorial/#Accessing-and-modifying-data","page":"Tutorial","title":"Accessing and modifying data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For any given MPI process, a PencilArray holds the data associated to its local partition in the global geometry. PencilArrays are accessed using local indices that start at 1, regardless of the location of the local process in the MPI topology. Note that PencilArrays, being based on regular Arrays, support both linear and Cartesian indexing (see the Julia docs for details).","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For convenience, the global_view function can be used to generate an OffsetArray wrapper that takes global indices.","category":"page"},{"location":"tutorial/#tutorial:output_data_layout","page":"Tutorial","title":"Output data layout","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"In memory, the dimensions of the transform output are by default reversed with respect to the input. That is, if the order of indices in the input data is (x, y, z), then the output has order (z, y, x) in memory. This detail is hidden from the user, and output arrays are always accessed in the same order as the input data, regardless of the underlying output dimension permutation. This applies to PencilArrays and to OffsetArrays returned by global_view.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The reasoning behind dimension permutations, is that they allow to always perform FFTs along the fastest array dimension and to avoid a local data transposition, resulting in performance gains. A similar approach is followed by other parallel FFT libraries. FFTW itself, in its distributed-memory routines, includes a flag that enables a similar behaviour. In PencilFFTs, index permutation is the default, but it can be disabled via the permute_dims flag of PencilFFTPlan.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"A great deal of work has been spent in making generic index permutations as efficient as possible, both in intermediate and in the output state of the multidimensional transforms. This has been achieved, in part, by making sure that permutations such as (3, 2, 1) are compile-time constants.","category":"page"},{"location":"tutorial/#Further-reading","page":"Tutorial","title":"Further reading","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For details on working with PencilArrays see the PencilArrays docs.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The examples on the sidebar further illustrate the use of transforms and provide an introduction to working with MPI-distributed data in the form of PencilArrays. In particular, the gradient example illustrates different ways of computing things using Fourier-transformed distributed arrays. Then, the incompressible Navier–Stokes example is a more advanced and complete example of a possible application of the PencilFFTs package.","category":"page"},{"location":"benchmarks/#Benchmarks","page":"Benchmarks","title":"Benchmarks","text":"","category":"section"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The performance of PencilFFTs.jl is comparable to that of other open-source parallel FFT libraries implemented in lower-level languages. Below, we show comparisons with the Fortran implementation of P3DFFT, possibly the most popular of these libraries. The benchmarks were performed on the Jean–Zay cluster of the IDRIS French computing centre (CNRS).","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The figure below shows strong scaling benchmarks of 3D real-to-complex FFTs using 2D (\"pencil\") decomposition. The benchmarks were run for input arrays of dimensions N_x N_y N_z = 512^3, 1024^3 and 2048^3. Each timing is averaged over 100 repetitions.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"
\n \n \n
","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"As seen above, PencilFFTs generally outperforms P3DFFT in its default setting. This is largely explained by the choice of using non-blocking point-to-point MPI communications (via MPI_Isend and MPI_Irecv), while P3DFFT uses collective MPI_Alltoallv calls. This enables PencilFFTs to perform data reordering operations on the partially received data while waiting for the incoming data, leading to better performance. Moreover, in contrast with P3DFFT, the high performance and scalability of PencilFFTs results from a highly generic code, handling decompositions in arbitrary dimensions and a relatively large (and extensible) variety of transformations.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"Note that PencilFFTs can optionally use collective communications (using MPI_Alltoallv) instead of point-to-point communications. For details, see the docs for PencilFFTPlan and for PencilArray transpositions. As seen above, collective communications generally perform worse than point-to-point ones, and runtimes are nearly indistinguishable from those of P3DFFT.","category":"page"},{"location":"benchmarks/#Benchmark-details","page":"Benchmarks","title":"Benchmark details","text":"","category":"section"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The benchmarks were performed using Julia 1.7-beta3 and Intel MPI 2019. We used PencilFFTs v0.12.5 with FFTW.jl v1.4.3 and MPI.jl v0.19.0. We used the Fortran implementation of P3DFFT, version 2.7.6, which was built with Intel 2019 compilers and linked to FFTW 3.3.8. The cluster where the benchmarks were run has Intel Cascade Lake 6248 processors with 2×20 cores per node.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The number of MPI processes along each decomposed dimension, P_1 and P_2, was automatically determined by a call to MPI_Dims_create, which tends to create a balanced decomposition with P_1 P_2. For instance, a total of 1024 processes is divided into P_1 = P_2 = 32. Different results may be obtained with other combinations, but this was not benchmarked.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The source files used to generate this benchmark, as well as the raw benchmark results, are all available in the PencilFFTs repo.","category":"page"},{"location":"GlobalFFTParams/#Global-FFT-parameters","page":"Global FFT parameters","title":"Global FFT parameters","text":"","category":"section"},{"location":"GlobalFFTParams/","page":"Global FFT parameters","title":"Global FFT parameters","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"GlobalFFTParams/","page":"Global FFT parameters","title":"Global FFT parameters","text":"GlobalFFTParams","category":"page"},{"location":"GlobalFFTParams/#PencilFFTs.GlobalFFTParams","page":"Global FFT parameters","title":"PencilFFTs.GlobalFFTParams","text":"GlobalFFTParams{T, N, inplace}\n\nSpecifies the global parameters for an N-dimensional distributed transform. These include the element type T and global data sizes of input and output data, as well as the transform types to be performed along each dimension.\n\n\n\nGlobalFFTParams(size_global, transforms, [real_type=Float64])\n\nDefine parameters for N-dimensional transform.\n\ntransforms must be a tuple of length N specifying the transforms to be applied along each dimension. Each element must be a subtype of Transforms.AbstractTransform. For all the possible transforms, see Transform types.\n\nThe element type must be a real type accepted by FFTW, i.e. either Float32 or Float64.\n\nNote that the transforms are applied one dimension at a time, with the leftmost dimension first for forward transforms.\n\nExample\n\nTo perform a 3D FFT of real data, first a real-to-complex FFT must be applied along the first dimension, followed by two complex-to-complex FFTs along the other dimensions:\n\njulia> size_global = (64, 32, 128); # size of real input data\n\njulia> transforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT());\n\njulia> fft_params = PencilFFTs.GlobalFFTParams(size_global, transforms)\nTransforms: (RFFT, FFT, FFT)\nInput type: Float64\nGlobal dimensions: (64, 32, 128) -> (33, 32, 128)\n\n\n\n\n\n","category":"type"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"EditURL = \"../../examples/in-place.jl\"","category":"page"},{"location":"generated/in-place/#In-place-transforms","page":"In-place transforms","title":"In-place transforms","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Complex-to-complex and real-to-real transforms can be performed in-place, enabling important memory savings. The procedure is very similar to that of out-of-place transforms described in the tutorial. The differences are illustrated in the sections below.","category":"page"},{"location":"generated/in-place/#Creating-a-domain-partition","page":"In-place transforms","title":"Creating a domain partition","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"We start by partitioning a domain of dimensions 163264 along all available MPI processes.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"using PencilFFTs\nusing MPI\nMPI.Init()\n\ndims_global = (16, 32, 64) # global dimensions","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Such a partitioning is described by a Pencil object. Here we choose to decompose the domain along the last two dimensions. In this case, the actual number of processes along each of these dimensions is chosen automatically.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"decomp_dims = (2, 3)\ncomm = MPI.COMM_WORLD\npen = Pencil(dims_global, decomp_dims, comm)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"warning: Allowed decompositions\nDistributed transforms using PencilFFTs.jl require that the first dimension is not decomposed. In other words, if one wants to perform transforms, then decomp_dims above must not contain 1.","category":"page"},{"location":"generated/in-place/#Creating-in-place-plans","page":"In-place transforms","title":"Creating in-place plans","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"To create an in-place plan, pass an in-place transform such as Transforms.FFT! or Transforms.R2R! to PencilFFTPlan. For instance:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"# Perform a 3D in-place complex-to-complex FFT.\ntransform = Transforms.FFT!()\n\n# Note that one can also combine different types of in-place transforms.\n# For instance:\n# transform = (\n# Transforms.R2R!(FFTW.REDFT01),\n# Transforms.FFT!(),\n# Transforms.R2R!(FFTW.DHT),\n# )","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"We can now create a distributed plan from the previously-created domain partition and the chosen transform.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan = PencilFFTPlan(pen, transform)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Note that in-place real-to-complex transforms are not currently supported. (In other words, the RFFT! transform type is not defined.)","category":"page"},{"location":"generated/in-place/#Allocating-data","page":"In-place transforms","title":"Allocating data","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"As with out-of-place plans, data should be allocated using allocate_input. The difference is that, for in-place plans, this function returns a ManyPencilArray object, which is a container holding multiple PencilArray views sharing the same memory space.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"# Allocate data for the plan.\n# Since `plan` is in-place, this returns a `ManyPencilArray` container.\nA = allocate_input(plan)\nsummary(A)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Note that allocate_output also works for in-place plans. In this case, it returns exactly the same thing as allocate_input.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"As shown in the next section, in-place plans must be applied on the returned ManyPencilArray. On the other hand, one usually wants to access and modify data, and for this one needs the PencilArray views contained in the ManyPencilArray. The input and output array views can be obtained by calling first(::ManyPencilArray) and last(::ManyPencilArray).","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"For instance, we can initialise the input array with some data before transforming:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"using Random\nu_in = first(A) # input data view\nrandn!(u_in)\nsummary(u_in)","category":"page"},{"location":"generated/in-place/#Applying-plans","page":"In-place transforms","title":"Applying plans","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Like in FFTW.jl, one can perform in-place transforms using the * and \\ operators. As mentioned above, in-place plans must be applied on the ManyPencilArray containers returned by allocate_input.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan * A; # performs in-place forward transform\nnothing #hide","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"After performing an in-place transform, data contained in u_in has been overwritten and has no \"physical\" meaning. In other words, u_in should not be used at this point. To access the transformed data, one should retrieve the output data view using last(A).","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"For instance, to compute the global sum of the transformed data:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"u_out = last(A) # output data view\nsum(u_out) # sum of transformed data (note that `sum` reduces over all processes)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Finally, we can perform a backward transform and do stuff with the input view:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan \\ A; # perform in-place backward transform\nnothing #hide","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"At this point, the data can be once again found in the input view u_in, while u_out should not be accessed.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"This page was generated using Literate.jl.","category":"page"},{"location":"Transforms/#Available-transforms","page":"Available transforms","title":"Available transforms","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"CurrentModule = PencilFFTs.Transforms","category":"page"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"Transforms","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms","page":"Available transforms","title":"PencilFFTs.Transforms","text":"Defines different one-dimensional FFT-based transforms.\n\nThe transforms are all subtypes of an AbstractTransform type.\n\nWhen possible, the names of the transforms are kept consistent with the functions exported by AbstractFFTs.jl and FFTW.jl.\n\n\n\n\n\n","category":"module"},{"location":"Transforms/#Transform-types","page":"Available transforms","title":"Transform types","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"FFT\nFFT!\nBFFT\nBFFT!\n\nRFFT\nRFFT!\nBRFFT\nBRFFT!\n\nR2R\nR2R!\n\nNoTransform\nNoTransform!","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.FFT","page":"Available transforms","title":"PencilFFTs.Transforms.FFT","text":"FFT()\n\nComplex-to-complex FFT.\n\nSee also AbstractFFTs.fft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.FFT!","page":"Available transforms","title":"PencilFFTs.Transforms.FFT!","text":"FFT!()\n\nIn-place version of FFT.\n\nSee also AbstractFFTs.fft!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BFFT","page":"Available transforms","title":"PencilFFTs.Transforms.BFFT","text":"BFFT()\n\nUnnormalised backward complex-to-complex FFT.\n\nLike AbstractFFTs.bfft, this transform is not normalised. To obtain the inverse transform, divide the output by the length of the transformed dimension.\n\nSee also AbstractFFTs.bfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.BFFT!","text":"BFFT()\n\nIn-place version of BFFT.\n\nSee also AbstractFFTs.bfft!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.RFFT","page":"Available transforms","title":"PencilFFTs.Transforms.RFFT","text":"RFFT()\n\nReal-to-complex FFT.\n\nSee also AbstractFFTs.rfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.RFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.RFFT!","text":"RFFT!()\n\nIn-place version of RFFT.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BRFFT","page":"Available transforms","title":"PencilFFTs.Transforms.BRFFT","text":"BRFFT(d::Integer)\nBRFFT((d1, d2, ..., dN))\n\nUnnormalised inverse of RFFT.\n\nTo obtain the inverse transform, divide the output by the length of the transformed dimension (of the real output array).\n\nAs described in the AbstractFFTs docs, the length of the output cannot be fully inferred from the input length. For this reason, the BRFFT constructor accepts an optional d argument indicating the output length.\n\nFor multidimensional datasets, a tuple of dimensions (d1, d2, ..., dN) may also be passed. This is equivalent to passing just dN. In this case, the last dimension (dN) is the one that changes size between the input and output. Note that this is the opposite of FFTW.brfft. The reason is that, in PencilFFTs, the last dimension is the one along which a complex-to-real transform is performed.\n\nSee also AbstractFFTs.brfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BRFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.BRFFT!","text":"BRFFT!(d::Integer)\nBRFFT!((d1, d2, ..., dN))\n\nIn-place version of BRFFT.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.R2R","page":"Available transforms","title":"PencilFFTs.Transforms.R2R","text":"R2R(kind)\n\nReal-to-real transform of type kind.\n\nThe possible values of kind are those described in the FFTW.r2r docs and the FFTW manual:\n\ndiscrete cosine transforms: FFTW.REDFT00, FFTW.REDFT01, FFTW.REDFFT10, FFTW.REDFFT11\ndiscrete sine transforms: FFTW.RODFT00, FFTW.RODFT01, FFTW.RODFFT10, FFTW.RODFFT11\ndiscrete Hartley transform: FFTW.DHT\n\nNote: half-complex format DFTs (FFTW.R2HC, FFTW.HC2R) are not currently supported.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.R2R!","page":"Available transforms","title":"PencilFFTs.Transforms.R2R!","text":"R2R!(kind)\n\nIn-place version of R2R.\n\nSee also FFTW.r2r!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.NoTransform","page":"Available transforms","title":"PencilFFTs.Transforms.NoTransform","text":"NoTransform()\n\nIdentity transform.\n\nSpecifies that no transformation should be applied.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.NoTransform!","page":"Available transforms","title":"PencilFFTs.Transforms.NoTransform!","text":"NoTransform!()\n\nIn-place version of NoTransform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#Internals","page":"Available transforms","title":"Internals","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"What follows is used internally in PencilFFTs.","category":"page"},{"location":"Transforms/#Types","page":"Available transforms","title":"Types","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"AbstractCustomPlan\nAbstractTransform\nIdentityPlan\nIdentityPlan!\nPlan","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.AbstractCustomPlan","page":"Available transforms","title":"PencilFFTs.Transforms.AbstractCustomPlan","text":"AbstractCustomPlan\n\nAbstract type defining a custom plan, to be used as an alternative to FFTW plans (FFTW.FFTWPlan).\n\nThe only custom plan defined in this module is IdentityPlan. The user can define other custom plans that are also subtypes of AbstractCustomPlan.\n\nNote that plan returns a subtype of either AbstractFFTs.Plan or AbstractCustomPlan.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.AbstractTransform","page":"Available transforms","title":"PencilFFTs.Transforms.AbstractTransform","text":"AbstractTransform\n\nSpecifies a one-dimensional FFT-based transform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.IdentityPlan","page":"Available transforms","title":"PencilFFTs.Transforms.IdentityPlan","text":"IdentityPlan\n\nType of plan associated to NoTransform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.IdentityPlan!","page":"Available transforms","title":"PencilFFTs.Transforms.IdentityPlan!","text":"IdentityPlan!\n\nType of plan associated to NoTransform!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.Plan","page":"Available transforms","title":"PencilFFTs.Transforms.Plan","text":"Plan = Union{AbstractFFTs.Plan, AbstractCustomPlan}\n\nUnion type representing any plan returned by plan.\n\nSee also AbstractCustomPlan.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#Functions","page":"Available transforms","title":"Functions","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"plan\n\nbinv\nscale_factor\n\neltype_input\neltype_output\nexpand_dims\nis_inplace\nkind\nlength_output","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.plan","page":"Available transforms","title":"PencilFFTs.Transforms.plan","text":"plan(transform::AbstractTransform, A, [dims];\n flags=FFTW.ESTIMATE, timelimit=Inf)\n\nCreate plan to transform array A along dimensions dims.\n\nIf dims is not specified, all dimensions of A are transformed.\n\nFor FFT plans, this function wraps the AbstractFFTs.jl and FFTW.jl plan creation functions. For more details on the function arguments, see AbstractFFTs.plan_fft.\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.binv","page":"Available transforms","title":"PencilFFTs.Transforms.binv","text":"binv(transform::AbstractTransform, d::Integer)\n\nReturns the backwards transform associated to the given transform.\n\nThe second argument must be the length of the first transformed dimension in the forward transform. It is used in particular when transform = RFFT(), to determine the length of the inverse (complex-to-real) transform. See the AbstractFFTs.irfft docs for details.\n\nThe backwards transform returned by this function is not normalised. The normalisation factor for a given array can be obtained by calling scale_factor.\n\nExample\n\njulia> binv(Transforms.FFT(), 42)\nBFFT\n\njulia> binv(Transforms.BRFFT(9), 42)\nRFFT\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.scale_factor","page":"Available transforms","title":"PencilFFTs.Transforms.scale_factor","text":"scale_factor(transform::AbstractTransform, A, [dims = 1:ndims(A)])\n\nGet factor required to normalise the given array after a transformation along dimensions dims (all dimensions by default).\n\nThe array A must have the dimensions of the transform input.\n\nImportant: the dimensions dims must be the same that were passed to plan.\n\nExamples\n\njulia> C = zeros(ComplexF32, 3, 4, 5);\n\njulia> scale_factor(Transforms.FFT(), C)\n60\n\njulia> scale_factor(Transforms.BFFT(), C)\n60\n\njulia> scale_factor(Transforms.BFFT(), C, 2:3)\n20\n\njulia> R = zeros(Float64, 3, 4, 5);\n\njulia> scale_factor(Transforms.RFFT(), R, 2)\n4\n\njulia> scale_factor(Transforms.RFFT(), R, 2:3)\n20\n\njulia> scale_factor(Transforms.BRFFT(8), C)\n96\n\njulia> scale_factor(Transforms.BRFFT(9), C)\n108\n\nThis will fail because the input of RFFT is real, and R is a complex array:\n\njulia> scale_factor(Transforms.RFFT(), C, 2:3)\nERROR: MethodError: no method matching scale_factor(::PencilFFTs.Transforms.RFFT, ::Array{ComplexF32, 3}, ::UnitRange{Int64})\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.eltype_input","page":"Available transforms","title":"PencilFFTs.Transforms.eltype_input","text":"eltype_input(transform::AbstractTransform, real_type<:AbstractFloat)\n\nDetermine input data type for a given transform given the floating point precision of the input data.\n\nSome transforms, such as R2R and NoTransform, can take both real and complex data. For those kinds of transforms, nothing is returned.\n\nExample\n\njulia> eltype_input(Transforms.FFT(), Float32)\nComplexF32 (alias for Complex{Float32})\n\njulia> eltype_input(Transforms.RFFT(), Float64)\nFloat64\n\njulia> eltype_input(Transforms.R2R(FFTW.REDFT01), Float64) # nothing\n\njulia> eltype_input(Transforms.NoTransform(), Float64) # nothing\n\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.eltype_output","page":"Available transforms","title":"PencilFFTs.Transforms.eltype_output","text":"eltype_output(transform::AbstractTransform, eltype_input)\n\nReturns the output data type for a given transform given the input type.\n\nThrows ArgumentError if the input data type is incompatible with the transform type.\n\nExample\n\njulia> eltype_output(Transforms.NoTransform(), Float32)\nFloat32\n\njulia> eltype_output(Transforms.RFFT(), Float64)\nComplexF64 (alias for Complex{Float64})\n\njulia> eltype_output(Transforms.BRFFT(4), ComplexF32)\nFloat32\n\njulia> eltype_output(Transforms.FFT(), Float64)\nERROR: ArgumentError: invalid input data type for PencilFFTs.Transforms.FFT: Float64\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.expand_dims","page":"Available transforms","title":"PencilFFTs.Transforms.expand_dims","text":"expand_dims(transform::AbstractTransform, Val(N))\n\nExpand a single multidimensional transform into one transform per dimension.\n\nExample\n\n# Expand a real-to-complex transform in 3 dimensions.\njulia> expand_dims(Transforms.RFFT(), Val(3))\n(RFFT, FFT, FFT)\n\njulia> expand_dims(Transforms.BRFFT(4), Val(3))\n(BFFT, BFFT, BRFFT{even})\n\njulia> expand_dims(Transforms.NoTransform(), Val(2))\n(NoTransform, NoTransform)\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.is_inplace","page":"Available transforms","title":"PencilFFTs.Transforms.is_inplace","text":"is_inplace(transform::AbstractTransform) -> Bool\nis_inplace(transforms::Vararg{AbtractTransform}) -> Union{Bool, Nothing}\n\nCheck whether a transform or a list of transforms is performed in-place.\n\nIf the list of transforms has a combination of in-place and out-of-place transforms, nothing is returned.\n\nExample\n\njulia> is_inplace(Transforms.RFFT())\nfalse\n\njulia> is_inplace(Transforms.NoTransform!())\ntrue\n\njulia> is_inplace(Transforms.FFT!(), Transforms.R2R!(FFTW.REDFT01))\ntrue\n\njulia> is_inplace(Transforms.FFT(), Transforms.R2R(FFTW.REDFT01))\nfalse\n\njulia> is_inplace(Transforms.FFT(), Transforms.R2R!(FFTW.REDFT01)) === nothing\ntrue\n\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.kind","page":"Available transforms","title":"PencilFFTs.Transforms.kind","text":"kind(transform::R2R)\n\nGet kind of real-to-real transform.\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.length_output","page":"Available transforms","title":"PencilFFTs.Transforms.length_output","text":"length_output(transform::AbstractTransform, length_in::Integer)\n\nReturns the length of the transform output, given the length of its input.\n\nThe input and output lengths are specified in terms of the respective input and output datatypes. For instance, for real-to-complex transforms, these are respectively the length of input real data and of output complex data.\n\n\n\n\n\n","category":"function"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"EditURL = \"../../examples/navier_stokes.jl\"","category":"page"},{"location":"generated/navier_stokes/#Navier–Stokes-equations","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"In this example, we numerically solve the incompressible Navier–Stokes equations","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"_t bmv + (bmv bm) bmv = -frac1ρ bm p + ν ^2 bmv\nquad bm bmv = 0","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where bmv(bmx t) and p(bmx t) are respectively the velocity and pressure fields, ν is the fluid kinematic viscosity and ρ is the fluid density.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.","category":"page"},{"location":"generated/navier_stokes/#First-steps","page":"Navier–Stokes equations","title":"First steps","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We start by loading the required packages, initialising MPI and setting the simulation parameters.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using MPI\nusing PencilFFTs\n\nMPI.Init()\ncomm = MPI.COMM_WORLD\nprocid = MPI.Comm_rank(comm) + 1\n\n# Simulation parameters\nNs = (64, 64, 64) # = (Nx, Ny, Nz)\nLs = (2π, 2π, 2π) # = (Lx, Ly, Lz)\n\n# Collocation points (\"global\" = over all processes).\n# We include the endpoint (length = N + 1) for convenience.\nxs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls) # = (x, y, z)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's check the number of MPI processes over which we're running our simulation:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"MPI.Comm_size(comm)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"pen = Pencil(Ns, comm)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"The subdomain associated to the local MPI process can be obtained using range_local:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"range_local(pen)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We now construct a distributed vector field that follows the decomposition configuration we just created:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v⃗₀ = (\n PencilArray{Float64}(undef, pen), # vx\n PencilArray{Float64}(undef, pen), # vy\n PencilArray{Float64}(undef, pen), # vz\n)\nsummary(v⃗₀[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We still need to fill this array with interesting values that represent a physical velocity field.","category":"page"},{"location":"generated/navier_stokes/#Initial-condition","page":"Navier–Stokes equations","title":"Initial condition","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"beginaligned\nv_x(x y z) = u₀ sin(k₀ x) cos(k₀ y) cos(k₀ z) \nv_y(x y z) = -u₀ cos(k₀ x) sin(k₀ y) cos(k₀ z) \nv_z(x y z) = 0\nendaligned","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where u₀ and k₀ are two parameters setting the amplitude and the period of the velocity field.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"grid = localgrid(pen, xs_global)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can use this to initialise the velocity field:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"u₀ = 1.0\nk₀ = 2π / Ls[1] # should be integer if L = 2π (to preserve periodicity)\n\n@. v⃗₀[1] = u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)\n@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)\n@. v⃗₀[3] = 0\nnothing # hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's plot a 2D slice of the velocity field managed by the local MPI process:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using GLMakie\n\n# Compute the norm of a vector field represented by a tuple of arrays.\nfunction vecnorm(v⃗::NTuple)\n vnorm = similar(v⃗[1])\n for n ∈ eachindex(v⃗[1])\n w = zero(eltype(vnorm))\n for v ∈ v⃗\n w += v[n]^2\n end\n vnorm[n] = sqrt(w)\n end\n vnorm\nend\n\n# This is useful for passing coordinates to Makie.contour!\nto_intervals(grid) = map(xs -> xs[begin]..xs[end], grid.coords)\n\nlet fig = Figure(size = (700, 600))\n ax = Axis3(fig[1, 1]; aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\")\n vnorm = parent(vecnorm(v⃗₀)) # use `parent` because Makie doesn't like custom array types...\n ct = contour!(\n ax, to_intervals(grid)..., vnorm;\n alpha = 0.2, levels = 4,\n colormap = :viridis,\n colorrange = (0.0, 1.0),\n highclip = (:red, 0.2), lowclip = (:green, 0.2),\n )\n cb = Colorbar(fig[1, 2], ct; label = \"Velocity magnitude\")\n fig\nend","category":"page"},{"location":"generated/navier_stokes/#Velocity-in-Fourier-space","page":"Navier–Stokes equations","title":"Velocity in Fourier space","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"bmv(bmx t) =\n_bmk hatbmv_bmk(t) e^i bmk bmx","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where bmk = (k_x k_y k_z) are the discrete wave numbers.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for k_x:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using AbstractFFTs: fftfreq, rfftfreq\n\nks_global = (\n rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]), # kx | real-to-complex\n fftfreq(Ns[2], 2π * Ns[2] / Ls[2]), # ky | complex-to-complex\n fftfreq(Ns[3], 2π * Ns[3] / Ls[3]), # kz | complex-to-complex\n)\n\nks_global[1]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ks_global[2]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ks_global[3]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"See PencilFFTPlan for details on creating plans and on optional keyword arguments.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients hatbmv_bmk:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v̂s = plan .* v⃗₀\nsummary(v̂s[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Note that, in Fourier space, the domain decomposition is performed along the directions x and y:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"pencil(v̂s[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"This is because the 3D FFTs are performed one dimension at a time, with the x direction first and the z direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers bmk associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"grid_fourier = localgrid(v̂s[1], ks_global)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, bmω = bm bmv. In Fourier space, this becomes hatbmω = i bmk hatbmv.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using StaticArrays: SVector\nusing LinearAlgebra: ×\n\nfunction curl_fourier!(\n ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,\n ) where {N}\n @inbounds for I ∈ eachindex(grid_fourier)\n # We use StaticArrays for the cross product between small vectors.\n ik⃗ = im * SVector(grid_fourier[I])\n v⃗ = SVector(getindex.(v̂s, Ref(I))) # = (v̂s[1][I], v̂s[2][I], ...)\n ω⃗ = ik⃗ × v⃗\n for n ∈ eachindex(ω⃗)\n ω̂s[n][I] = ω⃗[n]\n end\n end\n ω̂s\nend\n\nω̂s = similar.(v̂s)\ncurl_fourier!(ω̂s, v̂s, grid_fourier);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We finally transform back to physical space and plot the result:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ωs = plan .\\ ω̂s\n\nlet fig = Figure(size = (700, 600))\n ax = Axis3(fig[1, 1]; aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\")\n ω_norm = parent(vecnorm(ωs))\n ct = contour!(\n ax, to_intervals(grid)..., ω_norm;\n alpha = 0.1, levels = 0.8:0.2:2.0,\n colormap = :viridis, colorrange = (0.8, 2.0),\n highclip = (:red, 0.2), lowclip = (:green, 0.2),\n )\n cb = Colorbar(fig[1, 2], ct; label = \"Vorticity magnitude\")\n fig\nend","category":"page"},{"location":"generated/navier_stokes/#Computing-the-non-linear-term","page":"Navier–Stokes equations","title":"Computing the non-linear term","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"_t hatbmv_bmk =\n- mathcalP_bmk left widehat(bmv bm) bmv right\n- ν bmk^2 hatbmv_bmk\nquad text with quad\nmathcalP_bmk(hatbmF_bmk) = left( I - fracbmk \nbmkbmk^2 right) hatbmF_bmk","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where mathcalP_bmk is a projection operator allowing to preserve the incompressibility condition bm bmv = 0. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Now that we have the wave numbers bmk, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients hatbmv_bmk of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, hatbmF_bmk = left widehat(bmv bm) bmv right_bmk. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Below we implement a function that computes the non-linear term in Fourier space based on its convective form (bmv bm) bmv = bm (bmv bmv). Note that this equivalence uses the incompressibility condition bm bmv = 0.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using LinearAlgebra: mul!, ldiv! # for applying FFT plans in-place\n\n# Compute non-linear term in Fourier space from velocity field in physical\n# space. Optional keyword arguments may be passed to avoid allocations.\nfunction ns_nonlinear!(\n F̂s, vs, plan, grid_fourier;\n vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),\n )\n # Compute F_i = ∂_j (v_i v_j) for each i.\n # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)\n w, ŵ = vbuf, v̂buf\n @inbounds for (i, F̂i) ∈ enumerate(F̂s)\n F̂i .= 0\n vi = vs[i]\n for (j, vj) ∈ enumerate(vs)\n w .= vi .* vj # w = v_i * v_j in physical space\n mul!(ŵ, plan, w) # same in Fourier space\n # Add derivative in Fourier space\n for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n kj = k⃗[j]\n F̂i[I] += im * kj * ŵ[I]\n end\n end\n end\n F̂s\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"As an example, let's use this function on our initial velocity field:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"F̂s = similar.(v̂s)\nns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)\n ks_max = maximum.(abs, ks_global) # maximum stored wave numbers (kx_max, ky_max, kz_max)\n ks_lim = (2 / 3) .* ks_max\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I]\n if any(abs.(k⃗) .> ks_lim)\n for ŵ ∈ ŵs\n ŵ[I] = 0\n end\n end\n end\n ŵs\nend\n\n# We can apply this on the previously computed non-linear term:\ndealias_twothirds!(F̂s, grid_fourier, ks_global);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Finally, we implement the projection associated to the incompressibility condition:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function project_divergence_free!(ûs, grid_fourier)\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I]\n k² = sum(abs2, k⃗)\n iszero(k²) && continue # avoid division by zero\n û = getindex.(ûs, Ref(I)) # (ûs[1][I], ûs[2][I], ...)\n for i ∈ eachindex(û)\n ŵ = û[i]\n for j ∈ eachindex(û)\n ŵ -= k⃗[i] * k⃗[j] * û[j] / k²\n end\n ûs[i][I] = ŵ\n end\n end\n ûs\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)\nv̂s_proj .≈ v̂s # the last one may be false because v_z = 0 initially","category":"page"},{"location":"generated/navier_stokes/#Putting-it-all-together","page":"Navier–Stokes equations","title":"Putting it all together","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function ns_rhs!(\n dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,\n ) where {N}\n # 1. Compute non-linear term and dealias it\n (; plan, cache, ks_global, grid_fourier) = p\n F̂s = cache.F̂s\n ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])\n dealias_twothirds!(F̂s, grid_fourier, ks_global)\n\n # 2. Project onto divergence-free space\n project_divergence_free!(F̂s, grid_fourier)\n\n # 3. Transform velocity to Fourier space\n v̂s = cache.v̂s\n map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)\n\n # 4. Add viscous term (and multiply projected non-linear term by -1)\n ν = p.ν\n for n ∈ eachindex(v̂s)\n v̂ = v̂s[n]\n F̂ = F̂s[n]\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n k² = sum(abs2, k⃗)\n F̂[I] = -F̂[I] - ν * k² * v̂[I]\n end\n end\n\n # 5. Transform RHS back to physical space\n map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)\n\n nothing\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using OrdinaryDiffEqLowOrderRK # includes RK4\nusing RecursiveArrayTools: ArrayPartition\n\nns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)\nvs_init_ode = ArrayPartition(v⃗₀)\nsummary(vs_init_ode)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We now define solver parameters and temporary variables, and initialise the problem:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"params = (;\n ν = 5e-3, # kinematic viscosity\n plan, grid_fourier, ks_global,\n cache = (\n v̂s = similar.(v̂s),\n F̂s = similar.(v̂s),\n )\n)\n\ntspan = (0.0, 10.0)\nprob = ODEProblem{true}(ns_rhs!, vs_init_ode, tspan, params)\nintegrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum E(k), to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity ν must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large k), while high enough to allow the small-scale motions to be correctly resolved.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function energy_spectrum!(Ek, ks, v̂s, grid_fourier)\n Nk = length(Ek)\n @assert Nk == length(ks)\n Ek .= 0\n for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n knorm = sqrt(sum(abs2, k⃗))\n i = searchsortedfirst(ks, knorm)\n i > Nk && continue\n v⃗ = getindex.(v̂s, Ref(I)) # = (v̂s[1][I], v̂s[2][I], ...)\n factor = k⃗[1] == 0 ? 1 : 2 # account for Hermitian symmetry and r2c transform\n Ek[i] += factor * sum(abs2, v⃗) / 2\n end\n MPI.Allreduce!(Ek, +, get_comm(v̂s[1])) # sum across all processes\n Ek\nend\n\nks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])\nEk = similar(ks)\nv̂s = plan .* integrator.u.x\nenergy_spectrum!(Ek, ks, v̂s, grid_fourier)\nEk ./= scale_factor(plan)^2 # rescale energy\n\ncurl_fourier!(ω̂s, v̂s, grid_fourier)\nldiv!.(ωs, plan, ω̂s)\nω⃗_plot = Observable(ωs)\nk_plot = @view ks[2:end]\nE_plot = Observable(@view Ek[2:end])\nt_plot = Observable(integrator.t)\n\nfig = let\n fig = Figure(size = (1200, 600))\n ax = Axis3(\n fig[1, 1][1, 1]; title = @lift(\"t = $(round($t_plot, digits = 3))\"),\n aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\",\n )\n ω_mag = @lift parent(vecnorm($ω⃗_plot))\n ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)\n ct = contour!(\n ax, to_intervals(grid)..., ω_mag_norm;\n alpha = 0.3, levels = 3,\n colormap = :viridis, colorrange = (0.0, 1.0),\n highclip = (:red, 0.2), lowclip = (:green, 0.2),\n )\n cb = Colorbar(fig[1, 1][1, 2], ct; label = \"Normalised vorticity magnitude\")\n ax_sp = Axis(\n fig[1, 2];\n xlabel = \"k\", ylabel = \"E(k)\", xscale = log2, yscale = log10,\n title = \"Kinetic energy spectrum\",\n )\n ylims!(ax_sp, 1e-8, 1e0)\n scatterlines!(ax_sp, k_plot, E_plot)\n ks_slope = exp.(range(log(2.5), log(25.0), length = 3))\n E_fivethirds = @. 0.3 * ks_slope^(-5/3)\n @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)\n text!(ax_sp, L\"k^{-5/3}\"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))\n fig\nend\n\nusing Printf # hide\nwith_xvfb::Bool = ENV[\"DISPLAY\"] == \":99\" # hide\nnstep::Int = 0 # hide\ntmpdir::String = mktempdir() # hide\nfilename_frame(procid, nstep) = joinpath(tmpdir, @sprintf(\"proc%d_%04d.png\", procid, nstep)) # hide\nrecord(fig, \"vorticity_proc$procid.mp4\"; framerate = 10) do io\n with_xvfb && recordframe!(io) # hide\n while integrator.t < 20\n dt = 0.001\n step!(integrator, dt)\n t_plot[] = integrator.t\n mul!.(v̂s, plan, integrator.u.x) # current velocity in Fourier space\n curl_fourier!(ω̂s, v̂s, grid_fourier)\n ldiv!.(ω⃗_plot[], plan, ω̂s)\n ω⃗_plot[] = ω⃗_plot[] # to force updating the plot\n energy_spectrum!(Ek, ks, v̂s, grid_fourier)\n Ek ./= scale_factor(plan)^2 # rescale energy\n E_plot[] = E_plot[]\n global nstep += 1 # hide\n with_xvfb ? # hide\n save(filename_frame(procid, nstep), fig) : # hide\n recordframe!(io)\n end\nend;\n\nif with_xvfb # hide\n run(pipeline(`ffmpeg -y -r 10 -i $tmpdir/proc$(procid)_%04d.png -c:v libx264 -vf \"fps=25,format=yuv420p\" vorticity_proc$procid.mp4`; stdout = \"ffmpeg.out\", stderr = \"ffmpeg.err\")) # hide\nend # hide\nnothing # hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"
\n \n
","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"This page was generated using Literate.jl.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"EditURL = \"../../examples/gradient.jl\"","category":"page"},{"location":"generated/gradient/#Gradient-of-a-scalar-field","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This example shows different methods to compute the gradient of a real-valued 3D scalar field θ(bmx) in Fourier space, where bmx = (x y z). It is assumed that the field is periodic with period L = 2π along all dimensions.","category":"page"},{"location":"generated/gradient/#General-procedure","page":"Gradient of a scalar field","title":"General procedure","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The discrete Fourier expansion of θ writes","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"θ(bmx) = _bmk Z^3 hatθ(bmk) e^i bmk bmx","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"where bmk = (k_x k_y k_z) are the Fourier wave numbers and hatθ is the discrete Fourier transform of θ. Then, the spatial derivatives of θ are given by","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"frac θ(bmx) x_i =\n_bmk Z^3 i k_i hatθ(bmk) e^i bmk bmx","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"where the subscript i denotes one of the spatial components x, y or z.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In other words, to compute bm θ = (_x θ _y θ _z θ), one has to:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"transform θ to Fourier space to obtain hatθ,\nmultiply hatθ by i bmk,\ntransform the result back to physical space to obtain bm θ.","category":"page"},{"location":"generated/gradient/#Preparation","page":"Gradient of a scalar field","title":"Preparation","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In this section, we initialise a random real-valued scalar field θ and compute its FFT. For more details see the Tutorial.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"using MPI\nusing PencilFFTs\nusing Random\n\nMPI.Init()\n\n# Input data dimensions (Nx × Ny × Nz)\ndims = (64, 32, 64)\n\n# Apply a 3D real-to-complex (r2c) FFT.\ntransform = Transforms.RFFT()\n\n# Automatically create decomposition configuration\ncomm = MPI.COMM_WORLD\npen = Pencil(dims, comm)\n\n# Create plan\nplan = PencilFFTPlan(pen, transform)\n\n# Allocate data and initialise field\nθ = allocate_input(plan)\nrandn!(θ)\n\n# Perform distributed FFT\nθ_hat = plan * θ\nnothing # hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, we initialise the output that will hold ∇θ in Fourier space. Noting that ∇θ is a vector field, we choose to store it as a tuple of 3 PencilArrays.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"∇θ_hat = allocate_output(plan, Val(3))\n\n# This is equivalent:\n# ∇θ_hat = ntuple(d -> similar(θ_hat), Val(3))\n\nsummary(∇θ_hat)","category":"page"},{"location":"generated/gradient/#Fourier-wave-numbers","page":"Gradient of a scalar field","title":"Fourier wave numbers","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In general, the Fourier wave numbers are of the form k_i = 0 frac2πL_i frac4πL_i frac6πL_i , where L_i is the period along dimension i. When a real-to-complex Fourier transform is applied, roughly half of these wave numbers are redundant due to the Hermitian symmetry of the complex Fourier coefficients. In practice, this means that for the fastest dimension x (along which a real-to-complex transform is performed), the negative wave numbers are dropped, i.e. k_x = 0 frac2πL_x frac4πL_x .","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The AbstractFFTs package provides a convenient way to generate the Fourier wave numbers, using the functions fftfreq and rfftfreq. We can use these functions to initialise a \"grid\" of wave numbers associated to our 3D real-to-complex transform:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"using AbstractFFTs: fftfreq, rfftfreq\n\nbox_size = (2π, 2π, 2π) # Lx, Ly, Lz\nsample_rate = 2π .* dims ./ box_size\n\n# In our case (Lx = 2π and Nx even), this gives kx = [0, 1, 2, ..., Nx/2].\nkx = rfftfreq(dims[1], sample_rate[1])\n\n# In our case (Ly = 2π and Ny even), this gives\n# ky = [0, 1, 2, ..., Ny/2-1, -Ny/2, -Ny/2+1, ..., -1] (and similarly for kz).\nky = fftfreq(dims[2], sample_rate[2])\nkz = fftfreq(dims[3], sample_rate[3])\n\nkvec = (kx, ky, kz)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that kvec now contains the wave numbers associated to the global domain. In the following, we will only need the wave numbers associated to the portion of the domain handled by the local MPI process.","category":"page"},{"location":"generated/gradient/#gradient_method_global","page":"Gradient of a scalar field","title":"Method 1: global views","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"PencilArrays, returned for instance by allocate_input and allocate_output, take indices that start at 1, regardless of the location of the subdomain associated to the local process on the global grid. (In other words, PencilArrays take local indices.) On the other hand, we have defined the wave number vector kvec which, for each MPI process, is defined over the global domain, and as such it takes global indices.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"One straightforward way of making data arrays compatible with wave numbers is to use global views, i.e. arrays that take global indices. These are generated from PencilArrays by calling the global_view function. Note that, in general, global indices do not start at 1 for a given MPI process. A given process will own a range of data given by indices in (i1:i2, j1:j2, k1:k2).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"θ_glob = global_view(θ_hat)\n∇θ_glob = global_view.(∇θ_hat)\nsummary(θ_glob)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Once we have global views, we can combine data and wave numbers using the portion of global indices owned by the local MPI process, as shown below. We can use CartesianIndices to iterate over the global indices associated to the local process.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"for I in CartesianIndices(θ_glob)\n i, j, k = Tuple(I) # unpack indices\n\n # Wave number vector associated to current Cartesian index.\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n # Compute gradient in Fourier space.\n # Note that modifying ∇θ_glob also modifies the original PencilArray ∇θ_hat.\n ∇θ_glob[1][I] = im * kx * θ_glob[I]\n ∇θ_glob[2][I] = im * ky * θ_glob[I]\n ∇θ_glob[3][I] = im * kz * θ_glob[I]\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The above loop can be written in a slightly more efficient manner by precomputing im * θ_glob[I]:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@inbounds for I in CartesianIndices(θ_glob)\n i, j, k = Tuple(I)\n\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n u = im * θ_glob[I]\n\n ∇θ_glob[1][I] = kx * u\n ∇θ_glob[2][I] = ky * u\n ∇θ_glob[3][I] = kz * u\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Also note that the above can be easily written in a more generic way, e.g. for arbitrary dimensions, thanks in part to the use of CartesianIndices. Moreover, in the above there is no notion of the dimension permutations discussed in the tutorial, as it is all hidden behind the implementation of PencilArrays. And as seen later in the benchmarks, these (hidden) permutations have zero cost, as the speed is identical to that of a function that explicitly takes into account these permutations.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, we can perform a backwards transform to obtain bm θ in physical space:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"∇θ = plan \\ ∇θ_hat;\nnothing #hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that the transform is automatically broadcast over the three fields of the ∇θ_hat vector, and the result ∇θ is also a tuple of three PencilArrays.","category":"page"},{"location":"generated/gradient/#gradient_method_global_explicit","page":"Gradient of a scalar field","title":"Method 2: explicit global indexing","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Sometimes, one does not need to write generic code. In our case, one often knows the dimensionality of the problem and the memory layout of the data (i.e. the underlying index permutation).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Below is a reimplementation of the above loop, using explicit indices instead of CartesianIndices, and assuming that the underlying index permutation is (3, 2, 1), that is, data is stored in (z y x) order. As discussed in the tutorial, this is the default for transformed arrays. This example also serves as a more explicit explanation for what is going on in the first method.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"# Get local data range in the global grid.\nrng = axes(θ_glob) # = (i1:i2, j1:j2, k1:k2)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"For the loop below, we're assuming that the permutation is (3, 2, 1). In other words, the fastest index is the last one, and not the first one as it is usually in Julia. If the permutation is not (3, 2, 1), things will still work (well, except for the assertion below!), but the loop order will not be optimal.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@assert permutation(θ_hat) === Permutation(3, 2, 1)\n\n@inbounds for i in rng[1], j in rng[2], k in rng[3]\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n # Note that we still access the arrays in (i, j, k) order.\n # (The permutation happens behind the scenes!)\n u = im * θ_glob[i, j, k]\n\n ∇θ_glob[1][i, j, k] = kx * u\n ∇θ_glob[2][i, j, k] = ky * u\n ∇θ_glob[3][i, j, k] = kz * u\nend","category":"page"},{"location":"generated/gradient/#gradient_method_local","page":"Gradient of a scalar field","title":"Method 3: using local indices","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Alternatively, we can avoid global views and work directly on PencilArrays using local indices that start at 1. In this case, part of the strategy is to construct a \"local\" grid of wave numbers that can also be accessed with local indices. This can be conveniently done using the localgrid function of the PencilArrays.jl package, which accepts a PencilArray (or its associated Pencil) and the global coordinates (here kvec):","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"grid_fourier = localgrid(θ_hat, kvec)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that one can directly iterate on the returned grid object:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@inbounds for I in CartesianIndices(grid_fourier)\n # Wave number vector associated to current Cartesian index.\n local k⃗ # hide\n k⃗ = grid_fourier[I]\n u = im * θ_hat[I]\n ∇θ_hat[1][I] = k⃗[1] * u\n ∇θ_hat[2][I] = k⃗[2] * u\n ∇θ_hat[3][I] = k⃗[3] * u\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This implementation is as efficient as the other examples, while being slightly shorter to write. Moreover, it is quite generic, and can be made independent of the number of dimensions with little effort.","category":"page"},{"location":"generated/gradient/#gradient_method_broadcast","page":"Gradient of a scalar field","title":"Method 4: using broadcasting","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, note that the local grid object returned by localgrid makes it is possible to compute the gradient using broadcasting, thus fully avoiding scalar indexing. This can be quite convenient in some cases, and can also be very useful if one is working on GPUs (where scalar indexing is prohibitively expensive). Using broadcasting, the above examples simply become:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@. ∇θ_hat[1] = im * grid_fourier[1] * θ_hat\n@. ∇θ_hat[2] = im * grid_fourier[2] * θ_hat\n@. ∇θ_hat[3] = im * grid_fourier[3] * θ_hat\nnothing # hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Once again, as shown in the benchmarks further below, this method performs quite similarly to the other ones.","category":"page"},{"location":"generated/gradient/#Summary","page":"Gradient of a scalar field","title":"Summary","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The PencilArrays module provides different alternatives to deal with MPI-distributed data that may be subject to dimension permutations. In particular, one can choose to work with global indices (first two examples), with local indices (third example), or to avoid scalar indexing altogether (fourth example).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"If one wants to stay generic, making sure that the same code will work for arbitrary dimensions and will be efficient regardless of the underlying dimension permutation, methods 1, 3 or 4 should be preferred. These use CartesianIndices and make no assumptions on possible dimension permutations, which are by default enabled in the output of PencilFFTs transforms. In fact, such permutations are completely invisible in the implementations.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The second method uses explicit (i, j, k) indices. It assumes that the underlying permutation is (3, 2, 1) to loop with i as the slowest index and k as the fastest, which is the optimal order in this case given the permutation. As such, the implementation is less generic than the others, and differences in performance are negligible with respect to more generic variants.","category":"page"},{"location":"generated/gradient/#gradient_benchmarks","page":"Gradient of a scalar field","title":"Benchmark results","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The following are the benchmark results obtained from running examples/gradient.jl on a laptop, using 2 MPI processes and Julia 1.7.2, with an input array of global dimensions 64 32 64. The different methods detailed above are marked on the right. The \"lazy\" marks indicate runs where the wave numbers were represented by lazy Frequencies objects (returned by rfftfreq and fftfreq). Otherwise, they were collected into Vectors. For some reason, plain Vectors are faster when working with grids generated by localgrid.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In the script, additional implementations can be found which rely on a more advanced understanding of permutations and on the internals of the PencilArrays package. For instance, gradient_local_parent! directly works with the raw data stored in Julia Arrays, while gradient_local_linear! completely avoids CartesianIndices while staying generic and efficient. Nevertheless, these display roughly the same performance as the above examples.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":" gradient_global_view!... 89.900 μs\n gradient_global_view! (lazy)... 92.060 μs [Method 1]\n gradient_global_view_explicit!... 88.958 μs\n gradient_global_view_explicit! (lazy)... 81.055 μs [Method 2]\n gradient_local!... 92.305 μs\n gradient_grid!... 92.770 μs\n gradient_grid! (lazy)... 101.388 μs [Method 3]\n gradient_grid_broadcast!... 88.606 μs\n gradient_grid_broadcast! (lazy)... 151.020 μs [Method 4]\n gradient_local_parent!... 92.248 μs\n gradient_local_linear!... 91.212 μs\n gradient_local_linear_explicit!... 90.992 μs","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This page was generated using Literate.jl.","category":"page"},{"location":"PencilFFTs_timers/#PencilFFTs.measuring_performance","page":"Measuring performance","title":"Measuring performance","text":"","category":"section"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"It is possible to measure the time spent in different sections of the distributed transforms using the TimerOutputs package. This has a (very small) performance overhead, so it is disabled by default. To enable time measurements, call TimerOutputs.enable_debug_timings after loading PencilFFTs (see below for an example). For more details see the TimerOutputs docs.","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"Minimal example:","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"using MPI\nusing PencilFFTs\nusing TimerOutputs\n\n# Enable timing of `PencilFFTs` functions\nTimerOutputs.enable_debug_timings(PencilFFTs)\nTimerOutputs.enable_debug_timings(PencilArrays)\nTimerOutputs.enable_debug_timings(Transpositions)\n\nMPI.Init()\n\nplan = PencilFFTPlan(#= args... =#)\n\n# [do stuff with `plan`...]\n\n# Retrieve and print timing data associated to `plan`\nto = timer(plan)\nprint_timer(to)","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"By default, each PencilFFTPlan has its own TimerOutput. If you already have a TimerOutput, you can pass it to the PencilFFTPlan constructor:","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"to = TimerOutput()\nplan = PencilFFTPlan(..., timer=to)\n\n# [do stuff with `plan`...]\n\nprint_timer(to)","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"#PencilFFTs","page":"Home","title":"PencilFFTs","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Fast Fourier transforms of MPI-distributed Julia arrays.","category":"page"},{"location":"#Introduction","page":"Home","title":"Introduction","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package provides multidimensional FFTs and related transforms on MPI-distributed Julia arrays via the PencilArrays package.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The name of this package originates from the decomposition of 3D domains along two out of three dimensions, sometimes called pencil decomposition. This is illustrated by the figure below,[1] where each coloured block is managed by a different MPI process. Typically, one wants to compute FFTs on a scalar or vector field along the three spatial dimensions. In the case of a pencil decomposition, 3D FFTs are performed one dimension at a time, along the non-decomposed direction. Transforms must then be interleaved with global data transpositions to switch between pencil configurations. In high-performance computing environments, such data transpositions are generally the most expensive part of a parallel FFT computation, due to the large cost of communications between computing nodes.","category":"page"},{"location":"","page":"Home","title":"Home","text":"
\n \n
","category":"page"},{"location":"","page":"Home","title":"Home","text":"More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension N. The decompositions can be performed along an arbitrary number M N of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.","category":"page"},{"location":"#Features","page":"Home","title":"Features","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;\nFFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;\nin-place and out-of-place transforms;\nhigh scalability up to (at least) tens of thousands of MPI processes.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"PencilFFTs can be installed using the Julia package manager:","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> ] add PencilFFTs","category":"page"},{"location":"#Similar-projects","page":"Home","title":"Similar projects","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.\nPFFT is a very general parallel FFT library written in C.\nP3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.\n2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[1]: Figure adapted from this PhD thesis.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[2]: For the pencil decomposition represented in the figure, N = 3 and M = 2.","category":"page"},{"location":"PencilFFTs/#Distributed-FFT-plans","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"Distributed FFTs are implemented in the PencilFFTs module, and are built on top of the PencilArrays package.","category":"page"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"PencilFFTs/#Creating-plans","page":"Distributed FFT plans","title":"Creating plans","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"PencilFFTPlan","category":"page"},{"location":"PencilFFTs/#PencilFFTs.PencilFFTPlan","page":"Distributed FFT plans","title":"PencilFFTs.PencilFFTPlan","text":"PencilFFTPlan{T,N} <: AbstractFFTs.Plan{T}\n\nPlan for N-dimensional FFT-based transform on MPI-distributed data, where input data has type T.\n\n\n\nPencilFFTPlan(p::Pencil, transforms; kwargs...)\n\nCreate a PencilFFTPlan for distributed arrays following a given Pencil configuration. See variant below for details on the specification of transforms and on possible keyword arguments.\n\n\n\nPencilFFTPlan(\n A::PencilArray, transforms;\n fftw_flags = FFTW.ESTIMATE,\n fftw_timelimit = FFTW.NO_TIMELIMIT,\n permute_dims = Val(true),\n transpose_method = Transpositions.PointToPoint(),\n timer = timer(pencil(A)),\n)\n\nCreate plan for N-dimensional transform on MPI-distributed PencilArrays.\n\nExtended help\n\nThis creates a PencilFFTPlan for arrays sharing the same properties as A (dimensions, MPI decomposition, memory layout, ...), which describe data on an N-dimensional domain.\n\nTransforms\n\nThe transforms to be applied along each dimension are specified by the transforms argument. Possible transforms are defined as subtypes of Transforms.AbstractTransform, and are listed in Transform types. This argument may be either:\n\na tuple of N transforms to be applied along each dimension. For instance, transforms = (Transforms.R2R(FFTW.REDFT01), Transforms.RFFT(), Transforms.FFT());\na single transform to be applied along all dimensions. The input is automatically expanded into N equivalent transforms. For instance, for a three-dimensional array, transforms = Transforms.RFFT() specifies a 3D real-to-complex transform, and is equivalent to passing (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT()).\n\nNote that forward transforms are applied from left to right. In the last example, this means that a real-to-complex transform (RFFT) is first performed along the first dimension. This is followed by complex-to-complex transforms (FFT) along the second and third dimensions.\n\nInput data layout\n\nThe input PencilArray must satisfy the following constraints:\n\narray dimensions must not be permuted. This is the default when constructing PencilArrays.\nfor an M-dimensional domain decomposition (with M < N), the input array must be decomposed along the last M dimensions. For example, for a 2D decomposition of 3D data, the decomposed dimensions must be (2, 3). In particular, the first array dimension must not be distributed among different MPI processes.\nIn the PencilArrays package, the decomposed dimensions are specified at the moment of constructing a Pencil.\nthe element type must be compatible with the specified transform. For instance, real-to-complex transforms (Transforms.RFFT) require the input to be real floating point values. Other transforms, such as Transforms.R2R, accept both real and complex data.\n\nKeyword arguments\n\nThe keyword arguments fftw_flags and fftw_timelimit are passed to the FFTW plan creation functions (see AbstractFFTs docs).\npermute_dims determines whether the indices of the output data should be reversed. For instance, if the input data has global dimensions (Nx, Ny, Nz), then the output of a complex-to-complex FFT would have dimensions (Nz, Ny, Nx). This enables FFTs to always be performed along the first (i.e. fastest) array dimension, which could lead to performance gains. This option is enabled by default. For type inference reasons, it must be a value type (Val(true) or Val(false)).\ntranspose_method allows to select between implementations of the global data transpositions. See PencilArrays docs docs for details.\ntimer should be a TimerOutput object. See Measuring performance for details.\n\n\n\nPencilFFTPlan(\n dims_global::Dims{N}, transforms, proc_dims::Dims{M}, comm::MPI.Comm,\n [real_type = Float64]; extra_dims = (), kws...\n)\n\nCreate plan for N-dimensional transform.\n\nExtended help\n\nInstead of taking a PencilArray or a Pencil, this constructor requires the global dimensions of the input data, passed via the size_global argument.\n\nThe data is distributed over the MPI processes in the comm communicator. The distribution is performed over M dimensions (with M < N) according to the values in proc_dims, which specifies the number of MPI processes to put along each dimension.\n\nPencilArrays that may be transformed with the returned plan can be created using allocate_input.\n\nOptional arguments\n\nThe floating point precision can be selected by setting real_type parameter, which is Float64 by default.\nextra_dims may be used to specify the sizes of one or more extra dimensions that should not be transformed. These dimensions will be added to the rightmost (i.e. slowest) indices of the arrays. See Extra dimensions below for usage hints.\nsee the other constructor for more keyword arguments.\n\nExtra dimensions\n\nOne possible application of extra_dims is for describing the components of a vector or tensor field. However, this means that different PencilFFTPlans would need to be created for each kind of field (scalar, vector, ...). To avoid the creation of multiple plans, a possibly better alternative is to create tuples (or arrays) of PencilArrays using allocate_input and allocate_output.\n\nAnother more legitimate usage of extra_dims is to specify one or more Cartesian dimensions that should not be transformed nor split among MPI processes.\n\nExample\n\nSuppose we want to perform a 3D FFT of real data. The data is to be decomposed along two dimensions, over 8 MPI processes:\n\nsize_global = (64, 32, 128) # size of real input data\n\n# Perform real-to-complex transform along the first dimension, then\n# complex-to-complex transforms along the other dimensions.\ntransforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())\n# transforms = Transforms.RFFT() # this is equivalent to the above line\n\nproc_dims = (4, 2) # 2D decomposition\ncomm = MPI.COMM_WORLD\n\nplan = PencilFFTPlan(size_global, transforms, proc_dims, comm)\n\n\n\n\n\n","category":"type"},{"location":"PencilFFTs/#Allocating-data","page":"Distributed FFT plans","title":"Allocating data","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"allocate_input\nallocate_output","category":"page"},{"location":"PencilFFTs/#PencilFFTs.allocate_input","page":"Distributed FFT plans","title":"PencilFFTs.allocate_input","text":"allocate_input(p::PencilFFTPlan) -> PencilArray\nallocate_input(p::PencilFFTPlan, dims...) -> Array{PencilArray}\nallocate_input(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}\n\nAllocate uninitialised PencilArray that can hold input data for the given plan.\n\nThe second and third forms respectively allocate an array of PencilArrays of size dims, and a tuple of N PencilArrays.\n\nnote: In-place plans\nIf p is an in-place real-to-real or complex-to-complex plan, a ManyPencilArray is allocated. If p is an in-place real-to-complex plan, a ManyPencilArrayRFFT! is allocated. These types hold PencilArray wrappers for the input and output transforms (as well as for intermediate transforms) which share the same space in memory. The input and output PencilArrays should be respectively accessed by calling first(::ManyPencilArray) and last(::ManyPencilArray).ExampleSuppose p is an in-place PencilFFTPlan. Then,@assert is_inplace(p)\nA = allocate_input(p) :: ManyPencilArray\nv_in = first(A) :: PencilArray # input data view\nv_out = last(A) :: PencilArray # output data viewAlso note that in-place plans must be performed directly on the returned ManyPencilArray, and not on the contained PencilArray views:p * A # perform forward transform in-place\np \\ A # perform backward transform in-place\n# p * v_in # not allowed!!\n\n\n\n\n\n","category":"function"},{"location":"PencilFFTs/#PencilFFTs.allocate_output","page":"Distributed FFT plans","title":"PencilFFTs.allocate_output","text":"allocate_output(p::PencilFFTPlan) -> PencilArray\nallocate_output(p::PencilFFTPlan, dims...) -> Array{PencilArray}\nallocate_output(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}\n\nAllocate uninitialised PencilArray that can hold output data for the given plan.\n\nIf p is an in-place plan, a ManyPencilArray is allocated.\n\nSee allocate_input for details.\n\n\n\n\n\n","category":"function"},{"location":"PencilFFTs/#Methods","page":"Distributed FFT plans","title":"Methods","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"get_comm(::PencilFFTPlan)\nscale_factor(::PencilFFTPlan)\ntimer(::PencilFFTPlan)\nis_inplace(::PencilFFTPlan)","category":"page"},{"location":"PencilFFTs/#PencilArrays.Pencils.MPITopologies.get_comm-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilArrays.Pencils.MPITopologies.get_comm","text":"get_comm(p::PencilFFTPlan)\n\nGet MPI communicator associated to a PencilFFTPlan.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilFFTs.Transforms.scale_factor-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilFFTs.Transforms.scale_factor","text":"scale_factor(p::PencilFFTPlan)\n\nGet scale factor associated to a PencilFFTPlan.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilArrays.Pencils.timer-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilArrays.Pencils.timer","text":"timer(p::PencilFFTPlan)\n\nGet TimerOutput attached to a PencilFFTPlan.\n\nSee Measuring performance for details.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilFFTs.Transforms.is_inplace-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilFFTs.Transforms.is_inplace","text":"Transforms.is_inplace(p::PencilFFTPlan)\n\nReturns true if the given plan operates in-place on the input data, false otherwise.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#Internals","page":"Distributed FFT plans","title":"Internals","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"ManyPencilArrayRFFT!","category":"page"},{"location":"PencilFFTs/#PencilFFTs.ManyPencilArrayRFFT!","page":"Distributed FFT plans","title":"PencilFFTs.ManyPencilArrayRFFT!","text":"ManyPencilArrayRFFT!{T,N,M} <: AbstractManyPencilArray{N,M}\n\nContainer holding M different PencilArray views to the same underlying data buffer. All views share the same dimensionality N. The element type T of the first view is real, that of subsequent views is Complex{T}. \n\nThis can be used to perform in-place real-to-complex plan, see alsoTransforms.RFFT!. It is used internally for such transforms by allocate_input and should not be constructed directly.\n\n\n\nManyPencilArrayRFFT!{T}(undef, pencils...; extra_dims=())\n\nCreate a ManyPencilArrayRFFT! container that can hold data of type T and Complex{T} associated to all the given Pencils.\n\nThe optional extra_dims argument is the same as for PencilArray.\n\nSee also ManyPencilArray\n\n\n\n\n\n","category":"type"}] } diff --git a/dev/tutorial/index.html b/dev/tutorial/index.html index 723c885b..216e9723 100644 --- a/dev/tutorial/index.html +++ b/dev/tutorial/index.html @@ -1,5 +1,5 @@ -Tutorial · PencilFFTs.jl
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Tutorial

The following tutorial shows how to perform a 3D FFT of real periodic data defined on a grid of $N_x × N_y × N_z$ points.

+Tutorial · PencilFFTs.jl
GitHub

Tutorial

The following tutorial shows how to perform a 3D FFT of real periodic data defined on a grid of $N_x × N_y × N_z$ points.

- Strong scaling of PencilFFTs -

As seen above, PencilFFTs generally outperforms P3DFFT in its default setting. This is largely explained by the choice of using non-blocking point-to-point MPI communications (via MPI_Isend and MPI_Irecv), while P3DFFT uses collective MPI_Alltoallv calls. This enables PencilFFTs to perform data reordering operations on the partially received data while waiting for the incoming data, leading to better performance. Moreover, in contrast with P3DFFT, the high performance and scalability of PencilFFTs results from a highly generic code, handling decompositions in arbitrary dimensions and a relatively large (and extensible) variety of transformations.

Note that PencilFFTs can optionally use collective communications (using MPI_Alltoallv) instead of point-to-point communications. For details, see the docs for PencilFFTPlan and for PencilArray transpositions. As seen above, collective communications generally perform worse than point-to-point ones, and runtimes are nearly indistinguishable from those of P3DFFT.

Benchmark details

The benchmarks were performed using Julia 1.7-beta3 and Intel MPI 2019. We used PencilFFTs v0.12.5 with FFTW.jl v1.4.3 and MPI.jl v0.19.0. We used the Fortran implementation of P3DFFT, version 2.7.6, which was built with Intel 2019 compilers and linked to FFTW 3.3.8. The cluster where the benchmarks were run has Intel Cascade Lake 6248 processors with 2×20 cores per node.

The number of MPI processes along each decomposed dimension, $P_1$ and $P_2$, was automatically determined by a call to MPI_Dims_create, which tends to create a balanced decomposition with $P_1 ≈ P_2$. For instance, a total of 1024 processes is divided into $P_1 = P_2 = 32$. Different results may be obtained with other combinations, but this was not benchmarked.

The source files used to generate this benchmark, as well as the raw benchmark results, are all available in the PencilFFTs repo.

diff --git a/previews/PR39/generated/ffmpeg.err b/previews/PR39/generated/ffmpeg.err deleted file mode 100644 index 575f5ec5..00000000 --- a/previews/PR39/generated/ffmpeg.err +++ /dev/null @@ -1,49 +0,0 @@ -ffmpeg version 4.2.4-1ubuntu0.1 Copyright (c) 2000-2020 the FFmpeg developers - built with gcc 9 (Ubuntu 9.3.0-10ubuntu2) - configuration: --prefix=/usr --extra-version=1ubuntu0.1 --toolchain=hardened --libdir=/usr/lib/x86_64-linux-gnu --incdir=/usr/include/x86_64-linux-gnu --arch=amd64 --enable-gpl --disable-stripping --enable-avresample --disable-filter=resample --enable-avisynth --enable-gnutls --enable-ladspa --enable-libaom --enable-libass --enable-libbluray --enable-libbs2b --enable-libcaca --enable-libcdio --enable-libcodec2 --enable-libflite --enable-libfontconfig --enable-libfreetype --enable-libfribidi --enable-libgme --enable-libgsm --enable-libjack --enable-libmp3lame --enable-libmysofa --enable-libopenjpeg --enable-libopenmpt --enable-libopus --enable-libpulse --enable-librsvg --enable-librubberband --enable-libshine --enable-libsnappy --enable-libsoxr --enable-libspeex --enable-libssh --enable-libtheora --enable-libtwolame --enable-libvidstab --enable-libvorbis --enable-libvpx --enable-libwavpack --enable-libwebp --enable-libx265 --enable-libxml2 --enable-libxvid --enable-libzmq --enable-libzvbi --enable-lv2 --enable-omx --enable-openal --enable-opencl --enable-opengl --enable-sdl2 --enable-libdc1394 --enable-libdrm --enable-libiec61883 --enable-nvenc --enable-chromaprint --enable-frei0r --enable-libx264 --enable-shared - libavutil 56. 31.100 / 56. 31.100 - libavcodec 58. 54.100 / 58. 54.100 - libavformat 58. 29.100 / 58. 29.100 - libavdevice 58. 8.100 / 58. 8.100 - libavfilter 7. 57.100 / 7. 57.100 - libavresample 4. 0. 0 / 4. 0. 0 - libswscale 5. 5.100 / 5. 5.100 - libswresample 3. 5.100 / 3. 5.100 - libpostproc 55. 5.100 / 55. 5.100 -Input #0, image2, from '/tmp/jl_nuVjHe/proc1_%04d.png': - Duration: 00:00:04.52, start: 0.000000, bitrate: N/A - Stream #0:0: Video: png, rgb24(pc), 1200x600, 25 fps, 25 tbr, 25 tbn, 25 tbc -Stream mapping: - Stream #0:0 -> #0:0 (png (native) -> h264 (libx264)) -Press [q] to stop, [?] for help -[libx264 @ 0x55ae4622a540] using cpu capabilities: MMX2 SSE2Fast SSSE3 SSE4.2 AVX FMA3 BMI2 AVX2 AVX512 -[libx264 @ 0x55ae4622a540] profile High, level 3.1 -[libx264 @ 0x55ae4622a540] 264 - core 155 r2917 0a84d98 - H.264/MPEG-4 AVC codec - Copyleft 2003-2018 - http://www.videolan.org/x264.html - options: cabac=1 ref=3 deblock=1:0:0 analyse=0x3:0x113 me=hex subme=7 psy=1 psy_rd=1.00:0.00 mixed_ref=1 me_range=16 chroma_me=1 trellis=1 8x8dct=1 cqm=0 deadzone=21,11 fast_pskip=1 chroma_qp_offset=-2 threads=3 lookahead_threads=1 sliced_threads=0 nr=0 decimate=1 interlaced=0 bluray_compat=0 constrained_intra=0 bframes=3 b_pyramid=2 b_adapt=1 b_bias=0 direct=1 weightb=1 open_gop=0 weightp=2 keyint=250 keyint_min=25 scenecut=40 intra_refresh=0 rc_lookahead=40 rc=crf mbtree=1 crf=23.0 qcomp=0.60 qpmin=0 qpmax=69 qpstep=4 ip_ratio=1.40 aq=1:1.00 -Output #0, mp4, to 'vorticity_proc1.mp4': - Metadata: - encoder : Lavf58.29.100 - Stream #0:0: Video: h264 (libx264) (avc1 / 0x31637661), yuv420p, 1200x600, q=-1--1, 25 fps, 12800 tbn, 25 tbc - Metadata: - encoder : Lavc58.54.100 libx264 - Side data: - cpb: bitrate max/min/avg: 0/0/0 buffer size: 0 vbv_delay: -1 -frame= 18 fps=0.0 q=0.0 size= 0kB time=00:00:00.00 bitrate=N/A speed= 0x frame= 37 fps= 36 q=0.0 size= 0kB time=00:00:00.00 bitrate=N/A speed= 0x frame= 49 fps= 30 q=28.0 size= 0kB time=-00:00:00.03 bitrate=N/A speed=N/A frame= 57 fps= 24 q=28.0 size= 0kB time=00:00:00.28 bitrate= 1.4kbits/s speed=0.118x frame= 61 fps= 21 q=28.0 size= 0kB time=00:00:00.44 bitrate= 0.9kbits/s speed=0.153x frame= 65 fps= 19 q=28.0 size= 0kB time=00:00:00.60 bitrate= 0.6kbits/s speed=0.174x frame= 69 fps= 17 q=28.0 size= 0kB time=00:00:00.76 bitrate= 0.5kbits/s speed=0.19x frame= 73 fps= 16 q=28.0 size= 0kB time=00:00:00.92 bitrate= 0.4kbits/s 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time=00:00:02.88 bitrate= 0.1kbits/s speed=0.225x frame= 126 fps=9.4 q=28.0 size= 0kB time=00:00:03.04 bitrate= 0.1kbits/s speed=0.226x frame= 130 fps=9.2 q=28.0 size= 256kB time=00:00:03.20 bitrate= 655.5kbits/s speed=0.226x frame= 134 fps=9.1 q=28.0 size= 256kB time=00:00:03.36 bitrate= 624.3kbits/s speed=0.228x frame= 138 fps=9.0 q=28.0 size= 256kB time=00:00:03.52 bitrate= 595.9kbits/s speed=0.231x frame= 142 fps=9.0 q=28.0 size= 256kB time=00:00:03.68 bitrate= 570.0kbits/s speed=0.233x frame= 147 fps=8.9 q=28.0 size= 256kB time=00:00:03.88 bitrate= 540.6kbits/s speed=0.235x frame= 153 fps=8.9 q=28.0 size= 256kB time=00:00:04.12 bitrate= 509.1kbits/s speed=0.241x frame= 158 fps=8.9 q=28.0 size= 256kB time=00:00:04.32 bitrate= 485.5kbits/s speed=0.242x frame= 163 fps=8.8 q=28.0 size= 256kB time=00:00:04.52 bitrate= 464.0kbits/s speed=0.245x frame= 167 fps=8.8 q=28.0 size= 256kB time=00:00:04.68 bitrate= 448.2kbits/s speed=0.245x frame= 173 fps=8.8 q=28.0 size= 256kB time=00:00:04.92 bitrate= 426.3kbits/s speed=0.25x frame= 178 fps=8.7 q=28.0 size= 256kB time=00:00:05.12 bitrate= 409.7kbits/s speed=0.25x frame= 182 fps=8.6 q=28.0 size= 256kB time=00:00:05.28 bitrate= 397.3kbits/s speed=0.25x frame= 186 fps=8.5 q=28.0 size= 256kB time=00:00:05.44 bitrate= 385.6kbits/s speed=0.249x frame= 190 fps=8.5 q=28.0 size= 256kB time=00:00:05.60 bitrate= 374.6kbits/s speed=0.25x frame= 194 fps=8.4 q=28.0 size= 256kB time=00:00:05.76 bitrate= 364.2kbits/s speed=0.251x frame= 198 fps=8.4 q=28.0 size= 256kB time=00:00:05.92 bitrate= 354.3kbits/s speed=0.251x frame= 204 fps=8.5 q=28.0 size= 256kB time=00:00:06.16 bitrate= 340.5kbits/s speed=0.255x frame= 208 fps=8.4 q=28.0 size= 256kB time=00:00:06.32 bitrate= 331.9kbits/s speed=0.255x frame= 212 fps=8.4 q=28.0 size= 256kB time=00:00:06.48 bitrate= 323.7kbits/s speed=0.256x frame= 216 fps=8.3 q=28.0 size= 256kB time=00:00:06.64 bitrate= 315.9kbits/s speed=0.257x frame= 220 fps=8.3 q=28.0 size= 256kB time=00:00:06.80 bitrate= 308.5kbits/s speed=0.258x frame= 224 fps=8.3 q=28.0 size= 512kB time=00:00:06.96 bitrate= 602.7kbits/s speed=0.259x frame= 229 fps=8.3 q=28.0 size= 512kB time=00:00:07.16 bitrate= 585.8kbits/s speed=0.261x frame= 233 fps=8.3 q=28.0 size= 512kB time=00:00:07.32 bitrate= 573.0kbits/s speed=0.261x frame= 237 fps=8.3 q=28.0 size= 512kB time=00:00:07.48 bitrate= 560.8kbits/s speed=0.262x frame= 241 fps=8.3 q=28.0 size= 512kB time=00:00:07.64 bitrate= 549.0kbits/s speed=0.263x frame= 245 fps=8.3 q=28.0 size= 512kB time=00:00:07.80 bitrate= 537.8kbits/s speed=0.263x frame= 249 fps=8.2 q=28.0 size= 512kB time=00:00:07.96 bitrate= 527.0kbits/s speed=0.263x frame= 253 fps=8.2 q=28.0 size= 512kB time=00:00:08.12 bitrate= 516.6kbits/s speed=0.264x frame= 258 fps=8.1 q=28.0 size= 512kB time=00:00:08.32 bitrate= 504.2kbits/s speed=0.262x frame= 262 fps=8.1 q=28.0 size= 512kB time=00:00:08.48 bitrate= 494.7kbits/s speed=0.263x frame= 266 fps=8.1 q=28.0 size= 512kB time=00:00:08.64 bitrate= 485.5kbits/s speed=0.262x frame= 270 fps=8.0 q=28.0 size= 512kB time=00:00:08.80 bitrate= 476.7kbits/s speed=0.262x frame= 274 fps=8.0 q=28.0 size= 512kB time=00:00:08.96 bitrate= 468.2kbits/s speed=0.263x frame= 278 fps=8.0 q=28.0 size= 512kB time=00:00:09.12 bitrate= 459.9kbits/s speed=0.263x frame= 282 fps=8.0 q=28.0 size= 512kB time=00:00:09.28 bitrate= 452.0kbits/s speed=0.262x frame= 283 fps=6.9 q=-1.0 Lsize= 826kB time=00:00:11.20 bitrate= 604.0kbits/s speed=0.274x -video:822kB audio:0kB subtitle:0kB other streams:0kB global headers:0kB muxing overhead: 0.500685% -[libx264 @ 0x55ae4622a540] frame I:2 Avg QP:17.03 size: 38421 -[libx264 @ 0x55ae4622a540] frame P:73 Avg QP:25.14 size: 8194 -[libx264 @ 0x55ae4622a540] frame B:208 Avg QP:18.90 size: 797 -[libx264 @ 0x55ae4622a540] consecutive B-frames: 0.7% 2.8% 3.2% 93.3% -[libx264 @ 0x55ae4622a540] mb I I16..4: 33.0% 39.9% 27.1% -[libx264 @ 0x55ae4622a540] mb P I16..4: 0.3% 1.5% 0.9% P16..4: 5.8% 6.7% 4.5% 0.0% 0.0% skip:80.1% -[libx264 @ 0x55ae4622a540] mb B I16..4: 0.1% 0.1% 0.0% B16..8: 6.9% 1.0% 0.4% direct: 0.3% skip:91.2% L0:44.6% L1:47.3% BI: 8.0% -[libx264 @ 0x55ae4622a540] 8x8 transform intra:46.1% inter:39.0% -[libx264 @ 0x55ae4622a540] coded y,uvDC,uvAC intra: 39.8% 41.5% 33.4% inter: 2.7% 2.2% 0.7% -[libx264 @ 0x55ae4622a540] i16 v,h,dc,p: 70% 17% 10% 3% -[libx264 @ 0x55ae4622a540] i8 v,h,dc,ddl,ddr,vr,hd,vl,hu: 23% 12% 34% 5% 5% 4% 6% 4% 6% -[libx264 @ 0x55ae4622a540] i4 v,h,dc,ddl,ddr,vr,hd,vl,hu: 29% 18% 17% 5% 6% 5% 9% 4% 8% -[libx264 @ 0x55ae4622a540] i8c dc,h,v,p: 70% 13% 12% 5% -[libx264 @ 0x55ae4622a540] Weighted P-Frames: Y:0.0% UV:0.0% -[libx264 @ 0x55ae4622a540] ref P L0: 69.2% 15.2% 10.6% 4.9% -[libx264 @ 0x55ae4622a540] ref B L0: 85.9% 11.3% 2.7% -[libx264 @ 0x55ae4622a540] ref B L1: 98.1% 1.9% -[libx264 @ 0x55ae4622a540] kb/s:594.18 diff --git a/previews/PR39/generated/gradient/index.html b/previews/PR39/generated/gradient/index.html deleted file mode 100644 index 5123cec7..00000000 --- a/previews/PR39/generated/gradient/index.html +++ /dev/null @@ -1,112 +0,0 @@ - -Gradient of a scalar field · PencilFFTs.jl
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Gradient of a scalar field

This example shows different methods to compute the gradient of a real-valued 3D scalar field $θ(\bm{x})$ in Fourier space, where $\bm{x} = (x, y, z)$. It is assumed that the field is periodic with period $L = 2π$ along all dimensions.

General procedure

The discrete Fourier expansion of $θ$ writes

\[θ(\bm{x}) = ∑_{\bm{k} ∈ \Z^3} \hat{θ}(\bm{k}) \, e^{i \bm{k} ⋅ \bm{x}},\]

where $\bm{k} = (k_x, k_y, k_z)$ are the Fourier wave numbers and $\hat{θ}$ is the discrete Fourier transform of $θ$. Then, the spatial derivatives of $θ$ are given by

\[\frac{∂ θ(\bm{x})}{∂ x_i} = -∑_{\bm{k} ∈ \Z^3} i k_i \hat{θ}(\bm{k}) \, e^{i \bm{k} ⋅ \bm{x}},\]

where the subscript $i$ denotes one of the spatial components $x$, $y$ or $z$.

In other words, to compute $\bm{∇} θ = (∂_x θ, ∂_y θ, ∂_z θ)$, one has to:

  1. transform $θ$ to Fourier space to obtain $\hat{θ}$,
  2. multiply $\hat{θ}$ by $i \bm{k}$,
  3. transform the result back to physical space to obtain $\bm{∇} θ$.

Preparation

In this section, we initialise a random real-valued scalar field $θ$ and compute its FFT. For more details see the Tutorial.

using MPI
-using PencilFFTs
-using Random
-
-MPI.Init()
-
-# Input data dimensions (Nx × Ny × Nz)
-dims = (64, 32, 64)
-
-# Apply a 3D real-to-complex (r2c) FFT.
-transform = Transforms.RFFT()
-
-# Automatically create decomposition configuration
-comm = MPI.COMM_WORLD
-pen = Pencil(dims, comm)
-
-# Create plan
-plan = PencilFFTPlan(pen, transform)
-
-# Allocate data and initialise field
-θ = allocate_input(plan)
-randn!(θ)
-
-# Perform distributed FFT
-θ_hat = plan * θ

Finally, we initialise the output that will hold ∇θ in Fourier space. Noting that ∇θ is a vector field, we choose to store it as a tuple of 3 PencilArrays.

∇θ_hat = allocate_output(plan, Val(3))
-
-# This is equivalent:
-# ∇θ_hat = ntuple(d -> similar(θ_hat), Val(3))
-
-summary(∇θ_hat)
-
-# Fourier wave numbers
"Tuple{PencilArray{ComplexF64, 3, Array{ComplexF64, 3}, 3, 0, Pencil{3, 2, Permutation{(3, 2, 1), 3}, Vector{UInt8}}}, PencilArray{ComplexF64, 3, Array{ComplexF64, 3}, 3, 0, Pencil{3, 2, Permutation{(3, 2, 1), 3}, Vector{UInt8}}}, PencilArray{ComplexF64, 3, Array{ComplexF64, 3}, 3, 0, Pencil{3, 2, Permutation{(3, 2, 1), 3}, Vector{UInt8}}}}"

In general, the Fourier wave numbers are of the form $k_i = 0, ±\frac{2π}{L_i}, ±\frac{4π}{L_i}, ±\frac{6π}{L_i}, …$, where $L_i$ is the period along dimension $i$. When a real-to-complex Fourier transform is applied, roughly half of these wave numbers are redundant due to the Hermitian symmetry of the complex Fourier coefficients. In practice, this means that for the fastest dimension $x$ (along which a real-to-complex transform is performed), the negative wave numbers are dropped, i.e. $k_x = 0, \frac{2π}{L_x}, \frac{4π}{L_x}, …$.

The AbstractFFTs package provides a convenient way to generate the Fourier wave numbers, using the functions fftfreq and rfftfreq. We can use these functions to initialise a "grid" of wave numbers associated to our 3D real-to-complex transform:

using AbstractFFTs: fftfreq, rfftfreq
-
-box_size = (2π, 2π, 2π)  # Lx, Ly, Lz
-sample_rate = 2π .* dims ./ box_size
-
-# In our case (Lx = 2π and Nx even), this gives kx = [0, 1, 2, ..., Nx/2].
-kx = rfftfreq(dims[1], sample_rate[1])
-
-# In our case (Ly = 2π and Ny even), this gives
-# ky = [0, 1, 2, ..., Ny/2-1, -Ny/2, -Ny/2+1, ..., -1] (and similarly for kz).
-ky = fftfreq(dims[2], sample_rate[2])
-kz = fftfreq(dims[3], sample_rate[3])
-
-kvec = (kx, ky, kz)
([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0  …  23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, 32.0], [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0  …  -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0], [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0  …  -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0])

Note that kvec now contains the wave numbers associated to the global domain. In the following, we will only need the wave numbers associated to the portion of the domain handled by the local MPI process.

Method 1: global views

PencilArrays, returned for instance by allocate_input and allocate_output, take indices that start at 1, regardless of the location of the subdomain associated to the local process on the global grid. (In other words, PencilArrays take local indices.) On the other hand, we have defined the wave number vector kvec which, for each MPI process, is defined over the global domain, and as such it takes global indices.

One straightforward way of making data arrays compatible with wave numbers is to use global views, i.e. arrays that take global indices. These are generated from PencilArrays by calling the global_view function. Note that, in general, global indices do not start at 1 for a given MPI process. A given process will own a range of data given by indices in (i1:i2, j1:j2, k1:k2).

θ_glob = global_view(θ_hat)
-∇θ_glob = global_view.(∇θ_hat)
-summary(θ_glob)
"16×32×64 OffsetArray(::PencilArray{ComplexF64, 3}, 1:16, 1:32, 1:64) with eltype ComplexF64 with indices 1:16×1:32×1:64"

Once we have global views, we can combine data and wave numbers using the portion of global indices owned by the local MPI process, as shown below. We can use CartesianIndices to iterate over the global indices associated to the local process.

for I in CartesianIndices(θ_glob)
-    i, j, k = Tuple(I)  # unpack indices
-
-    # Wave number vector associated to current Cartesian index.
-    kx = kvec[1][i]
-    ky = kvec[2][j]
-    kz = kvec[3][k]
-
-    # Compute gradient in Fourier space.
-    # Note that modifying ∇θ_glob also modifies the original PencilArray ∇θ_hat.
-    ∇θ_glob[1][I] = im * kx * θ_glob[I]
-    ∇θ_glob[2][I] = im * ky * θ_glob[I]
-    ∇θ_glob[3][I] = im * kz * θ_glob[I]
-end

The above loop can be written in a slightly more efficient manner by precomputing im * θ_glob[I]:

@inbounds for I in CartesianIndices(θ_glob)
-    i, j, k = Tuple(I)
-
-    kx = kvec[1][i]
-    ky = kvec[2][j]
-    kz = kvec[3][k]
-
-    u = im * θ_glob[I]
-
-    ∇θ_glob[1][I] = kx * u
-    ∇θ_glob[2][I] = ky * u
-    ∇θ_glob[3][I] = kz * u
-end

Also note that the above can be easily written in a more generic way, e.g. for arbitrary dimensions, thanks in part to the use of CartesianIndices. Moreover, in the above there is no notion of the dimension permutations discussed in the tutorial, as it is all hidden behind the implementation of PencilArrays. And as seen later in the benchmarks, these (hidden) permutations have zero cost, as the speed is identical to that of a function that explicitly takes into account these permutations.

Finally, we can perform a backwards transform to obtain $\bm{∇} θ$ in physical space:

∇θ = plan \ ∇θ_hat;

Note that the transform is automatically broadcast over the three fields of the ∇θ_hat vector, and the result ∇θ is also a tuple of three PencilArrays.

Method 2: explicit global indexing

Sometimes, one does not need to write generic code. In our case, one often knows the dimensionality of the problem and the memory layout of the data (i.e. the underlying index permutation).

Below is a reimplementation of the above loop, using explicit indices instead of CartesianIndices, and assuming that the underlying index permutation is (3, 2, 1), that is, data is stored in $(z, y, x)$ order. As discussed in the tutorial, this is the default for transformed arrays. This example also serves as a more explicit explanation for what is going on in the first method.

# Get local data range in the global grid.
-rng = axes(θ_glob)  # = (i1:i2, j1:j2, k1:k2)
(OffsetArrays.IdOffsetRange(values=1:16, indices=1:16), OffsetArrays.IdOffsetRange(values=1:32, indices=1:32), OffsetArrays.IdOffsetRange(values=1:64, indices=1:64))

For the loop below, we're assuming that the permutation is (3, 2, 1). In other words, the fastest index is the last one, and not the first one as it is usually in Julia. If the permutation is not (3, 2, 1), things will still work (well, except for the assertion below!), but the loop order will not be optimal.

@assert permutation(θ_hat) === Permutation(3, 2, 1)
-
-@inbounds for i in rng[1], j in rng[2], k in rng[3]
-    kx = kvec[1][i]
-    ky = kvec[2][j]
-    kz = kvec[3][k]
-
-    # Note that we still access the arrays in (i, j, k) order.
-    # (The permutation happens behind the scenes!)
-    u = im * θ_glob[i, j, k]
-
-    ∇θ_glob[1][i, j, k] = kx * u
-    ∇θ_glob[2][i, j, k] = ky * u
-    ∇θ_glob[3][i, j, k] = kz * u
-end

Method 3: using local indices

Alternatively, we can avoid global views and work directly on PencilArrays using local indices that start at 1. In this case, part of the strategy is to construct a "local" grid of wave numbers that can also be accessed with local indices. This can be conveniently done using the localgrid function of the PencilArrays.jl package, which accepts a PencilArray (or its associated Pencil) and the global coordinates (here kvec):

grid_fourier = localgrid(θ_hat, kvec)
LocalRectilinearGrid{3} with Permutation(3, 2, 1) and coordinates:
- (1) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0]
- (2) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]
- (3) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]

Note that one can directly iterate on the returned grid object:

@inbounds for I in CartesianIndices(grid_fourier)
-    # Wave number vector associated to current Cartesian index.
-    k⃗ = grid_fourier[I]
-    u = im * θ_hat[I]
-    ∇θ_hat[1][I] = k⃗[1] * u
-    ∇θ_hat[2][I] = k⃗[2] * u
-    ∇θ_hat[3][I] = k⃗[3] * u
-end

This implementation is as efficient as the other examples, while being slightly shorter to write. Moreover, it is quite generic, and can be made independent of the number of dimensions with little effort.

Method 4: using broadcasting

Finally, note that the local grid object returned by localgrid makes it is possible to compute the gradient using broadcasting, thus fully avoiding scalar indexing. This can be quite convenient in some cases, and can also be very useful if one is working on GPUs (where scalar indexing is prohibitively expensive). Using broadcasting, the above examples simply become:

@. ∇θ_hat[1] = im * grid_fourier[1] * θ_hat
-@. ∇θ_hat[2] = im * grid_fourier[2] * θ_hat
-@. ∇θ_hat[3] = im * grid_fourier[3] * θ_hat

Once again, as shown in the benchmarks further below, this method performs quite similarly to the other ones.

Summary

The PencilArrays module provides different alternatives to deal with MPI-distributed data that may be subject to dimension permutations. In particular, one can choose to work with global indices (first two examples) or with local indices (third example).

If one wants to stay generic, making sure that the same code will work for arbitrary dimensions and will be efficient regardless of the underlying dimension permutation, methods 1 and 3 should be preferred. These use CartesianIndices and make no assumptions on the permutations (actually, permutations are completely invisible in the implementations).

The second method uses explicit (i, j, k) indices. It assumes that the underlying permutation is (3, 2, 1) to loop with i as the slowest index and k as the fastest, which is the optimal order in this case given the permutation. As such, the implementation is less generic than the others, and differences in performance are negligible with respect to more generic variants.

Benchmark results

The following are the benchmark results obtained from running examples/gradient.jl on a laptop, using 2 MPI processes and Julia 1.7.2, with an input array of global dimensions $64 × 32 × 64$. The different methods detailed above are marked on the right. The "lazy" marks indicate runs where the wave numbers were represented by lazy Frequencies objects (returned by rfftfreq and fftfreq). Otherwise, they were collected into Vectors. For some reason, plain Vectors are faster when working with grids generated by localgrid.

In the script, additional implementations can be found which rely on a more advanced understanding of permutations and on the internals of the PencilArrays package. For instance, gradient_local_parent! directly works with the raw data stored in Julia Arrays, while gradient_local_linear! completely avoids CartesianIndices while staying generic and efficient. Nevertheless, these display roughly the same performance as the above examples.

    gradient_global_view!...                  89.900 μs
-    gradient_global_view! (lazy)...           92.060 μs  [Method 1]
-    gradient_global_view_explicit!...         88.958 μs
-    gradient_global_view_explicit! (lazy)...  81.055 μs  [Method 2]
-    gradient_local!...                        92.305 μs
-    gradient_grid!...                         92.770 μs
-    gradient_grid! (lazy)...                  101.388 μs  [Method 3]
-    gradient_grid_broadcast!...               88.606 μs
-    gradient_grid_broadcast! (lazy)...        151.020 μs  [Method 4]
-    gradient_local_parent!...                 92.248 μs
-    gradient_local_linear!...                 91.212 μs
-    gradient_local_linear_explicit!...        90.992 μs

This page was generated using Literate.jl.

diff --git a/previews/PR39/generated/in-place/index.html b/previews/PR39/generated/in-place/index.html deleted file mode 100644 index 2fc85bcd..00000000 --- a/previews/PR39/generated/in-place/index.html +++ /dev/null @@ -1,36 +0,0 @@ - -In-place transforms · PencilFFTs.jl
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In-place transforms

Complex-to-complex and real-to-real transforms can be performed in-place, enabling important memory savings. The procedure is very similar to that of out-of-place transforms described in the tutorial. The differences are illustrated in the sections below.

Creating in-place plans

To create an in-place plan, pass an in-place transform such as Transforms.FFT! or Transforms.R2R! to PencilFFTPlan. For instance:

using PencilFFTs
-using MPI
-MPI.Init()
-
-dims = (16, 32, 64)
-
-# Perform a 3D in-place complex-to-complex FFT.
-transform = Transforms.FFT!()
-
-# Note that one can also combine different types of in-place transforms.
-# For instance:
-# transform = (
-#     Transforms.R2R!(FFTW.REDFT01),
-#     Transforms.FFT!(),
-#     Transforms.R2R!(FFTW.DHT),
-# )
-
-comm = MPI.COMM_WORLD
-Nproc = MPI.Comm_size(comm)
-proc_dims = (Nproc, )  # let's perform a 1D decomposition
-
-# Create plan
-plan = PencilFFTPlan(dims, transform, proc_dims, comm)
Transforms: (FFT!, FFT!, FFT!)
-Input type: ComplexF64
-Global dimensions: (16, 32, 64) -> (16, 32, 64)
-MPI topology: 1D decomposition (2 processes)

Note that in-place real-to-complex transforms are not currently supported. (In other words, the RFFT! transform type is not defined.)

Allocating data

As with out-of-place plans, data should be allocated using allocate_input. The difference is that, for in-place plans, this function returns a ManyPencilArray object, which is a container holding multiple PencilArray views sharing the same memory space.

# Allocate data for the plan.
-# Since `plan` is in-place, this returns a `ManyPencilArray` container.
-A = allocate_input(plan)
-summary(A)
"ManyPencilArray{ComplexF64, 3, 3, Tuple{PencilArray{ComplexF64, 3, Base.ReshapedArray{ComplexF64, 3, SubArray{ComplexF64, 1, Vector{ComplexF64}, Tuple{Base.OneTo{Int64}}, true}, Tuple{}}, 3, 0, Pencil{3, 1, NoPermutation, Vector{UInt8}}}, PencilArray{ComplexF64, 3, Base." ⋯ 140 bytes ⋯ "tation{(2, 1, 3), 3}, Vector{UInt8}}}, PencilArray{ComplexF64, 3, Base.ReshapedArray{ComplexF64, 3, SubArray{ComplexF64, 1, Vector{ComplexF64}, Tuple{Base.OneTo{Int64}}, true}, Tuple{}}, 3, 0, Pencil{3, 1, Permutation{(3, 2, 1), 3}, Vector{UInt8}}}}, Vector{ComplexF64}}"

Note that allocate_output also works for in-place plans, but it returns exactly the same thing as allocate_input.

As shown in the next section, in-place plans must be applied on the returned ManyPencilArray. On the other hand, one usually wants to access and modify data, and for this one needs the PencilArray views contained in the ManyPencilArray. The input and output array views can be obtained by calling first(::ManyPencilArray) and last(::ManyPencilArray).

For instance, we can initialise the input array with some data before transforming:

using Random
-u_in = first(A)  # input data view
-randn!(u_in)
-summary(u_in)
"16×32×32 PencilArray{ComplexF64, 3}(::Pencil{3, 1, NoPermutation, Array})"

Applying plans

Like in FFTW.jl, one can perform in-place transforms using the * and \ operators. As mentioned above, in-place plans must be applied on the ManyPencilArray containers returned by allocate_input.

plan * A;  # performs in-place forward transform

After performing an in-place transform, we usually want to do operations on the output data. For instance, let's compute the global sum of the transformed data:

u_out = last(A)  # output data view
-sum(u_out)       # sum of transformed data (note that `sum` reduces over all processes)
-16704.57297999713 - 26075.211129702257im

Finally, we can perform a backward transform and do stuff with the input view:

plan \ A;  # perform in-place backward transform
-
-# Now we can again do stuff with the input view `u_in`...
ManyPencilArray{ComplexF64, 3, 3, Tuple{PencilArray{ComplexF64, 3, Base.ReshapedArray{ComplexF64, 3, SubArray{ComplexF64, 1, Vector{ComplexF64}, Tuple{Base.OneTo{Int64}}, true}, Tuple{}}, 3, 0, Pencil{3, 1, NoPermutation, Vector{UInt8}}}, PencilArray{ComplexF64, 3, Base.ReshapedArray{ComplexF64, 3, SubArray{ComplexF64, 1, Vector{ComplexF64}, Tuple{Base.OneTo{Int64}}, true}, Tuple{}}, 3, 0, Pencil{3, 1, Permutation{(2, 1, 3), 3}, Vector{UInt8}}}, PencilArray{ComplexF64, 3, Base.ReshapedArray{ComplexF64, 3, SubArray{ComplexF64, 1, Vector{ComplexF64}, Tuple{Base.OneTo{Int64}}, true}, Tuple{}}, 3, 0, Pencil{3, 1, Permutation{(3, 2, 1), 3}, Vector{UInt8}}}}, Vector{ComplexF64}}(ComplexF64[-0.5097831109618262 - 0.7957522927765581im, 0.2202417749947463 - 0.785289975140012im, -0.5353057064885618 - 0.618943778382149im, -0.16679586291930193 + 1.1000625284499685im, -0.06287059098850775 + 0.3940657680992792im, -0.0676488663339263 - 0.3358693630685692im, -0.2665024483393818 - 0.36485631157592097im, -1.640474947719681 + 1.1209919803398318im, -0.1620869066677505 + 0.4605435413563161im, 0.5344169798666101 - 1.3932704573725625im  …  5.3294284e-316 + 5.21007717e-316im, 1.265e-321 + 5.02651163e-316im, 5.92418306e-316 + 1.265e-321im, 4.3646771e-316 + 5.0967061e-316im, 1.265e-321 + 5.33080706e-316im, 4.4210757e-316 + 1.265e-321im, 4.97987658e-316 + 4.8043386e-316im, 1.265e-321 + 4.96401983e-316im, 4.95763414e-316 + 1.265e-321im, 5.81217166e-316 + 4.67940956e-316im], ([-0.5097831109618262 - 0.7957522927765581im 0.12716706665418276 - 0.12875654706424944im … 0.17859453865834973 - 0.4217411044605556im -0.7944312606968564 - 0.09253164979146439im; 0.2202417749947463 - 0.785289975140012im 1.3093359072910185 + 0.059837311346692235im … 0.15432387221848942 + 0.22375757031683552im -0.49151750300253505 + 0.4157900244240209im; … ; 0.042164854407231334 + 0.9480424996648198im -0.5657929057915633 - 1.2688915997220356im … 1.5222571984718352 - 0.4712287049185293im 0.6006306661836718 - 0.03671433586171843im; -0.8791989022928659 - 1.487209158231464im -0.5051798750512 - 0.3346722947550943im … -0.9267129426223744 - 2.082207717821656im -1.1025492503378733 - 0.736132213420922im;;; -0.47794955698651426 + 0.06773651363538613im -1.3612201826795562 - 0.606392035392882im … -1.3088405026567709 - 0.2454739288589592im 0.5451683887822878 + 0.06503546434319307im; 0.29846234033603747 - 0.4132002714595261im 0.19210759879248707 + 1.3186620840651826im … 1.2714465427902613 - 0.4851454942695776im -1.0098026042123331 + 0.07024208341594904im; … ; -0.24824027797839351 - 0.04876116643429018im 0.41856060988233357 - 0.2715312948619732im … -1.1542211975954328 - 0.2873503351005518im 0.04570629143031518 - 0.7337384111903973im; 0.057270677639515793 + 0.24286804481236302im 0.41763626512151386 + 0.1304600823497969im … 1.0751384766764471 - 0.43758397787668285im 0.7933850403908033 + 0.3155437320542597im;;; -0.3487178201545347 - 1.1744200212787064im 0.5300874854872044 + 0.45873755294434376im … 0.22018434365044864 + 0.5109141714654811im 1.4708595398482207 + 0.2102314382032323im; -1.3602198958437413 + 0.5194692493821277im -0.01651667467020554 - 0.5692114351026699im … -0.5339730235925209 - 0.32506596674103816im 0.20933920987031568 + 0.6966594136412245im; … ; -0.4445046048485678 + 0.23492246050343196im -1.0981741604399797 + 0.7352116886076658im … 0.7693307751458044 + 0.6322266502109275im -1.70572695639484 - 0.42849926222920676im; 1.397098928657436 - 0.4799533950483356im -0.42075652092933447 + 0.4211417874293205im … -0.8239786701946163 - 0.08744545858510563im 0.5093645590018816 + 1.102632627856858im;;; … ;;; 0.47993519099777593 + 0.4581389986290574im 0.027560232851720157 - 0.729184780024793im … 0.654245211168236 + 0.6253777919197556im -1.0532610533535296 + 0.7153492127026945im; 0.0682645835244819 - 0.6263560497119439im 0.9572169724027622 - 0.4588775359215391im … -0.3443466239281733 + 0.7359741404393567im 1.1262037932564863 + 1.1133595179132616im; … ; 0.5188219456300835 - 0.5424389560002282im 1.1419686100769366 - 1.0292534586754853im … 0.3505568594944395 - 0.3391121885707313im -0.6026229061302698 - 0.22293665709389957im; 0.15074215756594228 + 0.3887574485649793im -1.3306164527855953 - 0.15814926696549433im … 0.11864385142683906 - 0.7091806170350403im -0.06311792038103259 + 0.045077374642394474im;;; -0.6536921185408382 - 0.7354082872314902im 0.874850658617963 - 0.16771815399094706im … -0.9138150232563286 - 0.93721093736461im -0.15836036264518266 + 0.2543453845920463im; 0.016116261087019357 + 0.8769788768661464im 0.2708171713169075 - 0.5046742938838706im … 0.6955878757236436 - 0.13696977894490758im 0.27067592891877135 + 0.5963128430544945im; … ; -0.8354330180746538 + 0.45852466534996633im -0.39114263901179885 - 0.3147102918770539im … -0.6808022548600332 - 0.23508958989133738im 0.9714163001003087 + 0.19394217785700588im; 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This page was generated using Literate.jl.

diff --git a/previews/PR39/generated/navier_stokes/index.html b/previews/PR39/generated/navier_stokes/index.html deleted file mode 100644 index db16a3bf..00000000 --- a/previews/PR39/generated/navier_stokes/index.html +++ /dev/null @@ -1,303 +0,0 @@ - -Navier–Stokes equations · PencilFFTs.jl
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Navier–Stokes equations

In this example, we numerically solve the incompressible Navier–Stokes equations

\[∂_t \bm{v} + (\bm{v} ⋅ \bm{∇}) \bm{v} = -\frac{1}{ρ} \bm{∇} p + ν ∇^2 \bm{v}, -\quad \bm{∇} ⋅ \bm{v} = 0,\]

where $\bm{v}(\bm{x}, t)$ and $p(\bm{x}, t)$ are respectively the velocity and pressure fields, $ν$ is the fluid kinematic viscosity and $ρ$ is the fluid density.

We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.

First steps

We start by loading the required packages, initialising MPI and setting the simulation parameters.

using MPI
-using PencilFFTs
-
-MPI.Init()
-comm = MPI.COMM_WORLD
-procid = MPI.Comm_rank(comm) + 1
-
-# Simulation parameters
-Ns = (64, 64, 64)  # = (Nx, Ny, Nz)
-Ls = (2π, 2π, 2π)  # = (Lx, Ly, Lz)
-
-# Collocation points ("global" = over all processes).
-# We include the endpoint (length = N + 1) for convenience.
-xs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls)  # = (x, y, z)
(0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586)

Let's check the number of MPI processes over which we're running our simulation:

MPI.Comm_size(comm)
2

We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:

pen = Pencil(Ns, comm)
Decomposition of 3D data
-    Data dimensions: (64, 64, 64)
-    Decomposed dimensions: (2, 3)
-    Data permutation: NoPermutation()
-    Array type: Array

The subdomain associated to the local MPI process can be obtained using range_local:

range_local(pen)
(1:64, 1:32, 1:64)

We now construct a distributed vector field that follows the decomposition configuration we just created:

v⃗₀ = (
-    PencilArray{Float64}(undef, pen),  # vx
-    PencilArray{Float64}(undef, pen),  # vy
-    PencilArray{Float64}(undef, pen),  # vz
-)
-summary(v⃗₀[1])
"64×32×64 PencilArray{Float64, 3}(::Pencil{3, 2, NoPermutation, Array})"

We still need to fill this array with interesting values that represent a physical velocity field.

Initial condition

Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:

\[\begin{aligned} -v_x(x, y, z) &= u₀ \sin(k₀ x) \cos(k₀ y) \cos(k₀ z) \\ -v_y(x, y, z) &= -u₀ \cos(k₀ x) \sin(k₀ y) \cos(k₀ z) \\ -v_z(x, y, z) &= 0 -\end{aligned}\]

where $u₀$ and $k₀$ are two parameters setting the amplitude and the period of the velocity field.

To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:

grid = localgrid(pen, xs_global)
LocalRectilinearGrid{3} with coordinates:
- (1) 0.0:0.09817477042468103:6.1850105367549055
- (2) 0.0:0.09817477042468103:3.043417883165112
- (3) 0.0:0.09817477042468103:6.1850105367549055

We can use this to initialise the velocity field:

u₀ = 1.0
-k₀ = 2π / Ls[1]  # should be integer if L = 2π (to preserve periodicity)
-
-@. v⃗₀[1] =  u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)
-@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)
-@. v⃗₀[3] =  0

Let's plot a 2D slice of the velocity field managed by the local MPI process:

using GLMakie
-
-# Compute the norm of a vector field represented by a tuple of arrays.
-function vecnorm(v⃗::NTuple)
-    vnorm = similar(v⃗[1])
-    for n ∈ eachindex(v⃗[1])
-        w = zero(eltype(vnorm))
-        for v ∈ v⃗
-            w += v[n]^2
-        end
-        vnorm[n] = sqrt(w)
-    end
-    vnorm
-end
-
-let fig = Figure(resolution = (700, 600))
-    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
-    vnorm = vecnorm(v⃗₀)
-    ct = contour!(
-        ax, grid.x, grid.y, grid.z, vnorm;
-        alpha = 0.2, levels = 4,
-        colormap = :viridis, colorrange = (0.0, 1.0),
-    )
-    cb = Colorbar(fig[1, 2], ct; label = "Velocity magnitude")
-    fig
-end

Velocity in Fourier space

In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series

\[\bm{v}(\bm{x}, t) = -∑_{\bm{k}} \hat{\bm{v}}_{\bm{k}}(t) \, e^{i \bm{k} ⋅ \bm{x}},\]

where $\bm{k} = (k_x, k_y, k_z)$ are the discrete wave numbers.

The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for $k_x$:

using AbstractFFTs: fftfreq, rfftfreq
-
-ks_global = (
-    rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]),  # kx | real-to-complex
-     fftfreq(Ns[2], 2π * Ns[2] / Ls[2]),  # ky | complex-to-complex
-     fftfreq(Ns[3], 2π * Ns[3] / Ls[3]),  # kz | complex-to-complex
-)
-
-ks_global[1]'
1×33 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
- 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  27.0  28.0  29.0  30.0  31.0  32.0
ks_global[2]'
1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
- 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0
ks_global[3]'
1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
- 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0

To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:

plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())
Transforms: (RFFT, FFT, FFT)
-Input type: Float64
-Global dimensions: (64, 64, 64) -> (33, 64, 64)
-MPI topology: 2D decomposition (2×1 processes)

See PencilFFTPlan for details on creating plans and on optional keyword arguments.

We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$:

v̂s = plan .* v⃗₀
-summary(v̂s[1])
"16×64×64 PencilArray{ComplexF64, 3}(::Pencil{3, 2, Permutation{(3, 2, 1), 3}, Array})"

Note that, in Fourier space, the domain decomposition is performed along the directions $x$ and $y$:

pencil(v̂s[1])
Decomposition of 3D data
-    Data dimensions: (33, 64, 64)
-    Decomposed dimensions: (1, 2)
-    Data permutation: Permutation(3, 2, 1)
-    Array type: Array

This is because the 3D FFTs are performed one dimension at a time, with the $x$ direction first and the $z$ direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.

To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers $\bm{k}$ associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:

grid_fourier = localgrid(v̂s[1], ks_global)
LocalRectilinearGrid{3} with Permutation(3, 2, 1) and coordinates:
- (1) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0]
- (2) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]
- (3) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]

As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, $\bm{ω} = \bm{∇} × \bm{v}$. In Fourier space, this becomes $\hat{\bm{ω}} = i \bm{k} × \hat{\bm{v}}$.

using StaticArrays: SVector
-using LinearAlgebra: ×
-
-function curl_fourier!(
-        ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,
-    ) where {N}
-    @inbounds for I ∈ eachindex(grid_fourier)
-        # We use StaticArrays for the cross product between small vectors.
-        ik⃗ = im * SVector(grid_fourier[I])
-        v⃗ = SVector(getindex.(v̂s, Ref(I)))  # = (v̂s[1][I], v̂s[2][I], ...)
-        ω⃗ = ik⃗ × v⃗
-        for n ∈ eachindex(ω⃗)
-            ω̂s[n][I] = ω⃗[n]
-        end
-    end
-    ω̂s
-end
-
-ω̂s = similar.(v̂s)
-curl_fourier!(ω̂s, v̂s, grid_fourier);

We finally transform back to physical space and plot the result:

ωs = plan .\ ω̂s
-
-let fig = Figure(resolution = (700, 600))
-    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
-    ω_norm = vecnorm(ωs)
-    ct = contour!(
-        ax, grid.x, grid.y, grid.z, ω_norm;
-        alpha = 0.1, levels = 0.8:0.2:2.0,
-        colormap = :viridis, colorrange = (0.8, 2.0),
-    )
-    cb = Colorbar(fig[1, 2], ct; label = "Vorticity magnitude")
-    fig
-end

Computing the non-linear term

One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as

\[∂_t \hat{\bm{v}}_{\bm{k}} = -- \mathcal{P}_{\bm{k}} \! \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right] -- ν |\bm{k}|^2 \hat{\bm{v}}_{\bm{k}} -\quad \text{ with } \quad -\mathcal{P}_{\bm{k}}(\hat{\bm{F}}_{\bm{k}}) = \left( I - \frac{\bm{k} ⊗ -\bm{k}}{|\bm{k}|^2} \right) \hat{\bm{F}}_{\bm{k}},\]

where $\mathcal{P}_{\bm{k}}$ is a projection operator allowing to preserve the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.

Now that we have the wave numbers $\bm{k}$, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$ of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, $\hat{\bm{F}}_{\bm{k}} = \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right]_{\bm{k}}$. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.

Below we implement a function that computes the non-linear term in Fourier space based on its convective form $(\bm{v} ⋅ \bm{∇}) \bm{v} = \bm{∇} ⋅ (\bm{v} ⊗ \bm{v})$. Note that this equivalence uses the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$.

using LinearAlgebra: mul!, ldiv!  # for applying FFT plans in-place
-
-# Compute non-linear term in Fourier space from velocity field in physical
-# space. Optional keyword arguments may be passed to avoid allocations.
-function ns_nonlinear!(
-        F̂s, vs, plan, grid_fourier;
-        vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),
-    )
-    # Compute F_i = ∂_j (v_i v_j) for each i.
-    # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)
-    w, ŵ = vbuf, v̂buf
-    @inbounds for (i, F̂i) ∈ enumerate(F̂s)
-        F̂i .= 0
-        vi = vs[i]
-        for (j, vj) ∈ enumerate(vs)
-            w .= vi .* vj     # w = v_i * v_j in physical space
-            mul!(ŵ, plan, w)  # same in Fourier space
-            # Add derivative in Fourier space
-            for I ∈ eachindex(grid_fourier)
-                k⃗ = grid_fourier[I]  # = (kx, ky, kz)
-                kj = k⃗[j]
-                F̂i[I] += im * kj * ŵ[I]
-            end
-        end
-    end
-    F̂s
-end
ns_nonlinear! (generic function with 1 method)

As an example, let's use this function on our initial velocity field:

F̂s = similar.(v̂s)
-ns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);

Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.

function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)
-    ks_max = maximum.(abs, ks_global)  # maximum stored wave numbers (kx_max, ky_max, kz_max)
-    ks_lim = (2 / 3) .* ks_max
-    @inbounds for I ∈ eachindex(grid_fourier)
-        k⃗ = grid_fourier[I]
-        if any(abs.(k⃗) .> ks_lim)
-            for ŵ ∈ ŵs
-                ŵ[I] = 0
-            end
-        end
-    end
-    ŵs
-end
-
-# We can apply this on the previously computed non-linear term:
-dealias_twothirds!(F̂s, grid_fourier, ks_global);

Finally, we implement the projection associated to the incompressibility condition:

function project_divergence_free!(ûs, grid_fourier)
-    @inbounds for I ∈ eachindex(grid_fourier)
-        k⃗ = grid_fourier[I]
-        k² = sum(abs2, k⃗)
-        iszero(k²) && continue  # avoid division by zero
-        û = getindex.(ûs, Ref(I))  # (ûs[1][I], ûs[2][I], ...)
-        for i ∈ eachindex(û)
-            ŵ = û[i]
-            for j ∈ eachindex(û)
-                ŵ -= k⃗[i] * k⃗[j] * û[j] / k²
-            end
-            ûs[i][I] = ŵ
-        end
-    end
-    ûs
-end
project_divergence_free! (generic function with 1 method)

We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:

v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)
-v̂s_proj .≈ v̂s  # the last one may be false because v_z = 0 initially
(true, true, false)

Putting it all together

To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:

function ns_rhs!(
-        dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,
-    ) where {N}
-    # 1. Compute non-linear term and dealias it
-    (; plan, cache, ks_global, grid_fourier) = p
-    F̂s = cache.F̂s
-    ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])
-    dealias_twothirds!(F̂s, grid_fourier, ks_global)
-
-    # 2. Project onto divergence-free space
-    project_divergence_free!(F̂s, grid_fourier)
-
-    # 3. Transform velocity to Fourier space
-    v̂s = cache.v̂s
-    map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)
-
-    # 4. Add viscous term (and multiply projected non-linear term by -1)
-    ν = p.ν
-    for n ∈ eachindex(v̂s)
-        v̂ = v̂s[n]
-        F̂ = F̂s[n]
-        @inbounds for I ∈ eachindex(grid_fourier)
-            k⃗ = grid_fourier[I]  # = (kx, ky, kz)
-            k² = sum(abs2, k⃗)
-            F̂[I] = -F̂[I] - ν * k² * v̂[I]
-        end
-    end
-
-    # 5. Transform RHS back to physical space
-    map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)
-
-    nothing
-end
ns_rhs! (generic function with 1 method)

For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.

using OrdinaryDiffEq
-using RecursiveArrayTools: ArrayPartition
-
-ns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)
-vs_init_ode = ArrayPartition(v⃗₀)
-summary(vs_init_ode)
"RecursiveArrayTools.ArrayPartition{Float64, Tuple{PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}}} with arrays:"

We now define solver parameters and temporary variables, and initialise the problem:

params = (;
-    ν = 5e-3,  # kinematic viscosity
-    plan, grid_fourier, ks_global,
-    cache = (
-        v̂s = similar.(v̂s),
-        F̂s = similar.(v̂s),
-    )
-)
-
-tspan = (0.0, 10.0)
-prob = ODEProblem(ns_rhs!, vs_init_ode, tspan, params)
-integrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);

We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum $E(k)$, to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity $ν$ must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large $k$), while high enough to allow the small-scale motions to be correctly resolved.

function energy_spectrum!(Ek, ks, v̂s, grid_fourier)
-    Nk = length(Ek)
-    @assert Nk == length(ks)
-    Ek .= 0
-    for I ∈ eachindex(grid_fourier)
-        k⃗ = grid_fourier[I]  # = (kx, ky, kz)
-        knorm = sqrt(sum(abs2, k⃗))
-        i = searchsortedfirst(ks, knorm)
-        i > Nk && continue
-        v⃗ = getindex.(v̂s, Ref(I))  # = (v̂s[1][I], v̂s[2][I], ...)
-        factor = k⃗[1] == 0 ? 1 : 2  # account for Hermitian symmetry and r2c transform
-        Ek[i] += factor * sum(abs2, v⃗) / 2
-    end
-    MPI.Allreduce!(Ek, +, get_comm(v̂s[1]))  # sum across all processes
-    Ek
-end
-
-ks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])
-Ek = similar(ks)
-v̂s = plan .* integrator.u.x
-energy_spectrum!(Ek, ks, v̂s, grid_fourier)
-Ek ./= scale_factor(plan)^2  # rescale energy
-
-curl_fourier!(ω̂s, v̂s, grid_fourier)
-ldiv!.(ωs, plan, ω̂s)
-ω⃗_plot = Observable(ωs)
-k_plot = @view ks[2:end]
-E_plot = Observable(@view Ek[2:end])
-t_plot = Observable(integrator.t)
-
-fig = let
-    fig = Figure(resolution = (1200, 600))
-    ax = Axis3(
-        fig[1, 1][1, 1]; title = @lift("t = $(round($t_plot, digits = 3))"),
-        aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z",
-    )
-    ω_mag = @lift vecnorm($ω⃗_plot)
-    ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)
-    ct = contour!(
-        ax, grid.x, grid.y, grid.z, ω_mag_norm;
-        alpha = 0.3, levels = 3,
-        colormap = :viridis, colorrange = (0.0, 1.0),
-    )
-    cb = Colorbar(fig[1, 1][1, 2], ct; label = "Normalised vorticity magnitude")
-    ax_sp = Axis(
-        fig[1, 2];
-        xlabel = "k", ylabel = "E(k)", xscale = log2, yscale = log10,
-        title = "Kinetic energy spectrum",
-    )
-    ylims!(ax_sp, 1e-8, 1e0)
-    scatterlines!(ax_sp, k_plot, E_plot)
-    ks_slope = exp.(range(log(2.5), log(25.0), length = 3))
-    E_fivethirds = @. 0.3 * ks_slope^(-5/3)
-    @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)
-    text!(ax_sp, L"k^{-5/3}"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))
-    fig
-end
-
-record(fig, "vorticity_proc$procid.mp4"; framerate = 10) do io
-    while integrator.t < 20
-        dt = 0.001
-        step!(integrator, dt)
-        t_plot[] = integrator.t
-        mul!.(v̂s, plan, integrator.u.x)  # current velocity in Fourier space
-        curl_fourier!(ω̂s, v̂s, grid_fourier)
-        ldiv!.(ω⃗_plot[], plan, ω̂s)
-        ω⃗_plot[] = ω⃗_plot[]  # to force updating the plot
-        energy_spectrum!(Ek, ks, v̂s, grid_fourier)
-        Ek ./= scale_factor(plan)^2  # rescale energy
-        E_plot[] = E_plot[]
-        recordframe!(io)
-    end
-end;
- -

This page was generated using Literate.jl.

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 diff --git a/previews/PR39/search/index.html b/previews/PR39/search/index.html deleted file mode 100644 index 2c868712..00000000 --- a/previews/PR39/search/index.html +++ /dev/null @@ -1,2 +0,0 @@ - -Search · PencilFFTs.jl

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    diff --git a/previews/PR39/search_index.js b/previews/PR39/search_index.js deleted file mode 100644 index 72c43a4b..00000000 --- a/previews/PR39/search_index.js +++ /dev/null @@ -1,3 +0,0 @@ -var documenterSearchIndex = {"docs": -[{"location":"tutorial/#Tutorial","page":"Tutorial","title":"Tutorial","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The following tutorial shows how to perform a 3D FFT of real periodic data defined on a grid of N_x N_y N_z points.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"
    \n \n \n
    ","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"By default, the domain is distributed on a 2D MPI topology of dimensions N_1 N_2. As an example, the above figure shows such a topology with N_1 = 4 and N_2 = 3, for a total of 12 MPI processes.","category":"page"},{"location":"tutorial/#tutorial:creating_plans","page":"Tutorial","title":"Creating plans","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The first thing to do is to create a domain decomposition configuration for the given dataset dimensions N_x N_y N_z. In the framework of PencilArrays, such a configuration is described by a Pencil object. As described in the PencilArrays docs, we can let the Pencil constructor automatically determine such a configuration. For this, only an MPI communicator and the dataset dimensions are needed:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using MPI\nusing PencilFFTs\n\nMPI.Init()\ncomm = MPI.COMM_WORLD\n\n# Input data dimensions (Nx × Ny × Nz)\ndims = (16, 32, 64)\npen = Pencil(dims, comm)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"By default this creates a 2D decomposition (for the case of a 3D dataset), but one can change this as detailed in the PencilArrays documentation linked above.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"We can now create a PencilFFTPlan, which requires information on decomposition configuration (the Pencil object) and on the transforms that will be applied:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# Apply a 3D real-to-complex (r2c) FFT.\ntransform = Transforms.RFFT()\n\n# Note that, for more control, one can instead separately specify the transforms along each dimension:\n# transform = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())\n\n# Create plan\nplan = PencilFFTPlan(pen, transform)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"See the PencilFFTPlan constructor for details on the accepted options, and the Transforms module for the possible transforms. It is also possible to enable fine-grained performance measurements via the TimerOutputs package, as described in Measuring performance.","category":"page"},{"location":"tutorial/#Allocating-data","page":"Tutorial","title":"Allocating data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Next, we want to apply the plan on some data. Transforms may only be applied on PencilArrays, which are array wrappers that include MPI decomposition information (in some sense, analogous to DistributedArrays in Julia's distributed computing approach). The helper function allocate_input can be used to allocate a PencilArray that is compatible with our plan:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# In our example, this returns a 3D PencilArray of real data (Float64).\nu = allocate_input(plan)\n\n# Fill the array with some (random) data\nusing Random\nrandn!(u)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"PencilArrays are a subtype of AbstractArray, and thus they support all common array operations.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Similarly, to preallocate output data, one can use allocate_output:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# In our example, this returns a 3D PencilArray of complex data (Complex{Float64}).\nv = allocate_output(plan)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"This is only required if one wants to apply the plans using a preallocated output (with mul!, see right below).","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The data types returned by allocate_input and allocate_output are slightly different when working with in-place transforms. See the in-place example for details.","category":"page"},{"location":"tutorial/#Applying-plans","page":"Tutorial","title":"Applying plans","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The interface to apply plans is consistent with that of AbstractFFTs. Namely, * and mul! are respectively used for forward transforms without and with preallocated output data. Similarly, \\ and ldiv! are used for backward transforms.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using LinearAlgebra # for mul!, ldiv!\n\n# Apply plan on `u` with `v` as an output\nmul!(v, plan, u)\n\n# Apply backward plan on `v` with `w` as an output\nw = similar(u)\nldiv!(w, plan, v) # now w ≈ u","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Note that, consistently with AbstractFFTs, normalisation is performed at the end of a backward transform, so that the original data is recovered when applying a forward followed by a backward transform.","category":"page"},{"location":"tutorial/#Accessing-and-modifying-data","page":"Tutorial","title":"Accessing and modifying data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For any given MPI process, a PencilArray holds the data associated to its local partition in the global geometry. PencilArrays are accessed using local indices that start at 1, regardless of the location of the local process in the MPI topology. Note that PencilArrays, being based on regular Arrays, support both linear and Cartesian indexing (see the Julia docs for details).","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For convenience, the global_view function can be used to generate an OffsetArray wrapper that takes global indices.","category":"page"},{"location":"tutorial/#tutorial:output_data_layout","page":"Tutorial","title":"Output data layout","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"In memory, the dimensions of the transform output are by default reversed with respect to the input. That is, if the order of indices in the input data is (x, y, z), then the output has order (z, y, x) in memory. This detail is hidden from the user, and output arrays are always accessed in the same order as the input data, regardless of the underlying output dimension permutation. This applies to PencilArrays and to OffsetArrays returned by global_view.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The reasoning behind dimension permutations, is that they allow to always perform FFTs along the fastest array dimension and to avoid a local data transposition, resulting in performance gains. A similar approach is followed by other parallel FFT libraries. FFTW itself, in its distributed-memory routines, includes a flag that enables a similar behaviour. In PencilFFTs, index permutation is the default, but it can be disabled via the permute_dims flag of PencilFFTPlan.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"A great deal of work has been spent in making generic index permutations as efficient as possible, both in intermediate and in the output state of the multidimensional transforms. This has been achieved, in part, by making sure that permutations such as (3, 2, 1) are compile-time constants.","category":"page"},{"location":"tutorial/#Further-reading","page":"Tutorial","title":"Further reading","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For details on working with PencilArrays see the PencilArrays docs.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The examples on the sidebar further illustrate the use of transforms and provide an introduction to working with MPI-distributed data in the form of PencilArrays. In particular, the gradient example illustrates different ways of computing things using Fourier-transformed distributed arrays. Then, the incompressible Navier–Stokes example is a more advanced and complete example of a possible application of the PencilFFTs package.","category":"page"},{"location":"benchmarks/#Benchmarks","page":"Benchmarks","title":"Benchmarks","text":"","category":"section"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The performance of PencilFFTs.jl is comparable to that of other open-source parallel FFT libraries implemented in lower-level languages. Below, we show comparisons with the Fortran implementation of P3DFFT, possibly the most popular of these libraries. The benchmarks were performed on the Jean–Zay cluster of the IDRIS French computing centre (CNRS).","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The figure below shows strong scaling benchmarks of 3D real-to-complex FFTs using 2D (\"pencil\") decomposition. The benchmarks were run for input arrays of dimensions N_x N_y N_z = 512^3, 1024^3 and 2048^3. Each timing is averaged over 100 repetitions.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"
    \n \n \n
    ","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"As seen above, PencilFFTs generally outperforms P3DFFT in its default setting. This is largely explained by the choice of using non-blocking point-to-point MPI communications (via MPI_Isend and MPI_Irecv), while P3DFFT uses collective MPI_Alltoallv calls. This enables PencilFFTs to perform data reordering operations on the partially received data while waiting for the incoming data, leading to better performance. Moreover, in contrast with P3DFFT, the high performance and scalability of PencilFFTs results from a highly generic code, handling decompositions in arbitrary dimensions and a relatively large (and extensible) variety of transformations.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"Note that PencilFFTs can optionally use collective communications (using MPI_Alltoallv) instead of point-to-point communications. For details, see the docs for PencilFFTPlan and for PencilArray transpositions. As seen above, collective communications generally perform worse than point-to-point ones, and runtimes are nearly indistinguishable from those of P3DFFT.","category":"page"},{"location":"benchmarks/#Benchmark-details","page":"Benchmarks","title":"Benchmark details","text":"","category":"section"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The benchmarks were performed using Julia 1.7-beta3 and Intel MPI 2019. We used PencilFFTs v0.12.5 with FFTW.jl v1.4.3 and MPI.jl v0.19.0. We used the Fortran implementation of P3DFFT, version 2.7.6, which was built with Intel 2019 compilers and linked to FFTW 3.3.8. The cluster where the benchmarks were run has Intel Cascade Lake 6248 processors with 2×20 cores per node.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The number of MPI processes along each decomposed dimension, P_1 and P_2, was automatically determined by a call to MPI_Dims_create, which tends to create a balanced decomposition with P_1 P_2. For instance, a total of 1024 processes is divided into P_1 = P_2 = 32. Different results may be obtained with other combinations, but this was not benchmarked.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The source files used to generate this benchmark, as well as the raw benchmark results, are all available in the PencilFFTs repo.","category":"page"},{"location":"GlobalFFTParams/#Global-FFT-parameters","page":"Global FFT parameters","title":"Global FFT parameters","text":"","category":"section"},{"location":"GlobalFFTParams/","page":"Global FFT parameters","title":"Global FFT parameters","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"GlobalFFTParams/","page":"Global FFT parameters","title":"Global FFT parameters","text":"GlobalFFTParams","category":"page"},{"location":"GlobalFFTParams/#PencilFFTs.GlobalFFTParams","page":"Global FFT parameters","title":"PencilFFTs.GlobalFFTParams","text":"GlobalFFTParams{T, N, inplace}\n\nSpecifies the global parameters for an N-dimensional distributed transform. These include the element type T and global data sizes of input and output data, as well as the transform types to be performed along each dimension.\n\n\n\nGlobalFFTParams(size_global, transforms, [real_type=Float64])\n\nDefine parameters for N-dimensional transform.\n\ntransforms must be a tuple of length N specifying the transforms to be applied along each dimension. Each element must be a subtype of Transforms.AbstractTransform. For all the possible transforms, see Transform types.\n\nThe element type must be a real type accepted by FFTW, i.e. either Float32 or Float64.\n\nNote that the transforms are applied one dimension at a time, with the leftmost dimension first for forward transforms.\n\nExample\n\nTo perform a 3D FFT of real data, first a real-to-complex FFT must be applied along the first dimension, followed by two complex-to-complex FFTs along the other dimensions:\n\njulia> size_global = (64, 32, 128); # size of real input data\n\njulia> transforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT());\n\njulia> fft_params = PencilFFTs.GlobalFFTParams(size_global, transforms)\nTransforms: (RFFT, FFT, FFT)\nInput type: Float64\nGlobal dimensions: (64, 32, 128) -> (33, 32, 128)\n\n\n\n\n\n","category":"type"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"EditURL = \"https://github.com/jipolanco/PencilFFTs.jl/blob/master/docs/examples/in-place.jl\"","category":"page"},{"location":"generated/in-place/#In-place-transforms","page":"In-place transforms","title":"In-place transforms","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Complex-to-complex and real-to-real transforms can be performed in-place, enabling important memory savings. The procedure is very similar to that of out-of-place transforms described in the tutorial. The differences are illustrated in the sections below.","category":"page"},{"location":"generated/in-place/#Creating-in-place-plans","page":"In-place transforms","title":"Creating in-place plans","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"To create an in-place plan, pass an in-place transform such as Transforms.FFT! or Transforms.R2R! to PencilFFTPlan. For instance:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"using PencilFFTs\nusing MPI\nMPI.Init()\n\ndims = (16, 32, 64)\n\n# Perform a 3D in-place complex-to-complex FFT.\ntransform = Transforms.FFT!()\n\n# Note that one can also combine different types of in-place transforms.\n# For instance:\n# transform = (\n# Transforms.R2R!(FFTW.REDFT01),\n# Transforms.FFT!(),\n# Transforms.R2R!(FFTW.DHT),\n# )\n\ncomm = MPI.COMM_WORLD\nNproc = MPI.Comm_size(comm)\nproc_dims = (Nproc, ) # let's perform a 1D decomposition\n\n# Create plan\nplan = PencilFFTPlan(dims, transform, proc_dims, comm)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Note that in-place real-to-complex transforms are not currently supported. (In other words, the RFFT! transform type is not defined.)","category":"page"},{"location":"generated/in-place/#Allocating-data","page":"In-place transforms","title":"Allocating data","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"As with out-of-place plans, data should be allocated using allocate_input. The difference is that, for in-place plans, this function returns a ManyPencilArray object, which is a container holding multiple PencilArray views sharing the same memory space.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"# Allocate data for the plan.\n# Since `plan` is in-place, this returns a `ManyPencilArray` container.\nA = allocate_input(plan)\nsummary(A)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Note that allocate_output also works for in-place plans, but it returns exactly the same thing as allocate_input.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"As shown in the next section, in-place plans must be applied on the returned ManyPencilArray. On the other hand, one usually wants to access and modify data, and for this one needs the PencilArray views contained in the ManyPencilArray. The input and output array views can be obtained by calling first(::ManyPencilArray) and last(::ManyPencilArray).","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"For instance, we can initialise the input array with some data before transforming:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"using Random\nu_in = first(A) # input data view\nrandn!(u_in)\nsummary(u_in)","category":"page"},{"location":"generated/in-place/#Applying-plans","page":"In-place transforms","title":"Applying plans","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Like in FFTW.jl, one can perform in-place transforms using the * and \\ operators. As mentioned above, in-place plans must be applied on the ManyPencilArray containers returned by allocate_input.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan * A; # performs in-place forward transform\nnothing #hide","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"After performing an in-place transform, we usually want to do operations on the output data. For instance, let's compute the global sum of the transformed data:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"u_out = last(A) # output data view\nsum(u_out) # sum of transformed data (note that `sum` reduces over all processes)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Finally, we can perform a backward transform and do stuff with the input view:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan \\ A; # perform in-place backward transform\n\n# Now we can again do stuff with the input view `u_in`...","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"This page was generated using Literate.jl.","category":"page"},{"location":"Transforms/#Available-transforms","page":"Available transforms","title":"Available transforms","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"CurrentModule = PencilFFTs.Transforms","category":"page"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"Transforms","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms","page":"Available transforms","title":"PencilFFTs.Transforms","text":"Defines different one-dimensional FFT-based transforms.\n\nThe transforms are all subtypes of an AbstractTransform type.\n\nWhen possible, the names of the transforms are kept consistent with the functions exported by AbstractFFTs.jl and FFTW.jl.\n\n\n\n\n\n","category":"module"},{"location":"Transforms/#Transform-types","page":"Available transforms","title":"Transform types","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"FFT\nFFT!\nBFFT\nBFFT!\n\nRFFT\nBRFFT\n\nR2R\nR2R!\n\nNoTransform\nNoTransform!","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.FFT","page":"Available transforms","title":"PencilFFTs.Transforms.FFT","text":"FFT()\n\nComplex-to-complex FFT.\n\nSee also AbstractFFTs.fft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.FFT!","page":"Available transforms","title":"PencilFFTs.Transforms.FFT!","text":"FFT!()\n\nIn-place version of FFT.\n\nSee also AbstractFFTs.fft!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BFFT","page":"Available transforms","title":"PencilFFTs.Transforms.BFFT","text":"BFFT()\n\nUnnormalised backward complex-to-complex FFT.\n\nLike AbstractFFTs.bfft, this transform is not normalised. To obtain the inverse transform, divide the output by the length of the transformed dimension.\n\nSee also AbstractFFTs.bfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.BFFT!","text":"BFFT()\n\nIn-place version of BFFT.\n\nSee also AbstractFFTs.bfft!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.RFFT","page":"Available transforms","title":"PencilFFTs.Transforms.RFFT","text":"RFFT()\n\nReal-to-complex FFT.\n\nSee also AbstractFFTs.rfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BRFFT","page":"Available transforms","title":"PencilFFTs.Transforms.BRFFT","text":"BRFFT(d::Integer)\nBRFFT((d1, d2, ..., dN))\n\nUnnormalised inverse of RFFT.\n\nTo obtain the inverse transform, divide the output by the length of the transformed dimension (of the real output array).\n\nAs described in the AbstractFFTs docs, the length of the output cannot be fully inferred from the input length. For this reason, the BRFFT constructor accepts an optional d argument indicating the output length.\n\nFor multidimensional datasets, a tuple of dimensions (d1, d2, ..., dN) may also be passed. This is equivalent to passing just dN. In this case, the last dimension (dN) is the one that changes size between the input and output. Note that this is the opposite of FFTW.brfft. The reason is that, in PencilFFTs, the last dimension is the one along which a complex-to-real transform is performed.\n\nSee also AbstractFFTs.brfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.R2R","page":"Available transforms","title":"PencilFFTs.Transforms.R2R","text":"R2R(kind)\n\nReal-to-real transform of type kind.\n\nThe possible values of kind are those described in the FFTW.r2r docs and the FFTW manual:\n\ndiscrete cosine transforms: FFTW.REDFT00, FFTW.REDFT01, FFTW.REDFFT10, FFTW.REDFFT11\ndiscrete sine transforms: FFTW.RODFT00, FFTW.RODFT01, FFTW.RODFFT10, FFTW.RODFFT11\ndiscrete Hartley transform: FFTW.DHT\n\nNote: half-complex format DFTs (FFTW.R2HC, FFTW.HC2R) are not currently supported.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.R2R!","page":"Available transforms","title":"PencilFFTs.Transforms.R2R!","text":"R2R!(kind)\n\nIn-place version of R2R.\n\nSee also FFTW.r2r!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.NoTransform","page":"Available transforms","title":"PencilFFTs.Transforms.NoTransform","text":"NoTransform()\n\nIdentity transform.\n\nSpecifies that no transformation should be applied.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.NoTransform!","page":"Available transforms","title":"PencilFFTs.Transforms.NoTransform!","text":"NoTransform!()\n\nIn-place version of NoTransform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#Internals","page":"Available transforms","title":"Internals","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"What follows is used internally in PencilFFTs.","category":"page"},{"location":"Transforms/#Types","page":"Available transforms","title":"Types","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"AbstractCustomPlan\nAbstractTransform\nIdentityPlan\nIdentityPlan!\nPlan","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.AbstractCustomPlan","page":"Available transforms","title":"PencilFFTs.Transforms.AbstractCustomPlan","text":"AbstractCustomPlan\n\nAbstract type defining a custom plan, to be used as an alternative to FFTW plans (FFTW.FFTWPlan).\n\nThe only custom plan defined in this module is IdentityPlan. The user can define other custom plans that are also subtypes of AbstractCustomPlan.\n\nNote that plan returns a subtype of either FFTW.FFTWPlan or AbstractCustomPlan.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.AbstractTransform","page":"Available transforms","title":"PencilFFTs.Transforms.AbstractTransform","text":"AbstractTransform\n\nSpecifies a one-dimensional FFT-based transform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.IdentityPlan","page":"Available transforms","title":"PencilFFTs.Transforms.IdentityPlan","text":"IdentityPlan\n\nType of plan associated to NoTransform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.IdentityPlan!","page":"Available transforms","title":"PencilFFTs.Transforms.IdentityPlan!","text":"IdentityPlan!\n\nType of plan associated to NoTransform!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.Plan","page":"Available transforms","title":"PencilFFTs.Transforms.Plan","text":"Plan = Union{FFTW.FFTWPlan, AbstractCustomPlan}\n\nUnion type representing any plan returned by plan.\n\nSee also AbstractCustomPlan.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#Functions","page":"Available transforms","title":"Functions","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"plan\n\nbinv\nscale_factor\n\neltype_input\neltype_output\nexpand_dims\nis_inplace\nkind\nlength_output","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.plan","page":"Available transforms","title":"PencilFFTs.Transforms.plan","text":"plan(transform::AbstractTransform, A, [dims];\n flags=FFTW.ESTIMATE, timelimit=Inf)\n\nCreate plan to transform array A along dimensions dims.\n\nIf dims is not specified, all dimensions of A are transformed.\n\nFor FFT plans, this function wraps the AbstractFFTs.jl and FFTW.jl plan creation functions. For more details on the function arguments, see AbstractFFTs.plan_fft.\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.binv","page":"Available transforms","title":"PencilFFTs.Transforms.binv","text":"binv(transform::AbstractTransform, d::Integer)\n\nReturns the backwards transform associated to the given transform.\n\nThe second argument must be the length of the first transformed dimension in the forward transform. It is used in particular when transform = RFFT(), to determine the length of the inverse (complex-to-real) transform. See the AbstractFFTs.irfft docs for details.\n\nThe backwards transform returned by this function is not normalised. The normalisation factor for a given array can be obtained by calling scale_factor.\n\nExample\n\njulia> binv(Transforms.FFT(), 42)\nBFFT\n\njulia> binv(Transforms.BRFFT(9), 42)\nRFFT\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.scale_factor","page":"Available transforms","title":"PencilFFTs.Transforms.scale_factor","text":"scale_factor(transform::AbstractTransform, A, [dims = 1:ndims(A)])\n\nGet factor required to normalise the given array after a transformation along dimensions dims (all dimensions by default).\n\nThe array A must have the dimensions of the transform input.\n\nImportant: the dimensions dims must be the same that were passed to plan.\n\nExamples\n\njulia> C = zeros(ComplexF32, 3, 4, 5);\n\njulia> scale_factor(Transforms.FFT(), C)\n60\n\njulia> scale_factor(Transforms.BFFT(), C)\n60\n\njulia> scale_factor(Transforms.BFFT(), C, 2:3)\n20\n\njulia> R = zeros(Float64, 3, 4, 5);\n\njulia> scale_factor(Transforms.RFFT(), R, 2)\n4\n\njulia> scale_factor(Transforms.RFFT(), R, 2:3)\n20\n\njulia> scale_factor(Transforms.BRFFT(8), C)\n96\n\njulia> scale_factor(Transforms.BRFFT(9), C)\n108\n\nThis will fail because the input of RFFT is real, and R is a complex array:\n\njulia> scale_factor(Transforms.RFFT(), C, 2:3)\nERROR: MethodError: no method matching scale_factor(::PencilFFTs.Transforms.RFFT, ::Array{ComplexF32, 3}, ::UnitRange{Int64})\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.eltype_input","page":"Available transforms","title":"PencilFFTs.Transforms.eltype_input","text":"eltype_input(transform::AbstractTransform, real_type<:AbstractFloat)\n\nDetermine input data type for a given transform given the floating point precision of the input data.\n\nSome transforms, such as R2R and NoTransform, can take both real and complex data. For those kinds of transforms, nothing is returned.\n\nExample\n\njulia> eltype_input(Transforms.FFT(), Float32)\nComplexF32 (alias for Complex{Float32})\n\njulia> eltype_input(Transforms.RFFT(), Float64)\nFloat64\n\njulia> eltype_input(Transforms.R2R(FFTW.REDFT01), Float64) # nothing\n\njulia> eltype_input(Transforms.NoTransform(), Float64) # nothing\n\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.eltype_output","page":"Available transforms","title":"PencilFFTs.Transforms.eltype_output","text":"eltype_output(transform::AbstractTransform, eltype_input)\n\nReturns the output data type for a given transform given the input type.\n\nThrows ArgumentError if the input data type is incompatible with the transform type.\n\nExample\n\njulia> eltype_output(Transforms.NoTransform(), Float32)\nFloat32\n\njulia> eltype_output(Transforms.RFFT(), Float64)\nComplexF64 (alias for Complex{Float64})\n\njulia> eltype_output(Transforms.BRFFT(4), ComplexF32)\nFloat32\n\njulia> eltype_output(Transforms.FFT(), Float64)\nERROR: ArgumentError: invalid input data type for PencilFFTs.Transforms.FFT: Float64\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.expand_dims","page":"Available transforms","title":"PencilFFTs.Transforms.expand_dims","text":"expand_dims(transform::AbstractTransform, Val(N))\n\nExpand a single multidimensional transform into one transform per dimension.\n\nExample\n\n# Expand a real-to-complex transform in 3 dimensions.\njulia> expand_dims(Transforms.RFFT(), Val(3))\n(RFFT, FFT, FFT)\n\njulia> expand_dims(Transforms.BRFFT(4), Val(3))\n(BFFT, BFFT, BRFFT{even})\n\njulia> expand_dims(Transforms.NoTransform(), Val(2))\n(NoTransform, NoTransform)\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.is_inplace","page":"Available transforms","title":"PencilFFTs.Transforms.is_inplace","text":"is_inplace(transform::AbstractTransform) -> Bool\nis_inplace(transforms::Vararg{AbtractTransform}) -> Union{Bool, Nothing}\n\nCheck whether a transform or a list of transforms is performed in-place.\n\nIf the list of transforms has a combination of in-place and out-of-place transforms, nothing is returned.\n\nExample\n\njulia> is_inplace(Transforms.RFFT())\nfalse\n\njulia> is_inplace(Transforms.NoTransform!())\ntrue\n\njulia> is_inplace(Transforms.FFT!(), Transforms.R2R!(FFTW.REDFT01))\ntrue\n\njulia> is_inplace(Transforms.FFT(), Transforms.R2R(FFTW.REDFT01))\nfalse\n\njulia> is_inplace(Transforms.FFT(), Transforms.R2R!(FFTW.REDFT01)) === nothing\ntrue\n\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.kind","page":"Available transforms","title":"PencilFFTs.Transforms.kind","text":"kind(transform::R2R)\n\nGet kind of real-to-real transform.\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.length_output","page":"Available transforms","title":"PencilFFTs.Transforms.length_output","text":"length_output(transform::AbstractTransform, length_in::Integer)\n\nReturns the length of the transform output, given the length of its input.\n\nThe input and output lengths are specified in terms of the respective input and output datatypes. For instance, for real-to-complex transforms, these are respectively the length of input real data and of output complex data.\n\n\n\n\n\n","category":"function"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"EditURL = \"https://github.com/jipolanco/PencilFFTs.jl/blob/master/docs/examples/navier_stokes.jl\"","category":"page"},{"location":"generated/navier_stokes/#Navier–Stokes-equations","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"In this example, we numerically solve the incompressible Navier–Stokes equations","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"_t bmv + (bmv bm) bmv = -frac1ρ bm p + ν ^2 bmv\nquad bm bmv = 0","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where bmv(bmx t) and p(bmx t) are respectively the velocity and pressure fields, ν is the fluid kinematic viscosity and ρ is the fluid density.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.","category":"page"},{"location":"generated/navier_stokes/#First-steps","page":"Navier–Stokes equations","title":"First steps","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We start by loading the required packages, initialising MPI and setting the simulation parameters.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using MPI\nusing PencilFFTs\n\nMPI.Init()\ncomm = MPI.COMM_WORLD\nprocid = MPI.Comm_rank(comm) + 1\n\n# Simulation parameters\nNs = (64, 64, 64) # = (Nx, Ny, Nz)\nLs = (2π, 2π, 2π) # = (Lx, Ly, Lz)\n\n# Collocation points (\"global\" = over all processes).\n# We include the endpoint (length = N + 1) for convenience.\nxs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls) # = (x, y, z)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's check the number of MPI processes over which we're running our simulation:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"MPI.Comm_size(comm)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"pen = Pencil(Ns, comm)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"The subdomain associated to the local MPI process can be obtained using range_local:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"range_local(pen)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We now construct a distributed vector field that follows the decomposition configuration we just created:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v⃗₀ = (\n PencilArray{Float64}(undef, pen), # vx\n PencilArray{Float64}(undef, pen), # vy\n PencilArray{Float64}(undef, pen), # vz\n)\nsummary(v⃗₀[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We still need to fill this array with interesting values that represent a physical velocity field.","category":"page"},{"location":"generated/navier_stokes/#Initial-condition","page":"Navier–Stokes equations","title":"Initial condition","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"beginaligned\nv_x(x y z) = u₀ sin(k₀ x) cos(k₀ y) cos(k₀ z) \nv_y(x y z) = -u₀ cos(k₀ x) sin(k₀ y) cos(k₀ z) \nv_z(x y z) = 0\nendaligned","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where u₀ and k₀ are two parameters setting the amplitude and the period of the velocity field.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"grid = localgrid(pen, xs_global)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can use this to initialise the velocity field:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"u₀ = 1.0\nk₀ = 2π / Ls[1] # should be integer if L = 2π (to preserve periodicity)\n\n@. v⃗₀[1] = u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)\n@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)\n@. v⃗₀[3] = 0\nnothing # hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's plot a 2D slice of the velocity field managed by the local MPI process:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using GLMakie\n\n# Compute the norm of a vector field represented by a tuple of arrays.\nfunction vecnorm(v⃗::NTuple)\n vnorm = similar(v⃗[1])\n for n ∈ eachindex(v⃗[1])\n w = zero(eltype(vnorm))\n for v ∈ v⃗\n w += v[n]^2\n end\n vnorm[n] = sqrt(w)\n end\n vnorm\nend\n\nlet fig = Figure(resolution = (700, 600))\n ax = Axis3(fig[1, 1]; aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\")\n vnorm = vecnorm(v⃗₀)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, vnorm;\n alpha = 0.2, levels = 4,\n colormap = :viridis, colorrange = (0.0, 1.0),\n )\n cb = Colorbar(fig[1, 2], ct; label = \"Velocity magnitude\")\n fig\nend","category":"page"},{"location":"generated/navier_stokes/#Velocity-in-Fourier-space","page":"Navier–Stokes equations","title":"Velocity in Fourier space","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"bmv(bmx t) =\n_bmk hatbmv_bmk(t) e^i bmk bmx","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where bmk = (k_x k_y k_z) are the discrete wave numbers.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for k_x:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using AbstractFFTs: fftfreq, rfftfreq\n\nks_global = (\n rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]), # kx | real-to-complex\n fftfreq(Ns[2], 2π * Ns[2] / Ls[2]), # ky | complex-to-complex\n fftfreq(Ns[3], 2π * Ns[3] / Ls[3]), # kz | complex-to-complex\n)\n\nks_global[1]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ks_global[2]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ks_global[3]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"See PencilFFTPlan for details on creating plans and on optional keyword arguments.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients hatbmv_bmk:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v̂s = plan .* v⃗₀\nsummary(v̂s[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Note that, in Fourier space, the domain decomposition is performed along the directions x and y:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"pencil(v̂s[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"This is because the 3D FFTs are performed one dimension at a time, with the x direction first and the z direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers bmk associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"grid_fourier = localgrid(v̂s[1], ks_global)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, bmω = bm bmv. In Fourier space, this becomes hatbmω = i bmk hatbmv.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using StaticArrays: SVector\nusing LinearAlgebra: ×\n\nfunction curl_fourier!(\n ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,\n ) where {N}\n @inbounds for I ∈ eachindex(grid_fourier)\n # We use StaticArrays for the cross product between small vectors.\n ik⃗ = im * SVector(grid_fourier[I])\n v⃗ = SVector(getindex.(v̂s, Ref(I))) # = (v̂s[1][I], v̂s[2][I], ...)\n ω⃗ = ik⃗ × v⃗\n for n ∈ eachindex(ω⃗)\n ω̂s[n][I] = ω⃗[n]\n end\n end\n ω̂s\nend\n\nω̂s = similar.(v̂s)\ncurl_fourier!(ω̂s, v̂s, grid_fourier);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We finally transform back to physical space and plot the result:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ωs = plan .\\ ω̂s\n\nlet fig = Figure(resolution = (700, 600))\n ax = Axis3(fig[1, 1]; aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\")\n ω_norm = vecnorm(ωs)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, ω_norm;\n alpha = 0.1, levels = 0.8:0.2:2.0,\n colormap = :viridis, colorrange = (0.8, 2.0),\n )\n cb = Colorbar(fig[1, 2], ct; label = \"Vorticity magnitude\")\n fig\nend","category":"page"},{"location":"generated/navier_stokes/#Computing-the-non-linear-term","page":"Navier–Stokes equations","title":"Computing the non-linear term","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"_t hatbmv_bmk =\n- mathcalP_bmk left widehat(bmv bm) bmv right\n- ν bmk^2 hatbmv_bmk\nquad text with quad\nmathcalP_bmk(hatbmF_bmk) = left( I - fracbmk \nbmkbmk^2 right) hatbmF_bmk","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where mathcalP_bmk is a projection operator allowing to preserve the incompressibility condition bm bmv = 0. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Now that we have the wave numbers bmk, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients hatbmv_bmk of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, hatbmF_bmk = left widehat(bmv bm) bmv right_bmk. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Below we implement a function that computes the non-linear term in Fourier space based on its convective form (bmv bm) bmv = bm (bmv bmv). Note that this equivalence uses the incompressibility condition bm bmv = 0.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using LinearAlgebra: mul!, ldiv! # for applying FFT plans in-place\n\n# Compute non-linear term in Fourier space from velocity field in physical\n# space. Optional keyword arguments may be passed to avoid allocations.\nfunction ns_nonlinear!(\n F̂s, vs, plan, grid_fourier;\n vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),\n )\n # Compute F_i = ∂_j (v_i v_j) for each i.\n # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)\n w, ŵ = vbuf, v̂buf\n @inbounds for (i, F̂i) ∈ enumerate(F̂s)\n F̂i .= 0\n vi = vs[i]\n for (j, vj) ∈ enumerate(vs)\n w .= vi .* vj # w = v_i * v_j in physical space\n mul!(ŵ, plan, w) # same in Fourier space\n # Add derivative in Fourier space\n for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n kj = k⃗[j]\n F̂i[I] += im * kj * ŵ[I]\n end\n end\n end\n F̂s\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"As an example, let's use this function on our initial velocity field:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"F̂s = similar.(v̂s)\nns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)\n ks_max = maximum.(abs, ks_global) # maximum stored wave numbers (kx_max, ky_max, kz_max)\n ks_lim = (2 / 3) .* ks_max\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I]\n if any(abs.(k⃗) .> ks_lim)\n for ŵ ∈ ŵs\n ŵ[I] = 0\n end\n end\n end\n ŵs\nend\n\n# We can apply this on the previously computed non-linear term:\ndealias_twothirds!(F̂s, grid_fourier, ks_global);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Finally, we implement the projection associated to the incompressibility condition:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function project_divergence_free!(ûs, grid_fourier)\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I]\n k² = sum(abs2, k⃗)\n iszero(k²) && continue # avoid division by zero\n û = getindex.(ûs, Ref(I)) # (ûs[1][I], ûs[2][I], ...)\n for i ∈ eachindex(û)\n ŵ = û[i]\n for j ∈ eachindex(û)\n ŵ -= k⃗[i] * k⃗[j] * û[j] / k²\n end\n ûs[i][I] = ŵ\n end\n end\n ûs\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)\nv̂s_proj .≈ v̂s # the last one may be false because v_z = 0 initially","category":"page"},{"location":"generated/navier_stokes/#Putting-it-all-together","page":"Navier–Stokes equations","title":"Putting it all together","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function ns_rhs!(\n dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,\n ) where {N}\n # 1. Compute non-linear term and dealias it\n (; plan, cache, ks_global, grid_fourier) = p\n F̂s = cache.F̂s\n ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])\n dealias_twothirds!(F̂s, grid_fourier, ks_global)\n\n # 2. Project onto divergence-free space\n project_divergence_free!(F̂s, grid_fourier)\n\n # 3. Transform velocity to Fourier space\n v̂s = cache.v̂s\n map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)\n\n # 4. Add viscous term (and multiply projected non-linear term by -1)\n ν = p.ν\n for n ∈ eachindex(v̂s)\n v̂ = v̂s[n]\n F̂ = F̂s[n]\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n k² = sum(abs2, k⃗)\n F̂[I] = -F̂[I] - ν * k² * v̂[I]\n end\n end\n\n # 5. Transform RHS back to physical space\n map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)\n\n nothing\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using OrdinaryDiffEq\nusing RecursiveArrayTools: ArrayPartition\n\nns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)\nvs_init_ode = ArrayPartition(v⃗₀)\nsummary(vs_init_ode)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We now define solver parameters and temporary variables, and initialise the problem:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"params = (;\n ν = 5e-3, # kinematic viscosity\n plan, grid_fourier, ks_global,\n cache = (\n v̂s = similar.(v̂s),\n F̂s = similar.(v̂s),\n )\n)\n\ntspan = (0.0, 10.0)\nprob = ODEProblem(ns_rhs!, vs_init_ode, tspan, params)\nintegrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum E(k), to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity ν must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large k), while high enough to allow the small-scale motions to be correctly resolved.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function energy_spectrum!(Ek, ks, v̂s, grid_fourier)\n Nk = length(Ek)\n @assert Nk == length(ks)\n Ek .= 0\n for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n knorm = sqrt(sum(abs2, k⃗))\n i = searchsortedfirst(ks, knorm)\n i > Nk && continue\n v⃗ = getindex.(v̂s, Ref(I)) # = (v̂s[1][I], v̂s[2][I], ...)\n factor = k⃗[1] == 0 ? 1 : 2 # account for Hermitian symmetry and r2c transform\n Ek[i] += factor * sum(abs2, v⃗) / 2\n end\n MPI.Allreduce!(Ek, +, get_comm(v̂s[1])) # sum across all processes\n Ek\nend\n\nks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])\nEk = similar(ks)\nv̂s = plan .* integrator.u.x\nenergy_spectrum!(Ek, ks, v̂s, grid_fourier)\nEk ./= scale_factor(plan)^2 # rescale energy\n\ncurl_fourier!(ω̂s, v̂s, grid_fourier)\nldiv!.(ωs, plan, ω̂s)\nω⃗_plot = Observable(ωs)\nk_plot = @view ks[2:end]\nE_plot = Observable(@view Ek[2:end])\nt_plot = Observable(integrator.t)\n\nfig = let\n fig = Figure(resolution = (1200, 600))\n ax = Axis3(\n fig[1, 1][1, 1]; title = @lift(\"t = $(round($t_plot, digits = 3))\"),\n aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\",\n )\n ω_mag = @lift vecnorm($ω⃗_plot)\n ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, ω_mag_norm;\n alpha = 0.3, levels = 3,\n colormap = :viridis, colorrange = (0.0, 1.0),\n )\n cb = Colorbar(fig[1, 1][1, 2], ct; label = \"Normalised vorticity magnitude\")\n ax_sp = Axis(\n fig[1, 2];\n xlabel = \"k\", ylabel = \"E(k)\", xscale = log2, yscale = log10,\n title = \"Kinetic energy spectrum\",\n )\n ylims!(ax_sp, 1e-8, 1e0)\n scatterlines!(ax_sp, k_plot, E_plot)\n ks_slope = exp.(range(log(2.5), log(25.0), length = 3))\n E_fivethirds = @. 0.3 * ks_slope^(-5/3)\n @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)\n text!(ax_sp, L\"k^{-5/3}\"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))\n fig\nend\n\nusing Printf # hide\nwith_xvfb = ENV[\"DISPLAY\"] == \":99\" # hide\nnstep = 0 # hide\nconst tmpdir = mktempdir() # hide\nfilename_frame(procid, nstep) = joinpath(tmpdir, @sprintf(\"proc%d_%04d.png\", procid, nstep)) # hide\nrecord(fig, \"vorticity_proc$procid.mp4\"; framerate = 10) do io\n with_xvfb && recordframe!(io) # hide\n while integrator.t < 20\n dt = 0.001\n step!(integrator, dt)\n t_plot[] = integrator.t\n mul!.(v̂s, plan, integrator.u.x) # current velocity in Fourier space\n curl_fourier!(ω̂s, v̂s, grid_fourier)\n ldiv!.(ω⃗_plot[], plan, ω̂s)\n ω⃗_plot[] = ω⃗_plot[] # to force updating the plot\n energy_spectrum!(Ek, ks, v̂s, grid_fourier)\n Ek ./= scale_factor(plan)^2 # rescale energy\n E_plot[] = E_plot[]\n global nstep += 1 # hide\n with_xvfb ? # hide\n save(filename_frame(procid, nstep), fig) : # hide\n recordframe!(io)\n end\nend;\n\nif with_xvfb # hide\n run(pipeline(`ffmpeg -y -r 10 -i $tmpdir/proc$(procid)_%04d.png -c:v libx264 -vf \"fps=25,format=yuv420p\" vorticity_proc$procid.mp4`; stdout = \"ffmpeg.out\", stderr = \"ffmpeg.err\")) # hide\nend # hide\nnothing # hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"
    \n \n
    ","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"This page was generated using Literate.jl.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"EditURL = \"https://github.com/jipolanco/PencilFFTs.jl/blob/master/docs/examples/gradient.jl\"","category":"page"},{"location":"generated/gradient/#Gradient-of-a-scalar-field","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This example shows different methods to compute the gradient of a real-valued 3D scalar field θ(bmx) in Fourier space, where bmx = (x y z). It is assumed that the field is periodic with period L = 2π along all dimensions.","category":"page"},{"location":"generated/gradient/#General-procedure","page":"Gradient of a scalar field","title":"General procedure","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The discrete Fourier expansion of θ writes","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"θ(bmx) = _bmk Z^3 hatθ(bmk) e^i bmk bmx","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"where bmk = (k_x k_y k_z) are the Fourier wave numbers and hatθ is the discrete Fourier transform of θ. Then, the spatial derivatives of θ are given by","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"frac θ(bmx) x_i =\n_bmk Z^3 i k_i hatθ(bmk) e^i bmk bmx","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"where the subscript i denotes one of the spatial components x, y or z.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In other words, to compute bm θ = (_x θ _y θ _z θ), one has to:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"transform θ to Fourier space to obtain hatθ,\nmultiply hatθ by i bmk,\ntransform the result back to physical space to obtain bm θ.","category":"page"},{"location":"generated/gradient/#Preparation","page":"Gradient of a scalar field","title":"Preparation","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In this section, we initialise a random real-valued scalar field θ and compute its FFT. For more details see the Tutorial.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"using MPI\nusing PencilFFTs\nusing Random\n\nMPI.Init()\n\n# Input data dimensions (Nx × Ny × Nz)\ndims = (64, 32, 64)\n\n# Apply a 3D real-to-complex (r2c) FFT.\ntransform = Transforms.RFFT()\n\n# Automatically create decomposition configuration\ncomm = MPI.COMM_WORLD\npen = Pencil(dims, comm)\n\n# Create plan\nplan = PencilFFTPlan(pen, transform)\n\n# Allocate data and initialise field\nθ = allocate_input(plan)\nrandn!(θ)\n\n# Perform distributed FFT\nθ_hat = plan * θ\nnothing # hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, we initialise the output that will hold ∇θ in Fourier space. Noting that ∇θ is a vector field, we choose to store it as a tuple of 3 PencilArrays.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"∇θ_hat = allocate_output(plan, Val(3))\n\n# This is equivalent:\n# ∇θ_hat = ntuple(d -> similar(θ_hat), Val(3))\n\nsummary(∇θ_hat)\n\n# Fourier wave numbers","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In general, the Fourier wave numbers are of the form k_i = 0 frac2πL_i frac4πL_i frac6πL_i , where L_i is the period along dimension i. When a real-to-complex Fourier transform is applied, roughly half of these wave numbers are redundant due to the Hermitian symmetry of the complex Fourier coefficients. In practice, this means that for the fastest dimension x (along which a real-to-complex transform is performed), the negative wave numbers are dropped, i.e. k_x = 0 frac2πL_x frac4πL_x .","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The AbstractFFTs package provides a convenient way to generate the Fourier wave numbers, using the functions fftfreq and rfftfreq. We can use these functions to initialise a \"grid\" of wave numbers associated to our 3D real-to-complex transform:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"using AbstractFFTs: fftfreq, rfftfreq\n\nbox_size = (2π, 2π, 2π) # Lx, Ly, Lz\nsample_rate = 2π .* dims ./ box_size\n\n# In our case (Lx = 2π and Nx even), this gives kx = [0, 1, 2, ..., Nx/2].\nkx = rfftfreq(dims[1], sample_rate[1])\n\n# In our case (Ly = 2π and Ny even), this gives\n# ky = [0, 1, 2, ..., Ny/2-1, -Ny/2, -Ny/2+1, ..., -1] (and similarly for kz).\nky = fftfreq(dims[2], sample_rate[2])\nkz = fftfreq(dims[3], sample_rate[3])\n\nkvec = (kx, ky, kz)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that kvec now contains the wave numbers associated to the global domain. In the following, we will only need the wave numbers associated to the portion of the domain handled by the local MPI process.","category":"page"},{"location":"generated/gradient/#gradient_method_global","page":"Gradient of a scalar field","title":"Method 1: global views","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"PencilArrays, returned for instance by allocate_input and allocate_output, take indices that start at 1, regardless of the location of the subdomain associated to the local process on the global grid. (In other words, PencilArrays take local indices.) On the other hand, we have defined the wave number vector kvec which, for each MPI process, is defined over the global domain, and as such it takes global indices.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"One straightforward way of making data arrays compatible with wave numbers is to use global views, i.e. arrays that take global indices. These are generated from PencilArrays by calling the global_view function. Note that, in general, global indices do not start at 1 for a given MPI process. A given process will own a range of data given by indices in (i1:i2, j1:j2, k1:k2).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"θ_glob = global_view(θ_hat)\n∇θ_glob = global_view.(∇θ_hat)\nsummary(θ_glob)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Once we have global views, we can combine data and wave numbers using the portion of global indices owned by the local MPI process, as shown below. We can use CartesianIndices to iterate over the global indices associated to the local process.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"for I in CartesianIndices(θ_glob)\n i, j, k = Tuple(I) # unpack indices\n\n # Wave number vector associated to current Cartesian index.\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n # Compute gradient in Fourier space.\n # Note that modifying ∇θ_glob also modifies the original PencilArray ∇θ_hat.\n ∇θ_glob[1][I] = im * kx * θ_glob[I]\n ∇θ_glob[2][I] = im * ky * θ_glob[I]\n ∇θ_glob[3][I] = im * kz * θ_glob[I]\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The above loop can be written in a slightly more efficient manner by precomputing im * θ_glob[I]:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@inbounds for I in CartesianIndices(θ_glob)\n i, j, k = Tuple(I)\n\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n u = im * θ_glob[I]\n\n ∇θ_glob[1][I] = kx * u\n ∇θ_glob[2][I] = ky * u\n ∇θ_glob[3][I] = kz * u\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Also note that the above can be easily written in a more generic way, e.g. for arbitrary dimensions, thanks in part to the use of CartesianIndices. Moreover, in the above there is no notion of the dimension permutations discussed in the tutorial, as it is all hidden behind the implementation of PencilArrays. And as seen later in the benchmarks, these (hidden) permutations have zero cost, as the speed is identical to that of a function that explicitly takes into account these permutations.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, we can perform a backwards transform to obtain bm θ in physical space:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"∇θ = plan \\ ∇θ_hat;\nnothing #hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that the transform is automatically broadcast over the three fields of the ∇θ_hat vector, and the result ∇θ is also a tuple of three PencilArrays.","category":"page"},{"location":"generated/gradient/#gradient_method_global_explicit","page":"Gradient of a scalar field","title":"Method 2: explicit global indexing","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Sometimes, one does not need to write generic code. In our case, one often knows the dimensionality of the problem and the memory layout of the data (i.e. the underlying index permutation).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Below is a reimplementation of the above loop, using explicit indices instead of CartesianIndices, and assuming that the underlying index permutation is (3, 2, 1), that is, data is stored in (z y x) order. As discussed in the tutorial, this is the default for transformed arrays. This example also serves as a more explicit explanation for what is going on in the first method.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"# Get local data range in the global grid.\nrng = axes(θ_glob) # = (i1:i2, j1:j2, k1:k2)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"For the loop below, we're assuming that the permutation is (3, 2, 1). In other words, the fastest index is the last one, and not the first one as it is usually in Julia. If the permutation is not (3, 2, 1), things will still work (well, except for the assertion below!), but the loop order will not be optimal.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@assert permutation(θ_hat) === Permutation(3, 2, 1)\n\n@inbounds for i in rng[1], j in rng[2], k in rng[3]\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n # Note that we still access the arrays in (i, j, k) order.\n # (The permutation happens behind the scenes!)\n u = im * θ_glob[i, j, k]\n\n ∇θ_glob[1][i, j, k] = kx * u\n ∇θ_glob[2][i, j, k] = ky * u\n ∇θ_glob[3][i, j, k] = kz * u\nend","category":"page"},{"location":"generated/gradient/#gradient_method_local","page":"Gradient of a scalar field","title":"Method 3: using local indices","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Alternatively, we can avoid global views and work directly on PencilArrays using local indices that start at 1. In this case, part of the strategy is to construct a \"local\" grid of wave numbers that can also be accessed with local indices. This can be conveniently done using the localgrid function of the PencilArrays.jl package, which accepts a PencilArray (or its associated Pencil) and the global coordinates (here kvec):","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"grid_fourier = localgrid(θ_hat, kvec)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that one can directly iterate on the returned grid object:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@inbounds for I in CartesianIndices(grid_fourier)\n # Wave number vector associated to current Cartesian index.\n local k⃗ # hide\n k⃗ = grid_fourier[I]\n u = im * θ_hat[I]\n ∇θ_hat[1][I] = k⃗[1] * u\n ∇θ_hat[2][I] = k⃗[2] * u\n ∇θ_hat[3][I] = k⃗[3] * u\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This implementation is as efficient as the other examples, while being slightly shorter to write. Moreover, it is quite generic, and can be made independent of the number of dimensions with little effort.","category":"page"},{"location":"generated/gradient/#Method-4:-using-broadcasting","page":"Gradient of a scalar field","title":"Method 4: using broadcasting","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, note that the local grid object returned by localgrid makes it is possible to compute the gradient using broadcasting, thus fully avoiding scalar indexing. This can be quite convenient in some cases, and can also be very useful if one is working on GPUs (where scalar indexing is prohibitively expensive). Using broadcasting, the above examples simply become:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@. ∇θ_hat[1] = im * grid_fourier[1] * θ_hat\n@. ∇θ_hat[2] = im * grid_fourier[2] * θ_hat\n@. ∇θ_hat[3] = im * grid_fourier[3] * θ_hat\nnothing # hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Once again, as shown in the benchmarks further below, this method performs quite similarly to the other ones.","category":"page"},{"location":"generated/gradient/#Summary","page":"Gradient of a scalar field","title":"Summary","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The PencilArrays module provides different alternatives to deal with MPI-distributed data that may be subject to dimension permutations. In particular, one can choose to work with global indices (first two examples) or with local indices (third example).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"If one wants to stay generic, making sure that the same code will work for arbitrary dimensions and will be efficient regardless of the underlying dimension permutation, methods 1 and 3 should be preferred. These use CartesianIndices and make no assumptions on the permutations (actually, permutations are completely invisible in the implementations).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The second method uses explicit (i, j, k) indices. It assumes that the underlying permutation is (3, 2, 1) to loop with i as the slowest index and k as the fastest, which is the optimal order in this case given the permutation. As such, the implementation is less generic than the others, and differences in performance are negligible with respect to more generic variants.","category":"page"},{"location":"generated/gradient/#gradient_benchmarks","page":"Gradient of a scalar field","title":"Benchmark results","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The following are the benchmark results obtained from running examples/gradient.jl on a laptop, using 2 MPI processes and Julia 1.7.2, with an input array of global dimensions 64 32 64. The different methods detailed above are marked on the right. The \"lazy\" marks indicate runs where the wave numbers were represented by lazy Frequencies objects (returned by rfftfreq and fftfreq). Otherwise, they were collected into Vectors. For some reason, plain Vectors are faster when working with grids generated by localgrid.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In the script, additional implementations can be found which rely on a more advanced understanding of permutations and on the internals of the PencilArrays package. For instance, gradient_local_parent! directly works with the raw data stored in Julia Arrays, while gradient_local_linear! completely avoids CartesianIndices while staying generic and efficient. Nevertheless, these display roughly the same performance as the above examples.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":" gradient_global_view!... 89.900 μs\n gradient_global_view! (lazy)... 92.060 μs [Method 1]\n gradient_global_view_explicit!... 88.958 μs\n gradient_global_view_explicit! (lazy)... 81.055 μs [Method 2]\n gradient_local!... 92.305 μs\n gradient_grid!... 92.770 μs\n gradient_grid! (lazy)... 101.388 μs [Method 3]\n gradient_grid_broadcast!... 88.606 μs\n gradient_grid_broadcast! (lazy)... 151.020 μs [Method 4]\n gradient_local_parent!... 92.248 μs\n gradient_local_linear!... 91.212 μs\n gradient_local_linear_explicit!... 90.992 μs","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This page was generated using Literate.jl.","category":"page"},{"location":"PencilFFTs_timers/#PencilFFTs.measuring_performance","page":"Measuring performance","title":"Measuring performance","text":"","category":"section"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"It is possible to measure the time spent in different sections of the distributed transforms using the TimerOutputs package. This has a (very small) performance overhead, so it is disabled by default. To enable time measurements, call TimerOutputs.enable_debug_timings after loading PencilFFTs (see below for an example). For more details see the TimerOutputs docs.","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"Minimal example:","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"using MPI\nusing PencilFFTs\nusing TimerOutputs\n\n# Enable timing of `PencilFFTs` functions\nTimerOutputs.enable_debug_timings(PencilFFTs)\nTimerOutputs.enable_debug_timings(PencilArrays)\nTimerOutputs.enable_debug_timings(Transpositions)\n\nMPI.Init()\n\nplan = PencilFFTPlan(#= args... =#)\n\n# [do stuff with `plan`...]\n\n# Retrieve and print timing data associated to `plan`\nto = timer(plan)\nprint_timer(to)","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"By default, each PencilFFTPlan has its own TimerOutput. If you already have a TimerOutput, you can pass it to the PencilFFTPlan constructor:","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"to = TimerOutput()\nplan = PencilFFTPlan(..., timer=to)\n\n# [do stuff with `plan`...]\n\nprint_timer(to)","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"#PencilFFTs","page":"Home","title":"PencilFFTs","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Fast Fourier transforms of MPI-distributed Julia arrays.","category":"page"},{"location":"#Introduction","page":"Home","title":"Introduction","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package provides multidimensional FFTs and related transforms on MPI-distributed Julia arrays via the PencilArrays package.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The name of this package originates from the decomposition of 3D domains along two out of three dimensions, sometimes called pencil decomposition. This is illustrated by the figure below,[1] where each coloured block is managed by a different MPI process. Typically, one wants to compute FFTs on a scalar or vector field along the three spatial dimensions. In the case of a pencil decomposition, 3D FFTs are performed one dimension at a time, along the non-decomposed direction. Transforms must then be interleaved with global data transpositions to switch between pencil configurations. In high-performance computing environments, such data transpositions are generally the most expensive part of a parallel FFT computation, due to the large cost of communications between computing nodes.","category":"page"},{"location":"","page":"Home","title":"Home","text":"
    \n \n
    ","category":"page"},{"location":"","page":"Home","title":"Home","text":"More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension N. The decompositions can be performed along an arbitrary number M N of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.","category":"page"},{"location":"#Features","page":"Home","title":"Features","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;\nFFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;\nin-place and out-of-place transforms;\nhigh scalability up to (at least) tens of thousands of MPI processes.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"PencilFFTs can be installed using the Julia package manager:","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> ] add PencilFFTs","category":"page"},{"location":"#Similar-projects","page":"Home","title":"Similar projects","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.\nPFFT is a very general parallel FFT library written in C.\nP3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.\n2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[1]: Figure adapted from this PhD thesis.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[2]: For the pencil decomposition represented in the figure, N = 3 and M = 2.","category":"page"},{"location":"PencilFFTs/#Distributed-FFT-plans","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"Distributed FFTs are implemented in the PencilFFTs module, and are built on top of the PencilArrays package.","category":"page"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"PencilFFTs/#Creating-plans","page":"Distributed FFT plans","title":"Creating plans","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"PencilFFTPlan","category":"page"},{"location":"PencilFFTs/#PencilFFTs.PencilFFTPlan","page":"Distributed FFT plans","title":"PencilFFTs.PencilFFTPlan","text":"PencilFFTPlan{T,N} <: AbstractFFTs.Plan{T}\n\nPlan for N-dimensional FFT-based transform on MPI-distributed data, where input data has type T.\n\n\n\nPencilFFTPlan(\n A::PencilArray, transforms;\n fftw_flags = FFTW.ESTIMATE,\n fftw_timelimit = FFTW.NO_TIMELIMIT,\n permute_dims = Val(true),\n transpose_method = Transpositions.PointToPoint(),\n timer = TimerOutput(),\n)\n\nPencilFFTPlan(p::Pencil, transforms; kwargs...)\n\nCreate plan for N-dimensional transform on MPI-distributed PencilArrays.\n\nExtended help\n\nThis creates a PencilFFTPlan for arrays sharing the same properties as A (dimensions, MPI decomposition, memory layout, ...), which describe data on an N-dimensional domain.\n\nAlternatively, the second form creates a PencilFFTPlan for distributed arrays following a given Pencil configuration.\n\nTransforms\n\nThe transforms to be applied along each dimension are specified by the transforms argument. Possible transforms are defined as subtypes of Transforms.AbstractTransform, and are listed in Transform types. This argument may be either:\n\na tuple of N transforms to be applied along each dimension. For instance, transforms = (Transforms.R2R(FFTW.REDFT01), Transforms.RFFT(), Transforms.FFT());\na single transform to be applied along all dimensions. The input is automatically expanded into N equivalent transforms. For instance, for a three-dimensional array, transforms = Transforms.RFFT() specifies a 3D real-to-complex transform, and is equivalent to passing (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT()).\n\nNote that forward transforms are applied from left to right. In the last example, this means that a real-to-complex transform (RFFT) is first performed along the first dimension. This is followed by complex-to-complex transforms (FFT) along the second and third dimensions.\n\nInput data layout\n\nThe input PencilArray must satisfy the following constraints:\n\narray dimensions must not be permuted. This is the default when constructing PencilArrays.\nfor an M-dimensional domain decomposition (with M < N), the input array must be decomposed along the last M dimensions. For example, for a 2D decomposition of 3D data, the decomposed dimensions must be (2, 3). In particular, the first array dimension must not be distributed among different MPI processes.\nIn the PencilArrays package, the decomposed dimensions are specified at the moment of constructing a Pencil.\nthe element type must be compatible with the specified transform. For instance, real-to-complex transforms (Transforms.RFFT) require the input to be real floating point values. Other transforms, such as Transforms.R2R, accept both real and complex data.\n\nKeyword arguments\n\nThe keyword arguments fftw_flags and fftw_timelimit are passed to the FFTW plan creation functions (see AbstractFFTs docs).\npermute_dims determines whether the indices of the output data should be reversed. For instance, if the input data has global dimensions (Nx, Ny, Nz), then the output of a complex-to-complex FFT would have dimensions (Nz, Ny, Nx). This enables FFTs to always be performed along the first (i.e. fastest) array dimension, which could lead to performance gains. This option is enabled by default. For type inference reasons, it must be a value type (Val(true) or Val(false)).\ntranspose_method allows to select between implementations of the global data transpositions. See PencilArrays docs docs for details.\ntimer should be a TimerOutput object. See Measuring performance for details.\n\n\n\nPencilFFTPlan(\n size_global::Dims{N}, transforms, proc_dims::Dims{M}, comm::MPI.Comm,\n [real_type = Float64]; extra_dims = (), kws...\n)\n\nCreate plan for N-dimensional transform.\n\nExtended help\n\nInstead of taking a PencilArray or a Pencil, this constructor requires the global dimensions of the input data, passed via the size_global argument.\n\nThe data is distributed over the MPI processes in the comm communicator. The distribution is performed over M dimensions (with M < N) according to the values in proc_dims, which specifies the number of MPI processes to put along each dimension.\n\nPencilArrays that may be transformed with the returned plan can be created using allocate_input.\n\nOptional arguments\n\nThe floating point precision can be selected by setting real_type parameter, which is Float64 by default.\nextra_dims may be used to specify the sizes of one or more extra dimensions that should not be transformed. These dimensions will be added to the rightmost (i.e. slowest) indices of the arrays. See Extra dimensions below for usage hints.\nsee the other constructor for more keyword arguments.\n\nExtra dimensions\n\nOne possible application of extra_dims is for describing the components of a vector or tensor field. However, this means that different PencilFFTPlans would need to be created for each kind of field (scalar, vector, ...). To avoid the creation of multiple plans, a possibly better alternative is to create tuples (or arrays) of PencilArrays using allocate_input and allocate_output.\n\nAnother more legitimate usage of extra_dims is to specify one or more Cartesian dimensions that should not be transformed nor split among MPI processes.\n\nExample\n\nSuppose we want to perform a 3D FFT of real data. The data is to be decomposed along two dimensions, over 8 MPI processes:\n\nsize_global = (64, 32, 128) # size of real input data\n\n# Perform real-to-complex transform along the first dimension, then\n# complex-to-complex transforms along the other dimensions.\ntransforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())\n# transforms = Transforms.RFFT() # this is equivalent to the above line\n\nproc_dims = (4, 2) # 2D decomposition\ncomm = MPI.COMM_WORLD\n\nplan = PencilFFTPlan(size_global, transforms, proc_dims, comm)\n\n\n\n\n\n","category":"type"},{"location":"PencilFFTs/#Allocating-data","page":"Distributed FFT plans","title":"Allocating data","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"allocate_input\nallocate_output","category":"page"},{"location":"PencilFFTs/#PencilFFTs.allocate_input","page":"Distributed FFT plans","title":"PencilFFTs.allocate_input","text":"allocate_input(p::PencilFFTPlan) -> PencilArray\nallocate_input(p::PencilFFTPlan, dims...) -> Array{PencilArray}\nallocate_input(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}\n\nAllocate uninitialised PencilArray that can hold input data for the given plan.\n\nThe second and third forms respectively allocate an array of PencilArrays of size dims, and a tuple of N PencilArrays.\n\nnote: In-place plans\nIf p is an in-place plan, a ManyPencilArray is allocated. This type holds PencilArray wrappers for the input and output transforms (as well as for intermediate transforms) which share the same space in memory. The input and output PencilArrays should be respectively accessed by calling first(::ManyPencilArray) and last(::ManyPencilArray).ExampleSuppose p is an in-place PencilFFTPlan. Then,@assert is_inplace(p)\nA = allocate_input(p) :: ManyPencilArray\nv_in = first(A) :: PencilArray # input data view\nv_out = last(A) :: PencilArray # output data viewAlso note that in-place plans must be performed directly on the returned ManyPencilArray, and not on the contained PencilArray views:p * A # perform forward transform in-place\np \\ A # perform backward transform in-place\n# p * v_in # not allowed!!\n\n\n\n\n\n","category":"function"},{"location":"PencilFFTs/#PencilFFTs.allocate_output","page":"Distributed FFT plans","title":"PencilFFTs.allocate_output","text":"allocate_output(p::PencilFFTPlan) -> PencilArray\nallocate_output(p::PencilFFTPlan, dims...) -> Array{PencilArray}\nallocate_output(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}\n\nAllocate uninitialised PencilArray that can hold output data for the given plan.\n\nIf p is an in-place plan, a ManyPencilArray is allocated.\n\nSee allocate_input for details.\n\n\n\n\n\n","category":"function"},{"location":"PencilFFTs/#Methods","page":"Distributed FFT plans","title":"Methods","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"get_comm(::PencilFFTPlan)\nscale_factor(::PencilFFTPlan)\ntimer(::PencilFFTPlan)\nis_inplace(::PencilFFTPlan)","category":"page"},{"location":"PencilFFTs/#PencilArrays.Pencils.MPITopologies.get_comm-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilArrays.Pencils.MPITopologies.get_comm","text":"get_comm(p::PencilFFTPlan)\n\nGet MPI communicator associated to a PencilFFTPlan.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilFFTs.Transforms.scale_factor-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilFFTs.Transforms.scale_factor","text":"scale_factor(p::PencilFFTPlan)\n\nGet scale factor associated to a PencilFFTPlan.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilArrays.Pencils.timer-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilArrays.Pencils.timer","text":"timer(p::PencilFFTPlan)\n\nGet TimerOutput attached to a PencilFFTPlan.\n\nSee Measuring performance for details.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilFFTs.Transforms.is_inplace-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilFFTs.Transforms.is_inplace","text":"Transforms.is_inplace(p::PencilFFTPlan)\n\nReturns true if the given plan operates in-place on the input data, false otherwise.\n\n\n\n\n\n","category":"method"}] -} diff --git a/previews/PR39/siteinfo.js b/previews/PR39/siteinfo.js deleted file mode 100644 index c08dc8b2..00000000 --- a/previews/PR39/siteinfo.js +++ /dev/null @@ -1 +0,0 @@ -var DOCUMENTER_CURRENT_VERSION = "previews/PR39"; diff --git a/previews/PR40/GlobalFFTParams/index.html b/previews/PR40/GlobalFFTParams/index.html deleted file mode 100644 index 6cb2e710..00000000 --- a/previews/PR40/GlobalFFTParams/index.html +++ /dev/null @@ -1,9 +0,0 @@ - -Global FFT parameters · PencilFFTs.jl
    Edit on GitHub

    Global FFT parameters

    PencilFFTs.GlobalFFTParams — Type
    GlobalFFTParams{T, N, inplace}

    Specifies the global parameters for an N-dimensional distributed transform. These include the element type T and global data sizes of input and output data, as well as the transform types to be performed along each dimension.

    GlobalFFTParams(size_global, transforms, [real_type=Float64])

    Define parameters for N-dimensional transform.

    transforms must be a tuple of length N specifying the transforms to be applied along each dimension. Each element must be a subtype of Transforms.AbstractTransform. For all the possible transforms, see Transform types.

    The element type must be a real type accepted by FFTW, i.e. either Float32 or Float64.

    Note that the transforms are applied one dimension at a time, with the leftmost dimension first for forward transforms.

    Example

    To perform a 3D FFT of real data, first a real-to-complex FFT must be applied along the first dimension, followed by two complex-to-complex FFTs along the other dimensions:

    julia> size_global = (64, 32, 128);  # size of real input data
-
-julia> transforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT());
-
-julia> fft_params = PencilFFTs.GlobalFFTParams(size_global, transforms)
-Transforms: (RFFT, FFT, FFT)
-Input type: Float64
-Global dimensions: (64, 32, 128) -> (33, 32, 128)
    source
    diff --git a/previews/PR40/PencilFFTs/index.html b/previews/PR40/PencilFFTs/index.html deleted file mode 100644 index 27093f76..00000000 --- a/previews/PR40/PencilFFTs/index.html +++ /dev/null @@ -1,31 +0,0 @@ - -Distributed FFT plans · PencilFFTs.jl
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    Distributed FFT plans

    Distributed FFTs are implemented in the PencilFFTs module, and are built on top of the PencilArrays package.

    Creating plans

    PencilFFTs.PencilFFTPlan — Type
    PencilFFTPlan{T,N} <: AbstractFFTs.Plan{T}

    Plan for N-dimensional FFT-based transform on MPI-distributed data, where input data has type T.

    PencilFFTPlan(p::Pencil, transforms; kwargs...)

    Create a PencilFFTPlan for distributed arrays following a given Pencil configuration. See variant below for details on the specification of transforms and on possible keyword arguments.

    PencilFFTPlan(
-    A::PencilArray, transforms;
-    fftw_flags = FFTW.ESTIMATE,
-    fftw_timelimit = FFTW.NO_TIMELIMIT,
-    permute_dims = Val(true),
-    transpose_method = Transpositions.PointToPoint(),
-    timer = TimerOutput(),
-)

    Create plan for N-dimensional transform on MPI-distributed PencilArrays.

    Extended help

    This creates a PencilFFTPlan for arrays sharing the same properties as A (dimensions, MPI decomposition, memory layout, ...), which describe data on an N-dimensional domain.

    Transforms

    The transforms to be applied along each dimension are specified by the transforms argument. Possible transforms are defined as subtypes of Transforms.AbstractTransform, and are listed in Transform types. This argument may be either:

    • a tuple of N transforms to be applied along each dimension. For instance, transforms = (Transforms.R2R(FFTW.REDFT01), Transforms.RFFT(), Transforms.FFT());

    • a single transform to be applied along all dimensions. The input is automatically expanded into N equivalent transforms. For instance, for a three-dimensional array, transforms = Transforms.RFFT() specifies a 3D real-to-complex transform, and is equivalent to passing (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT()).

    Note that forward transforms are applied from left to right. In the last example, this means that a real-to-complex transform (RFFT) is first performed along the first dimension. This is followed by complex-to-complex transforms (FFT) along the second and third dimensions.

    Input data layout

    The input PencilArray must satisfy the following constraints:

    • array dimensions must not be permuted. This is the default when constructing PencilArrays.

    • for an M-dimensional domain decomposition (with M < N), the input array must be decomposed along the last M dimensions. For example, for a 2D decomposition of 3D data, the decomposed dimensions must be (2, 3). In particular, the first array dimension must not be distributed among different MPI processes.

      In the PencilArrays package, the decomposed dimensions are specified at the moment of constructing a Pencil.

    • the element type must be compatible with the specified transform. For instance, real-to-complex transforms (Transforms.RFFT) require the input to be real floating point values. Other transforms, such as Transforms.R2R, accept both real and complex data.

    Keyword arguments

    • The keyword arguments fftw_flags and fftw_timelimit are passed to the FFTW plan creation functions (see AbstractFFTs docs).

    • permute_dims determines whether the indices of the output data should be reversed. For instance, if the input data has global dimensions (Nx, Ny, Nz), then the output of a complex-to-complex FFT would have dimensions (Nz, Ny, Nx). This enables FFTs to always be performed along the first (i.e. fastest) array dimension, which could lead to performance gains. This option is enabled by default. For type inference reasons, it must be a value type (Val(true) or Val(false)).

    • transpose_method allows to select between implementations of the global data transpositions. See PencilArrays docs docs for details.

    • timer should be a TimerOutput object. See Measuring performance for details.

    PencilFFTPlan(
-    dims_global::Dims{N}, transforms, proc_dims::Dims{M}, comm::MPI.Comm,
-    [real_type = Float64]; extra_dims = (), kws...
-)

    Create plan for N-dimensional transform.

    Extended help

    Instead of taking a PencilArray or a Pencil, this constructor requires the global dimensions of the input data, passed via the size_global argument.

    The data is distributed over the MPI processes in the comm communicator. The distribution is performed over M dimensions (with M < N) according to the values in proc_dims, which specifies the number of MPI processes to put along each dimension.

    PencilArrays that may be transformed with the returned plan can be created using allocate_input.

    Optional arguments

    • The floating point precision can be selected by setting real_type parameter, which is Float64 by default.

    • extra_dims may be used to specify the sizes of one or more extra dimensions that should not be transformed. These dimensions will be added to the rightmost (i.e. slowest) indices of the arrays. See Extra dimensions below for usage hints.

    • see the other constructor for more keyword arguments.

    Extra dimensions

    One possible application of extra_dims is for describing the components of a vector or tensor field. However, this means that different PencilFFTPlans would need to be created for each kind of field (scalar, vector, ...). To avoid the creation of multiple plans, a possibly better alternative is to create tuples (or arrays) of PencilArrays using allocate_input and allocate_output.

    Another more legitimate usage of extra_dims is to specify one or more Cartesian dimensions that should not be transformed nor split among MPI processes.

    Example

    Suppose we want to perform a 3D FFT of real data. The data is to be decomposed along two dimensions, over 8 MPI processes:

    size_global = (64, 32, 128)  # size of real input data
-
-# Perform real-to-complex transform along the first dimension, then
-# complex-to-complex transforms along the other dimensions.
-transforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())
-# transforms = Transforms.RFFT()  # this is equivalent to the above line
-
-proc_dims = (4, 2)  # 2D decomposition
-comm = MPI.COMM_WORLD
-
-plan = PencilFFTPlan(size_global, transforms, proc_dims, comm)
    source

    Allocating data

    PencilFFTs.allocate_input — Function
    allocate_input(p::PencilFFTPlan)          -> PencilArray
-allocate_input(p::PencilFFTPlan, dims...) -> Array{PencilArray}
-allocate_input(p::PencilFFTPlan, Val(N))  -> NTuple{N, PencilArray}

    Allocate uninitialised PencilArray that can hold input data for the given plan.

    The second and third forms respectively allocate an array of PencilArrays of size dims, and a tuple of N PencilArrays.

    In-place plans

    If p is an in-place plan, a ManyPencilArray is allocated. This type holds PencilArray wrappers for the input and output transforms (as well as for intermediate transforms) which share the same space in memory. The input and output PencilArrays should be respectively accessed by calling first(::ManyPencilArray) and last(::ManyPencilArray).

    Example

    Suppose p is an in-place PencilFFTPlan. Then,

    @assert is_inplace(p)
-A = allocate_input(p) :: ManyPencilArray
-v_in = first(A)       :: PencilArray  # input data view
-v_out = last(A)       :: PencilArray  # output data view

    Also note that in-place plans must be performed directly on the returned ManyPencilArray, and not on the contained PencilArray views:

    p * A       # perform forward transform in-place
-p \ A       # perform backward transform in-place
-# p * v_in  # not allowed!!
    source
    PencilFFTs.allocate_output — Function
    allocate_output(p::PencilFFTPlan)          -> PencilArray
-allocate_output(p::PencilFFTPlan, dims...) -> Array{PencilArray}
-allocate_output(p::PencilFFTPlan, Val(N))  -> NTuple{N, PencilArray}

    Allocate uninitialised PencilArray that can hold output data for the given plan.

    If p is an in-place plan, a ManyPencilArray is allocated.

    See allocate_input for details.

    source

    Methods

    diff --git a/previews/PR40/PencilFFTs_timers/index.html b/previews/PR40/PencilFFTs_timers/index.html deleted file mode 100644 index b677bcbd..00000000 --- a/previews/PR40/PencilFFTs_timers/index.html +++ /dev/null @@ -1,24 +0,0 @@ - -Measuring performance · PencilFFTs.jl
    Edit on GitHub

    Measuring performance

    It is possible to measure the time spent in different sections of the distributed transforms using the TimerOutputs package. This has a (very small) performance overhead, so it is disabled by default. To enable time measurements, call TimerOutputs.enable_debug_timings after loading PencilFFTs (see below for an example). For more details see the TimerOutputs docs.

    Minimal example:

    using MPI
-using PencilFFTs
-using TimerOutputs
-
-# Enable timing of `PencilFFTs` functions
-TimerOutputs.enable_debug_timings(PencilFFTs)
-TimerOutputs.enable_debug_timings(PencilArrays)
-TimerOutputs.enable_debug_timings(Transpositions)
-
-MPI.Init()
-
-plan = PencilFFTPlan(#= args... =#)
-
-# [do stuff with `plan`...]
-
-# Retrieve and print timing data associated to `plan`
-to = timer(plan)
-print_timer(to)

    By default, each PencilFFTPlan has its own TimerOutput. If you already have a TimerOutput, you can pass it to the PencilFFTPlan constructor:

    to = TimerOutput()
-plan = PencilFFTPlan(..., timer=to)
-
-# [do stuff with `plan`...]
-
-print_timer(to)
    diff --git a/previews/PR40/Transforms/index.html b/previews/PR40/Transforms/index.html deleted file mode 100644 index ee20610c..00000000 --- a/previews/PR40/Transforms/index.html +++ /dev/null @@ -1,74 +0,0 @@ - -Available transforms · PencilFFTs.jl
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    Available transforms

    Transform types

    PencilFFTs.Transforms.BFFT — Type
    BFFT()

    Unnormalised backward complex-to-complex FFT.

    Like AbstractFFTs.bfft, this transform is not normalised. To obtain the inverse transform, divide the output by the length of the transformed dimension.

    See also AbstractFFTs.bfft.

    source
    PencilFFTs.Transforms.BRFFT — Type
    BRFFT(d::Integer)
-BRFFT((d1, d2, ..., dN))

    Unnormalised inverse of RFFT.

    To obtain the inverse transform, divide the output by the length of the transformed dimension (of the real output array).

    As described in the AbstractFFTs docs, the length of the output cannot be fully inferred from the input length. For this reason, the BRFFT constructor accepts an optional d argument indicating the output length.

    For multidimensional datasets, a tuple of dimensions (d1, d2, ..., dN) may also be passed. This is equivalent to passing just dN. In this case, the last dimension (dN) is the one that changes size between the input and output. Note that this is the opposite of FFTW.brfft. The reason is that, in PencilFFTs, the last dimension is the one along which a complex-to-real transform is performed.

    See also AbstractFFTs.brfft.

    source

    Internals

    What follows is used internally in PencilFFTs.

    Types

    PencilFFTs.Transforms.AbstractCustomPlan — Type
    AbstractCustomPlan

    Abstract type defining a custom plan, to be used as an alternative to FFTW plans (FFTW.FFTWPlan).

    The only custom plan defined in this module is IdentityPlan. The user can define other custom plans that are also subtypes of AbstractCustomPlan.

    Note that plan returns a subtype of either FFTW.FFTWPlan or AbstractCustomPlan.

    source

    Functions

    PencilFFTs.Transforms.plan — Function
    plan(transform::AbstractTransform, A, [dims];
-     flags=FFTW.ESTIMATE, timelimit=Inf)

    Create plan to transform array A along dimensions dims.

    If dims is not specified, all dimensions of A are transformed.

    For FFT plans, this function wraps the AbstractFFTs.jl and FFTW.jl plan creation functions. For more details on the function arguments, see AbstractFFTs.plan_fft.

    source
    PencilFFTs.Transforms.binv — Function
    binv(transform::AbstractTransform, d::Integer)

    Returns the backwards transform associated to the given transform.

    The second argument must be the length of the first transformed dimension in the forward transform. It is used in particular when transform = RFFT(), to determine the length of the inverse (complex-to-real) transform. See the AbstractFFTs.irfft docs for details.

    The backwards transform returned by this function is not normalised. The normalisation factor for a given array can be obtained by calling scale_factor.

    Example

    julia> binv(Transforms.FFT(), 42)
-BFFT
-
-julia> binv(Transforms.BRFFT(9), 42)
-RFFT
    source
    PencilFFTs.Transforms.scale_factor — Function
    scale_factor(transform::AbstractTransform, A, [dims = 1:ndims(A)])

    Get factor required to normalise the given array after a transformation along dimensions dims (all dimensions by default).

    The array A must have the dimensions of the transform input.

    Important: the dimensions dims must be the same that were passed to plan.

    Examples

    julia> C = zeros(ComplexF32, 3, 4, 5);
-
-julia> scale_factor(Transforms.FFT(), C)
-60
-
-julia> scale_factor(Transforms.BFFT(), C)
-60
-
-julia> scale_factor(Transforms.BFFT(), C, 2:3)
-20
-
-julia> R = zeros(Float64, 3, 4, 5);
-
-julia> scale_factor(Transforms.RFFT(), R, 2)
-4
-
-julia> scale_factor(Transforms.RFFT(), R, 2:3)
-20
-
-julia> scale_factor(Transforms.BRFFT(8), C)
-96
-
-julia> scale_factor(Transforms.BRFFT(9), C)
-108

    This will fail because the input of RFFT is real, and R is a complex array:

    julia> scale_factor(Transforms.RFFT(), C, 2:3)
-ERROR: MethodError: no method matching scale_factor(::PencilFFTs.Transforms.RFFT, ::Array{ComplexF32, 3}, ::UnitRange{Int64})
    source
    PencilFFTs.Transforms.eltype_input — Function
    eltype_input(transform::AbstractTransform, real_type<:AbstractFloat)

    Determine input data type for a given transform given the floating point precision of the input data.

    Some transforms, such as R2R and NoTransform, can take both real and complex data. For those kinds of transforms, nothing is returned.

    Example

    julia> eltype_input(Transforms.FFT(), Float32)
-ComplexF32 (alias for Complex{Float32})
-
-julia> eltype_input(Transforms.RFFT(), Float64)
-Float64
-
-julia> eltype_input(Transforms.R2R(FFTW.REDFT01), Float64)  # nothing
-
-julia> eltype_input(Transforms.NoTransform(), Float64)  # nothing
-
    source
    PencilFFTs.Transforms.eltype_output — Function
    eltype_output(transform::AbstractTransform, eltype_input)

    Returns the output data type for a given transform given the input type.

    Throws ArgumentError if the input data type is incompatible with the transform type.

    Example

    julia> eltype_output(Transforms.NoTransform(), Float32)
-Float32
-
-julia> eltype_output(Transforms.RFFT(), Float64)
-ComplexF64 (alias for Complex{Float64})
-
-julia> eltype_output(Transforms.BRFFT(4), ComplexF32)
-Float32
-
-julia> eltype_output(Transforms.FFT(), Float64)
-ERROR: ArgumentError: invalid input data type for PencilFFTs.Transforms.FFT: Float64
    source
    PencilFFTs.Transforms.expand_dims — Function
    expand_dims(transform::AbstractTransform, Val(N))

    Expand a single multidimensional transform into one transform per dimension.

    Example

    # Expand a real-to-complex transform in 3 dimensions.
-julia> expand_dims(Transforms.RFFT(), Val(3))
-(RFFT, FFT, FFT)
-
-julia> expand_dims(Transforms.BRFFT(4), Val(3))
-(BFFT, BFFT, BRFFT{even})
-
-julia> expand_dims(Transforms.NoTransform(), Val(2))
-(NoTransform, NoTransform)
    source
    PencilFFTs.Transforms.is_inplace — Function
    is_inplace(transform::AbstractTransform)         -> Bool
-is_inplace(transforms::Vararg{AbtractTransform}) -> Union{Bool, Nothing}

    Check whether a transform or a list of transforms is performed in-place.

    If the list of transforms has a combination of in-place and out-of-place transforms, nothing is returned.

    Example

    julia> is_inplace(Transforms.RFFT())
-false
-
-julia> is_inplace(Transforms.NoTransform!())
-true
-
-julia> is_inplace(Transforms.FFT!(), Transforms.R2R!(FFTW.REDFT01))
-true
-
-julia> is_inplace(Transforms.FFT(), Transforms.R2R(FFTW.REDFT01))
-false
-
-julia> is_inplace(Transforms.FFT(), Transforms.R2R!(FFTW.REDFT01)) === nothing
-true
-
    source
    PencilFFTs.Transforms.length_output — Function
    length_output(transform::AbstractTransform, length_in::Integer)

    Returns the length of the transform output, given the length of its input.

    The input and output lengths are specified in terms of the respective input and output datatypes. For instance, for real-to-complex transforms, these are respectively the length of input real data and of output complex data.

    source
    diff --git a/previews/PR40/assets/documenter.js b/previews/PR40/assets/documenter.js deleted file mode 100644 index a1ada46f..00000000 --- a/previews/PR40/assets/documenter.js +++ /dev/null @@ -1,325 +0,0 @@ -// Generated by Documenter.jl -requirejs.config({ - paths: { - 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.0.1/languages/julia.min', - 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', - 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.12.1/jquery-ui.min', - 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.11/contrib/auto-render.min', - 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.6.0/jquery.min', - 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', - 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.11/katex.min', - 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.0.1/highlight.min', - 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.0.1/languages/julia-repl.min', - }, - shim: { - "highlight-julia": { - "deps": [ - "highlight" - ] - }, - "katex-auto-render": { - "deps": [ - "katex" - ] - }, - "headroom-jquery": { - "deps": [ - "jquery", - "headroom" - ] - }, - "highlight-julia-repl": { - "deps": [ - "highlight" - ] - } -} -}); -//////////////////////////////////////////////////////////////////////////////// -require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) { -$(document).ready(function() { - renderMathInElement( - document.body, - { - "delimiters": [ - { - "left": "$", - "right": "$", - "display": false - }, - { - "left": "$$", - "right": "$$", - "display": true - }, - { - "left": "\\[", - "right": "\\]", - "display": true - } - ] -} - - ); -}) - -}) -//////////////////////////////////////////////////////////////////////////////// -require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) { -$(document).ready(function() { - hljs.highlightAll(); -}) - -}) -//////////////////////////////////////////////////////////////////////////////// -require([], function() { -function addCopyButtonCallbacks() { - for (const el of document.getElementsByTagName("pre")) { - const button = document.createElement("button"); - button.classList.add("copy-button", "fas", "fa-copy"); - el.appendChild(button); - - const success = function () { - button.classList.add("success", "fa-check"); - button.classList.remove("fa-copy"); - }; - - const failure = function () { - button.classList.add("error", "fa-times"); - button.classList.remove("fa-copy"); - }; - - button.addEventListener("click", function () { - copyToClipboard(el.innerText).then(success, failure); - - setTimeout(function () { - button.classList.add("fa-copy"); - button.classList.remove("success", "fa-check", "fa-times"); - }, 5000); - }); - } -} - -function copyToClipboard(text) { - // clipboard API is only available in secure contexts - if (window.navigator && window.navigator.clipboard) { - return window.navigator.clipboard.writeText(text); - } else { - return new Promise(function (resolve, reject) { - try { - const el = document.createElement("textarea"); - el.textContent = text; - el.style.position = "fixed"; - el.style.opacity = 0; - document.body.appendChild(el); - el.select(); - document.execCommand("copy"); - - resolve(); - } catch (err) { - reject(err); - } finally { - document.body.removeChild(el); - } - }); - } -} - -if (document.readyState === "loading") { - document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks); -} else { - addCopyButtonCallbacks(); -} - -}) -//////////////////////////////////////////////////////////////////////////////// -require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { - -// Manages the top navigation bar (hides it when the user starts scrolling down on the -// mobile). -window.Headroom = Headroom; // work around buggy module loading? -$(document).ready(function() { - $('#documenter .docs-navbar').headroom({ - "tolerance": {"up": 10, "down": 10}, - }); -}) - -}) -//////////////////////////////////////////////////////////////////////////////// -require(['jquery'], function($) { - -// Modal settings dialog -$(document).ready(function() { - var settings = $('#documenter-settings'); - $('#documenter-settings-button').click(function(){ - settings.toggleClass('is-active'); - }); - // Close the dialog if X is clicked - $('#documenter-settings button.delete').click(function(){ - settings.removeClass('is-active'); - }); - // Close dialog if ESC is pressed - $(document).keyup(function(e) { - if (e.keyCode == 27) settings.removeClass('is-active'); - }); -}); - -}) -//////////////////////////////////////////////////////////////////////////////// -require(['jquery'], function($) { - -// Manages the showing and hiding of the sidebar. -$(document).ready(function() { - var sidebar = $("#documenter > .docs-sidebar"); - var sidebar_button = $("#documenter-sidebar-button") - sidebar_button.click(function(ev) { - ev.preventDefault(); - sidebar.toggleClass('visible'); - if (sidebar.hasClass('visible')) { - // Makes sure that the current menu item is visible in the sidebar. - $("#documenter .docs-menu a.is-active").focus(); - } - }); - $("#documenter > .docs-main").bind('click', function(ev) { - if ($(ev.target).is(sidebar_button)) { - return; - } - if (sidebar.hasClass('visible')) { - sidebar.removeClass('visible'); - } - }); -}) - -// Resizes the package name / sitename in the sidebar if it is too wide. -// Inspired by: https://github.com/davatron5000/FitText.js -$(document).ready(function() { - e = $("#documenter .docs-autofit"); - function resize() { - var L = parseInt(e.css('max-width'), 10); - var L0 = e.width(); - if(L0 > L) { - var h0 = parseInt(e.css('font-size'), 10); - e.css('font-size', L * h0 / L0); - // TODO: make sure it survives resizes? - } - } - // call once and then register events - resize(); - $(window).resize(resize); - $(window).on('orientationchange', resize); -}); - -// Scroll the navigation bar to the currently selected menu item -$(document).ready(function() { - var sidebar = $("#documenter .docs-menu").get(0); - var active = $("#documenter .docs-menu .is-active").get(0); - if(typeof active !== 'undefined') { - sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; - } -}) - -}) -//////////////////////////////////////////////////////////////////////////////// -require(['jquery'], function($) { - -function set_theme(theme) { - var active = null; - var disabled = []; - for (var i = 0; i < document.styleSheets.length; i++) { - var ss = document.styleSheets[i]; - var themename = ss.ownerNode.getAttribute("data-theme-name"); - if(themename === null) continue; // ignore non-theme stylesheets - // Find the active theme - if(themename === theme) active = ss; - else disabled.push(ss); - } - if(active !== null) { - active.disabled = false; - if(active.ownerNode.getAttribute("data-theme-primary") === null) { - document.getElementsByTagName('html')[0].className = "theme--" + theme; - } else { - document.getElementsByTagName('html')[0].className = ""; - } - disabled.forEach(function(ss){ - ss.disabled = true; - }); - } - - // Store the theme in localStorage - if(typeof(window.localStorage) !== "undefined") { - window.localStorage.setItem("documenter-theme", theme); - } else { - console.error("Browser does not support window.localStorage"); - } -} - -// Theme picker setup -$(document).ready(function() { - // onchange callback - $('#documenter-themepicker').change(function themepick_callback(ev){ - var themename = $('#documenter-themepicker option:selected').attr('value'); - set_theme(themename); - }); - - // Make sure that the themepicker displays the correct theme when the theme is retrieved - // from localStorage - if(typeof(window.localStorage) !== "undefined") { - var theme = window.localStorage.getItem("documenter-theme"); - if(theme !== null) { - $('#documenter-themepicker option').each(function(i,e) { - e.selected = (e.value === theme); - }) - } else { - $('#documenter-themepicker option').each(function(i,e) { - e.selected = $("html").hasClass(`theme--${e.value}`); - }) - } - } -}) - -}) -//////////////////////////////////////////////////////////////////////////////// -require(['jquery'], function($) { - -// update the version selector with info from the siteinfo.js and ../versions.js files -$(document).ready(function() { - var version_selector = $("#documenter .docs-version-selector"); - var version_selector_select = $("#documenter .docs-version-selector select"); - - version_selector_select.change(function(x) { - target_href = version_selector_select.children("option:selected").get(0).value; - window.location.href = target_href; - }); - - // add the current version to the selector based on siteinfo.js, but only if the selector is empty - if (typeof DOCUMENTER_CURRENT_VERSION !== 'undefined' && $('#version-selector > option').length == 0) { - var option = $(""); - version_selector_select.append(option); - } - - if (typeof DOC_VERSIONS !== 'undefined') { - var existing_versions = version_selector_select.children("option"); - var existing_versions_texts = existing_versions.map(function(i,x){return x.text}); - DOC_VERSIONS.forEach(function(each) { - var version_url = documenterBaseURL + "/../" + each; - var existing_id = $.inArray(each, existing_versions_texts); - // if not already in the version selector, add it as a new option, - // otherwise update the old option with the URL and enable it - if (existing_id == -1) { - var option = $(""); - version_selector_select.append(option); - } else { - var option = existing_versions[existing_id]; - option.value = version_url; - option.disabled = false; - } - }); - } - - // only show the version selector if the selector has been populated - if (version_selector_select.children("option").length > 0) { - version_selector.toggleClass("visible"); - } -}) - -}) diff --git a/previews/PR40/assets/search.js b/previews/PR40/assets/search.js deleted file mode 100644 index 1a514547..00000000 --- a/previews/PR40/assets/search.js +++ /dev/null @@ -1,251 +0,0 @@ -// Generated by Documenter.jl -requirejs.config({ - paths: { - 'lunr': 'https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min', - 'lodash': 'https://cdnjs.cloudflare.com/ajax/libs/lodash.js/4.17.21/lodash.min', - 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.6.0/jquery.min', - } -}); -//////////////////////////////////////////////////////////////////////////////// -require(['jquery', 'lunr', 'lodash'], function($, lunr, _) { - -$(document).ready(function() { - // parseUri 1.2.2 - // (c) Steven Levithan - // MIT License - function parseUri (str) { - var o = parseUri.options, - m = o.parser[o.strictMode ? "strict" : "loose"].exec(str), - uri = {}, - i = 14; - - while (i--) uri[o.key[i]] = m[i] || ""; - - uri[o.q.name] = {}; - uri[o.key[12]].replace(o.q.parser, function ($0, $1, $2) { - if ($1) uri[o.q.name][$1] = $2; - }); - - return uri; - }; - parseUri.options = { - strictMode: false, - key: ["source","protocol","authority","userInfo","user","password","host","port","relative","path","directory","file","query","anchor"], - q: { - name: "queryKey", - parser: /(?:^|&)([^&=]*)=?([^&]*)/g - }, - parser: { - strict: /^(?:([^:\/?#]+):)?(?:\/\/((?:(([^:@]*)(?::([^:@]*))?)?@)?([^:\/?#]*)(?::(\d*))?))?((((?:[^?#\/]*\/)*)([^?#]*))(?:\?([^#]*))?(?:#(.*))?)/, - loose: /^(?:(?![^:@]+:[^:@\/]*@)([^:\/?#.]+):)?(?:\/\/)?((?:(([^:@]*)(?::([^:@]*))?)?@)?([^:\/?#]*)(?::(\d*))?)(((\/(?:[^?#](?![^?#\/]*\.[^?#\/.]+(?:[?#]|$)))*\/?)?([^?#\/]*))(?:\?([^#]*))?(?:#(.*))?)/ - } - }; - - $("#search-form").submit(function(e) { - e.preventDefault() - }) - - // list below is the lunr 2.1.3 list minus the intersect with names(Base) - // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) - // ideally we'd just filter the original list but it's not available as a variable - lunr.stopWordFilter = lunr.generateStopWordFilter([ - 'a', - 'able', - 'about', - 'across', - 'after', - 'almost', - 'also', - 'am', - 'among', - 'an', - 'and', - 'are', - 'as', - 'at', - 'be', - 'because', - 'been', - 'but', - 'by', - 'can', - 'cannot', - 'could', - 'dear', - 'did', - 'does', - 'either', - 'ever', - 'every', - 'from', - 'got', - 'had', - 'has', - 'have', - 'he', - 'her', - 'hers', - 'him', - 'his', - 'how', - 'however', - 'i', - 'if', - 'into', - 'it', - 'its', - 'just', - 'least', - 'like', - 'likely', - 'may', - 'me', - 'might', - 'most', - 'must', - 'my', - 'neither', - 'no', - 'nor', - 'not', - 'of', - 'off', - 'often', - 'on', - 'or', - 'other', - 'our', - 'own', - 'rather', - 'said', - 'say', - 'says', - 'she', - 'should', - 'since', - 'so', - 'some', - 'than', - 'that', - 'the', - 'their', - 'them', - 'then', - 'there', - 'these', - 'they', - 'this', - 'tis', - 'to', - 'too', - 'twas', - 'us', - 'wants', - 'was', - 'we', - 'were', - 'what', - 'when', - 'who', - 'whom', - 'why', - 'will', - 'would', - 'yet', - 'you', - 'your' - ]) - - // add . as a separator, because otherwise "title": "Documenter.Anchors.add!" - // would not find anything if searching for "add!", only for the entire qualification - lunr.tokenizer.separator = /[\s\-\.]+/ - - // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names - lunr.trimmer = function (token) { - return token.update(function (s) { - return s.replace(/^[^a-zA-Z0-9@!]+/, '').replace(/[^a-zA-Z0-9@!]+$/, '') - }) - } - - lunr.Pipeline.registerFunction(lunr.stopWordFilter, 'juliaStopWordFilter') - lunr.Pipeline.registerFunction(lunr.trimmer, 'juliaTrimmer') - - var index = lunr(function () { - this.ref('location') - this.field('title',{boost: 100}) - this.field('text') - documenterSearchIndex['docs'].forEach(function(e) { - this.add(e) - }, this) - }) - var store = {} - - documenterSearchIndex['docs'].forEach(function(e) { - store[e.location] = {title: e.title, category: e.category, page: e.page} - }) - - $(function(){ - searchresults = $('#documenter-search-results'); - searchinfo = $('#documenter-search-info'); - searchbox = $('#documenter-search-query'); - function update_search(querystring) { - tokens = lunr.tokenizer(querystring) - results = index.query(function (q) { - tokens.forEach(function (t) { - q.term(t.toString(), { - fields: ["title"], - boost: 100, - usePipeline: true, - editDistance: 0, - wildcard: lunr.Query.wildcard.NONE - }) - q.term(t.toString(), { - fields: ["title"], - boost: 10, - usePipeline: true, - editDistance: 2, - wildcard: lunr.Query.wildcard.NONE - }) - q.term(t.toString(), { - fields: ["text"], - boost: 1, - usePipeline: true, - editDistance: 0, - wildcard: lunr.Query.wildcard.NONE - }) - }) - }) - searchinfo.text("Number of results: " + results.length) - searchresults.empty() - results.forEach(function(result) { - data = store[result.ref] - link = $(''+data.title+'') - link.attr('href', documenterBaseURL+'/'+result.ref) - if (data.category != "page"){ - cat = $('('+data.category+', '+data.page+')') - } else { - cat = $('('+data.category+')') - } - li = $('
  • ').append(link).append(" ").append(cat) - searchresults.append(li) - }) - } - - function update_search_box() { - querystring = searchbox.val() - update_search(querystring) - } - - searchbox.keyup(_.debounce(update_search_box, 250)) - searchbox.change(update_search_box) - - search_query_uri = parseUri(window.location).queryKey["q"] - if(search_query_uri !== undefined) { - search_query = decodeURIComponent(search_query_uri.replace(/\+/g, '%20')) - searchbox.val(search_query) - } - update_search_box(); - }) -}) - -}) diff --git a/previews/PR40/assets/themes/documenter-dark.css b/previews/PR40/assets/themes/documenter-dark.css deleted file mode 100644 index 36081413..00000000 --- a/previews/PR40/assets/themes/documenter-dark.css +++ /dev/null @@ -1,7702 +0,0 @@ -@charset "UTF-8"; -/* Font Awesome 5 mixin. Can be included in any rule that should render Font Awesome icons. */ -@keyframes spinAround { - from { - transform: rotate(0deg); } - to { - transform: rotate(359deg); } } - -html.theme--documenter-dark .tabs, html.theme--documenter-dark .pagination-previous, -html.theme--documenter-dark .pagination-next, -html.theme--documenter-dark .pagination-link, -html.theme--documenter-dark .pagination-ellipsis, html.theme--documenter-dark .breadcrumb, html.theme--documenter-dark .file, html.theme--documenter-dark .button, .is-unselectable, html.theme--documenter-dark .modal-close, html.theme--documenter-dark .delete { - -webkit-touch-callout: none; - -webkit-user-select: none; - -moz-user-select: none; - -ms-user-select: none; - user-select: none; } - -html.theme--documenter-dark .navbar-link:not(.is-arrowless)::after, html.theme--documenter-dark .select:not(.is-multiple):not(.is-loading)::after { - border: 3px solid transparent; - border-radius: 2px; - border-right: 0; - border-top: 0; - content: " "; - display: block; - height: 0.625em; - margin-top: -0.4375em; - pointer-events: none; - position: absolute; - top: 50%; - transform: rotate(-45deg); - transform-origin: center; - width: 0.625em; } - -html.theme--documenter-dark .admonition:not(:last-child), html.theme--documenter-dark .tabs:not(:last-child), html.theme--documenter-dark .message:not(:last-child), html.theme--documenter-dark .list:not(:last-child), html.theme--documenter-dark .level:not(:last-child), html.theme--documenter-dark .breadcrumb:not(:last-child), html.theme--documenter-dark .highlight:not(:last-child), html.theme--documenter-dark .block:not(:last-child), html.theme--documenter-dark .title:not(:last-child), -html.theme--documenter-dark .subtitle:not(:last-child), html.theme--documenter-dark .table-container:not(:last-child), html.theme--documenter-dark .table:not(:last-child), html.theme--documenter-dark .progress:not(:last-child), html.theme--documenter-dark .notification:not(:last-child), html.theme--documenter-dark .content:not(:last-child), html.theme--documenter-dark .box:not(:last-child) { - margin-bottom: 1.5rem; } - -html.theme--documenter-dark .modal-close, html.theme--documenter-dark .delete { - -moz-appearance: none; - -webkit-appearance: none; - background-color: rgba(10, 10, 10, 0.2); - border: none; - border-radius: 290486px; - cursor: pointer; - pointer-events: auto; - display: inline-block; - flex-grow: 0; - flex-shrink: 0; - font-size: 0; - height: 20px; - max-height: 20px; - max-width: 20px; - min-height: 20px; - min-width: 20px; - outline: none; - position: relative; - vertical-align: top; - width: 20px; } - html.theme--documenter-dark .modal-close::before, html.theme--documenter-dark .delete::before, html.theme--documenter-dark .modal-close::after, html.theme--documenter-dark .delete::after { - background-color: white; - content: ""; - display: block; - left: 50%; - position: absolute; - top: 50%; - transform: translateX(-50%) translateY(-50%) rotate(45deg); - transform-origin: center center; } - html.theme--documenter-dark .modal-close::before, html.theme--documenter-dark .delete::before { - height: 2px; - width: 50%; } - html.theme--documenter-dark .modal-close::after, html.theme--documenter-dark .delete::after { - height: 50%; - width: 2px; } - html.theme--documenter-dark .modal-close:hover, html.theme--documenter-dark .delete:hover, html.theme--documenter-dark .modal-close:focus, html.theme--documenter-dark .delete:focus { - background-color: rgba(10, 10, 10, 0.3); } - html.theme--documenter-dark .modal-close:active, html.theme--documenter-dark .delete:active { - background-color: rgba(10, 10, 10, 0.4); } - html.theme--documenter-dark .is-small.modal-close, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.modal-close, html.theme--documenter-dark .is-small.delete, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.delete { - height: 16px; - max-height: 16px; - max-width: 16px; - min-height: 16px; - min-width: 16px; - width: 16px; } - html.theme--documenter-dark .is-medium.modal-close, html.theme--documenter-dark .is-medium.delete { - height: 24px; - max-height: 24px; - max-width: 24px; - min-height: 24px; - min-width: 24px; - width: 24px; } - html.theme--documenter-dark .is-large.modal-close, html.theme--documenter-dark .is-large.delete { - height: 32px; - max-height: 32px; - max-width: 32px; - min-height: 32px; - min-width: 32px; - width: 32px; } - -html.theme--documenter-dark .control.is-loading::after, html.theme--documenter-dark .select.is-loading::after, html.theme--documenter-dark .loader, html.theme--documenter-dark .button.is-loading::after { - animation: spinAround 500ms infinite linear; - border: 2px solid #dbdee0; - border-radius: 290486px; - border-right-color: transparent; - border-top-color: transparent; - content: ""; - display: block; - height: 1em; - position: relative; - width: 1em; } - -html.theme--documenter-dark .hero-video, html.theme--documenter-dark .modal-background, html.theme--documenter-dark .modal, html.theme--documenter-dark .image.is-square img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-square img, -html.theme--documenter-dark .image.is-square .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-square .has-ratio, html.theme--documenter-dark .image.is-1by1 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by1 img, -html.theme--documenter-dark .image.is-1by1 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by1 .has-ratio, html.theme--documenter-dark .image.is-5by4 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by4 img, -html.theme--documenter-dark .image.is-5by4 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by4 .has-ratio, html.theme--documenter-dark .image.is-4by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by3 img, -html.theme--documenter-dark .image.is-4by3 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by3 .has-ratio, html.theme--documenter-dark .image.is-3by2 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by2 img, -html.theme--documenter-dark .image.is-3by2 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by2 .has-ratio, html.theme--documenter-dark .image.is-5by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by3 img, -html.theme--documenter-dark .image.is-5by3 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by3 .has-ratio, html.theme--documenter-dark .image.is-16by9 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-16by9 img, -html.theme--documenter-dark .image.is-16by9 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-16by9 .has-ratio, html.theme--documenter-dark .image.is-2by1 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by1 img, -html.theme--documenter-dark .image.is-2by1 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by1 .has-ratio, html.theme--documenter-dark .image.is-3by1 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by1 img, -html.theme--documenter-dark .image.is-3by1 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by1 .has-ratio, html.theme--documenter-dark .image.is-4by5 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by5 img, -html.theme--documenter-dark .image.is-4by5 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by5 .has-ratio, html.theme--documenter-dark .image.is-3by4 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by4 img, -html.theme--documenter-dark .image.is-3by4 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by4 .has-ratio, html.theme--documenter-dark .image.is-2by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by3 img, -html.theme--documenter-dark .image.is-2by3 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by3 .has-ratio, html.theme--documenter-dark .image.is-3by5 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by5 img, -html.theme--documenter-dark .image.is-3by5 .has-ratio, -html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by5 .has-ratio, html.theme--documenter-dark .image.is-9by16 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-9by16 img, -html.theme--documenter-dark .image.is-9by16 .has-ratio, 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(min-width: 769px) and (max-width: 1055px) { - .is-flex-tablet-only { - display: flex !important; } } - -@media screen and (max-width: 1055px) { - .is-flex-touch { - display: flex !important; } } - -@media screen and (min-width: 1056px) { - .is-flex-desktop { - display: flex !important; } } - -@media screen and (min-width: 1056px) and (max-width: 1215px) { - .is-flex-desktop-only { - display: flex !important; } } - -@media screen and (min-width: 1216px) { - .is-flex-widescreen { - display: flex !important; } } - -@media screen and (min-width: 1216px) and (max-width: 1407px) { - .is-flex-widescreen-only { - display: flex !important; } } - -@media screen and (min-width: 1408px) { - .is-flex-fullhd { - display: flex !important; } } - -.is-inline { - display: inline !important; } - -@media screen and (max-width: 768px) { - .is-inline-mobile { - display: inline !important; } } - -@media screen and (min-width: 769px), print { - .is-inline-tablet { - display: inline !important; } } - -@media screen and (min-width: 769px) and (max-width: 1055px) { - .is-inline-tablet-only { - display: inline !important; } } - -@media screen and (max-width: 1055px) { - .is-inline-touch { - display: inline !important; } } - -@media screen and (min-width: 1056px) { - .is-inline-desktop { - display: inline !important; } } - -@media screen and (min-width: 1056px) and (max-width: 1215px) { - .is-inline-desktop-only { - display: inline !important; } } - -@media screen and (min-width: 1216px) { - .is-inline-widescreen { - display: inline !important; } } - -@media screen and (min-width: 1216px) and (max-width: 1407px) { - .is-inline-widescreen-only { - display: inline !important; } } - -@media screen and (min-width: 1408px) { - .is-inline-fullhd { - display: inline !important; } } - -.is-inline-block { - display: inline-block !important; } - -@media screen and (max-width: 768px) { - .is-inline-block-mobile { - display: inline-block !important; } } - -@media screen and (min-width: 769px), print { - .is-inline-block-tablet { - display: inline-block !important; } } - -@media screen and (min-width: 769px) and (max-width: 1055px) { - .is-inline-block-tablet-only { - display: inline-block !important; } } - -@media screen and (max-width: 1055px) { - .is-inline-block-touch { - display: inline-block !important; } } - -@media screen and (min-width: 1056px) { - .is-inline-block-desktop { - display: inline-block !important; } } - -@media screen and (min-width: 1056px) and (max-width: 1215px) { - .is-inline-block-desktop-only { - display: inline-block !important; } } - -@media screen and (min-width: 1216px) { - .is-inline-block-widescreen { - display: inline-block !important; } } - -@media screen and (min-width: 1216px) and (max-width: 1407px) { - .is-inline-block-widescreen-only { - display: inline-block !important; } } - -@media screen and (min-width: 1408px) { - .is-inline-block-fullhd { - display: inline-block !important; } } - -.is-inline-flex { - display: inline-flex !important; } - -@media screen and (max-width: 768px) { - .is-inline-flex-mobile { - display: inline-flex !important; } } - -@media screen and (min-width: 769px), print { - .is-inline-flex-tablet { - display: inline-flex !important; } } - -@media screen and (min-width: 769px) and (max-width: 1055px) { - .is-inline-flex-tablet-only { - display: inline-flex !important; } } - -@media screen and (max-width: 1055px) { - .is-inline-flex-touch { - display: inline-flex !important; } } - -@media screen and (min-width: 1056px) { - .is-inline-flex-desktop { - display: inline-flex !important; } } - -@media screen and (min-width: 1056px) and (max-width: 1215px) { - .is-inline-flex-desktop-only { - display: inline-flex !important; } } - -@media screen and (min-width: 1216px) { - .is-inline-flex-widescreen { - display: inline-flex !important; } } - -@media screen and (min-width: 1216px) and (max-width: 1407px) { - .is-inline-flex-widescreen-only { - display: inline-flex !important; } } - -@media screen and (min-width: 1408px) { - .is-inline-flex-fullhd { - display: inline-flex !important; } } - -.is-hidden { - display: none !important; } - -.is-sr-only { - border: none !important; - clip: rect(0, 0, 0, 0) !important; - height: 0.01em !important; - overflow: hidden !important; - padding: 0 !important; - position: absolute !important; - white-space: nowrap !important; - width: 0.01em !important; } - -@media screen and (max-width: 768px) { - .is-hidden-mobile { - display: none !important; } } - -@media screen and (min-width: 769px), print { - .is-hidden-tablet { - display: none !important; } } - -@media screen and (min-width: 769px) and (max-width: 1055px) { - .is-hidden-tablet-only { - display: none !important; } } - -@media screen and (max-width: 1055px) { - .is-hidden-touch { - display: none !important; } } - -@media screen and (min-width: 1056px) { - .is-hidden-desktop { - display: none !important; } } - -@media screen and (min-width: 1056px) and (max-width: 1215px) { - .is-hidden-desktop-only { - display: none !important; } } - -@media screen and (min-width: 1216px) { - .is-hidden-widescreen { - display: none !important; } } - -@media screen and (min-width: 1216px) and (max-width: 1407px) { - .is-hidden-widescreen-only { - display: none !important; } } - -@media screen and (min-width: 1408px) { - .is-hidden-fullhd { - display: none !important; } } - -.is-invisible { - visibility: hidden !important; } - -@media screen and (max-width: 768px) { - .is-invisible-mobile { - visibility: hidden !important; } } - -@media screen and (min-width: 769px), print { - .is-invisible-tablet { - visibility: hidden !important; } } - -@media screen and (min-width: 769px) and (max-width: 1055px) { - .is-invisible-tablet-only { - visibility: hidden !important; } } - -@media screen and (max-width: 1055px) { - .is-invisible-touch { - visibility: hidden !important; } } - -@media screen and (min-width: 1056px) { - .is-invisible-desktop { - visibility: hidden !important; } } - -@media screen and (min-width: 1056px) and (max-width: 1215px) { - .is-invisible-desktop-only { - visibility: hidden !important; } } - -@media screen and (min-width: 1216px) { - .is-invisible-widescreen { - visibility: hidden !important; } } - -@media screen and (min-width: 1216px) and (max-width: 1407px) { - .is-invisible-widescreen-only { - visibility: hidden !important; } } - -@media screen and (min-width: 1408px) { - .is-invisible-fullhd { - visibility: hidden !important; } } - -.is-marginless { - margin: 0 !important; } - -.is-paddingless { - padding: 0 !important; } - -.is-radiusless { - border-radius: 0 !important; } - -.is-shadowless { - box-shadow: none !important; } - -.is-relative { - position: relative !important; } - -html.theme--documenter-dark { - /* This file contain the overall layout. - * - * The main container is
    that is identified by id #documenter. - */ - /*! - Theme: a11y-dark - Author: @ericwbailey - Maintainer: @ericwbailey - - Based on the Tomorrow Night Eighties theme: https://github.com/isagalaev/highlight.js/blob/master/src/styles/tomorrow-night-eighties.css -*/ - /* Comment */ - /* Red */ - /* Orange */ - /* Yellow */ - /* Green */ - /* Blue */ - /* Purple */ } - html.theme--documenter-dark html { - background-color: #1f2424; - font-size: 16px; - -moz-osx-font-smoothing: grayscale; - -webkit-font-smoothing: antialiased; - min-width: 300px; - overflow-x: auto; - overflow-y: scroll; - text-rendering: optimizeLegibility; - text-size-adjust: 100%; } - html.theme--documenter-dark article, - html.theme--documenter-dark aside, - html.theme--documenter-dark figure, - html.theme--documenter-dark footer, - html.theme--documenter-dark header, - html.theme--documenter-dark hgroup, - html.theme--documenter-dark section { - display: block; } - html.theme--documenter-dark body, - html.theme--documenter-dark button, - html.theme--documenter-dark input, - html.theme--documenter-dark select, - html.theme--documenter-dark textarea { - font-family: "Lato Medium", -apple-system, BlinkMacSystemFont, "Segoe UI", "Helvetica Neue", "Helvetica", "Arial", sans-serif; } - html.theme--documenter-dark code, - html.theme--documenter-dark pre { - -moz-osx-font-smoothing: auto; - -webkit-font-smoothing: auto; - font-family: "JuliaMono", "SFMono-Regular", "Menlo", "Consolas", "Liberation Mono", "DejaVu Sans Mono", monospace; } - html.theme--documenter-dark body { - color: #fff; - font-size: 1em; - font-weight: 400; - line-height: 1.5; } - html.theme--documenter-dark a { - color: #1abc9c; - cursor: pointer; - text-decoration: none; } - html.theme--documenter-dark a strong { - color: currentColor; } - html.theme--documenter-dark a:hover { - color: #1dd2af; } - html.theme--documenter-dark code { - background-color: rgba(255, 255, 255, 0.05); - color: #ececec; - font-size: 0.875em; - font-weight: normal; - padding: 0.1em; } - html.theme--documenter-dark hr { - background-color: #282f2f; - border: none; - display: block; - height: 2px; - margin: 1.5rem 0; } - html.theme--documenter-dark img { - height: auto; - max-width: 100%; } - html.theme--documenter-dark input[type="checkbox"], - html.theme--documenter-dark input[type="radio"] { - vertical-align: baseline; } - html.theme--documenter-dark small { - font-size: 0.875em; } - html.theme--documenter-dark span { - font-style: inherit; - font-weight: inherit; } - html.theme--documenter-dark strong { - color: #f2f2f2; - font-weight: 700; } - html.theme--documenter-dark fieldset { - border: none; } - html.theme--documenter-dark pre { - -webkit-overflow-scrolling: touch; - background-color: #282f2f; - color: #fff; - font-size: 0.875em; - overflow-x: auto; - padding: 1.25rem 1.5rem; - white-space: pre; - word-wrap: normal; } - html.theme--documenter-dark pre code { - background-color: transparent; - color: currentColor; - font-size: 1em; - padding: 0; } - html.theme--documenter-dark table td, - html.theme--documenter-dark table th { - vertical-align: top; } - html.theme--documenter-dark table td:not([align]), - html.theme--documenter-dark table th:not([align]) { - text-align: left; } - html.theme--documenter-dark table th { - color: #f2f2f2; } - html.theme--documenter-dark .box { - background-color: #343c3d; - border-radius: 8px; - box-shadow: none; - color: #fff; - display: block; - padding: 1.25rem; } - html.theme--documenter-dark a.box:hover, html.theme--documenter-dark a.box:focus { - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px #1abc9c; } - html.theme--documenter-dark a.box:active { - box-shadow: inset 0 1px 2px rgba(10, 10, 10, 0.2), 0 0 0 1px #1abc9c; } - html.theme--documenter-dark .button { - background-color: #282f2f; - border-color: #4c5759; - border-width: 1px; - color: #375a7f; - cursor: pointer; - justify-content: center; - padding-bottom: calc(0.375em - 1px); - padding-left: 0.75em; - padding-right: 0.75em; - padding-top: calc(0.375em - 1px); - text-align: center; - white-space: nowrap; } - html.theme--documenter-dark .button strong { - color: inherit; } - html.theme--documenter-dark .button .icon, html.theme--documenter-dark .button .icon.is-small, html.theme--documenter-dark .button #documenter .docs-sidebar form.docs-search > input.icon, html.theme--documenter-dark #documenter .docs-sidebar .button form.docs-search > input.icon, html.theme--documenter-dark .button .icon.is-medium, html.theme--documenter-dark .button .icon.is-large { - height: 1.5em; - width: 1.5em; } - html.theme--documenter-dark .button .icon:first-child:not(:last-child) { - margin-left: calc(-0.375em - 1px); - margin-right: 0.1875em; } - html.theme--documenter-dark .button .icon:last-child:not(:first-child) { - margin-left: 0.1875em; - margin-right: calc(-0.375em - 1px); } - html.theme--documenter-dark .button .icon:first-child:last-child { - margin-left: calc(-0.375em - 1px); - margin-right: calc(-0.375em - 1px); } - html.theme--documenter-dark .button:hover, html.theme--documenter-dark .button.is-hovered { - border-color: #8c9b9d; - color: #f2f2f2; } - html.theme--documenter-dark .button:focus, html.theme--documenter-dark .button.is-focused { - border-color: #8c9b9d; - color: #17a689; } - html.theme--documenter-dark .button:focus:not(:active), html.theme--documenter-dark .button.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(26, 188, 156, 0.25); } - html.theme--documenter-dark .button:active, html.theme--documenter-dark .button.is-active { - border-color: #343c3d; - color: #f2f2f2; } - html.theme--documenter-dark .button.is-text { - background-color: transparent; - border-color: transparent; - color: #fff; - text-decoration: underline; } - html.theme--documenter-dark .button.is-text:hover, html.theme--documenter-dark .button.is-text.is-hovered, html.theme--documenter-dark .button.is-text:focus, html.theme--documenter-dark .button.is-text.is-focused { - background-color: #282f2f; - color: #f2f2f2; } - html.theme--documenter-dark .button.is-text:active, html.theme--documenter-dark .button.is-text.is-active { - background-color: #1d2122; - color: #f2f2f2; } - html.theme--documenter-dark .button.is-text[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-text { - background-color: transparent; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-white { - background-color: white; - border-color: transparent; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-white:hover, html.theme--documenter-dark .button.is-white.is-hovered { - background-color: #f9f9f9; - border-color: transparent; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-white:focus, html.theme--documenter-dark .button.is-white.is-focused { - border-color: transparent; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-white:focus:not(:active), html.theme--documenter-dark .button.is-white.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); } - html.theme--documenter-dark .button.is-white:active, html.theme--documenter-dark .button.is-white.is-active { - background-color: #f2f2f2; - border-color: transparent; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-white[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-white { - background-color: white; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-white.is-inverted { - background-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .button.is-white.is-inverted:hover, html.theme--documenter-dark .button.is-white.is-inverted.is-hovered { - background-color: black; } - html.theme--documenter-dark .button.is-white.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-white.is-inverted { - background-color: #0a0a0a; - border-color: transparent; - box-shadow: none; - color: white; } - html.theme--documenter-dark .button.is-white.is-loading::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - html.theme--documenter-dark .button.is-white.is-outlined { - background-color: transparent; - border-color: white; - color: white; } - html.theme--documenter-dark .button.is-white.is-outlined:hover, html.theme--documenter-dark .button.is-white.is-outlined.is-hovered, html.theme--documenter-dark .button.is-white.is-outlined:focus, html.theme--documenter-dark .button.is-white.is-outlined.is-focused { - background-color: white; - border-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-white.is-outlined.is-loading::after { - border-color: transparent transparent white white !important; } - html.theme--documenter-dark .button.is-white.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-white.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-white.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-white.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - html.theme--documenter-dark .button.is-white.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-white.is-outlined { - background-color: transparent; - border-color: white; - box-shadow: none; - color: white; } - html.theme--documenter-dark .button.is-white.is-inverted.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-white.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-focused { - background-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent white white !important; } - html.theme--documenter-dark .button.is-white.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-white.is-inverted.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - box-shadow: none; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-black { - background-color: #0a0a0a; - border-color: transparent; - color: white; } - html.theme--documenter-dark .button.is-black:hover, html.theme--documenter-dark .button.is-black.is-hovered { - background-color: #040404; - border-color: transparent; - color: white; } - html.theme--documenter-dark .button.is-black:focus, html.theme--documenter-dark .button.is-black.is-focused { - border-color: transparent; - color: white; } - html.theme--documenter-dark .button.is-black:focus:not(:active), html.theme--documenter-dark .button.is-black.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); } - html.theme--documenter-dark .button.is-black:active, html.theme--documenter-dark .button.is-black.is-active { - background-color: black; - border-color: transparent; - color: white; } - html.theme--documenter-dark .button.is-black[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-black { - background-color: #0a0a0a; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-black.is-inverted { - background-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-black.is-inverted:hover, html.theme--documenter-dark .button.is-black.is-inverted.is-hovered { - background-color: #f2f2f2; } - html.theme--documenter-dark .button.is-black.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted { - background-color: white; - border-color: transparent; - box-shadow: none; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-black.is-loading::after { - border-color: transparent transparent white white !important; } - html.theme--documenter-dark .button.is-black.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-black.is-outlined:hover, html.theme--documenter-dark .button.is-black.is-outlined.is-hovered, html.theme--documenter-dark .button.is-black.is-outlined:focus, html.theme--documenter-dark .button.is-black.is-outlined.is-focused { - background-color: #0a0a0a; - border-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .button.is-black.is-outlined.is-loading::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - html.theme--documenter-dark .button.is-black.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-black.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent white white !important; } - html.theme--documenter-dark .button.is-black.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-black.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - box-shadow: none; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-black.is-inverted.is-outlined { - background-color: transparent; - border-color: white; - color: white; } - html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-focused { - background-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - html.theme--documenter-dark .button.is-black.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted.is-outlined { - background-color: transparent; - border-color: white; - box-shadow: none; - color: white; } - html.theme--documenter-dark .button.is-light { - background-color: #ecf0f1; - border-color: transparent; - color: #282f2f; } - html.theme--documenter-dark .button.is-light:hover, html.theme--documenter-dark .button.is-light.is-hovered { - background-color: #e5eaec; - border-color: transparent; - color: #282f2f; } - html.theme--documenter-dark .button.is-light:focus, html.theme--documenter-dark .button.is-light.is-focused { - border-color: transparent; - color: #282f2f; } - html.theme--documenter-dark .button.is-light:focus:not(:active), html.theme--documenter-dark .button.is-light.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(236, 240, 241, 0.25); } - html.theme--documenter-dark .button.is-light:active, html.theme--documenter-dark .button.is-light.is-active { - background-color: #dde4e6; - border-color: transparent; - color: #282f2f; } - html.theme--documenter-dark .button.is-light[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-light { - background-color: #ecf0f1; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-light.is-inverted { - background-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-light.is-inverted:hover, html.theme--documenter-dark .button.is-light.is-inverted.is-hovered { - background-color: #1d2122; } - html.theme--documenter-dark .button.is-light.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted { - background-color: #282f2f; - border-color: transparent; - box-shadow: none; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-light.is-loading::after { - border-color: transparent transparent #282f2f #282f2f !important; } - html.theme--documenter-dark .button.is-light.is-outlined { - background-color: transparent; - border-color: #ecf0f1; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-light.is-outlined:hover, html.theme--documenter-dark .button.is-light.is-outlined.is-hovered, html.theme--documenter-dark .button.is-light.is-outlined:focus, html.theme--documenter-dark .button.is-light.is-outlined.is-focused { - background-color: #ecf0f1; - border-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .button.is-light.is-outlined.is-loading::after { - border-color: transparent transparent #ecf0f1 #ecf0f1 !important; } - html.theme--documenter-dark .button.is-light.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-light.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #282f2f #282f2f !important; } - html.theme--documenter-dark .button.is-light.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-light.is-outlined { - background-color: transparent; - border-color: #ecf0f1; - box-shadow: none; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-light.is-inverted.is-outlined { - background-color: transparent; - border-color: #282f2f; - color: #282f2f; } - html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-focused { - background-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #ecf0f1 #ecf0f1 !important; } - html.theme--documenter-dark .button.is-light.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted.is-outlined { - background-color: transparent; - border-color: #282f2f; - box-shadow: none; - color: #282f2f; } - html.theme--documenter-dark .button.is-dark, html.theme--documenter-dark .content kbd.button { - background-color: #282f2f; - border-color: transparent; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-dark:hover, html.theme--documenter-dark .content kbd.button:hover, html.theme--documenter-dark .button.is-dark.is-hovered, html.theme--documenter-dark .content kbd.button.is-hovered { - background-color: #232829; - border-color: transparent; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-dark:focus, html.theme--documenter-dark .content kbd.button:focus, html.theme--documenter-dark .button.is-dark.is-focused, html.theme--documenter-dark .content kbd.button.is-focused { - border-color: transparent; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-dark:focus:not(:active), html.theme--documenter-dark .content kbd.button:focus:not(:active), html.theme--documenter-dark .button.is-dark.is-focused:not(:active), html.theme--documenter-dark .content kbd.button.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(40, 47, 47, 0.25); } - html.theme--documenter-dark .button.is-dark:active, html.theme--documenter-dark .content kbd.button:active, html.theme--documenter-dark .button.is-dark.is-active, html.theme--documenter-dark .content kbd.button.is-active { - background-color: #1d2122; - border-color: transparent; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-dark[disabled], html.theme--documenter-dark .content kbd.button[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-dark, fieldset[disabled] html.theme--documenter-dark .content kbd.button { - background-color: #282f2f; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-dark.is-inverted, html.theme--documenter-dark .content kbd.button.is-inverted { - background-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .button.is-dark.is-inverted:hover, html.theme--documenter-dark .content kbd.button.is-inverted:hover, html.theme--documenter-dark .button.is-dark.is-inverted.is-hovered, html.theme--documenter-dark .content kbd.button.is-inverted.is-hovered { - background-color: #dde4e6; } - html.theme--documenter-dark .button.is-dark.is-inverted[disabled], html.theme--documenter-dark .content kbd.button.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted, fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted { - background-color: #ecf0f1; - border-color: transparent; - box-shadow: none; - color: #282f2f; } - html.theme--documenter-dark .button.is-dark.is-loading::after, html.theme--documenter-dark .content kbd.button.is-loading::after { - border-color: transparent transparent #ecf0f1 #ecf0f1 !important; } - html.theme--documenter-dark .button.is-dark.is-outlined, html.theme--documenter-dark .content kbd.button.is-outlined { - background-color: transparent; - border-color: #282f2f; - color: #282f2f; } - html.theme--documenter-dark .button.is-dark.is-outlined:hover, html.theme--documenter-dark .content kbd.button.is-outlined:hover, html.theme--documenter-dark .button.is-dark.is-outlined.is-hovered, html.theme--documenter-dark .content kbd.button.is-outlined.is-hovered, html.theme--documenter-dark .button.is-dark.is-outlined:focus, html.theme--documenter-dark .content kbd.button.is-outlined:focus, html.theme--documenter-dark .button.is-dark.is-outlined.is-focused, html.theme--documenter-dark .content kbd.button.is-outlined.is-focused { - background-color: #282f2f; - border-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-dark.is-outlined.is-loading::after, html.theme--documenter-dark .content kbd.button.is-outlined.is-loading::after { - border-color: transparent transparent #282f2f #282f2f !important; } - html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:hover::after, html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after, html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after, html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #ecf0f1 #ecf0f1 !important; } - html.theme--documenter-dark .button.is-dark.is-outlined[disabled], html.theme--documenter-dark .content kbd.button.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined, fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined { - background-color: transparent; - border-color: #282f2f; - box-shadow: none; - color: #282f2f; } - html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined { - background-color: transparent; - border-color: #ecf0f1; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-focused { - background-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #282f2f #282f2f !important; } - html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined[disabled], html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined, fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined { - background-color: transparent; - border-color: #ecf0f1; - box-shadow: none; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-primary, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink { - background-color: #375a7f; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-primary:hover, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:hover, html.theme--documenter-dark .button.is-primary.is-hovered, html.theme--documenter-dark .docstring > section > a.button.is-hovered.docs-sourcelink { - background-color: #335476; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-primary:focus, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:focus, html.theme--documenter-dark .button.is-primary.is-focused, html.theme--documenter-dark .docstring > section > a.button.is-focused.docs-sourcelink { - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-primary:focus:not(:active), html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:focus:not(:active), html.theme--documenter-dark .button.is-primary.is-focused:not(:active), html.theme--documenter-dark .docstring > section > a.button.is-focused.docs-sourcelink:not(:active) { - box-shadow: 0 0 0 0.125em rgba(55, 90, 127, 0.25); } - html.theme--documenter-dark .button.is-primary:active, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:active, html.theme--documenter-dark .button.is-primary.is-active, html.theme--documenter-dark .docstring > section > a.button.is-active.docs-sourcelink { - background-color: #2f4d6d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-primary[disabled], html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-primary, fieldset[disabled] html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink { - background-color: #375a7f; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-primary.is-inverted, html.theme--documenter-dark .docstring > section > a.button.is-inverted.docs-sourcelink { - background-color: #fff; - color: #375a7f; } - html.theme--documenter-dark .button.is-primary.is-inverted:hover, html.theme--documenter-dark .docstring > section > a.button.is-inverted.docs-sourcelink:hover, html.theme--documenter-dark .button.is-primary.is-inverted.is-hovered, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-hovered.docs-sourcelink { - background-color: #f2f2f2; } - html.theme--documenter-dark .button.is-primary.is-inverted[disabled], html.theme--documenter-dark .docstring > section > a.button.is-inverted.docs-sourcelink[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-primary.is-inverted, fieldset[disabled] html.theme--documenter-dark .docstring > section > a.button.is-inverted.docs-sourcelink { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #375a7f; } - html.theme--documenter-dark .button.is-primary.is-loading::after, html.theme--documenter-dark .docstring > section > a.button.is-loading.docs-sourcelink::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-primary.is-outlined, html.theme--documenter-dark .docstring > section > a.button.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #375a7f; - color: #375a7f; } - html.theme--documenter-dark .button.is-primary.is-outlined:hover, html.theme--documenter-dark .docstring > section > a.button.is-outlined.docs-sourcelink:hover, html.theme--documenter-dark .button.is-primary.is-outlined.is-hovered, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-hovered.docs-sourcelink, html.theme--documenter-dark .button.is-primary.is-outlined:focus, html.theme--documenter-dark .docstring > section > a.button.is-outlined.docs-sourcelink:focus, html.theme--documenter-dark .button.is-primary.is-outlined.is-focused, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-focused.docs-sourcelink { - background-color: #375a7f; - border-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .button.is-primary.is-outlined.is-loading::after, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink::after { - border-color: transparent transparent #375a7f #375a7f !important; } - html.theme--documenter-dark .button.is-primary.is-outlined.is-loading:hover::after, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink:hover::after, html.theme--documenter-dark .button.is-primary.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-loading.is-hovered.docs-sourcelink::after, html.theme--documenter-dark .button.is-primary.is-outlined.is-loading:focus::after, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink:focus::after, html.theme--documenter-dark .button.is-primary.is-outlined.is-loading.is-focused::after, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-loading.is-focused.docs-sourcelink::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-primary.is-outlined[disabled], html.theme--documenter-dark .docstring > section > a.button.is-outlined.docs-sourcelink[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-primary.is-outlined, fieldset[disabled] html.theme--documenter-dark .docstring > section > a.button.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #375a7f; - box-shadow: none; - color: #375a7f; } - html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #fff; - color: #fff; } - html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined:hover, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink:hover, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-hovered.docs-sourcelink, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined:focus, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink:focus, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-focused, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-focused.docs-sourcelink { - background-color: #fff; - color: #375a7f; } - html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:hover::after, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-loading.is-hovered.docs-sourcelink::after, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:focus::after, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-loading.is-focused::after, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-loading.is-focused.docs-sourcelink::after { - border-color: transparent transparent #375a7f #375a7f !important; } - html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined[disabled], html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined, fieldset[disabled] html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - html.theme--documenter-dark .button.is-link { - background-color: #1abc9c; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-link:hover, html.theme--documenter-dark .button.is-link.is-hovered { - background-color: #18b193; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-link:focus, html.theme--documenter-dark .button.is-link.is-focused { - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-link:focus:not(:active), html.theme--documenter-dark .button.is-link.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(26, 188, 156, 0.25); } - html.theme--documenter-dark .button.is-link:active, html.theme--documenter-dark .button.is-link.is-active { - background-color: #17a689; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-link[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-link { - background-color: #1abc9c; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-link.is-inverted { - background-color: #fff; - color: #1abc9c; } - html.theme--documenter-dark .button.is-link.is-inverted:hover, html.theme--documenter-dark .button.is-link.is-inverted.is-hovered { - background-color: #f2f2f2; } - html.theme--documenter-dark .button.is-link.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-link.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #1abc9c; } - html.theme--documenter-dark .button.is-link.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-link.is-outlined { - background-color: transparent; - border-color: #1abc9c; - color: #1abc9c; } - html.theme--documenter-dark .button.is-link.is-outlined:hover, html.theme--documenter-dark .button.is-link.is-outlined.is-hovered, html.theme--documenter-dark .button.is-link.is-outlined:focus, html.theme--documenter-dark .button.is-link.is-outlined.is-focused { - background-color: #1abc9c; - border-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .button.is-link.is-outlined.is-loading::after { - border-color: transparent transparent #1abc9c #1abc9c !important; } - html.theme--documenter-dark .button.is-link.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-link.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-link.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-link.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-link.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-link.is-outlined { - background-color: transparent; - border-color: #1abc9c; - box-shadow: none; - color: #1abc9c; } - html.theme--documenter-dark .button.is-link.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - html.theme--documenter-dark .button.is-link.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #1abc9c; } - html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #1abc9c #1abc9c !important; } - html.theme--documenter-dark .button.is-link.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-link.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - html.theme--documenter-dark .button.is-info { - background-color: #024c7d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-info:hover, html.theme--documenter-dark .button.is-info.is-hovered { - background-color: #024470; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-info:focus, html.theme--documenter-dark .button.is-info.is-focused { - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-info:focus:not(:active), html.theme--documenter-dark .button.is-info.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(2, 76, 125, 0.25); } - html.theme--documenter-dark .button.is-info:active, html.theme--documenter-dark .button.is-info.is-active { - background-color: #023d64; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-info[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-info { - background-color: #024c7d; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-info.is-inverted { - background-color: #fff; - color: #024c7d; } - html.theme--documenter-dark .button.is-info.is-inverted:hover, html.theme--documenter-dark .button.is-info.is-inverted.is-hovered { - background-color: #f2f2f2; } - html.theme--documenter-dark .button.is-info.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-info.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #024c7d; } - html.theme--documenter-dark .button.is-info.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-info.is-outlined { - background-color: transparent; - border-color: #024c7d; - color: #024c7d; } - html.theme--documenter-dark .button.is-info.is-outlined:hover, html.theme--documenter-dark .button.is-info.is-outlined.is-hovered, html.theme--documenter-dark .button.is-info.is-outlined:focus, html.theme--documenter-dark .button.is-info.is-outlined.is-focused { - background-color: #024c7d; - border-color: #024c7d; - color: #fff; } - html.theme--documenter-dark .button.is-info.is-outlined.is-loading::after { - border-color: transparent transparent #024c7d #024c7d !important; } - html.theme--documenter-dark .button.is-info.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-info.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-info.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-info.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-info.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-info.is-outlined { - background-color: transparent; - border-color: #024c7d; - box-shadow: none; - color: #024c7d; } - html.theme--documenter-dark .button.is-info.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - html.theme--documenter-dark .button.is-info.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #024c7d; } - html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #024c7d #024c7d !important; } - html.theme--documenter-dark .button.is-info.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-info.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - html.theme--documenter-dark .button.is-success { - background-color: #008438; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-success:hover, html.theme--documenter-dark .button.is-success.is-hovered { - background-color: #007733; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-success:focus, html.theme--documenter-dark .button.is-success.is-focused { - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-success:focus:not(:active), html.theme--documenter-dark .button.is-success.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(0, 132, 56, 0.25); } - html.theme--documenter-dark .button.is-success:active, html.theme--documenter-dark .button.is-success.is-active { - background-color: #006b2d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-success[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-success { - background-color: #008438; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-success.is-inverted { - background-color: #fff; - color: #008438; } - html.theme--documenter-dark .button.is-success.is-inverted:hover, html.theme--documenter-dark .button.is-success.is-inverted.is-hovered { - background-color: #f2f2f2; } - html.theme--documenter-dark .button.is-success.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-success.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #008438; } - html.theme--documenter-dark .button.is-success.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-success.is-outlined { - background-color: transparent; - border-color: #008438; - color: #008438; } - html.theme--documenter-dark .button.is-success.is-outlined:hover, html.theme--documenter-dark .button.is-success.is-outlined.is-hovered, html.theme--documenter-dark .button.is-success.is-outlined:focus, html.theme--documenter-dark .button.is-success.is-outlined.is-focused { - background-color: #008438; - border-color: #008438; - color: #fff; } - html.theme--documenter-dark .button.is-success.is-outlined.is-loading::after { - border-color: transparent transparent #008438 #008438 !important; } - html.theme--documenter-dark .button.is-success.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-success.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-success.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-success.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-success.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-success.is-outlined { - background-color: transparent; - border-color: #008438; - box-shadow: none; - color: #008438; } - html.theme--documenter-dark .button.is-success.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - html.theme--documenter-dark .button.is-success.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #008438; } - html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #008438 #008438 !important; } - html.theme--documenter-dark .button.is-success.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-success.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - html.theme--documenter-dark .button.is-warning { - background-color: #ad8100; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-warning:hover, html.theme--documenter-dark .button.is-warning.is-hovered { - background-color: #a07700; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-warning:focus, html.theme--documenter-dark .button.is-warning.is-focused { - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-warning:focus:not(:active), html.theme--documenter-dark .button.is-warning.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(173, 129, 0, 0.25); } - html.theme--documenter-dark .button.is-warning:active, html.theme--documenter-dark .button.is-warning.is-active { - background-color: #946e00; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-warning[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-warning { - background-color: #ad8100; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-warning.is-inverted { - background-color: #fff; - color: #ad8100; } - html.theme--documenter-dark .button.is-warning.is-inverted:hover, html.theme--documenter-dark .button.is-warning.is-inverted.is-hovered { - background-color: #f2f2f2; } - html.theme--documenter-dark .button.is-warning.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #ad8100; } - html.theme--documenter-dark .button.is-warning.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-warning.is-outlined { - background-color: transparent; - border-color: #ad8100; - color: #ad8100; } - html.theme--documenter-dark .button.is-warning.is-outlined:hover, html.theme--documenter-dark .button.is-warning.is-outlined.is-hovered, html.theme--documenter-dark .button.is-warning.is-outlined:focus, html.theme--documenter-dark .button.is-warning.is-outlined.is-focused { - background-color: #ad8100; - border-color: #ad8100; - color: #fff; } - html.theme--documenter-dark .button.is-warning.is-outlined.is-loading::after { - border-color: transparent transparent #ad8100 #ad8100 !important; } - html.theme--documenter-dark .button.is-warning.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-warning.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-warning.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-warning.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-warning.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-outlined { - background-color: transparent; - border-color: #ad8100; - box-shadow: none; - color: #ad8100; } - html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #ad8100; } - html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #ad8100 #ad8100 !important; } - html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - html.theme--documenter-dark .button.is-danger { - background-color: #9e1b0d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-danger:hover, html.theme--documenter-dark .button.is-danger.is-hovered { - background-color: #92190c; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-danger:focus, html.theme--documenter-dark .button.is-danger.is-focused { - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-danger:focus:not(:active), html.theme--documenter-dark .button.is-danger.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(158, 27, 13, 0.25); } - html.theme--documenter-dark .button.is-danger:active, html.theme--documenter-dark .button.is-danger.is-active { - background-color: #86170b; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-danger[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-danger { - background-color: #9e1b0d; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-danger.is-inverted { - background-color: #fff; - color: #9e1b0d; } - html.theme--documenter-dark .button.is-danger.is-inverted:hover, html.theme--documenter-dark .button.is-danger.is-inverted.is-hovered { - background-color: #f2f2f2; } - html.theme--documenter-dark .button.is-danger.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #9e1b0d; } - html.theme--documenter-dark .button.is-danger.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-danger.is-outlined { - background-color: transparent; - border-color: #9e1b0d; - color: #9e1b0d; } - html.theme--documenter-dark .button.is-danger.is-outlined:hover, html.theme--documenter-dark .button.is-danger.is-outlined.is-hovered, html.theme--documenter-dark .button.is-danger.is-outlined:focus, html.theme--documenter-dark .button.is-danger.is-outlined.is-focused { - background-color: #9e1b0d; - border-color: #9e1b0d; - color: #fff; } - html.theme--documenter-dark .button.is-danger.is-outlined.is-loading::after { - border-color: transparent transparent #9e1b0d #9e1b0d !important; } - html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-danger.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-outlined { - background-color: transparent; - border-color: #9e1b0d; - box-shadow: none; - color: #9e1b0d; } - html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #9e1b0d; } - html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #9e1b0d #9e1b0d !important; } - html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - html.theme--documenter-dark .button.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.button { - border-radius: 3px; - font-size: 0.85em; } - html.theme--documenter-dark .button.is-normal { - font-size: 15px; } - html.theme--documenter-dark .button.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .button.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .button[disabled], fieldset[disabled] html.theme--documenter-dark .button { - background-color: #8c9b9d; - border-color: #dbdee0; - box-shadow: none; - opacity: 0.5; } - html.theme--documenter-dark .button.is-fullwidth { - display: flex; - width: 100%; } - html.theme--documenter-dark .button.is-loading { - color: transparent !important; - pointer-events: none; } - html.theme--documenter-dark .button.is-loading::after { - position: absolute; - left: calc(50% - (1em / 2)); - top: calc(50% - (1em / 2)); - position: absolute !important; } - html.theme--documenter-dark .button.is-static { - background-color: #282f2f; - border-color: #5e6d6f; - color: #dbdee0; - box-shadow: none; - pointer-events: none; } - html.theme--documenter-dark .button.is-rounded, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.button { - border-radius: 290486px; - padding-left: 1em; - padding-right: 1em; } - html.theme--documenter-dark .buttons { - align-items: center; - display: flex; - flex-wrap: wrap; - justify-content: flex-start; } - html.theme--documenter-dark .buttons .button { - margin-bottom: 0.5rem; } - html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth) { - margin-right: 0.5rem; } - html.theme--documenter-dark .buttons:last-child { - margin-bottom: -0.5rem; } - html.theme--documenter-dark .buttons:not(:last-child) { - margin-bottom: 1rem; } - html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large) { - border-radius: 3px; - font-size: 0.85em; } - html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large) { - font-size: 1.25rem; } - html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium) { - font-size: 1.5rem; } - html.theme--documenter-dark .buttons.has-addons .button:not(:first-child) { - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - html.theme--documenter-dark .buttons.has-addons .button:not(:last-child) { - border-bottom-right-radius: 0; - border-top-right-radius: 0; - margin-right: -1px; } - html.theme--documenter-dark .buttons.has-addons .button:last-child { - margin-right: 0; } - html.theme--documenter-dark .buttons.has-addons .button:hover, html.theme--documenter-dark .buttons.has-addons .button.is-hovered { - z-index: 2; } - html.theme--documenter-dark .buttons.has-addons .button:focus, html.theme--documenter-dark .buttons.has-addons .button.is-focused, html.theme--documenter-dark .buttons.has-addons .button:active, html.theme--documenter-dark .buttons.has-addons .button.is-active, html.theme--documenter-dark .buttons.has-addons .button.is-selected { - z-index: 3; } - html.theme--documenter-dark .buttons.has-addons .button:focus:hover, html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover, html.theme--documenter-dark .buttons.has-addons .button:active:hover, html.theme--documenter-dark .buttons.has-addons .button.is-active:hover, html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover { - z-index: 4; } - html.theme--documenter-dark .buttons.has-addons .button.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - html.theme--documenter-dark .buttons.is-centered { - justify-content: center; } - html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth) { - margin-left: 0.25rem; - margin-right: 0.25rem; } - html.theme--documenter-dark .buttons.is-right { - justify-content: flex-end; } - html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth) { - margin-left: 0.25rem; - margin-right: 0.25rem; } - html.theme--documenter-dark .container { - flex-grow: 1; - margin: 0 auto; - position: relative; - width: auto; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .container { - max-width: 992px; } - html.theme--documenter-dark .container.is-fluid { - margin-left: 32px; - margin-right: 32px; - max-width: none; } } - @media screen and (max-width: 1215px) { - html.theme--documenter-dark .container.is-widescreen { - max-width: 1152px; } } - @media screen and (max-width: 1407px) { - html.theme--documenter-dark .container.is-fullhd { - max-width: 1344px; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .container { - max-width: 1152px; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .container { - max-width: 1344px; } } - html.theme--documenter-dark .content li + li { - margin-top: 0.25em; } - html.theme--documenter-dark .content p:not(:last-child), - html.theme--documenter-dark .content dl:not(:last-child), - html.theme--documenter-dark .content ol:not(:last-child), - html.theme--documenter-dark .content ul:not(:last-child), - html.theme--documenter-dark .content blockquote:not(:last-child), - html.theme--documenter-dark .content pre:not(:last-child), - html.theme--documenter-dark .content table:not(:last-child) { - margin-bottom: 1em; } - html.theme--documenter-dark .content h1, - html.theme--documenter-dark .content h2, - html.theme--documenter-dark .content h3, - html.theme--documenter-dark .content h4, - html.theme--documenter-dark .content h5, - html.theme--documenter-dark .content h6 { - color: #f2f2f2; - font-weight: 600; - line-height: 1.125; } - html.theme--documenter-dark .content h1 { - font-size: 2em; - margin-bottom: 0.5em; } - html.theme--documenter-dark .content h1:not(:first-child) { - margin-top: 1em; } - html.theme--documenter-dark .content h2 { - font-size: 1.75em; - margin-bottom: 0.5714em; } - html.theme--documenter-dark .content h2:not(:first-child) { - margin-top: 1.1428em; } - html.theme--documenter-dark .content h3 { - font-size: 1.5em; - margin-bottom: 0.6666em; } - html.theme--documenter-dark .content h3:not(:first-child) { - margin-top: 1.3333em; } - html.theme--documenter-dark .content h4 { - font-size: 1.25em; - margin-bottom: 0.8em; } - html.theme--documenter-dark .content h5 { - font-size: 1.125em; - margin-bottom: 0.8888em; } - html.theme--documenter-dark .content h6 { - font-size: 1em; - margin-bottom: 1em; } - html.theme--documenter-dark .content blockquote { - background-color: #282f2f; - border-left: 5px solid #5e6d6f; - padding: 1.25em 1.5em; } - html.theme--documenter-dark .content ol { - list-style-position: outside; - margin-left: 2em; - margin-top: 1em; } - html.theme--documenter-dark .content ol:not([type]) { - list-style-type: decimal; } - html.theme--documenter-dark .content ol.is-lower-alpha:not([type]) { - list-style-type: lower-alpha; } - html.theme--documenter-dark .content ol.is-lower-roman:not([type]) { - list-style-type: lower-roman; } - html.theme--documenter-dark .content ol.is-upper-alpha:not([type]) { - list-style-type: upper-alpha; } - html.theme--documenter-dark .content ol.is-upper-roman:not([type]) { - list-style-type: upper-roman; } - html.theme--documenter-dark .content ul { - list-style: disc outside; - margin-left: 2em; - margin-top: 1em; } - html.theme--documenter-dark .content ul ul { - list-style-type: circle; - margin-top: 0.5em; } - html.theme--documenter-dark .content ul ul ul { - list-style-type: square; } - html.theme--documenter-dark .content dd { - margin-left: 2em; } - html.theme--documenter-dark .content figure { - margin-left: 2em; - margin-right: 2em; - text-align: center; } - html.theme--documenter-dark .content figure:not(:first-child) { - margin-top: 2em; } - html.theme--documenter-dark .content figure:not(:last-child) { - margin-bottom: 2em; } - html.theme--documenter-dark .content figure img { - display: inline-block; } - html.theme--documenter-dark .content figure figcaption { - font-style: italic; } - html.theme--documenter-dark .content pre { - -webkit-overflow-scrolling: touch; - overflow-x: auto; - padding: 0; - white-space: pre; - word-wrap: normal; } - html.theme--documenter-dark .content sup, - html.theme--documenter-dark .content sub { - font-size: 75%; } - html.theme--documenter-dark .content table { - width: 100%; } - html.theme--documenter-dark .content table td, - html.theme--documenter-dark .content table th { - border: 1px solid #5e6d6f; - border-width: 0 0 1px; - padding: 0.5em 0.75em; - vertical-align: top; } - html.theme--documenter-dark .content table th { - color: #f2f2f2; } - html.theme--documenter-dark .content table th:not([align]) { - text-align: left; } - html.theme--documenter-dark .content table thead td, - html.theme--documenter-dark .content table thead th { - border-width: 0 0 2px; - color: #f2f2f2; } - html.theme--documenter-dark .content table tfoot td, - html.theme--documenter-dark .content table tfoot th { - border-width: 2px 0 0; - color: #f2f2f2; } - html.theme--documenter-dark .content table tbody tr:last-child td, - html.theme--documenter-dark .content table tbody tr:last-child th { - border-bottom-width: 0; } - html.theme--documenter-dark .content .tabs li + li { - margin-top: 0; } - html.theme--documenter-dark .content.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.content { - font-size: 0.85em; } - html.theme--documenter-dark .content.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .content.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .icon { - align-items: center; - display: inline-flex; - justify-content: center; - height: 1.5rem; - width: 1.5rem; } - html.theme--documenter-dark .icon.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.icon { - height: 1rem; - width: 1rem; } - html.theme--documenter-dark .icon.is-medium { - height: 2rem; - width: 2rem; } - html.theme--documenter-dark .icon.is-large { - height: 3rem; - width: 3rem; } - html.theme--documenter-dark .image, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img { - display: block; - position: relative; } - html.theme--documenter-dark .image img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img img { - display: block; - height: auto; - width: 100%; } - html.theme--documenter-dark .image img.is-rounded, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img img.is-rounded { - border-radius: 290486px; } - html.theme--documenter-dark .image.is-square img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-square img, - html.theme--documenter-dark .image.is-square .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-square .has-ratio, html.theme--documenter-dark .image.is-1by1 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by1 img, - html.theme--documenter-dark .image.is-1by1 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by1 .has-ratio, html.theme--documenter-dark .image.is-5by4 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by4 img, - html.theme--documenter-dark .image.is-5by4 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by4 .has-ratio, html.theme--documenter-dark .image.is-4by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by3 img, - html.theme--documenter-dark .image.is-4by3 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by3 .has-ratio, html.theme--documenter-dark .image.is-3by2 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by2 img, - html.theme--documenter-dark .image.is-3by2 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by2 .has-ratio, html.theme--documenter-dark .image.is-5by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by3 img, - html.theme--documenter-dark .image.is-5by3 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by3 .has-ratio, html.theme--documenter-dark .image.is-16by9 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-16by9 img, - html.theme--documenter-dark .image.is-16by9 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-16by9 .has-ratio, html.theme--documenter-dark .image.is-2by1 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by1 img, - html.theme--documenter-dark .image.is-2by1 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by1 .has-ratio, html.theme--documenter-dark .image.is-3by1 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by1 img, - html.theme--documenter-dark .image.is-3by1 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by1 .has-ratio, html.theme--documenter-dark .image.is-4by5 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by5 img, - html.theme--documenter-dark .image.is-4by5 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by5 .has-ratio, html.theme--documenter-dark .image.is-3by4 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by4 img, - html.theme--documenter-dark .image.is-3by4 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by4 .has-ratio, html.theme--documenter-dark .image.is-2by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by3 img, - html.theme--documenter-dark .image.is-2by3 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by3 .has-ratio, html.theme--documenter-dark .image.is-3by5 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by5 img, - html.theme--documenter-dark .image.is-3by5 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by5 .has-ratio, html.theme--documenter-dark .image.is-9by16 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-9by16 img, - html.theme--documenter-dark .image.is-9by16 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-9by16 .has-ratio, html.theme--documenter-dark .image.is-1by2 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by2 img, - html.theme--documenter-dark .image.is-1by2 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by2 .has-ratio, html.theme--documenter-dark .image.is-1by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by3 img, - html.theme--documenter-dark .image.is-1by3 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by3 .has-ratio { - height: 100%; - width: 100%; } - html.theme--documenter-dark .image.is-square, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-square, html.theme--documenter-dark .image.is-1by1, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by1 { - padding-top: 100%; } - html.theme--documenter-dark .image.is-5by4, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by4 { - padding-top: 80%; } - html.theme--documenter-dark .image.is-4by3, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by3 { - padding-top: 75%; } - html.theme--documenter-dark .image.is-3by2, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by2 { - padding-top: 66.6666%; } - html.theme--documenter-dark .image.is-5by3, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by3 { - padding-top: 60%; } - html.theme--documenter-dark .image.is-16by9, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-16by9 { - padding-top: 56.25%; } - html.theme--documenter-dark .image.is-2by1, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by1 { - padding-top: 50%; } - html.theme--documenter-dark .image.is-3by1, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by1 { - padding-top: 33.3333%; } - html.theme--documenter-dark .image.is-4by5, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by5 { - padding-top: 125%; } - html.theme--documenter-dark .image.is-3by4, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by4 { - padding-top: 133.3333%; } - html.theme--documenter-dark .image.is-2by3, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by3 { - padding-top: 150%; } - html.theme--documenter-dark .image.is-3by5, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by5 { - padding-top: 166.6666%; } - html.theme--documenter-dark .image.is-9by16, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-9by16 { - padding-top: 177.7777%; } - html.theme--documenter-dark .image.is-1by2, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by2 { - padding-top: 200%; } - html.theme--documenter-dark .image.is-1by3, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by3 { - padding-top: 300%; } - html.theme--documenter-dark .image.is-16x16, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-16x16 { - height: 16px; - width: 16px; } - html.theme--documenter-dark .image.is-24x24, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-24x24 { - height: 24px; - width: 24px; } - html.theme--documenter-dark .image.is-32x32, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-32x32 { - height: 32px; - width: 32px; } - html.theme--documenter-dark .image.is-48x48, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-48x48 { - height: 48px; - width: 48px; } - html.theme--documenter-dark .image.is-64x64, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-64x64 { - height: 64px; - width: 64px; } - html.theme--documenter-dark .image.is-96x96, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-96x96 { - height: 96px; - width: 96px; } - html.theme--documenter-dark .image.is-128x128, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-128x128 { - height: 128px; - width: 128px; } - html.theme--documenter-dark .notification { - background-color: #282f2f; - border-radius: 0.4em; - padding: 1.25rem 2.5rem 1.25rem 1.5rem; - position: relative; } - html.theme--documenter-dark .notification a:not(.button):not(.dropdown-item) { - color: currentColor; - text-decoration: underline; } - html.theme--documenter-dark .notification strong { - color: currentColor; } - html.theme--documenter-dark .notification code, - html.theme--documenter-dark .notification pre { - background: white; } - html.theme--documenter-dark .notification pre code { - background: transparent; } - html.theme--documenter-dark .notification > .delete { - position: absolute; - right: 0.5rem; - top: 0.5rem; } - html.theme--documenter-dark .notification .title, - html.theme--documenter-dark .notification .subtitle, - html.theme--documenter-dark .notification .content { - color: currentColor; } - html.theme--documenter-dark .notification.is-white { - background-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .notification.is-black { - background-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .notification.is-light { - background-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .notification.is-dark, html.theme--documenter-dark .content kbd.notification { - background-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .notification.is-primary, html.theme--documenter-dark .docstring > section > a.notification.docs-sourcelink { - background-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .notification.is-link { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .notification.is-info { - background-color: #024c7d; - color: #fff; } - html.theme--documenter-dark .notification.is-success { - background-color: #008438; - color: #fff; } - html.theme--documenter-dark .notification.is-warning { - background-color: #ad8100; - color: #fff; } - html.theme--documenter-dark .notification.is-danger { - background-color: #9e1b0d; - color: #fff; } - html.theme--documenter-dark .progress { - -moz-appearance: none; - -webkit-appearance: none; - border: none; - border-radius: 290486px; - display: block; - height: 15px; - overflow: hidden; - padding: 0; - width: 100%; } - html.theme--documenter-dark .progress::-webkit-progress-bar { - background-color: #5e6d6f; } - html.theme--documenter-dark .progress::-webkit-progress-value { - background-color: #dbdee0; } - html.theme--documenter-dark .progress::-moz-progress-bar { - background-color: #dbdee0; } - html.theme--documenter-dark .progress::-ms-fill { - background-color: #dbdee0; - border: none; } - html.theme--documenter-dark .progress.is-white::-webkit-progress-value { - background-color: white; } - html.theme--documenter-dark .progress.is-white::-moz-progress-bar { - background-color: white; } - html.theme--documenter-dark .progress.is-white::-ms-fill { - background-color: white; } - html.theme--documenter-dark .progress.is-white:indeterminate { - background-image: linear-gradient(to right, white 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-black::-webkit-progress-value { - background-color: #0a0a0a; } - html.theme--documenter-dark .progress.is-black::-moz-progress-bar { - background-color: #0a0a0a; } - html.theme--documenter-dark .progress.is-black::-ms-fill { - background-color: #0a0a0a; } - html.theme--documenter-dark .progress.is-black:indeterminate { - background-image: linear-gradient(to right, #0a0a0a 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-light::-webkit-progress-value { - background-color: #ecf0f1; } - html.theme--documenter-dark .progress.is-light::-moz-progress-bar { - background-color: #ecf0f1; } - html.theme--documenter-dark .progress.is-light::-ms-fill { - background-color: #ecf0f1; } - html.theme--documenter-dark .progress.is-light:indeterminate { - background-image: linear-gradient(to right, #ecf0f1 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-dark::-webkit-progress-value, html.theme--documenter-dark .content kbd.progress::-webkit-progress-value { - background-color: #282f2f; } - html.theme--documenter-dark .progress.is-dark::-moz-progress-bar, html.theme--documenter-dark .content kbd.progress::-moz-progress-bar { - background-color: #282f2f; } - html.theme--documenter-dark .progress.is-dark::-ms-fill, html.theme--documenter-dark .content kbd.progress::-ms-fill { - background-color: #282f2f; } - html.theme--documenter-dark .progress.is-dark:indeterminate, html.theme--documenter-dark .content kbd.progress:indeterminate { - background-image: linear-gradient(to right, #282f2f 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-primary::-webkit-progress-value, html.theme--documenter-dark .docstring > section > a.progress.docs-sourcelink::-webkit-progress-value { - background-color: #375a7f; } - html.theme--documenter-dark .progress.is-primary::-moz-progress-bar, html.theme--documenter-dark .docstring > section > a.progress.docs-sourcelink::-moz-progress-bar { - background-color: #375a7f; } - html.theme--documenter-dark .progress.is-primary::-ms-fill, html.theme--documenter-dark .docstring > section > a.progress.docs-sourcelink::-ms-fill { - background-color: #375a7f; } - html.theme--documenter-dark .progress.is-primary:indeterminate, html.theme--documenter-dark .docstring > section > a.progress.docs-sourcelink:indeterminate { - background-image: linear-gradient(to right, #375a7f 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-link::-webkit-progress-value { - background-color: #1abc9c; } - html.theme--documenter-dark .progress.is-link::-moz-progress-bar { - background-color: #1abc9c; } - html.theme--documenter-dark .progress.is-link::-ms-fill { - background-color: #1abc9c; } - html.theme--documenter-dark .progress.is-link:indeterminate { - background-image: linear-gradient(to right, #1abc9c 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-info::-webkit-progress-value { - background-color: #024c7d; } - html.theme--documenter-dark .progress.is-info::-moz-progress-bar { - background-color: #024c7d; } - html.theme--documenter-dark .progress.is-info::-ms-fill { - background-color: #024c7d; } - html.theme--documenter-dark .progress.is-info:indeterminate { - background-image: linear-gradient(to right, #024c7d 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-success::-webkit-progress-value { - background-color: #008438; } - html.theme--documenter-dark .progress.is-success::-moz-progress-bar { - background-color: #008438; } - html.theme--documenter-dark .progress.is-success::-ms-fill { - background-color: #008438; } - html.theme--documenter-dark .progress.is-success:indeterminate { - background-image: linear-gradient(to right, #008438 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-warning::-webkit-progress-value { - background-color: #ad8100; } - html.theme--documenter-dark .progress.is-warning::-moz-progress-bar { - background-color: #ad8100; } - html.theme--documenter-dark .progress.is-warning::-ms-fill { - background-color: #ad8100; } - html.theme--documenter-dark .progress.is-warning:indeterminate { - background-image: linear-gradient(to right, #ad8100 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-danger::-webkit-progress-value { - background-color: #9e1b0d; } - html.theme--documenter-dark .progress.is-danger::-moz-progress-bar { - background-color: #9e1b0d; } - html.theme--documenter-dark .progress.is-danger::-ms-fill { - background-color: #9e1b0d; } - html.theme--documenter-dark .progress.is-danger:indeterminate { - background-image: linear-gradient(to right, #9e1b0d 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress:indeterminate { - animation-duration: 1.5s; - animation-iteration-count: infinite; - animation-name: moveIndeterminate; - animation-timing-function: linear; - background-color: #5e6d6f; - background-image: linear-gradient(to right, #fff 30%, #5e6d6f 30%); - background-position: top left; - background-repeat: no-repeat; - background-size: 150% 150%; } - html.theme--documenter-dark .progress:indeterminate::-webkit-progress-bar { - background-color: transparent; } - html.theme--documenter-dark .progress:indeterminate::-moz-progress-bar { - background-color: transparent; } - html.theme--documenter-dark .progress.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.progress { - height: 0.85em; } - html.theme--documenter-dark .progress.is-medium { - height: 1.25rem; } - html.theme--documenter-dark .progress.is-large { - height: 1.5rem; } - -@keyframes moveIndeterminate { - from { - background-position: 200% 0; } - to { - background-position: -200% 0; } } - html.theme--documenter-dark .table { - background-color: #343c3d; - color: #fff; } - html.theme--documenter-dark .table td, - html.theme--documenter-dark .table th { - border: 1px solid #5e6d6f; - border-width: 0 0 1px; - padding: 0.5em 0.75em; - vertical-align: top; } - html.theme--documenter-dark .table td.is-white, - html.theme--documenter-dark .table th.is-white { - background-color: white; - border-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .table td.is-black, - html.theme--documenter-dark .table th.is-black { - background-color: #0a0a0a; - border-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .table td.is-light, - html.theme--documenter-dark .table th.is-light { - background-color: #ecf0f1; - border-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .table td.is-dark, - html.theme--documenter-dark .table th.is-dark { - background-color: #282f2f; - border-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .table td.is-primary, - html.theme--documenter-dark .table th.is-primary { - background-color: #375a7f; - border-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .table td.is-link, - html.theme--documenter-dark .table th.is-link { - background-color: #1abc9c; - border-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .table td.is-info, - html.theme--documenter-dark .table th.is-info { - background-color: #024c7d; - border-color: #024c7d; - color: #fff; } - html.theme--documenter-dark .table td.is-success, - html.theme--documenter-dark .table th.is-success { - background-color: #008438; - border-color: #008438; - color: #fff; } - html.theme--documenter-dark .table td.is-warning, - html.theme--documenter-dark .table th.is-warning { - background-color: #ad8100; - border-color: #ad8100; - color: #fff; } - html.theme--documenter-dark .table td.is-danger, - html.theme--documenter-dark .table th.is-danger { - background-color: #9e1b0d; - border-color: #9e1b0d; - color: #fff; } - html.theme--documenter-dark .table td.is-narrow, - html.theme--documenter-dark .table th.is-narrow { - white-space: nowrap; - width: 1%; } - html.theme--documenter-dark .table td.is-selected, - html.theme--documenter-dark .table th.is-selected { - background-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .table td.is-selected a, - html.theme--documenter-dark .table td.is-selected strong, - html.theme--documenter-dark .table th.is-selected a, - html.theme--documenter-dark .table th.is-selected strong { - color: currentColor; } - html.theme--documenter-dark .table th { - color: #f2f2f2; } - html.theme--documenter-dark .table th:not([align]) { - text-align: left; } - html.theme--documenter-dark .table tr.is-selected { - background-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .table tr.is-selected a, - html.theme--documenter-dark .table tr.is-selected strong { - color: currentColor; } - html.theme--documenter-dark .table tr.is-selected td, - html.theme--documenter-dark .table tr.is-selected th { - border-color: #fff; - color: currentColor; } - html.theme--documenter-dark .table thead { - background-color: transparent; } - html.theme--documenter-dark .table thead td, - html.theme--documenter-dark .table thead th { - border-width: 0 0 2px; - color: #f2f2f2; } - html.theme--documenter-dark .table tfoot { - background-color: transparent; } - html.theme--documenter-dark .table tfoot td, - html.theme--documenter-dark .table tfoot th { - border-width: 2px 0 0; - color: #f2f2f2; } - html.theme--documenter-dark .table tbody { - background-color: transparent; } - html.theme--documenter-dark .table tbody tr:last-child td, - html.theme--documenter-dark .table tbody tr:last-child th { - border-bottom-width: 0; } - html.theme--documenter-dark .table.is-bordered td, - html.theme--documenter-dark .table.is-bordered th { - border-width: 1px; } - html.theme--documenter-dark .table.is-bordered tr:last-child td, - html.theme--documenter-dark .table.is-bordered tr:last-child th { - border-bottom-width: 1px; } - html.theme--documenter-dark .table.is-fullwidth { - width: 100%; } - html.theme--documenter-dark .table.is-hoverable tbody tr:not(.is-selected):hover { - background-color: #282f2f; } - html.theme--documenter-dark .table.is-hoverable.is-striped tbody tr:not(.is-selected):hover { - background-color: #282f2f; } - html.theme--documenter-dark .table.is-hoverable.is-striped tbody tr:not(.is-selected):hover:nth-child(even) { - background-color: #2d3435; } - html.theme--documenter-dark .table.is-narrow td, - html.theme--documenter-dark .table.is-narrow th { - padding: 0.25em 0.5em; } - html.theme--documenter-dark .table.is-striped tbody tr:not(.is-selected):nth-child(even) { - background-color: #282f2f; } - html.theme--documenter-dark .table-container { - -webkit-overflow-scrolling: touch; - overflow: auto; - overflow-y: hidden; - max-width: 100%; } - html.theme--documenter-dark .tags { - align-items: center; - display: flex; - flex-wrap: wrap; - justify-content: flex-start; } - html.theme--documenter-dark .tags .tag, html.theme--documenter-dark .tags .content kbd, html.theme--documenter-dark .content .tags kbd, html.theme--documenter-dark .tags .docstring > section > a.docs-sourcelink { - margin-bottom: 0.5rem; } - html.theme--documenter-dark .tags .tag:not(:last-child), html.theme--documenter-dark .tags .content kbd:not(:last-child), html.theme--documenter-dark .content .tags kbd:not(:last-child), html.theme--documenter-dark .tags .docstring > section > a.docs-sourcelink:not(:last-child) { - margin-right: 0.5rem; } - html.theme--documenter-dark .tags:last-child { - margin-bottom: -0.5rem; } - html.theme--documenter-dark .tags:not(:last-child) { - margin-bottom: 1rem; } - html.theme--documenter-dark .tags.are-medium .tag:not(.is-normal):not(.is-large), html.theme--documenter-dark .tags.are-medium .content kbd:not(.is-normal):not(.is-large), html.theme--documenter-dark .content .tags.are-medium kbd:not(.is-normal):not(.is-large), html.theme--documenter-dark .tags.are-medium .docstring > section > a.docs-sourcelink:not(.is-normal):not(.is-large) { - font-size: 15px; } - html.theme--documenter-dark .tags.are-large .tag:not(.is-normal):not(.is-medium), html.theme--documenter-dark .tags.are-large .content kbd:not(.is-normal):not(.is-medium), html.theme--documenter-dark .content .tags.are-large kbd:not(.is-normal):not(.is-medium), html.theme--documenter-dark .tags.are-large .docstring > section > a.docs-sourcelink:not(.is-normal):not(.is-medium) { - font-size: 1.25rem; } - html.theme--documenter-dark .tags.is-centered { - justify-content: center; } - html.theme--documenter-dark .tags.is-centered .tag, html.theme--documenter-dark .tags.is-centered .content kbd, html.theme--documenter-dark .content .tags.is-centered kbd, html.theme--documenter-dark .tags.is-centered .docstring > section > a.docs-sourcelink { - margin-right: 0.25rem; - margin-left: 0.25rem; } - html.theme--documenter-dark .tags.is-right { - justify-content: flex-end; } - html.theme--documenter-dark .tags.is-right .tag:not(:first-child), html.theme--documenter-dark .tags.is-right .content kbd:not(:first-child), html.theme--documenter-dark .content .tags.is-right kbd:not(:first-child), html.theme--documenter-dark .tags.is-right .docstring > section > a.docs-sourcelink:not(:first-child) { - margin-left: 0.5rem; } - html.theme--documenter-dark .tags.is-right .tag:not(:last-child), html.theme--documenter-dark .tags.is-right .content kbd:not(:last-child), html.theme--documenter-dark .content .tags.is-right kbd:not(:last-child), html.theme--documenter-dark .tags.is-right .docstring > section > a.docs-sourcelink:not(:last-child) { - margin-right: 0; } - html.theme--documenter-dark .tags.has-addons .tag, html.theme--documenter-dark .tags.has-addons .content kbd, html.theme--documenter-dark .content .tags.has-addons kbd, html.theme--documenter-dark .tags.has-addons .docstring > section > a.docs-sourcelink { - margin-right: 0; } - html.theme--documenter-dark .tags.has-addons .tag:not(:first-child), html.theme--documenter-dark .tags.has-addons .content kbd:not(:first-child), html.theme--documenter-dark .content .tags.has-addons kbd:not(:first-child), html.theme--documenter-dark .tags.has-addons .docstring > section > a.docs-sourcelink:not(:first-child) { - margin-left: 0; - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - html.theme--documenter-dark .tags.has-addons .tag:not(:last-child), html.theme--documenter-dark .tags.has-addons .content kbd:not(:last-child), html.theme--documenter-dark .content .tags.has-addons kbd:not(:last-child), html.theme--documenter-dark .tags.has-addons .docstring > section > a.docs-sourcelink:not(:last-child) { - border-bottom-right-radius: 0; - border-top-right-radius: 0; } - html.theme--documenter-dark .tag:not(body), html.theme--documenter-dark .content kbd:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body) { - align-items: center; - background-color: #282f2f; - border-radius: 0.4em; - color: #fff; - display: inline-flex; - font-size: 0.85em; - height: 2em; - justify-content: center; - line-height: 1.5; - padding-left: 0.75em; - padding-right: 0.75em; - white-space: nowrap; } - html.theme--documenter-dark .tag:not(body) .delete, html.theme--documenter-dark .content kbd:not(body) .delete, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body) .delete { - margin-left: 0.25rem; - margin-right: -0.375rem; } - html.theme--documenter-dark .tag.is-white:not(body), html.theme--documenter-dark .content kbd.is-white:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-white:not(body) { - background-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .tag.is-black:not(body), html.theme--documenter-dark .content kbd.is-black:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-black:not(body) { - background-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .tag.is-light:not(body), html.theme--documenter-dark .content kbd.is-light:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-light:not(body) { - background-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .tag.is-dark:not(body), html.theme--documenter-dark .content kbd:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-dark:not(body), html.theme--documenter-dark .content .docstring > section > kbd:not(body) { - background-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .tag.is-primary:not(body), html.theme--documenter-dark .content kbd.is-primary:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body) { - background-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .tag.is-link:not(body), html.theme--documenter-dark .content kbd.is-link:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-link:not(body) { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .tag.is-info:not(body), html.theme--documenter-dark .content kbd.is-info:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-info:not(body) { - background-color: #024c7d; - color: #fff; } - html.theme--documenter-dark .tag.is-success:not(body), html.theme--documenter-dark .content kbd.is-success:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-success:not(body) { - background-color: #008438; - color: #fff; } - html.theme--documenter-dark .tag.is-warning:not(body), html.theme--documenter-dark .content kbd.is-warning:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-warning:not(body) { - background-color: #ad8100; - color: #fff; } - html.theme--documenter-dark .tag.is-danger:not(body), html.theme--documenter-dark .content kbd.is-danger:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-danger:not(body) { - background-color: #9e1b0d; - color: #fff; } - html.theme--documenter-dark .tag.is-normal:not(body), html.theme--documenter-dark .content kbd.is-normal:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-normal:not(body) { - font-size: 0.85em; } - html.theme--documenter-dark .tag.is-medium:not(body), html.theme--documenter-dark .content kbd.is-medium:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-medium:not(body) { - font-size: 15px; } - html.theme--documenter-dark .tag.is-large:not(body), html.theme--documenter-dark .content kbd.is-large:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-large:not(body) { - font-size: 1.25rem; } - html.theme--documenter-dark .tag:not(body) .icon:first-child:not(:last-child), html.theme--documenter-dark .content kbd:not(body) .icon:first-child:not(:last-child), html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body) .icon:first-child:not(:last-child) { - margin-left: -0.375em; - margin-right: 0.1875em; } - html.theme--documenter-dark .tag:not(body) .icon:last-child:not(:first-child), html.theme--documenter-dark .content kbd:not(body) .icon:last-child:not(:first-child), html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body) .icon:last-child:not(:first-child) { - margin-left: 0.1875em; - margin-right: -0.375em; } - html.theme--documenter-dark .tag:not(body) .icon:first-child:last-child, html.theme--documenter-dark .content kbd:not(body) .icon:first-child:last-child, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body) .icon:first-child:last-child { - margin-left: -0.375em; - margin-right: -0.375em; } - html.theme--documenter-dark .tag.is-delete:not(body), html.theme--documenter-dark .content kbd.is-delete:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body) { - margin-left: 1px; - padding: 0; - position: relative; - width: 2em; } - html.theme--documenter-dark .tag.is-delete:not(body)::before, html.theme--documenter-dark .content kbd.is-delete:not(body)::before, html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body)::before, html.theme--documenter-dark .tag.is-delete:not(body)::after, html.theme--documenter-dark .content kbd.is-delete:not(body)::after, html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body)::after { - background-color: currentColor; - content: ""; - display: block; - left: 50%; - position: absolute; - top: 50%; - transform: translateX(-50%) translateY(-50%) rotate(45deg); - transform-origin: center center; } - html.theme--documenter-dark .tag.is-delete:not(body)::before, html.theme--documenter-dark .content kbd.is-delete:not(body)::before, html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body)::before { - height: 1px; - width: 50%; } - html.theme--documenter-dark .tag.is-delete:not(body)::after, html.theme--documenter-dark .content kbd.is-delete:not(body)::after, html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body)::after { - height: 50%; - width: 1px; } - html.theme--documenter-dark .tag.is-delete:not(body):hover, html.theme--documenter-dark .content kbd.is-delete:not(body):hover, html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body):hover, html.theme--documenter-dark .tag.is-delete:not(body):focus, html.theme--documenter-dark .content kbd.is-delete:not(body):focus, html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body):focus { - background-color: #1d2122; } - html.theme--documenter-dark .tag.is-delete:not(body):active, html.theme--documenter-dark .content kbd.is-delete:not(body):active, html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body):active { - background-color: #111414; } - html.theme--documenter-dark .tag.is-rounded:not(body), html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:not(body), html.theme--documenter-dark .content kbd.is-rounded:not(body), html.theme--documenter-dark #documenter .docs-sidebar .content form.docs-search > input:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-rounded:not(body) { - border-radius: 290486px; } - html.theme--documenter-dark a.tag:hover, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:hover { - text-decoration: underline; } - html.theme--documenter-dark .title, - html.theme--documenter-dark .subtitle { - word-break: break-word; } - html.theme--documenter-dark .title em, - html.theme--documenter-dark .title span, - html.theme--documenter-dark .subtitle em, - html.theme--documenter-dark .subtitle span { - font-weight: inherit; } - html.theme--documenter-dark .title sub, - html.theme--documenter-dark .subtitle sub { - font-size: 0.75em; } - html.theme--documenter-dark .title sup, - html.theme--documenter-dark .subtitle sup { - font-size: 0.75em; } - html.theme--documenter-dark .title .tag, html.theme--documenter-dark .title .content kbd, html.theme--documenter-dark .content .title kbd, html.theme--documenter-dark .title .docstring > section > a.docs-sourcelink, - html.theme--documenter-dark .subtitle .tag, - html.theme--documenter-dark .subtitle .content kbd, - html.theme--documenter-dark .content .subtitle kbd, - html.theme--documenter-dark .subtitle .docstring > section > a.docs-sourcelink { - vertical-align: middle; } - html.theme--documenter-dark .title { - color: #fff; - font-size: 2rem; - font-weight: 500; - line-height: 1.125; } - html.theme--documenter-dark .title strong { - color: inherit; - font-weight: inherit; } - html.theme--documenter-dark .title + .highlight { - margin-top: -0.75rem; } - html.theme--documenter-dark .title:not(.is-spaced) + .subtitle { - margin-top: -1.25rem; } - html.theme--documenter-dark .title.is-1 { - font-size: 3rem; } - html.theme--documenter-dark .title.is-2 { - font-size: 2.5rem; } - html.theme--documenter-dark .title.is-3 { - font-size: 2rem; } - html.theme--documenter-dark .title.is-4 { - font-size: 1.5rem; } - html.theme--documenter-dark .title.is-5 { - font-size: 1.25rem; } - html.theme--documenter-dark .title.is-6 { - font-size: 15px; } - html.theme--documenter-dark .title.is-7 { - font-size: 0.85em; } - html.theme--documenter-dark .subtitle { - color: #8c9b9d; - font-size: 1.25rem; - font-weight: 400; - line-height: 1.25; } - html.theme--documenter-dark .subtitle strong { - color: #8c9b9d; - font-weight: 600; } - html.theme--documenter-dark .subtitle:not(.is-spaced) + .title { - margin-top: -1.25rem; } - html.theme--documenter-dark .subtitle.is-1 { - font-size: 3rem; } - html.theme--documenter-dark .subtitle.is-2 { - font-size: 2.5rem; } - html.theme--documenter-dark .subtitle.is-3 { - font-size: 2rem; } - html.theme--documenter-dark .subtitle.is-4 { - font-size: 1.5rem; } - html.theme--documenter-dark .subtitle.is-5 { - font-size: 1.25rem; } - html.theme--documenter-dark .subtitle.is-6 { - font-size: 15px; } - html.theme--documenter-dark .subtitle.is-7 { - font-size: 0.85em; } - html.theme--documenter-dark .heading { - display: block; - font-size: 11px; - letter-spacing: 1px; - margin-bottom: 5px; - text-transform: uppercase; } - html.theme--documenter-dark .highlight { - font-weight: 400; - max-width: 100%; - overflow: hidden; - padding: 0; } - html.theme--documenter-dark .highlight pre { - overflow: auto; - max-width: 100%; } - html.theme--documenter-dark .number { - align-items: center; - background-color: #282f2f; - border-radius: 290486px; - display: inline-flex; - font-size: 1.25rem; - height: 2em; - justify-content: center; - margin-right: 1.5rem; - min-width: 2.5em; - padding: 0.25rem 0.5rem; - text-align: center; - vertical-align: top; } - html.theme--documenter-dark .select select, html.theme--documenter-dark .textarea, html.theme--documenter-dark .input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input { - background-color: #1f2424; - border-color: #5e6d6f; - border-radius: 0.4em; - color: #dbdee0; } - html.theme--documenter-dark .select select::-moz-placeholder, html.theme--documenter-dark .textarea::-moz-placeholder, html.theme--documenter-dark .input::-moz-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input::-moz-placeholder { - color: rgba(219, 222, 224, 0.3); } - html.theme--documenter-dark .select select::-webkit-input-placeholder, html.theme--documenter-dark .textarea::-webkit-input-placeholder, html.theme--documenter-dark .input::-webkit-input-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input::-webkit-input-placeholder { - color: rgba(219, 222, 224, 0.3); } - html.theme--documenter-dark .select select:-moz-placeholder, html.theme--documenter-dark .textarea:-moz-placeholder, html.theme--documenter-dark .input:-moz-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:-moz-placeholder { - color: rgba(219, 222, 224, 0.3); } - html.theme--documenter-dark .select select:-ms-input-placeholder, html.theme--documenter-dark .textarea:-ms-input-placeholder, html.theme--documenter-dark .input:-ms-input-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:-ms-input-placeholder { - color: rgba(219, 222, 224, 0.3); } - html.theme--documenter-dark .select select:hover, html.theme--documenter-dark .textarea:hover, html.theme--documenter-dark .input:hover, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:hover, html.theme--documenter-dark .select select.is-hovered, html.theme--documenter-dark .is-hovered.textarea, html.theme--documenter-dark .is-hovered.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-hovered { - border-color: #8c9b9d; } - html.theme--documenter-dark .select select:focus, html.theme--documenter-dark .textarea:focus, html.theme--documenter-dark .input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:focus, html.theme--documenter-dark .select select.is-focused, html.theme--documenter-dark .is-focused.textarea, html.theme--documenter-dark .is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .select select:active, html.theme--documenter-dark .textarea:active, html.theme--documenter-dark .input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:active, html.theme--documenter-dark .select select.is-active, html.theme--documenter-dark .is-active.textarea, html.theme--documenter-dark .is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - border-color: #1abc9c; - box-shadow: 0 0 0 0.125em rgba(26, 188, 156, 0.25); } - html.theme--documenter-dark .select select[disabled], html.theme--documenter-dark .textarea[disabled], html.theme--documenter-dark .input[disabled], html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[disabled], fieldset[disabled] html.theme--documenter-dark .select select, fieldset[disabled] html.theme--documenter-dark .textarea, fieldset[disabled] html.theme--documenter-dark .input, fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input { - background-color: #8c9b9d; - border-color: #282f2f; - box-shadow: none; - color: white; } - html.theme--documenter-dark .select select[disabled]::-moz-placeholder, html.theme--documenter-dark .textarea[disabled]::-moz-placeholder, html.theme--documenter-dark .input[disabled]::-moz-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[disabled]::-moz-placeholder, fieldset[disabled] html.theme--documenter-dark .select select::-moz-placeholder, fieldset[disabled] html.theme--documenter-dark .textarea::-moz-placeholder, fieldset[disabled] html.theme--documenter-dark .input::-moz-placeholder, fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input::-moz-placeholder { - color: rgba(255, 255, 255, 0.3); } - html.theme--documenter-dark .select select[disabled]::-webkit-input-placeholder, html.theme--documenter-dark .textarea[disabled]::-webkit-input-placeholder, html.theme--documenter-dark .input[disabled]::-webkit-input-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[disabled]::-webkit-input-placeholder, fieldset[disabled] html.theme--documenter-dark .select select::-webkit-input-placeholder, fieldset[disabled] html.theme--documenter-dark .textarea::-webkit-input-placeholder, fieldset[disabled] html.theme--documenter-dark .input::-webkit-input-placeholder, fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input::-webkit-input-placeholder { - color: rgba(255, 255, 255, 0.3); } - html.theme--documenter-dark .select select[disabled]:-moz-placeholder, html.theme--documenter-dark .textarea[disabled]:-moz-placeholder, html.theme--documenter-dark .input[disabled]:-moz-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[disabled]:-moz-placeholder, fieldset[disabled] html.theme--documenter-dark .select select:-moz-placeholder, fieldset[disabled] html.theme--documenter-dark .textarea:-moz-placeholder, fieldset[disabled] html.theme--documenter-dark .input:-moz-placeholder, fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:-moz-placeholder { - color: rgba(255, 255, 255, 0.3); } - html.theme--documenter-dark .select select[disabled]:-ms-input-placeholder, html.theme--documenter-dark .textarea[disabled]:-ms-input-placeholder, html.theme--documenter-dark .input[disabled]:-ms-input-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[disabled]:-ms-input-placeholder, fieldset[disabled] html.theme--documenter-dark .select select:-ms-input-placeholder, fieldset[disabled] html.theme--documenter-dark .textarea:-ms-input-placeholder, fieldset[disabled] html.theme--documenter-dark .input:-ms-input-placeholder, fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:-ms-input-placeholder { - color: rgba(255, 255, 255, 0.3); } - html.theme--documenter-dark .textarea, html.theme--documenter-dark .input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input { - box-shadow: inset 0 1px 2px rgba(10, 10, 10, 0.1); - max-width: 100%; - width: 100%; } - html.theme--documenter-dark .textarea[readonly], html.theme--documenter-dark .input[readonly], html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[readonly] { - box-shadow: none; } - html.theme--documenter-dark .is-white.textarea, html.theme--documenter-dark .is-white.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-white { - border-color: white; } - html.theme--documenter-dark .is-white.textarea:focus, html.theme--documenter-dark .is-white.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-white:focus, html.theme--documenter-dark .is-white.is-focused.textarea, html.theme--documenter-dark .is-white.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-white.textarea:active, html.theme--documenter-dark .is-white.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-white:active, html.theme--documenter-dark .is-white.is-active.textarea, html.theme--documenter-dark .is-white.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); } - html.theme--documenter-dark .is-black.textarea, html.theme--documenter-dark .is-black.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-black { - border-color: #0a0a0a; } - html.theme--documenter-dark .is-black.textarea:focus, html.theme--documenter-dark .is-black.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-black:focus, html.theme--documenter-dark .is-black.is-focused.textarea, html.theme--documenter-dark .is-black.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-black.textarea:active, html.theme--documenter-dark .is-black.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-black:active, html.theme--documenter-dark .is-black.is-active.textarea, html.theme--documenter-dark .is-black.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); } - html.theme--documenter-dark .is-light.textarea, html.theme--documenter-dark .is-light.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-light { - border-color: #ecf0f1; } - html.theme--documenter-dark .is-light.textarea:focus, html.theme--documenter-dark .is-light.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-light:focus, html.theme--documenter-dark .is-light.is-focused.textarea, html.theme--documenter-dark .is-light.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-light.textarea:active, html.theme--documenter-dark .is-light.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-light:active, html.theme--documenter-dark .is-light.is-active.textarea, html.theme--documenter-dark .is-light.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(236, 240, 241, 0.25); } - html.theme--documenter-dark .is-dark.textarea, html.theme--documenter-dark .content kbd.textarea, html.theme--documenter-dark .is-dark.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-dark, html.theme--documenter-dark .content kbd.input { - border-color: #282f2f; } - html.theme--documenter-dark .is-dark.textarea:focus, html.theme--documenter-dark .content kbd.textarea:focus, html.theme--documenter-dark .is-dark.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-dark:focus, html.theme--documenter-dark .content kbd.input:focus, html.theme--documenter-dark .is-dark.is-focused.textarea, html.theme--documenter-dark .content kbd.is-focused.textarea, html.theme--documenter-dark .is-dark.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .content kbd.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar .content form.docs-search > input.is-focused, html.theme--documenter-dark .is-dark.textarea:active, html.theme--documenter-dark .content kbd.textarea:active, html.theme--documenter-dark .is-dark.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-dark:active, html.theme--documenter-dark .content kbd.input:active, html.theme--documenter-dark .is-dark.is-active.textarea, html.theme--documenter-dark .content kbd.is-active.textarea, html.theme--documenter-dark .is-dark.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active, html.theme--documenter-dark .content kbd.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar .content form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(40, 47, 47, 0.25); } - html.theme--documenter-dark .is-primary.textarea, html.theme--documenter-dark .docstring > section > a.textarea.docs-sourcelink, html.theme--documenter-dark .is-primary.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-primary, html.theme--documenter-dark .docstring > section > a.input.docs-sourcelink { - border-color: #375a7f; } - html.theme--documenter-dark .is-primary.textarea:focus, html.theme--documenter-dark .docstring > section > a.textarea.docs-sourcelink:focus, html.theme--documenter-dark .is-primary.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-primary:focus, html.theme--documenter-dark .docstring > section > a.input.docs-sourcelink:focus, html.theme--documenter-dark .is-primary.is-focused.textarea, html.theme--documenter-dark .docstring > section > a.is-focused.textarea.docs-sourcelink, html.theme--documenter-dark .is-primary.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .docstring > section > a.is-focused.input.docs-sourcelink, html.theme--documenter-dark .is-primary.textarea:active, html.theme--documenter-dark .docstring > section > a.textarea.docs-sourcelink:active, html.theme--documenter-dark .is-primary.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-primary:active, html.theme--documenter-dark .docstring > section > a.input.docs-sourcelink:active, html.theme--documenter-dark .is-primary.is-active.textarea, html.theme--documenter-dark .docstring > section > a.is-active.textarea.docs-sourcelink, html.theme--documenter-dark .is-primary.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active, html.theme--documenter-dark .docstring > section > a.is-active.input.docs-sourcelink { - box-shadow: 0 0 0 0.125em rgba(55, 90, 127, 0.25); } - html.theme--documenter-dark .is-link.textarea, html.theme--documenter-dark .is-link.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-link { - border-color: #1abc9c; } - html.theme--documenter-dark .is-link.textarea:focus, html.theme--documenter-dark .is-link.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-link:focus, html.theme--documenter-dark .is-link.is-focused.textarea, html.theme--documenter-dark .is-link.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-link.textarea:active, html.theme--documenter-dark .is-link.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-link:active, html.theme--documenter-dark .is-link.is-active.textarea, html.theme--documenter-dark .is-link.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(26, 188, 156, 0.25); } - html.theme--documenter-dark .is-info.textarea, html.theme--documenter-dark .is-info.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-info { - border-color: #024c7d; } - html.theme--documenter-dark .is-info.textarea:focus, html.theme--documenter-dark .is-info.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-info:focus, html.theme--documenter-dark .is-info.is-focused.textarea, html.theme--documenter-dark .is-info.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-info.textarea:active, html.theme--documenter-dark .is-info.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-info:active, html.theme--documenter-dark .is-info.is-active.textarea, html.theme--documenter-dark .is-info.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(2, 76, 125, 0.25); } - html.theme--documenter-dark .is-success.textarea, html.theme--documenter-dark .is-success.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-success { - border-color: #008438; } - html.theme--documenter-dark .is-success.textarea:focus, html.theme--documenter-dark .is-success.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-success:focus, html.theme--documenter-dark .is-success.is-focused.textarea, html.theme--documenter-dark .is-success.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-success.textarea:active, html.theme--documenter-dark .is-success.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-success:active, html.theme--documenter-dark .is-success.is-active.textarea, html.theme--documenter-dark .is-success.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(0, 132, 56, 0.25); } - html.theme--documenter-dark .is-warning.textarea, html.theme--documenter-dark .is-warning.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-warning { - border-color: #ad8100; } - html.theme--documenter-dark .is-warning.textarea:focus, html.theme--documenter-dark .is-warning.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-warning:focus, html.theme--documenter-dark .is-warning.is-focused.textarea, html.theme--documenter-dark .is-warning.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-warning.textarea:active, html.theme--documenter-dark .is-warning.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-warning:active, html.theme--documenter-dark .is-warning.is-active.textarea, html.theme--documenter-dark .is-warning.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(173, 129, 0, 0.25); } - html.theme--documenter-dark .is-danger.textarea, html.theme--documenter-dark .is-danger.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-danger { - border-color: #9e1b0d; } - html.theme--documenter-dark .is-danger.textarea:focus, html.theme--documenter-dark .is-danger.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-danger:focus, html.theme--documenter-dark .is-danger.is-focused.textarea, html.theme--documenter-dark .is-danger.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-danger.textarea:active, html.theme--documenter-dark .is-danger.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-danger:active, html.theme--documenter-dark .is-danger.is-active.textarea, html.theme--documenter-dark .is-danger.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(158, 27, 13, 0.25); } - html.theme--documenter-dark .is-small.textarea, html.theme--documenter-dark .is-small.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input { - border-radius: 3px; - font-size: 0.85em; } - html.theme--documenter-dark .is-medium.textarea, html.theme--documenter-dark .is-medium.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .is-large.textarea, html.theme--documenter-dark .is-large.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .is-fullwidth.textarea, html.theme--documenter-dark .is-fullwidth.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-fullwidth { - display: block; - width: 100%; } - html.theme--documenter-dark .is-inline.textarea, html.theme--documenter-dark .is-inline.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-inline { - display: inline; - width: auto; } - html.theme--documenter-dark .input.is-rounded, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input { - border-radius: 290486px; - padding-left: 1em; - padding-right: 1em; } - html.theme--documenter-dark .input.is-static, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-static { - background-color: transparent; - border-color: transparent; - box-shadow: none; - padding-left: 0; - padding-right: 0; } - html.theme--documenter-dark .textarea { - display: block; - max-width: 100%; - min-width: 100%; - padding: 0.625em; - resize: vertical; } - html.theme--documenter-dark .textarea:not([rows]) { - max-height: 600px; - min-height: 120px; } - html.theme--documenter-dark .textarea[rows] { - height: initial; } - html.theme--documenter-dark .textarea.has-fixed-size { - resize: none; } - html.theme--documenter-dark .radio, html.theme--documenter-dark .checkbox { - cursor: pointer; - display: inline-block; - line-height: 1.25; - position: relative; } - html.theme--documenter-dark .radio input, html.theme--documenter-dark .checkbox input { - cursor: pointer; } - html.theme--documenter-dark .radio:hover, html.theme--documenter-dark .checkbox:hover { - color: #8c9b9d; } - html.theme--documenter-dark .radio[disabled], html.theme--documenter-dark .checkbox[disabled], fieldset[disabled] html.theme--documenter-dark .radio, fieldset[disabled] html.theme--documenter-dark .checkbox { - color: white; - cursor: not-allowed; } - html.theme--documenter-dark .radio + .radio { - margin-left: 0.5em; } - html.theme--documenter-dark .select { - display: inline-block; - max-width: 100%; - position: relative; - vertical-align: top; } - html.theme--documenter-dark .select:not(.is-multiple) { - height: 2.25em; } - html.theme--documenter-dark .select:not(.is-multiple):not(.is-loading)::after { - border-color: #1abc9c; - right: 1.125em; - z-index: 4; } - html.theme--documenter-dark .select.is-rounded select, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.select select { - border-radius: 290486px; - padding-left: 1em; } - html.theme--documenter-dark .select select { - cursor: pointer; - display: block; - font-size: 1em; - max-width: 100%; - outline: none; } - html.theme--documenter-dark .select select::-ms-expand { - display: none; } - html.theme--documenter-dark .select select[disabled]:hover, fieldset[disabled] html.theme--documenter-dark .select select:hover { - border-color: #282f2f; } - html.theme--documenter-dark .select select:not([multiple]) { - padding-right: 2.5em; } - html.theme--documenter-dark .select select[multiple] { - height: auto; - padding: 0; } - html.theme--documenter-dark .select select[multiple] option { - padding: 0.5em 1em; } - html.theme--documenter-dark .select:not(.is-multiple):not(.is-loading):hover::after { - border-color: #8c9b9d; } - html.theme--documenter-dark .select.is-white:not(:hover)::after { - border-color: white; } - html.theme--documenter-dark .select.is-white select { - border-color: white; } - html.theme--documenter-dark .select.is-white select:hover, html.theme--documenter-dark .select.is-white select.is-hovered { - border-color: #f2f2f2; } - html.theme--documenter-dark .select.is-white select:focus, html.theme--documenter-dark .select.is-white select.is-focused, html.theme--documenter-dark .select.is-white select:active, html.theme--documenter-dark .select.is-white select.is-active { - box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); } - html.theme--documenter-dark .select.is-black:not(:hover)::after { - border-color: #0a0a0a; } - html.theme--documenter-dark .select.is-black select { - border-color: #0a0a0a; } - html.theme--documenter-dark .select.is-black select:hover, html.theme--documenter-dark .select.is-black select.is-hovered { - border-color: black; } - html.theme--documenter-dark .select.is-black select:focus, html.theme--documenter-dark .select.is-black select.is-focused, html.theme--documenter-dark .select.is-black select:active, html.theme--documenter-dark .select.is-black select.is-active { - box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); } - html.theme--documenter-dark .select.is-light:not(:hover)::after { - border-color: #ecf0f1; } - html.theme--documenter-dark .select.is-light select { - border-color: #ecf0f1; } - html.theme--documenter-dark .select.is-light select:hover, html.theme--documenter-dark .select.is-light select.is-hovered { - border-color: #dde4e6; } - html.theme--documenter-dark .select.is-light select:focus, html.theme--documenter-dark .select.is-light select.is-focused, html.theme--documenter-dark .select.is-light select:active, html.theme--documenter-dark .select.is-light select.is-active { - box-shadow: 0 0 0 0.125em rgba(236, 240, 241, 0.25); } - html.theme--documenter-dark .select.is-dark:not(:hover)::after, html.theme--documenter-dark .content kbd.select:not(:hover)::after { - border-color: #282f2f; } - html.theme--documenter-dark .select.is-dark select, html.theme--documenter-dark .content kbd.select select { - border-color: #282f2f; } - html.theme--documenter-dark .select.is-dark select:hover, html.theme--documenter-dark .content kbd.select select:hover, html.theme--documenter-dark .select.is-dark select.is-hovered, html.theme--documenter-dark .content kbd.select select.is-hovered { - border-color: #1d2122; } - html.theme--documenter-dark .select.is-dark select:focus, html.theme--documenter-dark .content kbd.select select:focus, html.theme--documenter-dark .select.is-dark select.is-focused, html.theme--documenter-dark .content kbd.select select.is-focused, html.theme--documenter-dark .select.is-dark select:active, html.theme--documenter-dark .content kbd.select select:active, html.theme--documenter-dark .select.is-dark select.is-active, html.theme--documenter-dark .content kbd.select select.is-active { - box-shadow: 0 0 0 0.125em rgba(40, 47, 47, 0.25); } - html.theme--documenter-dark .select.is-primary:not(:hover)::after, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink:not(:hover)::after { - border-color: #375a7f; } - html.theme--documenter-dark .select.is-primary select, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select { - border-color: #375a7f; } - html.theme--documenter-dark .select.is-primary select:hover, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select:hover, html.theme--documenter-dark .select.is-primary select.is-hovered, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select.is-hovered { - border-color: #2f4d6d; } - html.theme--documenter-dark .select.is-primary select:focus, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select:focus, html.theme--documenter-dark .select.is-primary select.is-focused, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select.is-focused, html.theme--documenter-dark .select.is-primary select:active, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select:active, html.theme--documenter-dark .select.is-primary select.is-active, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select.is-active { - box-shadow: 0 0 0 0.125em rgba(55, 90, 127, 0.25); } - html.theme--documenter-dark .select.is-link:not(:hover)::after { - border-color: #1abc9c; } - html.theme--documenter-dark .select.is-link select { - border-color: #1abc9c; } - html.theme--documenter-dark .select.is-link select:hover, html.theme--documenter-dark .select.is-link select.is-hovered { - border-color: #17a689; } - html.theme--documenter-dark .select.is-link select:focus, html.theme--documenter-dark .select.is-link select.is-focused, html.theme--documenter-dark .select.is-link select:active, html.theme--documenter-dark .select.is-link select.is-active { - box-shadow: 0 0 0 0.125em rgba(26, 188, 156, 0.25); } - html.theme--documenter-dark .select.is-info:not(:hover)::after { - border-color: #024c7d; } - html.theme--documenter-dark .select.is-info select { - border-color: #024c7d; } - html.theme--documenter-dark .select.is-info select:hover, html.theme--documenter-dark .select.is-info select.is-hovered { - border-color: #023d64; } - html.theme--documenter-dark .select.is-info select:focus, html.theme--documenter-dark .select.is-info select.is-focused, html.theme--documenter-dark .select.is-info select:active, html.theme--documenter-dark .select.is-info select.is-active { - box-shadow: 0 0 0 0.125em rgba(2, 76, 125, 0.25); } - html.theme--documenter-dark .select.is-success:not(:hover)::after { - border-color: #008438; } - html.theme--documenter-dark .select.is-success select { - border-color: #008438; } - html.theme--documenter-dark .select.is-success select:hover, html.theme--documenter-dark .select.is-success select.is-hovered { - border-color: #006b2d; } - html.theme--documenter-dark .select.is-success select:focus, html.theme--documenter-dark .select.is-success select.is-focused, html.theme--documenter-dark .select.is-success select:active, html.theme--documenter-dark .select.is-success select.is-active { - box-shadow: 0 0 0 0.125em rgba(0, 132, 56, 0.25); } - html.theme--documenter-dark .select.is-warning:not(:hover)::after { - border-color: #ad8100; } - html.theme--documenter-dark .select.is-warning select { - border-color: #ad8100; } - html.theme--documenter-dark .select.is-warning select:hover, html.theme--documenter-dark .select.is-warning select.is-hovered { - border-color: #946e00; } - html.theme--documenter-dark .select.is-warning select:focus, html.theme--documenter-dark .select.is-warning select.is-focused, html.theme--documenter-dark .select.is-warning select:active, html.theme--documenter-dark .select.is-warning select.is-active { - box-shadow: 0 0 0 0.125em rgba(173, 129, 0, 0.25); } - html.theme--documenter-dark .select.is-danger:not(:hover)::after { - border-color: #9e1b0d; } - html.theme--documenter-dark .select.is-danger select { - border-color: #9e1b0d; } - html.theme--documenter-dark .select.is-danger select:hover, html.theme--documenter-dark .select.is-danger select.is-hovered { - border-color: #86170b; } - html.theme--documenter-dark .select.is-danger select:focus, html.theme--documenter-dark .select.is-danger select.is-focused, html.theme--documenter-dark .select.is-danger select:active, html.theme--documenter-dark .select.is-danger select.is-active { - box-shadow: 0 0 0 0.125em rgba(158, 27, 13, 0.25); } - html.theme--documenter-dark .select.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.select { - border-radius: 3px; - font-size: 0.85em; } - html.theme--documenter-dark .select.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .select.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .select.is-disabled::after { - border-color: white; } - html.theme--documenter-dark .select.is-fullwidth { - width: 100%; } - html.theme--documenter-dark .select.is-fullwidth select { - width: 100%; } - html.theme--documenter-dark .select.is-loading::after { - margin-top: 0; - position: absolute; - right: 0.625em; - top: 0.625em; - transform: none; } - html.theme--documenter-dark .select.is-loading.is-small:after, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-loading:after { - font-size: 0.85em; } - html.theme--documenter-dark .select.is-loading.is-medium:after { - font-size: 1.25rem; } - html.theme--documenter-dark .select.is-loading.is-large:after { - font-size: 1.5rem; } - html.theme--documenter-dark .file { - align-items: stretch; - display: flex; - justify-content: flex-start; - position: relative; } - html.theme--documenter-dark .file.is-white .file-cta { - background-color: white; - border-color: transparent; - color: #0a0a0a; } - html.theme--documenter-dark .file.is-white:hover .file-cta, html.theme--documenter-dark .file.is-white.is-hovered .file-cta { - background-color: #f9f9f9; - border-color: transparent; - color: #0a0a0a; } - html.theme--documenter-dark .file.is-white:focus .file-cta, html.theme--documenter-dark .file.is-white.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(255, 255, 255, 0.25); - color: #0a0a0a; } - html.theme--documenter-dark .file.is-white:active .file-cta, html.theme--documenter-dark .file.is-white.is-active .file-cta { - background-color: #f2f2f2; - border-color: transparent; - color: #0a0a0a; } - html.theme--documenter-dark .file.is-black .file-cta { - background-color: #0a0a0a; - border-color: transparent; - color: white; } - html.theme--documenter-dark .file.is-black:hover .file-cta, html.theme--documenter-dark .file.is-black.is-hovered .file-cta { - background-color: #040404; - border-color: transparent; - color: white; } - html.theme--documenter-dark .file.is-black:focus .file-cta, html.theme--documenter-dark .file.is-black.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(10, 10, 10, 0.25); - color: white; } - html.theme--documenter-dark .file.is-black:active .file-cta, html.theme--documenter-dark .file.is-black.is-active .file-cta { - background-color: black; - border-color: transparent; - color: white; } - html.theme--documenter-dark .file.is-light .file-cta { - background-color: #ecf0f1; - border-color: transparent; - color: #282f2f; } - html.theme--documenter-dark .file.is-light:hover .file-cta, html.theme--documenter-dark .file.is-light.is-hovered .file-cta { - background-color: #e5eaec; - border-color: transparent; - color: #282f2f; } - html.theme--documenter-dark .file.is-light:focus .file-cta, html.theme--documenter-dark .file.is-light.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(236, 240, 241, 0.25); - color: #282f2f; } - html.theme--documenter-dark .file.is-light:active .file-cta, html.theme--documenter-dark .file.is-light.is-active .file-cta { - background-color: #dde4e6; - border-color: transparent; - color: #282f2f; } - html.theme--documenter-dark .file.is-dark .file-cta, html.theme--documenter-dark .content kbd.file .file-cta { - background-color: #282f2f; - border-color: transparent; - color: #ecf0f1; } - html.theme--documenter-dark .file.is-dark:hover .file-cta, html.theme--documenter-dark .content kbd.file:hover .file-cta, html.theme--documenter-dark .file.is-dark.is-hovered .file-cta, html.theme--documenter-dark .content kbd.file.is-hovered .file-cta { - background-color: #232829; - border-color: transparent; - color: #ecf0f1; } - html.theme--documenter-dark .file.is-dark:focus .file-cta, html.theme--documenter-dark .content kbd.file:focus .file-cta, html.theme--documenter-dark .file.is-dark.is-focused .file-cta, html.theme--documenter-dark .content kbd.file.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(40, 47, 47, 0.25); - color: #ecf0f1; } - html.theme--documenter-dark .file.is-dark:active .file-cta, html.theme--documenter-dark .content kbd.file:active .file-cta, html.theme--documenter-dark .file.is-dark.is-active .file-cta, html.theme--documenter-dark .content kbd.file.is-active .file-cta { - background-color: #1d2122; - border-color: transparent; - color: #ecf0f1; } - html.theme--documenter-dark .file.is-primary .file-cta, html.theme--documenter-dark .docstring > section > a.file.docs-sourcelink .file-cta { - background-color: #375a7f; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-primary:hover .file-cta, html.theme--documenter-dark .docstring > section > a.file.docs-sourcelink:hover .file-cta, html.theme--documenter-dark .file.is-primary.is-hovered .file-cta, html.theme--documenter-dark .docstring > section > a.file.is-hovered.docs-sourcelink .file-cta { - background-color: #335476; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-primary:focus .file-cta, html.theme--documenter-dark .docstring > section > a.file.docs-sourcelink:focus .file-cta, html.theme--documenter-dark .file.is-primary.is-focused .file-cta, html.theme--documenter-dark .docstring > section > a.file.is-focused.docs-sourcelink .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(55, 90, 127, 0.25); - color: #fff; } - html.theme--documenter-dark .file.is-primary:active .file-cta, html.theme--documenter-dark .docstring > section > a.file.docs-sourcelink:active .file-cta, html.theme--documenter-dark .file.is-primary.is-active .file-cta, html.theme--documenter-dark .docstring > section > a.file.is-active.docs-sourcelink .file-cta { - background-color: #2f4d6d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-link .file-cta { - background-color: #1abc9c; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-link:hover .file-cta, html.theme--documenter-dark .file.is-link.is-hovered .file-cta { - background-color: #18b193; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-link:focus .file-cta, html.theme--documenter-dark .file.is-link.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(26, 188, 156, 0.25); - color: #fff; } - html.theme--documenter-dark .file.is-link:active .file-cta, html.theme--documenter-dark .file.is-link.is-active .file-cta { - background-color: #17a689; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-info .file-cta { - background-color: #024c7d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-info:hover .file-cta, html.theme--documenter-dark .file.is-info.is-hovered .file-cta { - background-color: #024470; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-info:focus .file-cta, html.theme--documenter-dark .file.is-info.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(2, 76, 125, 0.25); - color: #fff; } - html.theme--documenter-dark .file.is-info:active .file-cta, html.theme--documenter-dark .file.is-info.is-active .file-cta { - background-color: #023d64; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-success .file-cta { - background-color: #008438; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-success:hover .file-cta, html.theme--documenter-dark .file.is-success.is-hovered .file-cta { - background-color: #007733; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-success:focus .file-cta, html.theme--documenter-dark .file.is-success.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(0, 132, 56, 0.25); - color: #fff; } - html.theme--documenter-dark .file.is-success:active .file-cta, html.theme--documenter-dark .file.is-success.is-active .file-cta { - background-color: #006b2d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-warning .file-cta { - background-color: #ad8100; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-warning:hover .file-cta, html.theme--documenter-dark .file.is-warning.is-hovered .file-cta { - background-color: #a07700; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-warning:focus .file-cta, html.theme--documenter-dark .file.is-warning.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(173, 129, 0, 0.25); - color: #fff; } - html.theme--documenter-dark .file.is-warning:active .file-cta, html.theme--documenter-dark .file.is-warning.is-active .file-cta { - background-color: #946e00; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-danger .file-cta { - background-color: #9e1b0d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-danger:hover .file-cta, html.theme--documenter-dark .file.is-danger.is-hovered .file-cta { - background-color: #92190c; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-danger:focus .file-cta, html.theme--documenter-dark .file.is-danger.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(158, 27, 13, 0.25); - color: #fff; } - html.theme--documenter-dark .file.is-danger:active .file-cta, html.theme--documenter-dark .file.is-danger.is-active .file-cta { - background-color: #86170b; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.file { - font-size: 0.85em; } - html.theme--documenter-dark .file.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .file.is-medium .file-icon .fa { - font-size: 21px; } - html.theme--documenter-dark .file.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .file.is-large .file-icon .fa { - font-size: 28px; } - html.theme--documenter-dark .file.has-name .file-cta { - border-bottom-right-radius: 0; - border-top-right-radius: 0; } - html.theme--documenter-dark .file.has-name .file-name { - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - html.theme--documenter-dark .file.has-name.is-empty .file-cta { - border-radius: 0.4em; } - html.theme--documenter-dark .file.has-name.is-empty .file-name { - display: none; } - html.theme--documenter-dark .file.is-boxed .file-label { - flex-direction: column; } - html.theme--documenter-dark .file.is-boxed .file-cta { - flex-direction: column; - height: auto; - padding: 1em 3em; } - html.theme--documenter-dark .file.is-boxed .file-name { - border-width: 0 1px 1px; } - html.theme--documenter-dark .file.is-boxed .file-icon { - height: 1.5em; - width: 1.5em; } - html.theme--documenter-dark .file.is-boxed .file-icon .fa { - font-size: 21px; } - html.theme--documenter-dark .file.is-boxed.is-small .file-icon .fa, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-boxed .file-icon .fa { - font-size: 14px; } - html.theme--documenter-dark .file.is-boxed.is-medium .file-icon .fa { - font-size: 28px; } - html.theme--documenter-dark .file.is-boxed.is-large .file-icon .fa { - font-size: 35px; } - html.theme--documenter-dark .file.is-boxed.has-name .file-cta { - border-radius: 0.4em 0.4em 0 0; } - html.theme--documenter-dark .file.is-boxed.has-name .file-name { - border-radius: 0 0 0.4em 0.4em; - border-width: 0 1px 1px; } - html.theme--documenter-dark .file.is-centered { - justify-content: center; } - html.theme--documenter-dark .file.is-fullwidth .file-label { - width: 100%; } - html.theme--documenter-dark .file.is-fullwidth .file-name { - flex-grow: 1; - max-width: none; } - html.theme--documenter-dark .file.is-right { - justify-content: flex-end; } - html.theme--documenter-dark .file.is-right .file-cta { - border-radius: 0 0.4em 0.4em 0; } - html.theme--documenter-dark .file.is-right .file-name { - border-radius: 0.4em 0 0 0.4em; - border-width: 1px 0 1px 1px; - order: -1; } - html.theme--documenter-dark .file-label { - align-items: stretch; - display: flex; - cursor: pointer; - justify-content: flex-start; - overflow: hidden; - position: relative; } - html.theme--documenter-dark .file-label:hover .file-cta { - background-color: #e5eaec; - color: #282f2f; } - html.theme--documenter-dark .file-label:hover .file-name { - border-color: #596668; } - html.theme--documenter-dark .file-label:active .file-cta { - background-color: #dde4e6; - color: #282f2f; } - html.theme--documenter-dark .file-label:active .file-name { - border-color: #535f61; } - html.theme--documenter-dark .file-input { - height: 100%; - left: 0; - opacity: 0; - outline: none; - position: absolute; - top: 0; - width: 100%; } - html.theme--documenter-dark .file-cta, - html.theme--documenter-dark .file-name { - border-color: #5e6d6f; - border-radius: 0.4em; - font-size: 1em; - padding-left: 1em; - padding-right: 1em; - white-space: nowrap; } - html.theme--documenter-dark .file-cta { - background-color: #ecf0f1; - color: #343c3d; } - html.theme--documenter-dark .file-name { - border-color: #5e6d6f; - border-style: solid; - border-width: 1px 1px 1px 0; - display: block; - max-width: 16em; - overflow: hidden; - text-align: left; - text-overflow: ellipsis; } - html.theme--documenter-dark .file-icon { - align-items: center; - display: flex; - height: 1em; - justify-content: center; - margin-right: 0.5em; - width: 1em; } - html.theme--documenter-dark .file-icon .fa { - font-size: 14px; } - html.theme--documenter-dark .label { - color: #282f2f; - display: block; - font-size: 15px; - font-weight: 700; } - html.theme--documenter-dark .label:not(:last-child) { - margin-bottom: 0.5em; } - html.theme--documenter-dark .label.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.label { - font-size: 0.85em; } - html.theme--documenter-dark .label.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .label.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .help { - display: block; - font-size: 0.85em; - margin-top: 0.25rem; } - html.theme--documenter-dark .help.is-white { - color: white; } - html.theme--documenter-dark .help.is-black { - color: #0a0a0a; } - html.theme--documenter-dark .help.is-light { - color: #ecf0f1; } - html.theme--documenter-dark .help.is-dark, html.theme--documenter-dark .content kbd.help { - color: #282f2f; } - html.theme--documenter-dark .help.is-primary, html.theme--documenter-dark .docstring > section > a.help.docs-sourcelink { - color: #375a7f; } - html.theme--documenter-dark .help.is-link { - color: #1abc9c; } - html.theme--documenter-dark .help.is-info { - color: #024c7d; } - html.theme--documenter-dark .help.is-success { - color: #008438; } - html.theme--documenter-dark .help.is-warning { - color: #ad8100; } - html.theme--documenter-dark .help.is-danger { - color: #9e1b0d; } - html.theme--documenter-dark .field:not(:last-child) { - margin-bottom: 0.75rem; } - html.theme--documenter-dark .field.has-addons { - display: flex; - justify-content: flex-start; } - html.theme--documenter-dark .field.has-addons .control:not(:last-child) { - margin-right: -1px; } - html.theme--documenter-dark .field.has-addons .control:not(:first-child):not(:last-child) .button, - html.theme--documenter-dark .field.has-addons .control:not(:first-child):not(:last-child) .input, - html.theme--documenter-dark .field.has-addons .control:not(:first-child):not(:last-child) #documenter .docs-sidebar form.docs-search > input, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control:not(:first-child):not(:last-child) form.docs-search > input, - html.theme--documenter-dark .field.has-addons .control:not(:first-child):not(:last-child) .select select { - border-radius: 0; } - html.theme--documenter-dark .field.has-addons .control:first-child:not(:only-child) .button, - html.theme--documenter-dark .field.has-addons .control:first-child:not(:only-child) .input, - html.theme--documenter-dark .field.has-addons .control:first-child:not(:only-child) #documenter .docs-sidebar form.docs-search > input, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control:first-child:not(:only-child) form.docs-search > input, - html.theme--documenter-dark .field.has-addons .control:first-child:not(:only-child) .select select { - border-bottom-right-radius: 0; - border-top-right-radius: 0; } - html.theme--documenter-dark .field.has-addons .control:last-child:not(:only-child) .button, - html.theme--documenter-dark .field.has-addons .control:last-child:not(:only-child) .input, - html.theme--documenter-dark .field.has-addons .control:last-child:not(:only-child) #documenter .docs-sidebar form.docs-search > input, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control:last-child:not(:only-child) form.docs-search > input, - html.theme--documenter-dark .field.has-addons .control:last-child:not(:only-child) .select select { - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - html.theme--documenter-dark .field.has-addons .control .button:not([disabled]):hover, html.theme--documenter-dark .field.has-addons .control .button.is-hovered:not([disabled]), - html.theme--documenter-dark .field.has-addons .control .input:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):hover, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control .input.is-hovered:not([disabled]), - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-hovered:not([disabled]), - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-hovered:not([disabled]), - html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control .select select.is-hovered:not([disabled]) { - z-index: 2; } - html.theme--documenter-dark .field.has-addons .control .button:not([disabled]):focus, html.theme--documenter-dark .field.has-addons .control .button.is-focused:not([disabled]), html.theme--documenter-dark .field.has-addons .control .button:not([disabled]):active, html.theme--documenter-dark .field.has-addons .control .button.is-active:not([disabled]), - html.theme--documenter-dark .field.has-addons .control .input:not([disabled]):focus, - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):focus, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):focus, - html.theme--documenter-dark .field.has-addons .control .input.is-focused:not([disabled]), - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-focused:not([disabled]), - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-focused:not([disabled]), - html.theme--documenter-dark .field.has-addons .control .input:not([disabled]):active, - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):active, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):active, - html.theme--documenter-dark .field.has-addons .control .input.is-active:not([disabled]), - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-active:not([disabled]), - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-active:not([disabled]), - html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]):focus, - html.theme--documenter-dark .field.has-addons .control .select select.is-focused:not([disabled]), - html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]):active, - html.theme--documenter-dark .field.has-addons .control .select select.is-active:not([disabled]) { - z-index: 3; } - html.theme--documenter-dark .field.has-addons .control .button:not([disabled]):focus:hover, html.theme--documenter-dark .field.has-addons .control .button.is-focused:not([disabled]):hover, html.theme--documenter-dark .field.has-addons .control .button:not([disabled]):active:hover, html.theme--documenter-dark .field.has-addons .control .button.is-active:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control .input:not([disabled]):focus:hover, - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):focus:hover, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):focus:hover, - html.theme--documenter-dark .field.has-addons .control .input.is-focused:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-focused:not([disabled]):hover, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-focused:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control .input:not([disabled]):active:hover, - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):active:hover, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):active:hover, - html.theme--documenter-dark .field.has-addons .control .input.is-active:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-active:not([disabled]):hover, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-active:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]):focus:hover, - html.theme--documenter-dark .field.has-addons .control .select select.is-focused:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]):active:hover, - html.theme--documenter-dark .field.has-addons .control .select select.is-active:not([disabled]):hover { - z-index: 4; } - html.theme--documenter-dark .field.has-addons .control.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - html.theme--documenter-dark .field.has-addons.has-addons-centered { - justify-content: center; } - html.theme--documenter-dark .field.has-addons.has-addons-right { - justify-content: flex-end; } - html.theme--documenter-dark .field.has-addons.has-addons-fullwidth .control { - flex-grow: 1; - flex-shrink: 0; } - html.theme--documenter-dark .field.is-grouped { - display: flex; - justify-content: flex-start; } - html.theme--documenter-dark .field.is-grouped > .control { - flex-shrink: 0; } - html.theme--documenter-dark .field.is-grouped > .control:not(:last-child) { - margin-bottom: 0; - margin-right: 0.75rem; } - html.theme--documenter-dark .field.is-grouped > .control.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - html.theme--documenter-dark .field.is-grouped.is-grouped-centered { - justify-content: center; } - html.theme--documenter-dark .field.is-grouped.is-grouped-right { - justify-content: flex-end; } - html.theme--documenter-dark .field.is-grouped.is-grouped-multiline { - flex-wrap: wrap; } - html.theme--documenter-dark .field.is-grouped.is-grouped-multiline > .control:last-child, html.theme--documenter-dark .field.is-grouped.is-grouped-multiline > .control:not(:last-child) { - margin-bottom: 0.75rem; } - html.theme--documenter-dark .field.is-grouped.is-grouped-multiline:last-child { - margin-bottom: -0.75rem; } - html.theme--documenter-dark .field.is-grouped.is-grouped-multiline:not(:last-child) { - margin-bottom: 0; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .field.is-horizontal { - display: flex; } } - html.theme--documenter-dark .field-label .label { - font-size: inherit; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .field-label { - margin-bottom: 0.5rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .field-label { - flex-basis: 0; - flex-grow: 1; - flex-shrink: 0; - margin-right: 1.5rem; - text-align: right; } - html.theme--documenter-dark .field-label.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.field-label { - font-size: 0.85em; - padding-top: 0.375em; } - html.theme--documenter-dark .field-label.is-normal { - padding-top: 0.375em; } - html.theme--documenter-dark .field-label.is-medium { - font-size: 1.25rem; - padding-top: 0.375em; } - html.theme--documenter-dark .field-label.is-large { - font-size: 1.5rem; - padding-top: 0.375em; } } - html.theme--documenter-dark .field-body .field .field { - margin-bottom: 0; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .field-body { - display: flex; - flex-basis: 0; - flex-grow: 5; - flex-shrink: 1; } - html.theme--documenter-dark .field-body .field { - margin-bottom: 0; } - html.theme--documenter-dark .field-body > .field { - flex-shrink: 1; } - html.theme--documenter-dark .field-body > .field:not(.is-narrow) { - flex-grow: 1; } - html.theme--documenter-dark .field-body > .field:not(:last-child) { - margin-right: 0.75rem; } } - html.theme--documenter-dark .control { - box-sizing: border-box; - clear: both; - font-size: 15px; - position: relative; - text-align: left; } - html.theme--documenter-dark .control.has-icons-left .input:focus ~ .icon, html.theme--documenter-dark .control.has-icons-left #documenter .docs-sidebar form.docs-search > input:focus ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-left form.docs-search > input:focus ~ .icon, - html.theme--documenter-dark .control.has-icons-left .select:focus ~ .icon, html.theme--documenter-dark .control.has-icons-right .input:focus ~ .icon, html.theme--documenter-dark .control.has-icons-right #documenter .docs-sidebar form.docs-search > input:focus ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-right form.docs-search > input:focus ~ .icon, - html.theme--documenter-dark .control.has-icons-right .select:focus ~ .icon { - color: #5e6d6f; } - html.theme--documenter-dark .control.has-icons-left .input.is-small ~ .icon, html.theme--documenter-dark .control.has-icons-left #documenter .docs-sidebar form.docs-search > input ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-left form.docs-search > input ~ .icon, - html.theme--documenter-dark .control.has-icons-left .select.is-small ~ .icon, html.theme--documenter-dark .control.has-icons-right .input.is-small ~ .icon, html.theme--documenter-dark .control.has-icons-right #documenter .docs-sidebar form.docs-search > input ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-right form.docs-search > input ~ .icon, - html.theme--documenter-dark .control.has-icons-right .select.is-small ~ .icon { - font-size: 0.85em; } - html.theme--documenter-dark .control.has-icons-left .input.is-medium ~ .icon, html.theme--documenter-dark .control.has-icons-left #documenter .docs-sidebar form.docs-search > input.is-medium ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-left form.docs-search > input.is-medium ~ .icon, - html.theme--documenter-dark .control.has-icons-left .select.is-medium ~ .icon, html.theme--documenter-dark .control.has-icons-right .input.is-medium ~ .icon, html.theme--documenter-dark .control.has-icons-right #documenter .docs-sidebar form.docs-search > input.is-medium ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-right form.docs-search > input.is-medium ~ .icon, - html.theme--documenter-dark .control.has-icons-right .select.is-medium ~ .icon { - font-size: 1.25rem; } - html.theme--documenter-dark .control.has-icons-left .input.is-large ~ .icon, html.theme--documenter-dark .control.has-icons-left #documenter .docs-sidebar form.docs-search > input.is-large ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-left form.docs-search > input.is-large ~ .icon, - html.theme--documenter-dark .control.has-icons-left .select.is-large ~ .icon, html.theme--documenter-dark .control.has-icons-right .input.is-large ~ .icon, html.theme--documenter-dark .control.has-icons-right #documenter .docs-sidebar form.docs-search > input.is-large ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-right form.docs-search > input.is-large ~ .icon, - html.theme--documenter-dark .control.has-icons-right .select.is-large ~ .icon { - font-size: 1.5rem; } - html.theme--documenter-dark .control.has-icons-left .icon, html.theme--documenter-dark .control.has-icons-right .icon { - color: #dbdee0; - height: 2.25em; - pointer-events: none; - position: absolute; - top: 0; - width: 2.25em; - z-index: 4; } - html.theme--documenter-dark .control.has-icons-left .input, html.theme--documenter-dark .control.has-icons-left #documenter .docs-sidebar form.docs-search > input, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-left form.docs-search > input, - html.theme--documenter-dark .control.has-icons-left .select select { - padding-left: 2.25em; } - html.theme--documenter-dark .control.has-icons-left .icon.is-left { - left: 0; } - html.theme--documenter-dark .control.has-icons-right .input, html.theme--documenter-dark .control.has-icons-right #documenter .docs-sidebar form.docs-search > input, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-right form.docs-search > input, - html.theme--documenter-dark .control.has-icons-right .select select { - padding-right: 2.25em; } - html.theme--documenter-dark .control.has-icons-right .icon.is-right { - right: 0; } - html.theme--documenter-dark .control.is-loading::after { - position: absolute !important; - right: 0.625em; - top: 0.625em; - z-index: 4; } - html.theme--documenter-dark .control.is-loading.is-small:after, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-loading:after { - font-size: 0.85em; } - html.theme--documenter-dark .control.is-loading.is-medium:after { - font-size: 1.25rem; } - html.theme--documenter-dark .control.is-loading.is-large:after { - font-size: 1.5rem; } - html.theme--documenter-dark .breadcrumb { - font-size: 15px; - white-space: nowrap; } - html.theme--documenter-dark .breadcrumb a { - align-items: center; - color: #1abc9c; - display: flex; - justify-content: center; - padding: 0 0.75em; } - html.theme--documenter-dark .breadcrumb a:hover { - color: #1dd2af; } - html.theme--documenter-dark .breadcrumb li { - align-items: center; - display: flex; } - html.theme--documenter-dark .breadcrumb li:first-child a { - padding-left: 0; } - html.theme--documenter-dark .breadcrumb li.is-active a { - color: #f2f2f2; - cursor: default; - pointer-events: none; } - html.theme--documenter-dark .breadcrumb li + li::before { - color: #8c9b9d; - content: "\0002f"; } - html.theme--documenter-dark .breadcrumb ul, - html.theme--documenter-dark .breadcrumb ol { - align-items: flex-start; - display: flex; - flex-wrap: wrap; - justify-content: flex-start; } - html.theme--documenter-dark .breadcrumb .icon:first-child { - margin-right: 0.5em; } - html.theme--documenter-dark .breadcrumb .icon:last-child { - margin-left: 0.5em; } - html.theme--documenter-dark .breadcrumb.is-centered ol, - html.theme--documenter-dark .breadcrumb.is-centered ul { - justify-content: center; } - html.theme--documenter-dark .breadcrumb.is-right ol, - html.theme--documenter-dark .breadcrumb.is-right ul { - justify-content: flex-end; } - html.theme--documenter-dark .breadcrumb.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.breadcrumb { - font-size: 0.85em; } - html.theme--documenter-dark .breadcrumb.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .breadcrumb.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .breadcrumb.has-arrow-separator li + li::before { - content: "\02192"; } - html.theme--documenter-dark .breadcrumb.has-bullet-separator li + li::before { - content: "\02022"; } - html.theme--documenter-dark .breadcrumb.has-dot-separator li + li::before { - content: "\000b7"; } - html.theme--documenter-dark .breadcrumb.has-succeeds-separator li + li::before { - content: "\0227B"; } - html.theme--documenter-dark .card { - background-color: white; - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); - color: #fff; - max-width: 100%; - position: relative; } - html.theme--documenter-dark .card-header { - background-color: transparent; - align-items: stretch; - box-shadow: 0 1px 2px rgba(10, 10, 10, 0.1); - display: flex; } - html.theme--documenter-dark .card-header-title { - align-items: center; - color: #f2f2f2; - display: flex; - flex-grow: 1; - font-weight: 700; - padding: 0.75rem; } - html.theme--documenter-dark .card-header-title.is-centered { - justify-content: center; } - html.theme--documenter-dark .card-header-icon { - align-items: center; - cursor: pointer; - display: flex; - justify-content: center; - padding: 0.75rem; } - html.theme--documenter-dark .card-image { - display: block; - position: relative; } - html.theme--documenter-dark .card-content { - background-color: transparent; - padding: 1rem 1.25rem; } - html.theme--documenter-dark .card-footer { - background-color: transparent; - border-top: 1px solid #5e6d6f; - align-items: stretch; - display: flex; } - html.theme--documenter-dark .card-footer-item { - align-items: center; - display: flex; - flex-basis: 0; - flex-grow: 1; - flex-shrink: 0; - justify-content: center; - padding: 0.75rem; } - html.theme--documenter-dark .card-footer-item:not(:last-child) { - border-right: 1px solid #5e6d6f; } - html.theme--documenter-dark .card .media:not(:last-child) { - margin-bottom: 1.5rem; } - html.theme--documenter-dark .dropdown { - display: inline-flex; - position: relative; - vertical-align: top; } - html.theme--documenter-dark .dropdown.is-active .dropdown-menu, html.theme--documenter-dark .dropdown.is-hoverable:hover .dropdown-menu { - display: block; } - html.theme--documenter-dark .dropdown.is-right .dropdown-menu { - left: auto; - right: 0; } - html.theme--documenter-dark .dropdown.is-up .dropdown-menu { - bottom: 100%; - padding-bottom: 4px; - padding-top: initial; - top: auto; } - html.theme--documenter-dark .dropdown-menu { - display: none; - left: 0; - min-width: 12rem; - padding-top: 4px; - position: absolute; - top: 100%; - z-index: 20; } - html.theme--documenter-dark .dropdown-content { - background-color: #282f2f; - border-radius: 0.4em; - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); - padding-bottom: 0.5rem; - padding-top: 0.5rem; } - html.theme--documenter-dark .dropdown-item { - color: #fff; - display: block; - font-size: 0.875rem; - line-height: 1.5; - padding: 0.375rem 1rem; - position: relative; } - html.theme--documenter-dark a.dropdown-item, - html.theme--documenter-dark button.dropdown-item { - padding-right: 3rem; - text-align: left; - white-space: nowrap; - width: 100%; } - html.theme--documenter-dark a.dropdown-item:hover, - html.theme--documenter-dark button.dropdown-item:hover { - background-color: #282f2f; - color: #0a0a0a; } - html.theme--documenter-dark a.dropdown-item.is-active, - html.theme--documenter-dark button.dropdown-item.is-active { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .dropdown-divider { - background-color: #5e6d6f; - border: none; - display: block; - height: 1px; - margin: 0.5rem 0; } - html.theme--documenter-dark .level { - align-items: center; - justify-content: space-between; } - html.theme--documenter-dark .level code { - border-radius: 0.4em; } - html.theme--documenter-dark .level img { - display: inline-block; - vertical-align: top; } - html.theme--documenter-dark .level.is-mobile { - display: flex; } - html.theme--documenter-dark .level.is-mobile .level-left, - html.theme--documenter-dark .level.is-mobile .level-right { - display: flex; } - html.theme--documenter-dark .level.is-mobile .level-left + .level-right { - margin-top: 0; } - html.theme--documenter-dark .level.is-mobile .level-item:not(:last-child) { - margin-bottom: 0; - margin-right: 0.75rem; } - html.theme--documenter-dark .level.is-mobile .level-item:not(.is-narrow) { - flex-grow: 1; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .level { - display: flex; } - html.theme--documenter-dark .level > .level-item:not(.is-narrow) { - flex-grow: 1; } } - html.theme--documenter-dark .level-item { - align-items: center; - display: flex; - flex-basis: auto; - flex-grow: 0; - flex-shrink: 0; - justify-content: center; } - html.theme--documenter-dark .level-item .title, - html.theme--documenter-dark .level-item .subtitle { - margin-bottom: 0; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .level-item:not(:last-child) { - margin-bottom: 0.75rem; } } - html.theme--documenter-dark .level-left, - html.theme--documenter-dark .level-right { - flex-basis: auto; - flex-grow: 0; - flex-shrink: 0; } - html.theme--documenter-dark .level-left .level-item.is-flexible, - html.theme--documenter-dark .level-right .level-item.is-flexible { - flex-grow: 1; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .level-left .level-item:not(:last-child), - html.theme--documenter-dark .level-right .level-item:not(:last-child) { - margin-right: 0.75rem; } } - html.theme--documenter-dark .level-left { - align-items: center; - justify-content: flex-start; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .level-left + .level-right { - margin-top: 1.5rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .level-left { - display: flex; } } - html.theme--documenter-dark .level-right { - align-items: center; - justify-content: flex-end; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .level-right { - display: flex; } } - html.theme--documenter-dark .list { - background-color: white; - border-radius: 0.4em; - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .list-item { - display: block; - padding: 0.5em 1em; } - html.theme--documenter-dark .list-item:not(a) { - color: #fff; } - html.theme--documenter-dark .list-item:first-child { - border-top-left-radius: 0.4em; - border-top-right-radius: 0.4em; } - html.theme--documenter-dark .list-item:last-child { - border-bottom-left-radius: 0.4em; - border-bottom-right-radius: 0.4em; } - html.theme--documenter-dark .list-item:not(:last-child) { - border-bottom: 1px solid #5e6d6f; } - html.theme--documenter-dark .list-item.is-active { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark a.list-item { - background-color: #282f2f; - cursor: pointer; } - html.theme--documenter-dark .media { - align-items: flex-start; - display: flex; - text-align: left; } - html.theme--documenter-dark .media .content:not(:last-child) { - margin-bottom: 0.75rem; } - html.theme--documenter-dark .media .media { - border-top: 1px solid rgba(94, 109, 111, 0.5); - display: flex; - padding-top: 0.75rem; } - html.theme--documenter-dark .media .media .content:not(:last-child), - html.theme--documenter-dark .media .media .control:not(:last-child) { - margin-bottom: 0.5rem; } - html.theme--documenter-dark .media .media .media { - padding-top: 0.5rem; } - html.theme--documenter-dark .media .media .media + .media { - margin-top: 0.5rem; } - html.theme--documenter-dark .media + .media { - border-top: 1px solid rgba(94, 109, 111, 0.5); - margin-top: 1rem; - padding-top: 1rem; } - html.theme--documenter-dark .media.is-large + .media { - margin-top: 1.5rem; - padding-top: 1.5rem; } - html.theme--documenter-dark .media-left, - html.theme--documenter-dark .media-right { - flex-basis: auto; - flex-grow: 0; - flex-shrink: 0; } - html.theme--documenter-dark .media-left { - margin-right: 1rem; } - html.theme--documenter-dark .media-right { - margin-left: 1rem; } - html.theme--documenter-dark .media-content { - flex-basis: auto; - flex-grow: 1; - flex-shrink: 1; - text-align: left; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .media-content { - overflow-x: auto; } } - html.theme--documenter-dark .menu { - font-size: 15px; } - html.theme--documenter-dark .menu.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.menu { - font-size: 0.85em; } - html.theme--documenter-dark .menu.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .menu.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .menu-list { - line-height: 1.25; } - html.theme--documenter-dark .menu-list a { - border-radius: 3px; - color: #fff; - display: block; - padding: 0.5em 0.75em; } - html.theme--documenter-dark .menu-list a:hover { - background-color: #282f2f; - color: #f2f2f2; } - html.theme--documenter-dark .menu-list a.is-active { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .menu-list li ul { - border-left: 1px solid #5e6d6f; - margin: 0.75em; - padding-left: 0.75em; } - html.theme--documenter-dark .menu-label { - color: white; - font-size: 0.75em; - letter-spacing: 0.1em; - text-transform: uppercase; } - html.theme--documenter-dark .menu-label:not(:first-child) { - margin-top: 1em; } - html.theme--documenter-dark .menu-label:not(:last-child) { - margin-bottom: 1em; } - html.theme--documenter-dark .message { - background-color: #282f2f; - border-radius: 0.4em; - font-size: 15px; } - html.theme--documenter-dark .message strong { - color: currentColor; } - html.theme--documenter-dark .message a:not(.button):not(.tag):not(.dropdown-item) { - color: currentColor; - text-decoration: underline; } - html.theme--documenter-dark .message.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.message { - font-size: 0.85em; } - html.theme--documenter-dark .message.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .message.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .message.is-white { - background-color: white; } - html.theme--documenter-dark .message.is-white .message-header { - background-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .message.is-white .message-body { - border-color: white; - color: #4d4d4d; } - html.theme--documenter-dark .message.is-black { - background-color: #fafafa; } - html.theme--documenter-dark .message.is-black .message-header { - background-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .message.is-black .message-body { - border-color: #0a0a0a; - color: #090909; } - html.theme--documenter-dark .message.is-light { - background-color: #f9fafb; } - html.theme--documenter-dark .message.is-light .message-header { - background-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .message.is-light .message-body { - border-color: #ecf0f1; - color: #505050; } - html.theme--documenter-dark .message.is-dark, html.theme--documenter-dark .content kbd.message { - background-color: #f9fafa; } - html.theme--documenter-dark .message.is-dark .message-header, html.theme--documenter-dark .content kbd.message .message-header { - background-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .message.is-dark .message-body, html.theme--documenter-dark .content kbd.message .message-body { - border-color: #282f2f; - color: #212526; } - html.theme--documenter-dark .message.is-primary, html.theme--documenter-dark .docstring > section > a.message.docs-sourcelink { - background-color: #f8fafc; } - html.theme--documenter-dark .message.is-primary .message-header, html.theme--documenter-dark .docstring > section > a.message.docs-sourcelink .message-header { - background-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .message.is-primary .message-body, html.theme--documenter-dark .docstring > section > a.message.docs-sourcelink .message-body { - border-color: #375a7f; - color: #2b4159; } - html.theme--documenter-dark .message.is-link { - background-color: #f6fefc; } - html.theme--documenter-dark .message.is-link .message-header { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .message.is-link .message-body { - border-color: #1abc9c; - color: #0b2f28; } - html.theme--documenter-dark .message.is-info { - background-color: #f5fbff; } - html.theme--documenter-dark .message.is-info .message-header { - background-color: #024c7d; - color: #fff; } - html.theme--documenter-dark .message.is-info .message-body { - border-color: #024c7d; - color: #033659; } - html.theme--documenter-dark .message.is-success { - background-color: #f5fff9; } - html.theme--documenter-dark .message.is-success .message-header { - background-color: #008438; - color: #fff; } - html.theme--documenter-dark .message.is-success .message-body { - border-color: #008438; - color: #023518; } - html.theme--documenter-dark .message.is-warning { - background-color: #fffcf5; } - html.theme--documenter-dark .message.is-warning .message-header { - background-color: #ad8100; - color: #fff; } - html.theme--documenter-dark .message.is-warning .message-body { - border-color: #ad8100; - color: #3d2e03; } - html.theme--documenter-dark .message.is-danger { - background-color: #fef6f6; } - html.theme--documenter-dark .message.is-danger .message-header { - background-color: #9e1b0d; - color: #fff; } - html.theme--documenter-dark .message.is-danger .message-body { - border-color: #9e1b0d; - color: #7a170c; } - html.theme--documenter-dark .message-header { - align-items: center; - background-color: #fff; - border-radius: 0.4em 0.4em 0 0; - color: rgba(0, 0, 0, 0.7); - display: flex; - font-weight: 700; - justify-content: space-between; - line-height: 1.25; - padding: 0.75em; - position: relative; } - html.theme--documenter-dark .message-header .delete { - flex-grow: 0; - flex-shrink: 0; - margin-left: 0.75em; } - html.theme--documenter-dark .message-header + .message-body { - border-width: 0; - border-top-left-radius: 0; - border-top-right-radius: 0; } - html.theme--documenter-dark .message-body { - border-color: #5e6d6f; - border-radius: 0.4em; - border-style: solid; - border-width: 0 0 0 4px; - color: #fff; - padding: 1em 1.25em; } - html.theme--documenter-dark .message-body code, - html.theme--documenter-dark .message-body pre { - background-color: white; } - html.theme--documenter-dark .message-body pre code { - background-color: transparent; } - html.theme--documenter-dark .modal { - align-items: center; - display: none; - flex-direction: column; - justify-content: center; - overflow: hidden; - position: fixed; - z-index: 40; } - html.theme--documenter-dark .modal.is-active { - display: flex; } - html.theme--documenter-dark .modal-background { - background-color: rgba(10, 10, 10, 0.86); } - html.theme--documenter-dark .modal-content, - html.theme--documenter-dark .modal-card { - margin: 0 20px; - max-height: calc(100vh - 160px); - overflow: auto; - position: relative; - width: 100%; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .modal-content, - html.theme--documenter-dark .modal-card { - margin: 0 auto; - max-height: calc(100vh - 40px); - width: 640px; } } - html.theme--documenter-dark .modal-close { - background: none; - height: 40px; - position: fixed; - right: 20px; - top: 20px; - width: 40px; } - html.theme--documenter-dark .modal-card { - display: flex; - flex-direction: column; - max-height: calc(100vh - 40px); - overflow: hidden; - -ms-overflow-y: visible; } - html.theme--documenter-dark .modal-card-head, - html.theme--documenter-dark .modal-card-foot { - align-items: center; - background-color: #282f2f; - display: flex; - flex-shrink: 0; - justify-content: flex-start; - padding: 20px; - position: relative; } - html.theme--documenter-dark .modal-card-head { - border-bottom: 1px solid #5e6d6f; - border-top-left-radius: 8px; - border-top-right-radius: 8px; } - html.theme--documenter-dark .modal-card-title { - color: #f2f2f2; - flex-grow: 1; - flex-shrink: 0; - font-size: 1.5rem; - line-height: 1; } - html.theme--documenter-dark .modal-card-foot { - border-bottom-left-radius: 8px; - border-bottom-right-radius: 8px; - border-top: 1px solid #5e6d6f; } - html.theme--documenter-dark .modal-card-foot .button:not(:last-child) { - margin-right: 0.5em; } - html.theme--documenter-dark .modal-card-body { - -webkit-overflow-scrolling: touch; - background-color: white; - flex-grow: 1; - flex-shrink: 1; - overflow: auto; - padding: 20px; } - html.theme--documenter-dark .navbar { - background-color: #375a7f; - min-height: 4rem; - position: relative; - z-index: 30; } - html.theme--documenter-dark .navbar.is-white { - background-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-white .navbar-brand .navbar-link { - color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-white .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-white .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-white .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-white .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-white .navbar-brand .navbar-link.is-active { - background-color: #f2f2f2; - color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-brand .navbar-link::after { - border-color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-burger { - color: #0a0a0a; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-white .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-white .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-white .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-white .navbar-end .navbar-link { - color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-white .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-white .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-white .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-white .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-white .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-white .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-white .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-white .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-white .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-white .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-white .navbar-end .navbar-link.is-active { - background-color: #f2f2f2; - color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-white .navbar-end .navbar-link::after { - border-color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-white .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-white .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #f2f2f2; - color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-dropdown a.navbar-item.is-active { - background-color: white; - color: #0a0a0a; } } - html.theme--documenter-dark .navbar.is-black { - background-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-black .navbar-brand .navbar-link { - color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-black .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-black .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-black .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-black .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-black .navbar-brand .navbar-link.is-active { - background-color: black; - color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-brand .navbar-link::after { - border-color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-burger { - color: white; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-black .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-black .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-black .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-black .navbar-end .navbar-link { - color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-black .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-black .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-black .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-black .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-black .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-black .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-black .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-black .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-black .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-black .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-black .navbar-end .navbar-link.is-active { - background-color: black; - color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-black .navbar-end .navbar-link::after { - border-color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-black .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-black .navbar-item.has-dropdown.is-active .navbar-link { - background-color: black; - color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-dropdown a.navbar-item.is-active { - background-color: #0a0a0a; - color: white; } } - html.theme--documenter-dark .navbar.is-light { - background-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-light .navbar-brand .navbar-link { - color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-light .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-light .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-light .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-light .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-light .navbar-brand .navbar-link.is-active { - background-color: #dde4e6; - color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-brand .navbar-link::after { - border-color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-burger { - color: #282f2f; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-light .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-light .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-light .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-light .navbar-end .navbar-link { - color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-light .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-light .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-light .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-light .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-light .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-light .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-light .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-light .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-light .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-light .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-light .navbar-end .navbar-link.is-active { - background-color: #dde4e6; - color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-light .navbar-end .navbar-link::after { - border-color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-light .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-light .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #dde4e6; - color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-dropdown a.navbar-item.is-active { - background-color: #ecf0f1; - color: #282f2f; } } - html.theme--documenter-dark .navbar.is-dark, html.theme--documenter-dark .content kbd.navbar { - background-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-brand > .navbar-item, html.theme--documenter-dark .content kbd.navbar .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-dark .navbar-brand .navbar-link, - html.theme--documenter-dark .content kbd.navbar .navbar-brand .navbar-link { - color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .content kbd.navbar .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-dark .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .content kbd.navbar .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-dark .navbar-brand > a.navbar-item.is-active, html.theme--documenter-dark .content kbd.navbar .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-dark .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .content kbd.navbar .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-dark .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .content kbd.navbar .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-dark .navbar-brand .navbar-link.is-active, - html.theme--documenter-dark .content kbd.navbar .navbar-brand .navbar-link.is-active { - background-color: #1d2122; - color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-brand .navbar-link::after, html.theme--documenter-dark .content kbd.navbar .navbar-brand .navbar-link::after { - border-color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-burger, html.theme--documenter-dark .content kbd.navbar .navbar-burger { - color: #ecf0f1; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-dark .navbar-start > .navbar-item, html.theme--documenter-dark .content kbd.navbar .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-dark .navbar-start .navbar-link, - html.theme--documenter-dark .content kbd.navbar .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-dark .navbar-end > .navbar-item, - html.theme--documenter-dark .content kbd.navbar .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-dark .navbar-end .navbar-link, - html.theme--documenter-dark .content kbd.navbar .navbar-end .navbar-link { - color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .content kbd.navbar .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-dark .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .content kbd.navbar .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-dark .navbar-start > a.navbar-item.is-active, html.theme--documenter-dark .content kbd.navbar .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-dark .navbar-start .navbar-link:focus, - html.theme--documenter-dark .content kbd.navbar .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-dark .navbar-start .navbar-link:hover, - html.theme--documenter-dark .content kbd.navbar .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-dark .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .content kbd.navbar .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-dark .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .content kbd.navbar .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-dark .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .content kbd.navbar .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-dark .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .content kbd.navbar .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-dark .navbar-end .navbar-link:focus, - html.theme--documenter-dark .content kbd.navbar .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-dark .navbar-end .navbar-link:hover, - html.theme--documenter-dark .content kbd.navbar .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-dark .navbar-end .navbar-link.is-active, - html.theme--documenter-dark .content kbd.navbar .navbar-end .navbar-link.is-active { - background-color: #1d2122; - color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-start .navbar-link::after, html.theme--documenter-dark .content kbd.navbar .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-dark .navbar-end .navbar-link::after, - html.theme--documenter-dark .content kbd.navbar .navbar-end .navbar-link::after { - border-color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-item.has-dropdown:focus .navbar-link, html.theme--documenter-dark .content kbd.navbar .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-dark .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .content kbd.navbar .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-dark .navbar-item.has-dropdown.is-active .navbar-link, - html.theme--documenter-dark .content kbd.navbar .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #1d2122; - color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-dropdown a.navbar-item.is-active, html.theme--documenter-dark .content kbd.navbar .navbar-dropdown a.navbar-item.is-active { - background-color: #282f2f; - color: #ecf0f1; } } - html.theme--documenter-dark .navbar.is-primary, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink { - background-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-brand > .navbar-item, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-primary .navbar-brand .navbar-link, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-primary .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-primary .navbar-brand > a.navbar-item.is-active, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-primary .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-primary .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-primary .navbar-brand .navbar-link.is-active, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link.is-active { - background-color: #2f4d6d; - color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-brand .navbar-link::after, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-burger, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-primary .navbar-start > .navbar-item, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-primary .navbar-start .navbar-link, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-primary .navbar-end > .navbar-item, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-primary .navbar-end .navbar-link, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-primary .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-primary .navbar-start > a.navbar-item.is-active, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-primary .navbar-start .navbar-link:focus, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-primary .navbar-start .navbar-link:hover, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-primary .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-primary .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-primary .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-primary .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-primary .navbar-end .navbar-link:focus, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-primary .navbar-end .navbar-link:hover, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-primary .navbar-end .navbar-link.is-active, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link.is-active { - background-color: #2f4d6d; - color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-start .navbar-link::after, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-primary .navbar-end .navbar-link::after, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-item.has-dropdown:focus .navbar-link, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-primary .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-primary .navbar-item.has-dropdown.is-active .navbar-link, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #2f4d6d; - color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-dropdown a.navbar-item.is-active, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-dropdown a.navbar-item.is-active { - background-color: #375a7f; - color: #fff; } } - html.theme--documenter-dark .navbar.is-link { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-link .navbar-brand .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-link .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-link .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-link .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-link .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-link .navbar-brand .navbar-link.is-active { - background-color: #17a689; - color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-brand .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-link .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-link .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-link .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-link .navbar-end .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-link .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-link .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-link .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-link .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-link .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-link .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-link .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-link .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-link .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-link .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-link .navbar-end .navbar-link.is-active { - background-color: #17a689; - color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-link .navbar-end .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-link .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-link .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #17a689; - color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-dropdown a.navbar-item.is-active { - background-color: #1abc9c; - color: #fff; } } - html.theme--documenter-dark .navbar.is-info { - background-color: #024c7d; - color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-info .navbar-brand .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-info .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-info .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-info .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-info .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-info .navbar-brand .navbar-link.is-active { - background-color: #023d64; - color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-brand .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-info .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-info .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-info .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-info .navbar-end .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-info .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-info .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-info .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-info .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-info .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-info .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-info .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-info .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-info .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-info .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-info .navbar-end .navbar-link.is-active { - background-color: #023d64; - color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-info .navbar-end .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-info .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-info .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #023d64; - color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-dropdown a.navbar-item.is-active { - background-color: #024c7d; - color: #fff; } } - html.theme--documenter-dark .navbar.is-success { - background-color: #008438; - color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-success .navbar-brand .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-success .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-success .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-success .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-success .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-success .navbar-brand .navbar-link.is-active { - background-color: #006b2d; - color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-brand .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-success .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-success .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-success .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-success .navbar-end .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-success .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-success .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-success .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-success .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-success .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-success .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-success .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-success .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-success .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-success .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-success .navbar-end .navbar-link.is-active { - background-color: #006b2d; - color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-success .navbar-end .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-success .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-success .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #006b2d; - color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-dropdown a.navbar-item.is-active { - background-color: #008438; - color: #fff; } } - html.theme--documenter-dark .navbar.is-warning { - background-color: #ad8100; - color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-warning .navbar-brand .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-warning .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-warning .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-warning .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-warning .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-warning .navbar-brand .navbar-link.is-active { - background-color: #946e00; - color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-brand .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-warning .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-warning .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-warning .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-warning .navbar-end .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-warning .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-warning .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-warning .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-warning .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-warning .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-warning .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-warning .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-warning .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-warning .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-warning .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-warning .navbar-end .navbar-link.is-active { - background-color: #946e00; - color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-warning .navbar-end .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-warning .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-warning .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #946e00; - color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-dropdown a.navbar-item.is-active { - background-color: #ad8100; - color: #fff; } } - html.theme--documenter-dark .navbar.is-danger { - background-color: #9e1b0d; - color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-danger .navbar-brand .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-danger .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-danger .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-danger .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-danger .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-danger .navbar-brand .navbar-link.is-active { - background-color: #86170b; - color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-brand .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-danger .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-danger .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-danger .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-danger .navbar-end .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-danger .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-danger .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-danger .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-danger .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-danger .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-danger .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-danger .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-danger .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-danger .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-danger .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-danger .navbar-end .navbar-link.is-active { - background-color: #86170b; - color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-danger .navbar-end .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-danger .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-danger .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #86170b; - color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-dropdown a.navbar-item.is-active { - background-color: #9e1b0d; - color: #fff; } } - html.theme--documenter-dark .navbar > .container { - align-items: stretch; - display: flex; - min-height: 4rem; - width: 100%; } - html.theme--documenter-dark .navbar.has-shadow { - box-shadow: 0 2px 0 0 #282f2f; } - html.theme--documenter-dark .navbar.is-fixed-bottom, html.theme--documenter-dark .navbar.is-fixed-top { - left: 0; - position: fixed; - right: 0; - z-index: 30; } - html.theme--documenter-dark .navbar.is-fixed-bottom { - bottom: 0; } - html.theme--documenter-dark .navbar.is-fixed-bottom.has-shadow { - box-shadow: 0 -2px 0 0 #282f2f; } - html.theme--documenter-dark .navbar.is-fixed-top { - top: 0; } - html.theme--documenter-dark html.has-navbar-fixed-top, - html.theme--documenter-dark body.has-navbar-fixed-top { - padding-top: 4rem; } - html.theme--documenter-dark html.has-navbar-fixed-bottom, - html.theme--documenter-dark body.has-navbar-fixed-bottom { - padding-bottom: 4rem; } - html.theme--documenter-dark .navbar-brand, - html.theme--documenter-dark .navbar-tabs { - align-items: stretch; - display: flex; - flex-shrink: 0; - min-height: 4rem; } - html.theme--documenter-dark .navbar-brand a.navbar-item:focus, html.theme--documenter-dark .navbar-brand a.navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .navbar-tabs { - -webkit-overflow-scrolling: touch; - max-width: 100vw; - overflow-x: auto; - overflow-y: hidden; } - html.theme--documenter-dark .navbar-burger { - color: #fff; - cursor: pointer; - display: block; - height: 4rem; - position: relative; - width: 4rem; - margin-left: auto; } - html.theme--documenter-dark .navbar-burger span { - background-color: currentColor; - display: block; - height: 1px; - left: calc(50% - 8px); - position: absolute; - transform-origin: center; - transition-duration: 86ms; - transition-property: background-color, opacity, transform; - transition-timing-function: ease-out; - width: 16px; } - html.theme--documenter-dark .navbar-burger span:nth-child(1) { - top: calc(50% - 6px); } - html.theme--documenter-dark .navbar-burger span:nth-child(2) { - top: calc(50% - 1px); } - html.theme--documenter-dark .navbar-burger span:nth-child(3) { - top: calc(50% + 4px); } - html.theme--documenter-dark .navbar-burger:hover { - background-color: rgba(0, 0, 0, 0.05); } - html.theme--documenter-dark .navbar-burger.is-active span:nth-child(1) { - transform: translateY(5px) rotate(45deg); } - html.theme--documenter-dark .navbar-burger.is-active span:nth-child(2) { - opacity: 0; } - html.theme--documenter-dark .navbar-burger.is-active span:nth-child(3) { - transform: translateY(-5px) rotate(-45deg); } - html.theme--documenter-dark .navbar-menu { - display: none; } - html.theme--documenter-dark .navbar-item, - html.theme--documenter-dark .navbar-link { - color: #fff; - display: block; - line-height: 1.5; - padding: 0.5rem 0.75rem; - position: relative; } - html.theme--documenter-dark .navbar-item .icon:only-child, - html.theme--documenter-dark .navbar-link .icon:only-child { - margin-left: -0.25rem; - margin-right: -0.25rem; } - html.theme--documenter-dark a.navbar-item, - html.theme--documenter-dark .navbar-link { - cursor: pointer; } - html.theme--documenter-dark a.navbar-item:focus, html.theme--documenter-dark a.navbar-item:focus-within, html.theme--documenter-dark a.navbar-item:hover, html.theme--documenter-dark a.navbar-item.is-active, - html.theme--documenter-dark .navbar-link:focus, - html.theme--documenter-dark .navbar-link:focus-within, - html.theme--documenter-dark .navbar-link:hover, - html.theme--documenter-dark .navbar-link.is-active { - background-color: transparent; - color: #1abc9c; } - html.theme--documenter-dark .navbar-item { - display: block; - flex-grow: 0; - flex-shrink: 0; } - html.theme--documenter-dark .navbar-item img { - max-height: 1.75rem; } - html.theme--documenter-dark .navbar-item.has-dropdown { - padding: 0; } - html.theme--documenter-dark .navbar-item.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - html.theme--documenter-dark .navbar-item.is-tab { - border-bottom: 1px solid transparent; - min-height: 4rem; - padding-bottom: calc(0.5rem - 1px); } - html.theme--documenter-dark .navbar-item.is-tab:focus, html.theme--documenter-dark .navbar-item.is-tab:hover { - background-color: transparent; - border-bottom-color: #1abc9c; } - html.theme--documenter-dark .navbar-item.is-tab.is-active { - background-color: transparent; - border-bottom-color: #1abc9c; - border-bottom-style: solid; - border-bottom-width: 3px; - color: #1abc9c; - padding-bottom: calc(0.5rem - 3px); } - html.theme--documenter-dark .navbar-content { - flex-grow: 1; - flex-shrink: 1; } - html.theme--documenter-dark .navbar-link:not(.is-arrowless) { - padding-right: 2.5em; } - html.theme--documenter-dark .navbar-link:not(.is-arrowless)::after { - border-color: #fff; - margin-top: -0.375em; - right: 1.125em; } - html.theme--documenter-dark .navbar-dropdown { - font-size: 0.875rem; - padding-bottom: 0.5rem; - padding-top: 0.5rem; } - html.theme--documenter-dark .navbar-dropdown .navbar-item { - padding-left: 1.5rem; - padding-right: 1.5rem; } - html.theme--documenter-dark .navbar-divider { - background-color: rgba(0, 0, 0, 0.2); - border: none; - display: none; - height: 2px; - margin: 0.5rem 0; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .navbar > .container { - display: block; } - html.theme--documenter-dark .navbar-brand .navbar-item, - html.theme--documenter-dark .navbar-tabs .navbar-item { - align-items: center; - display: flex; } - html.theme--documenter-dark .navbar-link::after { - display: none; } - html.theme--documenter-dark .navbar-menu { - background-color: #375a7f; - box-shadow: 0 8px 16px rgba(10, 10, 10, 0.1); - padding: 0.5rem 0; } - html.theme--documenter-dark .navbar-menu.is-active { - display: block; } - html.theme--documenter-dark .navbar.is-fixed-bottom-touch, html.theme--documenter-dark .navbar.is-fixed-top-touch { - left: 0; - position: fixed; - right: 0; - z-index: 30; } - html.theme--documenter-dark .navbar.is-fixed-bottom-touch { - bottom: 0; } - html.theme--documenter-dark .navbar.is-fixed-bottom-touch.has-shadow { - box-shadow: 0 -2px 3px rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .navbar.is-fixed-top-touch { - top: 0; } - html.theme--documenter-dark .navbar.is-fixed-top .navbar-menu, html.theme--documenter-dark .navbar.is-fixed-top-touch .navbar-menu { - -webkit-overflow-scrolling: touch; - max-height: calc(100vh - 4rem); - overflow: auto; } - html.theme--documenter-dark html.has-navbar-fixed-top-touch, - html.theme--documenter-dark body.has-navbar-fixed-top-touch { - padding-top: 4rem; } - html.theme--documenter-dark html.has-navbar-fixed-bottom-touch, - html.theme--documenter-dark body.has-navbar-fixed-bottom-touch { - padding-bottom: 4rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar, - html.theme--documenter-dark .navbar-menu, - html.theme--documenter-dark .navbar-start, - html.theme--documenter-dark .navbar-end { - align-items: stretch; - display: flex; } - html.theme--documenter-dark .navbar { - min-height: 4rem; } - html.theme--documenter-dark .navbar.is-spaced { - padding: 1rem 2rem; } - html.theme--documenter-dark .navbar.is-spaced .navbar-start, - html.theme--documenter-dark .navbar.is-spaced .navbar-end { - align-items: center; } - html.theme--documenter-dark .navbar.is-spaced a.navbar-item, - html.theme--documenter-dark .navbar.is-spaced .navbar-link { - border-radius: 0.4em; } - html.theme--documenter-dark .navbar.is-transparent a.navbar-item:focus, html.theme--documenter-dark .navbar.is-transparent a.navbar-item:hover, html.theme--documenter-dark .navbar.is-transparent a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-transparent .navbar-link:focus, - html.theme--documenter-dark .navbar.is-transparent .navbar-link:hover, - html.theme--documenter-dark .navbar.is-transparent .navbar-link.is-active { - background-color: transparent !important; } - html.theme--documenter-dark .navbar.is-transparent .navbar-item.has-dropdown.is-active .navbar-link, html.theme--documenter-dark .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:focus .navbar-link, html.theme--documenter-dark .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:focus-within .navbar-link, html.theme--documenter-dark .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:hover .navbar-link { - background-color: transparent !important; } - html.theme--documenter-dark .navbar.is-transparent .navbar-dropdown a.navbar-item:focus, html.theme--documenter-dark .navbar.is-transparent .navbar-dropdown a.navbar-item:hover { - background-color: transparent; - color: #dbdee0; } - html.theme--documenter-dark .navbar.is-transparent .navbar-dropdown a.navbar-item.is-active { - background-color: transparent; - color: #1abc9c; } - html.theme--documenter-dark .navbar-burger { - display: none; } - html.theme--documenter-dark .navbar-item, - html.theme--documenter-dark .navbar-link { - align-items: center; - display: flex; } - html.theme--documenter-dark .navbar-item { - display: flex; } - html.theme--documenter-dark .navbar-item.has-dropdown { - align-items: stretch; } - html.theme--documenter-dark .navbar-item.has-dropdown-up .navbar-link::after { - transform: rotate(135deg) translate(0.25em, -0.25em); } - html.theme--documenter-dark .navbar-item.has-dropdown-up .navbar-dropdown { - border-bottom: 1px solid rgba(0, 0, 0, 0.2); - border-radius: 8px 8px 0 0; - border-top: none; - bottom: 100%; - box-shadow: 0 -8px 8px rgba(10, 10, 10, 0.1); - top: auto; } - html.theme--documenter-dark .navbar-item.is-active .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:focus .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:focus-within .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:hover .navbar-dropdown { - display: block; } - .navbar.is-spaced html.theme--documenter-dark .navbar-item.is-active .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-active .navbar-dropdown.is-boxed, .navbar.is-spaced html.theme--documenter-dark .navbar-item.is-hoverable:focus .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:focus .navbar-dropdown.is-boxed, .navbar.is-spaced html.theme--documenter-dark .navbar-item.is-hoverable:focus-within .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:focus-within .navbar-dropdown.is-boxed, .navbar.is-spaced html.theme--documenter-dark .navbar-item.is-hoverable:hover .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:hover .navbar-dropdown.is-boxed { - opacity: 1; - pointer-events: auto; - transform: translateY(0); } - html.theme--documenter-dark .navbar-menu { - flex-grow: 1; - flex-shrink: 0; } - html.theme--documenter-dark .navbar-start { - justify-content: flex-start; - margin-right: auto; } - html.theme--documenter-dark .navbar-end { - justify-content: flex-end; - margin-left: auto; } - html.theme--documenter-dark .navbar-dropdown { - background-color: #375a7f; - border-bottom-left-radius: 8px; - border-bottom-right-radius: 8px; - border-top: 1px solid rgba(0, 0, 0, 0.2); - box-shadow: 0 8px 8px rgba(10, 10, 10, 0.1); - display: none; - font-size: 0.875rem; - left: 0; - min-width: 100%; - position: absolute; - top: 100%; - z-index: 20; } - html.theme--documenter-dark .navbar-dropdown .navbar-item { - padding: 0.375rem 1rem; - white-space: nowrap; } - html.theme--documenter-dark .navbar-dropdown a.navbar-item { - padding-right: 3rem; } - html.theme--documenter-dark .navbar-dropdown a.navbar-item:focus, html.theme--documenter-dark .navbar-dropdown a.navbar-item:hover { - background-color: transparent; - color: #dbdee0; } - html.theme--documenter-dark .navbar-dropdown a.navbar-item.is-active { - background-color: transparent; - color: #1abc9c; } - .navbar.is-spaced html.theme--documenter-dark .navbar-dropdown, html.theme--documenter-dark .navbar-dropdown.is-boxed { - border-radius: 8px; - border-top: none; - box-shadow: 0 8px 8px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); - display: block; - opacity: 0; - pointer-events: none; - top: calc(100% + (-4px)); - transform: translateY(-5px); - transition-duration: 86ms; - transition-property: opacity, transform; } - html.theme--documenter-dark .navbar-dropdown.is-right { - left: auto; - right: 0; } - html.theme--documenter-dark .navbar-divider { - display: block; } - html.theme--documenter-dark .navbar > .container .navbar-brand, - html.theme--documenter-dark .container > .navbar .navbar-brand { - margin-left: -.75rem; } - html.theme--documenter-dark .navbar > .container .navbar-menu, - html.theme--documenter-dark .container > .navbar .navbar-menu { - margin-right: -.75rem; } - html.theme--documenter-dark .navbar.is-fixed-bottom-desktop, html.theme--documenter-dark .navbar.is-fixed-top-desktop { - left: 0; - position: fixed; - right: 0; - z-index: 30; } - html.theme--documenter-dark .navbar.is-fixed-bottom-desktop { - bottom: 0; } - html.theme--documenter-dark .navbar.is-fixed-bottom-desktop.has-shadow { - box-shadow: 0 -2px 3px rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .navbar.is-fixed-top-desktop { - top: 0; } - html.theme--documenter-dark html.has-navbar-fixed-top-desktop, - html.theme--documenter-dark body.has-navbar-fixed-top-desktop { - padding-top: 4rem; } - html.theme--documenter-dark html.has-navbar-fixed-bottom-desktop, - html.theme--documenter-dark body.has-navbar-fixed-bottom-desktop { - padding-bottom: 4rem; } - html.theme--documenter-dark html.has-spaced-navbar-fixed-top, - html.theme--documenter-dark body.has-spaced-navbar-fixed-top { - padding-top: 6rem; } - html.theme--documenter-dark html.has-spaced-navbar-fixed-bottom, - html.theme--documenter-dark body.has-spaced-navbar-fixed-bottom { - padding-bottom: 6rem; } - html.theme--documenter-dark a.navbar-item.is-active, - html.theme--documenter-dark .navbar-link.is-active { - color: #1abc9c; } - html.theme--documenter-dark a.navbar-item.is-active:not(:focus):not(:hover), - html.theme--documenter-dark .navbar-link.is-active:not(:focus):not(:hover) { - background-color: transparent; } - html.theme--documenter-dark .navbar-item.has-dropdown:focus .navbar-link, html.theme--documenter-dark .navbar-item.has-dropdown:hover .navbar-link, html.theme--documenter-dark .navbar-item.has-dropdown.is-active .navbar-link { - background-color: transparent; } } - html.theme--documenter-dark .hero.is-fullheight-with-navbar { - min-height: calc(100vh - 4rem); } - html.theme--documenter-dark .pagination { - font-size: 15px; - margin: -0.25rem; } - html.theme--documenter-dark .pagination.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.pagination { - font-size: 0.85em; } - html.theme--documenter-dark .pagination.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .pagination.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .pagination.is-rounded .pagination-previous, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.pagination .pagination-previous, - html.theme--documenter-dark .pagination.is-rounded .pagination-next, - html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.pagination .pagination-next { - padding-left: 1em; - padding-right: 1em; - border-radius: 290486px; } - html.theme--documenter-dark .pagination.is-rounded .pagination-link, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.pagination .pagination-link { - border-radius: 290486px; } - html.theme--documenter-dark .pagination, - html.theme--documenter-dark .pagination-list { - align-items: center; - display: flex; - justify-content: center; - text-align: center; } - html.theme--documenter-dark .pagination-previous, - html.theme--documenter-dark .pagination-next, - html.theme--documenter-dark .pagination-link, - html.theme--documenter-dark .pagination-ellipsis { - font-size: 1em; - justify-content: center; - margin: 0.25rem; - padding-left: 0.5em; - padding-right: 0.5em; - text-align: center; } - html.theme--documenter-dark .pagination-previous, - html.theme--documenter-dark .pagination-next, - html.theme--documenter-dark .pagination-link { - border-color: #5e6d6f; - color: #1abc9c; - min-width: 2.25em; } - html.theme--documenter-dark .pagination-previous:hover, - html.theme--documenter-dark .pagination-next:hover, - html.theme--documenter-dark .pagination-link:hover { - border-color: #8c9b9d; - color: #1dd2af; } - html.theme--documenter-dark .pagination-previous:focus, - html.theme--documenter-dark .pagination-next:focus, - html.theme--documenter-dark .pagination-link:focus { - border-color: #8c9b9d; } - html.theme--documenter-dark .pagination-previous:active, - html.theme--documenter-dark .pagination-next:active, - html.theme--documenter-dark .pagination-link:active { - box-shadow: inset 0 1px 2px rgba(10, 10, 10, 0.2); } - html.theme--documenter-dark .pagination-previous[disabled], - html.theme--documenter-dark .pagination-next[disabled], - html.theme--documenter-dark .pagination-link[disabled] { - background-color: #dbdee0; - border-color: #dbdee0; - box-shadow: none; - color: #5e6d6f; - opacity: 0.5; } - html.theme--documenter-dark .pagination-previous, - html.theme--documenter-dark .pagination-next { - padding-left: 0.75em; - padding-right: 0.75em; - white-space: nowrap; } - html.theme--documenter-dark .pagination-link.is-current { - background-color: #1abc9c; - border-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .pagination-ellipsis { - color: #8c9b9d; - pointer-events: none; } - html.theme--documenter-dark .pagination-list { - flex-wrap: wrap; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .pagination { - flex-wrap: wrap; } - html.theme--documenter-dark .pagination-previous, - html.theme--documenter-dark .pagination-next { - flex-grow: 1; - flex-shrink: 1; } - html.theme--documenter-dark .pagination-list li { - flex-grow: 1; - flex-shrink: 1; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .pagination-list { - flex-grow: 1; - flex-shrink: 1; - justify-content: flex-start; - order: 1; } - html.theme--documenter-dark .pagination-previous { - order: 2; } - html.theme--documenter-dark .pagination-next { - order: 3; } - html.theme--documenter-dark .pagination { - justify-content: space-between; } - html.theme--documenter-dark .pagination.is-centered .pagination-previous { - order: 1; } - html.theme--documenter-dark .pagination.is-centered .pagination-list { - justify-content: center; - order: 2; } - html.theme--documenter-dark .pagination.is-centered .pagination-next { - order: 3; } - html.theme--documenter-dark .pagination.is-right .pagination-previous { - order: 1; } - html.theme--documenter-dark .pagination.is-right .pagination-next { - order: 2; } - html.theme--documenter-dark .pagination.is-right .pagination-list { - justify-content: flex-end; - order: 3; } } - html.theme--documenter-dark .panel { - font-size: 15px; } - html.theme--documenter-dark .panel:not(:last-child) { - margin-bottom: 1.5rem; } - html.theme--documenter-dark .panel-heading, - html.theme--documenter-dark .panel-tabs, - html.theme--documenter-dark .panel-block { - border-bottom: 1px solid #5e6d6f; - border-left: 1px solid #5e6d6f; - border-right: 1px solid #5e6d6f; } - html.theme--documenter-dark .panel-heading:first-child, - html.theme--documenter-dark .panel-tabs:first-child, - html.theme--documenter-dark .panel-block:first-child { - border-top: 1px solid #5e6d6f; } - html.theme--documenter-dark .panel-heading { - background-color: #282f2f; - border-radius: 0.4em 0.4em 0 0; - color: #f2f2f2; - font-size: 1.25em; - font-weight: 300; - line-height: 1.25; - padding: 0.5em 0.75em; } - html.theme--documenter-dark .panel-tabs { - align-items: flex-end; - display: flex; - font-size: 0.875em; - justify-content: center; } - html.theme--documenter-dark .panel-tabs a { - border-bottom: 1px solid #5e6d6f; - margin-bottom: -1px; - padding: 0.5em; } - html.theme--documenter-dark .panel-tabs a.is-active { - border-bottom-color: #343c3d; - color: #17a689; } - html.theme--documenter-dark .panel-list a { - color: #fff; } - html.theme--documenter-dark .panel-list a:hover { - color: #1abc9c; } - html.theme--documenter-dark .panel-block { - align-items: center; - color: #f2f2f2; - display: flex; - justify-content: flex-start; - padding: 0.5em 0.75em; } - html.theme--documenter-dark .panel-block input[type="checkbox"] { - margin-right: 0.75em; } - html.theme--documenter-dark .panel-block > .control { - flex-grow: 1; - flex-shrink: 1; - width: 100%; } - html.theme--documenter-dark .panel-block.is-wrapped { - flex-wrap: wrap; } - html.theme--documenter-dark .panel-block.is-active { - border-left-color: #1abc9c; - color: #17a689; } - html.theme--documenter-dark .panel-block.is-active .panel-icon { - color: #1abc9c; } - html.theme--documenter-dark a.panel-block, - html.theme--documenter-dark label.panel-block { - cursor: pointer; } - html.theme--documenter-dark a.panel-block:hover, - html.theme--documenter-dark label.panel-block:hover { - background-color: #282f2f; } - html.theme--documenter-dark .panel-icon { - display: inline-block; - font-size: 14px; - height: 1em; - line-height: 1em; - text-align: center; - vertical-align: top; - width: 1em; - color: white; - margin-right: 0.75em; } - html.theme--documenter-dark .panel-icon .fa { - font-size: inherit; - line-height: inherit; } - html.theme--documenter-dark .tabs { - -webkit-overflow-scrolling: touch; - align-items: stretch; - display: flex; - font-size: 15px; - justify-content: space-between; - overflow: hidden; - overflow-x: auto; - white-space: nowrap; } - html.theme--documenter-dark .tabs a { - align-items: center; - border-bottom-color: #5e6d6f; - border-bottom-style: solid; - border-bottom-width: 1px; - color: #fff; - display: flex; - justify-content: center; - margin-bottom: -1px; - padding: 0.5em 1em; - vertical-align: top; } - html.theme--documenter-dark .tabs a:hover { - border-bottom-color: #f2f2f2; - color: #f2f2f2; } - html.theme--documenter-dark .tabs li { - display: block; } - html.theme--documenter-dark .tabs li.is-active a { - border-bottom-color: #1abc9c; - color: #1abc9c; } - html.theme--documenter-dark .tabs ul { - align-items: center; - border-bottom-color: #5e6d6f; - border-bottom-style: solid; - border-bottom-width: 1px; - display: flex; - flex-grow: 1; - flex-shrink: 0; - justify-content: flex-start; } - html.theme--documenter-dark .tabs ul.is-left { - padding-right: 0.75em; } - html.theme--documenter-dark .tabs ul.is-center { - flex: none; - justify-content: center; - padding-left: 0.75em; - padding-right: 0.75em; } - html.theme--documenter-dark .tabs ul.is-right { - justify-content: flex-end; - padding-left: 0.75em; } - html.theme--documenter-dark .tabs .icon:first-child { - margin-right: 0.5em; } - html.theme--documenter-dark .tabs .icon:last-child { - margin-left: 0.5em; } - html.theme--documenter-dark .tabs.is-centered ul { - justify-content: center; } - html.theme--documenter-dark .tabs.is-right ul { - justify-content: flex-end; } - html.theme--documenter-dark .tabs.is-boxed a { - border: 1px solid transparent; - border-radius: 0.4em 0.4em 0 0; } - html.theme--documenter-dark .tabs.is-boxed a:hover { - background-color: #282f2f; - border-bottom-color: #5e6d6f; } - html.theme--documenter-dark .tabs.is-boxed li.is-active a { - background-color: white; - border-color: #5e6d6f; - border-bottom-color: transparent !important; } - html.theme--documenter-dark .tabs.is-fullwidth li { - flex-grow: 1; - flex-shrink: 0; } - html.theme--documenter-dark .tabs.is-toggle a { - border-color: #5e6d6f; - border-style: solid; - border-width: 1px; - margin-bottom: 0; - position: relative; } - html.theme--documenter-dark .tabs.is-toggle a:hover { - background-color: #282f2f; - border-color: #8c9b9d; - z-index: 2; } - html.theme--documenter-dark .tabs.is-toggle li + li { - margin-left: -1px; } - html.theme--documenter-dark .tabs.is-toggle li:first-child a { - border-radius: 0.4em 0 0 0.4em; } - html.theme--documenter-dark .tabs.is-toggle li:last-child a { - border-radius: 0 0.4em 0.4em 0; } - html.theme--documenter-dark .tabs.is-toggle li.is-active a { - background-color: #1abc9c; - border-color: #1abc9c; - color: #fff; - z-index: 1; } - html.theme--documenter-dark .tabs.is-toggle ul { - border-bottom: none; } - html.theme--documenter-dark .tabs.is-toggle.is-toggle-rounded li:first-child a { - border-bottom-left-radius: 290486px; - border-top-left-radius: 290486px; - padding-left: 1.25em; } - html.theme--documenter-dark .tabs.is-toggle.is-toggle-rounded li:last-child a { - border-bottom-right-radius: 290486px; - border-top-right-radius: 290486px; - padding-right: 1.25em; } - html.theme--documenter-dark .tabs.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.tabs { - font-size: 0.85em; } - html.theme--documenter-dark .tabs.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .tabs.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .column { - display: block; - flex-basis: 0; - flex-grow: 1; - flex-shrink: 1; - padding: 0.75rem; } - .columns.is-mobile > html.theme--documenter-dark .column.is-narrow { - flex: none; } - .columns.is-mobile > html.theme--documenter-dark .column.is-full { - flex: none; - width: 100%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-three-quarters { - flex: none; - width: 75%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-two-thirds { - flex: none; - width: 66.6666%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-half { - flex: none; - width: 50%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-one-third { - flex: none; - width: 33.3333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-one-quarter { - flex: none; - width: 25%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-one-fifth { - flex: none; - width: 20%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-two-fifths { - flex: none; - width: 40%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-three-fifths { - flex: none; - width: 60%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-four-fifths { - flex: none; - width: 80%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-three-quarters { - margin-left: 75%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-two-thirds { - margin-left: 66.6666%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-half { - margin-left: 50%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-one-third { - margin-left: 33.3333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-one-quarter { - margin-left: 25%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-one-fifth { - margin-left: 20%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-two-fifths { - margin-left: 40%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-three-fifths { - margin-left: 60%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-four-fifths { - margin-left: 80%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-0 { - flex: none; - width: 0%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-0 { - margin-left: 0%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-1 { - flex: none; - width: 8.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-1 { - margin-left: 8.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-2 { - flex: none; - width: 16.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-2 { - margin-left: 16.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-3 { - flex: none; - width: 25%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-3 { - margin-left: 25%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-4 { - flex: none; - width: 33.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-4 { - margin-left: 33.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-5 { - flex: none; - width: 41.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-5 { - margin-left: 41.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-6 { - flex: none; - width: 50%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-6 { - margin-left: 50%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-7 { - flex: none; - width: 58.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-7 { - margin-left: 58.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-8 { - flex: none; - width: 66.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-8 { - margin-left: 66.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-9 { - flex: none; - width: 75%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-9 { - margin-left: 75%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-10 { - flex: none; - width: 83.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-10 { - margin-left: 83.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-11 { - flex: none; - width: 91.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-11 { - margin-left: 91.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-12 { - flex: none; - width: 100%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-12 { - margin-left: 100%; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .column.is-narrow-mobile { - flex: none; } - html.theme--documenter-dark .column.is-full-mobile { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-three-quarters-mobile { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-two-thirds-mobile { - flex: none; - width: 66.6666%; } - html.theme--documenter-dark .column.is-half-mobile { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-one-third-mobile { - flex: none; - width: 33.3333%; } - html.theme--documenter-dark .column.is-one-quarter-mobile { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-one-fifth-mobile { - flex: none; - width: 20%; } - html.theme--documenter-dark .column.is-two-fifths-mobile { - flex: none; - width: 40%; } - html.theme--documenter-dark .column.is-three-fifths-mobile { - flex: none; - width: 60%; } - html.theme--documenter-dark .column.is-four-fifths-mobile { - flex: none; - width: 80%; } - html.theme--documenter-dark .column.is-offset-three-quarters-mobile { - margin-left: 75%; } - html.theme--documenter-dark .column.is-offset-two-thirds-mobile { - margin-left: 66.6666%; } - html.theme--documenter-dark .column.is-offset-half-mobile { - margin-left: 50%; } - html.theme--documenter-dark .column.is-offset-one-third-mobile { - margin-left: 33.3333%; } - html.theme--documenter-dark .column.is-offset-one-quarter-mobile { - margin-left: 25%; } - html.theme--documenter-dark .column.is-offset-one-fifth-mobile { - margin-left: 20%; } - html.theme--documenter-dark .column.is-offset-two-fifths-mobile { - margin-left: 40%; } - html.theme--documenter-dark .column.is-offset-three-fifths-mobile { - margin-left: 60%; } - html.theme--documenter-dark .column.is-offset-four-fifths-mobile { - margin-left: 80%; } - html.theme--documenter-dark .column.is-0-mobile { - flex: none; - width: 0%; } - html.theme--documenter-dark .column.is-offset-0-mobile { - margin-left: 0%; } - html.theme--documenter-dark .column.is-1-mobile { - flex: none; - width: 8.3333333333%; } - html.theme--documenter-dark .column.is-offset-1-mobile { - margin-left: 8.3333333333%; } - html.theme--documenter-dark .column.is-2-mobile { - flex: none; - width: 16.6666666667%; } - html.theme--documenter-dark .column.is-offset-2-mobile { - margin-left: 16.6666666667%; } - html.theme--documenter-dark .column.is-3-mobile { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-offset-3-mobile { - margin-left: 25%; } - html.theme--documenter-dark .column.is-4-mobile { - flex: none; - width: 33.3333333333%; } - html.theme--documenter-dark .column.is-offset-4-mobile { - margin-left: 33.3333333333%; } - html.theme--documenter-dark .column.is-5-mobile { - flex: none; - width: 41.6666666667%; } - html.theme--documenter-dark .column.is-offset-5-mobile { - margin-left: 41.6666666667%; } - html.theme--documenter-dark .column.is-6-mobile { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-offset-6-mobile { - margin-left: 50%; } - html.theme--documenter-dark .column.is-7-mobile { - flex: none; - width: 58.3333333333%; } - html.theme--documenter-dark .column.is-offset-7-mobile { - margin-left: 58.3333333333%; } - html.theme--documenter-dark .column.is-8-mobile { - flex: none; - width: 66.6666666667%; } - html.theme--documenter-dark .column.is-offset-8-mobile { - margin-left: 66.6666666667%; } - html.theme--documenter-dark .column.is-9-mobile { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-offset-9-mobile { - margin-left: 75%; } - html.theme--documenter-dark .column.is-10-mobile { - flex: none; - width: 83.3333333333%; } - html.theme--documenter-dark .column.is-offset-10-mobile { - margin-left: 83.3333333333%; } - html.theme--documenter-dark .column.is-11-mobile { - flex: none; - width: 91.6666666667%; } - html.theme--documenter-dark .column.is-offset-11-mobile { - margin-left: 91.6666666667%; } - html.theme--documenter-dark .column.is-12-mobile { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-offset-12-mobile { - margin-left: 100%; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .column.is-narrow, html.theme--documenter-dark .column.is-narrow-tablet { - flex: none; } - html.theme--documenter-dark .column.is-full, html.theme--documenter-dark .column.is-full-tablet { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-three-quarters, html.theme--documenter-dark .column.is-three-quarters-tablet { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-two-thirds, html.theme--documenter-dark .column.is-two-thirds-tablet { - flex: none; - width: 66.6666%; } - html.theme--documenter-dark .column.is-half, html.theme--documenter-dark .column.is-half-tablet { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-one-third, html.theme--documenter-dark .column.is-one-third-tablet { - flex: none; - width: 33.3333%; } - html.theme--documenter-dark .column.is-one-quarter, html.theme--documenter-dark .column.is-one-quarter-tablet { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-one-fifth, html.theme--documenter-dark .column.is-one-fifth-tablet { - flex: none; - width: 20%; } - html.theme--documenter-dark .column.is-two-fifths, html.theme--documenter-dark .column.is-two-fifths-tablet { - flex: none; - width: 40%; } - html.theme--documenter-dark .column.is-three-fifths, html.theme--documenter-dark .column.is-three-fifths-tablet { - flex: none; - width: 60%; } - html.theme--documenter-dark .column.is-four-fifths, html.theme--documenter-dark .column.is-four-fifths-tablet { - flex: none; - width: 80%; } - html.theme--documenter-dark .column.is-offset-three-quarters, html.theme--documenter-dark .column.is-offset-three-quarters-tablet { - margin-left: 75%; } - html.theme--documenter-dark .column.is-offset-two-thirds, html.theme--documenter-dark .column.is-offset-two-thirds-tablet { - margin-left: 66.6666%; } - html.theme--documenter-dark .column.is-offset-half, html.theme--documenter-dark .column.is-offset-half-tablet { - margin-left: 50%; } - html.theme--documenter-dark .column.is-offset-one-third, html.theme--documenter-dark .column.is-offset-one-third-tablet { - margin-left: 33.3333%; } - html.theme--documenter-dark .column.is-offset-one-quarter, html.theme--documenter-dark .column.is-offset-one-quarter-tablet { - margin-left: 25%; } - html.theme--documenter-dark .column.is-offset-one-fifth, html.theme--documenter-dark .column.is-offset-one-fifth-tablet { - margin-left: 20%; } - html.theme--documenter-dark .column.is-offset-two-fifths, html.theme--documenter-dark .column.is-offset-two-fifths-tablet { - margin-left: 40%; } - html.theme--documenter-dark .column.is-offset-three-fifths, html.theme--documenter-dark .column.is-offset-three-fifths-tablet { - margin-left: 60%; } - html.theme--documenter-dark .column.is-offset-four-fifths, html.theme--documenter-dark .column.is-offset-four-fifths-tablet { - margin-left: 80%; } - html.theme--documenter-dark .column.is-0, html.theme--documenter-dark .column.is-0-tablet { - flex: none; - width: 0%; } - html.theme--documenter-dark .column.is-offset-0, html.theme--documenter-dark .column.is-offset-0-tablet { - margin-left: 0%; } - html.theme--documenter-dark .column.is-1, html.theme--documenter-dark .column.is-1-tablet { - flex: none; - width: 8.3333333333%; } - html.theme--documenter-dark .column.is-offset-1, html.theme--documenter-dark .column.is-offset-1-tablet { - margin-left: 8.3333333333%; } - html.theme--documenter-dark .column.is-2, html.theme--documenter-dark .column.is-2-tablet { - flex: none; - width: 16.6666666667%; } - html.theme--documenter-dark .column.is-offset-2, html.theme--documenter-dark .column.is-offset-2-tablet { - margin-left: 16.6666666667%; } - html.theme--documenter-dark .column.is-3, html.theme--documenter-dark .column.is-3-tablet { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-offset-3, html.theme--documenter-dark .column.is-offset-3-tablet { - margin-left: 25%; } - html.theme--documenter-dark .column.is-4, html.theme--documenter-dark .column.is-4-tablet { - flex: none; - width: 33.3333333333%; } - html.theme--documenter-dark .column.is-offset-4, html.theme--documenter-dark .column.is-offset-4-tablet { - margin-left: 33.3333333333%; } - html.theme--documenter-dark .column.is-5, html.theme--documenter-dark .column.is-5-tablet { - flex: none; - width: 41.6666666667%; } - html.theme--documenter-dark .column.is-offset-5, html.theme--documenter-dark .column.is-offset-5-tablet { - margin-left: 41.6666666667%; } - html.theme--documenter-dark .column.is-6, html.theme--documenter-dark .column.is-6-tablet { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-offset-6, html.theme--documenter-dark .column.is-offset-6-tablet { - margin-left: 50%; } - html.theme--documenter-dark .column.is-7, html.theme--documenter-dark .column.is-7-tablet { - flex: none; - width: 58.3333333333%; } - html.theme--documenter-dark .column.is-offset-7, html.theme--documenter-dark .column.is-offset-7-tablet { - margin-left: 58.3333333333%; } - html.theme--documenter-dark .column.is-8, html.theme--documenter-dark .column.is-8-tablet { - flex: none; - width: 66.6666666667%; } - html.theme--documenter-dark .column.is-offset-8, html.theme--documenter-dark .column.is-offset-8-tablet { - margin-left: 66.6666666667%; } - html.theme--documenter-dark .column.is-9, html.theme--documenter-dark .column.is-9-tablet { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-offset-9, html.theme--documenter-dark .column.is-offset-9-tablet { - margin-left: 75%; } - html.theme--documenter-dark .column.is-10, html.theme--documenter-dark .column.is-10-tablet { - flex: none; - width: 83.3333333333%; } - html.theme--documenter-dark .column.is-offset-10, html.theme--documenter-dark .column.is-offset-10-tablet { - margin-left: 83.3333333333%; } - html.theme--documenter-dark .column.is-11, html.theme--documenter-dark .column.is-11-tablet { - flex: none; - width: 91.6666666667%; } - html.theme--documenter-dark .column.is-offset-11, html.theme--documenter-dark .column.is-offset-11-tablet { - margin-left: 91.6666666667%; } - html.theme--documenter-dark .column.is-12, html.theme--documenter-dark .column.is-12-tablet { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-offset-12, html.theme--documenter-dark .column.is-offset-12-tablet { - margin-left: 100%; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .column.is-narrow-touch { - flex: none; } - html.theme--documenter-dark .column.is-full-touch { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-three-quarters-touch { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-two-thirds-touch { - flex: none; - width: 66.6666%; } - html.theme--documenter-dark .column.is-half-touch { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-one-third-touch { - flex: none; - width: 33.3333%; } - html.theme--documenter-dark .column.is-one-quarter-touch { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-one-fifth-touch { - flex: none; - width: 20%; } - html.theme--documenter-dark .column.is-two-fifths-touch { - flex: none; - width: 40%; } - html.theme--documenter-dark .column.is-three-fifths-touch { - flex: none; - width: 60%; } - html.theme--documenter-dark .column.is-four-fifths-touch { - flex: none; - width: 80%; } - html.theme--documenter-dark .column.is-offset-three-quarters-touch { - margin-left: 75%; } - html.theme--documenter-dark .column.is-offset-two-thirds-touch { - margin-left: 66.6666%; } - html.theme--documenter-dark .column.is-offset-half-touch { - margin-left: 50%; } - html.theme--documenter-dark .column.is-offset-one-third-touch { - margin-left: 33.3333%; } - html.theme--documenter-dark .column.is-offset-one-quarter-touch { - margin-left: 25%; } - html.theme--documenter-dark .column.is-offset-one-fifth-touch { - margin-left: 20%; } - html.theme--documenter-dark .column.is-offset-two-fifths-touch { - margin-left: 40%; } - html.theme--documenter-dark .column.is-offset-three-fifths-touch { - margin-left: 60%; } - html.theme--documenter-dark .column.is-offset-four-fifths-touch { - margin-left: 80%; } - html.theme--documenter-dark .column.is-0-touch { - flex: none; - width: 0%; } - html.theme--documenter-dark .column.is-offset-0-touch { - margin-left: 0%; } - html.theme--documenter-dark .column.is-1-touch { - flex: none; - width: 8.3333333333%; } - html.theme--documenter-dark .column.is-offset-1-touch { - margin-left: 8.3333333333%; } - html.theme--documenter-dark .column.is-2-touch { - flex: none; - width: 16.6666666667%; } - html.theme--documenter-dark .column.is-offset-2-touch { - margin-left: 16.6666666667%; } - html.theme--documenter-dark .column.is-3-touch { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-offset-3-touch { - margin-left: 25%; } - html.theme--documenter-dark .column.is-4-touch { - flex: none; - width: 33.3333333333%; } - html.theme--documenter-dark .column.is-offset-4-touch { - margin-left: 33.3333333333%; } - html.theme--documenter-dark .column.is-5-touch { - flex: none; - width: 41.6666666667%; } - html.theme--documenter-dark .column.is-offset-5-touch { - margin-left: 41.6666666667%; } - html.theme--documenter-dark .column.is-6-touch { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-offset-6-touch { - margin-left: 50%; } - html.theme--documenter-dark .column.is-7-touch { - flex: none; - width: 58.3333333333%; } - html.theme--documenter-dark .column.is-offset-7-touch { - margin-left: 58.3333333333%; } - html.theme--documenter-dark .column.is-8-touch { - flex: none; - width: 66.6666666667%; } - html.theme--documenter-dark .column.is-offset-8-touch { - margin-left: 66.6666666667%; } - html.theme--documenter-dark .column.is-9-touch { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-offset-9-touch { - margin-left: 75%; } - html.theme--documenter-dark .column.is-10-touch { - flex: none; - width: 83.3333333333%; } - html.theme--documenter-dark .column.is-offset-10-touch { - margin-left: 83.3333333333%; } - html.theme--documenter-dark .column.is-11-touch { - flex: none; - width: 91.6666666667%; } - html.theme--documenter-dark .column.is-offset-11-touch { - margin-left: 91.6666666667%; } - html.theme--documenter-dark .column.is-12-touch { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-offset-12-touch { - margin-left: 100%; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .column.is-narrow-desktop { - flex: none; } - html.theme--documenter-dark .column.is-full-desktop { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-three-quarters-desktop { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-two-thirds-desktop { - flex: none; - width: 66.6666%; } - html.theme--documenter-dark .column.is-half-desktop { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-one-third-desktop { - flex: none; - width: 33.3333%; } - html.theme--documenter-dark .column.is-one-quarter-desktop { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-one-fifth-desktop { - flex: none; - width: 20%; } - html.theme--documenter-dark .column.is-two-fifths-desktop { - flex: none; - width: 40%; } - html.theme--documenter-dark .column.is-three-fifths-desktop { - flex: none; - width: 60%; } - html.theme--documenter-dark .column.is-four-fifths-desktop { - flex: none; - width: 80%; } - html.theme--documenter-dark .column.is-offset-three-quarters-desktop { - margin-left: 75%; } - html.theme--documenter-dark .column.is-offset-two-thirds-desktop { - margin-left: 66.6666%; } - html.theme--documenter-dark .column.is-offset-half-desktop { - margin-left: 50%; } - html.theme--documenter-dark .column.is-offset-one-third-desktop { - margin-left: 33.3333%; } - html.theme--documenter-dark .column.is-offset-one-quarter-desktop { - margin-left: 25%; } - html.theme--documenter-dark .column.is-offset-one-fifth-desktop { - margin-left: 20%; } - html.theme--documenter-dark .column.is-offset-two-fifths-desktop { - margin-left: 40%; } - html.theme--documenter-dark .column.is-offset-three-fifths-desktop { - margin-left: 60%; } - html.theme--documenter-dark .column.is-offset-four-fifths-desktop { - margin-left: 80%; } - html.theme--documenter-dark .column.is-0-desktop { - flex: none; - width: 0%; } - html.theme--documenter-dark .column.is-offset-0-desktop { - margin-left: 0%; } - html.theme--documenter-dark .column.is-1-desktop { - flex: none; - width: 8.3333333333%; } - html.theme--documenter-dark .column.is-offset-1-desktop { - margin-left: 8.3333333333%; } - html.theme--documenter-dark .column.is-2-desktop { - flex: none; - width: 16.6666666667%; } - html.theme--documenter-dark .column.is-offset-2-desktop { - margin-left: 16.6666666667%; } - html.theme--documenter-dark .column.is-3-desktop { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-offset-3-desktop { - margin-left: 25%; } - html.theme--documenter-dark .column.is-4-desktop { - flex: none; - width: 33.3333333333%; } - html.theme--documenter-dark .column.is-offset-4-desktop { - margin-left: 33.3333333333%; } - html.theme--documenter-dark .column.is-5-desktop { - flex: none; - width: 41.6666666667%; } - html.theme--documenter-dark .column.is-offset-5-desktop { - margin-left: 41.6666666667%; } - html.theme--documenter-dark .column.is-6-desktop { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-offset-6-desktop { - margin-left: 50%; } - html.theme--documenter-dark .column.is-7-desktop { - flex: none; - width: 58.3333333333%; } - html.theme--documenter-dark .column.is-offset-7-desktop { - margin-left: 58.3333333333%; } - html.theme--documenter-dark .column.is-8-desktop { - flex: none; - width: 66.6666666667%; } - html.theme--documenter-dark .column.is-offset-8-desktop { - margin-left: 66.6666666667%; } - html.theme--documenter-dark .column.is-9-desktop { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-offset-9-desktop { - margin-left: 75%; } - html.theme--documenter-dark .column.is-10-desktop { - flex: none; - width: 83.3333333333%; } - html.theme--documenter-dark .column.is-offset-10-desktop { - margin-left: 83.3333333333%; } - html.theme--documenter-dark .column.is-11-desktop { - flex: none; - width: 91.6666666667%; } - html.theme--documenter-dark .column.is-offset-11-desktop { - margin-left: 91.6666666667%; } - html.theme--documenter-dark .column.is-12-desktop { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-offset-12-desktop { - margin-left: 100%; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .column.is-narrow-widescreen { - flex: none; } - html.theme--documenter-dark .column.is-full-widescreen { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-three-quarters-widescreen { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-two-thirds-widescreen { - flex: none; - width: 66.6666%; } - html.theme--documenter-dark .column.is-half-widescreen { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-one-third-widescreen { - flex: none; - width: 33.3333%; } - html.theme--documenter-dark .column.is-one-quarter-widescreen { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-one-fifth-widescreen { - flex: none; - width: 20%; } - html.theme--documenter-dark .column.is-two-fifths-widescreen { - flex: none; - width: 40%; } - html.theme--documenter-dark .column.is-three-fifths-widescreen { - flex: none; - width: 60%; } - html.theme--documenter-dark .column.is-four-fifths-widescreen { - flex: none; - width: 80%; } - html.theme--documenter-dark .column.is-offset-three-quarters-widescreen { - margin-left: 75%; } - html.theme--documenter-dark .column.is-offset-two-thirds-widescreen { - margin-left: 66.6666%; } - html.theme--documenter-dark .column.is-offset-half-widescreen { - margin-left: 50%; } - html.theme--documenter-dark .column.is-offset-one-third-widescreen { - margin-left: 33.3333%; } - html.theme--documenter-dark .column.is-offset-one-quarter-widescreen { - margin-left: 25%; } - html.theme--documenter-dark .column.is-offset-one-fifth-widescreen { - margin-left: 20%; } - html.theme--documenter-dark .column.is-offset-two-fifths-widescreen { - margin-left: 40%; } - html.theme--documenter-dark .column.is-offset-three-fifths-widescreen { - margin-left: 60%; } - html.theme--documenter-dark .column.is-offset-four-fifths-widescreen { - margin-left: 80%; } - html.theme--documenter-dark .column.is-0-widescreen { - flex: none; - width: 0%; } - html.theme--documenter-dark .column.is-offset-0-widescreen { - margin-left: 0%; } - html.theme--documenter-dark .column.is-1-widescreen { - flex: none; - width: 8.3333333333%; } - html.theme--documenter-dark .column.is-offset-1-widescreen { - margin-left: 8.3333333333%; } - html.theme--documenter-dark .column.is-2-widescreen { - flex: none; - width: 16.6666666667%; } - html.theme--documenter-dark .column.is-offset-2-widescreen { - margin-left: 16.6666666667%; } - html.theme--documenter-dark .column.is-3-widescreen { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-offset-3-widescreen { - margin-left: 25%; } - html.theme--documenter-dark .column.is-4-widescreen { - flex: none; - width: 33.3333333333%; } - html.theme--documenter-dark .column.is-offset-4-widescreen { - margin-left: 33.3333333333%; } - html.theme--documenter-dark .column.is-5-widescreen { - flex: none; - width: 41.6666666667%; } - html.theme--documenter-dark .column.is-offset-5-widescreen { - margin-left: 41.6666666667%; } - html.theme--documenter-dark .column.is-6-widescreen { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-offset-6-widescreen { - margin-left: 50%; } - html.theme--documenter-dark .column.is-7-widescreen { - flex: none; - width: 58.3333333333%; } - html.theme--documenter-dark .column.is-offset-7-widescreen { - margin-left: 58.3333333333%; } - html.theme--documenter-dark .column.is-8-widescreen { - flex: none; - width: 66.6666666667%; } - html.theme--documenter-dark .column.is-offset-8-widescreen { - margin-left: 66.6666666667%; } - html.theme--documenter-dark .column.is-9-widescreen { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-offset-9-widescreen { - margin-left: 75%; } - html.theme--documenter-dark .column.is-10-widescreen { - flex: none; - width: 83.3333333333%; } - html.theme--documenter-dark .column.is-offset-10-widescreen { - margin-left: 83.3333333333%; } - html.theme--documenter-dark .column.is-11-widescreen { - flex: none; - width: 91.6666666667%; } - html.theme--documenter-dark .column.is-offset-11-widescreen { - margin-left: 91.6666666667%; } - html.theme--documenter-dark .column.is-12-widescreen { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-offset-12-widescreen { - margin-left: 100%; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .column.is-narrow-fullhd { - flex: none; } - html.theme--documenter-dark .column.is-full-fullhd { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-three-quarters-fullhd { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-two-thirds-fullhd { - flex: none; - width: 66.6666%; } - html.theme--documenter-dark .column.is-half-fullhd { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-one-third-fullhd { - flex: none; - width: 33.3333%; } - html.theme--documenter-dark .column.is-one-quarter-fullhd { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-one-fifth-fullhd { - flex: none; - width: 20%; } - html.theme--documenter-dark .column.is-two-fifths-fullhd { - flex: none; - width: 40%; } - html.theme--documenter-dark .column.is-three-fifths-fullhd { - flex: none; - width: 60%; } - html.theme--documenter-dark .column.is-four-fifths-fullhd { - flex: none; - width: 80%; } - html.theme--documenter-dark .column.is-offset-three-quarters-fullhd { - margin-left: 75%; } - html.theme--documenter-dark .column.is-offset-two-thirds-fullhd { - margin-left: 66.6666%; } - html.theme--documenter-dark .column.is-offset-half-fullhd { - margin-left: 50%; } - html.theme--documenter-dark .column.is-offset-one-third-fullhd { - margin-left: 33.3333%; } - html.theme--documenter-dark .column.is-offset-one-quarter-fullhd { - margin-left: 25%; } - html.theme--documenter-dark .column.is-offset-one-fifth-fullhd { - margin-left: 20%; } - html.theme--documenter-dark .column.is-offset-two-fifths-fullhd { - margin-left: 40%; } - html.theme--documenter-dark .column.is-offset-three-fifths-fullhd { - margin-left: 60%; } - html.theme--documenter-dark .column.is-offset-four-fifths-fullhd { - margin-left: 80%; } - html.theme--documenter-dark .column.is-0-fullhd { - flex: none; - width: 0%; } - html.theme--documenter-dark .column.is-offset-0-fullhd { - margin-left: 0%; } - html.theme--documenter-dark .column.is-1-fullhd { - flex: none; - width: 8.3333333333%; } - html.theme--documenter-dark .column.is-offset-1-fullhd { - margin-left: 8.3333333333%; } - html.theme--documenter-dark .column.is-2-fullhd { - flex: none; - width: 16.6666666667%; } - html.theme--documenter-dark .column.is-offset-2-fullhd { - margin-left: 16.6666666667%; } - html.theme--documenter-dark .column.is-3-fullhd { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-offset-3-fullhd { - margin-left: 25%; } - html.theme--documenter-dark .column.is-4-fullhd { - flex: none; - width: 33.3333333333%; } - html.theme--documenter-dark .column.is-offset-4-fullhd { - margin-left: 33.3333333333%; } - html.theme--documenter-dark .column.is-5-fullhd { - flex: none; - width: 41.6666666667%; } - html.theme--documenter-dark .column.is-offset-5-fullhd { - margin-left: 41.6666666667%; } - html.theme--documenter-dark .column.is-6-fullhd { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-offset-6-fullhd { - margin-left: 50%; } - html.theme--documenter-dark .column.is-7-fullhd { - flex: none; - width: 58.3333333333%; } - html.theme--documenter-dark .column.is-offset-7-fullhd { - margin-left: 58.3333333333%; } - html.theme--documenter-dark .column.is-8-fullhd { - flex: none; - width: 66.6666666667%; } - html.theme--documenter-dark .column.is-offset-8-fullhd { - margin-left: 66.6666666667%; } - html.theme--documenter-dark .column.is-9-fullhd { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-offset-9-fullhd { - margin-left: 75%; } - html.theme--documenter-dark .column.is-10-fullhd { - flex: none; - width: 83.3333333333%; } - html.theme--documenter-dark .column.is-offset-10-fullhd { - margin-left: 83.3333333333%; } - html.theme--documenter-dark .column.is-11-fullhd { - flex: none; - width: 91.6666666667%; } - html.theme--documenter-dark .column.is-offset-11-fullhd { - margin-left: 91.6666666667%; } - html.theme--documenter-dark .column.is-12-fullhd { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-offset-12-fullhd { - margin-left: 100%; } } - html.theme--documenter-dark .columns { - margin-left: -0.75rem; - margin-right: -0.75rem; - margin-top: -0.75rem; } - html.theme--documenter-dark .columns:last-child { - margin-bottom: -0.75rem; } - html.theme--documenter-dark .columns:not(:last-child) { - margin-bottom: calc(1.5rem - 0.75rem); } - html.theme--documenter-dark .columns.is-centered { - justify-content: center; } - html.theme--documenter-dark .columns.is-gapless { - margin-left: 0; - margin-right: 0; - margin-top: 0; } - html.theme--documenter-dark .columns.is-gapless > .column { - margin: 0; - padding: 0 !important; } - html.theme--documenter-dark .columns.is-gapless:not(:last-child) { - margin-bottom: 1.5rem; } - html.theme--documenter-dark .columns.is-gapless:last-child { - margin-bottom: 0; } - html.theme--documenter-dark .columns.is-mobile { - display: flex; } - html.theme--documenter-dark .columns.is-multiline { - flex-wrap: wrap; } - html.theme--documenter-dark .columns.is-vcentered { - align-items: center; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns:not(.is-desktop) { - display: flex; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-desktop { - display: flex; } } - html.theme--documenter-dark .columns.is-variable { - --columnGap: 0.75rem; - margin-left: calc(-1 * var(--columnGap)); - margin-right: calc(-1 * var(--columnGap)); } - html.theme--documenter-dark .columns.is-variable .column { - padding-left: var(--columnGap); - padding-right: var(--columnGap); } - html.theme--documenter-dark .columns.is-variable.is-0 { - --columnGap: 0rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-0-mobile { - --columnGap: 0rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-0-tablet { - --columnGap: 0rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-0-tablet-only { - --columnGap: 0rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-0-touch { - --columnGap: 0rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-0-desktop { - --columnGap: 0rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-0-desktop-only { - --columnGap: 0rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-0-widescreen { - --columnGap: 0rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-0-widescreen-only { - --columnGap: 0rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-0-fullhd { - --columnGap: 0rem; } } - html.theme--documenter-dark .columns.is-variable.is-1 { - --columnGap: 0.25rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-1-mobile { - --columnGap: 0.25rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-1-tablet { - --columnGap: 0.25rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-1-tablet-only { - --columnGap: 0.25rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-1-touch { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-1-desktop { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-1-desktop-only { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-1-widescreen { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-1-widescreen-only { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-1-fullhd { - --columnGap: 0.25rem; } } - html.theme--documenter-dark .columns.is-variable.is-2 { - --columnGap: 0.5rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-2-mobile { - --columnGap: 0.5rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-2-tablet { - --columnGap: 0.5rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-2-tablet-only { - --columnGap: 0.5rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-2-touch { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-2-desktop { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-2-desktop-only { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-2-widescreen { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-2-widescreen-only { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-2-fullhd { - --columnGap: 0.5rem; } } - html.theme--documenter-dark .columns.is-variable.is-3 { - --columnGap: 0.75rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-3-mobile { - --columnGap: 0.75rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-3-tablet { - --columnGap: 0.75rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-3-tablet-only { - --columnGap: 0.75rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-3-touch { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-3-desktop { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-3-desktop-only { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-3-widescreen { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-3-widescreen-only { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-3-fullhd { - --columnGap: 0.75rem; } } - html.theme--documenter-dark .columns.is-variable.is-4 { - --columnGap: 1rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-4-mobile { - --columnGap: 1rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-4-tablet { - --columnGap: 1rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-4-tablet-only { - --columnGap: 1rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-4-touch { - --columnGap: 1rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-4-desktop { - --columnGap: 1rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-4-desktop-only { - --columnGap: 1rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-4-widescreen { - --columnGap: 1rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-4-widescreen-only { - --columnGap: 1rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-4-fullhd { - --columnGap: 1rem; } } - html.theme--documenter-dark .columns.is-variable.is-5 { - --columnGap: 1.25rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-5-mobile { - --columnGap: 1.25rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-5-tablet { - --columnGap: 1.25rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-5-tablet-only { - --columnGap: 1.25rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-5-touch { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-5-desktop { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-5-desktop-only { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-5-widescreen { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-5-widescreen-only { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-5-fullhd { - --columnGap: 1.25rem; } } - html.theme--documenter-dark .columns.is-variable.is-6 { - --columnGap: 1.5rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-6-mobile { - --columnGap: 1.5rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-6-tablet { - --columnGap: 1.5rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-6-tablet-only { - --columnGap: 1.5rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-6-touch { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-6-desktop { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-6-desktop-only { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-6-widescreen { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-6-widescreen-only { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-6-fullhd { - --columnGap: 1.5rem; } } - html.theme--documenter-dark .columns.is-variable.is-7 { - --columnGap: 1.75rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-7-mobile { - --columnGap: 1.75rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-7-tablet { - --columnGap: 1.75rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-7-tablet-only { - --columnGap: 1.75rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-7-touch { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-7-desktop { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-7-desktop-only { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-7-widescreen { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-7-widescreen-only { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-7-fullhd { - --columnGap: 1.75rem; } } - html.theme--documenter-dark .columns.is-variable.is-8 { - --columnGap: 2rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-8-mobile { - --columnGap: 2rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-8-tablet { - --columnGap: 2rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-8-tablet-only { - --columnGap: 2rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-8-touch { - --columnGap: 2rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-8-desktop { - --columnGap: 2rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-8-desktop-only { - --columnGap: 2rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-8-widescreen { - --columnGap: 2rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-8-widescreen-only { - --columnGap: 2rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-8-fullhd { - --columnGap: 2rem; } } - html.theme--documenter-dark .tile { - align-items: stretch; - display: block; - flex-basis: 0; - flex-grow: 1; - flex-shrink: 1; - min-height: min-content; } - html.theme--documenter-dark .tile.is-ancestor { - margin-left: -0.75rem; - margin-right: -0.75rem; - margin-top: -0.75rem; } - html.theme--documenter-dark .tile.is-ancestor:last-child { - margin-bottom: -0.75rem; } - html.theme--documenter-dark .tile.is-ancestor:not(:last-child) { - margin-bottom: 0.75rem; } - html.theme--documenter-dark .tile.is-child { - margin: 0 !important; } - html.theme--documenter-dark .tile.is-parent { - padding: 0.75rem; } - html.theme--documenter-dark .tile.is-vertical { - flex-direction: column; } - html.theme--documenter-dark .tile.is-vertical > .tile.is-child:not(:last-child) { - margin-bottom: 1.5rem !important; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .tile:not(.is-child) { - display: flex; } - html.theme--documenter-dark .tile.is-1 { - flex: none; - width: 8.3333333333%; } - html.theme--documenter-dark .tile.is-2 { - flex: none; - width: 16.6666666667%; } - html.theme--documenter-dark .tile.is-3 { - flex: none; - width: 25%; } - html.theme--documenter-dark .tile.is-4 { - flex: none; - width: 33.3333333333%; } - html.theme--documenter-dark .tile.is-5 { - flex: none; - width: 41.6666666667%; } - html.theme--documenter-dark .tile.is-6 { - flex: none; - width: 50%; } - html.theme--documenter-dark .tile.is-7 { - flex: none; - width: 58.3333333333%; } - html.theme--documenter-dark .tile.is-8 { - flex: none; - width: 66.6666666667%; } - html.theme--documenter-dark .tile.is-9 { - flex: none; - width: 75%; } - html.theme--documenter-dark .tile.is-10 { - flex: none; - width: 83.3333333333%; } - html.theme--documenter-dark .tile.is-11 { - flex: none; - width: 91.6666666667%; } - html.theme--documenter-dark .tile.is-12 { - flex: none; - width: 100%; } } - html.theme--documenter-dark .hero { - align-items: stretch; - display: flex; - flex-direction: column; - justify-content: space-between; } - html.theme--documenter-dark .hero .navbar { - background: none; } - html.theme--documenter-dark .hero .tabs ul { - border-bottom: none; } - html.theme--documenter-dark .hero.is-white { - background-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .hero.is-white a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-white strong { - color: inherit; } - html.theme--documenter-dark .hero.is-white .title { - color: #0a0a0a; } - html.theme--documenter-dark .hero.is-white .subtitle { - color: rgba(10, 10, 10, 0.9); } - html.theme--documenter-dark .hero.is-white .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-white .subtitle strong { - color: #0a0a0a; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-white .navbar-menu { - background-color: white; } } - html.theme--documenter-dark .hero.is-white .navbar-item, - html.theme--documenter-dark .hero.is-white .navbar-link { - color: rgba(10, 10, 10, 0.7); } - html.theme--documenter-dark .hero.is-white a.navbar-item:hover, html.theme--documenter-dark .hero.is-white a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-white .navbar-link:hover, - html.theme--documenter-dark .hero.is-white .navbar-link.is-active { - background-color: #f2f2f2; - color: #0a0a0a; } - html.theme--documenter-dark .hero.is-white .tabs a { - color: #0a0a0a; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-white .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-white .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-white .tabs.is-boxed a, html.theme--documenter-dark .hero.is-white .tabs.is-toggle a { - color: #0a0a0a; } - html.theme--documenter-dark .hero.is-white .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-white .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-white .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-white .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-white .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-white .tabs.is-toggle li.is-active a:hover { - background-color: #0a0a0a; - border-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .hero.is-white.is-bold { - background-image: linear-gradient(141deg, #e8e3e4 0%, white 71%, white 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-white.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #e8e3e4 0%, white 71%, white 100%); } } - html.theme--documenter-dark .hero.is-black { - background-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .hero.is-black a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-black strong { - color: inherit; } - html.theme--documenter-dark .hero.is-black .title { - color: white; } - html.theme--documenter-dark .hero.is-black .subtitle { - color: rgba(255, 255, 255, 0.9); } - html.theme--documenter-dark .hero.is-black .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-black .subtitle strong { - color: white; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-black .navbar-menu { - background-color: #0a0a0a; } } - html.theme--documenter-dark .hero.is-black .navbar-item, - html.theme--documenter-dark .hero.is-black .navbar-link { - color: rgba(255, 255, 255, 0.7); } - html.theme--documenter-dark .hero.is-black a.navbar-item:hover, html.theme--documenter-dark .hero.is-black a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-black .navbar-link:hover, - html.theme--documenter-dark .hero.is-black .navbar-link.is-active { - background-color: black; - color: white; } - html.theme--documenter-dark .hero.is-black .tabs a { - color: white; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-black .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-black .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-black .tabs.is-boxed a, html.theme--documenter-dark .hero.is-black .tabs.is-toggle a { - color: white; } - html.theme--documenter-dark .hero.is-black .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-black .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-black .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-black .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-black .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-black .tabs.is-toggle li.is-active a:hover { - background-color: white; - border-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .hero.is-black.is-bold { - background-image: linear-gradient(141deg, black 0%, #0a0a0a 71%, #181616 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-black.is-bold .navbar-menu { - background-image: linear-gradient(141deg, black 0%, #0a0a0a 71%, #181616 100%); } } - html.theme--documenter-dark .hero.is-light { - background-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .hero.is-light a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-light strong { - color: inherit; } - html.theme--documenter-dark .hero.is-light .title { - color: #282f2f; } - html.theme--documenter-dark .hero.is-light .subtitle { - color: rgba(40, 47, 47, 0.9); } - html.theme--documenter-dark .hero.is-light .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-light .subtitle strong { - color: #282f2f; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-light .navbar-menu { - background-color: #ecf0f1; } } - html.theme--documenter-dark .hero.is-light .navbar-item, - html.theme--documenter-dark .hero.is-light .navbar-link { - color: rgba(40, 47, 47, 0.7); } - html.theme--documenter-dark .hero.is-light a.navbar-item:hover, html.theme--documenter-dark .hero.is-light a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-light .navbar-link:hover, - html.theme--documenter-dark .hero.is-light .navbar-link.is-active { - background-color: #dde4e6; - color: #282f2f; } - html.theme--documenter-dark .hero.is-light .tabs a { - color: #282f2f; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-light .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-light .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-light .tabs.is-boxed a, html.theme--documenter-dark .hero.is-light .tabs.is-toggle a { - color: #282f2f; } - html.theme--documenter-dark .hero.is-light .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-light .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-light .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-light .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-light .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-light .tabs.is-toggle li.is-active a:hover { - background-color: #282f2f; - border-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .hero.is-light.is-bold { - background-image: linear-gradient(141deg, #cadfe0 0%, #ecf0f1 71%, #fafbfc 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-light.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #cadfe0 0%, #ecf0f1 71%, #fafbfc 100%); } } - html.theme--documenter-dark .hero.is-dark, html.theme--documenter-dark .content kbd.hero { - background-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .hero.is-dark a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), html.theme--documenter-dark .content kbd.hero a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-dark strong, - html.theme--documenter-dark .content kbd.hero strong { - color: inherit; } - html.theme--documenter-dark .hero.is-dark .title, html.theme--documenter-dark .content kbd.hero .title { - color: #ecf0f1; } - html.theme--documenter-dark .hero.is-dark .subtitle, html.theme--documenter-dark .content kbd.hero .subtitle { - color: rgba(236, 240, 241, 0.9); } - html.theme--documenter-dark .hero.is-dark .subtitle a:not(.button), html.theme--documenter-dark .content kbd.hero .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-dark .subtitle strong, - html.theme--documenter-dark .content kbd.hero .subtitle strong { - color: #ecf0f1; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-dark .navbar-menu, html.theme--documenter-dark .content kbd.hero .navbar-menu { - background-color: #282f2f; } } - html.theme--documenter-dark .hero.is-dark .navbar-item, html.theme--documenter-dark .content kbd.hero .navbar-item, - html.theme--documenter-dark .hero.is-dark .navbar-link, - html.theme--documenter-dark .content kbd.hero .navbar-link { - color: rgba(236, 240, 241, 0.7); } - html.theme--documenter-dark .hero.is-dark a.navbar-item:hover, html.theme--documenter-dark .content kbd.hero a.navbar-item:hover, html.theme--documenter-dark .hero.is-dark a.navbar-item.is-active, html.theme--documenter-dark .content kbd.hero a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-dark .navbar-link:hover, - html.theme--documenter-dark .content kbd.hero .navbar-link:hover, - html.theme--documenter-dark .hero.is-dark .navbar-link.is-active, - html.theme--documenter-dark .content kbd.hero .navbar-link.is-active { - background-color: #1d2122; - color: #ecf0f1; } - html.theme--documenter-dark .hero.is-dark .tabs a, html.theme--documenter-dark .content kbd.hero .tabs a { - color: #ecf0f1; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-dark .tabs a:hover, html.theme--documenter-dark .content kbd.hero .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-dark .tabs li.is-active a, html.theme--documenter-dark .content kbd.hero .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-dark .tabs.is-boxed a, html.theme--documenter-dark .content kbd.hero .tabs.is-boxed a, html.theme--documenter-dark .hero.is-dark .tabs.is-toggle a, html.theme--documenter-dark .content kbd.hero .tabs.is-toggle a { - color: #ecf0f1; } - html.theme--documenter-dark .hero.is-dark .tabs.is-boxed a:hover, html.theme--documenter-dark .content kbd.hero .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-dark .tabs.is-toggle a:hover, html.theme--documenter-dark .content kbd.hero .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-dark .tabs.is-boxed li.is-active a, html.theme--documenter-dark .content kbd.hero .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-dark .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-dark .tabs.is-toggle li.is-active a, html.theme--documenter-dark .content kbd.hero .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-dark .tabs.is-toggle li.is-active a:hover { - background-color: #ecf0f1; - border-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .hero.is-dark.is-bold, html.theme--documenter-dark .content kbd.hero.is-bold { - background-image: linear-gradient(141deg, #0f1615 0%, #282f2f 71%, #313c40 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-dark.is-bold .navbar-menu, html.theme--documenter-dark .content kbd.hero.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #0f1615 0%, #282f2f 71%, #313c40 100%); } } - html.theme--documenter-dark .hero.is-primary, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink { - background-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .hero.is-primary a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-primary strong, - html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink strong { - color: inherit; } - html.theme--documenter-dark .hero.is-primary .title, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .title { - color: #fff; } - html.theme--documenter-dark .hero.is-primary .subtitle, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .subtitle { - color: rgba(255, 255, 255, 0.9); } - html.theme--documenter-dark .hero.is-primary .subtitle a:not(.button), html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-primary .subtitle strong, - html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-primary .navbar-menu, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar-menu { - background-color: #375a7f; } } - html.theme--documenter-dark .hero.is-primary .navbar-item, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar-item, - html.theme--documenter-dark .hero.is-primary .navbar-link, - html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar-link { - color: rgba(255, 255, 255, 0.7); } - html.theme--documenter-dark .hero.is-primary a.navbar-item:hover, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink a.navbar-item:hover, html.theme--documenter-dark .hero.is-primary a.navbar-item.is-active, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-primary .navbar-link:hover, - html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar-link:hover, - html.theme--documenter-dark .hero.is-primary .navbar-link.is-active, - html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar-link.is-active { - background-color: #2f4d6d; - color: #fff; } - html.theme--documenter-dark .hero.is-primary .tabs a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs a { - color: #fff; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-primary .tabs a:hover, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-primary .tabs li.is-active a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-primary .tabs.is-boxed a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed a, html.theme--documenter-dark .hero.is-primary .tabs.is-toggle a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle a { - color: #fff; } - html.theme--documenter-dark .hero.is-primary .tabs.is-boxed a:hover, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-primary .tabs.is-toggle a:hover, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-primary .tabs.is-boxed li.is-active a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-primary .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-primary .tabs.is-toggle li.is-active a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-primary .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #375a7f; } - html.theme--documenter-dark .hero.is-primary.is-bold, html.theme--documenter-dark .docstring > section > a.hero.is-bold.docs-sourcelink { - background-image: linear-gradient(141deg, #214b62 0%, #375a7f 71%, #3a5796 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-primary.is-bold .navbar-menu, html.theme--documenter-dark .docstring > section > a.hero.is-bold.docs-sourcelink .navbar-menu { - background-image: linear-gradient(141deg, #214b62 0%, #375a7f 71%, #3a5796 100%); } } - html.theme--documenter-dark .hero.is-link { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .hero.is-link a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-link strong { - color: inherit; } - html.theme--documenter-dark .hero.is-link .title { - color: #fff; } - html.theme--documenter-dark .hero.is-link .subtitle { - color: rgba(255, 255, 255, 0.9); } - html.theme--documenter-dark .hero.is-link .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-link .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-link .navbar-menu { - background-color: #1abc9c; } } - html.theme--documenter-dark .hero.is-link .navbar-item, - html.theme--documenter-dark .hero.is-link .navbar-link { - color: rgba(255, 255, 255, 0.7); } - html.theme--documenter-dark .hero.is-link a.navbar-item:hover, html.theme--documenter-dark .hero.is-link a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-link .navbar-link:hover, - html.theme--documenter-dark .hero.is-link .navbar-link.is-active { - background-color: #17a689; - color: #fff; } - html.theme--documenter-dark .hero.is-link .tabs a { - color: #fff; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-link .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-link .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-link .tabs.is-boxed a, html.theme--documenter-dark .hero.is-link .tabs.is-toggle a { - color: #fff; } - html.theme--documenter-dark .hero.is-link .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-link .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-link .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-link .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-link .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-link .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #1abc9c; } - html.theme--documenter-dark .hero.is-link.is-bold { - background-image: linear-gradient(141deg, #0c9764 0%, #1abc9c 71%, #17d8d2 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-link.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #0c9764 0%, #1abc9c 71%, #17d8d2 100%); } } - html.theme--documenter-dark .hero.is-info { - background-color: #024c7d; - color: #fff; } - html.theme--documenter-dark .hero.is-info a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-info strong { - color: inherit; } - html.theme--documenter-dark .hero.is-info .title { - color: #fff; } - html.theme--documenter-dark .hero.is-info .subtitle { - color: rgba(255, 255, 255, 0.9); } - html.theme--documenter-dark .hero.is-info .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-info .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-info .navbar-menu { - background-color: #024c7d; } } - html.theme--documenter-dark .hero.is-info .navbar-item, - html.theme--documenter-dark .hero.is-info .navbar-link { - color: rgba(255, 255, 255, 0.7); } - html.theme--documenter-dark .hero.is-info a.navbar-item:hover, html.theme--documenter-dark .hero.is-info a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-info .navbar-link:hover, - html.theme--documenter-dark .hero.is-info .navbar-link.is-active { - background-color: #023d64; - color: #fff; } - html.theme--documenter-dark .hero.is-info .tabs a { - color: #fff; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-info .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-info .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-info .tabs.is-boxed a, html.theme--documenter-dark .hero.is-info .tabs.is-toggle a { - color: #fff; } - html.theme--documenter-dark .hero.is-info .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-info .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-info .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-info .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-info .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-info .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #024c7d; } - html.theme--documenter-dark .hero.is-info.is-bold { - background-image: linear-gradient(141deg, #003a4c 0%, #024c7d 71%, #004299 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-info.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #003a4c 0%, #024c7d 71%, #004299 100%); } } - html.theme--documenter-dark .hero.is-success { - background-color: #008438; - color: #fff; } - html.theme--documenter-dark .hero.is-success a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-success strong { - color: inherit; } - html.theme--documenter-dark .hero.is-success .title { - color: #fff; } - html.theme--documenter-dark .hero.is-success .subtitle { - color: rgba(255, 255, 255, 0.9); } - html.theme--documenter-dark .hero.is-success .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-success .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-success .navbar-menu { - background-color: #008438; } } - html.theme--documenter-dark .hero.is-success .navbar-item, - html.theme--documenter-dark .hero.is-success .navbar-link { - color: rgba(255, 255, 255, 0.7); } - html.theme--documenter-dark .hero.is-success a.navbar-item:hover, html.theme--documenter-dark .hero.is-success a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-success .navbar-link:hover, - html.theme--documenter-dark .hero.is-success .navbar-link.is-active { - background-color: #006b2d; - color: #fff; } - html.theme--documenter-dark .hero.is-success .tabs a { - color: #fff; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-success .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-success .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-success .tabs.is-boxed a, html.theme--documenter-dark .hero.is-success .tabs.is-toggle a { - color: #fff; } - html.theme--documenter-dark .hero.is-success .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-success .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-success .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-success .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-success .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-success .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #008438; } - html.theme--documenter-dark .hero.is-success.is-bold { - background-image: linear-gradient(141deg, #005115 0%, #008438 71%, #009e5d 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-success.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #005115 0%, #008438 71%, #009e5d 100%); } } - html.theme--documenter-dark .hero.is-warning { - background-color: #ad8100; - color: #fff; } - html.theme--documenter-dark .hero.is-warning a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-warning strong { - color: inherit; } - html.theme--documenter-dark .hero.is-warning .title { - color: #fff; } - html.theme--documenter-dark .hero.is-warning .subtitle { - color: rgba(255, 255, 255, 0.9); } - html.theme--documenter-dark .hero.is-warning .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-warning .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-warning .navbar-menu { - background-color: #ad8100; } } - html.theme--documenter-dark .hero.is-warning .navbar-item, - html.theme--documenter-dark .hero.is-warning .navbar-link { - color: rgba(255, 255, 255, 0.7); } - html.theme--documenter-dark .hero.is-warning a.navbar-item:hover, html.theme--documenter-dark .hero.is-warning a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-warning .navbar-link:hover, - html.theme--documenter-dark .hero.is-warning .navbar-link.is-active { - background-color: #946e00; - color: #fff; } - html.theme--documenter-dark .hero.is-warning .tabs a { - color: #fff; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-warning .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-warning .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-warning .tabs.is-boxed a, html.theme--documenter-dark .hero.is-warning .tabs.is-toggle a { - color: #fff; } - html.theme--documenter-dark .hero.is-warning .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-warning .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-warning .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-warning .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-warning .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-warning .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #ad8100; } - html.theme--documenter-dark .hero.is-warning.is-bold { - background-image: linear-gradient(141deg, #7a4700 0%, #ad8100 71%, #c7b500 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-warning.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #7a4700 0%, #ad8100 71%, #c7b500 100%); } } - html.theme--documenter-dark .hero.is-danger { - background-color: #9e1b0d; - color: #fff; } - html.theme--documenter-dark .hero.is-danger a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-danger strong { - color: inherit; } - html.theme--documenter-dark .hero.is-danger .title { - color: #fff; } - html.theme--documenter-dark .hero.is-danger .subtitle { - color: rgba(255, 255, 255, 0.9); } - html.theme--documenter-dark .hero.is-danger .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-danger .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-danger .navbar-menu { - background-color: #9e1b0d; } } - html.theme--documenter-dark .hero.is-danger .navbar-item, - html.theme--documenter-dark .hero.is-danger .navbar-link { - color: rgba(255, 255, 255, 0.7); } - html.theme--documenter-dark .hero.is-danger a.navbar-item:hover, html.theme--documenter-dark .hero.is-danger a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-danger .navbar-link:hover, - html.theme--documenter-dark .hero.is-danger .navbar-link.is-active { - background-color: #86170b; - color: #fff; } - html.theme--documenter-dark .hero.is-danger .tabs a { - color: #fff; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-danger .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-danger .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-danger .tabs.is-boxed a, html.theme--documenter-dark .hero.is-danger .tabs.is-toggle a { - color: #fff; } - html.theme--documenter-dark .hero.is-danger .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-danger .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-danger .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-danger .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-danger .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-danger .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #9e1b0d; } - html.theme--documenter-dark .hero.is-danger.is-bold { - background-image: linear-gradient(141deg, #75030b 0%, #9e1b0d 71%, #ba380a 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-danger.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #75030b 0%, #9e1b0d 71%, #ba380a 100%); } } - html.theme--documenter-dark .hero.is-small .hero-body, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.hero .hero-body { - padding-bottom: 1.5rem; - padding-top: 1.5rem; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .hero.is-medium .hero-body { - padding-bottom: 9rem; - padding-top: 9rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .hero.is-large .hero-body { - padding-bottom: 18rem; - padding-top: 18rem; } } - html.theme--documenter-dark .hero.is-halfheight .hero-body, html.theme--documenter-dark .hero.is-fullheight .hero-body, html.theme--documenter-dark .hero.is-fullheight-with-navbar .hero-body { - align-items: center; - display: flex; } - html.theme--documenter-dark .hero.is-halfheight .hero-body > .container, html.theme--documenter-dark .hero.is-fullheight .hero-body > .container, html.theme--documenter-dark .hero.is-fullheight-with-navbar .hero-body > .container { - flex-grow: 1; - flex-shrink: 1; } - html.theme--documenter-dark .hero.is-halfheight { - min-height: 50vh; } - html.theme--documenter-dark .hero.is-fullheight { - min-height: 100vh; } - html.theme--documenter-dark .hero-video { - overflow: hidden; } - html.theme--documenter-dark .hero-video video { - left: 50%; - min-height: 100%; - min-width: 100%; - position: absolute; - top: 50%; - transform: translate3d(-50%, -50%, 0); } - html.theme--documenter-dark .hero-video.is-transparent { - opacity: 0.3; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero-video { - display: none; } } - html.theme--documenter-dark .hero-buttons { - margin-top: 1.5rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero-buttons .button { - display: flex; } - html.theme--documenter-dark .hero-buttons .button:not(:last-child) { - margin-bottom: 0.75rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .hero-buttons { - display: flex; - justify-content: center; } - html.theme--documenter-dark .hero-buttons .button:not(:last-child) { - margin-right: 1.5rem; } } - html.theme--documenter-dark .hero-head, - html.theme--documenter-dark .hero-foot { - flex-grow: 0; - flex-shrink: 0; } - html.theme--documenter-dark .hero-body { - flex-grow: 1; - flex-shrink: 0; - padding: 3rem 1.5rem; } - html.theme--documenter-dark .section { - padding: 3rem 1.5rem; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .section.is-medium { - padding: 9rem 1.5rem; } - html.theme--documenter-dark .section.is-large { - padding: 18rem 1.5rem; } } - html.theme--documenter-dark .footer { - background-color: #282f2f; - padding: 3rem 1.5rem 6rem; } - html.theme--documenter-dark hr { - height: 1px; } - html.theme--documenter-dark h6 { - text-transform: uppercase; - letter-spacing: 0.5px; } - html.theme--documenter-dark .hero { - background-color: #343c3d; } - html.theme--documenter-dark a { - transition: all 200ms ease; } - html.theme--documenter-dark .button { - transition: all 200ms ease; - border-width: 1px; - color: white; } - html.theme--documenter-dark .button.is-active, html.theme--documenter-dark .button.is-focused, html.theme--documenter-dark .button:active, html.theme--documenter-dark .button:focus { - box-shadow: 0 0 0 2px rgba(140, 155, 157, 0.5); } - html.theme--documenter-dark .button.is-white.is-hovered, html.theme--documenter-dark .button.is-white:hover { - background-color: white; } - html.theme--documenter-dark .button.is-white.is-active, html.theme--documenter-dark .button.is-white.is-focused, html.theme--documenter-dark .button.is-white:active, html.theme--documenter-dark .button.is-white:focus { - border-color: white; - box-shadow: 0 0 0 2px rgba(255, 255, 255, 0.5); } - html.theme--documenter-dark .button.is-black.is-hovered, html.theme--documenter-dark .button.is-black:hover { - background-color: #1d1d1d; } - html.theme--documenter-dark .button.is-black.is-active, html.theme--documenter-dark .button.is-black.is-focused, html.theme--documenter-dark .button.is-black:active, html.theme--documenter-dark .button.is-black:focus { - border-color: #0a0a0a; - box-shadow: 0 0 0 2px rgba(10, 10, 10, 0.5); } - html.theme--documenter-dark .button.is-light.is-hovered, html.theme--documenter-dark .button.is-light:hover { - background-color: white; } - html.theme--documenter-dark .button.is-light.is-active, html.theme--documenter-dark .button.is-light.is-focused, html.theme--documenter-dark .button.is-light:active, html.theme--documenter-dark .button.is-light:focus { - border-color: #ecf0f1; - box-shadow: 0 0 0 2px rgba(236, 240, 241, 0.5); } - html.theme--documenter-dark .button.is-dark.is-hovered, html.theme--documenter-dark .content kbd.button.is-hovered, html.theme--documenter-dark .button.is-dark:hover, html.theme--documenter-dark .content kbd.button:hover { - background-color: #3a4344; } - html.theme--documenter-dark .button.is-dark.is-active, html.theme--documenter-dark .content kbd.button.is-active, html.theme--documenter-dark .button.is-dark.is-focused, html.theme--documenter-dark .content kbd.button.is-focused, html.theme--documenter-dark .button.is-dark:active, html.theme--documenter-dark .content kbd.button:active, html.theme--documenter-dark .button.is-dark:focus, html.theme--documenter-dark .content kbd.button:focus { - border-color: #282f2f; - box-shadow: 0 0 0 2px rgba(40, 47, 47, 0.5); } - html.theme--documenter-dark .button.is-primary.is-hovered, html.theme--documenter-dark .docstring > section > a.button.is-hovered.docs-sourcelink, html.theme--documenter-dark .button.is-primary:hover, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:hover { - background-color: #436d9a; } - html.theme--documenter-dark .button.is-primary.is-active, html.theme--documenter-dark .docstring > section > a.button.is-active.docs-sourcelink, html.theme--documenter-dark .button.is-primary.is-focused, html.theme--documenter-dark .docstring > section > a.button.is-focused.docs-sourcelink, html.theme--documenter-dark .button.is-primary:active, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:active, html.theme--documenter-dark .button.is-primary:focus, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:focus { - border-color: #375a7f; - box-shadow: 0 0 0 2px rgba(55, 90, 127, 0.5); } - html.theme--documenter-dark .button.is-link.is-hovered, html.theme--documenter-dark .button.is-link:hover { - background-color: #1fdeb8; } - html.theme--documenter-dark .button.is-link.is-active, html.theme--documenter-dark .button.is-link.is-focused, html.theme--documenter-dark .button.is-link:active, html.theme--documenter-dark .button.is-link:focus { - border-color: #1abc9c; - box-shadow: 0 0 0 2px rgba(26, 188, 156, 0.5); } - html.theme--documenter-dark .button.is-info.is-hovered, html.theme--documenter-dark .button.is-info:hover { - background-color: #0363a3; } - html.theme--documenter-dark .button.is-info.is-active, html.theme--documenter-dark .button.is-info.is-focused, html.theme--documenter-dark .button.is-info:active, html.theme--documenter-dark .button.is-info:focus { - border-color: #024c7d; - box-shadow: 0 0 0 2px rgba(2, 76, 125, 0.5); } - html.theme--documenter-dark .button.is-success.is-hovered, html.theme--documenter-dark .button.is-success:hover { - background-color: #00aa48; } - html.theme--documenter-dark .button.is-success.is-active, html.theme--documenter-dark .button.is-success.is-focused, html.theme--documenter-dark .button.is-success:active, html.theme--documenter-dark .button.is-success:focus { - border-color: #008438; - box-shadow: 0 0 0 2px rgba(0, 132, 56, 0.5); } - html.theme--documenter-dark .button.is-warning.is-hovered, html.theme--documenter-dark .button.is-warning:hover { - background-color: #d39e00; } - html.theme--documenter-dark .button.is-warning.is-active, html.theme--documenter-dark .button.is-warning.is-focused, html.theme--documenter-dark .button.is-warning:active, html.theme--documenter-dark .button.is-warning:focus { - border-color: #ad8100; - box-shadow: 0 0 0 2px rgba(173, 129, 0, 0.5); } - html.theme--documenter-dark .button.is-danger.is-hovered, html.theme--documenter-dark .button.is-danger:hover { - background-color: #c12110; } - html.theme--documenter-dark .button.is-danger.is-active, html.theme--documenter-dark .button.is-danger.is-focused, html.theme--documenter-dark .button.is-danger:active, html.theme--documenter-dark .button.is-danger:focus { - border-color: #9e1b0d; - box-shadow: 0 0 0 2px rgba(158, 27, 13, 0.5); } - html.theme--documenter-dark .label { - color: #dbdee0; } - html.theme--documenter-dark .button, - html.theme--documenter-dark .control.has-icons-left .icon, - html.theme--documenter-dark .control.has-icons-right .icon, - html.theme--documenter-dark .input, - html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input, - html.theme--documenter-dark .pagination-ellipsis, - html.theme--documenter-dark .pagination-link, - html.theme--documenter-dark .pagination-next, - html.theme--documenter-dark .pagination-previous, - html.theme--documenter-dark .select, - html.theme--documenter-dark .select select, - html.theme--documenter-dark .textarea { - height: 2.5em; } - html.theme--documenter-dark .input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input, - html.theme--documenter-dark .textarea { - transition: all 200ms ease; - box-shadow: none; - border-width: 1px; - padding-left: 1em; - padding-right: 1em; } - html.theme--documenter-dark .select:after, - html.theme--documenter-dark .select select { - border-width: 1px; } - html.theme--documenter-dark .control.has-addons .button, - html.theme--documenter-dark .control.has-addons .input, - html.theme--documenter-dark .control.has-addons #documenter .docs-sidebar form.docs-search > input, - html.theme--documenter-dark #documenter .docs-sidebar .control.has-addons form.docs-search > input, - html.theme--documenter-dark .control.has-addons .select { - margin-right: -1px; } - html.theme--documenter-dark .notification { - background-color: #343c3d; } - html.theme--documenter-dark .card { - box-shadow: none; - border: 1px solid #343c3d; - background-color: #282f2f; - border-radius: 0.4em; } - html.theme--documenter-dark .card .card-image img { - border-radius: 0.4em 0.4em 0 0; } - html.theme--documenter-dark .card .card-header { - box-shadow: none; - background-color: rgba(18, 18, 18, 0.2); - border-radius: 0.4em 0.4em 0 0; } - html.theme--documenter-dark .card .card-footer { - background-color: rgba(18, 18, 18, 0.2); } - html.theme--documenter-dark .card .card-footer, - html.theme--documenter-dark .card .card-footer-item { - border-width: 1px; - border-color: #343c3d; } - html.theme--documenter-dark .notification.is-white a:not(.button) { - color: #0a0a0a; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-black a:not(.button) { - color: white; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-light a:not(.button) { - color: #282f2f; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-dark a:not(.button), html.theme--documenter-dark .content kbd.notification a:not(.button) { - color: #ecf0f1; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-primary a:not(.button), html.theme--documenter-dark .docstring > section > a.notification.docs-sourcelink a:not(.button) { - color: #fff; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-link a:not(.button) { - color: #fff; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-info a:not(.button) { - color: #fff; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-success a:not(.button) { - color: #fff; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-warning a:not(.button) { - color: #fff; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-danger a:not(.button) { - color: #fff; - text-decoration: underline; } - html.theme--documenter-dark .tag, html.theme--documenter-dark .content kbd, html.theme--documenter-dark .docstring > section > a.docs-sourcelink { - border-radius: 0.4em; } - html.theme--documenter-dark .menu-list a { - transition: all 300ms ease; } - html.theme--documenter-dark .modal-card-body { - background-color: #282f2f; } - html.theme--documenter-dark .modal-card-foot, - html.theme--documenter-dark .modal-card-head { - border-color: #343c3d; } - html.theme--documenter-dark .message-header { - font-weight: 700; - background-color: #343c3d; - color: white; } - html.theme--documenter-dark .message-body { - border-width: 1px; - border-color: #343c3d; } - html.theme--documenter-dark .navbar { - border-radius: 0.4em; } - html.theme--documenter-dark .navbar.is-transparent { - background: none; } - html.theme--documenter-dark .navbar.is-primary .navbar-dropdown a.navbar-item.is-active, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-dropdown a.navbar-item.is-active { - background-color: #1abc9c; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .navbar .navbar-menu { - background-color: #375a7f; - border-radius: 0 0 0.4em 0.4em; } } - html.theme--documenter-dark .hero .navbar, - html.theme--documenter-dark body > .navbar { - border-radius: 0; } - html.theme--documenter-dark .pagination-link, - html.theme--documenter-dark .pagination-next, - html.theme--documenter-dark .pagination-previous { - border-width: 1px; } - html.theme--documenter-dark .panel-block, - html.theme--documenter-dark .panel-heading, - html.theme--documenter-dark .panel-tabs { - border-width: 1px; } - html.theme--documenter-dark .panel-block:first-child, - html.theme--documenter-dark .panel-heading:first-child, - html.theme--documenter-dark .panel-tabs:first-child { - border-top-width: 1px; } - html.theme--documenter-dark .panel-heading { - font-weight: 700; } - html.theme--documenter-dark .panel-tabs a { - border-width: 1px; - margin-bottom: -1px; } - html.theme--documenter-dark .panel-tabs a.is-active { - border-bottom-color: #17a689; } - html.theme--documenter-dark .panel-block:hover { - color: #1dd2af; } - html.theme--documenter-dark .panel-block:hover .panel-icon { - color: #1dd2af; } - html.theme--documenter-dark .panel-block.is-active .panel-icon { - color: #17a689; } - html.theme--documenter-dark .tabs a { - border-bottom-width: 1px; - margin-bottom: -1px; } - html.theme--documenter-dark .tabs ul { - border-bottom-width: 1px; } - html.theme--documenter-dark .tabs.is-boxed a { - border-width: 1px; } - html.theme--documenter-dark .tabs.is-boxed li.is-active a { - background-color: #1f2424; } - html.theme--documenter-dark .tabs.is-toggle li a { - border-width: 1px; - margin-bottom: 0; } - html.theme--documenter-dark .tabs.is-toggle li + li { - margin-left: -1px; } - html.theme--documenter-dark .hero.is-white .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-black .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-light .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-dark .navbar .navbar-dropdown .navbar-item:hover, html.theme--documenter-dark .content kbd.hero .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-primary .navbar .navbar-dropdown .navbar-item:hover, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-link .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-info .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-success .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-warning .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-danger .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark h1 .docs-heading-anchor, html.theme--documenter-dark h1 .docs-heading-anchor:hover, html.theme--documenter-dark h1 .docs-heading-anchor:visited, html.theme--documenter-dark h2 .docs-heading-anchor, html.theme--documenter-dark h2 .docs-heading-anchor:hover, html.theme--documenter-dark h2 .docs-heading-anchor:visited, html.theme--documenter-dark h3 .docs-heading-anchor, html.theme--documenter-dark h3 .docs-heading-anchor:hover, html.theme--documenter-dark h3 .docs-heading-anchor:visited, html.theme--documenter-dark h4 .docs-heading-anchor, html.theme--documenter-dark h4 .docs-heading-anchor:hover, html.theme--documenter-dark h4 .docs-heading-anchor:visited, html.theme--documenter-dark h5 .docs-heading-anchor, html.theme--documenter-dark h5 .docs-heading-anchor:hover, html.theme--documenter-dark h5 .docs-heading-anchor:visited, html.theme--documenter-dark h6 .docs-heading-anchor, html.theme--documenter-dark h6 .docs-heading-anchor:hover, html.theme--documenter-dark h6 .docs-heading-anchor:visited { - color: #f2f2f2; } - html.theme--documenter-dark h1 .docs-heading-anchor-permalink, html.theme--documenter-dark h2 .docs-heading-anchor-permalink, html.theme--documenter-dark h3 .docs-heading-anchor-permalink, html.theme--documenter-dark h4 .docs-heading-anchor-permalink, html.theme--documenter-dark h5 .docs-heading-anchor-permalink, html.theme--documenter-dark h6 .docs-heading-anchor-permalink { - visibility: hidden; - vertical-align: middle; - margin-left: 0.5em; - font-size: 0.7rem; } - html.theme--documenter-dark h1 .docs-heading-anchor-permalink::before, html.theme--documenter-dark h2 .docs-heading-anchor-permalink::before, html.theme--documenter-dark h3 .docs-heading-anchor-permalink::before, html.theme--documenter-dark h4 .docs-heading-anchor-permalink::before, html.theme--documenter-dark h5 .docs-heading-anchor-permalink::before, html.theme--documenter-dark h6 .docs-heading-anchor-permalink::before { - font-family: "Font Awesome 5 Free"; - font-weight: 900; - content: "\f0c1"; } - html.theme--documenter-dark h1:hover .docs-heading-anchor-permalink, html.theme--documenter-dark h2:hover .docs-heading-anchor-permalink, html.theme--documenter-dark h3:hover .docs-heading-anchor-permalink, html.theme--documenter-dark h4:hover .docs-heading-anchor-permalink, html.theme--documenter-dark h5:hover .docs-heading-anchor-permalink, html.theme--documenter-dark h6:hover .docs-heading-anchor-permalink { - visibility: visible; } - html.theme--documenter-dark .docs-light-only { - display: none !important; } - html.theme--documenter-dark pre { - position: relative; - overflow: hidden; } - html.theme--documenter-dark pre code, html.theme--documenter-dark pre code.hljs { - padding: 0 0.5rem !important; - overflow: auto; - display: block; } - html.theme--documenter-dark pre code:first-of-type, html.theme--documenter-dark pre code.hljs:first-of-type { - padding-top: 0.7rem !important; } - html.theme--documenter-dark pre code:last-of-type, html.theme--documenter-dark pre code.hljs:last-of-type { - padding-bottom: 0.7rem !important; } - html.theme--documenter-dark pre .copy-button { - opacity: 0.2; - transition: opacity 0.2s; - position: absolute; - right: 0em; - top: 0em; - padding: 0.5em; - width: 2.5em; - height: 2.5em; - background: transparent; - border: none; - font-family: "Font Awesome 5 Free"; - color: #fff; - cursor: pointer; - text-align: center; } - html.theme--documenter-dark pre .copy-button:focus, html.theme--documenter-dark pre .copy-button:hover { - opacity: 1; - background: rgba(255, 255, 255, 0.1); - color: #1abc9c; } - html.theme--documenter-dark pre .copy-button.success { - color: #259a12; - opacity: 1; } - html.theme--documenter-dark pre .copy-button.error { - color: #cb3c33; - opacity: 1; } - html.theme--documenter-dark pre:hover .copy-button { - opacity: 1; } - html.theme--documenter-dark .admonition { - background-color: #282f2f; - border-style: solid; - border-width: 1px; - border-color: #5e6d6f; - border-radius: 0.4em; - font-size: 15px; } - html.theme--documenter-dark .admonition strong { - color: currentColor; } - html.theme--documenter-dark .admonition.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.admonition { - font-size: 0.85em; } - html.theme--documenter-dark .admonition.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .admonition.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .admonition.is-default { - background-color: #282f2f; - border-color: #5e6d6f; } - html.theme--documenter-dark .admonition.is-default > .admonition-header { - background-color: #5e6d6f; } - html.theme--documenter-dark .admonition.is-info { - background-color: #282f2f; - border-color: #024c7d; } - html.theme--documenter-dark .admonition.is-info > .admonition-header { - background-color: #024c7d; } - html.theme--documenter-dark .admonition.is-success { - background-color: #282f2f; - border-color: #008438; } - html.theme--documenter-dark .admonition.is-success > .admonition-header { - background-color: #008438; } - html.theme--documenter-dark .admonition.is-warning { - background-color: #282f2f; - border-color: #ad8100; } - html.theme--documenter-dark .admonition.is-warning > .admonition-header { - background-color: #ad8100; } - html.theme--documenter-dark .admonition.is-danger { - background-color: #282f2f; - border-color: #9e1b0d; } - html.theme--documenter-dark .admonition.is-danger > .admonition-header { - background-color: #9e1b0d; } - html.theme--documenter-dark .admonition.is-compat { - background-color: #282f2f; - border-color: #137886; } - html.theme--documenter-dark .admonition.is-compat > .admonition-header { - background-color: #137886; } - html.theme--documenter-dark .admonition-header { - background-color: #5e6d6f; - align-items: center; - font-weight: 700; - justify-content: space-between; - line-height: 1.25; - padding: 0.75em; - position: relative; } - html.theme--documenter-dark .admonition-header:before { - font-family: "Font Awesome 5 Free"; - font-weight: 900; - margin-right: 0.75em; - content: "\f06a"; } - html.theme--documenter-dark .admonition-body { - color: #fff; - padding: 1em 1.25em; } - html.theme--documenter-dark .admonition-body pre { - background-color: #282f2f; } - html.theme--documenter-dark .admonition-body code { - background-color: rgba(255, 255, 255, 0.05); } - html.theme--documenter-dark .docstring { - margin-bottom: 1em; - background-color: transparent; - border: 1px solid #5e6d6f; - box-shadow: none; - max-width: 100%; } - html.theme--documenter-dark .docstring > header { - display: flex; - flex-grow: 1; - align-items: stretch; - padding: 0.75rem; - background-color: #282f2f; - box-shadow: 0 1px 2px rgba(10, 10, 10, 0.1); - box-shadow: none; - border-bottom: 1px solid #5e6d6f; } - html.theme--documenter-dark .docstring > header code { - background-color: transparent; } - html.theme--documenter-dark .docstring > header .docstring-binding { - margin-right: 0.3em; } - html.theme--documenter-dark .docstring > header .docstring-category { - margin-left: 0.3em; } - html.theme--documenter-dark .docstring > section { - position: relative; - padding: 1rem 1.25rem; - border-bottom: 1px solid #5e6d6f; } - html.theme--documenter-dark .docstring > section:last-child { - border-bottom: none; } - html.theme--documenter-dark .docstring > section > a.docs-sourcelink { - transition: opacity 0.3s; - opacity: 0; - position: absolute; - right: 0.625rem; - bottom: 0.5rem; } - html.theme--documenter-dark .docstring > section > a.docs-sourcelink:focus { - opacity: 1 !important; } - html.theme--documenter-dark .docstring:hover > section > a.docs-sourcelink { - opacity: 0.2; } - html.theme--documenter-dark .docstring:focus-within > section > a.docs-sourcelink { - opacity: 0.2; } - html.theme--documenter-dark .docstring > section:hover a.docs-sourcelink { - opacity: 1; } - html.theme--documenter-dark .documenter-example-output { - background-color: #1f2424; } - html.theme--documenter-dark .outdated-warning-overlay { - position: fixed; - top: 0; - left: 0; - right: 0; - box-shadow: 0 0 10px rgba(0, 0, 0, 0.3); - z-index: 999; - background-color: #282f2f; - border-bottom: 3px solid #9e1b0d; - padding: 10px 35px; - text-align: center; - font-size: 15px; } - html.theme--documenter-dark .outdated-warning-overlay .outdated-warning-closer { - position: absolute; - top: calc(50% - 10px); - right: 18px; - cursor: pointer; - width: 12px; } - html.theme--documenter-dark .outdated-warning-overlay a { - color: #1abc9c; } - html.theme--documenter-dark .outdated-warning-overlay a:hover { - color: #1dd2af; } - html.theme--documenter-dark .content pre { - border: 1px solid #5e6d6f; } - html.theme--documenter-dark .content code { - font-weight: inherit; } - html.theme--documenter-dark .content a code { - color: #1abc9c; } - html.theme--documenter-dark .content h1 code, html.theme--documenter-dark .content h2 code, html.theme--documenter-dark .content h3 code, html.theme--documenter-dark .content h4 code, html.theme--documenter-dark .content h5 code, html.theme--documenter-dark .content h6 code { - color: #f2f2f2; } - html.theme--documenter-dark .content table { - display: block; - width: initial; - max-width: 100%; - overflow-x: auto; } - html.theme--documenter-dark .content blockquote > ul:first-child, html.theme--documenter-dark .content blockquote > ol:first-child, html.theme--documenter-dark .content .admonition-body > ul:first-child, html.theme--documenter-dark .content .admonition-body > ol:first-child { - margin-top: 0; } - html.theme--documenter-dark pre, html.theme--documenter-dark code { - font-variant-ligatures: no-contextual; } - html.theme--documenter-dark .breadcrumb a.is-disabled { - cursor: default; - pointer-events: none; } - html.theme--documenter-dark .breadcrumb a.is-disabled, html.theme--documenter-dark .breadcrumb a.is-disabled:hover { - color: #f2f2f2; } - html.theme--documenter-dark .hljs { - background: initial !important; } - html.theme--documenter-dark .katex .katex-mathml { - top: 0; - right: 0; } - html.theme--documenter-dark .katex-display, html.theme--documenter-dark mjx-container, html.theme--documenter-dark .MathJax_Display { - margin: 0.5em 0 !important; } - html.theme--documenter-dark html { - -moz-osx-font-smoothing: auto; - -webkit-font-smoothing: auto; } - html.theme--documenter-dark li.no-marker { - list-style: none; } - html.theme--documenter-dark #documenter .docs-main > article { - overflow-wrap: break-word; } - html.theme--documenter-dark #documenter .docs-main > article .math-container { - overflow-x: auto; - overflow-y: hidden; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark #documenter .docs-main { - max-width: 52rem; - margin-left: 20rem; - padding-right: 1rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark #documenter .docs-main { - width: 100%; } - html.theme--documenter-dark #documenter .docs-main > article { - max-width: 52rem; - margin-left: auto; - margin-right: auto; - margin-bottom: 1rem; - padding: 0 1rem; } - html.theme--documenter-dark #documenter .docs-main > header, html.theme--documenter-dark #documenter .docs-main > nav { - max-width: 100%; - width: 100%; - margin: 0; } } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar { - background-color: #1f2424; - border-bottom: 1px solid #5e6d6f; - z-index: 2; - min-height: 4rem; - margin-bottom: 1rem; - display: flex; } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar .breadcrumb { - flex-grow: 1; } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right { - display: flex; - white-space: nowrap; } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-icon, html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-label, html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-sidebar-button { - display: inline-block; } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-label { - padding: 0; - margin-left: 0.3em; } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-settings-button { - margin: auto 0 auto 1rem; } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-sidebar-button { - font-size: 1.5rem; - margin: auto 0 auto 1rem; } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar > * { - margin: auto 0; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark #documenter .docs-main header.docs-navbar { - position: sticky; - top: 0; - padding: 0 1rem; - /* For Headroom.js */ - transition-property: top, box-shadow; - -webkit-transition-property: top, box-shadow; - /* Safari */ - transition-duration: 0.3s; - -webkit-transition-duration: 0.3s; - /* Safari */ } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar.headroom--not-top { - box-shadow: 0.2rem 0rem 0.4rem #171717; - transition-duration: 0.7s; - -webkit-transition-duration: 0.7s; - /* Safari */ } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar.headroom--unpinned.headroom--not-top.headroom--not-bottom { - top: -4.5rem; - transition-duration: 0.7s; - -webkit-transition-duration: 0.7s; - /* Safari */ } } - html.theme--documenter-dark #documenter .docs-main section.footnotes { - border-top: 1px solid #5e6d6f; } - html.theme--documenter-dark #documenter .docs-main section.footnotes li .tag:first-child, html.theme--documenter-dark #documenter .docs-main section.footnotes li .docstring > section > a.docs-sourcelink:first-child, html.theme--documenter-dark #documenter .docs-main section.footnotes li .content kbd:first-child, html.theme--documenter-dark .content #documenter .docs-main section.footnotes li kbd:first-child { - margin-right: 1em; - margin-bottom: 0.4em; } - html.theme--documenter-dark #documenter .docs-main .docs-footer { - display: flex; - flex-wrap: wrap; - margin-left: 0; - margin-right: 0; - border-top: 1px solid #5e6d6f; - padding-top: 1rem; - padding-bottom: 1rem; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark #documenter .docs-main .docs-footer { - padding-left: 1rem; - padding-right: 1rem; } } - html.theme--documenter-dark #documenter .docs-main .docs-footer .docs-footer-nextpage, html.theme--documenter-dark #documenter .docs-main .docs-footer .docs-footer-prevpage { - flex-grow: 1; } - html.theme--documenter-dark #documenter .docs-main .docs-footer .docs-footer-nextpage { - text-align: right; } - html.theme--documenter-dark #documenter .docs-main .docs-footer .flexbox-break { - flex-basis: 100%; - height: 0; } - html.theme--documenter-dark #documenter .docs-main .docs-footer .footer-message { - font-size: 0.8em; - margin: 0.5em auto 0 auto; - text-align: center; } - html.theme--documenter-dark #documenter .docs-sidebar { - display: flex; - flex-direction: column; - color: #fff; - background-color: #282f2f; - border-right: 1px solid #5e6d6f; - padding: 0; - flex: 0 0 18rem; - z-index: 5; - font-size: 15px; - position: fixed; - left: -18rem; - width: 18rem; - height: 100%; - transition: left 0.3s; - /* Setting up a nicer theme style for the scrollbar */ } - html.theme--documenter-dark #documenter .docs-sidebar.visible { - left: 0; - box-shadow: 0.4rem 0rem 0.8rem #171717; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark #documenter .docs-sidebar.visible { - box-shadow: none; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark #documenter .docs-sidebar { - left: 0; - top: 0; } } - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo { - margin-top: 1rem; - padding: 0 1rem; } - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img { - max-height: 6rem; - margin: auto; } - html.theme--documenter-dark #documenter .docs-sidebar .docs-package-name { - flex-shrink: 0; - font-size: 1.5rem; - font-weight: 700; - text-align: center; - white-space: nowrap; - overflow: hidden; - padding: 0.5rem 0; } - html.theme--documenter-dark #documenter .docs-sidebar .docs-package-name .docs-autofit { - max-width: 16.2rem; } - html.theme--documenter-dark #documenter .docs-sidebar .docs-package-name a, html.theme--documenter-dark #documenter .docs-sidebar .docs-package-name a:hover { - color: #fff; } - html.theme--documenter-dark #documenter .docs-sidebar .docs-version-selector { - border-top: 1px solid #5e6d6f; - display: none; - padding: 0.5rem; } - html.theme--documenter-dark #documenter .docs-sidebar .docs-version-selector.visible { - display: flex; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu { - flex-grow: 1; - user-select: none; - border-top: 1px solid #5e6d6f; - padding-bottom: 1.5rem; - /* Managing collapsible submenus */ } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu > li > .tocitem { - font-weight: bold; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu > li li { - font-size: 14.25px; - margin-left: 1em; - border-left: 1px solid #5e6d6f; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu input.collapse-toggle { - display: none; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.collapsed { - display: none; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu input:checked ~ ul.collapsed { - display: block; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu label.tocitem { - display: flex; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu label.tocitem .docs-label { - flex-grow: 2; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu label.tocitem .docs-chevron { - display: inline-block; - font-style: normal; - font-variant: normal; - text-rendering: auto; - line-height: 1; - font-size: 11.25px; - margin-left: 1rem; - margin-top: auto; - margin-bottom: auto; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu label.tocitem .docs-chevron::before { - font-family: "Font Awesome 5 Free"; - font-weight: 900; - content: "\f054"; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu input:checked ~ label.tocitem .docs-chevron::before { - content: "\f078"; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu .tocitem { - display: block; - padding: 0.5rem 0.5rem; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu .tocitem, html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu .tocitem:hover { - color: #fff; - background: #282f2f; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu a.tocitem:hover, html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu label.tocitem:hover { - color: #fff; - background-color: #32393a; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu li.is-active { - border-top: 1px solid #5e6d6f; - border-bottom: 1px solid #5e6d6f; - background-color: #1f2424; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu li.is-active .tocitem, html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu li.is-active .tocitem:hover { - background-color: #1f2424; - color: #fff; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu li.is-active ul.internal .tocitem:hover { - background-color: #32393a; - color: #fff; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu > li.is-active:first-child { - border-top: none; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.internal { - margin: 0 0.5rem 0.5rem; - border-top: 1px solid #5e6d6f; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.internal li { - font-size: 12.75px; - border-left: none; - margin-left: 0; - margin-top: 0.5rem; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.internal .tocitem { - width: 100%; - padding: 0; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.internal .tocitem::before { - content: "⚬"; - margin-right: 0.4em; } - html.theme--documenter-dark #documenter .docs-sidebar form.docs-search { - margin: auto; - margin-top: 0.5rem; - margin-bottom: 0.5rem; } - html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input { - width: 14.4rem; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu { - overflow-y: auto; - -webkit-overflow-scroll: touch; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu::-webkit-scrollbar { - width: .3rem; - background: none; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu::-webkit-scrollbar-thumb { - border-radius: 5px 0px 0px 5px; - background: #3b4445; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu::-webkit-scrollbar-thumb:hover { - background: #4e5a5c; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark #documenter .docs-sidebar { - overflow-y: auto; - -webkit-overflow-scroll: touch; } - html.theme--documenter-dark #documenter .docs-sidebar::-webkit-scrollbar { - width: .3rem; - background: none; } - html.theme--documenter-dark #documenter .docs-sidebar::-webkit-scrollbar-thumb { - border-radius: 5px 0px 0px 5px; - background: #3b4445; } - html.theme--documenter-dark #documenter .docs-sidebar::-webkit-scrollbar-thumb:hover { - background: #4e5a5c; } } - html.theme--documenter-dark #documenter .docs-main #documenter-search-info { - margin-bottom: 1rem; } - html.theme--documenter-dark #documenter .docs-main #documenter-search-results { - list-style-type: circle; - list-style-position: outside; } - html.theme--documenter-dark #documenter .docs-main #documenter-search-results li { - margin-left: 2rem; } - html.theme--documenter-dark #documenter .docs-main #documenter-search-results .docs-highlight { - background-color: yellow; } - html.theme--documenter-dark { - background-color: #1f2424; - font-size: 16px; - min-width: 300px; - overflow-x: auto; - overflow-y: scroll; - text-rendering: optimizeLegibility; - text-size-adjust: 100%; } - html.theme--documenter-dark .ansi span.sgr1 { - font-weight: bolder; } - html.theme--documenter-dark .ansi span.sgr2 { - font-weight: lighter; } - html.theme--documenter-dark .ansi span.sgr3 { - font-style: italic; } - html.theme--documenter-dark .ansi span.sgr4 { - text-decoration: underline; } - html.theme--documenter-dark .ansi span.sgr7 { - color: #1f2424; - background-color: #fff; } - html.theme--documenter-dark .ansi span.sgr8 { - color: transparent; } - html.theme--documenter-dark .ansi span.sgr8 span { - color: transparent; } - html.theme--documenter-dark .ansi span.sgr9 { - text-decoration: line-through; } - html.theme--documenter-dark .ansi span.sgr30 { - color: #242424; } - html.theme--documenter-dark .ansi span.sgr31 { - color: #f6705f; } - html.theme--documenter-dark .ansi span.sgr32 { - color: #4fb43a; } - html.theme--documenter-dark .ansi span.sgr33 { - color: #f4c72f; } - html.theme--documenter-dark .ansi span.sgr34 { - color: #7587f0; } - html.theme--documenter-dark .ansi span.sgr35 { - color: #bc89d3; } - html.theme--documenter-dark .ansi span.sgr36 { - color: #49b6ca; } - html.theme--documenter-dark .ansi span.sgr37 { - color: #b3bdbe; } - html.theme--documenter-dark .ansi span.sgr40 { - background-color: #242424; } - html.theme--documenter-dark .ansi span.sgr41 { - background-color: #f6705f; } - html.theme--documenter-dark .ansi span.sgr42 { - background-color: #4fb43a; } - html.theme--documenter-dark .ansi span.sgr43 { - background-color: #f4c72f; } - html.theme--documenter-dark .ansi span.sgr44 { - background-color: #7587f0; } - html.theme--documenter-dark .ansi span.sgr45 { - background-color: #bc89d3; } - html.theme--documenter-dark .ansi span.sgr46 { - background-color: #49b6ca; } - html.theme--documenter-dark .ansi span.sgr47 { - background-color: #b3bdbe; } - html.theme--documenter-dark .ansi span.sgr90 { - color: #92a0a2; } - html.theme--documenter-dark .ansi span.sgr91 { - color: #ff8674; } - html.theme--documenter-dark .ansi span.sgr92 { - color: #79d462; } - html.theme--documenter-dark .ansi span.sgr93 { - color: #ffe76b; } - html.theme--documenter-dark .ansi span.sgr94 { - color: #8a98ff; } - html.theme--documenter-dark .ansi span.sgr95 { - color: #d2a4e6; } - html.theme--documenter-dark .ansi span.sgr96 { - color: #6bc8db; } - html.theme--documenter-dark .ansi span.sgr97 { - color: #ecf0f1; } - html.theme--documenter-dark .ansi span.sgr100 { - background-color: #92a0a2; } - html.theme--documenter-dark .ansi span.sgr101 { - background-color: #ff8674; } - html.theme--documenter-dark .ansi span.sgr102 { - background-color: #79d462; } - html.theme--documenter-dark .ansi span.sgr103 { - background-color: #ffe76b; } - html.theme--documenter-dark .ansi span.sgr104 { - background-color: #8a98ff; } - html.theme--documenter-dark .ansi span.sgr105 { - background-color: #d2a4e6; } - html.theme--documenter-dark .ansi span.sgr106 { - background-color: #6bc8db; } - html.theme--documenter-dark .ansi span.sgr107 { - background-color: #ecf0f1; } - html.theme--documenter-dark code.language-julia-repl > span.hljs-meta { - color: #4fb43a; - font-weight: bolder; } - html.theme--documenter-dark .hljs { - background: #2b2b2b; - color: #f8f8f2; } - html.theme--documenter-dark .hljs-comment, - html.theme--documenter-dark .hljs-quote { - color: #d4d0ab; } - html.theme--documenter-dark .hljs-variable, - html.theme--documenter-dark .hljs-template-variable, - html.theme--documenter-dark .hljs-tag, - html.theme--documenter-dark .hljs-name, - html.theme--documenter-dark .hljs-selector-id, - html.theme--documenter-dark .hljs-selector-class, - html.theme--documenter-dark .hljs-regexp, - html.theme--documenter-dark .hljs-deletion { - color: #ffa07a; } - html.theme--documenter-dark .hljs-number, - html.theme--documenter-dark .hljs-built_in, - html.theme--documenter-dark .hljs-literal, - html.theme--documenter-dark .hljs-type, - html.theme--documenter-dark .hljs-params, - html.theme--documenter-dark .hljs-meta, - html.theme--documenter-dark .hljs-link { - color: #f5ab35; } - html.theme--documenter-dark .hljs-attribute { - color: #ffd700; } - html.theme--documenter-dark .hljs-string, - html.theme--documenter-dark .hljs-symbol, - html.theme--documenter-dark .hljs-bullet, - html.theme--documenter-dark .hljs-addition { - color: #abe338; } - html.theme--documenter-dark .hljs-title, - html.theme--documenter-dark .hljs-section { - color: #00e0e0; } - html.theme--documenter-dark .hljs-keyword, - html.theme--documenter-dark .hljs-selector-tag { - color: #dcc6e0; } - html.theme--documenter-dark .hljs-emphasis { - font-style: italic; } - html.theme--documenter-dark .hljs-strong { - font-weight: bold; } - @media screen and (-ms-high-contrast: active) { - html.theme--documenter-dark .hljs-addition, - html.theme--documenter-dark .hljs-attribute, - html.theme--documenter-dark .hljs-built_in, - html.theme--documenter-dark .hljs-bullet, - html.theme--documenter-dark .hljs-comment, - html.theme--documenter-dark .hljs-link, - html.theme--documenter-dark .hljs-literal, - html.theme--documenter-dark .hljs-meta, - html.theme--documenter-dark .hljs-number, - html.theme--documenter-dark .hljs-params, - html.theme--documenter-dark .hljs-string, - html.theme--documenter-dark .hljs-symbol, - html.theme--documenter-dark .hljs-type, - html.theme--documenter-dark .hljs-quote { - color: highlight; } - html.theme--documenter-dark .hljs-keyword, - html.theme--documenter-dark .hljs-selector-tag { - font-weight: bold; } } - html.theme--documenter-dark .hljs-subst { - color: #f8f8f2; } diff --git a/previews/PR40/assets/themes/documenter-light.css b/previews/PR40/assets/themes/documenter-light.css deleted file mode 100644 index f3e70f4b..00000000 --- a/previews/PR40/assets/themes/documenter-light.css +++ /dev/null @@ -1,7738 +0,0 @@ -@charset "UTF-8"; -/* Font Awesome 5 mixin. Can be included in any rule that should render Font Awesome icons. */ -@keyframes spinAround { - from { - transform: rotate(0deg); } - to { - transform: rotate(359deg); } } - -.tabs, .pagination-previous, -.pagination-next, -.pagination-link, -.pagination-ellipsis, .breadcrumb, .file, .button, .is-unselectable, .modal-close, .delete { - -webkit-touch-callout: none; - -webkit-user-select: none; - -moz-user-select: none; - -ms-user-select: none; - user-select: none; } - -.navbar-link:not(.is-arrowless)::after, .select:not(.is-multiple):not(.is-loading)::after { - border: 3px solid transparent; - border-radius: 2px; - border-right: 0; - border-top: 0; - content: " "; - display: block; - height: 0.625em; - margin-top: -0.4375em; - pointer-events: none; - position: absolute; - top: 50%; - transform: rotate(-45deg); - transform-origin: center; - width: 0.625em; } - -.admonition:not(:last-child), .tabs:not(:last-child), .message:not(:last-child), .list:not(:last-child), .level:not(:last-child), .breadcrumb:not(:last-child), .highlight:not(:last-child), .block:not(:last-child), .title:not(:last-child), -.subtitle:not(:last-child), .table-container:not(:last-child), .table:not(:last-child), .progress:not(:last-child), .notification:not(:last-child), .content:not(:last-child), .box:not(:last-child) { - margin-bottom: 1.5rem; } - -.modal-close, .delete { - -moz-appearance: none; - -webkit-appearance: none; - background-color: rgba(10, 10, 10, 0.2); - border: none; - border-radius: 290486px; - cursor: pointer; - pointer-events: auto; - display: inline-block; - flex-grow: 0; - flex-shrink: 0; - font-size: 0; - height: 20px; - max-height: 20px; - max-width: 20px; - min-height: 20px; - min-width: 20px; - outline: none; - position: relative; - vertical-align: top; - width: 20px; } - .modal-close::before, .delete::before, .modal-close::after, .delete::after { - background-color: white; - content: ""; - display: block; - left: 50%; - position: absolute; - top: 50%; - transform: translateX(-50%) translateY(-50%) rotate(45deg); - transform-origin: center center; } - .modal-close::before, .delete::before { - height: 2px; - width: 50%; } - .modal-close::after, .delete::after { - height: 50%; - width: 2px; } - .modal-close:hover, .delete:hover, .modal-close:focus, .delete:focus { - background-color: rgba(10, 10, 10, 0.3); } - .modal-close:active, .delete:active { - background-color: rgba(10, 10, 10, 0.4); } - .is-small.modal-close, #documenter .docs-sidebar form.docs-search > input.modal-close, .is-small.delete, #documenter .docs-sidebar form.docs-search > input.delete { - height: 16px; - max-height: 16px; - max-width: 16px; - min-height: 16px; - min-width: 16px; - width: 16px; } - .is-medium.modal-close, .is-medium.delete { - height: 24px; - max-height: 24px; - max-width: 24px; - min-height: 24px; - min-width: 24px; - width: 24px; } - .is-large.modal-close, .is-large.delete { - height: 32px; - max-height: 32px; - max-width: 32px; - min-height: 32px; - min-width: 32px; - width: 32px; } - -.control.is-loading::after, .select.is-loading::after, .loader, .button.is-loading::after { - animation: spinAround 500ms infinite linear; - border: 2px solid #dbdbdb; - border-radius: 290486px; - border-right-color: transparent; - border-top-color: transparent; - content: ""; - display: block; - height: 1em; - position: relative; - width: 1em; } - -.hero-video, .modal-background, .modal, .image.is-square img, #documenter .docs-sidebar .docs-logo > img.is-square img, -.image.is-square .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-square .has-ratio, .image.is-1by1 img, #documenter .docs-sidebar .docs-logo > img.is-1by1 img, -.image.is-1by1 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-1by1 .has-ratio, .image.is-5by4 img, #documenter .docs-sidebar .docs-logo > img.is-5by4 img, -.image.is-5by4 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-5by4 .has-ratio, .image.is-4by3 img, #documenter .docs-sidebar .docs-logo > img.is-4by3 img, -.image.is-4by3 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-4by3 .has-ratio, .image.is-3by2 img, #documenter .docs-sidebar .docs-logo > img.is-3by2 img, -.image.is-3by2 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-3by2 .has-ratio, .image.is-5by3 img, #documenter .docs-sidebar .docs-logo > img.is-5by3 img, -.image.is-5by3 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-5by3 .has-ratio, .image.is-16by9 img, #documenter .docs-sidebar .docs-logo > img.is-16by9 img, -.image.is-16by9 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-16by9 .has-ratio, .image.is-2by1 img, #documenter .docs-sidebar .docs-logo > img.is-2by1 img, -.image.is-2by1 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-2by1 .has-ratio, .image.is-3by1 img, #documenter .docs-sidebar .docs-logo > img.is-3by1 img, -.image.is-3by1 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-3by1 .has-ratio, .image.is-4by5 img, #documenter .docs-sidebar .docs-logo > img.is-4by5 img, -.image.is-4by5 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-4by5 .has-ratio, .image.is-3by4 img, #documenter .docs-sidebar .docs-logo > img.is-3by4 img, -.image.is-3by4 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-3by4 .has-ratio, .image.is-2by3 img, #documenter .docs-sidebar .docs-logo > img.is-2by3 img, -.image.is-2by3 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-2by3 .has-ratio, .image.is-3by5 img, #documenter .docs-sidebar .docs-logo > img.is-3by5 img, -.image.is-3by5 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-3by5 .has-ratio, .image.is-9by16 img, #documenter .docs-sidebar .docs-logo > img.is-9by16 img, -.image.is-9by16 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-9by16 .has-ratio, .image.is-1by2 img, #documenter .docs-sidebar .docs-logo > img.is-1by2 img, -.image.is-1by2 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-1by2 .has-ratio, .image.is-1by3 img, #documenter .docs-sidebar .docs-logo > img.is-1by3 img, -.image.is-1by3 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-1by3 .has-ratio, .is-overlay { - bottom: 0; - left: 0; - position: absolute; - right: 0; - top: 0; } - -.pagination-previous, -.pagination-next, -.pagination-link, -.pagination-ellipsis, .file-cta, -.file-name, .select select, .textarea, .input, #documenter .docs-sidebar form.docs-search > input, .button { - -moz-appearance: none; - -webkit-appearance: none; - align-items: center; - border: 1px solid transparent; - border-radius: 4px; - box-shadow: none; - display: inline-flex; - font-size: 1rem; - height: 2.25em; - justify-content: flex-start; - line-height: 1.5; - padding-bottom: calc(0.375em - 1px); - padding-left: calc(0.625em - 1px); - padding-right: calc(0.625em - 1px); - padding-top: calc(0.375em - 1px); - position: relative; - vertical-align: top; } - .pagination-previous:focus, - .pagination-next:focus, - .pagination-link:focus, - .pagination-ellipsis:focus, .file-cta:focus, - .file-name:focus, .select select:focus, .textarea:focus, .input:focus, #documenter .docs-sidebar form.docs-search > input:focus, .button:focus, .is-focused.pagination-previous, - .is-focused.pagination-next, - .is-focused.pagination-link, - .is-focused.pagination-ellipsis, .is-focused.file-cta, - .is-focused.file-name, .select select.is-focused, .is-focused.textarea, .is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-focused.button, .pagination-previous:active, - .pagination-next:active, - .pagination-link:active, - .pagination-ellipsis:active, .file-cta:active, - .file-name:active, .select select:active, .textarea:active, .input:active, #documenter .docs-sidebar form.docs-search > input:active, .button:active, .is-active.pagination-previous, - .is-active.pagination-next, - .is-active.pagination-link, - .is-active.pagination-ellipsis, .is-active.file-cta, - .is-active.file-name, .select select.is-active, .is-active.textarea, .is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active, .is-active.button { - outline: none; } - .pagination-previous[disabled], - .pagination-next[disabled], - .pagination-link[disabled], - .pagination-ellipsis[disabled], .file-cta[disabled], - .file-name[disabled], .select select[disabled], .textarea[disabled], .input[disabled], #documenter .docs-sidebar form.docs-search > input[disabled], .button[disabled], fieldset[disabled] .pagination-previous, - fieldset[disabled] .pagination-next, - fieldset[disabled] .pagination-link, - fieldset[disabled] .pagination-ellipsis, fieldset[disabled] .file-cta, - fieldset[disabled] .file-name, fieldset[disabled] .select select, .select fieldset[disabled] select, fieldset[disabled] .textarea, fieldset[disabled] .input, fieldset[disabled] #documenter .docs-sidebar form.docs-search > input, #documenter .docs-sidebar fieldset[disabled] form.docs-search > input, fieldset[disabled] .button { - cursor: not-allowed; } - -/*! minireset.css v0.0.4 | MIT License | github.com/jgthms/minireset.css */ -html, -body, -p, -ol, -ul, -li, -dl, -dt, -dd, -blockquote, -figure, -fieldset, -legend, -textarea, -pre, -iframe, -hr, -h1, -h2, -h3, -h4, -h5, -h6 { - margin: 0; - padding: 0; } - -h1, -h2, -h3, -h4, -h5, -h6 { - font-size: 100%; - font-weight: normal; } - -ul { - list-style: none; } - -button, -input, -select, -textarea { - margin: 0; } - -html { - box-sizing: border-box; } - -*, *::before, *::after { - box-sizing: inherit; } - -img, -embed, -iframe, -object, -video { - height: auto; - max-width: 100%; } - -audio { - max-width: 100%; } - -iframe { - border: 0; } - -table { - border-collapse: collapse; - border-spacing: 0; } - -td, -th { - padding: 0; } - td:not([align]), - th:not([align]) { - text-align: left; } - -html { - background-color: white; - font-size: 16px; - -moz-osx-font-smoothing: grayscale; - -webkit-font-smoothing: antialiased; - min-width: 300px; - overflow-x: auto; - overflow-y: scroll; - text-rendering: optimizeLegibility; - text-size-adjust: 100%; } - -article, -aside, -figure, -footer, -header, -hgroup, -section { - display: block; } - -body, -button, -input, -select, -textarea { - font-family: "Lato Medium", -apple-system, BlinkMacSystemFont, "Segoe UI", "Helvetica Neue", "Helvetica", "Arial", sans-serif; } - -code, -pre { - -moz-osx-font-smoothing: auto; - -webkit-font-smoothing: auto; - font-family: "JuliaMono", "SFMono-Regular", "Menlo", "Consolas", "Liberation Mono", "DejaVu Sans Mono", monospace; } - -body { - color: #222222; - font-size: 1em; - font-weight: 400; - line-height: 1.5; } - -a { - color: #2e63b8; - cursor: pointer; - text-decoration: none; } - a strong { - color: currentColor; } - a:hover { - color: #363636; } - -code { - background-color: rgba(0, 0, 0, 0.05); - color: #000000; - font-size: 0.875em; - font-weight: normal; - padding: 0.1em; } - -hr { - background-color: whitesmoke; - border: none; - display: block; - height: 2px; - margin: 1.5rem 0; } - -img { - height: auto; - max-width: 100%; } - -input[type="checkbox"], -input[type="radio"] { - vertical-align: baseline; } - -small { - font-size: 0.875em; } - -span { - font-style: inherit; - font-weight: inherit; } - -strong { - color: #222222; - font-weight: 700; } - -fieldset { - border: none; } - -pre { - -webkit-overflow-scrolling: touch; - background-color: whitesmoke; - color: #222222; - font-size: 0.875em; - overflow-x: auto; - padding: 1.25rem 1.5rem; - white-space: pre; - word-wrap: normal; } - pre code { - background-color: transparent; - color: currentColor; - font-size: 1em; - padding: 0; } - -table td, -table th { - vertical-align: top; } - table td:not([align]), - table th:not([align]) { - text-align: left; } - -table th { - color: #222222; } - -.is-clearfix::after { - clear: both; - content: " "; - display: table; } - -.is-pulled-left { - float: left !important; } - -.is-pulled-right { - float: right !important; } - -.is-clipped { - overflow: hidden !important; } - -.is-size-1 { - font-size: 3rem !important; } - -.is-size-2 { - font-size: 2.5rem !important; } - -.is-size-3 { - font-size: 2rem !important; } - -.is-size-4 { - font-size: 1.5rem !important; } - -.is-size-5 { - font-size: 1.25rem !important; } - -.is-size-6 { - font-size: 1rem !important; } - -.is-size-7, .docstring > section > a.docs-sourcelink { - font-size: 0.75rem !important; 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} - .is-size-3-touch { - font-size: 2rem !important; } - .is-size-4-touch { - font-size: 1.5rem !important; } - .is-size-5-touch { - font-size: 1.25rem !important; } - .is-size-6-touch { - font-size: 1rem !important; } - .is-size-7-touch { - font-size: 0.75rem !important; } } - -@media screen and (min-width: 1056px) { - .is-size-1-desktop { - font-size: 3rem !important; } - .is-size-2-desktop { - font-size: 2.5rem !important; } - .is-size-3-desktop { - font-size: 2rem !important; } - .is-size-4-desktop { - font-size: 1.5rem !important; } - .is-size-5-desktop { - font-size: 1.25rem !important; } - .is-size-6-desktop { - font-size: 1rem !important; } - .is-size-7-desktop { - font-size: 0.75rem !important; } } - -@media screen and (min-width: 1216px) { - .is-size-1-widescreen { - font-size: 3rem !important; } - .is-size-2-widescreen { - font-size: 2.5rem !important; } - .is-size-3-widescreen { - font-size: 2rem !important; } - .is-size-4-widescreen { - font-size: 1.5rem !important; } - .is-size-5-widescreen { - font-size: 1.25rem !important; } - .is-size-6-widescreen { - font-size: 1rem !important; } - .is-size-7-widescreen { - font-size: 0.75rem !important; } } - -@media screen and (min-width: 1408px) { - .is-size-1-fullhd { - font-size: 3rem !important; } - .is-size-2-fullhd { - font-size: 2.5rem !important; } - .is-size-3-fullhd { - font-size: 2rem !important; } - .is-size-4-fullhd { - font-size: 1.5rem !important; } - .is-size-5-fullhd { - font-size: 1.25rem !important; } - .is-size-6-fullhd { - font-size: 1rem !important; } - .is-size-7-fullhd { - font-size: 0.75rem !important; } } - -.has-text-centered { - text-align: center !important; } - -.has-text-justified { - text-align: justify !important; } - -.has-text-left { - text-align: left !important; } - -.has-text-right { - text-align: right !important; } - -@media screen and (max-width: 768px) { - .has-text-centered-mobile { - text-align: center !important; } } - -@media screen and (min-width: 769px), print { - .has-text-centered-tablet { - text-align: center !important; } } - -@media screen and (min-width: 769px) and (max-width: 1055px) { - .has-text-centered-tablet-only { - text-align: center !important; } } - -@media screen and (max-width: 1055px) { - .has-text-centered-touch { - text-align: center !important; } } - -@media screen and (min-width: 1056px) { - .has-text-centered-desktop { - text-align: center !important; } } - -@media screen and (min-width: 1056px) and (max-width: 1215px) { - .has-text-centered-desktop-only { - text-align: center !important; } } - -@media screen and (min-width: 1216px) { - .has-text-centered-widescreen { - text-align: center !important; } } - -@media screen and (min-width: 1216px) and (max-width: 1407px) { - .has-text-centered-widescreen-only { - text-align: center !important; } } - -@media screen and (min-width: 1408px) { - .has-text-centered-fullhd { - text-align: center !important; } } - -@media screen and (max-width: 768px) { - .has-text-justified-mobile { - text-align: justify !important; } } - -@media screen and (min-width: 769px), print { - .has-text-justified-tablet { - text-align: justify !important; } } - -@media screen and (min-width: 769px) and (max-width: 1055px) { - .has-text-justified-tablet-only { - text-align: justify !important; } } - -@media screen and (max-width: 1055px) { - .has-text-justified-touch { - text-align: justify !important; } } - -@media screen and (min-width: 1056px) { - .has-text-justified-desktop { - text-align: justify !important; } } - -@media screen and (min-width: 1056px) and (max-width: 1215px) { - .has-text-justified-desktop-only { - text-align: justify !important; } } - -@media screen and (min-width: 1216px) { - .has-text-justified-widescreen { - text-align: justify !important; } } - -@media screen and (min-width: 1216px) and (max-width: 1407px) { - .has-text-justified-widescreen-only { - text-align: justify !important; } } - -@media screen and (min-width: 1408px) { - .has-text-justified-fullhd { - text-align: justify !important; } } - -@media screen and (max-width: 768px) { - .has-text-left-mobile { - text-align: left !important; } } - -@media screen and (min-width: 769px), print { - .has-text-left-tablet { - text-align: left !important; } } - -@media screen and (min-width: 769px) and (max-width: 1055px) { - .has-text-left-tablet-only { - text-align: left !important; } } - -@media screen and (max-width: 1055px) { - .has-text-left-touch { - text-align: left !important; } } - -@media screen and (min-width: 1056px) { - .has-text-left-desktop { - text-align: left !important; } } - -@media screen and (min-width: 1056px) and (max-width: 1215px) { - .has-text-left-desktop-only { - text-align: left !important; } } - -@media screen and (min-width: 1216px) { - .has-text-left-widescreen { - text-align: left !important; } } - -@media screen and (min-width: 1216px) and (max-width: 1407px) { - .has-text-left-widescreen-only { - text-align: left !important; } } - -@media screen and (min-width: 1408px) { - .has-text-left-fullhd { - text-align: left !important; } } - -@media screen and (max-width: 768px) { - .has-text-right-mobile { - text-align: right !important; } } - -@media screen and (min-width: 769px), print { - .has-text-right-tablet { - text-align: right !important; } } - -@media screen and (min-width: 769px) and (max-width: 1055px) { - .has-text-right-tablet-only { - text-align: right !important; } } - -@media screen and (max-width: 1055px) { - .has-text-right-touch { - text-align: right !important; } } - -@media screen and (min-width: 1056px) { - .has-text-right-desktop { - text-align: right !important; } } - -@media screen and (min-width: 1056px) and (max-width: 1215px) { - .has-text-right-desktop-only { - text-align: right !important; } } - -@media screen and (min-width: 1216px) { - .has-text-right-widescreen { - text-align: right !important; } } - -@media screen and (min-width: 1216px) and (max-width: 1407px) { - .has-text-right-widescreen-only { - text-align: right !important; } } - -@media screen and (min-width: 1408px) { - .has-text-right-fullhd { - text-align: right !important; } } - -.is-capitalized { - text-transform: capitalize !important; } - -.is-lowercase { - text-transform: lowercase !important; } - -.is-uppercase { - text-transform: uppercase !important; 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- width: 1.5em; } - .button .icon:first-child:not(:last-child) { - margin-left: calc(-0.375em - 1px); - margin-right: 0.1875em; } - .button .icon:last-child:not(:first-child) { - margin-left: 0.1875em; - margin-right: calc(-0.375em - 1px); } - .button .icon:first-child:last-child { - margin-left: calc(-0.375em - 1px); - margin-right: calc(-0.375em - 1px); } - .button:hover, .button.is-hovered { - border-color: #b5b5b5; - color: #363636; } - .button:focus, .button.is-focused { - border-color: #3c5dcd; - color: #363636; } - .button:focus:not(:active), .button.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(46, 99, 184, 0.25); } - .button:active, .button.is-active { - border-color: #4a4a4a; - color: #363636; } - .button.is-text { - background-color: transparent; - border-color: transparent; - color: #222222; - text-decoration: underline; } - .button.is-text:hover, .button.is-text.is-hovered, .button.is-text:focus, .button.is-text.is-focused { - background-color: whitesmoke; - color: #222222; } - .button.is-text:active, .button.is-text.is-active { - background-color: #e8e8e8; - color: #222222; } - .button.is-text[disabled], fieldset[disabled] .button.is-text { - background-color: transparent; - border-color: transparent; - box-shadow: none; } - .button.is-white { - background-color: white; - border-color: transparent; - color: #0a0a0a; } - .button.is-white:hover, .button.is-white.is-hovered { - background-color: #f9f9f9; - border-color: transparent; - color: #0a0a0a; } - .button.is-white:focus, .button.is-white.is-focused { - border-color: transparent; - color: #0a0a0a; } - .button.is-white:focus:not(:active), .button.is-white.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); } - .button.is-white:active, .button.is-white.is-active { - background-color: #f2f2f2; - border-color: transparent; - color: #0a0a0a; } - .button.is-white[disabled], fieldset[disabled] .button.is-white { - background-color: white; - border-color: transparent; - box-shadow: none; } - .button.is-white.is-inverted { - background-color: #0a0a0a; - color: white; } - .button.is-white.is-inverted:hover, .button.is-white.is-inverted.is-hovered { - background-color: black; } - .button.is-white.is-inverted[disabled], fieldset[disabled] .button.is-white.is-inverted { - background-color: #0a0a0a; - border-color: transparent; - box-shadow: none; - color: white; } - .button.is-white.is-loading::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - .button.is-white.is-outlined { - background-color: transparent; - border-color: white; - color: white; } - .button.is-white.is-outlined:hover, .button.is-white.is-outlined.is-hovered, .button.is-white.is-outlined:focus, .button.is-white.is-outlined.is-focused { - background-color: white; - border-color: white; - color: #0a0a0a; } - .button.is-white.is-outlined.is-loading::after { - border-color: transparent transparent white white !important; } - .button.is-white.is-outlined.is-loading:hover::after, .button.is-white.is-outlined.is-loading.is-hovered::after, .button.is-white.is-outlined.is-loading:focus::after, .button.is-white.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - .button.is-white.is-outlined[disabled], fieldset[disabled] .button.is-white.is-outlined { - background-color: transparent; - border-color: white; - box-shadow: none; - color: white; } - .button.is-white.is-inverted.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - color: #0a0a0a; } - .button.is-white.is-inverted.is-outlined:hover, .button.is-white.is-inverted.is-outlined.is-hovered, .button.is-white.is-inverted.is-outlined:focus, .button.is-white.is-inverted.is-outlined.is-focused { - background-color: #0a0a0a; - color: white; } - .button.is-white.is-inverted.is-outlined.is-loading:hover::after, .button.is-white.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-white.is-inverted.is-outlined.is-loading:focus::after, .button.is-white.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent white white !important; } - .button.is-white.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-white.is-inverted.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - box-shadow: none; - color: #0a0a0a; } - .button.is-black { - background-color: #0a0a0a; - border-color: transparent; - color: white; } - .button.is-black:hover, .button.is-black.is-hovered { - background-color: #040404; - border-color: transparent; - color: white; } - .button.is-black:focus, .button.is-black.is-focused { - border-color: transparent; - color: white; } - .button.is-black:focus:not(:active), .button.is-black.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); } - .button.is-black:active, .button.is-black.is-active { - background-color: black; - border-color: transparent; - color: white; } - .button.is-black[disabled], fieldset[disabled] .button.is-black { - background-color: #0a0a0a; - border-color: transparent; - box-shadow: none; } - .button.is-black.is-inverted { - background-color: white; - color: #0a0a0a; } - .button.is-black.is-inverted:hover, .button.is-black.is-inverted.is-hovered { - background-color: #f2f2f2; } - .button.is-black.is-inverted[disabled], fieldset[disabled] .button.is-black.is-inverted { - background-color: white; - border-color: transparent; - box-shadow: none; - color: #0a0a0a; } - .button.is-black.is-loading::after { - border-color: transparent transparent white white !important; } - .button.is-black.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - color: #0a0a0a; } - .button.is-black.is-outlined:hover, .button.is-black.is-outlined.is-hovered, .button.is-black.is-outlined:focus, .button.is-black.is-outlined.is-focused { - background-color: #0a0a0a; - border-color: #0a0a0a; - color: white; } - .button.is-black.is-outlined.is-loading::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - .button.is-black.is-outlined.is-loading:hover::after, .button.is-black.is-outlined.is-loading.is-hovered::after, .button.is-black.is-outlined.is-loading:focus::after, .button.is-black.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent white white !important; } - .button.is-black.is-outlined[disabled], fieldset[disabled] .button.is-black.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - box-shadow: none; - color: #0a0a0a; } - .button.is-black.is-inverted.is-outlined { - background-color: transparent; - border-color: white; - color: white; } - .button.is-black.is-inverted.is-outlined:hover, .button.is-black.is-inverted.is-outlined.is-hovered, .button.is-black.is-inverted.is-outlined:focus, .button.is-black.is-inverted.is-outlined.is-focused { - background-color: white; - color: #0a0a0a; } - .button.is-black.is-inverted.is-outlined.is-loading:hover::after, .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-black.is-inverted.is-outlined.is-loading:focus::after, .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - .button.is-black.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-black.is-inverted.is-outlined { - background-color: transparent; - border-color: white; - box-shadow: none; - color: white; } - .button.is-light { - background-color: whitesmoke; - border-color: transparent; - color: #363636; } - .button.is-light:hover, .button.is-light.is-hovered { - background-color: #eeeeee; - border-color: transparent; - color: #363636; } - .button.is-light:focus, .button.is-light.is-focused { - border-color: transparent; - color: #363636; } - .button.is-light:focus:not(:active), .button.is-light.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(245, 245, 245, 0.25); } - .button.is-light:active, .button.is-light.is-active { - background-color: #e8e8e8; - border-color: transparent; - color: #363636; } - .button.is-light[disabled], fieldset[disabled] .button.is-light { - background-color: whitesmoke; - border-color: transparent; - box-shadow: none; } - .button.is-light.is-inverted { - background-color: #363636; - color: whitesmoke; } - .button.is-light.is-inverted:hover, .button.is-light.is-inverted.is-hovered { - background-color: #292929; } - .button.is-light.is-inverted[disabled], fieldset[disabled] .button.is-light.is-inverted { - background-color: #363636; - border-color: transparent; - box-shadow: none; - color: whitesmoke; } - .button.is-light.is-loading::after { - border-color: transparent transparent #363636 #363636 !important; } - .button.is-light.is-outlined { - background-color: transparent; - border-color: whitesmoke; - color: whitesmoke; } - .button.is-light.is-outlined:hover, .button.is-light.is-outlined.is-hovered, .button.is-light.is-outlined:focus, .button.is-light.is-outlined.is-focused { - background-color: whitesmoke; - border-color: whitesmoke; - color: #363636; } - .button.is-light.is-outlined.is-loading::after { - border-color: transparent transparent whitesmoke whitesmoke !important; } - .button.is-light.is-outlined.is-loading:hover::after, .button.is-light.is-outlined.is-loading.is-hovered::after, .button.is-light.is-outlined.is-loading:focus::after, .button.is-light.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #363636 #363636 !important; } - .button.is-light.is-outlined[disabled], fieldset[disabled] .button.is-light.is-outlined { - background-color: transparent; - border-color: whitesmoke; - box-shadow: none; - color: whitesmoke; } - .button.is-light.is-inverted.is-outlined { - background-color: transparent; - border-color: #363636; - color: #363636; } - .button.is-light.is-inverted.is-outlined:hover, .button.is-light.is-inverted.is-outlined.is-hovered, .button.is-light.is-inverted.is-outlined:focus, .button.is-light.is-inverted.is-outlined.is-focused { - background-color: #363636; - color: whitesmoke; } - .button.is-light.is-inverted.is-outlined.is-loading:hover::after, .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-light.is-inverted.is-outlined.is-loading:focus::after, .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent whitesmoke whitesmoke !important; } - .button.is-light.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-light.is-inverted.is-outlined { - background-color: transparent; - border-color: #363636; - box-shadow: none; - color: #363636; } - .button.is-dark, .content kbd.button { - background-color: #363636; - border-color: transparent; - color: whitesmoke; } - .button.is-dark:hover, .content kbd.button:hover, .button.is-dark.is-hovered, .content kbd.button.is-hovered { - background-color: #2f2f2f; - border-color: transparent; - color: whitesmoke; } - .button.is-dark:focus, .content kbd.button:focus, .button.is-dark.is-focused, .content kbd.button.is-focused { - border-color: transparent; - color: whitesmoke; } - .button.is-dark:focus:not(:active), .content kbd.button:focus:not(:active), .button.is-dark.is-focused:not(:active), .content kbd.button.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(54, 54, 54, 0.25); } - .button.is-dark:active, .content kbd.button:active, .button.is-dark.is-active, .content kbd.button.is-active { - background-color: #292929; - border-color: transparent; - color: whitesmoke; } - .button.is-dark[disabled], .content kbd.button[disabled], fieldset[disabled] .button.is-dark, fieldset[disabled] .content kbd.button, .content fieldset[disabled] kbd.button { - background-color: #363636; - border-color: transparent; - box-shadow: none; } - .button.is-dark.is-inverted, .content kbd.button.is-inverted { - background-color: whitesmoke; - color: #363636; } - .button.is-dark.is-inverted:hover, .content kbd.button.is-inverted:hover, .button.is-dark.is-inverted.is-hovered, .content kbd.button.is-inverted.is-hovered { - background-color: #e8e8e8; } - .button.is-dark.is-inverted[disabled], .content kbd.button.is-inverted[disabled], fieldset[disabled] .button.is-dark.is-inverted, fieldset[disabled] .content kbd.button.is-inverted, .content fieldset[disabled] kbd.button.is-inverted { - background-color: whitesmoke; - border-color: transparent; - box-shadow: none; - color: #363636; } - .button.is-dark.is-loading::after, .content kbd.button.is-loading::after { - border-color: transparent transparent whitesmoke whitesmoke !important; } - .button.is-dark.is-outlined, .content kbd.button.is-outlined { - background-color: transparent; - border-color: #363636; - color: #363636; } - .button.is-dark.is-outlined:hover, .content kbd.button.is-outlined:hover, .button.is-dark.is-outlined.is-hovered, .content kbd.button.is-outlined.is-hovered, .button.is-dark.is-outlined:focus, .content kbd.button.is-outlined:focus, .button.is-dark.is-outlined.is-focused, .content kbd.button.is-outlined.is-focused { - background-color: #363636; - border-color: #363636; - color: whitesmoke; } - .button.is-dark.is-outlined.is-loading::after, .content kbd.button.is-outlined.is-loading::after { - border-color: transparent transparent #363636 #363636 !important; } - .button.is-dark.is-outlined.is-loading:hover::after, .content kbd.button.is-outlined.is-loading:hover::after, .button.is-dark.is-outlined.is-loading.is-hovered::after, .content kbd.button.is-outlined.is-loading.is-hovered::after, .button.is-dark.is-outlined.is-loading:focus::after, .content kbd.button.is-outlined.is-loading:focus::after, .button.is-dark.is-outlined.is-loading.is-focused::after, .content kbd.button.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent whitesmoke whitesmoke !important; } - .button.is-dark.is-outlined[disabled], .content kbd.button.is-outlined[disabled], fieldset[disabled] .button.is-dark.is-outlined, fieldset[disabled] .content kbd.button.is-outlined, .content fieldset[disabled] kbd.button.is-outlined { - background-color: transparent; - border-color: #363636; - box-shadow: none; - color: #363636; } - .button.is-dark.is-inverted.is-outlined, .content kbd.button.is-inverted.is-outlined { - background-color: transparent; - border-color: whitesmoke; - color: whitesmoke; } - .button.is-dark.is-inverted.is-outlined:hover, .content kbd.button.is-inverted.is-outlined:hover, .button.is-dark.is-inverted.is-outlined.is-hovered, .content kbd.button.is-inverted.is-outlined.is-hovered, .button.is-dark.is-inverted.is-outlined:focus, .content kbd.button.is-inverted.is-outlined:focus, .button.is-dark.is-inverted.is-outlined.is-focused, .content kbd.button.is-inverted.is-outlined.is-focused { - background-color: whitesmoke; - color: #363636; } - .button.is-dark.is-inverted.is-outlined.is-loading:hover::after, .content kbd.button.is-inverted.is-outlined.is-loading:hover::after, .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after, .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-dark.is-inverted.is-outlined.is-loading:focus::after, .content kbd.button.is-inverted.is-outlined.is-loading:focus::after, .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after, .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #363636 #363636 !important; } - .button.is-dark.is-inverted.is-outlined[disabled], .content kbd.button.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-dark.is-inverted.is-outlined, fieldset[disabled] .content kbd.button.is-inverted.is-outlined, .content fieldset[disabled] kbd.button.is-inverted.is-outlined { - background-color: transparent; - border-color: whitesmoke; - box-shadow: none; - color: whitesmoke; } - .button.is-primary, .docstring > section > a.button.docs-sourcelink { - background-color: #4eb5de; - border-color: transparent; - color: #fff; } - .button.is-primary:hover, .docstring > section > a.button.docs-sourcelink:hover, .button.is-primary.is-hovered, .docstring > section > a.button.is-hovered.docs-sourcelink { - background-color: #43b1dc; - border-color: transparent; - color: #fff; } - .button.is-primary:focus, .docstring > section > a.button.docs-sourcelink:focus, .button.is-primary.is-focused, .docstring > section > a.button.is-focused.docs-sourcelink { - border-color: transparent; - color: #fff; } - .button.is-primary:focus:not(:active), .docstring > section > a.button.docs-sourcelink:focus:not(:active), .button.is-primary.is-focused:not(:active), .docstring > section > a.button.is-focused.docs-sourcelink:not(:active) { - box-shadow: 0 0 0 0.125em rgba(78, 181, 222, 0.25); } - .button.is-primary:active, .docstring > section > a.button.docs-sourcelink:active, .button.is-primary.is-active, .docstring > section > a.button.is-active.docs-sourcelink { - background-color: #39acda; - border-color: transparent; - color: #fff; } - .button.is-primary[disabled], .docstring > section > a.button.docs-sourcelink[disabled], fieldset[disabled] .button.is-primary, fieldset[disabled] .docstring > section > a.button.docs-sourcelink { - background-color: #4eb5de; - border-color: transparent; - box-shadow: none; } - .button.is-primary.is-inverted, .docstring > section > a.button.is-inverted.docs-sourcelink { - background-color: #fff; - color: #4eb5de; } - .button.is-primary.is-inverted:hover, .docstring > section > a.button.is-inverted.docs-sourcelink:hover, .button.is-primary.is-inverted.is-hovered, .docstring > section > a.button.is-inverted.is-hovered.docs-sourcelink { - background-color: #f2f2f2; } - .button.is-primary.is-inverted[disabled], .docstring > section > a.button.is-inverted.docs-sourcelink[disabled], fieldset[disabled] .button.is-primary.is-inverted, fieldset[disabled] .docstring > section > a.button.is-inverted.docs-sourcelink { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #4eb5de; } - .button.is-primary.is-loading::after, .docstring > section > a.button.is-loading.docs-sourcelink::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-primary.is-outlined, .docstring > section > a.button.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #4eb5de; - color: #4eb5de; } - .button.is-primary.is-outlined:hover, .docstring > section > a.button.is-outlined.docs-sourcelink:hover, .button.is-primary.is-outlined.is-hovered, .docstring > section > a.button.is-outlined.is-hovered.docs-sourcelink, .button.is-primary.is-outlined:focus, .docstring > section > a.button.is-outlined.docs-sourcelink:focus, .button.is-primary.is-outlined.is-focused, .docstring > section > a.button.is-outlined.is-focused.docs-sourcelink { - background-color: #4eb5de; - border-color: #4eb5de; - color: #fff; } - .button.is-primary.is-outlined.is-loading::after, .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink::after { - border-color: transparent transparent #4eb5de #4eb5de !important; } - .button.is-primary.is-outlined.is-loading:hover::after, .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink:hover::after, .button.is-primary.is-outlined.is-loading.is-hovered::after, .docstring > section > a.button.is-outlined.is-loading.is-hovered.docs-sourcelink::after, .button.is-primary.is-outlined.is-loading:focus::after, .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink:focus::after, .button.is-primary.is-outlined.is-loading.is-focused::after, .docstring > section > a.button.is-outlined.is-loading.is-focused.docs-sourcelink::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-primary.is-outlined[disabled], .docstring > section > a.button.is-outlined.docs-sourcelink[disabled], fieldset[disabled] .button.is-primary.is-outlined, fieldset[disabled] .docstring > section > a.button.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #4eb5de; - box-shadow: none; - color: #4eb5de; } - .button.is-primary.is-inverted.is-outlined, .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #fff; - color: #fff; } - .button.is-primary.is-inverted.is-outlined:hover, .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink:hover, .button.is-primary.is-inverted.is-outlined.is-hovered, .docstring > section > a.button.is-inverted.is-outlined.is-hovered.docs-sourcelink, .button.is-primary.is-inverted.is-outlined:focus, .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink:focus, .button.is-primary.is-inverted.is-outlined.is-focused, .docstring > section > a.button.is-inverted.is-outlined.is-focused.docs-sourcelink { - background-color: #fff; - color: #4eb5de; } - .button.is-primary.is-inverted.is-outlined.is-loading:hover::after, .docstring > section > a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:hover::after, .button.is-primary.is-inverted.is-outlined.is-loading.is-hovered::after, .docstring > section > a.button.is-inverted.is-outlined.is-loading.is-hovered.docs-sourcelink::after, .button.is-primary.is-inverted.is-outlined.is-loading:focus::after, .docstring > section > a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:focus::after, .button.is-primary.is-inverted.is-outlined.is-loading.is-focused::after, .docstring > section > a.button.is-inverted.is-outlined.is-loading.is-focused.docs-sourcelink::after { - border-color: transparent transparent #4eb5de #4eb5de !important; } - .button.is-primary.is-inverted.is-outlined[disabled], .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink[disabled], fieldset[disabled] .button.is-primary.is-inverted.is-outlined, fieldset[disabled] .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - .button.is-link { - background-color: #2e63b8; - border-color: transparent; - color: #fff; } - .button.is-link:hover, .button.is-link.is-hovered { - background-color: #2b5eae; - border-color: transparent; - color: #fff; } - .button.is-link:focus, .button.is-link.is-focused { - border-color: transparent; - color: #fff; } - .button.is-link:focus:not(:active), .button.is-link.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(46, 99, 184, 0.25); } - .button.is-link:active, .button.is-link.is-active { - background-color: #2958a4; - border-color: transparent; - color: #fff; } - .button.is-link[disabled], fieldset[disabled] .button.is-link { - background-color: #2e63b8; - border-color: transparent; - box-shadow: none; } - .button.is-link.is-inverted { - background-color: #fff; - color: #2e63b8; } - .button.is-link.is-inverted:hover, .button.is-link.is-inverted.is-hovered { - background-color: #f2f2f2; } - .button.is-link.is-inverted[disabled], fieldset[disabled] .button.is-link.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #2e63b8; } - .button.is-link.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-link.is-outlined { - background-color: transparent; - border-color: #2e63b8; - color: #2e63b8; } - .button.is-link.is-outlined:hover, .button.is-link.is-outlined.is-hovered, .button.is-link.is-outlined:focus, .button.is-link.is-outlined.is-focused { - background-color: #2e63b8; - border-color: #2e63b8; - color: #fff; } - .button.is-link.is-outlined.is-loading::after { - border-color: transparent transparent #2e63b8 #2e63b8 !important; } - .button.is-link.is-outlined.is-loading:hover::after, .button.is-link.is-outlined.is-loading.is-hovered::after, .button.is-link.is-outlined.is-loading:focus::after, .button.is-link.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-link.is-outlined[disabled], fieldset[disabled] .button.is-link.is-outlined { - background-color: transparent; - border-color: #2e63b8; - box-shadow: none; - color: #2e63b8; } - .button.is-link.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - .button.is-link.is-inverted.is-outlined:hover, .button.is-link.is-inverted.is-outlined.is-hovered, .button.is-link.is-inverted.is-outlined:focus, .button.is-link.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #2e63b8; } - .button.is-link.is-inverted.is-outlined.is-loading:hover::after, .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-link.is-inverted.is-outlined.is-loading:focus::after, .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #2e63b8 #2e63b8 !important; } - .button.is-link.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-link.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - .button.is-info { - background-color: #209cee; - border-color: transparent; - color: #fff; } - .button.is-info:hover, .button.is-info.is-hovered { - background-color: #1497ed; - border-color: transparent; - color: #fff; } - .button.is-info:focus, .button.is-info.is-focused { - border-color: transparent; - color: #fff; } - .button.is-info:focus:not(:active), .button.is-info.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(32, 156, 238, 0.25); } - .button.is-info:active, .button.is-info.is-active { - background-color: #1190e3; - border-color: transparent; - color: #fff; } - .button.is-info[disabled], fieldset[disabled] .button.is-info { - background-color: #209cee; - border-color: transparent; - box-shadow: none; } - .button.is-info.is-inverted { - background-color: #fff; - color: #209cee; } - .button.is-info.is-inverted:hover, .button.is-info.is-inverted.is-hovered { - background-color: #f2f2f2; } - .button.is-info.is-inverted[disabled], fieldset[disabled] .button.is-info.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #209cee; } - .button.is-info.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-info.is-outlined { - background-color: transparent; - border-color: #209cee; - color: #209cee; } - .button.is-info.is-outlined:hover, .button.is-info.is-outlined.is-hovered, .button.is-info.is-outlined:focus, .button.is-info.is-outlined.is-focused { - background-color: #209cee; - border-color: #209cee; - color: #fff; } - .button.is-info.is-outlined.is-loading::after { - border-color: transparent transparent #209cee #209cee !important; } - .button.is-info.is-outlined.is-loading:hover::after, .button.is-info.is-outlined.is-loading.is-hovered::after, .button.is-info.is-outlined.is-loading:focus::after, .button.is-info.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-info.is-outlined[disabled], fieldset[disabled] .button.is-info.is-outlined { - background-color: transparent; - border-color: #209cee; - box-shadow: none; - color: #209cee; } - .button.is-info.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - .button.is-info.is-inverted.is-outlined:hover, .button.is-info.is-inverted.is-outlined.is-hovered, .button.is-info.is-inverted.is-outlined:focus, .button.is-info.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #209cee; } - .button.is-info.is-inverted.is-outlined.is-loading:hover::after, .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-info.is-inverted.is-outlined.is-loading:focus::after, .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #209cee #209cee !important; } - .button.is-info.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-info.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - .button.is-success { - background-color: #22c35b; - border-color: transparent; - color: #fff; } - .button.is-success:hover, .button.is-success.is-hovered { - background-color: #20b856; - border-color: transparent; - color: #fff; } - .button.is-success:focus, .button.is-success.is-focused { - border-color: transparent; - color: #fff; } - .button.is-success:focus:not(:active), .button.is-success.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(34, 195, 91, 0.25); } - .button.is-success:active, .button.is-success.is-active { - background-color: #1ead51; - border-color: transparent; - color: #fff; } - .button.is-success[disabled], fieldset[disabled] .button.is-success { - background-color: #22c35b; - border-color: transparent; - box-shadow: none; } - .button.is-success.is-inverted { - background-color: #fff; - color: #22c35b; } - .button.is-success.is-inverted:hover, .button.is-success.is-inverted.is-hovered { - background-color: #f2f2f2; } - .button.is-success.is-inverted[disabled], fieldset[disabled] .button.is-success.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #22c35b; } - .button.is-success.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-success.is-outlined { - background-color: transparent; - border-color: #22c35b; - color: #22c35b; } - .button.is-success.is-outlined:hover, .button.is-success.is-outlined.is-hovered, .button.is-success.is-outlined:focus, .button.is-success.is-outlined.is-focused { - background-color: #22c35b; - border-color: #22c35b; - color: #fff; } - .button.is-success.is-outlined.is-loading::after { - border-color: transparent transparent #22c35b #22c35b !important; } - .button.is-success.is-outlined.is-loading:hover::after, .button.is-success.is-outlined.is-loading.is-hovered::after, .button.is-success.is-outlined.is-loading:focus::after, .button.is-success.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-success.is-outlined[disabled], fieldset[disabled] .button.is-success.is-outlined { - background-color: transparent; - border-color: #22c35b; - box-shadow: none; - color: #22c35b; } - .button.is-success.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - .button.is-success.is-inverted.is-outlined:hover, .button.is-success.is-inverted.is-outlined.is-hovered, .button.is-success.is-inverted.is-outlined:focus, .button.is-success.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #22c35b; } - .button.is-success.is-inverted.is-outlined.is-loading:hover::after, .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-success.is-inverted.is-outlined.is-loading:focus::after, .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #22c35b #22c35b !important; } - .button.is-success.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-success.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - .button.is-warning { - background-color: #ffdd57; - border-color: transparent; - color: rgba(0, 0, 0, 0.7); } - .button.is-warning:hover, .button.is-warning.is-hovered { - background-color: #ffda4a; - border-color: transparent; - color: rgba(0, 0, 0, 0.7); } - .button.is-warning:focus, .button.is-warning.is-focused { - border-color: transparent; - color: rgba(0, 0, 0, 0.7); } - .button.is-warning:focus:not(:active), .button.is-warning.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(255, 221, 87, 0.25); } - .button.is-warning:active, .button.is-warning.is-active { - background-color: #ffd83e; - border-color: transparent; - color: rgba(0, 0, 0, 0.7); } - .button.is-warning[disabled], fieldset[disabled] .button.is-warning { - background-color: #ffdd57; - border-color: transparent; - box-shadow: none; } - .button.is-warning.is-inverted { - background-color: rgba(0, 0, 0, 0.7); - color: #ffdd57; } - .button.is-warning.is-inverted:hover, .button.is-warning.is-inverted.is-hovered { - background-color: rgba(0, 0, 0, 0.7); } - .button.is-warning.is-inverted[disabled], fieldset[disabled] .button.is-warning.is-inverted { - background-color: rgba(0, 0, 0, 0.7); - border-color: transparent; - box-shadow: none; - color: #ffdd57; } - .button.is-warning.is-loading::after { - border-color: transparent transparent rgba(0, 0, 0, 0.7) rgba(0, 0, 0, 0.7) !important; } - .button.is-warning.is-outlined { - background-color: transparent; - border-color: #ffdd57; - color: #ffdd57; } - .button.is-warning.is-outlined:hover, .button.is-warning.is-outlined.is-hovered, .button.is-warning.is-outlined:focus, .button.is-warning.is-outlined.is-focused { - background-color: #ffdd57; - border-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .button.is-warning.is-outlined.is-loading::after { - border-color: transparent transparent #ffdd57 #ffdd57 !important; } - .button.is-warning.is-outlined.is-loading:hover::after, .button.is-warning.is-outlined.is-loading.is-hovered::after, .button.is-warning.is-outlined.is-loading:focus::after, .button.is-warning.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent rgba(0, 0, 0, 0.7) rgba(0, 0, 0, 0.7) !important; } - .button.is-warning.is-outlined[disabled], fieldset[disabled] .button.is-warning.is-outlined { - background-color: transparent; - border-color: #ffdd57; - box-shadow: none; - color: #ffdd57; } - .button.is-warning.is-inverted.is-outlined { - background-color: transparent; - border-color: rgba(0, 0, 0, 0.7); - color: rgba(0, 0, 0, 0.7); } - .button.is-warning.is-inverted.is-outlined:hover, .button.is-warning.is-inverted.is-outlined.is-hovered, .button.is-warning.is-inverted.is-outlined:focus, .button.is-warning.is-inverted.is-outlined.is-focused { - background-color: rgba(0, 0, 0, 0.7); - color: #ffdd57; } - .button.is-warning.is-inverted.is-outlined.is-loading:hover::after, .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-warning.is-inverted.is-outlined.is-loading:focus::after, .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #ffdd57 #ffdd57 !important; } - .button.is-warning.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-warning.is-inverted.is-outlined { - background-color: transparent; - border-color: rgba(0, 0, 0, 0.7); - box-shadow: none; - color: rgba(0, 0, 0, 0.7); } - .button.is-danger { - background-color: #da0b00; - border-color: transparent; - color: #fff; } - .button.is-danger:hover, .button.is-danger.is-hovered { - background-color: #cd0a00; - border-color: transparent; - color: #fff; } - .button.is-danger:focus, .button.is-danger.is-focused { - border-color: transparent; - color: #fff; } - .button.is-danger:focus:not(:active), .button.is-danger.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(218, 11, 0, 0.25); } - .button.is-danger:active, .button.is-danger.is-active { - background-color: #c10a00; - border-color: transparent; - color: #fff; } - .button.is-danger[disabled], fieldset[disabled] .button.is-danger { - background-color: #da0b00; - border-color: transparent; - box-shadow: none; } - .button.is-danger.is-inverted { - background-color: #fff; - color: #da0b00; } - .button.is-danger.is-inverted:hover, .button.is-danger.is-inverted.is-hovered { - background-color: #f2f2f2; } - .button.is-danger.is-inverted[disabled], fieldset[disabled] .button.is-danger.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #da0b00; } - .button.is-danger.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-danger.is-outlined { - background-color: transparent; - border-color: #da0b00; - color: #da0b00; } - .button.is-danger.is-outlined:hover, .button.is-danger.is-outlined.is-hovered, .button.is-danger.is-outlined:focus, .button.is-danger.is-outlined.is-focused { - background-color: #da0b00; - border-color: #da0b00; - color: #fff; } - .button.is-danger.is-outlined.is-loading::after { - border-color: transparent transparent #da0b00 #da0b00 !important; } - .button.is-danger.is-outlined.is-loading:hover::after, .button.is-danger.is-outlined.is-loading.is-hovered::after, .button.is-danger.is-outlined.is-loading:focus::after, .button.is-danger.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-danger.is-outlined[disabled], fieldset[disabled] .button.is-danger.is-outlined { - background-color: transparent; - border-color: #da0b00; - box-shadow: none; - color: #da0b00; } - .button.is-danger.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - .button.is-danger.is-inverted.is-outlined:hover, .button.is-danger.is-inverted.is-outlined.is-hovered, .button.is-danger.is-inverted.is-outlined:focus, .button.is-danger.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #da0b00; } - .button.is-danger.is-inverted.is-outlined.is-loading:hover::after, .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-danger.is-inverted.is-outlined.is-loading:focus::after, .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #da0b00 #da0b00 !important; } - .button.is-danger.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-danger.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - .button.is-small, #documenter .docs-sidebar form.docs-search > input.button { - border-radius: 2px; - font-size: 0.75rem; } - .button.is-normal { - font-size: 1rem; } - .button.is-medium { - font-size: 1.25rem; } - .button.is-large { - font-size: 1.5rem; } - .button[disabled], fieldset[disabled] .button { - background-color: white; - border-color: #dbdbdb; - box-shadow: none; - opacity: 0.5; } - .button.is-fullwidth { - display: flex; - width: 100%; } - .button.is-loading { - color: transparent !important; - pointer-events: none; } - .button.is-loading::after { - position: absolute; - left: calc(50% - (1em / 2)); - top: calc(50% - (1em / 2)); - position: absolute !important; } - .button.is-static { - background-color: whitesmoke; - border-color: #dbdbdb; - color: #6b6b6b; - box-shadow: none; - pointer-events: none; } - .button.is-rounded, #documenter .docs-sidebar form.docs-search > input.button { - border-radius: 290486px; - padding-left: 1em; - padding-right: 1em; } - -.buttons { - align-items: center; - display: flex; - flex-wrap: wrap; - justify-content: flex-start; } - .buttons .button { - margin-bottom: 0.5rem; } - .buttons .button:not(:last-child):not(.is-fullwidth) { - margin-right: 0.5rem; } - .buttons:last-child { - margin-bottom: -0.5rem; } - .buttons:not(:last-child) { - margin-bottom: 1rem; } - .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large) { - border-radius: 2px; - font-size: 0.75rem; } - .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large) { - font-size: 1.25rem; } - .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium) { - font-size: 1.5rem; } - .buttons.has-addons .button:not(:first-child) { - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - .buttons.has-addons .button:not(:last-child) { - border-bottom-right-radius: 0; - border-top-right-radius: 0; - margin-right: -1px; } - .buttons.has-addons .button:last-child { - margin-right: 0; } - .buttons.has-addons .button:hover, .buttons.has-addons .button.is-hovered { - z-index: 2; } - .buttons.has-addons .button:focus, .buttons.has-addons .button.is-focused, .buttons.has-addons .button:active, .buttons.has-addons .button.is-active, .buttons.has-addons .button.is-selected { - z-index: 3; } - .buttons.has-addons .button:focus:hover, .buttons.has-addons .button.is-focused:hover, .buttons.has-addons .button:active:hover, .buttons.has-addons .button.is-active:hover, .buttons.has-addons .button.is-selected:hover { - z-index: 4; } - .buttons.has-addons .button.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - .buttons.is-centered { - justify-content: center; } - .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth) { - margin-left: 0.25rem; - margin-right: 0.25rem; } - .buttons.is-right { - justify-content: flex-end; } - .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth) { - margin-left: 0.25rem; - margin-right: 0.25rem; } - -.container { - flex-grow: 1; - margin: 0 auto; - position: relative; - width: auto; } - @media screen and (min-width: 1056px) { - .container { - max-width: 992px; } - .container.is-fluid { - margin-left: 32px; - margin-right: 32px; - max-width: none; } } - @media screen and (max-width: 1215px) { - .container.is-widescreen { - max-width: 1152px; } } - @media screen and (max-width: 1407px) { - .container.is-fullhd { - max-width: 1344px; } } - @media screen and (min-width: 1216px) { - .container { - max-width: 1152px; } } - @media screen and (min-width: 1408px) { - .container { - max-width: 1344px; } } -.content li + li { - margin-top: 0.25em; } - -.content p:not(:last-child), -.content dl:not(:last-child), -.content ol:not(:last-child), -.content ul:not(:last-child), -.content blockquote:not(:last-child), -.content pre:not(:last-child), -.content table:not(:last-child) { - margin-bottom: 1em; } - -.content h1, -.content h2, -.content h3, -.content h4, -.content h5, -.content h6 { - color: #222222; - font-weight: 600; - line-height: 1.125; } - -.content h1 { - font-size: 2em; - margin-bottom: 0.5em; } - .content h1:not(:first-child) { - margin-top: 1em; } - -.content h2 { - font-size: 1.75em; - margin-bottom: 0.5714em; } - .content h2:not(:first-child) { - margin-top: 1.1428em; } - -.content h3 { - font-size: 1.5em; - margin-bottom: 0.6666em; } - .content h3:not(:first-child) { - margin-top: 1.3333em; } - -.content h4 { - font-size: 1.25em; - margin-bottom: 0.8em; } - -.content h5 { - font-size: 1.125em; - margin-bottom: 0.8888em; } - -.content h6 { - font-size: 1em; - margin-bottom: 1em; } - -.content blockquote { - background-color: whitesmoke; - border-left: 5px solid #dbdbdb; - padding: 1.25em 1.5em; } - -.content ol { - list-style-position: outside; - margin-left: 2em; - margin-top: 1em; } - .content ol:not([type]) { - list-style-type: decimal; } - .content ol.is-lower-alpha:not([type]) { - list-style-type: lower-alpha; } - .content ol.is-lower-roman:not([type]) { - list-style-type: lower-roman; } - .content ol.is-upper-alpha:not([type]) { - list-style-type: upper-alpha; } - .content ol.is-upper-roman:not([type]) { - list-style-type: upper-roman; } - -.content ul { - list-style: disc outside; - margin-left: 2em; - margin-top: 1em; } - .content ul ul { - list-style-type: circle; - margin-top: 0.5em; } - .content ul ul ul { - list-style-type: square; } - -.content dd { - margin-left: 2em; } - -.content figure { - margin-left: 2em; - margin-right: 2em; - text-align: center; } - .content figure:not(:first-child) { - margin-top: 2em; } - .content figure:not(:last-child) { - margin-bottom: 2em; } - .content figure img { - display: inline-block; } - .content figure figcaption { - font-style: italic; } - -.content pre { - -webkit-overflow-scrolling: touch; - overflow-x: auto; - padding: 0; - white-space: pre; - word-wrap: normal; } - -.content sup, -.content sub { - font-size: 75%; } - -.content table { - width: 100%; } - .content table td, - .content table th { - border: 1px solid #dbdbdb; - border-width: 0 0 1px; - padding: 0.5em 0.75em; - vertical-align: top; } - .content table th { - color: #222222; } - .content table th:not([align]) { - text-align: left; } - .content table thead td, - .content table thead th { - border-width: 0 0 2px; - color: #222222; } - .content table tfoot td, - .content table tfoot th { - border-width: 2px 0 0; - color: #222222; } - .content table tbody tr:last-child td, - .content table tbody tr:last-child th { - border-bottom-width: 0; } - -.content .tabs li + li { - margin-top: 0; } - -.content.is-small, #documenter .docs-sidebar form.docs-search > input.content { - font-size: 0.75rem; } - -.content.is-medium { - font-size: 1.25rem; } - -.content.is-large { - font-size: 1.5rem; } - -.icon { - align-items: center; - display: inline-flex; - justify-content: center; - height: 1.5rem; - width: 1.5rem; } - .icon.is-small, #documenter .docs-sidebar form.docs-search > input.icon { - height: 1rem; - width: 1rem; } - .icon.is-medium { - height: 2rem; - width: 2rem; } - .icon.is-large { - height: 3rem; - width: 3rem; } - -.image, #documenter .docs-sidebar .docs-logo > img { - display: block; - position: relative; } - .image img, #documenter .docs-sidebar .docs-logo > img img { - display: block; - height: auto; - width: 100%; } - .image img.is-rounded, #documenter .docs-sidebar .docs-logo > img img.is-rounded { - border-radius: 290486px; } - .image.is-square img, #documenter .docs-sidebar .docs-logo > img.is-square img, - .image.is-square .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-square .has-ratio, .image.is-1by1 img, #documenter .docs-sidebar .docs-logo > img.is-1by1 img, - .image.is-1by1 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-1by1 .has-ratio, .image.is-5by4 img, #documenter .docs-sidebar .docs-logo > img.is-5by4 img, - .image.is-5by4 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-5by4 .has-ratio, .image.is-4by3 img, #documenter .docs-sidebar .docs-logo > img.is-4by3 img, - .image.is-4by3 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-4by3 .has-ratio, .image.is-3by2 img, #documenter .docs-sidebar .docs-logo > img.is-3by2 img, - .image.is-3by2 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-3by2 .has-ratio, .image.is-5by3 img, #documenter .docs-sidebar .docs-logo > img.is-5by3 img, - .image.is-5by3 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-5by3 .has-ratio, .image.is-16by9 img, #documenter .docs-sidebar .docs-logo > img.is-16by9 img, - .image.is-16by9 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-16by9 .has-ratio, .image.is-2by1 img, #documenter .docs-sidebar .docs-logo > img.is-2by1 img, - .image.is-2by1 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-2by1 .has-ratio, .image.is-3by1 img, #documenter .docs-sidebar .docs-logo > img.is-3by1 img, - .image.is-3by1 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-3by1 .has-ratio, .image.is-4by5 img, #documenter .docs-sidebar .docs-logo > img.is-4by5 img, - .image.is-4by5 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-4by5 .has-ratio, .image.is-3by4 img, #documenter .docs-sidebar .docs-logo > img.is-3by4 img, - .image.is-3by4 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-3by4 .has-ratio, .image.is-2by3 img, #documenter .docs-sidebar .docs-logo > img.is-2by3 img, - .image.is-2by3 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-2by3 .has-ratio, .image.is-3by5 img, #documenter .docs-sidebar .docs-logo > img.is-3by5 img, - .image.is-3by5 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-3by5 .has-ratio, .image.is-9by16 img, #documenter .docs-sidebar .docs-logo > img.is-9by16 img, - .image.is-9by16 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-9by16 .has-ratio, .image.is-1by2 img, #documenter .docs-sidebar .docs-logo > img.is-1by2 img, - .image.is-1by2 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-1by2 .has-ratio, .image.is-1by3 img, #documenter .docs-sidebar .docs-logo > img.is-1by3 img, - .image.is-1by3 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-1by3 .has-ratio { - height: 100%; - width: 100%; } - .image.is-square, #documenter .docs-sidebar .docs-logo > img.is-square, .image.is-1by1, #documenter .docs-sidebar .docs-logo > img.is-1by1 { - padding-top: 100%; } - .image.is-5by4, #documenter .docs-sidebar .docs-logo > img.is-5by4 { - padding-top: 80%; } - .image.is-4by3, #documenter .docs-sidebar .docs-logo > img.is-4by3 { - padding-top: 75%; } - .image.is-3by2, #documenter .docs-sidebar .docs-logo > img.is-3by2 { - padding-top: 66.6666%; } - .image.is-5by3, #documenter .docs-sidebar .docs-logo > img.is-5by3 { - padding-top: 60%; } - .image.is-16by9, #documenter .docs-sidebar .docs-logo > img.is-16by9 { - padding-top: 56.25%; } - .image.is-2by1, #documenter .docs-sidebar .docs-logo > img.is-2by1 { - padding-top: 50%; } - .image.is-3by1, #documenter .docs-sidebar .docs-logo > img.is-3by1 { - padding-top: 33.3333%; } - .image.is-4by5, #documenter .docs-sidebar .docs-logo > img.is-4by5 { - padding-top: 125%; } - .image.is-3by4, #documenter .docs-sidebar .docs-logo > img.is-3by4 { - padding-top: 133.3333%; } - .image.is-2by3, #documenter .docs-sidebar .docs-logo > img.is-2by3 { - padding-top: 150%; } - .image.is-3by5, #documenter .docs-sidebar .docs-logo > img.is-3by5 { - padding-top: 166.6666%; } - .image.is-9by16, #documenter .docs-sidebar .docs-logo > img.is-9by16 { - padding-top: 177.7777%; } - .image.is-1by2, #documenter .docs-sidebar .docs-logo > img.is-1by2 { - padding-top: 200%; } - .image.is-1by3, #documenter .docs-sidebar .docs-logo > img.is-1by3 { - padding-top: 300%; } - .image.is-16x16, #documenter .docs-sidebar .docs-logo > img.is-16x16 { - height: 16px; - width: 16px; } - .image.is-24x24, #documenter .docs-sidebar .docs-logo > img.is-24x24 { - height: 24px; - width: 24px; } - .image.is-32x32, #documenter .docs-sidebar .docs-logo > img.is-32x32 { - height: 32px; - width: 32px; } - .image.is-48x48, #documenter .docs-sidebar .docs-logo > img.is-48x48 { - height: 48px; - width: 48px; } - .image.is-64x64, #documenter .docs-sidebar .docs-logo > img.is-64x64 { - height: 64px; - width: 64px; } - .image.is-96x96, #documenter .docs-sidebar .docs-logo > img.is-96x96 { - height: 96px; - width: 96px; } - .image.is-128x128, #documenter .docs-sidebar .docs-logo > img.is-128x128 { - height: 128px; - width: 128px; } - -.notification { - background-color: whitesmoke; - border-radius: 4px; - padding: 1.25rem 2.5rem 1.25rem 1.5rem; - position: relative; } - .notification a:not(.button):not(.dropdown-item) { - color: currentColor; - text-decoration: underline; } - .notification strong { - color: currentColor; } - .notification code, - .notification pre { - background: white; } - .notification pre code { - background: transparent; } - .notification > .delete { - position: absolute; - right: 0.5rem; - top: 0.5rem; } - .notification .title, - .notification .subtitle, - .notification .content { - color: currentColor; } - .notification.is-white { - background-color: white; - color: #0a0a0a; } - .notification.is-black { - background-color: #0a0a0a; - color: white; } - .notification.is-light { - background-color: whitesmoke; - color: #363636; } - .notification.is-dark, .content kbd.notification { - background-color: #363636; - color: whitesmoke; } - .notification.is-primary, .docstring > section > a.notification.docs-sourcelink { - background-color: #4eb5de; - color: #fff; } - .notification.is-link { - background-color: #2e63b8; - color: #fff; } - .notification.is-info { - background-color: #209cee; - color: #fff; } - .notification.is-success { - background-color: #22c35b; - color: #fff; } - .notification.is-warning { - background-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .notification.is-danger { - background-color: #da0b00; - color: #fff; } - -.progress { - -moz-appearance: none; - -webkit-appearance: none; - border: none; - border-radius: 290486px; - display: block; - height: 1rem; - overflow: hidden; - padding: 0; - width: 100%; } - .progress::-webkit-progress-bar { - background-color: #dbdbdb; } - .progress::-webkit-progress-value { - background-color: #222222; } - .progress::-moz-progress-bar { - background-color: #222222; } - .progress::-ms-fill { - background-color: #222222; - border: none; } - .progress.is-white::-webkit-progress-value { - background-color: white; } - .progress.is-white::-moz-progress-bar { - background-color: white; } - .progress.is-white::-ms-fill { - background-color: white; } - .progress.is-white:indeterminate { - background-image: linear-gradient(to right, white 30%, #dbdbdb 30%); } - .progress.is-black::-webkit-progress-value { - background-color: #0a0a0a; } - .progress.is-black::-moz-progress-bar { - background-color: #0a0a0a; } - .progress.is-black::-ms-fill { - background-color: #0a0a0a; } - .progress.is-black:indeterminate { - background-image: linear-gradient(to right, #0a0a0a 30%, #dbdbdb 30%); } - .progress.is-light::-webkit-progress-value { - background-color: whitesmoke; } - .progress.is-light::-moz-progress-bar { - background-color: whitesmoke; } - .progress.is-light::-ms-fill { - background-color: whitesmoke; } - .progress.is-light:indeterminate { - background-image: linear-gradient(to right, whitesmoke 30%, #dbdbdb 30%); } - .progress.is-dark::-webkit-progress-value, .content kbd.progress::-webkit-progress-value { - background-color: #363636; } - .progress.is-dark::-moz-progress-bar, .content kbd.progress::-moz-progress-bar { - background-color: #363636; } - .progress.is-dark::-ms-fill, .content kbd.progress::-ms-fill { - background-color: #363636; } - .progress.is-dark:indeterminate, .content kbd.progress:indeterminate { - background-image: linear-gradient(to right, #363636 30%, #dbdbdb 30%); } - .progress.is-primary::-webkit-progress-value, .docstring > section > a.progress.docs-sourcelink::-webkit-progress-value { - background-color: #4eb5de; } - .progress.is-primary::-moz-progress-bar, .docstring > section > a.progress.docs-sourcelink::-moz-progress-bar { - background-color: #4eb5de; } - .progress.is-primary::-ms-fill, .docstring > section > a.progress.docs-sourcelink::-ms-fill { - background-color: #4eb5de; } - .progress.is-primary:indeterminate, .docstring > section > a.progress.docs-sourcelink:indeterminate { - background-image: linear-gradient(to right, #4eb5de 30%, #dbdbdb 30%); } - .progress.is-link::-webkit-progress-value { - background-color: #2e63b8; } - .progress.is-link::-moz-progress-bar { - background-color: #2e63b8; } - .progress.is-link::-ms-fill { - background-color: #2e63b8; } - .progress.is-link:indeterminate { - background-image: linear-gradient(to right, #2e63b8 30%, #dbdbdb 30%); } - .progress.is-info::-webkit-progress-value { - background-color: #209cee; } - .progress.is-info::-moz-progress-bar { - background-color: #209cee; } - .progress.is-info::-ms-fill { - background-color: #209cee; } - .progress.is-info:indeterminate { - background-image: linear-gradient(to right, #209cee 30%, #dbdbdb 30%); } - .progress.is-success::-webkit-progress-value { - background-color: #22c35b; } - .progress.is-success::-moz-progress-bar { - background-color: #22c35b; } - .progress.is-success::-ms-fill { - background-color: #22c35b; } - .progress.is-success:indeterminate { - background-image: linear-gradient(to right, #22c35b 30%, #dbdbdb 30%); } - .progress.is-warning::-webkit-progress-value { - background-color: #ffdd57; } - .progress.is-warning::-moz-progress-bar { - background-color: #ffdd57; } - .progress.is-warning::-ms-fill { - background-color: #ffdd57; } - .progress.is-warning:indeterminate { - background-image: linear-gradient(to right, #ffdd57 30%, #dbdbdb 30%); } - .progress.is-danger::-webkit-progress-value { - background-color: #da0b00; } - .progress.is-danger::-moz-progress-bar { - background-color: #da0b00; } - .progress.is-danger::-ms-fill { - background-color: #da0b00; } - .progress.is-danger:indeterminate { - background-image: linear-gradient(to right, #da0b00 30%, #dbdbdb 30%); } - .progress:indeterminate { - animation-duration: 1.5s; - animation-iteration-count: infinite; - animation-name: moveIndeterminate; - animation-timing-function: linear; - background-color: #dbdbdb; - background-image: linear-gradient(to right, #222222 30%, #dbdbdb 30%); - background-position: top left; - background-repeat: no-repeat; - background-size: 150% 150%; } - .progress:indeterminate::-webkit-progress-bar { - background-color: transparent; } - .progress:indeterminate::-moz-progress-bar { - background-color: transparent; } - .progress.is-small, #documenter .docs-sidebar form.docs-search > input.progress { - height: 0.75rem; } - .progress.is-medium { - height: 1.25rem; } - .progress.is-large { - height: 1.5rem; } - -@keyframes moveIndeterminate { - from { - background-position: 200% 0; } - to { - background-position: -200% 0; } } - -.table { - background-color: white; - color: #363636; } - .table td, - .table th { - border: 1px solid #dbdbdb; - border-width: 0 0 1px; - padding: 0.5em 0.75em; - vertical-align: top; } - .table td.is-white, - .table th.is-white { - background-color: white; - border-color: white; - color: #0a0a0a; } - .table td.is-black, - .table th.is-black { - background-color: #0a0a0a; - border-color: #0a0a0a; - color: white; } - .table td.is-light, - .table th.is-light { - background-color: whitesmoke; - border-color: whitesmoke; - color: #363636; } - .table td.is-dark, - .table th.is-dark { - background-color: #363636; - border-color: #363636; - color: whitesmoke; } - .table td.is-primary, - .table th.is-primary { - background-color: #4eb5de; - border-color: #4eb5de; - color: #fff; } - .table td.is-link, - .table th.is-link { - background-color: #2e63b8; - border-color: #2e63b8; - color: #fff; } - .table td.is-info, - .table th.is-info { - background-color: #209cee; - border-color: #209cee; - color: #fff; } - .table td.is-success, - .table th.is-success { - background-color: #22c35b; - border-color: #22c35b; - color: #fff; } - .table td.is-warning, - .table th.is-warning { - background-color: #ffdd57; - border-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .table td.is-danger, - .table th.is-danger { - background-color: #da0b00; - border-color: #da0b00; - color: #fff; } - .table td.is-narrow, - .table th.is-narrow { - white-space: nowrap; - width: 1%; } - .table td.is-selected, - .table th.is-selected { - background-color: #4eb5de; - color: #fff; } - .table td.is-selected a, - .table td.is-selected strong, - .table th.is-selected a, - .table th.is-selected strong { - color: currentColor; } - .table th { - color: #222222; } - .table th:not([align]) { - text-align: left; } - .table tr.is-selected { - background-color: #4eb5de; - color: #fff; } - .table tr.is-selected a, - .table tr.is-selected strong { - color: currentColor; } - .table tr.is-selected td, - .table tr.is-selected th { - border-color: #fff; - color: currentColor; } - .table thead { - background-color: transparent; } - .table thead td, - .table thead th { - border-width: 0 0 2px; - color: #222222; } - .table tfoot { - background-color: transparent; } - .table tfoot td, - .table tfoot th { - border-width: 2px 0 0; - color: #222222; } - .table tbody { - background-color: transparent; } - .table tbody tr:last-child td, - .table tbody tr:last-child th { - border-bottom-width: 0; } - .table.is-bordered td, - .table.is-bordered th { - border-width: 1px; } - .table.is-bordered tr:last-child td, - .table.is-bordered tr:last-child th { - border-bottom-width: 1px; } - .table.is-fullwidth { - width: 100%; } - .table.is-hoverable tbody tr:not(.is-selected):hover { - background-color: #fafafa; } - .table.is-hoverable.is-striped tbody tr:not(.is-selected):hover { - background-color: #fafafa; } - .table.is-hoverable.is-striped tbody tr:not(.is-selected):hover:nth-child(even) { - background-color: whitesmoke; } - .table.is-narrow td, - .table.is-narrow th { - padding: 0.25em 0.5em; } - .table.is-striped tbody tr:not(.is-selected):nth-child(even) { - background-color: #fafafa; } - -.table-container { - -webkit-overflow-scrolling: touch; - overflow: auto; - overflow-y: hidden; - max-width: 100%; } - -.tags { - align-items: center; - display: flex; - flex-wrap: wrap; - justify-content: flex-start; } - .tags .tag, .tags .content kbd, .content .tags kbd, .tags .docstring > section > a.docs-sourcelink { - margin-bottom: 0.5rem; } - .tags .tag:not(:last-child), .tags .content kbd:not(:last-child), .content .tags kbd:not(:last-child), .tags .docstring > section > a.docs-sourcelink:not(:last-child) { - margin-right: 0.5rem; } - .tags:last-child { - margin-bottom: -0.5rem; } - .tags:not(:last-child) { - margin-bottom: 1rem; } - .tags.are-medium .tag:not(.is-normal):not(.is-large), .tags.are-medium .content kbd:not(.is-normal):not(.is-large), .content .tags.are-medium kbd:not(.is-normal):not(.is-large), .tags.are-medium .docstring > section > a.docs-sourcelink:not(.is-normal):not(.is-large) { - font-size: 1rem; } - .tags.are-large .tag:not(.is-normal):not(.is-medium), .tags.are-large .content kbd:not(.is-normal):not(.is-medium), .content .tags.are-large kbd:not(.is-normal):not(.is-medium), .tags.are-large .docstring > section > a.docs-sourcelink:not(.is-normal):not(.is-medium) { - font-size: 1.25rem; } - .tags.is-centered { - justify-content: center; } - .tags.is-centered .tag, .tags.is-centered .content kbd, .content .tags.is-centered kbd, .tags.is-centered .docstring > section > a.docs-sourcelink { - margin-right: 0.25rem; - margin-left: 0.25rem; } - .tags.is-right { - justify-content: flex-end; } - .tags.is-right .tag:not(:first-child), .tags.is-right .content kbd:not(:first-child), .content .tags.is-right kbd:not(:first-child), .tags.is-right .docstring > section > a.docs-sourcelink:not(:first-child) { - margin-left: 0.5rem; } - .tags.is-right .tag:not(:last-child), .tags.is-right .content kbd:not(:last-child), .content .tags.is-right kbd:not(:last-child), .tags.is-right .docstring > section > a.docs-sourcelink:not(:last-child) { - margin-right: 0; } - .tags.has-addons .tag, .tags.has-addons .content kbd, .content .tags.has-addons kbd, .tags.has-addons .docstring > section > a.docs-sourcelink { - margin-right: 0; } - .tags.has-addons .tag:not(:first-child), .tags.has-addons .content kbd:not(:first-child), .content .tags.has-addons kbd:not(:first-child), .tags.has-addons .docstring > section > a.docs-sourcelink:not(:first-child) { - margin-left: 0; - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - .tags.has-addons .tag:not(:last-child), .tags.has-addons .content kbd:not(:last-child), .content .tags.has-addons kbd:not(:last-child), .tags.has-addons .docstring > section > a.docs-sourcelink:not(:last-child) { - border-bottom-right-radius: 0; - border-top-right-radius: 0; } - -.tag:not(body), .content kbd:not(body), .docstring > section > a.docs-sourcelink:not(body) { - align-items: center; - background-color: whitesmoke; - border-radius: 4px; - color: #222222; - display: inline-flex; - font-size: 0.75rem; - height: 2em; - justify-content: center; - line-height: 1.5; - padding-left: 0.75em; - padding-right: 0.75em; - white-space: nowrap; } - .tag:not(body) .delete, .content kbd:not(body) .delete, .docstring > section > a.docs-sourcelink:not(body) .delete { - margin-left: 0.25rem; - margin-right: -0.375rem; } - .tag.is-white:not(body), .content kbd.is-white:not(body), .docstring > section > a.docs-sourcelink.is-white:not(body) { - background-color: white; - color: #0a0a0a; } - .tag.is-black:not(body), .content kbd.is-black:not(body), .docstring > section > a.docs-sourcelink.is-black:not(body) { - background-color: #0a0a0a; - color: white; } - .tag.is-light:not(body), .content kbd.is-light:not(body), .docstring > section > a.docs-sourcelink.is-light:not(body) { - background-color: whitesmoke; - color: #363636; } - .tag.is-dark:not(body), .content kbd:not(body), .docstring > section > a.docs-sourcelink.is-dark:not(body), .content .docstring > section > kbd:not(body) { - background-color: #363636; - color: whitesmoke; } - .tag.is-primary:not(body), .content kbd.is-primary:not(body), .docstring > section > a.docs-sourcelink:not(body) { - background-color: #4eb5de; - color: #fff; } - .tag.is-link:not(body), .content kbd.is-link:not(body), .docstring > section > a.docs-sourcelink.is-link:not(body) { - background-color: #2e63b8; - color: #fff; } - .tag.is-info:not(body), .content kbd.is-info:not(body), .docstring > section > a.docs-sourcelink.is-info:not(body) { - background-color: #209cee; - color: #fff; } - .tag.is-success:not(body), .content kbd.is-success:not(body), .docstring > section > a.docs-sourcelink.is-success:not(body) { - background-color: #22c35b; - color: #fff; } - .tag.is-warning:not(body), .content kbd.is-warning:not(body), .docstring > section > a.docs-sourcelink.is-warning:not(body) { - background-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .tag.is-danger:not(body), .content kbd.is-danger:not(body), .docstring > section > a.docs-sourcelink.is-danger:not(body) { - background-color: #da0b00; - color: #fff; } - .tag.is-normal:not(body), .content kbd.is-normal:not(body), .docstring > section > a.docs-sourcelink.is-normal:not(body) { - font-size: 0.75rem; } - .tag.is-medium:not(body), .content kbd.is-medium:not(body), .docstring > section > a.docs-sourcelink.is-medium:not(body) { - font-size: 1rem; } - .tag.is-large:not(body), .content kbd.is-large:not(body), .docstring > section > a.docs-sourcelink.is-large:not(body) { - font-size: 1.25rem; } - .tag:not(body) .icon:first-child:not(:last-child), .content kbd:not(body) .icon:first-child:not(:last-child), .docstring > section > a.docs-sourcelink:not(body) .icon:first-child:not(:last-child) { - margin-left: -0.375em; - margin-right: 0.1875em; } - .tag:not(body) .icon:last-child:not(:first-child), .content kbd:not(body) .icon:last-child:not(:first-child), .docstring > section > a.docs-sourcelink:not(body) .icon:last-child:not(:first-child) { - margin-left: 0.1875em; - margin-right: -0.375em; } - .tag:not(body) .icon:first-child:last-child, .content kbd:not(body) .icon:first-child:last-child, .docstring > section > a.docs-sourcelink:not(body) .icon:first-child:last-child { - margin-left: -0.375em; - margin-right: -0.375em; } - .tag.is-delete:not(body), .content kbd.is-delete:not(body), .docstring > section > a.docs-sourcelink.is-delete:not(body) { - margin-left: 1px; - padding: 0; - position: relative; - width: 2em; } - .tag.is-delete:not(body)::before, .content kbd.is-delete:not(body)::before, .docstring > section > a.docs-sourcelink.is-delete:not(body)::before, .tag.is-delete:not(body)::after, .content kbd.is-delete:not(body)::after, .docstring > section > a.docs-sourcelink.is-delete:not(body)::after { - background-color: currentColor; - content: ""; - display: block; - left: 50%; - position: absolute; - top: 50%; - transform: translateX(-50%) translateY(-50%) rotate(45deg); - transform-origin: center center; } - .tag.is-delete:not(body)::before, .content kbd.is-delete:not(body)::before, .docstring > section > a.docs-sourcelink.is-delete:not(body)::before { - height: 1px; - width: 50%; } - .tag.is-delete:not(body)::after, .content kbd.is-delete:not(body)::after, .docstring > section > a.docs-sourcelink.is-delete:not(body)::after { - height: 50%; - width: 1px; } - .tag.is-delete:not(body):hover, .content kbd.is-delete:not(body):hover, .docstring > section > a.docs-sourcelink.is-delete:not(body):hover, .tag.is-delete:not(body):focus, .content kbd.is-delete:not(body):focus, .docstring > section > a.docs-sourcelink.is-delete:not(body):focus { - background-color: #e8e8e8; } - .tag.is-delete:not(body):active, .content kbd.is-delete:not(body):active, .docstring > section > a.docs-sourcelink.is-delete:not(body):active { - background-color: #dbdbdb; } - .tag.is-rounded:not(body), #documenter .docs-sidebar form.docs-search > input:not(body), .content kbd.is-rounded:not(body), #documenter .docs-sidebar .content form.docs-search > input:not(body), .docstring > section > a.docs-sourcelink.is-rounded:not(body) { - border-radius: 290486px; } - -a.tag:hover, .docstring > section > a.docs-sourcelink:hover { - text-decoration: underline; } - -.title, -.subtitle { - word-break: break-word; } - .title em, - .title span, - .subtitle em, - .subtitle span { - font-weight: inherit; } - .title sub, - .subtitle sub { - font-size: 0.75em; } - .title sup, - .subtitle sup { - font-size: 0.75em; } - .title .tag, .title .content kbd, .content .title kbd, .title .docstring > section > a.docs-sourcelink, - .subtitle .tag, - .subtitle .content kbd, - .content .subtitle kbd, - .subtitle .docstring > section > a.docs-sourcelink { - vertical-align: middle; } - -.title { - color: #363636; - font-size: 2rem; - font-weight: 600; - line-height: 1.125; } - .title strong { - color: inherit; - font-weight: inherit; } - .title + .highlight { - margin-top: -0.75rem; } - .title:not(.is-spaced) + .subtitle { - margin-top: -1.25rem; } - .title.is-1 { - font-size: 3rem; } - .title.is-2 { - font-size: 2.5rem; } - .title.is-3 { - font-size: 2rem; } - .title.is-4 { - font-size: 1.5rem; } - .title.is-5 { - font-size: 1.25rem; } - .title.is-6 { - font-size: 1rem; } - .title.is-7 { - font-size: 0.75rem; } - -.subtitle { - color: #4a4a4a; - font-size: 1.25rem; - font-weight: 400; - line-height: 1.25; } - .subtitle strong { - color: #363636; - font-weight: 600; } - .subtitle:not(.is-spaced) + .title { - margin-top: -1.25rem; } - .subtitle.is-1 { - font-size: 3rem; } - .subtitle.is-2 { - font-size: 2.5rem; } - .subtitle.is-3 { - font-size: 2rem; } - .subtitle.is-4 { - font-size: 1.5rem; } - .subtitle.is-5 { - font-size: 1.25rem; } - .subtitle.is-6 { - font-size: 1rem; } - .subtitle.is-7 { - font-size: 0.75rem; } - -.heading { - display: block; - font-size: 11px; - letter-spacing: 1px; - margin-bottom: 5px; - text-transform: uppercase; } - -.highlight { - font-weight: 400; - max-width: 100%; - overflow: hidden; - padding: 0; } - .highlight pre { - overflow: auto; - max-width: 100%; } - -.number { - align-items: center; - background-color: whitesmoke; - border-radius: 290486px; - display: inline-flex; - font-size: 1.25rem; - height: 2em; - justify-content: center; - margin-right: 1.5rem; - min-width: 2.5em; - padding: 0.25rem 0.5rem; - text-align: center; - vertical-align: top; } - -.select select, .textarea, .input, #documenter .docs-sidebar form.docs-search > input { - background-color: white; - border-color: #dbdbdb; - border-radius: 4px; - color: #363636; } - .select select::-moz-placeholder, .textarea::-moz-placeholder, .input::-moz-placeholder, #documenter .docs-sidebar form.docs-search > input::-moz-placeholder { - color: rgba(54, 54, 54, 0.3); } - .select select::-webkit-input-placeholder, .textarea::-webkit-input-placeholder, .input::-webkit-input-placeholder, #documenter .docs-sidebar form.docs-search > input::-webkit-input-placeholder { - color: rgba(54, 54, 54, 0.3); } - .select select:-moz-placeholder, .textarea:-moz-placeholder, .input:-moz-placeholder, #documenter .docs-sidebar form.docs-search > input:-moz-placeholder { - color: rgba(54, 54, 54, 0.3); } - .select select:-ms-input-placeholder, .textarea:-ms-input-placeholder, .input:-ms-input-placeholder, #documenter .docs-sidebar form.docs-search > input:-ms-input-placeholder { - color: rgba(54, 54, 54, 0.3); } - .select select:hover, .textarea:hover, .input:hover, #documenter .docs-sidebar form.docs-search > input:hover, .select select.is-hovered, .is-hovered.textarea, .is-hovered.input, #documenter .docs-sidebar form.docs-search > input.is-hovered { - border-color: #b5b5b5; } - .select select:focus, .textarea:focus, .input:focus, #documenter .docs-sidebar form.docs-search > input:focus, .select select.is-focused, .is-focused.textarea, .is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .select select:active, .textarea:active, .input:active, #documenter .docs-sidebar form.docs-search > input:active, .select select.is-active, .is-active.textarea, .is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - border-color: #2e63b8; - box-shadow: 0 0 0 0.125em rgba(46, 99, 184, 0.25); } - .select select[disabled], .textarea[disabled], .input[disabled], #documenter .docs-sidebar form.docs-search > input[disabled], fieldset[disabled] .select select, .select fieldset[disabled] select, fieldset[disabled] .textarea, fieldset[disabled] .input, fieldset[disabled] #documenter .docs-sidebar form.docs-search > input, #documenter .docs-sidebar fieldset[disabled] form.docs-search > input { - background-color: whitesmoke; - border-color: whitesmoke; - box-shadow: none; - color: #6b6b6b; } - .select select[disabled]::-moz-placeholder, .textarea[disabled]::-moz-placeholder, .input[disabled]::-moz-placeholder, #documenter .docs-sidebar form.docs-search > input[disabled]::-moz-placeholder, fieldset[disabled] .select select::-moz-placeholder, .select fieldset[disabled] select::-moz-placeholder, fieldset[disabled] .textarea::-moz-placeholder, fieldset[disabled] .input::-moz-placeholder, fieldset[disabled] #documenter .docs-sidebar form.docs-search > input::-moz-placeholder, #documenter .docs-sidebar fieldset[disabled] form.docs-search > input::-moz-placeholder { - color: rgba(107, 107, 107, 0.3); } - .select select[disabled]::-webkit-input-placeholder, .textarea[disabled]::-webkit-input-placeholder, .input[disabled]::-webkit-input-placeholder, #documenter .docs-sidebar form.docs-search > input[disabled]::-webkit-input-placeholder, fieldset[disabled] .select select::-webkit-input-placeholder, .select fieldset[disabled] select::-webkit-input-placeholder, fieldset[disabled] .textarea::-webkit-input-placeholder, fieldset[disabled] .input::-webkit-input-placeholder, fieldset[disabled] #documenter .docs-sidebar form.docs-search > input::-webkit-input-placeholder, #documenter .docs-sidebar fieldset[disabled] form.docs-search > input::-webkit-input-placeholder { - color: rgba(107, 107, 107, 0.3); } - .select select[disabled]:-moz-placeholder, .textarea[disabled]:-moz-placeholder, .input[disabled]:-moz-placeholder, #documenter .docs-sidebar form.docs-search > input[disabled]:-moz-placeholder, fieldset[disabled] .select select:-moz-placeholder, .select fieldset[disabled] select:-moz-placeholder, fieldset[disabled] .textarea:-moz-placeholder, fieldset[disabled] .input:-moz-placeholder, fieldset[disabled] #documenter .docs-sidebar form.docs-search > input:-moz-placeholder, #documenter .docs-sidebar fieldset[disabled] form.docs-search > input:-moz-placeholder { - color: rgba(107, 107, 107, 0.3); } - .select select[disabled]:-ms-input-placeholder, .textarea[disabled]:-ms-input-placeholder, .input[disabled]:-ms-input-placeholder, #documenter .docs-sidebar form.docs-search > input[disabled]:-ms-input-placeholder, fieldset[disabled] .select select:-ms-input-placeholder, .select fieldset[disabled] select:-ms-input-placeholder, fieldset[disabled] .textarea:-ms-input-placeholder, fieldset[disabled] .input:-ms-input-placeholder, fieldset[disabled] #documenter .docs-sidebar form.docs-search > input:-ms-input-placeholder, #documenter .docs-sidebar fieldset[disabled] form.docs-search > input:-ms-input-placeholder { - color: rgba(107, 107, 107, 0.3); } - -.textarea, .input, #documenter .docs-sidebar form.docs-search > input { - box-shadow: inset 0 1px 2px rgba(10, 10, 10, 0.1); - max-width: 100%; - width: 100%; } - .textarea[readonly], .input[readonly], #documenter .docs-sidebar form.docs-search > input[readonly] { - box-shadow: none; } - .is-white.textarea, .is-white.input, #documenter .docs-sidebar form.docs-search > input.is-white { - border-color: white; } - .is-white.textarea:focus, .is-white.input:focus, #documenter .docs-sidebar form.docs-search > input.is-white:focus, .is-white.is-focused.textarea, .is-white.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-white.textarea:active, .is-white.input:active, #documenter .docs-sidebar form.docs-search > input.is-white:active, .is-white.is-active.textarea, .is-white.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); } - .is-black.textarea, .is-black.input, #documenter .docs-sidebar form.docs-search > input.is-black { - border-color: #0a0a0a; } - .is-black.textarea:focus, .is-black.input:focus, #documenter .docs-sidebar form.docs-search > input.is-black:focus, .is-black.is-focused.textarea, .is-black.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-black.textarea:active, .is-black.input:active, #documenter .docs-sidebar form.docs-search > input.is-black:active, .is-black.is-active.textarea, .is-black.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); } - .is-light.textarea, .is-light.input, #documenter .docs-sidebar form.docs-search > input.is-light { - border-color: whitesmoke; } - .is-light.textarea:focus, .is-light.input:focus, #documenter .docs-sidebar form.docs-search > input.is-light:focus, .is-light.is-focused.textarea, .is-light.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-light.textarea:active, .is-light.input:active, #documenter .docs-sidebar form.docs-search > input.is-light:active, .is-light.is-active.textarea, .is-light.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(245, 245, 245, 0.25); } - .is-dark.textarea, .content kbd.textarea, .is-dark.input, #documenter .docs-sidebar form.docs-search > input.is-dark, .content kbd.input { - border-color: #363636; } - .is-dark.textarea:focus, .content kbd.textarea:focus, .is-dark.input:focus, #documenter .docs-sidebar form.docs-search > input.is-dark:focus, .content kbd.input:focus, .is-dark.is-focused.textarea, .content kbd.is-focused.textarea, .is-dark.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .content kbd.is-focused.input, #documenter .docs-sidebar .content form.docs-search > input.is-focused, .is-dark.textarea:active, .content kbd.textarea:active, .is-dark.input:active, #documenter .docs-sidebar form.docs-search > input.is-dark:active, .content kbd.input:active, .is-dark.is-active.textarea, .content kbd.is-active.textarea, .is-dark.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active, .content kbd.is-active.input, #documenter .docs-sidebar .content form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(54, 54, 54, 0.25); } - .is-primary.textarea, .docstring > section > a.textarea.docs-sourcelink, .is-primary.input, #documenter .docs-sidebar form.docs-search > input.is-primary, .docstring > section > a.input.docs-sourcelink { - border-color: #4eb5de; } - .is-primary.textarea:focus, .docstring > section > a.textarea.docs-sourcelink:focus, .is-primary.input:focus, #documenter .docs-sidebar form.docs-search > input.is-primary:focus, .docstring > section > a.input.docs-sourcelink:focus, .is-primary.is-focused.textarea, .docstring > section > a.is-focused.textarea.docs-sourcelink, .is-primary.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .docstring > section > a.is-focused.input.docs-sourcelink, .is-primary.textarea:active, .docstring > section > a.textarea.docs-sourcelink:active, .is-primary.input:active, #documenter .docs-sidebar form.docs-search > input.is-primary:active, .docstring > section > a.input.docs-sourcelink:active, .is-primary.is-active.textarea, .docstring > section > a.is-active.textarea.docs-sourcelink, .is-primary.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active, .docstring > section > a.is-active.input.docs-sourcelink { - box-shadow: 0 0 0 0.125em rgba(78, 181, 222, 0.25); } - .is-link.textarea, .is-link.input, #documenter .docs-sidebar form.docs-search > input.is-link { - border-color: #2e63b8; } - .is-link.textarea:focus, .is-link.input:focus, #documenter .docs-sidebar form.docs-search > input.is-link:focus, .is-link.is-focused.textarea, .is-link.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-link.textarea:active, .is-link.input:active, #documenter .docs-sidebar form.docs-search > input.is-link:active, .is-link.is-active.textarea, .is-link.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(46, 99, 184, 0.25); } - .is-info.textarea, .is-info.input, #documenter .docs-sidebar form.docs-search > input.is-info { - border-color: #209cee; } - .is-info.textarea:focus, .is-info.input:focus, #documenter .docs-sidebar form.docs-search > input.is-info:focus, .is-info.is-focused.textarea, .is-info.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-info.textarea:active, .is-info.input:active, #documenter .docs-sidebar form.docs-search > input.is-info:active, .is-info.is-active.textarea, .is-info.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(32, 156, 238, 0.25); } - .is-success.textarea, .is-success.input, #documenter .docs-sidebar form.docs-search > input.is-success { - border-color: #22c35b; } - .is-success.textarea:focus, .is-success.input:focus, #documenter .docs-sidebar form.docs-search > input.is-success:focus, .is-success.is-focused.textarea, .is-success.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-success.textarea:active, .is-success.input:active, #documenter .docs-sidebar form.docs-search > input.is-success:active, .is-success.is-active.textarea, .is-success.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(34, 195, 91, 0.25); } - .is-warning.textarea, .is-warning.input, #documenter .docs-sidebar form.docs-search > input.is-warning { - border-color: #ffdd57; } - .is-warning.textarea:focus, .is-warning.input:focus, #documenter .docs-sidebar form.docs-search > input.is-warning:focus, .is-warning.is-focused.textarea, .is-warning.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-warning.textarea:active, .is-warning.input:active, #documenter .docs-sidebar form.docs-search > input.is-warning:active, .is-warning.is-active.textarea, .is-warning.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(255, 221, 87, 0.25); } - .is-danger.textarea, .is-danger.input, #documenter .docs-sidebar form.docs-search > input.is-danger { - border-color: #da0b00; } - .is-danger.textarea:focus, .is-danger.input:focus, #documenter .docs-sidebar form.docs-search > input.is-danger:focus, .is-danger.is-focused.textarea, .is-danger.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-danger.textarea:active, .is-danger.input:active, #documenter .docs-sidebar form.docs-search > input.is-danger:active, .is-danger.is-active.textarea, .is-danger.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(218, 11, 0, 0.25); } - .is-small.textarea, .is-small.input, #documenter .docs-sidebar form.docs-search > input { - border-radius: 2px; - font-size: 0.75rem; } - .is-medium.textarea, .is-medium.input, #documenter .docs-sidebar form.docs-search > input.is-medium { - font-size: 1.25rem; } - .is-large.textarea, .is-large.input, #documenter .docs-sidebar form.docs-search > input.is-large { - font-size: 1.5rem; } - .is-fullwidth.textarea, .is-fullwidth.input, #documenter .docs-sidebar form.docs-search > input.is-fullwidth { - display: block; - width: 100%; } - .is-inline.textarea, .is-inline.input, #documenter .docs-sidebar form.docs-search > input.is-inline { - display: inline; - width: auto; } - -.input.is-rounded, #documenter .docs-sidebar form.docs-search > input { - border-radius: 290486px; - padding-left: 1em; - padding-right: 1em; } - -.input.is-static, #documenter .docs-sidebar form.docs-search > input.is-static { - background-color: transparent; - border-color: transparent; - box-shadow: none; - padding-left: 0; - padding-right: 0; } - -.textarea { - display: block; - max-width: 100%; - min-width: 100%; - padding: 0.625em; - resize: vertical; } - .textarea:not([rows]) { - max-height: 600px; - min-height: 120px; } - .textarea[rows] { - height: initial; } - .textarea.has-fixed-size { - resize: none; } - -.radio, .checkbox { - cursor: pointer; - display: inline-block; - line-height: 1.25; - position: relative; } - .radio input, .checkbox input { - cursor: pointer; } - .radio:hover, .checkbox:hover { - color: #363636; } - .radio[disabled], .checkbox[disabled], fieldset[disabled] .radio, fieldset[disabled] .checkbox { - color: #6b6b6b; - cursor: not-allowed; } - -.radio + .radio { - margin-left: 0.5em; } - -.select { - display: inline-block; - max-width: 100%; - position: relative; - vertical-align: top; } - .select:not(.is-multiple) { - height: 2.25em; } - .select:not(.is-multiple):not(.is-loading)::after { - border-color: #2e63b8; - right: 1.125em; - z-index: 4; } - .select.is-rounded select, #documenter .docs-sidebar form.docs-search > input.select select { - border-radius: 290486px; - padding-left: 1em; } - .select select { - cursor: pointer; - display: block; - font-size: 1em; - max-width: 100%; - outline: none; } - .select select::-ms-expand { - display: none; } - .select select[disabled]:hover, fieldset[disabled] .select select:hover { - border-color: whitesmoke; } - .select select:not([multiple]) { - padding-right: 2.5em; } - .select select[multiple] { - height: auto; - padding: 0; } - .select select[multiple] option { - padding: 0.5em 1em; } - .select:not(.is-multiple):not(.is-loading):hover::after { - border-color: #363636; } - .select.is-white:not(:hover)::after { - border-color: white; } - .select.is-white select { - border-color: white; } - .select.is-white select:hover, .select.is-white select.is-hovered { - border-color: #f2f2f2; } - .select.is-white select:focus, .select.is-white select.is-focused, .select.is-white select:active, .select.is-white select.is-active { - box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); } - .select.is-black:not(:hover)::after { - border-color: #0a0a0a; } - .select.is-black select { - border-color: #0a0a0a; } - .select.is-black select:hover, .select.is-black select.is-hovered { - border-color: black; } - .select.is-black select:focus, .select.is-black select.is-focused, .select.is-black select:active, .select.is-black select.is-active { - box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); } - .select.is-light:not(:hover)::after { - border-color: whitesmoke; } - .select.is-light select { - border-color: whitesmoke; } - .select.is-light select:hover, .select.is-light select.is-hovered { - border-color: #e8e8e8; } - .select.is-light select:focus, .select.is-light select.is-focused, .select.is-light select:active, .select.is-light select.is-active { - box-shadow: 0 0 0 0.125em rgba(245, 245, 245, 0.25); } - .select.is-dark:not(:hover)::after, .content kbd.select:not(:hover)::after { - border-color: #363636; } - .select.is-dark select, .content kbd.select select { - border-color: #363636; } - .select.is-dark select:hover, .content kbd.select select:hover, .select.is-dark select.is-hovered, .content kbd.select select.is-hovered { - border-color: #292929; } - .select.is-dark select:focus, .content kbd.select select:focus, .select.is-dark select.is-focused, .content kbd.select select.is-focused, .select.is-dark select:active, .content kbd.select select:active, .select.is-dark select.is-active, .content kbd.select select.is-active { - box-shadow: 0 0 0 0.125em rgba(54, 54, 54, 0.25); } - .select.is-primary:not(:hover)::after, .docstring > section > a.select.docs-sourcelink:not(:hover)::after { - border-color: #4eb5de; } - .select.is-primary select, .docstring > section > a.select.docs-sourcelink select { - border-color: #4eb5de; } - .select.is-primary select:hover, .docstring > section > a.select.docs-sourcelink select:hover, .select.is-primary select.is-hovered, .docstring > section > a.select.docs-sourcelink select.is-hovered { - border-color: #39acda; } - .select.is-primary select:focus, .docstring > section > a.select.docs-sourcelink select:focus, .select.is-primary select.is-focused, .docstring > section > a.select.docs-sourcelink select.is-focused, .select.is-primary select:active, .docstring > section > a.select.docs-sourcelink select:active, .select.is-primary select.is-active, .docstring > section > a.select.docs-sourcelink select.is-active { - box-shadow: 0 0 0 0.125em rgba(78, 181, 222, 0.25); } - .select.is-link:not(:hover)::after { - border-color: #2e63b8; } - .select.is-link select { - border-color: #2e63b8; } - .select.is-link select:hover, .select.is-link select.is-hovered { - border-color: #2958a4; } - .select.is-link select:focus, .select.is-link select.is-focused, .select.is-link select:active, .select.is-link select.is-active { - box-shadow: 0 0 0 0.125em rgba(46, 99, 184, 0.25); } - .select.is-info:not(:hover)::after { - border-color: #209cee; } - .select.is-info select { - border-color: #209cee; } - .select.is-info select:hover, .select.is-info select.is-hovered { - border-color: #1190e3; } - .select.is-info select:focus, .select.is-info select.is-focused, .select.is-info select:active, .select.is-info select.is-active { - box-shadow: 0 0 0 0.125em rgba(32, 156, 238, 0.25); } - .select.is-success:not(:hover)::after { - border-color: #22c35b; } - .select.is-success select { - border-color: #22c35b; } - .select.is-success select:hover, .select.is-success select.is-hovered { - border-color: #1ead51; } - .select.is-success select:focus, .select.is-success select.is-focused, .select.is-success select:active, .select.is-success select.is-active { - box-shadow: 0 0 0 0.125em rgba(34, 195, 91, 0.25); } - .select.is-warning:not(:hover)::after { - border-color: #ffdd57; } - .select.is-warning select { - border-color: #ffdd57; } - .select.is-warning select:hover, .select.is-warning select.is-hovered { - border-color: #ffd83e; } - .select.is-warning select:focus, .select.is-warning select.is-focused, .select.is-warning select:active, .select.is-warning select.is-active { - box-shadow: 0 0 0 0.125em rgba(255, 221, 87, 0.25); } - .select.is-danger:not(:hover)::after { - border-color: #da0b00; } - .select.is-danger select { - border-color: #da0b00; } - .select.is-danger select:hover, .select.is-danger select.is-hovered { - border-color: #c10a00; } - .select.is-danger select:focus, .select.is-danger select.is-focused, .select.is-danger select:active, .select.is-danger select.is-active { - box-shadow: 0 0 0 0.125em rgba(218, 11, 0, 0.25); } - .select.is-small, #documenter .docs-sidebar form.docs-search > input.select { - border-radius: 2px; - font-size: 0.75rem; } - .select.is-medium { - font-size: 1.25rem; } - .select.is-large { - font-size: 1.5rem; } - .select.is-disabled::after { - border-color: #6b6b6b; } - .select.is-fullwidth { - width: 100%; } - .select.is-fullwidth select { - width: 100%; } - .select.is-loading::after { - margin-top: 0; - position: absolute; - right: 0.625em; - top: 0.625em; - transform: none; } - .select.is-loading.is-small:after, #documenter .docs-sidebar form.docs-search > input.is-loading:after { - font-size: 0.75rem; } - .select.is-loading.is-medium:after { - font-size: 1.25rem; } - .select.is-loading.is-large:after { - font-size: 1.5rem; } - -.file { - align-items: stretch; - display: flex; - justify-content: flex-start; - position: relative; } - .file.is-white .file-cta { - background-color: white; - border-color: transparent; - color: #0a0a0a; } - .file.is-white:hover .file-cta, .file.is-white.is-hovered .file-cta { - background-color: #f9f9f9; - border-color: transparent; - color: #0a0a0a; } - .file.is-white:focus .file-cta, .file.is-white.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(255, 255, 255, 0.25); - color: #0a0a0a; } - .file.is-white:active .file-cta, .file.is-white.is-active .file-cta { - background-color: #f2f2f2; - border-color: transparent; - color: #0a0a0a; } - .file.is-black .file-cta { - background-color: #0a0a0a; - border-color: transparent; - color: white; } - .file.is-black:hover .file-cta, .file.is-black.is-hovered .file-cta { - background-color: #040404; - border-color: transparent; - color: white; } - .file.is-black:focus .file-cta, .file.is-black.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(10, 10, 10, 0.25); - color: white; } - .file.is-black:active .file-cta, .file.is-black.is-active .file-cta { - background-color: black; - border-color: transparent; - color: white; } - .file.is-light .file-cta { - background-color: whitesmoke; - border-color: transparent; - color: #363636; } - .file.is-light:hover .file-cta, .file.is-light.is-hovered .file-cta { - background-color: #eeeeee; - border-color: transparent; - color: #363636; } - .file.is-light:focus .file-cta, .file.is-light.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(245, 245, 245, 0.25); - color: #363636; } - .file.is-light:active .file-cta, .file.is-light.is-active .file-cta { - background-color: #e8e8e8; - border-color: transparent; - color: #363636; } - .file.is-dark .file-cta, .content kbd.file .file-cta { - background-color: #363636; - border-color: transparent; - color: whitesmoke; } - .file.is-dark:hover .file-cta, .content kbd.file:hover .file-cta, .file.is-dark.is-hovered .file-cta, .content kbd.file.is-hovered .file-cta { - background-color: #2f2f2f; - border-color: transparent; - color: whitesmoke; } - .file.is-dark:focus .file-cta, .content kbd.file:focus .file-cta, .file.is-dark.is-focused .file-cta, .content kbd.file.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(54, 54, 54, 0.25); - color: whitesmoke; } - .file.is-dark:active .file-cta, .content kbd.file:active .file-cta, .file.is-dark.is-active .file-cta, .content kbd.file.is-active .file-cta { - background-color: #292929; - border-color: transparent; - color: whitesmoke; } - .file.is-primary .file-cta, .docstring > section > a.file.docs-sourcelink .file-cta { - background-color: #4eb5de; - border-color: transparent; - color: #fff; } - .file.is-primary:hover .file-cta, .docstring > section > a.file.docs-sourcelink:hover .file-cta, .file.is-primary.is-hovered .file-cta, .docstring > section > a.file.is-hovered.docs-sourcelink .file-cta { - background-color: #43b1dc; - border-color: transparent; - color: #fff; } - .file.is-primary:focus .file-cta, .docstring > section > a.file.docs-sourcelink:focus .file-cta, .file.is-primary.is-focused .file-cta, .docstring > section > a.file.is-focused.docs-sourcelink .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(78, 181, 222, 0.25); - color: #fff; } - .file.is-primary:active .file-cta, .docstring > section > a.file.docs-sourcelink:active .file-cta, .file.is-primary.is-active .file-cta, .docstring > section > a.file.is-active.docs-sourcelink .file-cta { - background-color: #39acda; - border-color: transparent; - color: #fff; } - .file.is-link .file-cta { - background-color: #2e63b8; - border-color: transparent; - color: #fff; } - .file.is-link:hover .file-cta, .file.is-link.is-hovered .file-cta { - background-color: #2b5eae; - border-color: transparent; - color: #fff; } - .file.is-link:focus .file-cta, .file.is-link.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(46, 99, 184, 0.25); - color: #fff; } - .file.is-link:active .file-cta, .file.is-link.is-active .file-cta { - background-color: #2958a4; - border-color: transparent; - color: #fff; } - .file.is-info .file-cta { - background-color: #209cee; - border-color: transparent; - color: #fff; } - .file.is-info:hover .file-cta, .file.is-info.is-hovered .file-cta { - background-color: #1497ed; - border-color: transparent; - color: #fff; } - .file.is-info:focus .file-cta, .file.is-info.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(32, 156, 238, 0.25); - color: #fff; } - .file.is-info:active .file-cta, .file.is-info.is-active .file-cta { - background-color: #1190e3; - border-color: transparent; - color: #fff; } - .file.is-success .file-cta { - background-color: #22c35b; - border-color: transparent; - color: #fff; } - .file.is-success:hover .file-cta, .file.is-success.is-hovered .file-cta { - background-color: #20b856; - border-color: transparent; - color: #fff; } - .file.is-success:focus .file-cta, .file.is-success.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(34, 195, 91, 0.25); - color: #fff; } - .file.is-success:active .file-cta, .file.is-success.is-active .file-cta { - background-color: #1ead51; - border-color: transparent; - color: #fff; } - .file.is-warning .file-cta { - background-color: #ffdd57; - border-color: transparent; - color: rgba(0, 0, 0, 0.7); } - .file.is-warning:hover .file-cta, .file.is-warning.is-hovered .file-cta { - background-color: #ffda4a; - border-color: transparent; - color: rgba(0, 0, 0, 0.7); } - .file.is-warning:focus .file-cta, .file.is-warning.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(255, 221, 87, 0.25); - color: rgba(0, 0, 0, 0.7); } - .file.is-warning:active .file-cta, .file.is-warning.is-active .file-cta { - background-color: #ffd83e; - border-color: transparent; - color: rgba(0, 0, 0, 0.7); } - .file.is-danger .file-cta { - background-color: #da0b00; - border-color: transparent; - color: #fff; } - .file.is-danger:hover .file-cta, .file.is-danger.is-hovered .file-cta { - background-color: #cd0a00; - border-color: transparent; - color: #fff; } - .file.is-danger:focus .file-cta, .file.is-danger.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(218, 11, 0, 0.25); - color: #fff; } - .file.is-danger:active .file-cta, .file.is-danger.is-active .file-cta { - background-color: #c10a00; - border-color: transparent; - color: #fff; } - .file.is-small, #documenter .docs-sidebar form.docs-search > input.file { - font-size: 0.75rem; } - .file.is-medium { - font-size: 1.25rem; } - .file.is-medium .file-icon .fa { - font-size: 21px; } - .file.is-large { - font-size: 1.5rem; } - .file.is-large .file-icon .fa { - font-size: 28px; } - .file.has-name .file-cta { - border-bottom-right-radius: 0; - border-top-right-radius: 0; } - .file.has-name .file-name { - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - .file.has-name.is-empty .file-cta { - border-radius: 4px; } - .file.has-name.is-empty .file-name { - display: none; } - .file.is-boxed .file-label { - flex-direction: column; } - .file.is-boxed .file-cta { - flex-direction: column; - height: auto; - padding: 1em 3em; } - .file.is-boxed .file-name { - border-width: 0 1px 1px; } - .file.is-boxed .file-icon { - height: 1.5em; - width: 1.5em; } - .file.is-boxed .file-icon .fa { - font-size: 21px; } - .file.is-boxed.is-small .file-icon .fa, #documenter .docs-sidebar form.docs-search > input.is-boxed .file-icon .fa { - font-size: 14px; } - .file.is-boxed.is-medium .file-icon .fa { - font-size: 28px; } - .file.is-boxed.is-large .file-icon .fa { - font-size: 35px; } - .file.is-boxed.has-name .file-cta { - border-radius: 4px 4px 0 0; } - .file.is-boxed.has-name .file-name { - border-radius: 0 0 4px 4px; - border-width: 0 1px 1px; } - .file.is-centered { - justify-content: center; } - .file.is-fullwidth .file-label { - width: 100%; } - .file.is-fullwidth .file-name { - flex-grow: 1; - max-width: none; } - .file.is-right { - justify-content: flex-end; } - .file.is-right .file-cta { - border-radius: 0 4px 4px 0; } - .file.is-right .file-name { - border-radius: 4px 0 0 4px; - border-width: 1px 0 1px 1px; - order: -1; } - -.file-label { - align-items: stretch; - display: flex; - cursor: pointer; - justify-content: flex-start; - overflow: hidden; - position: relative; } - .file-label:hover .file-cta { - background-color: #eeeeee; - color: #363636; } - .file-label:hover .file-name { - border-color: #d5d5d5; } - .file-label:active .file-cta { - background-color: #e8e8e8; - color: #363636; } - .file-label:active .file-name { - border-color: #cfcfcf; } - -.file-input { - height: 100%; - left: 0; - opacity: 0; - outline: none; - position: absolute; - top: 0; - width: 100%; } - -.file-cta, -.file-name { - border-color: #dbdbdb; - border-radius: 4px; - font-size: 1em; - padding-left: 1em; - padding-right: 1em; - white-space: nowrap; } - -.file-cta { - background-color: whitesmoke; - color: #4a4a4a; } - -.file-name { - border-color: #dbdbdb; - border-style: solid; - border-width: 1px 1px 1px 0; - display: block; - max-width: 16em; - overflow: hidden; - text-align: left; - text-overflow: ellipsis; } - -.file-icon { - align-items: center; - display: flex; - height: 1em; - justify-content: center; - margin-right: 0.5em; - width: 1em; } - .file-icon .fa { - font-size: 14px; } - -.label { - color: #363636; - display: block; - font-size: 1rem; - font-weight: 700; } - .label:not(:last-child) { - margin-bottom: 0.5em; } - .label.is-small, #documenter .docs-sidebar form.docs-search > input.label { - font-size: 0.75rem; } - .label.is-medium { - font-size: 1.25rem; } - .label.is-large { - font-size: 1.5rem; } - -.help { - display: block; - font-size: 0.75rem; - margin-top: 0.25rem; } - .help.is-white { - color: white; } - .help.is-black { - color: #0a0a0a; } - .help.is-light { - color: whitesmoke; } - .help.is-dark, .content kbd.help { - color: #363636; } - .help.is-primary, .docstring > section > a.help.docs-sourcelink { - color: #4eb5de; } - .help.is-link { - color: #2e63b8; } - .help.is-info { - color: #209cee; } - .help.is-success { - color: #22c35b; } - .help.is-warning { - color: #ffdd57; } - .help.is-danger { - color: #da0b00; } - -.field:not(:last-child) { - margin-bottom: 0.75rem; } - -.field.has-addons { - display: flex; - justify-content: flex-start; } - .field.has-addons .control:not(:last-child) { - margin-right: -1px; } - .field.has-addons .control:not(:first-child):not(:last-child) .button, - .field.has-addons .control:not(:first-child):not(:last-child) .input, - .field.has-addons .control:not(:first-child):not(:last-child) #documenter .docs-sidebar form.docs-search > input, - #documenter .docs-sidebar .field.has-addons .control:not(:first-child):not(:last-child) form.docs-search > input, - .field.has-addons .control:not(:first-child):not(:last-child) .select select { - border-radius: 0; } - .field.has-addons .control:first-child:not(:only-child) .button, - .field.has-addons .control:first-child:not(:only-child) .input, - .field.has-addons .control:first-child:not(:only-child) #documenter .docs-sidebar form.docs-search > input, - #documenter .docs-sidebar .field.has-addons .control:first-child:not(:only-child) form.docs-search > input, - .field.has-addons .control:first-child:not(:only-child) .select select { - border-bottom-right-radius: 0; - border-top-right-radius: 0; } - .field.has-addons .control:last-child:not(:only-child) .button, - .field.has-addons .control:last-child:not(:only-child) .input, - .field.has-addons .control:last-child:not(:only-child) #documenter .docs-sidebar form.docs-search > input, - #documenter .docs-sidebar .field.has-addons .control:last-child:not(:only-child) form.docs-search > input, - .field.has-addons .control:last-child:not(:only-child) .select select { - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - .field.has-addons .control .button:not([disabled]):hover, .field.has-addons .control .button.is-hovered:not([disabled]), - .field.has-addons .control .input:not([disabled]):hover, - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):hover, - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):hover, - .field.has-addons .control .input.is-hovered:not([disabled]), - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-hovered:not([disabled]), - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-hovered:not([disabled]), - .field.has-addons .control .select select:not([disabled]):hover, - .field.has-addons .control .select select.is-hovered:not([disabled]) { - z-index: 2; } - .field.has-addons .control .button:not([disabled]):focus, .field.has-addons .control .button.is-focused:not([disabled]), .field.has-addons .control .button:not([disabled]):active, .field.has-addons .control .button.is-active:not([disabled]), - .field.has-addons .control .input:not([disabled]):focus, - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):focus, - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):focus, - .field.has-addons .control .input.is-focused:not([disabled]), - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-focused:not([disabled]), - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-focused:not([disabled]), - .field.has-addons .control .input:not([disabled]):active, - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):active, - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):active, - .field.has-addons .control .input.is-active:not([disabled]), - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-active:not([disabled]), - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-active:not([disabled]), - .field.has-addons .control .select select:not([disabled]):focus, - .field.has-addons .control .select select.is-focused:not([disabled]), - .field.has-addons .control .select select:not([disabled]):active, - .field.has-addons .control .select select.is-active:not([disabled]) { - z-index: 3; } - .field.has-addons .control .button:not([disabled]):focus:hover, .field.has-addons .control .button.is-focused:not([disabled]):hover, .field.has-addons .control .button:not([disabled]):active:hover, .field.has-addons .control .button.is-active:not([disabled]):hover, - .field.has-addons .control .input:not([disabled]):focus:hover, - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):focus:hover, - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):focus:hover, - .field.has-addons .control .input.is-focused:not([disabled]):hover, - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-focused:not([disabled]):hover, - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-focused:not([disabled]):hover, - .field.has-addons .control .input:not([disabled]):active:hover, - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):active:hover, - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):active:hover, - .field.has-addons .control .input.is-active:not([disabled]):hover, - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-active:not([disabled]):hover, - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-active:not([disabled]):hover, - .field.has-addons .control .select select:not([disabled]):focus:hover, - .field.has-addons .control .select select.is-focused:not([disabled]):hover, - .field.has-addons .control .select select:not([disabled]):active:hover, - .field.has-addons .control .select select.is-active:not([disabled]):hover { - z-index: 4; } - .field.has-addons .control.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - .field.has-addons.has-addons-centered { - justify-content: center; } - .field.has-addons.has-addons-right { - justify-content: flex-end; } - .field.has-addons.has-addons-fullwidth .control { - flex-grow: 1; - flex-shrink: 0; } - -.field.is-grouped { - display: flex; - justify-content: flex-start; } - .field.is-grouped > .control { - flex-shrink: 0; } - .field.is-grouped > .control:not(:last-child) { - margin-bottom: 0; - margin-right: 0.75rem; } - .field.is-grouped > .control.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - .field.is-grouped.is-grouped-centered { - justify-content: center; } - .field.is-grouped.is-grouped-right { - justify-content: flex-end; } - .field.is-grouped.is-grouped-multiline { - flex-wrap: wrap; } - .field.is-grouped.is-grouped-multiline > .control:last-child, .field.is-grouped.is-grouped-multiline > .control:not(:last-child) { - margin-bottom: 0.75rem; } - .field.is-grouped.is-grouped-multiline:last-child { - margin-bottom: -0.75rem; } - .field.is-grouped.is-grouped-multiline:not(:last-child) { - margin-bottom: 0; } - -@media screen and (min-width: 769px), print { - .field.is-horizontal { - display: flex; } } - -.field-label .label { - font-size: inherit; } - -@media screen and (max-width: 768px) { - .field-label { - margin-bottom: 0.5rem; } } - -@media screen and (min-width: 769px), print { - .field-label { - flex-basis: 0; - flex-grow: 1; - flex-shrink: 0; - margin-right: 1.5rem; - text-align: right; } - .field-label.is-small, #documenter .docs-sidebar form.docs-search > input.field-label { - font-size: 0.75rem; - padding-top: 0.375em; } - .field-label.is-normal { - padding-top: 0.375em; } - .field-label.is-medium { - font-size: 1.25rem; - padding-top: 0.375em; } - .field-label.is-large { - font-size: 1.5rem; - padding-top: 0.375em; } } - -.field-body .field .field { - margin-bottom: 0; } - -@media screen and (min-width: 769px), print { - .field-body { - display: flex; - flex-basis: 0; - flex-grow: 5; - flex-shrink: 1; } - .field-body .field { - margin-bottom: 0; } - .field-body > .field { - flex-shrink: 1; } - .field-body > .field:not(.is-narrow) { - flex-grow: 1; } - .field-body > .field:not(:last-child) { - margin-right: 0.75rem; } } - -.control { - box-sizing: border-box; - clear: both; - font-size: 1rem; - position: relative; - text-align: left; } - .control.has-icons-left .input:focus ~ .icon, .control.has-icons-left #documenter .docs-sidebar form.docs-search > input:focus ~ .icon, #documenter .docs-sidebar .control.has-icons-left form.docs-search > input:focus ~ .icon, - .control.has-icons-left .select:focus ~ .icon, .control.has-icons-right .input:focus ~ .icon, .control.has-icons-right #documenter .docs-sidebar form.docs-search > input:focus ~ .icon, #documenter .docs-sidebar .control.has-icons-right form.docs-search > input:focus ~ .icon, - .control.has-icons-right .select:focus ~ .icon { - color: #6b6b6b; } - .control.has-icons-left .input.is-small ~ .icon, .control.has-icons-left #documenter .docs-sidebar form.docs-search > input ~ .icon, #documenter .docs-sidebar .control.has-icons-left form.docs-search > input ~ .icon, - .control.has-icons-left .select.is-small ~ .icon, .control.has-icons-right .input.is-small ~ .icon, .control.has-icons-right #documenter .docs-sidebar form.docs-search > input ~ .icon, #documenter .docs-sidebar .control.has-icons-right form.docs-search > input ~ .icon, - .control.has-icons-right .select.is-small ~ .icon { - font-size: 0.75rem; } - .control.has-icons-left .input.is-medium ~ .icon, .control.has-icons-left #documenter .docs-sidebar form.docs-search > input.is-medium ~ .icon, #documenter .docs-sidebar .control.has-icons-left form.docs-search > input.is-medium ~ .icon, - .control.has-icons-left .select.is-medium ~ .icon, .control.has-icons-right .input.is-medium ~ .icon, .control.has-icons-right #documenter .docs-sidebar form.docs-search > input.is-medium ~ .icon, #documenter .docs-sidebar .control.has-icons-right form.docs-search > input.is-medium ~ .icon, - .control.has-icons-right .select.is-medium ~ .icon { - font-size: 1.25rem; } - .control.has-icons-left .input.is-large ~ .icon, .control.has-icons-left #documenter .docs-sidebar form.docs-search > input.is-large ~ .icon, #documenter .docs-sidebar .control.has-icons-left form.docs-search > input.is-large ~ .icon, - .control.has-icons-left .select.is-large ~ .icon, .control.has-icons-right .input.is-large ~ .icon, .control.has-icons-right #documenter .docs-sidebar form.docs-search > input.is-large ~ .icon, #documenter .docs-sidebar .control.has-icons-right form.docs-search > input.is-large ~ .icon, - .control.has-icons-right .select.is-large ~ .icon { - font-size: 1.5rem; } - .control.has-icons-left .icon, .control.has-icons-right .icon { - color: #dbdbdb; - height: 2.25em; - pointer-events: none; - position: absolute; - top: 0; - width: 2.25em; - z-index: 4; } - .control.has-icons-left .input, .control.has-icons-left #documenter .docs-sidebar form.docs-search > input, #documenter .docs-sidebar .control.has-icons-left form.docs-search > input, - .control.has-icons-left .select select { - padding-left: 2.25em; } - .control.has-icons-left .icon.is-left { - left: 0; } - .control.has-icons-right .input, .control.has-icons-right #documenter .docs-sidebar form.docs-search > input, #documenter .docs-sidebar .control.has-icons-right form.docs-search > input, - .control.has-icons-right .select select { - padding-right: 2.25em; } - .control.has-icons-right .icon.is-right { - right: 0; } - .control.is-loading::after { - position: absolute !important; - right: 0.625em; - top: 0.625em; - z-index: 4; } - .control.is-loading.is-small:after, #documenter .docs-sidebar form.docs-search > input.is-loading:after { - font-size: 0.75rem; } - .control.is-loading.is-medium:after { - font-size: 1.25rem; } - .control.is-loading.is-large:after { - font-size: 1.5rem; } - -.breadcrumb { - font-size: 1rem; - white-space: nowrap; } - .breadcrumb a { - align-items: center; - color: #2e63b8; - display: flex; - justify-content: center; - padding: 0 0.75em; } - .breadcrumb a:hover { - color: #363636; } - .breadcrumb li { - align-items: center; - display: flex; } - .breadcrumb li:first-child a { - padding-left: 0; } - .breadcrumb li.is-active a { - color: #222222; - cursor: default; - pointer-events: none; } - .breadcrumb li + li::before { - color: #b5b5b5; - content: "\0002f"; } - .breadcrumb ul, - .breadcrumb ol { - align-items: flex-start; - display: flex; - flex-wrap: wrap; - justify-content: flex-start; } - .breadcrumb .icon:first-child { - margin-right: 0.5em; } - .breadcrumb .icon:last-child { - margin-left: 0.5em; } - .breadcrumb.is-centered ol, - .breadcrumb.is-centered ul { - justify-content: center; } - .breadcrumb.is-right ol, - .breadcrumb.is-right ul { - justify-content: flex-end; } - .breadcrumb.is-small, #documenter .docs-sidebar form.docs-search > input.breadcrumb { - font-size: 0.75rem; } - .breadcrumb.is-medium { - font-size: 1.25rem; } - .breadcrumb.is-large { - font-size: 1.5rem; } - .breadcrumb.has-arrow-separator li + li::before { - content: "\02192"; } - .breadcrumb.has-bullet-separator li + li::before { - content: "\02022"; } - .breadcrumb.has-dot-separator li + li::before { - content: "\000b7"; } - .breadcrumb.has-succeeds-separator li + li::before { - content: "\0227B"; } - -.card { - background-color: white; - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); - color: #222222; - max-width: 100%; - position: relative; } - -.card-header { - background-color: transparent; - align-items: stretch; - box-shadow: 0 1px 2px rgba(10, 10, 10, 0.1); - display: flex; } - -.card-header-title { - align-items: center; - color: #222222; - display: flex; - flex-grow: 1; - font-weight: 700; - padding: 0.75rem; } - .card-header-title.is-centered { - justify-content: center; } - -.card-header-icon { - align-items: center; - cursor: pointer; - display: flex; - justify-content: center; - padding: 0.75rem; } - -.card-image { - display: block; - position: relative; } - -.card-content { - background-color: transparent; - padding: 1rem 1.25rem; } - -.card-footer { - background-color: transparent; - border-top: 1px solid #dbdbdb; - align-items: stretch; - display: flex; } - -.card-footer-item { - align-items: center; - display: flex; - flex-basis: 0; - flex-grow: 1; - flex-shrink: 0; - justify-content: center; - padding: 0.75rem; } - .card-footer-item:not(:last-child) { - border-right: 1px solid #dbdbdb; } - -.card .media:not(:last-child) { - margin-bottom: 1.5rem; } - -.dropdown { - display: inline-flex; - position: relative; - vertical-align: top; } - .dropdown.is-active .dropdown-menu, .dropdown.is-hoverable:hover .dropdown-menu { - display: block; } - .dropdown.is-right .dropdown-menu { - left: auto; - right: 0; } - .dropdown.is-up .dropdown-menu { - bottom: 100%; - padding-bottom: 4px; - padding-top: initial; - top: auto; } - -.dropdown-menu { - display: none; - left: 0; - min-width: 12rem; - padding-top: 4px; - position: absolute; - top: 100%; - z-index: 20; } - -.dropdown-content { - background-color: white; - border-radius: 4px; - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); - padding-bottom: 0.5rem; - padding-top: 0.5rem; } - -.dropdown-item { - color: #4a4a4a; - display: block; - font-size: 0.875rem; - line-height: 1.5; - padding: 0.375rem 1rem; - position: relative; } - -a.dropdown-item, -button.dropdown-item { - padding-right: 3rem; - text-align: left; - white-space: nowrap; - width: 100%; } - a.dropdown-item:hover, - button.dropdown-item:hover { - background-color: whitesmoke; - color: #0a0a0a; } - a.dropdown-item.is-active, - button.dropdown-item.is-active { - background-color: #2e63b8; - color: #fff; } - -.dropdown-divider { - background-color: #dbdbdb; - border: none; - display: block; - height: 1px; - margin: 0.5rem 0; } - -.level { - align-items: center; - justify-content: space-between; } - .level code { - border-radius: 4px; } - .level img { - display: inline-block; - vertical-align: top; } - .level.is-mobile { - display: flex; } - .level.is-mobile .level-left, - .level.is-mobile .level-right { - display: flex; } - .level.is-mobile .level-left + .level-right { - margin-top: 0; } - .level.is-mobile .level-item:not(:last-child) { - margin-bottom: 0; - margin-right: 0.75rem; } - .level.is-mobile .level-item:not(.is-narrow) { - flex-grow: 1; } - @media screen and (min-width: 769px), print { - .level { - display: flex; } - .level > .level-item:not(.is-narrow) { - flex-grow: 1; } } -.level-item { - align-items: center; - display: flex; - flex-basis: auto; - flex-grow: 0; - flex-shrink: 0; - justify-content: center; } - .level-item .title, - .level-item .subtitle { - margin-bottom: 0; } - @media screen and (max-width: 768px) { - .level-item:not(:last-child) { - margin-bottom: 0.75rem; } } -.level-left, -.level-right { - flex-basis: auto; - flex-grow: 0; - flex-shrink: 0; } - .level-left .level-item.is-flexible, - .level-right .level-item.is-flexible { - flex-grow: 1; } - @media screen and (min-width: 769px), print { - .level-left .level-item:not(:last-child), - .level-right .level-item:not(:last-child) { - margin-right: 0.75rem; } } -.level-left { - align-items: center; - justify-content: flex-start; } - @media screen and (max-width: 768px) { - .level-left + .level-right { - margin-top: 1.5rem; } } - @media screen and (min-width: 769px), print { - .level-left { - display: flex; } } -.level-right { - align-items: center; - justify-content: flex-end; } - @media screen and (min-width: 769px), print { - .level-right { - display: flex; } } -.list { - background-color: white; - border-radius: 4px; - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); } - -.list-item { - display: block; - padding: 0.5em 1em; } - .list-item:not(a) { - color: #222222; } - .list-item:first-child { - border-top-left-radius: 4px; - border-top-right-radius: 4px; } - .list-item:last-child { - border-bottom-left-radius: 4px; - border-bottom-right-radius: 4px; } - .list-item:not(:last-child) { - border-bottom: 1px solid #dbdbdb; } - .list-item.is-active { - background-color: #2e63b8; - color: #fff; } - -a.list-item { - background-color: whitesmoke; - cursor: pointer; } - -.media { - align-items: flex-start; - display: flex; - text-align: left; } - .media .content:not(:last-child) { - margin-bottom: 0.75rem; } - .media .media { - border-top: 1px solid rgba(219, 219, 219, 0.5); - display: flex; - padding-top: 0.75rem; } - .media .media .content:not(:last-child), - .media .media .control:not(:last-child) { - margin-bottom: 0.5rem; } - .media .media .media { - padding-top: 0.5rem; } - .media .media .media + .media { - margin-top: 0.5rem; } - .media + .media { - border-top: 1px solid rgba(219, 219, 219, 0.5); - margin-top: 1rem; - padding-top: 1rem; } - .media.is-large + .media { - margin-top: 1.5rem; - padding-top: 1.5rem; } - -.media-left, -.media-right { - flex-basis: auto; - flex-grow: 0; - flex-shrink: 0; } - -.media-left { - margin-right: 1rem; } - -.media-right { - margin-left: 1rem; } - -.media-content { - flex-basis: auto; - flex-grow: 1; - flex-shrink: 1; - text-align: left; } - -@media screen and (max-width: 768px) { - .media-content { - overflow-x: auto; } } - -.menu { - font-size: 1rem; } - .menu.is-small, #documenter .docs-sidebar form.docs-search > input.menu { - font-size: 0.75rem; } - .menu.is-medium { - font-size: 1.25rem; } - .menu.is-large { - font-size: 1.5rem; } - -.menu-list { - line-height: 1.25; } - .menu-list a { - border-radius: 2px; - color: #222222; - display: block; - padding: 0.5em 0.75em; } - .menu-list a:hover { - background-color: whitesmoke; - color: #222222; } - .menu-list a.is-active { - background-color: #2e63b8; - color: #fff; } - .menu-list li ul { - border-left: 1px solid #dbdbdb; - margin: 0.75em; - padding-left: 0.75em; } - -.menu-label { - color: #6b6b6b; - font-size: 0.75em; - letter-spacing: 0.1em; - text-transform: uppercase; } - .menu-label:not(:first-child) { - margin-top: 1em; } - .menu-label:not(:last-child) { - margin-bottom: 1em; } - -.message { - background-color: whitesmoke; - border-radius: 4px; - font-size: 1rem; } - .message strong { - color: currentColor; } - .message a:not(.button):not(.tag):not(.dropdown-item) { - color: currentColor; - text-decoration: underline; } - .message.is-small, #documenter .docs-sidebar form.docs-search > input.message { - font-size: 0.75rem; } - .message.is-medium { - font-size: 1.25rem; } - .message.is-large { - font-size: 1.5rem; } - .message.is-white { - background-color: white; } - .message.is-white .message-header { - background-color: white; - color: #0a0a0a; } - .message.is-white .message-body { - border-color: white; - color: #4d4d4d; } - .message.is-black { - background-color: #fafafa; } - .message.is-black .message-header { - background-color: #0a0a0a; - color: white; } - .message.is-black .message-body { - border-color: #0a0a0a; - color: #090909; } - .message.is-light { - background-color: #fafafa; } - .message.is-light .message-header { - background-color: whitesmoke; - color: #363636; } - .message.is-light .message-body { - border-color: whitesmoke; - color: #505050; } - .message.is-dark, .content kbd.message { - background-color: #fafafa; } - .message.is-dark .message-header, .content kbd.message .message-header { - background-color: #363636; - color: whitesmoke; } - .message.is-dark .message-body, .content kbd.message .message-body { - border-color: #363636; - color: #2a2a2a; } - .message.is-primary, .docstring > section > a.message.docs-sourcelink { - background-color: #f6fbfd; } - .message.is-primary .message-header, .docstring > section > a.message.docs-sourcelink .message-header { - background-color: #4eb5de; - color: #fff; } - .message.is-primary .message-body, .docstring > section > a.message.docs-sourcelink .message-body { - border-color: #4eb5de; - color: #1f556a; } - .message.is-link { - background-color: #f7f9fd; } - .message.is-link .message-header { - background-color: #2e63b8; - color: #fff; } - .message.is-link .message-body { - border-color: #2e63b8; - color: #264981; } - .message.is-info { - background-color: #f6fbfe; } - .message.is-info .message-header { - background-color: #209cee; - color: #fff; } - .message.is-info .message-body { - border-color: #209cee; - color: #12537d; } - .message.is-success { - background-color: #f6fdf9; } - .message.is-success .message-header { - background-color: #22c35b; - color: #fff; } - .message.is-success .message-body { - border-color: #22c35b; - color: #0f361d; } - .message.is-warning { - background-color: #fffdf5; } - .message.is-warning .message-header { - background-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .message.is-warning .message-body { - border-color: #ffdd57; - color: #3c3108; } - .message.is-danger { - background-color: #fff5f5; } - .message.is-danger .message-header { - background-color: #da0b00; - color: #fff; } - .message.is-danger .message-body { - border-color: #da0b00; - color: #9b0c04; } - -.message-header { - align-items: center; - background-color: #222222; - border-radius: 4px 4px 0 0; - color: #fff; - display: flex; - font-weight: 700; - justify-content: space-between; - line-height: 1.25; - padding: 0.75em; - position: relative; } - .message-header .delete { - flex-grow: 0; - flex-shrink: 0; - margin-left: 0.75em; } - .message-header + .message-body { - border-width: 0; - border-top-left-radius: 0; - border-top-right-radius: 0; } - -.message-body { - border-color: #dbdbdb; - border-radius: 4px; - border-style: solid; - border-width: 0 0 0 4px; - color: #222222; - padding: 1em 1.25em; } - .message-body code, - .message-body pre { - background-color: white; } - .message-body pre code { - background-color: transparent; } - -.modal { - align-items: center; - display: none; - flex-direction: column; - justify-content: center; - overflow: hidden; - position: fixed; - z-index: 40; } - .modal.is-active { - display: flex; } - -.modal-background { - background-color: rgba(10, 10, 10, 0.86); } - -.modal-content, -.modal-card { - margin: 0 20px; - max-height: calc(100vh - 160px); - overflow: auto; - position: relative; - width: 100%; } - @media screen and (min-width: 769px), print { - .modal-content, - .modal-card { - margin: 0 auto; - max-height: calc(100vh - 40px); - width: 640px; } } -.modal-close { - background: none; - height: 40px; - position: fixed; - right: 20px; - top: 20px; - width: 40px; } - -.modal-card { - display: flex; - flex-direction: column; - max-height: calc(100vh - 40px); - overflow: hidden; - -ms-overflow-y: visible; } - -.modal-card-head, -.modal-card-foot { - align-items: center; - background-color: whitesmoke; - display: flex; - flex-shrink: 0; - justify-content: flex-start; - padding: 20px; - position: relative; } - -.modal-card-head { - border-bottom: 1px solid #dbdbdb; - border-top-left-radius: 6px; - border-top-right-radius: 6px; } - -.modal-card-title { - color: #222222; - flex-grow: 1; - flex-shrink: 0; - font-size: 1.5rem; - line-height: 1; } - -.modal-card-foot { - border-bottom-left-radius: 6px; - border-bottom-right-radius: 6px; - border-top: 1px solid #dbdbdb; } - .modal-card-foot .button:not(:last-child) { - margin-right: 0.5em; } - -.modal-card-body { - -webkit-overflow-scrolling: touch; - background-color: white; - flex-grow: 1; - flex-shrink: 1; - overflow: auto; - padding: 20px; } - -.navbar { - background-color: white; - min-height: 3.25rem; - position: relative; - z-index: 30; } - .navbar.is-white { - background-color: white; - color: #0a0a0a; } - .navbar.is-white .navbar-brand > .navbar-item, - .navbar.is-white .navbar-brand .navbar-link { - color: #0a0a0a; } - .navbar.is-white .navbar-brand > a.navbar-item:focus, .navbar.is-white .navbar-brand > a.navbar-item:hover, .navbar.is-white .navbar-brand > a.navbar-item.is-active, - .navbar.is-white .navbar-brand .navbar-link:focus, - .navbar.is-white .navbar-brand .navbar-link:hover, - .navbar.is-white .navbar-brand .navbar-link.is-active { - background-color: #f2f2f2; - color: #0a0a0a; } - .navbar.is-white .navbar-brand .navbar-link::after { - border-color: #0a0a0a; } - .navbar.is-white .navbar-burger { - color: #0a0a0a; } - @media screen and (min-width: 1056px) { - .navbar.is-white .navbar-start > .navbar-item, - .navbar.is-white .navbar-start .navbar-link, - .navbar.is-white .navbar-end > .navbar-item, - .navbar.is-white .navbar-end .navbar-link { - color: #0a0a0a; } - .navbar.is-white .navbar-start > a.navbar-item:focus, .navbar.is-white .navbar-start > a.navbar-item:hover, .navbar.is-white .navbar-start > a.navbar-item.is-active, - .navbar.is-white .navbar-start .navbar-link:focus, - .navbar.is-white .navbar-start .navbar-link:hover, - .navbar.is-white .navbar-start .navbar-link.is-active, - .navbar.is-white .navbar-end > a.navbar-item:focus, - .navbar.is-white .navbar-end > a.navbar-item:hover, - .navbar.is-white .navbar-end > a.navbar-item.is-active, - .navbar.is-white .navbar-end .navbar-link:focus, - .navbar.is-white .navbar-end .navbar-link:hover, - .navbar.is-white .navbar-end .navbar-link.is-active { - background-color: #f2f2f2; - color: #0a0a0a; } - .navbar.is-white .navbar-start .navbar-link::after, - .navbar.is-white .navbar-end .navbar-link::after { - border-color: #0a0a0a; } - .navbar.is-white .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-white .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-white .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #f2f2f2; - color: #0a0a0a; } - .navbar.is-white .navbar-dropdown a.navbar-item.is-active { - background-color: white; - color: #0a0a0a; } } - .navbar.is-black { - background-color: #0a0a0a; - color: white; } - .navbar.is-black .navbar-brand > .navbar-item, - .navbar.is-black .navbar-brand .navbar-link { - color: white; } - .navbar.is-black .navbar-brand > a.navbar-item:focus, .navbar.is-black .navbar-brand > a.navbar-item:hover, .navbar.is-black .navbar-brand > a.navbar-item.is-active, - .navbar.is-black .navbar-brand .navbar-link:focus, - .navbar.is-black .navbar-brand .navbar-link:hover, - .navbar.is-black .navbar-brand .navbar-link.is-active { - background-color: black; - color: white; } - .navbar.is-black .navbar-brand .navbar-link::after { - border-color: white; } - .navbar.is-black .navbar-burger { - color: white; } - @media screen and (min-width: 1056px) { - .navbar.is-black .navbar-start > .navbar-item, - .navbar.is-black .navbar-start .navbar-link, - .navbar.is-black .navbar-end > .navbar-item, - .navbar.is-black .navbar-end .navbar-link { - color: white; } - .navbar.is-black .navbar-start > a.navbar-item:focus, .navbar.is-black .navbar-start > a.navbar-item:hover, .navbar.is-black .navbar-start > a.navbar-item.is-active, - .navbar.is-black .navbar-start .navbar-link:focus, - .navbar.is-black .navbar-start .navbar-link:hover, - .navbar.is-black .navbar-start .navbar-link.is-active, - .navbar.is-black .navbar-end > a.navbar-item:focus, - .navbar.is-black .navbar-end > a.navbar-item:hover, - .navbar.is-black .navbar-end > a.navbar-item.is-active, - .navbar.is-black .navbar-end .navbar-link:focus, - .navbar.is-black .navbar-end .navbar-link:hover, - .navbar.is-black .navbar-end .navbar-link.is-active { - background-color: black; - color: white; } - .navbar.is-black .navbar-start .navbar-link::after, - .navbar.is-black .navbar-end .navbar-link::after { - border-color: white; } - .navbar.is-black .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-black .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-black .navbar-item.has-dropdown.is-active .navbar-link { - background-color: black; - color: white; } - .navbar.is-black .navbar-dropdown a.navbar-item.is-active { - background-color: #0a0a0a; - color: white; } } - .navbar.is-light { - background-color: whitesmoke; - color: #363636; } - .navbar.is-light .navbar-brand > .navbar-item, - .navbar.is-light .navbar-brand .navbar-link { - color: #363636; } - .navbar.is-light .navbar-brand > a.navbar-item:focus, .navbar.is-light .navbar-brand > a.navbar-item:hover, .navbar.is-light .navbar-brand > a.navbar-item.is-active, - .navbar.is-light .navbar-brand .navbar-link:focus, - .navbar.is-light .navbar-brand .navbar-link:hover, - .navbar.is-light .navbar-brand .navbar-link.is-active { - background-color: #e8e8e8; - color: #363636; } - .navbar.is-light .navbar-brand .navbar-link::after { - border-color: #363636; } - .navbar.is-light .navbar-burger { - color: #363636; } - @media screen and (min-width: 1056px) { - .navbar.is-light .navbar-start > .navbar-item, - .navbar.is-light .navbar-start .navbar-link, - .navbar.is-light .navbar-end > .navbar-item, - .navbar.is-light .navbar-end .navbar-link { - color: #363636; } - .navbar.is-light .navbar-start > a.navbar-item:focus, .navbar.is-light .navbar-start > a.navbar-item:hover, .navbar.is-light .navbar-start > a.navbar-item.is-active, - .navbar.is-light .navbar-start .navbar-link:focus, - .navbar.is-light .navbar-start .navbar-link:hover, - .navbar.is-light .navbar-start .navbar-link.is-active, - .navbar.is-light .navbar-end > a.navbar-item:focus, - .navbar.is-light .navbar-end > a.navbar-item:hover, - .navbar.is-light .navbar-end > a.navbar-item.is-active, - .navbar.is-light .navbar-end .navbar-link:focus, - .navbar.is-light .navbar-end .navbar-link:hover, - .navbar.is-light .navbar-end .navbar-link.is-active { - background-color: #e8e8e8; - color: #363636; } - .navbar.is-light .navbar-start .navbar-link::after, - .navbar.is-light .navbar-end .navbar-link::after { - border-color: #363636; } - .navbar.is-light .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-light .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-light .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #e8e8e8; - color: #363636; } - .navbar.is-light .navbar-dropdown a.navbar-item.is-active { - background-color: whitesmoke; - color: #363636; } } - .navbar.is-dark, .content kbd.navbar { - background-color: #363636; - color: whitesmoke; } - .navbar.is-dark .navbar-brand > .navbar-item, .content kbd.navbar .navbar-brand > .navbar-item, - .navbar.is-dark .navbar-brand .navbar-link, - .content kbd.navbar .navbar-brand .navbar-link { - color: whitesmoke; } - .navbar.is-dark .navbar-brand > a.navbar-item:focus, .content kbd.navbar .navbar-brand > a.navbar-item:focus, .navbar.is-dark .navbar-brand > a.navbar-item:hover, .content kbd.navbar .navbar-brand > a.navbar-item:hover, .navbar.is-dark .navbar-brand > a.navbar-item.is-active, .content kbd.navbar .navbar-brand > a.navbar-item.is-active, - .navbar.is-dark .navbar-brand .navbar-link:focus, - .content kbd.navbar .navbar-brand .navbar-link:focus, - .navbar.is-dark .navbar-brand .navbar-link:hover, - .content kbd.navbar .navbar-brand .navbar-link:hover, - .navbar.is-dark .navbar-brand .navbar-link.is-active, - .content kbd.navbar .navbar-brand .navbar-link.is-active { - background-color: #292929; - color: whitesmoke; } - .navbar.is-dark .navbar-brand .navbar-link::after, .content kbd.navbar .navbar-brand .navbar-link::after { - border-color: whitesmoke; } - .navbar.is-dark .navbar-burger, .content kbd.navbar .navbar-burger { - color: whitesmoke; } - @media screen and (min-width: 1056px) { - .navbar.is-dark .navbar-start > .navbar-item, .content kbd.navbar .navbar-start > .navbar-item, - .navbar.is-dark .navbar-start .navbar-link, - .content kbd.navbar .navbar-start .navbar-link, - .navbar.is-dark .navbar-end > .navbar-item, - .content kbd.navbar .navbar-end > .navbar-item, - .navbar.is-dark .navbar-end .navbar-link, - .content kbd.navbar .navbar-end .navbar-link { - color: whitesmoke; } - .navbar.is-dark .navbar-start > a.navbar-item:focus, .content kbd.navbar .navbar-start > a.navbar-item:focus, .navbar.is-dark .navbar-start > a.navbar-item:hover, .content kbd.navbar .navbar-start > a.navbar-item:hover, .navbar.is-dark .navbar-start > a.navbar-item.is-active, .content kbd.navbar .navbar-start > a.navbar-item.is-active, - .navbar.is-dark .navbar-start .navbar-link:focus, - .content kbd.navbar .navbar-start .navbar-link:focus, - .navbar.is-dark .navbar-start .navbar-link:hover, - .content kbd.navbar .navbar-start .navbar-link:hover, - .navbar.is-dark .navbar-start .navbar-link.is-active, - .content kbd.navbar .navbar-start .navbar-link.is-active, - .navbar.is-dark .navbar-end > a.navbar-item:focus, - .content kbd.navbar .navbar-end > a.navbar-item:focus, - .navbar.is-dark .navbar-end > a.navbar-item:hover, - .content kbd.navbar .navbar-end > a.navbar-item:hover, - .navbar.is-dark .navbar-end > a.navbar-item.is-active, - .content kbd.navbar .navbar-end > a.navbar-item.is-active, - .navbar.is-dark .navbar-end .navbar-link:focus, - .content kbd.navbar .navbar-end .navbar-link:focus, - .navbar.is-dark .navbar-end .navbar-link:hover, - .content kbd.navbar .navbar-end .navbar-link:hover, - .navbar.is-dark .navbar-end .navbar-link.is-active, - .content kbd.navbar .navbar-end .navbar-link.is-active { - background-color: #292929; - color: whitesmoke; } - .navbar.is-dark .navbar-start .navbar-link::after, .content kbd.navbar .navbar-start .navbar-link::after, - .navbar.is-dark .navbar-end .navbar-link::after, - .content kbd.navbar .navbar-end .navbar-link::after { - border-color: whitesmoke; } - .navbar.is-dark .navbar-item.has-dropdown:focus .navbar-link, .content kbd.navbar .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-dark .navbar-item.has-dropdown:hover .navbar-link, - .content kbd.navbar .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-dark .navbar-item.has-dropdown.is-active .navbar-link, - .content kbd.navbar .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #292929; - color: whitesmoke; } - .navbar.is-dark .navbar-dropdown a.navbar-item.is-active, .content kbd.navbar .navbar-dropdown a.navbar-item.is-active { - background-color: #363636; - color: whitesmoke; } } - .navbar.is-primary, .docstring > section > a.navbar.docs-sourcelink { - background-color: #4eb5de; - color: #fff; } - .navbar.is-primary .navbar-brand > .navbar-item, .docstring > section > a.navbar.docs-sourcelink .navbar-brand > .navbar-item, - .navbar.is-primary .navbar-brand .navbar-link, - .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link { - color: #fff; } - .navbar.is-primary .navbar-brand > a.navbar-item:focus, .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item:focus, .navbar.is-primary .navbar-brand > a.navbar-item:hover, .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item:hover, .navbar.is-primary .navbar-brand > a.navbar-item.is-active, .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item.is-active, - .navbar.is-primary .navbar-brand .navbar-link:focus, - .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link:focus, - .navbar.is-primary .navbar-brand .navbar-link:hover, - .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link:hover, - .navbar.is-primary .navbar-brand .navbar-link.is-active, - .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link.is-active { - background-color: #39acda; - color: #fff; } - .navbar.is-primary .navbar-brand .navbar-link::after, .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link::after { - border-color: #fff; } - .navbar.is-primary .navbar-burger, .docstring > section > a.navbar.docs-sourcelink .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - .navbar.is-primary .navbar-start > .navbar-item, .docstring > section > a.navbar.docs-sourcelink .navbar-start > .navbar-item, - .navbar.is-primary .navbar-start .navbar-link, - .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link, - .navbar.is-primary .navbar-end > .navbar-item, - .docstring > section > a.navbar.docs-sourcelink .navbar-end > .navbar-item, - .navbar.is-primary .navbar-end .navbar-link, - .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link { - color: #fff; } - .navbar.is-primary .navbar-start > a.navbar-item:focus, .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item:focus, .navbar.is-primary .navbar-start > a.navbar-item:hover, .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item:hover, .navbar.is-primary .navbar-start > a.navbar-item.is-active, .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item.is-active, - .navbar.is-primary .navbar-start .navbar-link:focus, - .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link:focus, - .navbar.is-primary .navbar-start .navbar-link:hover, - .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link:hover, - .navbar.is-primary .navbar-start .navbar-link.is-active, - .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link.is-active, - .navbar.is-primary .navbar-end > a.navbar-item:focus, - .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item:focus, - .navbar.is-primary .navbar-end > a.navbar-item:hover, - .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item:hover, - .navbar.is-primary .navbar-end > a.navbar-item.is-active, - .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item.is-active, - .navbar.is-primary .navbar-end .navbar-link:focus, - .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link:focus, - .navbar.is-primary .navbar-end .navbar-link:hover, - .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link:hover, - .navbar.is-primary .navbar-end .navbar-link.is-active, - .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link.is-active { - background-color: #39acda; - color: #fff; } - .navbar.is-primary .navbar-start .navbar-link::after, .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link::after, - .navbar.is-primary .navbar-end .navbar-link::after, - .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link::after { - border-color: #fff; } - .navbar.is-primary .navbar-item.has-dropdown:focus .navbar-link, .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-primary .navbar-item.has-dropdown:hover .navbar-link, - .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-primary .navbar-item.has-dropdown.is-active .navbar-link, - .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #39acda; - color: #fff; } - .navbar.is-primary .navbar-dropdown a.navbar-item.is-active, .docstring > section > a.navbar.docs-sourcelink .navbar-dropdown a.navbar-item.is-active { - background-color: #4eb5de; - color: #fff; } } - .navbar.is-link { - background-color: #2e63b8; - color: #fff; } - .navbar.is-link .navbar-brand > .navbar-item, - .navbar.is-link .navbar-brand .navbar-link { - color: #fff; } - .navbar.is-link .navbar-brand > a.navbar-item:focus, .navbar.is-link .navbar-brand > a.navbar-item:hover, .navbar.is-link .navbar-brand > a.navbar-item.is-active, - .navbar.is-link .navbar-brand .navbar-link:focus, - .navbar.is-link .navbar-brand .navbar-link:hover, - .navbar.is-link .navbar-brand .navbar-link.is-active { - background-color: #2958a4; - color: #fff; } - .navbar.is-link .navbar-brand .navbar-link::after { - border-color: #fff; } - .navbar.is-link .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - .navbar.is-link .navbar-start > .navbar-item, - .navbar.is-link .navbar-start .navbar-link, - .navbar.is-link .navbar-end > .navbar-item, - .navbar.is-link .navbar-end .navbar-link { - color: #fff; } - .navbar.is-link .navbar-start > a.navbar-item:focus, .navbar.is-link .navbar-start > a.navbar-item:hover, .navbar.is-link .navbar-start > a.navbar-item.is-active, - .navbar.is-link .navbar-start .navbar-link:focus, - .navbar.is-link .navbar-start .navbar-link:hover, - .navbar.is-link .navbar-start .navbar-link.is-active, - .navbar.is-link .navbar-end > a.navbar-item:focus, - .navbar.is-link .navbar-end > a.navbar-item:hover, - .navbar.is-link .navbar-end > a.navbar-item.is-active, - .navbar.is-link .navbar-end .navbar-link:focus, - .navbar.is-link .navbar-end .navbar-link:hover, - .navbar.is-link .navbar-end .navbar-link.is-active { - background-color: #2958a4; - color: #fff; } - .navbar.is-link .navbar-start .navbar-link::after, - .navbar.is-link .navbar-end .navbar-link::after { - border-color: #fff; } - .navbar.is-link .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-link .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-link .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #2958a4; - color: #fff; } - .navbar.is-link .navbar-dropdown a.navbar-item.is-active { - background-color: #2e63b8; - color: #fff; } } - .navbar.is-info { - background-color: #209cee; - color: #fff; } - .navbar.is-info .navbar-brand > .navbar-item, - .navbar.is-info .navbar-brand .navbar-link { - color: #fff; } - .navbar.is-info .navbar-brand > a.navbar-item:focus, .navbar.is-info .navbar-brand > a.navbar-item:hover, .navbar.is-info .navbar-brand > a.navbar-item.is-active, - .navbar.is-info .navbar-brand .navbar-link:focus, - .navbar.is-info .navbar-brand .navbar-link:hover, - .navbar.is-info .navbar-brand .navbar-link.is-active { - background-color: #1190e3; - color: #fff; } - .navbar.is-info .navbar-brand .navbar-link::after { - border-color: #fff; } - .navbar.is-info .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - .navbar.is-info .navbar-start > .navbar-item, - .navbar.is-info .navbar-start .navbar-link, - .navbar.is-info .navbar-end > .navbar-item, - .navbar.is-info .navbar-end .navbar-link { - color: #fff; } - .navbar.is-info .navbar-start > a.navbar-item:focus, .navbar.is-info .navbar-start > a.navbar-item:hover, .navbar.is-info .navbar-start > a.navbar-item.is-active, - .navbar.is-info .navbar-start .navbar-link:focus, - .navbar.is-info .navbar-start .navbar-link:hover, - .navbar.is-info .navbar-start .navbar-link.is-active, - .navbar.is-info .navbar-end > a.navbar-item:focus, - .navbar.is-info .navbar-end > a.navbar-item:hover, - .navbar.is-info .navbar-end > a.navbar-item.is-active, - .navbar.is-info .navbar-end .navbar-link:focus, - .navbar.is-info .navbar-end .navbar-link:hover, - .navbar.is-info .navbar-end .navbar-link.is-active { - background-color: #1190e3; - color: #fff; } - .navbar.is-info .navbar-start .navbar-link::after, - .navbar.is-info .navbar-end .navbar-link::after { - border-color: #fff; } - .navbar.is-info .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-info .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-info .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #1190e3; - color: #fff; } - .navbar.is-info .navbar-dropdown a.navbar-item.is-active { - background-color: #209cee; - color: #fff; } } - .navbar.is-success { - background-color: #22c35b; - color: #fff; } - .navbar.is-success .navbar-brand > .navbar-item, - .navbar.is-success .navbar-brand .navbar-link { - color: #fff; } - .navbar.is-success .navbar-brand > a.navbar-item:focus, .navbar.is-success .navbar-brand > a.navbar-item:hover, .navbar.is-success .navbar-brand > a.navbar-item.is-active, - .navbar.is-success .navbar-brand .navbar-link:focus, - .navbar.is-success .navbar-brand .navbar-link:hover, - .navbar.is-success .navbar-brand .navbar-link.is-active { - background-color: #1ead51; - color: #fff; } - .navbar.is-success .navbar-brand .navbar-link::after { - border-color: #fff; } - .navbar.is-success .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - .navbar.is-success .navbar-start > .navbar-item, - .navbar.is-success .navbar-start .navbar-link, - .navbar.is-success .navbar-end > .navbar-item, - .navbar.is-success .navbar-end .navbar-link { - color: #fff; } - .navbar.is-success .navbar-start > a.navbar-item:focus, .navbar.is-success .navbar-start > a.navbar-item:hover, .navbar.is-success .navbar-start > a.navbar-item.is-active, - .navbar.is-success .navbar-start .navbar-link:focus, - .navbar.is-success .navbar-start .navbar-link:hover, - .navbar.is-success .navbar-start .navbar-link.is-active, - .navbar.is-success .navbar-end > a.navbar-item:focus, - .navbar.is-success .navbar-end > a.navbar-item:hover, - .navbar.is-success .navbar-end > a.navbar-item.is-active, - .navbar.is-success .navbar-end .navbar-link:focus, - .navbar.is-success .navbar-end .navbar-link:hover, - .navbar.is-success .navbar-end .navbar-link.is-active { - background-color: #1ead51; - color: #fff; } - .navbar.is-success .navbar-start .navbar-link::after, - .navbar.is-success .navbar-end .navbar-link::after { - border-color: #fff; } - .navbar.is-success .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-success .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-success .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #1ead51; - color: #fff; } - .navbar.is-success .navbar-dropdown a.navbar-item.is-active { - background-color: #22c35b; - color: #fff; } } - .navbar.is-warning { - background-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-brand > .navbar-item, - .navbar.is-warning .navbar-brand .navbar-link { - color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-brand > a.navbar-item:focus, .navbar.is-warning .navbar-brand > a.navbar-item:hover, .navbar.is-warning .navbar-brand > a.navbar-item.is-active, - .navbar.is-warning .navbar-brand .navbar-link:focus, - .navbar.is-warning .navbar-brand .navbar-link:hover, - .navbar.is-warning .navbar-brand .navbar-link.is-active { - background-color: #ffd83e; - color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-brand .navbar-link::after { - border-color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-burger { - color: rgba(0, 0, 0, 0.7); } - @media screen and (min-width: 1056px) { - .navbar.is-warning .navbar-start > .navbar-item, - .navbar.is-warning .navbar-start .navbar-link, - .navbar.is-warning .navbar-end > .navbar-item, - .navbar.is-warning .navbar-end .navbar-link { - color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-start > a.navbar-item:focus, .navbar.is-warning .navbar-start > a.navbar-item:hover, .navbar.is-warning .navbar-start > a.navbar-item.is-active, - .navbar.is-warning .navbar-start .navbar-link:focus, - .navbar.is-warning .navbar-start .navbar-link:hover, - .navbar.is-warning .navbar-start .navbar-link.is-active, - .navbar.is-warning .navbar-end > a.navbar-item:focus, - .navbar.is-warning .navbar-end > a.navbar-item:hover, - .navbar.is-warning .navbar-end > a.navbar-item.is-active, - .navbar.is-warning .navbar-end .navbar-link:focus, - .navbar.is-warning .navbar-end .navbar-link:hover, - .navbar.is-warning .navbar-end .navbar-link.is-active { - background-color: #ffd83e; - color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-start .navbar-link::after, - .navbar.is-warning .navbar-end .navbar-link::after { - border-color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-warning .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-warning .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #ffd83e; - color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-dropdown a.navbar-item.is-active { - background-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } } - .navbar.is-danger { - background-color: #da0b00; - color: #fff; } - .navbar.is-danger .navbar-brand > .navbar-item, - .navbar.is-danger .navbar-brand .navbar-link { - color: #fff; } - .navbar.is-danger .navbar-brand > a.navbar-item:focus, .navbar.is-danger .navbar-brand > a.navbar-item:hover, .navbar.is-danger .navbar-brand > a.navbar-item.is-active, - .navbar.is-danger .navbar-brand .navbar-link:focus, - .navbar.is-danger .navbar-brand .navbar-link:hover, - .navbar.is-danger .navbar-brand .navbar-link.is-active { - background-color: #c10a00; - color: #fff; } - .navbar.is-danger .navbar-brand .navbar-link::after { - border-color: #fff; } - .navbar.is-danger .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - .navbar.is-danger .navbar-start > .navbar-item, - .navbar.is-danger .navbar-start .navbar-link, - .navbar.is-danger .navbar-end > .navbar-item, - .navbar.is-danger .navbar-end .navbar-link { - color: #fff; } - .navbar.is-danger .navbar-start > a.navbar-item:focus, .navbar.is-danger .navbar-start > a.navbar-item:hover, .navbar.is-danger .navbar-start > a.navbar-item.is-active, - .navbar.is-danger .navbar-start .navbar-link:focus, - .navbar.is-danger .navbar-start .navbar-link:hover, - .navbar.is-danger .navbar-start .navbar-link.is-active, - .navbar.is-danger .navbar-end > a.navbar-item:focus, - .navbar.is-danger .navbar-end > a.navbar-item:hover, - .navbar.is-danger .navbar-end > a.navbar-item.is-active, - .navbar.is-danger .navbar-end .navbar-link:focus, - .navbar.is-danger .navbar-end .navbar-link:hover, - .navbar.is-danger .navbar-end .navbar-link.is-active { - background-color: #c10a00; - color: #fff; } - .navbar.is-danger .navbar-start .navbar-link::after, - .navbar.is-danger .navbar-end .navbar-link::after { - border-color: #fff; } - .navbar.is-danger .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-danger .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-danger .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #c10a00; - color: #fff; } - .navbar.is-danger .navbar-dropdown a.navbar-item.is-active { - background-color: #da0b00; - color: #fff; } } - .navbar > .container { - align-items: stretch; - display: flex; - min-height: 3.25rem; - width: 100%; } - .navbar.has-shadow { - box-shadow: 0 2px 0 0 whitesmoke; } - .navbar.is-fixed-bottom, .navbar.is-fixed-top { - left: 0; - position: fixed; - right: 0; - z-index: 30; } - .navbar.is-fixed-bottom { - bottom: 0; } - .navbar.is-fixed-bottom.has-shadow { - box-shadow: 0 -2px 0 0 whitesmoke; } - .navbar.is-fixed-top { - top: 0; } - -html.has-navbar-fixed-top, -body.has-navbar-fixed-top { - padding-top: 3.25rem; } - -html.has-navbar-fixed-bottom, -body.has-navbar-fixed-bottom { - padding-bottom: 3.25rem; } - -.navbar-brand, -.navbar-tabs { - align-items: stretch; - display: flex; - flex-shrink: 0; - min-height: 3.25rem; } - -.navbar-brand a.navbar-item:focus, .navbar-brand a.navbar-item:hover { - background-color: transparent; } - -.navbar-tabs { - -webkit-overflow-scrolling: touch; - max-width: 100vw; - overflow-x: auto; - overflow-y: hidden; } - -.navbar-burger { - color: #4a4a4a; - cursor: pointer; - display: block; - height: 3.25rem; - position: relative; - width: 3.25rem; - margin-left: auto; } - .navbar-burger span { - background-color: currentColor; - display: block; - height: 1px; - left: calc(50% - 8px); - position: absolute; - transform-origin: center; - transition-duration: 86ms; - transition-property: background-color, opacity, transform; - transition-timing-function: ease-out; - width: 16px; } - .navbar-burger span:nth-child(1) { - top: calc(50% - 6px); } - .navbar-burger span:nth-child(2) { - top: calc(50% - 1px); } - .navbar-burger span:nth-child(3) { - top: calc(50% + 4px); } - .navbar-burger:hover { - background-color: rgba(0, 0, 0, 0.05); } - .navbar-burger.is-active span:nth-child(1) { - transform: translateY(5px) rotate(45deg); } - .navbar-burger.is-active span:nth-child(2) { - opacity: 0; } - .navbar-burger.is-active span:nth-child(3) { - transform: translateY(-5px) rotate(-45deg); } - -.navbar-menu { - display: none; } - -.navbar-item, -.navbar-link { - color: #4a4a4a; - display: block; - line-height: 1.5; - padding: 0.5rem 0.75rem; - position: relative; } - .navbar-item .icon:only-child, - .navbar-link .icon:only-child { - margin-left: -0.25rem; - margin-right: -0.25rem; } - -a.navbar-item, -.navbar-link { - cursor: pointer; } - a.navbar-item:focus, a.navbar-item:focus-within, a.navbar-item:hover, a.navbar-item.is-active, - .navbar-link:focus, - .navbar-link:focus-within, - .navbar-link:hover, - .navbar-link.is-active { - background-color: #fafafa; - color: #2e63b8; } - -.navbar-item { - display: block; - flex-grow: 0; - flex-shrink: 0; } - .navbar-item img { - max-height: 1.75rem; } - .navbar-item.has-dropdown { - padding: 0; } - .navbar-item.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - .navbar-item.is-tab { - border-bottom: 1px solid transparent; - min-height: 3.25rem; - padding-bottom: calc(0.5rem - 1px); } - .navbar-item.is-tab:focus, .navbar-item.is-tab:hover { - background-color: transparent; - border-bottom-color: #2e63b8; } - .navbar-item.is-tab.is-active { - background-color: transparent; - border-bottom-color: #2e63b8; - border-bottom-style: solid; - border-bottom-width: 3px; - color: #2e63b8; - padding-bottom: calc(0.5rem - 3px); } - -.navbar-content { - flex-grow: 1; - flex-shrink: 1; } - -.navbar-link:not(.is-arrowless) { - padding-right: 2.5em; } - .navbar-link:not(.is-arrowless)::after { - border-color: #2e63b8; - margin-top: -0.375em; - right: 1.125em; } - -.navbar-dropdown { - font-size: 0.875rem; - padding-bottom: 0.5rem; - padding-top: 0.5rem; } - .navbar-dropdown .navbar-item { - padding-left: 1.5rem; - padding-right: 1.5rem; } - -.navbar-divider { - background-color: whitesmoke; - border: none; - display: none; - height: 2px; - margin: 0.5rem 0; } - -@media screen and (max-width: 1055px) { - .navbar > .container { - display: block; } - .navbar-brand .navbar-item, - .navbar-tabs .navbar-item { - align-items: center; - display: flex; } - .navbar-link::after { - display: none; } - .navbar-menu { - background-color: white; - box-shadow: 0 8px 16px rgba(10, 10, 10, 0.1); - padding: 0.5rem 0; } - .navbar-menu.is-active { - display: block; } - .navbar.is-fixed-bottom-touch, .navbar.is-fixed-top-touch { - left: 0; - position: fixed; - right: 0; - z-index: 30; } - .navbar.is-fixed-bottom-touch { - bottom: 0; } - .navbar.is-fixed-bottom-touch.has-shadow { - box-shadow: 0 -2px 3px rgba(10, 10, 10, 0.1); } - .navbar.is-fixed-top-touch { - top: 0; } - .navbar.is-fixed-top .navbar-menu, .navbar.is-fixed-top-touch .navbar-menu { - -webkit-overflow-scrolling: touch; - max-height: calc(100vh - 3.25rem); - overflow: auto; } - html.has-navbar-fixed-top-touch, - body.has-navbar-fixed-top-touch { - padding-top: 3.25rem; } - html.has-navbar-fixed-bottom-touch, - body.has-navbar-fixed-bottom-touch { - padding-bottom: 3.25rem; } } - -@media screen and (min-width: 1056px) { - .navbar, - .navbar-menu, - .navbar-start, - .navbar-end { - align-items: stretch; - display: flex; } - .navbar { - min-height: 3.25rem; } - .navbar.is-spaced { - padding: 1rem 2rem; } - .navbar.is-spaced .navbar-start, - .navbar.is-spaced .navbar-end { - align-items: center; } - .navbar.is-spaced a.navbar-item, - .navbar.is-spaced .navbar-link { - border-radius: 4px; } - .navbar.is-transparent a.navbar-item:focus, .navbar.is-transparent a.navbar-item:hover, .navbar.is-transparent a.navbar-item.is-active, - .navbar.is-transparent .navbar-link:focus, - .navbar.is-transparent .navbar-link:hover, - .navbar.is-transparent .navbar-link.is-active { - background-color: transparent !important; } - .navbar.is-transparent .navbar-item.has-dropdown.is-active .navbar-link, .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:focus .navbar-link, .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:focus-within .navbar-link, .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:hover .navbar-link { - background-color: transparent !important; } - .navbar.is-transparent .navbar-dropdown a.navbar-item:focus, .navbar.is-transparent .navbar-dropdown a.navbar-item:hover { - background-color: whitesmoke; - color: #0a0a0a; } - .navbar.is-transparent .navbar-dropdown a.navbar-item.is-active { - background-color: whitesmoke; - color: #2e63b8; } - .navbar-burger { - display: none; } - .navbar-item, - .navbar-link { - align-items: center; - display: flex; } - .navbar-item { - display: flex; } - .navbar-item.has-dropdown { - align-items: stretch; } - .navbar-item.has-dropdown-up .navbar-link::after { - transform: rotate(135deg) translate(0.25em, -0.25em); } - .navbar-item.has-dropdown-up .navbar-dropdown { - border-bottom: 2px solid #dbdbdb; - border-radius: 6px 6px 0 0; - border-top: none; - bottom: 100%; - box-shadow: 0 -8px 8px rgba(10, 10, 10, 0.1); - top: auto; } - .navbar-item.is-active .navbar-dropdown, .navbar-item.is-hoverable:focus .navbar-dropdown, .navbar-item.is-hoverable:focus-within .navbar-dropdown, .navbar-item.is-hoverable:hover .navbar-dropdown { - display: block; } - .navbar.is-spaced .navbar-item.is-active .navbar-dropdown, .navbar-item.is-active .navbar-dropdown.is-boxed, .navbar.is-spaced .navbar-item.is-hoverable:focus .navbar-dropdown, .navbar-item.is-hoverable:focus .navbar-dropdown.is-boxed, .navbar.is-spaced .navbar-item.is-hoverable:focus-within .navbar-dropdown, .navbar-item.is-hoverable:focus-within .navbar-dropdown.is-boxed, .navbar.is-spaced .navbar-item.is-hoverable:hover .navbar-dropdown, .navbar-item.is-hoverable:hover .navbar-dropdown.is-boxed { - opacity: 1; - pointer-events: auto; - transform: translateY(0); } - .navbar-menu { - flex-grow: 1; - flex-shrink: 0; } - .navbar-start { - justify-content: flex-start; - margin-right: auto; } - .navbar-end { - justify-content: flex-end; - margin-left: auto; } - .navbar-dropdown { - background-color: white; - border-bottom-left-radius: 6px; - border-bottom-right-radius: 6px; - border-top: 2px solid #dbdbdb; - box-shadow: 0 8px 8px rgba(10, 10, 10, 0.1); - display: none; - font-size: 0.875rem; - left: 0; - min-width: 100%; - position: absolute; - top: 100%; - z-index: 20; } - .navbar-dropdown .navbar-item { - padding: 0.375rem 1rem; - white-space: nowrap; } - .navbar-dropdown a.navbar-item { - padding-right: 3rem; } - .navbar-dropdown a.navbar-item:focus, .navbar-dropdown a.navbar-item:hover { - background-color: whitesmoke; - color: #0a0a0a; } - .navbar-dropdown a.navbar-item.is-active { - background-color: whitesmoke; - color: #2e63b8; } - .navbar.is-spaced .navbar-dropdown, .navbar-dropdown.is-boxed { - border-radius: 6px; - border-top: none; - box-shadow: 0 8px 8px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); - display: block; - opacity: 0; - pointer-events: none; - top: calc(100% + (-4px)); - transform: translateY(-5px); - transition-duration: 86ms; - transition-property: opacity, transform; } - .navbar-dropdown.is-right { - left: auto; - right: 0; } - .navbar-divider { - display: block; } - .navbar > .container .navbar-brand, - .container > .navbar .navbar-brand { - margin-left: -.75rem; } - .navbar > .container .navbar-menu, - .container > .navbar .navbar-menu { - margin-right: -.75rem; } - .navbar.is-fixed-bottom-desktop, .navbar.is-fixed-top-desktop { - left: 0; - position: fixed; - right: 0; - z-index: 30; } - .navbar.is-fixed-bottom-desktop { - bottom: 0; } - .navbar.is-fixed-bottom-desktop.has-shadow { - box-shadow: 0 -2px 3px rgba(10, 10, 10, 0.1); } - .navbar.is-fixed-top-desktop { - top: 0; } - html.has-navbar-fixed-top-desktop, - body.has-navbar-fixed-top-desktop { - padding-top: 3.25rem; } - html.has-navbar-fixed-bottom-desktop, - body.has-navbar-fixed-bottom-desktop { - padding-bottom: 3.25rem; } - html.has-spaced-navbar-fixed-top, - body.has-spaced-navbar-fixed-top { - padding-top: 5.25rem; } - html.has-spaced-navbar-fixed-bottom, - body.has-spaced-navbar-fixed-bottom { - padding-bottom: 5.25rem; } - a.navbar-item.is-active, - .navbar-link.is-active { - color: #0a0a0a; } - a.navbar-item.is-active:not(:focus):not(:hover), - .navbar-link.is-active:not(:focus):not(:hover) { - background-color: transparent; } - .navbar-item.has-dropdown:focus .navbar-link, .navbar-item.has-dropdown:hover .navbar-link, .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #fafafa; } } - -.hero.is-fullheight-with-navbar { - min-height: calc(100vh - 3.25rem); } - -.pagination { - font-size: 1rem; - margin: -0.25rem; } - .pagination.is-small, #documenter .docs-sidebar form.docs-search > input.pagination { - font-size: 0.75rem; } - .pagination.is-medium { - font-size: 1.25rem; } - .pagination.is-large { - font-size: 1.5rem; } - .pagination.is-rounded .pagination-previous, #documenter .docs-sidebar form.docs-search > input.pagination .pagination-previous, - .pagination.is-rounded .pagination-next, - #documenter .docs-sidebar form.docs-search > input.pagination .pagination-next { - padding-left: 1em; - padding-right: 1em; - border-radius: 290486px; } - .pagination.is-rounded .pagination-link, #documenter .docs-sidebar form.docs-search > input.pagination .pagination-link { - border-radius: 290486px; } - -.pagination, -.pagination-list { - align-items: center; - display: flex; - justify-content: center; - text-align: center; } - -.pagination-previous, -.pagination-next, -.pagination-link, -.pagination-ellipsis { - font-size: 1em; - justify-content: center; - margin: 0.25rem; - padding-left: 0.5em; - padding-right: 0.5em; - text-align: center; } - -.pagination-previous, -.pagination-next, -.pagination-link { - border-color: #dbdbdb; - color: #363636; - min-width: 2.25em; } - .pagination-previous:hover, - .pagination-next:hover, - .pagination-link:hover { - border-color: #b5b5b5; - color: #363636; } - .pagination-previous:focus, - .pagination-next:focus, - .pagination-link:focus { - border-color: #3c5dcd; } - .pagination-previous:active, - .pagination-next:active, - .pagination-link:active { - box-shadow: inset 0 1px 2px rgba(10, 10, 10, 0.2); } - .pagination-previous[disabled], - .pagination-next[disabled], - .pagination-link[disabled] { - background-color: #dbdbdb; - border-color: #dbdbdb; - box-shadow: none; - color: #6b6b6b; - opacity: 0.5; } - -.pagination-previous, -.pagination-next { - padding-left: 0.75em; - padding-right: 0.75em; - white-space: nowrap; } - -.pagination-link.is-current { - background-color: #2e63b8; - border-color: #2e63b8; - color: #fff; } - -.pagination-ellipsis { - color: #b5b5b5; - pointer-events: none; } - -.pagination-list { - flex-wrap: wrap; } - -@media screen and (max-width: 768px) { - .pagination { - flex-wrap: wrap; } - .pagination-previous, - .pagination-next { - flex-grow: 1; - flex-shrink: 1; } - .pagination-list li { - flex-grow: 1; - flex-shrink: 1; } } - -@media screen and (min-width: 769px), print { - .pagination-list { - flex-grow: 1; - flex-shrink: 1; - justify-content: flex-start; - order: 1; } - .pagination-previous { - order: 2; } - .pagination-next { - order: 3; } - .pagination { - justify-content: space-between; } - .pagination.is-centered .pagination-previous { - order: 1; } - .pagination.is-centered .pagination-list { - justify-content: center; - order: 2; } - .pagination.is-centered .pagination-next { - order: 3; } - .pagination.is-right .pagination-previous { - order: 1; } - .pagination.is-right .pagination-next { - order: 2; } - .pagination.is-right .pagination-list { - justify-content: flex-end; - order: 3; } } - -.panel { - font-size: 1rem; } - .panel:not(:last-child) { - margin-bottom: 1.5rem; } - -.panel-heading, -.panel-tabs, -.panel-block { - border-bottom: 1px solid #dbdbdb; - border-left: 1px solid #dbdbdb; - border-right: 1px solid #dbdbdb; } - .panel-heading:first-child, - .panel-tabs:first-child, - .panel-block:first-child { - border-top: 1px solid #dbdbdb; } - -.panel-heading { - background-color: whitesmoke; - border-radius: 4px 4px 0 0; - color: #222222; - font-size: 1.25em; - font-weight: 300; - line-height: 1.25; - padding: 0.5em 0.75em; } - -.panel-tabs { - align-items: flex-end; - display: flex; - font-size: 0.875em; - justify-content: center; } - .panel-tabs a { - border-bottom: 1px solid #dbdbdb; - margin-bottom: -1px; - padding: 0.5em; } - .panel-tabs a.is-active { - border-bottom-color: #4a4a4a; - color: #363636; } - -.panel-list a { - color: #222222; } - .panel-list a:hover { - color: #2e63b8; } - -.panel-block { - align-items: center; - color: #222222; - display: flex; - justify-content: flex-start; - padding: 0.5em 0.75em; } - .panel-block input[type="checkbox"] { - margin-right: 0.75em; } - .panel-block > .control { - flex-grow: 1; - flex-shrink: 1; - width: 100%; } - .panel-block.is-wrapped { - flex-wrap: wrap; } - .panel-block.is-active { - border-left-color: #2e63b8; - color: #363636; } - .panel-block.is-active .panel-icon { - color: #2e63b8; } - -a.panel-block, -label.panel-block { - cursor: pointer; } - a.panel-block:hover, - label.panel-block:hover { - background-color: whitesmoke; } - -.panel-icon { - display: inline-block; - font-size: 14px; - height: 1em; - line-height: 1em; - text-align: center; - vertical-align: top; - width: 1em; - color: #6b6b6b; - margin-right: 0.75em; } - .panel-icon .fa { - font-size: inherit; - line-height: inherit; } - -.tabs { - -webkit-overflow-scrolling: touch; - align-items: stretch; - display: flex; - font-size: 1rem; - justify-content: space-between; - overflow: hidden; - overflow-x: auto; - white-space: nowrap; } - .tabs a { - align-items: center; - border-bottom-color: #dbdbdb; - border-bottom-style: solid; - border-bottom-width: 1px; - color: #222222; - display: flex; - justify-content: center; - margin-bottom: -1px; - padding: 0.5em 1em; - vertical-align: top; } - .tabs a:hover { - border-bottom-color: #222222; - color: #222222; } - .tabs li { - display: block; } - .tabs li.is-active a { - border-bottom-color: #2e63b8; - color: #2e63b8; } - .tabs ul { - align-items: center; - border-bottom-color: #dbdbdb; - border-bottom-style: solid; - border-bottom-width: 1px; - display: flex; - flex-grow: 1; - flex-shrink: 0; - justify-content: flex-start; } - .tabs ul.is-left { - padding-right: 0.75em; } - .tabs ul.is-center { - flex: none; - justify-content: center; - padding-left: 0.75em; - padding-right: 0.75em; } - .tabs ul.is-right { - justify-content: flex-end; - padding-left: 0.75em; } - .tabs .icon:first-child { - margin-right: 0.5em; } - .tabs .icon:last-child { - margin-left: 0.5em; } - .tabs.is-centered ul { - justify-content: center; } - .tabs.is-right ul { - justify-content: flex-end; } - .tabs.is-boxed a { - border: 1px solid transparent; - border-radius: 4px 4px 0 0; } - .tabs.is-boxed a:hover { - background-color: whitesmoke; - border-bottom-color: #dbdbdb; } - .tabs.is-boxed li.is-active a { - background-color: white; - border-color: #dbdbdb; - border-bottom-color: transparent !important; } - .tabs.is-fullwidth li { - flex-grow: 1; - flex-shrink: 0; } - .tabs.is-toggle a { - border-color: #dbdbdb; - border-style: solid; - border-width: 1px; - margin-bottom: 0; - position: relative; } - .tabs.is-toggle a:hover { - background-color: whitesmoke; - border-color: #b5b5b5; - z-index: 2; } - .tabs.is-toggle li + li { - margin-left: -1px; } - .tabs.is-toggle li:first-child a { - border-radius: 4px 0 0 4px; } - .tabs.is-toggle li:last-child a { - border-radius: 0 4px 4px 0; } - .tabs.is-toggle li.is-active a { - background-color: #2e63b8; - border-color: #2e63b8; - color: #fff; - z-index: 1; } - .tabs.is-toggle ul { - border-bottom: none; } - .tabs.is-toggle.is-toggle-rounded li:first-child a { - border-bottom-left-radius: 290486px; - border-top-left-radius: 290486px; - padding-left: 1.25em; } - .tabs.is-toggle.is-toggle-rounded li:last-child a { - border-bottom-right-radius: 290486px; - border-top-right-radius: 290486px; - padding-right: 1.25em; } - .tabs.is-small, #documenter .docs-sidebar form.docs-search > input.tabs { - font-size: 0.75rem; } - .tabs.is-medium { - font-size: 1.25rem; } - .tabs.is-large { - font-size: 1.5rem; } - -.column { - display: block; - flex-basis: 0; - flex-grow: 1; - flex-shrink: 1; - padding: 0.75rem; } - .columns.is-mobile > .column.is-narrow { - flex: none; } - .columns.is-mobile > .column.is-full { - flex: none; - width: 100%; } - .columns.is-mobile > .column.is-three-quarters { - flex: none; - width: 75%; } - .columns.is-mobile > .column.is-two-thirds { - flex: none; - width: 66.6666%; } - .columns.is-mobile > .column.is-half { - flex: none; - width: 50%; } - .columns.is-mobile > .column.is-one-third { - flex: none; - width: 33.3333%; } - .columns.is-mobile > .column.is-one-quarter { - flex: none; - width: 25%; } - .columns.is-mobile > .column.is-one-fifth { - flex: none; - width: 20%; } - .columns.is-mobile > .column.is-two-fifths { - flex: none; - width: 40%; } - .columns.is-mobile > .column.is-three-fifths { - flex: none; - width: 60%; } - .columns.is-mobile > .column.is-four-fifths { - flex: none; - width: 80%; } - .columns.is-mobile > .column.is-offset-three-quarters { - margin-left: 75%; } - .columns.is-mobile > .column.is-offset-two-thirds { - margin-left: 66.6666%; } - .columns.is-mobile > .column.is-offset-half { - margin-left: 50%; } - .columns.is-mobile > .column.is-offset-one-third { - margin-left: 33.3333%; } - .columns.is-mobile > .column.is-offset-one-quarter { - margin-left: 25%; } - .columns.is-mobile > .column.is-offset-one-fifth { - margin-left: 20%; } - .columns.is-mobile > .column.is-offset-two-fifths { - margin-left: 40%; } - .columns.is-mobile > .column.is-offset-three-fifths { - margin-left: 60%; } - .columns.is-mobile > .column.is-offset-four-fifths { - margin-left: 80%; } - .columns.is-mobile > .column.is-0 { - flex: none; - width: 0%; } - .columns.is-mobile > .column.is-offset-0 { - margin-left: 0%; } - .columns.is-mobile > .column.is-1 { - flex: none; - width: 8.3333333333%; } - .columns.is-mobile > .column.is-offset-1 { - margin-left: 8.3333333333%; } - .columns.is-mobile > .column.is-2 { - flex: none; - width: 16.6666666667%; } - .columns.is-mobile > .column.is-offset-2 { - margin-left: 16.6666666667%; } - .columns.is-mobile > .column.is-3 { - flex: none; - width: 25%; } - .columns.is-mobile > .column.is-offset-3 { - margin-left: 25%; } - .columns.is-mobile > .column.is-4 { - flex: none; - width: 33.3333333333%; } - .columns.is-mobile > .column.is-offset-4 { - margin-left: 33.3333333333%; } - .columns.is-mobile > .column.is-5 { - flex: none; - width: 41.6666666667%; } - .columns.is-mobile > .column.is-offset-5 { - margin-left: 41.6666666667%; } - .columns.is-mobile > .column.is-6 { - flex: none; - width: 50%; } - .columns.is-mobile > .column.is-offset-6 { - margin-left: 50%; } - .columns.is-mobile > .column.is-7 { - flex: none; - width: 58.3333333333%; } - .columns.is-mobile > .column.is-offset-7 { - margin-left: 58.3333333333%; } - .columns.is-mobile > .column.is-8 { - flex: none; - width: 66.6666666667%; } - .columns.is-mobile > .column.is-offset-8 { - margin-left: 66.6666666667%; } - .columns.is-mobile > .column.is-9 { - flex: none; - width: 75%; } - .columns.is-mobile > .column.is-offset-9 { - margin-left: 75%; } - .columns.is-mobile > .column.is-10 { - flex: none; - width: 83.3333333333%; } - .columns.is-mobile > .column.is-offset-10 { - margin-left: 83.3333333333%; } - .columns.is-mobile > .column.is-11 { - flex: none; - width: 91.6666666667%; } - .columns.is-mobile > .column.is-offset-11 { - margin-left: 91.6666666667%; } - .columns.is-mobile > .column.is-12 { - flex: none; - width: 100%; } - .columns.is-mobile > .column.is-offset-12 { - margin-left: 100%; } - @media screen and (max-width: 768px) { - .column.is-narrow-mobile { - flex: none; } - .column.is-full-mobile { - flex: none; - width: 100%; } - .column.is-three-quarters-mobile { - flex: none; - width: 75%; } - .column.is-two-thirds-mobile { - flex: none; - width: 66.6666%; } - .column.is-half-mobile { - flex: none; - width: 50%; } - .column.is-one-third-mobile { - flex: none; - width: 33.3333%; } - .column.is-one-quarter-mobile { - flex: none; - width: 25%; } - .column.is-one-fifth-mobile { - flex: none; - width: 20%; } - .column.is-two-fifths-mobile { - flex: none; - width: 40%; } - .column.is-three-fifths-mobile { - flex: none; - width: 60%; } - .column.is-four-fifths-mobile { - flex: none; - width: 80%; } - .column.is-offset-three-quarters-mobile { - margin-left: 75%; } - .column.is-offset-two-thirds-mobile { - margin-left: 66.6666%; } - .column.is-offset-half-mobile { - margin-left: 50%; } - .column.is-offset-one-third-mobile { - margin-left: 33.3333%; } - .column.is-offset-one-quarter-mobile { - margin-left: 25%; } - .column.is-offset-one-fifth-mobile { - margin-left: 20%; } - .column.is-offset-two-fifths-mobile { - margin-left: 40%; } - .column.is-offset-three-fifths-mobile { - margin-left: 60%; } - .column.is-offset-four-fifths-mobile { - margin-left: 80%; } - .column.is-0-mobile { - flex: none; - width: 0%; } - .column.is-offset-0-mobile { - margin-left: 0%; } - .column.is-1-mobile { - flex: none; - width: 8.3333333333%; } - .column.is-offset-1-mobile { - margin-left: 8.3333333333%; } - .column.is-2-mobile { - flex: none; - width: 16.6666666667%; } - .column.is-offset-2-mobile { - margin-left: 16.6666666667%; } - .column.is-3-mobile { - flex: none; - width: 25%; } - .column.is-offset-3-mobile { - margin-left: 25%; } - .column.is-4-mobile { - flex: none; - width: 33.3333333333%; } - .column.is-offset-4-mobile { - margin-left: 33.3333333333%; } - .column.is-5-mobile { - flex: none; - width: 41.6666666667%; } - .column.is-offset-5-mobile { - margin-left: 41.6666666667%; } - .column.is-6-mobile { - flex: none; - width: 50%; } - .column.is-offset-6-mobile { - margin-left: 50%; } - .column.is-7-mobile { - flex: none; - width: 58.3333333333%; } - .column.is-offset-7-mobile { - margin-left: 58.3333333333%; } - .column.is-8-mobile { - flex: none; - width: 66.6666666667%; } - .column.is-offset-8-mobile { - margin-left: 66.6666666667%; } - .column.is-9-mobile { - flex: none; - width: 75%; } - .column.is-offset-9-mobile { - margin-left: 75%; } - .column.is-10-mobile { - flex: none; - width: 83.3333333333%; } - .column.is-offset-10-mobile { - margin-left: 83.3333333333%; } - .column.is-11-mobile { - flex: none; - width: 91.6666666667%; } - .column.is-offset-11-mobile { - margin-left: 91.6666666667%; } - .column.is-12-mobile { - flex: none; - width: 100%; } - .column.is-offset-12-mobile { - margin-left: 100%; } } - @media screen and (min-width: 769px), print { - .column.is-narrow, .column.is-narrow-tablet { - flex: none; } - .column.is-full, .column.is-full-tablet { - flex: none; - width: 100%; } - .column.is-three-quarters, .column.is-three-quarters-tablet { - flex: none; - width: 75%; } - .column.is-two-thirds, .column.is-two-thirds-tablet { - flex: none; - width: 66.6666%; } - .column.is-half, .column.is-half-tablet { - flex: none; - width: 50%; } - .column.is-one-third, .column.is-one-third-tablet { - flex: none; - width: 33.3333%; } - .column.is-one-quarter, .column.is-one-quarter-tablet { - flex: none; - width: 25%; } - .column.is-one-fifth, .column.is-one-fifth-tablet { - flex: none; - width: 20%; } - .column.is-two-fifths, .column.is-two-fifths-tablet { - flex: none; - width: 40%; } - .column.is-three-fifths, .column.is-three-fifths-tablet { - flex: none; - width: 60%; } - .column.is-four-fifths, .column.is-four-fifths-tablet { - flex: none; - width: 80%; } - .column.is-offset-three-quarters, .column.is-offset-three-quarters-tablet { - margin-left: 75%; } - .column.is-offset-two-thirds, .column.is-offset-two-thirds-tablet { - margin-left: 66.6666%; } - .column.is-offset-half, .column.is-offset-half-tablet { - margin-left: 50%; } - .column.is-offset-one-third, .column.is-offset-one-third-tablet { - margin-left: 33.3333%; } - .column.is-offset-one-quarter, .column.is-offset-one-quarter-tablet { - margin-left: 25%; } - .column.is-offset-one-fifth, .column.is-offset-one-fifth-tablet { - margin-left: 20%; } - .column.is-offset-two-fifths, .column.is-offset-two-fifths-tablet { - margin-left: 40%; } - .column.is-offset-three-fifths, .column.is-offset-three-fifths-tablet { - margin-left: 60%; } - .column.is-offset-four-fifths, .column.is-offset-four-fifths-tablet { - margin-left: 80%; } - .column.is-0, .column.is-0-tablet { - flex: none; - width: 0%; } - .column.is-offset-0, .column.is-offset-0-tablet { - margin-left: 0%; } - .column.is-1, .column.is-1-tablet { - flex: none; - width: 8.3333333333%; } - .column.is-offset-1, .column.is-offset-1-tablet { - margin-left: 8.3333333333%; } - .column.is-2, .column.is-2-tablet { - flex: none; - width: 16.6666666667%; } - .column.is-offset-2, .column.is-offset-2-tablet { - margin-left: 16.6666666667%; } - .column.is-3, .column.is-3-tablet { - flex: none; - width: 25%; } - .column.is-offset-3, .column.is-offset-3-tablet { - margin-left: 25%; } - .column.is-4, .column.is-4-tablet { - flex: none; - width: 33.3333333333%; } - .column.is-offset-4, .column.is-offset-4-tablet { - margin-left: 33.3333333333%; } - .column.is-5, .column.is-5-tablet { - flex: none; - width: 41.6666666667%; } - .column.is-offset-5, .column.is-offset-5-tablet { - margin-left: 41.6666666667%; } - .column.is-6, .column.is-6-tablet { - flex: none; - width: 50%; } - .column.is-offset-6, .column.is-offset-6-tablet { - margin-left: 50%; } - .column.is-7, .column.is-7-tablet { - flex: none; - width: 58.3333333333%; } - .column.is-offset-7, .column.is-offset-7-tablet { - margin-left: 58.3333333333%; } - .column.is-8, .column.is-8-tablet { - flex: none; - width: 66.6666666667%; } - .column.is-offset-8, .column.is-offset-8-tablet { - margin-left: 66.6666666667%; } - .column.is-9, .column.is-9-tablet { - flex: none; - width: 75%; } - .column.is-offset-9, .column.is-offset-9-tablet { - margin-left: 75%; } - .column.is-10, .column.is-10-tablet { - flex: none; - width: 83.3333333333%; } - .column.is-offset-10, .column.is-offset-10-tablet { - margin-left: 83.3333333333%; } - .column.is-11, .column.is-11-tablet { - flex: none; - width: 91.6666666667%; } - .column.is-offset-11, .column.is-offset-11-tablet { - margin-left: 91.6666666667%; } - .column.is-12, .column.is-12-tablet { - flex: none; - width: 100%; } - .column.is-offset-12, .column.is-offset-12-tablet { - margin-left: 100%; } } - @media screen and (max-width: 1055px) { - .column.is-narrow-touch { - flex: none; } - .column.is-full-touch { - flex: none; - width: 100%; } - .column.is-three-quarters-touch { - flex: none; - width: 75%; } - .column.is-two-thirds-touch { - flex: none; - width: 66.6666%; } - .column.is-half-touch { - flex: none; - width: 50%; } - .column.is-one-third-touch { - flex: none; - width: 33.3333%; } - .column.is-one-quarter-touch { - flex: none; - width: 25%; } - .column.is-one-fifth-touch { - flex: none; - width: 20%; } - .column.is-two-fifths-touch { - flex: none; - width: 40%; } - .column.is-three-fifths-touch { - flex: none; - width: 60%; } - .column.is-four-fifths-touch { - flex: none; - width: 80%; } - .column.is-offset-three-quarters-touch { - margin-left: 75%; } - .column.is-offset-two-thirds-touch { - margin-left: 66.6666%; } - .column.is-offset-half-touch { - margin-left: 50%; } - .column.is-offset-one-third-touch { - margin-left: 33.3333%; } - .column.is-offset-one-quarter-touch { - margin-left: 25%; } - .column.is-offset-one-fifth-touch { - margin-left: 20%; } - .column.is-offset-two-fifths-touch { - margin-left: 40%; } - .column.is-offset-three-fifths-touch { - margin-left: 60%; } - .column.is-offset-four-fifths-touch { - margin-left: 80%; } - .column.is-0-touch { - flex: none; - width: 0%; } - .column.is-offset-0-touch { - margin-left: 0%; } - .column.is-1-touch { - flex: none; - width: 8.3333333333%; } - .column.is-offset-1-touch { - margin-left: 8.3333333333%; } - .column.is-2-touch { - flex: none; - width: 16.6666666667%; } - .column.is-offset-2-touch { - margin-left: 16.6666666667%; } - .column.is-3-touch { - flex: none; - width: 25%; } - .column.is-offset-3-touch { - margin-left: 25%; } - .column.is-4-touch { - flex: none; - width: 33.3333333333%; } - .column.is-offset-4-touch { - margin-left: 33.3333333333%; } - .column.is-5-touch { - flex: none; - width: 41.6666666667%; } - .column.is-offset-5-touch { - margin-left: 41.6666666667%; } - .column.is-6-touch { - flex: none; - width: 50%; } - .column.is-offset-6-touch { - margin-left: 50%; } - .column.is-7-touch { - flex: none; - width: 58.3333333333%; } - .column.is-offset-7-touch { - margin-left: 58.3333333333%; } - .column.is-8-touch { - flex: none; - width: 66.6666666667%; } - .column.is-offset-8-touch { - margin-left: 66.6666666667%; } - .column.is-9-touch { - flex: none; - width: 75%; } - .column.is-offset-9-touch { - margin-left: 75%; } - .column.is-10-touch { - flex: none; - width: 83.3333333333%; } - .column.is-offset-10-touch { - margin-left: 83.3333333333%; } - .column.is-11-touch { - flex: none; - width: 91.6666666667%; } - .column.is-offset-11-touch { - margin-left: 91.6666666667%; } - .column.is-12-touch { - flex: none; - width: 100%; } - .column.is-offset-12-touch { - margin-left: 100%; } } - @media screen and (min-width: 1056px) { - .column.is-narrow-desktop { - flex: none; } - .column.is-full-desktop { - flex: none; - width: 100%; } - .column.is-three-quarters-desktop { - flex: none; - width: 75%; } - .column.is-two-thirds-desktop { - flex: none; - width: 66.6666%; } - .column.is-half-desktop { - flex: none; - width: 50%; } - .column.is-one-third-desktop { - flex: none; - width: 33.3333%; } - .column.is-one-quarter-desktop { - flex: none; - width: 25%; } - .column.is-one-fifth-desktop { - flex: none; - width: 20%; } - .column.is-two-fifths-desktop { - flex: none; - width: 40%; } - .column.is-three-fifths-desktop { - flex: none; - width: 60%; } - .column.is-four-fifths-desktop { - flex: none; - width: 80%; } - .column.is-offset-three-quarters-desktop { - margin-left: 75%; } - .column.is-offset-two-thirds-desktop { - margin-left: 66.6666%; } - .column.is-offset-half-desktop { - margin-left: 50%; } - .column.is-offset-one-third-desktop { - margin-left: 33.3333%; } - .column.is-offset-one-quarter-desktop { - margin-left: 25%; } - .column.is-offset-one-fifth-desktop { - margin-left: 20%; } - .column.is-offset-two-fifths-desktop { - margin-left: 40%; } - .column.is-offset-three-fifths-desktop { - margin-left: 60%; } - .column.is-offset-four-fifths-desktop { - margin-left: 80%; } - .column.is-0-desktop { - flex: none; - width: 0%; } - .column.is-offset-0-desktop { - margin-left: 0%; } - .column.is-1-desktop { - flex: none; - width: 8.3333333333%; } - .column.is-offset-1-desktop { - margin-left: 8.3333333333%; } - .column.is-2-desktop { - flex: none; - width: 16.6666666667%; } - .column.is-offset-2-desktop { - margin-left: 16.6666666667%; } - .column.is-3-desktop { - flex: none; - width: 25%; } - .column.is-offset-3-desktop { - margin-left: 25%; } - .column.is-4-desktop { - flex: none; - width: 33.3333333333%; } - .column.is-offset-4-desktop { - margin-left: 33.3333333333%; } - .column.is-5-desktop { - flex: none; - width: 41.6666666667%; } - .column.is-offset-5-desktop { - margin-left: 41.6666666667%; } - .column.is-6-desktop { - flex: none; - width: 50%; } - .column.is-offset-6-desktop { - margin-left: 50%; } - .column.is-7-desktop { - flex: none; - width: 58.3333333333%; } - .column.is-offset-7-desktop { - margin-left: 58.3333333333%; } - .column.is-8-desktop { - flex: none; - width: 66.6666666667%; } - .column.is-offset-8-desktop { - margin-left: 66.6666666667%; } - .column.is-9-desktop { - flex: none; - width: 75%; } - .column.is-offset-9-desktop { - margin-left: 75%; } - .column.is-10-desktop { - flex: none; - width: 83.3333333333%; } - .column.is-offset-10-desktop { - margin-left: 83.3333333333%; } - .column.is-11-desktop { - flex: none; - width: 91.6666666667%; } - .column.is-offset-11-desktop { - margin-left: 91.6666666667%; } - .column.is-12-desktop { - flex: none; - width: 100%; } - .column.is-offset-12-desktop { - margin-left: 100%; } } - @media screen and (min-width: 1216px) { - .column.is-narrow-widescreen { - flex: none; } - .column.is-full-widescreen { - flex: none; - width: 100%; } - .column.is-three-quarters-widescreen { - flex: none; - width: 75%; } - .column.is-two-thirds-widescreen { - flex: none; - width: 66.6666%; } - .column.is-half-widescreen { - flex: none; - width: 50%; } - .column.is-one-third-widescreen { - flex: none; - width: 33.3333%; } - .column.is-one-quarter-widescreen { - flex: none; - width: 25%; } - .column.is-one-fifth-widescreen { - flex: none; - width: 20%; } - .column.is-two-fifths-widescreen { - flex: none; - width: 40%; } - .column.is-three-fifths-widescreen { - flex: none; - width: 60%; } - .column.is-four-fifths-widescreen { - flex: none; - width: 80%; } - .column.is-offset-three-quarters-widescreen { - margin-left: 75%; } - .column.is-offset-two-thirds-widescreen { - margin-left: 66.6666%; } - .column.is-offset-half-widescreen { - margin-left: 50%; } - .column.is-offset-one-third-widescreen { - margin-left: 33.3333%; } - .column.is-offset-one-quarter-widescreen { - margin-left: 25%; } - .column.is-offset-one-fifth-widescreen { - margin-left: 20%; } - .column.is-offset-two-fifths-widescreen { - margin-left: 40%; } - .column.is-offset-three-fifths-widescreen { - margin-left: 60%; } - .column.is-offset-four-fifths-widescreen { - margin-left: 80%; } - .column.is-0-widescreen { - flex: none; - width: 0%; } - .column.is-offset-0-widescreen { - margin-left: 0%; } - .column.is-1-widescreen { - flex: none; - width: 8.3333333333%; } - .column.is-offset-1-widescreen { - margin-left: 8.3333333333%; } - .column.is-2-widescreen { - flex: none; - width: 16.6666666667%; } - .column.is-offset-2-widescreen { - margin-left: 16.6666666667%; } - .column.is-3-widescreen { - flex: none; - width: 25%; } - .column.is-offset-3-widescreen { - margin-left: 25%; } - .column.is-4-widescreen { - flex: none; - width: 33.3333333333%; } - .column.is-offset-4-widescreen { - margin-left: 33.3333333333%; } - .column.is-5-widescreen { - flex: none; - width: 41.6666666667%; } - .column.is-offset-5-widescreen { - margin-left: 41.6666666667%; } - .column.is-6-widescreen { - flex: none; - width: 50%; } - .column.is-offset-6-widescreen { - margin-left: 50%; } - .column.is-7-widescreen { - flex: none; - width: 58.3333333333%; } - .column.is-offset-7-widescreen { - margin-left: 58.3333333333%; } - .column.is-8-widescreen { - flex: none; - width: 66.6666666667%; } - .column.is-offset-8-widescreen { - margin-left: 66.6666666667%; } - .column.is-9-widescreen { - flex: none; - width: 75%; } - .column.is-offset-9-widescreen { - margin-left: 75%; } - .column.is-10-widescreen { - flex: none; - width: 83.3333333333%; } - .column.is-offset-10-widescreen { - margin-left: 83.3333333333%; } - .column.is-11-widescreen { - flex: none; - width: 91.6666666667%; } - .column.is-offset-11-widescreen { - margin-left: 91.6666666667%; } - .column.is-12-widescreen { - flex: none; - width: 100%; } - .column.is-offset-12-widescreen { - margin-left: 100%; } } - @media screen and (min-width: 1408px) { - .column.is-narrow-fullhd { - flex: none; } - .column.is-full-fullhd { - flex: none; - width: 100%; } - .column.is-three-quarters-fullhd { - flex: none; - width: 75%; } - .column.is-two-thirds-fullhd { - flex: none; - width: 66.6666%; } - .column.is-half-fullhd { - flex: none; - width: 50%; } - .column.is-one-third-fullhd { - flex: none; - width: 33.3333%; } - .column.is-one-quarter-fullhd { - flex: none; - width: 25%; } - .column.is-one-fifth-fullhd { - flex: none; - width: 20%; } - .column.is-two-fifths-fullhd { - flex: none; - width: 40%; } - .column.is-three-fifths-fullhd { - flex: none; - width: 60%; } - .column.is-four-fifths-fullhd { - flex: none; - width: 80%; } - .column.is-offset-three-quarters-fullhd { - margin-left: 75%; } - .column.is-offset-two-thirds-fullhd { - margin-left: 66.6666%; } - .column.is-offset-half-fullhd { - margin-left: 50%; } - .column.is-offset-one-third-fullhd { - margin-left: 33.3333%; } - .column.is-offset-one-quarter-fullhd { - margin-left: 25%; } - .column.is-offset-one-fifth-fullhd { - margin-left: 20%; } - .column.is-offset-two-fifths-fullhd { - margin-left: 40%; } - .column.is-offset-three-fifths-fullhd { - margin-left: 60%; } - .column.is-offset-four-fifths-fullhd { - margin-left: 80%; } - .column.is-0-fullhd { - flex: none; - width: 0%; } - .column.is-offset-0-fullhd { - margin-left: 0%; } - .column.is-1-fullhd { - flex: none; - width: 8.3333333333%; } - .column.is-offset-1-fullhd { - margin-left: 8.3333333333%; } - .column.is-2-fullhd { - flex: none; - width: 16.6666666667%; } - .column.is-offset-2-fullhd { - margin-left: 16.6666666667%; } - .column.is-3-fullhd { - flex: none; - width: 25%; } - .column.is-offset-3-fullhd { - margin-left: 25%; } - .column.is-4-fullhd { - flex: none; - width: 33.3333333333%; } - .column.is-offset-4-fullhd { - margin-left: 33.3333333333%; } - .column.is-5-fullhd { - flex: none; - width: 41.6666666667%; } - .column.is-offset-5-fullhd { - margin-left: 41.6666666667%; } - .column.is-6-fullhd { - flex: none; - width: 50%; } - .column.is-offset-6-fullhd { - margin-left: 50%; } - .column.is-7-fullhd { - flex: none; - width: 58.3333333333%; } - .column.is-offset-7-fullhd { - margin-left: 58.3333333333%; } - .column.is-8-fullhd { - flex: none; - width: 66.6666666667%; } - .column.is-offset-8-fullhd { - margin-left: 66.6666666667%; } - .column.is-9-fullhd { - flex: none; - width: 75%; } - .column.is-offset-9-fullhd { - margin-left: 75%; } - .column.is-10-fullhd { - flex: none; - width: 83.3333333333%; } - .column.is-offset-10-fullhd { - margin-left: 83.3333333333%; } - .column.is-11-fullhd { - flex: none; - width: 91.6666666667%; } - .column.is-offset-11-fullhd { - margin-left: 91.6666666667%; } - .column.is-12-fullhd { - flex: none; - width: 100%; } - .column.is-offset-12-fullhd { - margin-left: 100%; } } -.columns { - margin-left: -0.75rem; - margin-right: -0.75rem; - margin-top: -0.75rem; } - .columns:last-child { - margin-bottom: -0.75rem; } - .columns:not(:last-child) { - margin-bottom: calc(1.5rem - 0.75rem); } - .columns.is-centered { - justify-content: center; } - .columns.is-gapless { - margin-left: 0; - margin-right: 0; - margin-top: 0; } - .columns.is-gapless > .column { - margin: 0; - padding: 0 !important; } - .columns.is-gapless:not(:last-child) { - margin-bottom: 1.5rem; } - .columns.is-gapless:last-child { - margin-bottom: 0; } - .columns.is-mobile { - display: flex; } - .columns.is-multiline { - flex-wrap: wrap; } - .columns.is-vcentered { - align-items: center; } - @media screen and (min-width: 769px), print { - .columns:not(.is-desktop) { - display: flex; } } - @media screen and (min-width: 1056px) { - .columns.is-desktop { - display: flex; } } -.columns.is-variable { - --columnGap: 0.75rem; - margin-left: calc(-1 * var(--columnGap)); - margin-right: calc(-1 * var(--columnGap)); } - .columns.is-variable .column { - padding-left: var(--columnGap); - padding-right: var(--columnGap); } - .columns.is-variable.is-0 { - --columnGap: 0rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-0-mobile { - --columnGap: 0rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-0-tablet { - --columnGap: 0rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-0-tablet-only { - --columnGap: 0rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-0-touch { - --columnGap: 0rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-0-desktop { - --columnGap: 0rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-0-desktop-only { - --columnGap: 0rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-0-widescreen { - --columnGap: 0rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-0-widescreen-only { - --columnGap: 0rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-0-fullhd { - --columnGap: 0rem; } } - .columns.is-variable.is-1 { - --columnGap: 0.25rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-1-mobile { - --columnGap: 0.25rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-1-tablet { - --columnGap: 0.25rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-1-tablet-only { - --columnGap: 0.25rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-1-touch { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-1-desktop { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-1-desktop-only { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-1-widescreen { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-1-widescreen-only { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-1-fullhd { - --columnGap: 0.25rem; } } - .columns.is-variable.is-2 { - --columnGap: 0.5rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-2-mobile { - --columnGap: 0.5rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-2-tablet { - --columnGap: 0.5rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-2-tablet-only { - --columnGap: 0.5rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-2-touch { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-2-desktop { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-2-desktop-only { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-2-widescreen { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-2-widescreen-only { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-2-fullhd { - --columnGap: 0.5rem; } } - .columns.is-variable.is-3 { - --columnGap: 0.75rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-3-mobile { - --columnGap: 0.75rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-3-tablet { - --columnGap: 0.75rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-3-tablet-only { - --columnGap: 0.75rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-3-touch { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-3-desktop { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-3-desktop-only { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-3-widescreen { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-3-widescreen-only { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-3-fullhd { - --columnGap: 0.75rem; } } - .columns.is-variable.is-4 { - --columnGap: 1rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-4-mobile { - --columnGap: 1rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-4-tablet { - --columnGap: 1rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-4-tablet-only { - --columnGap: 1rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-4-touch { - --columnGap: 1rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-4-desktop { - --columnGap: 1rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-4-desktop-only { - --columnGap: 1rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-4-widescreen { - --columnGap: 1rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-4-widescreen-only { - --columnGap: 1rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-4-fullhd { - --columnGap: 1rem; } } - .columns.is-variable.is-5 { - --columnGap: 1.25rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-5-mobile { - --columnGap: 1.25rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-5-tablet { - --columnGap: 1.25rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-5-tablet-only { - --columnGap: 1.25rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-5-touch { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-5-desktop { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-5-desktop-only { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-5-widescreen { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-5-widescreen-only { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-5-fullhd { - --columnGap: 1.25rem; } } - .columns.is-variable.is-6 { - --columnGap: 1.5rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-6-mobile { - --columnGap: 1.5rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-6-tablet { - --columnGap: 1.5rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-6-tablet-only { - --columnGap: 1.5rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-6-touch { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-6-desktop { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-6-desktop-only { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-6-widescreen { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-6-widescreen-only { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-6-fullhd { - --columnGap: 1.5rem; } } - .columns.is-variable.is-7 { - --columnGap: 1.75rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-7-mobile { - --columnGap: 1.75rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-7-tablet { - --columnGap: 1.75rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-7-tablet-only { - --columnGap: 1.75rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-7-touch { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-7-desktop { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-7-desktop-only { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-7-widescreen { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-7-widescreen-only { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-7-fullhd { - --columnGap: 1.75rem; } } - .columns.is-variable.is-8 { - --columnGap: 2rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-8-mobile { - --columnGap: 2rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-8-tablet { - --columnGap: 2rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-8-tablet-only { - --columnGap: 2rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-8-touch { - --columnGap: 2rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-8-desktop { - --columnGap: 2rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-8-desktop-only { - --columnGap: 2rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-8-widescreen { - --columnGap: 2rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-8-widescreen-only { - --columnGap: 2rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-8-fullhd { - --columnGap: 2rem; } } -.tile { - align-items: stretch; - display: block; - flex-basis: 0; - flex-grow: 1; - flex-shrink: 1; - min-height: min-content; } - .tile.is-ancestor { - margin-left: -0.75rem; - margin-right: -0.75rem; - margin-top: -0.75rem; } - .tile.is-ancestor:last-child { - margin-bottom: -0.75rem; } - .tile.is-ancestor:not(:last-child) { - margin-bottom: 0.75rem; } - .tile.is-child { - margin: 0 !important; } - .tile.is-parent { - padding: 0.75rem; } - .tile.is-vertical { - flex-direction: column; } - .tile.is-vertical > .tile.is-child:not(:last-child) { - margin-bottom: 1.5rem !important; } - @media screen and (min-width: 769px), print { - .tile:not(.is-child) { - display: flex; } - .tile.is-1 { - flex: none; - width: 8.3333333333%; } - .tile.is-2 { - flex: none; - width: 16.6666666667%; } - .tile.is-3 { - flex: none; - width: 25%; } - .tile.is-4 { - flex: none; - width: 33.3333333333%; } - .tile.is-5 { - flex: none; - width: 41.6666666667%; } - .tile.is-6 { - flex: none; - width: 50%; } - .tile.is-7 { - flex: none; - width: 58.3333333333%; } - .tile.is-8 { - flex: none; - width: 66.6666666667%; } - .tile.is-9 { - flex: none; - width: 75%; } - .tile.is-10 { - flex: none; - width: 83.3333333333%; } - .tile.is-11 { - flex: none; - width: 91.6666666667%; } - .tile.is-12 { - flex: none; - width: 100%; } } -.hero { - align-items: stretch; - display: flex; - flex-direction: column; - justify-content: space-between; } - .hero .navbar { - background: none; } - .hero .tabs ul { - border-bottom: none; } - .hero.is-white { - background-color: white; - color: #0a0a0a; } - .hero.is-white a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-white strong { - color: inherit; } - .hero.is-white .title { - color: #0a0a0a; } - .hero.is-white .subtitle { - color: rgba(10, 10, 10, 0.9); } - .hero.is-white .subtitle a:not(.button), - .hero.is-white .subtitle strong { - color: #0a0a0a; } - @media screen and (max-width: 1055px) { - .hero.is-white .navbar-menu { - background-color: white; } } - .hero.is-white .navbar-item, - .hero.is-white .navbar-link { - color: rgba(10, 10, 10, 0.7); } - .hero.is-white a.navbar-item:hover, .hero.is-white a.navbar-item.is-active, - .hero.is-white .navbar-link:hover, - .hero.is-white .navbar-link.is-active { - background-color: #f2f2f2; - color: #0a0a0a; } - .hero.is-white .tabs a { - color: #0a0a0a; - opacity: 0.9; } - .hero.is-white .tabs a:hover { - opacity: 1; } - .hero.is-white .tabs li.is-active a { - opacity: 1; } - .hero.is-white .tabs.is-boxed a, .hero.is-white .tabs.is-toggle a { - color: #0a0a0a; } - .hero.is-white .tabs.is-boxed a:hover, .hero.is-white .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-white .tabs.is-boxed li.is-active a, .hero.is-white .tabs.is-boxed li.is-active a:hover, .hero.is-white .tabs.is-toggle li.is-active a, .hero.is-white .tabs.is-toggle li.is-active a:hover { - background-color: #0a0a0a; - border-color: #0a0a0a; - color: white; } - .hero.is-white.is-bold { - background-image: linear-gradient(141deg, #e8e3e4 0%, white 71%, white 100%); } - @media screen and (max-width: 768px) { - .hero.is-white.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #e8e3e4 0%, white 71%, white 100%); } } - .hero.is-black { - background-color: #0a0a0a; - color: white; } - .hero.is-black a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-black strong { - color: inherit; } - .hero.is-black .title { - color: white; } - .hero.is-black .subtitle { - color: rgba(255, 255, 255, 0.9); } - .hero.is-black .subtitle a:not(.button), - .hero.is-black .subtitle strong { - color: white; } - @media screen and (max-width: 1055px) { - .hero.is-black .navbar-menu { - background-color: #0a0a0a; } } - .hero.is-black .navbar-item, - .hero.is-black .navbar-link { - color: rgba(255, 255, 255, 0.7); } - .hero.is-black a.navbar-item:hover, .hero.is-black a.navbar-item.is-active, - .hero.is-black .navbar-link:hover, - .hero.is-black .navbar-link.is-active { - background-color: black; - color: white; } - .hero.is-black .tabs a { - color: white; - opacity: 0.9; } - .hero.is-black .tabs a:hover { - opacity: 1; } - .hero.is-black .tabs li.is-active a { - opacity: 1; } - .hero.is-black .tabs.is-boxed a, .hero.is-black .tabs.is-toggle a { - color: white; } - .hero.is-black .tabs.is-boxed a:hover, .hero.is-black .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-black .tabs.is-boxed li.is-active a, .hero.is-black .tabs.is-boxed li.is-active a:hover, .hero.is-black .tabs.is-toggle li.is-active a, .hero.is-black .tabs.is-toggle li.is-active a:hover { - background-color: white; - border-color: white; - color: #0a0a0a; } - .hero.is-black.is-bold { - background-image: linear-gradient(141deg, black 0%, #0a0a0a 71%, #181616 100%); } - @media screen and (max-width: 768px) { - .hero.is-black.is-bold .navbar-menu { - background-image: linear-gradient(141deg, black 0%, #0a0a0a 71%, #181616 100%); } } - .hero.is-light { - background-color: whitesmoke; - color: #363636; } - .hero.is-light a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-light strong { - color: inherit; } - .hero.is-light .title { - color: #363636; } - .hero.is-light .subtitle { - color: rgba(54, 54, 54, 0.9); } - .hero.is-light .subtitle a:not(.button), - .hero.is-light .subtitle strong { - color: #363636; } - @media screen and (max-width: 1055px) { - .hero.is-light .navbar-menu { - background-color: whitesmoke; } } - .hero.is-light .navbar-item, - .hero.is-light .navbar-link { - color: rgba(54, 54, 54, 0.7); } - .hero.is-light a.navbar-item:hover, .hero.is-light a.navbar-item.is-active, - .hero.is-light .navbar-link:hover, - .hero.is-light .navbar-link.is-active { - background-color: #e8e8e8; - color: #363636; } - .hero.is-light .tabs a { - color: #363636; - opacity: 0.9; } - .hero.is-light .tabs a:hover { - opacity: 1; } - .hero.is-light .tabs li.is-active a { - opacity: 1; } - .hero.is-light .tabs.is-boxed a, .hero.is-light .tabs.is-toggle a { - color: #363636; } - .hero.is-light .tabs.is-boxed a:hover, .hero.is-light .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-light .tabs.is-boxed li.is-active a, .hero.is-light .tabs.is-boxed li.is-active a:hover, .hero.is-light .tabs.is-toggle li.is-active a, .hero.is-light .tabs.is-toggle li.is-active a:hover { - background-color: #363636; - border-color: #363636; - color: whitesmoke; } - .hero.is-light.is-bold { - background-image: linear-gradient(141deg, #dfd8d9 0%, whitesmoke 71%, white 100%); } - @media screen and (max-width: 768px) { - .hero.is-light.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #dfd8d9 0%, whitesmoke 71%, white 100%); } } - .hero.is-dark, .content kbd.hero { - background-color: #363636; - color: whitesmoke; } - .hero.is-dark a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), .content kbd.hero a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-dark strong, - .content kbd.hero strong { - color: inherit; } - .hero.is-dark .title, .content kbd.hero .title { - color: whitesmoke; } - .hero.is-dark .subtitle, .content kbd.hero .subtitle { - color: rgba(245, 245, 245, 0.9); } - .hero.is-dark .subtitle a:not(.button), .content kbd.hero .subtitle a:not(.button), - .hero.is-dark .subtitle strong, - .content kbd.hero .subtitle strong { - color: whitesmoke; } - @media screen and (max-width: 1055px) { - .hero.is-dark .navbar-menu, .content kbd.hero .navbar-menu { - background-color: #363636; } } - .hero.is-dark .navbar-item, .content kbd.hero .navbar-item, - .hero.is-dark .navbar-link, - .content kbd.hero .navbar-link { - color: rgba(245, 245, 245, 0.7); } - .hero.is-dark a.navbar-item:hover, .content kbd.hero a.navbar-item:hover, .hero.is-dark a.navbar-item.is-active, .content kbd.hero a.navbar-item.is-active, - .hero.is-dark .navbar-link:hover, - .content kbd.hero .navbar-link:hover, - .hero.is-dark .navbar-link.is-active, - .content kbd.hero .navbar-link.is-active { - background-color: #292929; - color: whitesmoke; } - .hero.is-dark .tabs a, .content kbd.hero .tabs a { - color: whitesmoke; - opacity: 0.9; } - .hero.is-dark .tabs a:hover, .content kbd.hero .tabs a:hover { - opacity: 1; } - .hero.is-dark .tabs li.is-active a, .content kbd.hero .tabs li.is-active a { - opacity: 1; } - .hero.is-dark .tabs.is-boxed a, .content kbd.hero .tabs.is-boxed a, .hero.is-dark .tabs.is-toggle a, .content kbd.hero .tabs.is-toggle a { - color: whitesmoke; } - .hero.is-dark .tabs.is-boxed a:hover, .content kbd.hero .tabs.is-boxed a:hover, .hero.is-dark .tabs.is-toggle a:hover, .content kbd.hero .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-dark .tabs.is-boxed li.is-active a, .content kbd.hero .tabs.is-boxed li.is-active a, .hero.is-dark .tabs.is-boxed li.is-active a:hover, .hero.is-dark .tabs.is-toggle li.is-active a, .content kbd.hero .tabs.is-toggle li.is-active a, .hero.is-dark .tabs.is-toggle li.is-active a:hover { - background-color: whitesmoke; - border-color: whitesmoke; - color: #363636; } - .hero.is-dark.is-bold, .content kbd.hero.is-bold { - background-image: linear-gradient(141deg, #1f191a 0%, #363636 71%, #46403f 100%); } - @media screen and (max-width: 768px) { - .hero.is-dark.is-bold .navbar-menu, .content kbd.hero.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #1f191a 0%, #363636 71%, #46403f 100%); } } - .hero.is-primary, .docstring > section > a.hero.docs-sourcelink { - background-color: #4eb5de; - color: #fff; } - .hero.is-primary a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), .docstring > section > a.hero.docs-sourcelink a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-primary strong, - .docstring > section > a.hero.docs-sourcelink strong { - color: inherit; } - .hero.is-primary .title, .docstring > section > a.hero.docs-sourcelink .title { - color: #fff; } - .hero.is-primary .subtitle, .docstring > section > a.hero.docs-sourcelink .subtitle { - color: rgba(255, 255, 255, 0.9); } - .hero.is-primary .subtitle a:not(.button), .docstring > section > a.hero.docs-sourcelink .subtitle a:not(.button), - .hero.is-primary .subtitle strong, - .docstring > section > a.hero.docs-sourcelink .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - .hero.is-primary .navbar-menu, .docstring > section > a.hero.docs-sourcelink .navbar-menu { - background-color: #4eb5de; } } - .hero.is-primary .navbar-item, .docstring > section > a.hero.docs-sourcelink .navbar-item, - .hero.is-primary .navbar-link, - .docstring > section > a.hero.docs-sourcelink .navbar-link { - color: rgba(255, 255, 255, 0.7); } - .hero.is-primary a.navbar-item:hover, .docstring > section > a.hero.docs-sourcelink a.navbar-item:hover, .hero.is-primary a.navbar-item.is-active, .docstring > section > a.hero.docs-sourcelink a.navbar-item.is-active, - .hero.is-primary .navbar-link:hover, - .docstring > section > a.hero.docs-sourcelink .navbar-link:hover, - .hero.is-primary .navbar-link.is-active, - .docstring > section > a.hero.docs-sourcelink .navbar-link.is-active { - background-color: #39acda; - color: #fff; } - .hero.is-primary .tabs a, .docstring > section > a.hero.docs-sourcelink .tabs a { - color: #fff; - opacity: 0.9; } - .hero.is-primary .tabs a:hover, .docstring > section > a.hero.docs-sourcelink .tabs a:hover { - opacity: 1; } - .hero.is-primary .tabs li.is-active a, .docstring > section > a.hero.docs-sourcelink .tabs li.is-active a { - opacity: 1; } - .hero.is-primary .tabs.is-boxed a, .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed a, .hero.is-primary .tabs.is-toggle a, .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle a { - color: #fff; } - .hero.is-primary .tabs.is-boxed a:hover, .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed a:hover, .hero.is-primary .tabs.is-toggle a:hover, .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-primary .tabs.is-boxed li.is-active a, .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed li.is-active a, .hero.is-primary .tabs.is-boxed li.is-active a:hover, .hero.is-primary .tabs.is-toggle li.is-active a, .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle li.is-active a, .hero.is-primary .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #4eb5de; } - .hero.is-primary.is-bold, .docstring > section > a.hero.is-bold.docs-sourcelink { - background-image: linear-gradient(141deg, #1bc7de 0%, #4eb5de 71%, #5fa9e7 100%); } - @media screen and (max-width: 768px) { - .hero.is-primary.is-bold .navbar-menu, .docstring > section > a.hero.is-bold.docs-sourcelink .navbar-menu { - background-image: linear-gradient(141deg, #1bc7de 0%, #4eb5de 71%, #5fa9e7 100%); } } - .hero.is-link { - background-color: #2e63b8; - color: #fff; } - .hero.is-link a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-link strong { - color: inherit; } - .hero.is-link .title { - color: #fff; } - .hero.is-link .subtitle { - color: rgba(255, 255, 255, 0.9); } - .hero.is-link .subtitle a:not(.button), - .hero.is-link .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - .hero.is-link .navbar-menu { - background-color: #2e63b8; } } - .hero.is-link .navbar-item, - .hero.is-link .navbar-link { - color: rgba(255, 255, 255, 0.7); } - .hero.is-link a.navbar-item:hover, .hero.is-link a.navbar-item.is-active, - .hero.is-link .navbar-link:hover, - .hero.is-link .navbar-link.is-active { - background-color: #2958a4; - color: #fff; } - .hero.is-link .tabs a { - color: #fff; - opacity: 0.9; } - .hero.is-link .tabs a:hover { - opacity: 1; } - .hero.is-link .tabs li.is-active a { - opacity: 1; } - .hero.is-link .tabs.is-boxed a, .hero.is-link .tabs.is-toggle a { - color: #fff; } - .hero.is-link .tabs.is-boxed a:hover, .hero.is-link .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-link .tabs.is-boxed li.is-active a, .hero.is-link .tabs.is-boxed li.is-active a:hover, .hero.is-link .tabs.is-toggle li.is-active a, .hero.is-link .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #2e63b8; } - .hero.is-link.is-bold { - background-image: linear-gradient(141deg, #1b6098 0%, #2e63b8 71%, #2d51d2 100%); } - @media screen and (max-width: 768px) { - .hero.is-link.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #1b6098 0%, #2e63b8 71%, #2d51d2 100%); } } - .hero.is-info { - background-color: #209cee; - color: #fff; } - .hero.is-info a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-info strong { - color: inherit; } - .hero.is-info .title { - color: #fff; } - .hero.is-info .subtitle { - color: rgba(255, 255, 255, 0.9); } - .hero.is-info .subtitle a:not(.button), - .hero.is-info .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - .hero.is-info .navbar-menu { - background-color: #209cee; } } - .hero.is-info .navbar-item, - .hero.is-info .navbar-link { - color: rgba(255, 255, 255, 0.7); } - .hero.is-info a.navbar-item:hover, .hero.is-info a.navbar-item.is-active, - .hero.is-info .navbar-link:hover, - .hero.is-info .navbar-link.is-active { - background-color: #1190e3; - color: #fff; } - .hero.is-info .tabs a { - color: #fff; - opacity: 0.9; } - .hero.is-info .tabs a:hover { - opacity: 1; } - .hero.is-info .tabs li.is-active a { - opacity: 1; } - .hero.is-info .tabs.is-boxed a, .hero.is-info .tabs.is-toggle a { - color: #fff; } - .hero.is-info .tabs.is-boxed a:hover, .hero.is-info .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-info .tabs.is-boxed li.is-active a, .hero.is-info .tabs.is-boxed li.is-active a:hover, .hero.is-info .tabs.is-toggle li.is-active a, .hero.is-info .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #209cee; } - .hero.is-info.is-bold { - background-image: linear-gradient(141deg, #05a6d6 0%, #209cee 71%, #3287f5 100%); } - @media screen and (max-width: 768px) { - .hero.is-info.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #05a6d6 0%, #209cee 71%, #3287f5 100%); } } - .hero.is-success { - background-color: #22c35b; - color: #fff; } - .hero.is-success a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-success strong { - color: inherit; } - .hero.is-success .title { - color: #fff; } - .hero.is-success .subtitle { - color: rgba(255, 255, 255, 0.9); } - .hero.is-success .subtitle a:not(.button), - .hero.is-success .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - .hero.is-success .navbar-menu { - background-color: #22c35b; } } - .hero.is-success .navbar-item, - .hero.is-success .navbar-link { - color: rgba(255, 255, 255, 0.7); } - .hero.is-success a.navbar-item:hover, .hero.is-success a.navbar-item.is-active, - .hero.is-success .navbar-link:hover, - .hero.is-success .navbar-link.is-active { - background-color: #1ead51; - color: #fff; } - .hero.is-success .tabs a { - color: #fff; - opacity: 0.9; } - .hero.is-success .tabs a:hover { - opacity: 1; } - .hero.is-success .tabs li.is-active a { - opacity: 1; } - .hero.is-success .tabs.is-boxed a, .hero.is-success .tabs.is-toggle a { - color: #fff; } - .hero.is-success .tabs.is-boxed a:hover, .hero.is-success .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-success .tabs.is-boxed li.is-active a, .hero.is-success .tabs.is-boxed li.is-active a:hover, .hero.is-success .tabs.is-toggle li.is-active a, .hero.is-success .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #22c35b; } - .hero.is-success.is-bold { - background-image: linear-gradient(141deg, #12a02c 0%, #22c35b 71%, #1fdf83 100%); } - @media screen and (max-width: 768px) { - .hero.is-success.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #12a02c 0%, #22c35b 71%, #1fdf83 100%); } } - .hero.is-warning { - background-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .hero.is-warning a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-warning strong { - color: inherit; } - .hero.is-warning .title { - color: rgba(0, 0, 0, 0.7); } - .hero.is-warning .subtitle { - color: rgba(0, 0, 0, 0.9); } - .hero.is-warning .subtitle a:not(.button), - .hero.is-warning .subtitle strong { - color: rgba(0, 0, 0, 0.7); } - @media screen and (max-width: 1055px) { - .hero.is-warning .navbar-menu { - background-color: #ffdd57; } } - .hero.is-warning .navbar-item, - .hero.is-warning .navbar-link { - color: rgba(0, 0, 0, 0.7); } - .hero.is-warning a.navbar-item:hover, .hero.is-warning a.navbar-item.is-active, - .hero.is-warning .navbar-link:hover, - .hero.is-warning .navbar-link.is-active { - background-color: #ffd83e; - color: rgba(0, 0, 0, 0.7); } - .hero.is-warning .tabs a { - color: rgba(0, 0, 0, 0.7); - opacity: 0.9; } - .hero.is-warning .tabs a:hover { - opacity: 1; } - .hero.is-warning .tabs li.is-active a { - opacity: 1; } - .hero.is-warning .tabs.is-boxed a, .hero.is-warning .tabs.is-toggle a { - color: rgba(0, 0, 0, 0.7); } - .hero.is-warning .tabs.is-boxed a:hover, .hero.is-warning .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-warning .tabs.is-boxed li.is-active a, .hero.is-warning .tabs.is-boxed li.is-active a:hover, .hero.is-warning .tabs.is-toggle li.is-active a, .hero.is-warning .tabs.is-toggle li.is-active a:hover { - background-color: rgba(0, 0, 0, 0.7); - border-color: rgba(0, 0, 0, 0.7); - color: #ffdd57; } - .hero.is-warning.is-bold { - background-image: linear-gradient(141deg, #ffae24 0%, #ffdd57 71%, #fffa71 100%); } - @media screen and (max-width: 768px) { - .hero.is-warning.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #ffae24 0%, #ffdd57 71%, #fffa71 100%); } } - .hero.is-danger { - background-color: #da0b00; - color: #fff; } - .hero.is-danger a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-danger strong { - color: inherit; } - .hero.is-danger .title { - color: #fff; } - .hero.is-danger .subtitle { - color: rgba(255, 255, 255, 0.9); } - .hero.is-danger .subtitle a:not(.button), - .hero.is-danger .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - .hero.is-danger .navbar-menu { - background-color: #da0b00; } } - .hero.is-danger .navbar-item, - .hero.is-danger .navbar-link { - color: rgba(255, 255, 255, 0.7); } - .hero.is-danger a.navbar-item:hover, .hero.is-danger a.navbar-item.is-active, - .hero.is-danger .navbar-link:hover, - .hero.is-danger .navbar-link.is-active { - background-color: #c10a00; - color: #fff; } - .hero.is-danger .tabs a { - color: #fff; - opacity: 0.9; } - .hero.is-danger .tabs a:hover { - opacity: 1; } - .hero.is-danger .tabs li.is-active a { - opacity: 1; } - .hero.is-danger .tabs.is-boxed a, .hero.is-danger .tabs.is-toggle a { - color: #fff; } - .hero.is-danger .tabs.is-boxed a:hover, .hero.is-danger .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-danger .tabs.is-boxed li.is-active a, .hero.is-danger .tabs.is-boxed li.is-active a:hover, .hero.is-danger .tabs.is-toggle li.is-active a, .hero.is-danger .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #da0b00; } - .hero.is-danger.is-bold { - background-image: linear-gradient(141deg, #a70013 0%, #da0b00 71%, #f43500 100%); } - @media screen and (max-width: 768px) { - .hero.is-danger.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #a70013 0%, #da0b00 71%, #f43500 100%); } } - .hero.is-small .hero-body, #documenter .docs-sidebar form.docs-search > input.hero .hero-body { - padding-bottom: 1.5rem; - padding-top: 1.5rem; } - @media screen and (min-width: 769px), print { - .hero.is-medium .hero-body { - padding-bottom: 9rem; - padding-top: 9rem; } } - @media screen and (min-width: 769px), print { - .hero.is-large .hero-body { - padding-bottom: 18rem; - padding-top: 18rem; } } - .hero.is-halfheight .hero-body, .hero.is-fullheight .hero-body, .hero.is-fullheight-with-navbar .hero-body { - align-items: center; - display: flex; } - .hero.is-halfheight .hero-body > .container, .hero.is-fullheight .hero-body > .container, .hero.is-fullheight-with-navbar .hero-body > .container { - flex-grow: 1; - flex-shrink: 1; } - .hero.is-halfheight { - min-height: 50vh; } - .hero.is-fullheight { - min-height: 100vh; } - -.hero-video { - overflow: hidden; } - .hero-video video { - left: 50%; - min-height: 100%; - min-width: 100%; - position: absolute; - top: 50%; - transform: translate3d(-50%, -50%, 0); } - .hero-video.is-transparent { - opacity: 0.3; } - @media screen and (max-width: 768px) { - .hero-video { - display: none; } } -.hero-buttons { - margin-top: 1.5rem; } - @media screen and (max-width: 768px) { - .hero-buttons .button { - display: flex; } - .hero-buttons .button:not(:last-child) { - margin-bottom: 0.75rem; } } - @media screen and (min-width: 769px), print { - .hero-buttons { - display: flex; - justify-content: center; } - .hero-buttons .button:not(:last-child) { - margin-right: 1.5rem; } } -.hero-head, -.hero-foot { - flex-grow: 0; - flex-shrink: 0; } - -.hero-body { - flex-grow: 1; - flex-shrink: 0; - padding: 3rem 1.5rem; } - -.section { - padding: 3rem 1.5rem; } - @media screen and (min-width: 1056px) { - .section.is-medium { - padding: 9rem 1.5rem; } - .section.is-large { - padding: 18rem 1.5rem; } } -.footer { - background-color: #fafafa; - padding: 3rem 1.5rem 6rem; } - -h1 .docs-heading-anchor, h1 .docs-heading-anchor:hover, h1 .docs-heading-anchor:visited, h2 .docs-heading-anchor, h2 .docs-heading-anchor:hover, h2 .docs-heading-anchor:visited, h3 .docs-heading-anchor, h3 .docs-heading-anchor:hover, h3 .docs-heading-anchor:visited, h4 .docs-heading-anchor, h4 .docs-heading-anchor:hover, h4 .docs-heading-anchor:visited, h5 .docs-heading-anchor, h5 .docs-heading-anchor:hover, h5 .docs-heading-anchor:visited, h6 .docs-heading-anchor, h6 .docs-heading-anchor:hover, h6 .docs-heading-anchor:visited { - color: #222222; } - -h1 .docs-heading-anchor-permalink, h2 .docs-heading-anchor-permalink, h3 .docs-heading-anchor-permalink, h4 .docs-heading-anchor-permalink, h5 .docs-heading-anchor-permalink, h6 .docs-heading-anchor-permalink { - visibility: hidden; - vertical-align: middle; - margin-left: 0.5em; - font-size: 0.7rem; } - h1 .docs-heading-anchor-permalink::before, h2 .docs-heading-anchor-permalink::before, h3 .docs-heading-anchor-permalink::before, h4 .docs-heading-anchor-permalink::before, h5 .docs-heading-anchor-permalink::before, h6 .docs-heading-anchor-permalink::before { - font-family: "Font Awesome 5 Free"; - font-weight: 900; - content: "\f0c1"; } - -h1:hover .docs-heading-anchor-permalink, h2:hover .docs-heading-anchor-permalink, h3:hover .docs-heading-anchor-permalink, h4:hover .docs-heading-anchor-permalink, h5:hover .docs-heading-anchor-permalink, h6:hover .docs-heading-anchor-permalink { - visibility: visible; } - -.docs-dark-only { - display: none !important; } - -pre { - position: relative; - overflow: hidden; } - pre code, pre code.hljs { - padding: 0 0.5rem !important; - overflow: auto; - display: block; } - pre code:first-of-type, pre code.hljs:first-of-type { - padding-top: 0.7rem !important; } - pre code:last-of-type, pre code.hljs:last-of-type { - padding-bottom: 0.7rem !important; } - pre .copy-button { - opacity: 0.2; - transition: opacity 0.2s; - position: absolute; - right: 0em; - top: 0em; - padding: 0.5em; - width: 2.5em; - height: 2.5em; - background: transparent; - border: none; - font-family: "Font Awesome 5 Free"; - color: #222222; - cursor: pointer; - text-align: center; } - pre .copy-button:focus, pre .copy-button:hover { - opacity: 1; - background: rgba(34, 34, 34, 0.1); - color: #2e63b8; } - pre .copy-button.success { - color: #259a12; - opacity: 1; } - pre .copy-button.error { - color: #cb3c33; - opacity: 1; } - pre:hover .copy-button { - opacity: 1; } - -.admonition { - background-color: #b5b5b5; - border-style: solid; - border-width: 1px; - border-color: #363636; - border-radius: 4px; - font-size: 1rem; } - .admonition strong { - color: currentColor; } - .admonition.is-small, #documenter .docs-sidebar form.docs-search > input.admonition { - font-size: 0.75rem; } - .admonition.is-medium { - font-size: 1.25rem; } - .admonition.is-large { - font-size: 1.5rem; } - .admonition.is-default { - background-color: #b5b5b5; - border-color: #363636; } - .admonition.is-default > .admonition-header { - background-color: #363636; - color: #fff; } - .admonition.is-default > .admonition-body { - color: #fff; } - .admonition.is-info { - background-color: #def0fc; - border-color: #209cee; } - .admonition.is-info > .admonition-header { - background-color: #209cee; - color: #fff; } - .admonition.is-info > .admonition-body { - color: rgba(0, 0, 0, 0.7); } - .admonition.is-success { - background-color: #bdf4d1; - border-color: #22c35b; } - .admonition.is-success > .admonition-header { - background-color: #22c35b; - color: #fff; } - .admonition.is-success > .admonition-body { - color: rgba(0, 0, 0, 0.7); } - .admonition.is-warning { - background-color: #fff3c5; - border-color: #ffdd57; } - .admonition.is-warning > .admonition-header { - background-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .admonition.is-warning > .admonition-body { - color: rgba(0, 0, 0, 0.7); } - .admonition.is-danger { - background-color: #ffaba7; - border-color: #da0b00; } - .admonition.is-danger > .admonition-header { - background-color: #da0b00; - color: #fff; } - .admonition.is-danger > .admonition-body { - color: rgba(0, 0, 0, 0.7); } - .admonition.is-compat { - background-color: #bdeff5; - border-color: #1db5c9; } - .admonition.is-compat > .admonition-header { - background-color: #1db5c9; - color: #fff; } - .admonition.is-compat > .admonition-body { - color: rgba(0, 0, 0, 0.7); } - -.admonition-header { - color: #fff; - background-color: #363636; - align-items: center; - font-weight: 700; - justify-content: space-between; - line-height: 1.25; - padding: 0.75em; - position: relative; } - .admonition-header:before { - font-family: "Font Awesome 5 Free"; - font-weight: 900; - margin-right: 0.75em; - content: "\f06a"; } - -.admonition-body { - color: #222222; - padding: 1em 1.25em; } - .admonition-body pre { - background-color: whitesmoke; } - .admonition-body code { - background-color: rgba(0, 0, 0, 0.05); } - -.docstring { - margin-bottom: 1em; - background-color: transparent; - border: 1px solid #dbdbdb; - box-shadow: 2px 2px 3px rgba(10, 10, 10, 0.1); - max-width: 100%; } - .docstring > header { - display: flex; - flex-grow: 1; - align-items: stretch; - padding: 0.75rem; - background-color: whitesmoke; - box-shadow: 0 1px 2px rgba(10, 10, 10, 0.1); - box-shadow: none; - border-bottom: 1px solid #dbdbdb; } - .docstring > header code { - background-color: transparent; } - .docstring > header .docstring-binding { - margin-right: 0.3em; } - .docstring > header .docstring-category { - margin-left: 0.3em; } - .docstring > section { - position: relative; - padding: 1rem 1.25rem; - border-bottom: 1px solid #dbdbdb; } - .docstring > section:last-child { - border-bottom: none; } - .docstring > section > a.docs-sourcelink { - transition: opacity 0.3s; - opacity: 0; - position: absolute; - right: 0.625rem; - bottom: 0.5rem; } - .docstring > section > a.docs-sourcelink:focus { - opacity: 1 !important; } - .docstring:hover > section > a.docs-sourcelink { - opacity: 0.2; } - .docstring:focus-within > section > a.docs-sourcelink { - opacity: 0.2; } - .docstring > section:hover a.docs-sourcelink { - opacity: 1; } - -.documenter-example-output { - background-color: white; } - -.outdated-warning-overlay { - position: fixed; - top: 0; - left: 0; - right: 0; - box-shadow: 0 0 10px rgba(0, 0, 0, 0.3); - z-index: 999; - background-color: #ffaba7; - color: rgba(0, 0, 0, 0.7); - border-bottom: 3px solid #da0b00; - padding: 10px 35px; - text-align: center; - font-size: 15px; } - .outdated-warning-overlay .outdated-warning-closer { - position: absolute; - top: calc(50% - 10px); - right: 18px; - cursor: pointer; - width: 12px; } - .outdated-warning-overlay a { - color: #2e63b8; } - .outdated-warning-overlay a:hover { - color: #363636; } - -.content pre { - border: 1px solid #dbdbdb; } - -.content code { - font-weight: inherit; } - -.content a code { - color: #2e63b8; } - -.content h1 code, .content h2 code, .content h3 code, .content h4 code, .content h5 code, .content h6 code { - color: #222222; } - -.content table { - display: block; - width: initial; - max-width: 100%; - overflow-x: auto; } - -.content blockquote > ul:first-child, .content blockquote > ol:first-child, .content .admonition-body > ul:first-child, .content .admonition-body > ol:first-child { - margin-top: 0; } - -pre, code { - font-variant-ligatures: no-contextual; } - -.breadcrumb a.is-disabled { - cursor: default; - pointer-events: none; } - .breadcrumb a.is-disabled, .breadcrumb a.is-disabled:hover { - color: #222222; } - -.hljs { - background: initial !important; } - -.katex .katex-mathml { - top: 0; - right: 0; } - -.katex-display, mjx-container, .MathJax_Display { - margin: 0.5em 0 !important; } - -html { - -moz-osx-font-smoothing: auto; - -webkit-font-smoothing: auto; } - -li.no-marker { - list-style: none; } - -/* This file contain the overall layout. - * - * The main container is
    that is identified by id #documenter. - */ -#documenter .docs-main > article { - overflow-wrap: break-word; } - #documenter .docs-main > article .math-container { - overflow-x: auto; - overflow-y: hidden; } - -@media screen and (min-width: 1056px) { - #documenter .docs-main { - max-width: 52rem; - margin-left: 20rem; - padding-right: 1rem; } } - -@media screen and (max-width: 1055px) { - #documenter .docs-main { - width: 100%; } - #documenter .docs-main > article { - max-width: 52rem; - margin-left: auto; - margin-right: auto; - margin-bottom: 1rem; - padding: 0 1rem; } - #documenter .docs-main > header, #documenter .docs-main > nav { - max-width: 100%; - width: 100%; - margin: 0; } } - -#documenter .docs-main header.docs-navbar { - background-color: white; - border-bottom: 1px solid #dbdbdb; - z-index: 2; - min-height: 4rem; - margin-bottom: 1rem; - display: flex; } - #documenter .docs-main header.docs-navbar .breadcrumb { - flex-grow: 1; } - #documenter .docs-main header.docs-navbar .docs-right { - display: flex; - white-space: nowrap; } - #documenter .docs-main header.docs-navbar .docs-right .docs-icon, #documenter .docs-main header.docs-navbar .docs-right .docs-label, #documenter .docs-main header.docs-navbar .docs-right .docs-sidebar-button { - display: inline-block; } - #documenter .docs-main header.docs-navbar .docs-right .docs-label { - padding: 0; - margin-left: 0.3em; } - #documenter .docs-main header.docs-navbar .docs-right .docs-settings-button { - margin: auto 0 auto 1rem; } - #documenter .docs-main header.docs-navbar .docs-right .docs-sidebar-button { - font-size: 1.5rem; - margin: auto 0 auto 1rem; } - #documenter .docs-main header.docs-navbar > * { - margin: auto 0; } - @media screen and (max-width: 1055px) { - #documenter .docs-main header.docs-navbar { - position: sticky; - top: 0; - padding: 0 1rem; - /* For Headroom.js */ - transition-property: top, box-shadow; - -webkit-transition-property: top, box-shadow; - /* Safari */ - transition-duration: 0.3s; - -webkit-transition-duration: 0.3s; - /* Safari */ } - #documenter .docs-main header.docs-navbar.headroom--not-top { - box-shadow: 0.2rem 0rem 0.4rem #bbb; - transition-duration: 0.7s; - -webkit-transition-duration: 0.7s; - /* Safari */ } - #documenter .docs-main header.docs-navbar.headroom--unpinned.headroom--not-top.headroom--not-bottom { - top: -4.5rem; - transition-duration: 0.7s; - -webkit-transition-duration: 0.7s; - /* Safari */ } } -#documenter .docs-main section.footnotes { - border-top: 1px solid #dbdbdb; } - #documenter .docs-main section.footnotes li .tag:first-child, #documenter .docs-main section.footnotes li .docstring > section > a.docs-sourcelink:first-child, #documenter .docs-main section.footnotes li .content kbd:first-child, .content #documenter .docs-main section.footnotes li kbd:first-child { - margin-right: 1em; - margin-bottom: 0.4em; } - -#documenter .docs-main .docs-footer { - display: flex; - flex-wrap: wrap; - margin-left: 0; - margin-right: 0; - border-top: 1px solid #dbdbdb; - padding-top: 1rem; - padding-bottom: 1rem; } - @media screen and (max-width: 1055px) { - #documenter .docs-main .docs-footer { - padding-left: 1rem; - padding-right: 1rem; } } - #documenter .docs-main .docs-footer .docs-footer-nextpage, #documenter .docs-main .docs-footer .docs-footer-prevpage { - flex-grow: 1; } - #documenter .docs-main .docs-footer .docs-footer-nextpage { - text-align: right; } - #documenter .docs-main .docs-footer .flexbox-break { - flex-basis: 100%; - height: 0; } - #documenter .docs-main .docs-footer .footer-message { - font-size: 0.8em; - margin: 0.5em auto 0 auto; - text-align: center; } - -#documenter .docs-sidebar { - display: flex; - flex-direction: column; - color: #0a0a0a; - background-color: whitesmoke; - border-right: 1px solid #dbdbdb; - padding: 0; - flex: 0 0 18rem; - z-index: 5; - font-size: 1rem; - position: fixed; - left: -18rem; - width: 18rem; - height: 100%; - transition: left 0.3s; - /* Setting up a nicer theme style for the scrollbar */ } - #documenter .docs-sidebar.visible { - left: 0; - box-shadow: 0.4rem 0rem 0.8rem #bbb; } - @media screen and (min-width: 1056px) { - #documenter .docs-sidebar.visible { - box-shadow: none; } } - @media screen and (min-width: 1056px) { - #documenter .docs-sidebar { - left: 0; - top: 0; } } - #documenter .docs-sidebar .docs-logo { - margin-top: 1rem; - padding: 0 1rem; } - #documenter .docs-sidebar .docs-logo > img { - max-height: 6rem; - margin: auto; } - #documenter .docs-sidebar .docs-package-name { - flex-shrink: 0; - font-size: 1.5rem; - font-weight: 700; - text-align: center; - white-space: nowrap; - overflow: hidden; - padding: 0.5rem 0; } - #documenter .docs-sidebar .docs-package-name .docs-autofit { - max-width: 16.2rem; } - #documenter .docs-sidebar .docs-package-name a, #documenter .docs-sidebar .docs-package-name a:hover { - color: #0a0a0a; } - #documenter .docs-sidebar .docs-version-selector { - border-top: 1px solid #dbdbdb; - display: none; - padding: 0.5rem; } - #documenter .docs-sidebar .docs-version-selector.visible { - display: flex; } - #documenter .docs-sidebar ul.docs-menu { - flex-grow: 1; - user-select: none; - border-top: 1px solid #dbdbdb; - padding-bottom: 1.5rem; - /* Managing collapsible submenus */ } - #documenter .docs-sidebar ul.docs-menu > li > .tocitem { - font-weight: bold; } - #documenter .docs-sidebar ul.docs-menu > li li { - font-size: 0.95rem; - margin-left: 1em; - border-left: 1px solid #dbdbdb; } - #documenter .docs-sidebar ul.docs-menu input.collapse-toggle { - display: none; } - #documenter .docs-sidebar ul.docs-menu ul.collapsed { - display: none; } - #documenter .docs-sidebar ul.docs-menu input:checked ~ ul.collapsed { - display: block; } - #documenter .docs-sidebar ul.docs-menu label.tocitem { - display: flex; } - #documenter .docs-sidebar ul.docs-menu label.tocitem .docs-label { - flex-grow: 2; } - #documenter .docs-sidebar ul.docs-menu label.tocitem .docs-chevron { - display: inline-block; - font-style: normal; - font-variant: normal; - text-rendering: auto; - line-height: 1; - font-size: 0.75rem; - margin-left: 1rem; - margin-top: auto; - margin-bottom: auto; } - #documenter .docs-sidebar ul.docs-menu label.tocitem .docs-chevron::before { - font-family: "Font Awesome 5 Free"; - font-weight: 900; - content: "\f054"; } - #documenter .docs-sidebar ul.docs-menu input:checked ~ label.tocitem .docs-chevron::before { - content: "\f078"; } - #documenter .docs-sidebar ul.docs-menu .tocitem { - display: block; - padding: 0.5rem 0.5rem; } - #documenter .docs-sidebar ul.docs-menu .tocitem, #documenter .docs-sidebar ul.docs-menu .tocitem:hover { - color: #0a0a0a; - background: whitesmoke; } - #documenter .docs-sidebar ul.docs-menu a.tocitem:hover, #documenter .docs-sidebar ul.docs-menu label.tocitem:hover { - color: #0a0a0a; - background-color: #ebebeb; } - #documenter .docs-sidebar ul.docs-menu li.is-active { - border-top: 1px solid #dbdbdb; - border-bottom: 1px solid #dbdbdb; - background-color: white; } - #documenter .docs-sidebar ul.docs-menu li.is-active .tocitem, #documenter .docs-sidebar ul.docs-menu li.is-active .tocitem:hover { - background-color: white; 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- - // Current version is not a version number, so we can't tell if it's the newest version. Abort. - if (!/v(\d+\.)*\d+/.test(window.DOCUMENTER_CURRENT_VERSION)) { - return - }; - - // Current version is newest version, so no need to add a warning. - if (window.DOCUMENTER_NEWEST === window.DOCUMENTER_CURRENT_VERSION) { - return - }; - - // Add a noindex meta tag (unless one exists) so that search engines don't index this version of the docs. - if (document.body.querySelector('meta[name="robots"]') === null) { - const meta = document.createElement('meta'); - meta.name = 'robots'; - meta.content = 'noindex'; - - document.getElementsByTagName('head')[0].appendChild(meta); - }; - - const div = document.createElement('div'); - div.classList.add('outdated-warning-overlay'); - const closer = document.createElement('button'); - closer.classList.add('outdated-warning-closer', 'delete'); - closer.addEventListener('click', function () { - document.body.removeChild(div); - }); - const href = window.documenterBaseURL + '/../' + window.DOCUMENTER_STABLE; - div.innerHTML = 'This documentation is not for the latest version.
    Go to the latest documentation.'; - div.appendChild(closer); - document.body.appendChild(div); -}; - -if (document.readyState === 'loading') { - document.addEventListener('DOMContentLoaded', maybeAddWarning); -} else { - maybeAddWarning(); -}; diff --git a/previews/PR40/benchmarks/index.html b/previews/PR40/benchmarks/index.html deleted file mode 100644 index 3c995350..00000000 --- a/previews/PR40/benchmarks/index.html +++ /dev/null @@ -1,12 +0,0 @@ - -Benchmarks · PencilFFTs.jl
    Edit on GitHub

    Benchmarks

    The performance of PencilFFTs.jl is comparable to that of other open-source parallel FFT libraries implemented in lower-level languages. Below, we show comparisons with the Fortran implementation of P3DFFT, possibly the most popular of these libraries. The benchmarks were performed on the Jean–Zay cluster of the IDRIS French computing centre (CNRS).

    The figure below shows strong scaling benchmarks of 3D real-to-complex FFTs using 2D ("pencil") decomposition. The benchmarks were run for input arrays of dimensions $N_x × N_y × N_z = 512^3$, $1024^3$ and $2048^3$. Each timing is averaged over 100 repetitions.

    - - Strong scaling of PencilFFTs -

    As seen above, PencilFFTs generally outperforms P3DFFT in its default setting. This is largely explained by the choice of using non-blocking point-to-point MPI communications (via MPI_Isend and MPI_Irecv), while P3DFFT uses collective MPI_Alltoallv calls. This enables PencilFFTs to perform data reordering operations on the partially received data while waiting for the incoming data, leading to better performance. Moreover, in contrast with P3DFFT, the high performance and scalability of PencilFFTs results from a highly generic code, handling decompositions in arbitrary dimensions and a relatively large (and extensible) variety of transformations.

    Note that PencilFFTs can optionally use collective communications (using MPI_Alltoallv) instead of point-to-point communications. For details, see the docs for PencilFFTPlan and for PencilArray transpositions. As seen above, collective communications generally perform worse than point-to-point ones, and runtimes are nearly indistinguishable from those of P3DFFT.

    Benchmark details

    The benchmarks were performed using Julia 1.7-beta3 and Intel MPI 2019. We used PencilFFTs v0.12.5 with FFTW.jl v1.4.3 and MPI.jl v0.19.0. We used the Fortran implementation of P3DFFT, version 2.7.6, which was built with Intel 2019 compilers and linked to FFTW 3.3.8. The cluster where the benchmarks were run has Intel Cascade Lake 6248 processors with 2×20 cores per node.

    The number of MPI processes along each decomposed dimension, $P_1$ and $P_2$, was automatically determined by a call to MPI_Dims_create, which tends to create a balanced decomposition with $P_1 ≈ P_2$. For instance, a total of 1024 processes is divided into $P_1 = P_2 = 32$. Different results may be obtained with other combinations, but this was not benchmarked.

    The source files used to generate this benchmark, as well as the raw benchmark results, are all available in the PencilFFTs repo.

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29% 18% 17% 5% 6% 5% 9% 4% 8% -[libx264 @ 0x55781631a540] i8c dc,h,v,p: 70% 13% 12% 5% -[libx264 @ 0x55781631a540] Weighted P-Frames: Y:0.0% UV:0.0% -[libx264 @ 0x55781631a540] ref P L0: 69.2% 15.2% 10.6% 4.9% -[libx264 @ 0x55781631a540] ref B L0: 85.9% 11.3% 2.7% -[libx264 @ 0x55781631a540] ref B L1: 98.1% 1.9% -[libx264 @ 0x55781631a540] kb/s:594.18 diff --git a/previews/PR40/generated/gradient/index.html b/previews/PR40/generated/gradient/index.html deleted file mode 100644 index cd4e5290..00000000 --- a/previews/PR40/generated/gradient/index.html +++ /dev/null @@ -1,112 +0,0 @@ - -Gradient of a scalar field · PencilFFTs.jl
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    Gradient of a scalar field

    This example shows different methods to compute the gradient of a real-valued 3D scalar field $θ(\bm{x})$ in Fourier space, where $\bm{x} = (x, y, z)$. It is assumed that the field is periodic with period $L = 2π$ along all dimensions.

    General procedure

    The discrete Fourier expansion of $θ$ writes

    \[θ(\bm{x}) = ∑_{\bm{k} ∈ \Z^3} \hat{θ}(\bm{k}) \, e^{i \bm{k} ⋅ \bm{x}},\]

    where $\bm{k} = (k_x, k_y, k_z)$ are the Fourier wave numbers and $\hat{θ}$ is the discrete Fourier transform of $θ$. Then, the spatial derivatives of $θ$ are given by

    \[\frac{∂ θ(\bm{x})}{∂ x_i} = -∑_{\bm{k} ∈ \Z^3} i k_i \hat{θ}(\bm{k}) \, e^{i \bm{k} ⋅ \bm{x}},\]

    where the subscript $i$ denotes one of the spatial components $x$, $y$ or $z$.

    In other words, to compute $\bm{∇} θ = (∂_x θ, ∂_y θ, ∂_z θ)$, one has to:

    1. transform $θ$ to Fourier space to obtain $\hat{θ}$,
    2. multiply $\hat{θ}$ by $i \bm{k}$,
    3. transform the result back to physical space to obtain $\bm{∇} θ$.

    Preparation

    In this section, we initialise a random real-valued scalar field $θ$ and compute its FFT. For more details see the Tutorial.

    using MPI
-using PencilFFTs
-using Random
-
-MPI.Init()
-
-# Input data dimensions (Nx × Ny × Nz)
-dims = (64, 32, 64)
-
-# Apply a 3D real-to-complex (r2c) FFT.
-transform = Transforms.RFFT()
-
-# Automatically create decomposition configuration
-comm = MPI.COMM_WORLD
-pen = Pencil(dims, comm)
-
-# Create plan
-plan = PencilFFTPlan(pen, transform)
-
-# Allocate data and initialise field
-θ = allocate_input(plan)
-randn!(θ)
-
-# Perform distributed FFT
-θ_hat = plan * θ

    Finally, we initialise the output that will hold ∇θ in Fourier space. Noting that ∇θ is a vector field, we choose to store it as a tuple of 3 PencilArrays.

    ∇θ_hat = allocate_output(plan, Val(3))
-
-# This is equivalent:
-# ∇θ_hat = ntuple(d -> similar(θ_hat), Val(3))
-
-summary(∇θ_hat)
-
-# Fourier wave numbers
    "Tuple{PencilArray{ComplexF64, 3, Array{ComplexF64, 3}, 3, 0, Pencil{3, 2, Permutation{(3, 2, 1), 3}, Vector{UInt8}}}, PencilArray{ComplexF64, 3, Array{ComplexF64, 3}, 3, 0, Pencil{3, 2, Permutation{(3, 2, 1), 3}, Vector{UInt8}}}, PencilArray{ComplexF64, 3, Array{ComplexF64, 3}, 3, 0, Pencil{3, 2, Permutation{(3, 2, 1), 3}, Vector{UInt8}}}}"

    In general, the Fourier wave numbers are of the form $k_i = 0, ±\frac{2π}{L_i}, ±\frac{4π}{L_i}, ±\frac{6π}{L_i}, …$, where $L_i$ is the period along dimension $i$. When a real-to-complex Fourier transform is applied, roughly half of these wave numbers are redundant due to the Hermitian symmetry of the complex Fourier coefficients. In practice, this means that for the fastest dimension $x$ (along which a real-to-complex transform is performed), the negative wave numbers are dropped, i.e. $k_x = 0, \frac{2π}{L_x}, \frac{4π}{L_x}, …$.

    The AbstractFFTs package provides a convenient way to generate the Fourier wave numbers, using the functions fftfreq and rfftfreq. We can use these functions to initialise a "grid" of wave numbers associated to our 3D real-to-complex transform:

    using AbstractFFTs: fftfreq, rfftfreq
-
-box_size = (2π, 2π, 2π)  # Lx, Ly, Lz
-sample_rate = 2π .* dims ./ box_size
-
-# In our case (Lx = 2π and Nx even), this gives kx = [0, 1, 2, ..., Nx/2].
-kx = rfftfreq(dims[1], sample_rate[1])
-
-# In our case (Ly = 2π and Ny even), this gives
-# ky = [0, 1, 2, ..., Ny/2-1, -Ny/2, -Ny/2+1, ..., -1] (and similarly for kz).
-ky = fftfreq(dims[2], sample_rate[2])
-kz = fftfreq(dims[3], sample_rate[3])
-
-kvec = (kx, ky, kz)
    ([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0  …  23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, 32.0], [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0  …  -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0], [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0  …  -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0])

    Note that kvec now contains the wave numbers associated to the global domain. In the following, we will only need the wave numbers associated to the portion of the domain handled by the local MPI process.

    Method 1: global views

    PencilArrays, returned for instance by allocate_input and allocate_output, take indices that start at 1, regardless of the location of the subdomain associated to the local process on the global grid. (In other words, PencilArrays take local indices.) On the other hand, we have defined the wave number vector kvec which, for each MPI process, is defined over the global domain, and as such it takes global indices.

    One straightforward way of making data arrays compatible with wave numbers is to use global views, i.e. arrays that take global indices. These are generated from PencilArrays by calling the global_view function. Note that, in general, global indices do not start at 1 for a given MPI process. A given process will own a range of data given by indices in (i1:i2, j1:j2, k1:k2).

    θ_glob = global_view(θ_hat)
-∇θ_glob = global_view.(∇θ_hat)
-summary(θ_glob)
    "16×32×64 OffsetArray(::PencilArray{ComplexF64, 3}, 1:16, 1:32, 1:64) with eltype ComplexF64 with indices 1:16×1:32×1:64"

    Once we have global views, we can combine data and wave numbers using the portion of global indices owned by the local MPI process, as shown below. We can use CartesianIndices to iterate over the global indices associated to the local process.

    for I in CartesianIndices(θ_glob)
-    i, j, k = Tuple(I)  # unpack indices
-
-    # Wave number vector associated to current Cartesian index.
-    kx = kvec[1][i]
-    ky = kvec[2][j]
-    kz = kvec[3][k]
-
-    # Compute gradient in Fourier space.
-    # Note that modifying ∇θ_glob also modifies the original PencilArray ∇θ_hat.
-    ∇θ_glob[1][I] = im * kx * θ_glob[I]
-    ∇θ_glob[2][I] = im * ky * θ_glob[I]
-    ∇θ_glob[3][I] = im * kz * θ_glob[I]
-end

    The above loop can be written in a slightly more efficient manner by precomputing im * θ_glob[I]:

    @inbounds for I in CartesianIndices(θ_glob)
-    i, j, k = Tuple(I)
-
-    kx = kvec[1][i]
-    ky = kvec[2][j]
-    kz = kvec[3][k]
-
-    u = im * θ_glob[I]
-
-    ∇θ_glob[1][I] = kx * u
-    ∇θ_glob[2][I] = ky * u
-    ∇θ_glob[3][I] = kz * u
-end

    Also note that the above can be easily written in a more generic way, e.g. for arbitrary dimensions, thanks in part to the use of CartesianIndices. Moreover, in the above there is no notion of the dimension permutations discussed in the tutorial, as it is all hidden behind the implementation of PencilArrays. And as seen later in the benchmarks, these (hidden) permutations have zero cost, as the speed is identical to that of a function that explicitly takes into account these permutations.

    Finally, we can perform a backwards transform to obtain $\bm{∇} θ$ in physical space:

    ∇θ = plan \ ∇θ_hat;

    Note that the transform is automatically broadcast over the three fields of the ∇θ_hat vector, and the result ∇θ is also a tuple of three PencilArrays.

    Method 2: explicit global indexing

    Sometimes, one does not need to write generic code. In our case, one often knows the dimensionality of the problem and the memory layout of the data (i.e. the underlying index permutation).

    Below is a reimplementation of the above loop, using explicit indices instead of CartesianIndices, and assuming that the underlying index permutation is (3, 2, 1), that is, data is stored in $(z, y, x)$ order. As discussed in the tutorial, this is the default for transformed arrays. This example also serves as a more explicit explanation for what is going on in the first method.

    # Get local data range in the global grid.
-rng = axes(θ_glob)  # = (i1:i2, j1:j2, k1:k2)
    (OffsetArrays.IdOffsetRange(values=1:16, indices=1:16), OffsetArrays.IdOffsetRange(values=1:32, indices=1:32), OffsetArrays.IdOffsetRange(values=1:64, indices=1:64))

    For the loop below, we're assuming that the permutation is (3, 2, 1). In other words, the fastest index is the last one, and not the first one as it is usually in Julia. If the permutation is not (3, 2, 1), things will still work (well, except for the assertion below!), but the loop order will not be optimal.

    @assert permutation(θ_hat) === Permutation(3, 2, 1)
-
-@inbounds for i in rng[1], j in rng[2], k in rng[3]
-    kx = kvec[1][i]
-    ky = kvec[2][j]
-    kz = kvec[3][k]
-
-    # Note that we still access the arrays in (i, j, k) order.
-    # (The permutation happens behind the scenes!)
-    u = im * θ_glob[i, j, k]
-
-    ∇θ_glob[1][i, j, k] = kx * u
-    ∇θ_glob[2][i, j, k] = ky * u
-    ∇θ_glob[3][i, j, k] = kz * u
-end

    Method 3: using local indices

    Alternatively, we can avoid global views and work directly on PencilArrays using local indices that start at 1. In this case, part of the strategy is to construct a "local" grid of wave numbers that can also be accessed with local indices. This can be conveniently done using the localgrid function of the PencilArrays.jl package, which accepts a PencilArray (or its associated Pencil) and the global coordinates (here kvec):

    grid_fourier = localgrid(θ_hat, kvec)
    LocalRectilinearGrid{3} with Permutation(3, 2, 1) and coordinates:
- (1) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0]
- (2) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]
- (3) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]

    Note that one can directly iterate on the returned grid object:

    @inbounds for I in CartesianIndices(grid_fourier)
-    # Wave number vector associated to current Cartesian index.
-    k⃗ = grid_fourier[I]
-    u = im * θ_hat[I]
-    ∇θ_hat[1][I] = k⃗[1] * u
-    ∇θ_hat[2][I] = k⃗[2] * u
-    ∇θ_hat[3][I] = k⃗[3] * u
-end

    This implementation is as efficient as the other examples, while being slightly shorter to write. Moreover, it is quite generic, and can be made independent of the number of dimensions with little effort.

    Method 4: using broadcasting

    Finally, note that the local grid object returned by localgrid makes it is possible to compute the gradient using broadcasting, thus fully avoiding scalar indexing. This can be quite convenient in some cases, and can also be very useful if one is working on GPUs (where scalar indexing is prohibitively expensive). Using broadcasting, the above examples simply become:

    @. ∇θ_hat[1] = im * grid_fourier[1] * θ_hat
-@. ∇θ_hat[2] = im * grid_fourier[2] * θ_hat
-@. ∇θ_hat[3] = im * grid_fourier[3] * θ_hat

    Once again, as shown in the benchmarks further below, this method performs quite similarly to the other ones.

    Summary

    The PencilArrays module provides different alternatives to deal with MPI-distributed data that may be subject to dimension permutations. In particular, one can choose to work with global indices (first two examples), with local indices (third example), or to avoid scalar indexing altogether (fourth example).

    If one wants to stay generic, making sure that the same code will work for arbitrary dimensions and will be efficient regardless of the underlying dimension permutation, methods 1, 3 or 4 should be preferred. These use CartesianIndices and make no assumptions on possible dimension permutations, which are by default enabled in the output of PencilFFTs transforms. In fact, such permutations are completely invisible in the implementations.

    The second method uses explicit (i, j, k) indices. It assumes that the underlying permutation is (3, 2, 1) to loop with i as the slowest index and k as the fastest, which is the optimal order in this case given the permutation. As such, the implementation is less generic than the others, and differences in performance are negligible with respect to more generic variants.

    Benchmark results

    The following are the benchmark results obtained from running examples/gradient.jl on a laptop, using 2 MPI processes and Julia 1.7.2, with an input array of global dimensions $64 × 32 × 64$. The different methods detailed above are marked on the right. The "lazy" marks indicate runs where the wave numbers were represented by lazy Frequencies objects (returned by rfftfreq and fftfreq). Otherwise, they were collected into Vectors. For some reason, plain Vectors are faster when working with grids generated by localgrid.

    In the script, additional implementations can be found which rely on a more advanced understanding of permutations and on the internals of the PencilArrays package. For instance, gradient_local_parent! directly works with the raw data stored in Julia Arrays, while gradient_local_linear! completely avoids CartesianIndices while staying generic and efficient. Nevertheless, these display roughly the same performance as the above examples.

        gradient_global_view!...                  89.900 μs
-    gradient_global_view! (lazy)...           92.060 μs  [Method 1]
-    gradient_global_view_explicit!...         88.958 μs
-    gradient_global_view_explicit! (lazy)...  81.055 μs  [Method 2]
-    gradient_local!...                        92.305 μs
-    gradient_grid!...                         92.770 μs
-    gradient_grid! (lazy)...                  101.388 μs  [Method 3]
-    gradient_grid_broadcast!...               88.606 μs
-    gradient_grid_broadcast! (lazy)...        151.020 μs  [Method 4]
-    gradient_local_parent!...                 92.248 μs
-    gradient_local_linear!...                 91.212 μs
-    gradient_local_linear_explicit!...        90.992 μs

    This page was generated using Literate.jl.

    diff --git a/previews/PR40/generated/navier_stokes/index.html b/previews/PR40/generated/navier_stokes/index.html deleted file mode 100644 index 0af0dd50..00000000 --- a/previews/PR40/generated/navier_stokes/index.html +++ /dev/null @@ -1,303 +0,0 @@ - -Navier–Stokes equations · PencilFFTs.jl
    Edit on GitHub

    Navier–Stokes equations

    In this example, we numerically solve the incompressible Navier–Stokes equations

    \[∂_t \bm{v} + (\bm{v} ⋅ \bm{∇}) \bm{v} = -\frac{1}{ρ} \bm{∇} p + ν ∇^2 \bm{v}, -\quad \bm{∇} ⋅ \bm{v} = 0,\]

    where $\bm{v}(\bm{x}, t)$ and $p(\bm{x}, t)$ are respectively the velocity and pressure fields, $ν$ is the fluid kinematic viscosity and $ρ$ is the fluid density.

    We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.

    First steps

    We start by loading the required packages, initialising MPI and setting the simulation parameters.

    using MPI
-using PencilFFTs
-
-MPI.Init()
-comm = MPI.COMM_WORLD
-procid = MPI.Comm_rank(comm) + 1
-
-# Simulation parameters
-Ns = (64, 64, 64)  # = (Nx, Ny, Nz)
-Ls = (2π, 2π, 2π)  # = (Lx, Ly, Lz)
-
-# Collocation points ("global" = over all processes).
-# We include the endpoint (length = N + 1) for convenience.
-xs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls)  # = (x, y, z)
    (0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586)

    Let's check the number of MPI processes over which we're running our simulation:

    MPI.Comm_size(comm)
    2

    We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:

    pen = Pencil(Ns, comm)
    Decomposition of 3D data
-    Data dimensions: (64, 64, 64)
-    Decomposed dimensions: (2, 3)
-    Data permutation: NoPermutation()
-    Array type: Array

    The subdomain associated to the local MPI process can be obtained using range_local:

    range_local(pen)
    (1:64, 1:32, 1:64)

    We now construct a distributed vector field that follows the decomposition configuration we just created:

    v⃗₀ = (
-    PencilArray{Float64}(undef, pen),  # vx
-    PencilArray{Float64}(undef, pen),  # vy
-    PencilArray{Float64}(undef, pen),  # vz
-)
-summary(v⃗₀[1])
    "64×32×64 PencilArray{Float64, 3}(::Pencil{3, 2, NoPermutation, Array})"

    We still need to fill this array with interesting values that represent a physical velocity field.

    Initial condition

    Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:

    \[\begin{aligned} -v_x(x, y, z) &= u₀ \sin(k₀ x) \cos(k₀ y) \cos(k₀ z) \\ -v_y(x, y, z) &= -u₀ \cos(k₀ x) \sin(k₀ y) \cos(k₀ z) \\ -v_z(x, y, z) &= 0 -\end{aligned}\]

    where $u₀$ and $k₀$ are two parameters setting the amplitude and the period of the velocity field.

    To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:

    grid = localgrid(pen, xs_global)
    LocalRectilinearGrid{3} with coordinates:
- (1) 0.0:0.09817477042468103:6.1850105367549055
- (2) 0.0:0.09817477042468103:3.043417883165112
- (3) 0.0:0.09817477042468103:6.1850105367549055

    We can use this to initialise the velocity field:

    u₀ = 1.0
-k₀ = 2π / Ls[1]  # should be integer if L = 2π (to preserve periodicity)
-
-@. v⃗₀[1] =  u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)
-@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)
-@. v⃗₀[3] =  0

    Let's plot a 2D slice of the velocity field managed by the local MPI process:

    using GLMakie
-
-# Compute the norm of a vector field represented by a tuple of arrays.
-function vecnorm(v⃗::NTuple)
-    vnorm = similar(v⃗[1])
-    for n ∈ eachindex(v⃗[1])
-        w = zero(eltype(vnorm))
-        for v ∈ v⃗
-            w += v[n]^2
-        end
-        vnorm[n] = sqrt(w)
-    end
-    vnorm
-end
-
-let fig = Figure(resolution = (700, 600))
-    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
-    vnorm = vecnorm(v⃗₀)
-    ct = contour!(
-        ax, grid.x, grid.y, grid.z, vnorm;
-        alpha = 0.2, levels = 4,
-        colormap = :viridis, colorrange = (0.0, 1.0),
-    )
-    cb = Colorbar(fig[1, 2], ct; label = "Velocity magnitude")
-    fig
-end

    Velocity in Fourier space

    In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series

    \[\bm{v}(\bm{x}, t) = -∑_{\bm{k}} \hat{\bm{v}}_{\bm{k}}(t) \, e^{i \bm{k} ⋅ \bm{x}},\]

    where $\bm{k} = (k_x, k_y, k_z)$ are the discrete wave numbers.

    The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for $k_x$:

    using AbstractFFTs: fftfreq, rfftfreq
-
-ks_global = (
-    rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]),  # kx | real-to-complex
-     fftfreq(Ns[2], 2π * Ns[2] / Ls[2]),  # ky | complex-to-complex
-     fftfreq(Ns[3], 2π * Ns[3] / Ls[3]),  # kz | complex-to-complex
-)
-
-ks_global[1]'
    1×33 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
- 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  27.0  28.0  29.0  30.0  31.0  32.0
    ks_global[2]'
    1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
- 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0
    ks_global[3]'
    1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
- 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0

    To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:

    plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())
    Transforms: (RFFT, FFT, FFT)
-Input type: Float64
-Global dimensions: (64, 64, 64) -> (33, 64, 64)
-MPI topology: 2D decomposition (2×1 processes)

    See PencilFFTPlan for details on creating plans and on optional keyword arguments.

    We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$:

    v̂s = plan .* v⃗₀
-summary(v̂s[1])
    "16×64×64 PencilArray{ComplexF64, 3}(::Pencil{3, 2, Permutation{(3, 2, 1), 3}, Array})"

    Note that, in Fourier space, the domain decomposition is performed along the directions $x$ and $y$:

    pencil(v̂s[1])
    Decomposition of 3D data
-    Data dimensions: (33, 64, 64)
-    Decomposed dimensions: (1, 2)
-    Data permutation: Permutation(3, 2, 1)
-    Array type: Array

    This is because the 3D FFTs are performed one dimension at a time, with the $x$ direction first and the $z$ direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.

    To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers $\bm{k}$ associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:

    grid_fourier = localgrid(v̂s[1], ks_global)
    LocalRectilinearGrid{3} with Permutation(3, 2, 1) and coordinates:
- (1) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0]
- (2) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]
- (3) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]

    As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, $\bm{ω} = \bm{∇} × \bm{v}$. In Fourier space, this becomes $\hat{\bm{ω}} = i \bm{k} × \hat{\bm{v}}$.

    using StaticArrays: SVector
-using LinearAlgebra: ×
-
-function curl_fourier!(
-        ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,
-    ) where {N}
-    @inbounds for I ∈ eachindex(grid_fourier)
-        # We use StaticArrays for the cross product between small vectors.
-        ik⃗ = im * SVector(grid_fourier[I])
-        v⃗ = SVector(getindex.(v̂s, Ref(I)))  # = (v̂s[1][I], v̂s[2][I], ...)
-        ω⃗ = ik⃗ × v⃗
-        for n ∈ eachindex(ω⃗)
-            ω̂s[n][I] = ω⃗[n]
-        end
-    end
-    ω̂s
-end
-
-ω̂s = similar.(v̂s)
-curl_fourier!(ω̂s, v̂s, grid_fourier);

    We finally transform back to physical space and plot the result:

    ωs = plan .\ ω̂s
-
-let fig = Figure(resolution = (700, 600))
-    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
-    ω_norm = vecnorm(ωs)
-    ct = contour!(
-        ax, grid.x, grid.y, grid.z, ω_norm;
-        alpha = 0.1, levels = 0.8:0.2:2.0,
-        colormap = :viridis, colorrange = (0.8, 2.0),
-    )
-    cb = Colorbar(fig[1, 2], ct; label = "Vorticity magnitude")
-    fig
-end

    Computing the non-linear term

    One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as

    \[∂_t \hat{\bm{v}}_{\bm{k}} = -- \mathcal{P}_{\bm{k}} \! \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right] -- ν |\bm{k}|^2 \hat{\bm{v}}_{\bm{k}} -\quad \text{ with } \quad -\mathcal{P}_{\bm{k}}(\hat{\bm{F}}_{\bm{k}}) = \left( I - \frac{\bm{k} ⊗ -\bm{k}}{|\bm{k}|^2} \right) \hat{\bm{F}}_{\bm{k}},\]

    where $\mathcal{P}_{\bm{k}}$ is a projection operator allowing to preserve the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.

    Now that we have the wave numbers $\bm{k}$, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$ of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, $\hat{\bm{F}}_{\bm{k}} = \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right]_{\bm{k}}$. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.

    Below we implement a function that computes the non-linear term in Fourier space based on its convective form $(\bm{v} ⋅ \bm{∇}) \bm{v} = \bm{∇} ⋅ (\bm{v} ⊗ \bm{v})$. Note that this equivalence uses the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$.

    using LinearAlgebra: mul!, ldiv!  # for applying FFT plans in-place
-
-# Compute non-linear term in Fourier space from velocity field in physical
-# space. Optional keyword arguments may be passed to avoid allocations.
-function ns_nonlinear!(
-        F̂s, vs, plan, grid_fourier;
-        vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),
-    )
-    # Compute F_i = ∂_j (v_i v_j) for each i.
-    # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)
-    w, ŵ = vbuf, v̂buf
-    @inbounds for (i, F̂i) ∈ enumerate(F̂s)
-        F̂i .= 0
-        vi = vs[i]
-        for (j, vj) ∈ enumerate(vs)
-            w .= vi .* vj     # w = v_i * v_j in physical space
-            mul!(ŵ, plan, w)  # same in Fourier space
-            # Add derivative in Fourier space
-            for I ∈ eachindex(grid_fourier)
-                k⃗ = grid_fourier[I]  # = (kx, ky, kz)
-                kj = k⃗[j]
-                F̂i[I] += im * kj * ŵ[I]
-            end
-        end
-    end
-    F̂s
-end
    ns_nonlinear! (generic function with 1 method)

    As an example, let's use this function on our initial velocity field:

    F̂s = similar.(v̂s)
-ns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);

    Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.

    function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)
-    ks_max = maximum.(abs, ks_global)  # maximum stored wave numbers (kx_max, ky_max, kz_max)
-    ks_lim = (2 / 3) .* ks_max
-    @inbounds for I ∈ eachindex(grid_fourier)
-        k⃗ = grid_fourier[I]
-        if any(abs.(k⃗) .> ks_lim)
-            for ŵ ∈ ŵs
-                ŵ[I] = 0
-            end
-        end
-    end
-    ŵs
-end
-
-# We can apply this on the previously computed non-linear term:
-dealias_twothirds!(F̂s, grid_fourier, ks_global);

    Finally, we implement the projection associated to the incompressibility condition:

    function project_divergence_free!(ûs, grid_fourier)
-    @inbounds for I ∈ eachindex(grid_fourier)
-        k⃗ = grid_fourier[I]
-        k² = sum(abs2, k⃗)
-        iszero(k²) && continue  # avoid division by zero
-        û = getindex.(ûs, Ref(I))  # (ûs[1][I], ûs[2][I], ...)
-        for i ∈ eachindex(û)
-            ŵ = û[i]
-            for j ∈ eachindex(û)
-                ŵ -= k⃗[i] * k⃗[j] * û[j] / k²
-            end
-            ûs[i][I] = ŵ
-        end
-    end
-    ûs
-end
    project_divergence_free! (generic function with 1 method)

    We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:

    v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)
-v̂s_proj .≈ v̂s  # the last one may be false because v_z = 0 initially
    (true, true, false)

    Putting it all together

    To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:

    function ns_rhs!(
-        dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,
-    ) where {N}
-    # 1. Compute non-linear term and dealias it
-    (; plan, cache, ks_global, grid_fourier) = p
-    F̂s = cache.F̂s
-    ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])
-    dealias_twothirds!(F̂s, grid_fourier, ks_global)
-
-    # 2. Project onto divergence-free space
-    project_divergence_free!(F̂s, grid_fourier)
-
-    # 3. Transform velocity to Fourier space
-    v̂s = cache.v̂s
-    map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)
-
-    # 4. Add viscous term (and multiply projected non-linear term by -1)
-    ν = p.ν
-    for n ∈ eachindex(v̂s)
-        v̂ = v̂s[n]
-        F̂ = F̂s[n]
-        @inbounds for I ∈ eachindex(grid_fourier)
-            k⃗ = grid_fourier[I]  # = (kx, ky, kz)
-            k² = sum(abs2, k⃗)
-            F̂[I] = -F̂[I] - ν * k² * v̂[I]
-        end
-    end
-
-    # 5. Transform RHS back to physical space
-    map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)
-
-    nothing
-end
    ns_rhs! (generic function with 1 method)

    For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.

    using OrdinaryDiffEq
-using RecursiveArrayTools: ArrayPartition
-
-ns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)
-vs_init_ode = ArrayPartition(v⃗₀)
-summary(vs_init_ode)
    "RecursiveArrayTools.ArrayPartition{Float64, Tuple{PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}}} with arrays:"

    We now define solver parameters and temporary variables, and initialise the problem:

    params = (;
-    ν = 5e-3,  # kinematic viscosity
-    plan, grid_fourier, ks_global,
-    cache = (
-        v̂s = similar.(v̂s),
-        F̂s = similar.(v̂s),
-    )
-)
-
-tspan = (0.0, 10.0)
-prob = ODEProblem(ns_rhs!, vs_init_ode, tspan, params)
-integrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);

    We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum $E(k)$, to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity $ν$ must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large $k$), while high enough to allow the small-scale motions to be correctly resolved.

    function energy_spectrum!(Ek, ks, v̂s, grid_fourier)
-    Nk = length(Ek)
-    @assert Nk == length(ks)
-    Ek .= 0
-    for I ∈ eachindex(grid_fourier)
-        k⃗ = grid_fourier[I]  # = (kx, ky, kz)
-        knorm = sqrt(sum(abs2, k⃗))
-        i = searchsortedfirst(ks, knorm)
-        i > Nk && continue
-        v⃗ = getindex.(v̂s, Ref(I))  # = (v̂s[1][I], v̂s[2][I], ...)
-        factor = k⃗[1] == 0 ? 1 : 2  # account for Hermitian symmetry and r2c transform
-        Ek[i] += factor * sum(abs2, v⃗) / 2
-    end
-    MPI.Allreduce!(Ek, +, get_comm(v̂s[1]))  # sum across all processes
-    Ek
-end
-
-ks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])
-Ek = similar(ks)
-v̂s = plan .* integrator.u.x
-energy_spectrum!(Ek, ks, v̂s, grid_fourier)
-Ek ./= scale_factor(plan)^2  # rescale energy
-
-curl_fourier!(ω̂s, v̂s, grid_fourier)
-ldiv!.(ωs, plan, ω̂s)
-ω⃗_plot = Observable(ωs)
-k_plot = @view ks[2:end]
-E_plot = Observable(@view Ek[2:end])
-t_plot = Observable(integrator.t)
-
-fig = let
-    fig = Figure(resolution = (1200, 600))
-    ax = Axis3(
-        fig[1, 1][1, 1]; title = @lift("t = $(round($t_plot, digits = 3))"),
-        aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z",
-    )
-    ω_mag = @lift vecnorm($ω⃗_plot)
-    ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)
-    ct = contour!(
-        ax, grid.x, grid.y, grid.z, ω_mag_norm;
-        alpha = 0.3, levels = 3,
-        colormap = :viridis, colorrange = (0.0, 1.0),
-    )
-    cb = Colorbar(fig[1, 1][1, 2], ct; label = "Normalised vorticity magnitude")
-    ax_sp = Axis(
-        fig[1, 2];
-        xlabel = "k", ylabel = "E(k)", xscale = log2, yscale = log10,
-        title = "Kinetic energy spectrum",
-    )
-    ylims!(ax_sp, 1e-8, 1e0)
-    scatterlines!(ax_sp, k_plot, E_plot)
-    ks_slope = exp.(range(log(2.5), log(25.0), length = 3))
-    E_fivethirds = @. 0.3 * ks_slope^(-5/3)
-    @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)
-    text!(ax_sp, L"k^{-5/3}"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))
-    fig
-end
-
-record(fig, "vorticity_proc$procid.mp4"; framerate = 10) do io
-    while integrator.t < 20
-        dt = 0.001
-        step!(integrator, dt)
-        t_plot[] = integrator.t
-        mul!.(v̂s, plan, integrator.u.x)  # current velocity in Fourier space
-        curl_fourier!(ω̂s, v̂s, grid_fourier)
-        ldiv!.(ω⃗_plot[], plan, ω̂s)
-        ω⃗_plot[] = ω⃗_plot[]  # to force updating the plot
-        energy_spectrum!(Ek, ks, v̂s, grid_fourier)
-        Ek ./= scale_factor(plan)^2  # rescale energy
-        E_plot[] = E_plot[]
-        recordframe!(io)
-    end
-end;
    - -

    This page was generated using Literate.jl.

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"strict" : "loose"].exec(str), - uri = {}, - i = 14; - - while (i--) uri[o.key[i]] = m[i] || ""; - - uri[o.q.name] = {}; - uri[o.key[12]].replace(o.q.parser, function ($0, $1, $2) { - if ($1) uri[o.q.name][$1] = $2; - }); - - return uri; - }; - parseUri.options = { - strictMode: false, - key: ["source","protocol","authority","userInfo","user","password","host","port","relative","path","directory","file","query","anchor"], - q: { - name: "queryKey", - parser: /(?:^|&)([^&=]*)=?([^&]*)/g - }, - parser: { - strict: /^(?:([^:\/?#]+):)?(?:\/\/((?:(([^:@]*)(?::([^:@]*))?)?@)?([^:\/?#]*)(?::(\d*))?))?((((?:[^?#\/]*\/)*)([^?#]*))(?:\?([^#]*))?(?:#(.*))?)/, - loose: /^(?:(?![^:@]+:[^:@\/]*@)([^:\/?#.]+):)?(?:\/\/)?((?:(([^:@]*)(?::([^:@]*))?)?@)?([^:\/?#]*)(?::(\d*))?)(((\/(?:[^?#](?![^?#\/]*\.[^?#\/.]+(?:[?#]|$)))*\/?)?([^?#\/]*))(?:\?([^#]*))?(?:#(.*))?)/ - } - }; - - $("#search-form").submit(function(e) { - e.preventDefault() - }) - - // list below is the lunr 2.1.3 list minus the intersect with names(Base) - // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) - // ideally we'd just filter the original list but it's not available as a variable - lunr.stopWordFilter = lunr.generateStopWordFilter([ - 'a', - 'able', - 'about', - 'across', - 'after', - 'almost', - 'also', - 'am', - 'among', - 'an', - 'and', - 'are', - 'as', - 'at', - 'be', - 'because', - 'been', - 'but', - 'by', - 'can', - 'cannot', - 'could', - 'dear', - 'did', - 'does', - 'either', - 'ever', - 'every', - 'from', - 'got', - 'had', - 'has', - 'have', - 'he', - 'her', - 'hers', - 'him', - 'his', - 'how', - 'however', - 'i', - 'if', - 'into', - 'it', - 'its', - 'just', - 'least', - 'like', - 'likely', - 'may', - 'me', - 'might', - 'most', - 'must', - 'my', - 'neither', - 'no', - 'nor', - 'not', - 'of', - 'off', - 'often', - 'on', - 'or', - 'other', - 'our', - 'own', - 'rather', - 'said', - 'say', - 'says', - 'she', - 'should', - 'since', - 'so', - 'some', - 'than', - 'that', - 'the', - 'their', - 'them', - 'then', - 'there', - 'these', - 'they', - 'this', - 'tis', - 'to', - 'too', - 'twas', - 'us', - 'wants', - 'was', - 'we', - 'were', - 'what', - 'when', - 'who', - 'whom', - 'why', - 'will', - 'would', - 'yet', - 'you', - 'your' - ]) - - // add . as a separator, because otherwise "title": "Documenter.Anchors.add!" - // would not find anything if searching for "add!", only for the entire qualification - lunr.tokenizer.separator = /[\s\-\.]+/ - - // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names - lunr.trimmer = function (token) { - return token.update(function (s) { - return s.replace(/^[^a-zA-Z0-9@!]+/, '').replace(/[^a-zA-Z0-9@!]+$/, '') - }) - } - - lunr.Pipeline.registerFunction(lunr.stopWordFilter, 'juliaStopWordFilter') - lunr.Pipeline.registerFunction(lunr.trimmer, 'juliaTrimmer') - - var index = lunr(function () { - this.ref('location') - this.field('title',{boost: 100}) - this.field('text') - documenterSearchIndex['docs'].forEach(function(e) { - this.add(e) - }, this) - }) - var store = {} - - documenterSearchIndex['docs'].forEach(function(e) { - store[e.location] = {title: e.title, category: e.category, page: e.page} - }) - - $(function(){ - searchresults = $('#documenter-search-results'); - searchinfo = $('#documenter-search-info'); - searchbox = $('#documenter-search-query'); - searchform = $('.docs-search'); - sidebar = $('.docs-sidebar'); - function update_search(querystring) { - tokens = lunr.tokenizer(querystring) - results = index.query(function (q) { - tokens.forEach(function (t) { - q.term(t.toString(), { - fields: ["title"], - boost: 100, - usePipeline: true, - editDistance: 0, - wildcard: lunr.Query.wildcard.NONE - }) - q.term(t.toString(), { - fields: ["title"], - boost: 10, - usePipeline: true, - editDistance: 2, - wildcard: lunr.Query.wildcard.NONE - }) - q.term(t.toString(), { - fields: ["text"], - boost: 1, - usePipeline: true, - editDistance: 0, - wildcard: lunr.Query.wildcard.NONE - }) - }) - }) - searchinfo.text("Number of results: " + results.length) - searchresults.empty() - results.forEach(function(result) { - data = store[result.ref] - link = $(''+data.title+'') - link.attr('href', documenterBaseURL+'/'+result.ref) - if (data.category != "page"){ - cat = $('('+data.category+', '+data.page+')') - } else { - cat = $('('+data.category+')') - } - li = $('
  • ').append(link).append(" ").append(cat) - searchresults.append(li) - }) - } - - function update_search_box() { - querystring = searchbox.val() - update_search(querystring) - } - - searchbox.keyup(_.debounce(update_search_box, 250)) - searchbox.change(update_search_box) - - // Disable enter-key form submission for the searchbox on the search page - // and just re-run search rather than refresh the whole page. - searchform.keypress( - function(event){ - if (event.which == '13') { - if (sidebar.hasClass('visible')) { - sidebar.removeClass('visible'); - } - update_search_box(); - event.preventDefault(); - } - } - ); - - search_query_uri = parseUri(window.location).queryKey["q"] - if(search_query_uri !== undefined) { - search_query = decodeURIComponent(search_query_uri.replace(/\+/g, '%20')) - searchbox.val(search_query) - } - update_search_box(); - }) -}) - -}) diff --git a/previews/PR62/assets/themes/documenter-dark.css b/previews/PR62/assets/themes/documenter-dark.css deleted file mode 100644 index 0bf1af7d..00000000 --- a/previews/PR62/assets/themes/documenter-dark.css +++ /dev/null @@ -1,7702 +0,0 @@ -@charset "UTF-8"; -/* Font Awesome 5 mixin. Can be included in any rule that should render Font Awesome icons. */ -@keyframes spinAround { - from { - transform: rotate(0deg); } - to { - transform: rotate(359deg); } } - -html.theme--documenter-dark .tabs, html.theme--documenter-dark .pagination-previous, -html.theme--documenter-dark .pagination-next, -html.theme--documenter-dark .pagination-link, -html.theme--documenter-dark .pagination-ellipsis, html.theme--documenter-dark .breadcrumb, html.theme--documenter-dark .file, html.theme--documenter-dark .button, .is-unselectable, html.theme--documenter-dark .modal-close, html.theme--documenter-dark .delete { - -webkit-touch-callout: none; - -webkit-user-select: none; - -moz-user-select: none; - -ms-user-select: none; - user-select: none; } - -html.theme--documenter-dark .navbar-link:not(.is-arrowless)::after, html.theme--documenter-dark .select:not(.is-multiple):not(.is-loading)::after { - border: 3px solid transparent; - border-radius: 2px; - border-right: 0; - border-top: 0; - content: " "; - display: block; - height: 0.625em; - margin-top: -0.4375em; - pointer-events: none; - position: absolute; - top: 50%; - transform: rotate(-45deg); - transform-origin: center; - width: 0.625em; } - -html.theme--documenter-dark .admonition:not(:last-child), html.theme--documenter-dark .tabs:not(:last-child), html.theme--documenter-dark .message:not(:last-child), html.theme--documenter-dark .list:not(:last-child), html.theme--documenter-dark .level:not(:last-child), html.theme--documenter-dark .breadcrumb:not(:last-child), html.theme--documenter-dark .highlight:not(:last-child), html.theme--documenter-dark .block:not(:last-child), html.theme--documenter-dark .title:not(:last-child), -html.theme--documenter-dark .subtitle:not(:last-child), html.theme--documenter-dark .table-container:not(:last-child), html.theme--documenter-dark .table:not(:last-child), html.theme--documenter-dark .progress:not(:last-child), html.theme--documenter-dark .notification:not(:last-child), html.theme--documenter-dark .content:not(:last-child), html.theme--documenter-dark .box:not(:last-child) { - margin-bottom: 1.5rem; } - -html.theme--documenter-dark .modal-close, html.theme--documenter-dark .delete { - -moz-appearance: none; - -webkit-appearance: none; - background-color: rgba(10, 10, 10, 0.2); - border: none; - border-radius: 290486px; - cursor: pointer; - pointer-events: auto; - display: inline-block; - flex-grow: 0; - flex-shrink: 0; - font-size: 0; - height: 20px; - max-height: 20px; - max-width: 20px; - min-height: 20px; - min-width: 20px; - outline: none; - position: relative; - vertical-align: top; - width: 20px; } - html.theme--documenter-dark .modal-close::before, html.theme--documenter-dark .delete::before, html.theme--documenter-dark .modal-close::after, html.theme--documenter-dark .delete::after { - background-color: white; - content: ""; - display: block; - left: 50%; - position: absolute; - top: 50%; - transform: translateX(-50%) translateY(-50%) 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} - html.theme--documenter-dark .button.is-white.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-white.is-inverted { - background-color: #0a0a0a; - border-color: transparent; - box-shadow: none; - color: white; } - html.theme--documenter-dark .button.is-white.is-loading::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - html.theme--documenter-dark .button.is-white.is-outlined { - background-color: transparent; - border-color: white; - color: white; } - html.theme--documenter-dark .button.is-white.is-outlined:hover, html.theme--documenter-dark .button.is-white.is-outlined.is-hovered, html.theme--documenter-dark .button.is-white.is-outlined:focus, html.theme--documenter-dark .button.is-white.is-outlined.is-focused { - background-color: white; - border-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-white.is-outlined.is-loading::after { - border-color: transparent transparent white white !important; } - html.theme--documenter-dark .button.is-white.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-white.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-white.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-white.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - html.theme--documenter-dark .button.is-white.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-white.is-outlined { - background-color: transparent; - border-color: white; - box-shadow: none; - color: white; } - html.theme--documenter-dark .button.is-white.is-inverted.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-white.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-focused { - background-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-white.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent white white !important; } - html.theme--documenter-dark .button.is-white.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-white.is-inverted.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - box-shadow: none; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-black { - background-color: #0a0a0a; - border-color: transparent; - color: white; } - html.theme--documenter-dark .button.is-black:hover, html.theme--documenter-dark .button.is-black.is-hovered { - background-color: #040404; - border-color: transparent; - color: white; } - html.theme--documenter-dark .button.is-black:focus, html.theme--documenter-dark .button.is-black.is-focused { - border-color: transparent; - color: white; } - html.theme--documenter-dark .button.is-black:focus:not(:active), html.theme--documenter-dark .button.is-black.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); } - html.theme--documenter-dark .button.is-black:active, html.theme--documenter-dark .button.is-black.is-active { - background-color: black; - border-color: transparent; - color: white; } - html.theme--documenter-dark .button.is-black[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-black { - background-color: #0a0a0a; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-black.is-inverted { - background-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-black.is-inverted:hover, html.theme--documenter-dark .button.is-black.is-inverted.is-hovered { - background-color: #f2f2f2; } - html.theme--documenter-dark .button.is-black.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted { - background-color: white; - border-color: transparent; - box-shadow: none; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-black.is-loading::after { - border-color: transparent transparent white white !important; } - html.theme--documenter-dark .button.is-black.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-black.is-outlined:hover, html.theme--documenter-dark .button.is-black.is-outlined.is-hovered, html.theme--documenter-dark .button.is-black.is-outlined:focus, html.theme--documenter-dark .button.is-black.is-outlined.is-focused { - background-color: #0a0a0a; - border-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .button.is-black.is-outlined.is-loading::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - html.theme--documenter-dark .button.is-black.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-black.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent white white !important; } - html.theme--documenter-dark .button.is-black.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-black.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - box-shadow: none; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-black.is-inverted.is-outlined { - background-color: transparent; - border-color: white; - color: white; } - html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-focused { - background-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - html.theme--documenter-dark .button.is-black.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted.is-outlined { - background-color: transparent; - border-color: white; - box-shadow: none; - color: white; } - html.theme--documenter-dark .button.is-light { - background-color: #ecf0f1; - border-color: transparent; - color: #282f2f; } - html.theme--documenter-dark .button.is-light:hover, html.theme--documenter-dark .button.is-light.is-hovered { - background-color: #e5eaec; - border-color: transparent; - color: #282f2f; } - html.theme--documenter-dark .button.is-light:focus, html.theme--documenter-dark .button.is-light.is-focused { - border-color: transparent; - color: #282f2f; } - html.theme--documenter-dark .button.is-light:focus:not(:active), html.theme--documenter-dark .button.is-light.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(236, 240, 241, 0.25); } - html.theme--documenter-dark .button.is-light:active, html.theme--documenter-dark .button.is-light.is-active { - background-color: #dde4e6; - border-color: transparent; - color: #282f2f; } - html.theme--documenter-dark .button.is-light[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-light { - background-color: #ecf0f1; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-light.is-inverted { - background-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-light.is-inverted:hover, html.theme--documenter-dark .button.is-light.is-inverted.is-hovered { - background-color: #1d2122; } - html.theme--documenter-dark .button.is-light.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted { - background-color: #282f2f; - border-color: transparent; - box-shadow: none; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-light.is-loading::after { - border-color: transparent transparent #282f2f #282f2f !important; } - html.theme--documenter-dark .button.is-light.is-outlined { - background-color: transparent; - border-color: #ecf0f1; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-light.is-outlined:hover, html.theme--documenter-dark .button.is-light.is-outlined.is-hovered, html.theme--documenter-dark .button.is-light.is-outlined:focus, html.theme--documenter-dark .button.is-light.is-outlined.is-focused { - background-color: #ecf0f1; - border-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .button.is-light.is-outlined.is-loading::after { - border-color: transparent transparent #ecf0f1 #ecf0f1 !important; } - html.theme--documenter-dark .button.is-light.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-light.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #282f2f #282f2f !important; } - html.theme--documenter-dark .button.is-light.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-light.is-outlined { - background-color: transparent; - border-color: #ecf0f1; - box-shadow: none; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-light.is-inverted.is-outlined { - background-color: transparent; - border-color: #282f2f; - color: #282f2f; } - html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-focused { - background-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #ecf0f1 #ecf0f1 !important; } - html.theme--documenter-dark .button.is-light.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted.is-outlined { - background-color: transparent; - border-color: #282f2f; - box-shadow: none; - color: #282f2f; } - html.theme--documenter-dark .button.is-dark, html.theme--documenter-dark .content kbd.button { - background-color: #282f2f; - border-color: transparent; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-dark:hover, html.theme--documenter-dark .content kbd.button:hover, html.theme--documenter-dark .button.is-dark.is-hovered, html.theme--documenter-dark .content kbd.button.is-hovered { - background-color: #232829; - border-color: transparent; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-dark:focus, html.theme--documenter-dark .content kbd.button:focus, html.theme--documenter-dark .button.is-dark.is-focused, html.theme--documenter-dark .content kbd.button.is-focused { - border-color: transparent; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-dark:focus:not(:active), html.theme--documenter-dark .content kbd.button:focus:not(:active), html.theme--documenter-dark .button.is-dark.is-focused:not(:active), html.theme--documenter-dark .content kbd.button.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(40, 47, 47, 0.25); } - html.theme--documenter-dark .button.is-dark:active, html.theme--documenter-dark .content kbd.button:active, html.theme--documenter-dark .button.is-dark.is-active, html.theme--documenter-dark .content kbd.button.is-active { - background-color: #1d2122; - border-color: transparent; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-dark[disabled], html.theme--documenter-dark .content kbd.button[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-dark, fieldset[disabled] html.theme--documenter-dark .content kbd.button { - background-color: #282f2f; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-dark.is-inverted, html.theme--documenter-dark .content kbd.button.is-inverted { - background-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .button.is-dark.is-inverted:hover, html.theme--documenter-dark .content kbd.button.is-inverted:hover, html.theme--documenter-dark .button.is-dark.is-inverted.is-hovered, html.theme--documenter-dark .content kbd.button.is-inverted.is-hovered { - background-color: #dde4e6; } - html.theme--documenter-dark .button.is-dark.is-inverted[disabled], html.theme--documenter-dark .content kbd.button.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted, fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted { - background-color: #ecf0f1; - border-color: transparent; - box-shadow: none; - color: #282f2f; } - html.theme--documenter-dark .button.is-dark.is-loading::after, html.theme--documenter-dark .content kbd.button.is-loading::after { - border-color: transparent transparent #ecf0f1 #ecf0f1 !important; } - html.theme--documenter-dark .button.is-dark.is-outlined, html.theme--documenter-dark .content kbd.button.is-outlined { - background-color: transparent; - border-color: #282f2f; - color: #282f2f; } - html.theme--documenter-dark .button.is-dark.is-outlined:hover, html.theme--documenter-dark .content kbd.button.is-outlined:hover, html.theme--documenter-dark .button.is-dark.is-outlined.is-hovered, html.theme--documenter-dark .content kbd.button.is-outlined.is-hovered, html.theme--documenter-dark .button.is-dark.is-outlined:focus, html.theme--documenter-dark .content kbd.button.is-outlined:focus, html.theme--documenter-dark .button.is-dark.is-outlined.is-focused, html.theme--documenter-dark .content kbd.button.is-outlined.is-focused { - background-color: #282f2f; - border-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-dark.is-outlined.is-loading::after, html.theme--documenter-dark .content kbd.button.is-outlined.is-loading::after { - border-color: transparent transparent #282f2f #282f2f !important; } - html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:hover::after, html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after, html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after, html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #ecf0f1 #ecf0f1 !important; } - html.theme--documenter-dark .button.is-dark.is-outlined[disabled], html.theme--documenter-dark .content kbd.button.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined, fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined { - background-color: transparent; - border-color: #282f2f; - box-shadow: none; - color: #282f2f; } - html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined { - background-color: transparent; - border-color: #ecf0f1; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-focused { - background-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after, html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #282f2f #282f2f !important; } - html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined[disabled], html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined, fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined { - background-color: transparent; - border-color: #ecf0f1; - box-shadow: none; - color: #ecf0f1; } - html.theme--documenter-dark .button.is-primary, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink { - background-color: #375a7f; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-primary:hover, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:hover, html.theme--documenter-dark .button.is-primary.is-hovered, html.theme--documenter-dark .docstring > section > a.button.is-hovered.docs-sourcelink { - background-color: #335476; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-primary:focus, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:focus, html.theme--documenter-dark .button.is-primary.is-focused, html.theme--documenter-dark .docstring > section > a.button.is-focused.docs-sourcelink { - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-primary:focus:not(:active), html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:focus:not(:active), html.theme--documenter-dark .button.is-primary.is-focused:not(:active), html.theme--documenter-dark .docstring > section > a.button.is-focused.docs-sourcelink:not(:active) { - box-shadow: 0 0 0 0.125em rgba(55, 90, 127, 0.25); } - html.theme--documenter-dark .button.is-primary:active, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:active, html.theme--documenter-dark .button.is-primary.is-active, html.theme--documenter-dark .docstring > section > a.button.is-active.docs-sourcelink { - background-color: #2f4d6d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-primary[disabled], html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-primary, fieldset[disabled] html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink { - background-color: #375a7f; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-primary.is-inverted, html.theme--documenter-dark .docstring > section > a.button.is-inverted.docs-sourcelink { - background-color: #fff; - color: #375a7f; } - html.theme--documenter-dark .button.is-primary.is-inverted:hover, html.theme--documenter-dark .docstring > section > a.button.is-inverted.docs-sourcelink:hover, html.theme--documenter-dark .button.is-primary.is-inverted.is-hovered, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-hovered.docs-sourcelink { - background-color: #f2f2f2; } - html.theme--documenter-dark .button.is-primary.is-inverted[disabled], html.theme--documenter-dark .docstring > section > a.button.is-inverted.docs-sourcelink[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-primary.is-inverted, fieldset[disabled] html.theme--documenter-dark .docstring > section > a.button.is-inverted.docs-sourcelink { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #375a7f; } - html.theme--documenter-dark .button.is-primary.is-loading::after, html.theme--documenter-dark .docstring > section > a.button.is-loading.docs-sourcelink::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-primary.is-outlined, html.theme--documenter-dark .docstring > section > a.button.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #375a7f; - color: #375a7f; } - html.theme--documenter-dark .button.is-primary.is-outlined:hover, html.theme--documenter-dark .docstring > section > a.button.is-outlined.docs-sourcelink:hover, html.theme--documenter-dark .button.is-primary.is-outlined.is-hovered, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-hovered.docs-sourcelink, html.theme--documenter-dark .button.is-primary.is-outlined:focus, html.theme--documenter-dark .docstring > section > a.button.is-outlined.docs-sourcelink:focus, html.theme--documenter-dark .button.is-primary.is-outlined.is-focused, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-focused.docs-sourcelink { - background-color: #375a7f; - border-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .button.is-primary.is-outlined.is-loading::after, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink::after { - border-color: transparent transparent #375a7f #375a7f !important; } - html.theme--documenter-dark .button.is-primary.is-outlined.is-loading:hover::after, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink:hover::after, html.theme--documenter-dark .button.is-primary.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-loading.is-hovered.docs-sourcelink::after, html.theme--documenter-dark .button.is-primary.is-outlined.is-loading:focus::after, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink:focus::after, html.theme--documenter-dark .button.is-primary.is-outlined.is-loading.is-focused::after, html.theme--documenter-dark .docstring > section > a.button.is-outlined.is-loading.is-focused.docs-sourcelink::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-primary.is-outlined[disabled], html.theme--documenter-dark .docstring > section > a.button.is-outlined.docs-sourcelink[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-primary.is-outlined, fieldset[disabled] html.theme--documenter-dark .docstring > section > a.button.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #375a7f; - box-shadow: none; - color: #375a7f; } - html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #fff; - color: #fff; } - html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined:hover, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink:hover, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-hovered.docs-sourcelink, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined:focus, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink:focus, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-focused, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-focused.docs-sourcelink { - background-color: #fff; - color: #375a7f; } - html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:hover::after, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-loading.is-hovered.docs-sourcelink::after, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:focus::after, html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined.is-loading.is-focused::after, html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.is-loading.is-focused.docs-sourcelink::after { - border-color: transparent transparent #375a7f #375a7f !important; } - html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined[disabled], html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-primary.is-inverted.is-outlined, fieldset[disabled] html.theme--documenter-dark .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - html.theme--documenter-dark .button.is-link { - background-color: #1abc9c; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-link:hover, html.theme--documenter-dark .button.is-link.is-hovered { - background-color: #18b193; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-link:focus, html.theme--documenter-dark .button.is-link.is-focused { - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-link:focus:not(:active), html.theme--documenter-dark .button.is-link.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(26, 188, 156, 0.25); } - html.theme--documenter-dark .button.is-link:active, html.theme--documenter-dark .button.is-link.is-active { - background-color: #17a689; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-link[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-link { - background-color: #1abc9c; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-link.is-inverted { - background-color: #fff; - color: #1abc9c; } - html.theme--documenter-dark .button.is-link.is-inverted:hover, html.theme--documenter-dark .button.is-link.is-inverted.is-hovered { - background-color: #f2f2f2; } - html.theme--documenter-dark .button.is-link.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-link.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #1abc9c; } - html.theme--documenter-dark .button.is-link.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-link.is-outlined { - background-color: transparent; - border-color: #1abc9c; - color: #1abc9c; } - html.theme--documenter-dark .button.is-link.is-outlined:hover, html.theme--documenter-dark .button.is-link.is-outlined.is-hovered, html.theme--documenter-dark .button.is-link.is-outlined:focus, html.theme--documenter-dark .button.is-link.is-outlined.is-focused { - background-color: #1abc9c; - border-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .button.is-link.is-outlined.is-loading::after { - border-color: transparent transparent #1abc9c #1abc9c !important; } - html.theme--documenter-dark .button.is-link.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-link.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-link.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-link.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-link.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-link.is-outlined { - background-color: transparent; - border-color: #1abc9c; - box-shadow: none; - color: #1abc9c; } - html.theme--documenter-dark .button.is-link.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - html.theme--documenter-dark .button.is-link.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #1abc9c; } - html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #1abc9c #1abc9c !important; } - html.theme--documenter-dark .button.is-link.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-link.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - html.theme--documenter-dark .button.is-info { - background-color: #024c7d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-info:hover, html.theme--documenter-dark .button.is-info.is-hovered { - background-color: #024470; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-info:focus, html.theme--documenter-dark .button.is-info.is-focused { - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-info:focus:not(:active), html.theme--documenter-dark .button.is-info.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(2, 76, 125, 0.25); } - html.theme--documenter-dark .button.is-info:active, html.theme--documenter-dark .button.is-info.is-active { - background-color: #023d64; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-info[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-info { - background-color: #024c7d; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-info.is-inverted { - background-color: #fff; - color: #024c7d; } - html.theme--documenter-dark .button.is-info.is-inverted:hover, html.theme--documenter-dark .button.is-info.is-inverted.is-hovered { - background-color: #f2f2f2; } - html.theme--documenter-dark .button.is-info.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-info.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #024c7d; } - html.theme--documenter-dark .button.is-info.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-info.is-outlined { - background-color: transparent; - border-color: #024c7d; - color: #024c7d; } - html.theme--documenter-dark .button.is-info.is-outlined:hover, html.theme--documenter-dark .button.is-info.is-outlined.is-hovered, html.theme--documenter-dark .button.is-info.is-outlined:focus, html.theme--documenter-dark .button.is-info.is-outlined.is-focused { - background-color: #024c7d; - border-color: #024c7d; - color: #fff; } - html.theme--documenter-dark .button.is-info.is-outlined.is-loading::after { - border-color: transparent transparent #024c7d #024c7d !important; } - html.theme--documenter-dark .button.is-info.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-info.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-info.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-info.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-info.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-info.is-outlined { - background-color: transparent; - border-color: #024c7d; - box-shadow: none; - color: #024c7d; } - html.theme--documenter-dark .button.is-info.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - html.theme--documenter-dark .button.is-info.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #024c7d; } - html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #024c7d #024c7d !important; } - html.theme--documenter-dark .button.is-info.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-info.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - html.theme--documenter-dark .button.is-success { - background-color: #008438; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-success:hover, html.theme--documenter-dark .button.is-success.is-hovered { - background-color: #007733; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-success:focus, html.theme--documenter-dark .button.is-success.is-focused { - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-success:focus:not(:active), html.theme--documenter-dark .button.is-success.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(0, 132, 56, 0.25); } - html.theme--documenter-dark .button.is-success:active, html.theme--documenter-dark .button.is-success.is-active { - background-color: #006b2d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-success[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-success { - background-color: #008438; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-success.is-inverted { - background-color: #fff; - color: #008438; } - html.theme--documenter-dark .button.is-success.is-inverted:hover, html.theme--documenter-dark .button.is-success.is-inverted.is-hovered { - background-color: #f2f2f2; } - html.theme--documenter-dark .button.is-success.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-success.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #008438; } - html.theme--documenter-dark .button.is-success.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-success.is-outlined { - background-color: transparent; - border-color: #008438; - color: #008438; } - html.theme--documenter-dark .button.is-success.is-outlined:hover, html.theme--documenter-dark .button.is-success.is-outlined.is-hovered, html.theme--documenter-dark .button.is-success.is-outlined:focus, html.theme--documenter-dark .button.is-success.is-outlined.is-focused { - background-color: #008438; - border-color: #008438; - color: #fff; } - html.theme--documenter-dark .button.is-success.is-outlined.is-loading::after { - border-color: transparent transparent #008438 #008438 !important; } - html.theme--documenter-dark .button.is-success.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-success.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-success.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-success.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-success.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-success.is-outlined { - background-color: transparent; - border-color: #008438; - box-shadow: none; - color: #008438; } - html.theme--documenter-dark .button.is-success.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - html.theme--documenter-dark .button.is-success.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #008438; } - html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #008438 #008438 !important; } - html.theme--documenter-dark .button.is-success.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-success.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - html.theme--documenter-dark .button.is-warning { - background-color: #ad8100; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-warning:hover, html.theme--documenter-dark .button.is-warning.is-hovered { - background-color: #a07700; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-warning:focus, html.theme--documenter-dark .button.is-warning.is-focused { - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-warning:focus:not(:active), html.theme--documenter-dark .button.is-warning.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(173, 129, 0, 0.25); } - html.theme--documenter-dark .button.is-warning:active, html.theme--documenter-dark .button.is-warning.is-active { - background-color: #946e00; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-warning[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-warning { - background-color: #ad8100; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-warning.is-inverted { - background-color: #fff; - color: #ad8100; } - html.theme--documenter-dark .button.is-warning.is-inverted:hover, html.theme--documenter-dark .button.is-warning.is-inverted.is-hovered { - background-color: #f2f2f2; } - html.theme--documenter-dark .button.is-warning.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #ad8100; } - html.theme--documenter-dark .button.is-warning.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-warning.is-outlined { - background-color: transparent; - border-color: #ad8100; - color: #ad8100; } - html.theme--documenter-dark .button.is-warning.is-outlined:hover, html.theme--documenter-dark .button.is-warning.is-outlined.is-hovered, html.theme--documenter-dark .button.is-warning.is-outlined:focus, html.theme--documenter-dark .button.is-warning.is-outlined.is-focused { - background-color: #ad8100; - border-color: #ad8100; - color: #fff; } - html.theme--documenter-dark .button.is-warning.is-outlined.is-loading::after { - border-color: transparent transparent #ad8100 #ad8100 !important; } - html.theme--documenter-dark .button.is-warning.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-warning.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-warning.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-warning.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-warning.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-outlined { - background-color: transparent; - border-color: #ad8100; - box-shadow: none; - color: #ad8100; } - html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #ad8100; } - html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #ad8100 #ad8100 !important; } - html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - html.theme--documenter-dark .button.is-danger { - background-color: #9e1b0d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-danger:hover, html.theme--documenter-dark .button.is-danger.is-hovered { - background-color: #92190c; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-danger:focus, html.theme--documenter-dark .button.is-danger.is-focused { - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-danger:focus:not(:active), html.theme--documenter-dark .button.is-danger.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(158, 27, 13, 0.25); } - html.theme--documenter-dark .button.is-danger:active, html.theme--documenter-dark .button.is-danger.is-active { - background-color: #86170b; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .button.is-danger[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-danger { - background-color: #9e1b0d; - border-color: transparent; - box-shadow: none; } - html.theme--documenter-dark .button.is-danger.is-inverted { - background-color: #fff; - color: #9e1b0d; } - html.theme--documenter-dark .button.is-danger.is-inverted:hover, html.theme--documenter-dark .button.is-danger.is-inverted.is-hovered { - background-color: #f2f2f2; } - html.theme--documenter-dark .button.is-danger.is-inverted[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #9e1b0d; } - html.theme--documenter-dark .button.is-danger.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-danger.is-outlined { - background-color: transparent; - border-color: #9e1b0d; - color: #9e1b0d; } - html.theme--documenter-dark .button.is-danger.is-outlined:hover, html.theme--documenter-dark .button.is-danger.is-outlined.is-hovered, html.theme--documenter-dark .button.is-danger.is-outlined:focus, html.theme--documenter-dark .button.is-danger.is-outlined.is-focused { - background-color: #9e1b0d; - border-color: #9e1b0d; - color: #fff; } - html.theme--documenter-dark .button.is-danger.is-outlined.is-loading::after { - border-color: transparent transparent #9e1b0d #9e1b0d !important; } - html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - html.theme--documenter-dark .button.is-danger.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-outlined { - background-color: transparent; - border-color: #9e1b0d; - box-shadow: none; - color: #9e1b0d; } - html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:hover, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-hovered, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:focus, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #9e1b0d; } - html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:hover::after, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:focus::after, html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #9e1b0d #9e1b0d !important; } - html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined[disabled], fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - html.theme--documenter-dark .button.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.button { - border-radius: 3px; - font-size: 0.85em; } - html.theme--documenter-dark .button.is-normal { - font-size: 15px; } - html.theme--documenter-dark .button.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .button.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .button[disabled], fieldset[disabled] html.theme--documenter-dark .button { - background-color: #8c9b9d; - border-color: #dbdee0; - box-shadow: none; - opacity: 0.5; } - html.theme--documenter-dark .button.is-fullwidth { - display: flex; - width: 100%; } - html.theme--documenter-dark .button.is-loading { - color: transparent !important; - pointer-events: none; } - html.theme--documenter-dark .button.is-loading::after { - position: absolute; - left: calc(50% - (1em / 2)); - top: calc(50% - (1em / 2)); - position: absolute !important; } - html.theme--documenter-dark .button.is-static { - background-color: #282f2f; - border-color: #5e6d6f; - color: #dbdee0; - box-shadow: none; - pointer-events: none; } - html.theme--documenter-dark .button.is-rounded, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.button { - border-radius: 290486px; - padding-left: 1em; - padding-right: 1em; } - html.theme--documenter-dark .buttons { - align-items: center; - display: flex; - flex-wrap: wrap; - justify-content: flex-start; } - html.theme--documenter-dark .buttons .button { - margin-bottom: 0.5rem; } - html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth) { - margin-right: 0.5rem; } - html.theme--documenter-dark .buttons:last-child { - margin-bottom: -0.5rem; } - html.theme--documenter-dark .buttons:not(:last-child) { - margin-bottom: 1rem; } - html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large) { - border-radius: 3px; - font-size: 0.85em; } - html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large) { - font-size: 1.25rem; } - html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium) { - font-size: 1.5rem; } - html.theme--documenter-dark .buttons.has-addons .button:not(:first-child) { - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - html.theme--documenter-dark .buttons.has-addons .button:not(:last-child) { - border-bottom-right-radius: 0; - border-top-right-radius: 0; - margin-right: -1px; } - html.theme--documenter-dark .buttons.has-addons .button:last-child { - margin-right: 0; } - html.theme--documenter-dark .buttons.has-addons .button:hover, html.theme--documenter-dark .buttons.has-addons .button.is-hovered { - z-index: 2; } - html.theme--documenter-dark .buttons.has-addons .button:focus, html.theme--documenter-dark .buttons.has-addons .button.is-focused, html.theme--documenter-dark .buttons.has-addons .button:active, html.theme--documenter-dark .buttons.has-addons .button.is-active, html.theme--documenter-dark .buttons.has-addons .button.is-selected { - z-index: 3; } - html.theme--documenter-dark .buttons.has-addons .button:focus:hover, html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover, html.theme--documenter-dark .buttons.has-addons .button:active:hover, html.theme--documenter-dark .buttons.has-addons .button.is-active:hover, html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover { - z-index: 4; } - html.theme--documenter-dark .buttons.has-addons .button.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - html.theme--documenter-dark .buttons.is-centered { - justify-content: center; } - html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth) { - margin-left: 0.25rem; - margin-right: 0.25rem; } - html.theme--documenter-dark .buttons.is-right { - justify-content: flex-end; } - html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth) { - margin-left: 0.25rem; - margin-right: 0.25rem; } - html.theme--documenter-dark .container { - flex-grow: 1; - margin: 0 auto; - position: relative; - width: auto; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .container { - max-width: 992px; } - html.theme--documenter-dark .container.is-fluid { - margin-left: 32px; - margin-right: 32px; - max-width: none; } } - @media screen and (max-width: 1215px) { - html.theme--documenter-dark .container.is-widescreen { - max-width: 1152px; } } - @media screen and (max-width: 1407px) { - html.theme--documenter-dark .container.is-fullhd { - max-width: 1344px; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .container { - max-width: 1152px; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .container { - max-width: 1344px; } } - html.theme--documenter-dark .content li + li { - margin-top: 0.25em; } - html.theme--documenter-dark .content p:not(:last-child), - html.theme--documenter-dark .content dl:not(:last-child), - html.theme--documenter-dark .content ol:not(:last-child), - html.theme--documenter-dark .content ul:not(:last-child), - html.theme--documenter-dark .content blockquote:not(:last-child), - html.theme--documenter-dark .content pre:not(:last-child), - html.theme--documenter-dark .content table:not(:last-child) { - margin-bottom: 1em; } - html.theme--documenter-dark .content h1, - html.theme--documenter-dark .content h2, - html.theme--documenter-dark .content h3, - html.theme--documenter-dark .content h4, - html.theme--documenter-dark .content h5, - html.theme--documenter-dark .content h6 { - color: #f2f2f2; - font-weight: 600; - line-height: 1.125; } - html.theme--documenter-dark .content h1 { - font-size: 2em; - margin-bottom: 0.5em; } - html.theme--documenter-dark .content h1:not(:first-child) { - margin-top: 1em; } - html.theme--documenter-dark .content h2 { - font-size: 1.75em; - margin-bottom: 0.5714em; } - html.theme--documenter-dark .content h2:not(:first-child) { - margin-top: 1.1428em; } - html.theme--documenter-dark .content h3 { - font-size: 1.5em; - margin-bottom: 0.6666em; } - html.theme--documenter-dark .content h3:not(:first-child) { - margin-top: 1.3333em; } - html.theme--documenter-dark .content h4 { - font-size: 1.25em; - margin-bottom: 0.8em; } - html.theme--documenter-dark .content h5 { - font-size: 1.125em; - margin-bottom: 0.8888em; } - html.theme--documenter-dark .content h6 { - font-size: 1em; - margin-bottom: 1em; } - html.theme--documenter-dark .content blockquote { - background-color: #282f2f; - border-left: 5px solid #5e6d6f; - padding: 1.25em 1.5em; } - html.theme--documenter-dark .content ol { - list-style-position: outside; - margin-left: 2em; - margin-top: 1em; } - html.theme--documenter-dark .content ol:not([type]) { - list-style-type: decimal; } - html.theme--documenter-dark .content ol.is-lower-alpha:not([type]) { - list-style-type: lower-alpha; } - html.theme--documenter-dark .content ol.is-lower-roman:not([type]) { - list-style-type: lower-roman; } - html.theme--documenter-dark .content ol.is-upper-alpha:not([type]) { - list-style-type: upper-alpha; } - html.theme--documenter-dark .content ol.is-upper-roman:not([type]) { - list-style-type: upper-roman; } - html.theme--documenter-dark .content ul { - list-style: disc outside; - margin-left: 2em; - margin-top: 1em; } - html.theme--documenter-dark .content ul ul { - list-style-type: circle; - margin-top: 0.5em; } - html.theme--documenter-dark .content ul ul ul { - list-style-type: square; } - html.theme--documenter-dark .content dd { - margin-left: 2em; } - html.theme--documenter-dark .content figure { - margin-left: 2em; - margin-right: 2em; - text-align: center; } - html.theme--documenter-dark .content figure:not(:first-child) { - margin-top: 2em; } - html.theme--documenter-dark .content figure:not(:last-child) { - margin-bottom: 2em; } - html.theme--documenter-dark .content figure img { - display: inline-block; } - html.theme--documenter-dark .content figure figcaption { - font-style: italic; } - html.theme--documenter-dark .content pre { - -webkit-overflow-scrolling: touch; - overflow-x: auto; - padding: 0; - white-space: pre; - word-wrap: normal; } - html.theme--documenter-dark .content sup, - html.theme--documenter-dark .content sub { - font-size: 75%; } - html.theme--documenter-dark .content table { - width: 100%; } - html.theme--documenter-dark .content table td, - html.theme--documenter-dark .content table th { - border: 1px solid #5e6d6f; - border-width: 0 0 1px; - padding: 0.5em 0.75em; - vertical-align: top; } - html.theme--documenter-dark .content table th { - color: #f2f2f2; } - html.theme--documenter-dark .content table th:not([align]) { - text-align: left; } - html.theme--documenter-dark .content table thead td, - html.theme--documenter-dark .content table thead th { - border-width: 0 0 2px; - color: #f2f2f2; } - html.theme--documenter-dark .content table tfoot td, - html.theme--documenter-dark .content table tfoot th { - border-width: 2px 0 0; - color: #f2f2f2; } - html.theme--documenter-dark .content table tbody tr:last-child td, - html.theme--documenter-dark .content table tbody tr:last-child th { - border-bottom-width: 0; } - html.theme--documenter-dark .content .tabs li + li { - margin-top: 0; } - html.theme--documenter-dark .content.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.content { - font-size: 0.85em; } - html.theme--documenter-dark .content.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .content.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .icon { - align-items: center; - display: inline-flex; - justify-content: center; - height: 1.5rem; - width: 1.5rem; } - html.theme--documenter-dark .icon.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.icon { - height: 1rem; - width: 1rem; } - html.theme--documenter-dark .icon.is-medium { - height: 2rem; - width: 2rem; } - html.theme--documenter-dark .icon.is-large { - height: 3rem; - width: 3rem; } - html.theme--documenter-dark .image, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img { - display: block; - position: relative; } - html.theme--documenter-dark .image img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img img { - display: block; - height: auto; - width: 100%; } - html.theme--documenter-dark .image img.is-rounded, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img img.is-rounded { - border-radius: 290486px; } - html.theme--documenter-dark .image.is-square img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-square img, - html.theme--documenter-dark .image.is-square .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-square .has-ratio, html.theme--documenter-dark .image.is-1by1 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by1 img, - html.theme--documenter-dark .image.is-1by1 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by1 .has-ratio, html.theme--documenter-dark .image.is-5by4 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by4 img, - html.theme--documenter-dark .image.is-5by4 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by4 .has-ratio, html.theme--documenter-dark .image.is-4by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by3 img, - html.theme--documenter-dark .image.is-4by3 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by3 .has-ratio, html.theme--documenter-dark .image.is-3by2 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by2 img, - html.theme--documenter-dark .image.is-3by2 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by2 .has-ratio, html.theme--documenter-dark .image.is-5by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by3 img, - html.theme--documenter-dark .image.is-5by3 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by3 .has-ratio, html.theme--documenter-dark .image.is-16by9 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-16by9 img, - html.theme--documenter-dark .image.is-16by9 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-16by9 .has-ratio, html.theme--documenter-dark .image.is-2by1 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by1 img, - html.theme--documenter-dark .image.is-2by1 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by1 .has-ratio, html.theme--documenter-dark .image.is-3by1 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by1 img, - html.theme--documenter-dark .image.is-3by1 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by1 .has-ratio, html.theme--documenter-dark .image.is-4by5 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by5 img, - html.theme--documenter-dark .image.is-4by5 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by5 .has-ratio, html.theme--documenter-dark .image.is-3by4 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by4 img, - html.theme--documenter-dark .image.is-3by4 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by4 .has-ratio, html.theme--documenter-dark .image.is-2by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by3 img, - html.theme--documenter-dark .image.is-2by3 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by3 .has-ratio, html.theme--documenter-dark .image.is-3by5 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by5 img, - html.theme--documenter-dark .image.is-3by5 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by5 .has-ratio, html.theme--documenter-dark .image.is-9by16 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-9by16 img, - html.theme--documenter-dark .image.is-9by16 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-9by16 .has-ratio, html.theme--documenter-dark .image.is-1by2 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by2 img, - html.theme--documenter-dark .image.is-1by2 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by2 .has-ratio, html.theme--documenter-dark .image.is-1by3 img, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by3 img, - html.theme--documenter-dark .image.is-1by3 .has-ratio, - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by3 .has-ratio { - height: 100%; - width: 100%; } - html.theme--documenter-dark .image.is-square, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-square, html.theme--documenter-dark .image.is-1by1, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by1 { - padding-top: 100%; } - html.theme--documenter-dark .image.is-5by4, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by4 { - padding-top: 80%; } - html.theme--documenter-dark .image.is-4by3, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by3 { - padding-top: 75%; } - html.theme--documenter-dark .image.is-3by2, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by2 { - padding-top: 66.6666%; } - html.theme--documenter-dark .image.is-5by3, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-5by3 { - padding-top: 60%; } - html.theme--documenter-dark .image.is-16by9, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-16by9 { - padding-top: 56.25%; } - html.theme--documenter-dark .image.is-2by1, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by1 { - padding-top: 50%; } - html.theme--documenter-dark .image.is-3by1, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by1 { - padding-top: 33.3333%; } - html.theme--documenter-dark .image.is-4by5, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-4by5 { - padding-top: 125%; } - html.theme--documenter-dark .image.is-3by4, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by4 { - padding-top: 133.3333%; } - html.theme--documenter-dark .image.is-2by3, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-2by3 { - padding-top: 150%; } - html.theme--documenter-dark .image.is-3by5, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-3by5 { - padding-top: 166.6666%; } - html.theme--documenter-dark .image.is-9by16, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-9by16 { - padding-top: 177.7777%; } - html.theme--documenter-dark .image.is-1by2, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by2 { - padding-top: 200%; } - html.theme--documenter-dark .image.is-1by3, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-1by3 { - padding-top: 300%; } - html.theme--documenter-dark .image.is-16x16, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-16x16 { - height: 16px; - width: 16px; } - html.theme--documenter-dark .image.is-24x24, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-24x24 { - height: 24px; - width: 24px; } - html.theme--documenter-dark .image.is-32x32, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-32x32 { - height: 32px; - width: 32px; } - html.theme--documenter-dark .image.is-48x48, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-48x48 { - height: 48px; - width: 48px; } - html.theme--documenter-dark .image.is-64x64, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-64x64 { - height: 64px; - width: 64px; } - html.theme--documenter-dark .image.is-96x96, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-96x96 { - height: 96px; - width: 96px; } - html.theme--documenter-dark .image.is-128x128, html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img.is-128x128 { - height: 128px; - width: 128px; } - html.theme--documenter-dark .notification { - background-color: #282f2f; - border-radius: 0.4em; - padding: 1.25rem 2.5rem 1.25rem 1.5rem; - position: relative; } - html.theme--documenter-dark .notification a:not(.button):not(.dropdown-item) { - color: currentColor; - text-decoration: underline; } - html.theme--documenter-dark .notification strong { - color: currentColor; } - html.theme--documenter-dark .notification code, - html.theme--documenter-dark .notification pre { - background: white; } - html.theme--documenter-dark .notification pre code { - background: transparent; } - html.theme--documenter-dark .notification > .delete { - position: absolute; - right: 0.5rem; - top: 0.5rem; } - html.theme--documenter-dark .notification .title, - html.theme--documenter-dark .notification .subtitle, - html.theme--documenter-dark .notification .content { - color: currentColor; } - html.theme--documenter-dark .notification.is-white { - background-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .notification.is-black { - background-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .notification.is-light { - background-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .notification.is-dark, html.theme--documenter-dark .content kbd.notification { - background-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .notification.is-primary, html.theme--documenter-dark .docstring > section > a.notification.docs-sourcelink { - background-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .notification.is-link { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .notification.is-info { - background-color: #024c7d; - color: #fff; } - html.theme--documenter-dark .notification.is-success { - background-color: #008438; - color: #fff; } - html.theme--documenter-dark .notification.is-warning { - background-color: #ad8100; - color: #fff; } - html.theme--documenter-dark .notification.is-danger { - background-color: #9e1b0d; - color: #fff; } - html.theme--documenter-dark .progress { - -moz-appearance: none; - -webkit-appearance: none; - border: none; - border-radius: 290486px; - display: block; - height: 15px; - overflow: hidden; - padding: 0; - width: 100%; } - html.theme--documenter-dark .progress::-webkit-progress-bar { - background-color: #5e6d6f; } - html.theme--documenter-dark .progress::-webkit-progress-value { - background-color: #dbdee0; } - html.theme--documenter-dark .progress::-moz-progress-bar { - background-color: #dbdee0; } - html.theme--documenter-dark .progress::-ms-fill { - background-color: #dbdee0; - border: none; } - html.theme--documenter-dark .progress.is-white::-webkit-progress-value { - background-color: white; } - html.theme--documenter-dark .progress.is-white::-moz-progress-bar { - background-color: white; } - html.theme--documenter-dark .progress.is-white::-ms-fill { - background-color: white; } - html.theme--documenter-dark .progress.is-white:indeterminate { - background-image: linear-gradient(to right, white 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-black::-webkit-progress-value { - background-color: #0a0a0a; } - html.theme--documenter-dark .progress.is-black::-moz-progress-bar { - background-color: #0a0a0a; } - html.theme--documenter-dark .progress.is-black::-ms-fill { - background-color: #0a0a0a; } - html.theme--documenter-dark .progress.is-black:indeterminate { - background-image: linear-gradient(to right, #0a0a0a 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-light::-webkit-progress-value { - background-color: #ecf0f1; } - html.theme--documenter-dark .progress.is-light::-moz-progress-bar { - background-color: #ecf0f1; } - html.theme--documenter-dark .progress.is-light::-ms-fill { - background-color: #ecf0f1; } - html.theme--documenter-dark .progress.is-light:indeterminate { - background-image: linear-gradient(to right, #ecf0f1 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-dark::-webkit-progress-value, html.theme--documenter-dark .content kbd.progress::-webkit-progress-value { - background-color: #282f2f; } - html.theme--documenter-dark .progress.is-dark::-moz-progress-bar, html.theme--documenter-dark .content kbd.progress::-moz-progress-bar { - background-color: #282f2f; } - html.theme--documenter-dark .progress.is-dark::-ms-fill, html.theme--documenter-dark .content kbd.progress::-ms-fill { - background-color: #282f2f; } - html.theme--documenter-dark .progress.is-dark:indeterminate, html.theme--documenter-dark .content kbd.progress:indeterminate { - background-image: linear-gradient(to right, #282f2f 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-primary::-webkit-progress-value, html.theme--documenter-dark .docstring > section > a.progress.docs-sourcelink::-webkit-progress-value { - background-color: #375a7f; } - html.theme--documenter-dark .progress.is-primary::-moz-progress-bar, html.theme--documenter-dark .docstring > section > a.progress.docs-sourcelink::-moz-progress-bar { - background-color: #375a7f; } - html.theme--documenter-dark .progress.is-primary::-ms-fill, html.theme--documenter-dark .docstring > section > a.progress.docs-sourcelink::-ms-fill { - background-color: #375a7f; } - html.theme--documenter-dark .progress.is-primary:indeterminate, html.theme--documenter-dark .docstring > section > a.progress.docs-sourcelink:indeterminate { - background-image: linear-gradient(to right, #375a7f 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-link::-webkit-progress-value { - background-color: #1abc9c; } - html.theme--documenter-dark .progress.is-link::-moz-progress-bar { - background-color: #1abc9c; } - html.theme--documenter-dark .progress.is-link::-ms-fill { - background-color: #1abc9c; } - html.theme--documenter-dark .progress.is-link:indeterminate { - background-image: linear-gradient(to right, #1abc9c 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-info::-webkit-progress-value { - background-color: #024c7d; } - html.theme--documenter-dark .progress.is-info::-moz-progress-bar { - background-color: #024c7d; } - html.theme--documenter-dark .progress.is-info::-ms-fill { - background-color: #024c7d; } - html.theme--documenter-dark .progress.is-info:indeterminate { - background-image: linear-gradient(to right, #024c7d 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-success::-webkit-progress-value { - background-color: #008438; } - html.theme--documenter-dark .progress.is-success::-moz-progress-bar { - background-color: #008438; } - html.theme--documenter-dark .progress.is-success::-ms-fill { - background-color: #008438; } - html.theme--documenter-dark .progress.is-success:indeterminate { - background-image: linear-gradient(to right, #008438 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-warning::-webkit-progress-value { - background-color: #ad8100; } - html.theme--documenter-dark .progress.is-warning::-moz-progress-bar { - background-color: #ad8100; } - html.theme--documenter-dark .progress.is-warning::-ms-fill { - background-color: #ad8100; } - html.theme--documenter-dark .progress.is-warning:indeterminate { - background-image: linear-gradient(to right, #ad8100 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress.is-danger::-webkit-progress-value { - background-color: #9e1b0d; } - html.theme--documenter-dark .progress.is-danger::-moz-progress-bar { - background-color: #9e1b0d; } - html.theme--documenter-dark .progress.is-danger::-ms-fill { - background-color: #9e1b0d; } - html.theme--documenter-dark .progress.is-danger:indeterminate { - background-image: linear-gradient(to right, #9e1b0d 30%, #5e6d6f 30%); } - html.theme--documenter-dark .progress:indeterminate { - animation-duration: 1.5s; - animation-iteration-count: infinite; - animation-name: moveIndeterminate; - animation-timing-function: linear; - background-color: #5e6d6f; - background-image: linear-gradient(to right, #fff 30%, #5e6d6f 30%); - background-position: top left; - background-repeat: no-repeat; - background-size: 150% 150%; } - html.theme--documenter-dark .progress:indeterminate::-webkit-progress-bar { - background-color: transparent; } - html.theme--documenter-dark .progress:indeterminate::-moz-progress-bar { - background-color: transparent; } - html.theme--documenter-dark .progress.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.progress { - height: 0.85em; } - html.theme--documenter-dark .progress.is-medium { - height: 1.25rem; } - html.theme--documenter-dark .progress.is-large { - height: 1.5rem; } - -@keyframes moveIndeterminate { - from { - background-position: 200% 0; } - to { - background-position: -200% 0; } } - html.theme--documenter-dark .table { - background-color: #343c3d; - color: #fff; } - html.theme--documenter-dark .table td, - html.theme--documenter-dark .table th { - border: 1px solid #5e6d6f; - border-width: 0 0 1px; - padding: 0.5em 0.75em; - vertical-align: top; } - html.theme--documenter-dark .table td.is-white, - html.theme--documenter-dark .table th.is-white { - background-color: white; - border-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .table td.is-black, - html.theme--documenter-dark .table th.is-black { - background-color: #0a0a0a; - border-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .table td.is-light, - html.theme--documenter-dark .table th.is-light { - background-color: #ecf0f1; - border-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .table td.is-dark, - html.theme--documenter-dark .table th.is-dark { - background-color: #282f2f; - border-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .table td.is-primary, - html.theme--documenter-dark .table th.is-primary { - background-color: #375a7f; - border-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .table td.is-link, - html.theme--documenter-dark .table th.is-link { - background-color: #1abc9c; - border-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .table td.is-info, - html.theme--documenter-dark .table th.is-info { - background-color: #024c7d; - border-color: #024c7d; - color: #fff; } - html.theme--documenter-dark .table td.is-success, - html.theme--documenter-dark .table th.is-success { - background-color: #008438; - border-color: #008438; - color: #fff; } - html.theme--documenter-dark .table td.is-warning, - html.theme--documenter-dark .table th.is-warning { - background-color: #ad8100; - border-color: #ad8100; - color: #fff; } - html.theme--documenter-dark .table td.is-danger, - html.theme--documenter-dark .table th.is-danger { - background-color: #9e1b0d; - border-color: #9e1b0d; - color: #fff; } - html.theme--documenter-dark .table td.is-narrow, - html.theme--documenter-dark .table th.is-narrow { - white-space: nowrap; - width: 1%; } - html.theme--documenter-dark .table td.is-selected, - html.theme--documenter-dark .table th.is-selected { - background-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .table td.is-selected a, - html.theme--documenter-dark .table td.is-selected strong, - html.theme--documenter-dark .table th.is-selected a, - html.theme--documenter-dark .table th.is-selected strong { - color: currentColor; } - html.theme--documenter-dark .table th { - color: #f2f2f2; } - html.theme--documenter-dark .table th:not([align]) { - text-align: left; } - html.theme--documenter-dark .table tr.is-selected { - background-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .table tr.is-selected a, - html.theme--documenter-dark .table tr.is-selected strong { - color: currentColor; } - html.theme--documenter-dark .table tr.is-selected td, - html.theme--documenter-dark .table tr.is-selected th { - border-color: #fff; - color: currentColor; } - html.theme--documenter-dark .table thead { - background-color: transparent; } - html.theme--documenter-dark .table thead td, - html.theme--documenter-dark .table thead th { - border-width: 0 0 2px; - color: #f2f2f2; } - html.theme--documenter-dark .table tfoot { - background-color: transparent; } - html.theme--documenter-dark .table tfoot td, - html.theme--documenter-dark .table tfoot th { - border-width: 2px 0 0; - color: #f2f2f2; } - html.theme--documenter-dark .table tbody { - background-color: transparent; } - html.theme--documenter-dark .table tbody tr:last-child td, - html.theme--documenter-dark .table tbody tr:last-child th { - border-bottom-width: 0; } - html.theme--documenter-dark .table.is-bordered td, - html.theme--documenter-dark .table.is-bordered th { - border-width: 1px; } - html.theme--documenter-dark .table.is-bordered tr:last-child td, - html.theme--documenter-dark .table.is-bordered tr:last-child th { - border-bottom-width: 1px; } - html.theme--documenter-dark .table.is-fullwidth { - width: 100%; } - html.theme--documenter-dark .table.is-hoverable tbody tr:not(.is-selected):hover { - background-color: #282f2f; } - html.theme--documenter-dark .table.is-hoverable.is-striped tbody tr:not(.is-selected):hover { - background-color: #282f2f; } - html.theme--documenter-dark .table.is-hoverable.is-striped tbody tr:not(.is-selected):hover:nth-child(even) { - background-color: #2d3435; } - html.theme--documenter-dark .table.is-narrow td, - html.theme--documenter-dark .table.is-narrow th { - padding: 0.25em 0.5em; } - html.theme--documenter-dark .table.is-striped tbody tr:not(.is-selected):nth-child(even) { - background-color: #282f2f; } - html.theme--documenter-dark .table-container { - -webkit-overflow-scrolling: touch; - overflow: auto; - overflow-y: hidden; - max-width: 100%; } - html.theme--documenter-dark .tags { - align-items: center; - display: flex; - flex-wrap: wrap; - justify-content: flex-start; } - html.theme--documenter-dark .tags .tag, html.theme--documenter-dark .tags .content kbd, html.theme--documenter-dark .content .tags kbd, html.theme--documenter-dark .tags .docstring > section > a.docs-sourcelink { - margin-bottom: 0.5rem; } - html.theme--documenter-dark .tags .tag:not(:last-child), html.theme--documenter-dark .tags .content kbd:not(:last-child), html.theme--documenter-dark .content .tags kbd:not(:last-child), html.theme--documenter-dark .tags .docstring > section > a.docs-sourcelink:not(:last-child) { - margin-right: 0.5rem; } - html.theme--documenter-dark .tags:last-child { - margin-bottom: -0.5rem; } - html.theme--documenter-dark .tags:not(:last-child) { - margin-bottom: 1rem; } - html.theme--documenter-dark .tags.are-medium .tag:not(.is-normal):not(.is-large), html.theme--documenter-dark .tags.are-medium .content kbd:not(.is-normal):not(.is-large), html.theme--documenter-dark .content .tags.are-medium kbd:not(.is-normal):not(.is-large), html.theme--documenter-dark .tags.are-medium .docstring > section > a.docs-sourcelink:not(.is-normal):not(.is-large) { - font-size: 15px; } - html.theme--documenter-dark .tags.are-large .tag:not(.is-normal):not(.is-medium), html.theme--documenter-dark .tags.are-large .content kbd:not(.is-normal):not(.is-medium), html.theme--documenter-dark .content .tags.are-large kbd:not(.is-normal):not(.is-medium), html.theme--documenter-dark .tags.are-large .docstring > section > a.docs-sourcelink:not(.is-normal):not(.is-medium) { - font-size: 1.25rem; } - html.theme--documenter-dark .tags.is-centered { - justify-content: center; } - html.theme--documenter-dark .tags.is-centered .tag, html.theme--documenter-dark .tags.is-centered .content kbd, html.theme--documenter-dark .content .tags.is-centered kbd, html.theme--documenter-dark .tags.is-centered .docstring > section > a.docs-sourcelink { - margin-right: 0.25rem; - margin-left: 0.25rem; } - html.theme--documenter-dark .tags.is-right { - justify-content: flex-end; } - html.theme--documenter-dark .tags.is-right .tag:not(:first-child), html.theme--documenter-dark .tags.is-right .content kbd:not(:first-child), html.theme--documenter-dark .content .tags.is-right kbd:not(:first-child), html.theme--documenter-dark .tags.is-right .docstring > section > a.docs-sourcelink:not(:first-child) { - margin-left: 0.5rem; } - html.theme--documenter-dark .tags.is-right .tag:not(:last-child), html.theme--documenter-dark .tags.is-right .content kbd:not(:last-child), html.theme--documenter-dark .content .tags.is-right kbd:not(:last-child), html.theme--documenter-dark .tags.is-right .docstring > section > a.docs-sourcelink:not(:last-child) { - margin-right: 0; } - html.theme--documenter-dark .tags.has-addons .tag, html.theme--documenter-dark .tags.has-addons .content kbd, html.theme--documenter-dark .content .tags.has-addons kbd, html.theme--documenter-dark .tags.has-addons .docstring > section > a.docs-sourcelink { - margin-right: 0; } - html.theme--documenter-dark .tags.has-addons .tag:not(:first-child), html.theme--documenter-dark .tags.has-addons .content kbd:not(:first-child), html.theme--documenter-dark .content .tags.has-addons kbd:not(:first-child), html.theme--documenter-dark .tags.has-addons .docstring > section > a.docs-sourcelink:not(:first-child) { - margin-left: 0; - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - html.theme--documenter-dark .tags.has-addons .tag:not(:last-child), html.theme--documenter-dark .tags.has-addons .content kbd:not(:last-child), html.theme--documenter-dark .content .tags.has-addons kbd:not(:last-child), html.theme--documenter-dark .tags.has-addons .docstring > section > a.docs-sourcelink:not(:last-child) { - border-bottom-right-radius: 0; - border-top-right-radius: 0; } - html.theme--documenter-dark .tag:not(body), html.theme--documenter-dark .content kbd:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body) { - align-items: center; - background-color: #282f2f; - border-radius: 0.4em; - color: #fff; - display: inline-flex; - font-size: 0.85em; - height: 2em; - justify-content: center; - line-height: 1.5; - padding-left: 0.75em; - padding-right: 0.75em; - white-space: nowrap; } - html.theme--documenter-dark .tag:not(body) .delete, html.theme--documenter-dark .content kbd:not(body) .delete, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body) .delete { - margin-left: 0.25rem; - margin-right: -0.375rem; } - html.theme--documenter-dark .tag.is-white:not(body), html.theme--documenter-dark .content kbd.is-white:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-white:not(body) { - background-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .tag.is-black:not(body), html.theme--documenter-dark .content kbd.is-black:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-black:not(body) { - background-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .tag.is-light:not(body), html.theme--documenter-dark .content kbd.is-light:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-light:not(body) { - background-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .tag.is-dark:not(body), html.theme--documenter-dark .content kbd:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-dark:not(body), html.theme--documenter-dark .content .docstring > section > kbd:not(body) { - background-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .tag.is-primary:not(body), html.theme--documenter-dark .content kbd.is-primary:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body) { - background-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .tag.is-link:not(body), html.theme--documenter-dark .content kbd.is-link:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-link:not(body) { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .tag.is-info:not(body), html.theme--documenter-dark .content kbd.is-info:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-info:not(body) { - background-color: #024c7d; - color: #fff; } - html.theme--documenter-dark .tag.is-success:not(body), html.theme--documenter-dark .content kbd.is-success:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-success:not(body) { - background-color: #008438; - color: #fff; } - html.theme--documenter-dark .tag.is-warning:not(body), html.theme--documenter-dark .content kbd.is-warning:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-warning:not(body) { - background-color: #ad8100; - color: #fff; } - html.theme--documenter-dark .tag.is-danger:not(body), html.theme--documenter-dark .content kbd.is-danger:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-danger:not(body) { - background-color: #9e1b0d; - color: #fff; } - html.theme--documenter-dark .tag.is-normal:not(body), html.theme--documenter-dark .content kbd.is-normal:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-normal:not(body) { - font-size: 0.85em; } - html.theme--documenter-dark .tag.is-medium:not(body), html.theme--documenter-dark .content kbd.is-medium:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-medium:not(body) { - font-size: 15px; } - html.theme--documenter-dark .tag.is-large:not(body), html.theme--documenter-dark .content kbd.is-large:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-large:not(body) { - font-size: 1.25rem; } - html.theme--documenter-dark .tag:not(body) .icon:first-child:not(:last-child), html.theme--documenter-dark .content kbd:not(body) .icon:first-child:not(:last-child), html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body) .icon:first-child:not(:last-child) { - margin-left: -0.375em; - margin-right: 0.1875em; } - html.theme--documenter-dark .tag:not(body) .icon:last-child:not(:first-child), html.theme--documenter-dark .content kbd:not(body) .icon:last-child:not(:first-child), html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body) .icon:last-child:not(:first-child) { - margin-left: 0.1875em; - margin-right: -0.375em; } - html.theme--documenter-dark .tag:not(body) .icon:first-child:last-child, html.theme--documenter-dark .content kbd:not(body) .icon:first-child:last-child, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:not(body) .icon:first-child:last-child { - margin-left: -0.375em; - margin-right: -0.375em; } - html.theme--documenter-dark .tag.is-delete:not(body), html.theme--documenter-dark .content kbd.is-delete:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body) { - margin-left: 1px; - padding: 0; - position: relative; - width: 2em; } - html.theme--documenter-dark .tag.is-delete:not(body)::before, html.theme--documenter-dark .content kbd.is-delete:not(body)::before, html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body)::before, html.theme--documenter-dark .tag.is-delete:not(body)::after, html.theme--documenter-dark .content kbd.is-delete:not(body)::after, html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body)::after { - background-color: currentColor; - content: ""; - display: block; - left: 50%; - position: absolute; - top: 50%; - transform: translateX(-50%) translateY(-50%) rotate(45deg); - transform-origin: center center; } - html.theme--documenter-dark .tag.is-delete:not(body)::before, html.theme--documenter-dark .content kbd.is-delete:not(body)::before, html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body)::before { - height: 1px; - width: 50%; } - html.theme--documenter-dark .tag.is-delete:not(body)::after, html.theme--documenter-dark .content kbd.is-delete:not(body)::after, html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body)::after { - height: 50%; - width: 1px; } - html.theme--documenter-dark .tag.is-delete:not(body):hover, html.theme--documenter-dark .content kbd.is-delete:not(body):hover, html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body):hover, html.theme--documenter-dark .tag.is-delete:not(body):focus, html.theme--documenter-dark .content kbd.is-delete:not(body):focus, html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body):focus { - background-color: #1d2122; } - html.theme--documenter-dark .tag.is-delete:not(body):active, html.theme--documenter-dark .content kbd.is-delete:not(body):active, html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-delete:not(body):active { - background-color: #111414; } - html.theme--documenter-dark .tag.is-rounded:not(body), html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:not(body), html.theme--documenter-dark .content kbd.is-rounded:not(body), html.theme--documenter-dark #documenter .docs-sidebar .content form.docs-search > input:not(body), html.theme--documenter-dark .docstring > section > a.docs-sourcelink.is-rounded:not(body) { - border-radius: 290486px; } - html.theme--documenter-dark a.tag:hover, html.theme--documenter-dark .docstring > section > a.docs-sourcelink:hover { - text-decoration: underline; } - html.theme--documenter-dark .title, - html.theme--documenter-dark .subtitle { - word-break: break-word; } - html.theme--documenter-dark .title em, - html.theme--documenter-dark .title span, - html.theme--documenter-dark .subtitle em, - html.theme--documenter-dark .subtitle span { - font-weight: inherit; } - html.theme--documenter-dark .title sub, - html.theme--documenter-dark .subtitle sub { - font-size: 0.75em; } - html.theme--documenter-dark .title sup, - html.theme--documenter-dark .subtitle sup { - font-size: 0.75em; } - html.theme--documenter-dark .title .tag, html.theme--documenter-dark .title .content kbd, html.theme--documenter-dark .content .title kbd, html.theme--documenter-dark .title .docstring > section > a.docs-sourcelink, - html.theme--documenter-dark .subtitle .tag, - html.theme--documenter-dark .subtitle .content kbd, - html.theme--documenter-dark .content .subtitle kbd, - html.theme--documenter-dark .subtitle .docstring > section > a.docs-sourcelink { - vertical-align: middle; } - html.theme--documenter-dark .title { - color: #fff; - font-size: 2rem; - font-weight: 500; - line-height: 1.125; } - html.theme--documenter-dark .title strong { - color: inherit; - font-weight: inherit; } - html.theme--documenter-dark .title + .highlight { - margin-top: -0.75rem; } - html.theme--documenter-dark .title:not(.is-spaced) + .subtitle { - margin-top: -1.25rem; } - html.theme--documenter-dark .title.is-1 { - font-size: 3rem; } - html.theme--documenter-dark .title.is-2 { - font-size: 2.5rem; } - html.theme--documenter-dark .title.is-3 { - font-size: 2rem; } - html.theme--documenter-dark .title.is-4 { - font-size: 1.5rem; } - html.theme--documenter-dark .title.is-5 { - font-size: 1.25rem; } - html.theme--documenter-dark .title.is-6 { - font-size: 15px; } - html.theme--documenter-dark .title.is-7 { - font-size: 0.85em; } - html.theme--documenter-dark .subtitle { - color: #8c9b9d; - font-size: 1.25rem; - font-weight: 400; - line-height: 1.25; } - html.theme--documenter-dark .subtitle strong { - color: #8c9b9d; - font-weight: 600; } - html.theme--documenter-dark .subtitle:not(.is-spaced) + .title { - margin-top: -1.25rem; } - html.theme--documenter-dark .subtitle.is-1 { - font-size: 3rem; } - html.theme--documenter-dark .subtitle.is-2 { - font-size: 2.5rem; } - html.theme--documenter-dark .subtitle.is-3 { - font-size: 2rem; } - html.theme--documenter-dark .subtitle.is-4 { - font-size: 1.5rem; } - html.theme--documenter-dark .subtitle.is-5 { - font-size: 1.25rem; } - html.theme--documenter-dark .subtitle.is-6 { - font-size: 15px; } - html.theme--documenter-dark .subtitle.is-7 { - font-size: 0.85em; } - html.theme--documenter-dark .heading { - display: block; - font-size: 11px; - letter-spacing: 1px; - margin-bottom: 5px; - text-transform: uppercase; } - html.theme--documenter-dark .highlight { - font-weight: 400; - max-width: 100%; - overflow: hidden; - padding: 0; } - html.theme--documenter-dark .highlight pre { - overflow: auto; - max-width: 100%; } - html.theme--documenter-dark .number { - align-items: center; - background-color: #282f2f; - border-radius: 290486px; - display: inline-flex; - font-size: 1.25rem; - height: 2em; - justify-content: center; - margin-right: 1.5rem; - min-width: 2.5em; - padding: 0.25rem 0.5rem; - text-align: center; - vertical-align: top; } - html.theme--documenter-dark .select select, html.theme--documenter-dark .textarea, html.theme--documenter-dark .input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input { - background-color: #1f2424; - border-color: #5e6d6f; - border-radius: 0.4em; - color: #dbdee0; } - html.theme--documenter-dark .select select::-moz-placeholder, html.theme--documenter-dark .textarea::-moz-placeholder, html.theme--documenter-dark .input::-moz-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input::-moz-placeholder { - color: rgba(219, 222, 224, 0.3); } - html.theme--documenter-dark .select select::-webkit-input-placeholder, html.theme--documenter-dark .textarea::-webkit-input-placeholder, html.theme--documenter-dark .input::-webkit-input-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input::-webkit-input-placeholder { - color: rgba(219, 222, 224, 0.3); } - html.theme--documenter-dark .select select:-moz-placeholder, html.theme--documenter-dark .textarea:-moz-placeholder, html.theme--documenter-dark .input:-moz-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:-moz-placeholder { - color: rgba(219, 222, 224, 0.3); } - html.theme--documenter-dark .select select:-ms-input-placeholder, html.theme--documenter-dark .textarea:-ms-input-placeholder, html.theme--documenter-dark .input:-ms-input-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:-ms-input-placeholder { - color: rgba(219, 222, 224, 0.3); } - html.theme--documenter-dark .select select:hover, html.theme--documenter-dark .textarea:hover, html.theme--documenter-dark .input:hover, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:hover, html.theme--documenter-dark .select select.is-hovered, html.theme--documenter-dark .is-hovered.textarea, html.theme--documenter-dark .is-hovered.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-hovered { - border-color: #8c9b9d; } - html.theme--documenter-dark .select select:focus, html.theme--documenter-dark .textarea:focus, html.theme--documenter-dark .input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:focus, html.theme--documenter-dark .select select.is-focused, html.theme--documenter-dark .is-focused.textarea, html.theme--documenter-dark .is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .select select:active, html.theme--documenter-dark .textarea:active, html.theme--documenter-dark .input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:active, html.theme--documenter-dark .select select.is-active, html.theme--documenter-dark .is-active.textarea, html.theme--documenter-dark .is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - border-color: #1abc9c; - box-shadow: 0 0 0 0.125em rgba(26, 188, 156, 0.25); } - html.theme--documenter-dark .select select[disabled], html.theme--documenter-dark .textarea[disabled], html.theme--documenter-dark .input[disabled], html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[disabled], fieldset[disabled] html.theme--documenter-dark .select select, fieldset[disabled] html.theme--documenter-dark .textarea, fieldset[disabled] html.theme--documenter-dark .input, fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input { - background-color: #8c9b9d; - border-color: #282f2f; - box-shadow: none; - color: white; } - html.theme--documenter-dark .select select[disabled]::-moz-placeholder, html.theme--documenter-dark .textarea[disabled]::-moz-placeholder, html.theme--documenter-dark .input[disabled]::-moz-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[disabled]::-moz-placeholder, fieldset[disabled] html.theme--documenter-dark .select select::-moz-placeholder, fieldset[disabled] html.theme--documenter-dark .textarea::-moz-placeholder, fieldset[disabled] html.theme--documenter-dark .input::-moz-placeholder, fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input::-moz-placeholder { - color: rgba(255, 255, 255, 0.3); } - html.theme--documenter-dark .select select[disabled]::-webkit-input-placeholder, html.theme--documenter-dark .textarea[disabled]::-webkit-input-placeholder, html.theme--documenter-dark .input[disabled]::-webkit-input-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[disabled]::-webkit-input-placeholder, fieldset[disabled] html.theme--documenter-dark .select select::-webkit-input-placeholder, fieldset[disabled] html.theme--documenter-dark .textarea::-webkit-input-placeholder, fieldset[disabled] html.theme--documenter-dark .input::-webkit-input-placeholder, fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input::-webkit-input-placeholder { - color: rgba(255, 255, 255, 0.3); } - html.theme--documenter-dark .select select[disabled]:-moz-placeholder, html.theme--documenter-dark .textarea[disabled]:-moz-placeholder, html.theme--documenter-dark .input[disabled]:-moz-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[disabled]:-moz-placeholder, fieldset[disabled] html.theme--documenter-dark .select select:-moz-placeholder, fieldset[disabled] html.theme--documenter-dark .textarea:-moz-placeholder, fieldset[disabled] html.theme--documenter-dark .input:-moz-placeholder, fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:-moz-placeholder { - color: rgba(255, 255, 255, 0.3); } - html.theme--documenter-dark .select select[disabled]:-ms-input-placeholder, html.theme--documenter-dark .textarea[disabled]:-ms-input-placeholder, html.theme--documenter-dark .input[disabled]:-ms-input-placeholder, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[disabled]:-ms-input-placeholder, fieldset[disabled] html.theme--documenter-dark .select select:-ms-input-placeholder, fieldset[disabled] html.theme--documenter-dark .textarea:-ms-input-placeholder, fieldset[disabled] html.theme--documenter-dark .input:-ms-input-placeholder, fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input:-ms-input-placeholder { - color: rgba(255, 255, 255, 0.3); } - html.theme--documenter-dark .textarea, html.theme--documenter-dark .input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input { - box-shadow: inset 0 1px 2px rgba(10, 10, 10, 0.1); - max-width: 100%; - width: 100%; } - html.theme--documenter-dark .textarea[readonly], html.theme--documenter-dark .input[readonly], html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input[readonly] { - box-shadow: none; } - html.theme--documenter-dark .is-white.textarea, html.theme--documenter-dark .is-white.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-white { - border-color: white; } - html.theme--documenter-dark .is-white.textarea:focus, html.theme--documenter-dark .is-white.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-white:focus, html.theme--documenter-dark .is-white.is-focused.textarea, html.theme--documenter-dark .is-white.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-white.textarea:active, html.theme--documenter-dark .is-white.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-white:active, html.theme--documenter-dark .is-white.is-active.textarea, html.theme--documenter-dark .is-white.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); } - html.theme--documenter-dark .is-black.textarea, html.theme--documenter-dark .is-black.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-black { - border-color: #0a0a0a; } - html.theme--documenter-dark .is-black.textarea:focus, html.theme--documenter-dark .is-black.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-black:focus, html.theme--documenter-dark .is-black.is-focused.textarea, html.theme--documenter-dark .is-black.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-black.textarea:active, html.theme--documenter-dark .is-black.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-black:active, html.theme--documenter-dark .is-black.is-active.textarea, html.theme--documenter-dark .is-black.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); } - html.theme--documenter-dark .is-light.textarea, html.theme--documenter-dark .is-light.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-light { - border-color: #ecf0f1; } - html.theme--documenter-dark .is-light.textarea:focus, html.theme--documenter-dark .is-light.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-light:focus, html.theme--documenter-dark .is-light.is-focused.textarea, html.theme--documenter-dark .is-light.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-light.textarea:active, html.theme--documenter-dark .is-light.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-light:active, html.theme--documenter-dark .is-light.is-active.textarea, html.theme--documenter-dark .is-light.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(236, 240, 241, 0.25); } - html.theme--documenter-dark .is-dark.textarea, html.theme--documenter-dark .content kbd.textarea, html.theme--documenter-dark .is-dark.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-dark, html.theme--documenter-dark .content kbd.input { - border-color: #282f2f; } - html.theme--documenter-dark .is-dark.textarea:focus, html.theme--documenter-dark .content kbd.textarea:focus, html.theme--documenter-dark .is-dark.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-dark:focus, html.theme--documenter-dark .content kbd.input:focus, html.theme--documenter-dark .is-dark.is-focused.textarea, html.theme--documenter-dark .content kbd.is-focused.textarea, html.theme--documenter-dark .is-dark.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .content kbd.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar .content form.docs-search > input.is-focused, html.theme--documenter-dark .is-dark.textarea:active, html.theme--documenter-dark .content kbd.textarea:active, html.theme--documenter-dark .is-dark.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-dark:active, html.theme--documenter-dark .content kbd.input:active, html.theme--documenter-dark .is-dark.is-active.textarea, html.theme--documenter-dark .content kbd.is-active.textarea, html.theme--documenter-dark .is-dark.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active, html.theme--documenter-dark .content kbd.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar .content form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(40, 47, 47, 0.25); } - html.theme--documenter-dark .is-primary.textarea, html.theme--documenter-dark .docstring > section > a.textarea.docs-sourcelink, html.theme--documenter-dark .is-primary.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-primary, html.theme--documenter-dark .docstring > section > a.input.docs-sourcelink { - border-color: #375a7f; } - html.theme--documenter-dark .is-primary.textarea:focus, html.theme--documenter-dark .docstring > section > a.textarea.docs-sourcelink:focus, html.theme--documenter-dark .is-primary.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-primary:focus, html.theme--documenter-dark .docstring > section > a.input.docs-sourcelink:focus, html.theme--documenter-dark .is-primary.is-focused.textarea, html.theme--documenter-dark .docstring > section > a.is-focused.textarea.docs-sourcelink, html.theme--documenter-dark .is-primary.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .docstring > section > a.is-focused.input.docs-sourcelink, html.theme--documenter-dark .is-primary.textarea:active, html.theme--documenter-dark .docstring > section > a.textarea.docs-sourcelink:active, html.theme--documenter-dark .is-primary.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-primary:active, html.theme--documenter-dark .docstring > section > a.input.docs-sourcelink:active, html.theme--documenter-dark .is-primary.is-active.textarea, html.theme--documenter-dark .docstring > section > a.is-active.textarea.docs-sourcelink, html.theme--documenter-dark .is-primary.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active, html.theme--documenter-dark .docstring > section > a.is-active.input.docs-sourcelink { - box-shadow: 0 0 0 0.125em rgba(55, 90, 127, 0.25); } - html.theme--documenter-dark .is-link.textarea, html.theme--documenter-dark .is-link.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-link { - border-color: #1abc9c; } - html.theme--documenter-dark .is-link.textarea:focus, html.theme--documenter-dark .is-link.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-link:focus, html.theme--documenter-dark .is-link.is-focused.textarea, html.theme--documenter-dark .is-link.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-link.textarea:active, html.theme--documenter-dark .is-link.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-link:active, html.theme--documenter-dark .is-link.is-active.textarea, html.theme--documenter-dark .is-link.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(26, 188, 156, 0.25); } - html.theme--documenter-dark .is-info.textarea, html.theme--documenter-dark .is-info.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-info { - border-color: #024c7d; } - html.theme--documenter-dark .is-info.textarea:focus, html.theme--documenter-dark .is-info.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-info:focus, html.theme--documenter-dark .is-info.is-focused.textarea, html.theme--documenter-dark .is-info.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-info.textarea:active, html.theme--documenter-dark .is-info.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-info:active, html.theme--documenter-dark .is-info.is-active.textarea, html.theme--documenter-dark .is-info.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(2, 76, 125, 0.25); } - html.theme--documenter-dark .is-success.textarea, html.theme--documenter-dark .is-success.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-success { - border-color: #008438; } - html.theme--documenter-dark .is-success.textarea:focus, html.theme--documenter-dark .is-success.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-success:focus, html.theme--documenter-dark .is-success.is-focused.textarea, html.theme--documenter-dark .is-success.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-success.textarea:active, html.theme--documenter-dark .is-success.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-success:active, html.theme--documenter-dark .is-success.is-active.textarea, html.theme--documenter-dark .is-success.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(0, 132, 56, 0.25); } - html.theme--documenter-dark .is-warning.textarea, html.theme--documenter-dark .is-warning.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-warning { - border-color: #ad8100; } - html.theme--documenter-dark .is-warning.textarea:focus, html.theme--documenter-dark .is-warning.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-warning:focus, html.theme--documenter-dark .is-warning.is-focused.textarea, html.theme--documenter-dark .is-warning.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-warning.textarea:active, html.theme--documenter-dark .is-warning.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-warning:active, html.theme--documenter-dark .is-warning.is-active.textarea, html.theme--documenter-dark .is-warning.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(173, 129, 0, 0.25); } - html.theme--documenter-dark .is-danger.textarea, html.theme--documenter-dark .is-danger.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-danger { - border-color: #9e1b0d; } - html.theme--documenter-dark .is-danger.textarea:focus, html.theme--documenter-dark .is-danger.input:focus, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-danger:focus, html.theme--documenter-dark .is-danger.is-focused.textarea, html.theme--documenter-dark .is-danger.is-focused.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-focused, html.theme--documenter-dark .is-danger.textarea:active, html.theme--documenter-dark .is-danger.input:active, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-danger:active, html.theme--documenter-dark .is-danger.is-active.textarea, html.theme--documenter-dark .is-danger.is-active.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(158, 27, 13, 0.25); } - html.theme--documenter-dark .is-small.textarea, html.theme--documenter-dark .is-small.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input { - border-radius: 3px; - font-size: 0.85em; } - html.theme--documenter-dark .is-medium.textarea, html.theme--documenter-dark .is-medium.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .is-large.textarea, html.theme--documenter-dark .is-large.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .is-fullwidth.textarea, html.theme--documenter-dark .is-fullwidth.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-fullwidth { - display: block; - width: 100%; } - html.theme--documenter-dark .is-inline.textarea, html.theme--documenter-dark .is-inline.input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-inline { - display: inline; - width: auto; } - html.theme--documenter-dark .input.is-rounded, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input { - border-radius: 290486px; - padding-left: 1em; - padding-right: 1em; } - html.theme--documenter-dark .input.is-static, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-static { - background-color: transparent; - border-color: transparent; - box-shadow: none; - padding-left: 0; - padding-right: 0; } - html.theme--documenter-dark .textarea { - display: block; - max-width: 100%; - min-width: 100%; - padding: 0.625em; - resize: vertical; } - html.theme--documenter-dark .textarea:not([rows]) { - max-height: 600px; - min-height: 120px; } - html.theme--documenter-dark .textarea[rows] { - height: initial; } - html.theme--documenter-dark .textarea.has-fixed-size { - resize: none; } - html.theme--documenter-dark .radio, html.theme--documenter-dark .checkbox { - cursor: pointer; - display: inline-block; - line-height: 1.25; - position: relative; } - html.theme--documenter-dark .radio input, html.theme--documenter-dark .checkbox input { - cursor: pointer; } - html.theme--documenter-dark .radio:hover, html.theme--documenter-dark .checkbox:hover { - color: #8c9b9d; } - html.theme--documenter-dark .radio[disabled], html.theme--documenter-dark .checkbox[disabled], fieldset[disabled] html.theme--documenter-dark .radio, fieldset[disabled] html.theme--documenter-dark .checkbox { - color: white; - cursor: not-allowed; } - html.theme--documenter-dark .radio + .radio { - margin-left: 0.5em; } - html.theme--documenter-dark .select { - display: inline-block; - max-width: 100%; - position: relative; - vertical-align: top; } - html.theme--documenter-dark .select:not(.is-multiple) { - height: 2.25em; } - html.theme--documenter-dark .select:not(.is-multiple):not(.is-loading)::after { - border-color: #1abc9c; - right: 1.125em; - z-index: 4; } - html.theme--documenter-dark .select.is-rounded select, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.select select { - border-radius: 290486px; - padding-left: 1em; } - html.theme--documenter-dark .select select { - cursor: pointer; - display: block; - font-size: 1em; - max-width: 100%; - outline: none; } - html.theme--documenter-dark .select select::-ms-expand { - display: none; } - html.theme--documenter-dark .select select[disabled]:hover, fieldset[disabled] html.theme--documenter-dark .select select:hover { - border-color: #282f2f; } - html.theme--documenter-dark .select select:not([multiple]) { - padding-right: 2.5em; } - html.theme--documenter-dark .select select[multiple] { - height: auto; - padding: 0; } - html.theme--documenter-dark .select select[multiple] option { - padding: 0.5em 1em; } - html.theme--documenter-dark .select:not(.is-multiple):not(.is-loading):hover::after { - border-color: #8c9b9d; } - html.theme--documenter-dark .select.is-white:not(:hover)::after { - border-color: white; } - html.theme--documenter-dark .select.is-white select { - border-color: white; } - html.theme--documenter-dark .select.is-white select:hover, html.theme--documenter-dark .select.is-white select.is-hovered { - border-color: #f2f2f2; } - html.theme--documenter-dark .select.is-white select:focus, html.theme--documenter-dark .select.is-white select.is-focused, html.theme--documenter-dark .select.is-white select:active, html.theme--documenter-dark .select.is-white select.is-active { - box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); } - html.theme--documenter-dark .select.is-black:not(:hover)::after { - border-color: #0a0a0a; } - html.theme--documenter-dark .select.is-black select { - border-color: #0a0a0a; } - html.theme--documenter-dark .select.is-black select:hover, html.theme--documenter-dark .select.is-black select.is-hovered { - border-color: black; } - html.theme--documenter-dark .select.is-black select:focus, html.theme--documenter-dark .select.is-black select.is-focused, html.theme--documenter-dark .select.is-black select:active, html.theme--documenter-dark .select.is-black select.is-active { - box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); } - html.theme--documenter-dark .select.is-light:not(:hover)::after { - border-color: #ecf0f1; } - html.theme--documenter-dark .select.is-light select { - border-color: #ecf0f1; } - html.theme--documenter-dark .select.is-light select:hover, html.theme--documenter-dark .select.is-light select.is-hovered { - border-color: #dde4e6; } - html.theme--documenter-dark .select.is-light select:focus, html.theme--documenter-dark .select.is-light select.is-focused, html.theme--documenter-dark .select.is-light select:active, html.theme--documenter-dark .select.is-light select.is-active { - box-shadow: 0 0 0 0.125em rgba(236, 240, 241, 0.25); } - html.theme--documenter-dark .select.is-dark:not(:hover)::after, html.theme--documenter-dark .content kbd.select:not(:hover)::after { - border-color: #282f2f; } - html.theme--documenter-dark .select.is-dark select, html.theme--documenter-dark .content kbd.select select { - border-color: #282f2f; } - html.theme--documenter-dark .select.is-dark select:hover, html.theme--documenter-dark .content kbd.select select:hover, html.theme--documenter-dark .select.is-dark select.is-hovered, html.theme--documenter-dark .content kbd.select select.is-hovered { - border-color: #1d2122; } - html.theme--documenter-dark .select.is-dark select:focus, html.theme--documenter-dark .content kbd.select select:focus, html.theme--documenter-dark .select.is-dark select.is-focused, html.theme--documenter-dark .content kbd.select select.is-focused, html.theme--documenter-dark .select.is-dark select:active, html.theme--documenter-dark .content kbd.select select:active, html.theme--documenter-dark .select.is-dark select.is-active, html.theme--documenter-dark .content kbd.select select.is-active { - box-shadow: 0 0 0 0.125em rgba(40, 47, 47, 0.25); } - html.theme--documenter-dark .select.is-primary:not(:hover)::after, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink:not(:hover)::after { - border-color: #375a7f; } - html.theme--documenter-dark .select.is-primary select, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select { - border-color: #375a7f; } - html.theme--documenter-dark .select.is-primary select:hover, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select:hover, html.theme--documenter-dark .select.is-primary select.is-hovered, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select.is-hovered { - border-color: #2f4d6d; } - html.theme--documenter-dark .select.is-primary select:focus, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select:focus, html.theme--documenter-dark .select.is-primary select.is-focused, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select.is-focused, html.theme--documenter-dark .select.is-primary select:active, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select:active, html.theme--documenter-dark .select.is-primary select.is-active, html.theme--documenter-dark .docstring > section > a.select.docs-sourcelink select.is-active { - box-shadow: 0 0 0 0.125em rgba(55, 90, 127, 0.25); } - html.theme--documenter-dark .select.is-link:not(:hover)::after { - border-color: #1abc9c; } - html.theme--documenter-dark .select.is-link select { - border-color: #1abc9c; } - html.theme--documenter-dark .select.is-link select:hover, html.theme--documenter-dark .select.is-link select.is-hovered { - border-color: #17a689; } - html.theme--documenter-dark .select.is-link select:focus, html.theme--documenter-dark .select.is-link select.is-focused, html.theme--documenter-dark .select.is-link select:active, html.theme--documenter-dark .select.is-link select.is-active { - box-shadow: 0 0 0 0.125em rgba(26, 188, 156, 0.25); } - html.theme--documenter-dark .select.is-info:not(:hover)::after { - border-color: #024c7d; } - html.theme--documenter-dark .select.is-info select { - border-color: #024c7d; } - html.theme--documenter-dark .select.is-info select:hover, html.theme--documenter-dark .select.is-info select.is-hovered { - border-color: #023d64; } - html.theme--documenter-dark .select.is-info select:focus, html.theme--documenter-dark .select.is-info select.is-focused, html.theme--documenter-dark .select.is-info select:active, html.theme--documenter-dark .select.is-info select.is-active { - box-shadow: 0 0 0 0.125em rgba(2, 76, 125, 0.25); } - html.theme--documenter-dark .select.is-success:not(:hover)::after { - border-color: #008438; } - html.theme--documenter-dark .select.is-success select { - border-color: #008438; } - html.theme--documenter-dark .select.is-success select:hover, html.theme--documenter-dark .select.is-success select.is-hovered { - border-color: #006b2d; } - html.theme--documenter-dark .select.is-success select:focus, html.theme--documenter-dark .select.is-success select.is-focused, html.theme--documenter-dark .select.is-success select:active, html.theme--documenter-dark .select.is-success select.is-active { - box-shadow: 0 0 0 0.125em rgba(0, 132, 56, 0.25); } - html.theme--documenter-dark .select.is-warning:not(:hover)::after { - border-color: #ad8100; } - html.theme--documenter-dark .select.is-warning select { - border-color: #ad8100; } - html.theme--documenter-dark .select.is-warning select:hover, html.theme--documenter-dark .select.is-warning select.is-hovered { - border-color: #946e00; } - html.theme--documenter-dark .select.is-warning select:focus, html.theme--documenter-dark .select.is-warning select.is-focused, html.theme--documenter-dark .select.is-warning select:active, html.theme--documenter-dark .select.is-warning select.is-active { - box-shadow: 0 0 0 0.125em rgba(173, 129, 0, 0.25); } - html.theme--documenter-dark .select.is-danger:not(:hover)::after { - border-color: #9e1b0d; } - html.theme--documenter-dark .select.is-danger select { - border-color: #9e1b0d; } - html.theme--documenter-dark .select.is-danger select:hover, html.theme--documenter-dark .select.is-danger select.is-hovered { - border-color: #86170b; } - html.theme--documenter-dark .select.is-danger select:focus, html.theme--documenter-dark .select.is-danger select.is-focused, html.theme--documenter-dark .select.is-danger select:active, html.theme--documenter-dark .select.is-danger select.is-active { - box-shadow: 0 0 0 0.125em rgba(158, 27, 13, 0.25); } - html.theme--documenter-dark .select.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.select { - border-radius: 3px; - font-size: 0.85em; } - html.theme--documenter-dark .select.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .select.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .select.is-disabled::after { - border-color: white; } - html.theme--documenter-dark .select.is-fullwidth { - width: 100%; } - html.theme--documenter-dark .select.is-fullwidth select { - width: 100%; } - html.theme--documenter-dark .select.is-loading::after { - margin-top: 0; - position: absolute; - right: 0.625em; - top: 0.625em; - transform: none; } - html.theme--documenter-dark .select.is-loading.is-small:after, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-loading:after { - font-size: 0.85em; } - html.theme--documenter-dark .select.is-loading.is-medium:after { - font-size: 1.25rem; } - html.theme--documenter-dark .select.is-loading.is-large:after { - font-size: 1.5rem; } - html.theme--documenter-dark .file { - align-items: stretch; - display: flex; - justify-content: flex-start; - position: relative; } - html.theme--documenter-dark .file.is-white .file-cta { - background-color: white; - border-color: transparent; - color: #0a0a0a; } - html.theme--documenter-dark .file.is-white:hover .file-cta, html.theme--documenter-dark .file.is-white.is-hovered .file-cta { - background-color: #f9f9f9; - border-color: transparent; - color: #0a0a0a; } - html.theme--documenter-dark .file.is-white:focus .file-cta, html.theme--documenter-dark .file.is-white.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(255, 255, 255, 0.25); - color: #0a0a0a; } - html.theme--documenter-dark .file.is-white:active .file-cta, html.theme--documenter-dark .file.is-white.is-active .file-cta { - background-color: #f2f2f2; - border-color: transparent; - color: #0a0a0a; } - html.theme--documenter-dark .file.is-black .file-cta { - background-color: #0a0a0a; - border-color: transparent; - color: white; } - html.theme--documenter-dark .file.is-black:hover .file-cta, html.theme--documenter-dark .file.is-black.is-hovered .file-cta { - background-color: #040404; - border-color: transparent; - color: white; } - html.theme--documenter-dark .file.is-black:focus .file-cta, html.theme--documenter-dark .file.is-black.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(10, 10, 10, 0.25); - color: white; } - html.theme--documenter-dark .file.is-black:active .file-cta, html.theme--documenter-dark .file.is-black.is-active .file-cta { - background-color: black; - border-color: transparent; - color: white; } - html.theme--documenter-dark .file.is-light .file-cta { - background-color: #ecf0f1; - border-color: transparent; - color: #282f2f; } - html.theme--documenter-dark .file.is-light:hover .file-cta, html.theme--documenter-dark .file.is-light.is-hovered .file-cta { - background-color: #e5eaec; - border-color: transparent; - color: #282f2f; } - html.theme--documenter-dark .file.is-light:focus .file-cta, html.theme--documenter-dark .file.is-light.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(236, 240, 241, 0.25); - color: #282f2f; } - html.theme--documenter-dark .file.is-light:active .file-cta, html.theme--documenter-dark .file.is-light.is-active .file-cta { - background-color: #dde4e6; - border-color: transparent; - color: #282f2f; } - html.theme--documenter-dark .file.is-dark .file-cta, html.theme--documenter-dark .content kbd.file .file-cta { - background-color: #282f2f; - border-color: transparent; - color: #ecf0f1; } - html.theme--documenter-dark .file.is-dark:hover .file-cta, html.theme--documenter-dark .content kbd.file:hover .file-cta, html.theme--documenter-dark .file.is-dark.is-hovered .file-cta, html.theme--documenter-dark .content kbd.file.is-hovered .file-cta { - background-color: #232829; - border-color: transparent; - color: #ecf0f1; } - html.theme--documenter-dark .file.is-dark:focus .file-cta, html.theme--documenter-dark .content kbd.file:focus .file-cta, html.theme--documenter-dark .file.is-dark.is-focused .file-cta, html.theme--documenter-dark .content kbd.file.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(40, 47, 47, 0.25); - color: #ecf0f1; } - html.theme--documenter-dark .file.is-dark:active .file-cta, html.theme--documenter-dark .content kbd.file:active .file-cta, html.theme--documenter-dark .file.is-dark.is-active .file-cta, html.theme--documenter-dark .content kbd.file.is-active .file-cta { - background-color: #1d2122; - border-color: transparent; - color: #ecf0f1; } - html.theme--documenter-dark .file.is-primary .file-cta, html.theme--documenter-dark .docstring > section > a.file.docs-sourcelink .file-cta { - background-color: #375a7f; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-primary:hover .file-cta, html.theme--documenter-dark .docstring > section > a.file.docs-sourcelink:hover .file-cta, html.theme--documenter-dark .file.is-primary.is-hovered .file-cta, html.theme--documenter-dark .docstring > section > a.file.is-hovered.docs-sourcelink .file-cta { - background-color: #335476; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-primary:focus .file-cta, html.theme--documenter-dark .docstring > section > a.file.docs-sourcelink:focus .file-cta, html.theme--documenter-dark .file.is-primary.is-focused .file-cta, html.theme--documenter-dark .docstring > section > a.file.is-focused.docs-sourcelink .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(55, 90, 127, 0.25); - color: #fff; } - html.theme--documenter-dark .file.is-primary:active .file-cta, html.theme--documenter-dark .docstring > section > a.file.docs-sourcelink:active .file-cta, html.theme--documenter-dark .file.is-primary.is-active .file-cta, html.theme--documenter-dark .docstring > section > a.file.is-active.docs-sourcelink .file-cta { - background-color: #2f4d6d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-link .file-cta { - background-color: #1abc9c; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-link:hover .file-cta, html.theme--documenter-dark .file.is-link.is-hovered .file-cta { - background-color: #18b193; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-link:focus .file-cta, html.theme--documenter-dark .file.is-link.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(26, 188, 156, 0.25); - color: #fff; } - html.theme--documenter-dark .file.is-link:active .file-cta, html.theme--documenter-dark .file.is-link.is-active .file-cta { - background-color: #17a689; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-info .file-cta { - background-color: #024c7d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-info:hover .file-cta, html.theme--documenter-dark .file.is-info.is-hovered .file-cta { - background-color: #024470; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-info:focus .file-cta, html.theme--documenter-dark .file.is-info.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(2, 76, 125, 0.25); - color: #fff; } - html.theme--documenter-dark .file.is-info:active .file-cta, html.theme--documenter-dark .file.is-info.is-active .file-cta { - background-color: #023d64; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-success .file-cta { - background-color: #008438; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-success:hover .file-cta, html.theme--documenter-dark .file.is-success.is-hovered .file-cta { - background-color: #007733; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-success:focus .file-cta, html.theme--documenter-dark .file.is-success.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(0, 132, 56, 0.25); - color: #fff; } - html.theme--documenter-dark .file.is-success:active .file-cta, html.theme--documenter-dark .file.is-success.is-active .file-cta { - background-color: #006b2d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-warning .file-cta { - background-color: #ad8100; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-warning:hover .file-cta, html.theme--documenter-dark .file.is-warning.is-hovered .file-cta { - background-color: #a07700; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-warning:focus .file-cta, html.theme--documenter-dark .file.is-warning.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(173, 129, 0, 0.25); - color: #fff; } - html.theme--documenter-dark .file.is-warning:active .file-cta, html.theme--documenter-dark .file.is-warning.is-active .file-cta { - background-color: #946e00; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-danger .file-cta { - background-color: #9e1b0d; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-danger:hover .file-cta, html.theme--documenter-dark .file.is-danger.is-hovered .file-cta { - background-color: #92190c; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-danger:focus .file-cta, html.theme--documenter-dark .file.is-danger.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(158, 27, 13, 0.25); - color: #fff; } - html.theme--documenter-dark .file.is-danger:active .file-cta, html.theme--documenter-dark .file.is-danger.is-active .file-cta { - background-color: #86170b; - border-color: transparent; - color: #fff; } - html.theme--documenter-dark .file.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.file { - font-size: 0.85em; } - html.theme--documenter-dark .file.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .file.is-medium .file-icon .fa { - font-size: 21px; } - html.theme--documenter-dark .file.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .file.is-large .file-icon .fa { - font-size: 28px; } - html.theme--documenter-dark .file.has-name .file-cta { - border-bottom-right-radius: 0; - border-top-right-radius: 0; } - html.theme--documenter-dark .file.has-name .file-name { - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - html.theme--documenter-dark .file.has-name.is-empty .file-cta { - border-radius: 0.4em; } - html.theme--documenter-dark .file.has-name.is-empty .file-name { - display: none; } - html.theme--documenter-dark .file.is-boxed .file-label { - flex-direction: column; } - html.theme--documenter-dark .file.is-boxed .file-cta { - flex-direction: column; - height: auto; - padding: 1em 3em; } - html.theme--documenter-dark .file.is-boxed .file-name { - border-width: 0 1px 1px; } - html.theme--documenter-dark .file.is-boxed .file-icon { - height: 1.5em; - width: 1.5em; } - html.theme--documenter-dark .file.is-boxed .file-icon .fa { - font-size: 21px; } - html.theme--documenter-dark .file.is-boxed.is-small .file-icon .fa, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-boxed .file-icon .fa { - font-size: 14px; } - html.theme--documenter-dark .file.is-boxed.is-medium .file-icon .fa { - font-size: 28px; } - html.theme--documenter-dark .file.is-boxed.is-large .file-icon .fa { - font-size: 35px; } - html.theme--documenter-dark .file.is-boxed.has-name .file-cta { - border-radius: 0.4em 0.4em 0 0; } - html.theme--documenter-dark .file.is-boxed.has-name .file-name { - border-radius: 0 0 0.4em 0.4em; - border-width: 0 1px 1px; } - html.theme--documenter-dark .file.is-centered { - justify-content: center; } - html.theme--documenter-dark .file.is-fullwidth .file-label { - width: 100%; } - html.theme--documenter-dark .file.is-fullwidth .file-name { - flex-grow: 1; - max-width: none; } - html.theme--documenter-dark .file.is-right { - justify-content: flex-end; } - html.theme--documenter-dark .file.is-right .file-cta { - border-radius: 0 0.4em 0.4em 0; } - html.theme--documenter-dark .file.is-right .file-name { - border-radius: 0.4em 0 0 0.4em; - border-width: 1px 0 1px 1px; - order: -1; } - html.theme--documenter-dark .file-label { - align-items: stretch; - display: flex; - cursor: pointer; - justify-content: flex-start; - overflow: hidden; - position: relative; } - html.theme--documenter-dark .file-label:hover .file-cta { - background-color: #e5eaec; - color: #282f2f; } - html.theme--documenter-dark .file-label:hover .file-name { - border-color: #596668; } - html.theme--documenter-dark .file-label:active .file-cta { - background-color: #dde4e6; - color: #282f2f; } - html.theme--documenter-dark .file-label:active .file-name { - border-color: #535f61; } - html.theme--documenter-dark .file-input { - height: 100%; - left: 0; - opacity: 0; - outline: none; - position: absolute; - top: 0; - width: 100%; } - html.theme--documenter-dark .file-cta, - html.theme--documenter-dark .file-name { - border-color: #5e6d6f; - border-radius: 0.4em; - font-size: 1em; - padding-left: 1em; - padding-right: 1em; - white-space: nowrap; } - html.theme--documenter-dark .file-cta { - background-color: #ecf0f1; - color: #343c3d; } - html.theme--documenter-dark .file-name { - border-color: #5e6d6f; - border-style: solid; - border-width: 1px 1px 1px 0; - display: block; - max-width: 16em; - overflow: hidden; - text-align: left; - text-overflow: ellipsis; } - html.theme--documenter-dark .file-icon { - align-items: center; - display: flex; - height: 1em; - justify-content: center; - margin-right: 0.5em; - width: 1em; } - html.theme--documenter-dark .file-icon .fa { - font-size: 14px; } - html.theme--documenter-dark .label { - color: #282f2f; - display: block; - font-size: 15px; - font-weight: 700; } - html.theme--documenter-dark .label:not(:last-child) { - margin-bottom: 0.5em; } - html.theme--documenter-dark .label.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.label { - font-size: 0.85em; } - html.theme--documenter-dark .label.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .label.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .help { - display: block; - font-size: 0.85em; - margin-top: 0.25rem; } - html.theme--documenter-dark .help.is-white { - color: white; } - html.theme--documenter-dark .help.is-black { - color: #0a0a0a; } - html.theme--documenter-dark .help.is-light { - color: #ecf0f1; } - html.theme--documenter-dark .help.is-dark, html.theme--documenter-dark .content kbd.help { - color: #282f2f; } - html.theme--documenter-dark .help.is-primary, html.theme--documenter-dark .docstring > section > a.help.docs-sourcelink { - color: #375a7f; } - html.theme--documenter-dark .help.is-link { - color: #1abc9c; } - html.theme--documenter-dark .help.is-info { - color: #024c7d; } - html.theme--documenter-dark .help.is-success { - color: #008438; } - html.theme--documenter-dark .help.is-warning { - color: #ad8100; } - html.theme--documenter-dark .help.is-danger { - color: #9e1b0d; } - html.theme--documenter-dark .field:not(:last-child) { - margin-bottom: 0.75rem; } - html.theme--documenter-dark .field.has-addons { - display: flex; - justify-content: flex-start; } - html.theme--documenter-dark .field.has-addons .control:not(:last-child) { - margin-right: -1px; } - html.theme--documenter-dark .field.has-addons .control:not(:first-child):not(:last-child) .button, - html.theme--documenter-dark .field.has-addons .control:not(:first-child):not(:last-child) .input, - html.theme--documenter-dark .field.has-addons .control:not(:first-child):not(:last-child) #documenter .docs-sidebar form.docs-search > input, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control:not(:first-child):not(:last-child) form.docs-search > input, - html.theme--documenter-dark .field.has-addons .control:not(:first-child):not(:last-child) .select select { - border-radius: 0; } - html.theme--documenter-dark .field.has-addons .control:first-child:not(:only-child) .button, - html.theme--documenter-dark .field.has-addons .control:first-child:not(:only-child) .input, - html.theme--documenter-dark .field.has-addons .control:first-child:not(:only-child) #documenter .docs-sidebar form.docs-search > input, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control:first-child:not(:only-child) form.docs-search > input, - html.theme--documenter-dark .field.has-addons .control:first-child:not(:only-child) .select select { - border-bottom-right-radius: 0; - border-top-right-radius: 0; } - html.theme--documenter-dark .field.has-addons .control:last-child:not(:only-child) .button, - html.theme--documenter-dark .field.has-addons .control:last-child:not(:only-child) .input, - html.theme--documenter-dark .field.has-addons .control:last-child:not(:only-child) #documenter .docs-sidebar form.docs-search > input, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control:last-child:not(:only-child) form.docs-search > input, - html.theme--documenter-dark .field.has-addons .control:last-child:not(:only-child) .select select { - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - html.theme--documenter-dark .field.has-addons .control .button:not([disabled]):hover, html.theme--documenter-dark .field.has-addons .control .button.is-hovered:not([disabled]), - html.theme--documenter-dark .field.has-addons .control .input:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):hover, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control .input.is-hovered:not([disabled]), - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-hovered:not([disabled]), - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-hovered:not([disabled]), - html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control .select select.is-hovered:not([disabled]) { - z-index: 2; } - html.theme--documenter-dark .field.has-addons .control .button:not([disabled]):focus, html.theme--documenter-dark .field.has-addons .control .button.is-focused:not([disabled]), html.theme--documenter-dark .field.has-addons .control .button:not([disabled]):active, html.theme--documenter-dark .field.has-addons .control .button.is-active:not([disabled]), - html.theme--documenter-dark .field.has-addons .control .input:not([disabled]):focus, - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):focus, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):focus, - html.theme--documenter-dark .field.has-addons .control .input.is-focused:not([disabled]), - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-focused:not([disabled]), - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-focused:not([disabled]), - html.theme--documenter-dark .field.has-addons .control .input:not([disabled]):active, - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):active, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):active, - html.theme--documenter-dark .field.has-addons .control .input.is-active:not([disabled]), - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-active:not([disabled]), - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-active:not([disabled]), - html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]):focus, - html.theme--documenter-dark .field.has-addons .control .select select.is-focused:not([disabled]), - html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]):active, - html.theme--documenter-dark .field.has-addons .control .select select.is-active:not([disabled]) { - z-index: 3; } - html.theme--documenter-dark .field.has-addons .control .button:not([disabled]):focus:hover, html.theme--documenter-dark .field.has-addons .control .button.is-focused:not([disabled]):hover, html.theme--documenter-dark .field.has-addons .control .button:not([disabled]):active:hover, html.theme--documenter-dark .field.has-addons .control .button.is-active:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control .input:not([disabled]):focus:hover, - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):focus:hover, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):focus:hover, - html.theme--documenter-dark .field.has-addons .control .input.is-focused:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-focused:not([disabled]):hover, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-focused:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control .input:not([disabled]):active:hover, - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):active:hover, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):active:hover, - html.theme--documenter-dark .field.has-addons .control .input.is-active:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-active:not([disabled]):hover, - html.theme--documenter-dark #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-active:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]):focus:hover, - html.theme--documenter-dark .field.has-addons .control .select select.is-focused:not([disabled]):hover, - html.theme--documenter-dark .field.has-addons .control .select select:not([disabled]):active:hover, - html.theme--documenter-dark .field.has-addons .control .select select.is-active:not([disabled]):hover { - z-index: 4; } - html.theme--documenter-dark .field.has-addons .control.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - html.theme--documenter-dark .field.has-addons.has-addons-centered { - justify-content: center; } - html.theme--documenter-dark .field.has-addons.has-addons-right { - justify-content: flex-end; } - html.theme--documenter-dark .field.has-addons.has-addons-fullwidth .control { - flex-grow: 1; - flex-shrink: 0; } - html.theme--documenter-dark .field.is-grouped { - display: flex; - justify-content: flex-start; } - html.theme--documenter-dark .field.is-grouped > .control { - flex-shrink: 0; } - html.theme--documenter-dark .field.is-grouped > .control:not(:last-child) { - margin-bottom: 0; - margin-right: 0.75rem; } - html.theme--documenter-dark .field.is-grouped > .control.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - html.theme--documenter-dark .field.is-grouped.is-grouped-centered { - justify-content: center; } - html.theme--documenter-dark .field.is-grouped.is-grouped-right { - justify-content: flex-end; } - html.theme--documenter-dark .field.is-grouped.is-grouped-multiline { - flex-wrap: wrap; } - html.theme--documenter-dark .field.is-grouped.is-grouped-multiline > .control:last-child, html.theme--documenter-dark .field.is-grouped.is-grouped-multiline > .control:not(:last-child) { - margin-bottom: 0.75rem; } - html.theme--documenter-dark .field.is-grouped.is-grouped-multiline:last-child { - margin-bottom: -0.75rem; } - html.theme--documenter-dark .field.is-grouped.is-grouped-multiline:not(:last-child) { - margin-bottom: 0; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .field.is-horizontal { - display: flex; } } - html.theme--documenter-dark .field-label .label { - font-size: inherit; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .field-label { - margin-bottom: 0.5rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .field-label { - flex-basis: 0; - flex-grow: 1; - flex-shrink: 0; - margin-right: 1.5rem; - text-align: right; } - html.theme--documenter-dark .field-label.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.field-label { - font-size: 0.85em; - padding-top: 0.375em; } - html.theme--documenter-dark .field-label.is-normal { - padding-top: 0.375em; } - html.theme--documenter-dark .field-label.is-medium { - font-size: 1.25rem; - padding-top: 0.375em; } - html.theme--documenter-dark .field-label.is-large { - font-size: 1.5rem; - padding-top: 0.375em; } } - html.theme--documenter-dark .field-body .field .field { - margin-bottom: 0; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .field-body { - display: flex; - flex-basis: 0; - flex-grow: 5; - flex-shrink: 1; } - html.theme--documenter-dark .field-body .field { - margin-bottom: 0; } - html.theme--documenter-dark .field-body > .field { - flex-shrink: 1; } - html.theme--documenter-dark .field-body > .field:not(.is-narrow) { - flex-grow: 1; } - html.theme--documenter-dark .field-body > .field:not(:last-child) { - margin-right: 0.75rem; } } - html.theme--documenter-dark .control { - box-sizing: border-box; - clear: both; - font-size: 15px; - position: relative; - text-align: left; } - html.theme--documenter-dark .control.has-icons-left .input:focus ~ .icon, html.theme--documenter-dark .control.has-icons-left #documenter .docs-sidebar form.docs-search > input:focus ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-left form.docs-search > input:focus ~ .icon, - html.theme--documenter-dark .control.has-icons-left .select:focus ~ .icon, html.theme--documenter-dark .control.has-icons-right .input:focus ~ .icon, html.theme--documenter-dark .control.has-icons-right #documenter .docs-sidebar form.docs-search > input:focus ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-right form.docs-search > input:focus ~ .icon, - html.theme--documenter-dark .control.has-icons-right .select:focus ~ .icon { - color: #5e6d6f; } - html.theme--documenter-dark .control.has-icons-left .input.is-small ~ .icon, html.theme--documenter-dark .control.has-icons-left #documenter .docs-sidebar form.docs-search > input ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-left form.docs-search > input ~ .icon, - html.theme--documenter-dark .control.has-icons-left .select.is-small ~ .icon, html.theme--documenter-dark .control.has-icons-right .input.is-small ~ .icon, html.theme--documenter-dark .control.has-icons-right #documenter .docs-sidebar form.docs-search > input ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-right form.docs-search > input ~ .icon, - html.theme--documenter-dark .control.has-icons-right .select.is-small ~ .icon { - font-size: 0.85em; } - html.theme--documenter-dark .control.has-icons-left .input.is-medium ~ .icon, html.theme--documenter-dark .control.has-icons-left #documenter .docs-sidebar form.docs-search > input.is-medium ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-left form.docs-search > input.is-medium ~ .icon, - html.theme--documenter-dark .control.has-icons-left .select.is-medium ~ .icon, html.theme--documenter-dark .control.has-icons-right .input.is-medium ~ .icon, html.theme--documenter-dark .control.has-icons-right #documenter .docs-sidebar form.docs-search > input.is-medium ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-right form.docs-search > input.is-medium ~ .icon, - html.theme--documenter-dark .control.has-icons-right .select.is-medium ~ .icon { - font-size: 1.25rem; } - html.theme--documenter-dark .control.has-icons-left .input.is-large ~ .icon, html.theme--documenter-dark .control.has-icons-left #documenter .docs-sidebar form.docs-search > input.is-large ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-left form.docs-search > input.is-large ~ .icon, - html.theme--documenter-dark .control.has-icons-left .select.is-large ~ .icon, html.theme--documenter-dark .control.has-icons-right .input.is-large ~ .icon, html.theme--documenter-dark .control.has-icons-right #documenter .docs-sidebar form.docs-search > input.is-large ~ .icon, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-right form.docs-search > input.is-large ~ .icon, - html.theme--documenter-dark .control.has-icons-right .select.is-large ~ .icon { - font-size: 1.5rem; } - html.theme--documenter-dark .control.has-icons-left .icon, html.theme--documenter-dark .control.has-icons-right .icon { - color: #dbdee0; - height: 2.25em; - pointer-events: none; - position: absolute; - top: 0; - width: 2.25em; - z-index: 4; } - html.theme--documenter-dark .control.has-icons-left .input, html.theme--documenter-dark .control.has-icons-left #documenter .docs-sidebar form.docs-search > input, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-left form.docs-search > input, - html.theme--documenter-dark .control.has-icons-left .select select { - padding-left: 2.25em; } - html.theme--documenter-dark .control.has-icons-left .icon.is-left { - left: 0; } - html.theme--documenter-dark .control.has-icons-right .input, html.theme--documenter-dark .control.has-icons-right #documenter .docs-sidebar form.docs-search > input, html.theme--documenter-dark #documenter .docs-sidebar .control.has-icons-right form.docs-search > input, - html.theme--documenter-dark .control.has-icons-right .select select { - padding-right: 2.25em; } - html.theme--documenter-dark .control.has-icons-right .icon.is-right { - right: 0; } - html.theme--documenter-dark .control.is-loading::after { - position: absolute !important; - right: 0.625em; - top: 0.625em; - z-index: 4; } - html.theme--documenter-dark .control.is-loading.is-small:after, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.is-loading:after { - font-size: 0.85em; } - html.theme--documenter-dark .control.is-loading.is-medium:after { - font-size: 1.25rem; } - html.theme--documenter-dark .control.is-loading.is-large:after { - font-size: 1.5rem; } - html.theme--documenter-dark .breadcrumb { - font-size: 15px; - white-space: nowrap; } - html.theme--documenter-dark .breadcrumb a { - align-items: center; - color: #1abc9c; - display: flex; - justify-content: center; - padding: 0 0.75em; } - html.theme--documenter-dark .breadcrumb a:hover { - color: #1dd2af; } - html.theme--documenter-dark .breadcrumb li { - align-items: center; - display: flex; } - html.theme--documenter-dark .breadcrumb li:first-child a { - padding-left: 0; } - html.theme--documenter-dark .breadcrumb li.is-active a { - color: #f2f2f2; - cursor: default; - pointer-events: none; } - html.theme--documenter-dark .breadcrumb li + li::before { - color: #8c9b9d; - content: "\0002f"; } - html.theme--documenter-dark .breadcrumb ul, - html.theme--documenter-dark .breadcrumb ol { - align-items: flex-start; - display: flex; - flex-wrap: wrap; - justify-content: flex-start; } - html.theme--documenter-dark .breadcrumb .icon:first-child { - margin-right: 0.5em; } - html.theme--documenter-dark .breadcrumb .icon:last-child { - margin-left: 0.5em; } - html.theme--documenter-dark .breadcrumb.is-centered ol, - html.theme--documenter-dark .breadcrumb.is-centered ul { - justify-content: center; } - html.theme--documenter-dark .breadcrumb.is-right ol, - html.theme--documenter-dark .breadcrumb.is-right ul { - justify-content: flex-end; } - html.theme--documenter-dark .breadcrumb.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.breadcrumb { - font-size: 0.85em; } - html.theme--documenter-dark .breadcrumb.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .breadcrumb.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .breadcrumb.has-arrow-separator li + li::before { - content: "\02192"; } - html.theme--documenter-dark .breadcrumb.has-bullet-separator li + li::before { - content: "\02022"; } - html.theme--documenter-dark .breadcrumb.has-dot-separator li + li::before { - content: "\000b7"; } - html.theme--documenter-dark .breadcrumb.has-succeeds-separator li + li::before { - content: "\0227B"; } - html.theme--documenter-dark .card { - background-color: white; - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); - color: #fff; - max-width: 100%; - position: relative; } - html.theme--documenter-dark .card-header { - background-color: transparent; - align-items: stretch; - box-shadow: 0 1px 2px rgba(10, 10, 10, 0.1); - display: flex; } - html.theme--documenter-dark .card-header-title { - align-items: center; - color: #f2f2f2; - display: flex; - flex-grow: 1; - font-weight: 700; - padding: 0.75rem; } - html.theme--documenter-dark .card-header-title.is-centered { - justify-content: center; } - html.theme--documenter-dark .card-header-icon { - align-items: center; - cursor: pointer; - display: flex; - justify-content: center; - padding: 0.75rem; } - html.theme--documenter-dark .card-image { - display: block; - position: relative; } - html.theme--documenter-dark .card-content { - background-color: transparent; - padding: 1.5rem; } - html.theme--documenter-dark .card-footer { - background-color: transparent; - border-top: 1px solid #5e6d6f; - align-items: stretch; - display: flex; } - html.theme--documenter-dark .card-footer-item { - align-items: center; - display: flex; - flex-basis: 0; - flex-grow: 1; - flex-shrink: 0; - justify-content: center; - padding: 0.75rem; } - html.theme--documenter-dark .card-footer-item:not(:last-child) { - border-right: 1px solid #5e6d6f; } - html.theme--documenter-dark .card .media:not(:last-child) { - margin-bottom: 1.5rem; } - html.theme--documenter-dark .dropdown { - display: inline-flex; - position: relative; - vertical-align: top; } - html.theme--documenter-dark .dropdown.is-active .dropdown-menu, html.theme--documenter-dark .dropdown.is-hoverable:hover .dropdown-menu { - display: block; } - html.theme--documenter-dark .dropdown.is-right .dropdown-menu { - left: auto; - right: 0; } - html.theme--documenter-dark .dropdown.is-up .dropdown-menu { - bottom: 100%; - padding-bottom: 4px; - padding-top: initial; - top: auto; } - html.theme--documenter-dark .dropdown-menu { - display: none; - left: 0; - min-width: 12rem; - padding-top: 4px; - position: absolute; - top: 100%; - z-index: 20; } - html.theme--documenter-dark .dropdown-content { - background-color: #282f2f; - border-radius: 0.4em; - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); - padding-bottom: 0.5rem; - padding-top: 0.5rem; } - html.theme--documenter-dark .dropdown-item { - color: #fff; - display: block; - font-size: 0.875rem; - line-height: 1.5; - padding: 0.375rem 1rem; - position: relative; } - html.theme--documenter-dark a.dropdown-item, - html.theme--documenter-dark button.dropdown-item { - padding-right: 3rem; - text-align: left; - white-space: nowrap; - width: 100%; } - html.theme--documenter-dark a.dropdown-item:hover, - html.theme--documenter-dark button.dropdown-item:hover { - background-color: #282f2f; - color: #0a0a0a; } - html.theme--documenter-dark a.dropdown-item.is-active, - html.theme--documenter-dark button.dropdown-item.is-active { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .dropdown-divider { - background-color: #5e6d6f; - border: none; - display: block; - height: 1px; - margin: 0.5rem 0; } - html.theme--documenter-dark .level { - align-items: center; - justify-content: space-between; } - html.theme--documenter-dark .level code { - border-radius: 0.4em; } - html.theme--documenter-dark .level img { - display: inline-block; - vertical-align: top; } - html.theme--documenter-dark .level.is-mobile { - display: flex; } - html.theme--documenter-dark .level.is-mobile .level-left, - html.theme--documenter-dark .level.is-mobile .level-right { - display: flex; } - html.theme--documenter-dark .level.is-mobile .level-left + .level-right { - margin-top: 0; } - html.theme--documenter-dark .level.is-mobile .level-item:not(:last-child) { - margin-bottom: 0; - margin-right: 0.75rem; } - html.theme--documenter-dark .level.is-mobile .level-item:not(.is-narrow) { - flex-grow: 1; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .level { - display: flex; } - html.theme--documenter-dark .level > .level-item:not(.is-narrow) { - flex-grow: 1; } } - html.theme--documenter-dark .level-item { - align-items: center; - display: flex; - flex-basis: auto; - flex-grow: 0; - flex-shrink: 0; - justify-content: center; } - html.theme--documenter-dark .level-item .title, - html.theme--documenter-dark .level-item .subtitle { - margin-bottom: 0; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .level-item:not(:last-child) { - margin-bottom: 0.75rem; } } - html.theme--documenter-dark .level-left, - html.theme--documenter-dark .level-right { - flex-basis: auto; - flex-grow: 0; - flex-shrink: 0; } - html.theme--documenter-dark .level-left .level-item.is-flexible, - html.theme--documenter-dark .level-right .level-item.is-flexible { - flex-grow: 1; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .level-left .level-item:not(:last-child), - html.theme--documenter-dark .level-right .level-item:not(:last-child) { - margin-right: 0.75rem; } } - html.theme--documenter-dark .level-left { - align-items: center; - justify-content: flex-start; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .level-left + .level-right { - margin-top: 1.5rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .level-left { - display: flex; } } - html.theme--documenter-dark .level-right { - align-items: center; - justify-content: flex-end; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .level-right { - display: flex; } } - html.theme--documenter-dark .list { - background-color: white; - border-radius: 0.4em; - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .list-item { - display: block; - padding: 0.5em 1em; } - html.theme--documenter-dark .list-item:not(a) { - color: #fff; } - html.theme--documenter-dark .list-item:first-child { - border-top-left-radius: 0.4em; - border-top-right-radius: 0.4em; } - html.theme--documenter-dark .list-item:last-child { - border-bottom-left-radius: 0.4em; - border-bottom-right-radius: 0.4em; } - html.theme--documenter-dark .list-item:not(:last-child) { - border-bottom: 1px solid #5e6d6f; } - html.theme--documenter-dark .list-item.is-active { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark a.list-item { - background-color: #282f2f; - cursor: pointer; } - html.theme--documenter-dark .media { - align-items: flex-start; - display: flex; - text-align: left; } - html.theme--documenter-dark .media .content:not(:last-child) { - margin-bottom: 0.75rem; } - html.theme--documenter-dark .media .media { - border-top: 1px solid rgba(94, 109, 111, 0.5); - display: flex; - padding-top: 0.75rem; } - html.theme--documenter-dark .media .media .content:not(:last-child), - html.theme--documenter-dark .media .media .control:not(:last-child) { - margin-bottom: 0.5rem; } - html.theme--documenter-dark .media .media .media { - padding-top: 0.5rem; } - html.theme--documenter-dark .media .media .media + .media { - margin-top: 0.5rem; } - html.theme--documenter-dark .media + .media { - border-top: 1px solid rgba(94, 109, 111, 0.5); - margin-top: 1rem; - padding-top: 1rem; } - html.theme--documenter-dark .media.is-large + .media { - margin-top: 1.5rem; - padding-top: 1.5rem; } - html.theme--documenter-dark .media-left, - html.theme--documenter-dark .media-right { - flex-basis: auto; - flex-grow: 0; - flex-shrink: 0; } - html.theme--documenter-dark .media-left { - margin-right: 1rem; } - html.theme--documenter-dark .media-right { - margin-left: 1rem; } - html.theme--documenter-dark .media-content { - flex-basis: auto; - flex-grow: 1; - flex-shrink: 1; - text-align: left; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .media-content { - overflow-x: auto; } } - html.theme--documenter-dark .menu { - font-size: 15px; } - html.theme--documenter-dark .menu.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.menu { - font-size: 0.85em; } - html.theme--documenter-dark .menu.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .menu.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .menu-list { - line-height: 1.25; } - html.theme--documenter-dark .menu-list a { - border-radius: 3px; - color: #fff; - display: block; - padding: 0.5em 0.75em; } - html.theme--documenter-dark .menu-list a:hover { - background-color: #282f2f; - color: #f2f2f2; } - html.theme--documenter-dark .menu-list a.is-active { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .menu-list li ul { - border-left: 1px solid #5e6d6f; - margin: 0.75em; - padding-left: 0.75em; } - html.theme--documenter-dark .menu-label { - color: white; - font-size: 0.75em; - letter-spacing: 0.1em; - text-transform: uppercase; } - html.theme--documenter-dark .menu-label:not(:first-child) { - margin-top: 1em; } - html.theme--documenter-dark .menu-label:not(:last-child) { - margin-bottom: 1em; } - html.theme--documenter-dark .message { - background-color: #282f2f; - border-radius: 0.4em; - font-size: 15px; } - html.theme--documenter-dark .message strong { - color: currentColor; } - html.theme--documenter-dark .message a:not(.button):not(.tag):not(.dropdown-item) { - color: currentColor; - text-decoration: underline; } - html.theme--documenter-dark .message.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.message { - font-size: 0.85em; } - html.theme--documenter-dark .message.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .message.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .message.is-white { - background-color: white; } - html.theme--documenter-dark .message.is-white .message-header { - background-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .message.is-white .message-body { - border-color: white; - color: #4d4d4d; } - html.theme--documenter-dark .message.is-black { - background-color: #fafafa; } - html.theme--documenter-dark .message.is-black .message-header { - background-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .message.is-black .message-body { - border-color: #0a0a0a; - color: #090909; } - html.theme--documenter-dark .message.is-light { - background-color: #f9fafb; } - html.theme--documenter-dark .message.is-light .message-header { - background-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .message.is-light .message-body { - border-color: #ecf0f1; - color: #505050; } - html.theme--documenter-dark .message.is-dark, html.theme--documenter-dark .content kbd.message { - background-color: #f9fafa; } - html.theme--documenter-dark .message.is-dark .message-header, html.theme--documenter-dark .content kbd.message .message-header { - background-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .message.is-dark .message-body, html.theme--documenter-dark .content kbd.message .message-body { - border-color: #282f2f; - color: #212526; } - html.theme--documenter-dark .message.is-primary, html.theme--documenter-dark .docstring > section > a.message.docs-sourcelink { - background-color: #f8fafc; } - html.theme--documenter-dark .message.is-primary .message-header, html.theme--documenter-dark .docstring > section > a.message.docs-sourcelink .message-header { - background-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .message.is-primary .message-body, html.theme--documenter-dark .docstring > section > a.message.docs-sourcelink .message-body { - border-color: #375a7f; - color: #2b4159; } - html.theme--documenter-dark .message.is-link { - background-color: #f6fefc; } - html.theme--documenter-dark .message.is-link .message-header { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .message.is-link .message-body { - border-color: #1abc9c; - color: #0b2f28; } - html.theme--documenter-dark .message.is-info { - background-color: #f5fbff; } - html.theme--documenter-dark .message.is-info .message-header { - background-color: #024c7d; - color: #fff; } - html.theme--documenter-dark .message.is-info .message-body { - border-color: #024c7d; - color: #033659; } - html.theme--documenter-dark .message.is-success { - background-color: #f5fff9; } - html.theme--documenter-dark .message.is-success .message-header { - background-color: #008438; - color: #fff; } - html.theme--documenter-dark .message.is-success .message-body { - border-color: #008438; - color: #023518; } - html.theme--documenter-dark .message.is-warning { - background-color: #fffcf5; } - html.theme--documenter-dark .message.is-warning .message-header { - background-color: #ad8100; - color: #fff; } - html.theme--documenter-dark .message.is-warning .message-body { - border-color: #ad8100; - color: #3d2e03; } - html.theme--documenter-dark .message.is-danger { - background-color: #fef6f6; } - html.theme--documenter-dark .message.is-danger .message-header { - background-color: #9e1b0d; - color: #fff; } - html.theme--documenter-dark .message.is-danger .message-body { - border-color: #9e1b0d; - color: #7a170c; } - html.theme--documenter-dark .message-header { - align-items: center; - background-color: #fff; - border-radius: 0.4em 0.4em 0 0; - color: rgba(0, 0, 0, 0.7); - display: flex; - font-weight: 700; - justify-content: space-between; - line-height: 1.25; - padding: 0.75em 1em; - position: relative; } - html.theme--documenter-dark .message-header .delete { - flex-grow: 0; - flex-shrink: 0; - margin-left: 0.75em; } - html.theme--documenter-dark .message-header + .message-body { - border-width: 0; - border-top-left-radius: 0; - border-top-right-radius: 0; } - html.theme--documenter-dark .message-body { - border-color: #5e6d6f; - border-radius: 0.4em; - border-style: solid; - border-width: 0 0 0 4px; - color: #fff; - padding: 1.25em 1.5em; } - html.theme--documenter-dark .message-body code, - html.theme--documenter-dark .message-body pre { - background-color: white; } - html.theme--documenter-dark .message-body pre code { - background-color: transparent; } - html.theme--documenter-dark .modal { - align-items: center; - display: none; - flex-direction: column; - justify-content: center; - overflow: hidden; - position: fixed; - z-index: 40; } - html.theme--documenter-dark .modal.is-active { - display: flex; } - html.theme--documenter-dark .modal-background { - background-color: rgba(10, 10, 10, 0.86); } - html.theme--documenter-dark .modal-content, - html.theme--documenter-dark .modal-card { - margin: 0 20px; - max-height: calc(100vh - 160px); - overflow: auto; - position: relative; - width: 100%; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .modal-content, - html.theme--documenter-dark .modal-card { - margin: 0 auto; - max-height: calc(100vh - 40px); - width: 640px; } } - html.theme--documenter-dark .modal-close { - background: none; - height: 40px; - position: fixed; - right: 20px; - top: 20px; - width: 40px; } - html.theme--documenter-dark .modal-card { - display: flex; - flex-direction: column; - max-height: calc(100vh - 40px); - overflow: hidden; - -ms-overflow-y: visible; } - html.theme--documenter-dark .modal-card-head, - html.theme--documenter-dark .modal-card-foot { - align-items: center; - background-color: #282f2f; - display: flex; - flex-shrink: 0; - justify-content: flex-start; - padding: 20px; - position: relative; } - html.theme--documenter-dark .modal-card-head { - border-bottom: 1px solid #5e6d6f; - border-top-left-radius: 8px; - border-top-right-radius: 8px; } - html.theme--documenter-dark .modal-card-title { - color: #f2f2f2; - flex-grow: 1; - flex-shrink: 0; - font-size: 1.5rem; - line-height: 1; } - html.theme--documenter-dark .modal-card-foot { - border-bottom-left-radius: 8px; - border-bottom-right-radius: 8px; - border-top: 1px solid #5e6d6f; } - html.theme--documenter-dark .modal-card-foot .button:not(:last-child) { - margin-right: 0.5em; } - html.theme--documenter-dark .modal-card-body { - -webkit-overflow-scrolling: touch; - background-color: white; - flex-grow: 1; - flex-shrink: 1; - overflow: auto; - padding: 20px; } - html.theme--documenter-dark .navbar { - background-color: #375a7f; - min-height: 4rem; - position: relative; - z-index: 30; } - html.theme--documenter-dark .navbar.is-white { - background-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-white .navbar-brand .navbar-link { - color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-white .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-white .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-white .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-white .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-white .navbar-brand .navbar-link.is-active { - background-color: #f2f2f2; - color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-brand .navbar-link::after { - border-color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-burger { - color: #0a0a0a; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-white .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-white .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-white .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-white .navbar-end .navbar-link { - color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-white .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-white .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-white .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-white .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-white .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-white .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-white .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-white .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-white .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-white .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-white .navbar-end .navbar-link.is-active { - background-color: #f2f2f2; - color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-white .navbar-end .navbar-link::after { - border-color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-white .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-white .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #f2f2f2; - color: #0a0a0a; } - html.theme--documenter-dark .navbar.is-white .navbar-dropdown a.navbar-item.is-active { - background-color: white; - color: #0a0a0a; } } - html.theme--documenter-dark .navbar.is-black { - background-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-black .navbar-brand .navbar-link { - color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-black .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-black .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-black .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-black .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-black .navbar-brand .navbar-link.is-active { - background-color: black; - color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-brand .navbar-link::after { - border-color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-burger { - color: white; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-black .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-black .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-black .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-black .navbar-end .navbar-link { - color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-black .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-black .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-black .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-black .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-black .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-black .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-black .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-black .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-black .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-black .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-black .navbar-end .navbar-link.is-active { - background-color: black; - color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-black .navbar-end .navbar-link::after { - border-color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-black .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-black .navbar-item.has-dropdown.is-active .navbar-link { - background-color: black; - color: white; } - html.theme--documenter-dark .navbar.is-black .navbar-dropdown a.navbar-item.is-active { - background-color: #0a0a0a; - color: white; } } - html.theme--documenter-dark .navbar.is-light { - background-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-light .navbar-brand .navbar-link { - color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-light .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-light .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-light .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-light .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-light .navbar-brand .navbar-link.is-active { - background-color: #dde4e6; - color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-brand .navbar-link::after { - border-color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-burger { - color: #282f2f; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-light .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-light .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-light .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-light .navbar-end .navbar-link { - color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-light .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-light .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-light .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-light .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-light .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-light .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-light .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-light .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-light .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-light .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-light .navbar-end .navbar-link.is-active { - background-color: #dde4e6; - color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-light .navbar-end .navbar-link::after { - border-color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-light .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-light .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #dde4e6; - color: #282f2f; } - html.theme--documenter-dark .navbar.is-light .navbar-dropdown a.navbar-item.is-active { - background-color: #ecf0f1; - color: #282f2f; } } - html.theme--documenter-dark .navbar.is-dark, html.theme--documenter-dark .content kbd.navbar { - background-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-brand > .navbar-item, html.theme--documenter-dark .content kbd.navbar .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-dark .navbar-brand .navbar-link, - html.theme--documenter-dark .content kbd.navbar .navbar-brand .navbar-link { - color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .content kbd.navbar .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-dark .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .content kbd.navbar .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-dark .navbar-brand > a.navbar-item.is-active, html.theme--documenter-dark .content kbd.navbar .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-dark .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .content kbd.navbar .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-dark .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .content kbd.navbar .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-dark .navbar-brand .navbar-link.is-active, - html.theme--documenter-dark .content kbd.navbar .navbar-brand .navbar-link.is-active { - background-color: #1d2122; - color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-brand .navbar-link::after, html.theme--documenter-dark .content kbd.navbar .navbar-brand .navbar-link::after { - border-color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-burger, html.theme--documenter-dark .content kbd.navbar .navbar-burger { - color: #ecf0f1; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-dark .navbar-start > .navbar-item, html.theme--documenter-dark .content kbd.navbar .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-dark .navbar-start .navbar-link, - html.theme--documenter-dark .content kbd.navbar .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-dark .navbar-end > .navbar-item, - html.theme--documenter-dark .content kbd.navbar .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-dark .navbar-end .navbar-link, - html.theme--documenter-dark .content kbd.navbar .navbar-end .navbar-link { - color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .content kbd.navbar .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-dark .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .content kbd.navbar .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-dark .navbar-start > a.navbar-item.is-active, html.theme--documenter-dark .content kbd.navbar .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-dark .navbar-start .navbar-link:focus, - html.theme--documenter-dark .content kbd.navbar .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-dark .navbar-start .navbar-link:hover, - html.theme--documenter-dark .content kbd.navbar .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-dark .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .content kbd.navbar .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-dark .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .content kbd.navbar .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-dark .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .content kbd.navbar .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-dark .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .content kbd.navbar .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-dark .navbar-end .navbar-link:focus, - html.theme--documenter-dark .content kbd.navbar .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-dark .navbar-end .navbar-link:hover, - html.theme--documenter-dark .content kbd.navbar .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-dark .navbar-end .navbar-link.is-active, - html.theme--documenter-dark .content kbd.navbar .navbar-end .navbar-link.is-active { - background-color: #1d2122; - color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-start .navbar-link::after, html.theme--documenter-dark .content kbd.navbar .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-dark .navbar-end .navbar-link::after, - html.theme--documenter-dark .content kbd.navbar .navbar-end .navbar-link::after { - border-color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-item.has-dropdown:focus .navbar-link, html.theme--documenter-dark .content kbd.navbar .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-dark .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .content kbd.navbar .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-dark .navbar-item.has-dropdown.is-active .navbar-link, - html.theme--documenter-dark .content kbd.navbar .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #1d2122; - color: #ecf0f1; } - html.theme--documenter-dark .navbar.is-dark .navbar-dropdown a.navbar-item.is-active, html.theme--documenter-dark .content kbd.navbar .navbar-dropdown a.navbar-item.is-active { - background-color: #282f2f; - color: #ecf0f1; } } - html.theme--documenter-dark .navbar.is-primary, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink { - background-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-brand > .navbar-item, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-primary .navbar-brand .navbar-link, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-primary .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-primary .navbar-brand > a.navbar-item.is-active, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-primary .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-primary .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-primary .navbar-brand .navbar-link.is-active, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link.is-active { - background-color: #2f4d6d; - color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-brand .navbar-link::after, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-burger, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-primary .navbar-start > .navbar-item, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-primary .navbar-start .navbar-link, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-primary .navbar-end > .navbar-item, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-primary .navbar-end .navbar-link, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-primary .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-primary .navbar-start > a.navbar-item.is-active, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-primary .navbar-start .navbar-link:focus, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-primary .navbar-start .navbar-link:hover, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-primary .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-primary .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-primary .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-primary .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-primary .navbar-end .navbar-link:focus, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-primary .navbar-end .navbar-link:hover, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-primary .navbar-end .navbar-link.is-active, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link.is-active { - background-color: #2f4d6d; - color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-start .navbar-link::after, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-primary .navbar-end .navbar-link::after, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-item.has-dropdown:focus .navbar-link, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-primary .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-primary .navbar-item.has-dropdown.is-active .navbar-link, - html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #2f4d6d; - color: #fff; } - html.theme--documenter-dark .navbar.is-primary .navbar-dropdown a.navbar-item.is-active, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-dropdown a.navbar-item.is-active { - background-color: #375a7f; - color: #fff; } } - html.theme--documenter-dark .navbar.is-link { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-link .navbar-brand .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-link .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-link .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-link .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-link .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-link .navbar-brand .navbar-link.is-active { - background-color: #17a689; - color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-brand .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-link .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-link .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-link .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-link .navbar-end .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-link .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-link .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-link .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-link .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-link .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-link .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-link .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-link .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-link .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-link .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-link .navbar-end .navbar-link.is-active { - background-color: #17a689; - color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-link .navbar-end .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-link .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-link .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #17a689; - color: #fff; } - html.theme--documenter-dark .navbar.is-link .navbar-dropdown a.navbar-item.is-active { - background-color: #1abc9c; - color: #fff; } } - html.theme--documenter-dark .navbar.is-info { - background-color: #024c7d; - color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-info .navbar-brand .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-info .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-info .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-info .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-info .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-info .navbar-brand .navbar-link.is-active { - background-color: #023d64; - color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-brand .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-info .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-info .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-info .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-info .navbar-end .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-info .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-info .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-info .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-info .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-info .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-info .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-info .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-info .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-info .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-info .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-info .navbar-end .navbar-link.is-active { - background-color: #023d64; - color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-info .navbar-end .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-info .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-info .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #023d64; - color: #fff; } - html.theme--documenter-dark .navbar.is-info .navbar-dropdown a.navbar-item.is-active { - background-color: #024c7d; - color: #fff; } } - html.theme--documenter-dark .navbar.is-success { - background-color: #008438; - color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-success .navbar-brand .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-success .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-success .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-success .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-success .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-success .navbar-brand .navbar-link.is-active { - background-color: #006b2d; - color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-brand .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-success .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-success .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-success .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-success .navbar-end .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-success .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-success .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-success .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-success .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-success .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-success .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-success .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-success .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-success .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-success .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-success .navbar-end .navbar-link.is-active { - background-color: #006b2d; - color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-success .navbar-end .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-success .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-success .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #006b2d; - color: #fff; } - html.theme--documenter-dark .navbar.is-success .navbar-dropdown a.navbar-item.is-active { - background-color: #008438; - color: #fff; } } - html.theme--documenter-dark .navbar.is-warning { - background-color: #ad8100; - color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-warning .navbar-brand .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-warning .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-warning .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-warning .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-warning .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-warning .navbar-brand .navbar-link.is-active { - background-color: #946e00; - color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-brand .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-warning .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-warning .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-warning .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-warning .navbar-end .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-warning .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-warning .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-warning .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-warning .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-warning .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-warning .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-warning .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-warning .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-warning .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-warning .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-warning .navbar-end .navbar-link.is-active { - background-color: #946e00; - color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-warning .navbar-end .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-warning .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-warning .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #946e00; - color: #fff; } - html.theme--documenter-dark .navbar.is-warning .navbar-dropdown a.navbar-item.is-active { - background-color: #ad8100; - color: #fff; } } - html.theme--documenter-dark .navbar.is-danger { - background-color: #9e1b0d; - color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-brand > .navbar-item, - html.theme--documenter-dark .navbar.is-danger .navbar-brand .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-brand > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-danger .navbar-brand > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-danger .navbar-brand > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-danger .navbar-brand .navbar-link:focus, - html.theme--documenter-dark .navbar.is-danger .navbar-brand .navbar-link:hover, - html.theme--documenter-dark .navbar.is-danger .navbar-brand .navbar-link.is-active { - background-color: #86170b; - color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-brand .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar.is-danger .navbar-start > .navbar-item, - html.theme--documenter-dark .navbar.is-danger .navbar-start .navbar-link, - html.theme--documenter-dark .navbar.is-danger .navbar-end > .navbar-item, - html.theme--documenter-dark .navbar.is-danger .navbar-end .navbar-link { - color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-start > a.navbar-item:focus, html.theme--documenter-dark .navbar.is-danger .navbar-start > a.navbar-item:hover, html.theme--documenter-dark .navbar.is-danger .navbar-start > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-danger .navbar-start .navbar-link:focus, - html.theme--documenter-dark .navbar.is-danger .navbar-start .navbar-link:hover, - html.theme--documenter-dark .navbar.is-danger .navbar-start .navbar-link.is-active, - html.theme--documenter-dark .navbar.is-danger .navbar-end > a.navbar-item:focus, - html.theme--documenter-dark .navbar.is-danger .navbar-end > a.navbar-item:hover, - html.theme--documenter-dark .navbar.is-danger .navbar-end > a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-danger .navbar-end .navbar-link:focus, - html.theme--documenter-dark .navbar.is-danger .navbar-end .navbar-link:hover, - html.theme--documenter-dark .navbar.is-danger .navbar-end .navbar-link.is-active { - background-color: #86170b; - color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-start .navbar-link::after, - html.theme--documenter-dark .navbar.is-danger .navbar-end .navbar-link::after { - border-color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-item.has-dropdown:focus .navbar-link, - html.theme--documenter-dark .navbar.is-danger .navbar-item.has-dropdown:hover .navbar-link, - html.theme--documenter-dark .navbar.is-danger .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #86170b; - color: #fff; } - html.theme--documenter-dark .navbar.is-danger .navbar-dropdown a.navbar-item.is-active { - background-color: #9e1b0d; - color: #fff; } } - html.theme--documenter-dark .navbar > .container { - align-items: stretch; - display: flex; - min-height: 4rem; - width: 100%; } - html.theme--documenter-dark .navbar.has-shadow { - box-shadow: 0 2px 0 0 #282f2f; } - html.theme--documenter-dark .navbar.is-fixed-bottom, html.theme--documenter-dark .navbar.is-fixed-top { - left: 0; - position: fixed; - right: 0; - z-index: 30; } - html.theme--documenter-dark .navbar.is-fixed-bottom { - bottom: 0; } - html.theme--documenter-dark .navbar.is-fixed-bottom.has-shadow { - box-shadow: 0 -2px 0 0 #282f2f; } - html.theme--documenter-dark .navbar.is-fixed-top { - top: 0; } - html.theme--documenter-dark html.has-navbar-fixed-top, - html.theme--documenter-dark body.has-navbar-fixed-top { - padding-top: 4rem; } - html.theme--documenter-dark html.has-navbar-fixed-bottom, - html.theme--documenter-dark body.has-navbar-fixed-bottom { - padding-bottom: 4rem; } - html.theme--documenter-dark .navbar-brand, - html.theme--documenter-dark .navbar-tabs { - align-items: stretch; - display: flex; - flex-shrink: 0; - min-height: 4rem; } - html.theme--documenter-dark .navbar-brand a.navbar-item:focus, html.theme--documenter-dark .navbar-brand a.navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .navbar-tabs { - -webkit-overflow-scrolling: touch; - max-width: 100vw; - overflow-x: auto; - overflow-y: hidden; } - html.theme--documenter-dark .navbar-burger { - color: #fff; - cursor: pointer; - display: block; - height: 4rem; - position: relative; - width: 4rem; - margin-left: auto; } - html.theme--documenter-dark .navbar-burger span { - background-color: currentColor; - display: block; - height: 1px; - left: calc(50% - 8px); - position: absolute; - transform-origin: center; - transition-duration: 86ms; - transition-property: background-color, opacity, transform; - transition-timing-function: ease-out; - width: 16px; } - html.theme--documenter-dark .navbar-burger span:nth-child(1) { - top: calc(50% - 6px); } - html.theme--documenter-dark .navbar-burger span:nth-child(2) { - top: calc(50% - 1px); } - html.theme--documenter-dark .navbar-burger span:nth-child(3) { - top: calc(50% + 4px); } - html.theme--documenter-dark .navbar-burger:hover { - background-color: rgba(0, 0, 0, 0.05); } - html.theme--documenter-dark .navbar-burger.is-active span:nth-child(1) { - transform: translateY(5px) rotate(45deg); } - html.theme--documenter-dark .navbar-burger.is-active span:nth-child(2) { - opacity: 0; } - html.theme--documenter-dark .navbar-burger.is-active span:nth-child(3) { - transform: translateY(-5px) rotate(-45deg); } - html.theme--documenter-dark .navbar-menu { - display: none; } - html.theme--documenter-dark .navbar-item, - html.theme--documenter-dark .navbar-link { - color: #fff; - display: block; - line-height: 1.5; - padding: 0.5rem 0.75rem; - position: relative; } - html.theme--documenter-dark .navbar-item .icon:only-child, - html.theme--documenter-dark .navbar-link .icon:only-child { - margin-left: -0.25rem; - margin-right: -0.25rem; } - html.theme--documenter-dark a.navbar-item, - html.theme--documenter-dark .navbar-link { - cursor: pointer; } - html.theme--documenter-dark a.navbar-item:focus, html.theme--documenter-dark a.navbar-item:focus-within, html.theme--documenter-dark a.navbar-item:hover, html.theme--documenter-dark a.navbar-item.is-active, - html.theme--documenter-dark .navbar-link:focus, - html.theme--documenter-dark .navbar-link:focus-within, - html.theme--documenter-dark .navbar-link:hover, - html.theme--documenter-dark .navbar-link.is-active { - background-color: transparent; - color: #1abc9c; } - html.theme--documenter-dark .navbar-item { - display: block; - flex-grow: 0; - flex-shrink: 0; } - html.theme--documenter-dark .navbar-item img { - max-height: 1.75rem; } - html.theme--documenter-dark .navbar-item.has-dropdown { - padding: 0; } - html.theme--documenter-dark .navbar-item.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - html.theme--documenter-dark .navbar-item.is-tab { - border-bottom: 1px solid transparent; - min-height: 4rem; - padding-bottom: calc(0.5rem - 1px); } - html.theme--documenter-dark .navbar-item.is-tab:focus, html.theme--documenter-dark .navbar-item.is-tab:hover { - background-color: transparent; - border-bottom-color: #1abc9c; } - html.theme--documenter-dark .navbar-item.is-tab.is-active { - background-color: transparent; - border-bottom-color: #1abc9c; - border-bottom-style: solid; - border-bottom-width: 3px; - color: #1abc9c; - padding-bottom: calc(0.5rem - 3px); } - html.theme--documenter-dark .navbar-content { - flex-grow: 1; - flex-shrink: 1; } - html.theme--documenter-dark .navbar-link:not(.is-arrowless) { - padding-right: 2.5em; } - html.theme--documenter-dark .navbar-link:not(.is-arrowless)::after { - border-color: #fff; - margin-top: -0.375em; - right: 1.125em; } - html.theme--documenter-dark .navbar-dropdown { - font-size: 0.875rem; - padding-bottom: 0.5rem; - padding-top: 0.5rem; } - html.theme--documenter-dark .navbar-dropdown .navbar-item { - padding-left: 1.5rem; - padding-right: 1.5rem; } - html.theme--documenter-dark .navbar-divider { - background-color: rgba(0, 0, 0, 0.2); - border: none; - display: none; - height: 2px; - margin: 0.5rem 0; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .navbar > .container { - display: block; } - html.theme--documenter-dark .navbar-brand .navbar-item, - html.theme--documenter-dark .navbar-tabs .navbar-item { - align-items: center; - display: flex; } - html.theme--documenter-dark .navbar-link::after { - display: none; } - html.theme--documenter-dark .navbar-menu { - background-color: #375a7f; - box-shadow: 0 8px 16px rgba(10, 10, 10, 0.1); - padding: 0.5rem 0; } - html.theme--documenter-dark .navbar-menu.is-active { - display: block; } - html.theme--documenter-dark .navbar.is-fixed-bottom-touch, html.theme--documenter-dark .navbar.is-fixed-top-touch { - left: 0; - position: fixed; - right: 0; - z-index: 30; } - html.theme--documenter-dark .navbar.is-fixed-bottom-touch { - bottom: 0; } - html.theme--documenter-dark .navbar.is-fixed-bottom-touch.has-shadow { - box-shadow: 0 -2px 3px rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .navbar.is-fixed-top-touch { - top: 0; } - html.theme--documenter-dark .navbar.is-fixed-top .navbar-menu, html.theme--documenter-dark .navbar.is-fixed-top-touch .navbar-menu { - -webkit-overflow-scrolling: touch; - max-height: calc(100vh - 4rem); - overflow: auto; } - html.theme--documenter-dark html.has-navbar-fixed-top-touch, - html.theme--documenter-dark body.has-navbar-fixed-top-touch { - padding-top: 4rem; } - html.theme--documenter-dark html.has-navbar-fixed-bottom-touch, - html.theme--documenter-dark body.has-navbar-fixed-bottom-touch { - padding-bottom: 4rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .navbar, - html.theme--documenter-dark .navbar-menu, - html.theme--documenter-dark .navbar-start, - html.theme--documenter-dark .navbar-end { - align-items: stretch; - display: flex; } - html.theme--documenter-dark .navbar { - min-height: 4rem; } - html.theme--documenter-dark .navbar.is-spaced { - padding: 1rem 2rem; } - html.theme--documenter-dark .navbar.is-spaced .navbar-start, - html.theme--documenter-dark .navbar.is-spaced .navbar-end { - align-items: center; } - html.theme--documenter-dark .navbar.is-spaced a.navbar-item, - html.theme--documenter-dark .navbar.is-spaced .navbar-link { - border-radius: 0.4em; } - html.theme--documenter-dark .navbar.is-transparent a.navbar-item:focus, html.theme--documenter-dark .navbar.is-transparent a.navbar-item:hover, html.theme--documenter-dark .navbar.is-transparent a.navbar-item.is-active, - html.theme--documenter-dark .navbar.is-transparent .navbar-link:focus, - html.theme--documenter-dark .navbar.is-transparent .navbar-link:hover, - html.theme--documenter-dark .navbar.is-transparent .navbar-link.is-active { - background-color: transparent !important; } - html.theme--documenter-dark .navbar.is-transparent .navbar-item.has-dropdown.is-active .navbar-link, html.theme--documenter-dark .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:focus .navbar-link, html.theme--documenter-dark .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:focus-within .navbar-link, html.theme--documenter-dark .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:hover .navbar-link { - background-color: transparent !important; } - html.theme--documenter-dark .navbar.is-transparent .navbar-dropdown a.navbar-item:focus, html.theme--documenter-dark .navbar.is-transparent .navbar-dropdown a.navbar-item:hover { - background-color: transparent; - color: #dbdee0; } - html.theme--documenter-dark .navbar.is-transparent .navbar-dropdown a.navbar-item.is-active { - background-color: transparent; - color: #1abc9c; } - html.theme--documenter-dark .navbar-burger { - display: none; } - html.theme--documenter-dark .navbar-item, - html.theme--documenter-dark .navbar-link { - align-items: center; - display: flex; } - html.theme--documenter-dark .navbar-item { - display: flex; } - html.theme--documenter-dark .navbar-item.has-dropdown { - align-items: stretch; } - html.theme--documenter-dark .navbar-item.has-dropdown-up .navbar-link::after { - transform: rotate(135deg) translate(0.25em, -0.25em); } - html.theme--documenter-dark .navbar-item.has-dropdown-up .navbar-dropdown { - border-bottom: 1px solid rgba(0, 0, 0, 0.2); - border-radius: 8px 8px 0 0; - border-top: none; - bottom: 100%; - box-shadow: 0 -8px 8px rgba(10, 10, 10, 0.1); - top: auto; } - html.theme--documenter-dark .navbar-item.is-active .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:focus .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:focus-within .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:hover .navbar-dropdown { - display: block; } - .navbar.is-spaced html.theme--documenter-dark .navbar-item.is-active .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-active .navbar-dropdown.is-boxed, .navbar.is-spaced html.theme--documenter-dark .navbar-item.is-hoverable:focus .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:focus .navbar-dropdown.is-boxed, .navbar.is-spaced html.theme--documenter-dark .navbar-item.is-hoverable:focus-within .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:focus-within .navbar-dropdown.is-boxed, .navbar.is-spaced html.theme--documenter-dark .navbar-item.is-hoverable:hover .navbar-dropdown, html.theme--documenter-dark .navbar-item.is-hoverable:hover .navbar-dropdown.is-boxed { - opacity: 1; - pointer-events: auto; - transform: translateY(0); } - html.theme--documenter-dark .navbar-menu { - flex-grow: 1; - flex-shrink: 0; } - html.theme--documenter-dark .navbar-start { - justify-content: flex-start; - margin-right: auto; } - html.theme--documenter-dark .navbar-end { - justify-content: flex-end; - margin-left: auto; } - html.theme--documenter-dark .navbar-dropdown { - background-color: #375a7f; - border-bottom-left-radius: 8px; - border-bottom-right-radius: 8px; - border-top: 1px solid rgba(0, 0, 0, 0.2); - box-shadow: 0 8px 8px rgba(10, 10, 10, 0.1); - display: none; - font-size: 0.875rem; - left: 0; - min-width: 100%; - position: absolute; - top: 100%; - z-index: 20; } - html.theme--documenter-dark .navbar-dropdown .navbar-item { - padding: 0.375rem 1rem; - white-space: nowrap; } - html.theme--documenter-dark .navbar-dropdown a.navbar-item { - padding-right: 3rem; } - html.theme--documenter-dark .navbar-dropdown a.navbar-item:focus, html.theme--documenter-dark .navbar-dropdown a.navbar-item:hover { - background-color: transparent; - color: #dbdee0; } - html.theme--documenter-dark .navbar-dropdown a.navbar-item.is-active { - background-color: transparent; - color: #1abc9c; } - .navbar.is-spaced html.theme--documenter-dark .navbar-dropdown, html.theme--documenter-dark .navbar-dropdown.is-boxed { - border-radius: 8px; - border-top: none; - box-shadow: 0 8px 8px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); - display: block; - opacity: 0; - pointer-events: none; - top: calc(100% + (-4px)); - transform: translateY(-5px); - transition-duration: 86ms; - transition-property: opacity, transform; } - html.theme--documenter-dark .navbar-dropdown.is-right { - left: auto; - right: 0; } - html.theme--documenter-dark .navbar-divider { - display: block; } - html.theme--documenter-dark .navbar > .container .navbar-brand, - html.theme--documenter-dark .container > .navbar .navbar-brand { - margin-left: -.75rem; } - html.theme--documenter-dark .navbar > .container .navbar-menu, - html.theme--documenter-dark .container > .navbar .navbar-menu { - margin-right: -.75rem; } - html.theme--documenter-dark .navbar.is-fixed-bottom-desktop, html.theme--documenter-dark .navbar.is-fixed-top-desktop { - left: 0; - position: fixed; - right: 0; - z-index: 30; } - html.theme--documenter-dark .navbar.is-fixed-bottom-desktop { - bottom: 0; } - html.theme--documenter-dark .navbar.is-fixed-bottom-desktop.has-shadow { - box-shadow: 0 -2px 3px rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .navbar.is-fixed-top-desktop { - top: 0; } - html.theme--documenter-dark html.has-navbar-fixed-top-desktop, - html.theme--documenter-dark body.has-navbar-fixed-top-desktop { - padding-top: 4rem; } - html.theme--documenter-dark html.has-navbar-fixed-bottom-desktop, - html.theme--documenter-dark body.has-navbar-fixed-bottom-desktop { - padding-bottom: 4rem; } - html.theme--documenter-dark html.has-spaced-navbar-fixed-top, - html.theme--documenter-dark body.has-spaced-navbar-fixed-top { - padding-top: 6rem; } - html.theme--documenter-dark html.has-spaced-navbar-fixed-bottom, - html.theme--documenter-dark body.has-spaced-navbar-fixed-bottom { - padding-bottom: 6rem; } - html.theme--documenter-dark a.navbar-item.is-active, - html.theme--documenter-dark .navbar-link.is-active { - color: #1abc9c; } - html.theme--documenter-dark a.navbar-item.is-active:not(:focus):not(:hover), - html.theme--documenter-dark .navbar-link.is-active:not(:focus):not(:hover) { - background-color: transparent; } - html.theme--documenter-dark .navbar-item.has-dropdown:focus .navbar-link, html.theme--documenter-dark .navbar-item.has-dropdown:hover .navbar-link, html.theme--documenter-dark .navbar-item.has-dropdown.is-active .navbar-link { - background-color: transparent; } } - html.theme--documenter-dark .hero.is-fullheight-with-navbar { - min-height: calc(100vh - 4rem); } - html.theme--documenter-dark .pagination { - font-size: 15px; - margin: -0.25rem; } - html.theme--documenter-dark .pagination.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.pagination { - font-size: 0.85em; } - html.theme--documenter-dark .pagination.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .pagination.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .pagination.is-rounded .pagination-previous, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.pagination .pagination-previous, - html.theme--documenter-dark .pagination.is-rounded .pagination-next, - html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.pagination .pagination-next { - padding-left: 1em; - padding-right: 1em; - border-radius: 290486px; } - html.theme--documenter-dark .pagination.is-rounded .pagination-link, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.pagination .pagination-link { - border-radius: 290486px; } - html.theme--documenter-dark .pagination, - html.theme--documenter-dark .pagination-list { - align-items: center; - display: flex; - justify-content: center; - text-align: center; } - html.theme--documenter-dark .pagination-previous, - html.theme--documenter-dark .pagination-next, - html.theme--documenter-dark .pagination-link, - html.theme--documenter-dark .pagination-ellipsis { - font-size: 1em; - justify-content: center; - margin: 0.25rem; - padding-left: 0.5em; - padding-right: 0.5em; - text-align: center; } - html.theme--documenter-dark .pagination-previous, - html.theme--documenter-dark .pagination-next, - html.theme--documenter-dark .pagination-link { - border-color: #5e6d6f; - color: #1abc9c; - min-width: 2.25em; } - html.theme--documenter-dark .pagination-previous:hover, - html.theme--documenter-dark .pagination-next:hover, - html.theme--documenter-dark .pagination-link:hover { - border-color: #8c9b9d; - color: #1dd2af; } - html.theme--documenter-dark .pagination-previous:focus, - html.theme--documenter-dark .pagination-next:focus, - html.theme--documenter-dark .pagination-link:focus { - border-color: #8c9b9d; } - html.theme--documenter-dark .pagination-previous:active, - html.theme--documenter-dark .pagination-next:active, - html.theme--documenter-dark .pagination-link:active { - box-shadow: inset 0 1px 2px rgba(10, 10, 10, 0.2); } - html.theme--documenter-dark .pagination-previous[disabled], - html.theme--documenter-dark .pagination-next[disabled], - html.theme--documenter-dark .pagination-link[disabled] { - background-color: #dbdee0; - border-color: #dbdee0; - box-shadow: none; - color: #5e6d6f; - opacity: 0.5; } - html.theme--documenter-dark .pagination-previous, - html.theme--documenter-dark .pagination-next { - padding-left: 0.75em; - padding-right: 0.75em; - white-space: nowrap; } - html.theme--documenter-dark .pagination-link.is-current { - background-color: #1abc9c; - border-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .pagination-ellipsis { - color: #8c9b9d; - pointer-events: none; } - html.theme--documenter-dark .pagination-list { - flex-wrap: wrap; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .pagination { - flex-wrap: wrap; } - html.theme--documenter-dark .pagination-previous, - html.theme--documenter-dark .pagination-next { - flex-grow: 1; - flex-shrink: 1; } - html.theme--documenter-dark .pagination-list li { - flex-grow: 1; - flex-shrink: 1; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .pagination-list { - flex-grow: 1; - flex-shrink: 1; - justify-content: flex-start; - order: 1; } - html.theme--documenter-dark .pagination-previous { - order: 2; } - html.theme--documenter-dark .pagination-next { - order: 3; } - html.theme--documenter-dark .pagination { - justify-content: space-between; } - html.theme--documenter-dark .pagination.is-centered .pagination-previous { - order: 1; } - html.theme--documenter-dark .pagination.is-centered .pagination-list { - justify-content: center; - order: 2; } - html.theme--documenter-dark .pagination.is-centered .pagination-next { - order: 3; } - html.theme--documenter-dark .pagination.is-right .pagination-previous { - order: 1; } - html.theme--documenter-dark .pagination.is-right .pagination-next { - order: 2; } - html.theme--documenter-dark .pagination.is-right .pagination-list { - justify-content: flex-end; - order: 3; } } - html.theme--documenter-dark .panel { - font-size: 15px; } - html.theme--documenter-dark .panel:not(:last-child) { - margin-bottom: 1.5rem; } - html.theme--documenter-dark .panel-heading, - html.theme--documenter-dark .panel-tabs, - html.theme--documenter-dark .panel-block { - border-bottom: 1px solid #5e6d6f; - border-left: 1px solid #5e6d6f; - border-right: 1px solid #5e6d6f; } - html.theme--documenter-dark .panel-heading:first-child, - html.theme--documenter-dark .panel-tabs:first-child, - html.theme--documenter-dark .panel-block:first-child { - border-top: 1px solid #5e6d6f; } - html.theme--documenter-dark .panel-heading { - background-color: #282f2f; - border-radius: 0.4em 0.4em 0 0; - color: #f2f2f2; - font-size: 1.25em; - font-weight: 300; - line-height: 1.25; - padding: 0.5em 0.75em; } - html.theme--documenter-dark .panel-tabs { - align-items: flex-end; - display: flex; - font-size: 0.875em; - justify-content: center; } - html.theme--documenter-dark .panel-tabs a { - border-bottom: 1px solid #5e6d6f; - margin-bottom: -1px; - padding: 0.5em; } - html.theme--documenter-dark .panel-tabs a.is-active { - border-bottom-color: #343c3d; - color: #17a689; } - html.theme--documenter-dark .panel-list a { - color: #fff; } - html.theme--documenter-dark .panel-list a:hover { - color: #1abc9c; } - html.theme--documenter-dark .panel-block { - align-items: center; - color: #f2f2f2; - display: flex; - justify-content: flex-start; - padding: 0.5em 0.75em; } - html.theme--documenter-dark .panel-block input[type="checkbox"] { - margin-right: 0.75em; } - html.theme--documenter-dark .panel-block > .control { - flex-grow: 1; - flex-shrink: 1; - width: 100%; } - html.theme--documenter-dark .panel-block.is-wrapped { - flex-wrap: wrap; } - html.theme--documenter-dark .panel-block.is-active { - border-left-color: #1abc9c; - color: #17a689; } - html.theme--documenter-dark .panel-block.is-active .panel-icon { - color: #1abc9c; } - html.theme--documenter-dark a.panel-block, - html.theme--documenter-dark label.panel-block { - cursor: pointer; } - html.theme--documenter-dark a.panel-block:hover, - html.theme--documenter-dark label.panel-block:hover { - background-color: #282f2f; } - html.theme--documenter-dark .panel-icon { - display: inline-block; - font-size: 14px; - height: 1em; - line-height: 1em; - text-align: center; - vertical-align: top; - width: 1em; - color: white; - margin-right: 0.75em; } - html.theme--documenter-dark .panel-icon .fa { - font-size: inherit; - line-height: inherit; } - html.theme--documenter-dark .tabs { - -webkit-overflow-scrolling: touch; - align-items: stretch; - display: flex; - font-size: 15px; - justify-content: space-between; - overflow: hidden; - overflow-x: auto; - white-space: nowrap; } - html.theme--documenter-dark .tabs a { - align-items: center; - border-bottom-color: #5e6d6f; - border-bottom-style: solid; - border-bottom-width: 1px; - color: #fff; - display: flex; - justify-content: center; - margin-bottom: -1px; - padding: 0.5em 1em; - vertical-align: top; } - html.theme--documenter-dark .tabs a:hover { - border-bottom-color: #f2f2f2; - color: #f2f2f2; } - html.theme--documenter-dark .tabs li { - display: block; } - html.theme--documenter-dark .tabs li.is-active a { - border-bottom-color: #1abc9c; - color: #1abc9c; } - html.theme--documenter-dark .tabs ul { - align-items: center; - border-bottom-color: #5e6d6f; - border-bottom-style: solid; - border-bottom-width: 1px; - display: flex; - flex-grow: 1; - flex-shrink: 0; - justify-content: flex-start; } - html.theme--documenter-dark .tabs ul.is-left { - padding-right: 0.75em; } - html.theme--documenter-dark .tabs ul.is-center { - flex: none; - justify-content: center; - padding-left: 0.75em; - padding-right: 0.75em; } - html.theme--documenter-dark .tabs ul.is-right { - justify-content: flex-end; - padding-left: 0.75em; } - html.theme--documenter-dark .tabs .icon:first-child { - margin-right: 0.5em; } - html.theme--documenter-dark .tabs .icon:last-child { - margin-left: 0.5em; } - html.theme--documenter-dark .tabs.is-centered ul { - justify-content: center; } - html.theme--documenter-dark .tabs.is-right ul { - justify-content: flex-end; } - html.theme--documenter-dark .tabs.is-boxed a { - border: 1px solid transparent; - border-radius: 0.4em 0.4em 0 0; } - html.theme--documenter-dark .tabs.is-boxed a:hover { - background-color: #282f2f; - border-bottom-color: #5e6d6f; } - html.theme--documenter-dark .tabs.is-boxed li.is-active a { - background-color: white; - border-color: #5e6d6f; - border-bottom-color: transparent !important; } - html.theme--documenter-dark .tabs.is-fullwidth li { - flex-grow: 1; - flex-shrink: 0; } - html.theme--documenter-dark .tabs.is-toggle a { - border-color: #5e6d6f; - border-style: solid; - border-width: 1px; - margin-bottom: 0; - position: relative; } - html.theme--documenter-dark .tabs.is-toggle a:hover { - background-color: #282f2f; - border-color: #8c9b9d; - z-index: 2; } - html.theme--documenter-dark .tabs.is-toggle li + li { - margin-left: -1px; } - html.theme--documenter-dark .tabs.is-toggle li:first-child a { - border-radius: 0.4em 0 0 0.4em; } - html.theme--documenter-dark .tabs.is-toggle li:last-child a { - border-radius: 0 0.4em 0.4em 0; } - html.theme--documenter-dark .tabs.is-toggle li.is-active a { - background-color: #1abc9c; - border-color: #1abc9c; - color: #fff; - z-index: 1; } - html.theme--documenter-dark .tabs.is-toggle ul { - border-bottom: none; } - html.theme--documenter-dark .tabs.is-toggle.is-toggle-rounded li:first-child a { - border-bottom-left-radius: 290486px; - border-top-left-radius: 290486px; - padding-left: 1.25em; } - html.theme--documenter-dark .tabs.is-toggle.is-toggle-rounded li:last-child a { - border-bottom-right-radius: 290486px; - border-top-right-radius: 290486px; - padding-right: 1.25em; } - html.theme--documenter-dark .tabs.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.tabs { - font-size: 0.85em; } - html.theme--documenter-dark .tabs.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .tabs.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .column { - display: block; - flex-basis: 0; - flex-grow: 1; - flex-shrink: 1; - padding: 0.75rem; } - .columns.is-mobile > html.theme--documenter-dark .column.is-narrow { - flex: none; } - .columns.is-mobile > html.theme--documenter-dark .column.is-full { - flex: none; - width: 100%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-three-quarters { - flex: none; - width: 75%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-two-thirds { - flex: none; - width: 66.6666%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-half { - flex: none; - width: 50%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-one-third { - flex: none; - width: 33.3333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-one-quarter { - flex: none; - width: 25%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-one-fifth { - flex: none; - width: 20%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-two-fifths { - flex: none; - width: 40%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-three-fifths { - flex: none; - width: 60%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-four-fifths { - flex: none; - width: 80%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-three-quarters { - margin-left: 75%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-two-thirds { - margin-left: 66.6666%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-half { - margin-left: 50%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-one-third { - margin-left: 33.3333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-one-quarter { - margin-left: 25%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-one-fifth { - margin-left: 20%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-two-fifths { - margin-left: 40%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-three-fifths { - margin-left: 60%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-four-fifths { - margin-left: 80%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-0 { - flex: none; - width: 0%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-0 { - margin-left: 0%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-1 { - flex: none; - width: 8.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-1 { - margin-left: 8.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-2 { - flex: none; - width: 16.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-2 { - margin-left: 16.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-3 { - flex: none; - width: 25%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-3 { - margin-left: 25%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-4 { - flex: none; - width: 33.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-4 { - margin-left: 33.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-5 { - flex: none; - width: 41.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-5 { - margin-left: 41.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-6 { - flex: none; - width: 50%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-6 { - margin-left: 50%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-7 { - flex: none; - width: 58.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-7 { - margin-left: 58.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-8 { - flex: none; - width: 66.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-8 { - margin-left: 66.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-9 { - flex: none; - width: 75%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-9 { - margin-left: 75%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-10 { - flex: none; - width: 83.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-10 { - margin-left: 83.3333333333%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-11 { - flex: none; - width: 91.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-11 { - margin-left: 91.6666666667%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-12 { - flex: none; - width: 100%; } - .columns.is-mobile > html.theme--documenter-dark .column.is-offset-12 { - margin-left: 100%; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .column.is-narrow-mobile { - flex: none; } - html.theme--documenter-dark .column.is-full-mobile { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-three-quarters-mobile { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-two-thirds-mobile { - flex: none; - width: 66.6666%; } - html.theme--documenter-dark .column.is-half-mobile { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-one-third-mobile { - flex: none; - width: 33.3333%; } - html.theme--documenter-dark .column.is-one-quarter-mobile { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-one-fifth-mobile { - flex: none; - width: 20%; } - html.theme--documenter-dark .column.is-two-fifths-mobile { - flex: none; - width: 40%; } - html.theme--documenter-dark .column.is-three-fifths-mobile { - flex: none; - width: 60%; } - html.theme--documenter-dark .column.is-four-fifths-mobile { - flex: none; - width: 80%; } - html.theme--documenter-dark .column.is-offset-three-quarters-mobile { - margin-left: 75%; } - html.theme--documenter-dark .column.is-offset-two-thirds-mobile { - margin-left: 66.6666%; } - html.theme--documenter-dark .column.is-offset-half-mobile { - margin-left: 50%; } - html.theme--documenter-dark .column.is-offset-one-third-mobile { - margin-left: 33.3333%; } - html.theme--documenter-dark .column.is-offset-one-quarter-mobile { - margin-left: 25%; } - html.theme--documenter-dark .column.is-offset-one-fifth-mobile { - margin-left: 20%; } - html.theme--documenter-dark .column.is-offset-two-fifths-mobile { - margin-left: 40%; } - html.theme--documenter-dark .column.is-offset-three-fifths-mobile { - margin-left: 60%; } - html.theme--documenter-dark .column.is-offset-four-fifths-mobile { - margin-left: 80%; } - html.theme--documenter-dark .column.is-0-mobile { - flex: none; - width: 0%; } - html.theme--documenter-dark .column.is-offset-0-mobile { - margin-left: 0%; } - html.theme--documenter-dark .column.is-1-mobile { - flex: none; - width: 8.3333333333%; } - html.theme--documenter-dark .column.is-offset-1-mobile { - margin-left: 8.3333333333%; } - html.theme--documenter-dark .column.is-2-mobile { - flex: none; - width: 16.6666666667%; } - html.theme--documenter-dark .column.is-offset-2-mobile { - margin-left: 16.6666666667%; } - html.theme--documenter-dark .column.is-3-mobile { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-offset-3-mobile { - margin-left: 25%; } - html.theme--documenter-dark .column.is-4-mobile { - flex: none; - width: 33.3333333333%; } - html.theme--documenter-dark .column.is-offset-4-mobile { - margin-left: 33.3333333333%; } - html.theme--documenter-dark .column.is-5-mobile { - flex: none; - width: 41.6666666667%; } - html.theme--documenter-dark .column.is-offset-5-mobile { - margin-left: 41.6666666667%; } - html.theme--documenter-dark .column.is-6-mobile { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-offset-6-mobile { - margin-left: 50%; } - html.theme--documenter-dark .column.is-7-mobile { - flex: none; - width: 58.3333333333%; } - html.theme--documenter-dark .column.is-offset-7-mobile { - margin-left: 58.3333333333%; } - html.theme--documenter-dark .column.is-8-mobile { - flex: none; - width: 66.6666666667%; } - html.theme--documenter-dark .column.is-offset-8-mobile { - margin-left: 66.6666666667%; } - html.theme--documenter-dark .column.is-9-mobile { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-offset-9-mobile { - margin-left: 75%; } - html.theme--documenter-dark .column.is-10-mobile { - flex: none; - width: 83.3333333333%; } - html.theme--documenter-dark .column.is-offset-10-mobile { - margin-left: 83.3333333333%; } - html.theme--documenter-dark .column.is-11-mobile { - flex: none; - width: 91.6666666667%; } - html.theme--documenter-dark .column.is-offset-11-mobile { - margin-left: 91.6666666667%; } - html.theme--documenter-dark .column.is-12-mobile { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-offset-12-mobile { - margin-left: 100%; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .column.is-narrow, html.theme--documenter-dark .column.is-narrow-tablet { - flex: none; } - html.theme--documenter-dark .column.is-full, html.theme--documenter-dark .column.is-full-tablet { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-three-quarters, html.theme--documenter-dark .column.is-three-quarters-tablet { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-two-thirds, html.theme--documenter-dark .column.is-two-thirds-tablet { - flex: none; - width: 66.6666%; } - html.theme--documenter-dark .column.is-half, html.theme--documenter-dark .column.is-half-tablet { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-one-third, html.theme--documenter-dark .column.is-one-third-tablet { - flex: none; - width: 33.3333%; } - html.theme--documenter-dark .column.is-one-quarter, html.theme--documenter-dark .column.is-one-quarter-tablet { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-one-fifth, html.theme--documenter-dark .column.is-one-fifth-tablet { - flex: none; - width: 20%; } - html.theme--documenter-dark .column.is-two-fifths, html.theme--documenter-dark .column.is-two-fifths-tablet { - flex: none; - width: 40%; } - html.theme--documenter-dark .column.is-three-fifths, html.theme--documenter-dark .column.is-three-fifths-tablet { - flex: none; - width: 60%; } - html.theme--documenter-dark .column.is-four-fifths, html.theme--documenter-dark .column.is-four-fifths-tablet { - flex: none; - width: 80%; } - html.theme--documenter-dark .column.is-offset-three-quarters, html.theme--documenter-dark .column.is-offset-three-quarters-tablet { - margin-left: 75%; } - html.theme--documenter-dark .column.is-offset-two-thirds, html.theme--documenter-dark .column.is-offset-two-thirds-tablet { - margin-left: 66.6666%; } - html.theme--documenter-dark .column.is-offset-half, html.theme--documenter-dark .column.is-offset-half-tablet { - margin-left: 50%; } - html.theme--documenter-dark .column.is-offset-one-third, html.theme--documenter-dark .column.is-offset-one-third-tablet { - margin-left: 33.3333%; } - html.theme--documenter-dark .column.is-offset-one-quarter, html.theme--documenter-dark .column.is-offset-one-quarter-tablet { - margin-left: 25%; } - html.theme--documenter-dark .column.is-offset-one-fifth, html.theme--documenter-dark .column.is-offset-one-fifth-tablet { - margin-left: 20%; } - html.theme--documenter-dark .column.is-offset-two-fifths, html.theme--documenter-dark .column.is-offset-two-fifths-tablet { - margin-left: 40%; } - html.theme--documenter-dark .column.is-offset-three-fifths, html.theme--documenter-dark .column.is-offset-three-fifths-tablet { - margin-left: 60%; } - html.theme--documenter-dark .column.is-offset-four-fifths, html.theme--documenter-dark .column.is-offset-four-fifths-tablet { - margin-left: 80%; } - html.theme--documenter-dark .column.is-0, html.theme--documenter-dark .column.is-0-tablet { - flex: none; - width: 0%; } - html.theme--documenter-dark .column.is-offset-0, html.theme--documenter-dark .column.is-offset-0-tablet { - margin-left: 0%; } - html.theme--documenter-dark .column.is-1, html.theme--documenter-dark .column.is-1-tablet { - flex: none; - width: 8.3333333333%; } - html.theme--documenter-dark .column.is-offset-1, html.theme--documenter-dark .column.is-offset-1-tablet { - margin-left: 8.3333333333%; } - html.theme--documenter-dark .column.is-2, html.theme--documenter-dark .column.is-2-tablet { - flex: none; - width: 16.6666666667%; } - html.theme--documenter-dark .column.is-offset-2, html.theme--documenter-dark .column.is-offset-2-tablet { - margin-left: 16.6666666667%; } - html.theme--documenter-dark .column.is-3, html.theme--documenter-dark .column.is-3-tablet { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-offset-3, html.theme--documenter-dark .column.is-offset-3-tablet { - margin-left: 25%; } - html.theme--documenter-dark .column.is-4, html.theme--documenter-dark .column.is-4-tablet { - flex: none; - width: 33.3333333333%; } - html.theme--documenter-dark .column.is-offset-4, html.theme--documenter-dark .column.is-offset-4-tablet { - margin-left: 33.3333333333%; } - html.theme--documenter-dark .column.is-5, html.theme--documenter-dark .column.is-5-tablet { - flex: none; - width: 41.6666666667%; } - html.theme--documenter-dark .column.is-offset-5, html.theme--documenter-dark .column.is-offset-5-tablet { - margin-left: 41.6666666667%; } - html.theme--documenter-dark .column.is-6, html.theme--documenter-dark .column.is-6-tablet { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-offset-6, html.theme--documenter-dark .column.is-offset-6-tablet { - margin-left: 50%; } - html.theme--documenter-dark .column.is-7, html.theme--documenter-dark .column.is-7-tablet { - flex: none; - width: 58.3333333333%; } - html.theme--documenter-dark .column.is-offset-7, html.theme--documenter-dark .column.is-offset-7-tablet { - margin-left: 58.3333333333%; } - html.theme--documenter-dark .column.is-8, html.theme--documenter-dark .column.is-8-tablet { - flex: none; - width: 66.6666666667%; } - html.theme--documenter-dark .column.is-offset-8, html.theme--documenter-dark .column.is-offset-8-tablet { - margin-left: 66.6666666667%; } - html.theme--documenter-dark .column.is-9, html.theme--documenter-dark .column.is-9-tablet { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-offset-9, html.theme--documenter-dark .column.is-offset-9-tablet { - margin-left: 75%; } - html.theme--documenter-dark .column.is-10, html.theme--documenter-dark .column.is-10-tablet { - flex: none; - width: 83.3333333333%; } - html.theme--documenter-dark .column.is-offset-10, html.theme--documenter-dark .column.is-offset-10-tablet { - margin-left: 83.3333333333%; } - html.theme--documenter-dark .column.is-11, html.theme--documenter-dark .column.is-11-tablet { - flex: none; - width: 91.6666666667%; } - html.theme--documenter-dark .column.is-offset-11, html.theme--documenter-dark .column.is-offset-11-tablet { - margin-left: 91.6666666667%; } - html.theme--documenter-dark .column.is-12, html.theme--documenter-dark .column.is-12-tablet { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-offset-12, html.theme--documenter-dark .column.is-offset-12-tablet { - margin-left: 100%; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .column.is-narrow-touch { - flex: none; } - html.theme--documenter-dark .column.is-full-touch { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-three-quarters-touch { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-two-thirds-touch { - flex: none; - width: 66.6666%; } - html.theme--documenter-dark .column.is-half-touch { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-one-third-touch { - flex: none; - width: 33.3333%; } - html.theme--documenter-dark .column.is-one-quarter-touch { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-one-fifth-touch { - flex: none; - width: 20%; } - html.theme--documenter-dark .column.is-two-fifths-touch { - flex: none; - width: 40%; } - html.theme--documenter-dark .column.is-three-fifths-touch { - flex: none; - width: 60%; } - html.theme--documenter-dark .column.is-four-fifths-touch { - flex: none; - width: 80%; } - html.theme--documenter-dark .column.is-offset-three-quarters-touch { - margin-left: 75%; } - html.theme--documenter-dark .column.is-offset-two-thirds-touch { - margin-left: 66.6666%; } - html.theme--documenter-dark .column.is-offset-half-touch { - margin-left: 50%; } - html.theme--documenter-dark .column.is-offset-one-third-touch { - margin-left: 33.3333%; } - html.theme--documenter-dark .column.is-offset-one-quarter-touch { - margin-left: 25%; } - html.theme--documenter-dark .column.is-offset-one-fifth-touch { - margin-left: 20%; } - html.theme--documenter-dark .column.is-offset-two-fifths-touch { - margin-left: 40%; } - html.theme--documenter-dark .column.is-offset-three-fifths-touch { - margin-left: 60%; } - html.theme--documenter-dark .column.is-offset-four-fifths-touch { - margin-left: 80%; } - html.theme--documenter-dark .column.is-0-touch { - flex: none; - width: 0%; } - html.theme--documenter-dark .column.is-offset-0-touch { - margin-left: 0%; } - html.theme--documenter-dark .column.is-1-touch { - flex: none; - width: 8.3333333333%; } - html.theme--documenter-dark .column.is-offset-1-touch { - margin-left: 8.3333333333%; } - html.theme--documenter-dark .column.is-2-touch { - flex: none; - width: 16.6666666667%; } - html.theme--documenter-dark .column.is-offset-2-touch { - margin-left: 16.6666666667%; } - html.theme--documenter-dark .column.is-3-touch { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-offset-3-touch { - margin-left: 25%; } - html.theme--documenter-dark .column.is-4-touch { - flex: none; - width: 33.3333333333%; } - html.theme--documenter-dark .column.is-offset-4-touch { - margin-left: 33.3333333333%; } - html.theme--documenter-dark .column.is-5-touch { - flex: none; - width: 41.6666666667%; } - html.theme--documenter-dark .column.is-offset-5-touch { - margin-left: 41.6666666667%; } - html.theme--documenter-dark .column.is-6-touch { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-offset-6-touch { - margin-left: 50%; } - html.theme--documenter-dark .column.is-7-touch { - flex: none; - width: 58.3333333333%; } - html.theme--documenter-dark .column.is-offset-7-touch { - margin-left: 58.3333333333%; } - html.theme--documenter-dark .column.is-8-touch { - flex: none; - width: 66.6666666667%; } - html.theme--documenter-dark .column.is-offset-8-touch { - margin-left: 66.6666666667%; } - html.theme--documenter-dark .column.is-9-touch { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-offset-9-touch { - margin-left: 75%; } - html.theme--documenter-dark .column.is-10-touch { - flex: none; - width: 83.3333333333%; } - html.theme--documenter-dark .column.is-offset-10-touch { - margin-left: 83.3333333333%; } - html.theme--documenter-dark .column.is-11-touch { - flex: none; - width: 91.6666666667%; } - html.theme--documenter-dark .column.is-offset-11-touch { - margin-left: 91.6666666667%; } - html.theme--documenter-dark .column.is-12-touch { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-offset-12-touch { - margin-left: 100%; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .column.is-narrow-desktop { - flex: none; } - html.theme--documenter-dark .column.is-full-desktop { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-three-quarters-desktop { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-two-thirds-desktop { - flex: none; - width: 66.6666%; } - html.theme--documenter-dark .column.is-half-desktop { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-one-third-desktop { - flex: none; - width: 33.3333%; } - html.theme--documenter-dark .column.is-one-quarter-desktop { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-one-fifth-desktop { - flex: none; - width: 20%; } - html.theme--documenter-dark .column.is-two-fifths-desktop { - flex: none; - width: 40%; } - html.theme--documenter-dark .column.is-three-fifths-desktop { - flex: none; - width: 60%; } - html.theme--documenter-dark .column.is-four-fifths-desktop { - flex: none; - width: 80%; } - html.theme--documenter-dark .column.is-offset-three-quarters-desktop { - margin-left: 75%; } - html.theme--documenter-dark .column.is-offset-two-thirds-desktop { - margin-left: 66.6666%; } - html.theme--documenter-dark .column.is-offset-half-desktop { - margin-left: 50%; } - html.theme--documenter-dark .column.is-offset-one-third-desktop { - margin-left: 33.3333%; } - html.theme--documenter-dark .column.is-offset-one-quarter-desktop { - margin-left: 25%; } - html.theme--documenter-dark .column.is-offset-one-fifth-desktop { - margin-left: 20%; } - html.theme--documenter-dark .column.is-offset-two-fifths-desktop { - margin-left: 40%; } - html.theme--documenter-dark .column.is-offset-three-fifths-desktop { - margin-left: 60%; } - html.theme--documenter-dark .column.is-offset-four-fifths-desktop { - margin-left: 80%; } - html.theme--documenter-dark .column.is-0-desktop { - flex: none; - width: 0%; } - html.theme--documenter-dark .column.is-offset-0-desktop { - margin-left: 0%; } - html.theme--documenter-dark .column.is-1-desktop { - flex: none; - width: 8.3333333333%; } - html.theme--documenter-dark .column.is-offset-1-desktop { - margin-left: 8.3333333333%; } - html.theme--documenter-dark .column.is-2-desktop { - flex: none; - width: 16.6666666667%; } - html.theme--documenter-dark .column.is-offset-2-desktop { - margin-left: 16.6666666667%; } - html.theme--documenter-dark .column.is-3-desktop { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-offset-3-desktop { - margin-left: 25%; } - html.theme--documenter-dark .column.is-4-desktop { - flex: none; - width: 33.3333333333%; } - html.theme--documenter-dark .column.is-offset-4-desktop { - margin-left: 33.3333333333%; } - html.theme--documenter-dark .column.is-5-desktop { - flex: none; - width: 41.6666666667%; } - html.theme--documenter-dark .column.is-offset-5-desktop { - margin-left: 41.6666666667%; } - html.theme--documenter-dark .column.is-6-desktop { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-offset-6-desktop { - margin-left: 50%; } - html.theme--documenter-dark .column.is-7-desktop { - flex: none; - width: 58.3333333333%; } - html.theme--documenter-dark .column.is-offset-7-desktop { - margin-left: 58.3333333333%; } - html.theme--documenter-dark .column.is-8-desktop { - flex: none; - width: 66.6666666667%; } - html.theme--documenter-dark .column.is-offset-8-desktop { - margin-left: 66.6666666667%; } - html.theme--documenter-dark .column.is-9-desktop { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-offset-9-desktop { - margin-left: 75%; } - html.theme--documenter-dark .column.is-10-desktop { - flex: none; - width: 83.3333333333%; } - html.theme--documenter-dark .column.is-offset-10-desktop { - margin-left: 83.3333333333%; } - html.theme--documenter-dark .column.is-11-desktop { - flex: none; - width: 91.6666666667%; } - html.theme--documenter-dark .column.is-offset-11-desktop { - margin-left: 91.6666666667%; } - html.theme--documenter-dark .column.is-12-desktop { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-offset-12-desktop { - margin-left: 100%; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .column.is-narrow-widescreen { - flex: none; } - html.theme--documenter-dark .column.is-full-widescreen { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-three-quarters-widescreen { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-two-thirds-widescreen { - flex: none; - width: 66.6666%; } - html.theme--documenter-dark .column.is-half-widescreen { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-one-third-widescreen { - flex: none; - width: 33.3333%; } - html.theme--documenter-dark .column.is-one-quarter-widescreen { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-one-fifth-widescreen { - flex: none; - width: 20%; } - html.theme--documenter-dark .column.is-two-fifths-widescreen { - flex: none; - width: 40%; } - html.theme--documenter-dark .column.is-three-fifths-widescreen { - flex: none; - width: 60%; } - html.theme--documenter-dark .column.is-four-fifths-widescreen { - flex: none; - width: 80%; } - html.theme--documenter-dark .column.is-offset-three-quarters-widescreen { - margin-left: 75%; } - html.theme--documenter-dark .column.is-offset-two-thirds-widescreen { - margin-left: 66.6666%; } - html.theme--documenter-dark .column.is-offset-half-widescreen { - margin-left: 50%; } - html.theme--documenter-dark .column.is-offset-one-third-widescreen { - margin-left: 33.3333%; } - html.theme--documenter-dark .column.is-offset-one-quarter-widescreen { - margin-left: 25%; } - html.theme--documenter-dark .column.is-offset-one-fifth-widescreen { - margin-left: 20%; } - html.theme--documenter-dark .column.is-offset-two-fifths-widescreen { - margin-left: 40%; } - html.theme--documenter-dark .column.is-offset-three-fifths-widescreen { - margin-left: 60%; } - html.theme--documenter-dark .column.is-offset-four-fifths-widescreen { - margin-left: 80%; } - html.theme--documenter-dark .column.is-0-widescreen { - flex: none; - width: 0%; } - html.theme--documenter-dark .column.is-offset-0-widescreen { - margin-left: 0%; } - html.theme--documenter-dark .column.is-1-widescreen { - flex: none; - width: 8.3333333333%; } - html.theme--documenter-dark .column.is-offset-1-widescreen { - margin-left: 8.3333333333%; } - html.theme--documenter-dark .column.is-2-widescreen { - flex: none; - width: 16.6666666667%; } - html.theme--documenter-dark .column.is-offset-2-widescreen { - margin-left: 16.6666666667%; } - html.theme--documenter-dark .column.is-3-widescreen { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-offset-3-widescreen { - margin-left: 25%; } - html.theme--documenter-dark .column.is-4-widescreen { - flex: none; - width: 33.3333333333%; } - html.theme--documenter-dark .column.is-offset-4-widescreen { - margin-left: 33.3333333333%; } - html.theme--documenter-dark .column.is-5-widescreen { - flex: none; - width: 41.6666666667%; } - html.theme--documenter-dark .column.is-offset-5-widescreen { - margin-left: 41.6666666667%; } - html.theme--documenter-dark .column.is-6-widescreen { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-offset-6-widescreen { - margin-left: 50%; } - html.theme--documenter-dark .column.is-7-widescreen { - flex: none; - width: 58.3333333333%; } - html.theme--documenter-dark .column.is-offset-7-widescreen { - margin-left: 58.3333333333%; } - html.theme--documenter-dark .column.is-8-widescreen { - flex: none; - width: 66.6666666667%; } - html.theme--documenter-dark .column.is-offset-8-widescreen { - margin-left: 66.6666666667%; } - html.theme--documenter-dark .column.is-9-widescreen { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-offset-9-widescreen { - margin-left: 75%; } - html.theme--documenter-dark .column.is-10-widescreen { - flex: none; - width: 83.3333333333%; } - html.theme--documenter-dark .column.is-offset-10-widescreen { - margin-left: 83.3333333333%; } - html.theme--documenter-dark .column.is-11-widescreen { - flex: none; - width: 91.6666666667%; } - html.theme--documenter-dark .column.is-offset-11-widescreen { - margin-left: 91.6666666667%; } - html.theme--documenter-dark .column.is-12-widescreen { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-offset-12-widescreen { - margin-left: 100%; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .column.is-narrow-fullhd { - flex: none; } - html.theme--documenter-dark .column.is-full-fullhd { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-three-quarters-fullhd { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-two-thirds-fullhd { - flex: none; - width: 66.6666%; } - html.theme--documenter-dark .column.is-half-fullhd { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-one-third-fullhd { - flex: none; - width: 33.3333%; } - html.theme--documenter-dark .column.is-one-quarter-fullhd { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-one-fifth-fullhd { - flex: none; - width: 20%; } - html.theme--documenter-dark .column.is-two-fifths-fullhd { - flex: none; - width: 40%; } - html.theme--documenter-dark .column.is-three-fifths-fullhd { - flex: none; - width: 60%; } - html.theme--documenter-dark .column.is-four-fifths-fullhd { - flex: none; - width: 80%; } - html.theme--documenter-dark .column.is-offset-three-quarters-fullhd { - margin-left: 75%; } - html.theme--documenter-dark .column.is-offset-two-thirds-fullhd { - margin-left: 66.6666%; } - html.theme--documenter-dark .column.is-offset-half-fullhd { - margin-left: 50%; } - html.theme--documenter-dark .column.is-offset-one-third-fullhd { - margin-left: 33.3333%; } - html.theme--documenter-dark .column.is-offset-one-quarter-fullhd { - margin-left: 25%; } - html.theme--documenter-dark .column.is-offset-one-fifth-fullhd { - margin-left: 20%; } - html.theme--documenter-dark .column.is-offset-two-fifths-fullhd { - margin-left: 40%; } - html.theme--documenter-dark .column.is-offset-three-fifths-fullhd { - margin-left: 60%; } - html.theme--documenter-dark .column.is-offset-four-fifths-fullhd { - margin-left: 80%; } - html.theme--documenter-dark .column.is-0-fullhd { - flex: none; - width: 0%; } - html.theme--documenter-dark .column.is-offset-0-fullhd { - margin-left: 0%; } - html.theme--documenter-dark .column.is-1-fullhd { - flex: none; - width: 8.3333333333%; } - html.theme--documenter-dark .column.is-offset-1-fullhd { - margin-left: 8.3333333333%; } - html.theme--documenter-dark .column.is-2-fullhd { - flex: none; - width: 16.6666666667%; } - html.theme--documenter-dark .column.is-offset-2-fullhd { - margin-left: 16.6666666667%; } - html.theme--documenter-dark .column.is-3-fullhd { - flex: none; - width: 25%; } - html.theme--documenter-dark .column.is-offset-3-fullhd { - margin-left: 25%; } - html.theme--documenter-dark .column.is-4-fullhd { - flex: none; - width: 33.3333333333%; } - html.theme--documenter-dark .column.is-offset-4-fullhd { - margin-left: 33.3333333333%; } - html.theme--documenter-dark .column.is-5-fullhd { - flex: none; - width: 41.6666666667%; } - html.theme--documenter-dark .column.is-offset-5-fullhd { - margin-left: 41.6666666667%; } - html.theme--documenter-dark .column.is-6-fullhd { - flex: none; - width: 50%; } - html.theme--documenter-dark .column.is-offset-6-fullhd { - margin-left: 50%; } - html.theme--documenter-dark .column.is-7-fullhd { - flex: none; - width: 58.3333333333%; } - html.theme--documenter-dark .column.is-offset-7-fullhd { - margin-left: 58.3333333333%; } - html.theme--documenter-dark .column.is-8-fullhd { - flex: none; - width: 66.6666666667%; } - html.theme--documenter-dark .column.is-offset-8-fullhd { - margin-left: 66.6666666667%; } - html.theme--documenter-dark .column.is-9-fullhd { - flex: none; - width: 75%; } - html.theme--documenter-dark .column.is-offset-9-fullhd { - margin-left: 75%; } - html.theme--documenter-dark .column.is-10-fullhd { - flex: none; - width: 83.3333333333%; } - html.theme--documenter-dark .column.is-offset-10-fullhd { - margin-left: 83.3333333333%; } - html.theme--documenter-dark .column.is-11-fullhd { - flex: none; - width: 91.6666666667%; } - html.theme--documenter-dark .column.is-offset-11-fullhd { - margin-left: 91.6666666667%; } - html.theme--documenter-dark .column.is-12-fullhd { - flex: none; - width: 100%; } - html.theme--documenter-dark .column.is-offset-12-fullhd { - margin-left: 100%; } } - html.theme--documenter-dark .columns { - margin-left: -0.75rem; - margin-right: -0.75rem; - margin-top: -0.75rem; } - html.theme--documenter-dark .columns:last-child { - margin-bottom: -0.75rem; } - html.theme--documenter-dark .columns:not(:last-child) { - margin-bottom: calc(1.5rem - 0.75rem); } - html.theme--documenter-dark .columns.is-centered { - justify-content: center; } - html.theme--documenter-dark .columns.is-gapless { - margin-left: 0; - margin-right: 0; - margin-top: 0; } - html.theme--documenter-dark .columns.is-gapless > .column { - margin: 0; - padding: 0 !important; } - html.theme--documenter-dark .columns.is-gapless:not(:last-child) { - margin-bottom: 1.5rem; } - html.theme--documenter-dark .columns.is-gapless:last-child { - margin-bottom: 0; } - html.theme--documenter-dark .columns.is-mobile { - display: flex; } - html.theme--documenter-dark .columns.is-multiline { - flex-wrap: wrap; } - html.theme--documenter-dark .columns.is-vcentered { - align-items: center; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns:not(.is-desktop) { - display: flex; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-desktop { - display: flex; } } - html.theme--documenter-dark .columns.is-variable { - --columnGap: 0.75rem; - margin-left: calc(-1 * var(--columnGap)); - margin-right: calc(-1 * var(--columnGap)); } - html.theme--documenter-dark .columns.is-variable .column { - padding-left: var(--columnGap); - padding-right: var(--columnGap); } - html.theme--documenter-dark .columns.is-variable.is-0 { - --columnGap: 0rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-0-mobile { - --columnGap: 0rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-0-tablet { - --columnGap: 0rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-0-tablet-only { - --columnGap: 0rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-0-touch { - --columnGap: 0rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-0-desktop { - --columnGap: 0rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-0-desktop-only { - --columnGap: 0rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-0-widescreen { - --columnGap: 0rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-0-widescreen-only { - --columnGap: 0rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-0-fullhd { - --columnGap: 0rem; } } - html.theme--documenter-dark .columns.is-variable.is-1 { - --columnGap: 0.25rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-1-mobile { - --columnGap: 0.25rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-1-tablet { - --columnGap: 0.25rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-1-tablet-only { - --columnGap: 0.25rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-1-touch { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-1-desktop { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-1-desktop-only { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-1-widescreen { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-1-widescreen-only { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-1-fullhd { - --columnGap: 0.25rem; } } - html.theme--documenter-dark .columns.is-variable.is-2 { - --columnGap: 0.5rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-2-mobile { - --columnGap: 0.5rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-2-tablet { - --columnGap: 0.5rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-2-tablet-only { - --columnGap: 0.5rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-2-touch { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-2-desktop { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-2-desktop-only { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-2-widescreen { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-2-widescreen-only { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-2-fullhd { - --columnGap: 0.5rem; } } - html.theme--documenter-dark .columns.is-variable.is-3 { - --columnGap: 0.75rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-3-mobile { - --columnGap: 0.75rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-3-tablet { - --columnGap: 0.75rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-3-tablet-only { - --columnGap: 0.75rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-3-touch { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-3-desktop { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-3-desktop-only { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-3-widescreen { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-3-widescreen-only { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-3-fullhd { - --columnGap: 0.75rem; } } - html.theme--documenter-dark .columns.is-variable.is-4 { - --columnGap: 1rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-4-mobile { - --columnGap: 1rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-4-tablet { - --columnGap: 1rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-4-tablet-only { - --columnGap: 1rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-4-touch { - --columnGap: 1rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-4-desktop { - --columnGap: 1rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-4-desktop-only { - --columnGap: 1rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-4-widescreen { - --columnGap: 1rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-4-widescreen-only { - --columnGap: 1rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-4-fullhd { - --columnGap: 1rem; } } - html.theme--documenter-dark .columns.is-variable.is-5 { - --columnGap: 1.25rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-5-mobile { - --columnGap: 1.25rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-5-tablet { - --columnGap: 1.25rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-5-tablet-only { - --columnGap: 1.25rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-5-touch { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-5-desktop { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-5-desktop-only { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-5-widescreen { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-5-widescreen-only { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-5-fullhd { - --columnGap: 1.25rem; } } - html.theme--documenter-dark .columns.is-variable.is-6 { - --columnGap: 1.5rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-6-mobile { - --columnGap: 1.5rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-6-tablet { - --columnGap: 1.5rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-6-tablet-only { - --columnGap: 1.5rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-6-touch { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-6-desktop { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-6-desktop-only { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-6-widescreen { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-6-widescreen-only { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-6-fullhd { - --columnGap: 1.5rem; } } - html.theme--documenter-dark .columns.is-variable.is-7 { - --columnGap: 1.75rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-7-mobile { - --columnGap: 1.75rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-7-tablet { - --columnGap: 1.75rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-7-tablet-only { - --columnGap: 1.75rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-7-touch { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-7-desktop { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-7-desktop-only { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-7-widescreen { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-7-widescreen-only { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-7-fullhd { - --columnGap: 1.75rem; } } - html.theme--documenter-dark .columns.is-variable.is-8 { - --columnGap: 2rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .columns.is-variable.is-8-mobile { - --columnGap: 2rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .columns.is-variable.is-8-tablet { - --columnGap: 2rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-8-tablet-only { - --columnGap: 2rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .columns.is-variable.is-8-touch { - --columnGap: 2rem; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .columns.is-variable.is-8-desktop { - --columnGap: 2rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - html.theme--documenter-dark .columns.is-variable.is-8-desktop-only { - --columnGap: 2rem; } } - @media screen and (min-width: 1216px) { - html.theme--documenter-dark .columns.is-variable.is-8-widescreen { - --columnGap: 2rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - html.theme--documenter-dark .columns.is-variable.is-8-widescreen-only { - --columnGap: 2rem; } } - @media screen and (min-width: 1408px) { - html.theme--documenter-dark .columns.is-variable.is-8-fullhd { - --columnGap: 2rem; } } - html.theme--documenter-dark .tile { - align-items: stretch; - display: block; - flex-basis: 0; - flex-grow: 1; - flex-shrink: 1; - min-height: min-content; } - html.theme--documenter-dark .tile.is-ancestor { - margin-left: -0.75rem; - margin-right: -0.75rem; - margin-top: -0.75rem; } - html.theme--documenter-dark .tile.is-ancestor:last-child { - margin-bottom: -0.75rem; } - html.theme--documenter-dark .tile.is-ancestor:not(:last-child) { - margin-bottom: 0.75rem; } - html.theme--documenter-dark .tile.is-child { - margin: 0 !important; } - html.theme--documenter-dark .tile.is-parent { - padding: 0.75rem; } - html.theme--documenter-dark .tile.is-vertical { - flex-direction: column; } - html.theme--documenter-dark .tile.is-vertical > .tile.is-child:not(:last-child) { - margin-bottom: 1.5rem !important; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .tile:not(.is-child) { - display: flex; } - html.theme--documenter-dark .tile.is-1 { - flex: none; - width: 8.3333333333%; } - html.theme--documenter-dark .tile.is-2 { - flex: none; - width: 16.6666666667%; } - html.theme--documenter-dark .tile.is-3 { - flex: none; - width: 25%; } - html.theme--documenter-dark .tile.is-4 { - flex: none; - width: 33.3333333333%; } - html.theme--documenter-dark .tile.is-5 { - flex: none; - width: 41.6666666667%; } - html.theme--documenter-dark .tile.is-6 { - flex: none; - width: 50%; } - html.theme--documenter-dark .tile.is-7 { - flex: none; - width: 58.3333333333%; } - html.theme--documenter-dark .tile.is-8 { - flex: none; - width: 66.6666666667%; } - html.theme--documenter-dark .tile.is-9 { - flex: none; - width: 75%; } - html.theme--documenter-dark .tile.is-10 { - flex: none; - width: 83.3333333333%; } - html.theme--documenter-dark .tile.is-11 { - flex: none; - width: 91.6666666667%; } - html.theme--documenter-dark .tile.is-12 { - flex: none; - width: 100%; } } - html.theme--documenter-dark .hero { - align-items: stretch; - display: flex; - flex-direction: column; - justify-content: space-between; } - html.theme--documenter-dark .hero .navbar { - background: none; } - html.theme--documenter-dark .hero .tabs ul { - border-bottom: none; } - html.theme--documenter-dark .hero.is-white { - background-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .hero.is-white a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-white strong { - color: inherit; } - html.theme--documenter-dark .hero.is-white .title { - color: #0a0a0a; } - html.theme--documenter-dark .hero.is-white .subtitle { - color: rgba(10, 10, 10, 0.9); } - html.theme--documenter-dark .hero.is-white .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-white .subtitle strong { - color: #0a0a0a; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-white .navbar-menu { - background-color: white; } } - html.theme--documenter-dark .hero.is-white .navbar-item, - html.theme--documenter-dark .hero.is-white .navbar-link { - color: rgba(10, 10, 10, 0.7); } - html.theme--documenter-dark .hero.is-white a.navbar-item:hover, html.theme--documenter-dark .hero.is-white a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-white .navbar-link:hover, - html.theme--documenter-dark .hero.is-white .navbar-link.is-active { - background-color: #f2f2f2; - color: #0a0a0a; } - html.theme--documenter-dark .hero.is-white .tabs a { - color: #0a0a0a; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-white .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-white .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-white .tabs.is-boxed a, html.theme--documenter-dark .hero.is-white .tabs.is-toggle a { - color: #0a0a0a; } - html.theme--documenter-dark .hero.is-white .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-white .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-white .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-white .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-white .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-white .tabs.is-toggle li.is-active a:hover { - background-color: #0a0a0a; - border-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .hero.is-white.is-bold { - background-image: linear-gradient(141deg, #e8e3e4 0%, white 71%, white 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-white.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #e8e3e4 0%, white 71%, white 100%); } } - html.theme--documenter-dark .hero.is-black { - background-color: #0a0a0a; - color: white; } - html.theme--documenter-dark .hero.is-black a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-black strong { - color: inherit; } - html.theme--documenter-dark .hero.is-black .title { - color: white; } - html.theme--documenter-dark .hero.is-black .subtitle { - color: rgba(255, 255, 255, 0.9); } - html.theme--documenter-dark .hero.is-black .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-black .subtitle strong { - color: white; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-black .navbar-menu { - background-color: #0a0a0a; } } - html.theme--documenter-dark .hero.is-black .navbar-item, - html.theme--documenter-dark .hero.is-black .navbar-link { - color: rgba(255, 255, 255, 0.7); } - html.theme--documenter-dark .hero.is-black a.navbar-item:hover, html.theme--documenter-dark .hero.is-black a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-black .navbar-link:hover, - html.theme--documenter-dark .hero.is-black .navbar-link.is-active { - background-color: black; - color: white; } - html.theme--documenter-dark .hero.is-black .tabs a { - color: white; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-black .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-black .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-black .tabs.is-boxed a, html.theme--documenter-dark .hero.is-black .tabs.is-toggle a { - color: white; } - html.theme--documenter-dark .hero.is-black .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-black .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-black .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-black .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-black .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-black .tabs.is-toggle li.is-active a:hover { - background-color: white; - border-color: white; - color: #0a0a0a; } - html.theme--documenter-dark .hero.is-black.is-bold { - background-image: linear-gradient(141deg, black 0%, #0a0a0a 71%, #181616 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-black.is-bold .navbar-menu { - background-image: linear-gradient(141deg, black 0%, #0a0a0a 71%, #181616 100%); } } - html.theme--documenter-dark .hero.is-light { - background-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .hero.is-light a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-light strong { - color: inherit; } - html.theme--documenter-dark .hero.is-light .title { - color: #282f2f; } - html.theme--documenter-dark .hero.is-light .subtitle { - color: rgba(40, 47, 47, 0.9); } - html.theme--documenter-dark .hero.is-light .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-light .subtitle strong { - color: #282f2f; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-light .navbar-menu { - background-color: #ecf0f1; } } - html.theme--documenter-dark .hero.is-light .navbar-item, - html.theme--documenter-dark .hero.is-light .navbar-link { - color: rgba(40, 47, 47, 0.7); } - html.theme--documenter-dark .hero.is-light a.navbar-item:hover, html.theme--documenter-dark .hero.is-light a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-light .navbar-link:hover, - html.theme--documenter-dark .hero.is-light .navbar-link.is-active { - background-color: #dde4e6; - color: #282f2f; } - html.theme--documenter-dark .hero.is-light .tabs a { - color: #282f2f; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-light .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-light .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-light .tabs.is-boxed a, html.theme--documenter-dark .hero.is-light .tabs.is-toggle a { - color: #282f2f; } - html.theme--documenter-dark .hero.is-light .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-light .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-light .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-light .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-light .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-light .tabs.is-toggle li.is-active a:hover { - background-color: #282f2f; - border-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .hero.is-light.is-bold { - background-image: linear-gradient(141deg, #cadfe0 0%, #ecf0f1 71%, #fafbfc 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-light.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #cadfe0 0%, #ecf0f1 71%, #fafbfc 100%); } } - html.theme--documenter-dark .hero.is-dark, html.theme--documenter-dark .content kbd.hero { - background-color: #282f2f; - color: #ecf0f1; } - html.theme--documenter-dark .hero.is-dark a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), html.theme--documenter-dark .content kbd.hero a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-dark strong, - html.theme--documenter-dark .content kbd.hero strong { - color: inherit; } - html.theme--documenter-dark .hero.is-dark .title, html.theme--documenter-dark .content kbd.hero .title { - color: #ecf0f1; } - html.theme--documenter-dark .hero.is-dark .subtitle, html.theme--documenter-dark .content kbd.hero .subtitle { - color: rgba(236, 240, 241, 0.9); } - html.theme--documenter-dark .hero.is-dark .subtitle a:not(.button), html.theme--documenter-dark .content kbd.hero .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-dark .subtitle strong, - html.theme--documenter-dark .content kbd.hero .subtitle strong { - color: #ecf0f1; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-dark .navbar-menu, html.theme--documenter-dark .content kbd.hero .navbar-menu { - background-color: #282f2f; } } - html.theme--documenter-dark .hero.is-dark .navbar-item, html.theme--documenter-dark .content kbd.hero .navbar-item, - html.theme--documenter-dark .hero.is-dark .navbar-link, - html.theme--documenter-dark .content kbd.hero .navbar-link { - color: rgba(236, 240, 241, 0.7); } - html.theme--documenter-dark .hero.is-dark a.navbar-item:hover, html.theme--documenter-dark .content kbd.hero a.navbar-item:hover, html.theme--documenter-dark .hero.is-dark a.navbar-item.is-active, html.theme--documenter-dark .content kbd.hero a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-dark .navbar-link:hover, - html.theme--documenter-dark .content kbd.hero .navbar-link:hover, - html.theme--documenter-dark .hero.is-dark .navbar-link.is-active, - html.theme--documenter-dark .content kbd.hero .navbar-link.is-active { - background-color: #1d2122; - color: #ecf0f1; } - html.theme--documenter-dark .hero.is-dark .tabs a, html.theme--documenter-dark .content kbd.hero .tabs a { - color: #ecf0f1; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-dark .tabs a:hover, html.theme--documenter-dark .content kbd.hero .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-dark .tabs li.is-active a, html.theme--documenter-dark .content kbd.hero .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-dark .tabs.is-boxed a, html.theme--documenter-dark .content kbd.hero .tabs.is-boxed a, html.theme--documenter-dark .hero.is-dark .tabs.is-toggle a, html.theme--documenter-dark .content kbd.hero .tabs.is-toggle a { - color: #ecf0f1; } - html.theme--documenter-dark .hero.is-dark .tabs.is-boxed a:hover, html.theme--documenter-dark .content kbd.hero .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-dark .tabs.is-toggle a:hover, html.theme--documenter-dark .content kbd.hero .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-dark .tabs.is-boxed li.is-active a, html.theme--documenter-dark .content kbd.hero .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-dark .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-dark .tabs.is-toggle li.is-active a, html.theme--documenter-dark .content kbd.hero .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-dark .tabs.is-toggle li.is-active a:hover { - background-color: #ecf0f1; - border-color: #ecf0f1; - color: #282f2f; } - html.theme--documenter-dark .hero.is-dark.is-bold, html.theme--documenter-dark .content kbd.hero.is-bold { - background-image: linear-gradient(141deg, #0f1615 0%, #282f2f 71%, #313c40 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-dark.is-bold .navbar-menu, html.theme--documenter-dark .content kbd.hero.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #0f1615 0%, #282f2f 71%, #313c40 100%); } } - html.theme--documenter-dark .hero.is-primary, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink { - background-color: #375a7f; - color: #fff; } - html.theme--documenter-dark .hero.is-primary a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-primary strong, - html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink strong { - color: inherit; } - html.theme--documenter-dark .hero.is-primary .title, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .title { - color: #fff; } - html.theme--documenter-dark .hero.is-primary .subtitle, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .subtitle { - color: rgba(255, 255, 255, 0.9); } - html.theme--documenter-dark .hero.is-primary .subtitle a:not(.button), html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-primary .subtitle strong, - html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-primary .navbar-menu, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar-menu { - background-color: #375a7f; } } - html.theme--documenter-dark .hero.is-primary .navbar-item, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar-item, - html.theme--documenter-dark .hero.is-primary .navbar-link, - html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar-link { - color: rgba(255, 255, 255, 0.7); } - html.theme--documenter-dark .hero.is-primary a.navbar-item:hover, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink a.navbar-item:hover, html.theme--documenter-dark .hero.is-primary a.navbar-item.is-active, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-primary .navbar-link:hover, - html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar-link:hover, - html.theme--documenter-dark .hero.is-primary .navbar-link.is-active, - html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar-link.is-active { - background-color: #2f4d6d; - color: #fff; } - html.theme--documenter-dark .hero.is-primary .tabs a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs a { - color: #fff; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-primary .tabs a:hover, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-primary .tabs li.is-active a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-primary .tabs.is-boxed a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed a, html.theme--documenter-dark .hero.is-primary .tabs.is-toggle a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle a { - color: #fff; } - html.theme--documenter-dark .hero.is-primary .tabs.is-boxed a:hover, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-primary .tabs.is-toggle a:hover, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-primary .tabs.is-boxed li.is-active a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-primary .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-primary .tabs.is-toggle li.is-active a, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-primary .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #375a7f; } - html.theme--documenter-dark .hero.is-primary.is-bold, html.theme--documenter-dark .docstring > section > a.hero.is-bold.docs-sourcelink { - background-image: linear-gradient(141deg, #214b62 0%, #375a7f 71%, #3a5796 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-primary.is-bold .navbar-menu, html.theme--documenter-dark .docstring > section > a.hero.is-bold.docs-sourcelink .navbar-menu { - background-image: linear-gradient(141deg, #214b62 0%, #375a7f 71%, #3a5796 100%); } } - html.theme--documenter-dark .hero.is-link { - background-color: #1abc9c; - color: #fff; } - html.theme--documenter-dark .hero.is-link a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-link strong { - color: inherit; } - html.theme--documenter-dark .hero.is-link .title { - color: #fff; } - html.theme--documenter-dark .hero.is-link .subtitle { - color: rgba(255, 255, 255, 0.9); } - html.theme--documenter-dark .hero.is-link .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-link .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-link .navbar-menu { - background-color: #1abc9c; } } - html.theme--documenter-dark .hero.is-link .navbar-item, - html.theme--documenter-dark .hero.is-link .navbar-link { - color: rgba(255, 255, 255, 0.7); } - html.theme--documenter-dark .hero.is-link a.navbar-item:hover, html.theme--documenter-dark .hero.is-link a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-link .navbar-link:hover, - html.theme--documenter-dark .hero.is-link .navbar-link.is-active { - background-color: #17a689; - color: #fff; } - html.theme--documenter-dark .hero.is-link .tabs a { - color: #fff; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-link .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-link .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-link .tabs.is-boxed a, html.theme--documenter-dark .hero.is-link .tabs.is-toggle a { - color: #fff; } - html.theme--documenter-dark .hero.is-link .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-link .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-link .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-link .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-link .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-link .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #1abc9c; } - html.theme--documenter-dark .hero.is-link.is-bold { - background-image: linear-gradient(141deg, #0c9764 0%, #1abc9c 71%, #17d8d2 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-link.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #0c9764 0%, #1abc9c 71%, #17d8d2 100%); } } - html.theme--documenter-dark .hero.is-info { - background-color: #024c7d; - color: #fff; } - html.theme--documenter-dark .hero.is-info a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-info strong { - color: inherit; } - html.theme--documenter-dark .hero.is-info .title { - color: #fff; } - html.theme--documenter-dark .hero.is-info .subtitle { - color: rgba(255, 255, 255, 0.9); } - html.theme--documenter-dark .hero.is-info .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-info .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-info .navbar-menu { - background-color: #024c7d; } } - html.theme--documenter-dark .hero.is-info .navbar-item, - html.theme--documenter-dark .hero.is-info .navbar-link { - color: rgba(255, 255, 255, 0.7); } - html.theme--documenter-dark .hero.is-info a.navbar-item:hover, html.theme--documenter-dark .hero.is-info a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-info .navbar-link:hover, - html.theme--documenter-dark .hero.is-info .navbar-link.is-active { - background-color: #023d64; - color: #fff; } - html.theme--documenter-dark .hero.is-info .tabs a { - color: #fff; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-info .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-info .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-info .tabs.is-boxed a, html.theme--documenter-dark .hero.is-info .tabs.is-toggle a { - color: #fff; } - html.theme--documenter-dark .hero.is-info .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-info .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-info .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-info .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-info .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-info .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #024c7d; } - html.theme--documenter-dark .hero.is-info.is-bold { - background-image: linear-gradient(141deg, #003a4c 0%, #024c7d 71%, #004299 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-info.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #003a4c 0%, #024c7d 71%, #004299 100%); } } - html.theme--documenter-dark .hero.is-success { - background-color: #008438; - color: #fff; } - html.theme--documenter-dark .hero.is-success a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-success strong { - color: inherit; } - html.theme--documenter-dark .hero.is-success .title { - color: #fff; } - html.theme--documenter-dark .hero.is-success .subtitle { - color: rgba(255, 255, 255, 0.9); } - html.theme--documenter-dark .hero.is-success .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-success .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-success .navbar-menu { - background-color: #008438; } } - html.theme--documenter-dark .hero.is-success .navbar-item, - html.theme--documenter-dark .hero.is-success .navbar-link { - color: rgba(255, 255, 255, 0.7); } - html.theme--documenter-dark .hero.is-success a.navbar-item:hover, html.theme--documenter-dark .hero.is-success a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-success .navbar-link:hover, - html.theme--documenter-dark .hero.is-success .navbar-link.is-active { - background-color: #006b2d; - color: #fff; } - html.theme--documenter-dark .hero.is-success .tabs a { - color: #fff; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-success .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-success .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-success .tabs.is-boxed a, html.theme--documenter-dark .hero.is-success .tabs.is-toggle a { - color: #fff; } - html.theme--documenter-dark .hero.is-success .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-success .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-success .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-success .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-success .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-success .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #008438; } - html.theme--documenter-dark .hero.is-success.is-bold { - background-image: linear-gradient(141deg, #005115 0%, #008438 71%, #009e5d 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-success.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #005115 0%, #008438 71%, #009e5d 100%); } } - html.theme--documenter-dark .hero.is-warning { - background-color: #ad8100; - color: #fff; } - html.theme--documenter-dark .hero.is-warning a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-warning strong { - color: inherit; } - html.theme--documenter-dark .hero.is-warning .title { - color: #fff; } - html.theme--documenter-dark .hero.is-warning .subtitle { - color: rgba(255, 255, 255, 0.9); } - html.theme--documenter-dark .hero.is-warning .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-warning .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-warning .navbar-menu { - background-color: #ad8100; } } - html.theme--documenter-dark .hero.is-warning .navbar-item, - html.theme--documenter-dark .hero.is-warning .navbar-link { - color: rgba(255, 255, 255, 0.7); } - html.theme--documenter-dark .hero.is-warning a.navbar-item:hover, html.theme--documenter-dark .hero.is-warning a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-warning .navbar-link:hover, - html.theme--documenter-dark .hero.is-warning .navbar-link.is-active { - background-color: #946e00; - color: #fff; } - html.theme--documenter-dark .hero.is-warning .tabs a { - color: #fff; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-warning .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-warning .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-warning .tabs.is-boxed a, html.theme--documenter-dark .hero.is-warning .tabs.is-toggle a { - color: #fff; } - html.theme--documenter-dark .hero.is-warning .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-warning .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-warning .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-warning .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-warning .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-warning .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #ad8100; } - html.theme--documenter-dark .hero.is-warning.is-bold { - background-image: linear-gradient(141deg, #7a4700 0%, #ad8100 71%, #c7b500 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-warning.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #7a4700 0%, #ad8100 71%, #c7b500 100%); } } - html.theme--documenter-dark .hero.is-danger { - background-color: #9e1b0d; - color: #fff; } - html.theme--documenter-dark .hero.is-danger a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - html.theme--documenter-dark .hero.is-danger strong { - color: inherit; } - html.theme--documenter-dark .hero.is-danger .title { - color: #fff; } - html.theme--documenter-dark .hero.is-danger .subtitle { - color: rgba(255, 255, 255, 0.9); } - html.theme--documenter-dark .hero.is-danger .subtitle a:not(.button), - html.theme--documenter-dark .hero.is-danger .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .hero.is-danger .navbar-menu { - background-color: #9e1b0d; } } - html.theme--documenter-dark .hero.is-danger .navbar-item, - html.theme--documenter-dark .hero.is-danger .navbar-link { - color: rgba(255, 255, 255, 0.7); } - html.theme--documenter-dark .hero.is-danger a.navbar-item:hover, html.theme--documenter-dark .hero.is-danger a.navbar-item.is-active, - html.theme--documenter-dark .hero.is-danger .navbar-link:hover, - html.theme--documenter-dark .hero.is-danger .navbar-link.is-active { - background-color: #86170b; - color: #fff; } - html.theme--documenter-dark .hero.is-danger .tabs a { - color: #fff; - opacity: 0.9; } - html.theme--documenter-dark .hero.is-danger .tabs a:hover { - opacity: 1; } - html.theme--documenter-dark .hero.is-danger .tabs li.is-active a { - opacity: 1; } - html.theme--documenter-dark .hero.is-danger .tabs.is-boxed a, html.theme--documenter-dark .hero.is-danger .tabs.is-toggle a { - color: #fff; } - html.theme--documenter-dark .hero.is-danger .tabs.is-boxed a:hover, html.theme--documenter-dark .hero.is-danger .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - html.theme--documenter-dark .hero.is-danger .tabs.is-boxed li.is-active a, html.theme--documenter-dark .hero.is-danger .tabs.is-boxed li.is-active a:hover, html.theme--documenter-dark .hero.is-danger .tabs.is-toggle li.is-active a, html.theme--documenter-dark .hero.is-danger .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #9e1b0d; } - html.theme--documenter-dark .hero.is-danger.is-bold { - background-image: linear-gradient(141deg, #75030b 0%, #9e1b0d 71%, #ba380a 100%); } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero.is-danger.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #75030b 0%, #9e1b0d 71%, #ba380a 100%); } } - html.theme--documenter-dark .hero.is-small .hero-body, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.hero .hero-body { - padding-bottom: 1.5rem; - padding-top: 1.5rem; } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .hero.is-medium .hero-body { - padding-bottom: 9rem; - padding-top: 9rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .hero.is-large .hero-body { - padding-bottom: 18rem; - padding-top: 18rem; } } - html.theme--documenter-dark .hero.is-halfheight .hero-body, html.theme--documenter-dark .hero.is-fullheight .hero-body, html.theme--documenter-dark .hero.is-fullheight-with-navbar .hero-body { - align-items: center; - display: flex; } - html.theme--documenter-dark .hero.is-halfheight .hero-body > .container, html.theme--documenter-dark .hero.is-fullheight .hero-body > .container, html.theme--documenter-dark .hero.is-fullheight-with-navbar .hero-body > .container { - flex-grow: 1; - flex-shrink: 1; } - html.theme--documenter-dark .hero.is-halfheight { - min-height: 50vh; } - html.theme--documenter-dark .hero.is-fullheight { - min-height: 100vh; } - html.theme--documenter-dark .hero-video { - overflow: hidden; } - html.theme--documenter-dark .hero-video video { - left: 50%; - min-height: 100%; - min-width: 100%; - position: absolute; - top: 50%; - transform: translate3d(-50%, -50%, 0); } - html.theme--documenter-dark .hero-video.is-transparent { - opacity: 0.3; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero-video { - display: none; } } - html.theme--documenter-dark .hero-buttons { - margin-top: 1.5rem; } - @media screen and (max-width: 768px) { - html.theme--documenter-dark .hero-buttons .button { - display: flex; } - html.theme--documenter-dark .hero-buttons .button:not(:last-child) { - margin-bottom: 0.75rem; } } - @media screen and (min-width: 769px), print { - html.theme--documenter-dark .hero-buttons { - display: flex; - justify-content: center; } - html.theme--documenter-dark .hero-buttons .button:not(:last-child) { - margin-right: 1.5rem; } } - html.theme--documenter-dark .hero-head, - html.theme--documenter-dark .hero-foot { - flex-grow: 0; - flex-shrink: 0; } - html.theme--documenter-dark .hero-body { - flex-grow: 1; - flex-shrink: 0; - padding: 3rem 1.5rem; } - html.theme--documenter-dark .section { - padding: 3rem 1.5rem; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark .section.is-medium { - padding: 9rem 1.5rem; } - html.theme--documenter-dark .section.is-large { - padding: 18rem 1.5rem; } } - html.theme--documenter-dark .footer { - background-color: #282f2f; - padding: 3rem 1.5rem 6rem; } - html.theme--documenter-dark hr { - height: 1px; } - html.theme--documenter-dark h6 { - text-transform: uppercase; - letter-spacing: 0.5px; } - html.theme--documenter-dark .hero { - background-color: #343c3d; } - html.theme--documenter-dark a { - transition: all 200ms ease; } - html.theme--documenter-dark .button { - transition: all 200ms ease; - border-width: 1px; - color: white; } - html.theme--documenter-dark .button.is-active, html.theme--documenter-dark .button.is-focused, html.theme--documenter-dark .button:active, html.theme--documenter-dark .button:focus { - box-shadow: 0 0 0 2px rgba(140, 155, 157, 0.5); } - html.theme--documenter-dark .button.is-white.is-hovered, html.theme--documenter-dark .button.is-white:hover { - background-color: white; } - html.theme--documenter-dark .button.is-white.is-active, html.theme--documenter-dark .button.is-white.is-focused, html.theme--documenter-dark .button.is-white:active, html.theme--documenter-dark .button.is-white:focus { - border-color: white; - box-shadow: 0 0 0 2px rgba(255, 255, 255, 0.5); } - html.theme--documenter-dark .button.is-black.is-hovered, html.theme--documenter-dark .button.is-black:hover { - background-color: #1d1d1d; } - html.theme--documenter-dark .button.is-black.is-active, html.theme--documenter-dark .button.is-black.is-focused, html.theme--documenter-dark .button.is-black:active, html.theme--documenter-dark .button.is-black:focus { - border-color: #0a0a0a; - box-shadow: 0 0 0 2px rgba(10, 10, 10, 0.5); } - html.theme--documenter-dark .button.is-light.is-hovered, html.theme--documenter-dark .button.is-light:hover { - background-color: white; } - html.theme--documenter-dark .button.is-light.is-active, html.theme--documenter-dark .button.is-light.is-focused, html.theme--documenter-dark .button.is-light:active, html.theme--documenter-dark .button.is-light:focus { - border-color: #ecf0f1; - box-shadow: 0 0 0 2px rgba(236, 240, 241, 0.5); } - html.theme--documenter-dark .button.is-dark.is-hovered, html.theme--documenter-dark .content kbd.button.is-hovered, html.theme--documenter-dark .button.is-dark:hover, html.theme--documenter-dark .content kbd.button:hover { - background-color: #3a4344; } - html.theme--documenter-dark .button.is-dark.is-active, html.theme--documenter-dark .content kbd.button.is-active, html.theme--documenter-dark .button.is-dark.is-focused, html.theme--documenter-dark .content kbd.button.is-focused, html.theme--documenter-dark .button.is-dark:active, html.theme--documenter-dark .content kbd.button:active, html.theme--documenter-dark .button.is-dark:focus, html.theme--documenter-dark .content kbd.button:focus { - border-color: #282f2f; - box-shadow: 0 0 0 2px rgba(40, 47, 47, 0.5); } - html.theme--documenter-dark .button.is-primary.is-hovered, html.theme--documenter-dark .docstring > section > a.button.is-hovered.docs-sourcelink, html.theme--documenter-dark .button.is-primary:hover, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:hover { - background-color: #436d9a; } - html.theme--documenter-dark .button.is-primary.is-active, html.theme--documenter-dark .docstring > section > a.button.is-active.docs-sourcelink, html.theme--documenter-dark .button.is-primary.is-focused, html.theme--documenter-dark .docstring > section > a.button.is-focused.docs-sourcelink, html.theme--documenter-dark .button.is-primary:active, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:active, html.theme--documenter-dark .button.is-primary:focus, html.theme--documenter-dark .docstring > section > a.button.docs-sourcelink:focus { - border-color: #375a7f; - box-shadow: 0 0 0 2px rgba(55, 90, 127, 0.5); } - html.theme--documenter-dark .button.is-link.is-hovered, html.theme--documenter-dark .button.is-link:hover { - background-color: #1fdeb8; } - html.theme--documenter-dark .button.is-link.is-active, html.theme--documenter-dark .button.is-link.is-focused, html.theme--documenter-dark .button.is-link:active, html.theme--documenter-dark .button.is-link:focus { - border-color: #1abc9c; - box-shadow: 0 0 0 2px rgba(26, 188, 156, 0.5); } - html.theme--documenter-dark .button.is-info.is-hovered, html.theme--documenter-dark .button.is-info:hover { - background-color: #0363a3; } - html.theme--documenter-dark .button.is-info.is-active, html.theme--documenter-dark .button.is-info.is-focused, html.theme--documenter-dark .button.is-info:active, html.theme--documenter-dark .button.is-info:focus { - border-color: #024c7d; - box-shadow: 0 0 0 2px rgba(2, 76, 125, 0.5); } - html.theme--documenter-dark .button.is-success.is-hovered, html.theme--documenter-dark .button.is-success:hover { - background-color: #00aa48; } - html.theme--documenter-dark .button.is-success.is-active, html.theme--documenter-dark .button.is-success.is-focused, html.theme--documenter-dark .button.is-success:active, html.theme--documenter-dark .button.is-success:focus { - border-color: #008438; - box-shadow: 0 0 0 2px rgba(0, 132, 56, 0.5); } - html.theme--documenter-dark .button.is-warning.is-hovered, html.theme--documenter-dark .button.is-warning:hover { - background-color: #d39e00; } - html.theme--documenter-dark .button.is-warning.is-active, html.theme--documenter-dark .button.is-warning.is-focused, html.theme--documenter-dark .button.is-warning:active, html.theme--documenter-dark .button.is-warning:focus { - border-color: #ad8100; - box-shadow: 0 0 0 2px rgba(173, 129, 0, 0.5); } - html.theme--documenter-dark .button.is-danger.is-hovered, html.theme--documenter-dark .button.is-danger:hover { - background-color: #c12110; } - html.theme--documenter-dark .button.is-danger.is-active, html.theme--documenter-dark .button.is-danger.is-focused, html.theme--documenter-dark .button.is-danger:active, html.theme--documenter-dark .button.is-danger:focus { - border-color: #9e1b0d; - box-shadow: 0 0 0 2px rgba(158, 27, 13, 0.5); } - html.theme--documenter-dark .label { - color: #dbdee0; } - html.theme--documenter-dark .button, - html.theme--documenter-dark .control.has-icons-left .icon, - html.theme--documenter-dark .control.has-icons-right .icon, - html.theme--documenter-dark .input, - html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input, - html.theme--documenter-dark .pagination-ellipsis, - html.theme--documenter-dark .pagination-link, - html.theme--documenter-dark .pagination-next, - html.theme--documenter-dark .pagination-previous, - html.theme--documenter-dark .select, - html.theme--documenter-dark .select select, - html.theme--documenter-dark .textarea { - height: 2.5em; } - html.theme--documenter-dark .input, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input, - html.theme--documenter-dark .textarea { - transition: all 200ms ease; - box-shadow: none; - border-width: 1px; - padding-left: 1em; - padding-right: 1em; } - html.theme--documenter-dark .select:after, - html.theme--documenter-dark .select select { - border-width: 1px; } - html.theme--documenter-dark .control.has-addons .button, - html.theme--documenter-dark .control.has-addons .input, - html.theme--documenter-dark .control.has-addons #documenter .docs-sidebar form.docs-search > input, - html.theme--documenter-dark #documenter .docs-sidebar .control.has-addons form.docs-search > input, - html.theme--documenter-dark .control.has-addons .select { - margin-right: -1px; } - html.theme--documenter-dark .notification { - background-color: #343c3d; } - html.theme--documenter-dark .card { - box-shadow: none; - border: 1px solid #343c3d; - background-color: #282f2f; - border-radius: 0.4em; } - html.theme--documenter-dark .card .card-image img { - border-radius: 0.4em 0.4em 0 0; } - html.theme--documenter-dark .card .card-header { - box-shadow: none; - background-color: rgba(18, 18, 18, 0.2); - border-radius: 0.4em 0.4em 0 0; } - html.theme--documenter-dark .card .card-footer { - background-color: rgba(18, 18, 18, 0.2); } - html.theme--documenter-dark .card .card-footer, - html.theme--documenter-dark .card .card-footer-item { - border-width: 1px; - border-color: #343c3d; } - html.theme--documenter-dark .notification.is-white a:not(.button) { - color: #0a0a0a; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-black a:not(.button) { - color: white; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-light a:not(.button) { - color: #282f2f; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-dark a:not(.button), html.theme--documenter-dark .content kbd.notification a:not(.button) { - color: #ecf0f1; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-primary a:not(.button), html.theme--documenter-dark .docstring > section > a.notification.docs-sourcelink a:not(.button) { - color: #fff; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-link a:not(.button) { - color: #fff; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-info a:not(.button) { - color: #fff; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-success a:not(.button) { - color: #fff; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-warning a:not(.button) { - color: #fff; - text-decoration: underline; } - html.theme--documenter-dark .notification.is-danger a:not(.button) { - color: #fff; - text-decoration: underline; } - html.theme--documenter-dark .tag, html.theme--documenter-dark .content kbd, html.theme--documenter-dark .docstring > section > a.docs-sourcelink { - border-radius: 0.4em; } - html.theme--documenter-dark .menu-list a { - transition: all 300ms ease; } - html.theme--documenter-dark .modal-card-body { - background-color: #282f2f; } - html.theme--documenter-dark .modal-card-foot, - html.theme--documenter-dark .modal-card-head { - border-color: #343c3d; } - html.theme--documenter-dark .message-header { - font-weight: 700; - background-color: #343c3d; - color: white; } - html.theme--documenter-dark .message-body { - border-width: 1px; - border-color: #343c3d; } - html.theme--documenter-dark .navbar { - border-radius: 0.4em; } - html.theme--documenter-dark .navbar.is-transparent { - background: none; } - html.theme--documenter-dark .navbar.is-primary .navbar-dropdown a.navbar-item.is-active, html.theme--documenter-dark .docstring > section > a.navbar.docs-sourcelink .navbar-dropdown a.navbar-item.is-active { - background-color: #1abc9c; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark .navbar .navbar-menu { - background-color: #375a7f; - border-radius: 0 0 0.4em 0.4em; } } - html.theme--documenter-dark .hero .navbar, - html.theme--documenter-dark body > .navbar { - border-radius: 0; } - html.theme--documenter-dark .pagination-link, - html.theme--documenter-dark .pagination-next, - html.theme--documenter-dark .pagination-previous { - border-width: 1px; } - html.theme--documenter-dark .panel-block, - html.theme--documenter-dark .panel-heading, - html.theme--documenter-dark .panel-tabs { - border-width: 1px; } - html.theme--documenter-dark .panel-block:first-child, - html.theme--documenter-dark .panel-heading:first-child, - html.theme--documenter-dark .panel-tabs:first-child { - border-top-width: 1px; } - html.theme--documenter-dark .panel-heading { - font-weight: 700; } - html.theme--documenter-dark .panel-tabs a { - border-width: 1px; - margin-bottom: -1px; } - html.theme--documenter-dark .panel-tabs a.is-active { - border-bottom-color: #17a689; } - html.theme--documenter-dark .panel-block:hover { - color: #1dd2af; } - html.theme--documenter-dark .panel-block:hover .panel-icon { - color: #1dd2af; } - html.theme--documenter-dark .panel-block.is-active .panel-icon { - color: #17a689; } - html.theme--documenter-dark .tabs a { - border-bottom-width: 1px; - margin-bottom: -1px; } - html.theme--documenter-dark .tabs ul { - border-bottom-width: 1px; } - html.theme--documenter-dark .tabs.is-boxed a { - border-width: 1px; } - html.theme--documenter-dark .tabs.is-boxed li.is-active a { - background-color: #1f2424; } - html.theme--documenter-dark .tabs.is-toggle li a { - border-width: 1px; - margin-bottom: 0; } - html.theme--documenter-dark .tabs.is-toggle li + li { - margin-left: -1px; } - html.theme--documenter-dark .hero.is-white .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-black .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-light .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-dark .navbar .navbar-dropdown .navbar-item:hover, html.theme--documenter-dark .content kbd.hero .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-primary .navbar .navbar-dropdown .navbar-item:hover, html.theme--documenter-dark .docstring > section > a.hero.docs-sourcelink .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-link .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-info .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-success .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-warning .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark .hero.is-danger .navbar .navbar-dropdown .navbar-item:hover { - background-color: transparent; } - html.theme--documenter-dark h1 .docs-heading-anchor, html.theme--documenter-dark h1 .docs-heading-anchor:hover, html.theme--documenter-dark h1 .docs-heading-anchor:visited, html.theme--documenter-dark h2 .docs-heading-anchor, html.theme--documenter-dark h2 .docs-heading-anchor:hover, html.theme--documenter-dark h2 .docs-heading-anchor:visited, html.theme--documenter-dark h3 .docs-heading-anchor, html.theme--documenter-dark h3 .docs-heading-anchor:hover, html.theme--documenter-dark h3 .docs-heading-anchor:visited, html.theme--documenter-dark h4 .docs-heading-anchor, html.theme--documenter-dark h4 .docs-heading-anchor:hover, html.theme--documenter-dark h4 .docs-heading-anchor:visited, html.theme--documenter-dark h5 .docs-heading-anchor, html.theme--documenter-dark h5 .docs-heading-anchor:hover, html.theme--documenter-dark h5 .docs-heading-anchor:visited, html.theme--documenter-dark h6 .docs-heading-anchor, html.theme--documenter-dark h6 .docs-heading-anchor:hover, html.theme--documenter-dark h6 .docs-heading-anchor:visited { - color: #f2f2f2; } - html.theme--documenter-dark h1 .docs-heading-anchor-permalink, html.theme--documenter-dark h2 .docs-heading-anchor-permalink, html.theme--documenter-dark h3 .docs-heading-anchor-permalink, html.theme--documenter-dark h4 .docs-heading-anchor-permalink, html.theme--documenter-dark h5 .docs-heading-anchor-permalink, html.theme--documenter-dark h6 .docs-heading-anchor-permalink { - visibility: hidden; - vertical-align: middle; - margin-left: 0.5em; - font-size: 0.7rem; } - html.theme--documenter-dark h1 .docs-heading-anchor-permalink::before, html.theme--documenter-dark h2 .docs-heading-anchor-permalink::before, html.theme--documenter-dark h3 .docs-heading-anchor-permalink::before, html.theme--documenter-dark h4 .docs-heading-anchor-permalink::before, html.theme--documenter-dark h5 .docs-heading-anchor-permalink::before, html.theme--documenter-dark h6 .docs-heading-anchor-permalink::before { - font-family: "Font Awesome 5 Free"; - font-weight: 900; - content: "\f0c1"; } - html.theme--documenter-dark h1:hover .docs-heading-anchor-permalink, html.theme--documenter-dark h2:hover .docs-heading-anchor-permalink, html.theme--documenter-dark h3:hover .docs-heading-anchor-permalink, html.theme--documenter-dark h4:hover .docs-heading-anchor-permalink, html.theme--documenter-dark h5:hover .docs-heading-anchor-permalink, html.theme--documenter-dark h6:hover .docs-heading-anchor-permalink { - visibility: visible; } - html.theme--documenter-dark .docs-light-only { - display: none !important; } - html.theme--documenter-dark pre { - position: relative; - overflow: hidden; } - html.theme--documenter-dark pre code, html.theme--documenter-dark pre code.hljs { - padding: 0 0.75rem !important; - overflow: auto; - display: block; } - html.theme--documenter-dark pre code:first-of-type, html.theme--documenter-dark pre code.hljs:first-of-type { - padding-top: 0.5rem !important; } - html.theme--documenter-dark pre code:last-of-type, html.theme--documenter-dark pre code.hljs:last-of-type { - padding-bottom: 0.5rem !important; } - html.theme--documenter-dark pre .copy-button { - opacity: 0.2; - transition: opacity 0.2s; - position: absolute; - right: 0em; - top: 0em; - padding: 0.5em; - width: 2.5em; - height: 2.5em; - background: transparent; - border: none; - font-family: "Font Awesome 5 Free"; - color: #fff; - cursor: pointer; - text-align: center; } - html.theme--documenter-dark pre .copy-button:focus, html.theme--documenter-dark pre .copy-button:hover { - opacity: 1; - background: rgba(255, 255, 255, 0.1); - color: #1abc9c; } - html.theme--documenter-dark pre .copy-button.success { - color: #259a12; - opacity: 1; } - html.theme--documenter-dark pre .copy-button.error { - color: #cb3c33; - opacity: 1; } - html.theme--documenter-dark pre:hover .copy-button { - opacity: 1; } - html.theme--documenter-dark .admonition { - background-color: #282f2f; - border-style: solid; - border-width: 1px; - border-color: #5e6d6f; - border-radius: 0.4em; - font-size: 15px; } - html.theme--documenter-dark .admonition strong { - color: currentColor; } - html.theme--documenter-dark .admonition.is-small, html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input.admonition { - font-size: 0.85em; } - html.theme--documenter-dark .admonition.is-medium { - font-size: 1.25rem; } - html.theme--documenter-dark .admonition.is-large { - font-size: 1.5rem; } - html.theme--documenter-dark .admonition.is-default { - background-color: #282f2f; - border-color: #5e6d6f; } - html.theme--documenter-dark .admonition.is-default > .admonition-header { - background-color: #5e6d6f; } - html.theme--documenter-dark .admonition.is-info { - background-color: #282f2f; - border-color: #024c7d; } - html.theme--documenter-dark .admonition.is-info > .admonition-header { - background-color: #024c7d; } - html.theme--documenter-dark .admonition.is-success { - background-color: #282f2f; - border-color: #008438; } - html.theme--documenter-dark .admonition.is-success > .admonition-header { - background-color: #008438; } - html.theme--documenter-dark .admonition.is-warning { - background-color: #282f2f; - border-color: #ad8100; } - html.theme--documenter-dark .admonition.is-warning > .admonition-header { - background-color: #ad8100; } - html.theme--documenter-dark .admonition.is-danger { - background-color: #282f2f; - border-color: #9e1b0d; } - html.theme--documenter-dark .admonition.is-danger > .admonition-header { - background-color: #9e1b0d; } - html.theme--documenter-dark .admonition.is-compat { - background-color: #282f2f; - border-color: #137886; } - html.theme--documenter-dark .admonition.is-compat > .admonition-header { - background-color: #137886; } - html.theme--documenter-dark .admonition-header { - background-color: #5e6d6f; - align-items: center; - font-weight: 700; - justify-content: space-between; - line-height: 1.25; - padding: 0.5rem 0.75rem; - position: relative; } - html.theme--documenter-dark .admonition-header:before { - font-family: "Font Awesome 5 Free"; - font-weight: 900; - margin-right: 0.75rem; - content: "\f06a"; } - html.theme--documenter-dark .admonition-body { - color: #fff; - padding: 0.5rem 0.75rem; } - html.theme--documenter-dark .admonition-body pre { - background-color: #282f2f; } - html.theme--documenter-dark .admonition-body code { - background-color: rgba(255, 255, 255, 0.05); } - html.theme--documenter-dark .docstring { - margin-bottom: 1em; - background-color: transparent; - border: 1px solid #5e6d6f; - box-shadow: none; - max-width: 100%; } - html.theme--documenter-dark .docstring > header { - display: flex; - flex-grow: 1; - align-items: stretch; - padding: 0.5rem 0.75rem; - background-color: #282f2f; - box-shadow: 0 1px 2px rgba(10, 10, 10, 0.1); - box-shadow: none; - border-bottom: 1px solid #5e6d6f; } - html.theme--documenter-dark .docstring > header code { - background-color: transparent; } - html.theme--documenter-dark .docstring > header .docstring-binding { - margin-right: 0.3em; } - html.theme--documenter-dark .docstring > header .docstring-category { - margin-left: 0.3em; } - html.theme--documenter-dark .docstring > section { - position: relative; - padding: 0.75rem 0.75rem; - border-bottom: 1px solid #5e6d6f; } - html.theme--documenter-dark .docstring > section:last-child { - border-bottom: none; } - html.theme--documenter-dark .docstring > section > a.docs-sourcelink { - transition: opacity 0.3s; - opacity: 0; - position: absolute; - right: 0.375rem; - bottom: 0.375rem; } - html.theme--documenter-dark .docstring > section > a.docs-sourcelink:focus { - opacity: 1 !important; } - html.theme--documenter-dark .docstring:hover > section > a.docs-sourcelink { - opacity: 0.2; } - html.theme--documenter-dark .docstring:focus-within > section > a.docs-sourcelink { - opacity: 0.2; } - html.theme--documenter-dark .docstring > section:hover a.docs-sourcelink { - opacity: 1; } - html.theme--documenter-dark .documenter-example-output { - background-color: #1f2424; } - html.theme--documenter-dark .outdated-warning-overlay { - position: fixed; - top: 0; - left: 0; - right: 0; - box-shadow: 0 0 10px rgba(0, 0, 0, 0.3); - z-index: 999; - background-color: #282f2f; - border-bottom: 3px solid #9e1b0d; - padding: 10px 35px; - text-align: center; - font-size: 15px; } - html.theme--documenter-dark .outdated-warning-overlay .outdated-warning-closer { - position: absolute; - top: calc(50% - 10px); - right: 18px; - cursor: pointer; - width: 12px; } - html.theme--documenter-dark .outdated-warning-overlay a { - color: #1abc9c; } - html.theme--documenter-dark .outdated-warning-overlay a:hover { - color: #1dd2af; } - html.theme--documenter-dark .content pre { - border: 1px solid #5e6d6f; } - html.theme--documenter-dark .content code { - font-weight: inherit; } - html.theme--documenter-dark .content a code { - color: #1abc9c; } - html.theme--documenter-dark .content h1 code, html.theme--documenter-dark .content h2 code, html.theme--documenter-dark .content h3 code, html.theme--documenter-dark .content h4 code, html.theme--documenter-dark .content h5 code, html.theme--documenter-dark .content h6 code { - color: #f2f2f2; } - html.theme--documenter-dark .content table { - display: block; - width: initial; - max-width: 100%; - overflow-x: auto; } - html.theme--documenter-dark .content blockquote > ul:first-child, html.theme--documenter-dark .content blockquote > ol:first-child, html.theme--documenter-dark .content .admonition-body > ul:first-child, html.theme--documenter-dark .content .admonition-body > ol:first-child { - margin-top: 0; } - html.theme--documenter-dark pre, html.theme--documenter-dark code { - font-variant-ligatures: no-contextual; } - html.theme--documenter-dark .breadcrumb a.is-disabled { - cursor: default; - pointer-events: none; } - html.theme--documenter-dark .breadcrumb a.is-disabled, html.theme--documenter-dark .breadcrumb a.is-disabled:hover { - color: #f2f2f2; } - html.theme--documenter-dark .hljs { - background: initial !important; } - html.theme--documenter-dark .katex .katex-mathml { - top: 0; - right: 0; } - html.theme--documenter-dark .katex-display, html.theme--documenter-dark mjx-container, html.theme--documenter-dark .MathJax_Display { - margin: 0.5em 0 !important; } - html.theme--documenter-dark html { - -moz-osx-font-smoothing: auto; - -webkit-font-smoothing: auto; } - html.theme--documenter-dark li.no-marker { - list-style: none; } - html.theme--documenter-dark #documenter .docs-main > article { - overflow-wrap: break-word; } - html.theme--documenter-dark #documenter .docs-main > article .math-container { - overflow-x: auto; - overflow-y: hidden; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark #documenter .docs-main { - max-width: 52rem; - margin-left: 20rem; - padding-right: 1rem; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark #documenter .docs-main { - width: 100%; } - html.theme--documenter-dark #documenter .docs-main > article { - max-width: 52rem; - margin-left: auto; - margin-right: auto; - margin-bottom: 1rem; - padding: 0 1rem; } - html.theme--documenter-dark #documenter .docs-main > header, html.theme--documenter-dark #documenter .docs-main > nav { - max-width: 100%; - width: 100%; - margin: 0; } } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar { - background-color: #1f2424; - border-bottom: 1px solid #5e6d6f; - z-index: 2; - min-height: 4rem; - margin-bottom: 1rem; - display: flex; } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar .breadcrumb { - flex-grow: 1; } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right { - display: flex; - white-space: nowrap; } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-icon, html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-label, html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-sidebar-button { - display: inline-block; } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-label { - padding: 0; - margin-left: 0.3em; } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-settings-button { - margin: auto 0 auto 1rem; } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar .docs-right .docs-sidebar-button { - font-size: 1.5rem; - margin: auto 0 auto 1rem; } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar > * { - margin: auto 0; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark #documenter .docs-main header.docs-navbar { - position: sticky; - top: 0; - padding: 0 1rem; - /* For Headroom.js */ - transition-property: top, box-shadow; - -webkit-transition-property: top, box-shadow; - /* Safari */ - transition-duration: 0.3s; - -webkit-transition-duration: 0.3s; - /* Safari */ } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar.headroom--not-top { - box-shadow: 0.2rem 0rem 0.4rem #171717; - transition-duration: 0.7s; - -webkit-transition-duration: 0.7s; - /* Safari */ } - html.theme--documenter-dark #documenter .docs-main header.docs-navbar.headroom--unpinned.headroom--not-top.headroom--not-bottom { - top: -4.5rem; - transition-duration: 0.7s; - -webkit-transition-duration: 0.7s; - /* Safari */ } } - html.theme--documenter-dark #documenter .docs-main section.footnotes { - border-top: 1px solid #5e6d6f; } - html.theme--documenter-dark #documenter .docs-main section.footnotes li .tag:first-child, html.theme--documenter-dark #documenter .docs-main section.footnotes li .docstring > section > a.docs-sourcelink:first-child, html.theme--documenter-dark #documenter .docs-main section.footnotes li .content kbd:first-child, html.theme--documenter-dark .content #documenter .docs-main section.footnotes li kbd:first-child { - margin-right: 1em; - margin-bottom: 0.4em; } - html.theme--documenter-dark #documenter .docs-main .docs-footer { - display: flex; - flex-wrap: wrap; - margin-left: 0; - margin-right: 0; - border-top: 1px solid #5e6d6f; - padding-top: 1rem; - padding-bottom: 1rem; } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark #documenter .docs-main .docs-footer { - padding-left: 1rem; - padding-right: 1rem; } } - html.theme--documenter-dark #documenter .docs-main .docs-footer .docs-footer-nextpage, html.theme--documenter-dark #documenter .docs-main .docs-footer .docs-footer-prevpage { - flex-grow: 1; } - html.theme--documenter-dark #documenter .docs-main .docs-footer .docs-footer-nextpage { - text-align: right; } - html.theme--documenter-dark #documenter .docs-main .docs-footer .flexbox-break { - flex-basis: 100%; - height: 0; } - html.theme--documenter-dark #documenter .docs-main .docs-footer .footer-message { - font-size: 0.8em; - margin: 0.5em auto 0 auto; - text-align: center; } - html.theme--documenter-dark #documenter .docs-sidebar { - display: flex; - flex-direction: column; - color: #fff; - background-color: #282f2f; - border-right: 1px solid #5e6d6f; - padding: 0; - flex: 0 0 18rem; - z-index: 5; - font-size: 15px; - position: fixed; - left: -18rem; - width: 18rem; - height: 100%; - transition: left 0.3s; - /* Setting up a nicer theme style for the scrollbar */ } - html.theme--documenter-dark #documenter .docs-sidebar.visible { - left: 0; - box-shadow: 0.4rem 0rem 0.8rem #171717; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark #documenter .docs-sidebar.visible { - box-shadow: none; } } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark #documenter .docs-sidebar { - left: 0; - top: 0; } } - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo { - margin-top: 1rem; - padding: 0 1rem; } - html.theme--documenter-dark #documenter .docs-sidebar .docs-logo > img { - max-height: 6rem; - margin: auto; } - html.theme--documenter-dark #documenter .docs-sidebar .docs-package-name { - flex-shrink: 0; - font-size: 1.5rem; - font-weight: 700; - text-align: center; - white-space: nowrap; - overflow: hidden; - padding: 0.5rem 0; } - html.theme--documenter-dark #documenter .docs-sidebar .docs-package-name .docs-autofit { - max-width: 16.2rem; } - html.theme--documenter-dark #documenter .docs-sidebar .docs-package-name a, html.theme--documenter-dark #documenter .docs-sidebar .docs-package-name a:hover { - color: #fff; } - html.theme--documenter-dark #documenter .docs-sidebar .docs-version-selector { - border-top: 1px solid #5e6d6f; - display: none; - padding: 0.5rem; } - html.theme--documenter-dark #documenter .docs-sidebar .docs-version-selector.visible { - display: flex; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu { - flex-grow: 1; - user-select: none; - border-top: 1px solid #5e6d6f; - padding-bottom: 1.5rem; - /* Managing collapsible submenus */ } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu > li > .tocitem { - font-weight: bold; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu > li li { - font-size: 14.25px; - margin-left: 1em; - border-left: 1px solid #5e6d6f; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu input.collapse-toggle { - display: none; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.collapsed { - display: none; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu input:checked ~ ul.collapsed { - display: block; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu label.tocitem { - display: flex; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu label.tocitem .docs-label { - flex-grow: 2; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu label.tocitem .docs-chevron { - display: inline-block; - font-style: normal; - font-variant: normal; - text-rendering: auto; - line-height: 1; - font-size: 11.25px; - margin-left: 1rem; - margin-top: auto; - margin-bottom: auto; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu label.tocitem .docs-chevron::before { - font-family: "Font Awesome 5 Free"; - font-weight: 900; - content: "\f054"; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu input:checked ~ label.tocitem .docs-chevron::before { - content: "\f078"; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu .tocitem { - display: block; - padding: 0.5rem 0.5rem; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu .tocitem, html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu .tocitem:hover { - color: #fff; - background: #282f2f; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu a.tocitem:hover, html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu label.tocitem:hover { - color: #fff; - background-color: #32393a; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu li.is-active { - border-top: 1px solid #5e6d6f; - border-bottom: 1px solid #5e6d6f; - background-color: #1f2424; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu li.is-active .tocitem, html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu li.is-active .tocitem:hover { - background-color: #1f2424; - color: #fff; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu li.is-active ul.internal .tocitem:hover { - background-color: #32393a; - color: #fff; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu > li.is-active:first-child { - border-top: none; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.internal { - margin: 0 0.5rem 0.5rem; - border-top: 1px solid #5e6d6f; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.internal li { - font-size: 12.75px; - border-left: none; - margin-left: 0; - margin-top: 0.5rem; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.internal .tocitem { - width: 100%; - padding: 0; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu ul.internal .tocitem::before { - content: "⚬"; - margin-right: 0.4em; } - html.theme--documenter-dark #documenter .docs-sidebar form.docs-search { - margin: auto; - margin-top: 0.5rem; - margin-bottom: 0.5rem; } - html.theme--documenter-dark #documenter .docs-sidebar form.docs-search > input { - width: 14.4rem; } - @media screen and (min-width: 1056px) { - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu { - overflow-y: auto; - -webkit-overflow-scroll: touch; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu::-webkit-scrollbar { - width: .3rem; - background: none; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu::-webkit-scrollbar-thumb { - border-radius: 5px 0px 0px 5px; - background: #3b4445; } - html.theme--documenter-dark #documenter .docs-sidebar ul.docs-menu::-webkit-scrollbar-thumb:hover { - background: #4e5a5c; } } - @media screen and (max-width: 1055px) { - html.theme--documenter-dark #documenter .docs-sidebar { - overflow-y: auto; - -webkit-overflow-scroll: touch; } - html.theme--documenter-dark #documenter .docs-sidebar::-webkit-scrollbar { - width: .3rem; - background: none; } - html.theme--documenter-dark #documenter .docs-sidebar::-webkit-scrollbar-thumb { - border-radius: 5px 0px 0px 5px; - background: #3b4445; } - html.theme--documenter-dark #documenter .docs-sidebar::-webkit-scrollbar-thumb:hover { - background: #4e5a5c; } } - html.theme--documenter-dark #documenter .docs-main #documenter-search-info { - margin-bottom: 1rem; } - html.theme--documenter-dark #documenter .docs-main #documenter-search-results { - list-style-type: circle; - list-style-position: outside; } - html.theme--documenter-dark #documenter .docs-main #documenter-search-results li { - margin-left: 2rem; } - html.theme--documenter-dark #documenter .docs-main #documenter-search-results .docs-highlight { - background-color: yellow; } - html.theme--documenter-dark { - background-color: #1f2424; - font-size: 16px; - min-width: 300px; - overflow-x: auto; - overflow-y: scroll; - text-rendering: optimizeLegibility; - text-size-adjust: 100%; } - html.theme--documenter-dark .ansi span.sgr1 { - font-weight: bolder; } - html.theme--documenter-dark .ansi span.sgr2 { - font-weight: lighter; } - html.theme--documenter-dark .ansi span.sgr3 { - font-style: italic; } - html.theme--documenter-dark .ansi span.sgr4 { - text-decoration: underline; } - html.theme--documenter-dark .ansi span.sgr7 { - color: #1f2424; - background-color: #fff; } - html.theme--documenter-dark .ansi span.sgr8 { - color: transparent; } - html.theme--documenter-dark .ansi span.sgr8 span { - color: transparent; } - html.theme--documenter-dark .ansi span.sgr9 { - text-decoration: line-through; } - html.theme--documenter-dark .ansi span.sgr30 { - color: #242424; } - html.theme--documenter-dark .ansi span.sgr31 { - color: #f6705f; } - html.theme--documenter-dark .ansi span.sgr32 { - color: #4fb43a; } - html.theme--documenter-dark .ansi span.sgr33 { - color: #f4c72f; } - html.theme--documenter-dark .ansi span.sgr34 { - color: #7587f0; } - html.theme--documenter-dark .ansi span.sgr35 { - color: #bc89d3; } - html.theme--documenter-dark .ansi span.sgr36 { - color: #49b6ca; } - html.theme--documenter-dark .ansi span.sgr37 { - color: #b3bdbe; } - html.theme--documenter-dark .ansi span.sgr40 { - background-color: #242424; } - html.theme--documenter-dark .ansi span.sgr41 { - background-color: #f6705f; } - html.theme--documenter-dark .ansi span.sgr42 { - background-color: #4fb43a; } - html.theme--documenter-dark .ansi span.sgr43 { - background-color: #f4c72f; } - html.theme--documenter-dark .ansi span.sgr44 { - background-color: #7587f0; } - html.theme--documenter-dark .ansi span.sgr45 { - background-color: #bc89d3; } - html.theme--documenter-dark .ansi span.sgr46 { - background-color: #49b6ca; } - html.theme--documenter-dark .ansi span.sgr47 { - background-color: #b3bdbe; } - html.theme--documenter-dark .ansi span.sgr90 { - color: #92a0a2; } - html.theme--documenter-dark .ansi span.sgr91 { - color: #ff8674; } - html.theme--documenter-dark .ansi span.sgr92 { - color: #79d462; } - html.theme--documenter-dark .ansi span.sgr93 { - color: #ffe76b; } - html.theme--documenter-dark .ansi span.sgr94 { - color: #8a98ff; } - html.theme--documenter-dark .ansi span.sgr95 { - color: #d2a4e6; } - html.theme--documenter-dark .ansi span.sgr96 { - color: #6bc8db; } - html.theme--documenter-dark .ansi span.sgr97 { - color: #ecf0f1; } - html.theme--documenter-dark .ansi span.sgr100 { - background-color: #92a0a2; } - html.theme--documenter-dark .ansi span.sgr101 { - background-color: #ff8674; } - html.theme--documenter-dark .ansi span.sgr102 { - background-color: #79d462; } - html.theme--documenter-dark .ansi span.sgr103 { - background-color: #ffe76b; } - html.theme--documenter-dark .ansi span.sgr104 { - background-color: #8a98ff; } - html.theme--documenter-dark .ansi span.sgr105 { - background-color: #d2a4e6; } - html.theme--documenter-dark .ansi span.sgr106 { - background-color: #6bc8db; } - html.theme--documenter-dark .ansi span.sgr107 { - background-color: #ecf0f1; } - html.theme--documenter-dark code.language-julia-repl > span.hljs-meta { - color: #4fb43a; - font-weight: bolder; } - html.theme--documenter-dark .hljs { - background: #2b2b2b; - color: #f8f8f2; } - html.theme--documenter-dark .hljs-comment, - html.theme--documenter-dark .hljs-quote { - color: #d4d0ab; } - html.theme--documenter-dark .hljs-variable, - html.theme--documenter-dark .hljs-template-variable, - html.theme--documenter-dark .hljs-tag, - html.theme--documenter-dark .hljs-name, - html.theme--documenter-dark .hljs-selector-id, - html.theme--documenter-dark .hljs-selector-class, - html.theme--documenter-dark .hljs-regexp, - html.theme--documenter-dark .hljs-deletion { - color: #ffa07a; } - html.theme--documenter-dark .hljs-number, - html.theme--documenter-dark .hljs-built_in, - html.theme--documenter-dark .hljs-literal, - html.theme--documenter-dark .hljs-type, - html.theme--documenter-dark .hljs-params, - html.theme--documenter-dark .hljs-meta, - html.theme--documenter-dark .hljs-link { - color: #f5ab35; } - html.theme--documenter-dark .hljs-attribute { - color: #ffd700; } - html.theme--documenter-dark .hljs-string, - html.theme--documenter-dark .hljs-symbol, - html.theme--documenter-dark .hljs-bullet, - html.theme--documenter-dark .hljs-addition { - color: #abe338; } - html.theme--documenter-dark .hljs-title, - html.theme--documenter-dark .hljs-section { - color: #00e0e0; } - html.theme--documenter-dark .hljs-keyword, - html.theme--documenter-dark .hljs-selector-tag { - color: #dcc6e0; } - html.theme--documenter-dark .hljs-emphasis { - font-style: italic; } - html.theme--documenter-dark .hljs-strong { - font-weight: bold; } - @media screen and (-ms-high-contrast: active) { - html.theme--documenter-dark .hljs-addition, - html.theme--documenter-dark .hljs-attribute, - html.theme--documenter-dark .hljs-built_in, - html.theme--documenter-dark .hljs-bullet, - html.theme--documenter-dark .hljs-comment, - html.theme--documenter-dark .hljs-link, - html.theme--documenter-dark .hljs-literal, - html.theme--documenter-dark .hljs-meta, - html.theme--documenter-dark .hljs-number, - html.theme--documenter-dark .hljs-params, - html.theme--documenter-dark .hljs-string, - html.theme--documenter-dark .hljs-symbol, - html.theme--documenter-dark .hljs-type, - html.theme--documenter-dark .hljs-quote { - color: highlight; } - html.theme--documenter-dark .hljs-keyword, - html.theme--documenter-dark .hljs-selector-tag { - font-weight: bold; } } - html.theme--documenter-dark .hljs-subst { - color: #f8f8f2; } diff --git a/previews/PR62/assets/themes/documenter-light.css b/previews/PR62/assets/themes/documenter-light.css deleted file mode 100644 index 9595d5b8..00000000 --- a/previews/PR62/assets/themes/documenter-light.css +++ /dev/null @@ -1,7738 +0,0 @@ -@charset "UTF-8"; -/* Font Awesome 5 mixin. Can be included in any rule that should render Font Awesome icons. */ -@keyframes spinAround { - from { - transform: rotate(0deg); } - to { - transform: rotate(359deg); } } - -.tabs, .pagination-previous, -.pagination-next, -.pagination-link, -.pagination-ellipsis, .breadcrumb, .file, .button, .is-unselectable, .modal-close, .delete { - -webkit-touch-callout: none; - -webkit-user-select: none; - -moz-user-select: none; - -ms-user-select: none; - user-select: none; } - -.navbar-link:not(.is-arrowless)::after, .select:not(.is-multiple):not(.is-loading)::after { - border: 3px solid transparent; - border-radius: 2px; - border-right: 0; - border-top: 0; - content: " "; - display: block; - height: 0.625em; - margin-top: -0.4375em; - pointer-events: none; - position: absolute; - top: 50%; - transform: rotate(-45deg); - transform-origin: center; - width: 0.625em; } - -.admonition:not(:last-child), .tabs:not(:last-child), .message:not(:last-child), .list:not(:last-child), .level:not(:last-child), .breadcrumb:not(:last-child), .highlight:not(:last-child), .block:not(:last-child), .title:not(:last-child), -.subtitle:not(:last-child), .table-container:not(:last-child), .table:not(:last-child), .progress:not(:last-child), .notification:not(:last-child), .content:not(:last-child), .box:not(:last-child) { - margin-bottom: 1.5rem; } - -.modal-close, .delete { - -moz-appearance: none; - -webkit-appearance: none; - background-color: rgba(10, 10, 10, 0.2); - border: none; - border-radius: 290486px; - cursor: pointer; - pointer-events: auto; - display: inline-block; - flex-grow: 0; - flex-shrink: 0; - font-size: 0; - height: 20px; - max-height: 20px; - max-width: 20px; - min-height: 20px; - min-width: 20px; - outline: none; - position: relative; - vertical-align: top; - width: 20px; } - .modal-close::before, .delete::before, .modal-close::after, .delete::after { - background-color: white; - content: ""; - display: block; - left: 50%; - position: absolute; - top: 50%; - transform: translateX(-50%) translateY(-50%) rotate(45deg); - transform-origin: center center; } - .modal-close::before, .delete::before { - height: 2px; - width: 50%; } - .modal-close::after, .delete::after { - height: 50%; - width: 2px; } - .modal-close:hover, .delete:hover, .modal-close:focus, .delete:focus { - background-color: rgba(10, 10, 10, 0.3); } - .modal-close:active, .delete:active { - background-color: rgba(10, 10, 10, 0.4); } - .is-small.modal-close, #documenter .docs-sidebar form.docs-search > input.modal-close, .is-small.delete, #documenter .docs-sidebar form.docs-search > input.delete { - height: 16px; - max-height: 16px; - max-width: 16px; - min-height: 16px; - min-width: 16px; - width: 16px; } - .is-medium.modal-close, .is-medium.delete { - height: 24px; - max-height: 24px; - max-width: 24px; - min-height: 24px; - min-width: 24px; - width: 24px; } - .is-large.modal-close, .is-large.delete { - height: 32px; - max-height: 32px; - max-width: 32px; - min-height: 32px; - min-width: 32px; - width: 32px; } - -.control.is-loading::after, .select.is-loading::after, .loader, .button.is-loading::after { - animation: spinAround 500ms infinite linear; - border: 2px solid #dbdbdb; - border-radius: 290486px; - border-right-color: transparent; - border-top-color: transparent; - content: ""; - display: block; - height: 1em; - position: relative; - width: 1em; } - -.hero-video, .modal-background, .modal, .image.is-square img, #documenter .docs-sidebar .docs-logo > img.is-square img, -.image.is-square .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-square .has-ratio, .image.is-1by1 img, #documenter .docs-sidebar .docs-logo > img.is-1by1 img, -.image.is-1by1 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-1by1 .has-ratio, .image.is-5by4 img, #documenter .docs-sidebar .docs-logo > img.is-5by4 img, -.image.is-5by4 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-5by4 .has-ratio, .image.is-4by3 img, #documenter .docs-sidebar .docs-logo > img.is-4by3 img, -.image.is-4by3 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-4by3 .has-ratio, .image.is-3by2 img, #documenter .docs-sidebar .docs-logo > img.is-3by2 img, -.image.is-3by2 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-3by2 .has-ratio, .image.is-5by3 img, #documenter .docs-sidebar .docs-logo > img.is-5by3 img, -.image.is-5by3 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-5by3 .has-ratio, .image.is-16by9 img, #documenter .docs-sidebar .docs-logo > img.is-16by9 img, -.image.is-16by9 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-16by9 .has-ratio, .image.is-2by1 img, #documenter .docs-sidebar .docs-logo > img.is-2by1 img, -.image.is-2by1 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-2by1 .has-ratio, .image.is-3by1 img, #documenter .docs-sidebar .docs-logo > img.is-3by1 img, -.image.is-3by1 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-3by1 .has-ratio, .image.is-4by5 img, #documenter .docs-sidebar .docs-logo > img.is-4by5 img, -.image.is-4by5 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-4by5 .has-ratio, .image.is-3by4 img, #documenter .docs-sidebar .docs-logo > img.is-3by4 img, -.image.is-3by4 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-3by4 .has-ratio, .image.is-2by3 img, #documenter .docs-sidebar .docs-logo > img.is-2by3 img, -.image.is-2by3 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-2by3 .has-ratio, .image.is-3by5 img, #documenter .docs-sidebar .docs-logo > img.is-3by5 img, -.image.is-3by5 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-3by5 .has-ratio, .image.is-9by16 img, #documenter .docs-sidebar .docs-logo > img.is-9by16 img, -.image.is-9by16 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-9by16 .has-ratio, .image.is-1by2 img, #documenter .docs-sidebar .docs-logo > img.is-1by2 img, -.image.is-1by2 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-1by2 .has-ratio, .image.is-1by3 img, #documenter .docs-sidebar .docs-logo > img.is-1by3 img, -.image.is-1by3 .has-ratio, -#documenter .docs-sidebar .docs-logo > img.is-1by3 .has-ratio, .is-overlay { - bottom: 0; - left: 0; - position: absolute; - right: 0; - top: 0; } - -.pagination-previous, -.pagination-next, -.pagination-link, -.pagination-ellipsis, .file-cta, -.file-name, .select select, .textarea, .input, #documenter .docs-sidebar form.docs-search > input, .button { - -moz-appearance: none; - -webkit-appearance: none; - align-items: center; - border: 1px solid transparent; - border-radius: 4px; - box-shadow: none; - display: inline-flex; - font-size: 1rem; - height: 2.25em; - justify-content: flex-start; - line-height: 1.5; - padding-bottom: calc(0.375em - 1px); - padding-left: calc(0.625em - 1px); - padding-right: calc(0.625em - 1px); - padding-top: calc(0.375em - 1px); - position: relative; - vertical-align: top; } - .pagination-previous:focus, - .pagination-next:focus, - .pagination-link:focus, - .pagination-ellipsis:focus, .file-cta:focus, - .file-name:focus, .select select:focus, .textarea:focus, .input:focus, #documenter .docs-sidebar form.docs-search > input:focus, .button:focus, .is-focused.pagination-previous, - .is-focused.pagination-next, - .is-focused.pagination-link, - .is-focused.pagination-ellipsis, .is-focused.file-cta, - .is-focused.file-name, .select select.is-focused, .is-focused.textarea, .is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-focused.button, .pagination-previous:active, - .pagination-next:active, - .pagination-link:active, - .pagination-ellipsis:active, .file-cta:active, - .file-name:active, .select select:active, .textarea:active, .input:active, #documenter .docs-sidebar form.docs-search > input:active, .button:active, .is-active.pagination-previous, - .is-active.pagination-next, - .is-active.pagination-link, - .is-active.pagination-ellipsis, .is-active.file-cta, - .is-active.file-name, .select select.is-active, .is-active.textarea, .is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active, .is-active.button { - outline: none; } - .pagination-previous[disabled], - .pagination-next[disabled], - .pagination-link[disabled], - .pagination-ellipsis[disabled], .file-cta[disabled], - .file-name[disabled], .select select[disabled], .textarea[disabled], .input[disabled], #documenter .docs-sidebar form.docs-search > input[disabled], .button[disabled], fieldset[disabled] .pagination-previous, - fieldset[disabled] .pagination-next, - fieldset[disabled] .pagination-link, - fieldset[disabled] .pagination-ellipsis, fieldset[disabled] .file-cta, - fieldset[disabled] .file-name, fieldset[disabled] .select select, .select fieldset[disabled] select, fieldset[disabled] .textarea, fieldset[disabled] .input, fieldset[disabled] #documenter .docs-sidebar form.docs-search > input, #documenter .docs-sidebar fieldset[disabled] form.docs-search > input, fieldset[disabled] .button { - cursor: not-allowed; } - -/*! minireset.css v0.0.4 | MIT License | github.com/jgthms/minireset.css */ -html, -body, -p, -ol, -ul, -li, -dl, -dt, -dd, -blockquote, -figure, -fieldset, -legend, -textarea, -pre, -iframe, -hr, -h1, -h2, -h3, -h4, -h5, -h6 { - margin: 0; - padding: 0; } - -h1, -h2, -h3, -h4, -h5, -h6 { - font-size: 100%; - font-weight: normal; } - -ul { - list-style: none; } - -button, -input, -select, -textarea { - margin: 0; } - -html { - box-sizing: border-box; } - -*, *::before, *::after { - box-sizing: inherit; } - -img, -embed, -iframe, -object, -video { - height: auto; - max-width: 100%; } - -audio { - max-width: 100%; } - -iframe { - border: 0; } - -table { - border-collapse: collapse; - border-spacing: 0; } - -td, -th { - padding: 0; } - td:not([align]), - th:not([align]) { - text-align: left; } - -html { - background-color: white; - font-size: 16px; - -moz-osx-font-smoothing: grayscale; - -webkit-font-smoothing: antialiased; - min-width: 300px; - overflow-x: auto; - overflow-y: scroll; - text-rendering: optimizeLegibility; - text-size-adjust: 100%; } - -article, -aside, -figure, -footer, -header, -hgroup, -section { - display: block; } - -body, -button, -input, -select, -textarea { - font-family: "Lato Medium", -apple-system, BlinkMacSystemFont, "Segoe UI", "Helvetica Neue", "Helvetica", "Arial", sans-serif; } - -code, -pre { - -moz-osx-font-smoothing: auto; - -webkit-font-smoothing: auto; - font-family: "JuliaMono", "SFMono-Regular", "Menlo", "Consolas", "Liberation Mono", "DejaVu Sans Mono", monospace; } - -body { - color: #222222; - font-size: 1em; - font-weight: 400; - line-height: 1.5; } - -a { - color: #2e63b8; - cursor: pointer; - text-decoration: none; } - a strong { - color: currentColor; } - a:hover { - color: #363636; } - -code { - background-color: rgba(0, 0, 0, 0.05); - color: #000000; - font-size: 0.875em; - font-weight: normal; - padding: 0.1em; } - -hr { - background-color: whitesmoke; - border: none; - display: block; - height: 2px; - margin: 1.5rem 0; } - -img { - height: auto; - max-width: 100%; } - -input[type="checkbox"], -input[type="radio"] { - vertical-align: baseline; } - -small { - font-size: 0.875em; } - -span { - font-style: inherit; - font-weight: inherit; } - -strong { - color: #222222; - font-weight: 700; } - -fieldset { - border: none; } - -pre { - -webkit-overflow-scrolling: touch; - background-color: whitesmoke; - color: #222222; - font-size: 0.875em; - overflow-x: auto; - padding: 1.25rem 1.5rem; - white-space: pre; - word-wrap: normal; } - pre code { - background-color: transparent; - color: currentColor; - font-size: 1em; - padding: 0; } - -table td, -table th { - vertical-align: top; } - table td:not([align]), - table th:not([align]) { - text-align: left; } - -table th { - color: #222222; } - -.is-clearfix::after { - clear: both; - content: " "; - display: table; } - -.is-pulled-left { - float: left !important; } - -.is-pulled-right { - float: right !important; } - -.is-clipped { - overflow: hidden !important; } - -.is-size-1 { - font-size: 3rem !important; } - -.is-size-2 { - font-size: 2.5rem !important; } - -.is-size-3 { - font-size: 2rem !important; } - -.is-size-4 { - font-size: 1.5rem !important; } - -.is-size-5 { - font-size: 1.25rem !important; } - -.is-size-6 { - font-size: 1rem !important; } - -.is-size-7, .docstring > section > a.docs-sourcelink { - font-size: 0.75rem !important; } - -@media screen and (max-width: 768px) { - .is-size-1-mobile { - font-size: 3rem !important; } - .is-size-2-mobile { - font-size: 2.5rem !important; } - .is-size-3-mobile { - font-size: 2rem !important; } - .is-size-4-mobile { - font-size: 1.5rem !important; } - .is-size-5-mobile { - font-size: 1.25rem !important; } - .is-size-6-mobile { - font-size: 1rem !important; } - .is-size-7-mobile { - font-size: 0.75rem !important; } } - -@media screen and (min-width: 769px), print { - .is-size-1-tablet { - font-size: 3rem !important; } - .is-size-2-tablet { - font-size: 2.5rem !important; } - .is-size-3-tablet { - font-size: 2rem !important; } - .is-size-4-tablet { - font-size: 1.5rem !important; } - .is-size-5-tablet { - font-size: 1.25rem !important; } - .is-size-6-tablet { - font-size: 1rem !important; } - .is-size-7-tablet { - font-size: 0.75rem !important; } } - -@media screen and (max-width: 1055px) { - .is-size-1-touch { - font-size: 3rem !important; } - .is-size-2-touch { - font-size: 2.5rem !important; } - .is-size-3-touch { - font-size: 2rem !important; } - .is-size-4-touch { - font-size: 1.5rem !important; } - .is-size-5-touch { - font-size: 1.25rem !important; } - .is-size-6-touch { - font-size: 1rem !important; } - .is-size-7-touch { - font-size: 0.75rem !important; } } - -@media screen and (min-width: 1056px) { - .is-size-1-desktop { - font-size: 3rem !important; } - .is-size-2-desktop { - font-size: 2.5rem !important; } - .is-size-3-desktop { - font-size: 2rem !important; } - .is-size-4-desktop { - font-size: 1.5rem !important; } - .is-size-5-desktop { - font-size: 1.25rem !important; } - .is-size-6-desktop { - font-size: 1rem !important; } - .is-size-7-desktop { - font-size: 0.75rem !important; } } - -@media screen and (min-width: 1216px) { - .is-size-1-widescreen { - font-size: 3rem !important; } - .is-size-2-widescreen { - font-size: 2.5rem !important; } - .is-size-3-widescreen { - font-size: 2rem !important; } - .is-size-4-widescreen { - font-size: 1.5rem !important; } - .is-size-5-widescreen { - font-size: 1.25rem !important; } - .is-size-6-widescreen { - font-size: 1rem !important; } - .is-size-7-widescreen { - font-size: 0.75rem !important; } } - -@media screen and (min-width: 1408px) { - .is-size-1-fullhd { - font-size: 3rem !important; } - .is-size-2-fullhd { - font-size: 2.5rem !important; } - .is-size-3-fullhd { - font-size: 2rem !important; } - .is-size-4-fullhd { - font-size: 1.5rem !important; } - .is-size-5-fullhd { - font-size: 1.25rem !important; } - .is-size-6-fullhd { - font-size: 1rem !important; } - .is-size-7-fullhd { - font-size: 0.75rem !important; } } - -.has-text-centered { - text-align: center !important; } - -.has-text-justified { - text-align: justify !important; } - -.has-text-left { - text-align: left !important; } - -.has-text-right { - text-align: right !important; } - -@media screen and (max-width: 768px) { - .has-text-centered-mobile { - text-align: center !important; } } - -@media screen and (min-width: 769px), print { - .has-text-centered-tablet { - text-align: center !important; } } - -@media screen and (min-width: 769px) and (max-width: 1055px) { - .has-text-centered-tablet-only { - text-align: center !important; } } - -@media screen and (max-width: 1055px) { - .has-text-centered-touch { - text-align: center !important; } } - -@media screen and (min-width: 1056px) { - .has-text-centered-desktop { - text-align: center !important; } } - -@media screen and (min-width: 1056px) and (max-width: 1215px) { - .has-text-centered-desktop-only { - text-align: center !important; } } - -@media screen and (min-width: 1216px) { - .has-text-centered-widescreen { - text-align: center !important; } } - -@media screen and (min-width: 1216px) and (max-width: 1407px) { - .has-text-centered-widescreen-only { - text-align: center !important; } } - -@media screen and (min-width: 1408px) { - .has-text-centered-fullhd { - text-align: center !important; } } - -@media screen and (max-width: 768px) { - .has-text-justified-mobile { - text-align: justify !important; } } - -@media screen and (min-width: 769px), print { - .has-text-justified-tablet { - text-align: justify !important; } } - -@media screen and (min-width: 769px) and (max-width: 1055px) { - .has-text-justified-tablet-only { - text-align: justify !important; } } - -@media screen and (max-width: 1055px) { - .has-text-justified-touch { - text-align: justify !important; } } - -@media screen and (min-width: 1056px) { - .has-text-justified-desktop { - text-align: justify !important; } } - -@media screen and (min-width: 1056px) and (max-width: 1215px) { - .has-text-justified-desktop-only { - text-align: justify !important; } } - -@media screen and (min-width: 1216px) { - .has-text-justified-widescreen { - text-align: justify !important; } } - -@media screen and (min-width: 1216px) and (max-width: 1407px) { - .has-text-justified-widescreen-only { - text-align: justify !important; } } - -@media screen and (min-width: 1408px) { - .has-text-justified-fullhd { - text-align: justify !important; } } - -@media screen and (max-width: 768px) { - .has-text-left-mobile { - text-align: left !important; } } - -@media screen and (min-width: 769px), print { - .has-text-left-tablet { - text-align: left !important; } } - -@media screen and (min-width: 769px) and (max-width: 1055px) { - .has-text-left-tablet-only { - text-align: left !important; } } - -@media screen and (max-width: 1055px) { - .has-text-left-touch { - text-align: left !important; } } - -@media screen and (min-width: 1056px) { - .has-text-left-desktop { - text-align: left !important; } } - -@media screen and (min-width: 1056px) and (max-width: 1215px) { - .has-text-left-desktop-only { - text-align: left !important; } } - -@media screen and (min-width: 1216px) { - .has-text-left-widescreen { - text-align: left !important; } } - -@media screen and (min-width: 1216px) and (max-width: 1407px) { - .has-text-left-widescreen-only { - text-align: left !important; } } - -@media screen and (min-width: 1408px) { - .has-text-left-fullhd { - text-align: left !important; } } - -@media screen and (max-width: 768px) { - .has-text-right-mobile { - text-align: right !important; } } - -@media screen and (min-width: 769px), print { - .has-text-right-tablet { - text-align: right !important; } } - -@media screen and (min-width: 769px) and (max-width: 1055px) { - .has-text-right-tablet-only { - text-align: right !important; } } - -@media screen and (max-width: 1055px) { - .has-text-right-touch { - text-align: right !important; } } - -@media screen and (min-width: 1056px) { - .has-text-right-desktop { - text-align: right !important; } } - -@media screen and (min-width: 1056px) and (max-width: 1215px) { - .has-text-right-desktop-only { - text-align: right !important; } } - -@media screen and (min-width: 1216px) { - .has-text-right-widescreen { - text-align: right !important; } } - -@media screen and (min-width: 1216px) and (max-width: 1407px) { - .has-text-right-widescreen-only { - text-align: right !important; } } - -@media screen and (min-width: 1408px) { - .has-text-right-fullhd { - text-align: right !important; } } - -.is-capitalized { - text-transform: capitalize !important; } - -.is-lowercase { - text-transform: lowercase !important; } - -.is-uppercase { - text-transform: uppercase !important; } - -.is-italic { - font-style: italic !important; } - -.has-text-white { - color: white !important; } - -a.has-text-white:hover, a.has-text-white:focus { - color: #e6e6e6 !important; } - -.has-background-white { - background-color: white !important; } - -.has-text-black { - color: #0a0a0a !important; } - -a.has-text-black:hover, a.has-text-black:focus { - color: black !important; } - -.has-background-black { - background-color: #0a0a0a !important; } - -.has-text-light { - color: whitesmoke !important; } - -a.has-text-light:hover, a.has-text-light:focus { - color: #dbdbdb !important; } - -.has-background-light { - background-color: whitesmoke !important; } - -.has-text-dark { - color: #363636 !important; } - -a.has-text-dark:hover, a.has-text-dark:focus { - color: #1c1c1c !important; } - -.has-background-dark { - background-color: #363636 !important; } - -.has-text-primary { - color: #4eb5de !important; } - -a.has-text-primary:hover, a.has-text-primary:focus { - color: #27a1d2 !important; } - -.has-background-primary { - background-color: #4eb5de !important; } - -.has-text-link { - color: #2e63b8 !important; } - -a.has-text-link:hover, a.has-text-link:focus { - color: #244d8f !important; } - -.has-background-link { - background-color: #2e63b8 !important; } - -.has-text-info { - color: #209cee !important; } - -a.has-text-info:hover, a.has-text-info:focus { - color: #1081cb !important; } - -.has-background-info { - background-color: #209cee !important; } - -.has-text-success { - color: #22c35b !important; } - -a.has-text-success:hover, a.has-text-success:focus { - color: #1a9847 !important; } - -.has-background-success { - background-color: #22c35b !important; } - -.has-text-warning { - color: #ffdd57 !important; } - -a.has-text-warning:hover, a.has-text-warning:focus { - color: #ffd324 !important; } - -.has-background-warning { - background-color: #ffdd57 !important; } - -.has-text-danger { - color: #da0b00 !important; } - -a.has-text-danger:hover, a.has-text-danger:focus { - color: #a70800 !important; } - -.has-background-danger { - background-color: #da0b00 !important; } - -.has-text-black-bis { - color: #121212 !important; } - -.has-background-black-bis { - background-color: #121212 !important; } - -.has-text-black-ter { - color: #242424 !important; } - -.has-background-black-ter { - background-color: #242424 !important; } - -.has-text-grey-darker { - color: #363636 !important; } - -.has-background-grey-darker { - background-color: #363636 !important; } - -.has-text-grey-dark { - color: #4a4a4a !important; } - -.has-background-grey-dark { - background-color: #4a4a4a !important; } - -.has-text-grey { - color: #6b6b6b !important; } - -.has-background-grey { - background-color: #6b6b6b !important; } - -.has-text-grey-light { - color: #b5b5b5 !important; } - -.has-background-grey-light { - background-color: #b5b5b5 !important; } - -.has-text-grey-lighter { - color: #dbdbdb !important; } - -.has-background-grey-lighter { - background-color: #dbdbdb !important; } - -.has-text-white-ter { - color: whitesmoke !important; } - -.has-background-white-ter { - background-color: whitesmoke !important; } - -.has-text-white-bis { - color: #fafafa !important; } - -.has-background-white-bis { - background-color: #fafafa !important; } - -.has-text-weight-light { - font-weight: 300 !important; } - -.has-text-weight-normal { - font-weight: 400 !important; } - -.has-text-weight-medium { - font-weight: 500 !important; } - -.has-text-weight-semibold { - font-weight: 600 !important; } - -.has-text-weight-bold { - font-weight: 700 !important; } - -.is-family-primary { - font-family: "Lato Medium", -apple-system, BlinkMacSystemFont, "Segoe UI", "Helvetica Neue", "Helvetica", "Arial", sans-serif !important; } - -.is-family-secondary { - font-family: "Lato Medium", -apple-system, BlinkMacSystemFont, "Segoe UI", "Helvetica Neue", "Helvetica", "Arial", sans-serif !important; } - -.is-family-sans-serif { - font-family: "Lato Medium", -apple-system, BlinkMacSystemFont, "Segoe UI", "Helvetica Neue", "Helvetica", "Arial", sans-serif !important; } - -.is-family-monospace { - font-family: "JuliaMono", "SFMono-Regular", "Menlo", "Consolas", "Liberation Mono", "DejaVu Sans Mono", monospace !important; } - -.is-family-code { - font-family: "JuliaMono", "SFMono-Regular", "Menlo", "Consolas", "Liberation Mono", "DejaVu Sans Mono", monospace !important; } - -.is-block { - display: block !important; } - -@media screen and (max-width: 768px) { - .is-block-mobile { - display: block !important; } } - -@media screen and (min-width: 769px), print { - .is-block-tablet { - display: block !important; } } - -@media screen and (min-width: 769px) and (max-width: 1055px) { - .is-block-tablet-only { - display: block !important; } } - -@media screen and (max-width: 1055px) { - .is-block-touch { - display: block !important; } } - -@media screen and (min-width: 1056px) { - .is-block-desktop { - display: block !important; } } - -@media screen and (min-width: 1056px) and (max-width: 1215px) { - .is-block-desktop-only { - display: block !important; } } - -@media screen and (min-width: 1216px) { - .is-block-widescreen { - display: block !important; } } - -@media screen and (min-width: 1216px) and (max-width: 1407px) { - .is-block-widescreen-only { - display: block !important; } } - -@media screen and (min-width: 1408px) { - .is-block-fullhd { - display: block !important; } } - -.is-flex { - display: flex !important; } - -@media screen and (max-width: 768px) { - .is-flex-mobile { - display: flex !important; } } - -@media screen and (min-width: 769px), print { - .is-flex-tablet { - display: flex !important; } } - -@media screen and (min-width: 769px) and (max-width: 1055px) { - .is-flex-tablet-only { - display: flex !important; } } - -@media screen and (max-width: 1055px) { - .is-flex-touch { - display: flex !important; } } - -@media screen and (min-width: 1056px) { - .is-flex-desktop { - display: flex !important; } } - -@media screen and (min-width: 1056px) and (max-width: 1215px) { - .is-flex-desktop-only { - display: flex !important; } } - -@media screen and (min-width: 1216px) { - .is-flex-widescreen { - display: flex !important; } } - -@media screen and (min-width: 1216px) and (max-width: 1407px) { - .is-flex-widescreen-only { - display: flex !important; } } - -@media screen and (min-width: 1408px) { - .is-flex-fullhd { - display: flex !important; } } - -.is-inline { - display: inline !important; } - -@media screen and (max-width: 768px) { - .is-inline-mobile { - display: inline !important; } } - -@media screen and (min-width: 769px), print { - .is-inline-tablet { - display: inline !important; } } - -@media screen and (min-width: 769px) and (max-width: 1055px) { - .is-inline-tablet-only { - display: inline !important; } } - -@media screen and (max-width: 1055px) { - .is-inline-touch { - display: inline !important; } } - -@media screen and (min-width: 1056px) { - .is-inline-desktop { - display: inline !important; } } - -@media screen and (min-width: 1056px) and (max-width: 1215px) { - .is-inline-desktop-only { - display: inline !important; } } - -@media screen and (min-width: 1216px) { - .is-inline-widescreen { - display: inline !important; } } - -@media screen and (min-width: 1216px) and (max-width: 1407px) { - .is-inline-widescreen-only { - display: inline !important; } } - -@media screen and (min-width: 1408px) { - .is-inline-fullhd { - display: inline !important; } } - -.is-inline-block { - display: inline-block !important; } - -@media screen and (max-width: 768px) { - .is-inline-block-mobile { - display: inline-block !important; } } - -@media screen and (min-width: 769px), print { - .is-inline-block-tablet { - display: inline-block !important; } } - -@media screen and (min-width: 769px) and (max-width: 1055px) { - .is-inline-block-tablet-only { - display: inline-block !important; } } - -@media screen and (max-width: 1055px) { - .is-inline-block-touch { - display: inline-block !important; } } - -@media screen and (min-width: 1056px) { - .is-inline-block-desktop { - display: inline-block !important; } } - -@media screen and (min-width: 1056px) and (max-width: 1215px) { - .is-inline-block-desktop-only { - display: inline-block !important; } } - -@media screen and (min-width: 1216px) { - .is-inline-block-widescreen { - display: inline-block !important; } } - -@media screen and (min-width: 1216px) and (max-width: 1407px) { - .is-inline-block-widescreen-only { - display: inline-block !important; } } - -@media screen and (min-width: 1408px) { - .is-inline-block-fullhd { - display: inline-block !important; } } - -.is-inline-flex { - display: inline-flex !important; } - -@media screen and (max-width: 768px) { - .is-inline-flex-mobile { - display: inline-flex !important; } } - -@media screen and (min-width: 769px), print { - .is-inline-flex-tablet { - display: inline-flex !important; } } - -@media screen and (min-width: 769px) and (max-width: 1055px) { - .is-inline-flex-tablet-only { - display: inline-flex !important; } } - -@media screen and (max-width: 1055px) { - .is-inline-flex-touch { - display: inline-flex !important; } } - -@media screen and (min-width: 1056px) { - .is-inline-flex-desktop { - display: inline-flex !important; } } - -@media screen and (min-width: 1056px) and (max-width: 1215px) { - .is-inline-flex-desktop-only { - display: inline-flex !important; } } - -@media screen and (min-width: 1216px) { - .is-inline-flex-widescreen { - display: inline-flex !important; } } - -@media screen and (min-width: 1216px) and (max-width: 1407px) { - .is-inline-flex-widescreen-only { - display: inline-flex !important; } } - -@media screen and (min-width: 1408px) { - .is-inline-flex-fullhd { - display: inline-flex !important; } } - -.is-hidden { - display: none !important; } - -.is-sr-only { - border: none !important; - clip: rect(0, 0, 0, 0) !important; - height: 0.01em !important; - overflow: hidden !important; - padding: 0 !important; - position: absolute !important; - white-space: nowrap !important; - width: 0.01em !important; } - -@media screen and (max-width: 768px) { - .is-hidden-mobile { - display: none !important; } } - -@media screen and (min-width: 769px), print { - .is-hidden-tablet { - display: none !important; } } - -@media screen and (min-width: 769px) and (max-width: 1055px) { - .is-hidden-tablet-only { - display: none !important; } } - -@media screen and (max-width: 1055px) { - .is-hidden-touch { - display: none !important; } } - -@media screen and (min-width: 1056px) { - .is-hidden-desktop { - display: none !important; } } - -@media screen and (min-width: 1056px) and (max-width: 1215px) { - .is-hidden-desktop-only { - display: none !important; } } - -@media screen and (min-width: 1216px) { - .is-hidden-widescreen { - display: none !important; } } - -@media screen and (min-width: 1216px) and (max-width: 1407px) { - .is-hidden-widescreen-only { - display: none !important; } } - -@media screen and (min-width: 1408px) { - .is-hidden-fullhd { - display: none !important; } } - -.is-invisible { - visibility: hidden !important; } - -@media screen and (max-width: 768px) { - .is-invisible-mobile { - visibility: hidden !important; } } - -@media screen and (min-width: 769px), print { - .is-invisible-tablet { - visibility: hidden !important; } } - -@media screen and (min-width: 769px) and (max-width: 1055px) { - .is-invisible-tablet-only { - visibility: hidden !important; } } - -@media screen and (max-width: 1055px) { - .is-invisible-touch { - visibility: hidden !important; } } - -@media screen and (min-width: 1056px) { - .is-invisible-desktop { - visibility: hidden !important; } } - -@media screen and (min-width: 1056px) and (max-width: 1215px) { - .is-invisible-desktop-only { - visibility: hidden !important; } } - -@media screen and (min-width: 1216px) { - .is-invisible-widescreen { - visibility: hidden !important; } } - -@media screen and (min-width: 1216px) and (max-width: 1407px) { - .is-invisible-widescreen-only { - visibility: hidden !important; } } - -@media screen and (min-width: 1408px) { - .is-invisible-fullhd { - visibility: hidden !important; } } - -.is-marginless { - margin: 0 !important; } - -.is-paddingless { - padding: 0 !important; } - -.is-radiusless { - border-radius: 0 !important; } - -.is-shadowless { - box-shadow: none !important; } - -.is-relative { - position: relative !important; } - -.box { - background-color: white; - border-radius: 6px; - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); - color: #222222; - display: block; - padding: 1.25rem; } - -a.box:hover, a.box:focus { - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px #2e63b8; } - -a.box:active { - box-shadow: inset 0 1px 2px rgba(10, 10, 10, 0.2), 0 0 0 1px #2e63b8; } - -.button { - background-color: white; - border-color: #dbdbdb; - border-width: 1px; - color: #363636; - cursor: pointer; - justify-content: center; - padding-bottom: calc(0.375em - 1px); - padding-left: 0.75em; - padding-right: 0.75em; - padding-top: calc(0.375em - 1px); - text-align: center; - white-space: nowrap; } - .button strong { - color: inherit; } - .button .icon, .button .icon.is-small, .button #documenter .docs-sidebar form.docs-search > input.icon, #documenter .docs-sidebar .button form.docs-search > input.icon, .button .icon.is-medium, .button .icon.is-large { - height: 1.5em; - width: 1.5em; } - .button .icon:first-child:not(:last-child) { - margin-left: calc(-0.375em - 1px); - margin-right: 0.1875em; } - .button .icon:last-child:not(:first-child) { - margin-left: 0.1875em; - margin-right: calc(-0.375em - 1px); } - .button .icon:first-child:last-child { - margin-left: calc(-0.375em - 1px); - margin-right: calc(-0.375em - 1px); } - .button:hover, .button.is-hovered { - border-color: #b5b5b5; - color: #363636; } - .button:focus, .button.is-focused { - border-color: #3c5dcd; - color: #363636; } - .button:focus:not(:active), .button.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(46, 99, 184, 0.25); } - .button:active, .button.is-active { - border-color: #4a4a4a; - color: #363636; } - .button.is-text { - background-color: transparent; - border-color: transparent; - color: #222222; - text-decoration: underline; } - .button.is-text:hover, .button.is-text.is-hovered, .button.is-text:focus, .button.is-text.is-focused { - background-color: whitesmoke; - color: #222222; } - .button.is-text:active, .button.is-text.is-active { - background-color: #e8e8e8; - color: #222222; } - .button.is-text[disabled], fieldset[disabled] .button.is-text { - background-color: transparent; - border-color: transparent; - box-shadow: none; } - .button.is-white { - background-color: white; - border-color: transparent; - color: #0a0a0a; } - .button.is-white:hover, .button.is-white.is-hovered { - background-color: #f9f9f9; - border-color: transparent; - color: #0a0a0a; } - .button.is-white:focus, .button.is-white.is-focused { - border-color: transparent; - color: #0a0a0a; } - .button.is-white:focus:not(:active), .button.is-white.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); } - .button.is-white:active, .button.is-white.is-active { - background-color: #f2f2f2; - border-color: transparent; - color: #0a0a0a; } - .button.is-white[disabled], fieldset[disabled] .button.is-white { - background-color: white; - border-color: transparent; - box-shadow: none; } - .button.is-white.is-inverted { - background-color: #0a0a0a; - color: white; } - .button.is-white.is-inverted:hover, .button.is-white.is-inverted.is-hovered { - background-color: black; } - .button.is-white.is-inverted[disabled], fieldset[disabled] .button.is-white.is-inverted { - background-color: #0a0a0a; - border-color: transparent; - box-shadow: none; - color: white; } - .button.is-white.is-loading::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - .button.is-white.is-outlined { - background-color: transparent; - border-color: white; - color: white; } - .button.is-white.is-outlined:hover, .button.is-white.is-outlined.is-hovered, .button.is-white.is-outlined:focus, .button.is-white.is-outlined.is-focused { - background-color: white; - border-color: white; - color: #0a0a0a; } - .button.is-white.is-outlined.is-loading::after { - border-color: transparent transparent white white !important; } - .button.is-white.is-outlined.is-loading:hover::after, .button.is-white.is-outlined.is-loading.is-hovered::after, .button.is-white.is-outlined.is-loading:focus::after, .button.is-white.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - .button.is-white.is-outlined[disabled], fieldset[disabled] .button.is-white.is-outlined { - background-color: transparent; - border-color: white; - box-shadow: none; - color: white; } - .button.is-white.is-inverted.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - color: #0a0a0a; } - .button.is-white.is-inverted.is-outlined:hover, .button.is-white.is-inverted.is-outlined.is-hovered, .button.is-white.is-inverted.is-outlined:focus, .button.is-white.is-inverted.is-outlined.is-focused { - background-color: #0a0a0a; - color: white; } - .button.is-white.is-inverted.is-outlined.is-loading:hover::after, .button.is-white.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-white.is-inverted.is-outlined.is-loading:focus::after, .button.is-white.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent white white !important; } - .button.is-white.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-white.is-inverted.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - box-shadow: none; - color: #0a0a0a; } - .button.is-black { - background-color: #0a0a0a; - border-color: transparent; - color: white; } - .button.is-black:hover, .button.is-black.is-hovered { - background-color: #040404; - border-color: transparent; - color: white; } - .button.is-black:focus, .button.is-black.is-focused { - border-color: transparent; - color: white; } - .button.is-black:focus:not(:active), .button.is-black.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); } - .button.is-black:active, .button.is-black.is-active { - background-color: black; - border-color: transparent; - color: white; } - .button.is-black[disabled], fieldset[disabled] .button.is-black { - background-color: #0a0a0a; - border-color: transparent; - box-shadow: none; } - .button.is-black.is-inverted { - background-color: white; - color: #0a0a0a; } - .button.is-black.is-inverted:hover, .button.is-black.is-inverted.is-hovered { - background-color: #f2f2f2; } - .button.is-black.is-inverted[disabled], fieldset[disabled] .button.is-black.is-inverted { - background-color: white; - border-color: transparent; - box-shadow: none; - color: #0a0a0a; } - .button.is-black.is-loading::after { - border-color: transparent transparent white white !important; } - .button.is-black.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - color: #0a0a0a; } - .button.is-black.is-outlined:hover, .button.is-black.is-outlined.is-hovered, .button.is-black.is-outlined:focus, .button.is-black.is-outlined.is-focused { - background-color: #0a0a0a; - border-color: #0a0a0a; - color: white; } - .button.is-black.is-outlined.is-loading::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - .button.is-black.is-outlined.is-loading:hover::after, .button.is-black.is-outlined.is-loading.is-hovered::after, .button.is-black.is-outlined.is-loading:focus::after, .button.is-black.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent white white !important; } - .button.is-black.is-outlined[disabled], fieldset[disabled] .button.is-black.is-outlined { - background-color: transparent; - border-color: #0a0a0a; - box-shadow: none; - color: #0a0a0a; } - .button.is-black.is-inverted.is-outlined { - background-color: transparent; - border-color: white; - color: white; } - .button.is-black.is-inverted.is-outlined:hover, .button.is-black.is-inverted.is-outlined.is-hovered, .button.is-black.is-inverted.is-outlined:focus, .button.is-black.is-inverted.is-outlined.is-focused { - background-color: white; - color: #0a0a0a; } - .button.is-black.is-inverted.is-outlined.is-loading:hover::after, .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-black.is-inverted.is-outlined.is-loading:focus::after, .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #0a0a0a #0a0a0a !important; } - .button.is-black.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-black.is-inverted.is-outlined { - background-color: transparent; - border-color: white; - box-shadow: none; - color: white; } - .button.is-light { - background-color: whitesmoke; - border-color: transparent; - color: #363636; } - .button.is-light:hover, .button.is-light.is-hovered { - background-color: #eeeeee; - border-color: transparent; - color: #363636; } - .button.is-light:focus, .button.is-light.is-focused { - border-color: transparent; - color: #363636; } - .button.is-light:focus:not(:active), .button.is-light.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(245, 245, 245, 0.25); } - .button.is-light:active, .button.is-light.is-active { - background-color: #e8e8e8; - border-color: transparent; - color: #363636; } - .button.is-light[disabled], fieldset[disabled] .button.is-light { - background-color: whitesmoke; - border-color: transparent; - box-shadow: none; } - .button.is-light.is-inverted { - background-color: #363636; - color: whitesmoke; } - .button.is-light.is-inverted:hover, .button.is-light.is-inverted.is-hovered { - background-color: #292929; } - .button.is-light.is-inverted[disabled], fieldset[disabled] .button.is-light.is-inverted { - background-color: #363636; - border-color: transparent; - box-shadow: none; - color: whitesmoke; } - .button.is-light.is-loading::after { - border-color: transparent transparent #363636 #363636 !important; } - .button.is-light.is-outlined { - background-color: transparent; - border-color: whitesmoke; - color: whitesmoke; } - .button.is-light.is-outlined:hover, .button.is-light.is-outlined.is-hovered, .button.is-light.is-outlined:focus, .button.is-light.is-outlined.is-focused { - background-color: whitesmoke; - border-color: whitesmoke; - color: #363636; } - .button.is-light.is-outlined.is-loading::after { - border-color: transparent transparent whitesmoke whitesmoke !important; } - .button.is-light.is-outlined.is-loading:hover::after, .button.is-light.is-outlined.is-loading.is-hovered::after, .button.is-light.is-outlined.is-loading:focus::after, .button.is-light.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #363636 #363636 !important; } - .button.is-light.is-outlined[disabled], fieldset[disabled] .button.is-light.is-outlined { - background-color: transparent; - border-color: whitesmoke; - box-shadow: none; - color: whitesmoke; } - .button.is-light.is-inverted.is-outlined { - background-color: transparent; - border-color: #363636; - color: #363636; } - .button.is-light.is-inverted.is-outlined:hover, .button.is-light.is-inverted.is-outlined.is-hovered, .button.is-light.is-inverted.is-outlined:focus, .button.is-light.is-inverted.is-outlined.is-focused { - background-color: #363636; - color: whitesmoke; } - .button.is-light.is-inverted.is-outlined.is-loading:hover::after, .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-light.is-inverted.is-outlined.is-loading:focus::after, .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent whitesmoke whitesmoke !important; } - .button.is-light.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-light.is-inverted.is-outlined { - background-color: transparent; - border-color: #363636; - box-shadow: none; - color: #363636; } - .button.is-dark, .content kbd.button { - background-color: #363636; - border-color: transparent; - color: whitesmoke; } - .button.is-dark:hover, .content kbd.button:hover, .button.is-dark.is-hovered, .content kbd.button.is-hovered { - background-color: #2f2f2f; - border-color: transparent; - color: whitesmoke; } - .button.is-dark:focus, .content kbd.button:focus, .button.is-dark.is-focused, .content kbd.button.is-focused { - border-color: transparent; - color: whitesmoke; } - .button.is-dark:focus:not(:active), .content kbd.button:focus:not(:active), .button.is-dark.is-focused:not(:active), .content kbd.button.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(54, 54, 54, 0.25); } - .button.is-dark:active, .content kbd.button:active, .button.is-dark.is-active, .content kbd.button.is-active { - background-color: #292929; - border-color: transparent; - color: whitesmoke; } - .button.is-dark[disabled], .content kbd.button[disabled], fieldset[disabled] .button.is-dark, fieldset[disabled] .content kbd.button, .content fieldset[disabled] kbd.button { - background-color: #363636; - border-color: transparent; - box-shadow: none; } - .button.is-dark.is-inverted, .content kbd.button.is-inverted { - background-color: whitesmoke; - color: #363636; } - .button.is-dark.is-inverted:hover, .content kbd.button.is-inverted:hover, .button.is-dark.is-inverted.is-hovered, .content kbd.button.is-inverted.is-hovered { - background-color: #e8e8e8; } - .button.is-dark.is-inverted[disabled], .content kbd.button.is-inverted[disabled], fieldset[disabled] .button.is-dark.is-inverted, fieldset[disabled] .content kbd.button.is-inverted, .content fieldset[disabled] kbd.button.is-inverted { - background-color: whitesmoke; - border-color: transparent; - box-shadow: none; - color: #363636; } - .button.is-dark.is-loading::after, .content kbd.button.is-loading::after { - border-color: transparent transparent whitesmoke whitesmoke !important; } - .button.is-dark.is-outlined, .content kbd.button.is-outlined { - background-color: transparent; - border-color: #363636; - color: #363636; } - .button.is-dark.is-outlined:hover, .content kbd.button.is-outlined:hover, .button.is-dark.is-outlined.is-hovered, .content kbd.button.is-outlined.is-hovered, .button.is-dark.is-outlined:focus, .content kbd.button.is-outlined:focus, .button.is-dark.is-outlined.is-focused, .content kbd.button.is-outlined.is-focused { - background-color: #363636; - border-color: #363636; - color: whitesmoke; } - .button.is-dark.is-outlined.is-loading::after, .content kbd.button.is-outlined.is-loading::after { - border-color: transparent transparent #363636 #363636 !important; } - .button.is-dark.is-outlined.is-loading:hover::after, .content kbd.button.is-outlined.is-loading:hover::after, .button.is-dark.is-outlined.is-loading.is-hovered::after, .content kbd.button.is-outlined.is-loading.is-hovered::after, .button.is-dark.is-outlined.is-loading:focus::after, .content kbd.button.is-outlined.is-loading:focus::after, .button.is-dark.is-outlined.is-loading.is-focused::after, .content kbd.button.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent whitesmoke whitesmoke !important; } - .button.is-dark.is-outlined[disabled], .content kbd.button.is-outlined[disabled], fieldset[disabled] .button.is-dark.is-outlined, fieldset[disabled] .content kbd.button.is-outlined, .content fieldset[disabled] kbd.button.is-outlined { - background-color: transparent; - border-color: #363636; - box-shadow: none; - color: #363636; } - .button.is-dark.is-inverted.is-outlined, .content kbd.button.is-inverted.is-outlined { - background-color: transparent; - border-color: whitesmoke; - color: whitesmoke; } - .button.is-dark.is-inverted.is-outlined:hover, .content kbd.button.is-inverted.is-outlined:hover, .button.is-dark.is-inverted.is-outlined.is-hovered, .content kbd.button.is-inverted.is-outlined.is-hovered, .button.is-dark.is-inverted.is-outlined:focus, .content kbd.button.is-inverted.is-outlined:focus, .button.is-dark.is-inverted.is-outlined.is-focused, .content kbd.button.is-inverted.is-outlined.is-focused { - background-color: whitesmoke; - color: #363636; } - .button.is-dark.is-inverted.is-outlined.is-loading:hover::after, .content kbd.button.is-inverted.is-outlined.is-loading:hover::after, .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after, .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-dark.is-inverted.is-outlined.is-loading:focus::after, .content kbd.button.is-inverted.is-outlined.is-loading:focus::after, .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after, .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #363636 #363636 !important; } - .button.is-dark.is-inverted.is-outlined[disabled], .content kbd.button.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-dark.is-inverted.is-outlined, fieldset[disabled] .content kbd.button.is-inverted.is-outlined, .content fieldset[disabled] kbd.button.is-inverted.is-outlined { - background-color: transparent; - border-color: whitesmoke; - box-shadow: none; - color: whitesmoke; } - .button.is-primary, .docstring > section > a.button.docs-sourcelink { - background-color: #4eb5de; - border-color: transparent; - color: #fff; } - .button.is-primary:hover, .docstring > section > a.button.docs-sourcelink:hover, .button.is-primary.is-hovered, .docstring > section > a.button.is-hovered.docs-sourcelink { - background-color: #43b1dc; - border-color: transparent; - color: #fff; } - .button.is-primary:focus, .docstring > section > a.button.docs-sourcelink:focus, .button.is-primary.is-focused, .docstring > section > a.button.is-focused.docs-sourcelink { - border-color: transparent; - color: #fff; } - .button.is-primary:focus:not(:active), .docstring > section > a.button.docs-sourcelink:focus:not(:active), .button.is-primary.is-focused:not(:active), .docstring > section > a.button.is-focused.docs-sourcelink:not(:active) { - box-shadow: 0 0 0 0.125em rgba(78, 181, 222, 0.25); } - .button.is-primary:active, .docstring > section > a.button.docs-sourcelink:active, .button.is-primary.is-active, .docstring > section > a.button.is-active.docs-sourcelink { - background-color: #39acda; - border-color: transparent; - color: #fff; } - .button.is-primary[disabled], .docstring > section > a.button.docs-sourcelink[disabled], fieldset[disabled] .button.is-primary, fieldset[disabled] .docstring > section > a.button.docs-sourcelink { - background-color: #4eb5de; - border-color: transparent; - box-shadow: none; } - .button.is-primary.is-inverted, .docstring > section > a.button.is-inverted.docs-sourcelink { - background-color: #fff; - color: #4eb5de; } - .button.is-primary.is-inverted:hover, .docstring > section > a.button.is-inverted.docs-sourcelink:hover, .button.is-primary.is-inverted.is-hovered, .docstring > section > a.button.is-inverted.is-hovered.docs-sourcelink { - background-color: #f2f2f2; } - .button.is-primary.is-inverted[disabled], .docstring > section > a.button.is-inverted.docs-sourcelink[disabled], fieldset[disabled] .button.is-primary.is-inverted, fieldset[disabled] .docstring > section > a.button.is-inverted.docs-sourcelink { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #4eb5de; } - .button.is-primary.is-loading::after, .docstring > section > a.button.is-loading.docs-sourcelink::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-primary.is-outlined, .docstring > section > a.button.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #4eb5de; - color: #4eb5de; } - .button.is-primary.is-outlined:hover, .docstring > section > a.button.is-outlined.docs-sourcelink:hover, .button.is-primary.is-outlined.is-hovered, .docstring > section > a.button.is-outlined.is-hovered.docs-sourcelink, .button.is-primary.is-outlined:focus, .docstring > section > a.button.is-outlined.docs-sourcelink:focus, .button.is-primary.is-outlined.is-focused, .docstring > section > a.button.is-outlined.is-focused.docs-sourcelink { - background-color: #4eb5de; - border-color: #4eb5de; - color: #fff; } - .button.is-primary.is-outlined.is-loading::after, .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink::after { - border-color: transparent transparent #4eb5de #4eb5de !important; } - .button.is-primary.is-outlined.is-loading:hover::after, .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink:hover::after, .button.is-primary.is-outlined.is-loading.is-hovered::after, .docstring > section > a.button.is-outlined.is-loading.is-hovered.docs-sourcelink::after, .button.is-primary.is-outlined.is-loading:focus::after, .docstring > section > a.button.is-outlined.is-loading.docs-sourcelink:focus::after, .button.is-primary.is-outlined.is-loading.is-focused::after, .docstring > section > a.button.is-outlined.is-loading.is-focused.docs-sourcelink::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-primary.is-outlined[disabled], .docstring > section > a.button.is-outlined.docs-sourcelink[disabled], fieldset[disabled] .button.is-primary.is-outlined, fieldset[disabled] .docstring > section > a.button.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #4eb5de; - box-shadow: none; - color: #4eb5de; } - .button.is-primary.is-inverted.is-outlined, .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #fff; - color: #fff; } - .button.is-primary.is-inverted.is-outlined:hover, .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink:hover, .button.is-primary.is-inverted.is-outlined.is-hovered, .docstring > section > a.button.is-inverted.is-outlined.is-hovered.docs-sourcelink, .button.is-primary.is-inverted.is-outlined:focus, .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink:focus, .button.is-primary.is-inverted.is-outlined.is-focused, .docstring > section > a.button.is-inverted.is-outlined.is-focused.docs-sourcelink { - background-color: #fff; - color: #4eb5de; } - .button.is-primary.is-inverted.is-outlined.is-loading:hover::after, .docstring > section > a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:hover::after, .button.is-primary.is-inverted.is-outlined.is-loading.is-hovered::after, .docstring > section > a.button.is-inverted.is-outlined.is-loading.is-hovered.docs-sourcelink::after, .button.is-primary.is-inverted.is-outlined.is-loading:focus::after, .docstring > section > a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:focus::after, .button.is-primary.is-inverted.is-outlined.is-loading.is-focused::after, .docstring > section > a.button.is-inverted.is-outlined.is-loading.is-focused.docs-sourcelink::after { - border-color: transparent transparent #4eb5de #4eb5de !important; } - .button.is-primary.is-inverted.is-outlined[disabled], .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink[disabled], fieldset[disabled] .button.is-primary.is-inverted.is-outlined, fieldset[disabled] .docstring > section > a.button.is-inverted.is-outlined.docs-sourcelink { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - .button.is-link { - background-color: #2e63b8; - border-color: transparent; - color: #fff; } - .button.is-link:hover, .button.is-link.is-hovered { - background-color: #2b5eae; - border-color: transparent; - color: #fff; } - .button.is-link:focus, .button.is-link.is-focused { - border-color: transparent; - color: #fff; } - .button.is-link:focus:not(:active), .button.is-link.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(46, 99, 184, 0.25); } - .button.is-link:active, .button.is-link.is-active { - background-color: #2958a4; - border-color: transparent; - color: #fff; } - .button.is-link[disabled], fieldset[disabled] .button.is-link { - background-color: #2e63b8; - border-color: transparent; - box-shadow: none; } - .button.is-link.is-inverted { - background-color: #fff; - color: #2e63b8; } - .button.is-link.is-inverted:hover, .button.is-link.is-inverted.is-hovered { - background-color: #f2f2f2; } - .button.is-link.is-inverted[disabled], fieldset[disabled] .button.is-link.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #2e63b8; } - .button.is-link.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-link.is-outlined { - background-color: transparent; - border-color: #2e63b8; - color: #2e63b8; } - .button.is-link.is-outlined:hover, .button.is-link.is-outlined.is-hovered, .button.is-link.is-outlined:focus, .button.is-link.is-outlined.is-focused { - background-color: #2e63b8; - border-color: #2e63b8; - color: #fff; } - .button.is-link.is-outlined.is-loading::after { - border-color: transparent transparent #2e63b8 #2e63b8 !important; } - .button.is-link.is-outlined.is-loading:hover::after, .button.is-link.is-outlined.is-loading.is-hovered::after, .button.is-link.is-outlined.is-loading:focus::after, .button.is-link.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-link.is-outlined[disabled], fieldset[disabled] .button.is-link.is-outlined { - background-color: transparent; - border-color: #2e63b8; - box-shadow: none; - color: #2e63b8; } - .button.is-link.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - .button.is-link.is-inverted.is-outlined:hover, .button.is-link.is-inverted.is-outlined.is-hovered, .button.is-link.is-inverted.is-outlined:focus, .button.is-link.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #2e63b8; } - .button.is-link.is-inverted.is-outlined.is-loading:hover::after, .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-link.is-inverted.is-outlined.is-loading:focus::after, .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #2e63b8 #2e63b8 !important; } - .button.is-link.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-link.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - .button.is-info { - background-color: #209cee; - border-color: transparent; - color: #fff; } - .button.is-info:hover, .button.is-info.is-hovered { - background-color: #1497ed; - border-color: transparent; - color: #fff; } - .button.is-info:focus, .button.is-info.is-focused { - border-color: transparent; - color: #fff; } - .button.is-info:focus:not(:active), .button.is-info.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(32, 156, 238, 0.25); } - .button.is-info:active, .button.is-info.is-active { - background-color: #1190e3; - border-color: transparent; - color: #fff; } - .button.is-info[disabled], fieldset[disabled] .button.is-info { - background-color: #209cee; - border-color: transparent; - box-shadow: none; } - .button.is-info.is-inverted { - background-color: #fff; - color: #209cee; } - .button.is-info.is-inverted:hover, .button.is-info.is-inverted.is-hovered { - background-color: #f2f2f2; } - .button.is-info.is-inverted[disabled], fieldset[disabled] .button.is-info.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #209cee; } - .button.is-info.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-info.is-outlined { - background-color: transparent; - border-color: #209cee; - color: #209cee; } - .button.is-info.is-outlined:hover, .button.is-info.is-outlined.is-hovered, .button.is-info.is-outlined:focus, .button.is-info.is-outlined.is-focused { - background-color: #209cee; - border-color: #209cee; - color: #fff; } - .button.is-info.is-outlined.is-loading::after { - border-color: transparent transparent #209cee #209cee !important; } - .button.is-info.is-outlined.is-loading:hover::after, .button.is-info.is-outlined.is-loading.is-hovered::after, .button.is-info.is-outlined.is-loading:focus::after, .button.is-info.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-info.is-outlined[disabled], fieldset[disabled] .button.is-info.is-outlined { - background-color: transparent; - border-color: #209cee; - box-shadow: none; - color: #209cee; } - .button.is-info.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - .button.is-info.is-inverted.is-outlined:hover, .button.is-info.is-inverted.is-outlined.is-hovered, .button.is-info.is-inverted.is-outlined:focus, .button.is-info.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #209cee; } - .button.is-info.is-inverted.is-outlined.is-loading:hover::after, .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-info.is-inverted.is-outlined.is-loading:focus::after, .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #209cee #209cee !important; } - .button.is-info.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-info.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - .button.is-success { - background-color: #22c35b; - border-color: transparent; - color: #fff; } - .button.is-success:hover, .button.is-success.is-hovered { - background-color: #20b856; - border-color: transparent; - color: #fff; } - .button.is-success:focus, .button.is-success.is-focused { - border-color: transparent; - color: #fff; } - .button.is-success:focus:not(:active), .button.is-success.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(34, 195, 91, 0.25); } - .button.is-success:active, .button.is-success.is-active { - background-color: #1ead51; - border-color: transparent; - color: #fff; } - .button.is-success[disabled], fieldset[disabled] .button.is-success { - background-color: #22c35b; - border-color: transparent; - box-shadow: none; } - .button.is-success.is-inverted { - background-color: #fff; - color: #22c35b; } - .button.is-success.is-inverted:hover, .button.is-success.is-inverted.is-hovered { - background-color: #f2f2f2; } - .button.is-success.is-inverted[disabled], fieldset[disabled] .button.is-success.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #22c35b; } - .button.is-success.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-success.is-outlined { - background-color: transparent; - border-color: #22c35b; - color: #22c35b; } - .button.is-success.is-outlined:hover, .button.is-success.is-outlined.is-hovered, .button.is-success.is-outlined:focus, .button.is-success.is-outlined.is-focused { - background-color: #22c35b; - border-color: #22c35b; - color: #fff; } - .button.is-success.is-outlined.is-loading::after { - border-color: transparent transparent #22c35b #22c35b !important; } - .button.is-success.is-outlined.is-loading:hover::after, .button.is-success.is-outlined.is-loading.is-hovered::after, .button.is-success.is-outlined.is-loading:focus::after, .button.is-success.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-success.is-outlined[disabled], fieldset[disabled] .button.is-success.is-outlined { - background-color: transparent; - border-color: #22c35b; - box-shadow: none; - color: #22c35b; } - .button.is-success.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - .button.is-success.is-inverted.is-outlined:hover, .button.is-success.is-inverted.is-outlined.is-hovered, .button.is-success.is-inverted.is-outlined:focus, .button.is-success.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #22c35b; } - .button.is-success.is-inverted.is-outlined.is-loading:hover::after, .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-success.is-inverted.is-outlined.is-loading:focus::after, .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #22c35b #22c35b !important; } - .button.is-success.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-success.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - .button.is-warning { - background-color: #ffdd57; - border-color: transparent; - color: rgba(0, 0, 0, 0.7); } - .button.is-warning:hover, .button.is-warning.is-hovered { - background-color: #ffda4a; - border-color: transparent; - color: rgba(0, 0, 0, 0.7); } - .button.is-warning:focus, .button.is-warning.is-focused { - border-color: transparent; - color: rgba(0, 0, 0, 0.7); } - .button.is-warning:focus:not(:active), .button.is-warning.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(255, 221, 87, 0.25); } - .button.is-warning:active, .button.is-warning.is-active { - background-color: #ffd83e; - border-color: transparent; - color: rgba(0, 0, 0, 0.7); } - .button.is-warning[disabled], fieldset[disabled] .button.is-warning { - background-color: #ffdd57; - border-color: transparent; - box-shadow: none; } - .button.is-warning.is-inverted { - background-color: rgba(0, 0, 0, 0.7); - color: #ffdd57; } - .button.is-warning.is-inverted:hover, .button.is-warning.is-inverted.is-hovered { - background-color: rgba(0, 0, 0, 0.7); } - .button.is-warning.is-inverted[disabled], fieldset[disabled] .button.is-warning.is-inverted { - background-color: rgba(0, 0, 0, 0.7); - border-color: transparent; - box-shadow: none; - color: #ffdd57; } - .button.is-warning.is-loading::after { - border-color: transparent transparent rgba(0, 0, 0, 0.7) rgba(0, 0, 0, 0.7) !important; } - .button.is-warning.is-outlined { - background-color: transparent; - border-color: #ffdd57; - color: #ffdd57; } - .button.is-warning.is-outlined:hover, .button.is-warning.is-outlined.is-hovered, .button.is-warning.is-outlined:focus, .button.is-warning.is-outlined.is-focused { - background-color: #ffdd57; - border-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .button.is-warning.is-outlined.is-loading::after { - border-color: transparent transparent #ffdd57 #ffdd57 !important; } - .button.is-warning.is-outlined.is-loading:hover::after, .button.is-warning.is-outlined.is-loading.is-hovered::after, .button.is-warning.is-outlined.is-loading:focus::after, .button.is-warning.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent rgba(0, 0, 0, 0.7) rgba(0, 0, 0, 0.7) !important; } - .button.is-warning.is-outlined[disabled], fieldset[disabled] .button.is-warning.is-outlined { - background-color: transparent; - border-color: #ffdd57; - box-shadow: none; - color: #ffdd57; } - .button.is-warning.is-inverted.is-outlined { - background-color: transparent; - border-color: rgba(0, 0, 0, 0.7); - color: rgba(0, 0, 0, 0.7); } - .button.is-warning.is-inverted.is-outlined:hover, .button.is-warning.is-inverted.is-outlined.is-hovered, .button.is-warning.is-inverted.is-outlined:focus, .button.is-warning.is-inverted.is-outlined.is-focused { - background-color: rgba(0, 0, 0, 0.7); - color: #ffdd57; } - .button.is-warning.is-inverted.is-outlined.is-loading:hover::after, .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-warning.is-inverted.is-outlined.is-loading:focus::after, .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #ffdd57 #ffdd57 !important; } - .button.is-warning.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-warning.is-inverted.is-outlined { - background-color: transparent; - border-color: rgba(0, 0, 0, 0.7); - box-shadow: none; - color: rgba(0, 0, 0, 0.7); } - .button.is-danger { - background-color: #da0b00; - border-color: transparent; - color: #fff; } - .button.is-danger:hover, .button.is-danger.is-hovered { - background-color: #cd0a00; - border-color: transparent; - color: #fff; } - .button.is-danger:focus, .button.is-danger.is-focused { - border-color: transparent; - color: #fff; } - .button.is-danger:focus:not(:active), .button.is-danger.is-focused:not(:active) { - box-shadow: 0 0 0 0.125em rgba(218, 11, 0, 0.25); } - .button.is-danger:active, .button.is-danger.is-active { - background-color: #c10a00; - border-color: transparent; - color: #fff; } - .button.is-danger[disabled], fieldset[disabled] .button.is-danger { - background-color: #da0b00; - border-color: transparent; - box-shadow: none; } - .button.is-danger.is-inverted { - background-color: #fff; - color: #da0b00; } - .button.is-danger.is-inverted:hover, .button.is-danger.is-inverted.is-hovered { - background-color: #f2f2f2; } - .button.is-danger.is-inverted[disabled], fieldset[disabled] .button.is-danger.is-inverted { - background-color: #fff; - border-color: transparent; - box-shadow: none; - color: #da0b00; } - .button.is-danger.is-loading::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-danger.is-outlined { - background-color: transparent; - border-color: #da0b00; - color: #da0b00; } - .button.is-danger.is-outlined:hover, .button.is-danger.is-outlined.is-hovered, .button.is-danger.is-outlined:focus, .button.is-danger.is-outlined.is-focused { - background-color: #da0b00; - border-color: #da0b00; - color: #fff; } - .button.is-danger.is-outlined.is-loading::after { - border-color: transparent transparent #da0b00 #da0b00 !important; } - .button.is-danger.is-outlined.is-loading:hover::after, .button.is-danger.is-outlined.is-loading.is-hovered::after, .button.is-danger.is-outlined.is-loading:focus::after, .button.is-danger.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #fff #fff !important; } - .button.is-danger.is-outlined[disabled], fieldset[disabled] .button.is-danger.is-outlined { - background-color: transparent; - border-color: #da0b00; - box-shadow: none; - color: #da0b00; } - .button.is-danger.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - color: #fff; } - .button.is-danger.is-inverted.is-outlined:hover, .button.is-danger.is-inverted.is-outlined.is-hovered, .button.is-danger.is-inverted.is-outlined:focus, .button.is-danger.is-inverted.is-outlined.is-focused { - background-color: #fff; - color: #da0b00; } - .button.is-danger.is-inverted.is-outlined.is-loading:hover::after, .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after, .button.is-danger.is-inverted.is-outlined.is-loading:focus::after, .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after { - border-color: transparent transparent #da0b00 #da0b00 !important; } - .button.is-danger.is-inverted.is-outlined[disabled], fieldset[disabled] .button.is-danger.is-inverted.is-outlined { - background-color: transparent; - border-color: #fff; - box-shadow: none; - color: #fff; } - .button.is-small, #documenter .docs-sidebar form.docs-search > input.button { - border-radius: 2px; - font-size: 0.75rem; } - .button.is-normal { - font-size: 1rem; } - .button.is-medium { - font-size: 1.25rem; } - .button.is-large { - font-size: 1.5rem; } - .button[disabled], fieldset[disabled] .button { - background-color: white; - border-color: #dbdbdb; - box-shadow: none; - opacity: 0.5; } - .button.is-fullwidth { - display: flex; - width: 100%; } - .button.is-loading { - color: transparent !important; - pointer-events: none; } - .button.is-loading::after { - position: absolute; - left: calc(50% - (1em / 2)); - top: calc(50% - (1em / 2)); - position: absolute !important; } - .button.is-static { - background-color: whitesmoke; - border-color: #dbdbdb; - color: #6b6b6b; - box-shadow: none; - pointer-events: none; } - .button.is-rounded, #documenter .docs-sidebar form.docs-search > input.button { - border-radius: 290486px; - padding-left: 1em; - padding-right: 1em; } - -.buttons { - align-items: center; - display: flex; - flex-wrap: wrap; - justify-content: flex-start; } - .buttons .button { - margin-bottom: 0.5rem; } - .buttons .button:not(:last-child):not(.is-fullwidth) { - margin-right: 0.5rem; } - .buttons:last-child { - margin-bottom: -0.5rem; } - .buttons:not(:last-child) { - margin-bottom: 1rem; } - .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large) { - border-radius: 2px; - font-size: 0.75rem; } - .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large) { - font-size: 1.25rem; } - .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium) { - font-size: 1.5rem; } - .buttons.has-addons .button:not(:first-child) { - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - .buttons.has-addons .button:not(:last-child) { - border-bottom-right-radius: 0; - border-top-right-radius: 0; - margin-right: -1px; } - .buttons.has-addons .button:last-child { - margin-right: 0; } - .buttons.has-addons .button:hover, .buttons.has-addons .button.is-hovered { - z-index: 2; } - .buttons.has-addons .button:focus, .buttons.has-addons .button.is-focused, .buttons.has-addons .button:active, .buttons.has-addons .button.is-active, .buttons.has-addons .button.is-selected { - z-index: 3; } - .buttons.has-addons .button:focus:hover, .buttons.has-addons .button.is-focused:hover, .buttons.has-addons .button:active:hover, .buttons.has-addons .button.is-active:hover, .buttons.has-addons .button.is-selected:hover { - z-index: 4; } - .buttons.has-addons .button.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - .buttons.is-centered { - justify-content: center; } - .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth) { - margin-left: 0.25rem; - margin-right: 0.25rem; } - .buttons.is-right { - justify-content: flex-end; } - .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth) { - margin-left: 0.25rem; - margin-right: 0.25rem; } - -.container { - flex-grow: 1; - margin: 0 auto; - position: relative; - width: auto; } - @media screen and (min-width: 1056px) { - .container { - max-width: 992px; } - .container.is-fluid { - margin-left: 32px; - margin-right: 32px; - max-width: none; } } - @media screen and (max-width: 1215px) { - .container.is-widescreen { - max-width: 1152px; } } - @media screen and (max-width: 1407px) { - .container.is-fullhd { - max-width: 1344px; } } - @media screen and (min-width: 1216px) { - .container { - max-width: 1152px; } } - @media screen and (min-width: 1408px) { - .container { - max-width: 1344px; } } -.content li + li { - margin-top: 0.25em; } - -.content p:not(:last-child), -.content dl:not(:last-child), -.content ol:not(:last-child), -.content ul:not(:last-child), -.content blockquote:not(:last-child), -.content pre:not(:last-child), -.content table:not(:last-child) { - margin-bottom: 1em; } - -.content h1, -.content h2, -.content h3, -.content h4, -.content h5, -.content h6 { - color: #222222; - font-weight: 600; - line-height: 1.125; } - -.content h1 { - font-size: 2em; - margin-bottom: 0.5em; } - .content h1:not(:first-child) { - margin-top: 1em; } - -.content h2 { - font-size: 1.75em; - margin-bottom: 0.5714em; } - .content h2:not(:first-child) { - margin-top: 1.1428em; } - -.content h3 { - font-size: 1.5em; - margin-bottom: 0.6666em; } - .content h3:not(:first-child) { - margin-top: 1.3333em; } - -.content h4 { - font-size: 1.25em; - margin-bottom: 0.8em; } - -.content h5 { - font-size: 1.125em; - margin-bottom: 0.8888em; } - -.content h6 { - font-size: 1em; - margin-bottom: 1em; } - -.content blockquote { - background-color: whitesmoke; - border-left: 5px solid #dbdbdb; - padding: 1.25em 1.5em; } - -.content ol { - list-style-position: outside; - margin-left: 2em; - margin-top: 1em; } - .content ol:not([type]) { - list-style-type: decimal; } - .content ol.is-lower-alpha:not([type]) { - list-style-type: lower-alpha; } - .content ol.is-lower-roman:not([type]) { - list-style-type: lower-roman; } - .content ol.is-upper-alpha:not([type]) { - list-style-type: upper-alpha; } - .content ol.is-upper-roman:not([type]) { - list-style-type: upper-roman; } - -.content ul { - list-style: disc outside; - margin-left: 2em; - margin-top: 1em; } - .content ul ul { - list-style-type: circle; - margin-top: 0.5em; } - .content ul ul ul { - list-style-type: square; } - -.content dd { - margin-left: 2em; } - -.content figure { - margin-left: 2em; - margin-right: 2em; - text-align: center; } - .content figure:not(:first-child) { - margin-top: 2em; } - .content figure:not(:last-child) { - margin-bottom: 2em; } - .content figure img { - display: inline-block; } - .content figure figcaption { - font-style: italic; } - -.content pre { - -webkit-overflow-scrolling: touch; - overflow-x: auto; - padding: 0; - white-space: pre; - word-wrap: normal; } - -.content sup, -.content sub { - font-size: 75%; } - -.content table { - width: 100%; } - .content table td, - .content table th { - border: 1px solid #dbdbdb; - border-width: 0 0 1px; - padding: 0.5em 0.75em; - vertical-align: top; } - .content table th { - color: #222222; } - .content table th:not([align]) { - text-align: left; } - .content table thead td, - .content table thead th { - border-width: 0 0 2px; - color: #222222; } - .content table tfoot td, - .content table tfoot th { - border-width: 2px 0 0; - color: #222222; } - .content table tbody tr:last-child td, - .content table tbody tr:last-child th { - border-bottom-width: 0; } - -.content .tabs li + li { - margin-top: 0; } - -.content.is-small, #documenter .docs-sidebar form.docs-search > input.content { - font-size: 0.75rem; } - -.content.is-medium { - font-size: 1.25rem; } - -.content.is-large { - font-size: 1.5rem; } - -.icon { - align-items: center; - display: inline-flex; - justify-content: center; - height: 1.5rem; - width: 1.5rem; } - .icon.is-small, #documenter .docs-sidebar form.docs-search > input.icon { - height: 1rem; - width: 1rem; } - .icon.is-medium { - height: 2rem; - width: 2rem; } - .icon.is-large { - height: 3rem; - width: 3rem; } - -.image, #documenter .docs-sidebar .docs-logo > img { - display: block; - position: relative; } - .image img, #documenter .docs-sidebar .docs-logo > img img { - display: block; - height: auto; - width: 100%; } - .image img.is-rounded, #documenter .docs-sidebar .docs-logo > img img.is-rounded { - border-radius: 290486px; } - .image.is-square img, #documenter .docs-sidebar .docs-logo > img.is-square img, - .image.is-square .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-square .has-ratio, .image.is-1by1 img, #documenter .docs-sidebar .docs-logo > img.is-1by1 img, - .image.is-1by1 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-1by1 .has-ratio, .image.is-5by4 img, #documenter .docs-sidebar .docs-logo > img.is-5by4 img, - .image.is-5by4 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-5by4 .has-ratio, .image.is-4by3 img, #documenter .docs-sidebar .docs-logo > img.is-4by3 img, - .image.is-4by3 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-4by3 .has-ratio, .image.is-3by2 img, #documenter .docs-sidebar .docs-logo > img.is-3by2 img, - .image.is-3by2 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-3by2 .has-ratio, .image.is-5by3 img, #documenter .docs-sidebar .docs-logo > img.is-5by3 img, - .image.is-5by3 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-5by3 .has-ratio, .image.is-16by9 img, #documenter .docs-sidebar .docs-logo > img.is-16by9 img, - .image.is-16by9 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-16by9 .has-ratio, .image.is-2by1 img, #documenter .docs-sidebar .docs-logo > img.is-2by1 img, - .image.is-2by1 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-2by1 .has-ratio, .image.is-3by1 img, #documenter .docs-sidebar .docs-logo > img.is-3by1 img, - .image.is-3by1 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-3by1 .has-ratio, .image.is-4by5 img, #documenter .docs-sidebar .docs-logo > img.is-4by5 img, - .image.is-4by5 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-4by5 .has-ratio, .image.is-3by4 img, #documenter .docs-sidebar .docs-logo > img.is-3by4 img, - .image.is-3by4 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-3by4 .has-ratio, .image.is-2by3 img, #documenter .docs-sidebar .docs-logo > img.is-2by3 img, - .image.is-2by3 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-2by3 .has-ratio, .image.is-3by5 img, #documenter .docs-sidebar .docs-logo > img.is-3by5 img, - .image.is-3by5 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-3by5 .has-ratio, .image.is-9by16 img, #documenter .docs-sidebar .docs-logo > img.is-9by16 img, - .image.is-9by16 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-9by16 .has-ratio, .image.is-1by2 img, #documenter .docs-sidebar .docs-logo > img.is-1by2 img, - .image.is-1by2 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-1by2 .has-ratio, .image.is-1by3 img, #documenter .docs-sidebar .docs-logo > img.is-1by3 img, - .image.is-1by3 .has-ratio, - #documenter .docs-sidebar .docs-logo > img.is-1by3 .has-ratio { - height: 100%; - width: 100%; } - .image.is-square, #documenter .docs-sidebar .docs-logo > img.is-square, .image.is-1by1, #documenter .docs-sidebar .docs-logo > img.is-1by1 { - padding-top: 100%; } - .image.is-5by4, #documenter .docs-sidebar .docs-logo > img.is-5by4 { - padding-top: 80%; } - .image.is-4by3, #documenter .docs-sidebar .docs-logo > img.is-4by3 { - padding-top: 75%; } - .image.is-3by2, #documenter .docs-sidebar .docs-logo > img.is-3by2 { - padding-top: 66.6666%; } - .image.is-5by3, #documenter .docs-sidebar .docs-logo > img.is-5by3 { - padding-top: 60%; } - .image.is-16by9, #documenter .docs-sidebar .docs-logo > img.is-16by9 { - padding-top: 56.25%; } - .image.is-2by1, #documenter .docs-sidebar .docs-logo > img.is-2by1 { - padding-top: 50%; } - .image.is-3by1, #documenter .docs-sidebar .docs-logo > img.is-3by1 { - padding-top: 33.3333%; } - .image.is-4by5, #documenter .docs-sidebar .docs-logo > img.is-4by5 { - padding-top: 125%; } - .image.is-3by4, #documenter .docs-sidebar .docs-logo > img.is-3by4 { - padding-top: 133.3333%; } - .image.is-2by3, #documenter .docs-sidebar .docs-logo > img.is-2by3 { - padding-top: 150%; } - .image.is-3by5, #documenter .docs-sidebar .docs-logo > img.is-3by5 { - padding-top: 166.6666%; } - .image.is-9by16, #documenter .docs-sidebar .docs-logo > img.is-9by16 { - padding-top: 177.7777%; } - .image.is-1by2, #documenter .docs-sidebar .docs-logo > img.is-1by2 { - padding-top: 200%; } - .image.is-1by3, #documenter .docs-sidebar .docs-logo > img.is-1by3 { - padding-top: 300%; } - .image.is-16x16, #documenter .docs-sidebar .docs-logo > img.is-16x16 { - height: 16px; - width: 16px; } - .image.is-24x24, #documenter .docs-sidebar .docs-logo > img.is-24x24 { - height: 24px; - width: 24px; } - .image.is-32x32, #documenter .docs-sidebar .docs-logo > img.is-32x32 { - height: 32px; - width: 32px; } - .image.is-48x48, #documenter .docs-sidebar .docs-logo > img.is-48x48 { - height: 48px; - width: 48px; } - .image.is-64x64, #documenter .docs-sidebar .docs-logo > img.is-64x64 { - height: 64px; - width: 64px; } - .image.is-96x96, #documenter .docs-sidebar .docs-logo > img.is-96x96 { - height: 96px; - width: 96px; } - .image.is-128x128, #documenter .docs-sidebar .docs-logo > img.is-128x128 { - height: 128px; - width: 128px; } - -.notification { - background-color: whitesmoke; - border-radius: 4px; - padding: 1.25rem 2.5rem 1.25rem 1.5rem; - position: relative; } - .notification a:not(.button):not(.dropdown-item) { - color: currentColor; - text-decoration: underline; } - .notification strong { - color: currentColor; } - .notification code, - .notification pre { - background: white; } - .notification pre code { - background: transparent; } - .notification > .delete { - position: absolute; - right: 0.5rem; - top: 0.5rem; } - .notification .title, - .notification .subtitle, - .notification .content { - color: currentColor; } - .notification.is-white { - background-color: white; - color: #0a0a0a; } - .notification.is-black { - background-color: #0a0a0a; - color: white; } - .notification.is-light { - background-color: whitesmoke; - color: #363636; } - .notification.is-dark, .content kbd.notification { - background-color: #363636; - color: whitesmoke; } - .notification.is-primary, .docstring > section > a.notification.docs-sourcelink { - background-color: #4eb5de; - color: #fff; } - .notification.is-link { - background-color: #2e63b8; - color: #fff; } - .notification.is-info { - background-color: #209cee; - color: #fff; } - .notification.is-success { - background-color: #22c35b; - color: #fff; } - .notification.is-warning { - background-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .notification.is-danger { - background-color: #da0b00; - color: #fff; } - -.progress { - -moz-appearance: none; - -webkit-appearance: none; - border: none; - border-radius: 290486px; - display: block; - height: 1rem; - overflow: hidden; - padding: 0; - width: 100%; } - .progress::-webkit-progress-bar { - background-color: #dbdbdb; } - .progress::-webkit-progress-value { - background-color: #222222; } - .progress::-moz-progress-bar { - background-color: #222222; } - .progress::-ms-fill { - background-color: #222222; - border: none; } - .progress.is-white::-webkit-progress-value { - background-color: white; } - .progress.is-white::-moz-progress-bar { - background-color: white; } - .progress.is-white::-ms-fill { - background-color: white; } - .progress.is-white:indeterminate { - background-image: linear-gradient(to right, white 30%, #dbdbdb 30%); } - .progress.is-black::-webkit-progress-value { - background-color: #0a0a0a; } - .progress.is-black::-moz-progress-bar { - background-color: #0a0a0a; } - .progress.is-black::-ms-fill { - background-color: #0a0a0a; } - .progress.is-black:indeterminate { - background-image: linear-gradient(to right, #0a0a0a 30%, #dbdbdb 30%); } - .progress.is-light::-webkit-progress-value { - background-color: whitesmoke; } - .progress.is-light::-moz-progress-bar { - background-color: whitesmoke; } - .progress.is-light::-ms-fill { - background-color: whitesmoke; } - .progress.is-light:indeterminate { - background-image: linear-gradient(to right, whitesmoke 30%, #dbdbdb 30%); } - .progress.is-dark::-webkit-progress-value, .content kbd.progress::-webkit-progress-value { - background-color: #363636; } - .progress.is-dark::-moz-progress-bar, .content kbd.progress::-moz-progress-bar { - background-color: #363636; } - .progress.is-dark::-ms-fill, .content kbd.progress::-ms-fill { - background-color: #363636; } - .progress.is-dark:indeterminate, .content kbd.progress:indeterminate { - background-image: linear-gradient(to right, #363636 30%, #dbdbdb 30%); } - .progress.is-primary::-webkit-progress-value, .docstring > section > a.progress.docs-sourcelink::-webkit-progress-value { - background-color: #4eb5de; } - .progress.is-primary::-moz-progress-bar, .docstring > section > a.progress.docs-sourcelink::-moz-progress-bar { - background-color: #4eb5de; } - .progress.is-primary::-ms-fill, .docstring > section > a.progress.docs-sourcelink::-ms-fill { - background-color: #4eb5de; } - .progress.is-primary:indeterminate, .docstring > section > a.progress.docs-sourcelink:indeterminate { - background-image: linear-gradient(to right, #4eb5de 30%, #dbdbdb 30%); } - .progress.is-link::-webkit-progress-value { - background-color: #2e63b8; } - .progress.is-link::-moz-progress-bar { - background-color: #2e63b8; } - .progress.is-link::-ms-fill { - background-color: #2e63b8; } - .progress.is-link:indeterminate { - background-image: linear-gradient(to right, #2e63b8 30%, #dbdbdb 30%); } - .progress.is-info::-webkit-progress-value { - background-color: #209cee; } - .progress.is-info::-moz-progress-bar { - background-color: #209cee; } - .progress.is-info::-ms-fill { - background-color: #209cee; } - .progress.is-info:indeterminate { - background-image: linear-gradient(to right, #209cee 30%, #dbdbdb 30%); } - .progress.is-success::-webkit-progress-value { - background-color: #22c35b; } - .progress.is-success::-moz-progress-bar { - background-color: #22c35b; } - .progress.is-success::-ms-fill { - background-color: #22c35b; } - .progress.is-success:indeterminate { - background-image: linear-gradient(to right, #22c35b 30%, #dbdbdb 30%); } - .progress.is-warning::-webkit-progress-value { - background-color: #ffdd57; } - .progress.is-warning::-moz-progress-bar { - background-color: #ffdd57; } - .progress.is-warning::-ms-fill { - background-color: #ffdd57; } - .progress.is-warning:indeterminate { - background-image: linear-gradient(to right, #ffdd57 30%, #dbdbdb 30%); } - .progress.is-danger::-webkit-progress-value { - background-color: #da0b00; } - .progress.is-danger::-moz-progress-bar { - background-color: #da0b00; } - .progress.is-danger::-ms-fill { - background-color: #da0b00; } - .progress.is-danger:indeterminate { - background-image: linear-gradient(to right, #da0b00 30%, #dbdbdb 30%); } - .progress:indeterminate { - animation-duration: 1.5s; - animation-iteration-count: infinite; - animation-name: moveIndeterminate; - animation-timing-function: linear; - background-color: #dbdbdb; - background-image: linear-gradient(to right, #222222 30%, #dbdbdb 30%); - background-position: top left; - background-repeat: no-repeat; - background-size: 150% 150%; } - .progress:indeterminate::-webkit-progress-bar { - background-color: transparent; } - .progress:indeterminate::-moz-progress-bar { - background-color: transparent; } - .progress.is-small, #documenter .docs-sidebar form.docs-search > input.progress { - height: 0.75rem; } - .progress.is-medium { - height: 1.25rem; } - .progress.is-large { - height: 1.5rem; } - -@keyframes moveIndeterminate { - from { - background-position: 200% 0; } - to { - background-position: -200% 0; } } - -.table { - background-color: white; - color: #363636; } - .table td, - .table th { - border: 1px solid #dbdbdb; - border-width: 0 0 1px; - padding: 0.5em 0.75em; - vertical-align: top; } - .table td.is-white, - .table th.is-white { - background-color: white; - border-color: white; - color: #0a0a0a; } - .table td.is-black, - .table th.is-black { - background-color: #0a0a0a; - border-color: #0a0a0a; - color: white; } - .table td.is-light, - .table th.is-light { - background-color: whitesmoke; - border-color: whitesmoke; - color: #363636; } - .table td.is-dark, - .table th.is-dark { - background-color: #363636; - border-color: #363636; - color: whitesmoke; } - .table td.is-primary, - .table th.is-primary { - background-color: #4eb5de; - border-color: #4eb5de; - color: #fff; } - .table td.is-link, - .table th.is-link { - background-color: #2e63b8; - border-color: #2e63b8; - color: #fff; } - .table td.is-info, - .table th.is-info { - background-color: #209cee; - border-color: #209cee; - color: #fff; } - .table td.is-success, - .table th.is-success { - background-color: #22c35b; - border-color: #22c35b; - color: #fff; } - .table td.is-warning, - .table th.is-warning { - background-color: #ffdd57; - border-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .table td.is-danger, - .table th.is-danger { - background-color: #da0b00; - border-color: #da0b00; - color: #fff; } - .table td.is-narrow, - .table th.is-narrow { - white-space: nowrap; - width: 1%; } - .table td.is-selected, - .table th.is-selected { - background-color: #4eb5de; - color: #fff; } - .table td.is-selected a, - .table td.is-selected strong, - .table th.is-selected a, - .table th.is-selected strong { - color: currentColor; } - .table th { - color: #222222; } - .table th:not([align]) { - text-align: left; } - .table tr.is-selected { - background-color: #4eb5de; - color: #fff; } - .table tr.is-selected a, - .table tr.is-selected strong { - color: currentColor; } - .table tr.is-selected td, - .table tr.is-selected th { - border-color: #fff; - color: currentColor; } - .table thead { - background-color: transparent; } - .table thead td, - .table thead th { - border-width: 0 0 2px; - color: #222222; } - .table tfoot { - background-color: transparent; } - .table tfoot td, - .table tfoot th { - border-width: 2px 0 0; - color: #222222; } - .table tbody { - background-color: transparent; } - .table tbody tr:last-child td, - .table tbody tr:last-child th { - border-bottom-width: 0; } - .table.is-bordered td, - .table.is-bordered th { - border-width: 1px; } - .table.is-bordered tr:last-child td, - .table.is-bordered tr:last-child th { - border-bottom-width: 1px; } - .table.is-fullwidth { - width: 100%; } - .table.is-hoverable tbody tr:not(.is-selected):hover { - background-color: #fafafa; } - .table.is-hoverable.is-striped tbody tr:not(.is-selected):hover { - background-color: #fafafa; } - .table.is-hoverable.is-striped tbody tr:not(.is-selected):hover:nth-child(even) { - background-color: whitesmoke; } - .table.is-narrow td, - .table.is-narrow th { - padding: 0.25em 0.5em; } - .table.is-striped tbody tr:not(.is-selected):nth-child(even) { - background-color: #fafafa; } - -.table-container { - -webkit-overflow-scrolling: touch; - overflow: auto; - overflow-y: hidden; - max-width: 100%; } - -.tags { - align-items: center; - display: flex; - flex-wrap: wrap; - justify-content: flex-start; } - .tags .tag, .tags .content kbd, .content .tags kbd, .tags .docstring > section > a.docs-sourcelink { - margin-bottom: 0.5rem; } - .tags .tag:not(:last-child), .tags .content kbd:not(:last-child), .content .tags kbd:not(:last-child), .tags .docstring > section > a.docs-sourcelink:not(:last-child) { - margin-right: 0.5rem; } - .tags:last-child { - margin-bottom: -0.5rem; } - .tags:not(:last-child) { - margin-bottom: 1rem; } - .tags.are-medium .tag:not(.is-normal):not(.is-large), .tags.are-medium .content kbd:not(.is-normal):not(.is-large), .content .tags.are-medium kbd:not(.is-normal):not(.is-large), .tags.are-medium .docstring > section > a.docs-sourcelink:not(.is-normal):not(.is-large) { - font-size: 1rem; } - .tags.are-large .tag:not(.is-normal):not(.is-medium), .tags.are-large .content kbd:not(.is-normal):not(.is-medium), .content .tags.are-large kbd:not(.is-normal):not(.is-medium), .tags.are-large .docstring > section > a.docs-sourcelink:not(.is-normal):not(.is-medium) { - font-size: 1.25rem; } - .tags.is-centered { - justify-content: center; } - .tags.is-centered .tag, .tags.is-centered .content kbd, .content .tags.is-centered kbd, .tags.is-centered .docstring > section > a.docs-sourcelink { - margin-right: 0.25rem; - margin-left: 0.25rem; } - .tags.is-right { - justify-content: flex-end; } - .tags.is-right .tag:not(:first-child), .tags.is-right .content kbd:not(:first-child), .content .tags.is-right kbd:not(:first-child), .tags.is-right .docstring > section > a.docs-sourcelink:not(:first-child) { - margin-left: 0.5rem; } - .tags.is-right .tag:not(:last-child), .tags.is-right .content kbd:not(:last-child), .content .tags.is-right kbd:not(:last-child), .tags.is-right .docstring > section > a.docs-sourcelink:not(:last-child) { - margin-right: 0; } - .tags.has-addons .tag, .tags.has-addons .content kbd, .content .tags.has-addons kbd, .tags.has-addons .docstring > section > a.docs-sourcelink { - margin-right: 0; } - .tags.has-addons .tag:not(:first-child), .tags.has-addons .content kbd:not(:first-child), .content .tags.has-addons kbd:not(:first-child), .tags.has-addons .docstring > section > a.docs-sourcelink:not(:first-child) { - margin-left: 0; - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - .tags.has-addons .tag:not(:last-child), .tags.has-addons .content kbd:not(:last-child), .content .tags.has-addons kbd:not(:last-child), .tags.has-addons .docstring > section > a.docs-sourcelink:not(:last-child) { - border-bottom-right-radius: 0; - border-top-right-radius: 0; } - -.tag:not(body), .content kbd:not(body), .docstring > section > a.docs-sourcelink:not(body) { - align-items: center; - background-color: whitesmoke; - border-radius: 4px; - color: #222222; - display: inline-flex; - font-size: 0.75rem; - height: 2em; - justify-content: center; - line-height: 1.5; - padding-left: 0.75em; - padding-right: 0.75em; - white-space: nowrap; } - .tag:not(body) .delete, .content kbd:not(body) .delete, .docstring > section > a.docs-sourcelink:not(body) .delete { - margin-left: 0.25rem; - margin-right: -0.375rem; } - .tag.is-white:not(body), .content kbd.is-white:not(body), .docstring > section > a.docs-sourcelink.is-white:not(body) { - background-color: white; - color: #0a0a0a; } - .tag.is-black:not(body), .content kbd.is-black:not(body), .docstring > section > a.docs-sourcelink.is-black:not(body) { - background-color: #0a0a0a; - color: white; } - .tag.is-light:not(body), .content kbd.is-light:not(body), .docstring > section > a.docs-sourcelink.is-light:not(body) { - background-color: whitesmoke; - color: #363636; } - .tag.is-dark:not(body), .content kbd:not(body), .docstring > section > a.docs-sourcelink.is-dark:not(body), .content .docstring > section > kbd:not(body) { - background-color: #363636; - color: whitesmoke; } - .tag.is-primary:not(body), .content kbd.is-primary:not(body), .docstring > section > a.docs-sourcelink:not(body) { - background-color: #4eb5de; - color: #fff; } - .tag.is-link:not(body), .content kbd.is-link:not(body), .docstring > section > a.docs-sourcelink.is-link:not(body) { - background-color: #2e63b8; - color: #fff; } - .tag.is-info:not(body), .content kbd.is-info:not(body), .docstring > section > a.docs-sourcelink.is-info:not(body) { - background-color: #209cee; - color: #fff; } - .tag.is-success:not(body), .content kbd.is-success:not(body), .docstring > section > a.docs-sourcelink.is-success:not(body) { - background-color: #22c35b; - color: #fff; } - .tag.is-warning:not(body), .content kbd.is-warning:not(body), .docstring > section > a.docs-sourcelink.is-warning:not(body) { - background-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .tag.is-danger:not(body), .content kbd.is-danger:not(body), .docstring > section > a.docs-sourcelink.is-danger:not(body) { - background-color: #da0b00; - color: #fff; } - .tag.is-normal:not(body), .content kbd.is-normal:not(body), .docstring > section > a.docs-sourcelink.is-normal:not(body) { - font-size: 0.75rem; } - .tag.is-medium:not(body), .content kbd.is-medium:not(body), .docstring > section > a.docs-sourcelink.is-medium:not(body) { - font-size: 1rem; } - .tag.is-large:not(body), .content kbd.is-large:not(body), .docstring > section > a.docs-sourcelink.is-large:not(body) { - font-size: 1.25rem; } - .tag:not(body) .icon:first-child:not(:last-child), .content kbd:not(body) .icon:first-child:not(:last-child), .docstring > section > a.docs-sourcelink:not(body) .icon:first-child:not(:last-child) { - margin-left: -0.375em; - margin-right: 0.1875em; } - .tag:not(body) .icon:last-child:not(:first-child), .content kbd:not(body) .icon:last-child:not(:first-child), .docstring > section > a.docs-sourcelink:not(body) .icon:last-child:not(:first-child) { - margin-left: 0.1875em; - margin-right: -0.375em; } - .tag:not(body) .icon:first-child:last-child, .content kbd:not(body) .icon:first-child:last-child, .docstring > section > a.docs-sourcelink:not(body) .icon:first-child:last-child { - margin-left: -0.375em; - margin-right: -0.375em; } - .tag.is-delete:not(body), .content kbd.is-delete:not(body), .docstring > section > a.docs-sourcelink.is-delete:not(body) { - margin-left: 1px; - padding: 0; - position: relative; - width: 2em; } - .tag.is-delete:not(body)::before, .content kbd.is-delete:not(body)::before, .docstring > section > a.docs-sourcelink.is-delete:not(body)::before, .tag.is-delete:not(body)::after, .content kbd.is-delete:not(body)::after, .docstring > section > a.docs-sourcelink.is-delete:not(body)::after { - background-color: currentColor; - content: ""; - display: block; - left: 50%; - position: absolute; - top: 50%; - transform: translateX(-50%) translateY(-50%) rotate(45deg); - transform-origin: center center; } - .tag.is-delete:not(body)::before, .content kbd.is-delete:not(body)::before, .docstring > section > a.docs-sourcelink.is-delete:not(body)::before { - height: 1px; - width: 50%; } - .tag.is-delete:not(body)::after, .content kbd.is-delete:not(body)::after, .docstring > section > a.docs-sourcelink.is-delete:not(body)::after { - height: 50%; - width: 1px; } - .tag.is-delete:not(body):hover, .content kbd.is-delete:not(body):hover, .docstring > section > a.docs-sourcelink.is-delete:not(body):hover, .tag.is-delete:not(body):focus, .content kbd.is-delete:not(body):focus, .docstring > section > a.docs-sourcelink.is-delete:not(body):focus { - background-color: #e8e8e8; } - .tag.is-delete:not(body):active, .content kbd.is-delete:not(body):active, .docstring > section > a.docs-sourcelink.is-delete:not(body):active { - background-color: #dbdbdb; } - .tag.is-rounded:not(body), #documenter .docs-sidebar form.docs-search > input:not(body), .content kbd.is-rounded:not(body), #documenter .docs-sidebar .content form.docs-search > input:not(body), .docstring > section > a.docs-sourcelink.is-rounded:not(body) { - border-radius: 290486px; } - -a.tag:hover, .docstring > section > a.docs-sourcelink:hover { - text-decoration: underline; } - -.title, -.subtitle { - word-break: break-word; } - .title em, - .title span, - .subtitle em, - .subtitle span { - font-weight: inherit; } - .title sub, - .subtitle sub { - font-size: 0.75em; } - .title sup, - .subtitle sup { - font-size: 0.75em; } - .title .tag, .title .content kbd, .content .title kbd, .title .docstring > section > a.docs-sourcelink, - .subtitle .tag, - .subtitle .content kbd, - .content .subtitle kbd, - .subtitle .docstring > section > a.docs-sourcelink { - vertical-align: middle; } - -.title { - color: #363636; - font-size: 2rem; - font-weight: 600; - line-height: 1.125; } - .title strong { - color: inherit; - font-weight: inherit; } - .title + .highlight { - margin-top: -0.75rem; } - .title:not(.is-spaced) + .subtitle { - margin-top: -1.25rem; } - .title.is-1 { - font-size: 3rem; } - .title.is-2 { - font-size: 2.5rem; } - .title.is-3 { - font-size: 2rem; } - .title.is-4 { - font-size: 1.5rem; } - .title.is-5 { - font-size: 1.25rem; } - .title.is-6 { - font-size: 1rem; } - .title.is-7 { - font-size: 0.75rem; } - -.subtitle { - color: #4a4a4a; - font-size: 1.25rem; - font-weight: 400; - line-height: 1.25; } - .subtitle strong { - color: #363636; - font-weight: 600; } - .subtitle:not(.is-spaced) + .title { - margin-top: -1.25rem; } - .subtitle.is-1 { - font-size: 3rem; } - .subtitle.is-2 { - font-size: 2.5rem; } - .subtitle.is-3 { - font-size: 2rem; } - .subtitle.is-4 { - font-size: 1.5rem; } - .subtitle.is-5 { - font-size: 1.25rem; } - .subtitle.is-6 { - font-size: 1rem; } - .subtitle.is-7 { - font-size: 0.75rem; } - -.heading { - display: block; - font-size: 11px; - letter-spacing: 1px; - margin-bottom: 5px; - text-transform: uppercase; } - -.highlight { - font-weight: 400; - max-width: 100%; - overflow: hidden; - padding: 0; } - .highlight pre { - overflow: auto; - max-width: 100%; } - -.number { - align-items: center; - background-color: whitesmoke; - border-radius: 290486px; - display: inline-flex; - font-size: 1.25rem; - height: 2em; - justify-content: center; - margin-right: 1.5rem; - min-width: 2.5em; - padding: 0.25rem 0.5rem; - text-align: center; - vertical-align: top; } - -.select select, .textarea, .input, #documenter .docs-sidebar form.docs-search > input { - background-color: white; - border-color: #dbdbdb; - border-radius: 4px; - color: #363636; } - .select select::-moz-placeholder, .textarea::-moz-placeholder, .input::-moz-placeholder, #documenter .docs-sidebar form.docs-search > input::-moz-placeholder { - color: rgba(54, 54, 54, 0.3); } - .select select::-webkit-input-placeholder, .textarea::-webkit-input-placeholder, .input::-webkit-input-placeholder, #documenter .docs-sidebar form.docs-search > input::-webkit-input-placeholder { - color: rgba(54, 54, 54, 0.3); } - .select select:-moz-placeholder, .textarea:-moz-placeholder, .input:-moz-placeholder, #documenter .docs-sidebar form.docs-search > input:-moz-placeholder { - color: rgba(54, 54, 54, 0.3); } - .select select:-ms-input-placeholder, .textarea:-ms-input-placeholder, .input:-ms-input-placeholder, #documenter .docs-sidebar form.docs-search > input:-ms-input-placeholder { - color: rgba(54, 54, 54, 0.3); } - .select select:hover, .textarea:hover, .input:hover, #documenter .docs-sidebar form.docs-search > input:hover, .select select.is-hovered, .is-hovered.textarea, .is-hovered.input, #documenter .docs-sidebar form.docs-search > input.is-hovered { - border-color: #b5b5b5; } - .select select:focus, .textarea:focus, .input:focus, #documenter .docs-sidebar form.docs-search > input:focus, .select select.is-focused, .is-focused.textarea, .is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .select select:active, .textarea:active, .input:active, #documenter .docs-sidebar form.docs-search > input:active, .select select.is-active, .is-active.textarea, .is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - border-color: #2e63b8; - box-shadow: 0 0 0 0.125em rgba(46, 99, 184, 0.25); } - .select select[disabled], .textarea[disabled], .input[disabled], #documenter .docs-sidebar form.docs-search > input[disabled], fieldset[disabled] .select select, .select fieldset[disabled] select, fieldset[disabled] .textarea, fieldset[disabled] .input, fieldset[disabled] #documenter .docs-sidebar form.docs-search > input, #documenter .docs-sidebar fieldset[disabled] form.docs-search > input { - background-color: whitesmoke; - border-color: whitesmoke; - box-shadow: none; - color: #6b6b6b; } - .select select[disabled]::-moz-placeholder, .textarea[disabled]::-moz-placeholder, .input[disabled]::-moz-placeholder, #documenter .docs-sidebar form.docs-search > input[disabled]::-moz-placeholder, fieldset[disabled] .select select::-moz-placeholder, .select fieldset[disabled] select::-moz-placeholder, fieldset[disabled] .textarea::-moz-placeholder, fieldset[disabled] .input::-moz-placeholder, fieldset[disabled] #documenter .docs-sidebar form.docs-search > input::-moz-placeholder, #documenter .docs-sidebar fieldset[disabled] form.docs-search > input::-moz-placeholder { - color: rgba(107, 107, 107, 0.3); } - .select select[disabled]::-webkit-input-placeholder, .textarea[disabled]::-webkit-input-placeholder, .input[disabled]::-webkit-input-placeholder, #documenter .docs-sidebar form.docs-search > input[disabled]::-webkit-input-placeholder, fieldset[disabled] .select select::-webkit-input-placeholder, .select fieldset[disabled] select::-webkit-input-placeholder, fieldset[disabled] .textarea::-webkit-input-placeholder, fieldset[disabled] .input::-webkit-input-placeholder, fieldset[disabled] #documenter .docs-sidebar form.docs-search > input::-webkit-input-placeholder, #documenter .docs-sidebar fieldset[disabled] form.docs-search > input::-webkit-input-placeholder { - color: rgba(107, 107, 107, 0.3); } - .select select[disabled]:-moz-placeholder, .textarea[disabled]:-moz-placeholder, .input[disabled]:-moz-placeholder, #documenter .docs-sidebar form.docs-search > input[disabled]:-moz-placeholder, fieldset[disabled] .select select:-moz-placeholder, .select fieldset[disabled] select:-moz-placeholder, fieldset[disabled] .textarea:-moz-placeholder, fieldset[disabled] .input:-moz-placeholder, fieldset[disabled] #documenter .docs-sidebar form.docs-search > input:-moz-placeholder, #documenter .docs-sidebar fieldset[disabled] form.docs-search > input:-moz-placeholder { - color: rgba(107, 107, 107, 0.3); } - .select select[disabled]:-ms-input-placeholder, .textarea[disabled]:-ms-input-placeholder, .input[disabled]:-ms-input-placeholder, #documenter .docs-sidebar form.docs-search > input[disabled]:-ms-input-placeholder, fieldset[disabled] .select select:-ms-input-placeholder, .select fieldset[disabled] select:-ms-input-placeholder, fieldset[disabled] .textarea:-ms-input-placeholder, fieldset[disabled] .input:-ms-input-placeholder, fieldset[disabled] #documenter .docs-sidebar form.docs-search > input:-ms-input-placeholder, #documenter .docs-sidebar fieldset[disabled] form.docs-search > input:-ms-input-placeholder { - color: rgba(107, 107, 107, 0.3); } - -.textarea, .input, #documenter .docs-sidebar form.docs-search > input { - box-shadow: inset 0 1px 2px rgba(10, 10, 10, 0.1); - max-width: 100%; - width: 100%; } - .textarea[readonly], .input[readonly], #documenter .docs-sidebar form.docs-search > input[readonly] { - box-shadow: none; } - .is-white.textarea, .is-white.input, #documenter .docs-sidebar form.docs-search > input.is-white { - border-color: white; } - .is-white.textarea:focus, .is-white.input:focus, #documenter .docs-sidebar form.docs-search > input.is-white:focus, .is-white.is-focused.textarea, .is-white.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-white.textarea:active, .is-white.input:active, #documenter .docs-sidebar form.docs-search > input.is-white:active, .is-white.is-active.textarea, .is-white.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); } - .is-black.textarea, .is-black.input, #documenter .docs-sidebar form.docs-search > input.is-black { - border-color: #0a0a0a; } - .is-black.textarea:focus, .is-black.input:focus, #documenter .docs-sidebar form.docs-search > input.is-black:focus, .is-black.is-focused.textarea, .is-black.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-black.textarea:active, .is-black.input:active, #documenter .docs-sidebar form.docs-search > input.is-black:active, .is-black.is-active.textarea, .is-black.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); } - .is-light.textarea, .is-light.input, #documenter .docs-sidebar form.docs-search > input.is-light { - border-color: whitesmoke; } - .is-light.textarea:focus, .is-light.input:focus, #documenter .docs-sidebar form.docs-search > input.is-light:focus, .is-light.is-focused.textarea, .is-light.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-light.textarea:active, .is-light.input:active, #documenter .docs-sidebar form.docs-search > input.is-light:active, .is-light.is-active.textarea, .is-light.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(245, 245, 245, 0.25); } - .is-dark.textarea, .content kbd.textarea, .is-dark.input, #documenter .docs-sidebar form.docs-search > input.is-dark, .content kbd.input { - border-color: #363636; } - .is-dark.textarea:focus, .content kbd.textarea:focus, .is-dark.input:focus, #documenter .docs-sidebar form.docs-search > input.is-dark:focus, .content kbd.input:focus, .is-dark.is-focused.textarea, .content kbd.is-focused.textarea, .is-dark.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .content kbd.is-focused.input, #documenter .docs-sidebar .content form.docs-search > input.is-focused, .is-dark.textarea:active, .content kbd.textarea:active, .is-dark.input:active, #documenter .docs-sidebar form.docs-search > input.is-dark:active, .content kbd.input:active, .is-dark.is-active.textarea, .content kbd.is-active.textarea, .is-dark.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active, .content kbd.is-active.input, #documenter .docs-sidebar .content form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(54, 54, 54, 0.25); } - .is-primary.textarea, .docstring > section > a.textarea.docs-sourcelink, .is-primary.input, #documenter .docs-sidebar form.docs-search > input.is-primary, .docstring > section > a.input.docs-sourcelink { - border-color: #4eb5de; } - .is-primary.textarea:focus, .docstring > section > a.textarea.docs-sourcelink:focus, .is-primary.input:focus, #documenter .docs-sidebar form.docs-search > input.is-primary:focus, .docstring > section > a.input.docs-sourcelink:focus, .is-primary.is-focused.textarea, .docstring > section > a.is-focused.textarea.docs-sourcelink, .is-primary.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .docstring > section > a.is-focused.input.docs-sourcelink, .is-primary.textarea:active, .docstring > section > a.textarea.docs-sourcelink:active, .is-primary.input:active, #documenter .docs-sidebar form.docs-search > input.is-primary:active, .docstring > section > a.input.docs-sourcelink:active, .is-primary.is-active.textarea, .docstring > section > a.is-active.textarea.docs-sourcelink, .is-primary.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active, .docstring > section > a.is-active.input.docs-sourcelink { - box-shadow: 0 0 0 0.125em rgba(78, 181, 222, 0.25); } - .is-link.textarea, .is-link.input, #documenter .docs-sidebar form.docs-search > input.is-link { - border-color: #2e63b8; } - .is-link.textarea:focus, .is-link.input:focus, #documenter .docs-sidebar form.docs-search > input.is-link:focus, .is-link.is-focused.textarea, .is-link.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-link.textarea:active, .is-link.input:active, #documenter .docs-sidebar form.docs-search > input.is-link:active, .is-link.is-active.textarea, .is-link.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(46, 99, 184, 0.25); } - .is-info.textarea, .is-info.input, #documenter .docs-sidebar form.docs-search > input.is-info { - border-color: #209cee; } - .is-info.textarea:focus, .is-info.input:focus, #documenter .docs-sidebar form.docs-search > input.is-info:focus, .is-info.is-focused.textarea, .is-info.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-info.textarea:active, .is-info.input:active, #documenter .docs-sidebar form.docs-search > input.is-info:active, .is-info.is-active.textarea, .is-info.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(32, 156, 238, 0.25); } - .is-success.textarea, .is-success.input, #documenter .docs-sidebar form.docs-search > input.is-success { - border-color: #22c35b; } - .is-success.textarea:focus, .is-success.input:focus, #documenter .docs-sidebar form.docs-search > input.is-success:focus, .is-success.is-focused.textarea, .is-success.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-success.textarea:active, .is-success.input:active, #documenter .docs-sidebar form.docs-search > input.is-success:active, .is-success.is-active.textarea, .is-success.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(34, 195, 91, 0.25); } - .is-warning.textarea, .is-warning.input, #documenter .docs-sidebar form.docs-search > input.is-warning { - border-color: #ffdd57; } - .is-warning.textarea:focus, .is-warning.input:focus, #documenter .docs-sidebar form.docs-search > input.is-warning:focus, .is-warning.is-focused.textarea, .is-warning.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-warning.textarea:active, .is-warning.input:active, #documenter .docs-sidebar form.docs-search > input.is-warning:active, .is-warning.is-active.textarea, .is-warning.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(255, 221, 87, 0.25); } - .is-danger.textarea, .is-danger.input, #documenter .docs-sidebar form.docs-search > input.is-danger { - border-color: #da0b00; } - .is-danger.textarea:focus, .is-danger.input:focus, #documenter .docs-sidebar form.docs-search > input.is-danger:focus, .is-danger.is-focused.textarea, .is-danger.is-focused.input, #documenter .docs-sidebar form.docs-search > input.is-focused, .is-danger.textarea:active, .is-danger.input:active, #documenter .docs-sidebar form.docs-search > input.is-danger:active, .is-danger.is-active.textarea, .is-danger.is-active.input, #documenter .docs-sidebar form.docs-search > input.is-active { - box-shadow: 0 0 0 0.125em rgba(218, 11, 0, 0.25); } - .is-small.textarea, .is-small.input, #documenter .docs-sidebar form.docs-search > input { - border-radius: 2px; - font-size: 0.75rem; } - .is-medium.textarea, .is-medium.input, #documenter .docs-sidebar form.docs-search > input.is-medium { - font-size: 1.25rem; } - .is-large.textarea, .is-large.input, #documenter .docs-sidebar form.docs-search > input.is-large { - font-size: 1.5rem; } - .is-fullwidth.textarea, .is-fullwidth.input, #documenter .docs-sidebar form.docs-search > input.is-fullwidth { - display: block; - width: 100%; } - .is-inline.textarea, .is-inline.input, #documenter .docs-sidebar form.docs-search > input.is-inline { - display: inline; - width: auto; } - -.input.is-rounded, #documenter .docs-sidebar form.docs-search > input { - border-radius: 290486px; - padding-left: 1em; - padding-right: 1em; } - -.input.is-static, #documenter .docs-sidebar form.docs-search > input.is-static { - background-color: transparent; - border-color: transparent; - box-shadow: none; - padding-left: 0; - padding-right: 0; } - -.textarea { - display: block; - max-width: 100%; - min-width: 100%; - padding: 0.625em; - resize: vertical; } - .textarea:not([rows]) { - max-height: 600px; - min-height: 120px; } - .textarea[rows] { - height: initial; } - .textarea.has-fixed-size { - resize: none; } - -.radio, .checkbox { - cursor: pointer; - display: inline-block; - line-height: 1.25; - position: relative; } - .radio input, .checkbox input { - cursor: pointer; } - .radio:hover, .checkbox:hover { - color: #363636; } - .radio[disabled], .checkbox[disabled], fieldset[disabled] .radio, fieldset[disabled] .checkbox { - color: #6b6b6b; - cursor: not-allowed; } - -.radio + .radio { - margin-left: 0.5em; } - -.select { - display: inline-block; - max-width: 100%; - position: relative; - vertical-align: top; } - .select:not(.is-multiple) { - height: 2.25em; } - .select:not(.is-multiple):not(.is-loading)::after { - border-color: #2e63b8; - right: 1.125em; - z-index: 4; } - .select.is-rounded select, #documenter .docs-sidebar form.docs-search > input.select select { - border-radius: 290486px; - padding-left: 1em; } - .select select { - cursor: pointer; - display: block; - font-size: 1em; - max-width: 100%; - outline: none; } - .select select::-ms-expand { - display: none; } - .select select[disabled]:hover, fieldset[disabled] .select select:hover { - border-color: whitesmoke; } - .select select:not([multiple]) { - padding-right: 2.5em; } - .select select[multiple] { - height: auto; - padding: 0; } - .select select[multiple] option { - padding: 0.5em 1em; } - .select:not(.is-multiple):not(.is-loading):hover::after { - border-color: #363636; } - .select.is-white:not(:hover)::after { - border-color: white; } - .select.is-white select { - border-color: white; } - .select.is-white select:hover, .select.is-white select.is-hovered { - border-color: #f2f2f2; } - .select.is-white select:focus, .select.is-white select.is-focused, .select.is-white select:active, .select.is-white select.is-active { - box-shadow: 0 0 0 0.125em rgba(255, 255, 255, 0.25); } - .select.is-black:not(:hover)::after { - border-color: #0a0a0a; } - .select.is-black select { - border-color: #0a0a0a; } - .select.is-black select:hover, .select.is-black select.is-hovered { - border-color: black; } - .select.is-black select:focus, .select.is-black select.is-focused, .select.is-black select:active, .select.is-black select.is-active { - box-shadow: 0 0 0 0.125em rgba(10, 10, 10, 0.25); } - .select.is-light:not(:hover)::after { - border-color: whitesmoke; } - .select.is-light select { - border-color: whitesmoke; } - .select.is-light select:hover, .select.is-light select.is-hovered { - border-color: #e8e8e8; } - .select.is-light select:focus, .select.is-light select.is-focused, .select.is-light select:active, .select.is-light select.is-active { - box-shadow: 0 0 0 0.125em rgba(245, 245, 245, 0.25); } - .select.is-dark:not(:hover)::after, .content kbd.select:not(:hover)::after { - border-color: #363636; } - .select.is-dark select, .content kbd.select select { - border-color: #363636; } - .select.is-dark select:hover, .content kbd.select select:hover, .select.is-dark select.is-hovered, .content kbd.select select.is-hovered { - border-color: #292929; } - .select.is-dark select:focus, .content kbd.select select:focus, .select.is-dark select.is-focused, .content kbd.select select.is-focused, .select.is-dark select:active, .content kbd.select select:active, .select.is-dark select.is-active, .content kbd.select select.is-active { - box-shadow: 0 0 0 0.125em rgba(54, 54, 54, 0.25); } - .select.is-primary:not(:hover)::after, .docstring > section > a.select.docs-sourcelink:not(:hover)::after { - border-color: #4eb5de; } - .select.is-primary select, .docstring > section > a.select.docs-sourcelink select { - border-color: #4eb5de; } - .select.is-primary select:hover, .docstring > section > a.select.docs-sourcelink select:hover, .select.is-primary select.is-hovered, .docstring > section > a.select.docs-sourcelink select.is-hovered { - border-color: #39acda; } - .select.is-primary select:focus, .docstring > section > a.select.docs-sourcelink select:focus, .select.is-primary select.is-focused, .docstring > section > a.select.docs-sourcelink select.is-focused, .select.is-primary select:active, .docstring > section > a.select.docs-sourcelink select:active, .select.is-primary select.is-active, .docstring > section > a.select.docs-sourcelink select.is-active { - box-shadow: 0 0 0 0.125em rgba(78, 181, 222, 0.25); } - .select.is-link:not(:hover)::after { - border-color: #2e63b8; } - .select.is-link select { - border-color: #2e63b8; } - .select.is-link select:hover, .select.is-link select.is-hovered { - border-color: #2958a4; } - .select.is-link select:focus, .select.is-link select.is-focused, .select.is-link select:active, .select.is-link select.is-active { - box-shadow: 0 0 0 0.125em rgba(46, 99, 184, 0.25); } - .select.is-info:not(:hover)::after { - border-color: #209cee; } - .select.is-info select { - border-color: #209cee; } - .select.is-info select:hover, .select.is-info select.is-hovered { - border-color: #1190e3; } - .select.is-info select:focus, .select.is-info select.is-focused, .select.is-info select:active, .select.is-info select.is-active { - box-shadow: 0 0 0 0.125em rgba(32, 156, 238, 0.25); } - .select.is-success:not(:hover)::after { - border-color: #22c35b; } - .select.is-success select { - border-color: #22c35b; } - .select.is-success select:hover, .select.is-success select.is-hovered { - border-color: #1ead51; } - .select.is-success select:focus, .select.is-success select.is-focused, .select.is-success select:active, .select.is-success select.is-active { - box-shadow: 0 0 0 0.125em rgba(34, 195, 91, 0.25); } - .select.is-warning:not(:hover)::after { - border-color: #ffdd57; } - .select.is-warning select { - border-color: #ffdd57; } - .select.is-warning select:hover, .select.is-warning select.is-hovered { - border-color: #ffd83e; } - .select.is-warning select:focus, .select.is-warning select.is-focused, .select.is-warning select:active, .select.is-warning select.is-active { - box-shadow: 0 0 0 0.125em rgba(255, 221, 87, 0.25); } - .select.is-danger:not(:hover)::after { - border-color: #da0b00; } - .select.is-danger select { - border-color: #da0b00; } - .select.is-danger select:hover, .select.is-danger select.is-hovered { - border-color: #c10a00; } - .select.is-danger select:focus, .select.is-danger select.is-focused, .select.is-danger select:active, .select.is-danger select.is-active { - box-shadow: 0 0 0 0.125em rgba(218, 11, 0, 0.25); } - .select.is-small, #documenter .docs-sidebar form.docs-search > input.select { - border-radius: 2px; - font-size: 0.75rem; } - .select.is-medium { - font-size: 1.25rem; } - .select.is-large { - font-size: 1.5rem; } - .select.is-disabled::after { - border-color: #6b6b6b; } - .select.is-fullwidth { - width: 100%; } - .select.is-fullwidth select { - width: 100%; } - .select.is-loading::after { - margin-top: 0; - position: absolute; - right: 0.625em; - top: 0.625em; - transform: none; } - .select.is-loading.is-small:after, #documenter .docs-sidebar form.docs-search > input.is-loading:after { - font-size: 0.75rem; } - .select.is-loading.is-medium:after { - font-size: 1.25rem; } - .select.is-loading.is-large:after { - font-size: 1.5rem; } - -.file { - align-items: stretch; - display: flex; - justify-content: flex-start; - position: relative; } - .file.is-white .file-cta { - background-color: white; - border-color: transparent; - color: #0a0a0a; } - .file.is-white:hover .file-cta, .file.is-white.is-hovered .file-cta { - background-color: #f9f9f9; - border-color: transparent; - color: #0a0a0a; } - .file.is-white:focus .file-cta, .file.is-white.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(255, 255, 255, 0.25); - color: #0a0a0a; } - .file.is-white:active .file-cta, .file.is-white.is-active .file-cta { - background-color: #f2f2f2; - border-color: transparent; - color: #0a0a0a; } - .file.is-black .file-cta { - background-color: #0a0a0a; - border-color: transparent; - color: white; } - .file.is-black:hover .file-cta, .file.is-black.is-hovered .file-cta { - background-color: #040404; - border-color: transparent; - color: white; } - .file.is-black:focus .file-cta, .file.is-black.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(10, 10, 10, 0.25); - color: white; } - .file.is-black:active .file-cta, .file.is-black.is-active .file-cta { - background-color: black; - border-color: transparent; - color: white; } - .file.is-light .file-cta { - background-color: whitesmoke; - border-color: transparent; - color: #363636; } - .file.is-light:hover .file-cta, .file.is-light.is-hovered .file-cta { - background-color: #eeeeee; - border-color: transparent; - color: #363636; } - .file.is-light:focus .file-cta, .file.is-light.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(245, 245, 245, 0.25); - color: #363636; } - .file.is-light:active .file-cta, .file.is-light.is-active .file-cta { - background-color: #e8e8e8; - border-color: transparent; - color: #363636; } - .file.is-dark .file-cta, .content kbd.file .file-cta { - background-color: #363636; - border-color: transparent; - color: whitesmoke; } - .file.is-dark:hover .file-cta, .content kbd.file:hover .file-cta, .file.is-dark.is-hovered .file-cta, .content kbd.file.is-hovered .file-cta { - background-color: #2f2f2f; - border-color: transparent; - color: whitesmoke; } - .file.is-dark:focus .file-cta, .content kbd.file:focus .file-cta, .file.is-dark.is-focused .file-cta, .content kbd.file.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(54, 54, 54, 0.25); - color: whitesmoke; } - .file.is-dark:active .file-cta, .content kbd.file:active .file-cta, .file.is-dark.is-active .file-cta, .content kbd.file.is-active .file-cta { - background-color: #292929; - border-color: transparent; - color: whitesmoke; } - .file.is-primary .file-cta, .docstring > section > a.file.docs-sourcelink .file-cta { - background-color: #4eb5de; - border-color: transparent; - color: #fff; } - .file.is-primary:hover .file-cta, .docstring > section > a.file.docs-sourcelink:hover .file-cta, .file.is-primary.is-hovered .file-cta, .docstring > section > a.file.is-hovered.docs-sourcelink .file-cta { - background-color: #43b1dc; - border-color: transparent; - color: #fff; } - .file.is-primary:focus .file-cta, .docstring > section > a.file.docs-sourcelink:focus .file-cta, .file.is-primary.is-focused .file-cta, .docstring > section > a.file.is-focused.docs-sourcelink .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(78, 181, 222, 0.25); - color: #fff; } - .file.is-primary:active .file-cta, .docstring > section > a.file.docs-sourcelink:active .file-cta, .file.is-primary.is-active .file-cta, .docstring > section > a.file.is-active.docs-sourcelink .file-cta { - background-color: #39acda; - border-color: transparent; - color: #fff; } - .file.is-link .file-cta { - background-color: #2e63b8; - border-color: transparent; - color: #fff; } - .file.is-link:hover .file-cta, .file.is-link.is-hovered .file-cta { - background-color: #2b5eae; - border-color: transparent; - color: #fff; } - .file.is-link:focus .file-cta, .file.is-link.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(46, 99, 184, 0.25); - color: #fff; } - .file.is-link:active .file-cta, .file.is-link.is-active .file-cta { - background-color: #2958a4; - border-color: transparent; - color: #fff; } - .file.is-info .file-cta { - background-color: #209cee; - border-color: transparent; - color: #fff; } - .file.is-info:hover .file-cta, .file.is-info.is-hovered .file-cta { - background-color: #1497ed; - border-color: transparent; - color: #fff; } - .file.is-info:focus .file-cta, .file.is-info.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(32, 156, 238, 0.25); - color: #fff; } - .file.is-info:active .file-cta, .file.is-info.is-active .file-cta { - background-color: #1190e3; - border-color: transparent; - color: #fff; } - .file.is-success .file-cta { - background-color: #22c35b; - border-color: transparent; - color: #fff; } - .file.is-success:hover .file-cta, .file.is-success.is-hovered .file-cta { - background-color: #20b856; - border-color: transparent; - color: #fff; } - .file.is-success:focus .file-cta, .file.is-success.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(34, 195, 91, 0.25); - color: #fff; } - .file.is-success:active .file-cta, .file.is-success.is-active .file-cta { - background-color: #1ead51; - border-color: transparent; - color: #fff; } - .file.is-warning .file-cta { - background-color: #ffdd57; - border-color: transparent; - color: rgba(0, 0, 0, 0.7); } - .file.is-warning:hover .file-cta, .file.is-warning.is-hovered .file-cta { - background-color: #ffda4a; - border-color: transparent; - color: rgba(0, 0, 0, 0.7); } - .file.is-warning:focus .file-cta, .file.is-warning.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(255, 221, 87, 0.25); - color: rgba(0, 0, 0, 0.7); } - .file.is-warning:active .file-cta, .file.is-warning.is-active .file-cta { - background-color: #ffd83e; - border-color: transparent; - color: rgba(0, 0, 0, 0.7); } - .file.is-danger .file-cta { - background-color: #da0b00; - border-color: transparent; - color: #fff; } - .file.is-danger:hover .file-cta, .file.is-danger.is-hovered .file-cta { - background-color: #cd0a00; - border-color: transparent; - color: #fff; } - .file.is-danger:focus .file-cta, .file.is-danger.is-focused .file-cta { - border-color: transparent; - box-shadow: 0 0 0.5em rgba(218, 11, 0, 0.25); - color: #fff; } - .file.is-danger:active .file-cta, .file.is-danger.is-active .file-cta { - background-color: #c10a00; - border-color: transparent; - color: #fff; } - .file.is-small, #documenter .docs-sidebar form.docs-search > input.file { - font-size: 0.75rem; } - .file.is-medium { - font-size: 1.25rem; } - .file.is-medium .file-icon .fa { - font-size: 21px; } - .file.is-large { - font-size: 1.5rem; } - .file.is-large .file-icon .fa { - font-size: 28px; } - .file.has-name .file-cta { - border-bottom-right-radius: 0; - border-top-right-radius: 0; } - .file.has-name .file-name { - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - .file.has-name.is-empty .file-cta { - border-radius: 4px; } - .file.has-name.is-empty .file-name { - display: none; } - .file.is-boxed .file-label { - flex-direction: column; } - .file.is-boxed .file-cta { - flex-direction: column; - height: auto; - padding: 1em 3em; } - .file.is-boxed .file-name { - border-width: 0 1px 1px; } - .file.is-boxed .file-icon { - height: 1.5em; - width: 1.5em; } - .file.is-boxed .file-icon .fa { - font-size: 21px; } - .file.is-boxed.is-small .file-icon .fa, #documenter .docs-sidebar form.docs-search > input.is-boxed .file-icon .fa { - font-size: 14px; } - .file.is-boxed.is-medium .file-icon .fa { - font-size: 28px; } - .file.is-boxed.is-large .file-icon .fa { - font-size: 35px; } - .file.is-boxed.has-name .file-cta { - border-radius: 4px 4px 0 0; } - .file.is-boxed.has-name .file-name { - border-radius: 0 0 4px 4px; - border-width: 0 1px 1px; } - .file.is-centered { - justify-content: center; } - .file.is-fullwidth .file-label { - width: 100%; } - .file.is-fullwidth .file-name { - flex-grow: 1; - max-width: none; } - .file.is-right { - justify-content: flex-end; } - .file.is-right .file-cta { - border-radius: 0 4px 4px 0; } - .file.is-right .file-name { - border-radius: 4px 0 0 4px; - border-width: 1px 0 1px 1px; - order: -1; } - -.file-label { - align-items: stretch; - display: flex; - cursor: pointer; - justify-content: flex-start; - overflow: hidden; - position: relative; } - .file-label:hover .file-cta { - background-color: #eeeeee; - color: #363636; } - .file-label:hover .file-name { - border-color: #d5d5d5; } - .file-label:active .file-cta { - background-color: #e8e8e8; - color: #363636; } - .file-label:active .file-name { - border-color: #cfcfcf; } - -.file-input { - height: 100%; - left: 0; - opacity: 0; - outline: none; - position: absolute; - top: 0; - width: 100%; } - -.file-cta, -.file-name { - border-color: #dbdbdb; - border-radius: 4px; - font-size: 1em; - padding-left: 1em; - padding-right: 1em; - white-space: nowrap; } - -.file-cta { - background-color: whitesmoke; - color: #4a4a4a; } - -.file-name { - border-color: #dbdbdb; - border-style: solid; - border-width: 1px 1px 1px 0; - display: block; - max-width: 16em; - overflow: hidden; - text-align: left; - text-overflow: ellipsis; } - -.file-icon { - align-items: center; - display: flex; - height: 1em; - justify-content: center; - margin-right: 0.5em; - width: 1em; } - .file-icon .fa { - font-size: 14px; } - -.label { - color: #363636; - display: block; - font-size: 1rem; - font-weight: 700; } - .label:not(:last-child) { - margin-bottom: 0.5em; } - .label.is-small, #documenter .docs-sidebar form.docs-search > input.label { - font-size: 0.75rem; } - .label.is-medium { - font-size: 1.25rem; } - .label.is-large { - font-size: 1.5rem; } - -.help { - display: block; - font-size: 0.75rem; - margin-top: 0.25rem; } - .help.is-white { - color: white; } - .help.is-black { - color: #0a0a0a; } - .help.is-light { - color: whitesmoke; } - .help.is-dark, .content kbd.help { - color: #363636; } - .help.is-primary, .docstring > section > a.help.docs-sourcelink { - color: #4eb5de; } - .help.is-link { - color: #2e63b8; } - .help.is-info { - color: #209cee; } - .help.is-success { - color: #22c35b; } - .help.is-warning { - color: #ffdd57; } - .help.is-danger { - color: #da0b00; } - -.field:not(:last-child) { - margin-bottom: 0.75rem; } - -.field.has-addons { - display: flex; - justify-content: flex-start; } - .field.has-addons .control:not(:last-child) { - margin-right: -1px; } - .field.has-addons .control:not(:first-child):not(:last-child) .button, - .field.has-addons .control:not(:first-child):not(:last-child) .input, - .field.has-addons .control:not(:first-child):not(:last-child) #documenter .docs-sidebar form.docs-search > input, - #documenter .docs-sidebar .field.has-addons .control:not(:first-child):not(:last-child) form.docs-search > input, - .field.has-addons .control:not(:first-child):not(:last-child) .select select { - border-radius: 0; } - .field.has-addons .control:first-child:not(:only-child) .button, - .field.has-addons .control:first-child:not(:only-child) .input, - .field.has-addons .control:first-child:not(:only-child) #documenter .docs-sidebar form.docs-search > input, - #documenter .docs-sidebar .field.has-addons .control:first-child:not(:only-child) form.docs-search > input, - .field.has-addons .control:first-child:not(:only-child) .select select { - border-bottom-right-radius: 0; - border-top-right-radius: 0; } - .field.has-addons .control:last-child:not(:only-child) .button, - .field.has-addons .control:last-child:not(:only-child) .input, - .field.has-addons .control:last-child:not(:only-child) #documenter .docs-sidebar form.docs-search > input, - #documenter .docs-sidebar .field.has-addons .control:last-child:not(:only-child) form.docs-search > input, - .field.has-addons .control:last-child:not(:only-child) .select select { - border-bottom-left-radius: 0; - border-top-left-radius: 0; } - .field.has-addons .control .button:not([disabled]):hover, .field.has-addons .control .button.is-hovered:not([disabled]), - .field.has-addons .control .input:not([disabled]):hover, - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):hover, - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):hover, - .field.has-addons .control .input.is-hovered:not([disabled]), - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-hovered:not([disabled]), - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-hovered:not([disabled]), - .field.has-addons .control .select select:not([disabled]):hover, - .field.has-addons .control .select select.is-hovered:not([disabled]) { - z-index: 2; } - .field.has-addons .control .button:not([disabled]):focus, .field.has-addons .control .button.is-focused:not([disabled]), .field.has-addons .control .button:not([disabled]):active, .field.has-addons .control .button.is-active:not([disabled]), - .field.has-addons .control .input:not([disabled]):focus, - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):focus, - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):focus, - .field.has-addons .control .input.is-focused:not([disabled]), - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-focused:not([disabled]), - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-focused:not([disabled]), - .field.has-addons .control .input:not([disabled]):active, - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):active, - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):active, - .field.has-addons .control .input.is-active:not([disabled]), - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-active:not([disabled]), - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-active:not([disabled]), - .field.has-addons .control .select select:not([disabled]):focus, - .field.has-addons .control .select select.is-focused:not([disabled]), - .field.has-addons .control .select select:not([disabled]):active, - .field.has-addons .control .select select.is-active:not([disabled]) { - z-index: 3; } - .field.has-addons .control .button:not([disabled]):focus:hover, .field.has-addons .control .button.is-focused:not([disabled]):hover, .field.has-addons .control .button:not([disabled]):active:hover, .field.has-addons .control .button.is-active:not([disabled]):hover, - .field.has-addons .control .input:not([disabled]):focus:hover, - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):focus:hover, - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):focus:hover, - .field.has-addons .control .input.is-focused:not([disabled]):hover, - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-focused:not([disabled]):hover, - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-focused:not([disabled]):hover, - .field.has-addons .control .input:not([disabled]):active:hover, - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input:not([disabled]):active:hover, - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input:not([disabled]):active:hover, - .field.has-addons .control .input.is-active:not([disabled]):hover, - .field.has-addons .control #documenter .docs-sidebar form.docs-search > input.is-active:not([disabled]):hover, - #documenter .docs-sidebar .field.has-addons .control form.docs-search > input.is-active:not([disabled]):hover, - .field.has-addons .control .select select:not([disabled]):focus:hover, - .field.has-addons .control .select select.is-focused:not([disabled]):hover, - .field.has-addons .control .select select:not([disabled]):active:hover, - .field.has-addons .control .select select.is-active:not([disabled]):hover { - z-index: 4; } - .field.has-addons .control.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - .field.has-addons.has-addons-centered { - justify-content: center; } - .field.has-addons.has-addons-right { - justify-content: flex-end; } - .field.has-addons.has-addons-fullwidth .control { - flex-grow: 1; - flex-shrink: 0; } - -.field.is-grouped { - display: flex; - justify-content: flex-start; } - .field.is-grouped > .control { - flex-shrink: 0; } - .field.is-grouped > .control:not(:last-child) { - margin-bottom: 0; - margin-right: 0.75rem; } - .field.is-grouped > .control.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - .field.is-grouped.is-grouped-centered { - justify-content: center; } - .field.is-grouped.is-grouped-right { - justify-content: flex-end; } - .field.is-grouped.is-grouped-multiline { - flex-wrap: wrap; } - .field.is-grouped.is-grouped-multiline > .control:last-child, .field.is-grouped.is-grouped-multiline > .control:not(:last-child) { - margin-bottom: 0.75rem; } - .field.is-grouped.is-grouped-multiline:last-child { - margin-bottom: -0.75rem; } - .field.is-grouped.is-grouped-multiline:not(:last-child) { - margin-bottom: 0; } - -@media screen and (min-width: 769px), print { - .field.is-horizontal { - display: flex; } } - -.field-label .label { - font-size: inherit; } - -@media screen and (max-width: 768px) { - .field-label { - margin-bottom: 0.5rem; } } - -@media screen and (min-width: 769px), print { - .field-label { - flex-basis: 0; - flex-grow: 1; - flex-shrink: 0; - margin-right: 1.5rem; - text-align: right; } - .field-label.is-small, #documenter .docs-sidebar form.docs-search > input.field-label { - font-size: 0.75rem; - padding-top: 0.375em; } - .field-label.is-normal { - padding-top: 0.375em; } - .field-label.is-medium { - font-size: 1.25rem; - padding-top: 0.375em; } - .field-label.is-large { - font-size: 1.5rem; - padding-top: 0.375em; } } - -.field-body .field .field { - margin-bottom: 0; } - -@media screen and (min-width: 769px), print { - .field-body { - display: flex; - flex-basis: 0; - flex-grow: 5; - flex-shrink: 1; } - .field-body .field { - margin-bottom: 0; } - .field-body > .field { - flex-shrink: 1; } - .field-body > .field:not(.is-narrow) { - flex-grow: 1; } - .field-body > .field:not(:last-child) { - margin-right: 0.75rem; } } - -.control { - box-sizing: border-box; - clear: both; - font-size: 1rem; - position: relative; - text-align: left; } - .control.has-icons-left .input:focus ~ .icon, .control.has-icons-left #documenter .docs-sidebar form.docs-search > input:focus ~ .icon, #documenter .docs-sidebar .control.has-icons-left form.docs-search > input:focus ~ .icon, - .control.has-icons-left .select:focus ~ .icon, .control.has-icons-right .input:focus ~ .icon, .control.has-icons-right #documenter .docs-sidebar form.docs-search > input:focus ~ .icon, #documenter .docs-sidebar .control.has-icons-right form.docs-search > input:focus ~ .icon, - .control.has-icons-right .select:focus ~ .icon { - color: #6b6b6b; } - .control.has-icons-left .input.is-small ~ .icon, .control.has-icons-left #documenter .docs-sidebar form.docs-search > input ~ .icon, #documenter .docs-sidebar .control.has-icons-left form.docs-search > input ~ .icon, - .control.has-icons-left .select.is-small ~ .icon, .control.has-icons-right .input.is-small ~ .icon, .control.has-icons-right #documenter .docs-sidebar form.docs-search > input ~ .icon, #documenter .docs-sidebar .control.has-icons-right form.docs-search > input ~ .icon, - .control.has-icons-right .select.is-small ~ .icon { - font-size: 0.75rem; } - .control.has-icons-left .input.is-medium ~ .icon, .control.has-icons-left #documenter .docs-sidebar form.docs-search > input.is-medium ~ .icon, #documenter .docs-sidebar .control.has-icons-left form.docs-search > input.is-medium ~ .icon, - .control.has-icons-left .select.is-medium ~ .icon, .control.has-icons-right .input.is-medium ~ .icon, .control.has-icons-right #documenter .docs-sidebar form.docs-search > input.is-medium ~ .icon, #documenter .docs-sidebar .control.has-icons-right form.docs-search > input.is-medium ~ .icon, - .control.has-icons-right .select.is-medium ~ .icon { - font-size: 1.25rem; } - .control.has-icons-left .input.is-large ~ .icon, .control.has-icons-left #documenter .docs-sidebar form.docs-search > input.is-large ~ .icon, #documenter .docs-sidebar .control.has-icons-left form.docs-search > input.is-large ~ .icon, - .control.has-icons-left .select.is-large ~ .icon, .control.has-icons-right .input.is-large ~ .icon, .control.has-icons-right #documenter .docs-sidebar form.docs-search > input.is-large ~ .icon, #documenter .docs-sidebar .control.has-icons-right form.docs-search > input.is-large ~ .icon, - .control.has-icons-right .select.is-large ~ .icon { - font-size: 1.5rem; } - .control.has-icons-left .icon, .control.has-icons-right .icon { - color: #dbdbdb; - height: 2.25em; - pointer-events: none; - position: absolute; - top: 0; - width: 2.25em; - z-index: 4; } - .control.has-icons-left .input, .control.has-icons-left #documenter .docs-sidebar form.docs-search > input, #documenter .docs-sidebar .control.has-icons-left form.docs-search > input, - .control.has-icons-left .select select { - padding-left: 2.25em; } - .control.has-icons-left .icon.is-left { - left: 0; } - .control.has-icons-right .input, .control.has-icons-right #documenter .docs-sidebar form.docs-search > input, #documenter .docs-sidebar .control.has-icons-right form.docs-search > input, - .control.has-icons-right .select select { - padding-right: 2.25em; } - .control.has-icons-right .icon.is-right { - right: 0; } - .control.is-loading::after { - position: absolute !important; - right: 0.625em; - top: 0.625em; - z-index: 4; } - .control.is-loading.is-small:after, #documenter .docs-sidebar form.docs-search > input.is-loading:after { - font-size: 0.75rem; } - .control.is-loading.is-medium:after { - font-size: 1.25rem; } - .control.is-loading.is-large:after { - font-size: 1.5rem; } - -.breadcrumb { - font-size: 1rem; - white-space: nowrap; } - .breadcrumb a { - align-items: center; - color: #2e63b8; - display: flex; - justify-content: center; - padding: 0 0.75em; } - .breadcrumb a:hover { - color: #363636; } - .breadcrumb li { - align-items: center; - display: flex; } - .breadcrumb li:first-child a { - padding-left: 0; } - .breadcrumb li.is-active a { - color: #222222; - cursor: default; - pointer-events: none; } - .breadcrumb li + li::before { - color: #b5b5b5; - content: "\0002f"; } - .breadcrumb ul, - .breadcrumb ol { - align-items: flex-start; - display: flex; - flex-wrap: wrap; - justify-content: flex-start; } - .breadcrumb .icon:first-child { - margin-right: 0.5em; } - .breadcrumb .icon:last-child { - margin-left: 0.5em; } - .breadcrumb.is-centered ol, - .breadcrumb.is-centered ul { - justify-content: center; } - .breadcrumb.is-right ol, - .breadcrumb.is-right ul { - justify-content: flex-end; } - .breadcrumb.is-small, #documenter .docs-sidebar form.docs-search > input.breadcrumb { - font-size: 0.75rem; } - .breadcrumb.is-medium { - font-size: 1.25rem; } - .breadcrumb.is-large { - font-size: 1.5rem; } - .breadcrumb.has-arrow-separator li + li::before { - content: "\02192"; } - .breadcrumb.has-bullet-separator li + li::before { - content: "\02022"; } - .breadcrumb.has-dot-separator li + li::before { - content: "\000b7"; } - .breadcrumb.has-succeeds-separator li + li::before { - content: "\0227B"; } - -.card { - background-color: white; - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); - color: #222222; - max-width: 100%; - position: relative; } - -.card-header { - background-color: transparent; - align-items: stretch; - box-shadow: 0 1px 2px rgba(10, 10, 10, 0.1); - display: flex; } - -.card-header-title { - align-items: center; - color: #222222; - display: flex; - flex-grow: 1; - font-weight: 700; - padding: 0.75rem; } - .card-header-title.is-centered { - justify-content: center; } - -.card-header-icon { - align-items: center; - cursor: pointer; - display: flex; - justify-content: center; - padding: 0.75rem; } - -.card-image { - display: block; - position: relative; } - -.card-content { - background-color: transparent; - padding: 1.5rem; } - -.card-footer { - background-color: transparent; - border-top: 1px solid #dbdbdb; - align-items: stretch; - display: flex; } - -.card-footer-item { - align-items: center; - display: flex; - flex-basis: 0; - flex-grow: 1; - flex-shrink: 0; - justify-content: center; - padding: 0.75rem; } - .card-footer-item:not(:last-child) { - border-right: 1px solid #dbdbdb; } - -.card .media:not(:last-child) { - margin-bottom: 1.5rem; } - -.dropdown { - display: inline-flex; - position: relative; - vertical-align: top; } - .dropdown.is-active .dropdown-menu, .dropdown.is-hoverable:hover .dropdown-menu { - display: block; } - .dropdown.is-right .dropdown-menu { - left: auto; - right: 0; } - .dropdown.is-up .dropdown-menu { - bottom: 100%; - padding-bottom: 4px; - padding-top: initial; - top: auto; } - -.dropdown-menu { - display: none; - left: 0; - min-width: 12rem; - padding-top: 4px; - position: absolute; - top: 100%; - z-index: 20; } - -.dropdown-content { - background-color: white; - border-radius: 4px; - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); - padding-bottom: 0.5rem; - padding-top: 0.5rem; } - -.dropdown-item { - color: #4a4a4a; - display: block; - font-size: 0.875rem; - line-height: 1.5; - padding: 0.375rem 1rem; - position: relative; } - -a.dropdown-item, -button.dropdown-item { - padding-right: 3rem; - text-align: left; - white-space: nowrap; - width: 100%; } - a.dropdown-item:hover, - button.dropdown-item:hover { - background-color: whitesmoke; - color: #0a0a0a; } - a.dropdown-item.is-active, - button.dropdown-item.is-active { - background-color: #2e63b8; - color: #fff; } - -.dropdown-divider { - background-color: #dbdbdb; - border: none; - display: block; - height: 1px; - margin: 0.5rem 0; } - -.level { - align-items: center; - justify-content: space-between; } - .level code { - border-radius: 4px; } - .level img { - display: inline-block; - vertical-align: top; } - .level.is-mobile { - display: flex; } - .level.is-mobile .level-left, - .level.is-mobile .level-right { - display: flex; } - .level.is-mobile .level-left + .level-right { - margin-top: 0; } - .level.is-mobile .level-item:not(:last-child) { - margin-bottom: 0; - margin-right: 0.75rem; } - .level.is-mobile .level-item:not(.is-narrow) { - flex-grow: 1; } - @media screen and (min-width: 769px), print { - .level { - display: flex; } - .level > .level-item:not(.is-narrow) { - flex-grow: 1; } } -.level-item { - align-items: center; - display: flex; - flex-basis: auto; - flex-grow: 0; - flex-shrink: 0; - justify-content: center; } - .level-item .title, - .level-item .subtitle { - margin-bottom: 0; } - @media screen and (max-width: 768px) { - .level-item:not(:last-child) { - margin-bottom: 0.75rem; } } -.level-left, -.level-right { - flex-basis: auto; - flex-grow: 0; - flex-shrink: 0; } - .level-left .level-item.is-flexible, - .level-right .level-item.is-flexible { - flex-grow: 1; } - @media screen and (min-width: 769px), print { - .level-left .level-item:not(:last-child), - .level-right .level-item:not(:last-child) { - margin-right: 0.75rem; } } -.level-left { - align-items: center; - justify-content: flex-start; } - @media screen and (max-width: 768px) { - .level-left + .level-right { - margin-top: 1.5rem; } } - @media screen and (min-width: 769px), print { - .level-left { - display: flex; } } -.level-right { - align-items: center; - justify-content: flex-end; } - @media screen and (min-width: 769px), print { - .level-right { - display: flex; } } -.list { - background-color: white; - border-radius: 4px; - box-shadow: 0 2px 3px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); } - -.list-item { - display: block; - padding: 0.5em 1em; } - .list-item:not(a) { - color: #222222; } - .list-item:first-child { - border-top-left-radius: 4px; - border-top-right-radius: 4px; } - .list-item:last-child { - border-bottom-left-radius: 4px; - border-bottom-right-radius: 4px; } - .list-item:not(:last-child) { - border-bottom: 1px solid #dbdbdb; } - .list-item.is-active { - background-color: #2e63b8; - color: #fff; } - -a.list-item { - background-color: whitesmoke; - cursor: pointer; } - -.media { - align-items: flex-start; - display: flex; - text-align: left; } - .media .content:not(:last-child) { - margin-bottom: 0.75rem; } - .media .media { - border-top: 1px solid rgba(219, 219, 219, 0.5); - display: flex; - padding-top: 0.75rem; } - .media .media .content:not(:last-child), - .media .media .control:not(:last-child) { - margin-bottom: 0.5rem; } - .media .media .media { - padding-top: 0.5rem; } - .media .media .media + .media { - margin-top: 0.5rem; } - .media + .media { - border-top: 1px solid rgba(219, 219, 219, 0.5); - margin-top: 1rem; - padding-top: 1rem; } - .media.is-large + .media { - margin-top: 1.5rem; - padding-top: 1.5rem; } - -.media-left, -.media-right { - flex-basis: auto; - flex-grow: 0; - flex-shrink: 0; } - -.media-left { - margin-right: 1rem; } - -.media-right { - margin-left: 1rem; } - -.media-content { - flex-basis: auto; - flex-grow: 1; - flex-shrink: 1; - text-align: left; } - -@media screen and (max-width: 768px) { - .media-content { - overflow-x: auto; } } - -.menu { - font-size: 1rem; } - .menu.is-small, #documenter .docs-sidebar form.docs-search > input.menu { - font-size: 0.75rem; } - .menu.is-medium { - font-size: 1.25rem; } - .menu.is-large { - font-size: 1.5rem; } - -.menu-list { - line-height: 1.25; } - .menu-list a { - border-radius: 2px; - color: #222222; - display: block; - padding: 0.5em 0.75em; } - .menu-list a:hover { - background-color: whitesmoke; - color: #222222; } - .menu-list a.is-active { - background-color: #2e63b8; - color: #fff; } - .menu-list li ul { - border-left: 1px solid #dbdbdb; - margin: 0.75em; - padding-left: 0.75em; } - -.menu-label { - color: #6b6b6b; - font-size: 0.75em; - letter-spacing: 0.1em; - text-transform: uppercase; } - .menu-label:not(:first-child) { - margin-top: 1em; } - .menu-label:not(:last-child) { - margin-bottom: 1em; } - -.message { - background-color: whitesmoke; - border-radius: 4px; - font-size: 1rem; } - .message strong { - color: currentColor; } - .message a:not(.button):not(.tag):not(.dropdown-item) { - color: currentColor; - text-decoration: underline; } - .message.is-small, #documenter .docs-sidebar form.docs-search > input.message { - font-size: 0.75rem; } - .message.is-medium { - font-size: 1.25rem; } - .message.is-large { - font-size: 1.5rem; } - .message.is-white { - background-color: white; } - .message.is-white .message-header { - background-color: white; - color: #0a0a0a; } - .message.is-white .message-body { - border-color: white; - color: #4d4d4d; } - .message.is-black { - background-color: #fafafa; } - .message.is-black .message-header { - background-color: #0a0a0a; - color: white; } - .message.is-black .message-body { - border-color: #0a0a0a; - color: #090909; } - .message.is-light { - background-color: #fafafa; } - .message.is-light .message-header { - background-color: whitesmoke; - color: #363636; } - .message.is-light .message-body { - border-color: whitesmoke; - color: #505050; } - .message.is-dark, .content kbd.message { - background-color: #fafafa; } - .message.is-dark .message-header, .content kbd.message .message-header { - background-color: #363636; - color: whitesmoke; } - .message.is-dark .message-body, .content kbd.message .message-body { - border-color: #363636; - color: #2a2a2a; } - .message.is-primary, .docstring > section > a.message.docs-sourcelink { - background-color: #f6fbfd; } - .message.is-primary .message-header, .docstring > section > a.message.docs-sourcelink .message-header { - background-color: #4eb5de; - color: #fff; } - .message.is-primary .message-body, .docstring > section > a.message.docs-sourcelink .message-body { - border-color: #4eb5de; - color: #1f556a; } - .message.is-link { - background-color: #f7f9fd; } - .message.is-link .message-header { - background-color: #2e63b8; - color: #fff; } - .message.is-link .message-body { - border-color: #2e63b8; - color: #264981; } - .message.is-info { - background-color: #f6fbfe; } - .message.is-info .message-header { - background-color: #209cee; - color: #fff; } - .message.is-info .message-body { - border-color: #209cee; - color: #12537d; } - .message.is-success { - background-color: #f6fdf9; } - .message.is-success .message-header { - background-color: #22c35b; - color: #fff; } - .message.is-success .message-body { - border-color: #22c35b; - color: #0f361d; } - .message.is-warning { - background-color: #fffdf5; } - .message.is-warning .message-header { - background-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .message.is-warning .message-body { - border-color: #ffdd57; - color: #3c3108; } - .message.is-danger { - background-color: #fff5f5; } - .message.is-danger .message-header { - background-color: #da0b00; - color: #fff; } - .message.is-danger .message-body { - border-color: #da0b00; - color: #9b0c04; } - -.message-header { - align-items: center; - background-color: #222222; - border-radius: 4px 4px 0 0; - color: #fff; - display: flex; - font-weight: 700; - justify-content: space-between; - line-height: 1.25; - padding: 0.75em 1em; - position: relative; } - .message-header .delete { - flex-grow: 0; - flex-shrink: 0; - margin-left: 0.75em; } - .message-header + .message-body { - border-width: 0; - border-top-left-radius: 0; - border-top-right-radius: 0; } - -.message-body { - border-color: #dbdbdb; - border-radius: 4px; - border-style: solid; - border-width: 0 0 0 4px; - color: #222222; - padding: 1.25em 1.5em; } - .message-body code, - .message-body pre { - background-color: white; } - .message-body pre code { - background-color: transparent; } - -.modal { - align-items: center; - display: none; - flex-direction: column; - justify-content: center; - overflow: hidden; - position: fixed; - z-index: 40; } - .modal.is-active { - display: flex; } - -.modal-background { - background-color: rgba(10, 10, 10, 0.86); } - -.modal-content, -.modal-card { - margin: 0 20px; - max-height: calc(100vh - 160px); - overflow: auto; - position: relative; - width: 100%; } - @media screen and (min-width: 769px), print { - .modal-content, - .modal-card { - margin: 0 auto; - max-height: calc(100vh - 40px); - width: 640px; } } -.modal-close { - background: none; - height: 40px; - position: fixed; - right: 20px; - top: 20px; - width: 40px; } - -.modal-card { - display: flex; - flex-direction: column; - max-height: calc(100vh - 40px); - overflow: hidden; - -ms-overflow-y: visible; } - -.modal-card-head, -.modal-card-foot { - align-items: center; - background-color: whitesmoke; - display: flex; - flex-shrink: 0; - justify-content: flex-start; - padding: 20px; - position: relative; } - -.modal-card-head { - border-bottom: 1px solid #dbdbdb; - border-top-left-radius: 6px; - border-top-right-radius: 6px; } - -.modal-card-title { - color: #222222; - flex-grow: 1; - flex-shrink: 0; - font-size: 1.5rem; - line-height: 1; } - -.modal-card-foot { - border-bottom-left-radius: 6px; - border-bottom-right-radius: 6px; - border-top: 1px solid #dbdbdb; } - .modal-card-foot .button:not(:last-child) { - margin-right: 0.5em; } - -.modal-card-body { - -webkit-overflow-scrolling: touch; - background-color: white; - flex-grow: 1; - flex-shrink: 1; - overflow: auto; - padding: 20px; } - -.navbar { - background-color: white; - min-height: 3.25rem; - position: relative; - z-index: 30; } - .navbar.is-white { - background-color: white; - color: #0a0a0a; } - .navbar.is-white .navbar-brand > .navbar-item, - .navbar.is-white .navbar-brand .navbar-link { - color: #0a0a0a; } - .navbar.is-white .navbar-brand > a.navbar-item:focus, .navbar.is-white .navbar-brand > a.navbar-item:hover, .navbar.is-white .navbar-brand > a.navbar-item.is-active, - .navbar.is-white .navbar-brand .navbar-link:focus, - .navbar.is-white .navbar-brand .navbar-link:hover, - .navbar.is-white .navbar-brand .navbar-link.is-active { - background-color: #f2f2f2; - color: #0a0a0a; } - .navbar.is-white .navbar-brand .navbar-link::after { - border-color: #0a0a0a; } - .navbar.is-white .navbar-burger { - color: #0a0a0a; } - @media screen and (min-width: 1056px) { - .navbar.is-white .navbar-start > .navbar-item, - .navbar.is-white .navbar-start .navbar-link, - .navbar.is-white .navbar-end > .navbar-item, - .navbar.is-white .navbar-end .navbar-link { - color: #0a0a0a; } - .navbar.is-white .navbar-start > a.navbar-item:focus, .navbar.is-white .navbar-start > a.navbar-item:hover, .navbar.is-white .navbar-start > a.navbar-item.is-active, - .navbar.is-white .navbar-start .navbar-link:focus, - .navbar.is-white .navbar-start .navbar-link:hover, - .navbar.is-white .navbar-start .navbar-link.is-active, - .navbar.is-white .navbar-end > a.navbar-item:focus, - .navbar.is-white .navbar-end > a.navbar-item:hover, - .navbar.is-white .navbar-end > a.navbar-item.is-active, - .navbar.is-white .navbar-end .navbar-link:focus, - .navbar.is-white .navbar-end .navbar-link:hover, - .navbar.is-white .navbar-end .navbar-link.is-active { - background-color: #f2f2f2; - color: #0a0a0a; } - .navbar.is-white .navbar-start .navbar-link::after, - .navbar.is-white .navbar-end .navbar-link::after { - border-color: #0a0a0a; } - .navbar.is-white .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-white .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-white .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #f2f2f2; - color: #0a0a0a; } - .navbar.is-white .navbar-dropdown a.navbar-item.is-active { - background-color: white; - color: #0a0a0a; } } - .navbar.is-black { - background-color: #0a0a0a; - color: white; } - .navbar.is-black .navbar-brand > .navbar-item, - .navbar.is-black .navbar-brand .navbar-link { - color: white; } - .navbar.is-black .navbar-brand > a.navbar-item:focus, .navbar.is-black .navbar-brand > a.navbar-item:hover, .navbar.is-black .navbar-brand > a.navbar-item.is-active, - .navbar.is-black .navbar-brand .navbar-link:focus, - .navbar.is-black .navbar-brand .navbar-link:hover, - .navbar.is-black .navbar-brand .navbar-link.is-active { - background-color: black; - color: white; } - .navbar.is-black .navbar-brand .navbar-link::after { - border-color: white; } - .navbar.is-black .navbar-burger { - color: white; } - @media screen and (min-width: 1056px) { - .navbar.is-black .navbar-start > .navbar-item, - .navbar.is-black .navbar-start .navbar-link, - .navbar.is-black .navbar-end > .navbar-item, - .navbar.is-black .navbar-end .navbar-link { - color: white; } - .navbar.is-black .navbar-start > a.navbar-item:focus, .navbar.is-black .navbar-start > a.navbar-item:hover, .navbar.is-black .navbar-start > a.navbar-item.is-active, - .navbar.is-black .navbar-start .navbar-link:focus, - .navbar.is-black .navbar-start .navbar-link:hover, - .navbar.is-black .navbar-start .navbar-link.is-active, - .navbar.is-black .navbar-end > a.navbar-item:focus, - .navbar.is-black .navbar-end > a.navbar-item:hover, - .navbar.is-black .navbar-end > a.navbar-item.is-active, - .navbar.is-black .navbar-end .navbar-link:focus, - .navbar.is-black .navbar-end .navbar-link:hover, - .navbar.is-black .navbar-end .navbar-link.is-active { - background-color: black; - color: white; } - .navbar.is-black .navbar-start .navbar-link::after, - .navbar.is-black .navbar-end .navbar-link::after { - border-color: white; } - .navbar.is-black .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-black .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-black .navbar-item.has-dropdown.is-active .navbar-link { - background-color: black; - color: white; } - .navbar.is-black .navbar-dropdown a.navbar-item.is-active { - background-color: #0a0a0a; - color: white; } } - .navbar.is-light { - background-color: whitesmoke; - color: #363636; } - .navbar.is-light .navbar-brand > .navbar-item, - .navbar.is-light .navbar-brand .navbar-link { - color: #363636; } - .navbar.is-light .navbar-brand > a.navbar-item:focus, .navbar.is-light .navbar-brand > a.navbar-item:hover, .navbar.is-light .navbar-brand > a.navbar-item.is-active, - .navbar.is-light .navbar-brand .navbar-link:focus, - .navbar.is-light .navbar-brand .navbar-link:hover, - .navbar.is-light .navbar-brand .navbar-link.is-active { - background-color: #e8e8e8; - color: #363636; } - .navbar.is-light .navbar-brand .navbar-link::after { - border-color: #363636; } - .navbar.is-light .navbar-burger { - color: #363636; } - @media screen and (min-width: 1056px) { - .navbar.is-light .navbar-start > .navbar-item, - .navbar.is-light .navbar-start .navbar-link, - .navbar.is-light .navbar-end > .navbar-item, - .navbar.is-light .navbar-end .navbar-link { - color: #363636; } - .navbar.is-light .navbar-start > a.navbar-item:focus, .navbar.is-light .navbar-start > a.navbar-item:hover, .navbar.is-light .navbar-start > a.navbar-item.is-active, - .navbar.is-light .navbar-start .navbar-link:focus, - .navbar.is-light .navbar-start .navbar-link:hover, - .navbar.is-light .navbar-start .navbar-link.is-active, - .navbar.is-light .navbar-end > a.navbar-item:focus, - .navbar.is-light .navbar-end > a.navbar-item:hover, - .navbar.is-light .navbar-end > a.navbar-item.is-active, - .navbar.is-light .navbar-end .navbar-link:focus, - .navbar.is-light .navbar-end .navbar-link:hover, - .navbar.is-light .navbar-end .navbar-link.is-active { - background-color: #e8e8e8; - color: #363636; } - .navbar.is-light .navbar-start .navbar-link::after, - .navbar.is-light .navbar-end .navbar-link::after { - border-color: #363636; } - .navbar.is-light .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-light .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-light .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #e8e8e8; - color: #363636; } - .navbar.is-light .navbar-dropdown a.navbar-item.is-active { - background-color: whitesmoke; - color: #363636; } } - .navbar.is-dark, .content kbd.navbar { - background-color: #363636; - color: whitesmoke; } - .navbar.is-dark .navbar-brand > .navbar-item, .content kbd.navbar .navbar-brand > .navbar-item, - .navbar.is-dark .navbar-brand .navbar-link, - .content kbd.navbar .navbar-brand .navbar-link { - color: whitesmoke; } - .navbar.is-dark .navbar-brand > a.navbar-item:focus, .content kbd.navbar .navbar-brand > a.navbar-item:focus, .navbar.is-dark .navbar-brand > a.navbar-item:hover, .content kbd.navbar .navbar-brand > a.navbar-item:hover, .navbar.is-dark .navbar-brand > a.navbar-item.is-active, .content kbd.navbar .navbar-brand > a.navbar-item.is-active, - .navbar.is-dark .navbar-brand .navbar-link:focus, - .content kbd.navbar .navbar-brand .navbar-link:focus, - .navbar.is-dark .navbar-brand .navbar-link:hover, - .content kbd.navbar .navbar-brand .navbar-link:hover, - .navbar.is-dark .navbar-brand .navbar-link.is-active, - .content kbd.navbar .navbar-brand .navbar-link.is-active { - background-color: #292929; - color: whitesmoke; } - .navbar.is-dark .navbar-brand .navbar-link::after, .content kbd.navbar .navbar-brand .navbar-link::after { - border-color: whitesmoke; } - .navbar.is-dark .navbar-burger, .content kbd.navbar .navbar-burger { - color: whitesmoke; } - @media screen and (min-width: 1056px) { - .navbar.is-dark .navbar-start > .navbar-item, .content kbd.navbar .navbar-start > .navbar-item, - .navbar.is-dark .navbar-start .navbar-link, - .content kbd.navbar .navbar-start .navbar-link, - .navbar.is-dark .navbar-end > .navbar-item, - .content kbd.navbar .navbar-end > .navbar-item, - .navbar.is-dark .navbar-end .navbar-link, - .content kbd.navbar .navbar-end .navbar-link { - color: whitesmoke; } - .navbar.is-dark .navbar-start > a.navbar-item:focus, .content kbd.navbar .navbar-start > a.navbar-item:focus, .navbar.is-dark .navbar-start > a.navbar-item:hover, .content kbd.navbar .navbar-start > a.navbar-item:hover, .navbar.is-dark .navbar-start > a.navbar-item.is-active, .content kbd.navbar .navbar-start > a.navbar-item.is-active, - .navbar.is-dark .navbar-start .navbar-link:focus, - .content kbd.navbar .navbar-start .navbar-link:focus, - .navbar.is-dark .navbar-start .navbar-link:hover, - .content kbd.navbar .navbar-start .navbar-link:hover, - .navbar.is-dark .navbar-start .navbar-link.is-active, - .content kbd.navbar .navbar-start .navbar-link.is-active, - .navbar.is-dark .navbar-end > a.navbar-item:focus, - .content kbd.navbar .navbar-end > a.navbar-item:focus, - .navbar.is-dark .navbar-end > a.navbar-item:hover, - .content kbd.navbar .navbar-end > a.navbar-item:hover, - .navbar.is-dark .navbar-end > a.navbar-item.is-active, - .content kbd.navbar .navbar-end > a.navbar-item.is-active, - .navbar.is-dark .navbar-end .navbar-link:focus, - .content kbd.navbar .navbar-end .navbar-link:focus, - .navbar.is-dark .navbar-end .navbar-link:hover, - .content kbd.navbar .navbar-end .navbar-link:hover, - .navbar.is-dark .navbar-end .navbar-link.is-active, - .content kbd.navbar .navbar-end .navbar-link.is-active { - background-color: #292929; - color: whitesmoke; } - .navbar.is-dark .navbar-start .navbar-link::after, .content kbd.navbar .navbar-start .navbar-link::after, - .navbar.is-dark .navbar-end .navbar-link::after, - .content kbd.navbar .navbar-end .navbar-link::after { - border-color: whitesmoke; } - .navbar.is-dark .navbar-item.has-dropdown:focus .navbar-link, .content kbd.navbar .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-dark .navbar-item.has-dropdown:hover .navbar-link, - .content kbd.navbar .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-dark .navbar-item.has-dropdown.is-active .navbar-link, - .content kbd.navbar .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #292929; - color: whitesmoke; } - .navbar.is-dark .navbar-dropdown a.navbar-item.is-active, .content kbd.navbar .navbar-dropdown a.navbar-item.is-active { - background-color: #363636; - color: whitesmoke; } } - .navbar.is-primary, .docstring > section > a.navbar.docs-sourcelink { - background-color: #4eb5de; - color: #fff; } - .navbar.is-primary .navbar-brand > .navbar-item, .docstring > section > a.navbar.docs-sourcelink .navbar-brand > .navbar-item, - .navbar.is-primary .navbar-brand .navbar-link, - .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link { - color: #fff; } - .navbar.is-primary .navbar-brand > a.navbar-item:focus, .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item:focus, .navbar.is-primary .navbar-brand > a.navbar-item:hover, .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item:hover, .navbar.is-primary .navbar-brand > a.navbar-item.is-active, .docstring > section > a.navbar.docs-sourcelink .navbar-brand > a.navbar-item.is-active, - .navbar.is-primary .navbar-brand .navbar-link:focus, - .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link:focus, - .navbar.is-primary .navbar-brand .navbar-link:hover, - .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link:hover, - .navbar.is-primary .navbar-brand .navbar-link.is-active, - .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link.is-active { - background-color: #39acda; - color: #fff; } - .navbar.is-primary .navbar-brand .navbar-link::after, .docstring > section > a.navbar.docs-sourcelink .navbar-brand .navbar-link::after { - border-color: #fff; } - .navbar.is-primary .navbar-burger, .docstring > section > a.navbar.docs-sourcelink .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - .navbar.is-primary .navbar-start > .navbar-item, .docstring > section > a.navbar.docs-sourcelink .navbar-start > .navbar-item, - .navbar.is-primary .navbar-start .navbar-link, - .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link, - .navbar.is-primary .navbar-end > .navbar-item, - .docstring > section > a.navbar.docs-sourcelink .navbar-end > .navbar-item, - .navbar.is-primary .navbar-end .navbar-link, - .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link { - color: #fff; } - .navbar.is-primary .navbar-start > a.navbar-item:focus, .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item:focus, .navbar.is-primary .navbar-start > a.navbar-item:hover, .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item:hover, .navbar.is-primary .navbar-start > a.navbar-item.is-active, .docstring > section > a.navbar.docs-sourcelink .navbar-start > a.navbar-item.is-active, - .navbar.is-primary .navbar-start .navbar-link:focus, - .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link:focus, - .navbar.is-primary .navbar-start .navbar-link:hover, - .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link:hover, - .navbar.is-primary .navbar-start .navbar-link.is-active, - .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link.is-active, - .navbar.is-primary .navbar-end > a.navbar-item:focus, - .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item:focus, - .navbar.is-primary .navbar-end > a.navbar-item:hover, - .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item:hover, - .navbar.is-primary .navbar-end > a.navbar-item.is-active, - .docstring > section > a.navbar.docs-sourcelink .navbar-end > a.navbar-item.is-active, - .navbar.is-primary .navbar-end .navbar-link:focus, - .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link:focus, - .navbar.is-primary .navbar-end .navbar-link:hover, - .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link:hover, - .navbar.is-primary .navbar-end .navbar-link.is-active, - .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link.is-active { - background-color: #39acda; - color: #fff; } - .navbar.is-primary .navbar-start .navbar-link::after, .docstring > section > a.navbar.docs-sourcelink .navbar-start .navbar-link::after, - .navbar.is-primary .navbar-end .navbar-link::after, - .docstring > section > a.navbar.docs-sourcelink .navbar-end .navbar-link::after { - border-color: #fff; } - .navbar.is-primary .navbar-item.has-dropdown:focus .navbar-link, .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-primary .navbar-item.has-dropdown:hover .navbar-link, - .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-primary .navbar-item.has-dropdown.is-active .navbar-link, - .docstring > section > a.navbar.docs-sourcelink .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #39acda; - color: #fff; } - .navbar.is-primary .navbar-dropdown a.navbar-item.is-active, .docstring > section > a.navbar.docs-sourcelink .navbar-dropdown a.navbar-item.is-active { - background-color: #4eb5de; - color: #fff; } } - .navbar.is-link { - background-color: #2e63b8; - color: #fff; } - .navbar.is-link .navbar-brand > .navbar-item, - .navbar.is-link .navbar-brand .navbar-link { - color: #fff; } - .navbar.is-link .navbar-brand > a.navbar-item:focus, .navbar.is-link .navbar-brand > a.navbar-item:hover, .navbar.is-link .navbar-brand > a.navbar-item.is-active, - .navbar.is-link .navbar-brand .navbar-link:focus, - .navbar.is-link .navbar-brand .navbar-link:hover, - .navbar.is-link .navbar-brand .navbar-link.is-active { - background-color: #2958a4; - color: #fff; } - .navbar.is-link .navbar-brand .navbar-link::after { - border-color: #fff; } - .navbar.is-link .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - .navbar.is-link .navbar-start > .navbar-item, - .navbar.is-link .navbar-start .navbar-link, - .navbar.is-link .navbar-end > .navbar-item, - .navbar.is-link .navbar-end .navbar-link { - color: #fff; } - .navbar.is-link .navbar-start > a.navbar-item:focus, .navbar.is-link .navbar-start > a.navbar-item:hover, .navbar.is-link .navbar-start > a.navbar-item.is-active, - .navbar.is-link .navbar-start .navbar-link:focus, - .navbar.is-link .navbar-start .navbar-link:hover, - .navbar.is-link .navbar-start .navbar-link.is-active, - .navbar.is-link .navbar-end > a.navbar-item:focus, - .navbar.is-link .navbar-end > a.navbar-item:hover, - .navbar.is-link .navbar-end > a.navbar-item.is-active, - .navbar.is-link .navbar-end .navbar-link:focus, - .navbar.is-link .navbar-end .navbar-link:hover, - .navbar.is-link .navbar-end .navbar-link.is-active { - background-color: #2958a4; - color: #fff; } - .navbar.is-link .navbar-start .navbar-link::after, - .navbar.is-link .navbar-end .navbar-link::after { - border-color: #fff; } - .navbar.is-link .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-link .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-link .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #2958a4; - color: #fff; } - .navbar.is-link .navbar-dropdown a.navbar-item.is-active { - background-color: #2e63b8; - color: #fff; } } - .navbar.is-info { - background-color: #209cee; - color: #fff; } - .navbar.is-info .navbar-brand > .navbar-item, - .navbar.is-info .navbar-brand .navbar-link { - color: #fff; } - .navbar.is-info .navbar-brand > a.navbar-item:focus, .navbar.is-info .navbar-brand > a.navbar-item:hover, .navbar.is-info .navbar-brand > a.navbar-item.is-active, - .navbar.is-info .navbar-brand .navbar-link:focus, - .navbar.is-info .navbar-brand .navbar-link:hover, - .navbar.is-info .navbar-brand .navbar-link.is-active { - background-color: #1190e3; - color: #fff; } - .navbar.is-info .navbar-brand .navbar-link::after { - border-color: #fff; } - .navbar.is-info .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - .navbar.is-info .navbar-start > .navbar-item, - .navbar.is-info .navbar-start .navbar-link, - .navbar.is-info .navbar-end > .navbar-item, - .navbar.is-info .navbar-end .navbar-link { - color: #fff; } - .navbar.is-info .navbar-start > a.navbar-item:focus, .navbar.is-info .navbar-start > a.navbar-item:hover, .navbar.is-info .navbar-start > a.navbar-item.is-active, - .navbar.is-info .navbar-start .navbar-link:focus, - .navbar.is-info .navbar-start .navbar-link:hover, - .navbar.is-info .navbar-start .navbar-link.is-active, - .navbar.is-info .navbar-end > a.navbar-item:focus, - .navbar.is-info .navbar-end > a.navbar-item:hover, - .navbar.is-info .navbar-end > a.navbar-item.is-active, - .navbar.is-info .navbar-end .navbar-link:focus, - .navbar.is-info .navbar-end .navbar-link:hover, - .navbar.is-info .navbar-end .navbar-link.is-active { - background-color: #1190e3; - color: #fff; } - .navbar.is-info .navbar-start .navbar-link::after, - .navbar.is-info .navbar-end .navbar-link::after { - border-color: #fff; } - .navbar.is-info .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-info .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-info .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #1190e3; - color: #fff; } - .navbar.is-info .navbar-dropdown a.navbar-item.is-active { - background-color: #209cee; - color: #fff; } } - .navbar.is-success { - background-color: #22c35b; - color: #fff; } - .navbar.is-success .navbar-brand > .navbar-item, - .navbar.is-success .navbar-brand .navbar-link { - color: #fff; } - .navbar.is-success .navbar-brand > a.navbar-item:focus, .navbar.is-success .navbar-brand > a.navbar-item:hover, .navbar.is-success .navbar-brand > a.navbar-item.is-active, - .navbar.is-success .navbar-brand .navbar-link:focus, - .navbar.is-success .navbar-brand .navbar-link:hover, - .navbar.is-success .navbar-brand .navbar-link.is-active { - background-color: #1ead51; - color: #fff; } - .navbar.is-success .navbar-brand .navbar-link::after { - border-color: #fff; } - .navbar.is-success .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - .navbar.is-success .navbar-start > .navbar-item, - .navbar.is-success .navbar-start .navbar-link, - .navbar.is-success .navbar-end > .navbar-item, - .navbar.is-success .navbar-end .navbar-link { - color: #fff; } - .navbar.is-success .navbar-start > a.navbar-item:focus, .navbar.is-success .navbar-start > a.navbar-item:hover, .navbar.is-success .navbar-start > a.navbar-item.is-active, - .navbar.is-success .navbar-start .navbar-link:focus, - .navbar.is-success .navbar-start .navbar-link:hover, - .navbar.is-success .navbar-start .navbar-link.is-active, - .navbar.is-success .navbar-end > a.navbar-item:focus, - .navbar.is-success .navbar-end > a.navbar-item:hover, - .navbar.is-success .navbar-end > a.navbar-item.is-active, - .navbar.is-success .navbar-end .navbar-link:focus, - .navbar.is-success .navbar-end .navbar-link:hover, - .navbar.is-success .navbar-end .navbar-link.is-active { - background-color: #1ead51; - color: #fff; } - .navbar.is-success .navbar-start .navbar-link::after, - .navbar.is-success .navbar-end .navbar-link::after { - border-color: #fff; } - .navbar.is-success .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-success .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-success .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #1ead51; - color: #fff; } - .navbar.is-success .navbar-dropdown a.navbar-item.is-active { - background-color: #22c35b; - color: #fff; } } - .navbar.is-warning { - background-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-brand > .navbar-item, - .navbar.is-warning .navbar-brand .navbar-link { - color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-brand > a.navbar-item:focus, .navbar.is-warning .navbar-brand > a.navbar-item:hover, .navbar.is-warning .navbar-brand > a.navbar-item.is-active, - .navbar.is-warning .navbar-brand .navbar-link:focus, - .navbar.is-warning .navbar-brand .navbar-link:hover, - .navbar.is-warning .navbar-brand .navbar-link.is-active { - background-color: #ffd83e; - color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-brand .navbar-link::after { - border-color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-burger { - color: rgba(0, 0, 0, 0.7); } - @media screen and (min-width: 1056px) { - .navbar.is-warning .navbar-start > .navbar-item, - .navbar.is-warning .navbar-start .navbar-link, - .navbar.is-warning .navbar-end > .navbar-item, - .navbar.is-warning .navbar-end .navbar-link { - color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-start > a.navbar-item:focus, .navbar.is-warning .navbar-start > a.navbar-item:hover, .navbar.is-warning .navbar-start > a.navbar-item.is-active, - .navbar.is-warning .navbar-start .navbar-link:focus, - .navbar.is-warning .navbar-start .navbar-link:hover, - .navbar.is-warning .navbar-start .navbar-link.is-active, - .navbar.is-warning .navbar-end > a.navbar-item:focus, - .navbar.is-warning .navbar-end > a.navbar-item:hover, - .navbar.is-warning .navbar-end > a.navbar-item.is-active, - .navbar.is-warning .navbar-end .navbar-link:focus, - .navbar.is-warning .navbar-end .navbar-link:hover, - .navbar.is-warning .navbar-end .navbar-link.is-active { - background-color: #ffd83e; - color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-start .navbar-link::after, - .navbar.is-warning .navbar-end .navbar-link::after { - border-color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-warning .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-warning .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #ffd83e; - color: rgba(0, 0, 0, 0.7); } - .navbar.is-warning .navbar-dropdown a.navbar-item.is-active { - background-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } } - .navbar.is-danger { - background-color: #da0b00; - color: #fff; } - .navbar.is-danger .navbar-brand > .navbar-item, - .navbar.is-danger .navbar-brand .navbar-link { - color: #fff; } - .navbar.is-danger .navbar-brand > a.navbar-item:focus, .navbar.is-danger .navbar-brand > a.navbar-item:hover, .navbar.is-danger .navbar-brand > a.navbar-item.is-active, - .navbar.is-danger .navbar-brand .navbar-link:focus, - .navbar.is-danger .navbar-brand .navbar-link:hover, - .navbar.is-danger .navbar-brand .navbar-link.is-active { - background-color: #c10a00; - color: #fff; } - .navbar.is-danger .navbar-brand .navbar-link::after { - border-color: #fff; } - .navbar.is-danger .navbar-burger { - color: #fff; } - @media screen and (min-width: 1056px) { - .navbar.is-danger .navbar-start > .navbar-item, - .navbar.is-danger .navbar-start .navbar-link, - .navbar.is-danger .navbar-end > .navbar-item, - .navbar.is-danger .navbar-end .navbar-link { - color: #fff; } - .navbar.is-danger .navbar-start > a.navbar-item:focus, .navbar.is-danger .navbar-start > a.navbar-item:hover, .navbar.is-danger .navbar-start > a.navbar-item.is-active, - .navbar.is-danger .navbar-start .navbar-link:focus, - .navbar.is-danger .navbar-start .navbar-link:hover, - .navbar.is-danger .navbar-start .navbar-link.is-active, - .navbar.is-danger .navbar-end > a.navbar-item:focus, - .navbar.is-danger .navbar-end > a.navbar-item:hover, - .navbar.is-danger .navbar-end > a.navbar-item.is-active, - .navbar.is-danger .navbar-end .navbar-link:focus, - .navbar.is-danger .navbar-end .navbar-link:hover, - .navbar.is-danger .navbar-end .navbar-link.is-active { - background-color: #c10a00; - color: #fff; } - .navbar.is-danger .navbar-start .navbar-link::after, - .navbar.is-danger .navbar-end .navbar-link::after { - border-color: #fff; } - .navbar.is-danger .navbar-item.has-dropdown:focus .navbar-link, - .navbar.is-danger .navbar-item.has-dropdown:hover .navbar-link, - .navbar.is-danger .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #c10a00; - color: #fff; } - .navbar.is-danger .navbar-dropdown a.navbar-item.is-active { - background-color: #da0b00; - color: #fff; } } - .navbar > .container { - align-items: stretch; - display: flex; - min-height: 3.25rem; - width: 100%; } - .navbar.has-shadow { - box-shadow: 0 2px 0 0 whitesmoke; } - .navbar.is-fixed-bottom, .navbar.is-fixed-top { - left: 0; - position: fixed; - right: 0; - z-index: 30; } - .navbar.is-fixed-bottom { - bottom: 0; } - .navbar.is-fixed-bottom.has-shadow { - box-shadow: 0 -2px 0 0 whitesmoke; } - .navbar.is-fixed-top { - top: 0; } - -html.has-navbar-fixed-top, -body.has-navbar-fixed-top { - padding-top: 3.25rem; } - -html.has-navbar-fixed-bottom, -body.has-navbar-fixed-bottom { - padding-bottom: 3.25rem; } - -.navbar-brand, -.navbar-tabs { - align-items: stretch; - display: flex; - flex-shrink: 0; - min-height: 3.25rem; } - -.navbar-brand a.navbar-item:focus, .navbar-brand a.navbar-item:hover { - background-color: transparent; } - -.navbar-tabs { - -webkit-overflow-scrolling: touch; - max-width: 100vw; - overflow-x: auto; - overflow-y: hidden; } - -.navbar-burger { - color: #4a4a4a; - cursor: pointer; - display: block; - height: 3.25rem; - position: relative; - width: 3.25rem; - margin-left: auto; } - .navbar-burger span { - background-color: currentColor; - display: block; - height: 1px; - left: calc(50% - 8px); - position: absolute; - transform-origin: center; - transition-duration: 86ms; - transition-property: background-color, opacity, transform; - transition-timing-function: ease-out; - width: 16px; } - .navbar-burger span:nth-child(1) { - top: calc(50% - 6px); } - .navbar-burger span:nth-child(2) { - top: calc(50% - 1px); } - .navbar-burger span:nth-child(3) { - top: calc(50% + 4px); } - .navbar-burger:hover { - background-color: rgba(0, 0, 0, 0.05); } - .navbar-burger.is-active span:nth-child(1) { - transform: translateY(5px) rotate(45deg); } - .navbar-burger.is-active span:nth-child(2) { - opacity: 0; } - .navbar-burger.is-active span:nth-child(3) { - transform: translateY(-5px) rotate(-45deg); } - -.navbar-menu { - display: none; } - -.navbar-item, -.navbar-link { - color: #4a4a4a; - display: block; - line-height: 1.5; - padding: 0.5rem 0.75rem; - position: relative; } - .navbar-item .icon:only-child, - .navbar-link .icon:only-child { - margin-left: -0.25rem; - margin-right: -0.25rem; } - -a.navbar-item, -.navbar-link { - cursor: pointer; } - a.navbar-item:focus, a.navbar-item:focus-within, a.navbar-item:hover, a.navbar-item.is-active, - .navbar-link:focus, - .navbar-link:focus-within, - .navbar-link:hover, - .navbar-link.is-active { - background-color: #fafafa; - color: #2e63b8; } - -.navbar-item { - display: block; - flex-grow: 0; - flex-shrink: 0; } - .navbar-item img { - max-height: 1.75rem; } - .navbar-item.has-dropdown { - padding: 0; } - .navbar-item.is-expanded { - flex-grow: 1; - flex-shrink: 1; } - .navbar-item.is-tab { - border-bottom: 1px solid transparent; - min-height: 3.25rem; - padding-bottom: calc(0.5rem - 1px); } - .navbar-item.is-tab:focus, .navbar-item.is-tab:hover { - background-color: transparent; - border-bottom-color: #2e63b8; } - .navbar-item.is-tab.is-active { - background-color: transparent; - border-bottom-color: #2e63b8; - border-bottom-style: solid; - border-bottom-width: 3px; - color: #2e63b8; - padding-bottom: calc(0.5rem - 3px); } - -.navbar-content { - flex-grow: 1; - flex-shrink: 1; } - -.navbar-link:not(.is-arrowless) { - padding-right: 2.5em; } - .navbar-link:not(.is-arrowless)::after { - border-color: #2e63b8; - margin-top: -0.375em; - right: 1.125em; } - -.navbar-dropdown { - font-size: 0.875rem; - padding-bottom: 0.5rem; - padding-top: 0.5rem; } - .navbar-dropdown .navbar-item { - padding-left: 1.5rem; - padding-right: 1.5rem; } - -.navbar-divider { - background-color: whitesmoke; - border: none; - display: none; - height: 2px; - margin: 0.5rem 0; } - -@media screen and (max-width: 1055px) { - .navbar > .container { - display: block; } - .navbar-brand .navbar-item, - .navbar-tabs .navbar-item { - align-items: center; - display: flex; } - .navbar-link::after { - display: none; } - .navbar-menu { - background-color: white; - box-shadow: 0 8px 16px rgba(10, 10, 10, 0.1); - padding: 0.5rem 0; } - .navbar-menu.is-active { - display: block; } - .navbar.is-fixed-bottom-touch, .navbar.is-fixed-top-touch { - left: 0; - position: fixed; - right: 0; - z-index: 30; } - .navbar.is-fixed-bottom-touch { - bottom: 0; } - .navbar.is-fixed-bottom-touch.has-shadow { - box-shadow: 0 -2px 3px rgba(10, 10, 10, 0.1); } - .navbar.is-fixed-top-touch { - top: 0; } - .navbar.is-fixed-top .navbar-menu, .navbar.is-fixed-top-touch .navbar-menu { - -webkit-overflow-scrolling: touch; - max-height: calc(100vh - 3.25rem); - overflow: auto; } - html.has-navbar-fixed-top-touch, - body.has-navbar-fixed-top-touch { - padding-top: 3.25rem; } - html.has-navbar-fixed-bottom-touch, - body.has-navbar-fixed-bottom-touch { - padding-bottom: 3.25rem; } } - -@media screen and (min-width: 1056px) { - .navbar, - .navbar-menu, - .navbar-start, - .navbar-end { - align-items: stretch; - display: flex; } - .navbar { - min-height: 3.25rem; } - .navbar.is-spaced { - padding: 1rem 2rem; } - .navbar.is-spaced .navbar-start, - .navbar.is-spaced .navbar-end { - align-items: center; } - .navbar.is-spaced a.navbar-item, - .navbar.is-spaced .navbar-link { - border-radius: 4px; } - .navbar.is-transparent a.navbar-item:focus, .navbar.is-transparent a.navbar-item:hover, .navbar.is-transparent a.navbar-item.is-active, - .navbar.is-transparent .navbar-link:focus, - .navbar.is-transparent .navbar-link:hover, - .navbar.is-transparent .navbar-link.is-active { - background-color: transparent !important; } - .navbar.is-transparent .navbar-item.has-dropdown.is-active .navbar-link, .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:focus .navbar-link, .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:focus-within .navbar-link, .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:hover .navbar-link { - background-color: transparent !important; } - .navbar.is-transparent .navbar-dropdown a.navbar-item:focus, .navbar.is-transparent .navbar-dropdown a.navbar-item:hover { - background-color: whitesmoke; - color: #0a0a0a; } - .navbar.is-transparent .navbar-dropdown a.navbar-item.is-active { - background-color: whitesmoke; - color: #2e63b8; } - .navbar-burger { - display: none; } - .navbar-item, - .navbar-link { - align-items: center; - display: flex; } - .navbar-item { - display: flex; } - .navbar-item.has-dropdown { - align-items: stretch; } - .navbar-item.has-dropdown-up .navbar-link::after { - transform: rotate(135deg) translate(0.25em, -0.25em); } - .navbar-item.has-dropdown-up .navbar-dropdown { - border-bottom: 2px solid #dbdbdb; - border-radius: 6px 6px 0 0; - border-top: none; - bottom: 100%; - box-shadow: 0 -8px 8px rgba(10, 10, 10, 0.1); - top: auto; } - .navbar-item.is-active .navbar-dropdown, .navbar-item.is-hoverable:focus .navbar-dropdown, .navbar-item.is-hoverable:focus-within .navbar-dropdown, .navbar-item.is-hoverable:hover .navbar-dropdown { - display: block; } - .navbar.is-spaced .navbar-item.is-active .navbar-dropdown, .navbar-item.is-active .navbar-dropdown.is-boxed, .navbar.is-spaced .navbar-item.is-hoverable:focus .navbar-dropdown, .navbar-item.is-hoverable:focus .navbar-dropdown.is-boxed, .navbar.is-spaced .navbar-item.is-hoverable:focus-within .navbar-dropdown, .navbar-item.is-hoverable:focus-within .navbar-dropdown.is-boxed, .navbar.is-spaced .navbar-item.is-hoverable:hover .navbar-dropdown, .navbar-item.is-hoverable:hover .navbar-dropdown.is-boxed { - opacity: 1; - pointer-events: auto; - transform: translateY(0); } - .navbar-menu { - flex-grow: 1; - flex-shrink: 0; } - .navbar-start { - justify-content: flex-start; - margin-right: auto; } - .navbar-end { - justify-content: flex-end; - margin-left: auto; } - .navbar-dropdown { - background-color: white; - border-bottom-left-radius: 6px; - border-bottom-right-radius: 6px; - border-top: 2px solid #dbdbdb; - box-shadow: 0 8px 8px rgba(10, 10, 10, 0.1); - display: none; - font-size: 0.875rem; - left: 0; - min-width: 100%; - position: absolute; - top: 100%; - z-index: 20; } - .navbar-dropdown .navbar-item { - padding: 0.375rem 1rem; - white-space: nowrap; } - .navbar-dropdown a.navbar-item { - padding-right: 3rem; } - .navbar-dropdown a.navbar-item:focus, .navbar-dropdown a.navbar-item:hover { - background-color: whitesmoke; - color: #0a0a0a; } - .navbar-dropdown a.navbar-item.is-active { - background-color: whitesmoke; - color: #2e63b8; } - .navbar.is-spaced .navbar-dropdown, .navbar-dropdown.is-boxed { - border-radius: 6px; - border-top: none; - box-shadow: 0 8px 8px rgba(10, 10, 10, 0.1), 0 0 0 1px rgba(10, 10, 10, 0.1); - display: block; - opacity: 0; - pointer-events: none; - top: calc(100% + (-4px)); - transform: translateY(-5px); - transition-duration: 86ms; - transition-property: opacity, transform; } - .navbar-dropdown.is-right { - left: auto; - right: 0; } - .navbar-divider { - display: block; } - .navbar > .container .navbar-brand, - .container > .navbar .navbar-brand { - margin-left: -.75rem; } - .navbar > .container .navbar-menu, - .container > .navbar .navbar-menu { - margin-right: -.75rem; } - .navbar.is-fixed-bottom-desktop, .navbar.is-fixed-top-desktop { - left: 0; - position: fixed; - right: 0; - z-index: 30; } - .navbar.is-fixed-bottom-desktop { - bottom: 0; } - .navbar.is-fixed-bottom-desktop.has-shadow { - box-shadow: 0 -2px 3px rgba(10, 10, 10, 0.1); } - .navbar.is-fixed-top-desktop { - top: 0; } - html.has-navbar-fixed-top-desktop, - body.has-navbar-fixed-top-desktop { - padding-top: 3.25rem; } - html.has-navbar-fixed-bottom-desktop, - body.has-navbar-fixed-bottom-desktop { - padding-bottom: 3.25rem; } - html.has-spaced-navbar-fixed-top, - body.has-spaced-navbar-fixed-top { - padding-top: 5.25rem; } - html.has-spaced-navbar-fixed-bottom, - body.has-spaced-navbar-fixed-bottom { - padding-bottom: 5.25rem; } - a.navbar-item.is-active, - .navbar-link.is-active { - color: #0a0a0a; } - a.navbar-item.is-active:not(:focus):not(:hover), - .navbar-link.is-active:not(:focus):not(:hover) { - background-color: transparent; } - .navbar-item.has-dropdown:focus .navbar-link, .navbar-item.has-dropdown:hover .navbar-link, .navbar-item.has-dropdown.is-active .navbar-link { - background-color: #fafafa; } } - -.hero.is-fullheight-with-navbar { - min-height: calc(100vh - 3.25rem); } - -.pagination { - font-size: 1rem; - margin: -0.25rem; } - .pagination.is-small, #documenter .docs-sidebar form.docs-search > input.pagination { - font-size: 0.75rem; } - .pagination.is-medium { - font-size: 1.25rem; } - .pagination.is-large { - font-size: 1.5rem; } - .pagination.is-rounded .pagination-previous, #documenter .docs-sidebar form.docs-search > input.pagination .pagination-previous, - .pagination.is-rounded .pagination-next, - #documenter .docs-sidebar form.docs-search > input.pagination .pagination-next { - padding-left: 1em; - padding-right: 1em; - border-radius: 290486px; } - .pagination.is-rounded .pagination-link, #documenter .docs-sidebar form.docs-search > input.pagination .pagination-link { - border-radius: 290486px; } - -.pagination, -.pagination-list { - align-items: center; - display: flex; - justify-content: center; - text-align: center; } - -.pagination-previous, -.pagination-next, -.pagination-link, -.pagination-ellipsis { - font-size: 1em; - justify-content: center; - margin: 0.25rem; - padding-left: 0.5em; - padding-right: 0.5em; - text-align: center; } - -.pagination-previous, -.pagination-next, -.pagination-link { - border-color: #dbdbdb; - color: #363636; - min-width: 2.25em; } - .pagination-previous:hover, - .pagination-next:hover, - .pagination-link:hover { - border-color: #b5b5b5; - color: #363636; } - .pagination-previous:focus, - .pagination-next:focus, - .pagination-link:focus { - border-color: #3c5dcd; } - .pagination-previous:active, - .pagination-next:active, - .pagination-link:active { - box-shadow: inset 0 1px 2px rgba(10, 10, 10, 0.2); } - .pagination-previous[disabled], - .pagination-next[disabled], - .pagination-link[disabled] { - background-color: #dbdbdb; - border-color: #dbdbdb; - box-shadow: none; - color: #6b6b6b; - opacity: 0.5; } - -.pagination-previous, -.pagination-next { - padding-left: 0.75em; - padding-right: 0.75em; - white-space: nowrap; } - -.pagination-link.is-current { - background-color: #2e63b8; - border-color: #2e63b8; - color: #fff; } - -.pagination-ellipsis { - color: #b5b5b5; - pointer-events: none; } - -.pagination-list { - flex-wrap: wrap; } - -@media screen and (max-width: 768px) { - .pagination { - flex-wrap: wrap; } - .pagination-previous, - .pagination-next { - flex-grow: 1; - flex-shrink: 1; } - .pagination-list li { - flex-grow: 1; - flex-shrink: 1; } } - -@media screen and (min-width: 769px), print { - .pagination-list { - flex-grow: 1; - flex-shrink: 1; - justify-content: flex-start; - order: 1; } - .pagination-previous { - order: 2; } - .pagination-next { - order: 3; } - .pagination { - justify-content: space-between; } - .pagination.is-centered .pagination-previous { - order: 1; } - .pagination.is-centered .pagination-list { - justify-content: center; - order: 2; } - .pagination.is-centered .pagination-next { - order: 3; } - .pagination.is-right .pagination-previous { - order: 1; } - .pagination.is-right .pagination-next { - order: 2; } - .pagination.is-right .pagination-list { - justify-content: flex-end; - order: 3; } } - -.panel { - font-size: 1rem; } - .panel:not(:last-child) { - margin-bottom: 1.5rem; } - -.panel-heading, -.panel-tabs, -.panel-block { - border-bottom: 1px solid #dbdbdb; - border-left: 1px solid #dbdbdb; - border-right: 1px solid #dbdbdb; } - .panel-heading:first-child, - .panel-tabs:first-child, - .panel-block:first-child { - border-top: 1px solid #dbdbdb; } - -.panel-heading { - background-color: whitesmoke; - border-radius: 4px 4px 0 0; - color: #222222; - font-size: 1.25em; - font-weight: 300; - line-height: 1.25; - padding: 0.5em 0.75em; } - -.panel-tabs { - align-items: flex-end; - display: flex; - font-size: 0.875em; - justify-content: center; } - .panel-tabs a { - border-bottom: 1px solid #dbdbdb; - margin-bottom: -1px; - padding: 0.5em; } - .panel-tabs a.is-active { - border-bottom-color: #4a4a4a; - color: #363636; } - -.panel-list a { - color: #222222; } - .panel-list a:hover { - color: #2e63b8; } - -.panel-block { - align-items: center; - color: #222222; - display: flex; - justify-content: flex-start; - padding: 0.5em 0.75em; } - .panel-block input[type="checkbox"] { - margin-right: 0.75em; } - .panel-block > .control { - flex-grow: 1; - flex-shrink: 1; - width: 100%; } - .panel-block.is-wrapped { - flex-wrap: wrap; } - .panel-block.is-active { - border-left-color: #2e63b8; - color: #363636; } - .panel-block.is-active .panel-icon { - color: #2e63b8; } - -a.panel-block, -label.panel-block { - cursor: pointer; } - a.panel-block:hover, - label.panel-block:hover { - background-color: whitesmoke; } - -.panel-icon { - display: inline-block; - font-size: 14px; - height: 1em; - line-height: 1em; - text-align: center; - vertical-align: top; - width: 1em; - color: #6b6b6b; - margin-right: 0.75em; } - .panel-icon .fa { - font-size: inherit; - line-height: inherit; } - -.tabs { - -webkit-overflow-scrolling: touch; - align-items: stretch; - display: flex; - font-size: 1rem; - justify-content: space-between; - overflow: hidden; - overflow-x: auto; - white-space: nowrap; } - .tabs a { - align-items: center; - border-bottom-color: #dbdbdb; - border-bottom-style: solid; - border-bottom-width: 1px; - color: #222222; - display: flex; - justify-content: center; - margin-bottom: -1px; - padding: 0.5em 1em; - vertical-align: top; } - .tabs a:hover { - border-bottom-color: #222222; - color: #222222; } - .tabs li { - display: block; } - .tabs li.is-active a { - border-bottom-color: #2e63b8; - color: #2e63b8; } - .tabs ul { - align-items: center; - border-bottom-color: #dbdbdb; - border-bottom-style: solid; - border-bottom-width: 1px; - display: flex; - flex-grow: 1; - flex-shrink: 0; - justify-content: flex-start; } - .tabs ul.is-left { - padding-right: 0.75em; } - .tabs ul.is-center { - flex: none; - justify-content: center; - padding-left: 0.75em; - padding-right: 0.75em; } - .tabs ul.is-right { - justify-content: flex-end; - padding-left: 0.75em; } - .tabs .icon:first-child { - margin-right: 0.5em; } - .tabs .icon:last-child { - margin-left: 0.5em; } - .tabs.is-centered ul { - justify-content: center; } - .tabs.is-right ul { - justify-content: flex-end; } - .tabs.is-boxed a { - border: 1px solid transparent; - border-radius: 4px 4px 0 0; } - .tabs.is-boxed a:hover { - background-color: whitesmoke; - border-bottom-color: #dbdbdb; } - .tabs.is-boxed li.is-active a { - background-color: white; - border-color: #dbdbdb; - border-bottom-color: transparent !important; } - .tabs.is-fullwidth li { - flex-grow: 1; - flex-shrink: 0; } - .tabs.is-toggle a { - border-color: #dbdbdb; - border-style: solid; - border-width: 1px; - margin-bottom: 0; - position: relative; } - .tabs.is-toggle a:hover { - background-color: whitesmoke; - border-color: #b5b5b5; - z-index: 2; } - .tabs.is-toggle li + li { - margin-left: -1px; } - .tabs.is-toggle li:first-child a { - border-radius: 4px 0 0 4px; } - .tabs.is-toggle li:last-child a { - border-radius: 0 4px 4px 0; } - .tabs.is-toggle li.is-active a { - background-color: #2e63b8; - border-color: #2e63b8; - color: #fff; - z-index: 1; } - .tabs.is-toggle ul { - border-bottom: none; } - .tabs.is-toggle.is-toggle-rounded li:first-child a { - border-bottom-left-radius: 290486px; - border-top-left-radius: 290486px; - padding-left: 1.25em; } - .tabs.is-toggle.is-toggle-rounded li:last-child a { - border-bottom-right-radius: 290486px; - border-top-right-radius: 290486px; - padding-right: 1.25em; } - .tabs.is-small, #documenter .docs-sidebar form.docs-search > input.tabs { - font-size: 0.75rem; } - .tabs.is-medium { - font-size: 1.25rem; } - .tabs.is-large { - font-size: 1.5rem; } - -.column { - display: block; - flex-basis: 0; - flex-grow: 1; - flex-shrink: 1; - padding: 0.75rem; } - .columns.is-mobile > .column.is-narrow { - flex: none; } - .columns.is-mobile > .column.is-full { - flex: none; - width: 100%; } - .columns.is-mobile > .column.is-three-quarters { - flex: none; - width: 75%; } - .columns.is-mobile > .column.is-two-thirds { - flex: none; - width: 66.6666%; } - .columns.is-mobile > .column.is-half { - flex: none; - width: 50%; } - .columns.is-mobile > .column.is-one-third { - flex: none; - width: 33.3333%; } - .columns.is-mobile > .column.is-one-quarter { - flex: none; - width: 25%; } - .columns.is-mobile > .column.is-one-fifth { - flex: none; - width: 20%; } - .columns.is-mobile > .column.is-two-fifths { - flex: none; - width: 40%; } - .columns.is-mobile > .column.is-three-fifths { - flex: none; - width: 60%; } - .columns.is-mobile > .column.is-four-fifths { - flex: none; - width: 80%; } - .columns.is-mobile > .column.is-offset-three-quarters { - margin-left: 75%; } - .columns.is-mobile > .column.is-offset-two-thirds { - margin-left: 66.6666%; } - .columns.is-mobile > .column.is-offset-half { - margin-left: 50%; } - .columns.is-mobile > .column.is-offset-one-third { - margin-left: 33.3333%; } - .columns.is-mobile > .column.is-offset-one-quarter { - margin-left: 25%; } - .columns.is-mobile > .column.is-offset-one-fifth { - margin-left: 20%; } - .columns.is-mobile > .column.is-offset-two-fifths { - margin-left: 40%; } - .columns.is-mobile > .column.is-offset-three-fifths { - margin-left: 60%; } - .columns.is-mobile > .column.is-offset-four-fifths { - margin-left: 80%; } - .columns.is-mobile > .column.is-0 { - flex: none; - width: 0%; } - .columns.is-mobile > .column.is-offset-0 { - margin-left: 0%; } - .columns.is-mobile > .column.is-1 { - flex: none; - width: 8.3333333333%; } - .columns.is-mobile > .column.is-offset-1 { - margin-left: 8.3333333333%; } - .columns.is-mobile > .column.is-2 { - flex: none; - width: 16.6666666667%; } - .columns.is-mobile > .column.is-offset-2 { - margin-left: 16.6666666667%; } - .columns.is-mobile > .column.is-3 { - flex: none; - width: 25%; } - .columns.is-mobile > .column.is-offset-3 { - margin-left: 25%; } - .columns.is-mobile > .column.is-4 { - flex: none; - width: 33.3333333333%; } - .columns.is-mobile > .column.is-offset-4 { - margin-left: 33.3333333333%; } - .columns.is-mobile > .column.is-5 { - flex: none; - width: 41.6666666667%; } - .columns.is-mobile > .column.is-offset-5 { - margin-left: 41.6666666667%; } - .columns.is-mobile > .column.is-6 { - flex: none; - width: 50%; } - .columns.is-mobile > .column.is-offset-6 { - margin-left: 50%; } - .columns.is-mobile > .column.is-7 { - flex: none; - width: 58.3333333333%; } - .columns.is-mobile > .column.is-offset-7 { - margin-left: 58.3333333333%; } - .columns.is-mobile > .column.is-8 { - flex: none; - width: 66.6666666667%; } - .columns.is-mobile > .column.is-offset-8 { - margin-left: 66.6666666667%; } - .columns.is-mobile > .column.is-9 { - flex: none; - width: 75%; } - .columns.is-mobile > .column.is-offset-9 { - margin-left: 75%; } - .columns.is-mobile > .column.is-10 { - flex: none; - width: 83.3333333333%; } - .columns.is-mobile > .column.is-offset-10 { - margin-left: 83.3333333333%; } - .columns.is-mobile > .column.is-11 { - flex: none; - width: 91.6666666667%; } - .columns.is-mobile > .column.is-offset-11 { - margin-left: 91.6666666667%; } - .columns.is-mobile > .column.is-12 { - flex: none; - width: 100%; } - .columns.is-mobile > .column.is-offset-12 { - margin-left: 100%; } - @media screen and (max-width: 768px) { - .column.is-narrow-mobile { - flex: none; } - .column.is-full-mobile { - flex: none; - width: 100%; } - .column.is-three-quarters-mobile { - flex: none; - width: 75%; } - .column.is-two-thirds-mobile { - flex: none; - width: 66.6666%; } - .column.is-half-mobile { - flex: none; - width: 50%; } - .column.is-one-third-mobile { - flex: none; - width: 33.3333%; } - .column.is-one-quarter-mobile { - flex: none; - width: 25%; } - .column.is-one-fifth-mobile { - flex: none; - width: 20%; } - .column.is-two-fifths-mobile { - flex: none; - width: 40%; } - .column.is-three-fifths-mobile { - flex: none; - width: 60%; } - .column.is-four-fifths-mobile { - flex: none; - width: 80%; } - .column.is-offset-three-quarters-mobile { - margin-left: 75%; } - .column.is-offset-two-thirds-mobile { - margin-left: 66.6666%; } - .column.is-offset-half-mobile { - margin-left: 50%; } - .column.is-offset-one-third-mobile { - margin-left: 33.3333%; } - .column.is-offset-one-quarter-mobile { - margin-left: 25%; } - .column.is-offset-one-fifth-mobile { - margin-left: 20%; } - .column.is-offset-two-fifths-mobile { - margin-left: 40%; } - .column.is-offset-three-fifths-mobile { - margin-left: 60%; } - .column.is-offset-four-fifths-mobile { - margin-left: 80%; } - .column.is-0-mobile { - flex: none; - width: 0%; } - .column.is-offset-0-mobile { - margin-left: 0%; } - .column.is-1-mobile { - flex: none; - width: 8.3333333333%; } - .column.is-offset-1-mobile { - margin-left: 8.3333333333%; } - .column.is-2-mobile { - flex: none; - width: 16.6666666667%; } - .column.is-offset-2-mobile { - margin-left: 16.6666666667%; } - .column.is-3-mobile { - flex: none; - width: 25%; } - .column.is-offset-3-mobile { - margin-left: 25%; } - .column.is-4-mobile { - flex: none; - width: 33.3333333333%; } - .column.is-offset-4-mobile { - margin-left: 33.3333333333%; } - .column.is-5-mobile { - flex: none; - width: 41.6666666667%; } - .column.is-offset-5-mobile { - margin-left: 41.6666666667%; } - .column.is-6-mobile { - flex: none; - width: 50%; } - .column.is-offset-6-mobile { - margin-left: 50%; } - .column.is-7-mobile { - flex: none; - width: 58.3333333333%; } - .column.is-offset-7-mobile { - margin-left: 58.3333333333%; } - .column.is-8-mobile { - flex: none; - width: 66.6666666667%; } - .column.is-offset-8-mobile { - margin-left: 66.6666666667%; } - .column.is-9-mobile { - flex: none; - width: 75%; } - .column.is-offset-9-mobile { - margin-left: 75%; } - .column.is-10-mobile { - flex: none; - width: 83.3333333333%; } - .column.is-offset-10-mobile { - margin-left: 83.3333333333%; } - .column.is-11-mobile { - flex: none; - width: 91.6666666667%; } - .column.is-offset-11-mobile { - margin-left: 91.6666666667%; } - .column.is-12-mobile { - flex: none; - width: 100%; } - .column.is-offset-12-mobile { - margin-left: 100%; } } - @media screen and (min-width: 769px), print { - .column.is-narrow, .column.is-narrow-tablet { - flex: none; } - .column.is-full, .column.is-full-tablet { - flex: none; - width: 100%; } - .column.is-three-quarters, .column.is-three-quarters-tablet { - flex: none; - width: 75%; } - .column.is-two-thirds, .column.is-two-thirds-tablet { - flex: none; - width: 66.6666%; } - .column.is-half, .column.is-half-tablet { - flex: none; - width: 50%; } - .column.is-one-third, .column.is-one-third-tablet { - flex: none; - width: 33.3333%; } - .column.is-one-quarter, .column.is-one-quarter-tablet { - flex: none; - width: 25%; } - .column.is-one-fifth, .column.is-one-fifth-tablet { - flex: none; - width: 20%; } - .column.is-two-fifths, .column.is-two-fifths-tablet { - flex: none; - width: 40%; } - .column.is-three-fifths, .column.is-three-fifths-tablet { - flex: none; - width: 60%; } - .column.is-four-fifths, .column.is-four-fifths-tablet { - flex: none; - width: 80%; } - .column.is-offset-three-quarters, .column.is-offset-three-quarters-tablet { - margin-left: 75%; } - .column.is-offset-two-thirds, .column.is-offset-two-thirds-tablet { - margin-left: 66.6666%; } - .column.is-offset-half, .column.is-offset-half-tablet { - margin-left: 50%; } - .column.is-offset-one-third, .column.is-offset-one-third-tablet { - margin-left: 33.3333%; } - .column.is-offset-one-quarter, .column.is-offset-one-quarter-tablet { - margin-left: 25%; } - .column.is-offset-one-fifth, .column.is-offset-one-fifth-tablet { - margin-left: 20%; } - .column.is-offset-two-fifths, .column.is-offset-two-fifths-tablet { - margin-left: 40%; } - .column.is-offset-three-fifths, .column.is-offset-three-fifths-tablet { - margin-left: 60%; } - .column.is-offset-four-fifths, .column.is-offset-four-fifths-tablet { - margin-left: 80%; } - .column.is-0, .column.is-0-tablet { - flex: none; - width: 0%; } - .column.is-offset-0, .column.is-offset-0-tablet { - margin-left: 0%; } - .column.is-1, .column.is-1-tablet { - flex: none; - width: 8.3333333333%; } - .column.is-offset-1, .column.is-offset-1-tablet { - margin-left: 8.3333333333%; } - .column.is-2, .column.is-2-tablet { - flex: none; - width: 16.6666666667%; } - .column.is-offset-2, .column.is-offset-2-tablet { - margin-left: 16.6666666667%; } - .column.is-3, .column.is-3-tablet { - flex: none; - width: 25%; } - .column.is-offset-3, .column.is-offset-3-tablet { - margin-left: 25%; } - .column.is-4, .column.is-4-tablet { - flex: none; - width: 33.3333333333%; } - .column.is-offset-4, .column.is-offset-4-tablet { - margin-left: 33.3333333333%; } - .column.is-5, .column.is-5-tablet { - flex: none; - width: 41.6666666667%; } - .column.is-offset-5, .column.is-offset-5-tablet { - margin-left: 41.6666666667%; } - .column.is-6, .column.is-6-tablet { - flex: none; - width: 50%; } - .column.is-offset-6, .column.is-offset-6-tablet { - margin-left: 50%; } - .column.is-7, .column.is-7-tablet { - flex: none; - width: 58.3333333333%; } - .column.is-offset-7, .column.is-offset-7-tablet { - margin-left: 58.3333333333%; } - .column.is-8, .column.is-8-tablet { - flex: none; - width: 66.6666666667%; } - .column.is-offset-8, .column.is-offset-8-tablet { - margin-left: 66.6666666667%; } - .column.is-9, .column.is-9-tablet { - flex: none; - width: 75%; } - .column.is-offset-9, .column.is-offset-9-tablet { - margin-left: 75%; } - .column.is-10, .column.is-10-tablet { - flex: none; - width: 83.3333333333%; } - .column.is-offset-10, .column.is-offset-10-tablet { - margin-left: 83.3333333333%; } - .column.is-11, .column.is-11-tablet { - flex: none; - width: 91.6666666667%; } - .column.is-offset-11, .column.is-offset-11-tablet { - margin-left: 91.6666666667%; } - .column.is-12, .column.is-12-tablet { - flex: none; - width: 100%; } - .column.is-offset-12, .column.is-offset-12-tablet { - margin-left: 100%; } } - @media screen and (max-width: 1055px) { - .column.is-narrow-touch { - flex: none; } - .column.is-full-touch { - flex: none; - width: 100%; } - .column.is-three-quarters-touch { - flex: none; - width: 75%; } - .column.is-two-thirds-touch { - flex: none; - width: 66.6666%; } - .column.is-half-touch { - flex: none; - width: 50%; } - .column.is-one-third-touch { - flex: none; - width: 33.3333%; } - .column.is-one-quarter-touch { - flex: none; - width: 25%; } - .column.is-one-fifth-touch { - flex: none; - width: 20%; } - .column.is-two-fifths-touch { - flex: none; - width: 40%; } - .column.is-three-fifths-touch { - flex: none; - width: 60%; } - .column.is-four-fifths-touch { - flex: none; - width: 80%; } - .column.is-offset-three-quarters-touch { - margin-left: 75%; } - .column.is-offset-two-thirds-touch { - margin-left: 66.6666%; } - .column.is-offset-half-touch { - margin-left: 50%; } - .column.is-offset-one-third-touch { - margin-left: 33.3333%; } - .column.is-offset-one-quarter-touch { - margin-left: 25%; } - .column.is-offset-one-fifth-touch { - margin-left: 20%; } - .column.is-offset-two-fifths-touch { - margin-left: 40%; } - .column.is-offset-three-fifths-touch { - margin-left: 60%; } - .column.is-offset-four-fifths-touch { - margin-left: 80%; } - .column.is-0-touch { - flex: none; - width: 0%; } - .column.is-offset-0-touch { - margin-left: 0%; } - .column.is-1-touch { - flex: none; - width: 8.3333333333%; } - .column.is-offset-1-touch { - margin-left: 8.3333333333%; } - .column.is-2-touch { - flex: none; - width: 16.6666666667%; } - .column.is-offset-2-touch { - margin-left: 16.6666666667%; } - .column.is-3-touch { - flex: none; - width: 25%; } - .column.is-offset-3-touch { - margin-left: 25%; } - .column.is-4-touch { - flex: none; - width: 33.3333333333%; } - .column.is-offset-4-touch { - margin-left: 33.3333333333%; } - .column.is-5-touch { - flex: none; - width: 41.6666666667%; } - .column.is-offset-5-touch { - margin-left: 41.6666666667%; } - .column.is-6-touch { - flex: none; - width: 50%; } - .column.is-offset-6-touch { - margin-left: 50%; } - .column.is-7-touch { - flex: none; - width: 58.3333333333%; } - .column.is-offset-7-touch { - margin-left: 58.3333333333%; } - .column.is-8-touch { - flex: none; - width: 66.6666666667%; } - .column.is-offset-8-touch { - margin-left: 66.6666666667%; } - .column.is-9-touch { - flex: none; - width: 75%; } - .column.is-offset-9-touch { - margin-left: 75%; } - .column.is-10-touch { - flex: none; - width: 83.3333333333%; } - .column.is-offset-10-touch { - margin-left: 83.3333333333%; } - .column.is-11-touch { - flex: none; - width: 91.6666666667%; } - .column.is-offset-11-touch { - margin-left: 91.6666666667%; } - .column.is-12-touch { - flex: none; - width: 100%; } - .column.is-offset-12-touch { - margin-left: 100%; } } - @media screen and (min-width: 1056px) { - .column.is-narrow-desktop { - flex: none; } - .column.is-full-desktop { - flex: none; - width: 100%; } - .column.is-three-quarters-desktop { - flex: none; - width: 75%; } - .column.is-two-thirds-desktop { - flex: none; - width: 66.6666%; } - .column.is-half-desktop { - flex: none; - width: 50%; } - .column.is-one-third-desktop { - flex: none; - width: 33.3333%; } - .column.is-one-quarter-desktop { - flex: none; - width: 25%; } - .column.is-one-fifth-desktop { - flex: none; - width: 20%; } - .column.is-two-fifths-desktop { - flex: none; - width: 40%; } - .column.is-three-fifths-desktop { - flex: none; - width: 60%; } - .column.is-four-fifths-desktop { - flex: none; - width: 80%; } - .column.is-offset-three-quarters-desktop { - margin-left: 75%; } - .column.is-offset-two-thirds-desktop { - margin-left: 66.6666%; } - .column.is-offset-half-desktop { - margin-left: 50%; } - .column.is-offset-one-third-desktop { - margin-left: 33.3333%; } - .column.is-offset-one-quarter-desktop { - margin-left: 25%; } - .column.is-offset-one-fifth-desktop { - margin-left: 20%; } - .column.is-offset-two-fifths-desktop { - margin-left: 40%; } - .column.is-offset-three-fifths-desktop { - margin-left: 60%; } - .column.is-offset-four-fifths-desktop { - margin-left: 80%; } - .column.is-0-desktop { - flex: none; - width: 0%; } - .column.is-offset-0-desktop { - margin-left: 0%; } - .column.is-1-desktop { - flex: none; - width: 8.3333333333%; } - .column.is-offset-1-desktop { - margin-left: 8.3333333333%; } - .column.is-2-desktop { - flex: none; - width: 16.6666666667%; } - .column.is-offset-2-desktop { - margin-left: 16.6666666667%; } - .column.is-3-desktop { - flex: none; - width: 25%; } - .column.is-offset-3-desktop { - margin-left: 25%; } - .column.is-4-desktop { - flex: none; - width: 33.3333333333%; } - .column.is-offset-4-desktop { - margin-left: 33.3333333333%; } - .column.is-5-desktop { - flex: none; - width: 41.6666666667%; } - .column.is-offset-5-desktop { - margin-left: 41.6666666667%; } - .column.is-6-desktop { - flex: none; - width: 50%; } - .column.is-offset-6-desktop { - margin-left: 50%; } - .column.is-7-desktop { - flex: none; - width: 58.3333333333%; } - .column.is-offset-7-desktop { - margin-left: 58.3333333333%; } - .column.is-8-desktop { - flex: none; - width: 66.6666666667%; } - .column.is-offset-8-desktop { - margin-left: 66.6666666667%; } - .column.is-9-desktop { - flex: none; - width: 75%; } - .column.is-offset-9-desktop { - margin-left: 75%; } - .column.is-10-desktop { - flex: none; - width: 83.3333333333%; } - .column.is-offset-10-desktop { - margin-left: 83.3333333333%; } - .column.is-11-desktop { - flex: none; - width: 91.6666666667%; } - .column.is-offset-11-desktop { - margin-left: 91.6666666667%; } - .column.is-12-desktop { - flex: none; - width: 100%; } - .column.is-offset-12-desktop { - margin-left: 100%; } } - @media screen and (min-width: 1216px) { - .column.is-narrow-widescreen { - flex: none; } - .column.is-full-widescreen { - flex: none; - width: 100%; } - .column.is-three-quarters-widescreen { - flex: none; - width: 75%; } - .column.is-two-thirds-widescreen { - flex: none; - width: 66.6666%; } - .column.is-half-widescreen { - flex: none; - width: 50%; } - .column.is-one-third-widescreen { - flex: none; - width: 33.3333%; } - .column.is-one-quarter-widescreen { - flex: none; - width: 25%; } - .column.is-one-fifth-widescreen { - flex: none; - width: 20%; } - .column.is-two-fifths-widescreen { - flex: none; - width: 40%; } - .column.is-three-fifths-widescreen { - flex: none; - width: 60%; } - .column.is-four-fifths-widescreen { - flex: none; - width: 80%; } - .column.is-offset-three-quarters-widescreen { - margin-left: 75%; } - .column.is-offset-two-thirds-widescreen { - margin-left: 66.6666%; } - .column.is-offset-half-widescreen { - margin-left: 50%; } - .column.is-offset-one-third-widescreen { - margin-left: 33.3333%; } - .column.is-offset-one-quarter-widescreen { - margin-left: 25%; } - .column.is-offset-one-fifth-widescreen { - margin-left: 20%; } - .column.is-offset-two-fifths-widescreen { - margin-left: 40%; } - .column.is-offset-three-fifths-widescreen { - margin-left: 60%; } - .column.is-offset-four-fifths-widescreen { - margin-left: 80%; } - .column.is-0-widescreen { - flex: none; - width: 0%; } - .column.is-offset-0-widescreen { - margin-left: 0%; } - .column.is-1-widescreen { - flex: none; - width: 8.3333333333%; } - .column.is-offset-1-widescreen { - margin-left: 8.3333333333%; } - .column.is-2-widescreen { - flex: none; - width: 16.6666666667%; } - .column.is-offset-2-widescreen { - margin-left: 16.6666666667%; } - .column.is-3-widescreen { - flex: none; - width: 25%; } - .column.is-offset-3-widescreen { - margin-left: 25%; } - .column.is-4-widescreen { - flex: none; - width: 33.3333333333%; } - .column.is-offset-4-widescreen { - margin-left: 33.3333333333%; } - .column.is-5-widescreen { - flex: none; - width: 41.6666666667%; } - .column.is-offset-5-widescreen { - margin-left: 41.6666666667%; } - .column.is-6-widescreen { - flex: none; - width: 50%; } - .column.is-offset-6-widescreen { - margin-left: 50%; } - .column.is-7-widescreen { - flex: none; - width: 58.3333333333%; } - .column.is-offset-7-widescreen { - margin-left: 58.3333333333%; } - .column.is-8-widescreen { - flex: none; - width: 66.6666666667%; } - .column.is-offset-8-widescreen { - margin-left: 66.6666666667%; } - .column.is-9-widescreen { - flex: none; - width: 75%; } - .column.is-offset-9-widescreen { - margin-left: 75%; } - .column.is-10-widescreen { - flex: none; - width: 83.3333333333%; } - .column.is-offset-10-widescreen { - margin-left: 83.3333333333%; } - .column.is-11-widescreen { - flex: none; - width: 91.6666666667%; } - .column.is-offset-11-widescreen { - margin-left: 91.6666666667%; } - .column.is-12-widescreen { - flex: none; - width: 100%; } - .column.is-offset-12-widescreen { - margin-left: 100%; } } - @media screen and (min-width: 1408px) { - .column.is-narrow-fullhd { - flex: none; } - .column.is-full-fullhd { - flex: none; - width: 100%; } - .column.is-three-quarters-fullhd { - flex: none; - width: 75%; } - .column.is-two-thirds-fullhd { - flex: none; - width: 66.6666%; } - .column.is-half-fullhd { - flex: none; - width: 50%; } - .column.is-one-third-fullhd { - flex: none; - width: 33.3333%; } - .column.is-one-quarter-fullhd { - flex: none; - width: 25%; } - .column.is-one-fifth-fullhd { - flex: none; - width: 20%; } - .column.is-two-fifths-fullhd { - flex: none; - width: 40%; } - .column.is-three-fifths-fullhd { - flex: none; - width: 60%; } - .column.is-four-fifths-fullhd { - flex: none; - width: 80%; } - .column.is-offset-three-quarters-fullhd { - margin-left: 75%; } - .column.is-offset-two-thirds-fullhd { - margin-left: 66.6666%; } - .column.is-offset-half-fullhd { - margin-left: 50%; } - .column.is-offset-one-third-fullhd { - margin-left: 33.3333%; } - .column.is-offset-one-quarter-fullhd { - margin-left: 25%; } - .column.is-offset-one-fifth-fullhd { - margin-left: 20%; } - .column.is-offset-two-fifths-fullhd { - margin-left: 40%; } - .column.is-offset-three-fifths-fullhd { - margin-left: 60%; } - .column.is-offset-four-fifths-fullhd { - margin-left: 80%; } - .column.is-0-fullhd { - flex: none; - width: 0%; } - .column.is-offset-0-fullhd { - margin-left: 0%; } - .column.is-1-fullhd { - flex: none; - width: 8.3333333333%; } - .column.is-offset-1-fullhd { - margin-left: 8.3333333333%; } - .column.is-2-fullhd { - flex: none; - width: 16.6666666667%; } - .column.is-offset-2-fullhd { - margin-left: 16.6666666667%; } - .column.is-3-fullhd { - flex: none; - width: 25%; } - .column.is-offset-3-fullhd { - margin-left: 25%; } - .column.is-4-fullhd { - flex: none; - width: 33.3333333333%; } - .column.is-offset-4-fullhd { - margin-left: 33.3333333333%; } - .column.is-5-fullhd { - flex: none; - width: 41.6666666667%; } - .column.is-offset-5-fullhd { - margin-left: 41.6666666667%; } - .column.is-6-fullhd { - flex: none; - width: 50%; } - .column.is-offset-6-fullhd { - margin-left: 50%; } - .column.is-7-fullhd { - flex: none; - width: 58.3333333333%; } - .column.is-offset-7-fullhd { - margin-left: 58.3333333333%; } - .column.is-8-fullhd { - flex: none; - width: 66.6666666667%; } - .column.is-offset-8-fullhd { - margin-left: 66.6666666667%; } - .column.is-9-fullhd { - flex: none; - width: 75%; } - .column.is-offset-9-fullhd { - margin-left: 75%; } - .column.is-10-fullhd { - flex: none; - width: 83.3333333333%; } - .column.is-offset-10-fullhd { - margin-left: 83.3333333333%; } - .column.is-11-fullhd { - flex: none; - width: 91.6666666667%; } - .column.is-offset-11-fullhd { - margin-left: 91.6666666667%; } - .column.is-12-fullhd { - flex: none; - width: 100%; } - .column.is-offset-12-fullhd { - margin-left: 100%; } } -.columns { - margin-left: -0.75rem; - margin-right: -0.75rem; - margin-top: -0.75rem; } - .columns:last-child { - margin-bottom: -0.75rem; } - .columns:not(:last-child) { - margin-bottom: calc(1.5rem - 0.75rem); } - .columns.is-centered { - justify-content: center; } - .columns.is-gapless { - margin-left: 0; - margin-right: 0; - margin-top: 0; } - .columns.is-gapless > .column { - margin: 0; - padding: 0 !important; } - .columns.is-gapless:not(:last-child) { - margin-bottom: 1.5rem; } - .columns.is-gapless:last-child { - margin-bottom: 0; } - .columns.is-mobile { - display: flex; } - .columns.is-multiline { - flex-wrap: wrap; } - .columns.is-vcentered { - align-items: center; } - @media screen and (min-width: 769px), print { - .columns:not(.is-desktop) { - display: flex; } } - @media screen and (min-width: 1056px) { - .columns.is-desktop { - display: flex; } } -.columns.is-variable { - --columnGap: 0.75rem; - margin-left: calc(-1 * var(--columnGap)); - margin-right: calc(-1 * var(--columnGap)); } - .columns.is-variable .column { - padding-left: var(--columnGap); - padding-right: var(--columnGap); } - .columns.is-variable.is-0 { - --columnGap: 0rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-0-mobile { - --columnGap: 0rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-0-tablet { - --columnGap: 0rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-0-tablet-only { - --columnGap: 0rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-0-touch { - --columnGap: 0rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-0-desktop { - --columnGap: 0rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-0-desktop-only { - --columnGap: 0rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-0-widescreen { - --columnGap: 0rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-0-widescreen-only { - --columnGap: 0rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-0-fullhd { - --columnGap: 0rem; } } - .columns.is-variable.is-1 { - --columnGap: 0.25rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-1-mobile { - --columnGap: 0.25rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-1-tablet { - --columnGap: 0.25rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-1-tablet-only { - --columnGap: 0.25rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-1-touch { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-1-desktop { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-1-desktop-only { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-1-widescreen { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-1-widescreen-only { - --columnGap: 0.25rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-1-fullhd { - --columnGap: 0.25rem; } } - .columns.is-variable.is-2 { - --columnGap: 0.5rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-2-mobile { - --columnGap: 0.5rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-2-tablet { - --columnGap: 0.5rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-2-tablet-only { - --columnGap: 0.5rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-2-touch { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-2-desktop { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-2-desktop-only { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-2-widescreen { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-2-widescreen-only { - --columnGap: 0.5rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-2-fullhd { - --columnGap: 0.5rem; } } - .columns.is-variable.is-3 { - --columnGap: 0.75rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-3-mobile { - --columnGap: 0.75rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-3-tablet { - --columnGap: 0.75rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-3-tablet-only { - --columnGap: 0.75rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-3-touch { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-3-desktop { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-3-desktop-only { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-3-widescreen { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-3-widescreen-only { - --columnGap: 0.75rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-3-fullhd { - --columnGap: 0.75rem; } } - .columns.is-variable.is-4 { - --columnGap: 1rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-4-mobile { - --columnGap: 1rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-4-tablet { - --columnGap: 1rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-4-tablet-only { - --columnGap: 1rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-4-touch { - --columnGap: 1rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-4-desktop { - --columnGap: 1rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-4-desktop-only { - --columnGap: 1rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-4-widescreen { - --columnGap: 1rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-4-widescreen-only { - --columnGap: 1rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-4-fullhd { - --columnGap: 1rem; } } - .columns.is-variable.is-5 { - --columnGap: 1.25rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-5-mobile { - --columnGap: 1.25rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-5-tablet { - --columnGap: 1.25rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-5-tablet-only { - --columnGap: 1.25rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-5-touch { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-5-desktop { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-5-desktop-only { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-5-widescreen { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-5-widescreen-only { - --columnGap: 1.25rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-5-fullhd { - --columnGap: 1.25rem; } } - .columns.is-variable.is-6 { - --columnGap: 1.5rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-6-mobile { - --columnGap: 1.5rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-6-tablet { - --columnGap: 1.5rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-6-tablet-only { - --columnGap: 1.5rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-6-touch { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-6-desktop { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-6-desktop-only { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-6-widescreen { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-6-widescreen-only { - --columnGap: 1.5rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-6-fullhd { - --columnGap: 1.5rem; } } - .columns.is-variable.is-7 { - --columnGap: 1.75rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-7-mobile { - --columnGap: 1.75rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-7-tablet { - --columnGap: 1.75rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-7-tablet-only { - --columnGap: 1.75rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-7-touch { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-7-desktop { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-7-desktop-only { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-7-widescreen { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-7-widescreen-only { - --columnGap: 1.75rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-7-fullhd { - --columnGap: 1.75rem; } } - .columns.is-variable.is-8 { - --columnGap: 2rem; } - @media screen and (max-width: 768px) { - .columns.is-variable.is-8-mobile { - --columnGap: 2rem; } } - @media screen and (min-width: 769px), print { - .columns.is-variable.is-8-tablet { - --columnGap: 2rem; } } - @media screen and (min-width: 769px) and (max-width: 1055px) { - .columns.is-variable.is-8-tablet-only { - --columnGap: 2rem; } } - @media screen and (max-width: 1055px) { - .columns.is-variable.is-8-touch { - --columnGap: 2rem; } } - @media screen and (min-width: 1056px) { - .columns.is-variable.is-8-desktop { - --columnGap: 2rem; } } - @media screen and (min-width: 1056px) and (max-width: 1215px) { - .columns.is-variable.is-8-desktop-only { - --columnGap: 2rem; } } - @media screen and (min-width: 1216px) { - .columns.is-variable.is-8-widescreen { - --columnGap: 2rem; } } - @media screen and (min-width: 1216px) and (max-width: 1407px) { - .columns.is-variable.is-8-widescreen-only { - --columnGap: 2rem; } } - @media screen and (min-width: 1408px) { - .columns.is-variable.is-8-fullhd { - --columnGap: 2rem; } } -.tile { - align-items: stretch; - display: block; - flex-basis: 0; - flex-grow: 1; - flex-shrink: 1; - min-height: min-content; } - .tile.is-ancestor { - margin-left: -0.75rem; - margin-right: -0.75rem; - margin-top: -0.75rem; } - .tile.is-ancestor:last-child { - margin-bottom: -0.75rem; } - .tile.is-ancestor:not(:last-child) { - margin-bottom: 0.75rem; } - .tile.is-child { - margin: 0 !important; } - .tile.is-parent { - padding: 0.75rem; } - .tile.is-vertical { - flex-direction: column; } - .tile.is-vertical > .tile.is-child:not(:last-child) { - margin-bottom: 1.5rem !important; } - @media screen and (min-width: 769px), print { - .tile:not(.is-child) { - display: flex; } - .tile.is-1 { - flex: none; - width: 8.3333333333%; } - .tile.is-2 { - flex: none; - width: 16.6666666667%; } - .tile.is-3 { - flex: none; - width: 25%; } - .tile.is-4 { - flex: none; - width: 33.3333333333%; } - .tile.is-5 { - flex: none; - width: 41.6666666667%; } - .tile.is-6 { - flex: none; - width: 50%; } - .tile.is-7 { - flex: none; - width: 58.3333333333%; } - .tile.is-8 { - flex: none; - width: 66.6666666667%; } - .tile.is-9 { - flex: none; - width: 75%; } - .tile.is-10 { - flex: none; - width: 83.3333333333%; } - .tile.is-11 { - flex: none; - width: 91.6666666667%; } - .tile.is-12 { - flex: none; - width: 100%; } } -.hero { - align-items: stretch; - display: flex; - flex-direction: column; - justify-content: space-between; } - .hero .navbar { - background: none; } - .hero .tabs ul { - border-bottom: none; } - .hero.is-white { - background-color: white; - color: #0a0a0a; } - .hero.is-white a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-white strong { - color: inherit; } - .hero.is-white .title { - color: #0a0a0a; } - .hero.is-white .subtitle { - color: rgba(10, 10, 10, 0.9); } - .hero.is-white .subtitle a:not(.button), - .hero.is-white .subtitle strong { - color: #0a0a0a; } - @media screen and (max-width: 1055px) { - .hero.is-white .navbar-menu { - background-color: white; } } - .hero.is-white .navbar-item, - .hero.is-white .navbar-link { - color: rgba(10, 10, 10, 0.7); } - .hero.is-white a.navbar-item:hover, .hero.is-white a.navbar-item.is-active, - .hero.is-white .navbar-link:hover, - .hero.is-white .navbar-link.is-active { - background-color: #f2f2f2; - color: #0a0a0a; } - .hero.is-white .tabs a { - color: #0a0a0a; - opacity: 0.9; } - .hero.is-white .tabs a:hover { - opacity: 1; } - .hero.is-white .tabs li.is-active a { - opacity: 1; } - .hero.is-white .tabs.is-boxed a, .hero.is-white .tabs.is-toggle a { - color: #0a0a0a; } - .hero.is-white .tabs.is-boxed a:hover, .hero.is-white .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-white .tabs.is-boxed li.is-active a, .hero.is-white .tabs.is-boxed li.is-active a:hover, .hero.is-white .tabs.is-toggle li.is-active a, .hero.is-white .tabs.is-toggle li.is-active a:hover { - background-color: #0a0a0a; - border-color: #0a0a0a; - color: white; } - .hero.is-white.is-bold { - background-image: linear-gradient(141deg, #e8e3e4 0%, white 71%, white 100%); } - @media screen and (max-width: 768px) { - .hero.is-white.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #e8e3e4 0%, white 71%, white 100%); } } - .hero.is-black { - background-color: #0a0a0a; - color: white; } - .hero.is-black a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-black strong { - color: inherit; } - .hero.is-black .title { - color: white; } - .hero.is-black .subtitle { - color: rgba(255, 255, 255, 0.9); } - .hero.is-black .subtitle a:not(.button), - .hero.is-black .subtitle strong { - color: white; } - @media screen and (max-width: 1055px) { - .hero.is-black .navbar-menu { - background-color: #0a0a0a; } } - .hero.is-black .navbar-item, - .hero.is-black .navbar-link { - color: rgba(255, 255, 255, 0.7); } - .hero.is-black a.navbar-item:hover, .hero.is-black a.navbar-item.is-active, - .hero.is-black .navbar-link:hover, - .hero.is-black .navbar-link.is-active { - background-color: black; - color: white; } - .hero.is-black .tabs a { - color: white; - opacity: 0.9; } - .hero.is-black .tabs a:hover { - opacity: 1; } - .hero.is-black .tabs li.is-active a { - opacity: 1; } - .hero.is-black .tabs.is-boxed a, .hero.is-black .tabs.is-toggle a { - color: white; } - .hero.is-black .tabs.is-boxed a:hover, .hero.is-black .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-black .tabs.is-boxed li.is-active a, .hero.is-black .tabs.is-boxed li.is-active a:hover, .hero.is-black .tabs.is-toggle li.is-active a, .hero.is-black .tabs.is-toggle li.is-active a:hover { - background-color: white; - border-color: white; - color: #0a0a0a; } - .hero.is-black.is-bold { - background-image: linear-gradient(141deg, black 0%, #0a0a0a 71%, #181616 100%); } - @media screen and (max-width: 768px) { - .hero.is-black.is-bold .navbar-menu { - background-image: linear-gradient(141deg, black 0%, #0a0a0a 71%, #181616 100%); } } - .hero.is-light { - background-color: whitesmoke; - color: #363636; } - .hero.is-light a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-light strong { - color: inherit; } - .hero.is-light .title { - color: #363636; } - .hero.is-light .subtitle { - color: rgba(54, 54, 54, 0.9); } - .hero.is-light .subtitle a:not(.button), - .hero.is-light .subtitle strong { - color: #363636; } - @media screen and (max-width: 1055px) { - .hero.is-light .navbar-menu { - background-color: whitesmoke; } } - .hero.is-light .navbar-item, - .hero.is-light .navbar-link { - color: rgba(54, 54, 54, 0.7); } - .hero.is-light a.navbar-item:hover, .hero.is-light a.navbar-item.is-active, - .hero.is-light .navbar-link:hover, - .hero.is-light .navbar-link.is-active { - background-color: #e8e8e8; - color: #363636; } - .hero.is-light .tabs a { - color: #363636; - opacity: 0.9; } - .hero.is-light .tabs a:hover { - opacity: 1; } - .hero.is-light .tabs li.is-active a { - opacity: 1; } - .hero.is-light .tabs.is-boxed a, .hero.is-light .tabs.is-toggle a { - color: #363636; } - .hero.is-light .tabs.is-boxed a:hover, .hero.is-light .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-light .tabs.is-boxed li.is-active a, .hero.is-light .tabs.is-boxed li.is-active a:hover, .hero.is-light .tabs.is-toggle li.is-active a, .hero.is-light .tabs.is-toggle li.is-active a:hover { - background-color: #363636; - border-color: #363636; - color: whitesmoke; } - .hero.is-light.is-bold { - background-image: linear-gradient(141deg, #dfd8d9 0%, whitesmoke 71%, white 100%); } - @media screen and (max-width: 768px) { - .hero.is-light.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #dfd8d9 0%, whitesmoke 71%, white 100%); } } - .hero.is-dark, .content kbd.hero { - background-color: #363636; - color: whitesmoke; } - .hero.is-dark a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), .content kbd.hero a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-dark strong, - .content kbd.hero strong { - color: inherit; } - .hero.is-dark .title, .content kbd.hero .title { - color: whitesmoke; } - .hero.is-dark .subtitle, .content kbd.hero .subtitle { - color: rgba(245, 245, 245, 0.9); } - .hero.is-dark .subtitle a:not(.button), .content kbd.hero .subtitle a:not(.button), - .hero.is-dark .subtitle strong, - .content kbd.hero .subtitle strong { - color: whitesmoke; } - @media screen and (max-width: 1055px) { - .hero.is-dark .navbar-menu, .content kbd.hero .navbar-menu { - background-color: #363636; } } - .hero.is-dark .navbar-item, .content kbd.hero .navbar-item, - .hero.is-dark .navbar-link, - .content kbd.hero .navbar-link { - color: rgba(245, 245, 245, 0.7); } - .hero.is-dark a.navbar-item:hover, .content kbd.hero a.navbar-item:hover, .hero.is-dark a.navbar-item.is-active, .content kbd.hero a.navbar-item.is-active, - .hero.is-dark .navbar-link:hover, - .content kbd.hero .navbar-link:hover, - .hero.is-dark .navbar-link.is-active, - .content kbd.hero .navbar-link.is-active { - background-color: #292929; - color: whitesmoke; } - .hero.is-dark .tabs a, .content kbd.hero .tabs a { - color: whitesmoke; - opacity: 0.9; } - .hero.is-dark .tabs a:hover, .content kbd.hero .tabs a:hover { - opacity: 1; } - .hero.is-dark .tabs li.is-active a, .content kbd.hero .tabs li.is-active a { - opacity: 1; } - .hero.is-dark .tabs.is-boxed a, .content kbd.hero .tabs.is-boxed a, .hero.is-dark .tabs.is-toggle a, .content kbd.hero .tabs.is-toggle a { - color: whitesmoke; } - .hero.is-dark .tabs.is-boxed a:hover, .content kbd.hero .tabs.is-boxed a:hover, .hero.is-dark .tabs.is-toggle a:hover, .content kbd.hero .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-dark .tabs.is-boxed li.is-active a, .content kbd.hero .tabs.is-boxed li.is-active a, .hero.is-dark .tabs.is-boxed li.is-active a:hover, .hero.is-dark .tabs.is-toggle li.is-active a, .content kbd.hero .tabs.is-toggle li.is-active a, .hero.is-dark .tabs.is-toggle li.is-active a:hover { - background-color: whitesmoke; - border-color: whitesmoke; - color: #363636; } - .hero.is-dark.is-bold, .content kbd.hero.is-bold { - background-image: linear-gradient(141deg, #1f191a 0%, #363636 71%, #46403f 100%); } - @media screen and (max-width: 768px) { - .hero.is-dark.is-bold .navbar-menu, .content kbd.hero.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #1f191a 0%, #363636 71%, #46403f 100%); } } - .hero.is-primary, .docstring > section > a.hero.docs-sourcelink { - background-color: #4eb5de; - color: #fff; } - .hero.is-primary a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), .docstring > section > a.hero.docs-sourcelink a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-primary strong, - .docstring > section > a.hero.docs-sourcelink strong { - color: inherit; } - .hero.is-primary .title, .docstring > section > a.hero.docs-sourcelink .title { - color: #fff; } - .hero.is-primary .subtitle, .docstring > section > a.hero.docs-sourcelink .subtitle { - color: rgba(255, 255, 255, 0.9); } - .hero.is-primary .subtitle a:not(.button), .docstring > section > a.hero.docs-sourcelink .subtitle a:not(.button), - .hero.is-primary .subtitle strong, - .docstring > section > a.hero.docs-sourcelink .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - .hero.is-primary .navbar-menu, .docstring > section > a.hero.docs-sourcelink .navbar-menu { - background-color: #4eb5de; } } - .hero.is-primary .navbar-item, .docstring > section > a.hero.docs-sourcelink .navbar-item, - .hero.is-primary .navbar-link, - .docstring > section > a.hero.docs-sourcelink .navbar-link { - color: rgba(255, 255, 255, 0.7); } - .hero.is-primary a.navbar-item:hover, .docstring > section > a.hero.docs-sourcelink a.navbar-item:hover, .hero.is-primary a.navbar-item.is-active, .docstring > section > a.hero.docs-sourcelink a.navbar-item.is-active, - .hero.is-primary .navbar-link:hover, - .docstring > section > a.hero.docs-sourcelink .navbar-link:hover, - .hero.is-primary .navbar-link.is-active, - .docstring > section > a.hero.docs-sourcelink .navbar-link.is-active { - background-color: #39acda; - color: #fff; } - .hero.is-primary .tabs a, .docstring > section > a.hero.docs-sourcelink .tabs a { - color: #fff; - opacity: 0.9; } - .hero.is-primary .tabs a:hover, .docstring > section > a.hero.docs-sourcelink .tabs a:hover { - opacity: 1; } - .hero.is-primary .tabs li.is-active a, .docstring > section > a.hero.docs-sourcelink .tabs li.is-active a { - opacity: 1; } - .hero.is-primary .tabs.is-boxed a, .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed a, .hero.is-primary .tabs.is-toggle a, .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle a { - color: #fff; } - .hero.is-primary .tabs.is-boxed a:hover, .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed a:hover, .hero.is-primary .tabs.is-toggle a:hover, .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-primary .tabs.is-boxed li.is-active a, .docstring > section > a.hero.docs-sourcelink .tabs.is-boxed li.is-active a, .hero.is-primary .tabs.is-boxed li.is-active a:hover, .hero.is-primary .tabs.is-toggle li.is-active a, .docstring > section > a.hero.docs-sourcelink .tabs.is-toggle li.is-active a, .hero.is-primary .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #4eb5de; } - .hero.is-primary.is-bold, .docstring > section > a.hero.is-bold.docs-sourcelink { - background-image: linear-gradient(141deg, #1bc7de 0%, #4eb5de 71%, #5fa9e7 100%); } - @media screen and (max-width: 768px) { - .hero.is-primary.is-bold .navbar-menu, .docstring > section > a.hero.is-bold.docs-sourcelink .navbar-menu { - background-image: linear-gradient(141deg, #1bc7de 0%, #4eb5de 71%, #5fa9e7 100%); } } - .hero.is-link { - background-color: #2e63b8; - color: #fff; } - .hero.is-link a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-link strong { - color: inherit; } - .hero.is-link .title { - color: #fff; } - .hero.is-link .subtitle { - color: rgba(255, 255, 255, 0.9); } - .hero.is-link .subtitle a:not(.button), - .hero.is-link .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - .hero.is-link .navbar-menu { - background-color: #2e63b8; } } - .hero.is-link .navbar-item, - .hero.is-link .navbar-link { - color: rgba(255, 255, 255, 0.7); } - .hero.is-link a.navbar-item:hover, .hero.is-link a.navbar-item.is-active, - .hero.is-link .navbar-link:hover, - .hero.is-link .navbar-link.is-active { - background-color: #2958a4; - color: #fff; } - .hero.is-link .tabs a { - color: #fff; - opacity: 0.9; } - .hero.is-link .tabs a:hover { - opacity: 1; } - .hero.is-link .tabs li.is-active a { - opacity: 1; } - .hero.is-link .tabs.is-boxed a, .hero.is-link .tabs.is-toggle a { - color: #fff; } - .hero.is-link .tabs.is-boxed a:hover, .hero.is-link .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-link .tabs.is-boxed li.is-active a, .hero.is-link .tabs.is-boxed li.is-active a:hover, .hero.is-link .tabs.is-toggle li.is-active a, .hero.is-link .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #2e63b8; } - .hero.is-link.is-bold { - background-image: linear-gradient(141deg, #1b6098 0%, #2e63b8 71%, #2d51d2 100%); } - @media screen and (max-width: 768px) { - .hero.is-link.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #1b6098 0%, #2e63b8 71%, #2d51d2 100%); } } - .hero.is-info { - background-color: #209cee; - color: #fff; } - .hero.is-info a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-info strong { - color: inherit; } - .hero.is-info .title { - color: #fff; } - .hero.is-info .subtitle { - color: rgba(255, 255, 255, 0.9); } - .hero.is-info .subtitle a:not(.button), - .hero.is-info .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - .hero.is-info .navbar-menu { - background-color: #209cee; } } - .hero.is-info .navbar-item, - .hero.is-info .navbar-link { - color: rgba(255, 255, 255, 0.7); } - .hero.is-info a.navbar-item:hover, .hero.is-info a.navbar-item.is-active, - .hero.is-info .navbar-link:hover, - .hero.is-info .navbar-link.is-active { - background-color: #1190e3; - color: #fff; } - .hero.is-info .tabs a { - color: #fff; - opacity: 0.9; } - .hero.is-info .tabs a:hover { - opacity: 1; } - .hero.is-info .tabs li.is-active a { - opacity: 1; } - .hero.is-info .tabs.is-boxed a, .hero.is-info .tabs.is-toggle a { - color: #fff; } - .hero.is-info .tabs.is-boxed a:hover, .hero.is-info .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-info .tabs.is-boxed li.is-active a, .hero.is-info .tabs.is-boxed li.is-active a:hover, .hero.is-info .tabs.is-toggle li.is-active a, .hero.is-info .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #209cee; } - .hero.is-info.is-bold { - background-image: linear-gradient(141deg, #05a6d6 0%, #209cee 71%, #3287f5 100%); } - @media screen and (max-width: 768px) { - .hero.is-info.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #05a6d6 0%, #209cee 71%, #3287f5 100%); } } - .hero.is-success { - background-color: #22c35b; - color: #fff; } - .hero.is-success a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-success strong { - color: inherit; } - .hero.is-success .title { - color: #fff; } - .hero.is-success .subtitle { - color: rgba(255, 255, 255, 0.9); } - .hero.is-success .subtitle a:not(.button), - .hero.is-success .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - .hero.is-success .navbar-menu { - background-color: #22c35b; } } - .hero.is-success .navbar-item, - .hero.is-success .navbar-link { - color: rgba(255, 255, 255, 0.7); } - .hero.is-success a.navbar-item:hover, .hero.is-success a.navbar-item.is-active, - .hero.is-success .navbar-link:hover, - .hero.is-success .navbar-link.is-active { - background-color: #1ead51; - color: #fff; } - .hero.is-success .tabs a { - color: #fff; - opacity: 0.9; } - .hero.is-success .tabs a:hover { - opacity: 1; } - .hero.is-success .tabs li.is-active a { - opacity: 1; } - .hero.is-success .tabs.is-boxed a, .hero.is-success .tabs.is-toggle a { - color: #fff; } - .hero.is-success .tabs.is-boxed a:hover, .hero.is-success .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-success .tabs.is-boxed li.is-active a, .hero.is-success .tabs.is-boxed li.is-active a:hover, .hero.is-success .tabs.is-toggle li.is-active a, .hero.is-success .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #22c35b; } - .hero.is-success.is-bold { - background-image: linear-gradient(141deg, #12a02c 0%, #22c35b 71%, #1fdf83 100%); } - @media screen and (max-width: 768px) { - .hero.is-success.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #12a02c 0%, #22c35b 71%, #1fdf83 100%); } } - .hero.is-warning { - background-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .hero.is-warning a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-warning strong { - color: inherit; } - .hero.is-warning .title { - color: rgba(0, 0, 0, 0.7); } - .hero.is-warning .subtitle { - color: rgba(0, 0, 0, 0.9); } - .hero.is-warning .subtitle a:not(.button), - .hero.is-warning .subtitle strong { - color: rgba(0, 0, 0, 0.7); } - @media screen and (max-width: 1055px) { - .hero.is-warning .navbar-menu { - background-color: #ffdd57; } } - .hero.is-warning .navbar-item, - .hero.is-warning .navbar-link { - color: rgba(0, 0, 0, 0.7); } - .hero.is-warning a.navbar-item:hover, .hero.is-warning a.navbar-item.is-active, - .hero.is-warning .navbar-link:hover, - .hero.is-warning .navbar-link.is-active { - background-color: #ffd83e; - color: rgba(0, 0, 0, 0.7); } - .hero.is-warning .tabs a { - color: rgba(0, 0, 0, 0.7); - opacity: 0.9; } - .hero.is-warning .tabs a:hover { - opacity: 1; } - .hero.is-warning .tabs li.is-active a { - opacity: 1; } - .hero.is-warning .tabs.is-boxed a, .hero.is-warning .tabs.is-toggle a { - color: rgba(0, 0, 0, 0.7); } - .hero.is-warning .tabs.is-boxed a:hover, .hero.is-warning .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-warning .tabs.is-boxed li.is-active a, .hero.is-warning .tabs.is-boxed li.is-active a:hover, .hero.is-warning .tabs.is-toggle li.is-active a, .hero.is-warning .tabs.is-toggle li.is-active a:hover { - background-color: rgba(0, 0, 0, 0.7); - border-color: rgba(0, 0, 0, 0.7); - color: #ffdd57; } - .hero.is-warning.is-bold { - background-image: linear-gradient(141deg, #ffae24 0%, #ffdd57 71%, #fffa71 100%); } - @media screen and (max-width: 768px) { - .hero.is-warning.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #ffae24 0%, #ffdd57 71%, #fffa71 100%); } } - .hero.is-danger { - background-color: #da0b00; - color: #fff; } - .hero.is-danger a:not(.button):not(.dropdown-item):not(.tag):not(.pagination-link.is-current), - .hero.is-danger strong { - color: inherit; } - .hero.is-danger .title { - color: #fff; } - .hero.is-danger .subtitle { - color: rgba(255, 255, 255, 0.9); } - .hero.is-danger .subtitle a:not(.button), - .hero.is-danger .subtitle strong { - color: #fff; } - @media screen and (max-width: 1055px) { - .hero.is-danger .navbar-menu { - background-color: #da0b00; } } - .hero.is-danger .navbar-item, - .hero.is-danger .navbar-link { - color: rgba(255, 255, 255, 0.7); } - .hero.is-danger a.navbar-item:hover, .hero.is-danger a.navbar-item.is-active, - .hero.is-danger .navbar-link:hover, - .hero.is-danger .navbar-link.is-active { - background-color: #c10a00; - color: #fff; } - .hero.is-danger .tabs a { - color: #fff; - opacity: 0.9; } - .hero.is-danger .tabs a:hover { - opacity: 1; } - .hero.is-danger .tabs li.is-active a { - opacity: 1; } - .hero.is-danger .tabs.is-boxed a, .hero.is-danger .tabs.is-toggle a { - color: #fff; } - .hero.is-danger .tabs.is-boxed a:hover, .hero.is-danger .tabs.is-toggle a:hover { - background-color: rgba(10, 10, 10, 0.1); } - .hero.is-danger .tabs.is-boxed li.is-active a, .hero.is-danger .tabs.is-boxed li.is-active a:hover, .hero.is-danger .tabs.is-toggle li.is-active a, .hero.is-danger .tabs.is-toggle li.is-active a:hover { - background-color: #fff; - border-color: #fff; - color: #da0b00; } - .hero.is-danger.is-bold { - background-image: linear-gradient(141deg, #a70013 0%, #da0b00 71%, #f43500 100%); } - @media screen and (max-width: 768px) { - .hero.is-danger.is-bold .navbar-menu { - background-image: linear-gradient(141deg, #a70013 0%, #da0b00 71%, #f43500 100%); } } - .hero.is-small .hero-body, #documenter .docs-sidebar form.docs-search > input.hero .hero-body { - padding-bottom: 1.5rem; - padding-top: 1.5rem; } - @media screen and (min-width: 769px), print { - .hero.is-medium .hero-body { - padding-bottom: 9rem; - padding-top: 9rem; } } - @media screen and (min-width: 769px), print { - .hero.is-large .hero-body { - padding-bottom: 18rem; - padding-top: 18rem; } } - .hero.is-halfheight .hero-body, .hero.is-fullheight .hero-body, .hero.is-fullheight-with-navbar .hero-body { - align-items: center; - display: flex; } - .hero.is-halfheight .hero-body > .container, .hero.is-fullheight .hero-body > .container, .hero.is-fullheight-with-navbar .hero-body > .container { - flex-grow: 1; - flex-shrink: 1; } - .hero.is-halfheight { - min-height: 50vh; } - .hero.is-fullheight { - min-height: 100vh; } - -.hero-video { - overflow: hidden; } - .hero-video video { - left: 50%; - min-height: 100%; - min-width: 100%; - position: absolute; - top: 50%; - transform: translate3d(-50%, -50%, 0); } - .hero-video.is-transparent { - opacity: 0.3; } - @media screen and (max-width: 768px) { - .hero-video { - display: none; } } -.hero-buttons { - margin-top: 1.5rem; } - @media screen and (max-width: 768px) { - .hero-buttons .button { - display: flex; } - .hero-buttons .button:not(:last-child) { - margin-bottom: 0.75rem; } } - @media screen and (min-width: 769px), print { - .hero-buttons { - display: flex; - justify-content: center; } - .hero-buttons .button:not(:last-child) { - margin-right: 1.5rem; } } -.hero-head, -.hero-foot { - flex-grow: 0; - flex-shrink: 0; } - -.hero-body { - flex-grow: 1; - flex-shrink: 0; - padding: 3rem 1.5rem; } - -.section { - padding: 3rem 1.5rem; } - @media screen and (min-width: 1056px) { - .section.is-medium { - padding: 9rem 1.5rem; } - .section.is-large { - padding: 18rem 1.5rem; } } -.footer { - background-color: #fafafa; - padding: 3rem 1.5rem 6rem; } - -h1 .docs-heading-anchor, h1 .docs-heading-anchor:hover, h1 .docs-heading-anchor:visited, h2 .docs-heading-anchor, h2 .docs-heading-anchor:hover, h2 .docs-heading-anchor:visited, h3 .docs-heading-anchor, h3 .docs-heading-anchor:hover, h3 .docs-heading-anchor:visited, h4 .docs-heading-anchor, h4 .docs-heading-anchor:hover, h4 .docs-heading-anchor:visited, h5 .docs-heading-anchor, h5 .docs-heading-anchor:hover, h5 .docs-heading-anchor:visited, h6 .docs-heading-anchor, h6 .docs-heading-anchor:hover, h6 .docs-heading-anchor:visited { - color: #222222; } - -h1 .docs-heading-anchor-permalink, h2 .docs-heading-anchor-permalink, h3 .docs-heading-anchor-permalink, h4 .docs-heading-anchor-permalink, h5 .docs-heading-anchor-permalink, h6 .docs-heading-anchor-permalink { - visibility: hidden; - vertical-align: middle; - margin-left: 0.5em; - font-size: 0.7rem; } - h1 .docs-heading-anchor-permalink::before, h2 .docs-heading-anchor-permalink::before, h3 .docs-heading-anchor-permalink::before, h4 .docs-heading-anchor-permalink::before, h5 .docs-heading-anchor-permalink::before, h6 .docs-heading-anchor-permalink::before { - font-family: "Font Awesome 5 Free"; - font-weight: 900; - content: "\f0c1"; } - -h1:hover .docs-heading-anchor-permalink, h2:hover .docs-heading-anchor-permalink, h3:hover .docs-heading-anchor-permalink, h4:hover .docs-heading-anchor-permalink, h5:hover .docs-heading-anchor-permalink, h6:hover .docs-heading-anchor-permalink { - visibility: visible; } - -.docs-dark-only { - display: none !important; } - -pre { - position: relative; - overflow: hidden; } - pre code, pre code.hljs { - padding: 0 0.75rem !important; - overflow: auto; - display: block; } - pre code:first-of-type, pre code.hljs:first-of-type { - padding-top: 0.5rem !important; } - pre code:last-of-type, pre code.hljs:last-of-type { - padding-bottom: 0.5rem !important; } - pre .copy-button { - opacity: 0.2; - transition: opacity 0.2s; - position: absolute; - right: 0em; - top: 0em; - padding: 0.5em; - width: 2.5em; - height: 2.5em; - background: transparent; - border: none; - font-family: "Font Awesome 5 Free"; - color: #222222; - cursor: pointer; - text-align: center; } - pre .copy-button:focus, pre .copy-button:hover { - opacity: 1; - background: rgba(34, 34, 34, 0.1); - color: #2e63b8; } - pre .copy-button.success { - color: #259a12; - opacity: 1; } - pre .copy-button.error { - color: #cb3c33; - opacity: 1; } - pre:hover .copy-button { - opacity: 1; } - -.admonition { - background-color: #b5b5b5; - border-style: solid; - border-width: 1px; - border-color: #363636; - border-radius: 4px; - font-size: 1rem; } - .admonition strong { - color: currentColor; } - .admonition.is-small, #documenter .docs-sidebar form.docs-search > input.admonition { - font-size: 0.75rem; } - .admonition.is-medium { - font-size: 1.25rem; } - .admonition.is-large { - font-size: 1.5rem; } - .admonition.is-default { - background-color: #b5b5b5; - border-color: #363636; } - .admonition.is-default > .admonition-header { - background-color: #363636; - color: #fff; } - .admonition.is-default > .admonition-body { - color: #fff; } - .admonition.is-info { - background-color: #def0fc; - border-color: #209cee; } - .admonition.is-info > .admonition-header { - background-color: #209cee; - color: #fff; } - .admonition.is-info > .admonition-body { - color: rgba(0, 0, 0, 0.7); } - .admonition.is-success { - background-color: #bdf4d1; - border-color: #22c35b; } - .admonition.is-success > .admonition-header { - background-color: #22c35b; - color: #fff; } - .admonition.is-success > .admonition-body { - color: rgba(0, 0, 0, 0.7); } - .admonition.is-warning { - background-color: #fff3c5; - border-color: #ffdd57; } - .admonition.is-warning > .admonition-header { - background-color: #ffdd57; - color: rgba(0, 0, 0, 0.7); } - .admonition.is-warning > .admonition-body { - color: rgba(0, 0, 0, 0.7); } - .admonition.is-danger { - background-color: #ffaba7; - border-color: #da0b00; } - .admonition.is-danger > .admonition-header { - background-color: #da0b00; - color: #fff; } - .admonition.is-danger > .admonition-body { - color: rgba(0, 0, 0, 0.7); } - .admonition.is-compat { - background-color: #bdeff5; - border-color: #1db5c9; } - .admonition.is-compat > .admonition-header { - background-color: #1db5c9; - color: #fff; } - .admonition.is-compat > .admonition-body { - color: rgba(0, 0, 0, 0.7); } - -.admonition-header { - color: #fff; - background-color: #363636; - align-items: center; - font-weight: 700; - justify-content: space-between; - line-height: 1.25; - padding: 0.5rem 0.75rem; - position: relative; } - .admonition-header:before { - font-family: "Font Awesome 5 Free"; - font-weight: 900; - margin-right: 0.75rem; - content: "\f06a"; } - -.admonition-body { - color: #222222; - padding: 0.5rem 0.75rem; } - .admonition-body pre { - background-color: whitesmoke; } - .admonition-body code { - background-color: rgba(0, 0, 0, 0.05); } - -.docstring { - margin-bottom: 1em; - background-color: transparent; - border: 1px solid #dbdbdb; - box-shadow: 2px 2px 3px rgba(10, 10, 10, 0.1); - max-width: 100%; } - .docstring > header { - display: flex; - flex-grow: 1; - align-items: stretch; - padding: 0.5rem 0.75rem; - background-color: whitesmoke; - box-shadow: 0 1px 2px rgba(10, 10, 10, 0.1); - box-shadow: none; - border-bottom: 1px solid #dbdbdb; } - .docstring > header code { - background-color: transparent; } - .docstring > header .docstring-binding { - margin-right: 0.3em; } - .docstring > header .docstring-category { - margin-left: 0.3em; } - .docstring > section { - position: relative; - padding: 0.75rem 0.75rem; - border-bottom: 1px solid #dbdbdb; } - .docstring > section:last-child { - border-bottom: none; } - .docstring > section > a.docs-sourcelink { - transition: opacity 0.3s; - opacity: 0; - position: absolute; - right: 0.375rem; - bottom: 0.375rem; } - .docstring > section > a.docs-sourcelink:focus { - opacity: 1 !important; } - .docstring:hover > section > a.docs-sourcelink { - opacity: 0.2; } - .docstring:focus-within > section > a.docs-sourcelink { - opacity: 0.2; } - .docstring > section:hover a.docs-sourcelink { - opacity: 1; } - -.documenter-example-output { - background-color: white; } - -.outdated-warning-overlay { - position: fixed; - top: 0; - left: 0; - right: 0; - box-shadow: 0 0 10px rgba(0, 0, 0, 0.3); - z-index: 999; - background-color: #ffaba7; - color: rgba(0, 0, 0, 0.7); - border-bottom: 3px solid #da0b00; - padding: 10px 35px; - text-align: center; - font-size: 15px; } - .outdated-warning-overlay .outdated-warning-closer { - position: absolute; - top: calc(50% - 10px); - right: 18px; - cursor: pointer; - width: 12px; } - .outdated-warning-overlay a { - color: #2e63b8; 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} - -html { - -moz-osx-font-smoothing: auto; - -webkit-font-smoothing: auto; } - -li.no-marker { - list-style: none; } - -/* This file contain the overall layout. - * - * The main container is
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Gets put into the tag.. -function set_theme_from_local_storage() { - // Intialize the theme to null, which means default - var theme = null; - // If the browser supports the localstorage and is not disabled then try to get the - // documenter theme - if(window.localStorage != null) { - // Get the user-picked theme from localStorage. May be `null`, which means the default - // theme. - theme = window.localStorage.getItem("documenter-theme"); - } - // Check if the browser supports user color preference - var darkPreference = false; - // Check if the users preference is for dark color scheme - if(window.matchMedia('(prefers-color-scheme: dark)').matches === true) { - darkPreference = true; - } - // Initialize a few variables for the loop: - // - // - active: will contain the index of the theme that should be active. Note that there - // is no guarantee that localStorage contains sane values. If `active` stays `null` - // we either could not find the theme or it is the default (primary) theme anyway. - // Either way, we then need to stick to the primary theme. - // - // - disabled: style sheets that should be disabled (i.e. all the theme style sheets - // that are not the currently active theme) - var active = null; var disabled = []; var darkTheme = null; - for (var i = 0; i < document.styleSheets.length; i++) { - var ss = document.styleSheets[i]; - // The tag of each style sheet is expected to have a data-theme-name attribute - // which must contain the name of the theme. The names in localStorage much match this. - var themename = ss.ownerNode.getAttribute("data-theme-name"); - // attribute not set => non-theme stylesheet => ignore - if(themename === null) continue; - // To distinguish the default (primary) theme, it needs to have the data-theme-primary - // attribute set. - var isprimary = (ss.ownerNode.getAttribute("data-theme-primary") !== null); - // Check if the theme is primary dark theme - var isDarkTheme = (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null); - // If ss is for dark theme then set the value of darkTheme to the name of the theme - if(isDarkTheme) darkTheme = themename; - // If we find a matching theme (and it's not the default), we'll set active to non-null - if(themename === theme) active = i; - // Store the style sheets of inactive themes so that we could disable them - if(themename !== theme) disabled.push(ss); - } - if(active !== null) { - // If we did find an active theme, we'll (1) add the theme--$(theme) class to - document.getElementsByTagName('html')[0].className = "theme--" + theme; - // and (2) disable all the other theme stylesheets - disabled.forEach(function(ss){ - ss.disabled = true; - }); - } - else if(darkTheme !== null && darkPreference === true) { - // If we did find an active theme, we'll (1) add the theme--$(theme) class to - document.getElementsByTagName('html')[0].className = "theme--" + darkTheme; - // and (2) disable all the other theme stylesheets - disabled.forEach(function(ss){ - if (ss.ownerNode.getAttribute("data-theme-name") !== darkTheme) { - ss.disabled = true; - } - }); - } -} -set_theme_from_local_storage(); diff --git a/previews/PR62/generated/ffmpeg.err b/previews/PR62/generated/ffmpeg.err deleted file mode 100644 index c85c27d1..00000000 --- a/previews/PR62/generated/ffmpeg.err +++ /dev/null @@ -1,48 +0,0 @@ -ffmpeg version 4.4.2-0ubuntu0.22.04.1 Copyright (c) 2000-2021 the FFmpeg developers - built with gcc 11 (Ubuntu 11.2.0-19ubuntu1) - configuration: --prefix=/usr --extra-version=0ubuntu0.22.04.1 --toolchain=hardened --libdir=/usr/lib/x86_64-linux-gnu --incdir=/usr/include/x86_64-linux-gnu --arch=amd64 --enable-gpl --disable-stripping --enable-gnutls --enable-ladspa --enable-libaom --enable-libass --enable-libbluray --enable-libbs2b --enable-libcaca --enable-libcdio --enable-libcodec2 --enable-libdav1d --enable-libflite --enable-libfontconfig --enable-libfreetype --enable-libfribidi --enable-libgme --enable-libgsm --enable-libjack --enable-libmp3lame --enable-libmysofa --enable-libopenjpeg --enable-libopenmpt --enable-libopus --enable-libpulse --enable-librabbitmq --enable-librubberband --enable-libshine --enable-libsnappy --enable-libsoxr --enable-libspeex --enable-libsrt --enable-libssh --enable-libtheora --enable-libtwolame --enable-libvidstab --enable-libvorbis --enable-libvpx --enable-libwebp --enable-libx265 --enable-libxml2 --enable-libxvid --enable-libzimg --enable-libzmq --enable-libzvbi --enable-lv2 --enable-omx --enable-openal --enable-opencl --enable-opengl --enable-sdl2 --enable-pocketsphinx --enable-librsvg --enable-libmfx --enable-libdc1394 --enable-libdrm --enable-libiec61883 --enable-chromaprint --enable-frei0r --enable-libx264 --enable-shared - 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-Navier–Stokes equations · PencilFFTs.jl
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    Navier–Stokes equations

    In this example, we numerically solve the incompressible Navier–Stokes equations

    \[∂_t \bm{v} + (\bm{v} ⋅ \bm{∇}) \bm{v} = -\frac{1}{ρ} \bm{∇} p + ν ∇^2 \bm{v}, -\quad \bm{∇} ⋅ \bm{v} = 0,\]

    where $\bm{v}(\bm{x}, t)$ and $p(\bm{x}, t)$ are respectively the velocity and pressure fields, $ν$ is the fluid kinematic viscosity and $ρ$ is the fluid density.

    We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.

    First steps

    We start by loading the required packages, initialising MPI and setting the simulation parameters.

    using MPI
-using PencilFFTs
-
-MPI.Init()
-comm = MPI.COMM_WORLD
-procid = MPI.Comm_rank(comm) + 1
-
-# Simulation parameters
-Ns = (64, 64, 64)  # = (Nx, Ny, Nz)
-Ls = (2π, 2π, 2π)  # = (Lx, Ly, Lz)
-
-# Collocation points ("global" = over all processes).
-# We include the endpoint (length = N + 1) for convenience.
-xs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls)  # = (x, y, z)
    (0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586)

    Let's check the number of MPI processes over which we're running our simulation:

    MPI.Comm_size(comm)
    2

    We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:

    pen = Pencil(Ns, comm)
    Decomposition of 3D data
-    Data dimensions: (64, 64, 64)
-    Decomposed dimensions: (2, 3)
-    Data permutation: NoPermutation()
-    Array type: Array

    The subdomain associated to the local MPI process can be obtained using range_local:

    range_local(pen)
    (1:64, 1:32, 1:64)

    We now construct a distributed vector field that follows the decomposition configuration we just created:

    v⃗₀ = (
-    PencilArray{Float64}(undef, pen),  # vx
-    PencilArray{Float64}(undef, pen),  # vy
-    PencilArray{Float64}(undef, pen),  # vz
-)
-summary(v⃗₀[1])
    "64×32×64 PencilArray{Float64, 3}(::Pencil{3, 2, NoPermutation, Array})"

    We still need to fill this array with interesting values that represent a physical velocity field.

    Initial condition

    Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:

    \[\begin{aligned} -v_x(x, y, z) &= u₀ \sin(k₀ x) \cos(k₀ y) \cos(k₀ z) \\ -v_y(x, y, z) &= -u₀ \cos(k₀ x) \sin(k₀ y) \cos(k₀ z) \\ -v_z(x, y, z) &= 0 -\end{aligned}\]

    where $u₀$ and $k₀$ are two parameters setting the amplitude and the period of the velocity field.

    To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:

    grid = localgrid(pen, xs_global)
    LocalRectilinearGrid{3} with coordinates:
- (1) 0.0:0.09817477042468103:6.1850105367549055
- (2) 0.0:0.09817477042468103:3.043417883165112
- (3) 0.0:0.09817477042468103:6.1850105367549055

    We can use this to initialise the velocity field:

    u₀ = 1.0
-k₀ = 2π / Ls[1]  # should be integer if L = 2π (to preserve periodicity)
-
-@. v⃗₀[1] =  u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)
-@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)
-@. v⃗₀[3] =  0

    Let's plot a 2D slice of the velocity field managed by the local MPI process:

    using GLMakie
-
-# Compute the norm of a vector field represented by a tuple of arrays.
-function vecnorm(v⃗::NTuple)
-    vnorm = similar(v⃗[1])
-    for n ∈ eachindex(v⃗[1])
-        w = zero(eltype(vnorm))
-        for v ∈ v⃗
-            w += v[n]^2
-        end
-        vnorm[n] = sqrt(w)
-    end
-    vnorm
-end
-
-let fig = Figure(resolution = (700, 600))
-    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
-    vnorm = vecnorm(v⃗₀)
-    ct = contour!(
-        ax, grid.x, grid.y, grid.z, vnorm;
-        alpha = 0.2, levels = 4,
-        colormap = :viridis, colorrange = (0.0, 1.0),
-    )
-    cb = Colorbar(fig[1, 2], ct; label = "Velocity magnitude")
-    fig
-end

    Velocity in Fourier space

    In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series

    \[\bm{v}(\bm{x}, t) = -∑_{\bm{k}} \hat{\bm{v}}_{\bm{k}}(t) \, e^{i \bm{k} ⋅ \bm{x}},\]

    where $\bm{k} = (k_x, k_y, k_z)$ are the discrete wave numbers.

    The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for $k_x$:

    using AbstractFFTs: fftfreq, rfftfreq
-
-ks_global = (
-    rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]),  # kx | real-to-complex
-     fftfreq(Ns[2], 2π * Ns[2] / Ls[2]),  # ky | complex-to-complex
-     fftfreq(Ns[3], 2π * Ns[3] / Ls[3]),  # kz | complex-to-complex
-)
-
-ks_global[1]'
    1×33 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
- 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  27.0  28.0  29.0  30.0  31.0  32.0
    ks_global[2]'
    1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
- 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0
    ks_global[3]'
    1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
- 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0

    To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:

    plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())
    Transforms: (RFFT, FFT, FFT)
-Input type: Float64
-Global dimensions: (64, 64, 64) -> (33, 64, 64)
-MPI topology: 2D decomposition (2×1 processes)

    See PencilFFTPlan for details on creating plans and on optional keyword arguments.

    We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$:

    v̂s = plan .* v⃗₀
-summary(v̂s[1])
    "16×64×64 PencilArray{ComplexF64, 3}(::Pencil{3, 2, Permutation{(3, 2, 1), 3}, Array})"

    Note that, in Fourier space, the domain decomposition is performed along the directions $x$ and $y$:

    pencil(v̂s[1])
    Decomposition of 3D data
-    Data dimensions: (33, 64, 64)
-    Decomposed dimensions: (1, 2)
-    Data permutation: Permutation(3, 2, 1)
-    Array type: Array

    This is because the 3D FFTs are performed one dimension at a time, with the $x$ direction first and the $z$ direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.

    To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers $\bm{k}$ associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:

    grid_fourier = localgrid(v̂s[1], ks_global)
    LocalRectilinearGrid{3} with Permutation(3, 2, 1) and coordinates:
- (1) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0]
- (2) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]
- (3) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]

    As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, $\bm{ω} = \bm{∇} × \bm{v}$. In Fourier space, this becomes $\hat{\bm{ω}} = i \bm{k} × \hat{\bm{v}}$.

    using StaticArrays: SVector
-using LinearAlgebra: ×
-
-function curl_fourier!(
-        ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,
-    ) where {N}
-    @inbounds for I ∈ eachindex(grid_fourier)
-        # We use StaticArrays for the cross product between small vectors.
-        ik⃗ = im * SVector(grid_fourier[I])
-        v⃗ = SVector(getindex.(v̂s, Ref(I)))  # = (v̂s[1][I], v̂s[2][I], ...)
-        ω⃗ = ik⃗ × v⃗
-        for n ∈ eachindex(ω⃗)
-            ω̂s[n][I] = ω⃗[n]
-        end
-    end
-    ω̂s
-end
-
-ω̂s = similar.(v̂s)
-curl_fourier!(ω̂s, v̂s, grid_fourier);

    We finally transform back to physical space and plot the result:

    ωs = plan .\ ω̂s
-
-let fig = Figure(resolution = (700, 600))
-    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
-    ω_norm = vecnorm(ωs)
-    ct = contour!(
-        ax, grid.x, grid.y, grid.z, ω_norm;
-        alpha = 0.1, levels = 0.8:0.2:2.0,
-        colormap = :viridis, colorrange = (0.8, 2.0),
-    )
-    cb = Colorbar(fig[1, 2], ct; label = "Vorticity magnitude")
-    fig
-end

    Computing the non-linear term

    One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as

    \[∂_t \hat{\bm{v}}_{\bm{k}} = -- \mathcal{P}_{\bm{k}} \! \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right] -- ν |\bm{k}|^2 \hat{\bm{v}}_{\bm{k}} -\quad \text{ with } \quad -\mathcal{P}_{\bm{k}}(\hat{\bm{F}}_{\bm{k}}) = \left( I - \frac{\bm{k} ⊗ -\bm{k}}{|\bm{k}|^2} \right) \hat{\bm{F}}_{\bm{k}},\]

    where $\mathcal{P}_{\bm{k}}$ is a projection operator allowing to preserve the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.

    Now that we have the wave numbers $\bm{k}$, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$ of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, $\hat{\bm{F}}_{\bm{k}} = \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right]_{\bm{k}}$. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.

    Below we implement a function that computes the non-linear term in Fourier space based on its convective form $(\bm{v} ⋅ \bm{∇}) \bm{v} = \bm{∇} ⋅ (\bm{v} ⊗ \bm{v})$. Note that this equivalence uses the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$.

    using LinearAlgebra: mul!, ldiv!  # for applying FFT plans in-place
-
-# Compute non-linear term in Fourier space from velocity field in physical
-# space. Optional keyword arguments may be passed to avoid allocations.
-function ns_nonlinear!(
-        F̂s, vs, plan, grid_fourier;
-        vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),
-    )
-    # Compute F_i = ∂_j (v_i v_j) for each i.
-    # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)
-    w, ŵ = vbuf, v̂buf
-    @inbounds for (i, F̂i) ∈ enumerate(F̂s)
-        F̂i .= 0
-        vi = vs[i]
-        for (j, vj) ∈ enumerate(vs)
-            w .= vi .* vj     # w = v_i * v_j in physical space
-            mul!(ŵ, plan, w)  # same in Fourier space
-            # Add derivative in Fourier space
-            for I ∈ eachindex(grid_fourier)
-                k⃗ = grid_fourier[I]  # = (kx, ky, kz)
-                kj = k⃗[j]
-                F̂i[I] += im * kj * ŵ[I]
-            end
-        end
-    end
-    F̂s
-end
    ns_nonlinear! (generic function with 1 method)

    As an example, let's use this function on our initial velocity field:

    F̂s = similar.(v̂s)
-ns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);

    Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.

    function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)
-    ks_max = maximum.(abs, ks_global)  # maximum stored wave numbers (kx_max, ky_max, kz_max)
-    ks_lim = (2 / 3) .* ks_max
-    @inbounds for I ∈ eachindex(grid_fourier)
-        k⃗ = grid_fourier[I]
-        if any(abs.(k⃗) .> ks_lim)
-            for ŵ ∈ ŵs
-                ŵ[I] = 0
-            end
-        end
-    end
-    ŵs
-end
-
-# We can apply this on the previously computed non-linear term:
-dealias_twothirds!(F̂s, grid_fourier, ks_global);

    Finally, we implement the projection associated to the incompressibility condition:

    function project_divergence_free!(ûs, grid_fourier)
-    @inbounds for I ∈ eachindex(grid_fourier)
-        k⃗ = grid_fourier[I]
-        k² = sum(abs2, k⃗)
-        iszero(k²) && continue  # avoid division by zero
-        û = getindex.(ûs, Ref(I))  # (ûs[1][I], ûs[2][I], ...)
-        for i ∈ eachindex(û)
-            ŵ = û[i]
-            for j ∈ eachindex(û)
-                ŵ -= k⃗[i] * k⃗[j] * û[j] / k²
-            end
-            ûs[i][I] = ŵ
-        end
-    end
-    ûs
-end
    project_divergence_free! (generic function with 1 method)

    We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:

    v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)
-v̂s_proj .≈ v̂s  # the last one may be false because v_z = 0 initially
    (true, true, false)

    Putting it all together

    To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:

    function ns_rhs!(
-        dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,
-    ) where {N}
-    # 1. Compute non-linear term and dealias it
-    (; plan, cache, ks_global, grid_fourier) = p
-    F̂s = cache.F̂s
-    ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])
-    dealias_twothirds!(F̂s, grid_fourier, ks_global)
-
-    # 2. Project onto divergence-free space
-    project_divergence_free!(F̂s, grid_fourier)
-
-    # 3. Transform velocity to Fourier space
-    v̂s = cache.v̂s
-    map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)
-
-    # 4. Add viscous term (and multiply projected non-linear term by -1)
-    ν = p.ν
-    for n ∈ eachindex(v̂s)
-        v̂ = v̂s[n]
-        F̂ = F̂s[n]
-        @inbounds for I ∈ eachindex(grid_fourier)
-            k⃗ = grid_fourier[I]  # = (kx, ky, kz)
-            k² = sum(abs2, k⃗)
-            F̂[I] = -F̂[I] - ν * k² * v̂[I]
-        end
-    end
-
-    # 5. Transform RHS back to physical space
-    map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)
-
-    nothing
-end
    ns_rhs! (generic function with 1 method)

    For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.

    using OrdinaryDiffEq
-using RecursiveArrayTools: ArrayPartition
-
-ns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)
-vs_init_ode = ArrayPartition(v⃗₀)
-summary(vs_init_ode)
    "RecursiveArrayTools.ArrayPartition{Float64, Tuple{PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}}} with arrays:"

    We now define solver parameters and temporary variables, and initialise the problem:

    params = (;
-    ν = 5e-3,  # kinematic viscosity
-    plan, grid_fourier, ks_global,
-    cache = (
-        v̂s = similar.(v̂s),
-        F̂s = similar.(v̂s),
-    )
-)
-
-tspan = (0.0, 10.0)
-prob = ODEProblem(ns_rhs!, vs_init_ode, tspan, params)
-integrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);

    We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum $E(k)$, to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity $ν$ must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large $k$), while high enough to allow the small-scale motions to be correctly resolved.

    function energy_spectrum!(Ek, ks, v̂s, grid_fourier)
-    Nk = length(Ek)
-    @assert Nk == length(ks)
-    Ek .= 0
-    for I ∈ eachindex(grid_fourier)
-        k⃗ = grid_fourier[I]  # = (kx, ky, kz)
-        knorm = sqrt(sum(abs2, k⃗))
-        i = searchsortedfirst(ks, knorm)
-        i > Nk && continue
-        v⃗ = getindex.(v̂s, Ref(I))  # = (v̂s[1][I], v̂s[2][I], ...)
-        factor = k⃗[1] == 0 ? 1 : 2  # account for Hermitian symmetry and r2c transform
-        Ek[i] += factor * sum(abs2, v⃗) / 2
-    end
-    MPI.Allreduce!(Ek, +, get_comm(v̂s[1]))  # sum across all processes
-    Ek
-end
-
-ks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])
-Ek = similar(ks)
-v̂s = plan .* integrator.u.x
-energy_spectrum!(Ek, ks, v̂s, grid_fourier)
-Ek ./= scale_factor(plan)^2  # rescale energy
-
-curl_fourier!(ω̂s, v̂s, grid_fourier)
-ldiv!.(ωs, plan, ω̂s)
-ω⃗_plot = Observable(ωs)
-k_plot = @view ks[2:end]
-E_plot = Observable(@view Ek[2:end])
-t_plot = Observable(integrator.t)
-
-fig = let
-    fig = Figure(resolution = (1200, 600))
-    ax = Axis3(
-        fig[1, 1][1, 1]; title = @lift("t = $(round($t_plot, digits = 3))"),
-        aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z",
-    )
-    ω_mag = @lift vecnorm($ω⃗_plot)
-    ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)
-    ct = contour!(
-        ax, grid.x, grid.y, grid.z, ω_mag_norm;
-        alpha = 0.3, levels = 3,
-        colormap = :viridis, colorrange = (0.0, 1.0),
-    )
-    cb = Colorbar(fig[1, 1][1, 2], ct; label = "Normalised vorticity magnitude")
-    ax_sp = Axis(
-        fig[1, 2];
-        xlabel = "k", ylabel = "E(k)", xscale = log2, yscale = log10,
-        title = "Kinetic energy spectrum",
-    )
-    ylims!(ax_sp, 1e-8, 1e0)
-    scatterlines!(ax_sp, k_plot, E_plot)
-    ks_slope = exp.(range(log(2.5), log(25.0), length = 3))
-    E_fivethirds = @. 0.3 * ks_slope^(-5/3)
-    @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)
-    text!(ax_sp, L"k^{-5/3}"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))
-    fig
-end
-
-record(fig, "vorticity_proc$procid.mp4"; framerate = 10) do io
-    while integrator.t < 20
-        dt = 0.001
-        step!(integrator, dt)
-        t_plot[] = integrator.t
-        mul!.(v̂s, plan, integrator.u.x)  # current velocity in Fourier space
-        curl_fourier!(ω̂s, v̂s, grid_fourier)
-        ldiv!.(ω⃗_plot[], plan, ω̂s)
-        ω⃗_plot[] = ω⃗_plot[]  # to force updating the plot
-        energy_spectrum!(Ek, ks, v̂s, grid_fourier)
-        Ek ./= scale_factor(plan)^2  # rescale energy
-        E_plot[] = E_plot[]
-        recordframe!(io)
-    end
-end;
    - -

    This page was generated using Literate.jl.

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    ","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"By default, the domain is distributed on a 2D MPI topology of dimensions N_1 N_2. As an example, the above figure shows such a topology with N_1 = 4 and N_2 = 3, for a total of 12 MPI processes.","category":"page"},{"location":"tutorial/#tutorial:creating_plans","page":"Tutorial","title":"Creating plans","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The first thing to do is to create a domain decomposition configuration for the given dataset dimensions N_x N_y N_z. In the framework of PencilArrays, such a configuration is described by a Pencil object. As described in the PencilArrays docs, we can let the Pencil constructor automatically determine such a configuration. For this, only an MPI communicator and the dataset dimensions are needed:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using MPI\nusing PencilFFTs\n\nMPI.Init()\ncomm = MPI.COMM_WORLD\n\n# Input data dimensions (Nx × Ny × Nz)\ndims = (16, 32, 64)\npen = Pencil(dims, comm)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"By default this creates a 2D decomposition (for the case of a 3D dataset), but one can change this as detailed in the PencilArrays documentation linked above.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"We can now create a PencilFFTPlan, which requires information on decomposition configuration (the Pencil object) and on the transforms that will be applied:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# Apply a 3D real-to-complex (r2c) FFT.\ntransform = Transforms.RFFT()\n\n# Note that, for more control, one can instead separately specify the transforms along each dimension:\n# transform = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())\n\n# Create plan\nplan = PencilFFTPlan(pen, transform)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"See the PencilFFTPlan constructor for details on the accepted options, and the Transforms module for the possible transforms. It is also possible to enable fine-grained performance measurements via the TimerOutputs package, as described in Measuring performance.","category":"page"},{"location":"tutorial/#Allocating-data","page":"Tutorial","title":"Allocating data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Next, we want to apply the plan on some data. Transforms may only be applied on PencilArrays, which are array wrappers that include MPI decomposition information (in some sense, analogous to DistributedArrays in Julia's distributed computing approach). The helper function allocate_input can be used to allocate a PencilArray that is compatible with our plan:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# In our example, this returns a 3D PencilArray of real data (Float64).\nu = allocate_input(plan)\n\n# Fill the array with some (random) data\nusing Random\nrandn!(u)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"PencilArrays are a subtype of AbstractArray, and thus they support all common array operations.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Similarly, to preallocate output data, one can use allocate_output:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# In our example, this returns a 3D PencilArray of complex data (Complex{Float64}).\nv = allocate_output(plan)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"This is only required if one wants to apply the plans using a preallocated output (with mul!, see right below).","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The data types returned by allocate_input and allocate_output are slightly different when working with in-place transforms. See the in-place example for details.","category":"page"},{"location":"tutorial/#Applying-plans","page":"Tutorial","title":"Applying plans","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The interface to apply plans is consistent with that of AbstractFFTs. Namely, * and mul! are respectively used for forward transforms without and with preallocated output data. Similarly, \\ and ldiv! are used for backward transforms.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using LinearAlgebra # for mul!, ldiv!\n\n# Apply plan on `u` with `v` as an output\nmul!(v, plan, u)\n\n# Apply backward plan on `v` with `w` as an output\nw = similar(u)\nldiv!(w, plan, v) # now w ≈ u","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Note that, consistently with AbstractFFTs, normalisation is performed at the end of a backward transform, so that the original data is recovered when applying a forward followed by a backward transform.","category":"page"},{"location":"tutorial/#Accessing-and-modifying-data","page":"Tutorial","title":"Accessing and modifying data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For any given MPI process, a PencilArray holds the data associated to its local partition in the global geometry. PencilArrays are accessed using local indices that start at 1, regardless of the location of the local process in the MPI topology. Note that PencilArrays, being based on regular Arrays, support both linear and Cartesian indexing (see the Julia docs for details).","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For convenience, the global_view function can be used to generate an OffsetArray wrapper that takes global indices.","category":"page"},{"location":"tutorial/#tutorial:output_data_layout","page":"Tutorial","title":"Output data layout","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"In memory, the dimensions of the transform output are by default reversed with respect to the input. That is, if the order of indices in the input data is (x, y, z), then the output has order (z, y, x) in memory. This detail is hidden from the user, and output arrays are always accessed in the same order as the input data, regardless of the underlying output dimension permutation. This applies to PencilArrays and to OffsetArrays returned by global_view.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The reasoning behind dimension permutations, is that they allow to always perform FFTs along the fastest array dimension and to avoid a local data transposition, resulting in performance gains. A similar approach is followed by other parallel FFT libraries. FFTW itself, in its distributed-memory routines, includes a flag that enables a similar behaviour. In PencilFFTs, index permutation is the default, but it can be disabled via the permute_dims flag of PencilFFTPlan.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"A great deal of work has been spent in making generic index permutations as efficient as possible, both in intermediate and in the output state of the multidimensional transforms. This has been achieved, in part, by making sure that permutations such as (3, 2, 1) are compile-time constants.","category":"page"},{"location":"tutorial/#Further-reading","page":"Tutorial","title":"Further reading","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For details on working with PencilArrays see the PencilArrays docs.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The examples on the sidebar further illustrate the use of transforms and provide an introduction to working with MPI-distributed data in the form of PencilArrays. In particular, the gradient example illustrates different ways of computing things using Fourier-transformed distributed arrays. Then, the incompressible Navier–Stokes example is a more advanced and complete example of a possible application of the PencilFFTs package.","category":"page"},{"location":"benchmarks/#Benchmarks","page":"Benchmarks","title":"Benchmarks","text":"","category":"section"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The performance of PencilFFTs.jl is comparable to that of other open-source parallel FFT libraries implemented in lower-level languages. Below, we show comparisons with the Fortran implementation of P3DFFT, possibly the most popular of these libraries. The benchmarks were performed on the Jean–Zay cluster of the IDRIS French computing centre (CNRS).","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The figure below shows strong scaling benchmarks of 3D real-to-complex FFTs using 2D (\"pencil\") decomposition. The benchmarks were run for input arrays of dimensions N_x N_y N_z = 512^3, 1024^3 and 2048^3. Each timing is averaged over 100 repetitions.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"
    \n \n \n
    ","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"As seen above, PencilFFTs generally outperforms P3DFFT in its default setting. This is largely explained by the choice of using non-blocking point-to-point MPI communications (via MPI_Isend and MPI_Irecv), while P3DFFT uses collective MPI_Alltoallv calls. This enables PencilFFTs to perform data reordering operations on the partially received data while waiting for the incoming data, leading to better performance. Moreover, in contrast with P3DFFT, the high performance and scalability of PencilFFTs results from a highly generic code, handling decompositions in arbitrary dimensions and a relatively large (and extensible) variety of transformations.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"Note that PencilFFTs can optionally use collective communications (using MPI_Alltoallv) instead of point-to-point communications. For details, see the docs for PencilFFTPlan and for PencilArray transpositions. As seen above, collective communications generally perform worse than point-to-point ones, and runtimes are nearly indistinguishable from those of P3DFFT.","category":"page"},{"location":"benchmarks/#Benchmark-details","page":"Benchmarks","title":"Benchmark details","text":"","category":"section"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The benchmarks were performed using Julia 1.7-beta3 and Intel MPI 2019. We used PencilFFTs v0.12.5 with FFTW.jl v1.4.3 and MPI.jl v0.19.0. We used the Fortran implementation of P3DFFT, version 2.7.6, which was built with Intel 2019 compilers and linked to FFTW 3.3.8. The cluster where the benchmarks were run has Intel Cascade Lake 6248 processors with 2×20 cores per node.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The number of MPI processes along each decomposed dimension, P_1 and P_2, was automatically determined by a call to MPI_Dims_create, which tends to create a balanced decomposition with P_1 P_2. For instance, a total of 1024 processes is divided into P_1 = P_2 = 32. Different results may be obtained with other combinations, but this was not benchmarked.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The source files used to generate this benchmark, as well as the raw benchmark results, are all available in the PencilFFTs repo.","category":"page"},{"location":"GlobalFFTParams/#Global-FFT-parameters","page":"Global FFT parameters","title":"Global FFT parameters","text":"","category":"section"},{"location":"GlobalFFTParams/","page":"Global FFT parameters","title":"Global FFT parameters","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"GlobalFFTParams/","page":"Global FFT parameters","title":"Global FFT parameters","text":"GlobalFFTParams","category":"page"},{"location":"GlobalFFTParams/#PencilFFTs.GlobalFFTParams","page":"Global FFT parameters","title":"PencilFFTs.GlobalFFTParams","text":"GlobalFFTParams{T, N, inplace}\n\nSpecifies the global parameters for an N-dimensional distributed transform. These include the element type T and global data sizes of input and output data, as well as the transform types to be performed along each dimension.\n\n\n\nGlobalFFTParams(size_global, transforms, [real_type=Float64])\n\nDefine parameters for N-dimensional transform.\n\ntransforms must be a tuple of length N specifying the transforms to be applied along each dimension. Each element must be a subtype of Transforms.AbstractTransform. For all the possible transforms, see Transform types.\n\nThe element type must be a real type accepted by FFTW, i.e. either Float32 or Float64.\n\nNote that the transforms are applied one dimension at a time, with the leftmost dimension first for forward transforms.\n\nExample\n\nTo perform a 3D FFT of real data, first a real-to-complex FFT must be applied along the first dimension, followed by two complex-to-complex FFTs along the other dimensions:\n\njulia> size_global = (64, 32, 128); # size of real input data\n\njulia> transforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT());\n\njulia> fft_params = PencilFFTs.GlobalFFTParams(size_global, transforms)\nTransforms: (RFFT, FFT, FFT)\nInput type: Float64\nGlobal dimensions: (64, 32, 128) -> (33, 32, 128)\n\n\n\n\n\n","category":"type"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"EditURL = \"https://github.com/jipolanco/PencilFFTs.jl/blob/master/docs/examples/in-place.jl\"","category":"page"},{"location":"generated/in-place/#In-place-transforms","page":"In-place transforms","title":"In-place transforms","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Complex-to-complex and real-to-real transforms can be performed in-place, enabling important memory savings. The procedure is very similar to that of out-of-place transforms described in the tutorial. The differences are illustrated in the sections below.","category":"page"},{"location":"generated/in-place/#Creating-a-domain-partition","page":"In-place transforms","title":"Creating a domain partition","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"We start by partitioning a domain of dimensions 163264 along all available MPI processes.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"using PencilFFTs\nusing MPI\nMPI.Init()\n\ndims_global = (16, 32, 64) # global dimensions","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Such a partitioning is described by a Pencil object. Here we choose to decompose the domain along the last two dimensions. In this case, the actual number of processes along each of these dimensions is chosen automatically.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"decomp_dims = (2, 3)\ncomm = MPI.COMM_WORLD\npen = Pencil(dims_global, decomp_dims, comm)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"warning: Allowed decompositions\nDistributed transforms using PencilFFTs.jl require that the first dimension is not decomposed. In other words, if one wants to perform transforms, then decomp_dims above must not contain 1.","category":"page"},{"location":"generated/in-place/#Creating-in-place-plans","page":"In-place transforms","title":"Creating in-place plans","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"To create an in-place plan, pass an in-place transform such as Transforms.FFT! or Transforms.R2R! to PencilFFTPlan. For instance:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"# Perform a 3D in-place complex-to-complex FFT.\ntransform = Transforms.FFT!()\n\n# Note that one can also combine different types of in-place transforms.\n# For instance:\n# transform = (\n# Transforms.R2R!(FFTW.REDFT01),\n# Transforms.FFT!(),\n# Transforms.R2R!(FFTW.DHT),\n# )","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"We can now create a distributed plan from the previously-created domain partition and the chosen transform.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan = PencilFFTPlan(pen, transform)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Note that in-place real-to-complex transforms are not currently supported. (In other words, the RFFT! transform type is not defined.)","category":"page"},{"location":"generated/in-place/#Allocating-data","page":"In-place transforms","title":"Allocating data","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"As with out-of-place plans, data should be allocated using allocate_input. The difference is that, for in-place plans, this function returns a ManyPencilArray object, which is a container holding multiple PencilArray views sharing the same memory space.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"# Allocate data for the plan.\n# Since `plan` is in-place, this returns a `ManyPencilArray` container.\nA = allocate_input(plan)\nsummary(A)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Note that allocate_output also works for in-place plans. In this case, it returns exactly the same thing as allocate_input.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"As shown in the next section, in-place plans must be applied on the returned ManyPencilArray. On the other hand, one usually wants to access and modify data, and for this one needs the PencilArray views contained in the ManyPencilArray. The input and output array views can be obtained by calling first(::ManyPencilArray) and last(::ManyPencilArray).","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"For instance, we can initialise the input array with some data before transforming:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"using Random\nu_in = first(A) # input data view\nrandn!(u_in)\nsummary(u_in)","category":"page"},{"location":"generated/in-place/#Applying-plans","page":"In-place transforms","title":"Applying plans","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Like in FFTW.jl, one can perform in-place transforms using the * and \\ operators. As mentioned above, in-place plans must be applied on the ManyPencilArray containers returned by allocate_input.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan * A; # performs in-place forward transform\nnothing #hide","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"After performing an in-place transform, data contained in u_in has been overwritten and has no \"physical\" meaning. In other words, u_in should not be used at this point. To access the transformed data, one should retrieve the output data view using last(A).","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"For instance, to compute the global sum of the transformed data:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"u_out = last(A) # output data view\nsum(u_out) # sum of transformed data (note that `sum` reduces over all processes)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Finally, we can perform a backward transform and do stuff with the input view:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan \\ A; # perform in-place backward transform\nnothing #hide","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"At this point, the data can be once again found in the input view u_in, while u_out should not be accessed.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"This page was generated using Literate.jl.","category":"page"},{"location":"Transforms/#Available-transforms","page":"Available transforms","title":"Available transforms","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"CurrentModule = PencilFFTs.Transforms","category":"page"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"Transforms","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms","page":"Available transforms","title":"PencilFFTs.Transforms","text":"Defines different one-dimensional FFT-based transforms.\n\nThe transforms are all subtypes of an AbstractTransform type.\n\nWhen possible, the names of the transforms are kept consistent with the functions exported by AbstractFFTs.jl and FFTW.jl.\n\n\n\n\n\n","category":"module"},{"location":"Transforms/#Transform-types","page":"Available transforms","title":"Transform types","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"FFT\nFFT!\nBFFT\nBFFT!\n\nRFFT\nBRFFT\n\nR2R\nR2R!\n\nNoTransform\nNoTransform!","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.FFT","page":"Available transforms","title":"PencilFFTs.Transforms.FFT","text":"FFT()\n\nComplex-to-complex FFT.\n\nSee also AbstractFFTs.fft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.FFT!","page":"Available transforms","title":"PencilFFTs.Transforms.FFT!","text":"FFT!()\n\nIn-place version of FFT.\n\nSee also AbstractFFTs.fft!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BFFT","page":"Available transforms","title":"PencilFFTs.Transforms.BFFT","text":"BFFT()\n\nUnnormalised backward complex-to-complex FFT.\n\nLike AbstractFFTs.bfft, this transform is not normalised. To obtain the inverse transform, divide the output by the length of the transformed dimension.\n\nSee also AbstractFFTs.bfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.BFFT!","text":"BFFT()\n\nIn-place version of BFFT.\n\nSee also AbstractFFTs.bfft!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.RFFT","page":"Available transforms","title":"PencilFFTs.Transforms.RFFT","text":"RFFT()\n\nReal-to-complex FFT.\n\nSee also AbstractFFTs.rfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BRFFT","page":"Available transforms","title":"PencilFFTs.Transforms.BRFFT","text":"BRFFT(d::Integer)\nBRFFT((d1, d2, ..., dN))\n\nUnnormalised inverse of RFFT.\n\nTo obtain the inverse transform, divide the output by the length of the transformed dimension (of the real output array).\n\nAs described in the AbstractFFTs docs, the length of the output cannot be fully inferred from the input length. For this reason, the BRFFT constructor accepts an optional d argument indicating the output length.\n\nFor multidimensional datasets, a tuple of dimensions (d1, d2, ..., dN) may also be passed. This is equivalent to passing just dN. In this case, the last dimension (dN) is the one that changes size between the input and output. Note that this is the opposite of FFTW.brfft. The reason is that, in PencilFFTs, the last dimension is the one along which a complex-to-real transform is performed.\n\nSee also AbstractFFTs.brfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.R2R","page":"Available transforms","title":"PencilFFTs.Transforms.R2R","text":"R2R(kind)\n\nReal-to-real transform of type kind.\n\nThe possible values of kind are those described in the FFTW.r2r docs and the FFTW manual:\n\ndiscrete cosine transforms: FFTW.REDFT00, FFTW.REDFT01, FFTW.REDFFT10, FFTW.REDFFT11\ndiscrete sine transforms: FFTW.RODFT00, FFTW.RODFT01, FFTW.RODFFT10, FFTW.RODFFT11\ndiscrete Hartley transform: FFTW.DHT\n\nNote: half-complex format DFTs (FFTW.R2HC, FFTW.HC2R) are not currently supported.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.R2R!","page":"Available transforms","title":"PencilFFTs.Transforms.R2R!","text":"R2R!(kind)\n\nIn-place version of R2R.\n\nSee also FFTW.r2r!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.NoTransform","page":"Available transforms","title":"PencilFFTs.Transforms.NoTransform","text":"NoTransform()\n\nIdentity transform.\n\nSpecifies that no transformation should be applied.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.NoTransform!","page":"Available transforms","title":"PencilFFTs.Transforms.NoTransform!","text":"NoTransform!()\n\nIn-place version of NoTransform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#Internals","page":"Available transforms","title":"Internals","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"What follows is used internally in PencilFFTs.","category":"page"},{"location":"Transforms/#Types","page":"Available transforms","title":"Types","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"AbstractCustomPlan\nAbstractTransform\nIdentityPlan\nIdentityPlan!\nPlan","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.AbstractCustomPlan","page":"Available transforms","title":"PencilFFTs.Transforms.AbstractCustomPlan","text":"AbstractCustomPlan\n\nAbstract type defining a custom plan, to be used as an alternative to FFTW plans (FFTW.FFTWPlan).\n\nThe only custom plan defined in this module is IdentityPlan. The user can define other custom plans that are also subtypes of AbstractCustomPlan.\n\nNote that plan returns a subtype of either AbstractFFTs.Plan or AbstractCustomPlan.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.AbstractTransform","page":"Available transforms","title":"PencilFFTs.Transforms.AbstractTransform","text":"AbstractTransform\n\nSpecifies a one-dimensional FFT-based transform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.IdentityPlan","page":"Available transforms","title":"PencilFFTs.Transforms.IdentityPlan","text":"IdentityPlan\n\nType of plan associated to NoTransform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.IdentityPlan!","page":"Available transforms","title":"PencilFFTs.Transforms.IdentityPlan!","text":"IdentityPlan!\n\nType of plan associated to NoTransform!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.Plan","page":"Available transforms","title":"PencilFFTs.Transforms.Plan","text":"Plan = Union{AbstractFFTs.Plan, AbstractCustomPlan}\n\nUnion type representing any plan returned by plan.\n\nSee also AbstractCustomPlan.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#Functions","page":"Available transforms","title":"Functions","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"plan\n\nbinv\nscale_factor\n\neltype_input\neltype_output\nexpand_dims\nis_inplace\nkind\nlength_output","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.plan","page":"Available transforms","title":"PencilFFTs.Transforms.plan","text":"plan(transform::AbstractTransform, A, [dims];\n flags=FFTW.ESTIMATE, timelimit=Inf)\n\nCreate plan to transform array A along dimensions dims.\n\nIf dims is not specified, all dimensions of A are transformed.\n\nFor FFT plans, this function wraps the AbstractFFTs.jl and FFTW.jl plan creation functions. For more details on the function arguments, see AbstractFFTs.plan_fft.\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.binv","page":"Available transforms","title":"PencilFFTs.Transforms.binv","text":"binv(transform::AbstractTransform, d::Integer)\n\nReturns the backwards transform associated to the given transform.\n\nThe second argument must be the length of the first transformed dimension in the forward transform. It is used in particular when transform = RFFT(), to determine the length of the inverse (complex-to-real) transform. See the AbstractFFTs.irfft docs for details.\n\nThe backwards transform returned by this function is not normalised. The normalisation factor for a given array can be obtained by calling scale_factor.\n\nExample\n\njulia> binv(Transforms.FFT(), 42)\nBFFT\n\njulia> binv(Transforms.BRFFT(9), 42)\nRFFT\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.scale_factor","page":"Available transforms","title":"PencilFFTs.Transforms.scale_factor","text":"scale_factor(transform::AbstractTransform, A, [dims = 1:ndims(A)])\n\nGet factor required to normalise the given array after a transformation along dimensions dims (all dimensions by default).\n\nThe array A must have the dimensions of the transform input.\n\nImportant: the dimensions dims must be the same that were passed to plan.\n\nExamples\n\njulia> C = zeros(ComplexF32, 3, 4, 5);\n\njulia> scale_factor(Transforms.FFT(), C)\n60\n\njulia> scale_factor(Transforms.BFFT(), C)\n60\n\njulia> scale_factor(Transforms.BFFT(), C, 2:3)\n20\n\njulia> R = zeros(Float64, 3, 4, 5);\n\njulia> scale_factor(Transforms.RFFT(), R, 2)\n4\n\njulia> scale_factor(Transforms.RFFT(), R, 2:3)\n20\n\njulia> scale_factor(Transforms.BRFFT(8), C)\n96\n\njulia> scale_factor(Transforms.BRFFT(9), C)\n108\n\nThis will fail because the input of RFFT is real, and R is a complex array:\n\njulia> scale_factor(Transforms.RFFT(), C, 2:3)\nERROR: MethodError: no method matching scale_factor(::PencilFFTs.Transforms.RFFT, ::Array{ComplexF32, 3}, ::UnitRange{Int64})\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.eltype_input","page":"Available transforms","title":"PencilFFTs.Transforms.eltype_input","text":"eltype_input(transform::AbstractTransform, real_type<:AbstractFloat)\n\nDetermine input data type for a given transform given the floating point precision of the input data.\n\nSome transforms, such as R2R and NoTransform, can take both real and complex data. For those kinds of transforms, nothing is returned.\n\nExample\n\njulia> eltype_input(Transforms.FFT(), Float32)\nComplexF32 (alias for Complex{Float32})\n\njulia> eltype_input(Transforms.RFFT(), Float64)\nFloat64\n\njulia> eltype_input(Transforms.R2R(FFTW.REDFT01), Float64) # nothing\n\njulia> eltype_input(Transforms.NoTransform(), Float64) # nothing\n\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.eltype_output","page":"Available transforms","title":"PencilFFTs.Transforms.eltype_output","text":"eltype_output(transform::AbstractTransform, eltype_input)\n\nReturns the output data type for a given transform given the input type.\n\nThrows ArgumentError if the input data type is incompatible with the transform type.\n\nExample\n\njulia> eltype_output(Transforms.NoTransform(), Float32)\nFloat32\n\njulia> eltype_output(Transforms.RFFT(), Float64)\nComplexF64 (alias for Complex{Float64})\n\njulia> eltype_output(Transforms.BRFFT(4), ComplexF32)\nFloat32\n\njulia> eltype_output(Transforms.FFT(), Float64)\nERROR: ArgumentError: invalid input data type for PencilFFTs.Transforms.FFT: Float64\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.expand_dims","page":"Available transforms","title":"PencilFFTs.Transforms.expand_dims","text":"expand_dims(transform::AbstractTransform, Val(N))\n\nExpand a single multidimensional transform into one transform per dimension.\n\nExample\n\n# Expand a real-to-complex transform in 3 dimensions.\njulia> expand_dims(Transforms.RFFT(), Val(3))\n(RFFT, FFT, FFT)\n\njulia> expand_dims(Transforms.BRFFT(4), Val(3))\n(BFFT, BFFT, BRFFT{even})\n\njulia> expand_dims(Transforms.NoTransform(), Val(2))\n(NoTransform, NoTransform)\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.is_inplace","page":"Available transforms","title":"PencilFFTs.Transforms.is_inplace","text":"is_inplace(transform::AbstractTransform) -> Bool\nis_inplace(transforms::Vararg{AbtractTransform}) -> Union{Bool, Nothing}\n\nCheck whether a transform or a list of transforms is performed in-place.\n\nIf the list of transforms has a combination of in-place and out-of-place transforms, nothing is returned.\n\nExample\n\njulia> is_inplace(Transforms.RFFT())\nfalse\n\njulia> is_inplace(Transforms.NoTransform!())\ntrue\n\njulia> is_inplace(Transforms.FFT!(), Transforms.R2R!(FFTW.REDFT01))\ntrue\n\njulia> is_inplace(Transforms.FFT(), Transforms.R2R(FFTW.REDFT01))\nfalse\n\njulia> is_inplace(Transforms.FFT(), Transforms.R2R!(FFTW.REDFT01)) === nothing\ntrue\n\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.kind","page":"Available transforms","title":"PencilFFTs.Transforms.kind","text":"kind(transform::R2R)\n\nGet kind of real-to-real transform.\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.length_output","page":"Available transforms","title":"PencilFFTs.Transforms.length_output","text":"length_output(transform::AbstractTransform, length_in::Integer)\n\nReturns the length of the transform output, given the length of its input.\n\nThe input and output lengths are specified in terms of the respective input and output datatypes. For instance, for real-to-complex transforms, these are respectively the length of input real data and of output complex data.\n\n\n\n\n\n","category":"function"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"EditURL = \"https://github.com/jipolanco/PencilFFTs.jl/blob/master/docs/examples/navier_stokes.jl\"","category":"page"},{"location":"generated/navier_stokes/#Navier–Stokes-equations","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"In this example, we numerically solve the incompressible Navier–Stokes equations","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"_t bmv + (bmv bm) bmv = -frac1ρ bm p + ν ^2 bmv\nquad bm bmv = 0","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where bmv(bmx t) and p(bmx t) are respectively the velocity and pressure fields, ν is the fluid kinematic viscosity and ρ is the fluid density.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.","category":"page"},{"location":"generated/navier_stokes/#First-steps","page":"Navier–Stokes equations","title":"First steps","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We start by loading the required packages, initialising MPI and setting the simulation parameters.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using MPI\nusing PencilFFTs\n\nMPI.Init()\ncomm = MPI.COMM_WORLD\nprocid = MPI.Comm_rank(comm) + 1\n\n# Simulation parameters\nNs = (64, 64, 64) # = (Nx, Ny, Nz)\nLs = (2π, 2π, 2π) # = (Lx, Ly, Lz)\n\n# Collocation points (\"global\" = over all processes).\n# We include the endpoint (length = N + 1) for convenience.\nxs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls) # = (x, y, z)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's check the number of MPI processes over which we're running our simulation:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"MPI.Comm_size(comm)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"pen = Pencil(Ns, comm)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"The subdomain associated to the local MPI process can be obtained using range_local:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"range_local(pen)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We now construct a distributed vector field that follows the decomposition configuration we just created:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v⃗₀ = (\n PencilArray{Float64}(undef, pen), # vx\n PencilArray{Float64}(undef, pen), # vy\n PencilArray{Float64}(undef, pen), # vz\n)\nsummary(v⃗₀[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We still need to fill this array with interesting values that represent a physical velocity field.","category":"page"},{"location":"generated/navier_stokes/#Initial-condition","page":"Navier–Stokes equations","title":"Initial condition","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"beginaligned\nv_x(x y z) = u₀ sin(k₀ x) cos(k₀ y) cos(k₀ z) \nv_y(x y z) = -u₀ cos(k₀ x) sin(k₀ y) cos(k₀ z) \nv_z(x y z) = 0\nendaligned","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where u₀ and k₀ are two parameters setting the amplitude and the period of the velocity field.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"grid = localgrid(pen, xs_global)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can use this to initialise the velocity field:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"u₀ = 1.0\nk₀ = 2π / Ls[1] # should be integer if L = 2π (to preserve periodicity)\n\n@. v⃗₀[1] = u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)\n@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)\n@. v⃗₀[3] = 0\nnothing # hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's plot a 2D slice of the velocity field managed by the local MPI process:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using GLMakie\n\n# Compute the norm of a vector field represented by a tuple of arrays.\nfunction vecnorm(v⃗::NTuple)\n vnorm = similar(v⃗[1])\n for n ∈ eachindex(v⃗[1])\n w = zero(eltype(vnorm))\n for v ∈ v⃗\n w += v[n]^2\n end\n vnorm[n] = sqrt(w)\n end\n vnorm\nend\n\nlet fig = Figure(resolution = (700, 600))\n ax = Axis3(fig[1, 1]; aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\")\n vnorm = vecnorm(v⃗₀)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, vnorm;\n alpha = 0.2, levels = 4,\n colormap = :viridis, colorrange = (0.0, 1.0),\n )\n cb = Colorbar(fig[1, 2], ct; label = \"Velocity magnitude\")\n fig\nend","category":"page"},{"location":"generated/navier_stokes/#Velocity-in-Fourier-space","page":"Navier–Stokes equations","title":"Velocity in Fourier space","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"bmv(bmx t) =\n_bmk hatbmv_bmk(t) e^i bmk bmx","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where bmk = (k_x k_y k_z) are the discrete wave numbers.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for k_x:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using AbstractFFTs: fftfreq, rfftfreq\n\nks_global = (\n rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]), # kx | real-to-complex\n fftfreq(Ns[2], 2π * Ns[2] / Ls[2]), # ky | complex-to-complex\n fftfreq(Ns[3], 2π * Ns[3] / Ls[3]), # kz | complex-to-complex\n)\n\nks_global[1]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ks_global[2]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ks_global[3]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"See PencilFFTPlan for details on creating plans and on optional keyword arguments.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients hatbmv_bmk:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v̂s = plan .* v⃗₀\nsummary(v̂s[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Note that, in Fourier space, the domain decomposition is performed along the directions x and y:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"pencil(v̂s[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"This is because the 3D FFTs are performed one dimension at a time, with the x direction first and the z direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers bmk associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"grid_fourier = localgrid(v̂s[1], ks_global)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, bmω = bm bmv. In Fourier space, this becomes hatbmω = i bmk hatbmv.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using StaticArrays: SVector\nusing LinearAlgebra: ×\n\nfunction curl_fourier!(\n ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,\n ) where {N}\n @inbounds for I ∈ eachindex(grid_fourier)\n # We use StaticArrays for the cross product between small vectors.\n ik⃗ = im * SVector(grid_fourier[I])\n v⃗ = SVector(getindex.(v̂s, Ref(I))) # = (v̂s[1][I], v̂s[2][I], ...)\n ω⃗ = ik⃗ × v⃗\n for n ∈ eachindex(ω⃗)\n ω̂s[n][I] = ω⃗[n]\n end\n end\n ω̂s\nend\n\nω̂s = similar.(v̂s)\ncurl_fourier!(ω̂s, v̂s, grid_fourier);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We finally transform back to physical space and plot the result:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ωs = plan .\\ ω̂s\n\nlet fig = Figure(resolution = (700, 600))\n ax = Axis3(fig[1, 1]; aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\")\n ω_norm = vecnorm(ωs)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, ω_norm;\n alpha = 0.1, levels = 0.8:0.2:2.0,\n colormap = :viridis, colorrange = (0.8, 2.0),\n )\n cb = Colorbar(fig[1, 2], ct; label = \"Vorticity magnitude\")\n fig\nend","category":"page"},{"location":"generated/navier_stokes/#Computing-the-non-linear-term","page":"Navier–Stokes equations","title":"Computing the non-linear term","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"_t hatbmv_bmk =\n- mathcalP_bmk left widehat(bmv bm) bmv right\n- ν bmk^2 hatbmv_bmk\nquad text with quad\nmathcalP_bmk(hatbmF_bmk) = left( I - fracbmk \nbmkbmk^2 right) hatbmF_bmk","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where mathcalP_bmk is a projection operator allowing to preserve the incompressibility condition bm bmv = 0. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Now that we have the wave numbers bmk, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients hatbmv_bmk of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, hatbmF_bmk = left widehat(bmv bm) bmv right_bmk. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Below we implement a function that computes the non-linear term in Fourier space based on its convective form (bmv bm) bmv = bm (bmv bmv). Note that this equivalence uses the incompressibility condition bm bmv = 0.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using LinearAlgebra: mul!, ldiv! # for applying FFT plans in-place\n\n# Compute non-linear term in Fourier space from velocity field in physical\n# space. Optional keyword arguments may be passed to avoid allocations.\nfunction ns_nonlinear!(\n F̂s, vs, plan, grid_fourier;\n vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),\n )\n # Compute F_i = ∂_j (v_i v_j) for each i.\n # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)\n w, ŵ = vbuf, v̂buf\n @inbounds for (i, F̂i) ∈ enumerate(F̂s)\n F̂i .= 0\n vi = vs[i]\n for (j, vj) ∈ enumerate(vs)\n w .= vi .* vj # w = v_i * v_j in physical space\n mul!(ŵ, plan, w) # same in Fourier space\n # Add derivative in Fourier space\n for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n kj = k⃗[j]\n F̂i[I] += im * kj * ŵ[I]\n end\n end\n end\n F̂s\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"As an example, let's use this function on our initial velocity field:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"F̂s = similar.(v̂s)\nns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)\n ks_max = maximum.(abs, ks_global) # maximum stored wave numbers (kx_max, ky_max, kz_max)\n ks_lim = (2 / 3) .* ks_max\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I]\n if any(abs.(k⃗) .> ks_lim)\n for ŵ ∈ ŵs\n ŵ[I] = 0\n end\n end\n end\n ŵs\nend\n\n# We can apply this on the previously computed non-linear term:\ndealias_twothirds!(F̂s, grid_fourier, ks_global);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Finally, we implement the projection associated to the incompressibility condition:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function project_divergence_free!(ûs, grid_fourier)\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I]\n k² = sum(abs2, k⃗)\n iszero(k²) && continue # avoid division by zero\n û = getindex.(ûs, Ref(I)) # (ûs[1][I], ûs[2][I], ...)\n for i ∈ eachindex(û)\n ŵ = û[i]\n for j ∈ eachindex(û)\n ŵ -= k⃗[i] * k⃗[j] * û[j] / k²\n end\n ûs[i][I] = ŵ\n end\n end\n ûs\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)\nv̂s_proj .≈ v̂s # the last one may be false because v_z = 0 initially","category":"page"},{"location":"generated/navier_stokes/#Putting-it-all-together","page":"Navier–Stokes equations","title":"Putting it all together","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function ns_rhs!(\n dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,\n ) where {N}\n # 1. Compute non-linear term and dealias it\n (; plan, cache, ks_global, grid_fourier) = p\n F̂s = cache.F̂s\n ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])\n dealias_twothirds!(F̂s, grid_fourier, ks_global)\n\n # 2. Project onto divergence-free space\n project_divergence_free!(F̂s, grid_fourier)\n\n # 3. Transform velocity to Fourier space\n v̂s = cache.v̂s\n map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)\n\n # 4. Add viscous term (and multiply projected non-linear term by -1)\n ν = p.ν\n for n ∈ eachindex(v̂s)\n v̂ = v̂s[n]\n F̂ = F̂s[n]\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n k² = sum(abs2, k⃗)\n F̂[I] = -F̂[I] - ν * k² * v̂[I]\n end\n end\n\n # 5. Transform RHS back to physical space\n map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)\n\n nothing\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using OrdinaryDiffEq\nusing RecursiveArrayTools: ArrayPartition\n\nns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)\nvs_init_ode = ArrayPartition(v⃗₀)\nsummary(vs_init_ode)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We now define solver parameters and temporary variables, and initialise the problem:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"params = (;\n ν = 5e-3, # kinematic viscosity\n plan, grid_fourier, ks_global,\n cache = (\n v̂s = similar.(v̂s),\n F̂s = similar.(v̂s),\n )\n)\n\ntspan = (0.0, 10.0)\nprob = ODEProblem(ns_rhs!, vs_init_ode, tspan, params)\nintegrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum E(k), to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity ν must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large k), while high enough to allow the small-scale motions to be correctly resolved.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function energy_spectrum!(Ek, ks, v̂s, grid_fourier)\n Nk = length(Ek)\n @assert Nk == length(ks)\n Ek .= 0\n for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n knorm = sqrt(sum(abs2, k⃗))\n i = searchsortedfirst(ks, knorm)\n i > Nk && continue\n v⃗ = getindex.(v̂s, Ref(I)) # = (v̂s[1][I], v̂s[2][I], ...)\n factor = k⃗[1] == 0 ? 1 : 2 # account for Hermitian symmetry and r2c transform\n Ek[i] += factor * sum(abs2, v⃗) / 2\n end\n MPI.Allreduce!(Ek, +, get_comm(v̂s[1])) # sum across all processes\n Ek\nend\n\nks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])\nEk = similar(ks)\nv̂s = plan .* integrator.u.x\nenergy_spectrum!(Ek, ks, v̂s, grid_fourier)\nEk ./= scale_factor(plan)^2 # rescale energy\n\ncurl_fourier!(ω̂s, v̂s, grid_fourier)\nldiv!.(ωs, plan, ω̂s)\nω⃗_plot = Observable(ωs)\nk_plot = @view ks[2:end]\nE_plot = Observable(@view Ek[2:end])\nt_plot = Observable(integrator.t)\n\nfig = let\n fig = Figure(resolution = (1200, 600))\n ax = Axis3(\n fig[1, 1][1, 1]; title = @lift(\"t = $(round($t_plot, digits = 3))\"),\n aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\",\n )\n ω_mag = @lift vecnorm($ω⃗_plot)\n ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, ω_mag_norm;\n alpha = 0.3, levels = 3,\n colormap = :viridis, colorrange = (0.0, 1.0),\n )\n cb = Colorbar(fig[1, 1][1, 2], ct; label = \"Normalised vorticity magnitude\")\n ax_sp = Axis(\n fig[1, 2];\n xlabel = \"k\", ylabel = \"E(k)\", xscale = log2, yscale = log10,\n title = \"Kinetic energy spectrum\",\n )\n ylims!(ax_sp, 1e-8, 1e0)\n scatterlines!(ax_sp, k_plot, E_plot)\n ks_slope = exp.(range(log(2.5), log(25.0), length = 3))\n E_fivethirds = @. 0.3 * ks_slope^(-5/3)\n @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)\n text!(ax_sp, L\"k^{-5/3}\"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))\n fig\nend\n\nusing Printf # hide\nwith_xvfb = ENV[\"DISPLAY\"] == \":99\" # hide\nnstep = 0 # hide\nconst tmpdir = mktempdir() # hide\nfilename_frame(procid, nstep) = joinpath(tmpdir, @sprintf(\"proc%d_%04d.png\", procid, nstep)) # hide\nrecord(fig, \"vorticity_proc$procid.mp4\"; framerate = 10) do io\n with_xvfb && recordframe!(io) # hide\n while integrator.t < 20\n dt = 0.001\n step!(integrator, dt)\n t_plot[] = integrator.t\n mul!.(v̂s, plan, integrator.u.x) # current velocity in Fourier space\n curl_fourier!(ω̂s, v̂s, grid_fourier)\n ldiv!.(ω⃗_plot[], plan, ω̂s)\n ω⃗_plot[] = ω⃗_plot[] # to force updating the plot\n energy_spectrum!(Ek, ks, v̂s, grid_fourier)\n Ek ./= scale_factor(plan)^2 # rescale energy\n E_plot[] = E_plot[]\n global nstep += 1 # hide\n with_xvfb ? # hide\n save(filename_frame(procid, nstep), fig) : # hide\n recordframe!(io)\n end\nend;\n\nif with_xvfb # hide\n run(pipeline(`ffmpeg -y -r 10 -i $tmpdir/proc$(procid)_%04d.png -c:v libx264 -vf \"fps=25,format=yuv420p\" vorticity_proc$procid.mp4`; stdout = \"ffmpeg.out\", stderr = \"ffmpeg.err\")) # hide\nend # hide\nnothing # hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"
    \n \n
    ","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"This page was generated using Literate.jl.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"EditURL = \"https://github.com/jipolanco/PencilFFTs.jl/blob/master/docs/examples/gradient.jl\"","category":"page"},{"location":"generated/gradient/#Gradient-of-a-scalar-field","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This example shows different methods to compute the gradient of a real-valued 3D scalar field θ(bmx) in Fourier space, where bmx = (x y z). It is assumed that the field is periodic with period L = 2π along all dimensions.","category":"page"},{"location":"generated/gradient/#General-procedure","page":"Gradient of a scalar field","title":"General procedure","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The discrete Fourier expansion of θ writes","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"θ(bmx) = _bmk Z^3 hatθ(bmk) e^i bmk bmx","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"where bmk = (k_x k_y k_z) are the Fourier wave numbers and hatθ is the discrete Fourier transform of θ. Then, the spatial derivatives of θ are given by","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"frac θ(bmx) x_i =\n_bmk Z^3 i k_i hatθ(bmk) e^i bmk bmx","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"where the subscript i denotes one of the spatial components x, y or z.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In other words, to compute bm θ = (_x θ _y θ _z θ), one has to:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"transform θ to Fourier space to obtain hatθ,\nmultiply hatθ by i bmk,\ntransform the result back to physical space to obtain bm θ.","category":"page"},{"location":"generated/gradient/#Preparation","page":"Gradient of a scalar field","title":"Preparation","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In this section, we initialise a random real-valued scalar field θ and compute its FFT. For more details see the Tutorial.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"using MPI\nusing PencilFFTs\nusing Random\n\nMPI.Init()\n\n# Input data dimensions (Nx × Ny × Nz)\ndims = (64, 32, 64)\n\n# Apply a 3D real-to-complex (r2c) FFT.\ntransform = Transforms.RFFT()\n\n# Automatically create decomposition configuration\ncomm = MPI.COMM_WORLD\npen = Pencil(dims, comm)\n\n# Create plan\nplan = PencilFFTPlan(pen, transform)\n\n# Allocate data and initialise field\nθ = allocate_input(plan)\nrandn!(θ)\n\n# Perform distributed FFT\nθ_hat = plan * θ\nnothing # hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, we initialise the output that will hold ∇θ in Fourier space. Noting that ∇θ is a vector field, we choose to store it as a tuple of 3 PencilArrays.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"∇θ_hat = allocate_output(plan, Val(3))\n\n# This is equivalent:\n# ∇θ_hat = ntuple(d -> similar(θ_hat), Val(3))\n\nsummary(∇θ_hat)","category":"page"},{"location":"generated/gradient/#Fourier-wave-numbers","page":"Gradient of a scalar field","title":"Fourier wave numbers","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In general, the Fourier wave numbers are of the form k_i = 0 frac2πL_i frac4πL_i frac6πL_i , where L_i is the period along dimension i. When a real-to-complex Fourier transform is applied, roughly half of these wave numbers are redundant due to the Hermitian symmetry of the complex Fourier coefficients. In practice, this means that for the fastest dimension x (along which a real-to-complex transform is performed), the negative wave numbers are dropped, i.e. k_x = 0 frac2πL_x frac4πL_x .","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The AbstractFFTs package provides a convenient way to generate the Fourier wave numbers, using the functions fftfreq and rfftfreq. We can use these functions to initialise a \"grid\" of wave numbers associated to our 3D real-to-complex transform:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"using AbstractFFTs: fftfreq, rfftfreq\n\nbox_size = (2π, 2π, 2π) # Lx, Ly, Lz\nsample_rate = 2π .* dims ./ box_size\n\n# In our case (Lx = 2π and Nx even), this gives kx = [0, 1, 2, ..., Nx/2].\nkx = rfftfreq(dims[1], sample_rate[1])\n\n# In our case (Ly = 2π and Ny even), this gives\n# ky = [0, 1, 2, ..., Ny/2-1, -Ny/2, -Ny/2+1, ..., -1] (and similarly for kz).\nky = fftfreq(dims[2], sample_rate[2])\nkz = fftfreq(dims[3], sample_rate[3])\n\nkvec = (kx, ky, kz)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that kvec now contains the wave numbers associated to the global domain. In the following, we will only need the wave numbers associated to the portion of the domain handled by the local MPI process.","category":"page"},{"location":"generated/gradient/#gradient_method_global","page":"Gradient of a scalar field","title":"Method 1: global views","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"PencilArrays, returned for instance by allocate_input and allocate_output, take indices that start at 1, regardless of the location of the subdomain associated to the local process on the global grid. (In other words, PencilArrays take local indices.) On the other hand, we have defined the wave number vector kvec which, for each MPI process, is defined over the global domain, and as such it takes global indices.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"One straightforward way of making data arrays compatible with wave numbers is to use global views, i.e. arrays that take global indices. These are generated from PencilArrays by calling the global_view function. Note that, in general, global indices do not start at 1 for a given MPI process. A given process will own a range of data given by indices in (i1:i2, j1:j2, k1:k2).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"θ_glob = global_view(θ_hat)\n∇θ_glob = global_view.(∇θ_hat)\nsummary(θ_glob)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Once we have global views, we can combine data and wave numbers using the portion of global indices owned by the local MPI process, as shown below. We can use CartesianIndices to iterate over the global indices associated to the local process.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"for I in CartesianIndices(θ_glob)\n i, j, k = Tuple(I) # unpack indices\n\n # Wave number vector associated to current Cartesian index.\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n # Compute gradient in Fourier space.\n # Note that modifying ∇θ_glob also modifies the original PencilArray ∇θ_hat.\n ∇θ_glob[1][I] = im * kx * θ_glob[I]\n ∇θ_glob[2][I] = im * ky * θ_glob[I]\n ∇θ_glob[3][I] = im * kz * θ_glob[I]\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The above loop can be written in a slightly more efficient manner by precomputing im * θ_glob[I]:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@inbounds for I in CartesianIndices(θ_glob)\n i, j, k = Tuple(I)\n\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n u = im * θ_glob[I]\n\n ∇θ_glob[1][I] = kx * u\n ∇θ_glob[2][I] = ky * u\n ∇θ_glob[3][I] = kz * u\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Also note that the above can be easily written in a more generic way, e.g. for arbitrary dimensions, thanks in part to the use of CartesianIndices. Moreover, in the above there is no notion of the dimension permutations discussed in the tutorial, as it is all hidden behind the implementation of PencilArrays. And as seen later in the benchmarks, these (hidden) permutations have zero cost, as the speed is identical to that of a function that explicitly takes into account these permutations.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, we can perform a backwards transform to obtain bm θ in physical space:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"∇θ = plan \\ ∇θ_hat;\nnothing #hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that the transform is automatically broadcast over the three fields of the ∇θ_hat vector, and the result ∇θ is also a tuple of three PencilArrays.","category":"page"},{"location":"generated/gradient/#gradient_method_global_explicit","page":"Gradient of a scalar field","title":"Method 2: explicit global indexing","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Sometimes, one does not need to write generic code. In our case, one often knows the dimensionality of the problem and the memory layout of the data (i.e. the underlying index permutation).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Below is a reimplementation of the above loop, using explicit indices instead of CartesianIndices, and assuming that the underlying index permutation is (3, 2, 1), that is, data is stored in (z y x) order. As discussed in the tutorial, this is the default for transformed arrays. This example also serves as a more explicit explanation for what is going on in the first method.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"# Get local data range in the global grid.\nrng = axes(θ_glob) # = (i1:i2, j1:j2, k1:k2)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"For the loop below, we're assuming that the permutation is (3, 2, 1). In other words, the fastest index is the last one, and not the first one as it is usually in Julia. If the permutation is not (3, 2, 1), things will still work (well, except for the assertion below!), but the loop order will not be optimal.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@assert permutation(θ_hat) === Permutation(3, 2, 1)\n\n@inbounds for i in rng[1], j in rng[2], k in rng[3]\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n # Note that we still access the arrays in (i, j, k) order.\n # (The permutation happens behind the scenes!)\n u = im * θ_glob[i, j, k]\n\n ∇θ_glob[1][i, j, k] = kx * u\n ∇θ_glob[2][i, j, k] = ky * u\n ∇θ_glob[3][i, j, k] = kz * u\nend","category":"page"},{"location":"generated/gradient/#gradient_method_local","page":"Gradient of a scalar field","title":"Method 3: using local indices","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Alternatively, we can avoid global views and work directly on PencilArrays using local indices that start at 1. In this case, part of the strategy is to construct a \"local\" grid of wave numbers that can also be accessed with local indices. This can be conveniently done using the localgrid function of the PencilArrays.jl package, which accepts a PencilArray (or its associated Pencil) and the global coordinates (here kvec):","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"grid_fourier = localgrid(θ_hat, kvec)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that one can directly iterate on the returned grid object:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@inbounds for I in CartesianIndices(grid_fourier)\n # Wave number vector associated to current Cartesian index.\n local k⃗ # hide\n k⃗ = grid_fourier[I]\n u = im * θ_hat[I]\n ∇θ_hat[1][I] = k⃗[1] * u\n ∇θ_hat[2][I] = k⃗[2] * u\n ∇θ_hat[3][I] = k⃗[3] * u\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This implementation is as efficient as the other examples, while being slightly shorter to write. Moreover, it is quite generic, and can be made independent of the number of dimensions with little effort.","category":"page"},{"location":"generated/gradient/#gradient_method_broadcast","page":"Gradient of a scalar field","title":"Method 4: using broadcasting","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, note that the local grid object returned by localgrid makes it is possible to compute the gradient using broadcasting, thus fully avoiding scalar indexing. This can be quite convenient in some cases, and can also be very useful if one is working on GPUs (where scalar indexing is prohibitively expensive). Using broadcasting, the above examples simply become:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@. ∇θ_hat[1] = im * grid_fourier[1] * θ_hat\n@. ∇θ_hat[2] = im * grid_fourier[2] * θ_hat\n@. ∇θ_hat[3] = im * grid_fourier[3] * θ_hat\nnothing # hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Once again, as shown in the benchmarks further below, this method performs quite similarly to the other ones.","category":"page"},{"location":"generated/gradient/#Summary","page":"Gradient of a scalar field","title":"Summary","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The PencilArrays module provides different alternatives to deal with MPI-distributed data that may be subject to dimension permutations. In particular, one can choose to work with global indices (first two examples), with local indices (third example), or to avoid scalar indexing altogether (fourth example).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"If one wants to stay generic, making sure that the same code will work for arbitrary dimensions and will be efficient regardless of the underlying dimension permutation, methods 1, 3 or 4 should be preferred. These use CartesianIndices and make no assumptions on possible dimension permutations, which are by default enabled in the output of PencilFFTs transforms. In fact, such permutations are completely invisible in the implementations.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The second method uses explicit (i, j, k) indices. It assumes that the underlying permutation is (3, 2, 1) to loop with i as the slowest index and k as the fastest, which is the optimal order in this case given the permutation. As such, the implementation is less generic than the others, and differences in performance are negligible with respect to more generic variants.","category":"page"},{"location":"generated/gradient/#gradient_benchmarks","page":"Gradient of a scalar field","title":"Benchmark results","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The following are the benchmark results obtained from running examples/gradient.jl on a laptop, using 2 MPI processes and Julia 1.7.2, with an input array of global dimensions 64 32 64. The different methods detailed above are marked on the right. The \"lazy\" marks indicate runs where the wave numbers were represented by lazy Frequencies objects (returned by rfftfreq and fftfreq). Otherwise, they were collected into Vectors. For some reason, plain Vectors are faster when working with grids generated by localgrid.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In the script, additional implementations can be found which rely on a more advanced understanding of permutations and on the internals of the PencilArrays package. For instance, gradient_local_parent! directly works with the raw data stored in Julia Arrays, while gradient_local_linear! completely avoids CartesianIndices while staying generic and efficient. Nevertheless, these display roughly the same performance as the above examples.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":" gradient_global_view!... 89.900 μs\n gradient_global_view! (lazy)... 92.060 μs [Method 1]\n gradient_global_view_explicit!... 88.958 μs\n gradient_global_view_explicit! (lazy)... 81.055 μs [Method 2]\n gradient_local!... 92.305 μs\n gradient_grid!... 92.770 μs\n gradient_grid! (lazy)... 101.388 μs [Method 3]\n gradient_grid_broadcast!... 88.606 μs\n gradient_grid_broadcast! (lazy)... 151.020 μs [Method 4]\n gradient_local_parent!... 92.248 μs\n gradient_local_linear!... 91.212 μs\n gradient_local_linear_explicit!... 90.992 μs","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This page was generated using Literate.jl.","category":"page"},{"location":"PencilFFTs_timers/#PencilFFTs.measuring_performance","page":"Measuring performance","title":"Measuring performance","text":"","category":"section"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"It is possible to measure the time spent in different sections of the distributed transforms using the TimerOutputs package. This has a (very small) performance overhead, so it is disabled by default. To enable time measurements, call TimerOutputs.enable_debug_timings after loading PencilFFTs (see below for an example). For more details see the TimerOutputs docs.","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"Minimal example:","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"using MPI\nusing PencilFFTs\nusing TimerOutputs\n\n# Enable timing of `PencilFFTs` functions\nTimerOutputs.enable_debug_timings(PencilFFTs)\nTimerOutputs.enable_debug_timings(PencilArrays)\nTimerOutputs.enable_debug_timings(Transpositions)\n\nMPI.Init()\n\nplan = PencilFFTPlan(#= args... =#)\n\n# [do stuff with `plan`...]\n\n# Retrieve and print timing data associated to `plan`\nto = timer(plan)\nprint_timer(to)","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"By default, each PencilFFTPlan has its own TimerOutput. If you already have a TimerOutput, you can pass it to the PencilFFTPlan constructor:","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"to = TimerOutput()\nplan = PencilFFTPlan(..., timer=to)\n\n# [do stuff with `plan`...]\n\nprint_timer(to)","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"#PencilFFTs","page":"Home","title":"PencilFFTs","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Fast Fourier transforms of MPI-distributed Julia arrays.","category":"page"},{"location":"#Introduction","page":"Home","title":"Introduction","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package provides multidimensional FFTs and related transforms on MPI-distributed Julia arrays via the PencilArrays package.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The name of this package originates from the decomposition of 3D domains along two out of three dimensions, sometimes called pencil decomposition. This is illustrated by the figure below,[1] where each coloured block is managed by a different MPI process. Typically, one wants to compute FFTs on a scalar or vector field along the three spatial dimensions. In the case of a pencil decomposition, 3D FFTs are performed one dimension at a time, along the non-decomposed direction. Transforms must then be interleaved with global data transpositions to switch between pencil configurations. In high-performance computing environments, such data transpositions are generally the most expensive part of a parallel FFT computation, due to the large cost of communications between computing nodes.","category":"page"},{"location":"","page":"Home","title":"Home","text":"
    \n \n
    ","category":"page"},{"location":"","page":"Home","title":"Home","text":"More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension N. The decompositions can be performed along an arbitrary number M N of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.","category":"page"},{"location":"#Features","page":"Home","title":"Features","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;\nFFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;\nin-place and out-of-place transforms;\nhigh scalability up to (at least) tens of thousands of MPI processes.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"PencilFFTs can be installed using the Julia package manager:","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> ] add PencilFFTs","category":"page"},{"location":"#Similar-projects","page":"Home","title":"Similar projects","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.\nPFFT is a very general parallel FFT library written in C.\nP3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.\n2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[1]: Figure adapted from this PhD thesis.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[2]: For the pencil decomposition represented in the figure, N = 3 and M = 2.","category":"page"},{"location":"PencilFFTs/#Distributed-FFT-plans","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"Distributed FFTs are implemented in the PencilFFTs module, and are built on top of the PencilArrays package.","category":"page"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"PencilFFTs/#Creating-plans","page":"Distributed FFT plans","title":"Creating plans","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"PencilFFTPlan","category":"page"},{"location":"PencilFFTs/#PencilFFTs.PencilFFTPlan","page":"Distributed FFT plans","title":"PencilFFTs.PencilFFTPlan","text":"PencilFFTPlan{T,N} <: AbstractFFTs.Plan{T}\n\nPlan for N-dimensional FFT-based transform on MPI-distributed data, where input data has type T.\n\n\n\nPencilFFTPlan(p::Pencil, transforms; kwargs...)\n\nCreate a PencilFFTPlan for distributed arrays following a given Pencil configuration. See variant below for details on the specification of transforms and on possible keyword arguments.\n\n\n\nPencilFFTPlan(\n A::PencilArray, transforms;\n fftw_flags = FFTW.ESTIMATE,\n fftw_timelimit = FFTW.NO_TIMELIMIT,\n permute_dims = Val(true),\n transpose_method = Transpositions.PointToPoint(),\n timer = timer(pencil(A)),\n)\n\nCreate plan for N-dimensional transform on MPI-distributed PencilArrays.\n\nExtended help\n\nThis creates a PencilFFTPlan for arrays sharing the same properties as A (dimensions, MPI decomposition, memory layout, ...), which describe data on an N-dimensional domain.\n\nTransforms\n\nThe transforms to be applied along each dimension are specified by the transforms argument. Possible transforms are defined as subtypes of Transforms.AbstractTransform, and are listed in Transform types. This argument may be either:\n\na tuple of N transforms to be applied along each dimension. For instance, transforms = (Transforms.R2R(FFTW.REDFT01), Transforms.RFFT(), Transforms.FFT());\na single transform to be applied along all dimensions. The input is automatically expanded into N equivalent transforms. For instance, for a three-dimensional array, transforms = Transforms.RFFT() specifies a 3D real-to-complex transform, and is equivalent to passing (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT()).\n\nNote that forward transforms are applied from left to right. In the last example, this means that a real-to-complex transform (RFFT) is first performed along the first dimension. This is followed by complex-to-complex transforms (FFT) along the second and third dimensions.\n\nInput data layout\n\nThe input PencilArray must satisfy the following constraints:\n\narray dimensions must not be permuted. This is the default when constructing PencilArrays.\nfor an M-dimensional domain decomposition (with M < N), the input array must be decomposed along the last M dimensions. For example, for a 2D decomposition of 3D data, the decomposed dimensions must be (2, 3). In particular, the first array dimension must not be distributed among different MPI processes.\nIn the PencilArrays package, the decomposed dimensions are specified at the moment of constructing a Pencil.\nthe element type must be compatible with the specified transform. For instance, real-to-complex transforms (Transforms.RFFT) require the input to be real floating point values. Other transforms, such as Transforms.R2R, accept both real and complex data.\n\nKeyword arguments\n\nThe keyword arguments fftw_flags and fftw_timelimit are passed to the FFTW plan creation functions (see AbstractFFTs docs).\npermute_dims determines whether the indices of the output data should be reversed. For instance, if the input data has global dimensions (Nx, Ny, Nz), then the output of a complex-to-complex FFT would have dimensions (Nz, Ny, Nx). This enables FFTs to always be performed along the first (i.e. fastest) array dimension, which could lead to performance gains. This option is enabled by default. For type inference reasons, it must be a value type (Val(true) or Val(false)).\ntranspose_method allows to select between implementations of the global data transpositions. See PencilArrays docs docs for details.\ntimer should be a TimerOutput object. See Measuring performance for details.\n\n\n\nPencilFFTPlan(\n dims_global::Dims{N}, transforms, proc_dims::Dims{M}, comm::MPI.Comm,\n [real_type = Float64]; extra_dims = (), kws...\n)\n\nCreate plan for N-dimensional transform.\n\nExtended help\n\nInstead of taking a PencilArray or a Pencil, this constructor requires the global dimensions of the input data, passed via the size_global argument.\n\nThe data is distributed over the MPI processes in the comm communicator. The distribution is performed over M dimensions (with M < N) according to the values in proc_dims, which specifies the number of MPI processes to put along each dimension.\n\nPencilArrays that may be transformed with the returned plan can be created using allocate_input.\n\nOptional arguments\n\nThe floating point precision can be selected by setting real_type parameter, which is Float64 by default.\nextra_dims may be used to specify the sizes of one or more extra dimensions that should not be transformed. These dimensions will be added to the rightmost (i.e. slowest) indices of the arrays. See Extra dimensions below for usage hints.\nsee the other constructor for more keyword arguments.\n\nExtra dimensions\n\nOne possible application of extra_dims is for describing the components of a vector or tensor field. However, this means that different PencilFFTPlans would need to be created for each kind of field (scalar, vector, ...). To avoid the creation of multiple plans, a possibly better alternative is to create tuples (or arrays) of PencilArrays using allocate_input and allocate_output.\n\nAnother more legitimate usage of extra_dims is to specify one or more Cartesian dimensions that should not be transformed nor split among MPI processes.\n\nExample\n\nSuppose we want to perform a 3D FFT of real data. The data is to be decomposed along two dimensions, over 8 MPI processes:\n\nsize_global = (64, 32, 128) # size of real input data\n\n# Perform real-to-complex transform along the first dimension, then\n# complex-to-complex transforms along the other dimensions.\ntransforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())\n# transforms = Transforms.RFFT() # this is equivalent to the above line\n\nproc_dims = (4, 2) # 2D decomposition\ncomm = MPI.COMM_WORLD\n\nplan = PencilFFTPlan(size_global, transforms, proc_dims, comm)\n\n\n\n\n\n","category":"type"},{"location":"PencilFFTs/#Allocating-data","page":"Distributed FFT plans","title":"Allocating data","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"allocate_input\nallocate_output","category":"page"},{"location":"PencilFFTs/#PencilFFTs.allocate_input","page":"Distributed FFT plans","title":"PencilFFTs.allocate_input","text":"allocate_input(p::PencilFFTPlan) -> PencilArray\nallocate_input(p::PencilFFTPlan, dims...) -> Array{PencilArray}\nallocate_input(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}\n\nAllocate uninitialised PencilArray that can hold input data for the given plan.\n\nThe second and third forms respectively allocate an array of PencilArrays of size dims, and a tuple of N PencilArrays.\n\nnote: In-place plans\nIf p is an in-place plan, a ManyPencilArray is allocated. This type holds PencilArray wrappers for the input and output transforms (as well as for intermediate transforms) which share the same space in memory. The input and output PencilArrays should be respectively accessed by calling first(::ManyPencilArray) and last(::ManyPencilArray).ExampleSuppose p is an in-place PencilFFTPlan. Then,@assert is_inplace(p)\nA = allocate_input(p) :: ManyPencilArray\nv_in = first(A) :: PencilArray # input data view\nv_out = last(A) :: PencilArray # output data viewAlso note that in-place plans must be performed directly on the returned ManyPencilArray, and not on the contained PencilArray views:p * A # perform forward transform in-place\np \\ A # perform backward transform in-place\n# p * v_in # not allowed!!\n\n\n\n\n\n","category":"function"},{"location":"PencilFFTs/#PencilFFTs.allocate_output","page":"Distributed FFT plans","title":"PencilFFTs.allocate_output","text":"allocate_output(p::PencilFFTPlan) -> PencilArray\nallocate_output(p::PencilFFTPlan, dims...) -> Array{PencilArray}\nallocate_output(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}\n\nAllocate uninitialised PencilArray that can hold output data for the given plan.\n\nIf p is an in-place plan, a ManyPencilArray is allocated.\n\nSee allocate_input for details.\n\n\n\n\n\n","category":"function"},{"location":"PencilFFTs/#Methods","page":"Distributed FFT plans","title":"Methods","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"get_comm(::PencilFFTPlan)\nscale_factor(::PencilFFTPlan)\ntimer(::PencilFFTPlan)\nis_inplace(::PencilFFTPlan)","category":"page"},{"location":"PencilFFTs/#PencilArrays.Pencils.MPITopologies.get_comm-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilArrays.Pencils.MPITopologies.get_comm","text":"get_comm(p::PencilFFTPlan)\n\nGet MPI communicator associated to a PencilFFTPlan.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilFFTs.Transforms.scale_factor-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilFFTs.Transforms.scale_factor","text":"scale_factor(p::PencilFFTPlan)\n\nGet scale factor associated to a PencilFFTPlan.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilArrays.Pencils.timer-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilArrays.Pencils.timer","text":"timer(p::PencilFFTPlan)\n\nGet TimerOutput attached to a PencilFFTPlan.\n\nSee Measuring performance for details.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilFFTs.Transforms.is_inplace-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilFFTs.Transforms.is_inplace","text":"Transforms.is_inplace(p::PencilFFTPlan)\n\nReturns true if the given plan operates in-place on the input data, false otherwise.\n\n\n\n\n\n","category":"method"}] -} diff --git a/previews/PR62/siteinfo.js b/previews/PR62/siteinfo.js deleted file mode 100644 index 34c224d6..00000000 --- a/previews/PR62/siteinfo.js +++ /dev/null @@ -1 +0,0 @@ -var DOCUMENTER_CURRENT_VERSION = "previews/PR62"; diff --git a/previews/PR59/GlobalFFTParams/index.html b/previews/PR63/GlobalFFTParams/index.html similarity index 95% rename from previews/PR59/GlobalFFTParams/index.html rename to previews/PR63/GlobalFFTParams/index.html index 233ced58..bd67cf0e 100644 --- a/previews/PR59/GlobalFFTParams/index.html +++ b/previews/PR63/GlobalFFTParams/index.html @@ -6,4 +6,4 @@ julia> fft_params = PencilFFTs.GlobalFFTParams(size_global, transforms) Transforms: (RFFT, FFT, FFT) Input type: Float64 -Global dimensions: (64, 32, 128) -> (33, 32, 128)
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     diff --git a/previews/PR40/search/index.html b/previews/PR40/search/index.html deleted file mode 100644 index f8670dcd..00000000 --- a/previews/PR40/search/index.html +++ /dev/null @@ -1,2 +0,0 @@ - -Search · PencilFFTs.jl

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      diff --git a/previews/PR40/search_index.js b/previews/PR40/search_index.js deleted file mode 100644 index 4146bd68..00000000 --- a/previews/PR40/search_index.js +++ /dev/null @@ -1,3 +0,0 @@ -var documenterSearchIndex = {"docs": -[{"location":"tutorial/#Tutorial","page":"Tutorial","title":"Tutorial","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The following tutorial shows how to perform a 3D FFT of real periodic data defined on a grid of N_x N_y N_z points.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"
      \n \n \n
      ","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"By default, the domain is distributed on a 2D MPI topology of dimensions N_1 N_2. As an example, the above figure shows such a topology with N_1 = 4 and N_2 = 3, for a total of 12 MPI processes.","category":"page"},{"location":"tutorial/#tutorial:creating_plans","page":"Tutorial","title":"Creating plans","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The first thing to do is to create a domain decomposition configuration for the given dataset dimensions N_x N_y N_z. In the framework of PencilArrays, such a configuration is described by a Pencil object. As described in the PencilArrays docs, we can let the Pencil constructor automatically determine such a configuration. For this, only an MPI communicator and the dataset dimensions are needed:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using MPI\nusing PencilFFTs\n\nMPI.Init()\ncomm = MPI.COMM_WORLD\n\n# Input data dimensions (Nx × Ny × Nz)\ndims = (16, 32, 64)\npen = Pencil(dims, comm)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"By default this creates a 2D decomposition (for the case of a 3D dataset), but one can change this as detailed in the PencilArrays documentation linked above.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"We can now create a PencilFFTPlan, which requires information on decomposition configuration (the Pencil object) and on the transforms that will be applied:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# Apply a 3D real-to-complex (r2c) FFT.\ntransform = Transforms.RFFT()\n\n# Note that, for more control, one can instead separately specify the transforms along each dimension:\n# transform = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())\n\n# Create plan\nplan = PencilFFTPlan(pen, transform)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"See the PencilFFTPlan constructor for details on the accepted options, and the Transforms module for the possible transforms. It is also possible to enable fine-grained performance measurements via the TimerOutputs package, as described in Measuring performance.","category":"page"},{"location":"tutorial/#Allocating-data","page":"Tutorial","title":"Allocating data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Next, we want to apply the plan on some data. Transforms may only be applied on PencilArrays, which are array wrappers that include MPI decomposition information (in some sense, analogous to DistributedArrays in Julia's distributed computing approach). The helper function allocate_input can be used to allocate a PencilArray that is compatible with our plan:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# In our example, this returns a 3D PencilArray of real data (Float64).\nu = allocate_input(plan)\n\n# Fill the array with some (random) data\nusing Random\nrandn!(u)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"PencilArrays are a subtype of AbstractArray, and thus they support all common array operations.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Similarly, to preallocate output data, one can use allocate_output:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# In our example, this returns a 3D PencilArray of complex data (Complex{Float64}).\nv = allocate_output(plan)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"This is only required if one wants to apply the plans using a preallocated output (with mul!, see right below).","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The data types returned by allocate_input and allocate_output are slightly different when working with in-place transforms. See the in-place example for details.","category":"page"},{"location":"tutorial/#Applying-plans","page":"Tutorial","title":"Applying plans","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The interface to apply plans is consistent with that of AbstractFFTs. Namely, * and mul! are respectively used for forward transforms without and with preallocated output data. Similarly, \\ and ldiv! are used for backward transforms.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using LinearAlgebra # for mul!, ldiv!\n\n# Apply plan on `u` with `v` as an output\nmul!(v, plan, u)\n\n# Apply backward plan on `v` with `w` as an output\nw = similar(u)\nldiv!(w, plan, v) # now w ≈ u","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Note that, consistently with AbstractFFTs, normalisation is performed at the end of a backward transform, so that the original data is recovered when applying a forward followed by a backward transform.","category":"page"},{"location":"tutorial/#Accessing-and-modifying-data","page":"Tutorial","title":"Accessing and modifying data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For any given MPI process, a PencilArray holds the data associated to its local partition in the global geometry. PencilArrays are accessed using local indices that start at 1, regardless of the location of the local process in the MPI topology. Note that PencilArrays, being based on regular Arrays, support both linear and Cartesian indexing (see the Julia docs for details).","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For convenience, the global_view function can be used to generate an OffsetArray wrapper that takes global indices.","category":"page"},{"location":"tutorial/#tutorial:output_data_layout","page":"Tutorial","title":"Output data layout","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"In memory, the dimensions of the transform output are by default reversed with respect to the input. That is, if the order of indices in the input data is (x, y, z), then the output has order (z, y, x) in memory. This detail is hidden from the user, and output arrays are always accessed in the same order as the input data, regardless of the underlying output dimension permutation. This applies to PencilArrays and to OffsetArrays returned by global_view.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The reasoning behind dimension permutations, is that they allow to always perform FFTs along the fastest array dimension and to avoid a local data transposition, resulting in performance gains. A similar approach is followed by other parallel FFT libraries. FFTW itself, in its distributed-memory routines, includes a flag that enables a similar behaviour. In PencilFFTs, index permutation is the default, but it can be disabled via the permute_dims flag of PencilFFTPlan.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"A great deal of work has been spent in making generic index permutations as efficient as possible, both in intermediate and in the output state of the multidimensional transforms. This has been achieved, in part, by making sure that permutations such as (3, 2, 1) are compile-time constants.","category":"page"},{"location":"tutorial/#Further-reading","page":"Tutorial","title":"Further reading","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For details on working with PencilArrays see the PencilArrays docs.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The examples on the sidebar further illustrate the use of transforms and provide an introduction to working with MPI-distributed data in the form of PencilArrays. In particular, the gradient example illustrates different ways of computing things using Fourier-transformed distributed arrays. Then, the incompressible Navier–Stokes example is a more advanced and complete example of a possible application of the PencilFFTs package.","category":"page"},{"location":"benchmarks/#Benchmarks","page":"Benchmarks","title":"Benchmarks","text":"","category":"section"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The performance of PencilFFTs.jl is comparable to that of other open-source parallel FFT libraries implemented in lower-level languages. Below, we show comparisons with the Fortran implementation of P3DFFT, possibly the most popular of these libraries. The benchmarks were performed on the Jean–Zay cluster of the IDRIS French computing centre (CNRS).","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The figure below shows strong scaling benchmarks of 3D real-to-complex FFTs using 2D (\"pencil\") decomposition. The benchmarks were run for input arrays of dimensions N_x N_y N_z = 512^3, 1024^3 and 2048^3. Each timing is averaged over 100 repetitions.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"
      \n \n \n
      ","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"As seen above, PencilFFTs generally outperforms P3DFFT in its default setting. This is largely explained by the choice of using non-blocking point-to-point MPI communications (via MPI_Isend and MPI_Irecv), while P3DFFT uses collective MPI_Alltoallv calls. This enables PencilFFTs to perform data reordering operations on the partially received data while waiting for the incoming data, leading to better performance. Moreover, in contrast with P3DFFT, the high performance and scalability of PencilFFTs results from a highly generic code, handling decompositions in arbitrary dimensions and a relatively large (and extensible) variety of transformations.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"Note that PencilFFTs can optionally use collective communications (using MPI_Alltoallv) instead of point-to-point communications. For details, see the docs for PencilFFTPlan and for PencilArray transpositions. As seen above, collective communications generally perform worse than point-to-point ones, and runtimes are nearly indistinguishable from those of P3DFFT.","category":"page"},{"location":"benchmarks/#Benchmark-details","page":"Benchmarks","title":"Benchmark details","text":"","category":"section"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The benchmarks were performed using Julia 1.7-beta3 and Intel MPI 2019. We used PencilFFTs v0.12.5 with FFTW.jl v1.4.3 and MPI.jl v0.19.0. We used the Fortran implementation of P3DFFT, version 2.7.6, which was built with Intel 2019 compilers and linked to FFTW 3.3.8. The cluster where the benchmarks were run has Intel Cascade Lake 6248 processors with 2×20 cores per node.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The number of MPI processes along each decomposed dimension, P_1 and P_2, was automatically determined by a call to MPI_Dims_create, which tends to create a balanced decomposition with P_1 P_2. For instance, a total of 1024 processes is divided into P_1 = P_2 = 32. Different results may be obtained with other combinations, but this was not benchmarked.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The source files used to generate this benchmark, as well as the raw benchmark results, are all available in the PencilFFTs repo.","category":"page"},{"location":"GlobalFFTParams/#Global-FFT-parameters","page":"Global FFT parameters","title":"Global FFT parameters","text":"","category":"section"},{"location":"GlobalFFTParams/","page":"Global FFT parameters","title":"Global FFT parameters","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"GlobalFFTParams/","page":"Global FFT parameters","title":"Global FFT parameters","text":"GlobalFFTParams","category":"page"},{"location":"GlobalFFTParams/#PencilFFTs.GlobalFFTParams","page":"Global FFT parameters","title":"PencilFFTs.GlobalFFTParams","text":"GlobalFFTParams{T, N, inplace}\n\nSpecifies the global parameters for an N-dimensional distributed transform. These include the element type T and global data sizes of input and output data, as well as the transform types to be performed along each dimension.\n\n\n\nGlobalFFTParams(size_global, transforms, [real_type=Float64])\n\nDefine parameters for N-dimensional transform.\n\ntransforms must be a tuple of length N specifying the transforms to be applied along each dimension. Each element must be a subtype of Transforms.AbstractTransform. For all the possible transforms, see Transform types.\n\nThe element type must be a real type accepted by FFTW, i.e. either Float32 or Float64.\n\nNote that the transforms are applied one dimension at a time, with the leftmost dimension first for forward transforms.\n\nExample\n\nTo perform a 3D FFT of real data, first a real-to-complex FFT must be applied along the first dimension, followed by two complex-to-complex FFTs along the other dimensions:\n\njulia> size_global = (64, 32, 128); # size of real input data\n\njulia> transforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT());\n\njulia> fft_params = PencilFFTs.GlobalFFTParams(size_global, transforms)\nTransforms: (RFFT, FFT, FFT)\nInput type: Float64\nGlobal dimensions: (64, 32, 128) -> (33, 32, 128)\n\n\n\n\n\n","category":"type"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"EditURL = \"https://github.com/jipolanco/PencilFFTs.jl/blob/master/docs/examples/in-place.jl\"","category":"page"},{"location":"generated/in-place/#In-place-transforms","page":"In-place transforms","title":"In-place transforms","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Complex-to-complex and real-to-real transforms can be performed in-place, enabling important memory savings. The procedure is very similar to that of out-of-place transforms described in the tutorial. The differences are illustrated in the sections below.","category":"page"},{"location":"generated/in-place/#Creating-a-domain-partition","page":"In-place transforms","title":"Creating a domain partition","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"We start by partitioning a domain of dimensions 163264 along all available MPI processes.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"using PencilFFTs\nusing MPI\nMPI.Init()\n\ndims_global = (16, 32, 64) # global dimensions","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Such a partitioning is described by a Pencil object. Here we choose to decompose the domain along the last two dimensions. In this case, the actual number of processes along each of these dimensions is chosen automatically.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"decomp_dims = (2, 3)\ncomm = MPI.COMM_WORLD\npen = Pencil(dims_global, decomp_dims, comm)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"warning: Allowed decompositions\nDistributed transforms using PencilFFTs.jl require that the first dimension is not decomposed. In other words, if one wants to perform transforms, then decomp_dims above must not contain 1.","category":"page"},{"location":"generated/in-place/#Creating-in-place-plans","page":"In-place transforms","title":"Creating in-place plans","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"To create an in-place plan, pass an in-place transform such as Transforms.FFT! or Transforms.R2R! to PencilFFTPlan. For instance:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"# Perform a 3D in-place complex-to-complex FFT.\ntransform = Transforms.FFT!()\n\n# Note that one can also combine different types of in-place transforms.\n# For instance:\n# transform = (\n# Transforms.R2R!(FFTW.REDFT01),\n# Transforms.FFT!(),\n# Transforms.R2R!(FFTW.DHT),\n# )","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"We can now create a distributed plan from the previously-created domain partition and the chosen transform.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan = PencilFFTPlan(pen, transform)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Note that in-place real-to-complex transforms are not currently supported. (In other words, the RFFT! transform type is not defined.)","category":"page"},{"location":"generated/in-place/#Allocating-data","page":"In-place transforms","title":"Allocating data","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"As with out-of-place plans, data should be allocated using allocate_input. The difference is that, for in-place plans, this function returns a ManyPencilArray object, which is a container holding multiple PencilArray views sharing the same memory space.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"# Allocate data for the plan.\n# Since `plan` is in-place, this returns a `ManyPencilArray` container.\nA = allocate_input(plan)\nsummary(A)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Note that allocate_output also works for in-place plans. In this case, it returns exactly the same thing as allocate_input.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"As shown in the next section, in-place plans must be applied on the returned ManyPencilArray. On the other hand, one usually wants to access and modify data, and for this one needs the PencilArray views contained in the ManyPencilArray. The input and output array views can be obtained by calling first(::ManyPencilArray) and last(::ManyPencilArray).","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"For instance, we can initialise the input array with some data before transforming:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"using Random\nu_in = first(A) # input data view\nrandn!(u_in)\nsummary(u_in)","category":"page"},{"location":"generated/in-place/#Applying-plans","page":"In-place transforms","title":"Applying plans","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Like in FFTW.jl, one can perform in-place transforms using the * and \\ operators. As mentioned above, in-place plans must be applied on the ManyPencilArray containers returned by allocate_input.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan * A; # performs in-place forward transform\nnothing #hide","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"After performing an in-place transform, data contained in u_in has been overwritten and has no \"physical\" meaning. In other words, u_in should not be used at this point. To access the transformed data, one should retrieve the output data view using last(A).","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"For instance, to compute the global sum of the transformed data:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"u_out = last(A) # output data view\nsum(u_out) # sum of transformed data (note that `sum` reduces over all processes)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Finally, we can perform a backward transform and do stuff with the input view:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan \\ A; # perform in-place backward transform\nnothing #hide","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"At this point, the data can be once again found in the input view u_in, while u_out should not be accessed.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"This page was generated using Literate.jl.","category":"page"},{"location":"Transforms/#Available-transforms","page":"Available transforms","title":"Available transforms","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"CurrentModule = PencilFFTs.Transforms","category":"page"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"Transforms","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms","page":"Available transforms","title":"PencilFFTs.Transforms","text":"Defines different one-dimensional FFT-based transforms.\n\nThe transforms are all subtypes of an AbstractTransform type.\n\nWhen possible, the names of the transforms are kept consistent with the functions exported by AbstractFFTs.jl and FFTW.jl.\n\n\n\n\n\n","category":"module"},{"location":"Transforms/#Transform-types","page":"Available transforms","title":"Transform types","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"FFT\nFFT!\nBFFT\nBFFT!\n\nRFFT\nBRFFT\n\nR2R\nR2R!\n\nNoTransform\nNoTransform!","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.FFT","page":"Available transforms","title":"PencilFFTs.Transforms.FFT","text":"FFT()\n\nComplex-to-complex FFT.\n\nSee also AbstractFFTs.fft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.FFT!","page":"Available transforms","title":"PencilFFTs.Transforms.FFT!","text":"FFT!()\n\nIn-place version of FFT.\n\nSee also AbstractFFTs.fft!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BFFT","page":"Available transforms","title":"PencilFFTs.Transforms.BFFT","text":"BFFT()\n\nUnnormalised backward complex-to-complex FFT.\n\nLike AbstractFFTs.bfft, this transform is not normalised. To obtain the inverse transform, divide the output by the length of the transformed dimension.\n\nSee also AbstractFFTs.bfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.BFFT!","text":"BFFT()\n\nIn-place version of BFFT.\n\nSee also AbstractFFTs.bfft!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.RFFT","page":"Available transforms","title":"PencilFFTs.Transforms.RFFT","text":"RFFT()\n\nReal-to-complex FFT.\n\nSee also AbstractFFTs.rfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BRFFT","page":"Available transforms","title":"PencilFFTs.Transforms.BRFFT","text":"BRFFT(d::Integer)\nBRFFT((d1, d2, ..., dN))\n\nUnnormalised inverse of RFFT.\n\nTo obtain the inverse transform, divide the output by the length of the transformed dimension (of the real output array).\n\nAs described in the AbstractFFTs docs, the length of the output cannot be fully inferred from the input length. For this reason, the BRFFT constructor accepts an optional d argument indicating the output length.\n\nFor multidimensional datasets, a tuple of dimensions (d1, d2, ..., dN) may also be passed. This is equivalent to passing just dN. In this case, the last dimension (dN) is the one that changes size between the input and output. Note that this is the opposite of FFTW.brfft. The reason is that, in PencilFFTs, the last dimension is the one along which a complex-to-real transform is performed.\n\nSee also AbstractFFTs.brfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.R2R","page":"Available transforms","title":"PencilFFTs.Transforms.R2R","text":"R2R(kind)\n\nReal-to-real transform of type kind.\n\nThe possible values of kind are those described in the FFTW.r2r docs and the FFTW manual:\n\ndiscrete cosine transforms: FFTW.REDFT00, FFTW.REDFT01, FFTW.REDFFT10, FFTW.REDFFT11\ndiscrete sine transforms: FFTW.RODFT00, FFTW.RODFT01, FFTW.RODFFT10, FFTW.RODFFT11\ndiscrete Hartley transform: FFTW.DHT\n\nNote: half-complex format DFTs (FFTW.R2HC, FFTW.HC2R) are not currently supported.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.R2R!","page":"Available transforms","title":"PencilFFTs.Transforms.R2R!","text":"R2R!(kind)\n\nIn-place version of R2R.\n\nSee also FFTW.r2r!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.NoTransform","page":"Available transforms","title":"PencilFFTs.Transforms.NoTransform","text":"NoTransform()\n\nIdentity transform.\n\nSpecifies that no transformation should be applied.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.NoTransform!","page":"Available transforms","title":"PencilFFTs.Transforms.NoTransform!","text":"NoTransform!()\n\nIn-place version of NoTransform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#Internals","page":"Available transforms","title":"Internals","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"What follows is used internally in PencilFFTs.","category":"page"},{"location":"Transforms/#Types","page":"Available transforms","title":"Types","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"AbstractCustomPlan\nAbstractTransform\nIdentityPlan\nIdentityPlan!\nPlan","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.AbstractCustomPlan","page":"Available transforms","title":"PencilFFTs.Transforms.AbstractCustomPlan","text":"AbstractCustomPlan\n\nAbstract type defining a custom plan, to be used as an alternative to FFTW plans (FFTW.FFTWPlan).\n\nThe only custom plan defined in this module is IdentityPlan. The user can define other custom plans that are also subtypes of AbstractCustomPlan.\n\nNote that plan returns a subtype of either FFTW.FFTWPlan or AbstractCustomPlan.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.AbstractTransform","page":"Available transforms","title":"PencilFFTs.Transforms.AbstractTransform","text":"AbstractTransform\n\nSpecifies a one-dimensional FFT-based transform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.IdentityPlan","page":"Available transforms","title":"PencilFFTs.Transforms.IdentityPlan","text":"IdentityPlan\n\nType of plan associated to NoTransform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.IdentityPlan!","page":"Available transforms","title":"PencilFFTs.Transforms.IdentityPlan!","text":"IdentityPlan!\n\nType of plan associated to NoTransform!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.Plan","page":"Available transforms","title":"PencilFFTs.Transforms.Plan","text":"Plan = Union{FFTW.FFTWPlan, AbstractCustomPlan}\n\nUnion type representing any plan returned by plan.\n\nSee also AbstractCustomPlan.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#Functions","page":"Available transforms","title":"Functions","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"plan\n\nbinv\nscale_factor\n\neltype_input\neltype_output\nexpand_dims\nis_inplace\nkind\nlength_output","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.plan","page":"Available transforms","title":"PencilFFTs.Transforms.plan","text":"plan(transform::AbstractTransform, A, [dims];\n flags=FFTW.ESTIMATE, timelimit=Inf)\n\nCreate plan to transform array A along dimensions dims.\n\nIf dims is not specified, all dimensions of A are transformed.\n\nFor FFT plans, this function wraps the AbstractFFTs.jl and FFTW.jl plan creation functions. For more details on the function arguments, see AbstractFFTs.plan_fft.\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.binv","page":"Available transforms","title":"PencilFFTs.Transforms.binv","text":"binv(transform::AbstractTransform, d::Integer)\n\nReturns the backwards transform associated to the given transform.\n\nThe second argument must be the length of the first transformed dimension in the forward transform. It is used in particular when transform = RFFT(), to determine the length of the inverse (complex-to-real) transform. See the AbstractFFTs.irfft docs for details.\n\nThe backwards transform returned by this function is not normalised. The normalisation factor for a given array can be obtained by calling scale_factor.\n\nExample\n\njulia> binv(Transforms.FFT(), 42)\nBFFT\n\njulia> binv(Transforms.BRFFT(9), 42)\nRFFT\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.scale_factor","page":"Available transforms","title":"PencilFFTs.Transforms.scale_factor","text":"scale_factor(transform::AbstractTransform, A, [dims = 1:ndims(A)])\n\nGet factor required to normalise the given array after a transformation along dimensions dims (all dimensions by default).\n\nThe array A must have the dimensions of the transform input.\n\nImportant: the dimensions dims must be the same that were passed to plan.\n\nExamples\n\njulia> C = zeros(ComplexF32, 3, 4, 5);\n\njulia> scale_factor(Transforms.FFT(), C)\n60\n\njulia> scale_factor(Transforms.BFFT(), C)\n60\n\njulia> scale_factor(Transforms.BFFT(), C, 2:3)\n20\n\njulia> R = zeros(Float64, 3, 4, 5);\n\njulia> scale_factor(Transforms.RFFT(), R, 2)\n4\n\njulia> scale_factor(Transforms.RFFT(), R, 2:3)\n20\n\njulia> scale_factor(Transforms.BRFFT(8), C)\n96\n\njulia> scale_factor(Transforms.BRFFT(9), C)\n108\n\nThis will fail because the input of RFFT is real, and R is a complex array:\n\njulia> scale_factor(Transforms.RFFT(), C, 2:3)\nERROR: MethodError: no method matching scale_factor(::PencilFFTs.Transforms.RFFT, ::Array{ComplexF32, 3}, ::UnitRange{Int64})\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.eltype_input","page":"Available transforms","title":"PencilFFTs.Transforms.eltype_input","text":"eltype_input(transform::AbstractTransform, real_type<:AbstractFloat)\n\nDetermine input data type for a given transform given the floating point precision of the input data.\n\nSome transforms, such as R2R and NoTransform, can take both real and complex data. For those kinds of transforms, nothing is returned.\n\nExample\n\njulia> eltype_input(Transforms.FFT(), Float32)\nComplexF32 (alias for Complex{Float32})\n\njulia> eltype_input(Transforms.RFFT(), Float64)\nFloat64\n\njulia> eltype_input(Transforms.R2R(FFTW.REDFT01), Float64) # nothing\n\njulia> eltype_input(Transforms.NoTransform(), Float64) # nothing\n\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.eltype_output","page":"Available transforms","title":"PencilFFTs.Transforms.eltype_output","text":"eltype_output(transform::AbstractTransform, eltype_input)\n\nReturns the output data type for a given transform given the input type.\n\nThrows ArgumentError if the input data type is incompatible with the transform type.\n\nExample\n\njulia> eltype_output(Transforms.NoTransform(), Float32)\nFloat32\n\njulia> eltype_output(Transforms.RFFT(), Float64)\nComplexF64 (alias for Complex{Float64})\n\njulia> eltype_output(Transforms.BRFFT(4), ComplexF32)\nFloat32\n\njulia> eltype_output(Transforms.FFT(), Float64)\nERROR: ArgumentError: invalid input data type for PencilFFTs.Transforms.FFT: Float64\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.expand_dims","page":"Available transforms","title":"PencilFFTs.Transforms.expand_dims","text":"expand_dims(transform::AbstractTransform, Val(N))\n\nExpand a single multidimensional transform into one transform per dimension.\n\nExample\n\n# Expand a real-to-complex transform in 3 dimensions.\njulia> expand_dims(Transforms.RFFT(), Val(3))\n(RFFT, FFT, FFT)\n\njulia> expand_dims(Transforms.BRFFT(4), Val(3))\n(BFFT, BFFT, BRFFT{even})\n\njulia> expand_dims(Transforms.NoTransform(), Val(2))\n(NoTransform, NoTransform)\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.is_inplace","page":"Available transforms","title":"PencilFFTs.Transforms.is_inplace","text":"is_inplace(transform::AbstractTransform) -> Bool\nis_inplace(transforms::Vararg{AbtractTransform}) -> Union{Bool, Nothing}\n\nCheck whether a transform or a list of transforms is performed in-place.\n\nIf the list of transforms has a combination of in-place and out-of-place transforms, nothing is returned.\n\nExample\n\njulia> is_inplace(Transforms.RFFT())\nfalse\n\njulia> is_inplace(Transforms.NoTransform!())\ntrue\n\njulia> is_inplace(Transforms.FFT!(), Transforms.R2R!(FFTW.REDFT01))\ntrue\n\njulia> is_inplace(Transforms.FFT(), Transforms.R2R(FFTW.REDFT01))\nfalse\n\njulia> is_inplace(Transforms.FFT(), Transforms.R2R!(FFTW.REDFT01)) === nothing\ntrue\n\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.kind","page":"Available transforms","title":"PencilFFTs.Transforms.kind","text":"kind(transform::R2R)\n\nGet kind of real-to-real transform.\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.length_output","page":"Available transforms","title":"PencilFFTs.Transforms.length_output","text":"length_output(transform::AbstractTransform, length_in::Integer)\n\nReturns the length of the transform output, given the length of its input.\n\nThe input and output lengths are specified in terms of the respective input and output datatypes. For instance, for real-to-complex transforms, these are respectively the length of input real data and of output complex data.\n\n\n\n\n\n","category":"function"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"EditURL = \"https://github.com/jipolanco/PencilFFTs.jl/blob/master/docs/examples/navier_stokes.jl\"","category":"page"},{"location":"generated/navier_stokes/#Navier–Stokes-equations","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"In this example, we numerically solve the incompressible Navier–Stokes equations","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"_t bmv + (bmv bm) bmv = -frac1ρ bm p + ν ^2 bmv\nquad bm bmv = 0","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where bmv(bmx t) and p(bmx t) are respectively the velocity and pressure fields, ν is the fluid kinematic viscosity and ρ is the fluid density.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.","category":"page"},{"location":"generated/navier_stokes/#First-steps","page":"Navier–Stokes equations","title":"First steps","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We start by loading the required packages, initialising MPI and setting the simulation parameters.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using MPI\nusing PencilFFTs\n\nMPI.Init()\ncomm = MPI.COMM_WORLD\nprocid = MPI.Comm_rank(comm) + 1\n\n# Simulation parameters\nNs = (64, 64, 64) # = (Nx, Ny, Nz)\nLs = (2π, 2π, 2π) # = (Lx, Ly, Lz)\n\n# Collocation points (\"global\" = over all processes).\n# We include the endpoint (length = N + 1) for convenience.\nxs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls) # = (x, y, z)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's check the number of MPI processes over which we're running our simulation:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"MPI.Comm_size(comm)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"pen = Pencil(Ns, comm)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"The subdomain associated to the local MPI process can be obtained using range_local:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"range_local(pen)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We now construct a distributed vector field that follows the decomposition configuration we just created:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v⃗₀ = (\n PencilArray{Float64}(undef, pen), # vx\n PencilArray{Float64}(undef, pen), # vy\n PencilArray{Float64}(undef, pen), # vz\n)\nsummary(v⃗₀[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We still need to fill this array with interesting values that represent a physical velocity field.","category":"page"},{"location":"generated/navier_stokes/#Initial-condition","page":"Navier–Stokes equations","title":"Initial condition","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"beginaligned\nv_x(x y z) = u₀ sin(k₀ x) cos(k₀ y) cos(k₀ z) \nv_y(x y z) = -u₀ cos(k₀ x) sin(k₀ y) cos(k₀ z) \nv_z(x y z) = 0\nendaligned","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where u₀ and k₀ are two parameters setting the amplitude and the period of the velocity field.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"grid = localgrid(pen, xs_global)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can use this to initialise the velocity field:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"u₀ = 1.0\nk₀ = 2π / Ls[1] # should be integer if L = 2π (to preserve periodicity)\n\n@. v⃗₀[1] = u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)\n@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)\n@. v⃗₀[3] = 0\nnothing # hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's plot a 2D slice of the velocity field managed by the local MPI process:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using GLMakie\n\n# Compute the norm of a vector field represented by a tuple of arrays.\nfunction vecnorm(v⃗::NTuple)\n vnorm = similar(v⃗[1])\n for n ∈ eachindex(v⃗[1])\n w = zero(eltype(vnorm))\n for v ∈ v⃗\n w += v[n]^2\n end\n vnorm[n] = sqrt(w)\n end\n vnorm\nend\n\nlet fig = Figure(resolution = (700, 600))\n ax = Axis3(fig[1, 1]; aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\")\n vnorm = vecnorm(v⃗₀)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, vnorm;\n alpha = 0.2, levels = 4,\n colormap = :viridis, colorrange = (0.0, 1.0),\n )\n cb = Colorbar(fig[1, 2], ct; label = \"Velocity magnitude\")\n fig\nend","category":"page"},{"location":"generated/navier_stokes/#Velocity-in-Fourier-space","page":"Navier–Stokes equations","title":"Velocity in Fourier space","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"bmv(bmx t) =\n_bmk hatbmv_bmk(t) e^i bmk bmx","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where bmk = (k_x k_y k_z) are the discrete wave numbers.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for k_x:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using AbstractFFTs: fftfreq, rfftfreq\n\nks_global = (\n rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]), # kx | real-to-complex\n fftfreq(Ns[2], 2π * Ns[2] / Ls[2]), # ky | complex-to-complex\n fftfreq(Ns[3], 2π * Ns[3] / Ls[3]), # kz | complex-to-complex\n)\n\nks_global[1]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ks_global[2]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ks_global[3]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"See PencilFFTPlan for details on creating plans and on optional keyword arguments.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients hatbmv_bmk:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v̂s = plan .* v⃗₀\nsummary(v̂s[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Note that, in Fourier space, the domain decomposition is performed along the directions x and y:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"pencil(v̂s[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"This is because the 3D FFTs are performed one dimension at a time, with the x direction first and the z direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers bmk associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"grid_fourier = localgrid(v̂s[1], ks_global)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, bmω = bm bmv. In Fourier space, this becomes hatbmω = i bmk hatbmv.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using StaticArrays: SVector\nusing LinearAlgebra: ×\n\nfunction curl_fourier!(\n ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,\n ) where {N}\n @inbounds for I ∈ eachindex(grid_fourier)\n # We use StaticArrays for the cross product between small vectors.\n ik⃗ = im * SVector(grid_fourier[I])\n v⃗ = SVector(getindex.(v̂s, Ref(I))) # = (v̂s[1][I], v̂s[2][I], ...)\n ω⃗ = ik⃗ × v⃗\n for n ∈ eachindex(ω⃗)\n ω̂s[n][I] = ω⃗[n]\n end\n end\n ω̂s\nend\n\nω̂s = similar.(v̂s)\ncurl_fourier!(ω̂s, v̂s, grid_fourier);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We finally transform back to physical space and plot the result:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ωs = plan .\\ ω̂s\n\nlet fig = Figure(resolution = (700, 600))\n ax = Axis3(fig[1, 1]; aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\")\n ω_norm = vecnorm(ωs)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, ω_norm;\n alpha = 0.1, levels = 0.8:0.2:2.0,\n colormap = :viridis, colorrange = (0.8, 2.0),\n )\n cb = Colorbar(fig[1, 2], ct; label = \"Vorticity magnitude\")\n fig\nend","category":"page"},{"location":"generated/navier_stokes/#Computing-the-non-linear-term","page":"Navier–Stokes equations","title":"Computing the non-linear term","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"_t hatbmv_bmk =\n- mathcalP_bmk left widehat(bmv bm) bmv right\n- ν bmk^2 hatbmv_bmk\nquad text with quad\nmathcalP_bmk(hatbmF_bmk) = left( I - fracbmk \nbmkbmk^2 right) hatbmF_bmk","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where mathcalP_bmk is a projection operator allowing to preserve the incompressibility condition bm bmv = 0. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Now that we have the wave numbers bmk, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients hatbmv_bmk of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, hatbmF_bmk = left widehat(bmv bm) bmv right_bmk. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Below we implement a function that computes the non-linear term in Fourier space based on its convective form (bmv bm) bmv = bm (bmv bmv). Note that this equivalence uses the incompressibility condition bm bmv = 0.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using LinearAlgebra: mul!, ldiv! # for applying FFT plans in-place\n\n# Compute non-linear term in Fourier space from velocity field in physical\n# space. Optional keyword arguments may be passed to avoid allocations.\nfunction ns_nonlinear!(\n F̂s, vs, plan, grid_fourier;\n vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),\n )\n # Compute F_i = ∂_j (v_i v_j) for each i.\n # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)\n w, ŵ = vbuf, v̂buf\n @inbounds for (i, F̂i) ∈ enumerate(F̂s)\n F̂i .= 0\n vi = vs[i]\n for (j, vj) ∈ enumerate(vs)\n w .= vi .* vj # w = v_i * v_j in physical space\n mul!(ŵ, plan, w) # same in Fourier space\n # Add derivative in Fourier space\n for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n kj = k⃗[j]\n F̂i[I] += im * kj * ŵ[I]\n end\n end\n end\n F̂s\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"As an example, let's use this function on our initial velocity field:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"F̂s = similar.(v̂s)\nns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)\n ks_max = maximum.(abs, ks_global) # maximum stored wave numbers (kx_max, ky_max, kz_max)\n ks_lim = (2 / 3) .* ks_max\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I]\n if any(abs.(k⃗) .> ks_lim)\n for ŵ ∈ ŵs\n ŵ[I] = 0\n end\n end\n end\n ŵs\nend\n\n# We can apply this on the previously computed non-linear term:\ndealias_twothirds!(F̂s, grid_fourier, ks_global);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Finally, we implement the projection associated to the incompressibility condition:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function project_divergence_free!(ûs, grid_fourier)\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I]\n k² = sum(abs2, k⃗)\n iszero(k²) && continue # avoid division by zero\n û = getindex.(ûs, Ref(I)) # (ûs[1][I], ûs[2][I], ...)\n for i ∈ eachindex(û)\n ŵ = û[i]\n for j ∈ eachindex(û)\n ŵ -= k⃗[i] * k⃗[j] * û[j] / k²\n end\n ûs[i][I] = ŵ\n end\n end\n ûs\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)\nv̂s_proj .≈ v̂s # the last one may be false because v_z = 0 initially","category":"page"},{"location":"generated/navier_stokes/#Putting-it-all-together","page":"Navier–Stokes equations","title":"Putting it all together","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function ns_rhs!(\n dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,\n ) where {N}\n # 1. Compute non-linear term and dealias it\n (; plan, cache, ks_global, grid_fourier) = p\n F̂s = cache.F̂s\n ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])\n dealias_twothirds!(F̂s, grid_fourier, ks_global)\n\n # 2. Project onto divergence-free space\n project_divergence_free!(F̂s, grid_fourier)\n\n # 3. Transform velocity to Fourier space\n v̂s = cache.v̂s\n map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)\n\n # 4. Add viscous term (and multiply projected non-linear term by -1)\n ν = p.ν\n for n ∈ eachindex(v̂s)\n v̂ = v̂s[n]\n F̂ = F̂s[n]\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n k² = sum(abs2, k⃗)\n F̂[I] = -F̂[I] - ν * k² * v̂[I]\n end\n end\n\n # 5. Transform RHS back to physical space\n map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)\n\n nothing\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using OrdinaryDiffEq\nusing RecursiveArrayTools: ArrayPartition\n\nns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)\nvs_init_ode = ArrayPartition(v⃗₀)\nsummary(vs_init_ode)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We now define solver parameters and temporary variables, and initialise the problem:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"params = (;\n ν = 5e-3, # kinematic viscosity\n plan, grid_fourier, ks_global,\n cache = (\n v̂s = similar.(v̂s),\n F̂s = similar.(v̂s),\n )\n)\n\ntspan = (0.0, 10.0)\nprob = ODEProblem(ns_rhs!, vs_init_ode, tspan, params)\nintegrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum E(k), to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity ν must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large k), while high enough to allow the small-scale motions to be correctly resolved.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function energy_spectrum!(Ek, ks, v̂s, grid_fourier)\n Nk = length(Ek)\n @assert Nk == length(ks)\n Ek .= 0\n for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n knorm = sqrt(sum(abs2, k⃗))\n i = searchsortedfirst(ks, knorm)\n i > Nk && continue\n v⃗ = getindex.(v̂s, Ref(I)) # = (v̂s[1][I], v̂s[2][I], ...)\n factor = k⃗[1] == 0 ? 1 : 2 # account for Hermitian symmetry and r2c transform\n Ek[i] += factor * sum(abs2, v⃗) / 2\n end\n MPI.Allreduce!(Ek, +, get_comm(v̂s[1])) # sum across all processes\n Ek\nend\n\nks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])\nEk = similar(ks)\nv̂s = plan .* integrator.u.x\nenergy_spectrum!(Ek, ks, v̂s, grid_fourier)\nEk ./= scale_factor(plan)^2 # rescale energy\n\ncurl_fourier!(ω̂s, v̂s, grid_fourier)\nldiv!.(ωs, plan, ω̂s)\nω⃗_plot = Observable(ωs)\nk_plot = @view ks[2:end]\nE_plot = Observable(@view Ek[2:end])\nt_plot = Observable(integrator.t)\n\nfig = let\n fig = Figure(resolution = (1200, 600))\n ax = Axis3(\n fig[1, 1][1, 1]; title = @lift(\"t = $(round($t_plot, digits = 3))\"),\n aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\",\n )\n ω_mag = @lift vecnorm($ω⃗_plot)\n ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, ω_mag_norm;\n alpha = 0.3, levels = 3,\n colormap = :viridis, colorrange = (0.0, 1.0),\n )\n cb = Colorbar(fig[1, 1][1, 2], ct; label = \"Normalised vorticity magnitude\")\n ax_sp = Axis(\n fig[1, 2];\n xlabel = \"k\", ylabel = \"E(k)\", xscale = log2, yscale = log10,\n title = \"Kinetic energy spectrum\",\n )\n ylims!(ax_sp, 1e-8, 1e0)\n scatterlines!(ax_sp, k_plot, E_plot)\n ks_slope = exp.(range(log(2.5), log(25.0), length = 3))\n E_fivethirds = @. 0.3 * ks_slope^(-5/3)\n @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)\n text!(ax_sp, L\"k^{-5/3}\"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))\n fig\nend\n\nusing Printf # hide\nwith_xvfb = ENV[\"DISPLAY\"] == \":99\" # hide\nnstep = 0 # hide\nconst tmpdir = mktempdir() # hide\nfilename_frame(procid, nstep) = joinpath(tmpdir, @sprintf(\"proc%d_%04d.png\", procid, nstep)) # hide\nrecord(fig, \"vorticity_proc$procid.mp4\"; framerate = 10) do io\n with_xvfb && recordframe!(io) # hide\n while integrator.t < 20\n dt = 0.001\n step!(integrator, dt)\n t_plot[] = integrator.t\n mul!.(v̂s, plan, integrator.u.x) # current velocity in Fourier space\n curl_fourier!(ω̂s, v̂s, grid_fourier)\n ldiv!.(ω⃗_plot[], plan, ω̂s)\n ω⃗_plot[] = ω⃗_plot[] # to force updating the plot\n energy_spectrum!(Ek, ks, v̂s, grid_fourier)\n Ek ./= scale_factor(plan)^2 # rescale energy\n E_plot[] = E_plot[]\n global nstep += 1 # hide\n with_xvfb ? # hide\n save(filename_frame(procid, nstep), fig) : # hide\n recordframe!(io)\n end\nend;\n\nif with_xvfb # hide\n run(pipeline(`ffmpeg -y -r 10 -i $tmpdir/proc$(procid)_%04d.png -c:v libx264 -vf \"fps=25,format=yuv420p\" vorticity_proc$procid.mp4`; stdout = \"ffmpeg.out\", stderr = \"ffmpeg.err\")) # hide\nend # hide\nnothing # hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"
      \n \n
      ","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"This page was generated using Literate.jl.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"EditURL = \"https://github.com/jipolanco/PencilFFTs.jl/blob/master/docs/examples/gradient.jl\"","category":"page"},{"location":"generated/gradient/#Gradient-of-a-scalar-field","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This example shows different methods to compute the gradient of a real-valued 3D scalar field θ(bmx) in Fourier space, where bmx = (x y z). It is assumed that the field is periodic with period L = 2π along all dimensions.","category":"page"},{"location":"generated/gradient/#General-procedure","page":"Gradient of a scalar field","title":"General procedure","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The discrete Fourier expansion of θ writes","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"θ(bmx) = _bmk Z^3 hatθ(bmk) e^i bmk bmx","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"where bmk = (k_x k_y k_z) are the Fourier wave numbers and hatθ is the discrete Fourier transform of θ. Then, the spatial derivatives of θ are given by","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"frac θ(bmx) x_i =\n_bmk Z^3 i k_i hatθ(bmk) e^i bmk bmx","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"where the subscript i denotes one of the spatial components x, y or z.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In other words, to compute bm θ = (_x θ _y θ _z θ), one has to:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"transform θ to Fourier space to obtain hatθ,\nmultiply hatθ by i bmk,\ntransform the result back to physical space to obtain bm θ.","category":"page"},{"location":"generated/gradient/#Preparation","page":"Gradient of a scalar field","title":"Preparation","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In this section, we initialise a random real-valued scalar field θ and compute its FFT. For more details see the Tutorial.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"using MPI\nusing PencilFFTs\nusing Random\n\nMPI.Init()\n\n# Input data dimensions (Nx × Ny × Nz)\ndims = (64, 32, 64)\n\n# Apply a 3D real-to-complex (r2c) FFT.\ntransform = Transforms.RFFT()\n\n# Automatically create decomposition configuration\ncomm = MPI.COMM_WORLD\npen = Pencil(dims, comm)\n\n# Create plan\nplan = PencilFFTPlan(pen, transform)\n\n# Allocate data and initialise field\nθ = allocate_input(plan)\nrandn!(θ)\n\n# Perform distributed FFT\nθ_hat = plan * θ\nnothing # hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, we initialise the output that will hold ∇θ in Fourier space. Noting that ∇θ is a vector field, we choose to store it as a tuple of 3 PencilArrays.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"∇θ_hat = allocate_output(plan, Val(3))\n\n# This is equivalent:\n# ∇θ_hat = ntuple(d -> similar(θ_hat), Val(3))\n\nsummary(∇θ_hat)\n\n# Fourier wave numbers","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In general, the Fourier wave numbers are of the form k_i = 0 frac2πL_i frac4πL_i frac6πL_i , where L_i is the period along dimension i. When a real-to-complex Fourier transform is applied, roughly half of these wave numbers are redundant due to the Hermitian symmetry of the complex Fourier coefficients. In practice, this means that for the fastest dimension x (along which a real-to-complex transform is performed), the negative wave numbers are dropped, i.e. k_x = 0 frac2πL_x frac4πL_x .","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The AbstractFFTs package provides a convenient way to generate the Fourier wave numbers, using the functions fftfreq and rfftfreq. We can use these functions to initialise a \"grid\" of wave numbers associated to our 3D real-to-complex transform:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"using AbstractFFTs: fftfreq, rfftfreq\n\nbox_size = (2π, 2π, 2π) # Lx, Ly, Lz\nsample_rate = 2π .* dims ./ box_size\n\n# In our case (Lx = 2π and Nx even), this gives kx = [0, 1, 2, ..., Nx/2].\nkx = rfftfreq(dims[1], sample_rate[1])\n\n# In our case (Ly = 2π and Ny even), this gives\n# ky = [0, 1, 2, ..., Ny/2-1, -Ny/2, -Ny/2+1, ..., -1] (and similarly for kz).\nky = fftfreq(dims[2], sample_rate[2])\nkz = fftfreq(dims[3], sample_rate[3])\n\nkvec = (kx, ky, kz)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that kvec now contains the wave numbers associated to the global domain. In the following, we will only need the wave numbers associated to the portion of the domain handled by the local MPI process.","category":"page"},{"location":"generated/gradient/#gradient_method_global","page":"Gradient of a scalar field","title":"Method 1: global views","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"PencilArrays, returned for instance by allocate_input and allocate_output, take indices that start at 1, regardless of the location of the subdomain associated to the local process on the global grid. (In other words, PencilArrays take local indices.) On the other hand, we have defined the wave number vector kvec which, for each MPI process, is defined over the global domain, and as such it takes global indices.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"One straightforward way of making data arrays compatible with wave numbers is to use global views, i.e. arrays that take global indices. These are generated from PencilArrays by calling the global_view function. Note that, in general, global indices do not start at 1 for a given MPI process. A given process will own a range of data given by indices in (i1:i2, j1:j2, k1:k2).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"θ_glob = global_view(θ_hat)\n∇θ_glob = global_view.(∇θ_hat)\nsummary(θ_glob)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Once we have global views, we can combine data and wave numbers using the portion of global indices owned by the local MPI process, as shown below. We can use CartesianIndices to iterate over the global indices associated to the local process.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"for I in CartesianIndices(θ_glob)\n i, j, k = Tuple(I) # unpack indices\n\n # Wave number vector associated to current Cartesian index.\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n # Compute gradient in Fourier space.\n # Note that modifying ∇θ_glob also modifies the original PencilArray ∇θ_hat.\n ∇θ_glob[1][I] = im * kx * θ_glob[I]\n ∇θ_glob[2][I] = im * ky * θ_glob[I]\n ∇θ_glob[3][I] = im * kz * θ_glob[I]\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The above loop can be written in a slightly more efficient manner by precomputing im * θ_glob[I]:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@inbounds for I in CartesianIndices(θ_glob)\n i, j, k = Tuple(I)\n\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n u = im * θ_glob[I]\n\n ∇θ_glob[1][I] = kx * u\n ∇θ_glob[2][I] = ky * u\n ∇θ_glob[3][I] = kz * u\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Also note that the above can be easily written in a more generic way, e.g. for arbitrary dimensions, thanks in part to the use of CartesianIndices. Moreover, in the above there is no notion of the dimension permutations discussed in the tutorial, as it is all hidden behind the implementation of PencilArrays. And as seen later in the benchmarks, these (hidden) permutations have zero cost, as the speed is identical to that of a function that explicitly takes into account these permutations.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, we can perform a backwards transform to obtain bm θ in physical space:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"∇θ = plan \\ ∇θ_hat;\nnothing #hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that the transform is automatically broadcast over the three fields of the ∇θ_hat vector, and the result ∇θ is also a tuple of three PencilArrays.","category":"page"},{"location":"generated/gradient/#gradient_method_global_explicit","page":"Gradient of a scalar field","title":"Method 2: explicit global indexing","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Sometimes, one does not need to write generic code. In our case, one often knows the dimensionality of the problem and the memory layout of the data (i.e. the underlying index permutation).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Below is a reimplementation of the above loop, using explicit indices instead of CartesianIndices, and assuming that the underlying index permutation is (3, 2, 1), that is, data is stored in (z y x) order. As discussed in the tutorial, this is the default for transformed arrays. This example also serves as a more explicit explanation for what is going on in the first method.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"# Get local data range in the global grid.\nrng = axes(θ_glob) # = (i1:i2, j1:j2, k1:k2)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"For the loop below, we're assuming that the permutation is (3, 2, 1). In other words, the fastest index is the last one, and not the first one as it is usually in Julia. If the permutation is not (3, 2, 1), things will still work (well, except for the assertion below!), but the loop order will not be optimal.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@assert permutation(θ_hat) === Permutation(3, 2, 1)\n\n@inbounds for i in rng[1], j in rng[2], k in rng[3]\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n # Note that we still access the arrays in (i, j, k) order.\n # (The permutation happens behind the scenes!)\n u = im * θ_glob[i, j, k]\n\n ∇θ_glob[1][i, j, k] = kx * u\n ∇θ_glob[2][i, j, k] = ky * u\n ∇θ_glob[3][i, j, k] = kz * u\nend","category":"page"},{"location":"generated/gradient/#gradient_method_local","page":"Gradient of a scalar field","title":"Method 3: using local indices","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Alternatively, we can avoid global views and work directly on PencilArrays using local indices that start at 1. In this case, part of the strategy is to construct a \"local\" grid of wave numbers that can also be accessed with local indices. This can be conveniently done using the localgrid function of the PencilArrays.jl package, which accepts a PencilArray (or its associated Pencil) and the global coordinates (here kvec):","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"grid_fourier = localgrid(θ_hat, kvec)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that one can directly iterate on the returned grid object:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@inbounds for I in CartesianIndices(grid_fourier)\n # Wave number vector associated to current Cartesian index.\n local k⃗ # hide\n k⃗ = grid_fourier[I]\n u = im * θ_hat[I]\n ∇θ_hat[1][I] = k⃗[1] * u\n ∇θ_hat[2][I] = k⃗[2] * u\n ∇θ_hat[3][I] = k⃗[3] * u\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This implementation is as efficient as the other examples, while being slightly shorter to write. Moreover, it is quite generic, and can be made independent of the number of dimensions with little effort.","category":"page"},{"location":"generated/gradient/#gradient_method_broadcast","page":"Gradient of a scalar field","title":"Method 4: using broadcasting","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, note that the local grid object returned by localgrid makes it is possible to compute the gradient using broadcasting, thus fully avoiding scalar indexing. This can be quite convenient in some cases, and can also be very useful if one is working on GPUs (where scalar indexing is prohibitively expensive). Using broadcasting, the above examples simply become:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@. ∇θ_hat[1] = im * grid_fourier[1] * θ_hat\n@. ∇θ_hat[2] = im * grid_fourier[2] * θ_hat\n@. ∇θ_hat[3] = im * grid_fourier[3] * θ_hat\nnothing # hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Once again, as shown in the benchmarks further below, this method performs quite similarly to the other ones.","category":"page"},{"location":"generated/gradient/#Summary","page":"Gradient of a scalar field","title":"Summary","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The PencilArrays module provides different alternatives to deal with MPI-distributed data that may be subject to dimension permutations. In particular, one can choose to work with global indices (first two examples), with local indices (third example), or to avoid scalar indexing altogether (fourth example).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"If one wants to stay generic, making sure that the same code will work for arbitrary dimensions and will be efficient regardless of the underlying dimension permutation, methods 1, 3 or 4 should be preferred. These use CartesianIndices and make no assumptions on possible dimension permutations, which are by default enabled in the output of PencilFFTs transforms. In fact, such permutations are completely invisible in the implementations.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The second method uses explicit (i, j, k) indices. It assumes that the underlying permutation is (3, 2, 1) to loop with i as the slowest index and k as the fastest, which is the optimal order in this case given the permutation. As such, the implementation is less generic than the others, and differences in performance are negligible with respect to more generic variants.","category":"page"},{"location":"generated/gradient/#gradient_benchmarks","page":"Gradient of a scalar field","title":"Benchmark results","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The following are the benchmark results obtained from running examples/gradient.jl on a laptop, using 2 MPI processes and Julia 1.7.2, with an input array of global dimensions 64 32 64. The different methods detailed above are marked on the right. The \"lazy\" marks indicate runs where the wave numbers were represented by lazy Frequencies objects (returned by rfftfreq and fftfreq). Otherwise, they were collected into Vectors. For some reason, plain Vectors are faster when working with grids generated by localgrid.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In the script, additional implementations can be found which rely on a more advanced understanding of permutations and on the internals of the PencilArrays package. For instance, gradient_local_parent! directly works with the raw data stored in Julia Arrays, while gradient_local_linear! completely avoids CartesianIndices while staying generic and efficient. Nevertheless, these display roughly the same performance as the above examples.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":" gradient_global_view!... 89.900 μs\n gradient_global_view! (lazy)... 92.060 μs [Method 1]\n gradient_global_view_explicit!... 88.958 μs\n gradient_global_view_explicit! (lazy)... 81.055 μs [Method 2]\n gradient_local!... 92.305 μs\n gradient_grid!... 92.770 μs\n gradient_grid! (lazy)... 101.388 μs [Method 3]\n gradient_grid_broadcast!... 88.606 μs\n gradient_grid_broadcast! (lazy)... 151.020 μs [Method 4]\n gradient_local_parent!... 92.248 μs\n gradient_local_linear!... 91.212 μs\n gradient_local_linear_explicit!... 90.992 μs","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This page was generated using Literate.jl.","category":"page"},{"location":"PencilFFTs_timers/#PencilFFTs.measuring_performance","page":"Measuring performance","title":"Measuring performance","text":"","category":"section"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"It is possible to measure the time spent in different sections of the distributed transforms using the TimerOutputs package. This has a (very small) performance overhead, so it is disabled by default. To enable time measurements, call TimerOutputs.enable_debug_timings after loading PencilFFTs (see below for an example). For more details see the TimerOutputs docs.","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"Minimal example:","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"using MPI\nusing PencilFFTs\nusing TimerOutputs\n\n# Enable timing of `PencilFFTs` functions\nTimerOutputs.enable_debug_timings(PencilFFTs)\nTimerOutputs.enable_debug_timings(PencilArrays)\nTimerOutputs.enable_debug_timings(Transpositions)\n\nMPI.Init()\n\nplan = PencilFFTPlan(#= args... =#)\n\n# [do stuff with `plan`...]\n\n# Retrieve and print timing data associated to `plan`\nto = timer(plan)\nprint_timer(to)","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"By default, each PencilFFTPlan has its own TimerOutput. If you already have a TimerOutput, you can pass it to the PencilFFTPlan constructor:","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"to = TimerOutput()\nplan = PencilFFTPlan(..., timer=to)\n\n# [do stuff with `plan`...]\n\nprint_timer(to)","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"#PencilFFTs","page":"Home","title":"PencilFFTs","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Fast Fourier transforms of MPI-distributed Julia arrays.","category":"page"},{"location":"#Introduction","page":"Home","title":"Introduction","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package provides multidimensional FFTs and related transforms on MPI-distributed Julia arrays via the PencilArrays package.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The name of this package originates from the decomposition of 3D domains along two out of three dimensions, sometimes called pencil decomposition. This is illustrated by the figure below,[1] where each coloured block is managed by a different MPI process. Typically, one wants to compute FFTs on a scalar or vector field along the three spatial dimensions. In the case of a pencil decomposition, 3D FFTs are performed one dimension at a time, along the non-decomposed direction. Transforms must then be interleaved with global data transpositions to switch between pencil configurations. In high-performance computing environments, such data transpositions are generally the most expensive part of a parallel FFT computation, due to the large cost of communications between computing nodes.","category":"page"},{"location":"","page":"Home","title":"Home","text":"
      \n \n
      ","category":"page"},{"location":"","page":"Home","title":"Home","text":"More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension N. The decompositions can be performed along an arbitrary number M N of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.","category":"page"},{"location":"#Features","page":"Home","title":"Features","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;\nFFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;\nin-place and out-of-place transforms;\nhigh scalability up to (at least) tens of thousands of MPI processes.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"PencilFFTs can be installed using the Julia package manager:","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> ] add PencilFFTs","category":"page"},{"location":"#Similar-projects","page":"Home","title":"Similar projects","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.\nPFFT is a very general parallel FFT library written in C.\nP3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.\n2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[1]: Figure adapted from this PhD thesis.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[2]: For the pencil decomposition represented in the figure, N = 3 and M = 2.","category":"page"},{"location":"PencilFFTs/#Distributed-FFT-plans","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"Distributed FFTs are implemented in the PencilFFTs module, and are built on top of the PencilArrays package.","category":"page"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"PencilFFTs/#Creating-plans","page":"Distributed FFT plans","title":"Creating plans","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"PencilFFTPlan","category":"page"},{"location":"PencilFFTs/#PencilFFTs.PencilFFTPlan","page":"Distributed FFT plans","title":"PencilFFTs.PencilFFTPlan","text":"PencilFFTPlan{T,N} <: AbstractFFTs.Plan{T}\n\nPlan for N-dimensional FFT-based transform on MPI-distributed data, where input data has type T.\n\n\n\nPencilFFTPlan(p::Pencil, transforms; kwargs...)\n\nCreate a PencilFFTPlan for distributed arrays following a given Pencil configuration. See variant below for details on the specification of transforms and on possible keyword arguments.\n\n\n\nPencilFFTPlan(\n A::PencilArray, transforms;\n fftw_flags = FFTW.ESTIMATE,\n fftw_timelimit = FFTW.NO_TIMELIMIT,\n permute_dims = Val(true),\n transpose_method = Transpositions.PointToPoint(),\n timer = TimerOutput(),\n)\n\nCreate plan for N-dimensional transform on MPI-distributed PencilArrays.\n\nExtended help\n\nThis creates a PencilFFTPlan for arrays sharing the same properties as A (dimensions, MPI decomposition, memory layout, ...), which describe data on an N-dimensional domain.\n\nTransforms\n\nThe transforms to be applied along each dimension are specified by the transforms argument. Possible transforms are defined as subtypes of Transforms.AbstractTransform, and are listed in Transform types. This argument may be either:\n\na tuple of N transforms to be applied along each dimension. For instance, transforms = (Transforms.R2R(FFTW.REDFT01), Transforms.RFFT(), Transforms.FFT());\na single transform to be applied along all dimensions. The input is automatically expanded into N equivalent transforms. For instance, for a three-dimensional array, transforms = Transforms.RFFT() specifies a 3D real-to-complex transform, and is equivalent to passing (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT()).\n\nNote that forward transforms are applied from left to right. In the last example, this means that a real-to-complex transform (RFFT) is first performed along the first dimension. This is followed by complex-to-complex transforms (FFT) along the second and third dimensions.\n\nInput data layout\n\nThe input PencilArray must satisfy the following constraints:\n\narray dimensions must not be permuted. This is the default when constructing PencilArrays.\nfor an M-dimensional domain decomposition (with M < N), the input array must be decomposed along the last M dimensions. For example, for a 2D decomposition of 3D data, the decomposed dimensions must be (2, 3). In particular, the first array dimension must not be distributed among different MPI processes.\nIn the PencilArrays package, the decomposed dimensions are specified at the moment of constructing a Pencil.\nthe element type must be compatible with the specified transform. For instance, real-to-complex transforms (Transforms.RFFT) require the input to be real floating point values. Other transforms, such as Transforms.R2R, accept both real and complex data.\n\nKeyword arguments\n\nThe keyword arguments fftw_flags and fftw_timelimit are passed to the FFTW plan creation functions (see AbstractFFTs docs).\npermute_dims determines whether the indices of the output data should be reversed. For instance, if the input data has global dimensions (Nx, Ny, Nz), then the output of a complex-to-complex FFT would have dimensions (Nz, Ny, Nx). This enables FFTs to always be performed along the first (i.e. fastest) array dimension, which could lead to performance gains. This option is enabled by default. For type inference reasons, it must be a value type (Val(true) or Val(false)).\ntranspose_method allows to select between implementations of the global data transpositions. See PencilArrays docs docs for details.\ntimer should be a TimerOutput object. See Measuring performance for details.\n\n\n\nPencilFFTPlan(\n dims_global::Dims{N}, transforms, proc_dims::Dims{M}, comm::MPI.Comm,\n [real_type = Float64]; extra_dims = (), kws...\n)\n\nCreate plan for N-dimensional transform.\n\nExtended help\n\nInstead of taking a PencilArray or a Pencil, this constructor requires the global dimensions of the input data, passed via the size_global argument.\n\nThe data is distributed over the MPI processes in the comm communicator. The distribution is performed over M dimensions (with M < N) according to the values in proc_dims, which specifies the number of MPI processes to put along each dimension.\n\nPencilArrays that may be transformed with the returned plan can be created using allocate_input.\n\nOptional arguments\n\nThe floating point precision can be selected by setting real_type parameter, which is Float64 by default.\nextra_dims may be used to specify the sizes of one or more extra dimensions that should not be transformed. These dimensions will be added to the rightmost (i.e. slowest) indices of the arrays. See Extra dimensions below for usage hints.\nsee the other constructor for more keyword arguments.\n\nExtra dimensions\n\nOne possible application of extra_dims is for describing the components of a vector or tensor field. However, this means that different PencilFFTPlans would need to be created for each kind of field (scalar, vector, ...). To avoid the creation of multiple plans, a possibly better alternative is to create tuples (or arrays) of PencilArrays using allocate_input and allocate_output.\n\nAnother more legitimate usage of extra_dims is to specify one or more Cartesian dimensions that should not be transformed nor split among MPI processes.\n\nExample\n\nSuppose we want to perform a 3D FFT of real data. The data is to be decomposed along two dimensions, over 8 MPI processes:\n\nsize_global = (64, 32, 128) # size of real input data\n\n# Perform real-to-complex transform along the first dimension, then\n# complex-to-complex transforms along the other dimensions.\ntransforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())\n# transforms = Transforms.RFFT() # this is equivalent to the above line\n\nproc_dims = (4, 2) # 2D decomposition\ncomm = MPI.COMM_WORLD\n\nplan = PencilFFTPlan(size_global, transforms, proc_dims, comm)\n\n\n\n\n\n","category":"type"},{"location":"PencilFFTs/#Allocating-data","page":"Distributed FFT plans","title":"Allocating data","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"allocate_input\nallocate_output","category":"page"},{"location":"PencilFFTs/#PencilFFTs.allocate_input","page":"Distributed FFT plans","title":"PencilFFTs.allocate_input","text":"allocate_input(p::PencilFFTPlan) -> PencilArray\nallocate_input(p::PencilFFTPlan, dims...) -> Array{PencilArray}\nallocate_input(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}\n\nAllocate uninitialised PencilArray that can hold input data for the given plan.\n\nThe second and third forms respectively allocate an array of PencilArrays of size dims, and a tuple of N PencilArrays.\n\nnote: In-place plans\nIf p is an in-place plan, a ManyPencilArray is allocated. This type holds PencilArray wrappers for the input and output transforms (as well as for intermediate transforms) which share the same space in memory. The input and output PencilArrays should be respectively accessed by calling first(::ManyPencilArray) and last(::ManyPencilArray).ExampleSuppose p is an in-place PencilFFTPlan. Then,@assert is_inplace(p)\nA = allocate_input(p) :: ManyPencilArray\nv_in = first(A) :: PencilArray # input data view\nv_out = last(A) :: PencilArray # output data viewAlso note that in-place plans must be performed directly on the returned ManyPencilArray, and not on the contained PencilArray views:p * A # perform forward transform in-place\np \\ A # perform backward transform in-place\n# p * v_in # not allowed!!\n\n\n\n\n\n","category":"function"},{"location":"PencilFFTs/#PencilFFTs.allocate_output","page":"Distributed FFT plans","title":"PencilFFTs.allocate_output","text":"allocate_output(p::PencilFFTPlan) -> PencilArray\nallocate_output(p::PencilFFTPlan, dims...) -> Array{PencilArray}\nallocate_output(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}\n\nAllocate uninitialised PencilArray that can hold output data for the given plan.\n\nIf p is an in-place plan, a ManyPencilArray is allocated.\n\nSee allocate_input for details.\n\n\n\n\n\n","category":"function"},{"location":"PencilFFTs/#Methods","page":"Distributed FFT plans","title":"Methods","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"get_comm(::PencilFFTPlan)\nscale_factor(::PencilFFTPlan)\ntimer(::PencilFFTPlan)\nis_inplace(::PencilFFTPlan)","category":"page"},{"location":"PencilFFTs/#PencilArrays.Pencils.MPITopologies.get_comm-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilArrays.Pencils.MPITopologies.get_comm","text":"get_comm(p::PencilFFTPlan)\n\nGet MPI communicator associated to a PencilFFTPlan.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilFFTs.Transforms.scale_factor-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilFFTs.Transforms.scale_factor","text":"scale_factor(p::PencilFFTPlan)\n\nGet scale factor associated to a PencilFFTPlan.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilArrays.Pencils.timer-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilArrays.Pencils.timer","text":"timer(p::PencilFFTPlan)\n\nGet TimerOutput attached to a PencilFFTPlan.\n\nSee Measuring performance for details.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilFFTs.Transforms.is_inplace-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilFFTs.Transforms.is_inplace","text":"Transforms.is_inplace(p::PencilFFTPlan)\n\nReturns true if the given plan operates in-place on the input data, false otherwise.\n\n\n\n\n\n","category":"method"}] -} diff --git a/previews/PR40/siteinfo.js b/previews/PR40/siteinfo.js deleted file mode 100644 index 46530e3e..00000000 --- a/previews/PR40/siteinfo.js +++ /dev/null @@ -1 +0,0 @@ -var DOCUMENTER_CURRENT_VERSION = "previews/PR40"; diff --git a/previews/PR48/GlobalFFTParams/index.html b/previews/PR48/GlobalFFTParams/index.html deleted file mode 100644 index 0b9d70ac..00000000 --- a/previews/PR48/GlobalFFTParams/index.html +++ /dev/null @@ -1,9 +0,0 @@ - -Global FFT parameters · PencilFFTs.jl
      Edit on GitHub

      Global FFT parameters

      PencilFFTs.GlobalFFTParams — Type
      GlobalFFTParams{T, N, inplace}

      Specifies the global parameters for an N-dimensional distributed transform. These include the element type T and global data sizes of input and output data, as well as the transform types to be performed along each dimension.

      GlobalFFTParams(size_global, transforms, [real_type=Float64])

      Define parameters for N-dimensional transform.

      transforms must be a tuple of length N specifying the transforms to be applied along each dimension. Each element must be a subtype of Transforms.AbstractTransform. For all the possible transforms, see Transform types.

      The element type must be a real type accepted by FFTW, i.e. either Float32 or Float64.

      Note that the transforms are applied one dimension at a time, with the leftmost dimension first for forward transforms.

      Example

      To perform a 3D FFT of real data, first a real-to-complex FFT must be applied along the first dimension, followed by two complex-to-complex FFTs along the other dimensions:

      julia> size_global = (64, 32, 128);  # size of real input data
-
-julia> transforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT());
-
-julia> fft_params = PencilFFTs.GlobalFFTParams(size_global, transforms)
-Transforms: (RFFT, FFT, FFT)
-Input type: Float64
-Global dimensions: (64, 32, 128) -> (33, 32, 128)
      source
      diff --git a/previews/PR48/PencilFFTs_timers/index.html b/previews/PR48/PencilFFTs_timers/index.html deleted file mode 100644 index b24ebfc7..00000000 --- a/previews/PR48/PencilFFTs_timers/index.html +++ /dev/null @@ -1,24 +0,0 @@ - -Measuring performance · PencilFFTs.jl
      Edit on GitHub

      Measuring performance

      It is possible to measure the time spent in different sections of the distributed transforms using the TimerOutputs package. This has a (very small) performance overhead, so it is disabled by default. To enable time measurements, call TimerOutputs.enable_debug_timings after loading PencilFFTs (see below for an example). For more details see the TimerOutputs docs.

      Minimal example:

      using MPI
-using PencilFFTs
-using TimerOutputs
-
-# Enable timing of `PencilFFTs` functions
-TimerOutputs.enable_debug_timings(PencilFFTs)
-TimerOutputs.enable_debug_timings(PencilArrays)
-TimerOutputs.enable_debug_timings(Transpositions)
-
-MPI.Init()
-
-plan = PencilFFTPlan(#= args... =#)
-
-# [do stuff with `plan`...]
-
-# Retrieve and print timing data associated to `plan`
-to = timer(plan)
-print_timer(to)

      By default, each PencilFFTPlan has its own TimerOutput. If you already have a TimerOutput, you can pass it to the PencilFFTPlan constructor:

      to = TimerOutput()
-plan = PencilFFTPlan(..., timer=to)
-
-# [do stuff with `plan`...]
-
-print_timer(to)
      diff --git a/previews/PR48/Transforms/index.html b/previews/PR48/Transforms/index.html deleted file mode 100644 index 1e1bd782..00000000 --- a/previews/PR48/Transforms/index.html +++ /dev/null @@ -1,74 +0,0 @@ - -Available transforms · PencilFFTs.jl
      Edit on GitHub

      Available transforms

      Transform types

      PencilFFTs.Transforms.BFFT — Type
      BFFT()

      Unnormalised backward complex-to-complex FFT.

      Like AbstractFFTs.bfft, this transform is not normalised. To obtain the inverse transform, divide the output by the length of the transformed dimension.

      See also AbstractFFTs.bfft.

      source
      PencilFFTs.Transforms.BRFFT — Type
      BRFFT(d::Integer)
-BRFFT((d1, d2, ..., dN))

      Unnormalised inverse of RFFT.

      To obtain the inverse transform, divide the output by the length of the transformed dimension (of the real output array).

      As described in the AbstractFFTs docs, the length of the output cannot be fully inferred from the input length. For this reason, the BRFFT constructor accepts an optional d argument indicating the output length.

      For multidimensional datasets, a tuple of dimensions (d1, d2, ..., dN) may also be passed. This is equivalent to passing just dN. In this case, the last dimension (dN) is the one that changes size between the input and output. Note that this is the opposite of FFTW.brfft. The reason is that, in PencilFFTs, the last dimension is the one along which a complex-to-real transform is performed.

      See also AbstractFFTs.brfft.

      source

      Internals

      What follows is used internally in PencilFFTs.

      Types

      PencilFFTs.Transforms.AbstractCustomPlan — Type
      AbstractCustomPlan

      Abstract type defining a custom plan, to be used as an alternative to FFTW plans (FFTW.FFTWPlan).

      The only custom plan defined in this module is IdentityPlan. The user can define other custom plans that are also subtypes of AbstractCustomPlan.

      Note that plan returns a subtype of either AbstractFFTs.Plan or AbstractCustomPlan.

      source

      Functions

      PencilFFTs.Transforms.plan — Function
      plan(transform::AbstractTransform, A, [dims];
-     flags=FFTW.ESTIMATE, timelimit=Inf)

      Create plan to transform array A along dimensions dims.

      If dims is not specified, all dimensions of A are transformed.

      For FFT plans, this function wraps the AbstractFFTs.jl and FFTW.jl plan creation functions. For more details on the function arguments, see AbstractFFTs.plan_fft.

      source
      PencilFFTs.Transforms.binv — Function
      binv(transform::AbstractTransform, d::Integer)

      Returns the backwards transform associated to the given transform.

      The second argument must be the length of the first transformed dimension in the forward transform. It is used in particular when transform = RFFT(), to determine the length of the inverse (complex-to-real) transform. See the AbstractFFTs.irfft docs for details.

      The backwards transform returned by this function is not normalised. The normalisation factor for a given array can be obtained by calling scale_factor.

      Example

      julia> binv(Transforms.FFT(), 42)
-BFFT
-
-julia> binv(Transforms.BRFFT(9), 42)
-RFFT
      source
      PencilFFTs.Transforms.scale_factor — Function
      scale_factor(transform::AbstractTransform, A, [dims = 1:ndims(A)])

      Get factor required to normalise the given array after a transformation along dimensions dims (all dimensions by default).

      The array A must have the dimensions of the transform input.

      Important: the dimensions dims must be the same that were passed to plan.

      Examples

      julia> C = zeros(ComplexF32, 3, 4, 5);
-
-julia> scale_factor(Transforms.FFT(), C)
-60
-
-julia> scale_factor(Transforms.BFFT(), C)
-60
-
-julia> scale_factor(Transforms.BFFT(), C, 2:3)
-20
-
-julia> R = zeros(Float64, 3, 4, 5);
-
-julia> scale_factor(Transforms.RFFT(), R, 2)
-4
-
-julia> scale_factor(Transforms.RFFT(), R, 2:3)
-20
-
-julia> scale_factor(Transforms.BRFFT(8), C)
-96
-
-julia> scale_factor(Transforms.BRFFT(9), C)
-108

      This will fail because the input of RFFT is real, and R is a complex array:

      julia> scale_factor(Transforms.RFFT(), C, 2:3)
-ERROR: MethodError: no method matching scale_factor(::PencilFFTs.Transforms.RFFT, ::Array{ComplexF32, 3}, ::UnitRange{Int64})
      source
      PencilFFTs.Transforms.eltype_input — Function
      eltype_input(transform::AbstractTransform, real_type<:AbstractFloat)

      Determine input data type for a given transform given the floating point precision of the input data.

      Some transforms, such as R2R and NoTransform, can take both real and complex data. For those kinds of transforms, nothing is returned.

      Example

      julia> eltype_input(Transforms.FFT(), Float32)
-ComplexF32 (alias for Complex{Float32})
-
-julia> eltype_input(Transforms.RFFT(), Float64)
-Float64
-
-julia> eltype_input(Transforms.R2R(FFTW.REDFT01), Float64)  # nothing
-
-julia> eltype_input(Transforms.NoTransform(), Float64)  # nothing
-
      source
      PencilFFTs.Transforms.eltype_output — Function
      eltype_output(transform::AbstractTransform, eltype_input)

      Returns the output data type for a given transform given the input type.

      Throws ArgumentError if the input data type is incompatible with the transform type.

      Example

      julia> eltype_output(Transforms.NoTransform(), Float32)
-Float32
-
-julia> eltype_output(Transforms.RFFT(), Float64)
-ComplexF64 (alias for Complex{Float64})
-
-julia> eltype_output(Transforms.BRFFT(4), ComplexF32)
-Float32
-
-julia> eltype_output(Transforms.FFT(), Float64)
-ERROR: ArgumentError: invalid input data type for PencilFFTs.Transforms.FFT: Float64
      source
      PencilFFTs.Transforms.expand_dims — Function
      expand_dims(transform::AbstractTransform, Val(N))

      Expand a single multidimensional transform into one transform per dimension.

      Example

      # Expand a real-to-complex transform in 3 dimensions.
-julia> expand_dims(Transforms.RFFT(), Val(3))
-(RFFT, FFT, FFT)
-
-julia> expand_dims(Transforms.BRFFT(4), Val(3))
-(BFFT, BFFT, BRFFT{even})
-
-julia> expand_dims(Transforms.NoTransform(), Val(2))
-(NoTransform, NoTransform)
      source
      PencilFFTs.Transforms.is_inplace — Function
      is_inplace(transform::AbstractTransform)         -> Bool
-is_inplace(transforms::Vararg{AbtractTransform}) -> Union{Bool, Nothing}

      Check whether a transform or a list of transforms is performed in-place.

      If the list of transforms has a combination of in-place and out-of-place transforms, nothing is returned.

      Example

      julia> is_inplace(Transforms.RFFT())
-false
-
-julia> is_inplace(Transforms.NoTransform!())
-true
-
-julia> is_inplace(Transforms.FFT!(), Transforms.R2R!(FFTW.REDFT01))
-true
-
-julia> is_inplace(Transforms.FFT(), Transforms.R2R(FFTW.REDFT01))
-false
-
-julia> is_inplace(Transforms.FFT(), Transforms.R2R!(FFTW.REDFT01)) === nothing
-true
-
      source
      PencilFFTs.Transforms.length_output — Function
      length_output(transform::AbstractTransform, length_in::Integer)

      Returns the length of the transform output, given the length of its input.

      The input and output lengths are specified in terms of the respective input and output datatypes. For instance, for real-to-complex transforms, these are respectively the length of input real data and of output complex data.

      source
      diff --git a/previews/PR48/benchmarks/index.html b/previews/PR48/benchmarks/index.html deleted file mode 100644 index 9520a9e3..00000000 --- a/previews/PR48/benchmarks/index.html +++ /dev/null @@ -1,12 +0,0 @@ - -Benchmarks · PencilFFTs.jl
      Edit on GitHub

      Benchmarks

      The performance of PencilFFTs.jl is comparable to that of other open-source parallel FFT libraries implemented in lower-level languages. Below, we show comparisons with the Fortran implementation of P3DFFT, possibly the most popular of these libraries. The benchmarks were performed on the Jean–Zay cluster of the IDRIS French computing centre (CNRS).

      The figure below shows strong scaling benchmarks of 3D real-to-complex FFTs using 2D ("pencil") decomposition. The benchmarks were run for input arrays of dimensions $N_x × N_y × N_z = 512^3$, $1024^3$ and $2048^3$. Each timing is averaged over 100 repetitions.

      - - Strong scaling of PencilFFTs -

      As seen above, PencilFFTs generally outperforms P3DFFT in its default setting. This is largely explained by the choice of using non-blocking point-to-point MPI communications (via MPI_Isend and MPI_Irecv), while P3DFFT uses collective MPI_Alltoallv calls. This enables PencilFFTs to perform data reordering operations on the partially received data while waiting for the incoming data, leading to better performance. Moreover, in contrast with P3DFFT, the high performance and scalability of PencilFFTs results from a highly generic code, handling decompositions in arbitrary dimensions and a relatively large (and extensible) variety of transformations.

      Note that PencilFFTs can optionally use collective communications (using MPI_Alltoallv) instead of point-to-point communications. For details, see the docs for PencilFFTPlan and for PencilArray transpositions. As seen above, collective communications generally perform worse than point-to-point ones, and runtimes are nearly indistinguishable from those of P3DFFT.

      Benchmark details

      The benchmarks were performed using Julia 1.7-beta3 and Intel MPI 2019. We used PencilFFTs v0.12.5 with FFTW.jl v1.4.3 and MPI.jl v0.19.0. We used the Fortran implementation of P3DFFT, version 2.7.6, which was built with Intel 2019 compilers and linked to FFTW 3.3.8. The cluster where the benchmarks were run has Intel Cascade Lake 6248 processors with 2×20 cores per node.

      The number of MPI processes along each decomposed dimension, $P_1$ and $P_2$, was automatically determined by a call to MPI_Dims_create, which tends to create a balanced decomposition with $P_1 ≈ P_2$. For instance, a total of 1024 processes is divided into $P_1 = P_2 = 32$. Different results may be obtained with other combinations, but this was not benchmarked.

      The source files used to generate this benchmark, as well as the raw benchmark results, are all available in the PencilFFTs repo.

      diff --git a/previews/PR48/generated/ffmpeg.err b/previews/PR48/generated/ffmpeg.err deleted file mode 100644 index e8f391d2..00000000 --- a/previews/PR48/generated/ffmpeg.err +++ /dev/null @@ -1,49 +0,0 @@ -ffmpeg version 4.2.7-0ubuntu0.1 Copyright (c) 2000-2022 the FFmpeg developers - built with gcc 9 (Ubuntu 9.4.0-1ubuntu1~20.04.1) - configuration: --prefix=/usr --extra-version=0ubuntu0.1 --toolchain=hardened --libdir=/usr/lib/x86_64-linux-gnu --incdir=/usr/include/x86_64-linux-gnu --arch=amd64 --enable-gpl --disable-stripping --enable-avresample --disable-filter=resample --enable-avisynth --enable-gnutls --enable-ladspa --enable-libaom --enable-libass --enable-libbluray --enable-libbs2b --enable-libcaca --enable-libcdio --enable-libcodec2 --enable-libflite --enable-libfontconfig --enable-libfreetype --enable-libfribidi --enable-libgme --enable-libgsm --enable-libjack --enable-libmp3lame --enable-libmysofa --enable-libopenjpeg --enable-libopenmpt --enable-libopus --enable-libpulse --enable-librsvg --enable-librubberband --enable-libshine --enable-libsnappy --enable-libsoxr --enable-libspeex --enable-libssh --enable-libtheora --enable-libtwolame --enable-libvidstab --enable-libvorbis --enable-libvpx --enable-libwavpack --enable-libwebp --enable-libx265 --enable-libxml2 --enable-libxvid --enable-libzmq --enable-libzvbi --enable-lv2 --enable-omx --enable-openal --enable-opencl --enable-opengl --enable-sdl2 --enable-libdc1394 --enable-libdrm --enable-libiec61883 --enable-nvenc --enable-chromaprint --enable-frei0r --enable-libx264 --enable-shared - libavutil 56. 31.100 / 56. 31.100 - libavcodec 58. 54.100 / 58. 54.100 - libavformat 58. 29.100 / 58. 29.100 - libavdevice 58. 8.100 / 58. 8.100 - libavfilter 7. 57.100 / 7. 57.100 - libavresample 4. 0. 0 / 4. 0. 0 - libswscale 5. 5.100 / 5. 5.100 - libswresample 3. 5.100 / 3. 5.100 - libpostproc 55. 5.100 / 55. 5.100 -Input #0, image2, from '/tmp/jl_PkIoGX/proc1_%04d.png': - Duration: 00:00:04.52, start: 0.000000, bitrate: N/A - Stream #0:0: Video: png, rgb24(pc), 1200x600, 25 fps, 25 tbr, 25 tbn, 25 tbc -Stream mapping: - Stream #0:0 -> #0:0 (png (native) -> h264 (libx264)) -Press [q] to stop, [?] for help -[libx264 @ 0x55e1dd26a540] using cpu capabilities: MMX2 SSE2Fast SSSE3 SSE4.2 AVX FMA3 BMI2 AVX2 AVX512 -[libx264 @ 0x55e1dd26a540] profile High, level 3.1 -[libx264 @ 0x55e1dd26a540] 264 - core 155 r2917 0a84d98 - H.264/MPEG-4 AVC codec - Copyleft 2003-2018 - http://www.videolan.org/x264.html - options: cabac=1 ref=3 deblock=1:0:0 analyse=0x3:0x113 me=hex subme=7 psy=1 psy_rd=1.00:0.00 mixed_ref=1 me_range=16 chroma_me=1 trellis=1 8x8dct=1 cqm=0 deadzone=21,11 fast_pskip=1 chroma_qp_offset=-2 threads=3 lookahead_threads=1 sliced_threads=0 nr=0 decimate=1 interlaced=0 bluray_compat=0 constrained_intra=0 bframes=3 b_pyramid=2 b_adapt=1 b_bias=0 direct=1 weightb=1 open_gop=0 weightp=2 keyint=250 keyint_min=25 scenecut=40 intra_refresh=0 rc_lookahead=40 rc=crf mbtree=1 crf=23.0 qcomp=0.60 qpmin=0 qpmax=69 qpstep=4 ip_ratio=1.40 aq=1:1.00 -Output #0, mp4, to 'vorticity_proc1.mp4': - 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-In-place transforms · PencilFFTs.jl
      Edit on GitHub

      In-place transforms

      Complex-to-complex and real-to-real transforms can be performed in-place, enabling important memory savings. The procedure is very similar to that of out-of-place transforms described in the tutorial. The differences are illustrated in the sections below.

      Creating a domain partition

      We start by partitioning a domain of dimensions $16×32×64$ along all available MPI processes.

      using PencilFFTs
-using MPI
-MPI.Init()
-
-dims_global = (16, 32, 64)  # global dimensions
      (16, 32, 64)

      Such a partitioning is described by a Pencil object. Here we choose to decompose the domain along the last two dimensions. In this case, the actual number of processes along each of these dimensions is chosen automatically.

      decomp_dims = (2, 3)
-comm = MPI.COMM_WORLD
-pen = Pencil(dims_global, decomp_dims, comm)
      Decomposition of 3D data
-    Data dimensions: (16, 32, 64)
-    Decomposed dimensions: (2, 3)
-    Data permutation: NoPermutation()
-    Array type: Array
      Allowed decompositions

      Distributed transforms using PencilFFTs.jl require that the first dimension is not decomposed. In other words, if one wants to perform transforms, then decomp_dims above must not contain 1.

      Creating in-place plans

      To create an in-place plan, pass an in-place transform such as Transforms.FFT! or Transforms.R2R! to PencilFFTPlan. For instance:

      # Perform a 3D in-place complex-to-complex FFT.
-transform = Transforms.FFT!()
-
-# Note that one can also combine different types of in-place transforms.
-# For instance:
-# transform = (
-#     Transforms.R2R!(FFTW.REDFT01),
-#     Transforms.FFT!(),
-#     Transforms.R2R!(FFTW.DHT),
-# )
      FFT!

      We can now create a distributed plan from the previously-created domain partition and the chosen transform.

      plan = PencilFFTPlan(pen, transform)
      Transforms: (FFT!, FFT!, FFT!)
-Input type: ComplexF64
-Global dimensions: (16, 32, 64) -> (16, 32, 64)
-MPI topology: 2D decomposition (2×1 processes)

      Note that in-place real-to-complex transforms are not currently supported. (In other words, the RFFT! transform type is not defined.)

      Allocating data

      As with out-of-place plans, data should be allocated using allocate_input. The difference is that, for in-place plans, this function returns a ManyPencilArray object, which is a container holding multiple PencilArray views sharing the same memory space.

      # Allocate data for the plan.
-# Since `plan` is in-place, this returns a `ManyPencilArray` container.
-A = allocate_input(plan)
-summary(A)
      "ManyPencilArray{ComplexF64, 3, 3, Tuple{PencilArray{ComplexF64, 3, Base.ReshapedArray{ComplexF64, 3, SubArray{ComplexF64, 1, Vector{ComplexF64}, Tuple{Base.OneTo{Int64}}, true}, Tuple{}}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{ComplexF64, 3, Base." ⋯ 140 bytes ⋯ "tation{(2, 1, 3), 3}, Vector{UInt8}}}, PencilArray{ComplexF64, 3, Base.ReshapedArray{ComplexF64, 3, SubArray{ComplexF64, 1, Vector{ComplexF64}, Tuple{Base.OneTo{Int64}}, true}, Tuple{}}, 3, 0, Pencil{3, 2, Permutation{(3, 2, 1), 3}, Vector{UInt8}}}}, Vector{ComplexF64}}"

      Note that allocate_output also works for in-place plans. In this case, it returns exactly the same thing as allocate_input.

      As shown in the next section, in-place plans must be applied on the returned ManyPencilArray. On the other hand, one usually wants to access and modify data, and for this one needs the PencilArray views contained in the ManyPencilArray. The input and output array views can be obtained by calling first(::ManyPencilArray) and last(::ManyPencilArray).

      For instance, we can initialise the input array with some data before transforming:

      using Random
-u_in = first(A)  # input data view
-randn!(u_in)
-summary(u_in)
      "16×16×64 PencilArray{ComplexF64, 3}(::Pencil{3, 2, NoPermutation, Array})"

      Applying plans

      Like in FFTW.jl, one can perform in-place transforms using the * and \ operators. As mentioned above, in-place plans must be applied on the ManyPencilArray containers returned by allocate_input.

      plan * A;  # performs in-place forward transform

      After performing an in-place transform, data contained in u_in has been overwritten and has no "physical" meaning. In other words, u_in should not be used at this point. To access the transformed data, one should retrieve the output data view using last(A).

      For instance, to compute the global sum of the transformed data:

      u_out = last(A)  # output data view
-sum(u_out)       # sum of transformed data (note that `sum` reduces over all processes)
      5342.662262046821 + 49991.35283143533im

      Finally, we can perform a backward transform and do stuff with the input view:

      plan \ A;  # perform in-place backward transform

      At this point, the data can be once again found in the input view u_in, while u_out should not be accessed.

      This page was generated using Literate.jl.

      diff --git a/previews/PR48/generated/navier_stokes/index.html b/previews/PR48/generated/navier_stokes/index.html deleted file mode 100644 index 6b9cb7e6..00000000 --- a/previews/PR48/generated/navier_stokes/index.html +++ /dev/null @@ -1,303 +0,0 @@ - -Navier–Stokes equations · PencilFFTs.jl
      Edit on GitHub

      Navier–Stokes equations

      In this example, we numerically solve the incompressible Navier–Stokes equations

      \[∂_t \bm{v} + (\bm{v} ⋅ \bm{∇}) \bm{v} = -\frac{1}{ρ} \bm{∇} p + ν ∇^2 \bm{v}, -\quad \bm{∇} ⋅ \bm{v} = 0,\]

      where $\bm{v}(\bm{x}, t)$ and $p(\bm{x}, t)$ are respectively the velocity and pressure fields, $ν$ is the fluid kinematic viscosity and $ρ$ is the fluid density.

      We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.

      First steps

      We start by loading the required packages, initialising MPI and setting the simulation parameters.

      using MPI
-using PencilFFTs
-
-MPI.Init()
-comm = MPI.COMM_WORLD
-procid = MPI.Comm_rank(comm) + 1
-
-# Simulation parameters
-Ns = (64, 64, 64)  # = (Nx, Ny, Nz)
-Ls = (2π, 2π, 2π)  # = (Lx, Ly, Lz)
-
-# Collocation points ("global" = over all processes).
-# We include the endpoint (length = N + 1) for convenience.
-xs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls)  # = (x, y, z)
      (0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586)

      Let's check the number of MPI processes over which we're running our simulation:

      MPI.Comm_size(comm)
      2

      We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:

      pen = Pencil(Ns, comm)
      Decomposition of 3D data
-    Data dimensions: (64, 64, 64)
-    Decomposed dimensions: (2, 3)
-    Data permutation: NoPermutation()
-    Array type: Array

      The subdomain associated to the local MPI process can be obtained using range_local:

      range_local(pen)
      (1:64, 1:32, 1:64)

      We now construct a distributed vector field that follows the decomposition configuration we just created:

      v⃗₀ = (
-    PencilArray{Float64}(undef, pen),  # vx
-    PencilArray{Float64}(undef, pen),  # vy
-    PencilArray{Float64}(undef, pen),  # vz
-)
-summary(v⃗₀[1])
      "64×32×64 PencilArray{Float64, 3}(::Pencil{3, 2, NoPermutation, Array})"

      We still need to fill this array with interesting values that represent a physical velocity field.

      Initial condition

      Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:

      \[\begin{aligned} -v_x(x, y, z) &= u₀ \sin(k₀ x) \cos(k₀ y) \cos(k₀ z) \\ -v_y(x, y, z) &= -u₀ \cos(k₀ x) \sin(k₀ y) \cos(k₀ z) \\ -v_z(x, y, z) &= 0 -\end{aligned}\]

      where $u₀$ and $k₀$ are two parameters setting the amplitude and the period of the velocity field.

      To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:

      grid = localgrid(pen, xs_global)
      LocalRectilinearGrid{3} with coordinates:
- (1) 0.0:0.09817477042468103:6.1850105367549055
- (2) 0.0:0.09817477042468103:3.043417883165112
- (3) 0.0:0.09817477042468103:6.1850105367549055

      We can use this to initialise the velocity field:

      u₀ = 1.0
-k₀ = 2π / Ls[1]  # should be integer if L = 2π (to preserve periodicity)
-
-@. v⃗₀[1] =  u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)
-@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)
-@. v⃗₀[3] =  0

      Let's plot a 2D slice of the velocity field managed by the local MPI process:

      using GLMakie
-
-# Compute the norm of a vector field represented by a tuple of arrays.
-function vecnorm(v⃗::NTuple)
-    vnorm = similar(v⃗[1])
-    for n ∈ eachindex(v⃗[1])
-        w = zero(eltype(vnorm))
-        for v ∈ v⃗
-            w += v[n]^2
-        end
-        vnorm[n] = sqrt(w)
-    end
-    vnorm
-end
-
-let fig = Figure(resolution = (700, 600))
-    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
-    vnorm = vecnorm(v⃗₀)
-    ct = contour!(
-        ax, grid.x, grid.y, grid.z, vnorm;
-        alpha = 0.2, levels = 4,
-        colormap = :viridis, colorrange = (0.0, 1.0),
-    )
-    cb = Colorbar(fig[1, 2], ct; label = "Velocity magnitude")
-    fig
-end

      Velocity in Fourier space

      In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series

      \[\bm{v}(\bm{x}, t) = -∑_{\bm{k}} \hat{\bm{v}}_{\bm{k}}(t) \, e^{i \bm{k} ⋅ \bm{x}},\]

      where $\bm{k} = (k_x, k_y, k_z)$ are the discrete wave numbers.

      The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for $k_x$:

      using AbstractFFTs: fftfreq, rfftfreq
-
-ks_global = (
-    rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]),  # kx | real-to-complex
-     fftfreq(Ns[2], 2π * Ns[2] / Ls[2]),  # ky | complex-to-complex
-     fftfreq(Ns[3], 2π * Ns[3] / Ls[3]),  # kz | complex-to-complex
-)
-
-ks_global[1]'
      1×33 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
- 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  27.0  28.0  29.0  30.0  31.0  32.0
      ks_global[2]'
      1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
- 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0
      ks_global[3]'
      1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
- 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0

      To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:

      plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())
      Transforms: (RFFT, FFT, FFT)
-Input type: Float64
-Global dimensions: (64, 64, 64) -> (33, 64, 64)
-MPI topology: 2D decomposition (2×1 processes)

      See PencilFFTPlan for details on creating plans and on optional keyword arguments.

      We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$:

      v̂s = plan .* v⃗₀
-summary(v̂s[1])
      "16×64×64 PencilArray{ComplexF64, 3}(::Pencil{3, 2, Permutation{(3, 2, 1), 3}, Array})"

      Note that, in Fourier space, the domain decomposition is performed along the directions $x$ and $y$:

      pencil(v̂s[1])
      Decomposition of 3D data
-    Data dimensions: (33, 64, 64)
-    Decomposed dimensions: (1, 2)
-    Data permutation: Permutation(3, 2, 1)
-    Array type: Array

      This is because the 3D FFTs are performed one dimension at a time, with the $x$ direction first and the $z$ direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.

      To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers $\bm{k}$ associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:

      grid_fourier = localgrid(v̂s[1], ks_global)
      LocalRectilinearGrid{3} with Permutation(3, 2, 1) and coordinates:
- (1) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0]
- (2) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]
- (3) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]

      As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, $\bm{ω} = \bm{∇} × \bm{v}$. In Fourier space, this becomes $\hat{\bm{ω}} = i \bm{k} × \hat{\bm{v}}$.

      using StaticArrays: SVector
-using LinearAlgebra: ×
-
-function curl_fourier!(
-        ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,
-    ) where {N}
-    @inbounds for I ∈ eachindex(grid_fourier)
-        # We use StaticArrays for the cross product between small vectors.
-        ik⃗ = im * SVector(grid_fourier[I])
-        v⃗ = SVector(getindex.(v̂s, Ref(I)))  # = (v̂s[1][I], v̂s[2][I], ...)
-        ω⃗ = ik⃗ × v⃗
-        for n ∈ eachindex(ω⃗)
-            ω̂s[n][I] = ω⃗[n]
-        end
-    end
-    ω̂s
-end
-
-ω̂s = similar.(v̂s)
-curl_fourier!(ω̂s, v̂s, grid_fourier);

      We finally transform back to physical space and plot the result:

      ωs = plan .\ ω̂s
-
-let fig = Figure(resolution = (700, 600))
-    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
-    ω_norm = vecnorm(ωs)
-    ct = contour!(
-        ax, grid.x, grid.y, grid.z, ω_norm;
-        alpha = 0.1, levels = 0.8:0.2:2.0,
-        colormap = :viridis, colorrange = (0.8, 2.0),
-    )
-    cb = Colorbar(fig[1, 2], ct; label = "Vorticity magnitude")
-    fig
-end

      Computing the non-linear term

      One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as

      \[∂_t \hat{\bm{v}}_{\bm{k}} = -- \mathcal{P}_{\bm{k}} \! \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right] -- ν |\bm{k}|^2 \hat{\bm{v}}_{\bm{k}} -\quad \text{ with } \quad -\mathcal{P}_{\bm{k}}(\hat{\bm{F}}_{\bm{k}}) = \left( I - \frac{\bm{k} ⊗ -\bm{k}}{|\bm{k}|^2} \right) \hat{\bm{F}}_{\bm{k}},\]

      where $\mathcal{P}_{\bm{k}}$ is a projection operator allowing to preserve the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.

      Now that we have the wave numbers $\bm{k}$, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$ of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, $\hat{\bm{F}}_{\bm{k}} = \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right]_{\bm{k}}$. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.

      Below we implement a function that computes the non-linear term in Fourier space based on its convective form $(\bm{v} ⋅ \bm{∇}) \bm{v} = \bm{∇} ⋅ (\bm{v} ⊗ \bm{v})$. Note that this equivalence uses the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$.

      using LinearAlgebra: mul!, ldiv!  # for applying FFT plans in-place
-
-# Compute non-linear term in Fourier space from velocity field in physical
-# space. Optional keyword arguments may be passed to avoid allocations.
-function ns_nonlinear!(
-        F̂s, vs, plan, grid_fourier;
-        vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),
-    )
-    # Compute F_i = ∂_j (v_i v_j) for each i.
-    # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)
-    w, ŵ = vbuf, v̂buf
-    @inbounds for (i, F̂i) ∈ enumerate(F̂s)
-        F̂i .= 0
-        vi = vs[i]
-        for (j, vj) ∈ enumerate(vs)
-            w .= vi .* vj     # w = v_i * v_j in physical space
-            mul!(ŵ, plan, w)  # same in Fourier space
-            # Add derivative in Fourier space
-            for I ∈ eachindex(grid_fourier)
-                k⃗ = grid_fourier[I]  # = (kx, ky, kz)
-                kj = k⃗[j]
-                F̂i[I] += im * kj * ŵ[I]
-            end
-        end
-    end
-    F̂s
-end
      ns_nonlinear! (generic function with 1 method)

      As an example, let's use this function on our initial velocity field:

      F̂s = similar.(v̂s)
-ns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);

      Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.

      function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)
-    ks_max = maximum.(abs, ks_global)  # maximum stored wave numbers (kx_max, ky_max, kz_max)
-    ks_lim = (2 / 3) .* ks_max
-    @inbounds for I ∈ eachindex(grid_fourier)
-        k⃗ = grid_fourier[I]
-        if any(abs.(k⃗) .> ks_lim)
-            for ŵ ∈ ŵs
-                ŵ[I] = 0
-            end
-        end
-    end
-    ŵs
-end
-
-# We can apply this on the previously computed non-linear term:
-dealias_twothirds!(F̂s, grid_fourier, ks_global);

      Finally, we implement the projection associated to the incompressibility condition:

      function project_divergence_free!(ûs, grid_fourier)
-    @inbounds for I ∈ eachindex(grid_fourier)
-        k⃗ = grid_fourier[I]
-        k² = sum(abs2, k⃗)
-        iszero(k²) && continue  # avoid division by zero
-        û = getindex.(ûs, Ref(I))  # (ûs[1][I], ûs[2][I], ...)
-        for i ∈ eachindex(û)
-            ŵ = û[i]
-            for j ∈ eachindex(û)
-                ŵ -= k⃗[i] * k⃗[j] * û[j] / k²
-            end
-            ûs[i][I] = ŵ
-        end
-    end
-    ûs
-end
      project_divergence_free! (generic function with 1 method)

      We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:

      v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)
-v̂s_proj .≈ v̂s  # the last one may be false because v_z = 0 initially
      (true, true, false)

      Putting it all together

      To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:

      function ns_rhs!(
-        dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,
-    ) where {N}
-    # 1. Compute non-linear term and dealias it
-    (; plan, cache, ks_global, grid_fourier) = p
-    F̂s = cache.F̂s
-    ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])
-    dealias_twothirds!(F̂s, grid_fourier, ks_global)
-
-    # 2. Project onto divergence-free space
-    project_divergence_free!(F̂s, grid_fourier)
-
-    # 3. Transform velocity to Fourier space
-    v̂s = cache.v̂s
-    map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)
-
-    # 4. Add viscous term (and multiply projected non-linear term by -1)
-    ν = p.ν
-    for n ∈ eachindex(v̂s)
-        v̂ = v̂s[n]
-        F̂ = F̂s[n]
-        @inbounds for I ∈ eachindex(grid_fourier)
-            k⃗ = grid_fourier[I]  # = (kx, ky, kz)
-            k² = sum(abs2, k⃗)
-            F̂[I] = -F̂[I] - ν * k² * v̂[I]
-        end
-    end
-
-    # 5. Transform RHS back to physical space
-    map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)
-
-    nothing
-end
      ns_rhs! (generic function with 1 method)

      For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.

      using OrdinaryDiffEq
-using RecursiveArrayTools: ArrayPartition
-
-ns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)
-vs_init_ode = ArrayPartition(v⃗₀)
-summary(vs_init_ode)
      "RecursiveArrayTools.ArrayPartition{Float64, Tuple{PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}}} with arrays:"

      We now define solver parameters and temporary variables, and initialise the problem:

      params = (;
-    ν = 5e-3,  # kinematic viscosity
-    plan, grid_fourier, ks_global,
-    cache = (
-        v̂s = similar.(v̂s),
-        F̂s = similar.(v̂s),
-    )
-)
-
-tspan = (0.0, 10.0)
-prob = ODEProblem(ns_rhs!, vs_init_ode, tspan, params)
-integrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);

      We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum $E(k)$, to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity $ν$ must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large $k$), while high enough to allow the small-scale motions to be correctly resolved.

      function energy_spectrum!(Ek, ks, v̂s, grid_fourier)
-    Nk = length(Ek)
-    @assert Nk == length(ks)
-    Ek .= 0
-    for I ∈ eachindex(grid_fourier)
-        k⃗ = grid_fourier[I]  # = (kx, ky, kz)
-        knorm = sqrt(sum(abs2, k⃗))
-        i = searchsortedfirst(ks, knorm)
-        i > Nk && continue
-        v⃗ = getindex.(v̂s, Ref(I))  # = (v̂s[1][I], v̂s[2][I], ...)
-        factor = k⃗[1] == 0 ? 1 : 2  # account for Hermitian symmetry and r2c transform
-        Ek[i] += factor * sum(abs2, v⃗) / 2
-    end
-    MPI.Allreduce!(Ek, +, get_comm(v̂s[1]))  # sum across all processes
-    Ek
-end
-
-ks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])
-Ek = similar(ks)
-v̂s = plan .* integrator.u.x
-energy_spectrum!(Ek, ks, v̂s, grid_fourier)
-Ek ./= scale_factor(plan)^2  # rescale energy
-
-curl_fourier!(ω̂s, v̂s, grid_fourier)
-ldiv!.(ωs, plan, ω̂s)
-ω⃗_plot = Observable(ωs)
-k_plot = @view ks[2:end]
-E_plot = Observable(@view Ek[2:end])
-t_plot = Observable(integrator.t)
-
-fig = let
-    fig = Figure(resolution = (1200, 600))
-    ax = Axis3(
-        fig[1, 1][1, 1]; title = @lift("t = $(round($t_plot, digits = 3))"),
-        aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z",
-    )
-    ω_mag = @lift vecnorm($ω⃗_plot)
-    ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)
-    ct = contour!(
-        ax, grid.x, grid.y, grid.z, ω_mag_norm;
-        alpha = 0.3, levels = 3,
-        colormap = :viridis, colorrange = (0.0, 1.0),
-    )
-    cb = Colorbar(fig[1, 1][1, 2], ct; label = "Normalised vorticity magnitude")
-    ax_sp = Axis(
-        fig[1, 2];
-        xlabel = "k", ylabel = "E(k)", xscale = log2, yscale = log10,
-        title = "Kinetic energy spectrum",
-    )
-    ylims!(ax_sp, 1e-8, 1e0)
-    scatterlines!(ax_sp, k_plot, E_plot)
-    ks_slope = exp.(range(log(2.5), log(25.0), length = 3))
-    E_fivethirds = @. 0.3 * ks_slope^(-5/3)
-    @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)
-    text!(ax_sp, L"k^{-5/3}"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))
-    fig
-end
-
-record(fig, "vorticity_proc$procid.mp4"; framerate = 10) do io
-    while integrator.t < 20
-        dt = 0.001
-        step!(integrator, dt)
-        t_plot[] = integrator.t
-        mul!.(v̂s, plan, integrator.u.x)  # current velocity in Fourier space
-        curl_fourier!(ω̂s, v̂s, grid_fourier)
-        ldiv!.(ω⃗_plot[], plan, ω̂s)
-        ω⃗_plot[] = ω⃗_plot[]  # to force updating the plot
-        energy_spectrum!(Ek, ks, v̂s, grid_fourier)
-        Ek ./= scale_factor(plan)^2  # rescale energy
-        E_plot[] = E_plot[]
-        recordframe!(io)
-    end
-end;
      - -

      This page was generated using Literate.jl.

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zAb$bS1dss&Q2_R$-~${O15h&v^zIUnZ@PiAK9Cm(L@NS}B`Lt)>>C8)KsrJKzy|<7 z%;=j#}Kmbet`~m+m0`uXm9+3YwAMoDBx*9+S0DcgNloYs^gBRd$ z^AP|64{+SS>3=)^rh&8bx4dHj>HyLJe{BNg-`XezZ~*+S0P_^w0N`&vdz-VEPR>Sv zfBp(!z&A4hLlTLI>kkHuR XugY&qL2?GJrW_v_S-BZmn3?|vi^KLc diff --git a/previews/PR48/index.html b/previews/PR48/index.html deleted file mode 100644 index eb219dba..00000000 --- a/previews/PR48/index.html +++ /dev/null @@ -1,7 +0,0 @@ - -Home · PencilFFTs.jl

      Edit on GitHub

      PencilFFTs

      Fast Fourier transforms of MPI-distributed Julia arrays.

      Introduction

      This package provides multidimensional FFTs and related transforms on MPI-distributed Julia arrays via the PencilArrays package.

      The name of this package originates from the decomposition of 3D domains along two out of three dimensions, sometimes called pencil decomposition. This is illustrated by the figure below,[1] where each coloured block is managed by a different MPI process. Typically, one wants to compute FFTs on a scalar or vector field along the three spatial dimensions. In the case of a pencil decomposition, 3D FFTs are performed one dimension at a time, along the non-decomposed direction. Transforms must then be interleaved with global data transpositions to switch between pencil configurations. In high-performance computing environments, such data transpositions are generally the most expensive part of a parallel FFT computation, due to the large cost of communications between computing nodes.

      - Pencil decomposition of 3D domains -

      More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension $N$. The decompositions can be performed along an arbitrary number $M < N$ of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.

      The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.

      Features

      • distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;

      • FFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;

      • in-place and out-of-place transforms;

      • high scalability up to (at least) tens of thousands of MPI processes.

      Installation

      PencilFFTs can be installed using the Julia package manager:

      julia> ] add PencilFFTs

      Similar projects

      • FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.

      • PFFT is a very general parallel FFT library written in C.

      • P3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.

      • 2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.

      • 1Figure adapted from this PhD thesis.
      • 2For the pencil decomposition represented in the figure, $N = 3$ and $M = 2$.
      diff --git a/previews/PR48/search/index.html b/previews/PR48/search/index.html deleted file mode 100644 index c861407d..00000000 --- a/previews/PR48/search/index.html +++ /dev/null @@ -1,2 +0,0 @@ - -Search · PencilFFTs.jl

      Loading search...

        diff --git a/previews/PR48/search_index.js b/previews/PR48/search_index.js deleted file mode 100644 index ebd1df56..00000000 --- a/previews/PR48/search_index.js +++ /dev/null @@ -1,3 +0,0 @@ -var documenterSearchIndex = {"docs": -[{"location":"tutorial/#Tutorial","page":"Tutorial","title":"Tutorial","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The following tutorial shows how to perform a 3D FFT of real periodic data defined on a grid of N_x N_y N_z points.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"
        \n \n \n
        ","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"By default, the domain is distributed on a 2D MPI topology of dimensions N_1 N_2. As an example, the above figure shows such a topology with N_1 = 4 and N_2 = 3, for a total of 12 MPI processes.","category":"page"},{"location":"tutorial/#tutorial:creating_plans","page":"Tutorial","title":"Creating plans","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The first thing to do is to create a domain decomposition configuration for the given dataset dimensions N_x N_y N_z. In the framework of PencilArrays, such a configuration is described by a Pencil object. As described in the PencilArrays docs, we can let the Pencil constructor automatically determine such a configuration. For this, only an MPI communicator and the dataset dimensions are needed:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using MPI\nusing PencilFFTs\n\nMPI.Init()\ncomm = MPI.COMM_WORLD\n\n# Input data dimensions (Nx × Ny × Nz)\ndims = (16, 32, 64)\npen = Pencil(dims, comm)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"By default this creates a 2D decomposition (for the case of a 3D dataset), but one can change this as detailed in the PencilArrays documentation linked above.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"We can now create a PencilFFTPlan, which requires information on decomposition configuration (the Pencil object) and on the transforms that will be applied:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# Apply a 3D real-to-complex (r2c) FFT.\ntransform = Transforms.RFFT()\n\n# Note that, for more control, one can instead separately specify the transforms along each dimension:\n# transform = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())\n\n# Create plan\nplan = PencilFFTPlan(pen, transform)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"See the PencilFFTPlan constructor for details on the accepted options, and the Transforms module for the possible transforms. It is also possible to enable fine-grained performance measurements via the TimerOutputs package, as described in Measuring performance.","category":"page"},{"location":"tutorial/#Allocating-data","page":"Tutorial","title":"Allocating data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Next, we want to apply the plan on some data. Transforms may only be applied on PencilArrays, which are array wrappers that include MPI decomposition information (in some sense, analogous to DistributedArrays in Julia's distributed computing approach). The helper function allocate_input can be used to allocate a PencilArray that is compatible with our plan:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# In our example, this returns a 3D PencilArray of real data (Float64).\nu = allocate_input(plan)\n\n# Fill the array with some (random) data\nusing Random\nrandn!(u)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"PencilArrays are a subtype of AbstractArray, and thus they support all common array operations.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Similarly, to preallocate output data, one can use allocate_output:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# In our example, this returns a 3D PencilArray of complex data (Complex{Float64}).\nv = allocate_output(plan)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"This is only required if one wants to apply the plans using a preallocated output (with mul!, see right below).","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The data types returned by allocate_input and allocate_output are slightly different when working with in-place transforms. See the in-place example for details.","category":"page"},{"location":"tutorial/#Applying-plans","page":"Tutorial","title":"Applying plans","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The interface to apply plans is consistent with that of AbstractFFTs. Namely, * and mul! are respectively used for forward transforms without and with preallocated output data. Similarly, \\ and ldiv! are used for backward transforms.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using LinearAlgebra # for mul!, ldiv!\n\n# Apply plan on `u` with `v` as an output\nmul!(v, plan, u)\n\n# Apply backward plan on `v` with `w` as an output\nw = similar(u)\nldiv!(w, plan, v) # now w ≈ u","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Note that, consistently with AbstractFFTs, normalisation is performed at the end of a backward transform, so that the original data is recovered when applying a forward followed by a backward transform.","category":"page"},{"location":"tutorial/#Accessing-and-modifying-data","page":"Tutorial","title":"Accessing and modifying data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For any given MPI process, a PencilArray holds the data associated to its local partition in the global geometry. PencilArrays are accessed using local indices that start at 1, regardless of the location of the local process in the MPI topology. Note that PencilArrays, being based on regular Arrays, support both linear and Cartesian indexing (see the Julia docs for details).","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For convenience, the global_view function can be used to generate an OffsetArray wrapper that takes global indices.","category":"page"},{"location":"tutorial/#tutorial:output_data_layout","page":"Tutorial","title":"Output data layout","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"In memory, the dimensions of the transform output are by default reversed with respect to the input. That is, if the order of indices in the input data is (x, y, z), then the output has order (z, y, x) in memory. This detail is hidden from the user, and output arrays are always accessed in the same order as the input data, regardless of the underlying output dimension permutation. This applies to PencilArrays and to OffsetArrays returned by global_view.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The reasoning behind dimension permutations, is that they allow to always perform FFTs along the fastest array dimension and to avoid a local data transposition, resulting in performance gains. A similar approach is followed by other parallel FFT libraries. FFTW itself, in its distributed-memory routines, includes a flag that enables a similar behaviour. In PencilFFTs, index permutation is the default, but it can be disabled via the permute_dims flag of PencilFFTPlan.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"A great deal of work has been spent in making generic index permutations as efficient as possible, both in intermediate and in the output state of the multidimensional transforms. This has been achieved, in part, by making sure that permutations such as (3, 2, 1) are compile-time constants.","category":"page"},{"location":"tutorial/#Further-reading","page":"Tutorial","title":"Further reading","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For details on working with PencilArrays see the PencilArrays docs.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The examples on the sidebar further illustrate the use of transforms and provide an introduction to working with MPI-distributed data in the form of PencilArrays. In particular, the gradient example illustrates different ways of computing things using Fourier-transformed distributed arrays. Then, the incompressible Navier–Stokes example is a more advanced and complete example of a possible application of the PencilFFTs package.","category":"page"},{"location":"benchmarks/#Benchmarks","page":"Benchmarks","title":"Benchmarks","text":"","category":"section"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The performance of PencilFFTs.jl is comparable to that of other open-source parallel FFT libraries implemented in lower-level languages. Below, we show comparisons with the Fortran implementation of P3DFFT, possibly the most popular of these libraries. The benchmarks were performed on the Jean–Zay cluster of the IDRIS French computing centre (CNRS).","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The figure below shows strong scaling benchmarks of 3D real-to-complex FFTs using 2D (\"pencil\") decomposition. The benchmarks were run for input arrays of dimensions N_x N_y N_z = 512^3, 1024^3 and 2048^3. Each timing is averaged over 100 repetitions.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"
        \n \n \n
        ","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"As seen above, PencilFFTs generally outperforms P3DFFT in its default setting. This is largely explained by the choice of using non-blocking point-to-point MPI communications (via MPI_Isend and MPI_Irecv), while P3DFFT uses collective MPI_Alltoallv calls. This enables PencilFFTs to perform data reordering operations on the partially received data while waiting for the incoming data, leading to better performance. Moreover, in contrast with P3DFFT, the high performance and scalability of PencilFFTs results from a highly generic code, handling decompositions in arbitrary dimensions and a relatively large (and extensible) variety of transformations.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"Note that PencilFFTs can optionally use collective communications (using MPI_Alltoallv) instead of point-to-point communications. For details, see the docs for PencilFFTPlan and for PencilArray transpositions. As seen above, collective communications generally perform worse than point-to-point ones, and runtimes are nearly indistinguishable from those of P3DFFT.","category":"page"},{"location":"benchmarks/#Benchmark-details","page":"Benchmarks","title":"Benchmark details","text":"","category":"section"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The benchmarks were performed using Julia 1.7-beta3 and Intel MPI 2019. We used PencilFFTs v0.12.5 with FFTW.jl v1.4.3 and MPI.jl v0.19.0. We used the Fortran implementation of P3DFFT, version 2.7.6, which was built with Intel 2019 compilers and linked to FFTW 3.3.8. The cluster where the benchmarks were run has Intel Cascade Lake 6248 processors with 2×20 cores per node.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The number of MPI processes along each decomposed dimension, P_1 and P_2, was automatically determined by a call to MPI_Dims_create, which tends to create a balanced decomposition with P_1 P_2. For instance, a total of 1024 processes is divided into P_1 = P_2 = 32. Different results may be obtained with other combinations, but this was not benchmarked.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The source files used to generate this benchmark, as well as the raw benchmark results, are all available in the PencilFFTs repo.","category":"page"},{"location":"GlobalFFTParams/#Global-FFT-parameters","page":"Global FFT parameters","title":"Global FFT parameters","text":"","category":"section"},{"location":"GlobalFFTParams/","page":"Global FFT parameters","title":"Global FFT parameters","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"GlobalFFTParams/","page":"Global FFT parameters","title":"Global FFT parameters","text":"GlobalFFTParams","category":"page"},{"location":"GlobalFFTParams/#PencilFFTs.GlobalFFTParams","page":"Global FFT parameters","title":"PencilFFTs.GlobalFFTParams","text":"GlobalFFTParams{T, N, inplace}\n\nSpecifies the global parameters for an N-dimensional distributed transform. These include the element type T and global data sizes of input and output data, as well as the transform types to be performed along each dimension.\n\n\n\nGlobalFFTParams(size_global, transforms, [real_type=Float64])\n\nDefine parameters for N-dimensional transform.\n\ntransforms must be a tuple of length N specifying the transforms to be applied along each dimension. Each element must be a subtype of Transforms.AbstractTransform. For all the possible transforms, see Transform types.\n\nThe element type must be a real type accepted by FFTW, i.e. either Float32 or Float64.\n\nNote that the transforms are applied one dimension at a time, with the leftmost dimension first for forward transforms.\n\nExample\n\nTo perform a 3D FFT of real data, first a real-to-complex FFT must be applied along the first dimension, followed by two complex-to-complex FFTs along the other dimensions:\n\njulia> size_global = (64, 32, 128); # size of real input data\n\njulia> transforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT());\n\njulia> fft_params = PencilFFTs.GlobalFFTParams(size_global, transforms)\nTransforms: (RFFT, FFT, FFT)\nInput type: Float64\nGlobal dimensions: (64, 32, 128) -> (33, 32, 128)\n\n\n\n\n\n","category":"type"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"EditURL = \"https://github.com/jipolanco/PencilFFTs.jl/blob/master/docs/examples/in-place.jl\"","category":"page"},{"location":"generated/in-place/#In-place-transforms","page":"In-place transforms","title":"In-place transforms","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Complex-to-complex and real-to-real transforms can be performed in-place, enabling important memory savings. The procedure is very similar to that of out-of-place transforms described in the tutorial. The differences are illustrated in the sections below.","category":"page"},{"location":"generated/in-place/#Creating-a-domain-partition","page":"In-place transforms","title":"Creating a domain partition","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"We start by partitioning a domain of dimensions 163264 along all available MPI processes.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"using PencilFFTs\nusing MPI\nMPI.Init()\n\ndims_global = (16, 32, 64) # global dimensions","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Such a partitioning is described by a Pencil object. Here we choose to decompose the domain along the last two dimensions. In this case, the actual number of processes along each of these dimensions is chosen automatically.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"decomp_dims = (2, 3)\ncomm = MPI.COMM_WORLD\npen = Pencil(dims_global, decomp_dims, comm)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"warning: Allowed decompositions\nDistributed transforms using PencilFFTs.jl require that the first dimension is not decomposed. In other words, if one wants to perform transforms, then decomp_dims above must not contain 1.","category":"page"},{"location":"generated/in-place/#Creating-in-place-plans","page":"In-place transforms","title":"Creating in-place plans","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"To create an in-place plan, pass an in-place transform such as Transforms.FFT! or Transforms.R2R! to PencilFFTPlan. For instance:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"# Perform a 3D in-place complex-to-complex FFT.\ntransform = Transforms.FFT!()\n\n# Note that one can also combine different types of in-place transforms.\n# For instance:\n# transform = (\n# Transforms.R2R!(FFTW.REDFT01),\n# Transforms.FFT!(),\n# Transforms.R2R!(FFTW.DHT),\n# )","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"We can now create a distributed plan from the previously-created domain partition and the chosen transform.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan = PencilFFTPlan(pen, transform)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Note that in-place real-to-complex transforms are not currently supported. (In other words, the RFFT! transform type is not defined.)","category":"page"},{"location":"generated/in-place/#Allocating-data","page":"In-place transforms","title":"Allocating data","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"As with out-of-place plans, data should be allocated using allocate_input. The difference is that, for in-place plans, this function returns a ManyPencilArray object, which is a container holding multiple PencilArray views sharing the same memory space.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"# Allocate data for the plan.\n# Since `plan` is in-place, this returns a `ManyPencilArray` container.\nA = allocate_input(plan)\nsummary(A)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Note that allocate_output also works for in-place plans. In this case, it returns exactly the same thing as allocate_input.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"As shown in the next section, in-place plans must be applied on the returned ManyPencilArray. On the other hand, one usually wants to access and modify data, and for this one needs the PencilArray views contained in the ManyPencilArray. The input and output array views can be obtained by calling first(::ManyPencilArray) and last(::ManyPencilArray).","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"For instance, we can initialise the input array with some data before transforming:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"using Random\nu_in = first(A) # input data view\nrandn!(u_in)\nsummary(u_in)","category":"page"},{"location":"generated/in-place/#Applying-plans","page":"In-place transforms","title":"Applying plans","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Like in FFTW.jl, one can perform in-place transforms using the * and \\ operators. As mentioned above, in-place plans must be applied on the ManyPencilArray containers returned by allocate_input.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan * A; # performs in-place forward transform\nnothing #hide","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"After performing an in-place transform, data contained in u_in has been overwritten and has no \"physical\" meaning. In other words, u_in should not be used at this point. To access the transformed data, one should retrieve the output data view using last(A).","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"For instance, to compute the global sum of the transformed data:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"u_out = last(A) # output data view\nsum(u_out) # sum of transformed data (note that `sum` reduces over all processes)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Finally, we can perform a backward transform and do stuff with the input view:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan \\ A; # perform in-place backward transform\nnothing #hide","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"At this point, the data can be once again found in the input view u_in, while u_out should not be accessed.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"This page was generated using Literate.jl.","category":"page"},{"location":"Transforms/#Available-transforms","page":"Available transforms","title":"Available transforms","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"CurrentModule = PencilFFTs.Transforms","category":"page"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"Transforms","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms","page":"Available transforms","title":"PencilFFTs.Transforms","text":"Defines different one-dimensional FFT-based transforms.\n\nThe transforms are all subtypes of an AbstractTransform type.\n\nWhen possible, the names of the transforms are kept consistent with the functions exported by AbstractFFTs.jl and FFTW.jl.\n\n\n\n\n\n","category":"module"},{"location":"Transforms/#Transform-types","page":"Available transforms","title":"Transform types","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"FFT\nFFT!\nBFFT\nBFFT!\n\nRFFT\nBRFFT\n\nR2R\nR2R!\n\nNoTransform\nNoTransform!","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.FFT","page":"Available transforms","title":"PencilFFTs.Transforms.FFT","text":"FFT()\n\nComplex-to-complex FFT.\n\nSee also AbstractFFTs.fft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.FFT!","page":"Available transforms","title":"PencilFFTs.Transforms.FFT!","text":"FFT!()\n\nIn-place version of FFT.\n\nSee also AbstractFFTs.fft!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BFFT","page":"Available transforms","title":"PencilFFTs.Transforms.BFFT","text":"BFFT()\n\nUnnormalised backward complex-to-complex FFT.\n\nLike AbstractFFTs.bfft, this transform is not normalised. To obtain the inverse transform, divide the output by the length of the transformed dimension.\n\nSee also AbstractFFTs.bfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.BFFT!","text":"BFFT()\n\nIn-place version of BFFT.\n\nSee also AbstractFFTs.bfft!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.RFFT","page":"Available transforms","title":"PencilFFTs.Transforms.RFFT","text":"RFFT()\n\nReal-to-complex FFT.\n\nSee also AbstractFFTs.rfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BRFFT","page":"Available transforms","title":"PencilFFTs.Transforms.BRFFT","text":"BRFFT(d::Integer)\nBRFFT((d1, d2, ..., dN))\n\nUnnormalised inverse of RFFT.\n\nTo obtain the inverse transform, divide the output by the length of the transformed dimension (of the real output array).\n\nAs described in the AbstractFFTs docs, the length of the output cannot be fully inferred from the input length. For this reason, the BRFFT constructor accepts an optional d argument indicating the output length.\n\nFor multidimensional datasets, a tuple of dimensions (d1, d2, ..., dN) may also be passed. This is equivalent to passing just dN. In this case, the last dimension (dN) is the one that changes size between the input and output. Note that this is the opposite of FFTW.brfft. The reason is that, in PencilFFTs, the last dimension is the one along which a complex-to-real transform is performed.\n\nSee also AbstractFFTs.brfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.R2R","page":"Available transforms","title":"PencilFFTs.Transforms.R2R","text":"R2R(kind)\n\nReal-to-real transform of type kind.\n\nThe possible values of kind are those described in the FFTW.r2r docs and the FFTW manual:\n\ndiscrete cosine transforms: FFTW.REDFT00, FFTW.REDFT01, FFTW.REDFFT10, FFTW.REDFFT11\ndiscrete sine transforms: FFTW.RODFT00, FFTW.RODFT01, FFTW.RODFFT10, FFTW.RODFFT11\ndiscrete Hartley transform: FFTW.DHT\n\nNote: half-complex format DFTs (FFTW.R2HC, FFTW.HC2R) are not currently supported.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.R2R!","page":"Available transforms","title":"PencilFFTs.Transforms.R2R!","text":"R2R!(kind)\n\nIn-place version of R2R.\n\nSee also FFTW.r2r!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.NoTransform","page":"Available transforms","title":"PencilFFTs.Transforms.NoTransform","text":"NoTransform()\n\nIdentity transform.\n\nSpecifies that no transformation should be applied.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.NoTransform!","page":"Available transforms","title":"PencilFFTs.Transforms.NoTransform!","text":"NoTransform!()\n\nIn-place version of NoTransform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#Internals","page":"Available transforms","title":"Internals","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"What follows is used internally in PencilFFTs.","category":"page"},{"location":"Transforms/#Types","page":"Available transforms","title":"Types","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"AbstractCustomPlan\nAbstractTransform\nIdentityPlan\nIdentityPlan!\nPlan","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.AbstractCustomPlan","page":"Available transforms","title":"PencilFFTs.Transforms.AbstractCustomPlan","text":"AbstractCustomPlan\n\nAbstract type defining a custom plan, to be used as an alternative to FFTW plans (FFTW.FFTWPlan).\n\nThe only custom plan defined in this module is IdentityPlan. The user can define other custom plans that are also subtypes of AbstractCustomPlan.\n\nNote that plan returns a subtype of either AbstractFFTs.Plan or AbstractCustomPlan.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.AbstractTransform","page":"Available transforms","title":"PencilFFTs.Transforms.AbstractTransform","text":"AbstractTransform\n\nSpecifies a one-dimensional FFT-based transform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.IdentityPlan","page":"Available transforms","title":"PencilFFTs.Transforms.IdentityPlan","text":"IdentityPlan\n\nType of plan associated to NoTransform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.IdentityPlan!","page":"Available transforms","title":"PencilFFTs.Transforms.IdentityPlan!","text":"IdentityPlan!\n\nType of plan associated to NoTransform!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.Plan","page":"Available transforms","title":"PencilFFTs.Transforms.Plan","text":"Plan = Union{AbstractFFTs.Plan, AbstractCustomPlan}\n\nUnion type representing any plan returned by plan.\n\nSee also AbstractCustomPlan.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#Functions","page":"Available transforms","title":"Functions","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"plan\n\nbinv\nscale_factor\n\neltype_input\neltype_output\nexpand_dims\nis_inplace\nkind\nlength_output","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.plan","page":"Available transforms","title":"PencilFFTs.Transforms.plan","text":"plan(transform::AbstractTransform, A, [dims];\n flags=FFTW.ESTIMATE, timelimit=Inf)\n\nCreate plan to transform array A along dimensions dims.\n\nIf dims is not specified, all dimensions of A are transformed.\n\nFor FFT plans, this function wraps the AbstractFFTs.jl and FFTW.jl plan creation functions. For more details on the function arguments, see AbstractFFTs.plan_fft.\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.binv","page":"Available transforms","title":"PencilFFTs.Transforms.binv","text":"binv(transform::AbstractTransform, d::Integer)\n\nReturns the backwards transform associated to the given transform.\n\nThe second argument must be the length of the first transformed dimension in the forward transform. It is used in particular when transform = RFFT(), to determine the length of the inverse (complex-to-real) transform. See the AbstractFFTs.irfft docs for details.\n\nThe backwards transform returned by this function is not normalised. The normalisation factor for a given array can be obtained by calling scale_factor.\n\nExample\n\njulia> binv(Transforms.FFT(), 42)\nBFFT\n\njulia> binv(Transforms.BRFFT(9), 42)\nRFFT\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.scale_factor","page":"Available transforms","title":"PencilFFTs.Transforms.scale_factor","text":"scale_factor(transform::AbstractTransform, A, [dims = 1:ndims(A)])\n\nGet factor required to normalise the given array after a transformation along dimensions dims (all dimensions by default).\n\nThe array A must have the dimensions of the transform input.\n\nImportant: the dimensions dims must be the same that were passed to plan.\n\nExamples\n\njulia> C = zeros(ComplexF32, 3, 4, 5);\n\njulia> scale_factor(Transforms.FFT(), C)\n60\n\njulia> scale_factor(Transforms.BFFT(), C)\n60\n\njulia> scale_factor(Transforms.BFFT(), C, 2:3)\n20\n\njulia> R = zeros(Float64, 3, 4, 5);\n\njulia> scale_factor(Transforms.RFFT(), R, 2)\n4\n\njulia> scale_factor(Transforms.RFFT(), R, 2:3)\n20\n\njulia> scale_factor(Transforms.BRFFT(8), C)\n96\n\njulia> scale_factor(Transforms.BRFFT(9), C)\n108\n\nThis will fail because the input of RFFT is real, and R is a complex array:\n\njulia> scale_factor(Transforms.RFFT(), C, 2:3)\nERROR: MethodError: no method matching scale_factor(::PencilFFTs.Transforms.RFFT, ::Array{ComplexF32, 3}, ::UnitRange{Int64})\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.eltype_input","page":"Available transforms","title":"PencilFFTs.Transforms.eltype_input","text":"eltype_input(transform::AbstractTransform, real_type<:AbstractFloat)\n\nDetermine input data type for a given transform given the floating point precision of the input data.\n\nSome transforms, such as R2R and NoTransform, can take both real and complex data. For those kinds of transforms, nothing is returned.\n\nExample\n\njulia> eltype_input(Transforms.FFT(), Float32)\nComplexF32 (alias for Complex{Float32})\n\njulia> eltype_input(Transforms.RFFT(), Float64)\nFloat64\n\njulia> eltype_input(Transforms.R2R(FFTW.REDFT01), Float64) # nothing\n\njulia> eltype_input(Transforms.NoTransform(), Float64) # nothing\n\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.eltype_output","page":"Available transforms","title":"PencilFFTs.Transforms.eltype_output","text":"eltype_output(transform::AbstractTransform, eltype_input)\n\nReturns the output data type for a given transform given the input type.\n\nThrows ArgumentError if the input data type is incompatible with the transform type.\n\nExample\n\njulia> eltype_output(Transforms.NoTransform(), Float32)\nFloat32\n\njulia> eltype_output(Transforms.RFFT(), Float64)\nComplexF64 (alias for Complex{Float64})\n\njulia> eltype_output(Transforms.BRFFT(4), ComplexF32)\nFloat32\n\njulia> eltype_output(Transforms.FFT(), Float64)\nERROR: ArgumentError: invalid input data type for PencilFFTs.Transforms.FFT: Float64\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.expand_dims","page":"Available transforms","title":"PencilFFTs.Transforms.expand_dims","text":"expand_dims(transform::AbstractTransform, Val(N))\n\nExpand a single multidimensional transform into one transform per dimension.\n\nExample\n\n# Expand a real-to-complex transform in 3 dimensions.\njulia> expand_dims(Transforms.RFFT(), Val(3))\n(RFFT, FFT, FFT)\n\njulia> expand_dims(Transforms.BRFFT(4), Val(3))\n(BFFT, BFFT, BRFFT{even})\n\njulia> expand_dims(Transforms.NoTransform(), Val(2))\n(NoTransform, NoTransform)\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.is_inplace","page":"Available transforms","title":"PencilFFTs.Transforms.is_inplace","text":"is_inplace(transform::AbstractTransform) -> Bool\nis_inplace(transforms::Vararg{AbtractTransform}) -> Union{Bool, Nothing}\n\nCheck whether a transform or a list of transforms is performed in-place.\n\nIf the list of transforms has a combination of in-place and out-of-place transforms, nothing is returned.\n\nExample\n\njulia> is_inplace(Transforms.RFFT())\nfalse\n\njulia> is_inplace(Transforms.NoTransform!())\ntrue\n\njulia> is_inplace(Transforms.FFT!(), Transforms.R2R!(FFTW.REDFT01))\ntrue\n\njulia> is_inplace(Transforms.FFT(), Transforms.R2R(FFTW.REDFT01))\nfalse\n\njulia> is_inplace(Transforms.FFT(), Transforms.R2R!(FFTW.REDFT01)) === nothing\ntrue\n\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.kind","page":"Available transforms","title":"PencilFFTs.Transforms.kind","text":"kind(transform::R2R)\n\nGet kind of real-to-real transform.\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.length_output","page":"Available transforms","title":"PencilFFTs.Transforms.length_output","text":"length_output(transform::AbstractTransform, length_in::Integer)\n\nReturns the length of the transform output, given the length of its input.\n\nThe input and output lengths are specified in terms of the respective input and output datatypes. For instance, for real-to-complex transforms, these are respectively the length of input real data and of output complex data.\n\n\n\n\n\n","category":"function"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"EditURL = \"https://github.com/jipolanco/PencilFFTs.jl/blob/master/docs/examples/navier_stokes.jl\"","category":"page"},{"location":"generated/navier_stokes/#Navier–Stokes-equations","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"In this example, we numerically solve the incompressible Navier–Stokes equations","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"_t bmv + (bmv bm) bmv = -frac1ρ bm p + ν ^2 bmv\nquad bm bmv = 0","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where bmv(bmx t) and p(bmx t) are respectively the velocity and pressure fields, ν is the fluid kinematic viscosity and ρ is the fluid density.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.","category":"page"},{"location":"generated/navier_stokes/#First-steps","page":"Navier–Stokes equations","title":"First steps","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We start by loading the required packages, initialising MPI and setting the simulation parameters.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using MPI\nusing PencilFFTs\n\nMPI.Init()\ncomm = MPI.COMM_WORLD\nprocid = MPI.Comm_rank(comm) + 1\n\n# Simulation parameters\nNs = (64, 64, 64) # = (Nx, Ny, Nz)\nLs = (2π, 2π, 2π) # = (Lx, Ly, Lz)\n\n# Collocation points (\"global\" = over all processes).\n# We include the endpoint (length = N + 1) for convenience.\nxs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls) # = (x, y, z)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's check the number of MPI processes over which we're running our simulation:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"MPI.Comm_size(comm)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"pen = Pencil(Ns, comm)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"The subdomain associated to the local MPI process can be obtained using range_local:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"range_local(pen)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We now construct a distributed vector field that follows the decomposition configuration we just created:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v⃗₀ = (\n PencilArray{Float64}(undef, pen), # vx\n PencilArray{Float64}(undef, pen), # vy\n PencilArray{Float64}(undef, pen), # vz\n)\nsummary(v⃗₀[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We still need to fill this array with interesting values that represent a physical velocity field.","category":"page"},{"location":"generated/navier_stokes/#Initial-condition","page":"Navier–Stokes equations","title":"Initial condition","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"beginaligned\nv_x(x y z) = u₀ sin(k₀ x) cos(k₀ y) cos(k₀ z) \nv_y(x y z) = -u₀ cos(k₀ x) sin(k₀ y) cos(k₀ z) \nv_z(x y z) = 0\nendaligned","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where u₀ and k₀ are two parameters setting the amplitude and the period of the velocity field.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"grid = localgrid(pen, xs_global)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can use this to initialise the velocity field:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"u₀ = 1.0\nk₀ = 2π / Ls[1] # should be integer if L = 2π (to preserve periodicity)\n\n@. v⃗₀[1] = u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)\n@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)\n@. v⃗₀[3] = 0\nnothing # hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's plot a 2D slice of the velocity field managed by the local MPI process:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using GLMakie\n\n# Compute the norm of a vector field represented by a tuple of arrays.\nfunction vecnorm(v⃗::NTuple)\n vnorm = similar(v⃗[1])\n for n ∈ eachindex(v⃗[1])\n w = zero(eltype(vnorm))\n for v ∈ v⃗\n w += v[n]^2\n end\n vnorm[n] = sqrt(w)\n end\n vnorm\nend\n\nlet fig = Figure(resolution = (700, 600))\n ax = Axis3(fig[1, 1]; aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\")\n vnorm = vecnorm(v⃗₀)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, vnorm;\n alpha = 0.2, levels = 4,\n colormap = :viridis, colorrange = (0.0, 1.0),\n )\n cb = Colorbar(fig[1, 2], ct; label = \"Velocity magnitude\")\n fig\nend","category":"page"},{"location":"generated/navier_stokes/#Velocity-in-Fourier-space","page":"Navier–Stokes equations","title":"Velocity in Fourier space","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"bmv(bmx t) =\n_bmk hatbmv_bmk(t) e^i bmk bmx","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where bmk = (k_x k_y k_z) are the discrete wave numbers.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for k_x:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using AbstractFFTs: fftfreq, rfftfreq\n\nks_global = (\n rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]), # kx | real-to-complex\n fftfreq(Ns[2], 2π * Ns[2] / Ls[2]), # ky | complex-to-complex\n fftfreq(Ns[3], 2π * Ns[3] / Ls[3]), # kz | complex-to-complex\n)\n\nks_global[1]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ks_global[2]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ks_global[3]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"See PencilFFTPlan for details on creating plans and on optional keyword arguments.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients hatbmv_bmk:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v̂s = plan .* v⃗₀\nsummary(v̂s[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Note that, in Fourier space, the domain decomposition is performed along the directions x and y:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"pencil(v̂s[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"This is because the 3D FFTs are performed one dimension at a time, with the x direction first and the z direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers bmk associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"grid_fourier = localgrid(v̂s[1], ks_global)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, bmω = bm bmv. In Fourier space, this becomes hatbmω = i bmk hatbmv.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using StaticArrays: SVector\nusing LinearAlgebra: ×\n\nfunction curl_fourier!(\n ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,\n ) where {N}\n @inbounds for I ∈ eachindex(grid_fourier)\n # We use StaticArrays for the cross product between small vectors.\n ik⃗ = im * SVector(grid_fourier[I])\n v⃗ = SVector(getindex.(v̂s, Ref(I))) # = (v̂s[1][I], v̂s[2][I], ...)\n ω⃗ = ik⃗ × v⃗\n for n ∈ eachindex(ω⃗)\n ω̂s[n][I] = ω⃗[n]\n end\n end\n ω̂s\nend\n\nω̂s = similar.(v̂s)\ncurl_fourier!(ω̂s, v̂s, grid_fourier);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We finally transform back to physical space and plot the result:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ωs = plan .\\ ω̂s\n\nlet fig = Figure(resolution = (700, 600))\n ax = Axis3(fig[1, 1]; aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\")\n ω_norm = vecnorm(ωs)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, ω_norm;\n alpha = 0.1, levels = 0.8:0.2:2.0,\n colormap = :viridis, colorrange = (0.8, 2.0),\n )\n cb = Colorbar(fig[1, 2], ct; label = \"Vorticity magnitude\")\n fig\nend","category":"page"},{"location":"generated/navier_stokes/#Computing-the-non-linear-term","page":"Navier–Stokes equations","title":"Computing the non-linear term","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"_t hatbmv_bmk =\n- mathcalP_bmk left widehat(bmv bm) bmv right\n- ν bmk^2 hatbmv_bmk\nquad text with quad\nmathcalP_bmk(hatbmF_bmk) = left( I - fracbmk \nbmkbmk^2 right) hatbmF_bmk","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where mathcalP_bmk is a projection operator allowing to preserve the incompressibility condition bm bmv = 0. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Now that we have the wave numbers bmk, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients hatbmv_bmk of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, hatbmF_bmk = left widehat(bmv bm) bmv right_bmk. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Below we implement a function that computes the non-linear term in Fourier space based on its convective form (bmv bm) bmv = bm (bmv bmv). Note that this equivalence uses the incompressibility condition bm bmv = 0.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using LinearAlgebra: mul!, ldiv! # for applying FFT plans in-place\n\n# Compute non-linear term in Fourier space from velocity field in physical\n# space. Optional keyword arguments may be passed to avoid allocations.\nfunction ns_nonlinear!(\n F̂s, vs, plan, grid_fourier;\n vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),\n )\n # Compute F_i = ∂_j (v_i v_j) for each i.\n # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)\n w, ŵ = vbuf, v̂buf\n @inbounds for (i, F̂i) ∈ enumerate(F̂s)\n F̂i .= 0\n vi = vs[i]\n for (j, vj) ∈ enumerate(vs)\n w .= vi .* vj # w = v_i * v_j in physical space\n mul!(ŵ, plan, w) # same in Fourier space\n # Add derivative in Fourier space\n for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n kj = k⃗[j]\n F̂i[I] += im * kj * ŵ[I]\n end\n end\n end\n F̂s\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"As an example, let's use this function on our initial velocity field:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"F̂s = similar.(v̂s)\nns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)\n ks_max = maximum.(abs, ks_global) # maximum stored wave numbers (kx_max, ky_max, kz_max)\n ks_lim = (2 / 3) .* ks_max\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I]\n if any(abs.(k⃗) .> ks_lim)\n for ŵ ∈ ŵs\n ŵ[I] = 0\n end\n end\n end\n ŵs\nend\n\n# We can apply this on the previously computed non-linear term:\ndealias_twothirds!(F̂s, grid_fourier, ks_global);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Finally, we implement the projection associated to the incompressibility condition:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function project_divergence_free!(ûs, grid_fourier)\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I]\n k² = sum(abs2, k⃗)\n iszero(k²) && continue # avoid division by zero\n û = getindex.(ûs, Ref(I)) # (ûs[1][I], ûs[2][I], ...)\n for i ∈ eachindex(û)\n ŵ = û[i]\n for j ∈ eachindex(û)\n ŵ -= k⃗[i] * k⃗[j] * û[j] / k²\n end\n ûs[i][I] = ŵ\n end\n end\n ûs\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)\nv̂s_proj .≈ v̂s # the last one may be false because v_z = 0 initially","category":"page"},{"location":"generated/navier_stokes/#Putting-it-all-together","page":"Navier–Stokes equations","title":"Putting it all together","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function ns_rhs!(\n dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,\n ) where {N}\n # 1. Compute non-linear term and dealias it\n (; plan, cache, ks_global, grid_fourier) = p\n F̂s = cache.F̂s\n ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])\n dealias_twothirds!(F̂s, grid_fourier, ks_global)\n\n # 2. Project onto divergence-free space\n project_divergence_free!(F̂s, grid_fourier)\n\n # 3. Transform velocity to Fourier space\n v̂s = cache.v̂s\n map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)\n\n # 4. Add viscous term (and multiply projected non-linear term by -1)\n ν = p.ν\n for n ∈ eachindex(v̂s)\n v̂ = v̂s[n]\n F̂ = F̂s[n]\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n k² = sum(abs2, k⃗)\n F̂[I] = -F̂[I] - ν * k² * v̂[I]\n end\n end\n\n # 5. Transform RHS back to physical space\n map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)\n\n nothing\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using OrdinaryDiffEq\nusing RecursiveArrayTools: ArrayPartition\n\nns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)\nvs_init_ode = ArrayPartition(v⃗₀)\nsummary(vs_init_ode)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We now define solver parameters and temporary variables, and initialise the problem:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"params = (;\n ν = 5e-3, # kinematic viscosity\n plan, grid_fourier, ks_global,\n cache = (\n v̂s = similar.(v̂s),\n F̂s = similar.(v̂s),\n )\n)\n\ntspan = (0.0, 10.0)\nprob = ODEProblem(ns_rhs!, vs_init_ode, tspan, params)\nintegrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum E(k), to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity ν must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large k), while high enough to allow the small-scale motions to be correctly resolved.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function energy_spectrum!(Ek, ks, v̂s, grid_fourier)\n Nk = length(Ek)\n @assert Nk == length(ks)\n Ek .= 0\n for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n knorm = sqrt(sum(abs2, k⃗))\n i = searchsortedfirst(ks, knorm)\n i > Nk && continue\n v⃗ = getindex.(v̂s, Ref(I)) # = (v̂s[1][I], v̂s[2][I], ...)\n factor = k⃗[1] == 0 ? 1 : 2 # account for Hermitian symmetry and r2c transform\n Ek[i] += factor * sum(abs2, v⃗) / 2\n end\n MPI.Allreduce!(Ek, +, get_comm(v̂s[1])) # sum across all processes\n Ek\nend\n\nks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])\nEk = similar(ks)\nv̂s = plan .* integrator.u.x\nenergy_spectrum!(Ek, ks, v̂s, grid_fourier)\nEk ./= scale_factor(plan)^2 # rescale energy\n\ncurl_fourier!(ω̂s, v̂s, grid_fourier)\nldiv!.(ωs, plan, ω̂s)\nω⃗_plot = Observable(ωs)\nk_plot = @view ks[2:end]\nE_plot = Observable(@view Ek[2:end])\nt_plot = Observable(integrator.t)\n\nfig = let\n fig = Figure(resolution = (1200, 600))\n ax = Axis3(\n fig[1, 1][1, 1]; title = @lift(\"t = $(round($t_plot, digits = 3))\"),\n aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\",\n )\n ω_mag = @lift vecnorm($ω⃗_plot)\n ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, ω_mag_norm;\n alpha = 0.3, levels = 3,\n colormap = :viridis, colorrange = (0.0, 1.0),\n )\n cb = Colorbar(fig[1, 1][1, 2], ct; label = \"Normalised vorticity magnitude\")\n ax_sp = Axis(\n fig[1, 2];\n xlabel = \"k\", ylabel = \"E(k)\", xscale = log2, yscale = log10,\n title = \"Kinetic energy spectrum\",\n )\n ylims!(ax_sp, 1e-8, 1e0)\n scatterlines!(ax_sp, k_plot, E_plot)\n ks_slope = exp.(range(log(2.5), log(25.0), length = 3))\n E_fivethirds = @. 0.3 * ks_slope^(-5/3)\n @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)\n text!(ax_sp, L\"k^{-5/3}\"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))\n fig\nend\n\nusing Printf # hide\nwith_xvfb = ENV[\"DISPLAY\"] == \":99\" # hide\nnstep = 0 # hide\nconst tmpdir = mktempdir() # hide\nfilename_frame(procid, nstep) = joinpath(tmpdir, @sprintf(\"proc%d_%04d.png\", procid, nstep)) # hide\nrecord(fig, \"vorticity_proc$procid.mp4\"; framerate = 10) do io\n with_xvfb && recordframe!(io) # hide\n while integrator.t < 20\n dt = 0.001\n step!(integrator, dt)\n t_plot[] = integrator.t\n mul!.(v̂s, plan, integrator.u.x) # current velocity in Fourier space\n curl_fourier!(ω̂s, v̂s, grid_fourier)\n ldiv!.(ω⃗_plot[], plan, ω̂s)\n ω⃗_plot[] = ω⃗_plot[] # to force updating the plot\n energy_spectrum!(Ek, ks, v̂s, grid_fourier)\n Ek ./= scale_factor(plan)^2 # rescale energy\n E_plot[] = E_plot[]\n global nstep += 1 # hide\n with_xvfb ? # hide\n save(filename_frame(procid, nstep), fig) : # hide\n recordframe!(io)\n end\nend;\n\nif with_xvfb # hide\n run(pipeline(`ffmpeg -y -r 10 -i $tmpdir/proc$(procid)_%04d.png -c:v libx264 -vf \"fps=25,format=yuv420p\" vorticity_proc$procid.mp4`; stdout = \"ffmpeg.out\", stderr = \"ffmpeg.err\")) # hide\nend # hide\nnothing # hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"
        \n \n
        ","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"This page was generated using Literate.jl.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"EditURL = \"https://github.com/jipolanco/PencilFFTs.jl/blob/master/docs/examples/gradient.jl\"","category":"page"},{"location":"generated/gradient/#Gradient-of-a-scalar-field","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This example shows different methods to compute the gradient of a real-valued 3D scalar field θ(bmx) in Fourier space, where bmx = (x y z). It is assumed that the field is periodic with period L = 2π along all dimensions.","category":"page"},{"location":"generated/gradient/#General-procedure","page":"Gradient of a scalar field","title":"General procedure","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The discrete Fourier expansion of θ writes","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"θ(bmx) = _bmk Z^3 hatθ(bmk) e^i bmk bmx","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"where bmk = (k_x k_y k_z) are the Fourier wave numbers and hatθ is the discrete Fourier transform of θ. Then, the spatial derivatives of θ are given by","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"frac θ(bmx) x_i =\n_bmk Z^3 i k_i hatθ(bmk) e^i bmk bmx","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"where the subscript i denotes one of the spatial components x, y or z.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In other words, to compute bm θ = (_x θ _y θ _z θ), one has to:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"transform θ to Fourier space to obtain hatθ,\nmultiply hatθ by i bmk,\ntransform the result back to physical space to obtain bm θ.","category":"page"},{"location":"generated/gradient/#Preparation","page":"Gradient of a scalar field","title":"Preparation","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In this section, we initialise a random real-valued scalar field θ and compute its FFT. For more details see the Tutorial.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"using MPI\nusing PencilFFTs\nusing Random\n\nMPI.Init()\n\n# Input data dimensions (Nx × Ny × Nz)\ndims = (64, 32, 64)\n\n# Apply a 3D real-to-complex (r2c) FFT.\ntransform = Transforms.RFFT()\n\n# Automatically create decomposition configuration\ncomm = MPI.COMM_WORLD\npen = Pencil(dims, comm)\n\n# Create plan\nplan = PencilFFTPlan(pen, transform)\n\n# Allocate data and initialise field\nθ = allocate_input(plan)\nrandn!(θ)\n\n# Perform distributed FFT\nθ_hat = plan * θ\nnothing # hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, we initialise the output that will hold ∇θ in Fourier space. Noting that ∇θ is a vector field, we choose to store it as a tuple of 3 PencilArrays.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"∇θ_hat = allocate_output(plan, Val(3))\n\n# This is equivalent:\n# ∇θ_hat = ntuple(d -> similar(θ_hat), Val(3))\n\nsummary(∇θ_hat)","category":"page"},{"location":"generated/gradient/#Fourier-wave-numbers","page":"Gradient of a scalar field","title":"Fourier wave numbers","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In general, the Fourier wave numbers are of the form k_i = 0 frac2πL_i frac4πL_i frac6πL_i , where L_i is the period along dimension i. When a real-to-complex Fourier transform is applied, roughly half of these wave numbers are redundant due to the Hermitian symmetry of the complex Fourier coefficients. In practice, this means that for the fastest dimension x (along which a real-to-complex transform is performed), the negative wave numbers are dropped, i.e. k_x = 0 frac2πL_x frac4πL_x .","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The AbstractFFTs package provides a convenient way to generate the Fourier wave numbers, using the functions fftfreq and rfftfreq. We can use these functions to initialise a \"grid\" of wave numbers associated to our 3D real-to-complex transform:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"using AbstractFFTs: fftfreq, rfftfreq\n\nbox_size = (2π, 2π, 2π) # Lx, Ly, Lz\nsample_rate = 2π .* dims ./ box_size\n\n# In our case (Lx = 2π and Nx even), this gives kx = [0, 1, 2, ..., Nx/2].\nkx = rfftfreq(dims[1], sample_rate[1])\n\n# In our case (Ly = 2π and Ny even), this gives\n# ky = [0, 1, 2, ..., Ny/2-1, -Ny/2, -Ny/2+1, ..., -1] (and similarly for kz).\nky = fftfreq(dims[2], sample_rate[2])\nkz = fftfreq(dims[3], sample_rate[3])\n\nkvec = (kx, ky, kz)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that kvec now contains the wave numbers associated to the global domain. In the following, we will only need the wave numbers associated to the portion of the domain handled by the local MPI process.","category":"page"},{"location":"generated/gradient/#gradient_method_global","page":"Gradient of a scalar field","title":"Method 1: global views","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"PencilArrays, returned for instance by allocate_input and allocate_output, take indices that start at 1, regardless of the location of the subdomain associated to the local process on the global grid. (In other words, PencilArrays take local indices.) On the other hand, we have defined the wave number vector kvec which, for each MPI process, is defined over the global domain, and as such it takes global indices.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"One straightforward way of making data arrays compatible with wave numbers is to use global views, i.e. arrays that take global indices. These are generated from PencilArrays by calling the global_view function. Note that, in general, global indices do not start at 1 for a given MPI process. A given process will own a range of data given by indices in (i1:i2, j1:j2, k1:k2).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"θ_glob = global_view(θ_hat)\n∇θ_glob = global_view.(∇θ_hat)\nsummary(θ_glob)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Once we have global views, we can combine data and wave numbers using the portion of global indices owned by the local MPI process, as shown below. We can use CartesianIndices to iterate over the global indices associated to the local process.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"for I in CartesianIndices(θ_glob)\n i, j, k = Tuple(I) # unpack indices\n\n # Wave number vector associated to current Cartesian index.\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n # Compute gradient in Fourier space.\n # Note that modifying ∇θ_glob also modifies the original PencilArray ∇θ_hat.\n ∇θ_glob[1][I] = im * kx * θ_glob[I]\n ∇θ_glob[2][I] = im * ky * θ_glob[I]\n ∇θ_glob[3][I] = im * kz * θ_glob[I]\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The above loop can be written in a slightly more efficient manner by precomputing im * θ_glob[I]:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@inbounds for I in CartesianIndices(θ_glob)\n i, j, k = Tuple(I)\n\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n u = im * θ_glob[I]\n\n ∇θ_glob[1][I] = kx * u\n ∇θ_glob[2][I] = ky * u\n ∇θ_glob[3][I] = kz * u\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Also note that the above can be easily written in a more generic way, e.g. for arbitrary dimensions, thanks in part to the use of CartesianIndices. Moreover, in the above there is no notion of the dimension permutations discussed in the tutorial, as it is all hidden behind the implementation of PencilArrays. And as seen later in the benchmarks, these (hidden) permutations have zero cost, as the speed is identical to that of a function that explicitly takes into account these permutations.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, we can perform a backwards transform to obtain bm θ in physical space:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"∇θ = plan \\ ∇θ_hat;\nnothing #hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that the transform is automatically broadcast over the three fields of the ∇θ_hat vector, and the result ∇θ is also a tuple of three PencilArrays.","category":"page"},{"location":"generated/gradient/#gradient_method_global_explicit","page":"Gradient of a scalar field","title":"Method 2: explicit global indexing","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Sometimes, one does not need to write generic code. In our case, one often knows the dimensionality of the problem and the memory layout of the data (i.e. the underlying index permutation).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Below is a reimplementation of the above loop, using explicit indices instead of CartesianIndices, and assuming that the underlying index permutation is (3, 2, 1), that is, data is stored in (z y x) order. As discussed in the tutorial, this is the default for transformed arrays. This example also serves as a more explicit explanation for what is going on in the first method.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"# Get local data range in the global grid.\nrng = axes(θ_glob) # = (i1:i2, j1:j2, k1:k2)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"For the loop below, we're assuming that the permutation is (3, 2, 1). In other words, the fastest index is the last one, and not the first one as it is usually in Julia. If the permutation is not (3, 2, 1), things will still work (well, except for the assertion below!), but the loop order will not be optimal.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@assert permutation(θ_hat) === Permutation(3, 2, 1)\n\n@inbounds for i in rng[1], j in rng[2], k in rng[3]\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n # Note that we still access the arrays in (i, j, k) order.\n # (The permutation happens behind the scenes!)\n u = im * θ_glob[i, j, k]\n\n ∇θ_glob[1][i, j, k] = kx * u\n ∇θ_glob[2][i, j, k] = ky * u\n ∇θ_glob[3][i, j, k] = kz * u\nend","category":"page"},{"location":"generated/gradient/#gradient_method_local","page":"Gradient of a scalar field","title":"Method 3: using local indices","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Alternatively, we can avoid global views and work directly on PencilArrays using local indices that start at 1. In this case, part of the strategy is to construct a \"local\" grid of wave numbers that can also be accessed with local indices. This can be conveniently done using the localgrid function of the PencilArrays.jl package, which accepts a PencilArray (or its associated Pencil) and the global coordinates (here kvec):","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"grid_fourier = localgrid(θ_hat, kvec)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that one can directly iterate on the returned grid object:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@inbounds for I in CartesianIndices(grid_fourier)\n # Wave number vector associated to current Cartesian index.\n local k⃗ # hide\n k⃗ = grid_fourier[I]\n u = im * θ_hat[I]\n ∇θ_hat[1][I] = k⃗[1] * u\n ∇θ_hat[2][I] = k⃗[2] * u\n ∇θ_hat[3][I] = k⃗[3] * u\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This implementation is as efficient as the other examples, while being slightly shorter to write. Moreover, it is quite generic, and can be made independent of the number of dimensions with little effort.","category":"page"},{"location":"generated/gradient/#gradient_method_broadcast","page":"Gradient of a scalar field","title":"Method 4: using broadcasting","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, note that the local grid object returned by localgrid makes it is possible to compute the gradient using broadcasting, thus fully avoiding scalar indexing. This can be quite convenient in some cases, and can also be very useful if one is working on GPUs (where scalar indexing is prohibitively expensive). Using broadcasting, the above examples simply become:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@. ∇θ_hat[1] = im * grid_fourier[1] * θ_hat\n@. ∇θ_hat[2] = im * grid_fourier[2] * θ_hat\n@. ∇θ_hat[3] = im * grid_fourier[3] * θ_hat\nnothing # hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Once again, as shown in the benchmarks further below, this method performs quite similarly to the other ones.","category":"page"},{"location":"generated/gradient/#Summary","page":"Gradient of a scalar field","title":"Summary","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The PencilArrays module provides different alternatives to deal with MPI-distributed data that may be subject to dimension permutations. In particular, one can choose to work with global indices (first two examples), with local indices (third example), or to avoid scalar indexing altogether (fourth example).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"If one wants to stay generic, making sure that the same code will work for arbitrary dimensions and will be efficient regardless of the underlying dimension permutation, methods 1, 3 or 4 should be preferred. These use CartesianIndices and make no assumptions on possible dimension permutations, which are by default enabled in the output of PencilFFTs transforms. In fact, such permutations are completely invisible in the implementations.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The second method uses explicit (i, j, k) indices. It assumes that the underlying permutation is (3, 2, 1) to loop with i as the slowest index and k as the fastest, which is the optimal order in this case given the permutation. As such, the implementation is less generic than the others, and differences in performance are negligible with respect to more generic variants.","category":"page"},{"location":"generated/gradient/#gradient_benchmarks","page":"Gradient of a scalar field","title":"Benchmark results","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The following are the benchmark results obtained from running examples/gradient.jl on a laptop, using 2 MPI processes and Julia 1.7.2, with an input array of global dimensions 64 32 64. The different methods detailed above are marked on the right. The \"lazy\" marks indicate runs where the wave numbers were represented by lazy Frequencies objects (returned by rfftfreq and fftfreq). Otherwise, they were collected into Vectors. For some reason, plain Vectors are faster when working with grids generated by localgrid.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In the script, additional implementations can be found which rely on a more advanced understanding of permutations and on the internals of the PencilArrays package. For instance, gradient_local_parent! directly works with the raw data stored in Julia Arrays, while gradient_local_linear! completely avoids CartesianIndices while staying generic and efficient. Nevertheless, these display roughly the same performance as the above examples.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":" gradient_global_view!... 89.900 μs\n gradient_global_view! (lazy)... 92.060 μs [Method 1]\n gradient_global_view_explicit!... 88.958 μs\n gradient_global_view_explicit! (lazy)... 81.055 μs [Method 2]\n gradient_local!... 92.305 μs\n gradient_grid!... 92.770 μs\n gradient_grid! (lazy)... 101.388 μs [Method 3]\n gradient_grid_broadcast!... 88.606 μs\n gradient_grid_broadcast! (lazy)... 151.020 μs [Method 4]\n gradient_local_parent!... 92.248 μs\n gradient_local_linear!... 91.212 μs\n gradient_local_linear_explicit!... 90.992 μs","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This page was generated using Literate.jl.","category":"page"},{"location":"PencilFFTs_timers/#PencilFFTs.measuring_performance","page":"Measuring performance","title":"Measuring performance","text":"","category":"section"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"It is possible to measure the time spent in different sections of the distributed transforms using the TimerOutputs package. This has a (very small) performance overhead, so it is disabled by default. To enable time measurements, call TimerOutputs.enable_debug_timings after loading PencilFFTs (see below for an example). For more details see the TimerOutputs docs.","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"Minimal example:","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"using MPI\nusing PencilFFTs\nusing TimerOutputs\n\n# Enable timing of `PencilFFTs` functions\nTimerOutputs.enable_debug_timings(PencilFFTs)\nTimerOutputs.enable_debug_timings(PencilArrays)\nTimerOutputs.enable_debug_timings(Transpositions)\n\nMPI.Init()\n\nplan = PencilFFTPlan(#= args... =#)\n\n# [do stuff with `plan`...]\n\n# Retrieve and print timing data associated to `plan`\nto = timer(plan)\nprint_timer(to)","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"By default, each PencilFFTPlan has its own TimerOutput. If you already have a TimerOutput, you can pass it to the PencilFFTPlan constructor:","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"to = TimerOutput()\nplan = PencilFFTPlan(..., timer=to)\n\n# [do stuff with `plan`...]\n\nprint_timer(to)","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"#PencilFFTs","page":"Home","title":"PencilFFTs","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Fast Fourier transforms of MPI-distributed Julia arrays.","category":"page"},{"location":"#Introduction","page":"Home","title":"Introduction","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package provides multidimensional FFTs and related transforms on MPI-distributed Julia arrays via the PencilArrays package.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The name of this package originates from the decomposition of 3D domains along two out of three dimensions, sometimes called pencil decomposition. This is illustrated by the figure below,[1] where each coloured block is managed by a different MPI process. Typically, one wants to compute FFTs on a scalar or vector field along the three spatial dimensions. In the case of a pencil decomposition, 3D FFTs are performed one dimension at a time, along the non-decomposed direction. Transforms must then be interleaved with global data transpositions to switch between pencil configurations. In high-performance computing environments, such data transpositions are generally the most expensive part of a parallel FFT computation, due to the large cost of communications between computing nodes.","category":"page"},{"location":"","page":"Home","title":"Home","text":"
        \n \n
        ","category":"page"},{"location":"","page":"Home","title":"Home","text":"More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension N. The decompositions can be performed along an arbitrary number M N of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.","category":"page"},{"location":"#Features","page":"Home","title":"Features","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;\nFFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;\nin-place and out-of-place transforms;\nhigh scalability up to (at least) tens of thousands of MPI processes.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"PencilFFTs can be installed using the Julia package manager:","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> ] add PencilFFTs","category":"page"},{"location":"#Similar-projects","page":"Home","title":"Similar projects","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.\nPFFT is a very general parallel FFT library written in C.\nP3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.\n2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[1]: Figure adapted from this PhD thesis.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[2]: For the pencil decomposition represented in the figure, N = 3 and M = 2.","category":"page"},{"location":"PencilFFTs/#Distributed-FFT-plans","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"Distributed FFTs are implemented in the PencilFFTs module, and are built on top of the PencilArrays package.","category":"page"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"PencilFFTs/#Creating-plans","page":"Distributed FFT plans","title":"Creating plans","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"PencilFFTPlan","category":"page"},{"location":"PencilFFTs/#PencilFFTs.PencilFFTPlan","page":"Distributed FFT plans","title":"PencilFFTs.PencilFFTPlan","text":"PencilFFTPlan{T,N} <: AbstractFFTs.Plan{T}\n\nPlan for N-dimensional FFT-based transform on MPI-distributed data, where input data has type T.\n\n\n\nPencilFFTPlan(p::Pencil, transforms; kwargs...)\n\nCreate a PencilFFTPlan for distributed arrays following a given Pencil configuration. See variant below for details on the specification of transforms and on possible keyword arguments.\n\n\n\nPencilFFTPlan(\n A::PencilArray, transforms;\n fftw_flags = FFTW.ESTIMATE,\n fftw_timelimit = FFTW.NO_TIMELIMIT,\n permute_dims = Val(true),\n transpose_method = Transpositions.PointToPoint(),\n timer = TimerOutput(),\n)\n\nCreate plan for N-dimensional transform on MPI-distributed PencilArrays.\n\nExtended help\n\nThis creates a PencilFFTPlan for arrays sharing the same properties as A (dimensions, MPI decomposition, memory layout, ...), which describe data on an N-dimensional domain.\n\nTransforms\n\nThe transforms to be applied along each dimension are specified by the transforms argument. Possible transforms are defined as subtypes of Transforms.AbstractTransform, and are listed in Transform types. This argument may be either:\n\na tuple of N transforms to be applied along each dimension. For instance, transforms = (Transforms.R2R(FFTW.REDFT01), Transforms.RFFT(), Transforms.FFT());\na single transform to be applied along all dimensions. The input is automatically expanded into N equivalent transforms. For instance, for a three-dimensional array, transforms = Transforms.RFFT() specifies a 3D real-to-complex transform, and is equivalent to passing (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT()).\n\nNote that forward transforms are applied from left to right. In the last example, this means that a real-to-complex transform (RFFT) is first performed along the first dimension. This is followed by complex-to-complex transforms (FFT) along the second and third dimensions.\n\nInput data layout\n\nThe input PencilArray must satisfy the following constraints:\n\narray dimensions must not be permuted. This is the default when constructing PencilArrays.\nfor an M-dimensional domain decomposition (with M < N), the input array must be decomposed along the last M dimensions. For example, for a 2D decomposition of 3D data, the decomposed dimensions must be (2, 3). In particular, the first array dimension must not be distributed among different MPI processes.\nIn the PencilArrays package, the decomposed dimensions are specified at the moment of constructing a Pencil.\nthe element type must be compatible with the specified transform. For instance, real-to-complex transforms (Transforms.RFFT) require the input to be real floating point values. Other transforms, such as Transforms.R2R, accept both real and complex data.\n\nKeyword arguments\n\nThe keyword arguments fftw_flags and fftw_timelimit are passed to the FFTW plan creation functions (see AbstractFFTs docs).\npermute_dims determines whether the indices of the output data should be reversed. For instance, if the input data has global dimensions (Nx, Ny, Nz), then the output of a complex-to-complex FFT would have dimensions (Nz, Ny, Nx). This enables FFTs to always be performed along the first (i.e. fastest) array dimension, which could lead to performance gains. This option is enabled by default. For type inference reasons, it must be a value type (Val(true) or Val(false)).\ntranspose_method allows to select between implementations of the global data transpositions. See PencilArrays docs docs for details.\ntimer should be a TimerOutput object. See Measuring performance for details.\n\n\n\nPencilFFTPlan(\n dims_global::Dims{N}, transforms, proc_dims::Dims{M}, comm::MPI.Comm,\n [real_type = Float64]; extra_dims = (), kws...\n)\n\nCreate plan for N-dimensional transform.\n\nExtended help\n\nInstead of taking a PencilArray or a Pencil, this constructor requires the global dimensions of the input data, passed via the size_global argument.\n\nThe data is distributed over the MPI processes in the comm communicator. The distribution is performed over M dimensions (with M < N) according to the values in proc_dims, which specifies the number of MPI processes to put along each dimension.\n\nPencilArrays that may be transformed with the returned plan can be created using allocate_input.\n\nOptional arguments\n\nThe floating point precision can be selected by setting real_type parameter, which is Float64 by default.\nextra_dims may be used to specify the sizes of one or more extra dimensions that should not be transformed. These dimensions will be added to the rightmost (i.e. slowest) indices of the arrays. See Extra dimensions below for usage hints.\nsee the other constructor for more keyword arguments.\n\nExtra dimensions\n\nOne possible application of extra_dims is for describing the components of a vector or tensor field. However, this means that different PencilFFTPlans would need to be created for each kind of field (scalar, vector, ...). To avoid the creation of multiple plans, a possibly better alternative is to create tuples (or arrays) of PencilArrays using allocate_input and allocate_output.\n\nAnother more legitimate usage of extra_dims is to specify one or more Cartesian dimensions that should not be transformed nor split among MPI processes.\n\nExample\n\nSuppose we want to perform a 3D FFT of real data. The data is to be decomposed along two dimensions, over 8 MPI processes:\n\nsize_global = (64, 32, 128) # size of real input data\n\n# Perform real-to-complex transform along the first dimension, then\n# complex-to-complex transforms along the other dimensions.\ntransforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())\n# transforms = Transforms.RFFT() # this is equivalent to the above line\n\nproc_dims = (4, 2) # 2D decomposition\ncomm = MPI.COMM_WORLD\n\nplan = PencilFFTPlan(size_global, transforms, proc_dims, comm)\n\n\n\n\n\n","category":"type"},{"location":"PencilFFTs/#Allocating-data","page":"Distributed FFT plans","title":"Allocating data","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"allocate_input\nallocate_output","category":"page"},{"location":"PencilFFTs/#PencilFFTs.allocate_input","page":"Distributed FFT plans","title":"PencilFFTs.allocate_input","text":"allocate_input(p::PencilFFTPlan) -> PencilArray\nallocate_input(p::PencilFFTPlan, dims...) -> Array{PencilArray}\nallocate_input(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}\n\nAllocate uninitialised PencilArray that can hold input data for the given plan.\n\nThe second and third forms respectively allocate an array of PencilArrays of size dims, and a tuple of N PencilArrays.\n\nnote: In-place plans\nIf p is an in-place plan, a ManyPencilArray is allocated. This type holds PencilArray wrappers for the input and output transforms (as well as for intermediate transforms) which share the same space in memory. The input and output PencilArrays should be respectively accessed by calling first(::ManyPencilArray) and last(::ManyPencilArray).ExampleSuppose p is an in-place PencilFFTPlan. Then,@assert is_inplace(p)\nA = allocate_input(p) :: ManyPencilArray\nv_in = first(A) :: PencilArray # input data view\nv_out = last(A) :: PencilArray # output data viewAlso note that in-place plans must be performed directly on the returned ManyPencilArray, and not on the contained PencilArray views:p * A # perform forward transform in-place\np \\ A # perform backward transform in-place\n# p * v_in # not allowed!!\n\n\n\n\n\n","category":"function"},{"location":"PencilFFTs/#PencilFFTs.allocate_output","page":"Distributed FFT plans","title":"PencilFFTs.allocate_output","text":"allocate_output(p::PencilFFTPlan) -> PencilArray\nallocate_output(p::PencilFFTPlan, dims...) -> Array{PencilArray}\nallocate_output(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}\n\nAllocate uninitialised PencilArray that can hold output data for the given plan.\n\nIf p is an in-place plan, a ManyPencilArray is allocated.\n\nSee allocate_input for details.\n\n\n\n\n\n","category":"function"},{"location":"PencilFFTs/#Methods","page":"Distributed FFT plans","title":"Methods","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"get_comm(::PencilFFTPlan)\nscale_factor(::PencilFFTPlan)\ntimer(::PencilFFTPlan)\nis_inplace(::PencilFFTPlan)","category":"page"},{"location":"PencilFFTs/#PencilArrays.Pencils.MPITopologies.get_comm-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilArrays.Pencils.MPITopologies.get_comm","text":"get_comm(p::PencilFFTPlan)\n\nGet MPI communicator associated to a PencilFFTPlan.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilFFTs.Transforms.scale_factor-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilFFTs.Transforms.scale_factor","text":"scale_factor(p::PencilFFTPlan)\n\nGet scale factor associated to a PencilFFTPlan.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilArrays.Pencils.timer-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilArrays.Pencils.timer","text":"timer(p::PencilFFTPlan)\n\nGet TimerOutput attached to a PencilFFTPlan.\n\nSee Measuring performance for details.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilFFTs.Transforms.is_inplace-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilFFTs.Transforms.is_inplace","text":"Transforms.is_inplace(p::PencilFFTPlan)\n\nReturns true if the given plan operates in-place on the input data, false otherwise.\n\n\n\n\n\n","category":"method"}] -} diff --git a/previews/PR48/siteinfo.js b/previews/PR48/siteinfo.js deleted file mode 100644 index d4d3f93b..00000000 --- a/previews/PR48/siteinfo.js +++ /dev/null @@ -1 +0,0 @@ -var DOCUMENTER_CURRENT_VERSION = "previews/PR48"; diff --git a/previews/PR55/GlobalFFTParams/index.html b/previews/PR55/GlobalFFTParams/index.html deleted file mode 100644 index 2e928d14..00000000 --- a/previews/PR55/GlobalFFTParams/index.html +++ /dev/null @@ -1,9 +0,0 @@ - -Global FFT parameters · PencilFFTs.jl
        Edit on GitHub

        Global FFT parameters

        PencilFFTs.GlobalFFTParams — Type
        GlobalFFTParams{T, N, inplace}

        Specifies the global parameters for an N-dimensional distributed transform. These include the element type T and global data sizes of input and output data, as well as the transform types to be performed along each dimension.

        GlobalFFTParams(size_global, transforms, [real_type=Float64])

        Define parameters for N-dimensional transform.

        transforms must be a tuple of length N specifying the transforms to be applied along each dimension. Each element must be a subtype of Transforms.AbstractTransform. For all the possible transforms, see Transform types.

        The element type must be a real type accepted by FFTW, i.e. either Float32 or Float64.

        Note that the transforms are applied one dimension at a time, with the leftmost dimension first for forward transforms.

        Example

        To perform a 3D FFT of real data, first a real-to-complex FFT must be applied along the first dimension, followed by two complex-to-complex FFTs along the other dimensions:

        julia> size_global = (64, 32, 128);  # size of real input data
-
-julia> transforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT());
-
-julia> fft_params = PencilFFTs.GlobalFFTParams(size_global, transforms)
-Transforms: (RFFT, FFT, FFT)
-Input type: Float64
-Global dimensions: (64, 32, 128) -> (33, 32, 128)
        source
        diff --git a/previews/PR55/PencilFFTs_timers/index.html b/previews/PR55/PencilFFTs_timers/index.html deleted file mode 100644 index eec146a9..00000000 --- a/previews/PR55/PencilFFTs_timers/index.html +++ /dev/null @@ -1,24 +0,0 @@ - -Measuring performance · PencilFFTs.jl
        Edit on GitHub

        Measuring performance

        It is possible to measure the time spent in different sections of the distributed transforms using the TimerOutputs package. This has a (very small) performance overhead, so it is disabled by default. To enable time measurements, call TimerOutputs.enable_debug_timings after loading PencilFFTs (see below for an example). For more details see the TimerOutputs docs.

        Minimal example:

        using MPI
-using PencilFFTs
-using TimerOutputs
-
-# Enable timing of `PencilFFTs` functions
-TimerOutputs.enable_debug_timings(PencilFFTs)
-TimerOutputs.enable_debug_timings(PencilArrays)
-TimerOutputs.enable_debug_timings(Transpositions)
-
-MPI.Init()
-
-plan = PencilFFTPlan(#= args... =#)
-
-# [do stuff with `plan`...]
-
-# Retrieve and print timing data associated to `plan`
-to = timer(plan)
-print_timer(to)

        By default, each PencilFFTPlan has its own TimerOutput. If you already have a TimerOutput, you can pass it to the PencilFFTPlan constructor:

        to = TimerOutput()
-plan = PencilFFTPlan(..., timer=to)
-
-# [do stuff with `plan`...]
-
-print_timer(to)
        diff --git a/previews/PR55/Transforms/index.html b/previews/PR55/Transforms/index.html deleted file mode 100644 index 867654d8..00000000 --- a/previews/PR55/Transforms/index.html +++ /dev/null @@ -1,74 +0,0 @@ - -Available transforms · PencilFFTs.jl
        Edit on GitHub

        Available transforms

        Transform types

        PencilFFTs.Transforms.BFFT — Type
        BFFT()

        Unnormalised backward complex-to-complex FFT.

        Like AbstractFFTs.bfft, this transform is not normalised. To obtain the inverse transform, divide the output by the length of the transformed dimension.

        See also AbstractFFTs.bfft.

        source
        PencilFFTs.Transforms.BRFFT — Type
        BRFFT(d::Integer)
-BRFFT((d1, d2, ..., dN))

        Unnormalised inverse of RFFT.

        To obtain the inverse transform, divide the output by the length of the transformed dimension (of the real output array).

        As described in the AbstractFFTs docs, the length of the output cannot be fully inferred from the input length. For this reason, the BRFFT constructor accepts an optional d argument indicating the output length.

        For multidimensional datasets, a tuple of dimensions (d1, d2, ..., dN) may also be passed. This is equivalent to passing just dN. In this case, the last dimension (dN) is the one that changes size between the input and output. Note that this is the opposite of FFTW.brfft. The reason is that, in PencilFFTs, the last dimension is the one along which a complex-to-real transform is performed.

        See also AbstractFFTs.brfft.

        source

        Internals

        What follows is used internally in PencilFFTs.

        Types

        PencilFFTs.Transforms.AbstractCustomPlan — Type
        AbstractCustomPlan

        Abstract type defining a custom plan, to be used as an alternative to FFTW plans (FFTW.FFTWPlan).

        The only custom plan defined in this module is IdentityPlan. The user can define other custom plans that are also subtypes of AbstractCustomPlan.

        Note that plan returns a subtype of either AbstractFFTs.Plan or AbstractCustomPlan.

        source

        Functions

        PencilFFTs.Transforms.plan — Function
        plan(transform::AbstractTransform, A, [dims];
-     flags=FFTW.ESTIMATE, timelimit=Inf)

        Create plan to transform array A along dimensions dims.

        If dims is not specified, all dimensions of A are transformed.

        For FFT plans, this function wraps the AbstractFFTs.jl and FFTW.jl plan creation functions. For more details on the function arguments, see AbstractFFTs.plan_fft.

        source
        PencilFFTs.Transforms.binv — Function
        binv(transform::AbstractTransform, d::Integer)

        Returns the backwards transform associated to the given transform.

        The second argument must be the length of the first transformed dimension in the forward transform. It is used in particular when transform = RFFT(), to determine the length of the inverse (complex-to-real) transform. See the AbstractFFTs.irfft docs for details.

        The backwards transform returned by this function is not normalised. The normalisation factor for a given array can be obtained by calling scale_factor.

        Example

        julia> binv(Transforms.FFT(), 42)
-BFFT
-
-julia> binv(Transforms.BRFFT(9), 42)
-RFFT
        source
        PencilFFTs.Transforms.scale_factor — Function
        scale_factor(transform::AbstractTransform, A, [dims = 1:ndims(A)])

        Get factor required to normalise the given array after a transformation along dimensions dims (all dimensions by default).

        The array A must have the dimensions of the transform input.

        Important: the dimensions dims must be the same that were passed to plan.

        Examples

        julia> C = zeros(ComplexF32, 3, 4, 5);
-
-julia> scale_factor(Transforms.FFT(), C)
-60
-
-julia> scale_factor(Transforms.BFFT(), C)
-60
-
-julia> scale_factor(Transforms.BFFT(), C, 2:3)
-20
-
-julia> R = zeros(Float64, 3, 4, 5);
-
-julia> scale_factor(Transforms.RFFT(), R, 2)
-4
-
-julia> scale_factor(Transforms.RFFT(), R, 2:3)
-20
-
-julia> scale_factor(Transforms.BRFFT(8), C)
-96
-
-julia> scale_factor(Transforms.BRFFT(9), C)
-108

        This will fail because the input of RFFT is real, and R is a complex array:

        julia> scale_factor(Transforms.RFFT(), C, 2:3)
-ERROR: MethodError: no method matching scale_factor(::PencilFFTs.Transforms.RFFT, ::Array{ComplexF32, 3}, ::UnitRange{Int64})
        source
        PencilFFTs.Transforms.eltype_input — Function
        eltype_input(transform::AbstractTransform, real_type<:AbstractFloat)

        Determine input data type for a given transform given the floating point precision of the input data.

        Some transforms, such as R2R and NoTransform, can take both real and complex data. For those kinds of transforms, nothing is returned.

        Example

        julia> eltype_input(Transforms.FFT(), Float32)
-ComplexF32 (alias for Complex{Float32})
-
-julia> eltype_input(Transforms.RFFT(), Float64)
-Float64
-
-julia> eltype_input(Transforms.R2R(FFTW.REDFT01), Float64)  # nothing
-
-julia> eltype_input(Transforms.NoTransform(), Float64)  # nothing
-
        source
        PencilFFTs.Transforms.eltype_output — Function
        eltype_output(transform::AbstractTransform, eltype_input)

        Returns the output data type for a given transform given the input type.

        Throws ArgumentError if the input data type is incompatible with the transform type.

        Example

        julia> eltype_output(Transforms.NoTransform(), Float32)
-Float32
-
-julia> eltype_output(Transforms.RFFT(), Float64)
-ComplexF64 (alias for Complex{Float64})
-
-julia> eltype_output(Transforms.BRFFT(4), ComplexF32)
-Float32
-
-julia> eltype_output(Transforms.FFT(), Float64)
-ERROR: ArgumentError: invalid input data type for PencilFFTs.Transforms.FFT: Float64
        source
        PencilFFTs.Transforms.expand_dims — Function
        expand_dims(transform::AbstractTransform, Val(N))

        Expand a single multidimensional transform into one transform per dimension.

        Example

        # Expand a real-to-complex transform in 3 dimensions.
-julia> expand_dims(Transforms.RFFT(), Val(3))
-(RFFT, FFT, FFT)
-
-julia> expand_dims(Transforms.BRFFT(4), Val(3))
-(BFFT, BFFT, BRFFT{even})
-
-julia> expand_dims(Transforms.NoTransform(), Val(2))
-(NoTransform, NoTransform)
        source
        PencilFFTs.Transforms.is_inplace — Function
        is_inplace(transform::AbstractTransform)         -> Bool
-is_inplace(transforms::Vararg{AbtractTransform}) -> Union{Bool, Nothing}

        Check whether a transform or a list of transforms is performed in-place.

        If the list of transforms has a combination of in-place and out-of-place transforms, nothing is returned.

        Example

        julia> is_inplace(Transforms.RFFT())
-false
-
-julia> is_inplace(Transforms.NoTransform!())
-true
-
-julia> is_inplace(Transforms.FFT!(), Transforms.R2R!(FFTW.REDFT01))
-true
-
-julia> is_inplace(Transforms.FFT(), Transforms.R2R(FFTW.REDFT01))
-false
-
-julia> is_inplace(Transforms.FFT(), Transforms.R2R!(FFTW.REDFT01)) === nothing
-true
-
        source
        PencilFFTs.Transforms.length_output — Function
        length_output(transform::AbstractTransform, length_in::Integer)

        Returns the length of the transform output, given the length of its input.

        The input and output lengths are specified in terms of the respective input and output datatypes. For instance, for real-to-complex transforms, these are respectively the length of input real data and of output complex data.

        source
        diff --git a/previews/PR55/benchmarks/index.html b/previews/PR55/benchmarks/index.html deleted file mode 100644 index 28040f29..00000000 --- a/previews/PR55/benchmarks/index.html +++ /dev/null @@ -1,12 +0,0 @@ - -Benchmarks · PencilFFTs.jl
        Edit on GitHub

        Benchmarks

        The performance of PencilFFTs.jl is comparable to that of other open-source parallel FFT libraries implemented in lower-level languages. Below, we show comparisons with the Fortran implementation of P3DFFT, possibly the most popular of these libraries. The benchmarks were performed on the Jean–Zay cluster of the IDRIS French computing centre (CNRS).

        The figure below shows strong scaling benchmarks of 3D real-to-complex FFTs using 2D ("pencil") decomposition. The benchmarks were run for input arrays of dimensions $N_x × N_y × N_z = 512^3$, $1024^3$ and $2048^3$. Each timing is averaged over 100 repetitions.

        - - Strong scaling of PencilFFTs -

        As seen above, PencilFFTs generally outperforms P3DFFT in its default setting. This is largely explained by the choice of using non-blocking point-to-point MPI communications (via MPI_Isend and MPI_Irecv), while P3DFFT uses collective MPI_Alltoallv calls. This enables PencilFFTs to perform data reordering operations on the partially received data while waiting for the incoming data, leading to better performance. Moreover, in contrast with P3DFFT, the high performance and scalability of PencilFFTs results from a highly generic code, handling decompositions in arbitrary dimensions and a relatively large (and extensible) variety of transformations.

        Note that PencilFFTs can optionally use collective communications (using MPI_Alltoallv) instead of point-to-point communications. For details, see the docs for PencilFFTPlan and for PencilArray transpositions. As seen above, collective communications generally perform worse than point-to-point ones, and runtimes are nearly indistinguishable from those of P3DFFT.

        Benchmark details

        The benchmarks were performed using Julia 1.7-beta3 and Intel MPI 2019. We used PencilFFTs v0.12.5 with FFTW.jl v1.4.3 and MPI.jl v0.19.0. We used the Fortran implementation of P3DFFT, version 2.7.6, which was built with Intel 2019 compilers and linked to FFTW 3.3.8. The cluster where the benchmarks were run has Intel Cascade Lake 6248 processors with 2×20 cores per node.

        The number of MPI processes along each decomposed dimension, $P_1$ and $P_2$, was automatically determined by a call to MPI_Dims_create, which tends to create a balanced decomposition with $P_1 ≈ P_2$. For instance, a total of 1024 processes is divided into $P_1 = P_2 = 32$. Different results may be obtained with other combinations, but this was not benchmarked.

        The source files used to generate this benchmark, as well as the raw benchmark results, are all available in the PencilFFTs repo.

        diff --git a/previews/PR55/generated/ffmpeg.err b/previews/PR55/generated/ffmpeg.err deleted file mode 100644 index d55f9107..00000000 --- a/previews/PR55/generated/ffmpeg.err +++ /dev/null @@ -1,49 +0,0 @@ -ffmpeg version 4.2.7-0ubuntu0.1 Copyright (c) 2000-2022 the FFmpeg developers - built with gcc 9 (Ubuntu 9.4.0-1ubuntu1~20.04.1) - configuration: --prefix=/usr --extra-version=0ubuntu0.1 --toolchain=hardened --libdir=/usr/lib/x86_64-linux-gnu --incdir=/usr/include/x86_64-linux-gnu --arch=amd64 --enable-gpl --disable-stripping --enable-avresample --disable-filter=resample --enable-avisynth --enable-gnutls --enable-ladspa --enable-libaom --enable-libass --enable-libbluray --enable-libbs2b --enable-libcaca --enable-libcdio --enable-libcodec2 --enable-libflite --enable-libfontconfig --enable-libfreetype --enable-libfribidi --enable-libgme --enable-libgsm --enable-libjack --enable-libmp3lame --enable-libmysofa --enable-libopenjpeg --enable-libopenmpt --enable-libopus --enable-libpulse --enable-librsvg --enable-librubberband --enable-libshine --enable-libsnappy --enable-libsoxr --enable-libspeex --enable-libssh --enable-libtheora --enable-libtwolame --enable-libvidstab --enable-libvorbis --enable-libvpx --enable-libwavpack --enable-libwebp --enable-libx265 --enable-libxml2 --enable-libxvid --enable-libzmq --enable-libzvbi --enable-lv2 --enable-omx --enable-openal --enable-opencl --enable-opengl --enable-sdl2 --enable-libdc1394 --enable-libdrm --enable-libiec61883 --enable-nvenc --enable-chromaprint --enable-frei0r --enable-libx264 --enable-shared - libavutil 56. 31.100 / 56. 31.100 - libavcodec 58. 54.100 / 58. 54.100 - libavformat 58. 29.100 / 58. 29.100 - libavdevice 58. 8.100 / 58. 8.100 - libavfilter 7. 57.100 / 7. 57.100 - libavresample 4. 0. 0 / 4. 0. 0 - libswscale 5. 5.100 / 5. 5.100 - libswresample 3. 5.100 / 3. 5.100 - libpostproc 55. 5.100 / 55. 5.100 -Input #0, image2, from '/tmp/jl_JjMFcO/proc1_%04d.png': - Duration: 00:00:04.52, start: 0.000000, bitrate: N/A - Stream #0:0: Video: png, rgb24(pc), 1200x600, 25 fps, 25 tbr, 25 tbn, 25 tbc -Stream mapping: - Stream #0:0 -> #0:0 (png (native) -> h264 (libx264)) -Press [q] to stop, [?] for help -[libx264 @ 0x560096220540] using cpu capabilities: MMX2 SSE2Fast SSSE3 SSE4.2 AVX FMA3 BMI2 AVX2 AVX512 -[libx264 @ 0x560096220540] profile High, level 3.1 -[libx264 @ 0x560096220540] 264 - core 155 r2917 0a84d98 - H.264/MPEG-4 AVC codec - Copyleft 2003-2018 - http://www.videolan.org/x264.html - options: cabac=1 ref=3 deblock=1:0:0 analyse=0x3:0x113 me=hex subme=7 psy=1 psy_rd=1.00:0.00 mixed_ref=1 me_range=16 chroma_me=1 trellis=1 8x8dct=1 cqm=0 deadzone=21,11 fast_pskip=1 chroma_qp_offset=-2 threads=3 lookahead_threads=1 sliced_threads=0 nr=0 decimate=1 interlaced=0 bluray_compat=0 constrained_intra=0 bframes=3 b_pyramid=2 b_adapt=1 b_bias=0 direct=1 weightb=1 open_gop=0 weightp=2 keyint=250 keyint_min=25 scenecut=40 intra_refresh=0 rc_lookahead=40 rc=crf mbtree=1 crf=23.0 qcomp=0.60 qpmin=0 qpmax=69 qpstep=4 ip_ratio=1.40 aq=1:1.00 -Output #0, mp4, to 'vorticity_proc1.mp4': - 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-Navier–Stokes equations · PencilFFTs.jl
        Edit on GitHub

        Navier–Stokes equations

        In this example, we numerically solve the incompressible Navier–Stokes equations

        \[∂_t \bm{v} + (\bm{v} ⋅ \bm{∇}) \bm{v} = -\frac{1}{ρ} \bm{∇} p + ν ∇^2 \bm{v}, -\quad \bm{∇} ⋅ \bm{v} = 0,\]

        where $\bm{v}(\bm{x}, t)$ and $p(\bm{x}, t)$ are respectively the velocity and pressure fields, $ν$ is the fluid kinematic viscosity and $ρ$ is the fluid density.

        We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.

        First steps

        We start by loading the required packages, initialising MPI and setting the simulation parameters.

        using MPI
-using PencilFFTs
-
-MPI.Init()
-comm = MPI.COMM_WORLD
-procid = MPI.Comm_rank(comm) + 1
-
-# Simulation parameters
-Ns = (64, 64, 64)  # = (Nx, Ny, Nz)
-Ls = (2π, 2π, 2π)  # = (Lx, Ly, Lz)
-
-# Collocation points ("global" = over all processes).
-# We include the endpoint (length = N + 1) for convenience.
-xs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls)  # = (x, y, z)
        (0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586)

        Let's check the number of MPI processes over which we're running our simulation:

        MPI.Comm_size(comm)
        2

        We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:

        pen = Pencil(Ns, comm)
        Decomposition of 3D data
-    Data dimensions: (64, 64, 64)
-    Decomposed dimensions: (2, 3)
-    Data permutation: NoPermutation()
-    Array type: Array

        The subdomain associated to the local MPI process can be obtained using range_local:

        range_local(pen)
        (1:64, 1:32, 1:64)

        We now construct a distributed vector field that follows the decomposition configuration we just created:

        v⃗₀ = (
-    PencilArray{Float64}(undef, pen),  # vx
-    PencilArray{Float64}(undef, pen),  # vy
-    PencilArray{Float64}(undef, pen),  # vz
-)
-summary(v⃗₀[1])
        "64×32×64 PencilArray{Float64, 3}(::Pencil{3, 2, NoPermutation, Array})"

        We still need to fill this array with interesting values that represent a physical velocity field.

        Initial condition

        Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:

        \[\begin{aligned} -v_x(x, y, z) &= u₀ \sin(k₀ x) \cos(k₀ y) \cos(k₀ z) \\ -v_y(x, y, z) &= -u₀ \cos(k₀ x) \sin(k₀ y) \cos(k₀ z) \\ -v_z(x, y, z) &= 0 -\end{aligned}\]

        where $u₀$ and $k₀$ are two parameters setting the amplitude and the period of the velocity field.

        To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:

        grid = localgrid(pen, xs_global)
        LocalRectilinearGrid{3} with coordinates:
- (1) 0.0:0.09817477042468103:6.1850105367549055
- (2) 0.0:0.09817477042468103:3.043417883165112
- (3) 0.0:0.09817477042468103:6.1850105367549055

        We can use this to initialise the velocity field:

        u₀ = 1.0
-k₀ = 2π / Ls[1]  # should be integer if L = 2π (to preserve periodicity)
-
-@. v⃗₀[1] =  u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)
-@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)
-@. v⃗₀[3] =  0

        Let's plot a 2D slice of the velocity field managed by the local MPI process:

        using GLMakie
-
-# Compute the norm of a vector field represented by a tuple of arrays.
-function vecnorm(v⃗::NTuple)
-    vnorm = similar(v⃗[1])
-    for n ∈ eachindex(v⃗[1])
-        w = zero(eltype(vnorm))
-        for v ∈ v⃗
-            w += v[n]^2
-        end
-        vnorm[n] = sqrt(w)
-    end
-    vnorm
-end
-
-let fig = Figure(resolution = (700, 600))
-    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
-    vnorm = vecnorm(v⃗₀)
-    ct = contour!(
-        ax, grid.x, grid.y, grid.z, vnorm;
-        alpha = 0.2, levels = 4,
-        colormap = :viridis, colorrange = (0.0, 1.0),
-    )
-    cb = Colorbar(fig[1, 2], ct; label = "Velocity magnitude")
-    fig
-end

        Velocity in Fourier space

        In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series

        \[\bm{v}(\bm{x}, t) = -∑_{\bm{k}} \hat{\bm{v}}_{\bm{k}}(t) \, e^{i \bm{k} ⋅ \bm{x}},\]

        where $\bm{k} = (k_x, k_y, k_z)$ are the discrete wave numbers.

        The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for $k_x$:

        using AbstractFFTs: fftfreq, rfftfreq
-
-ks_global = (
-    rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]),  # kx | real-to-complex
-     fftfreq(Ns[2], 2π * Ns[2] / Ls[2]),  # ky | complex-to-complex
-     fftfreq(Ns[3], 2π * Ns[3] / Ls[3]),  # kz | complex-to-complex
-)
-
-ks_global[1]'
        1×33 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
- 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  27.0  28.0  29.0  30.0  31.0  32.0
        ks_global[2]'
        1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
- 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0
        ks_global[3]'
        1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
- 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0

        To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:

        plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())
        Transforms: (RFFT, FFT, FFT)
-Input type: Float64
-Global dimensions: (64, 64, 64) -> (33, 64, 64)
-MPI topology: 2D decomposition (2×1 processes)

        See PencilFFTPlan for details on creating plans and on optional keyword arguments.

        We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$:

        v̂s = plan .* v⃗₀
-summary(v̂s[1])
        "16×64×64 PencilArray{ComplexF64, 3}(::Pencil{3, 2, Permutation{(3, 2, 1), 3}, Array})"

        Note that, in Fourier space, the domain decomposition is performed along the directions $x$ and $y$:

        pencil(v̂s[1])
        Decomposition of 3D data
-    Data dimensions: (33, 64, 64)
-    Decomposed dimensions: (1, 2)
-    Data permutation: Permutation(3, 2, 1)
-    Array type: Array

        This is because the 3D FFTs are performed one dimension at a time, with the $x$ direction first and the $z$ direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.

        To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers $\bm{k}$ associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:

        grid_fourier = localgrid(v̂s[1], ks_global)
        LocalRectilinearGrid{3} with Permutation(3, 2, 1) and coordinates:
- (1) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0]
- (2) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]
- (3) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]

        As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, $\bm{ω} = \bm{∇} × \bm{v}$. In Fourier space, this becomes $\hat{\bm{ω}} = i \bm{k} × \hat{\bm{v}}$.

        using StaticArrays: SVector
-using LinearAlgebra: ×
-
-function curl_fourier!(
-        ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,
-    ) where {N}
-    @inbounds for I ∈ eachindex(grid_fourier)
-        # We use StaticArrays for the cross product between small vectors.
-        ik⃗ = im * SVector(grid_fourier[I])
-        v⃗ = SVector(getindex.(v̂s, Ref(I)))  # = (v̂s[1][I], v̂s[2][I], ...)
-        ω⃗ = ik⃗ × v⃗
-        for n ∈ eachindex(ω⃗)
-            ω̂s[n][I] = ω⃗[n]
-        end
-    end
-    ω̂s
-end
-
-ω̂s = similar.(v̂s)
-curl_fourier!(ω̂s, v̂s, grid_fourier);

        We finally transform back to physical space and plot the result:

        ωs = plan .\ ω̂s
-
-let fig = Figure(resolution = (700, 600))
-    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
-    ω_norm = vecnorm(ωs)
-    ct = contour!(
-        ax, grid.x, grid.y, grid.z, ω_norm;
-        alpha = 0.1, levels = 0.8:0.2:2.0,
-        colormap = :viridis, colorrange = (0.8, 2.0),
-    )
-    cb = Colorbar(fig[1, 2], ct; label = "Vorticity magnitude")
-    fig
-end

        Computing the non-linear term

        One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as

        \[∂_t \hat{\bm{v}}_{\bm{k}} = -- \mathcal{P}_{\bm{k}} \! \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right] -- ν |\bm{k}|^2 \hat{\bm{v}}_{\bm{k}} -\quad \text{ with } \quad -\mathcal{P}_{\bm{k}}(\hat{\bm{F}}_{\bm{k}}) = \left( I - \frac{\bm{k} ⊗ -\bm{k}}{|\bm{k}|^2} \right) \hat{\bm{F}}_{\bm{k}},\]

        where $\mathcal{P}_{\bm{k}}$ is a projection operator allowing to preserve the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.

        Now that we have the wave numbers $\bm{k}$, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$ of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, $\hat{\bm{F}}_{\bm{k}} = \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right]_{\bm{k}}$. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.

        Below we implement a function that computes the non-linear term in Fourier space based on its convective form $(\bm{v} ⋅ \bm{∇}) \bm{v} = \bm{∇} ⋅ (\bm{v} ⊗ \bm{v})$. Note that this equivalence uses the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$.

        using LinearAlgebra: mul!, ldiv!  # for applying FFT plans in-place
-
-# Compute non-linear term in Fourier space from velocity field in physical
-# space. Optional keyword arguments may be passed to avoid allocations.
-function ns_nonlinear!(
-        F̂s, vs, plan, grid_fourier;
-        vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),
-    )
-    # Compute F_i = ∂_j (v_i v_j) for each i.
-    # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)
-    w, ŵ = vbuf, v̂buf
-    @inbounds for (i, F̂i) ∈ enumerate(F̂s)
-        F̂i .= 0
-        vi = vs[i]
-        for (j, vj) ∈ enumerate(vs)
-            w .= vi .* vj     # w = v_i * v_j in physical space
-            mul!(ŵ, plan, w)  # same in Fourier space
-            # Add derivative in Fourier space
-            for I ∈ eachindex(grid_fourier)
-                k⃗ = grid_fourier[I]  # = (kx, ky, kz)
-                kj = k⃗[j]
-                F̂i[I] += im * kj * ŵ[I]
-            end
-        end
-    end
-    F̂s
-end
        ns_nonlinear! (generic function with 1 method)

        As an example, let's use this function on our initial velocity field:

        F̂s = similar.(v̂s)
-ns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);

        Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.

        function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)
-    ks_max = maximum.(abs, ks_global)  # maximum stored wave numbers (kx_max, ky_max, kz_max)
-    ks_lim = (2 / 3) .* ks_max
-    @inbounds for I ∈ eachindex(grid_fourier)
-        k⃗ = grid_fourier[I]
-        if any(abs.(k⃗) .> ks_lim)
-            for ŵ ∈ ŵs
-                ŵ[I] = 0
-            end
-        end
-    end
-    ŵs
-end
-
-# We can apply this on the previously computed non-linear term:
-dealias_twothirds!(F̂s, grid_fourier, ks_global);

        Finally, we implement the projection associated to the incompressibility condition:

        function project_divergence_free!(ûs, grid_fourier)
-    @inbounds for I ∈ eachindex(grid_fourier)
-        k⃗ = grid_fourier[I]
-        k² = sum(abs2, k⃗)
-        iszero(k²) && continue  # avoid division by zero
-        û = getindex.(ûs, Ref(I))  # (ûs[1][I], ûs[2][I], ...)
-        for i ∈ eachindex(û)
-            ŵ = û[i]
-            for j ∈ eachindex(û)
-                ŵ -= k⃗[i] * k⃗[j] * û[j] / k²
-            end
-            ûs[i][I] = ŵ
-        end
-    end
-    ûs
-end
        project_divergence_free! (generic function with 1 method)

        We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:

        v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)
-v̂s_proj .≈ v̂s  # the last one may be false because v_z = 0 initially
        (true, true, false)

        Putting it all together

        To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:

        function ns_rhs!(
-        dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,
-    ) where {N}
-    # 1. Compute non-linear term and dealias it
-    (; plan, cache, ks_global, grid_fourier) = p
-    F̂s = cache.F̂s
-    ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])
-    dealias_twothirds!(F̂s, grid_fourier, ks_global)
-
-    # 2. Project onto divergence-free space
-    project_divergence_free!(F̂s, grid_fourier)
-
-    # 3. Transform velocity to Fourier space
-    v̂s = cache.v̂s
-    map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)
-
-    # 4. Add viscous term (and multiply projected non-linear term by -1)
-    ν = p.ν
-    for n ∈ eachindex(v̂s)
-        v̂ = v̂s[n]
-        F̂ = F̂s[n]
-        @inbounds for I ∈ eachindex(grid_fourier)
-            k⃗ = grid_fourier[I]  # = (kx, ky, kz)
-            k² = sum(abs2, k⃗)
-            F̂[I] = -F̂[I] - ν * k² * v̂[I]
-        end
-    end
-
-    # 5. Transform RHS back to physical space
-    map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)
-
-    nothing
-end
        ns_rhs! (generic function with 1 method)

        For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.

        using OrdinaryDiffEq
-using RecursiveArrayTools: ArrayPartition
-
-ns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)
-vs_init_ode = ArrayPartition(v⃗₀)
-summary(vs_init_ode)
        "RecursiveArrayTools.ArrayPartition{Float64, Tuple{PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}}} with arrays:"

        We now define solver parameters and temporary variables, and initialise the problem:

        params = (;
-    ν = 5e-3,  # kinematic viscosity
-    plan, grid_fourier, ks_global,
-    cache = (
-        v̂s = similar.(v̂s),
-        F̂s = similar.(v̂s),
-    )
-)
-
-tspan = (0.0, 10.0)
-prob = ODEProblem(ns_rhs!, vs_init_ode, tspan, params)
-integrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);

        We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum $E(k)$, to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity $ν$ must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large $k$), while high enough to allow the small-scale motions to be correctly resolved.

        function energy_spectrum!(Ek, ks, v̂s, grid_fourier)
-    Nk = length(Ek)
-    @assert Nk == length(ks)
-    Ek .= 0
-    for I ∈ eachindex(grid_fourier)
-        k⃗ = grid_fourier[I]  # = (kx, ky, kz)
-        knorm = sqrt(sum(abs2, k⃗))
-        i = searchsortedfirst(ks, knorm)
-        i > Nk && continue
-        v⃗ = getindex.(v̂s, Ref(I))  # = (v̂s[1][I], v̂s[2][I], ...)
-        factor = k⃗[1] == 0 ? 1 : 2  # account for Hermitian symmetry and r2c transform
-        Ek[i] += factor * sum(abs2, v⃗) / 2
-    end
-    MPI.Allreduce!(Ek, +, get_comm(v̂s[1]))  # sum across all processes
-    Ek
-end
-
-ks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])
-Ek = similar(ks)
-v̂s = plan .* integrator.u.x
-energy_spectrum!(Ek, ks, v̂s, grid_fourier)
-Ek ./= scale_factor(plan)^2  # rescale energy
-
-curl_fourier!(ω̂s, v̂s, grid_fourier)
-ldiv!.(ωs, plan, ω̂s)
-ω⃗_plot = Observable(ωs)
-k_plot = @view ks[2:end]
-E_plot = Observable(@view Ek[2:end])
-t_plot = Observable(integrator.t)
-
-fig = let
-    fig = Figure(resolution = (1200, 600))
-    ax = Axis3(
-        fig[1, 1][1, 1]; title = @lift("t = $(round($t_plot, digits = 3))"),
-        aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z",
-    )
-    ω_mag = @lift vecnorm($ω⃗_plot)
-    ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)
-    ct = contour!(
-        ax, grid.x, grid.y, grid.z, ω_mag_norm;
-        alpha = 0.3, levels = 3,
-        colormap = :viridis, colorrange = (0.0, 1.0),
-    )
-    cb = Colorbar(fig[1, 1][1, 2], ct; label = "Normalised vorticity magnitude")
-    ax_sp = Axis(
-        fig[1, 2];
-        xlabel = "k", ylabel = "E(k)", xscale = log2, yscale = log10,
-        title = "Kinetic energy spectrum",
-    )
-    ylims!(ax_sp, 1e-8, 1e0)
-    scatterlines!(ax_sp, k_plot, E_plot)
-    ks_slope = exp.(range(log(2.5), log(25.0), length = 3))
-    E_fivethirds = @. 0.3 * ks_slope^(-5/3)
-    @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)
-    text!(ax_sp, L"k^{-5/3}"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))
-    fig
-end
-
-record(fig, "vorticity_proc$procid.mp4"; framerate = 10) do io
-    while integrator.t < 20
-        dt = 0.001
-        step!(integrator, dt)
-        t_plot[] = integrator.t
-        mul!.(v̂s, plan, integrator.u.x)  # current velocity in Fourier space
-        curl_fourier!(ω̂s, v̂s, grid_fourier)
-        ldiv!.(ω⃗_plot[], plan, ω̂s)
-        ω⃗_plot[] = ω⃗_plot[]  # to force updating the plot
-        energy_spectrum!(Ek, ks, v̂s, grid_fourier)
-        Ek ./= scale_factor(plan)^2  # rescale energy
-        E_plot[] = E_plot[]
-        recordframe!(io)
-    end
-end;
        - -

        This page was generated using Literate.jl.

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        PencilFFTs

        Fast Fourier transforms of MPI-distributed Julia arrays.

        Introduction

        This package provides multidimensional FFTs and related transforms on MPI-distributed Julia arrays via the PencilArrays package.

        The name of this package originates from the decomposition of 3D domains along two out of three dimensions, sometimes called pencil decomposition. This is illustrated by the figure below,[1] where each coloured block is managed by a different MPI process. Typically, one wants to compute FFTs on a scalar or vector field along the three spatial dimensions. In the case of a pencil decomposition, 3D FFTs are performed one dimension at a time, along the non-decomposed direction. Transforms must then be interleaved with global data transpositions to switch between pencil configurations. In high-performance computing environments, such data transpositions are generally the most expensive part of a parallel FFT computation, due to the large cost of communications between computing nodes.

        - Pencil decomposition of 3D domains -

        More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension $N$. The decompositions can be performed along an arbitrary number $M < N$ of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.

        The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.

        Features

        • distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;

        • FFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;

        • in-place and out-of-place transforms;

        • high scalability up to (at least) tens of thousands of MPI processes.

        Installation

        PencilFFTs can be installed using the Julia package manager:

        julia> ] add PencilFFTs

        Similar projects

        • FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.

        • PFFT is a very general parallel FFT library written in C.

        • P3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.

        • 2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.

        • 1Figure adapted from this PhD thesis.
        • 2For the pencil decomposition represented in the figure, $N = 3$ and $M = 2$.
        diff --git a/previews/PR55/search/index.html b/previews/PR55/search/index.html deleted file mode 100644 index 4d3c7abe..00000000 --- a/previews/PR55/search/index.html +++ /dev/null @@ -1,2 +0,0 @@ - -Search · PencilFFTs.jl

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-Navier–Stokes equations · PencilFFTs.jl
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          Navier–Stokes equations

          In this example, we numerically solve the incompressible Navier–Stokes equations

          \[∂_t \bm{v} + (\bm{v} ⋅ \bm{∇}) \bm{v} = -\frac{1}{ρ} \bm{∇} p + ν ∇^2 \bm{v}, -\quad \bm{∇} ⋅ \bm{v} = 0,\]

          where $\bm{v}(\bm{x}, t)$ and $p(\bm{x}, t)$ are respectively the velocity and pressure fields, $ν$ is the fluid kinematic viscosity and $ρ$ is the fluid density.

          We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.

          First steps

          We start by loading the required packages, initialising MPI and setting the simulation parameters.

          using MPI
-using PencilFFTs
-
-MPI.Init()
-comm = MPI.COMM_WORLD
-procid = MPI.Comm_rank(comm) + 1
-
-# Simulation parameters
-Ns = (64, 64, 64)  # = (Nx, Ny, Nz)
-Ls = (2π, 2π, 2π)  # = (Lx, Ly, Lz)
-
-# Collocation points ("global" = over all processes).
-# We include the endpoint (length = N + 1) for convenience.
-xs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls)  # = (x, y, z)
          (0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586)

          Let's check the number of MPI processes over which we're running our simulation:

          MPI.Comm_size(comm)
          2

          We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:

          pen = Pencil(Ns, comm)
          Decomposition of 3D data
-    Data dimensions: (64, 64, 64)
-    Decomposed dimensions: (2, 3)
-    Data permutation: NoPermutation()
-    Array type: Array

          The subdomain associated to the local MPI process can be obtained using range_local:

          range_local(pen)
          (1:64, 1:32, 1:64)

          We now construct a distributed vector field that follows the decomposition configuration we just created:

          v⃗₀ = (
-    PencilArray{Float64}(undef, pen),  # vx
-    PencilArray{Float64}(undef, pen),  # vy
-    PencilArray{Float64}(undef, pen),  # vz
-)
-summary(v⃗₀[1])
          "64×32×64 PencilArray{Float64, 3}(::Pencil{3, 2, NoPermutation, Array})"

          We still need to fill this array with interesting values that represent a physical velocity field.

          Initial condition

          Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:

          \[\begin{aligned} -v_x(x, y, z) &= u₀ \sin(k₀ x) \cos(k₀ y) \cos(k₀ z) \\ -v_y(x, y, z) &= -u₀ \cos(k₀ x) \sin(k₀ y) \cos(k₀ z) \\ -v_z(x, y, z) &= 0 -\end{aligned}\]

          where $u₀$ and $k₀$ are two parameters setting the amplitude and the period of the velocity field.

          To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:

          grid = localgrid(pen, xs_global)
          LocalRectilinearGrid{3} with coordinates:
- (1) 0.0:0.09817477042468103:6.1850105367549055
- (2) 0.0:0.09817477042468103:3.043417883165112
- (3) 0.0:0.09817477042468103:6.1850105367549055

          We can use this to initialise the velocity field:

          u₀ = 1.0
-k₀ = 2π / Ls[1]  # should be integer if L = 2π (to preserve periodicity)
-
-@. v⃗₀[1] =  u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)
-@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)
-@. v⃗₀[3] =  0

          Let's plot a 2D slice of the velocity field managed by the local MPI process:

          using GLMakie
-
-# Compute the norm of a vector field represented by a tuple of arrays.
-function vecnorm(v⃗::NTuple)
-    vnorm = similar(v⃗[1])
-    for n ∈ eachindex(v⃗[1])
-        w = zero(eltype(vnorm))
-        for v ∈ v⃗
-            w += v[n]^2
-        end
-        vnorm[n] = sqrt(w)
-    end
-    vnorm
-end
-
-let fig = Figure(resolution = (700, 600))
-    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
-    vnorm = vecnorm(v⃗₀)
-    ct = contour!(
-        ax, grid.x, grid.y, grid.z, vnorm;
-        alpha = 0.2, levels = 4,
-        colormap = :viridis, colorrange = (0.0, 1.0),
-    )
-    cb = Colorbar(fig[1, 2], ct; label = "Velocity magnitude")
-    fig
-end

          Velocity in Fourier space

          In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series

          \[\bm{v}(\bm{x}, t) = -∑_{\bm{k}} \hat{\bm{v}}_{\bm{k}}(t) \, e^{i \bm{k} ⋅ \bm{x}},\]

          where $\bm{k} = (k_x, k_y, k_z)$ are the discrete wave numbers.

          The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for $k_x$:

          using AbstractFFTs: fftfreq, rfftfreq
-
-ks_global = (
-    rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]),  # kx | real-to-complex
-     fftfreq(Ns[2], 2π * Ns[2] / Ls[2]),  # ky | complex-to-complex
-     fftfreq(Ns[3], 2π * Ns[3] / Ls[3]),  # kz | complex-to-complex
-)
-
-ks_global[1]'
          1×33 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
- 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  27.0  28.0  29.0  30.0  31.0  32.0
          ks_global[2]'
          1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
- 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0
          ks_global[3]'
          1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
- 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0

          To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:

          plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())
          Transforms: (RFFT, FFT, FFT)
-Input type: Float64
-Global dimensions: (64, 64, 64) -> (33, 64, 64)
-MPI topology: 2D decomposition (2×1 processes)

          See PencilFFTPlan for details on creating plans and on optional keyword arguments.

          We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$:

          v̂s = plan .* v⃗₀
-summary(v̂s[1])
          "16×64×64 PencilArray{ComplexF64, 3}(::Pencil{3, 2, Permutation{(3, 2, 1), 3}, Array})"

          Note that, in Fourier space, the domain decomposition is performed along the directions $x$ and $y$:

          pencil(v̂s[1])
          Decomposition of 3D data
-    Data dimensions: (33, 64, 64)
-    Decomposed dimensions: (1, 2)
-    Data permutation: Permutation(3, 2, 1)
-    Array type: Array

          This is because the 3D FFTs are performed one dimension at a time, with the $x$ direction first and the $z$ direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.

          To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers $\bm{k}$ associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:

          grid_fourier = localgrid(v̂s[1], ks_global)
          LocalRectilinearGrid{3} with Permutation(3, 2, 1) and coordinates:
- (1) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0]
- (2) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]
- (3) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]

          As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, $\bm{ω} = \bm{∇} × \bm{v}$. In Fourier space, this becomes $\hat{\bm{ω}} = i \bm{k} × \hat{\bm{v}}$.

          using StaticArrays: SVector
-using LinearAlgebra: ×
-
-function curl_fourier!(
-        ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,
-    ) where {N}
-    @inbounds for I ∈ eachindex(grid_fourier)
-        # We use StaticArrays for the cross product between small vectors.
-        ik⃗ = im * SVector(grid_fourier[I])
-        v⃗ = SVector(getindex.(v̂s, Ref(I)))  # = (v̂s[1][I], v̂s[2][I], ...)
-        ω⃗ = ik⃗ × v⃗
-        for n ∈ eachindex(ω⃗)
-            ω̂s[n][I] = ω⃗[n]
-        end
-    end
-    ω̂s
-end
-
-ω̂s = similar.(v̂s)
-curl_fourier!(ω̂s, v̂s, grid_fourier);

          We finally transform back to physical space and plot the result:

          ωs = plan .\ ω̂s
-
-let fig = Figure(resolution = (700, 600))
-    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
-    ω_norm = vecnorm(ωs)
-    ct = contour!(
-        ax, grid.x, grid.y, grid.z, ω_norm;
-        alpha = 0.1, levels = 0.8:0.2:2.0,
-        colormap = :viridis, colorrange = (0.8, 2.0),
-    )
-    cb = Colorbar(fig[1, 2], ct; label = "Vorticity magnitude")
-    fig
-end

          Computing the non-linear term

          One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as

          \[∂_t \hat{\bm{v}}_{\bm{k}} = -- \mathcal{P}_{\bm{k}} \! \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right] -- ν |\bm{k}|^2 \hat{\bm{v}}_{\bm{k}} -\quad \text{ with } \quad -\mathcal{P}_{\bm{k}}(\hat{\bm{F}}_{\bm{k}}) = \left( I - \frac{\bm{k} ⊗ -\bm{k}}{|\bm{k}|^2} \right) \hat{\bm{F}}_{\bm{k}},\]

          where $\mathcal{P}_{\bm{k}}$ is a projection operator allowing to preserve the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.

          Now that we have the wave numbers $\bm{k}$, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$ of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, $\hat{\bm{F}}_{\bm{k}} = \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right]_{\bm{k}}$. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.

          Below we implement a function that computes the non-linear term in Fourier space based on its convective form $(\bm{v} ⋅ \bm{∇}) \bm{v} = \bm{∇} ⋅ (\bm{v} ⊗ \bm{v})$. Note that this equivalence uses the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$.

          using LinearAlgebra: mul!, ldiv!  # for applying FFT plans in-place
-
-# Compute non-linear term in Fourier space from velocity field in physical
-# space. Optional keyword arguments may be passed to avoid allocations.
-function ns_nonlinear!(
-        F̂s, vs, plan, grid_fourier;
-        vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),
-    )
-    # Compute F_i = ∂_j (v_i v_j) for each i.
-    # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)
-    w, ŵ = vbuf, v̂buf
-    @inbounds for (i, F̂i) ∈ enumerate(F̂s)
-        F̂i .= 0
-        vi = vs[i]
-        for (j, vj) ∈ enumerate(vs)
-            w .= vi .* vj     # w = v_i * v_j in physical space
-            mul!(ŵ, plan, w)  # same in Fourier space
-            # Add derivative in Fourier space
-            for I ∈ eachindex(grid_fourier)
-                k⃗ = grid_fourier[I]  # = (kx, ky, kz)
-                kj = k⃗[j]
-                F̂i[I] += im * kj * ŵ[I]
-            end
-        end
-    end
-    F̂s
-end
          ns_nonlinear! (generic function with 1 method)

          As an example, let's use this function on our initial velocity field:

          F̂s = similar.(v̂s)
-ns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);

          Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.

          function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)
-    ks_max = maximum.(abs, ks_global)  # maximum stored wave numbers (kx_max, ky_max, kz_max)
-    ks_lim = (2 / 3) .* ks_max
-    @inbounds for I ∈ eachindex(grid_fourier)
-        k⃗ = grid_fourier[I]
-        if any(abs.(k⃗) .> ks_lim)
-            for ŵ ∈ ŵs
-                ŵ[I] = 0
-            end
-        end
-    end
-    ŵs
-end
-
-# We can apply this on the previously computed non-linear term:
-dealias_twothirds!(F̂s, grid_fourier, ks_global);

          Finally, we implement the projection associated to the incompressibility condition:

          function project_divergence_free!(ûs, grid_fourier)
-    @inbounds for I ∈ eachindex(grid_fourier)
-        k⃗ = grid_fourier[I]
-        k² = sum(abs2, k⃗)
-        iszero(k²) && continue  # avoid division by zero
-        û = getindex.(ûs, Ref(I))  # (ûs[1][I], ûs[2][I], ...)
-        for i ∈ eachindex(û)
-            ŵ = û[i]
-            for j ∈ eachindex(û)
-                ŵ -= k⃗[i] * k⃗[j] * û[j] / k²
-            end
-            ûs[i][I] = ŵ
-        end
-    end
-    ûs
-end
          project_divergence_free! (generic function with 1 method)

          We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:

          v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)
-v̂s_proj .≈ v̂s  # the last one may be false because v_z = 0 initially
          (true, true, false)

          Putting it all together

          To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:

          function ns_rhs!(
-        dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,
-    ) where {N}
-    # 1. Compute non-linear term and dealias it
-    (; plan, cache, ks_global, grid_fourier) = p
-    F̂s = cache.F̂s
-    ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])
-    dealias_twothirds!(F̂s, grid_fourier, ks_global)
-
-    # 2. Project onto divergence-free space
-    project_divergence_free!(F̂s, grid_fourier)
-
-    # 3. Transform velocity to Fourier space
-    v̂s = cache.v̂s
-    map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)
-
-    # 4. Add viscous term (and multiply projected non-linear term by -1)
-    ν = p.ν
-    for n ∈ eachindex(v̂s)
-        v̂ = v̂s[n]
-        F̂ = F̂s[n]
-        @inbounds for I ∈ eachindex(grid_fourier)
-            k⃗ = grid_fourier[I]  # = (kx, ky, kz)
-            k² = sum(abs2, k⃗)
-            F̂[I] = -F̂[I] - ν * k² * v̂[I]
-        end
-    end
-
-    # 5. Transform RHS back to physical space
-    map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)
-
-    nothing
-end
          ns_rhs! (generic function with 1 method)

          For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.

          using OrdinaryDiffEq
-using RecursiveArrayTools: ArrayPartition
-
-ns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)
-vs_init_ode = ArrayPartition(v⃗₀)
-summary(vs_init_ode)
          "RecursiveArrayTools.ArrayPartition{Float64, Tuple{PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}}} with arrays:"

          We now define solver parameters and temporary variables, and initialise the problem:

          params = (;
-    ν = 5e-3,  # kinematic viscosity
-    plan, grid_fourier, ks_global,
-    cache = (
-        v̂s = similar.(v̂s),
-        F̂s = similar.(v̂s),
-    )
-)
-
-tspan = (0.0, 10.0)
-prob = ODEProblem(ns_rhs!, vs_init_ode, tspan, params)
-integrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);

          We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum $E(k)$, to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity $ν$ must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large $k$), while high enough to allow the small-scale motions to be correctly resolved.

          function energy_spectrum!(Ek, ks, v̂s, grid_fourier)
-    Nk = length(Ek)
-    @assert Nk == length(ks)
-    Ek .= 0
-    for I ∈ eachindex(grid_fourier)
-        k⃗ = grid_fourier[I]  # = (kx, ky, kz)
-        knorm = sqrt(sum(abs2, k⃗))
-        i = searchsortedfirst(ks, knorm)
-        i > Nk && continue
-        v⃗ = getindex.(v̂s, Ref(I))  # = (v̂s[1][I], v̂s[2][I], ...)
-        factor = k⃗[1] == 0 ? 1 : 2  # account for Hermitian symmetry and r2c transform
-        Ek[i] += factor * sum(abs2, v⃗) / 2
-    end
-    MPI.Allreduce!(Ek, +, get_comm(v̂s[1]))  # sum across all processes
-    Ek
-end
-
-ks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])
-Ek = similar(ks)
-v̂s = plan .* integrator.u.x
-energy_spectrum!(Ek, ks, v̂s, grid_fourier)
-Ek ./= scale_factor(plan)^2  # rescale energy
-
-curl_fourier!(ω̂s, v̂s, grid_fourier)
-ldiv!.(ωs, plan, ω̂s)
-ω⃗_plot = Observable(ωs)
-k_plot = @view ks[2:end]
-E_plot = Observable(@view Ek[2:end])
-t_plot = Observable(integrator.t)
-
-fig = let
-    fig = Figure(resolution = (1200, 600))
-    ax = Axis3(
-        fig[1, 1][1, 1]; title = @lift("t = $(round($t_plot, digits = 3))"),
-        aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z",
-    )
-    ω_mag = @lift vecnorm($ω⃗_plot)
-    ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)
-    ct = contour!(
-        ax, grid.x, grid.y, grid.z, ω_mag_norm;
-        alpha = 0.3, levels = 3,
-        colormap = :viridis, colorrange = (0.0, 1.0),
-    )
-    cb = Colorbar(fig[1, 1][1, 2], ct; label = "Normalised vorticity magnitude")
-    ax_sp = Axis(
-        fig[1, 2];
-        xlabel = "k", ylabel = "E(k)", xscale = log2, yscale = log10,
-        title = "Kinetic energy spectrum",
-    )
-    ylims!(ax_sp, 1e-8, 1e0)
-    scatterlines!(ax_sp, k_plot, E_plot)
-    ks_slope = exp.(range(log(2.5), log(25.0), length = 3))
-    E_fivethirds = @. 0.3 * ks_slope^(-5/3)
-    @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)
-    text!(ax_sp, L"k^{-5/3}"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))
-    fig
-end
-
-record(fig, "vorticity_proc$procid.mp4"; framerate = 10) do io
-    while integrator.t < 20
-        dt = 0.001
-        step!(integrator, dt)
-        t_plot[] = integrator.t
-        mul!.(v̂s, plan, integrator.u.x)  # current velocity in Fourier space
-        curl_fourier!(ω̂s, v̂s, grid_fourier)
-        ldiv!.(ω⃗_plot[], plan, ω̂s)
-        ω⃗_plot[] = ω⃗_plot[]  # to force updating the plot
-        energy_spectrum!(Ek, ks, v̂s, grid_fourier)
-        Ek ./= scale_factor(plan)^2  # rescale energy
-        E_plot[] = E_plot[]
-        recordframe!(io)
-    end
-end;
          - -

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          Edit on GitHub

          PencilFFTs

          Fast Fourier transforms of MPI-distributed Julia arrays.

          Introduction

          This package provides multidimensional FFTs and related transforms on MPI-distributed Julia arrays via the PencilArrays package.

          The name of this package originates from the decomposition of 3D domains along two out of three dimensions, sometimes called pencil decomposition. This is illustrated by the figure below,[1] where each coloured block is managed by a different MPI process. Typically, one wants to compute FFTs on a scalar or vector field along the three spatial dimensions. In the case of a pencil decomposition, 3D FFTs are performed one dimension at a time, along the non-decomposed direction. Transforms must then be interleaved with global data transpositions to switch between pencil configurations. In high-performance computing environments, such data transpositions are generally the most expensive part of a parallel FFT computation, due to the large cost of communications between computing nodes.

          - Pencil decomposition of 3D domains -

          More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension $N$. The decompositions can be performed along an arbitrary number $M < N$ of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.

          The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.

          Features

          • distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;

          • FFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;

          • in-place and out-of-place transforms;

          • high scalability up to (at least) tens of thousands of MPI processes.

          Installation

          PencilFFTs can be installed using the Julia package manager:

          julia> ] add PencilFFTs

          Similar projects

          • FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.

          • PFFT is a very general parallel FFT library written in C.

          • P3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.

          • 2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.

          • 1Figure adapted from this PhD thesis.
          • 2For the pencil decomposition represented in the figure, $N = 3$ and $M = 2$.
          • Edit on GitHub

          PencilFFTs

          Fast Fourier transforms of MPI-distributed Julia arrays.

          Introduction

          This package provides multidimensional FFTs and related transforms on MPI-distributed Julia arrays via the PencilArrays package.

          The name of this package originates from the decomposition of 3D domains along two out of three dimensions, sometimes called pencil decomposition. This is illustrated by the figure below,[1] where each coloured block is managed by a different MPI process. Typically, one wants to compute FFTs on a scalar or vector field along the three spatial dimensions. In the case of a pencil decomposition, 3D FFTs are performed one dimension at a time, along the non-decomposed direction. Transforms must then be interleaved with global data transpositions to switch between pencil configurations. In high-performance computing environments, such data transpositions are generally the most expensive part of a parallel FFT computation, due to the large cost of communications between computing nodes.

          - Pencil decomposition of 3D domains -

          More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension $N$. The decompositions can be performed along an arbitrary number $M < N$ of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.

          The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.

          Features

          • distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;

          • FFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;

          • in-place and out-of-place transforms;

          • high scalability up to (at least) tens of thousands of MPI processes.

          Installation

          PencilFFTs can be installed using the Julia package manager:

          julia> ] add PencilFFTs

          Similar projects

          • FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.

          • PFFT is a very general parallel FFT library written in C.

          • P3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.

          • 2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.

          • 1Figure adapted from this PhD thesis.
          • 2For the pencil decomposition represented in the figure, $N = 3$ and $M = 2$.
          Edit on GitHub

          PencilFFTs

          Fast Fourier transforms of MPI-distributed Julia arrays.

          Introduction

          This package provides multidimensional FFTs and related transforms on MPI-distributed Julia arrays via the PencilArrays package.

          The name of this package originates from the decomposition of 3D domains along two out of three dimensions, sometimes called pencil decomposition. This is illustrated by the figure below,[1] where each coloured block is managed by a different MPI process. Typically, one wants to compute FFTs on a scalar or vector field along the three spatial dimensions. In the case of a pencil decomposition, 3D FFTs are performed one dimension at a time, along the non-decomposed direction. Transforms must then be interleaved with global data transpositions to switch between pencil configurations. In high-performance computing environments, such data transpositions are generally the most expensive part of a parallel FFT computation, due to the large cost of communications between computing nodes.

          +Home · PencilFFTs.jl
          GitHub

          PencilFFTs

          Fast Fourier transforms of MPI-distributed Julia arrays.

          Introduction

          This package provides multidimensional FFTs and related transforms on MPI-distributed Julia arrays via the PencilArrays package.

          The name of this package originates from the decomposition of 3D domains along two out of three dimensions, sometimes called pencil decomposition. This is illustrated by the figure below,[1] where each coloured block is managed by a different MPI process. Typically, one wants to compute FFTs on a scalar or vector field along the three spatial dimensions. In the case of a pencil decomposition, 3D FFTs are performed one dimension at a time, along the non-decomposed direction. Transforms must then be interleaved with global data transpositions to switch between pencil configurations. In high-performance computing environments, such data transpositions are generally the most expensive part of a parallel FFT computation, due to the large cost of communications between computing nodes.

          Pencil decomposition of 3D domains -

          More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension $N$. The decompositions can be performed along an arbitrary number $M < N$ of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.

          The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.

          Features

          • distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;

          • FFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;

          • in-place and out-of-place transforms;

          • high scalability up to (at least) tens of thousands of MPI processes.

          Installation

          PencilFFTs can be installed using the Julia package manager:

          julia> ] add PencilFFTs

          Similar projects

          • FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.

          • PFFT is a very general parallel FFT library written in C.

          • P3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.

          • 2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.

          • 1Figure adapted from this PhD thesis.
          • 2For the pencil decomposition represented in the figure, $N = 3$ and $M = 2$.
          +

          More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension $N$. The decompositions can be performed along an arbitrary number $M < N$ of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.

          The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.

          Features

          • distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;

          • FFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;

          • in-place and out-of-place transforms;

          • high scalability up to (at least) tens of thousands of MPI processes.

          Installation

          PencilFFTs can be installed using the Julia package manager:

          julia> ] add PencilFFTs

          Similar projects

          • FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.

          • PFFT is a very general parallel FFT library written in C.

          • P3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.

          • 2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.

          • 1Figure adapted from this PhD thesis.
          • 2For the pencil decomposition represented in the figure, $N = 3$ and $M = 2$.

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          R0DDeZ3ExECdKh4v@yG8h0qnz?1N$1dTb zgneU!Ef?i#?b=LFAbxX_GP0&;An`*68D;}8-0+8A(AeK6%<+FE1LCI2irhVdc_KYV zY`yn*?Lkt^3W{gPAu1`3go5TOKrX@x-a zz{SL;HBt+HFVt2R0wIJn36dCtFnQnN*vki$c)MAylQlx{vY%uBCcsZX=bIhV+8H-@ zD;S%YR`VHG1MwCkdXfrQ0VC!B%P|xKCaJluyE1cWFitpABk}jS4GSsE#u>m&>mS>H zL~LjqdckINLCQ`EBNu7dkHly`9sde**_tkZB_xa zXEGd%@A`_5yLm&Ur^(L)>gA zjU^Pix#gf`8{;1KL)`e-XujBP_+BxU0RpjzDA*aaGi`<7z+ETO6T4*57BWVB{eu3210TD))$|Lsd?4JGF#b31R3=tU>C51 zhRb2hvxFtF+mx929n6#CVdiPE@}}VG6L019d#Z30@LQaT8nXg#AHPnQ9sNp#PBRjq z75aFY`jiECu+_o6{!xUAfCEAJ<;K(E5QK0ynyFwE2Ozs4VY~F7>nr1i0sFHk*kUv2 zC~^B4dB|22#$WJFfLzEJUbgUCexsiLJKBh3oBX6`r3Oz7EKF&>?N64-4Crc;TGpbY z#0S)aNzPU>jIdX7PyyKJU;choMRp)AP`aco#;RqV7V>EMN0Vegsfu@Hkc1ErwK-_i ze%y#GjTO$Sr<~5wh5mc9Wfc;y!kGGWSDcGJEzq`1VL3t*df;XJS^`I0tID`@yvOhO zCzXfKP3fIW3G6Yd!c6id+OOrwW|fNAg3LFC`C9hh%7`l}f_>PlwNC{`9f${JN1Aip z0U#)I{MNX2*xioG&bxydbGugmMIwe^e@>2;Uu3+i4o7t$qu_W}sL7Krdqu|Eh5G{O zG2F4mNbD@A#nQ6PV?ANsvBmcG=CByKpj5!V$_ajc;YUIn?d?gDCLVfKV>Rv=Y}=yk zsxZ1gldElQ%rc$Aq90rtRyj~QbcaKoqw#e_ECw{f^B-K!Oz*}p(haC^MN&rt<~Y{H z?^ZapFGM*we6SMhZzH^&U<@*Hhat2jl|9;!s1F@`V6aeWXviCw1nMWu zBH~gubklBO5|YP?=J!E)Zsv9=mY(x(c+cwDZ*B9F!JHv_g{N{ytpaS33eOWnOqdRD z()UMqD*z841{TK0g5`EB*0_BgT+jYU=i3h8K3ypWQg{1kaaj^mJjWBR;cZHAw656c ze|eL^28Ec|ZV%IepIzq#!^Cc@Har9_tr;@yjn=-9LEwXOqbi??;XH*G85djA&9<(G%CGj4;E*Gjt5BuS5hBIsQ#

          +Global dimensions: (64, 32, 128) -> (33, 32, 128)
          source
          diff --git a/previews/PR59/PencilFFTs/index.html b/previews/PR63/PencilFFTs/index.html similarity index 95% rename from previews/PR59/PencilFFTs/index.html rename to previews/PR63/PencilFFTs/index.html index 3d7757c4..1efba4cb 100644 --- a/previews/PR59/PencilFFTs/index.html +++ b/previews/PR63/PencilFFTs/index.html @@ -19,13 +19,13 @@ proc_dims = (4, 2) # 2D decomposition comm = MPI.COMM_WORLD -plan = PencilFFTPlan(size_global, transforms, proc_dims, comm)
          source

          Allocating data

          PencilFFTs.allocate_input — Function
          allocate_input(p::PencilFFTPlan)          -> PencilArray
+plan = PencilFFTPlan(size_global, transforms, proc_dims, comm)
          source

          Allocating data

          PencilFFTs.allocate_input — Function
          allocate_input(p::PencilFFTPlan)          -> PencilArray
 allocate_input(p::PencilFFTPlan, dims...) -> Array{PencilArray}
 allocate_input(p::PencilFFTPlan, Val(N))  -> NTuple{N, PencilArray}

          Allocate uninitialised PencilArray that can hold input data for the given plan.

          The second and third forms respectively allocate an array of PencilArrays of size dims, and a tuple of N PencilArrays.

          In-place plans

          If p is an in-place plan, a ManyPencilArray is allocated. This type holds PencilArray wrappers for the input and output transforms (as well as for intermediate transforms) which share the same space in memory. The input and output PencilArrays should be respectively accessed by calling first(::ManyPencilArray) and last(::ManyPencilArray).

          Example

          Suppose p is an in-place PencilFFTPlan. Then,

          @assert is_inplace(p)
 A = allocate_input(p) :: ManyPencilArray
 v_in = first(A)       :: PencilArray  # input data view
 v_out = last(A)       :: PencilArray  # output data view

          Also note that in-place plans must be performed directly on the returned ManyPencilArray, and not on the contained PencilArray views:

          p * A       # perform forward transform in-place
 p \ A       # perform backward transform in-place
-# p * v_in  # not allowed!!
          source
          source
          PencilFFTs.allocate_output — Function
          allocate_output(p::PencilFFTPlan)          -> PencilArray
 allocate_output(p::PencilFFTPlan, dims...) -> Array{PencilArray}
-allocate_output(p::PencilFFTPlan, Val(N))  -> NTuple{N, PencilArray}

          Allocate uninitialised PencilArray that can hold output data for the given plan.

          If p is an in-place plan, a ManyPencilArray is allocated.

          See allocate_input for details.

          source

          Methods

          +allocate_output(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}

          Allocate uninitialised PencilArray that can hold output data for the given plan.

          If p is an in-place plan, a ManyPencilArray is allocated.

          See allocate_input for details.

          source

          Methods

          diff --git a/previews/PR62/PencilFFTs_timers/index.html b/previews/PR63/PencilFFTs_timers/index.html similarity index 97% rename from previews/PR62/PencilFFTs_timers/index.html rename to previews/PR63/PencilFFTs_timers/index.html index 8bde6088..87516c45 100644 --- a/previews/PR62/PencilFFTs_timers/index.html +++ b/previews/PR63/PencilFFTs_timers/index.html @@ -21,4 +21,4 @@ # [do stuff with `plan`...] -print_timer(to) +print_timer(to) diff --git a/previews/PR62/Transforms/index.html b/previews/PR63/Transforms/index.html similarity index 91% rename from previews/PR62/Transforms/index.html rename to previews/PR63/Transforms/index.html index db4ef7e8..481884da 100644 --- a/previews/PR62/Transforms/index.html +++ b/previews/PR63/Transforms/index.html @@ -1,11 +1,11 @@ -Available transforms · PencilFFTs.jl
          Edit on GitHub

          Available transforms

          Transform types

          PencilFFTs.Transforms.BFFT — Type
          BFFT()

          Unnormalised backward complex-to-complex FFT.

          Like AbstractFFTs.bfft, this transform is not normalised. To obtain the inverse transform, divide the output by the length of the transformed dimension.

          See also AbstractFFTs.bfft.

          source
          PencilFFTs.Transforms.BRFFT — Type
          BRFFT(d::Integer)
-BRFFT((d1, d2, ..., dN))

          Unnormalised inverse of RFFT.

          To obtain the inverse transform, divide the output by the length of the transformed dimension (of the real output array).

          As described in the AbstractFFTs docs, the length of the output cannot be fully inferred from the input length. For this reason, the BRFFT constructor accepts an optional d argument indicating the output length.

          For multidimensional datasets, a tuple of dimensions (d1, d2, ..., dN) may also be passed. This is equivalent to passing just dN. In this case, the last dimension (dN) is the one that changes size between the input and output. Note that this is the opposite of FFTW.brfft. The reason is that, in PencilFFTs, the last dimension is the one along which a complex-to-real transform is performed.

          See also AbstractFFTs.brfft.

          source

          Internals

          What follows is used internally in PencilFFTs.

          Types

          PencilFFTs.Transforms.AbstractCustomPlan — Type
          AbstractCustomPlan

          Abstract type defining a custom plan, to be used as an alternative to FFTW plans (FFTW.FFTWPlan).

          The only custom plan defined in this module is IdentityPlan. The user can define other custom plans that are also subtypes of AbstractCustomPlan.

          Note that plan returns a subtype of either AbstractFFTs.Plan or AbstractCustomPlan.

          source

          Functions

          PencilFFTs.Transforms.plan — Function
          plan(transform::AbstractTransform, A, [dims];
-     flags=FFTW.ESTIMATE, timelimit=Inf)

          Create plan to transform array A along dimensions dims.

          If dims is not specified, all dimensions of A are transformed.

          For FFT plans, this function wraps the AbstractFFTs.jl and FFTW.jl plan creation functions. For more details on the function arguments, see AbstractFFTs.plan_fft.

          source
          PencilFFTs.Transforms.binv — Function
          binv(transform::AbstractTransform, d::Integer)

          Returns the backwards transform associated to the given transform.

          The second argument must be the length of the first transformed dimension in the forward transform. It is used in particular when transform = RFFT(), to determine the length of the inverse (complex-to-real) transform. See the AbstractFFTs.irfft docs for details.

          The backwards transform returned by this function is not normalised. The normalisation factor for a given array can be obtained by calling scale_factor.

          Example

          julia> binv(Transforms.FFT(), 42)
+Available transforms · PencilFFTs.jl
          Edit on GitHub
          Available transforms
          Transform types
          PencilFFTs.Transforms.BFFT — Type
          BFFT()
          Unnormalised backward complex-to-complex FFT.
          Like AbstractFFTs.bfft, this transform is not normalised. To obtain the inverse transform, divide the output by the length of the transformed dimension.
          See also AbstractFFTs.bfft.
          source
          PencilFFTs.Transforms.BRFFT — Type
          BRFFT(d::Integer)
+BRFFT((d1, d2, ..., dN))
          Unnormalised inverse of RFFT.
          To obtain the inverse transform, divide the output by the length of the transformed dimension (of the real output array).
          As described in the AbstractFFTs docs, the length of the output cannot be fully inferred from the input length. For this reason, the BRFFT constructor accepts an optional d argument indicating the output length.
          For multidimensional datasets, a tuple of dimensions (d1, d2, ..., dN) may also be passed. This is equivalent to passing just dN. In this case, the last dimension (dN) is the one that changes size between the input and output. Note that this is the opposite of FFTW.brfft. The reason is that, in PencilFFTs, the last dimension is the one along which a complex-to-real transform is performed.
          See also AbstractFFTs.brfft.
          source
          Internals
          What follows is used internally in PencilFFTs.
          Types
          PencilFFTs.Transforms.AbstractCustomPlan — Type
          AbstractCustomPlan
          Abstract type defining a custom plan, to be used as an alternative to FFTW plans (FFTW.FFTWPlan).
          The only custom plan defined in this module is IdentityPlan. The user can define other custom plans that are also subtypes of AbstractCustomPlan.
          Note that plan returns a subtype of either AbstractFFTs.Plan or AbstractCustomPlan.
          source
          Functions
          PencilFFTs.Transforms.plan — Function
          plan(transform::AbstractTransform, A, [dims];
+     flags=FFTW.ESTIMATE, timelimit=Inf)
          Create plan to transform array A along dimensions dims.
          If dims is not specified, all dimensions of A are transformed.
          For FFT plans, this function wraps the AbstractFFTs.jl and FFTW.jl plan creation functions. For more details on the function arguments, see AbstractFFTs.plan_fft.
          source
          PencilFFTs.Transforms.binv — Function
          binv(transform::AbstractTransform, d::Integer)
          Returns the backwards transform associated to the given transform.
          The second argument must be the length of the first transformed dimension in the forward transform. It is used in particular when transform = RFFT(), to determine the length of the inverse (complex-to-real) transform. See the AbstractFFTs.irfft docs for details.
          The backwards transform returned by this function is not normalised. The normalisation factor for a given array can be obtained by calling scale_factor.
          Example
          julia> binv(Transforms.FFT(), 42)
 BFFT
 
 julia> binv(Transforms.BRFFT(9), 42)
-RFFT
          source
          PencilFFTs.Transforms.scale_factor — Function
          scale_factor(transform::AbstractTransform, A, [dims = 1:ndims(A)])
          Get factor required to normalise the given array after a transformation along dimensions dims (all dimensions by default).
          The array A must have the dimensions of the transform input.
          Important: the dimensions dims must be the same that were passed to plan.
          Examples
          julia> C = zeros(ComplexF32, 3, 4, 5);
+RFFT
          source
          PencilFFTs.Transforms.scale_factor — Function
          scale_factor(transform::AbstractTransform, A, [dims = 1:ndims(A)])
          Get factor required to normalise the given array after a transformation along dimensions dims (all dimensions by default).
          The array A must have the dimensions of the transform input.
          Important: the dimensions dims must be the same that were passed to plan.
          Examples
          julia> C = zeros(ComplexF32, 3, 4, 5);
 
 julia> scale_factor(Transforms.FFT(), C)
 60
@@ -29,7 +29,7 @@
 
 julia> scale_factor(Transforms.BRFFT(9), C)
 108
          This will fail because the input of RFFT is real, and R is a complex array:
          julia> scale_factor(Transforms.RFFT(), C, 2:3)
-ERROR: MethodError: no method matching scale_factor(::PencilFFTs.Transforms.RFFT, ::Array{ComplexF32, 3}, ::UnitRange{Int64})
          source
          PencilFFTs.Transforms.eltype_input — Function
          eltype_input(transform::AbstractTransform, real_type<:AbstractFloat)
          Determine input data type for a given transform given the floating point precision of the input data.
          Some transforms, such as R2R and NoTransform, can take both real and complex data. For those kinds of transforms, nothing is returned.
          Example
          julia> eltype_input(Transforms.FFT(), Float32)
+ERROR: MethodError: no method matching scale_factor(::PencilFFTs.Transforms.RFFT, ::Array{ComplexF32, 3}, ::UnitRange{Int64})
          source
          PencilFFTs.Transforms.eltype_input — Function
          eltype_input(transform::AbstractTransform, real_type<:AbstractFloat)
          Determine input data type for a given transform given the floating point precision of the input data.
          Some transforms, such as R2R and NoTransform, can take both real and complex data. For those kinds of transforms, nothing is returned.
          Example
          julia> eltype_input(Transforms.FFT(), Float32)
 ComplexF32 (alias for Complex{Float32})
 
 julia> eltype_input(Transforms.RFFT(), Float64)
@@ -38,7 +38,7 @@
 julia> eltype_input(Transforms.R2R(FFTW.REDFT01), Float64)  # nothing
 
 julia> eltype_input(Transforms.NoTransform(), Float64)  # nothing
-
          source
          PencilFFTs.Transforms.eltype_output — Function
          eltype_output(transform::AbstractTransform, eltype_input)
          Returns the output data type for a given transform given the input type.
          Throws ArgumentError if the input data type is incompatible with the transform type.
          Example
          julia> eltype_output(Transforms.NoTransform(), Float32)
+
          source
          PencilFFTs.Transforms.eltype_output — Function
          eltype_output(transform::AbstractTransform, eltype_input)
          Returns the output data type for a given transform given the input type.
          Throws ArgumentError if the input data type is incompatible with the transform type.
          Example
          julia> eltype_output(Transforms.NoTransform(), Float32)
 Float32
 
 julia> eltype_output(Transforms.RFFT(), Float64)
@@ -48,7 +48,7 @@
 Float32
 
 julia> eltype_output(Transforms.FFT(), Float64)
-ERROR: ArgumentError: invalid input data type for PencilFFTs.Transforms.FFT: Float64
          source
          PencilFFTs.Transforms.expand_dims — Function
          expand_dims(transform::AbstractTransform, Val(N))
          Expand a single multidimensional transform into one transform per dimension.
          Example
          # Expand a real-to-complex transform in 3 dimensions.
+ERROR: ArgumentError: invalid input data type for PencilFFTs.Transforms.FFT: Float64
          source
          PencilFFTs.Transforms.expand_dims — Function
          expand_dims(transform::AbstractTransform, Val(N))
          Expand a single multidimensional transform into one transform per dimension.
          Example
          # Expand a real-to-complex transform in 3 dimensions.
 julia> expand_dims(Transforms.RFFT(), Val(3))
 (RFFT, FFT, FFT)
 
@@ -56,7 +56,7 @@
 (BFFT, BFFT, BRFFT{even})
 
 julia> expand_dims(Transforms.NoTransform(), Val(2))
-(NoTransform, NoTransform)
          source
          PencilFFTs.Transforms.is_inplace — Function
          is_inplace(transform::AbstractTransform)         -> Bool
 is_inplace(transforms::Vararg{AbtractTransform}) -> Union{Bool, Nothing}
          Check whether a transform or a list of transforms is performed in-place.
          If the list of transforms has a combination of in-place and out-of-place transforms, nothing is returned.
          Example
          julia> is_inplace(Transforms.RFFT())
 false
 
@@ -71,4 +71,4 @@
 
 julia> is_inplace(Transforms.FFT(), Transforms.R2R!(FFTW.REDFT01)) === nothing
 true
-
          source
          PencilFFTs.Transforms.length_output — Function
          length_output(transform::AbstractTransform, length_in::Integer)
          Returns the length of the transform output, given the length of its input.
          The input and output lengths are specified in terms of the respective input and output datatypes. For instance, for real-to-complex transforms, these are respectively the length of input real data and of output complex data.
          source
          
+
          source
          PencilFFTs.Transforms.length_output — Function
          length_output(transform::AbstractTransform, length_in::Integer)

          Returns the length of the transform output, given the length of its input.

          The input and output lengths are specified in terms of the respective input and output datatypes. For instance, for real-to-complex transforms, these are respectively the length of input real data and of output complex data.

          source
          diff --git a/previews/PR39/assets/custom.css b/previews/PR63/assets/custom.css similarity index 100% rename from previews/PR39/assets/custom.css rename to previews/PR63/assets/custom.css diff --git a/previews/PR59/assets/documenter.js b/previews/PR63/assets/documenter.js similarity index 100% rename from previews/PR59/assets/documenter.js rename to previews/PR63/assets/documenter.js diff --git a/previews/PR39/assets/logo.svg b/previews/PR63/assets/logo.svg similarity index 100% rename from previews/PR39/assets/logo.svg rename to previews/PR63/assets/logo.svg diff --git a/dev/assets/search.js b/previews/PR63/assets/search.js similarity index 100% rename from dev/assets/search.js rename to previews/PR63/assets/search.js diff --git a/previews/PR48/assets/themes/documenter-dark.css b/previews/PR63/assets/themes/documenter-dark.css similarity index 100% rename from previews/PR48/assets/themes/documenter-dark.css rename to previews/PR63/assets/themes/documenter-dark.css diff --git a/previews/PR48/assets/themes/documenter-light.css b/previews/PR63/assets/themes/documenter-light.css similarity index 100% rename from previews/PR48/assets/themes/documenter-light.css rename to previews/PR63/assets/themes/documenter-light.css diff --git a/previews/PR39/assets/themeswap.js b/previews/PR63/assets/themeswap.js similarity index 100% rename from previews/PR39/assets/themeswap.js rename to previews/PR63/assets/themeswap.js diff --git a/previews/PR39/assets/tomate.js b/previews/PR63/assets/tomate.js similarity index 100% rename from previews/PR39/assets/tomate.js rename to previews/PR63/assets/tomate.js diff --git a/previews/PR48/assets/warner.js b/previews/PR63/assets/warner.js similarity index 100% rename from previews/PR48/assets/warner.js rename to previews/PR63/assets/warner.js diff --git a/previews/PR62/benchmarks/index.html b/previews/PR63/benchmarks/index.html similarity index 97% rename from previews/PR62/benchmarks/index.html rename to previews/PR63/benchmarks/index.html index dd0010bc..ec6c0359 100644 --- a/previews/PR62/benchmarks/index.html +++ b/previews/PR63/benchmarks/index.html @@ -9,4 +9,4 @@ width="75%" src="../img/benchmark_idris.svg" alt="Strong scaling of PencilFFTs"> -

          As seen above, PencilFFTs generally outperforms P3DFFT in its default setting. This is largely explained by the choice of using non-blocking point-to-point MPI communications (via MPI_Isend and MPI_Irecv), while P3DFFT uses collective MPI_Alltoallv calls. This enables PencilFFTs to perform data reordering operations on the partially received data while waiting for the incoming data, leading to better performance. Moreover, in contrast with P3DFFT, the high performance and scalability of PencilFFTs results from a highly generic code, handling decompositions in arbitrary dimensions and a relatively large (and extensible) variety of transformations.

          Note that PencilFFTs can optionally use collective communications (using MPI_Alltoallv) instead of point-to-point communications. For details, see the docs for PencilFFTPlan and for PencilArray transpositions. As seen above, collective communications generally perform worse than point-to-point ones, and runtimes are nearly indistinguishable from those of P3DFFT.

          Benchmark details

          The benchmarks were performed using Julia 1.7-beta3 and Intel MPI 2019. We used PencilFFTs v0.12.5 with FFTW.jl v1.4.3 and MPI.jl v0.19.0. We used the Fortran implementation of P3DFFT, version 2.7.6, which was built with Intel 2019 compilers and linked to FFTW 3.3.8. The cluster where the benchmarks were run has Intel Cascade Lake 6248 processors with 2×20 cores per node.

          The number of MPI processes along each decomposed dimension, $P_1$ and $P_2$, was automatically determined by a call to MPI_Dims_create, which tends to create a balanced decomposition with $P_1 ≈ P_2$. For instance, a total of 1024 processes is divided into $P_1 = P_2 = 32$. Different results may be obtained with other combinations, but this was not benchmarked.

          The source files used to generate this benchmark, as well as the raw benchmark results, are all available in the PencilFFTs repo.

          +

          As seen above, PencilFFTs generally outperforms P3DFFT in its default setting. This is largely explained by the choice of using non-blocking point-to-point MPI communications (via MPI_Isend and MPI_Irecv), while P3DFFT uses collective MPI_Alltoallv calls. This enables PencilFFTs to perform data reordering operations on the partially received data while waiting for the incoming data, leading to better performance. Moreover, in contrast with P3DFFT, the high performance and scalability of PencilFFTs results from a highly generic code, handling decompositions in arbitrary dimensions and a relatively large (and extensible) variety of transformations.

          Note that PencilFFTs can optionally use collective communications (using MPI_Alltoallv) instead of point-to-point communications. For details, see the docs for PencilFFTPlan and for PencilArray transpositions. As seen above, collective communications generally perform worse than point-to-point ones, and runtimes are nearly indistinguishable from those of P3DFFT.

          Benchmark details

          The benchmarks were performed using Julia 1.7-beta3 and Intel MPI 2019. We used PencilFFTs v0.12.5 with FFTW.jl v1.4.3 and MPI.jl v0.19.0. We used the Fortran implementation of P3DFFT, version 2.7.6, which was built with Intel 2019 compilers and linked to FFTW 3.3.8. The cluster where the benchmarks were run has Intel Cascade Lake 6248 processors with 2×20 cores per node.

          The number of MPI processes along each decomposed dimension, $P_1$ and $P_2$, was automatically determined by a call to MPI_Dims_create, which tends to create a balanced decomposition with $P_1 ≈ P_2$. For instance, a total of 1024 processes is divided into $P_1 = P_2 = 32$. Different results may be obtained with other combinations, but this was not benchmarked.

          The source files used to generate this benchmark, as well as the raw benchmark results, are all available in the PencilFFTs repo.

          diff --git a/previews/PR63/generated/ffmpeg.err b/previews/PR63/generated/ffmpeg.err new file mode 100644 index 00000000..b7f5acd9 --- /dev/null +++ b/previews/PR63/generated/ffmpeg.err @@ -0,0 +1,48 @@ +ffmpeg version 4.4.2-0ubuntu0.22.04.1 Copyright (c) 2000-2021 the FFmpeg developers + built with gcc 11 (Ubuntu 11.2.0-19ubuntu1) + configuration: --prefix=/usr --extra-version=0ubuntu0.22.04.1 --toolchain=hardened --libdir=/usr/lib/x86_64-linux-gnu --incdir=/usr/include/x86_64-linux-gnu --arch=amd64 --enable-gpl --disable-stripping --enable-gnutls --enable-ladspa --enable-libaom --enable-libass --enable-libbluray --enable-libbs2b --enable-libcaca --enable-libcdio --enable-libcodec2 --enable-libdav1d --enable-libflite --enable-libfontconfig --enable-libfreetype --enable-libfribidi --enable-libgme --enable-libgsm --enable-libjack --enable-libmp3lame --enable-libmysofa --enable-libopenjpeg --enable-libopenmpt --enable-libopus --enable-libpulse --enable-librabbitmq --enable-librubberband --enable-libshine --enable-libsnappy --enable-libsoxr --enable-libspeex --enable-libsrt --enable-libssh --enable-libtheora --enable-libtwolame --enable-libvidstab --enable-libvorbis --enable-libvpx --enable-libwebp --enable-libx265 --enable-libxml2 --enable-libxvid --enable-libzimg --enable-libzmq --enable-libzvbi --enable-lv2 --enable-omx --enable-openal --enable-opencl --enable-opengl --enable-sdl2 --enable-pocketsphinx --enable-librsvg --enable-libmfx --enable-libdc1394 --enable-libdrm --enable-libiec61883 --enable-chromaprint --enable-frei0r --enable-libx264 --enable-shared + libavutil 56. 70.100 / 56. 70.100 + libavcodec 58.134.100 / 58.134.100 + libavformat 58. 76.100 / 58. 76.100 + libavdevice 58. 13.100 / 58. 13.100 + libavfilter 7.110.100 / 7.110.100 + libswscale 5. 9.100 / 5. 9.100 + libswresample 3. 9.100 / 3. 9.100 + libpostproc 55. 9.100 / 55. 9.100 +Input #0, image2, from '/tmp/jl_2AvgDR/proc1_%04d.png': + Duration: 00:00:04.52, start: 0.000000, bitrate: N/A + Stream #0:0: Video: png, rgb24(pc), 1200x600, 25 fps, 25 tbr, 25 tbn, 25 tbc +Stream mapping: + Stream #0:0 -> #0:0 (png (native) -> h264 (libx264)) +Press [q] to stop, [?] for help +[libx264 @ 0x558a6d62e700] using cpu capabilities: MMX2 SSE2Fast SSSE3 SSE4.2 AVX FMA3 BMI2 AVX2 AVX512 +[libx264 @ 0x558a6d62e700] profile High, level 3.1, 4:2:0, 8-bit +[libx264 @ 0x558a6d62e700] 264 - core 163 r3060 5db6aa6 - H.264/MPEG-4 AVC codec - Copyleft 2003-2021 - http://www.videolan.org/x264.html - options: cabac=1 ref=3 deblock=1:0:0 analyse=0x3:0x113 me=hex subme=7 psy=1 psy_rd=1.00:0.00 mixed_ref=1 me_range=16 chroma_me=1 trellis=1 8x8dct=1 cqm=0 deadzone=21,11 fast_pskip=1 chroma_qp_offset=-2 threads=3 lookahead_threads=1 sliced_threads=0 nr=0 decimate=1 interlaced=0 bluray_compat=0 constrained_intra=0 bframes=3 b_pyramid=2 b_adapt=1 b_bias=0 direct=1 weightb=1 open_gop=0 weightp=2 keyint=250 keyint_min=25 scenecut=40 intra_refresh=0 rc_lookahead=40 rc=crf mbtree=1 crf=23.0 qcomp=0.60 qpmin=0 qpmax=69 qpstep=4 ip_ratio=1.40 aq=1:1.00 +Output #0, mp4, to 'vorticity_proc1.mp4': + Metadata: + encoder : Lavf58.76.100 + Stream #0:0: Video: h264 (avc1 / 0x31637661), yuv420p(tv, progressive), 1200x600, q=2-31, 25 fps, 12800 tbn + Metadata: + encoder : Lavc58.134.100 libx264 + Side data: + cpb: bitrate max/min/avg: 0/0/0 buffer size: 0 vbv_delay: N/A +frame= 3 fps=0.0 q=0.0 size= 0kB time=00:00:00.00 bitrate=N/A speed= 0x frame= 50 fps=0.0 q=28.0 size= 0kB time=00:00:00.00 bitrate=4923.1kbits/s speed=0.000119x frame= 75 fps= 63 q=28.0 size= 0kB time=00:00:01.00 bitrate= 0.4kbits/s speed=0.844x frame= 98 fps= 57 q=28.0 size= 0kB time=00:00:01.92 bitrate= 0.2kbits/s speed=1.13x frame= 120 fps= 54 q=28.0 size= 0kB time=00:00:02.80 bitrate= 0.1kbits/s speed=1.26x frame= 148 fps= 54 q=28.0 size= 256kB time=00:00:03.92 bitrate= 535.1kbits/s speed=1.42x frame= 175 fps= 53 q=28.0 size= 256kB time=00:00:05.00 bitrate= 419.5kbits/s speed=1.52x frame= 198 fps= 52 q=28.0 size= 256kB time=00:00:05.92 bitrate= 354.3kbits/s speed=1.55x frame= 225 fps= 51 q=28.0 size= 256kB time=00:00:07.00 bitrate= 299.6kbits/s speed=1.59x frame= 250 fps= 50 q=28.0 size= 512kB time=00:00:08.00 bitrate= 524.3kbits/s speed= 1.6x frame= 275 fps= 50 q=28.0 size= 512kB time=00:00:09.00 bitrate= 466.1kbits/s speed=1.62x frame= 283 fps= 44 q=-1.0 Lsize= 821kB time=00:00:11.20 bitrate= 600.2kbits/s speed=1.76x +video:816kB audio:0kB subtitle:0kB other streams:0kB global headers:0kB muxing overhead: 0.506766% +[libx264 @ 0x558a6d62e700] frame I:2 Avg QP:16.82 size: 37443 +[libx264 @ 0x558a6d62e700] frame P:74 Avg QP:25.03 size: 8126 +[libx264 @ 0x558a6d62e700] frame B:207 Avg QP:19.11 size: 769 +[libx264 @ 0x558a6d62e700] consecutive B-frames: 0.7% 4.2% 3.2% 91.9% +[libx264 @ 0x558a6d62e700] mb I I16..4: 32.2% 41.8% 26.0% +[libx264 @ 0x558a6d62e700] mb P I16..4: 0.4% 1.5% 0.9% P16..4: 5.7% 6.7% 4.5% 0.0% 0.0% skip:80.3% +[libx264 @ 0x558a6d62e700] mb B I16..4: 0.1% 0.1% 0.0% B16..8: 6.6% 0.9% 0.4% direct: 0.3% skip:91.6% L0:44.2% L1:47.5% BI: 8.3% +[libx264 @ 0x558a6d62e700] 8x8 transform intra:46.8% inter:37.9% +[libx264 @ 0x558a6d62e700] coded y,uvDC,uvAC intra: 39.3% 42.0% 33.9% inter: 2.7% 2.2% 0.7% +[libx264 @ 0x558a6d62e700] i16 v,h,dc,p: 68% 19% 10% 3% +[libx264 @ 0x558a6d62e700] i8 v,h,dc,ddl,ddr,vr,hd,vl,hu: 27% 11% 32% 5% 5% 4% 6% 4% 6% +[libx264 @ 0x558a6d62e700] i4 v,h,dc,ddl,ddr,vr,hd,vl,hu: 29% 18% 17% 5% 6% 5% 9% 4% 7% +[libx264 @ 0x558a6d62e700] i8c dc,h,v,p: 68% 15% 13% 5% +[libx264 @ 0x558a6d62e700] Weighted P-Frames: Y:0.0% UV:0.0% +[libx264 @ 0x558a6d62e700] ref P L0: 69.7% 15.2% 10.0% 5.1% +[libx264 @ 0x558a6d62e700] ref B L0: 86.3% 11.0% 2.7% +[libx264 @ 0x558a6d62e700] ref B L1: 98.3% 1.7% +[libx264 @ 0x558a6d62e700] kb/s:590.39 diff --git a/previews/PR39/generated/ffmpeg.out b/previews/PR63/generated/ffmpeg.out similarity index 100% rename from previews/PR39/generated/ffmpeg.out rename to previews/PR63/generated/ffmpeg.out diff --git a/previews/PR62/generated/gradient/index.html b/previews/PR63/generated/gradient/index.html similarity index 99% rename from previews/PR62/generated/gradient/index.html rename to previews/PR63/generated/gradient/index.html index c89d74e1..44882b1b 100644 --- a/previews/PR62/generated/gradient/index.html +++ b/previews/PR63/generated/gradient/index.html @@ -107,4 +107,4 @@ gradient_grid_broadcast! (lazy)... 151.020 μs [Method 4] gradient_local_parent!... 92.248 μs gradient_local_linear!... 91.212 μs - gradient_local_linear_explicit!... 90.992 μs

          This page was generated using Literate.jl.

          + gradient_local_linear_explicit!... 90.992 μs

          This page was generated using Literate.jl.

          diff --git a/previews/PR59/generated/in-place/index.html b/previews/PR63/generated/in-place/index.html similarity index 97% rename from previews/PR59/generated/in-place/index.html rename to previews/PR63/generated/in-place/index.html index b03642a2..8973c27a 100644 --- a/previews/PR59/generated/in-place/index.html +++ b/previews/PR63/generated/in-place/index.html @@ -28,4 +28,4 @@ u_in = first(A) # input data view randn!(u_in) summary(u_in)
          "16×16×64 PencilArray{ComplexF64, 3}(::Pencil{3, 2, NoPermutation, Array})"

          Applying plans

          Like in FFTW.jl, one can perform in-place transforms using the * and \ operators. As mentioned above, in-place plans must be applied on the ManyPencilArray containers returned by allocate_input.

          plan * A;  # performs in-place forward transform

          After performing an in-place transform, data contained in u_in has been overwritten and has no "physical" meaning. In other words, u_in should not be used at this point. To access the transformed data, one should retrieve the output data view using last(A).

          For instance, to compute the global sum of the transformed data:

          u_out = last(A)  # output data view
-sum(u_out)       # sum of transformed data (note that `sum` reduces over all processes)
          -39261.87119917473 - 9182.012054695917im

          Finally, we can perform a backward transform and do stuff with the input view:

          plan \ A;  # perform in-place backward transform

          At this point, the data can be once again found in the input view u_in, while u_out should not be accessed.

          This page was generated using Literate.jl.

          +sum(u_out) # sum of transformed data (note that `sum` reduces over all processes)
          961.7857574484242 - 11248.700238838526im

          Finally, we can perform a backward transform and do stuff with the input view:

          plan \ A;  # perform in-place backward transform

          At this point, the data can be once again found in the input view u_in, while u_out should not be accessed.

          This page was generated using Literate.jl.

          diff --git a/previews/PR63/generated/navier_stokes/index.html b/previews/PR63/generated/navier_stokes/index.html new file mode 100644 index 00000000..174f13eb --- /dev/null +++ b/previews/PR63/generated/navier_stokes/index.html @@ -0,0 +1,303 @@ + +Navier–Stokes equations · PencilFFTs.jl
          Edit on GitHub

          Navier–Stokes equations

          In this example, we numerically solve the incompressible Navier–Stokes equations

          \[∂_t \bm{v} + (\bm{v} ⋅ \bm{∇}) \bm{v} = -\frac{1}{ρ} \bm{∇} p + ν ∇^2 \bm{v}, +\quad \bm{∇} ⋅ \bm{v} = 0,\]

          where $\bm{v}(\bm{x}, t)$ and $p(\bm{x}, t)$ are respectively the velocity and pressure fields, $ν$ is the fluid kinematic viscosity and $ρ$ is the fluid density.

          We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.

          First steps

          We start by loading the required packages, initialising MPI and setting the simulation parameters.

          using MPI
+using PencilFFTs
+
+MPI.Init()
+comm = MPI.COMM_WORLD
+procid = MPI.Comm_rank(comm) + 1
+
+# Simulation parameters
+Ns = (64, 64, 64)  # = (Nx, Ny, Nz)
+Ls = (2π, 2π, 2π)  # = (Lx, Ly, Lz)
+
+# Collocation points ("global" = over all processes).
+# We include the endpoint (length = N + 1) for convenience.
+xs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls)  # = (x, y, z)
          (0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586)

          Let's check the number of MPI processes over which we're running our simulation:

          MPI.Comm_size(comm)
          2

          We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:

          pen = Pencil(Ns, comm)
          Decomposition of 3D data
+    Data dimensions: (64, 64, 64)
+    Decomposed dimensions: (2, 3)
+    Data permutation: NoPermutation()
+    Array type: Array

          The subdomain associated to the local MPI process can be obtained using range_local:

          range_local(pen)
          (1:64, 1:32, 1:64)

          We now construct a distributed vector field that follows the decomposition configuration we just created:

          v⃗₀ = (
+    PencilArray{Float64}(undef, pen),  # vx
+    PencilArray{Float64}(undef, pen),  # vy
+    PencilArray{Float64}(undef, pen),  # vz
+)
+summary(v⃗₀[1])
          "64×32×64 PencilArray{Float64, 3}(::Pencil{3, 2, NoPermutation, Array})"

          We still need to fill this array with interesting values that represent a physical velocity field.

          Initial condition

          Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:

          \[\begin{aligned} +v_x(x, y, z) &= u₀ \sin(k₀ x) \cos(k₀ y) \cos(k₀ z) \\ +v_y(x, y, z) &= -u₀ \cos(k₀ x) \sin(k₀ y) \cos(k₀ z) \\ +v_z(x, y, z) &= 0 +\end{aligned}\]

          where $u₀$ and $k₀$ are two parameters setting the amplitude and the period of the velocity field.

          To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:

          grid = localgrid(pen, xs_global)
          LocalRectilinearGrid{3} with coordinates:
+ (1) 0.0:0.09817477042468103:6.1850105367549055
+ (2) 0.0:0.09817477042468103:3.043417883165112
+ (3) 0.0:0.09817477042468103:6.1850105367549055

          We can use this to initialise the velocity field:

          u₀ = 1.0
+k₀ = 2π / Ls[1]  # should be integer if L = 2π (to preserve periodicity)
+
+@. v⃗₀[1] =  u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)
+@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)
+@. v⃗₀[3] =  0

          Let's plot a 2D slice of the velocity field managed by the local MPI process:

          using GLMakie
+
+# Compute the norm of a vector field represented by a tuple of arrays.
+function vecnorm(v⃗::NTuple)
+    vnorm = similar(v⃗[1])
+    for n ∈ eachindex(v⃗[1])
+        w = zero(eltype(vnorm))
+        for v ∈ v⃗
+            w += v[n]^2
+        end
+        vnorm[n] = sqrt(w)
+    end
+    vnorm
+end
+
+let fig = Figure(resolution = (700, 600))
+    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
+    vnorm = vecnorm(v⃗₀)
+    ct = contour!(
+        ax, grid.x, grid.y, grid.z, vnorm;
+        alpha = 0.2, levels = 4,
+        colormap = :viridis, colorrange = (0.0, 1.0),
+    )
+    cb = Colorbar(fig[1, 2], ct; label = "Velocity magnitude")
+    fig
+end
          Figure()

          Velocity in Fourier space

          In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series

          \[\bm{v}(\bm{x}, t) = +∑_{\bm{k}} \hat{\bm{v}}_{\bm{k}}(t) \, e^{i \bm{k} ⋅ \bm{x}},\]

          where $\bm{k} = (k_x, k_y, k_z)$ are the discrete wave numbers.

          The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for $k_x$:

          using AbstractFFTs: fftfreq, rfftfreq
+
+ks_global = (
+    rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]),  # kx | real-to-complex
+     fftfreq(Ns[2], 2π * Ns[2] / Ls[2]),  # ky | complex-to-complex
+     fftfreq(Ns[3], 2π * Ns[3] / Ls[3]),  # kz | complex-to-complex
+)
+
+ks_global[1]'
          1×33 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
+ 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  27.0  28.0  29.0  30.0  31.0  32.0
          ks_global[2]'
          1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
+ 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0
          ks_global[3]'
          1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
+ 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0

          To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:

          plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())
          Transforms: (RFFT, FFT, FFT)
+Input type: Float64
+Global dimensions: (64, 64, 64) -> (33, 64, 64)
+MPI topology: 2D decomposition (2×1 processes)

          See PencilFFTPlan for details on creating plans and on optional keyword arguments.

          We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$:

          v̂s = plan .* v⃗₀
+summary(v̂s[1])
          "16×64×64 PencilArray{ComplexF64, 3}(::Pencil{3, 2, Permutation{(3, 2, 1), 3}, Array})"

          Note that, in Fourier space, the domain decomposition is performed along the directions $x$ and $y$:

          pencil(v̂s[1])
          Decomposition of 3D data
+    Data dimensions: (33, 64, 64)
+    Decomposed dimensions: (1, 2)
+    Data permutation: Permutation(3, 2, 1)
+    Array type: Array

          This is because the 3D FFTs are performed one dimension at a time, with the $x$ direction first and the $z$ direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.

          To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers $\bm{k}$ associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:

          grid_fourier = localgrid(v̂s[1], ks_global)
          LocalRectilinearGrid{3} with Permutation(3, 2, 1) and coordinates:
+ (1) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0]
+ (2) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]
+ (3) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]

          As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, $\bm{ω} = \bm{∇} × \bm{v}$. In Fourier space, this becomes $\hat{\bm{ω}} = i \bm{k} × \hat{\bm{v}}$.

          using StaticArrays: SVector
+using LinearAlgebra: ×
+
+function curl_fourier!(
+        ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,
+    ) where {N}
+    @inbounds for I ∈ eachindex(grid_fourier)
+        # We use StaticArrays for the cross product between small vectors.
+        ik⃗ = im * SVector(grid_fourier[I])
+        v⃗ = SVector(getindex.(v̂s, Ref(I)))  # = (v̂s[1][I], v̂s[2][I], ...)
+        ω⃗ = ik⃗ × v⃗
+        for n ∈ eachindex(ω⃗)
+            ω̂s[n][I] = ω⃗[n]
+        end
+    end
+    ω̂s
+end
+
+ω̂s = similar.(v̂s)
+curl_fourier!(ω̂s, v̂s, grid_fourier);

          We finally transform back to physical space and plot the result:

          ωs = plan .\ ω̂s
+
+let fig = Figure(resolution = (700, 600))
+    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
+    ω_norm = vecnorm(ωs)
+    ct = contour!(
+        ax, grid.x, grid.y, grid.z, ω_norm;
+        alpha = 0.1, levels = 0.8:0.2:2.0,
+        colormap = :viridis, colorrange = (0.8, 2.0),
+    )
+    cb = Colorbar(fig[1, 2], ct; label = "Vorticity magnitude")
+    fig
+end
          Figure()

          Computing the non-linear term

          One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as

          \[∂_t \hat{\bm{v}}_{\bm{k}} = +- \mathcal{P}_{\bm{k}} \! \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right] +- ν |\bm{k}|^2 \hat{\bm{v}}_{\bm{k}} +\quad \text{ with } \quad +\mathcal{P}_{\bm{k}}(\hat{\bm{F}}_{\bm{k}}) = \left( I - \frac{\bm{k} ⊗ +\bm{k}}{|\bm{k}|^2} \right) \hat{\bm{F}}_{\bm{k}},\]

          where $\mathcal{P}_{\bm{k}}$ is a projection operator allowing to preserve the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.

          Now that we have the wave numbers $\bm{k}$, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$ of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, $\hat{\bm{F}}_{\bm{k}} = \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right]_{\bm{k}}$. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.

          Below we implement a function that computes the non-linear term in Fourier space based on its convective form $(\bm{v} ⋅ \bm{∇}) \bm{v} = \bm{∇} ⋅ (\bm{v} ⊗ \bm{v})$. Note that this equivalence uses the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$.

          using LinearAlgebra: mul!, ldiv!  # for applying FFT plans in-place
+
+# Compute non-linear term in Fourier space from velocity field in physical
+# space. Optional keyword arguments may be passed to avoid allocations.
+function ns_nonlinear!(
+        F̂s, vs, plan, grid_fourier;
+        vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),
+    )
+    # Compute F_i = ∂_j (v_i v_j) for each i.
+    # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)
+    w, ŵ = vbuf, v̂buf
+    @inbounds for (i, F̂i) ∈ enumerate(F̂s)
+        F̂i .= 0
+        vi = vs[i]
+        for (j, vj) ∈ enumerate(vs)
+            w .= vi .* vj     # w = v_i * v_j in physical space
+            mul!(ŵ, plan, w)  # same in Fourier space
+            # Add derivative in Fourier space
+            for I ∈ eachindex(grid_fourier)
+                k⃗ = grid_fourier[I]  # = (kx, ky, kz)
+                kj = k⃗[j]
+                F̂i[I] += im * kj * ŵ[I]
+            end
+        end
+    end
+    F̂s
+end
          ns_nonlinear! (generic function with 1 method)

          As an example, let's use this function on our initial velocity field:

          F̂s = similar.(v̂s)
+ns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);

          Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.

          function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)
+    ks_max = maximum.(abs, ks_global)  # maximum stored wave numbers (kx_max, ky_max, kz_max)
+    ks_lim = (2 / 3) .* ks_max
+    @inbounds for I ∈ eachindex(grid_fourier)
+        k⃗ = grid_fourier[I]
+        if any(abs.(k⃗) .> ks_lim)
+            for ŵ ∈ ŵs
+                ŵ[I] = 0
+            end
+        end
+    end
+    ŵs
+end
+
+# We can apply this on the previously computed non-linear term:
+dealias_twothirds!(F̂s, grid_fourier, ks_global);

          Finally, we implement the projection associated to the incompressibility condition:

          function project_divergence_free!(ûs, grid_fourier)
+    @inbounds for I ∈ eachindex(grid_fourier)
+        k⃗ = grid_fourier[I]
+        k² = sum(abs2, k⃗)
+        iszero(k²) && continue  # avoid division by zero
+        û = getindex.(ûs, Ref(I))  # (ûs[1][I], ûs[2][I], ...)
+        for i ∈ eachindex(û)
+            ŵ = û[i]
+            for j ∈ eachindex(û)
+                ŵ -= k⃗[i] * k⃗[j] * û[j] / k²
+            end
+            ûs[i][I] = ŵ
+        end
+    end
+    ûs
+end
          project_divergence_free! (generic function with 1 method)

          We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:

          v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)
+v̂s_proj .≈ v̂s  # the last one may be false because v_z = 0 initially
          (true, true, false)

          Putting it all together

          To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:

          function ns_rhs!(
+        dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,
+    ) where {N}
+    # 1. Compute non-linear term and dealias it
+    (; plan, cache, ks_global, grid_fourier) = p
+    F̂s = cache.F̂s
+    ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])
+    dealias_twothirds!(F̂s, grid_fourier, ks_global)
+
+    # 2. Project onto divergence-free space
+    project_divergence_free!(F̂s, grid_fourier)
+
+    # 3. Transform velocity to Fourier space
+    v̂s = cache.v̂s
+    map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)
+
+    # 4. Add viscous term (and multiply projected non-linear term by -1)
+    ν = p.ν
+    for n ∈ eachindex(v̂s)
+        v̂ = v̂s[n]
+        F̂ = F̂s[n]
+        @inbounds for I ∈ eachindex(grid_fourier)
+            k⃗ = grid_fourier[I]  # = (kx, ky, kz)
+            k² = sum(abs2, k⃗)
+            F̂[I] = -F̂[I] - ν * k² * v̂[I]
+        end
+    end
+
+    # 5. Transform RHS back to physical space
+    map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)
+
+    nothing
+end
          ns_rhs! (generic function with 1 method)

          For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.

          using OrdinaryDiffEq
+using RecursiveArrayTools: ArrayPartition
+
+ns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)
+vs_init_ode = ArrayPartition(v⃗₀)
+summary(vs_init_ode)
          "RecursiveArrayTools.ArrayPartition{Float64, Tuple{PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}}} with arrays:"

          We now define solver parameters and temporary variables, and initialise the problem:

          params = (;
+    ν = 5e-3,  # kinematic viscosity
+    plan, grid_fourier, ks_global,
+    cache = (
+        v̂s = similar.(v̂s),
+        F̂s = similar.(v̂s),
+    )
+)
+
+tspan = (0.0, 10.0)
+prob = ODEProblem(ns_rhs!, vs_init_ode, tspan, params)
+integrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);

          We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum $E(k)$, to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity $ν$ must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large $k$), while high enough to allow the small-scale motions to be correctly resolved.

          function energy_spectrum!(Ek, ks, v̂s, grid_fourier)
+    Nk = length(Ek)
+    @assert Nk == length(ks)
+    Ek .= 0
+    for I ∈ eachindex(grid_fourier)
+        k⃗ = grid_fourier[I]  # = (kx, ky, kz)
+        knorm = sqrt(sum(abs2, k⃗))
+        i = searchsortedfirst(ks, knorm)
+        i > Nk && continue
+        v⃗ = getindex.(v̂s, Ref(I))  # = (v̂s[1][I], v̂s[2][I], ...)
+        factor = k⃗[1] == 0 ? 1 : 2  # account for Hermitian symmetry and r2c transform
+        Ek[i] += factor * sum(abs2, v⃗) / 2
+    end
+    MPI.Allreduce!(Ek, +, get_comm(v̂s[1]))  # sum across all processes
+    Ek
+end
+
+ks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])
+Ek = similar(ks)
+v̂s = plan .* integrator.u.x
+energy_spectrum!(Ek, ks, v̂s, grid_fourier)
+Ek ./= scale_factor(plan)^2  # rescale energy
+
+curl_fourier!(ω̂s, v̂s, grid_fourier)
+ldiv!.(ωs, plan, ω̂s)
+ω⃗_plot = Observable(ωs)
+k_plot = @view ks[2:end]
+E_plot = Observable(@view Ek[2:end])
+t_plot = Observable(integrator.t)
+
+fig = let
+    fig = Figure(resolution = (1200, 600))
+    ax = Axis3(
+        fig[1, 1][1, 1]; title = @lift("t = $(round($t_plot, digits = 3))"),
+        aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z",
+    )
+    ω_mag = @lift vecnorm($ω⃗_plot)
+    ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)
+    ct = contour!(
+        ax, grid.x, grid.y, grid.z, ω_mag_norm;
+        alpha = 0.3, levels = 3,
+        colormap = :viridis, colorrange = (0.0, 1.0),
+    )
+    cb = Colorbar(fig[1, 1][1, 2], ct; label = "Normalised vorticity magnitude")
+    ax_sp = Axis(
+        fig[1, 2];
+        xlabel = "k", ylabel = "E(k)", xscale = log2, yscale = log10,
+        title = "Kinetic energy spectrum",
+    )
+    ylims!(ax_sp, 1e-8, 1e0)
+    scatterlines!(ax_sp, k_plot, E_plot)
+    ks_slope = exp.(range(log(2.5), log(25.0), length = 3))
+    E_fivethirds = @. 0.3 * ks_slope^(-5/3)
+    @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)
+    text!(ax_sp, L"k^{-5/3}"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))
+    fig
+end
+
+record(fig, "vorticity_proc$procid.mp4"; framerate = 10) do io
+    while integrator.t < 20
+        dt = 0.001
+        step!(integrator, dt)
+        t_plot[] = integrator.t
+        mul!.(v̂s, plan, integrator.u.x)  # current velocity in Fourier space
+        curl_fourier!(ω̂s, v̂s, grid_fourier)
+        ldiv!.(ω⃗_plot[], plan, ω̂s)
+        ω⃗_plot[] = ω⃗_plot[]  # to force updating the plot
+        energy_spectrum!(Ek, ks, v̂s, grid_fourier)
+        Ek ./= scale_factor(plan)^2  # rescale energy
+        E_plot[] = E_plot[]
+        recordframe!(io)
+    end
+end;
          + +

          This page was generated using Literate.jl.

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z(Es?1@?vLdWdyX5+L``)>}-G{E|A*KzeX}zn%X-5qXXn_WeQ%g|7TGLwjHF5>`iUW zzX+wXuI>!Y3j}R!}H4Xsq8VD?*{ed$5@~*cy*ZxU=jpE0F0A33q7X)Gl1oqFrIcNy5qYCgj0BQh$IEaM+`5S;1Fjg3_1GWk1 z1yzASj4i-e2Y`cPs0M*ZC;$l_D+AyM2*eEdPYC!5DhB|JBfSNDK>&Cc04ETL?gMbv z0?y?+0KjLCa6lg>z&ikRfk2eFfCO>}k_8YCY~KSp*atY^7dS_R7Jvl%-UtBjo0hU^V*vEg1GbR101oH{9Rc>iv1|a)1OV7%90TG3``ZQJH;@m@fB1Jm>VQC0|egffkh=$uqJed-0RZv}0(`@m2KqAq9NZt6 z8yLV(NEm>F;{n%Uxwx7*01{jv0I&csb~AM~0;IN`* -

          The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.

          Features

          • distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;

          • FFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;

          • in-place and out-of-place transforms;

          • high scalability up to (at least) tens of thousands of MPI processes.

          Installation

          PencilFFTs can be installed using the Julia package manager:

          julia> ] add PencilFFTs

          Similar projects

          • FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.

          • PFFT is a very general parallel FFT library written in C.

          • P3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.

          • 2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.

          • 1Figure adapted from this PhD thesis.
          • 2For the pencil decomposition represented in the figure, $N = 3$ and $M = 2$.
          +

          More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension $N$. The decompositions can be performed along an arbitrary number $M < N$ of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.

          The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.

          Features

          • distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;

          • FFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;

          • in-place and out-of-place transforms;

          • high scalability up to (at least) tens of thousands of MPI processes.

          Installation

          PencilFFTs can be installed using the Julia package manager:

          julia> ] add PencilFFTs

          Similar projects

          • FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.

          • PFFT is a very general parallel FFT library written in C.

          • P3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.

          • 2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.

          • 1Figure adapted from this PhD thesis.
          • 2For the pencil decomposition represented in the figure, $N = 3$ and $M = 2$.
          diff --git a/previews/PR59/search/index.html b/previews/PR63/search/index.html similarity index 94% rename from previews/PR59/search/index.html rename to previews/PR63/search/index.html index 707ead7f..bb9013b7 100644 --- a/previews/PR59/search/index.html +++ b/previews/PR63/search/index.html @@ -1,2 +1,2 @@ -Search · PencilFFTs.jl

          Loading search...

            +Search · PencilFFTs.jl

            Loading search...

              diff --git a/previews/PR55/search_index.js b/previews/PR63/search_index.js similarity index 100% rename from previews/PR55/search_index.js rename to previews/PR63/search_index.js diff --git a/previews/PR63/siteinfo.js b/previews/PR63/siteinfo.js new file mode 100644 index 00000000..f031a292 --- /dev/null +++ b/previews/PR63/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "previews/PR63"; diff --git a/previews/PR62/tutorial/index.html b/previews/PR63/tutorial/index.html similarity index 98% rename from previews/PR62/tutorial/index.html rename to previews/PR63/tutorial/index.html index 0af4e242..cad960c6 100644 --- a/previews/PR62/tutorial/index.html +++ b/previews/PR63/tutorial/index.html @@ -37,4 +37,4 @@ # Apply backward plan on `v` with `w` as an output w = similar(u) -ldiv!(w, plan, v) # now w ≈ u

              Note that, consistently with AbstractFFTs, normalisation is performed at the end of a backward transform, so that the original data is recovered when applying a forward followed by a backward transform.

              Accessing and modifying data

              For any given MPI process, a PencilArray holds the data associated to its local partition in the global geometry. PencilArrays are accessed using local indices that start at 1, regardless of the location of the local process in the MPI topology. Note that PencilArrays, being based on regular Arrays, support both linear and Cartesian indexing (see the Julia docs for details).

              For convenience, the global_view function can be used to generate an OffsetArray wrapper that takes global indices.

              Output data layout

              In memory, the dimensions of the transform output are by default reversed with respect to the input. That is, if the order of indices in the input data is (x, y, z), then the output has order (z, y, x) in memory. This detail is hidden from the user, and output arrays are always accessed in the same order as the input data, regardless of the underlying output dimension permutation. This applies to PencilArrays and to OffsetArrays returned by global_view.

              The reasoning behind dimension permutations, is that they allow to always perform FFTs along the fastest array dimension and to avoid a local data transposition, resulting in performance gains. A similar approach is followed by other parallel FFT libraries. FFTW itself, in its distributed-memory routines, includes a flag that enables a similar behaviour. In PencilFFTs, index permutation is the default, but it can be disabled via the permute_dims flag of PencilFFTPlan.

              A great deal of work has been spent in making generic index permutations as efficient as possible, both in intermediate and in the output state of the multidimensional transforms. This has been achieved, in part, by making sure that permutations such as (3, 2, 1) are compile-time constants.

              Further reading

              For details on working with PencilArrays see the PencilArrays docs.

              The examples on the sidebar further illustrate the use of transforms and provide an introduction to working with MPI-distributed data in the form of PencilArrays. In particular, the gradient example illustrates different ways of computing things using Fourier-transformed distributed arrays. Then, the incompressible Navier–Stokes example is a more advanced and complete example of a possible application of the PencilFFTs package.

              +ldiv!(w, plan, v) # now w ≈ u

              Note that, consistently with AbstractFFTs, normalisation is performed at the end of a backward transform, so that the original data is recovered when applying a forward followed by a backward transform.

              Accessing and modifying data

              For any given MPI process, a PencilArray holds the data associated to its local partition in the global geometry. PencilArrays are accessed using local indices that start at 1, regardless of the location of the local process in the MPI topology. Note that PencilArrays, being based on regular Arrays, support both linear and Cartesian indexing (see the Julia docs for details).

              For convenience, the global_view function can be used to generate an OffsetArray wrapper that takes global indices.

              Output data layout

              In memory, the dimensions of the transform output are by default reversed with respect to the input. That is, if the order of indices in the input data is (x, y, z), then the output has order (z, y, x) in memory. This detail is hidden from the user, and output arrays are always accessed in the same order as the input data, regardless of the underlying output dimension permutation. This applies to PencilArrays and to OffsetArrays returned by global_view.

              The reasoning behind dimension permutations, is that they allow to always perform FFTs along the fastest array dimension and to avoid a local data transposition, resulting in performance gains. A similar approach is followed by other parallel FFT libraries. FFTW itself, in its distributed-memory routines, includes a flag that enables a similar behaviour. In PencilFFTs, index permutation is the default, but it can be disabled via the permute_dims flag of PencilFFTPlan.

              A great deal of work has been spent in making generic index permutations as efficient as possible, both in intermediate and in the output state of the multidimensional transforms. This has been achieved, in part, by making sure that permutations such as (3, 2, 1) are compile-time constants.

              Further reading

              For details on working with PencilArrays see the PencilArrays docs.

              The examples on the sidebar further illustrate the use of transforms and provide an introduction to working with MPI-distributed data in the form of PencilArrays. In particular, the gradient example illustrates different ways of computing things using Fourier-transformed distributed arrays. Then, the incompressible Navier–Stokes example is a more advanced and complete example of a possible application of the PencilFFTs package.

              diff --git a/previews/PR68/.documenter-siteinfo.json b/previews/PR68/.documenter-siteinfo.json new file mode 100644 index 00000000..7e116849 --- /dev/null +++ b/previews/PR68/.documenter-siteinfo.json @@ -0,0 +1 @@ +{"documenter":{"julia_version":"1.9.3","generation_timestamp":"2023-09-18T13:25:03","documenter_version":"1.0.1"}} \ No newline at end of file diff --git a/previews/PR68/GlobalFFTParams/index.html b/previews/PR68/GlobalFFTParams/index.html new file mode 100644 index 00000000..0fc32cc8 --- /dev/null +++ b/previews/PR68/GlobalFFTParams/index.html @@ -0,0 +1,9 @@ + +Global FFT parameters · PencilFFTs.jl
              GitHub

              Global FFT parameters

              PencilFFTs.GlobalFFTParams — Type
              GlobalFFTParams{T, N, inplace}

              Specifies the global parameters for an N-dimensional distributed transform. These include the element type T and global data sizes of input and output data, as well as the transform types to be performed along each dimension.

              GlobalFFTParams(size_global, transforms, [real_type=Float64])

              Define parameters for N-dimensional transform.

              transforms must be a tuple of length N specifying the transforms to be applied along each dimension. Each element must be a subtype of Transforms.AbstractTransform. For all the possible transforms, see Transform types.

              The element type must be a real type accepted by FFTW, i.e. either Float32 or Float64.

              Note that the transforms are applied one dimension at a time, with the leftmost dimension first for forward transforms.

              Example

              To perform a 3D FFT of real data, first a real-to-complex FFT must be applied along the first dimension, followed by two complex-to-complex FFTs along the other dimensions:

              julia> size_global = (64, 32, 128);  # size of real input data
+
+julia> transforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT());
+
+julia> fft_params = PencilFFTs.GlobalFFTParams(size_global, transforms)
+Transforms: (RFFT, FFT, FFT)
+Input type: Float64
+Global dimensions: (64, 32, 128) -> (33, 32, 128)
              source
              diff --git a/previews/PR68/PencilFFTs/index.html b/previews/PR68/PencilFFTs/index.html new file mode 100644 index 00000000..df90388d --- /dev/null +++ b/previews/PR68/PencilFFTs/index.html @@ -0,0 +1,31 @@ + +Distributed FFT plans · PencilFFTs.jl
              GitHub

              Distributed FFT plans

              Distributed FFTs are implemented in the PencilFFTs module, and are built on top of the PencilArrays package.

              Creating plans

              PencilFFTs.PencilFFTPlan — Type
              PencilFFTPlan{T,N} <: AbstractFFTs.Plan{T}

              Plan for N-dimensional FFT-based transform on MPI-distributed data, where input data has type T.

              PencilFFTPlan(p::Pencil, transforms; kwargs...)

              Create a PencilFFTPlan for distributed arrays following a given Pencil configuration. See variant below for details on the specification of transforms and on possible keyword arguments.

              PencilFFTPlan(
+    A::PencilArray, transforms;
+    fftw_flags = FFTW.ESTIMATE,
+    fftw_timelimit = FFTW.NO_TIMELIMIT,
+    permute_dims = Val(true),
+    transpose_method = Transpositions.PointToPoint(),
+    timer = timer(pencil(A)),
+)

              Create plan for N-dimensional transform on MPI-distributed PencilArrays.

              Extended help

              This creates a PencilFFTPlan for arrays sharing the same properties as A (dimensions, MPI decomposition, memory layout, ...), which describe data on an N-dimensional domain.

              Transforms

              The transforms to be applied along each dimension are specified by the transforms argument. Possible transforms are defined as subtypes of Transforms.AbstractTransform, and are listed in Transform types. This argument may be either:

              • a tuple of N transforms to be applied along each dimension. For instance, transforms = (Transforms.R2R(FFTW.REDFT01), Transforms.RFFT(), Transforms.FFT());

              • a single transform to be applied along all dimensions. The input is automatically expanded into N equivalent transforms. For instance, for a three-dimensional array, transforms = Transforms.RFFT() specifies a 3D real-to-complex transform, and is equivalent to passing (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT()).

              Note that forward transforms are applied from left to right. In the last example, this means that a real-to-complex transform (RFFT) is first performed along the first dimension. This is followed by complex-to-complex transforms (FFT) along the second and third dimensions.

              Input data layout

              The input PencilArray must satisfy the following constraints:

              • array dimensions must not be permuted. This is the default when constructing PencilArrays.

              • for an M-dimensional domain decomposition (with M < N), the input array must be decomposed along the last M dimensions. For example, for a 2D decomposition of 3D data, the decomposed dimensions must be (2, 3). In particular, the first array dimension must not be distributed among different MPI processes.

                In the PencilArrays package, the decomposed dimensions are specified at the moment of constructing a Pencil.

              • the element type must be compatible with the specified transform. For instance, real-to-complex transforms (Transforms.RFFT) require the input to be real floating point values. Other transforms, such as Transforms.R2R, accept both real and complex data.

              Keyword arguments

              • The keyword arguments fftw_flags and fftw_timelimit are passed to the FFTW plan creation functions (see AbstractFFTs docs).

              • permute_dims determines whether the indices of the output data should be reversed. For instance, if the input data has global dimensions (Nx, Ny, Nz), then the output of a complex-to-complex FFT would have dimensions (Nz, Ny, Nx). This enables FFTs to always be performed along the first (i.e. fastest) array dimension, which could lead to performance gains. This option is enabled by default. For type inference reasons, it must be a value type (Val(true) or Val(false)).

              • transpose_method allows to select between implementations of the global data transpositions. See PencilArrays docs docs for details.

              • timer should be a TimerOutput object. See Measuring performance for details.

              PencilFFTPlan(
+    dims_global::Dims{N}, transforms, proc_dims::Dims{M}, comm::MPI.Comm,
+    [real_type = Float64]; extra_dims = (), kws...
+)

              Create plan for N-dimensional transform.

              Extended help

              Instead of taking a PencilArray or a Pencil, this constructor requires the global dimensions of the input data, passed via the size_global argument.

              The data is distributed over the MPI processes in the comm communicator. The distribution is performed over M dimensions (with M < N) according to the values in proc_dims, which specifies the number of MPI processes to put along each dimension.

              PencilArrays that may be transformed with the returned plan can be created using allocate_input.

              Optional arguments

              • The floating point precision can be selected by setting real_type parameter, which is Float64 by default.

              • extra_dims may be used to specify the sizes of one or more extra dimensions that should not be transformed. These dimensions will be added to the rightmost (i.e. slowest) indices of the arrays. See Extra dimensions below for usage hints.

              • see the other constructor for more keyword arguments.

              Extra dimensions

              One possible application of extra_dims is for describing the components of a vector or tensor field. However, this means that different PencilFFTPlans would need to be created for each kind of field (scalar, vector, ...). To avoid the creation of multiple plans, a possibly better alternative is to create tuples (or arrays) of PencilArrays using allocate_input and allocate_output.

              Another more legitimate usage of extra_dims is to specify one or more Cartesian dimensions that should not be transformed nor split among MPI processes.

              Example

              Suppose we want to perform a 3D FFT of real data. The data is to be decomposed along two dimensions, over 8 MPI processes:

              size_global = (64, 32, 128)  # size of real input data
+
+# Perform real-to-complex transform along the first dimension, then
+# complex-to-complex transforms along the other dimensions.
+transforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())
+# transforms = Transforms.RFFT()  # this is equivalent to the above line
+
+proc_dims = (4, 2)  # 2D decomposition
+comm = MPI.COMM_WORLD
+
+plan = PencilFFTPlan(size_global, transforms, proc_dims, comm)
              source

              Allocating data

              PencilFFTs.allocate_input — Function
              allocate_input(p::PencilFFTPlan)          -> PencilArray
+allocate_input(p::PencilFFTPlan, dims...) -> Array{PencilArray}
+allocate_input(p::PencilFFTPlan, Val(N))  -> NTuple{N, PencilArray}

              Allocate uninitialised PencilArray that can hold input data for the given plan.

              The second and third forms respectively allocate an array of PencilArrays of size dims, and a tuple of N PencilArrays.

              In-place plans

              If p is an in-place real-to-real or complex-to-complex plan, a ManyPencilArray is allocated. If p is an in-place real-to-complex plan, a ManyPencilArrayRFFT! is allocated.

              These types hold PencilArray wrappers for the input and output transforms (as well as for intermediate transforms) which share the same space in memory. The input and output PencilArrays should be respectively accessed by calling first(::ManyPencilArray) and last(::ManyPencilArray).

              Example

              Suppose p is an in-place PencilFFTPlan. Then,

              @assert is_inplace(p)
+A = allocate_input(p) :: ManyPencilArray
+v_in = first(A)       :: PencilArray  # input data view
+v_out = last(A)       :: PencilArray  # output data view

              Also note that in-place plans must be performed directly on the returned ManyPencilArray, and not on the contained PencilArray views:

              p * A       # perform forward transform in-place
+p \ A       # perform backward transform in-place
+# p * v_in  # not allowed!!
              source
              PencilFFTs.allocate_output — Function
              allocate_output(p::PencilFFTPlan)          -> PencilArray
+allocate_output(p::PencilFFTPlan, dims...) -> Array{PencilArray}
+allocate_output(p::PencilFFTPlan, Val(N))  -> NTuple{N, PencilArray}

              Allocate uninitialised PencilArray that can hold output data for the given plan.

              If p is an in-place plan, a ManyPencilArray is allocated.

              See allocate_input for details.

              source

              Methods

              Internals

              PencilFFTs.ManyPencilArrayRFFT! — Type
              ManyPencilArrayRFFT!{T,N,M} <: AbstractManyPencilArray{N,M}

              Container holding M different PencilArray views to the same underlying data buffer. All views share the same and dimensionality N. The element type T of the first view is real, that of subsequent views is Complex{T}.

              This can be used to perform in-place real-to-complex plan, see alsoTransforms.RFFT!. It is used internally for such transforms by allocate_input and should not be constructed directly.

              ManyPencilArrayRFFT!{T}(undef, pencils...; extra_dims=())

              Create a ManyPencilArrayRFFT! container that can hold data of type T and Complex{T} associated to all the given Pencils.

              The optional extra_dims argument is the same as for PencilArray.

              See also ManyPencilArray

              source
              diff --git a/previews/PR68/PencilFFTs_timers/index.html b/previews/PR68/PencilFFTs_timers/index.html new file mode 100644 index 00000000..2896bb8f --- /dev/null +++ b/previews/PR68/PencilFFTs_timers/index.html @@ -0,0 +1,24 @@ + +Measuring performance · PencilFFTs.jl
              GitHub

              Measuring performance

              It is possible to measure the time spent in different sections of the distributed transforms using the TimerOutputs package. This has a (very small) performance overhead, so it is disabled by default. To enable time measurements, call TimerOutputs.enable_debug_timings after loading PencilFFTs (see below for an example). For more details see the TimerOutputs docs.

              Minimal example:

              using MPI
+using PencilFFTs
+using TimerOutputs
+
+# Enable timing of `PencilFFTs` functions
+TimerOutputs.enable_debug_timings(PencilFFTs)
+TimerOutputs.enable_debug_timings(PencilArrays)
+TimerOutputs.enable_debug_timings(Transpositions)
+
+MPI.Init()
+
+plan = PencilFFTPlan(#= args... =#)
+
+# [do stuff with `plan`...]
+
+# Retrieve and print timing data associated to `plan`
+to = timer(plan)
+print_timer(to)

              By default, each PencilFFTPlan has its own TimerOutput. If you already have a TimerOutput, you can pass it to the PencilFFTPlan constructor:

              to = TimerOutput()
+plan = PencilFFTPlan(..., timer=to)
+
+# [do stuff with `plan`...]
+
+print_timer(to)
              diff --git a/previews/PR68/Transforms/index.html b/previews/PR68/Transforms/index.html new file mode 100644 index 00000000..9cc385ef --- /dev/null +++ b/previews/PR68/Transforms/index.html @@ -0,0 +1,75 @@ + +Available transforms · PencilFFTs.jl
              GitHub

              Available transforms

              Transform types

              PencilFFTs.Transforms.BFFT — Type
              BFFT()

              Unnormalised backward complex-to-complex FFT.

              Like AbstractFFTs.bfft, this transform is not normalised. To obtain the inverse transform, divide the output by the length of the transformed dimension.

              See also AbstractFFTs.bfft.

              source
              PencilFFTs.Transforms.BRFFT — Type
              BRFFT(d::Integer)
+BRFFT((d1, d2, ..., dN))

              Unnormalised inverse of RFFT.

              To obtain the inverse transform, divide the output by the length of the transformed dimension (of the real output array).

              As described in the AbstractFFTs docs, the length of the output cannot be fully inferred from the input length. For this reason, the BRFFT constructor accepts an optional d argument indicating the output length.

              For multidimensional datasets, a tuple of dimensions (d1, d2, ..., dN) may also be passed. This is equivalent to passing just dN. In this case, the last dimension (dN) is the one that changes size between the input and output. Note that this is the opposite of FFTW.brfft. The reason is that, in PencilFFTs, the last dimension is the one along which a complex-to-real transform is performed.

              See also AbstractFFTs.brfft.

              source

              Internals

              What follows is used internally in PencilFFTs.

              Types

              PencilFFTs.Transforms.AbstractCustomPlan — Type
              AbstractCustomPlan

              Abstract type defining a custom plan, to be used as an alternative to FFTW plans (FFTW.FFTWPlan).

              The only custom plan defined in this module is IdentityPlan. The user can define other custom plans that are also subtypes of AbstractCustomPlan.

              Note that plan returns a subtype of either AbstractFFTs.Plan or AbstractCustomPlan.

              source

              Functions

              PencilFFTs.Transforms.plan — Function
              plan(transform::AbstractTransform, A, [dims];
+     flags=FFTW.ESTIMATE, timelimit=Inf)

              Create plan to transform array A along dimensions dims.

              If dims is not specified, all dimensions of A are transformed.

              For FFT plans, this function wraps the AbstractFFTs.jl and FFTW.jl plan creation functions. For more details on the function arguments, see AbstractFFTs.plan_fft.

              source
              PencilFFTs.Transforms.binv — Function
              binv(transform::AbstractTransform, d::Integer)

              Returns the backwards transform associated to the given transform.

              The second argument must be the length of the first transformed dimension in the forward transform. It is used in particular when transform = RFFT(), to determine the length of the inverse (complex-to-real) transform. See the AbstractFFTs.irfft docs for details.

              The backwards transform returned by this function is not normalised. The normalisation factor for a given array can be obtained by calling scale_factor.

              Example

              julia> binv(Transforms.FFT(), 42)
+BFFT
+
+julia> binv(Transforms.BRFFT(9), 42)
+RFFT
              source
              PencilFFTs.Transforms.scale_factor — Function
              scale_factor(transform::AbstractTransform, A, [dims = 1:ndims(A)])

              Get factor required to normalise the given array after a transformation along dimensions dims (all dimensions by default).

              The array A must have the dimensions of the transform input.

              Important: the dimensions dims must be the same that were passed to plan.

              Examples

              julia> C = zeros(ComplexF32, 3, 4, 5);
+
+julia> scale_factor(Transforms.FFT(), C)
+60
+
+julia> scale_factor(Transforms.BFFT(), C)
+60
+
+julia> scale_factor(Transforms.BFFT(), C, 2:3)
+20
+
+julia> R = zeros(Float64, 3, 4, 5);
+
+julia> scale_factor(Transforms.RFFT(), R, 2)
+4
+
+julia> scale_factor(Transforms.RFFT(), R, 2:3)
+20
+
+julia> scale_factor(Transforms.BRFFT(8), C)
+96
+
+julia> scale_factor(Transforms.BRFFT(9), C)
+108

              This will fail because the input of RFFT is real, and R is a complex array:

              julia> scale_factor(Transforms.RFFT(), C, 2:3)
+ERROR: MethodError: no method matching scale_factor(::PencilFFTs.Transforms.RFFT, ::Array{ComplexF32, 3}, ::UnitRange{Int64})
              source
              PencilFFTs.Transforms.eltype_input — Function
              eltype_input(transform::AbstractTransform, real_type<:AbstractFloat)

              Determine input data type for a given transform given the floating point precision of the input data.

              Some transforms, such as R2R and NoTransform, can take both real and complex data. For those kinds of transforms, nothing is returned.

              Example

              julia> eltype_input(Transforms.FFT(), Float32)
+ComplexF32 (alias for Complex{Float32})
+
+julia> eltype_input(Transforms.RFFT(), Float64)
+Float64
+
+julia> eltype_input(Transforms.R2R(FFTW.REDFT01), Float64)  # nothing
+
+julia> eltype_input(Transforms.NoTransform(), Float64)  # nothing
+
              source
              PencilFFTs.Transforms.eltype_output — Function
              eltype_output(transform::AbstractTransform, eltype_input)

              Returns the output data type for a given transform given the input type.

              Throws ArgumentError if the input data type is incompatible with the transform type.

              Example

              julia> eltype_output(Transforms.NoTransform(), Float32)
+Float32
+
+julia> eltype_output(Transforms.RFFT(), Float64)
+ComplexF64 (alias for Complex{Float64})
+
+julia> eltype_output(Transforms.BRFFT(4), ComplexF32)
+Float32
+
+julia> eltype_output(Transforms.FFT(), Float64)
+ERROR: ArgumentError: invalid input data type for PencilFFTs.Transforms.FFT: Float64
              source
              PencilFFTs.Transforms.expand_dims — Function
              expand_dims(transform::AbstractTransform, Val(N))

              Expand a single multidimensional transform into one transform per dimension.

              Example

              # Expand a real-to-complex transform in 3 dimensions.
+julia> expand_dims(Transforms.RFFT(), Val(3))
+(RFFT, FFT, FFT)
+
+julia> expand_dims(Transforms.BRFFT(4), Val(3))
+(BFFT, BFFT, BRFFT{even})
+
+julia> expand_dims(Transforms.NoTransform(), Val(2))
+(NoTransform, NoTransform)
              source
              PencilFFTs.Transforms.is_inplace — Function
              is_inplace(transform::AbstractTransform)         -> Bool
+is_inplace(transforms::Vararg{AbtractTransform}) -> Union{Bool, Nothing}

              Check whether a transform or a list of transforms is performed in-place.

              If the list of transforms has a combination of in-place and out-of-place transforms, nothing is returned.

              Example

              julia> is_inplace(Transforms.RFFT())
+false
+
+julia> is_inplace(Transforms.NoTransform!())
+true
+
+julia> is_inplace(Transforms.FFT!(), Transforms.R2R!(FFTW.REDFT01))
+true
+
+julia> is_inplace(Transforms.FFT(), Transforms.R2R(FFTW.REDFT01))
+false
+
+julia> is_inplace(Transforms.FFT(), Transforms.R2R!(FFTW.REDFT01)) === nothing
+true
+
              source
              PencilFFTs.Transforms.length_output — Function
              length_output(transform::AbstractTransform, length_in::Integer)

              Returns the length of the transform output, given the length of its input.

              The input and output lengths are specified in terms of the respective input and output datatypes. For instance, for real-to-complex transforms, these are respectively the length of input real data and of output complex data.

              source
              diff --git a/previews/PR40/assets/custom.css b/previews/PR68/assets/custom.css similarity index 100% rename from previews/PR40/assets/custom.css rename to previews/PR68/assets/custom.css diff --git a/previews/PR68/assets/documenter.js b/previews/PR68/assets/documenter.js new file mode 100644 index 00000000..7002e251 --- /dev/null +++ b/previews/PR68/assets/documenter.js @@ -0,0 +1,874 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min', + 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min', + 'minisearch': 'https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min', + 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min', + }, + shim: { + "highlight-julia": { + "deps": [ + "highlight" + ] + }, + "katex-auto-render": { + "deps": [ + "katex" + ] + }, + "headroom-jquery": { + "deps": [ + "jquery", + "headroom" + ] + }, + "highlight-julia-repl": { + "deps": [ + "highlight" + ] + } +} +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) { +$(document).ready(function() { + renderMathInElement( + document.body, + { + "delimiters": [ + { + "left": "$", + "right": "$", + "display": false + }, + { + "left": "$$", + "right": "$$", + "display": true + }, + { + "left": "\\[", + "right": "\\]", + "display": true + } + ] +} + + ); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) { +$(document).ready(function() { + hljs.highlightAll(); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +var isExpanded = true; + +$(document).on("click", ".docstring header", function () { + let articleToggleTitle = "Expand docstring"; + + if ($(this).siblings("section").is(":visible")) { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + } else { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + articleToggleTitle = "Collapse docstring"; + } + + $(this) + .find(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + $(this).siblings("section").slideToggle(); +}); + +$(document).on("click", ".docs-article-toggle-button", function () { + let articleToggleTitle = "Expand docstring"; + let navArticleToggleTitle = "Expand all docstrings"; + + if (isExpanded) { + $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + + isExpanded = false; + + $(".docstring section").slideUp(); + } else { + $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + isExpanded = true; + articleToggleTitle = "Collapse docstring"; + navArticleToggleTitle = "Collapse all docstrings"; + + $(".docstring section").slideDown(); + } + + $(this).prop("title", navArticleToggleTitle); + $(".docstring-article-toggle-button").prop("title", articleToggleTitle); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require([], function() { +function addCopyButtonCallbacks() { + for (const el of document.getElementsByTagName("pre")) { + const button = document.createElement("button"); + button.classList.add("copy-button", "fa-solid", "fa-copy"); + button.setAttribute("aria-label", "Copy this code block"); + button.setAttribute("title", "Copy"); + + el.appendChild(button); + + const success = function () { + button.classList.add("success", "fa-check"); + button.classList.remove("fa-copy"); + }; + + const failure = function () { + button.classList.add("error", "fa-xmark"); + button.classList.remove("fa-copy"); + }; + + button.addEventListener("click", function () { + copyToClipboard(el.innerText).then(success, failure); + + setTimeout(function () { + button.classList.add("fa-copy"); + button.classList.remove("success", "fa-check", "fa-xmark"); + }, 5000); + }); + } +} + +function copyToClipboard(text) { + // clipboard API is only available in secure contexts + if (window.navigator && window.navigator.clipboard) { + return window.navigator.clipboard.writeText(text); + } else { + return new Promise(function (resolve, reject) { + try { + const el = document.createElement("textarea"); + el.textContent = text; + el.style.position = "fixed"; + el.style.opacity = 0; + document.body.appendChild(el); + el.select(); + document.execCommand("copy"); + + resolve(); + } catch (err) { + reject(err); + } finally { + document.body.removeChild(el); + } + }); + } +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks); +} else { + addCopyButtonCallbacks(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { + +// Manages the top navigation bar (hides it when the user starts scrolling down on the +// mobile). +window.Headroom = Headroom; // work around buggy module loading? +$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'minisearch'], function($, minisearch) { + +// In general, most search related things will have "search" as a prefix. +// To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +let results = []; +let timer = undefined; + +let data = documenterSearchIndex["docs"].map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; +}); + +// list below is the lunr 2.1.3 list minus the intersect with names(Base) +// (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) +// ideally we'd just filter the original list but it's not available as a variable +const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", +]); + +let index = new minisearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with search results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + boost: { title: 100 }, + fuzzy: 2, + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + } + + return word ?? null; + }, + tokenize: (string) => string.split(/[\s\-\.]+/), + }, +}); + +index.addAll(data); + +let filters = [...new Set(data.map((x) => x.category))]; +var modal_filters = make_modal_body_filters(filters); +var filter_results = []; + +$(document).on("keyup", ".documenter-search-input", function (event) { + // Adding a debounce to prevent disruptions from super-speed typing! + debounce(() => update_search(filter_results), 300); +}); + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + $(this).removeClass("search-filter-selected"); + } else { + $(this).addClass("search-filter-selected"); + } + + // Adding a debounce to prevent disruptions from crazy clicking! + debounce(() => get_filters(), 300); +}); + +/** + * A debounce function, takes a function and an optional timeout in milliseconds + * + * @function callback + * @param {number} timeout + */ +function debounce(callback, timeout = 300) { + clearTimeout(timer); + timer = setTimeout(callback, timeout); +} + +/** + * Make/Update the search component + * + * @param {string[]} selected_filters + */ +function update_search(selected_filters = []) { + let initial_search_body = ` +
              Type something to get started!
              + `; + + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + results = index.search(querystring, { + filter: (result) => { + // Filtering results + if (selected_filters.length === 0) { + return result.score >= 1; + } else { + return ( + result.score >= 1 && selected_filters.includes(result.category) + ); + } + }, + }); + + let search_result_container = ``; + let search_divider = `
              `; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + results.forEach(function (result) { + if (result.location) { + // Checking for duplication of results for the same page + if (!links.includes(result.location)) { + search_results += make_search_result(result, querystring); + count++; + } + + links.push(result.location); + } + }); + + let result_count = `
              ${count} result(s)
              `; + + search_result_container = ` +
              + ${modal_filters} + ${search_divider} + ${result_count} +
              + ${search_results} +
              +
              + `; + } else { + search_result_container = ` +
              + ${modal_filters} + ${search_divider} +
              0 result(s)
              +
              +
              No result found!
              + `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + filter_results = []; + modal_filters = make_modal_body_filters(filters, filter_results); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(initial_search_body); + } +} + +/** + * Make the modal filter html + * + * @param {string[]} filters + * @param {string[]} selected_filters + * @returns string + */ +function make_modal_body_filters(filters, selected_filters = []) { + let str = ``; + + filters.forEach((val) => { + if (selected_filters.includes(val)) { + str += `${val}`; + } else { + str += `${val}`; + } + }); + + let filter_html = ` +
              + Filters: + ${str} +
              + `; + + return filter_html; +} + +/** + * Make the result component given a minisearch result data object and the value of the search input as queryString. + * To view the result object structure, refer: https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ +function make_search_result(result, querystring) { + let search_divider = `
              `; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + + let textindex = new RegExp(`\\b${querystring}\\b`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`\\b${querystring}\\b`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
              +
              ${result.title}
              +
              ${result.category}
              +
              +

              + ${display_result} +

              +
              + ${display_link} +
              +               + ${search_divider} + `; + + return result_div; +} + +/** + * Get selected filters, remake the filter html and lastly update the search modal + */ +function get_filters() { + let ele = $(".search-filters .search-filter-selected").get(); + filter_results = ele.map((x) => $(x).text().toLowerCase()); + modal_filters = make_modal_body_filters(filters, filter_results); + update_search(filter_results); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); + }); + // Close the dialog if X is clicked + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); + }); + // Close dialog if ESC is pressed + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let search_modal_header = ` +
              +
              +

              + + + + +

              +
              +
              + +
              +
              +`; + +let initial_search_body = ` +
              Type something to get started!
              +`; + +let search_modal_footer = ` +
              + + Ctrl + + / to search + + esc to close +
              +`; + +$(document.body).append( + ` + + ` +); + +document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); +}); + +document.querySelector(".close-search-modal").addEventListener("click", () => { + closeModal(); +}); + +$(document).on("click", ".search-result-link", function () { + closeModal(); +}); + +document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; +}); + +// Functions to open and close a modal +function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); +} + +function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
              Type something to get started!
              + `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); +} + +document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function () { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { + ev.preventDefault(); + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind("click", function (ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); + } + }); +}); + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function () { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css("max-width"), 10); + var L0 = e.width(); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on("orientationchange", resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function () { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if (typeof active !== "undefined") { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Theme picker setup +$(document).ready(function () { + // onchange callback + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); + } + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function () { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== "undefined") { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $( + "" + ); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}); + +}) diff --git a/previews/PR40/assets/logo.svg b/previews/PR68/assets/logo.svg similarity index 100% rename from previews/PR40/assets/logo.svg rename to previews/PR68/assets/logo.svg diff --git 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html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark 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.button.is-warning.is-active{background-color:#946e00;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-warning[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning{background-color:#ad8100;border-color:#ad8100;box-shadow:none}html.theme--documenter-dark .button.is-warning.is-inverted{background-color:#fff;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark 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.button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #9e1b0d #9e1b0d !important}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fdeeec;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#fce3e0;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#fcd8d5;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--documenter-dark .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--documenter-dark .buttons .button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--documenter-dark .buttons.has-addons .button:last-child{margin-right:0}html.theme--documenter-dark .buttons.has-addons .button:hover,html.theme--documenter-dark .buttons.has-addons .button.is-hovered{z-index:2}html.theme--documenter-dark .buttons.has-addons .button:focus,html.theme--documenter-dark .buttons.has-addons .button.is-focused,html.theme--documenter-dark .buttons.has-addons .button:active,html.theme--documenter-dark .buttons.has-addons .button.is-active,html.theme--documenter-dark .buttons.has-addons .button.is-selected{z-index:3}html.theme--documenter-dark .buttons.has-addons .button:focus:hover,html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover,html.theme--documenter-dark .buttons.has-addons .button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons 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+ Maintainer: @highlightjs/core-team + Website: https://highlightjs.org/ + License: see project LICENSE + Touched: 2021 +*/pre code.hljs{display:block;overflow-x:auto;padding:1em}code.hljs{padding:3px 5px}.hljs{background:#F3F3F3;color:#444}.hljs-comment{color:#697070}.hljs-tag,.hljs-punctuation{color:#444a}.hljs-tag .hljs-name,.hljs-tag .hljs-attr{color:#444}.hljs-keyword,.hljs-attribute,.hljs-selector-tag,.hljs-meta .hljs-keyword,.hljs-doctag,.hljs-name{font-weight:bold}.hljs-type,.hljs-string,.hljs-number,.hljs-selector-id,.hljs-selector-class,.hljs-quote,.hljs-template-tag,.hljs-deletion{color:#880000}.hljs-title,.hljs-section{color:#880000;font-weight:bold}.hljs-regexp,.hljs-symbol,.hljs-variable,.hljs-template-variable,.hljs-link,.hljs-selector-attr,.hljs-operator,.hljs-selector-pseudo{color:#ab5656}.hljs-literal{color:#695}.hljs-built_in,.hljs-bullet,.hljs-code,.hljs-addition{color:#397300}.hljs-meta{color:#1f7199}.hljs-meta .hljs-string{color:#38a}.hljs-emphasis{font-style:italic}.hljs-strong{font-weight:bold} diff --git a/previews/PR68/assets/themeswap.js b/previews/PR68/assets/themeswap.js new file mode 100644 index 00000000..9f5eebe6 --- /dev/null +++ b/previews/PR68/assets/themeswap.js @@ -0,0 +1,84 @@ +// Small function to quickly swap out themes. Gets put into the tag.. +function set_theme_from_local_storage() { + // Initialize the theme to null, which means default + var theme = null; + // If the browser supports the localstorage and is not disabled then try to get the + // documenter theme + if (window.localStorage != null) { + // Get the user-picked theme from localStorage. May be `null`, which means the default + // theme. + theme = window.localStorage.getItem("documenter-theme"); + } + // Check if the users preference is for dark color scheme + var darkPreference = + window.matchMedia("(prefers-color-scheme: dark)").matches === true; + // Initialize a few variables for the loop: + // + // - active: will contain the index of the theme that should be active. Note that there + // is no guarantee that localStorage contains sane values. If `active` stays `null` + // we either could not find the theme or it is the default (primary) theme anyway. + // Either way, we then need to stick to the primary theme. + // + // - disabled: style sheets that should be disabled (i.e. all the theme style sheets + // that are not the currently active theme) + var active = null; + var disabled = []; + var primaryLightTheme = null; + var primaryDarkTheme = null; + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // To distinguish the default (primary) theme, it needs to have the data-theme-primary + // attribute set. + if (ss.ownerNode.getAttribute("data-theme-primary") !== null) { + primaryLightTheme = themename; + } + // Check if the theme is primary dark theme so that we could store its name in darkTheme + if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) { + primaryDarkTheme = themename; + } + // If we find a matching theme (and it's not the default), we'll set active to non-null + if (themename === theme) active = i; + // Store the style sheets of inactive themes so that we could disable them + if (themename !== theme) disabled.push(ss); + } + var activeTheme = null; + if (active !== null) { + // If we did find an active theme, we'll (1) add the theme--$(theme) class to + document.getElementsByTagName("html")[0].className = "theme--" + theme; + activeTheme = theme; + } else { + // If we did _not_ find an active theme, then we need to fall back to the primary theme + // which can either be dark or light, depending on the user's OS preference. + var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme; + // In case it somehow happens that the relevant primary theme was not found in the + // preceding loop, we abort without doing anything. + if (activeTheme === null) { + console.error("Unable to determine primary theme."); + return; + } + // When switching to the primary light theme, then we must not have a class name + // for the tag. That's only for non-primary or the primary dark theme. + if (darkPreference) { + document.getElementsByTagName("html")[0].className = + "theme--" + activeTheme; + } else { + document.getElementsByTagName("html")[0].className = ""; + } + } + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // we'll disable all the stylesheets, except for the active one + ss.disabled = !(themename == activeTheme); + } +} +set_theme_from_local_storage(); diff --git a/previews/PR40/assets/tomate.js b/previews/PR68/assets/tomate.js similarity index 100% rename from previews/PR40/assets/tomate.js rename to previews/PR68/assets/tomate.js diff --git a/previews/PR68/assets/warner.js b/previews/PR68/assets/warner.js new file mode 100644 index 00000000..3f6f5d00 --- /dev/null +++ b/previews/PR68/assets/warner.js @@ -0,0 +1,52 @@ +function maybeAddWarning() { + // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE + // in siteinfo.js. + // If either of these are undefined something went horribly wrong, so we abort. + if ( + window.DOCUMENTER_NEWEST === undefined || + window.DOCUMENTER_CURRENT_VERSION === undefined || + window.DOCUMENTER_STABLE === undefined + ) { + return; + } + + // Current version is not a version number, so we can't tell if it's the newest version. Abort. + if (!/v(\d+\.)*\d+/.test(window.DOCUMENTER_CURRENT_VERSION)) { + return; + } + + // Current version is newest version, so no need to add a warning. + if (window.DOCUMENTER_NEWEST === window.DOCUMENTER_CURRENT_VERSION) { + return; + } + + // Add a noindex meta tag (unless one exists) so that search engines don't index this version of the docs. + if (document.body.querySelector('meta[name="robots"]') === null) { + const meta = document.createElement("meta"); + meta.name = "robots"; + meta.content = "noindex"; + + document.getElementsByTagName("head")[0].appendChild(meta); + } + + const div = document.createElement("div"); + div.classList.add("outdated-warning-overlay"); + const closer = document.createElement("button"); + closer.classList.add("outdated-warning-closer", "delete"); + closer.addEventListener("click", function () { + document.body.removeChild(div); + }); + const href = window.documenterBaseURL + "/../" + window.DOCUMENTER_STABLE; + div.innerHTML = + 'This documentation is not for the latest stable release, but for either the development version or an older release.
              Click here to go to the documentation for the latest stable release.'; + div.appendChild(closer); + document.body.appendChild(div); +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", maybeAddWarning); +} else { + maybeAddWarning(); +} diff --git a/previews/PR68/benchmarks/index.html b/previews/PR68/benchmarks/index.html new file mode 100644 index 00000000..bdcca257 --- /dev/null +++ b/previews/PR68/benchmarks/index.html @@ -0,0 +1,12 @@ + +Benchmarks · PencilFFTs.jl
              GitHub

              Benchmarks

              The performance of PencilFFTs.jl is comparable to that of other open-source parallel FFT libraries implemented in lower-level languages. Below, we show comparisons with the Fortran implementation of P3DFFT, possibly the most popular of these libraries. The benchmarks were performed on the Jean–Zay cluster of the IDRIS French computing centre (CNRS).

              The figure below shows strong scaling benchmarks of 3D real-to-complex FFTs using 2D ("pencil") decomposition. The benchmarks were run for input arrays of dimensions $N_x × N_y × N_z = 512^3$, $1024^3$ and $2048^3$. Each timing is averaged over 100 repetitions.

              + + Strong scaling of PencilFFTs +

              As seen above, PencilFFTs generally outperforms P3DFFT in its default setting. This is largely explained by the choice of using non-blocking point-to-point MPI communications (via MPI_Isend and MPI_Irecv), while P3DFFT uses collective MPI_Alltoallv calls. This enables PencilFFTs to perform data reordering operations on the partially received data while waiting for the incoming data, leading to better performance. Moreover, in contrast with P3DFFT, the high performance and scalability of PencilFFTs results from a highly generic code, handling decompositions in arbitrary dimensions and a relatively large (and extensible) variety of transformations.

              Note that PencilFFTs can optionally use collective communications (using MPI_Alltoallv) instead of point-to-point communications. For details, see the docs for PencilFFTPlan and for PencilArray transpositions. As seen above, collective communications generally perform worse than point-to-point ones, and runtimes are nearly indistinguishable from those of P3DFFT.

              Benchmark details

              The benchmarks were performed using Julia 1.7-beta3 and Intel MPI 2019. We used PencilFFTs v0.12.5 with FFTW.jl v1.4.3 and MPI.jl v0.19.0. We used the Fortran implementation of P3DFFT, version 2.7.6, which was built with Intel 2019 compilers and linked to FFTW 3.3.8. The cluster where the benchmarks were run has Intel Cascade Lake 6248 processors with 2×20 cores per node.

              The number of MPI processes along each decomposed dimension, $P_1$ and $P_2$, was automatically determined by a call to MPI_Dims_create, which tends to create a balanced decomposition with $P_1 ≈ P_2$. For instance, a total of 1024 processes is divided into $P_1 = P_2 = 32$. Different results may be obtained with other combinations, but this was not benchmarked.

              The source files used to generate this benchmark, as well as the raw benchmark results, are all available in the PencilFFTs repo.

              diff --git a/previews/PR68/generated/ffmpeg.err b/previews/PR68/generated/ffmpeg.err new file mode 100644 index 00000000..17927c05 --- /dev/null +++ b/previews/PR68/generated/ffmpeg.err @@ -0,0 +1,48 @@ +ffmpeg version 4.4.2-0ubuntu0.22.04.1 Copyright (c) 2000-2021 the FFmpeg developers + built with gcc 11 (Ubuntu 11.2.0-19ubuntu1) + configuration: --prefix=/usr --extra-version=0ubuntu0.22.04.1 --toolchain=hardened --libdir=/usr/lib/x86_64-linux-gnu --incdir=/usr/include/x86_64-linux-gnu --arch=amd64 --enable-gpl --disable-stripping --enable-gnutls --enable-ladspa --enable-libaom --enable-libass --enable-libbluray --enable-libbs2b --enable-libcaca --enable-libcdio --enable-libcodec2 --enable-libdav1d --enable-libflite --enable-libfontconfig --enable-libfreetype --enable-libfribidi --enable-libgme --enable-libgsm --enable-libjack --enable-libmp3lame --enable-libmysofa --enable-libopenjpeg --enable-libopenmpt --enable-libopus --enable-libpulse --enable-librabbitmq --enable-librubberband --enable-libshine --enable-libsnappy --enable-libsoxr --enable-libspeex --enable-libsrt --enable-libssh --enable-libtheora --enable-libtwolame --enable-libvidstab --enable-libvorbis --enable-libvpx --enable-libwebp --enable-libx265 --enable-libxml2 --enable-libxvid --enable-libzimg --enable-libzmq --enable-libzvbi --enable-lv2 --enable-omx --enable-openal --enable-opencl --enable-opengl --enable-sdl2 --enable-pocketsphinx --enable-librsvg --enable-libmfx --enable-libdc1394 --enable-libdrm --enable-libiec61883 --enable-chromaprint --enable-frei0r --enable-libx264 --enable-shared + libavutil 56. 70.100 / 56. 70.100 + libavcodec 58.134.100 / 58.134.100 + libavformat 58. 76.100 / 58. 76.100 + libavdevice 58. 13.100 / 58. 13.100 + libavfilter 7.110.100 / 7.110.100 + libswscale 5. 9.100 / 5. 9.100 + libswresample 3. 9.100 / 3. 9.100 + libpostproc 55. 9.100 / 55. 9.100 +Input #0, image2, from '/tmp/jl_54Mz0j/proc1_%04d.png': + Duration: 00:00:04.52, start: 0.000000, bitrate: N/A + Stream #0:0: Video: png, rgb24(pc), 1200x600, 25 fps, 25 tbr, 25 tbn, 25 tbc +Stream mapping: + Stream #0:0 -> #0:0 (png (native) -> h264 (libx264)) +Press [q] to stop, [?] for help +[libx264 @ 0x555562f1a700] using cpu capabilities: MMX2 SSE2Fast SSSE3 SSE4.2 AVX FMA3 BMI2 AVX2 +[libx264 @ 0x555562f1a700] profile High, level 3.1, 4:2:0, 8-bit +[libx264 @ 0x555562f1a700] 264 - core 163 r3060 5db6aa6 - H.264/MPEG-4 AVC codec - Copyleft 2003-2021 - http://www.videolan.org/x264.html - options: cabac=1 ref=3 deblock=1:0:0 analyse=0x3:0x113 me=hex subme=7 psy=1 psy_rd=1.00:0.00 mixed_ref=1 me_range=16 chroma_me=1 trellis=1 8x8dct=1 cqm=0 deadzone=21,11 fast_pskip=1 chroma_qp_offset=-2 threads=3 lookahead_threads=1 sliced_threads=0 nr=0 decimate=1 interlaced=0 bluray_compat=0 constrained_intra=0 bframes=3 b_pyramid=2 b_adapt=1 b_bias=0 direct=1 weightb=1 open_gop=0 weightp=2 keyint=250 keyint_min=25 scenecut=40 intra_refresh=0 rc_lookahead=40 rc=crf mbtree=1 crf=23.0 qcomp=0.60 qpmin=0 qpmax=69 qpstep=4 ip_ratio=1.40 aq=1:1.00 +Output #0, mp4, to 'vorticity_proc1.mp4': + Metadata: + encoder : Lavf58.76.100 + Stream #0:0: Video: h264 (avc1 / 0x31637661), yuv420p(tv, progressive), 1200x600, q=2-31, 25 fps, 12800 tbn + Metadata: + encoder : Lavc58.134.100 libx264 + Side data: + cpb: bitrate max/min/avg: 0/0/0 buffer size: 0 vbv_delay: N/A +frame= 3 fps=0.0 q=0.0 size= 0kB time=00:00:00.00 bitrate=N/A speed= 0x frame= 40 fps=0.0 q=0.0 size= 0kB time=00:00:00.00 bitrate=N/A speed= 0x frame= 58 fps= 55 q=28.0 size= 0kB time=00:00:00.32 bitrate= 1.2kbits/s speed=0.302x frame= 73 fps= 47 q=28.0 size= 0kB time=00:00:00.92 bitrate= 0.4kbits/s speed=0.589x frame= 88 fps= 42 q=28.0 size= 0kB time=00:00:01.52 bitrate= 0.3kbits/s speed=0.733x frame= 100 fps= 38 q=28.0 size= 0kB time=00:00:02.00 bitrate= 0.2kbits/s speed=0.767x frame= 113 fps= 36 q=28.0 size= 0kB time=00:00:02.52 bitrate= 0.2kbits/s speed=0.805x frame= 128 fps= 35 q=28.0 size= 0kB time=00:00:03.12 bitrate= 0.1kbits/s speed=0.857x frame= 143 fps= 34 q=28.0 size= 256kB time=00:00:03.72 bitrate= 563.8kbits/s speed=0.888x frame= 163 fps= 35 q=28.0 size= 256kB time=00:00:04.52 bitrate= 464.0kbits/s speed=0.96x frame= 180 fps= 34 q=28.0 size= 256kB time=00:00:05.20 bitrate= 403.4kbits/s speed=0.995x frame= 195 fps= 34 q=28.0 size= 256kB time=00:00:05.80 bitrate= 361.6kbits/s speed=1.01x frame= 213 fps= 34 q=28.0 size= 256kB time=00:00:06.52 bitrate= 321.7kbits/s speed=1.03x frame= 233 fps= 34 q=28.0 size= 512kB time=00:00:07.32 bitrate= 573.0kbits/s speed=1.06x frame= 250 fps= 34 q=28.0 size= 512kB time=00:00:08.00 bitrate= 524.3kbits/s speed=1.08x frame= 268 fps= 34 q=28.0 size= 512kB time=00:00:08.72 bitrate= 481.0kbits/s speed= 1.1x frame= 283 fps= 33 q=28.0 size= 512kB time=00:00:09.32 bitrate= 450.1kbits/s speed= 1.1x frame= 283 fps= 31 q=-1.0 Lsize= 823kB time=00:00:11.20 bitrate= 602.2kbits/s speed=1.22x +video:819kB audio:0kB subtitle:0kB other streams:0kB global headers:0kB muxing overhead: 0.505071% +[libx264 @ 0x555562f1a700] frame I:2 Avg QP:16.74 size: 37874 +[libx264 @ 0x555562f1a700] frame P:74 Avg QP:25.00 size: 8141 +[libx264 @ 0x555562f1a700] frame B:207 Avg QP:18.64 size: 773 +[libx264 @ 0x555562f1a700] consecutive B-frames: 0.7% 4.2% 3.2% 91.9% +[libx264 @ 0x555562f1a700] mb I I16..4: 31.9% 42.3% 25.8% +[libx264 @ 0x555562f1a700] mb P I16..4: 0.4% 1.5% 0.9% P16..4: 5.6% 6.8% 4.5% 0.0% 0.0% skip:80.3% +[libx264 @ 0x555562f1a700] mb B I16..4: 0.1% 0.1% 0.0% B16..8: 6.8% 0.9% 0.3% direct: 0.3% skip:91.5% L0:45.3% L1:46.7% BI: 8.1% +[libx264 @ 0x555562f1a700] 8x8 transform intra:47.4% inter:37.5% +[libx264 @ 0x555562f1a700] coded y,uvDC,uvAC intra: 40.4% 43.2% 34.8% inter: 2.7% 2.2% 0.6% +[libx264 @ 0x555562f1a700] i16 v,h,dc,p: 68% 20% 9% 3% +[libx264 @ 0x555562f1a700] i8 v,h,dc,ddl,ddr,vr,hd,vl,hu: 29% 11% 29% 5% 5% 4% 6% 4% 7% +[libx264 @ 0x555562f1a700] i4 v,h,dc,ddl,ddr,vr,hd,vl,hu: 29% 18% 17% 5% 6% 5% 8% 4% 7% +[libx264 @ 0x555562f1a700] i8c dc,h,v,p: 68% 15% 12% 5% +[libx264 @ 0x555562f1a700] Weighted P-Frames: Y:0.0% UV:0.0% +[libx264 @ 0x555562f1a700] ref P L0: 69.4% 15.3% 10.3% 5.1% +[libx264 @ 0x555562f1a700] ref B L0: 85.2% 11.8% 3.0% +[libx264 @ 0x555562f1a700] ref B L1: 98.3% 1.7% +[libx264 @ 0x555562f1a700] kb/s:592.37 diff --git a/previews/PR40/generated/ffmpeg.out b/previews/PR68/generated/ffmpeg.out similarity index 100% rename from previews/PR40/generated/ffmpeg.out rename to previews/PR68/generated/ffmpeg.out diff --git a/previews/PR68/generated/gradient/index.html b/previews/PR68/generated/gradient/index.html new file mode 100644 index 00000000..70a7eec4 --- /dev/null +++ b/previews/PR68/generated/gradient/index.html @@ -0,0 +1,110 @@ + +Gradient of a scalar field · PencilFFTs.jl
              GitHub

              Gradient of a scalar field

              This example shows different methods to compute the gradient of a real-valued 3D scalar field $θ(\bm{x})$ in Fourier space, where $\bm{x} = (x, y, z)$. It is assumed that the field is periodic with period $L = 2π$ along all dimensions.

              General procedure

              The discrete Fourier expansion of $θ$ writes

              \[θ(\bm{x}) = ∑_{\bm{k} ∈ \Z^3} \hat{θ}(\bm{k}) \, e^{i \bm{k} ⋅ \bm{x}},\]

              where $\bm{k} = (k_x, k_y, k_z)$ are the Fourier wave numbers and $\hat{θ}$ is the discrete Fourier transform of $θ$. Then, the spatial derivatives of $θ$ are given by

              \[\frac{∂ θ(\bm{x})}{∂ x_i} = +∑_{\bm{k} ∈ \Z^3} i k_i \hat{θ}(\bm{k}) \, e^{i \bm{k} ⋅ \bm{x}},\]

              where the subscript $i$ denotes one of the spatial components $x$, $y$ or $z$.

              In other words, to compute $\bm{∇} θ = (∂_x θ, ∂_y θ, ∂_z θ)$, one has to:

              1. transform $θ$ to Fourier space to obtain $\hat{θ}$,
              2. multiply $\hat{θ}$ by $i \bm{k}$,
              3. transform the result back to physical space to obtain $\bm{∇} θ$.

              Preparation

              In this section, we initialise a random real-valued scalar field $θ$ and compute its FFT. For more details see the Tutorial.

              using MPI
+using PencilFFTs
+using Random
+
+MPI.Init()
+
+# Input data dimensions (Nx × Ny × Nz)
+dims = (64, 32, 64)
+
+# Apply a 3D real-to-complex (r2c) FFT.
+transform = Transforms.RFFT()
+
+# Automatically create decomposition configuration
+comm = MPI.COMM_WORLD
+pen = Pencil(dims, comm)
+
+# Create plan
+plan = PencilFFTPlan(pen, transform)
+
+# Allocate data and initialise field
+θ = allocate_input(plan)
+randn!(θ)
+
+# Perform distributed FFT
+θ_hat = plan * θ

              Finally, we initialise the output that will hold ∇θ in Fourier space. Noting that ∇θ is a vector field, we choose to store it as a tuple of 3 PencilArrays.

              ∇θ_hat = allocate_output(plan, Val(3))
+
+# This is equivalent:
+# ∇θ_hat = ntuple(d -> similar(θ_hat), Val(3))
+
+summary(∇θ_hat)
              "Tuple{PencilArray{ComplexF64, 3, Array{ComplexF64, 3}, 3, 0, Pencil{3, 2, Permutation{(3, 2, 1), 3}, Vector{UInt8}}}, PencilArray{ComplexF64, 3, Array{ComplexF64, 3}, 3, 0, Pencil{3, 2, Permutation{(3, 2, 1), 3}, Vector{UInt8}}}, PencilArray{ComplexF64, 3, Array{ComplexF64, 3}, 3, 0, Pencil{3, 2, Permutation{(3, 2, 1), 3}, Vector{UInt8}}}}"

              Fourier wave numbers

              In general, the Fourier wave numbers are of the form $k_i = 0, ±\frac{2π}{L_i}, ±\frac{4π}{L_i}, ±\frac{6π}{L_i}, …$, where $L_i$ is the period along dimension $i$. When a real-to-complex Fourier transform is applied, roughly half of these wave numbers are redundant due to the Hermitian symmetry of the complex Fourier coefficients. In practice, this means that for the fastest dimension $x$ (along which a real-to-complex transform is performed), the negative wave numbers are dropped, i.e. $k_x = 0, \frac{2π}{L_x}, \frac{4π}{L_x}, …$.

              The AbstractFFTs package provides a convenient way to generate the Fourier wave numbers, using the functions fftfreq and rfftfreq. We can use these functions to initialise a "grid" of wave numbers associated to our 3D real-to-complex transform:

              using AbstractFFTs: fftfreq, rfftfreq
+
+box_size = (2π, 2π, 2π)  # Lx, Ly, Lz
+sample_rate = 2π .* dims ./ box_size
+
+# In our case (Lx = 2π and Nx even), this gives kx = [0, 1, 2, ..., Nx/2].
+kx = rfftfreq(dims[1], sample_rate[1])
+
+# In our case (Ly = 2π and Ny even), this gives
+# ky = [0, 1, 2, ..., Ny/2-1, -Ny/2, -Ny/2+1, ..., -1] (and similarly for kz).
+ky = fftfreq(dims[2], sample_rate[2])
+kz = fftfreq(dims[3], sample_rate[3])
+
+kvec = (kx, ky, kz)
              ([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0  …  23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, 32.0], [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0  …  -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0], [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0  …  -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0])

              Note that kvec now contains the wave numbers associated to the global domain. In the following, we will only need the wave numbers associated to the portion of the domain handled by the local MPI process.

              Method 1: global views

              PencilArrays, returned for instance by allocate_input and allocate_output, take indices that start at 1, regardless of the location of the subdomain associated to the local process on the global grid. (In other words, PencilArrays take local indices.) On the other hand, we have defined the wave number vector kvec which, for each MPI process, is defined over the global domain, and as such it takes global indices.

              One straightforward way of making data arrays compatible with wave numbers is to use global views, i.e. arrays that take global indices. These are generated from PencilArrays by calling the global_view function. Note that, in general, global indices do not start at 1 for a given MPI process. A given process will own a range of data given by indices in (i1:i2, j1:j2, k1:k2).

              θ_glob = global_view(θ_hat)
+∇θ_glob = global_view.(∇θ_hat)
+summary(θ_glob)
              "16×32×64 OffsetArray(::PencilArray{ComplexF64, 3}, 1:16, 1:32, 1:64) with eltype ComplexF64 with indices 1:16×1:32×1:64"

              Once we have global views, we can combine data and wave numbers using the portion of global indices owned by the local MPI process, as shown below. We can use CartesianIndices to iterate over the global indices associated to the local process.

              for I in CartesianIndices(θ_glob)
+    i, j, k = Tuple(I)  # unpack indices
+
+    # Wave number vector associated to current Cartesian index.
+    kx = kvec[1][i]
+    ky = kvec[2][j]
+    kz = kvec[3][k]
+
+    # Compute gradient in Fourier space.
+    # Note that modifying ∇θ_glob also modifies the original PencilArray ∇θ_hat.
+    ∇θ_glob[1][I] = im * kx * θ_glob[I]
+    ∇θ_glob[2][I] = im * ky * θ_glob[I]
+    ∇θ_glob[3][I] = im * kz * θ_glob[I]
+end

              The above loop can be written in a slightly more efficient manner by precomputing im * θ_glob[I]:

              @inbounds for I in CartesianIndices(θ_glob)
+    i, j, k = Tuple(I)
+
+    kx = kvec[1][i]
+    ky = kvec[2][j]
+    kz = kvec[3][k]
+
+    u = im * θ_glob[I]
+
+    ∇θ_glob[1][I] = kx * u
+    ∇θ_glob[2][I] = ky * u
+    ∇θ_glob[3][I] = kz * u
+end

              Also note that the above can be easily written in a more generic way, e.g. for arbitrary dimensions, thanks in part to the use of CartesianIndices. Moreover, in the above there is no notion of the dimension permutations discussed in the tutorial, as it is all hidden behind the implementation of PencilArrays. And as seen later in the benchmarks, these (hidden) permutations have zero cost, as the speed is identical to that of a function that explicitly takes into account these permutations.

              Finally, we can perform a backwards transform to obtain $\bm{∇} θ$ in physical space:

              ∇θ = plan \ ∇θ_hat;

              Note that the transform is automatically broadcast over the three fields of the ∇θ_hat vector, and the result ∇θ is also a tuple of three PencilArrays.

              Method 2: explicit global indexing

              Sometimes, one does not need to write generic code. In our case, one often knows the dimensionality of the problem and the memory layout of the data (i.e. the underlying index permutation).

              Below is a reimplementation of the above loop, using explicit indices instead of CartesianIndices, and assuming that the underlying index permutation is (3, 2, 1), that is, data is stored in $(z, y, x)$ order. As discussed in the tutorial, this is the default for transformed arrays. This example also serves as a more explicit explanation for what is going on in the first method.

              # Get local data range in the global grid.
+rng = axes(θ_glob)  # = (i1:i2, j1:j2, k1:k2)
              (OffsetArrays.IdOffsetRange(values=1:16, indices=1:16), OffsetArrays.IdOffsetRange(values=1:32, indices=1:32), OffsetArrays.IdOffsetRange(values=1:64, indices=1:64))

              For the loop below, we're assuming that the permutation is (3, 2, 1). In other words, the fastest index is the last one, and not the first one as it is usually in Julia. If the permutation is not (3, 2, 1), things will still work (well, except for the assertion below!), but the loop order will not be optimal.

              @assert permutation(θ_hat) === Permutation(3, 2, 1)
+
+@inbounds for i in rng[1], j in rng[2], k in rng[3]
+    kx = kvec[1][i]
+    ky = kvec[2][j]
+    kz = kvec[3][k]
+
+    # Note that we still access the arrays in (i, j, k) order.
+    # (The permutation happens behind the scenes!)
+    u = im * θ_glob[i, j, k]
+
+    ∇θ_glob[1][i, j, k] = kx * u
+    ∇θ_glob[2][i, j, k] = ky * u
+    ∇θ_glob[3][i, j, k] = kz * u
+end

              Method 3: using local indices

              Alternatively, we can avoid global views and work directly on PencilArrays using local indices that start at 1. In this case, part of the strategy is to construct a "local" grid of wave numbers that can also be accessed with local indices. This can be conveniently done using the localgrid function of the PencilArrays.jl package, which accepts a PencilArray (or its associated Pencil) and the global coordinates (here kvec):

              grid_fourier = localgrid(θ_hat, kvec)
              LocalRectilinearGrid{3} with Permutation(3, 2, 1) and coordinates:
+ (1) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0]
+ (2) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]
+ (3) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]

              Note that one can directly iterate on the returned grid object:

              @inbounds for I in CartesianIndices(grid_fourier)
+    # Wave number vector associated to current Cartesian index.
+    k⃗ = grid_fourier[I]
+    u = im * θ_hat[I]
+    ∇θ_hat[1][I] = k⃗[1] * u
+    ∇θ_hat[2][I] = k⃗[2] * u
+    ∇θ_hat[3][I] = k⃗[3] * u
+end

              This implementation is as efficient as the other examples, while being slightly shorter to write. Moreover, it is quite generic, and can be made independent of the number of dimensions with little effort.

              Method 4: using broadcasting

              Finally, note that the local grid object returned by localgrid makes it is possible to compute the gradient using broadcasting, thus fully avoiding scalar indexing. This can be quite convenient in some cases, and can also be very useful if one is working on GPUs (where scalar indexing is prohibitively expensive). Using broadcasting, the above examples simply become:

              @. ∇θ_hat[1] = im * grid_fourier[1] * θ_hat
+@. ∇θ_hat[2] = im * grid_fourier[2] * θ_hat
+@. ∇θ_hat[3] = im * grid_fourier[3] * θ_hat

              Once again, as shown in the benchmarks further below, this method performs quite similarly to the other ones.

              Summary

              The PencilArrays module provides different alternatives to deal with MPI-distributed data that may be subject to dimension permutations. In particular, one can choose to work with global indices (first two examples), with local indices (third example), or to avoid scalar indexing altogether (fourth example).

              If one wants to stay generic, making sure that the same code will work for arbitrary dimensions and will be efficient regardless of the underlying dimension permutation, methods 1, 3 or 4 should be preferred. These use CartesianIndices and make no assumptions on possible dimension permutations, which are by default enabled in the output of PencilFFTs transforms. In fact, such permutations are completely invisible in the implementations.

              The second method uses explicit (i, j, k) indices. It assumes that the underlying permutation is (3, 2, 1) to loop with i as the slowest index and k as the fastest, which is the optimal order in this case given the permutation. As such, the implementation is less generic than the others, and differences in performance are negligible with respect to more generic variants.

              Benchmark results

              The following are the benchmark results obtained from running examples/gradient.jl on a laptop, using 2 MPI processes and Julia 1.7.2, with an input array of global dimensions $64 × 32 × 64$. The different methods detailed above are marked on the right. The "lazy" marks indicate runs where the wave numbers were represented by lazy Frequencies objects (returned by rfftfreq and fftfreq). Otherwise, they were collected into Vectors. For some reason, plain Vectors are faster when working with grids generated by localgrid.

              In the script, additional implementations can be found which rely on a more advanced understanding of permutations and on the internals of the PencilArrays package. For instance, gradient_local_parent! directly works with the raw data stored in Julia Arrays, while gradient_local_linear! completely avoids CartesianIndices while staying generic and efficient. Nevertheless, these display roughly the same performance as the above examples.

                  gradient_global_view!...                  89.900 μs
+    gradient_global_view! (lazy)...           92.060 μs  [Method 1]
+    gradient_global_view_explicit!...         88.958 μs
+    gradient_global_view_explicit! (lazy)...  81.055 μs  [Method 2]
+    gradient_local!...                        92.305 μs
+    gradient_grid!...                         92.770 μs
+    gradient_grid! (lazy)...                  101.388 μs  [Method 3]
+    gradient_grid_broadcast!...               88.606 μs
+    gradient_grid_broadcast! (lazy)...        151.020 μs  [Method 4]
+    gradient_local_parent!...                 92.248 μs
+    gradient_local_linear!...                 91.212 μs
+    gradient_local_linear_explicit!...        90.992 μs

              This page was generated using Literate.jl.

              diff --git a/previews/PR40/generated/in-place/index.html b/previews/PR68/generated/in-place/index.html similarity index 53% rename from previews/PR40/generated/in-place/index.html rename to previews/PR68/generated/in-place/index.html index 9844d1df..5c6912af 100644 --- a/previews/PR40/generated/in-place/index.html +++ b/previews/PR68/generated/in-place/index.html @@ -1,5 +1,5 @@ -In-place transforms · PencilFFTs.jl
              Edit on GitHub

              In-place transforms

              Complex-to-complex and real-to-real transforms can be performed in-place, enabling important memory savings. The procedure is very similar to that of out-of-place transforms described in the tutorial. The differences are illustrated in the sections below.

              Creating a domain partition

              We start by partitioning a domain of dimensions $16×32×64$ along all available MPI processes.

              using PencilFFTs
+In-place transforms · PencilFFTs.jl
              GitHub
              In-place transforms
              Complex-to-complex and real-to-real transforms can be performed in-place, enabling important memory savings. The procedure is very similar to that of out-of-place transforms described in the tutorial. The differences are illustrated in the sections below.
              Creating a domain partition
              We start by partitioning a domain of dimensions $16×32×64$ along all available MPI processes.
              using PencilFFTs
 using MPI
 MPI.Init()
 
@@ -28,4 +28,4 @@
 u_in = first(A)  # input data view
 randn!(u_in)
 summary(u_in)
              "16×16×64 PencilArray{ComplexF64, 3}(::Pencil{3, 2, NoPermutation, Array})"
              Applying plans
              Like in FFTW.jl, one can perform in-place transforms using the * and \ operators. As mentioned above, in-place plans must be applied on the ManyPencilArray containers returned by allocate_input.
              plan * A;  # performs in-place forward transform
              After performing an in-place transform, data contained in u_in has been overwritten and has no "physical" meaning. In other words, u_in should not be used at this point. To access the transformed data, one should retrieve the output data view using last(A).
              For instance, to compute the global sum of the transformed data:
              u_out = last(A)  # output data view
-sum(u_out)       # sum of transformed data (note that `sum` reduces over all processes)
              -365.138291212882 + 16428.95317060907im
              Finally, we can perform a backward transform and do stuff with the input view:
              plan \ A;  # perform in-place backward transform
              At this point, the data can be once again found in the input view u_in, while u_out should not be accessed.
              This page was generated using Literate.jl.
              
+sum(u_out)       # sum of transformed data (note that `sum` reduces over all processes)
              552.5900431208138 - 33470.15869664838im

              Finally, we can perform a backward transform and do stuff with the input view:

              plan \ A;  # perform in-place backward transform

              At this point, the data can be once again found in the input view u_in, while u_out should not be accessed.

              This page was generated using Literate.jl.

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b/previews/PR68/generated/navier_stokes/index.html new file mode 100644 index 00000000..cbd56170 --- /dev/null +++ b/previews/PR68/generated/navier_stokes/index.html @@ -0,0 +1,307 @@ + +Navier–Stokes equations · PencilFFTs.jl

              GitHub

              Navier–Stokes equations

              In this example, we numerically solve the incompressible Navier–Stokes equations

              \[∂_t \bm{v} + (\bm{v} ⋅ \bm{∇}) \bm{v} = -\frac{1}{ρ} \bm{∇} p + ν ∇^2 \bm{v}, +\quad \bm{∇} ⋅ \bm{v} = 0,\]

              where $\bm{v}(\bm{x}, t)$ and $p(\bm{x}, t)$ are respectively the velocity and pressure fields, $ν$ is the fluid kinematic viscosity and $ρ$ is the fluid density.

              We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.

              First steps

              We start by loading the required packages, initialising MPI and setting the simulation parameters.

              using MPI
+using PencilFFTs
+
+MPI.Init()
+comm = MPI.COMM_WORLD
+procid = MPI.Comm_rank(comm) + 1
+
+# Simulation parameters
+Ns = (64, 64, 64)  # = (Nx, Ny, Nz)
+Ls = (2π, 2π, 2π)  # = (Lx, Ly, Lz)
+
+# Collocation points ("global" = over all processes).
+# We include the endpoint (length = N + 1) for convenience.
+xs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls)  # = (x, y, z)
              (0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586)

              Let's check the number of MPI processes over which we're running our simulation:

              MPI.Comm_size(comm)
              2

              We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:

              pen = Pencil(Ns, comm)
              Decomposition of 3D data
+    Data dimensions: (64, 64, 64)
+    Decomposed dimensions: (2, 3)
+    Data permutation: NoPermutation()
+    Array type: Array

              The subdomain associated to the local MPI process can be obtained using range_local:

              range_local(pen)
              (1:64, 1:32, 1:64)

              We now construct a distributed vector field that follows the decomposition configuration we just created:

              v⃗₀ = (
+    PencilArray{Float64}(undef, pen),  # vx
+    PencilArray{Float64}(undef, pen),  # vy
+    PencilArray{Float64}(undef, pen),  # vz
+)
+summary(v⃗₀[1])
              "64×32×64 PencilArray{Float64, 3}(::Pencil{3, 2, NoPermutation, Array})"

              We still need to fill this array with interesting values that represent a physical velocity field.

              Initial condition

              Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:

              \[\begin{aligned} +v_x(x, y, z) &= u₀ \sin(k₀ x) \cos(k₀ y) \cos(k₀ z) \\ +v_y(x, y, z) &= -u₀ \cos(k₀ x) \sin(k₀ y) \cos(k₀ z) \\ +v_z(x, y, z) &= 0 +\end{aligned}\]

              where $u₀$ and $k₀$ are two parameters setting the amplitude and the period of the velocity field.

              To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:

              grid = localgrid(pen, xs_global)
              LocalRectilinearGrid{3} with coordinates:
+ (1) 0.0:0.09817477042468103:6.1850105367549055
+ (2) 0.0:0.09817477042468103:3.043417883165112
+ (3) 0.0:0.09817477042468103:6.1850105367549055

              We can use this to initialise the velocity field:

              u₀ = 1.0
+k₀ = 2π / Ls[1]  # should be integer if L = 2π (to preserve periodicity)
+
+@. v⃗₀[1] =  u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)
+@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)
+@. v⃗₀[3] =  0

              Let's plot a 2D slice of the velocity field managed by the local MPI process:

              using GLMakie
+
+# Compute the norm of a vector field represented by a tuple of arrays.
+function vecnorm(v⃗::NTuple)
+    vnorm = similar(v⃗[1])
+    for n ∈ eachindex(v⃗[1])
+        w = zero(eltype(vnorm))
+        for v ∈ v⃗
+            w += v[n]^2
+        end
+        vnorm[n] = sqrt(w)
+    end
+    vnorm
+end
+
+let fig = Figure(resolution = (700, 600))
+    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
+    vnorm = parent(vecnorm(v⃗₀))  # use `parent` because Makie doesn't like custom array types...
+    ct = contour!(
+        ax, grid.x, grid.y, grid.z, vnorm;
+        alpha = 0.2, levels = 4,
+        colormap = :viridis,
+        colorrange = (0.0, 1.0),
+        highclip = (:red, 0.2), lowclip = (:green, 0.2),
+    )
+    cb = Colorbar(fig[1, 2], ct; label = "Velocity magnitude")
+    fig
+end
              Example block output

              Velocity in Fourier space

              In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series

              \[\bm{v}(\bm{x}, t) = +∑_{\bm{k}} \hat{\bm{v}}_{\bm{k}}(t) \, e^{i \bm{k} ⋅ \bm{x}},\]

              where $\bm{k} = (k_x, k_y, k_z)$ are the discrete wave numbers.

              The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for $k_x$:

              using AbstractFFTs: fftfreq, rfftfreq
+
+ks_global = (
+    rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]),  # kx | real-to-complex
+     fftfreq(Ns[2], 2π * Ns[2] / Ls[2]),  # ky | complex-to-complex
+     fftfreq(Ns[3], 2π * Ns[3] / Ls[3]),  # kz | complex-to-complex
+)
+
+ks_global[1]'
              1×33 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
+ 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  27.0  28.0  29.0  30.0  31.0  32.0
              ks_global[2]'
              1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
+ 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0
              ks_global[3]'
              1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
+ 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0

              To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:

              plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())
              Transforms: (RFFT, FFT, FFT)
+Input type: Float64
+Global dimensions: (64, 64, 64) -> (33, 64, 64)
+MPI topology: 2D decomposition (2×1 processes)

              See PencilFFTPlan for details on creating plans and on optional keyword arguments.

              We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$:

              v̂s = plan .* v⃗₀
+summary(v̂s[1])
              "16×64×64 PencilArray{ComplexF64, 3}(::Pencil{3, 2, Permutation{(3, 2, 1), 3}, Array})"

              Note that, in Fourier space, the domain decomposition is performed along the directions $x$ and $y$:

              pencil(v̂s[1])
              Decomposition of 3D data
+    Data dimensions: (33, 64, 64)
+    Decomposed dimensions: (1, 2)
+    Data permutation: Permutation(3, 2, 1)
+    Array type: Array

              This is because the 3D FFTs are performed one dimension at a time, with the $x$ direction first and the $z$ direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.

              To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers $\bm{k}$ associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:

              grid_fourier = localgrid(v̂s[1], ks_global)
              LocalRectilinearGrid{3} with Permutation(3, 2, 1) and coordinates:
+ (1) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0]
+ (2) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]
+ (3) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]

              As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, $\bm{ω} = \bm{∇} × \bm{v}$. In Fourier space, this becomes $\hat{\bm{ω}} = i \bm{k} × \hat{\bm{v}}$.

              using StaticArrays: SVector
+using LinearAlgebra: ×
+
+function curl_fourier!(
+        ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,
+    ) where {N}
+    @inbounds for I ∈ eachindex(grid_fourier)
+        # We use StaticArrays for the cross product between small vectors.
+        ik⃗ = im * SVector(grid_fourier[I])
+        v⃗ = SVector(getindex.(v̂s, Ref(I)))  # = (v̂s[1][I], v̂s[2][I], ...)
+        ω⃗ = ik⃗ × v⃗
+        for n ∈ eachindex(ω⃗)
+            ω̂s[n][I] = ω⃗[n]
+        end
+    end
+    ω̂s
+end
+
+ω̂s = similar.(v̂s)
+curl_fourier!(ω̂s, v̂s, grid_fourier);

              We finally transform back to physical space and plot the result:

              ωs = plan .\ ω̂s
+
+let fig = Figure(resolution = (700, 600))
+    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
+    ω_norm = parent(vecnorm(ωs))
+    ct = contour!(
+        ax, grid.x, grid.y, grid.z, ω_norm;
+        alpha = 0.1, levels = 0.8:0.2:2.0,
+        colormap = :viridis, colorrange = (0.8, 2.0),
+        highclip = (:red, 0.2), lowclip = (:green, 0.2),
+    )
+    cb = Colorbar(fig[1, 2], ct; label = "Vorticity magnitude")
+    fig
+end
              Example block output

              Computing the non-linear term

              One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as

              \[∂_t \hat{\bm{v}}_{\bm{k}} = +- \mathcal{P}_{\bm{k}} \! \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right] +- ν |\bm{k}|^2 \hat{\bm{v}}_{\bm{k}} +\quad \text{ with } \quad +\mathcal{P}_{\bm{k}}(\hat{\bm{F}}_{\bm{k}}) = \left( I - \frac{\bm{k} ⊗ +\bm{k}}{|\bm{k}|^2} \right) \hat{\bm{F}}_{\bm{k}},\]

              where $\mathcal{P}_{\bm{k}}$ is a projection operator allowing to preserve the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.

              Now that we have the wave numbers $\bm{k}$, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$ of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, $\hat{\bm{F}}_{\bm{k}} = \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right]_{\bm{k}}$. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.

              Below we implement a function that computes the non-linear term in Fourier space based on its convective form $(\bm{v} ⋅ \bm{∇}) \bm{v} = \bm{∇} ⋅ (\bm{v} ⊗ \bm{v})$. Note that this equivalence uses the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$.

              using LinearAlgebra: mul!, ldiv!  # for applying FFT plans in-place
+
+# Compute non-linear term in Fourier space from velocity field in physical
+# space. Optional keyword arguments may be passed to avoid allocations.
+function ns_nonlinear!(
+        F̂s, vs, plan, grid_fourier;
+        vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),
+    )
+    # Compute F_i = ∂_j (v_i v_j) for each i.
+    # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)
+    w, ŵ = vbuf, v̂buf
+    @inbounds for (i, F̂i) ∈ enumerate(F̂s)
+        F̂i .= 0
+        vi = vs[i]
+        for (j, vj) ∈ enumerate(vs)
+            w .= vi .* vj     # w = v_i * v_j in physical space
+            mul!(ŵ, plan, w)  # same in Fourier space
+            # Add derivative in Fourier space
+            for I ∈ eachindex(grid_fourier)
+                k⃗ = grid_fourier[I]  # = (kx, ky, kz)
+                kj = k⃗[j]
+                F̂i[I] += im * kj * ŵ[I]
+            end
+        end
+    end
+    F̂s
+end
              ns_nonlinear! (generic function with 1 method)

              As an example, let's use this function on our initial velocity field:

              F̂s = similar.(v̂s)
+ns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);

              Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.

              function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)
+    ks_max = maximum.(abs, ks_global)  # maximum stored wave numbers (kx_max, ky_max, kz_max)
+    ks_lim = (2 / 3) .* ks_max
+    @inbounds for I ∈ eachindex(grid_fourier)
+        k⃗ = grid_fourier[I]
+        if any(abs.(k⃗) .> ks_lim)
+            for ŵ ∈ ŵs
+                ŵ[I] = 0
+            end
+        end
+    end
+    ŵs
+end
+
+# We can apply this on the previously computed non-linear term:
+dealias_twothirds!(F̂s, grid_fourier, ks_global);

              Finally, we implement the projection associated to the incompressibility condition:

              function project_divergence_free!(ûs, grid_fourier)
+    @inbounds for I ∈ eachindex(grid_fourier)
+        k⃗ = grid_fourier[I]
+        k² = sum(abs2, k⃗)
+        iszero(k²) && continue  # avoid division by zero
+        û = getindex.(ûs, Ref(I))  # (ûs[1][I], ûs[2][I], ...)
+        for i ∈ eachindex(û)
+            ŵ = û[i]
+            for j ∈ eachindex(û)
+                ŵ -= k⃗[i] * k⃗[j] * û[j] / k²
+            end
+            ûs[i][I] = ŵ
+        end
+    end
+    ûs
+end
              project_divergence_free! (generic function with 1 method)

              We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:

              v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)
+v̂s_proj .≈ v̂s  # the last one may be false because v_z = 0 initially
              (true, true, false)

              Putting it all together

              To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:

              function ns_rhs!(
+        dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,
+    ) where {N}
+    # 1. Compute non-linear term and dealias it
+    (; plan, cache, ks_global, grid_fourier) = p
+    F̂s = cache.F̂s
+    ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])
+    dealias_twothirds!(F̂s, grid_fourier, ks_global)
+
+    # 2. Project onto divergence-free space
+    project_divergence_free!(F̂s, grid_fourier)
+
+    # 3. Transform velocity to Fourier space
+    v̂s = cache.v̂s
+    map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)
+
+    # 4. Add viscous term (and multiply projected non-linear term by -1)
+    ν = p.ν
+    for n ∈ eachindex(v̂s)
+        v̂ = v̂s[n]
+        F̂ = F̂s[n]
+        @inbounds for I ∈ eachindex(grid_fourier)
+            k⃗ = grid_fourier[I]  # = (kx, ky, kz)
+            k² = sum(abs2, k⃗)
+            F̂[I] = -F̂[I] - ν * k² * v̂[I]
+        end
+    end
+
+    # 5. Transform RHS back to physical space
+    map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)
+
+    nothing
+end
              ns_rhs! (generic function with 1 method)

              For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.

              using OrdinaryDiffEq
+using RecursiveArrayTools: ArrayPartition
+
+ns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)
+vs_init_ode = ArrayPartition(v⃗₀)
+summary(vs_init_ode)
              "RecursiveArrayTools.ArrayPartition{Float64, Tuple{PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}}} with arrays:"

              We now define solver parameters and temporary variables, and initialise the problem:

              params = (;
+    ν = 5e-3,  # kinematic viscosity
+    plan, grid_fourier, ks_global,
+    cache = (
+        v̂s = similar.(v̂s),
+        F̂s = similar.(v̂s),
+    )
+)
+
+tspan = (0.0, 10.0)
+prob = ODEProblem{true}(ns_rhs!, vs_init_ode, tspan, params)
+integrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);

              We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum $E(k)$, to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity $ν$ must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large $k$), while high enough to allow the small-scale motions to be correctly resolved.

              function energy_spectrum!(Ek, ks, v̂s, grid_fourier)
+    Nk = length(Ek)
+    @assert Nk == length(ks)
+    Ek .= 0
+    for I ∈ eachindex(grid_fourier)
+        k⃗ = grid_fourier[I]  # = (kx, ky, kz)
+        knorm = sqrt(sum(abs2, k⃗))
+        i = searchsortedfirst(ks, knorm)
+        i > Nk && continue
+        v⃗ = getindex.(v̂s, Ref(I))  # = (v̂s[1][I], v̂s[2][I], ...)
+        factor = k⃗[1] == 0 ? 1 : 2  # account for Hermitian symmetry and r2c transform
+        Ek[i] += factor * sum(abs2, v⃗) / 2
+    end
+    MPI.Allreduce!(Ek, +, get_comm(v̂s[1]))  # sum across all processes
+    Ek
+end
+
+ks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])
+Ek = similar(ks)
+v̂s = plan .* integrator.u.x
+energy_spectrum!(Ek, ks, v̂s, grid_fourier)
+Ek ./= scale_factor(plan)^2  # rescale energy
+
+curl_fourier!(ω̂s, v̂s, grid_fourier)
+ldiv!.(ωs, plan, ω̂s)
+ω⃗_plot = Observable(ωs)
+k_plot = @view ks[2:end]
+E_plot = Observable(@view Ek[2:end])
+t_plot = Observable(integrator.t)
+
+fig = let
+    fig = Figure(resolution = (1200, 600))
+    ax = Axis3(
+        fig[1, 1][1, 1]; title = @lift("t = $(round($t_plot, digits = 3))"),
+        aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z",
+    )
+    ω_mag = @lift parent(vecnorm($ω⃗_plot))
+    ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)
+    ct = contour!(
+        ax, grid.x, grid.y, grid.z, ω_mag_norm;
+        alpha = 0.3, levels = 3,
+        colormap = :viridis, colorrange = (0.0, 1.0),
+        highclip = (:red, 0.2), lowclip = (:green, 0.2),
+    )
+    cb = Colorbar(fig[1, 1][1, 2], ct; label = "Normalised vorticity magnitude")
+    ax_sp = Axis(
+        fig[1, 2];
+        xlabel = "k", ylabel = "E(k)", xscale = log2, yscale = log10,
+        title = "Kinetic energy spectrum",
+    )
+    ylims!(ax_sp, 1e-8, 1e0)
+    scatterlines!(ax_sp, k_plot, E_plot)
+    ks_slope = exp.(range(log(2.5), log(25.0), length = 3))
+    E_fivethirds = @. 0.3 * ks_slope^(-5/3)
+    @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)
+    text!(ax_sp, L"k^{-5/3}"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))
+    fig
+end
+
+record(fig, "vorticity_proc$procid.mp4"; framerate = 10) do io
+    while integrator.t < 20
+        dt = 0.001
+        step!(integrator, dt)
+        t_plot[] = integrator.t
+        mul!.(v̂s, plan, integrator.u.x)  # current velocity in Fourier space
+        curl_fourier!(ω̂s, v̂s, grid_fourier)
+        ldiv!.(ω⃗_plot[], plan, ω̂s)
+        ω⃗_plot[] = ω⃗_plot[]  # to force updating the plot
+        energy_spectrum!(Ek, ks, v̂s, grid_fourier)
+        Ek ./= scale_factor(plan)^2  # rescale energy
+        E_plot[] = E_plot[]
+        recordframe!(io)
+    end
+end;
              + +

              This page was generated using Literate.jl.

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b/previews/PR68/img/benchmark_idris.svg similarity index 100% rename from previews/PR40/img/benchmark_idris.svg rename to previews/PR68/img/benchmark_idris.svg diff --git a/previews/PR55/img/pencils.svg b/previews/PR68/img/pencils.svg similarity index 100% rename from previews/PR55/img/pencils.svg rename to previews/PR68/img/pencils.svg diff --git a/previews/PR68/index.html b/previews/PR68/index.html new file mode 100644 index 00000000..3679b3d7 --- /dev/null +++ b/previews/PR68/index.html @@ -0,0 +1,7 @@ + +Home · PencilFFTs.jl

              GitHub

              PencilFFTs

              Fast Fourier transforms of MPI-distributed Julia arrays.

              Introduction

              This package provides multidimensional FFTs and related transforms on MPI-distributed Julia arrays via the PencilArrays package.

              The name of this package originates from the decomposition of 3D domains along two out of three dimensions, sometimes called pencil decomposition. This is illustrated by the figure below,[1] where each coloured block is managed by a different MPI process. Typically, one wants to compute FFTs on a scalar or vector field along the three spatial dimensions. In the case of a pencil decomposition, 3D FFTs are performed one dimension at a time, along the non-decomposed direction. Transforms must then be interleaved with global data transpositions to switch between pencil configurations. In high-performance computing environments, such data transpositions are generally the most expensive part of a parallel FFT computation, due to the large cost of communications between computing nodes.

              + Pencil decomposition of 3D domains +

              More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension $N$. The decompositions can be performed along an arbitrary number $M < N$ of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.

              The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.

              Features

              • distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;

              • FFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;

              • in-place and out-of-place transforms;

              • high scalability up to (at least) tens of thousands of MPI processes.

              Installation

              PencilFFTs can be installed using the Julia package manager:

              julia> ] add PencilFFTs

              Similar projects

              • FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.

              • PFFT is a very general parallel FFT library written in C.

              • P3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.

              • 2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.

              • 1Figure adapted from this PhD thesis.
              • 2For the pencil decomposition represented in the figure, $N = 3$ and $M = 2$.
              diff --git a/previews/PR68/search_index.js b/previews/PR68/search_index.js new file mode 100644 index 00000000..f2bc3873 --- /dev/null +++ b/previews/PR68/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"tutorial/#Tutorial","page":"Tutorial","title":"Tutorial","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The following tutorial shows how to perform a 3D FFT of real periodic data defined on a grid of N_x N_y N_z points.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"
              \n \n \n
              ","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"By default, the domain is distributed on a 2D MPI topology of dimensions N_1 N_2. As an example, the above figure shows such a topology with N_1 = 4 and N_2 = 3, for a total of 12 MPI processes.","category":"page"},{"location":"tutorial/#tutorial:creating_plans","page":"Tutorial","title":"Creating plans","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The first thing to do is to create a domain decomposition configuration for the given dataset dimensions N_x N_y N_z. In the framework of PencilArrays, such a configuration is described by a Pencil object. As described in the PencilArrays docs, we can let the Pencil constructor automatically determine such a configuration. For this, only an MPI communicator and the dataset dimensions are needed:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using MPI\nusing PencilFFTs\n\nMPI.Init()\ncomm = MPI.COMM_WORLD\n\n# Input data dimensions (Nx × Ny × Nz)\ndims = (16, 32, 64)\npen = Pencil(dims, comm)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"By default this creates a 2D decomposition (for the case of a 3D dataset), but one can change this as detailed in the PencilArrays documentation linked above.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"We can now create a PencilFFTPlan, which requires information on decomposition configuration (the Pencil object) and on the transforms that will be applied:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# Apply a 3D real-to-complex (r2c) FFT.\ntransform = Transforms.RFFT()\n\n# Note that, for more control, one can instead separately specify the transforms along each dimension:\n# transform = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())\n\n# Create plan\nplan = PencilFFTPlan(pen, transform)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"See the PencilFFTPlan constructor for details on the accepted options, and the Transforms module for the possible transforms. It is also possible to enable fine-grained performance measurements via the TimerOutputs package, as described in Measuring performance.","category":"page"},{"location":"tutorial/#Allocating-data","page":"Tutorial","title":"Allocating data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Next, we want to apply the plan on some data. Transforms may only be applied on PencilArrays, which are array wrappers that include MPI decomposition information (in some sense, analogous to DistributedArrays in Julia's distributed computing approach). The helper function allocate_input can be used to allocate a PencilArray that is compatible with our plan:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# In our example, this returns a 3D PencilArray of real data (Float64).\nu = allocate_input(plan)\n\n# Fill the array with some (random) data\nusing Random\nrandn!(u)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"PencilArrays are a subtype of AbstractArray, and thus they support all common array operations.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Similarly, to preallocate output data, one can use allocate_output:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# In our example, this returns a 3D PencilArray of complex data (Complex{Float64}).\nv = allocate_output(plan)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"This is only required if one wants to apply the plans using a preallocated output (with mul!, see right below).","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The data types returned by allocate_input and allocate_output are slightly different when working with in-place transforms. See the in-place example for details.","category":"page"},{"location":"tutorial/#Applying-plans","page":"Tutorial","title":"Applying plans","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The interface to apply plans is consistent with that of AbstractFFTs. Namely, * and mul! are respectively used for forward transforms without and with preallocated output data. Similarly, \\ and ldiv! are used for backward transforms.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using LinearAlgebra # for mul!, ldiv!\n\n# Apply plan on `u` with `v` as an output\nmul!(v, plan, u)\n\n# Apply backward plan on `v` with `w` as an output\nw = similar(u)\nldiv!(w, plan, v) # now w ≈ u","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Note that, consistently with AbstractFFTs, normalisation is performed at the end of a backward transform, so that the original data is recovered when applying a forward followed by a backward transform.","category":"page"},{"location":"tutorial/#Accessing-and-modifying-data","page":"Tutorial","title":"Accessing and modifying data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For any given MPI process, a PencilArray holds the data associated to its local partition in the global geometry. PencilArrays are accessed using local indices that start at 1, regardless of the location of the local process in the MPI topology. Note that PencilArrays, being based on regular Arrays, support both linear and Cartesian indexing (see the Julia docs for details).","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For convenience, the global_view function can be used to generate an OffsetArray wrapper that takes global indices.","category":"page"},{"location":"tutorial/#tutorial:output_data_layout","page":"Tutorial","title":"Output data layout","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"In memory, the dimensions of the transform output are by default reversed with respect to the input. That is, if the order of indices in the input data is (x, y, z), then the output has order (z, y, x) in memory. This detail is hidden from the user, and output arrays are always accessed in the same order as the input data, regardless of the underlying output dimension permutation. This applies to PencilArrays and to OffsetArrays returned by global_view.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The reasoning behind dimension permutations, is that they allow to always perform FFTs along the fastest array dimension and to avoid a local data transposition, resulting in performance gains. A similar approach is followed by other parallel FFT libraries. FFTW itself, in its distributed-memory routines, includes a flag that enables a similar behaviour. In PencilFFTs, index permutation is the default, but it can be disabled via the permute_dims flag of PencilFFTPlan.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"A great deal of work has been spent in making generic index permutations as efficient as possible, both in intermediate and in the output state of the multidimensional transforms. This has been achieved, in part, by making sure that permutations such as (3, 2, 1) are compile-time constants.","category":"page"},{"location":"tutorial/#Further-reading","page":"Tutorial","title":"Further reading","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For details on working with PencilArrays see the PencilArrays docs.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The examples on the sidebar further illustrate the use of transforms and provide an introduction to working with MPI-distributed data in the form of PencilArrays. In particular, the gradient example illustrates different ways of computing things using Fourier-transformed distributed arrays. Then, the incompressible Navier–Stokes example is a more advanced and complete example of a possible application of the PencilFFTs package.","category":"page"},{"location":"benchmarks/#Benchmarks","page":"Benchmarks","title":"Benchmarks","text":"","category":"section"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The performance of PencilFFTs.jl is comparable to that of other open-source parallel FFT libraries implemented in lower-level languages. Below, we show comparisons with the Fortran implementation of P3DFFT, possibly the most popular of these libraries. The benchmarks were performed on the Jean–Zay cluster of the IDRIS French computing centre (CNRS).","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The figure below shows strong scaling benchmarks of 3D real-to-complex FFTs using 2D (\"pencil\") decomposition. The benchmarks were run for input arrays of dimensions N_x N_y N_z = 512^3, 1024^3 and 2048^3. Each timing is averaged over 100 repetitions.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"
              \n \n \n
              ","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"As seen above, PencilFFTs generally outperforms P3DFFT in its default setting. This is largely explained by the choice of using non-blocking point-to-point MPI communications (via MPI_Isend and MPI_Irecv), while P3DFFT uses collective MPI_Alltoallv calls. This enables PencilFFTs to perform data reordering operations on the partially received data while waiting for the incoming data, leading to better performance. Moreover, in contrast with P3DFFT, the high performance and scalability of PencilFFTs results from a highly generic code, handling decompositions in arbitrary dimensions and a relatively large (and extensible) variety of transformations.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"Note that PencilFFTs can optionally use collective communications (using MPI_Alltoallv) instead of point-to-point communications. For details, see the docs for PencilFFTPlan and for PencilArray transpositions. As seen above, collective communications generally perform worse than point-to-point ones, and runtimes are nearly indistinguishable from those of P3DFFT.","category":"page"},{"location":"benchmarks/#Benchmark-details","page":"Benchmarks","title":"Benchmark details","text":"","category":"section"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The benchmarks were performed using Julia 1.7-beta3 and Intel MPI 2019. We used PencilFFTs v0.12.5 with FFTW.jl v1.4.3 and MPI.jl v0.19.0. We used the Fortran implementation of P3DFFT, version 2.7.6, which was built with Intel 2019 compilers and linked to FFTW 3.3.8. The cluster where the benchmarks were run has Intel Cascade Lake 6248 processors with 2×20 cores per node.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The number of MPI processes along each decomposed dimension, P_1 and P_2, was automatically determined by a call to MPI_Dims_create, which tends to create a balanced decomposition with P_1 P_2. For instance, a total of 1024 processes is divided into P_1 = P_2 = 32. Different results may be obtained with other combinations, but this was not benchmarked.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The source files used to generate this benchmark, as well as the raw benchmark results, are all available in the PencilFFTs repo.","category":"page"},{"location":"GlobalFFTParams/#Global-FFT-parameters","page":"Global FFT parameters","title":"Global FFT parameters","text":"","category":"section"},{"location":"GlobalFFTParams/","page":"Global FFT parameters","title":"Global FFT parameters","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"GlobalFFTParams/","page":"Global FFT parameters","title":"Global FFT parameters","text":"GlobalFFTParams","category":"page"},{"location":"GlobalFFTParams/#PencilFFTs.GlobalFFTParams","page":"Global FFT parameters","title":"PencilFFTs.GlobalFFTParams","text":"GlobalFFTParams{T, N, inplace}\n\nSpecifies the global parameters for an N-dimensional distributed transform. These include the element type T and global data sizes of input and output data, as well as the transform types to be performed along each dimension.\n\n\n\nGlobalFFTParams(size_global, transforms, [real_type=Float64])\n\nDefine parameters for N-dimensional transform.\n\ntransforms must be a tuple of length N specifying the transforms to be applied along each dimension. Each element must be a subtype of Transforms.AbstractTransform. For all the possible transforms, see Transform types.\n\nThe element type must be a real type accepted by FFTW, i.e. either Float32 or Float64.\n\nNote that the transforms are applied one dimension at a time, with the leftmost dimension first for forward transforms.\n\nExample\n\nTo perform a 3D FFT of real data, first a real-to-complex FFT must be applied along the first dimension, followed by two complex-to-complex FFTs along the other dimensions:\n\njulia> size_global = (64, 32, 128); # size of real input data\n\njulia> transforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT());\n\njulia> fft_params = PencilFFTs.GlobalFFTParams(size_global, transforms)\nTransforms: (RFFT, FFT, FFT)\nInput type: Float64\nGlobal dimensions: (64, 32, 128) -> (33, 32, 128)\n\n\n\n\n\n","category":"type"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"EditURL = \"../../examples/in-place.jl\"","category":"page"},{"location":"generated/in-place/#In-place-transforms","page":"In-place transforms","title":"In-place transforms","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Complex-to-complex and real-to-real transforms can be performed in-place, enabling important memory savings. The procedure is very similar to that of out-of-place transforms described in the tutorial. The differences are illustrated in the sections below.","category":"page"},{"location":"generated/in-place/#Creating-a-domain-partition","page":"In-place transforms","title":"Creating a domain partition","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"We start by partitioning a domain of dimensions 163264 along all available MPI processes.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"using PencilFFTs\nusing MPI\nMPI.Init()\n\ndims_global = (16, 32, 64) # global dimensions","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Such a partitioning is described by a Pencil object. Here we choose to decompose the domain along the last two dimensions. In this case, the actual number of processes along each of these dimensions is chosen automatically.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"decomp_dims = (2, 3)\ncomm = MPI.COMM_WORLD\npen = Pencil(dims_global, decomp_dims, comm)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"warning: Allowed decompositions\nDistributed transforms using PencilFFTs.jl require that the first dimension is not decomposed. In other words, if one wants to perform transforms, then decomp_dims above must not contain 1.","category":"page"},{"location":"generated/in-place/#Creating-in-place-plans","page":"In-place transforms","title":"Creating in-place plans","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"To create an in-place plan, pass an in-place transform such as Transforms.FFT! or Transforms.R2R! to PencilFFTPlan. For instance:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"# Perform a 3D in-place complex-to-complex FFT.\ntransform = Transforms.FFT!()\n\n# Note that one can also combine different types of in-place transforms.\n# For instance:\n# transform = (\n# Transforms.R2R!(FFTW.REDFT01),\n# Transforms.FFT!(),\n# Transforms.R2R!(FFTW.DHT),\n# )","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"We can now create a distributed plan from the previously-created domain partition and the chosen transform.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan = PencilFFTPlan(pen, transform)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Note that in-place real-to-complex transforms are not currently supported. (In other words, the RFFT! transform type is not defined.)","category":"page"},{"location":"generated/in-place/#Allocating-data","page":"In-place transforms","title":"Allocating data","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"As with out-of-place plans, data should be allocated using allocate_input. The difference is that, for in-place plans, this function returns a ManyPencilArray object, which is a container holding multiple PencilArray views sharing the same memory space.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"# Allocate data for the plan.\n# Since `plan` is in-place, this returns a `ManyPencilArray` container.\nA = allocate_input(plan)\nsummary(A)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Note that allocate_output also works for in-place plans. In this case, it returns exactly the same thing as allocate_input.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"As shown in the next section, in-place plans must be applied on the returned ManyPencilArray. On the other hand, one usually wants to access and modify data, and for this one needs the PencilArray views contained in the ManyPencilArray. The input and output array views can be obtained by calling first(::ManyPencilArray) and last(::ManyPencilArray).","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"For instance, we can initialise the input array with some data before transforming:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"using Random\nu_in = first(A) # input data view\nrandn!(u_in)\nsummary(u_in)","category":"page"},{"location":"generated/in-place/#Applying-plans","page":"In-place transforms","title":"Applying plans","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Like in FFTW.jl, one can perform in-place transforms using the * and \\ operators. As mentioned above, in-place plans must be applied on the ManyPencilArray containers returned by allocate_input.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan * A; # performs in-place forward transform\nnothing #hide","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"After performing an in-place transform, data contained in u_in has been overwritten and has no \"physical\" meaning. In other words, u_in should not be used at this point. To access the transformed data, one should retrieve the output data view using last(A).","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"For instance, to compute the global sum of the transformed data:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"u_out = last(A) # output data view\nsum(u_out) # sum of transformed data (note that `sum` reduces over all processes)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Finally, we can perform a backward transform and do stuff with the input view:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan \\ A; # perform in-place backward transform\nnothing #hide","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"At this point, the data can be once again found in the input view u_in, while u_out should not be accessed.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"This page was generated using Literate.jl.","category":"page"},{"location":"Transforms/#Available-transforms","page":"Available transforms","title":"Available transforms","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"CurrentModule = PencilFFTs.Transforms","category":"page"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"Transforms","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms","page":"Available transforms","title":"PencilFFTs.Transforms","text":"Defines different one-dimensional FFT-based transforms.\n\nThe transforms are all subtypes of an AbstractTransform type.\n\nWhen possible, the names of the transforms are kept consistent with the functions exported by AbstractFFTs.jl and FFTW.jl.\n\n\n\n\n\n","category":"module"},{"location":"Transforms/#Transform-types","page":"Available transforms","title":"Transform types","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"FFT\nFFT!\nBFFT\nBFFT!\n\nRFFT\nRFFT!\nBRFFT\nBRFFT!\n\nR2R\nR2R!\n\nNoTransform\nNoTransform!","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.FFT","page":"Available transforms","title":"PencilFFTs.Transforms.FFT","text":"FFT()\n\nComplex-to-complex FFT.\n\nSee also AbstractFFTs.fft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.FFT!","page":"Available transforms","title":"PencilFFTs.Transforms.FFT!","text":"FFT!()\n\nIn-place version of FFT.\n\nSee also AbstractFFTs.fft!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BFFT","page":"Available transforms","title":"PencilFFTs.Transforms.BFFT","text":"BFFT()\n\nUnnormalised backward complex-to-complex FFT.\n\nLike AbstractFFTs.bfft, this transform is not normalised. To obtain the inverse transform, divide the output by the length of the transformed dimension.\n\nSee also AbstractFFTs.bfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.BFFT!","text":"BFFT()\n\nIn-place version of BFFT.\n\nSee also AbstractFFTs.bfft!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.RFFT","page":"Available transforms","title":"PencilFFTs.Transforms.RFFT","text":"RFFT()\n\nReal-to-complex FFT.\n\nSee also AbstractFFTs.rfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.RFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.RFFT!","text":"RFFT!()\n\nIn-place version of RFFT.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BRFFT","page":"Available transforms","title":"PencilFFTs.Transforms.BRFFT","text":"BRFFT(d::Integer)\nBRFFT((d1, d2, ..., dN))\n\nUnnormalised inverse of RFFT.\n\nTo obtain the inverse transform, divide the output by the length of the transformed dimension (of the real output array).\n\nAs described in the AbstractFFTs docs, the length of the output cannot be fully inferred from the input length. For this reason, the BRFFT constructor accepts an optional d argument indicating the output length.\n\nFor multidimensional datasets, a tuple of dimensions (d1, d2, ..., dN) may also be passed. This is equivalent to passing just dN. In this case, the last dimension (dN) is the one that changes size between the input and output. Note that this is the opposite of FFTW.brfft. The reason is that, in PencilFFTs, the last dimension is the one along which a complex-to-real transform is performed.\n\nSee also AbstractFFTs.brfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BRFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.BRFFT!","text":"BRFFT!(d::Integer)\nBRFFT!((d1, d2, ..., dN))\n\nIn-place version of BRFFT.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.R2R","page":"Available transforms","title":"PencilFFTs.Transforms.R2R","text":"R2R(kind)\n\nReal-to-real transform of type kind.\n\nThe possible values of kind are those described in the FFTW.r2r docs and the FFTW manual:\n\ndiscrete cosine transforms: FFTW.REDFT00, FFTW.REDFT01, FFTW.REDFFT10, FFTW.REDFFT11\ndiscrete sine transforms: FFTW.RODFT00, FFTW.RODFT01, FFTW.RODFFT10, FFTW.RODFFT11\ndiscrete Hartley transform: FFTW.DHT\n\nNote: half-complex format DFTs (FFTW.R2HC, FFTW.HC2R) are not currently supported.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.R2R!","page":"Available transforms","title":"PencilFFTs.Transforms.R2R!","text":"R2R!(kind)\n\nIn-place version of R2R.\n\nSee also FFTW.r2r!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.NoTransform","page":"Available transforms","title":"PencilFFTs.Transforms.NoTransform","text":"NoTransform()\n\nIdentity transform.\n\nSpecifies that no transformation should be applied.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.NoTransform!","page":"Available transforms","title":"PencilFFTs.Transforms.NoTransform!","text":"NoTransform!()\n\nIn-place version of NoTransform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#Internals","page":"Available transforms","title":"Internals","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"What follows is used internally in PencilFFTs.","category":"page"},{"location":"Transforms/#Types","page":"Available transforms","title":"Types","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"AbstractCustomPlan\nAbstractTransform\nIdentityPlan\nIdentityPlan!\nPlan","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.AbstractCustomPlan","page":"Available transforms","title":"PencilFFTs.Transforms.AbstractCustomPlan","text":"AbstractCustomPlan\n\nAbstract type defining a custom plan, to be used as an alternative to FFTW plans (FFTW.FFTWPlan).\n\nThe only custom plan defined in this module is IdentityPlan. The user can define other custom plans that are also subtypes of AbstractCustomPlan.\n\nNote that plan returns a subtype of either AbstractFFTs.Plan or AbstractCustomPlan.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.AbstractTransform","page":"Available transforms","title":"PencilFFTs.Transforms.AbstractTransform","text":"AbstractTransform\n\nSpecifies a one-dimensional FFT-based transform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.IdentityPlan","page":"Available transforms","title":"PencilFFTs.Transforms.IdentityPlan","text":"IdentityPlan\n\nType of plan associated to NoTransform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.IdentityPlan!","page":"Available transforms","title":"PencilFFTs.Transforms.IdentityPlan!","text":"IdentityPlan!\n\nType of plan associated to NoTransform!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.Plan","page":"Available transforms","title":"PencilFFTs.Transforms.Plan","text":"Plan = Union{AbstractFFTs.Plan, AbstractCustomPlan}\n\nUnion type representing any plan returned by plan.\n\nSee also AbstractCustomPlan.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#Functions","page":"Available transforms","title":"Functions","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"plan\n\nbinv\nscale_factor\n\neltype_input\neltype_output\nexpand_dims\nis_inplace\nkind\nlength_output","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.plan","page":"Available transforms","title":"PencilFFTs.Transforms.plan","text":"plan(transform::AbstractTransform, A, [dims];\n flags=FFTW.ESTIMATE, timelimit=Inf)\n\nCreate plan to transform array A along dimensions dims.\n\nIf dims is not specified, all dimensions of A are transformed.\n\nFor FFT plans, this function wraps the AbstractFFTs.jl and FFTW.jl plan creation functions. For more details on the function arguments, see AbstractFFTs.plan_fft.\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.binv","page":"Available transforms","title":"PencilFFTs.Transforms.binv","text":"binv(transform::AbstractTransform, d::Integer)\n\nReturns the backwards transform associated to the given transform.\n\nThe second argument must be the length of the first transformed dimension in the forward transform. It is used in particular when transform = RFFT(), to determine the length of the inverse (complex-to-real) transform. See the AbstractFFTs.irfft docs for details.\n\nThe backwards transform returned by this function is not normalised. The normalisation factor for a given array can be obtained by calling scale_factor.\n\nExample\n\njulia> binv(Transforms.FFT(), 42)\nBFFT\n\njulia> binv(Transforms.BRFFT(9), 42)\nRFFT\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.scale_factor","page":"Available transforms","title":"PencilFFTs.Transforms.scale_factor","text":"scale_factor(transform::AbstractTransform, A, [dims = 1:ndims(A)])\n\nGet factor required to normalise the given array after a transformation along dimensions dims (all dimensions by default).\n\nThe array A must have the dimensions of the transform input.\n\nImportant: the dimensions dims must be the same that were passed to plan.\n\nExamples\n\njulia> C = zeros(ComplexF32, 3, 4, 5);\n\njulia> scale_factor(Transforms.FFT(), C)\n60\n\njulia> scale_factor(Transforms.BFFT(), C)\n60\n\njulia> scale_factor(Transforms.BFFT(), C, 2:3)\n20\n\njulia> R = zeros(Float64, 3, 4, 5);\n\njulia> scale_factor(Transforms.RFFT(), R, 2)\n4\n\njulia> scale_factor(Transforms.RFFT(), R, 2:3)\n20\n\njulia> scale_factor(Transforms.BRFFT(8), C)\n96\n\njulia> scale_factor(Transforms.BRFFT(9), C)\n108\n\nThis will fail because the input of RFFT is real, and R is a complex array:\n\njulia> scale_factor(Transforms.RFFT(), C, 2:3)\nERROR: MethodError: no method matching scale_factor(::PencilFFTs.Transforms.RFFT, ::Array{ComplexF32, 3}, ::UnitRange{Int64})\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.eltype_input","page":"Available transforms","title":"PencilFFTs.Transforms.eltype_input","text":"eltype_input(transform::AbstractTransform, real_type<:AbstractFloat)\n\nDetermine input data type for a given transform given the floating point precision of the input data.\n\nSome transforms, such as R2R and NoTransform, can take both real and complex data. For those kinds of transforms, nothing is returned.\n\nExample\n\njulia> eltype_input(Transforms.FFT(), Float32)\nComplexF32 (alias for Complex{Float32})\n\njulia> eltype_input(Transforms.RFFT(), Float64)\nFloat64\n\njulia> eltype_input(Transforms.R2R(FFTW.REDFT01), Float64) # nothing\n\njulia> eltype_input(Transforms.NoTransform(), Float64) # nothing\n\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.eltype_output","page":"Available transforms","title":"PencilFFTs.Transforms.eltype_output","text":"eltype_output(transform::AbstractTransform, eltype_input)\n\nReturns the output data type for a given transform given the input type.\n\nThrows ArgumentError if the input data type is incompatible with the transform type.\n\nExample\n\njulia> eltype_output(Transforms.NoTransform(), Float32)\nFloat32\n\njulia> eltype_output(Transforms.RFFT(), Float64)\nComplexF64 (alias for Complex{Float64})\n\njulia> eltype_output(Transforms.BRFFT(4), ComplexF32)\nFloat32\n\njulia> eltype_output(Transforms.FFT(), Float64)\nERROR: ArgumentError: invalid input data type for PencilFFTs.Transforms.FFT: Float64\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.expand_dims","page":"Available transforms","title":"PencilFFTs.Transforms.expand_dims","text":"expand_dims(transform::AbstractTransform, Val(N))\n\nExpand a single multidimensional transform into one transform per dimension.\n\nExample\n\n# Expand a real-to-complex transform in 3 dimensions.\njulia> expand_dims(Transforms.RFFT(), Val(3))\n(RFFT, FFT, FFT)\n\njulia> expand_dims(Transforms.BRFFT(4), Val(3))\n(BFFT, BFFT, BRFFT{even})\n\njulia> expand_dims(Transforms.NoTransform(), Val(2))\n(NoTransform, NoTransform)\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.is_inplace","page":"Available transforms","title":"PencilFFTs.Transforms.is_inplace","text":"is_inplace(transform::AbstractTransform) -> Bool\nis_inplace(transforms::Vararg{AbtractTransform}) -> Union{Bool, Nothing}\n\nCheck whether a transform or a list of transforms is performed in-place.\n\nIf the list of transforms has a combination of in-place and out-of-place transforms, nothing is returned.\n\nExample\n\njulia> is_inplace(Transforms.RFFT())\nfalse\n\njulia> is_inplace(Transforms.NoTransform!())\ntrue\n\njulia> is_inplace(Transforms.FFT!(), Transforms.R2R!(FFTW.REDFT01))\ntrue\n\njulia> is_inplace(Transforms.FFT(), Transforms.R2R(FFTW.REDFT01))\nfalse\n\njulia> is_inplace(Transforms.FFT(), Transforms.R2R!(FFTW.REDFT01)) === nothing\ntrue\n\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.kind","page":"Available transforms","title":"PencilFFTs.Transforms.kind","text":"kind(transform::R2R)\n\nGet kind of real-to-real transform.\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.length_output","page":"Available transforms","title":"PencilFFTs.Transforms.length_output","text":"length_output(transform::AbstractTransform, length_in::Integer)\n\nReturns the length of the transform output, given the length of its input.\n\nThe input and output lengths are specified in terms of the respective input and output datatypes. For instance, for real-to-complex transforms, these are respectively the length of input real data and of output complex data.\n\n\n\n\n\n","category":"function"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"EditURL = \"../../examples/navier_stokes.jl\"","category":"page"},{"location":"generated/navier_stokes/#Navier–Stokes-equations","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"In this example, we numerically solve the incompressible Navier–Stokes equations","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"_t bmv + (bmv bm) bmv = -frac1ρ bm p + ν ^2 bmv\nquad bm bmv = 0","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where bmv(bmx t) and p(bmx t) are respectively the velocity and pressure fields, ν is the fluid kinematic viscosity and ρ is the fluid density.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.","category":"page"},{"location":"generated/navier_stokes/#First-steps","page":"Navier–Stokes equations","title":"First steps","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We start by loading the required packages, initialising MPI and setting the simulation parameters.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using MPI\nusing PencilFFTs\n\nMPI.Init()\ncomm = MPI.COMM_WORLD\nprocid = MPI.Comm_rank(comm) + 1\n\n# Simulation parameters\nNs = (64, 64, 64) # = (Nx, Ny, Nz)\nLs = (2π, 2π, 2π) # = (Lx, Ly, Lz)\n\n# Collocation points (\"global\" = over all processes).\n# We include the endpoint (length = N + 1) for convenience.\nxs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls) # = (x, y, z)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's check the number of MPI processes over which we're running our simulation:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"MPI.Comm_size(comm)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"pen = Pencil(Ns, comm)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"The subdomain associated to the local MPI process can be obtained using range_local:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"range_local(pen)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We now construct a distributed vector field that follows the decomposition configuration we just created:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v⃗₀ = (\n PencilArray{Float64}(undef, pen), # vx\n PencilArray{Float64}(undef, pen), # vy\n PencilArray{Float64}(undef, pen), # vz\n)\nsummary(v⃗₀[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We still need to fill this array with interesting values that represent a physical velocity field.","category":"page"},{"location":"generated/navier_stokes/#Initial-condition","page":"Navier–Stokes equations","title":"Initial condition","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"beginaligned\nv_x(x y z) = u₀ sin(k₀ x) cos(k₀ y) cos(k₀ z) \nv_y(x y z) = -u₀ cos(k₀ x) sin(k₀ y) cos(k₀ z) \nv_z(x y z) = 0\nendaligned","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where u₀ and k₀ are two parameters setting the amplitude and the period of the velocity field.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"grid = localgrid(pen, xs_global)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can use this to initialise the velocity field:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"u₀ = 1.0\nk₀ = 2π / Ls[1] # should be integer if L = 2π (to preserve periodicity)\n\n@. v⃗₀[1] = u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)\n@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)\n@. v⃗₀[3] = 0\nnothing # hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's plot a 2D slice of the velocity field managed by the local MPI process:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using GLMakie\n\n# Compute the norm of a vector field represented by a tuple of arrays.\nfunction vecnorm(v⃗::NTuple)\n vnorm = similar(v⃗[1])\n for n ∈ eachindex(v⃗[1])\n w = zero(eltype(vnorm))\n for v ∈ v⃗\n w += v[n]^2\n end\n vnorm[n] = sqrt(w)\n end\n vnorm\nend\n\nlet fig = Figure(resolution = (700, 600))\n ax = Axis3(fig[1, 1]; aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\")\n vnorm = parent(vecnorm(v⃗₀)) # use `parent` because Makie doesn't like custom array types...\n ct = contour!(\n ax, grid.x, grid.y, grid.z, vnorm;\n alpha = 0.2, levels = 4,\n colormap = :viridis,\n colorrange = (0.0, 1.0),\n highclip = (:red, 0.2), lowclip = (:green, 0.2),\n )\n cb = Colorbar(fig[1, 2], ct; label = \"Velocity magnitude\")\n fig\nend","category":"page"},{"location":"generated/navier_stokes/#Velocity-in-Fourier-space","page":"Navier–Stokes equations","title":"Velocity in Fourier space","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"bmv(bmx t) =\n_bmk hatbmv_bmk(t) e^i bmk bmx","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where bmk = (k_x k_y k_z) are the discrete wave numbers.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for k_x:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using AbstractFFTs: fftfreq, rfftfreq\n\nks_global = (\n rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]), # kx | real-to-complex\n fftfreq(Ns[2], 2π * Ns[2] / Ls[2]), # ky | complex-to-complex\n fftfreq(Ns[3], 2π * Ns[3] / Ls[3]), # kz | complex-to-complex\n)\n\nks_global[1]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ks_global[2]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ks_global[3]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"See PencilFFTPlan for details on creating plans and on optional keyword arguments.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients hatbmv_bmk:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v̂s = plan .* v⃗₀\nsummary(v̂s[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Note that, in Fourier space, the domain decomposition is performed along the directions x and y:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"pencil(v̂s[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"This is because the 3D FFTs are performed one dimension at a time, with the x direction first and the z direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers bmk associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"grid_fourier = localgrid(v̂s[1], ks_global)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, bmω = bm bmv. In Fourier space, this becomes hatbmω = i bmk hatbmv.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using StaticArrays: SVector\nusing LinearAlgebra: ×\n\nfunction curl_fourier!(\n ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,\n ) where {N}\n @inbounds for I ∈ eachindex(grid_fourier)\n # We use StaticArrays for the cross product between small vectors.\n ik⃗ = im * SVector(grid_fourier[I])\n v⃗ = SVector(getindex.(v̂s, Ref(I))) # = (v̂s[1][I], v̂s[2][I], ...)\n ω⃗ = ik⃗ × v⃗\n for n ∈ eachindex(ω⃗)\n ω̂s[n][I] = ω⃗[n]\n end\n end\n ω̂s\nend\n\nω̂s = similar.(v̂s)\ncurl_fourier!(ω̂s, v̂s, grid_fourier);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We finally transform back to physical space and plot the result:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ωs = plan .\\ ω̂s\n\nlet fig = Figure(resolution = (700, 600))\n ax = Axis3(fig[1, 1]; aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\")\n ω_norm = parent(vecnorm(ωs))\n ct = contour!(\n ax, grid.x, grid.y, grid.z, ω_norm;\n alpha = 0.1, levels = 0.8:0.2:2.0,\n colormap = :viridis, colorrange = (0.8, 2.0),\n highclip = (:red, 0.2), lowclip = (:green, 0.2),\n )\n cb = Colorbar(fig[1, 2], ct; label = \"Vorticity magnitude\")\n fig\nend","category":"page"},{"location":"generated/navier_stokes/#Computing-the-non-linear-term","page":"Navier–Stokes equations","title":"Computing the non-linear term","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"_t hatbmv_bmk =\n- mathcalP_bmk left widehat(bmv bm) bmv right\n- ν bmk^2 hatbmv_bmk\nquad text with quad\nmathcalP_bmk(hatbmF_bmk) = left( I - fracbmk \nbmkbmk^2 right) hatbmF_bmk","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where mathcalP_bmk is a projection operator allowing to preserve the incompressibility condition bm bmv = 0. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Now that we have the wave numbers bmk, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients hatbmv_bmk of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, hatbmF_bmk = left widehat(bmv bm) bmv right_bmk. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Below we implement a function that computes the non-linear term in Fourier space based on its convective form (bmv bm) bmv = bm (bmv bmv). Note that this equivalence uses the incompressibility condition bm bmv = 0.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using LinearAlgebra: mul!, ldiv! # for applying FFT plans in-place\n\n# Compute non-linear term in Fourier space from velocity field in physical\n# space. Optional keyword arguments may be passed to avoid allocations.\nfunction ns_nonlinear!(\n F̂s, vs, plan, grid_fourier;\n vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),\n )\n # Compute F_i = ∂_j (v_i v_j) for each i.\n # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)\n w, ŵ = vbuf, v̂buf\n @inbounds for (i, F̂i) ∈ enumerate(F̂s)\n F̂i .= 0\n vi = vs[i]\n for (j, vj) ∈ enumerate(vs)\n w .= vi .* vj # w = v_i * v_j in physical space\n mul!(ŵ, plan, w) # same in Fourier space\n # Add derivative in Fourier space\n for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n kj = k⃗[j]\n F̂i[I] += im * kj * ŵ[I]\n end\n end\n end\n F̂s\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"As an example, let's use this function on our initial velocity field:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"F̂s = similar.(v̂s)\nns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)\n ks_max = maximum.(abs, ks_global) # maximum stored wave numbers (kx_max, ky_max, kz_max)\n ks_lim = (2 / 3) .* ks_max\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I]\n if any(abs.(k⃗) .> ks_lim)\n for ŵ ∈ ŵs\n ŵ[I] = 0\n end\n end\n end\n ŵs\nend\n\n# We can apply this on the previously computed non-linear term:\ndealias_twothirds!(F̂s, grid_fourier, ks_global);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Finally, we implement the projection associated to the incompressibility condition:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function project_divergence_free!(ûs, grid_fourier)\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I]\n k² = sum(abs2, k⃗)\n iszero(k²) && continue # avoid division by zero\n û = getindex.(ûs, Ref(I)) # (ûs[1][I], ûs[2][I], ...)\n for i ∈ eachindex(û)\n ŵ = û[i]\n for j ∈ eachindex(û)\n ŵ -= k⃗[i] * k⃗[j] * û[j] / k²\n end\n ûs[i][I] = ŵ\n end\n end\n ûs\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)\nv̂s_proj .≈ v̂s # the last one may be false because v_z = 0 initially","category":"page"},{"location":"generated/navier_stokes/#Putting-it-all-together","page":"Navier–Stokes equations","title":"Putting it all together","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function ns_rhs!(\n dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,\n ) where {N}\n # 1. Compute non-linear term and dealias it\n (; plan, cache, ks_global, grid_fourier) = p\n F̂s = cache.F̂s\n ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])\n dealias_twothirds!(F̂s, grid_fourier, ks_global)\n\n # 2. Project onto divergence-free space\n project_divergence_free!(F̂s, grid_fourier)\n\n # 3. Transform velocity to Fourier space\n v̂s = cache.v̂s\n map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)\n\n # 4. Add viscous term (and multiply projected non-linear term by -1)\n ν = p.ν\n for n ∈ eachindex(v̂s)\n v̂ = v̂s[n]\n F̂ = F̂s[n]\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n k² = sum(abs2, k⃗)\n F̂[I] = -F̂[I] - ν * k² * v̂[I]\n end\n end\n\n # 5. Transform RHS back to physical space\n map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)\n\n nothing\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using OrdinaryDiffEq\nusing RecursiveArrayTools: ArrayPartition\n\nns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)\nvs_init_ode = ArrayPartition(v⃗₀)\nsummary(vs_init_ode)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We now define solver parameters and temporary variables, and initialise the problem:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"params = (;\n ν = 5e-3, # kinematic viscosity\n plan, grid_fourier, ks_global,\n cache = (\n v̂s = similar.(v̂s),\n F̂s = similar.(v̂s),\n )\n)\n\ntspan = (0.0, 10.0)\nprob = ODEProblem{true}(ns_rhs!, vs_init_ode, tspan, params)\nintegrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum E(k), to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity ν must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large k), while high enough to allow the small-scale motions to be correctly resolved.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function energy_spectrum!(Ek, ks, v̂s, grid_fourier)\n Nk = length(Ek)\n @assert Nk == length(ks)\n Ek .= 0\n for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n knorm = sqrt(sum(abs2, k⃗))\n i = searchsortedfirst(ks, knorm)\n i > Nk && continue\n v⃗ = getindex.(v̂s, Ref(I)) # = (v̂s[1][I], v̂s[2][I], ...)\n factor = k⃗[1] == 0 ? 1 : 2 # account for Hermitian symmetry and r2c transform\n Ek[i] += factor * sum(abs2, v⃗) / 2\n end\n MPI.Allreduce!(Ek, +, get_comm(v̂s[1])) # sum across all processes\n Ek\nend\n\nks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])\nEk = similar(ks)\nv̂s = plan .* integrator.u.x\nenergy_spectrum!(Ek, ks, v̂s, grid_fourier)\nEk ./= scale_factor(plan)^2 # rescale energy\n\ncurl_fourier!(ω̂s, v̂s, grid_fourier)\nldiv!.(ωs, plan, ω̂s)\nω⃗_plot = Observable(ωs)\nk_plot = @view ks[2:end]\nE_plot = Observable(@view Ek[2:end])\nt_plot = Observable(integrator.t)\n\nfig = let\n fig = Figure(resolution = (1200, 600))\n ax = Axis3(\n fig[1, 1][1, 1]; title = @lift(\"t = $(round($t_plot, digits = 3))\"),\n aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\",\n )\n ω_mag = @lift parent(vecnorm($ω⃗_plot))\n ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, ω_mag_norm;\n alpha = 0.3, levels = 3,\n colormap = :viridis, colorrange = (0.0, 1.0),\n highclip = (:red, 0.2), lowclip = (:green, 0.2),\n )\n cb = Colorbar(fig[1, 1][1, 2], ct; label = \"Normalised vorticity magnitude\")\n ax_sp = Axis(\n fig[1, 2];\n xlabel = \"k\", ylabel = \"E(k)\", xscale = log2, yscale = log10,\n title = \"Kinetic energy spectrum\",\n )\n ylims!(ax_sp, 1e-8, 1e0)\n scatterlines!(ax_sp, k_plot, E_plot)\n ks_slope = exp.(range(log(2.5), log(25.0), length = 3))\n E_fivethirds = @. 0.3 * ks_slope^(-5/3)\n @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)\n text!(ax_sp, L\"k^{-5/3}\"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))\n fig\nend\n\nusing Printf # hide\nwith_xvfb = ENV[\"DISPLAY\"] == \":99\" # hide\nnstep = 0 # hide\nconst tmpdir = mktempdir() # hide\nfilename_frame(procid, nstep) = joinpath(tmpdir, @sprintf(\"proc%d_%04d.png\", procid, nstep)) # hide\nrecord(fig, \"vorticity_proc$procid.mp4\"; framerate = 10) do io\n with_xvfb && recordframe!(io) # hide\n while integrator.t < 20\n dt = 0.001\n step!(integrator, dt)\n t_plot[] = integrator.t\n mul!.(v̂s, plan, integrator.u.x) # current velocity in Fourier space\n curl_fourier!(ω̂s, v̂s, grid_fourier)\n ldiv!.(ω⃗_plot[], plan, ω̂s)\n ω⃗_plot[] = ω⃗_plot[] # to force updating the plot\n energy_spectrum!(Ek, ks, v̂s, grid_fourier)\n Ek ./= scale_factor(plan)^2 # rescale energy\n E_plot[] = E_plot[]\n global nstep += 1 # hide\n with_xvfb ? # hide\n save(filename_frame(procid, nstep), fig) : # hide\n recordframe!(io)\n end\nend;\n\nif with_xvfb # hide\n run(pipeline(`ffmpeg -y -r 10 -i $tmpdir/proc$(procid)_%04d.png -c:v libx264 -vf \"fps=25,format=yuv420p\" vorticity_proc$procid.mp4`; stdout = \"ffmpeg.out\", stderr = \"ffmpeg.err\")) # hide\nend # hide\nnothing # hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"
              \n \n
              ","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"This page was generated using Literate.jl.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"EditURL = \"../../examples/gradient.jl\"","category":"page"},{"location":"generated/gradient/#Gradient-of-a-scalar-field","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This example shows different methods to compute the gradient of a real-valued 3D scalar field θ(bmx) in Fourier space, where bmx = (x y z). It is assumed that the field is periodic with period L = 2π along all dimensions.","category":"page"},{"location":"generated/gradient/#General-procedure","page":"Gradient of a scalar field","title":"General procedure","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The discrete Fourier expansion of θ writes","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"θ(bmx) = _bmk Z^3 hatθ(bmk) e^i bmk bmx","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"where bmk = (k_x k_y k_z) are the Fourier wave numbers and hatθ is the discrete Fourier transform of θ. Then, the spatial derivatives of θ are given by","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"frac θ(bmx) x_i =\n_bmk Z^3 i k_i hatθ(bmk) e^i bmk bmx","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"where the subscript i denotes one of the spatial components x, y or z.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In other words, to compute bm θ = (_x θ _y θ _z θ), one has to:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"transform θ to Fourier space to obtain hatθ,\nmultiply hatθ by i bmk,\ntransform the result back to physical space to obtain bm θ.","category":"page"},{"location":"generated/gradient/#Preparation","page":"Gradient of a scalar field","title":"Preparation","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In this section, we initialise a random real-valued scalar field θ and compute its FFT. For more details see the Tutorial.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"using MPI\nusing PencilFFTs\nusing Random\n\nMPI.Init()\n\n# Input data dimensions (Nx × Ny × Nz)\ndims = (64, 32, 64)\n\n# Apply a 3D real-to-complex (r2c) FFT.\ntransform = Transforms.RFFT()\n\n# Automatically create decomposition configuration\ncomm = MPI.COMM_WORLD\npen = Pencil(dims, comm)\n\n# Create plan\nplan = PencilFFTPlan(pen, transform)\n\n# Allocate data and initialise field\nθ = allocate_input(plan)\nrandn!(θ)\n\n# Perform distributed FFT\nθ_hat = plan * θ\nnothing # hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, we initialise the output that will hold ∇θ in Fourier space. Noting that ∇θ is a vector field, we choose to store it as a tuple of 3 PencilArrays.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"∇θ_hat = allocate_output(plan, Val(3))\n\n# This is equivalent:\n# ∇θ_hat = ntuple(d -> similar(θ_hat), Val(3))\n\nsummary(∇θ_hat)","category":"page"},{"location":"generated/gradient/#Fourier-wave-numbers","page":"Gradient of a scalar field","title":"Fourier wave numbers","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In general, the Fourier wave numbers are of the form k_i = 0 frac2πL_i frac4πL_i frac6πL_i , where L_i is the period along dimension i. When a real-to-complex Fourier transform is applied, roughly half of these wave numbers are redundant due to the Hermitian symmetry of the complex Fourier coefficients. In practice, this means that for the fastest dimension x (along which a real-to-complex transform is performed), the negative wave numbers are dropped, i.e. k_x = 0 frac2πL_x frac4πL_x .","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The AbstractFFTs package provides a convenient way to generate the Fourier wave numbers, using the functions fftfreq and rfftfreq. We can use these functions to initialise a \"grid\" of wave numbers associated to our 3D real-to-complex transform:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"using AbstractFFTs: fftfreq, rfftfreq\n\nbox_size = (2π, 2π, 2π) # Lx, Ly, Lz\nsample_rate = 2π .* dims ./ box_size\n\n# In our case (Lx = 2π and Nx even), this gives kx = [0, 1, 2, ..., Nx/2].\nkx = rfftfreq(dims[1], sample_rate[1])\n\n# In our case (Ly = 2π and Ny even), this gives\n# ky = [0, 1, 2, ..., Ny/2-1, -Ny/2, -Ny/2+1, ..., -1] (and similarly for kz).\nky = fftfreq(dims[2], sample_rate[2])\nkz = fftfreq(dims[3], sample_rate[3])\n\nkvec = (kx, ky, kz)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that kvec now contains the wave numbers associated to the global domain. In the following, we will only need the wave numbers associated to the portion of the domain handled by the local MPI process.","category":"page"},{"location":"generated/gradient/#gradient_method_global","page":"Gradient of a scalar field","title":"Method 1: global views","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"PencilArrays, returned for instance by allocate_input and allocate_output, take indices that start at 1, regardless of the location of the subdomain associated to the local process on the global grid. (In other words, PencilArrays take local indices.) On the other hand, we have defined the wave number vector kvec which, for each MPI process, is defined over the global domain, and as such it takes global indices.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"One straightforward way of making data arrays compatible with wave numbers is to use global views, i.e. arrays that take global indices. These are generated from PencilArrays by calling the global_view function. Note that, in general, global indices do not start at 1 for a given MPI process. A given process will own a range of data given by indices in (i1:i2, j1:j2, k1:k2).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"θ_glob = global_view(θ_hat)\n∇θ_glob = global_view.(∇θ_hat)\nsummary(θ_glob)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Once we have global views, we can combine data and wave numbers using the portion of global indices owned by the local MPI process, as shown below. We can use CartesianIndices to iterate over the global indices associated to the local process.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"for I in CartesianIndices(θ_glob)\n i, j, k = Tuple(I) # unpack indices\n\n # Wave number vector associated to current Cartesian index.\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n # Compute gradient in Fourier space.\n # Note that modifying ∇θ_glob also modifies the original PencilArray ∇θ_hat.\n ∇θ_glob[1][I] = im * kx * θ_glob[I]\n ∇θ_glob[2][I] = im * ky * θ_glob[I]\n ∇θ_glob[3][I] = im * kz * θ_glob[I]\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The above loop can be written in a slightly more efficient manner by precomputing im * θ_glob[I]:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@inbounds for I in CartesianIndices(θ_glob)\n i, j, k = Tuple(I)\n\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n u = im * θ_glob[I]\n\n ∇θ_glob[1][I] = kx * u\n ∇θ_glob[2][I] = ky * u\n ∇θ_glob[3][I] = kz * u\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Also note that the above can be easily written in a more generic way, e.g. for arbitrary dimensions, thanks in part to the use of CartesianIndices. Moreover, in the above there is no notion of the dimension permutations discussed in the tutorial, as it is all hidden behind the implementation of PencilArrays. And as seen later in the benchmarks, these (hidden) permutations have zero cost, as the speed is identical to that of a function that explicitly takes into account these permutations.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, we can perform a backwards transform to obtain bm θ in physical space:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"∇θ = plan \\ ∇θ_hat;\nnothing #hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that the transform is automatically broadcast over the three fields of the ∇θ_hat vector, and the result ∇θ is also a tuple of three PencilArrays.","category":"page"},{"location":"generated/gradient/#gradient_method_global_explicit","page":"Gradient of a scalar field","title":"Method 2: explicit global indexing","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Sometimes, one does not need to write generic code. In our case, one often knows the dimensionality of the problem and the memory layout of the data (i.e. the underlying index permutation).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Below is a reimplementation of the above loop, using explicit indices instead of CartesianIndices, and assuming that the underlying index permutation is (3, 2, 1), that is, data is stored in (z y x) order. As discussed in the tutorial, this is the default for transformed arrays. This example also serves as a more explicit explanation for what is going on in the first method.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"# Get local data range in the global grid.\nrng = axes(θ_glob) # = (i1:i2, j1:j2, k1:k2)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"For the loop below, we're assuming that the permutation is (3, 2, 1). In other words, the fastest index is the last one, and not the first one as it is usually in Julia. If the permutation is not (3, 2, 1), things will still work (well, except for the assertion below!), but the loop order will not be optimal.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@assert permutation(θ_hat) === Permutation(3, 2, 1)\n\n@inbounds for i in rng[1], j in rng[2], k in rng[3]\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n # Note that we still access the arrays in (i, j, k) order.\n # (The permutation happens behind the scenes!)\n u = im * θ_glob[i, j, k]\n\n ∇θ_glob[1][i, j, k] = kx * u\n ∇θ_glob[2][i, j, k] = ky * u\n ∇θ_glob[3][i, j, k] = kz * u\nend","category":"page"},{"location":"generated/gradient/#gradient_method_local","page":"Gradient of a scalar field","title":"Method 3: using local indices","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Alternatively, we can avoid global views and work directly on PencilArrays using local indices that start at 1. In this case, part of the strategy is to construct a \"local\" grid of wave numbers that can also be accessed with local indices. This can be conveniently done using the localgrid function of the PencilArrays.jl package, which accepts a PencilArray (or its associated Pencil) and the global coordinates (here kvec):","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"grid_fourier = localgrid(θ_hat, kvec)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that one can directly iterate on the returned grid object:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@inbounds for I in CartesianIndices(grid_fourier)\n # Wave number vector associated to current Cartesian index.\n local k⃗ # hide\n k⃗ = grid_fourier[I]\n u = im * θ_hat[I]\n ∇θ_hat[1][I] = k⃗[1] * u\n ∇θ_hat[2][I] = k⃗[2] * u\n ∇θ_hat[3][I] = k⃗[3] * u\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This implementation is as efficient as the other examples, while being slightly shorter to write. Moreover, it is quite generic, and can be made independent of the number of dimensions with little effort.","category":"page"},{"location":"generated/gradient/#gradient_method_broadcast","page":"Gradient of a scalar field","title":"Method 4: using broadcasting","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, note that the local grid object returned by localgrid makes it is possible to compute the gradient using broadcasting, thus fully avoiding scalar indexing. This can be quite convenient in some cases, and can also be very useful if one is working on GPUs (where scalar indexing is prohibitively expensive). Using broadcasting, the above examples simply become:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@. ∇θ_hat[1] = im * grid_fourier[1] * θ_hat\n@. ∇θ_hat[2] = im * grid_fourier[2] * θ_hat\n@. ∇θ_hat[3] = im * grid_fourier[3] * θ_hat\nnothing # hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Once again, as shown in the benchmarks further below, this method performs quite similarly to the other ones.","category":"page"},{"location":"generated/gradient/#Summary","page":"Gradient of a scalar field","title":"Summary","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The PencilArrays module provides different alternatives to deal with MPI-distributed data that may be subject to dimension permutations. In particular, one can choose to work with global indices (first two examples), with local indices (third example), or to avoid scalar indexing altogether (fourth example).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"If one wants to stay generic, making sure that the same code will work for arbitrary dimensions and will be efficient regardless of the underlying dimension permutation, methods 1, 3 or 4 should be preferred. These use CartesianIndices and make no assumptions on possible dimension permutations, which are by default enabled in the output of PencilFFTs transforms. In fact, such permutations are completely invisible in the implementations.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The second method uses explicit (i, j, k) indices. It assumes that the underlying permutation is (3, 2, 1) to loop with i as the slowest index and k as the fastest, which is the optimal order in this case given the permutation. As such, the implementation is less generic than the others, and differences in performance are negligible with respect to more generic variants.","category":"page"},{"location":"generated/gradient/#gradient_benchmarks","page":"Gradient of a scalar field","title":"Benchmark results","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The following are the benchmark results obtained from running examples/gradient.jl on a laptop, using 2 MPI processes and Julia 1.7.2, with an input array of global dimensions 64 32 64. The different methods detailed above are marked on the right. The \"lazy\" marks indicate runs where the wave numbers were represented by lazy Frequencies objects (returned by rfftfreq and fftfreq). Otherwise, they were collected into Vectors. For some reason, plain Vectors are faster when working with grids generated by localgrid.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In the script, additional implementations can be found which rely on a more advanced understanding of permutations and on the internals of the PencilArrays package. For instance, gradient_local_parent! directly works with the raw data stored in Julia Arrays, while gradient_local_linear! completely avoids CartesianIndices while staying generic and efficient. Nevertheless, these display roughly the same performance as the above examples.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":" gradient_global_view!... 89.900 μs\n gradient_global_view! (lazy)... 92.060 μs [Method 1]\n gradient_global_view_explicit!... 88.958 μs\n gradient_global_view_explicit! (lazy)... 81.055 μs [Method 2]\n gradient_local!... 92.305 μs\n gradient_grid!... 92.770 μs\n gradient_grid! (lazy)... 101.388 μs [Method 3]\n gradient_grid_broadcast!... 88.606 μs\n gradient_grid_broadcast! (lazy)... 151.020 μs [Method 4]\n gradient_local_parent!... 92.248 μs\n gradient_local_linear!... 91.212 μs\n gradient_local_linear_explicit!... 90.992 μs","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This page was generated using Literate.jl.","category":"page"},{"location":"PencilFFTs_timers/#PencilFFTs.measuring_performance","page":"Measuring performance","title":"Measuring performance","text":"","category":"section"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"It is possible to measure the time spent in different sections of the distributed transforms using the TimerOutputs package. This has a (very small) performance overhead, so it is disabled by default. To enable time measurements, call TimerOutputs.enable_debug_timings after loading PencilFFTs (see below for an example). For more details see the TimerOutputs docs.","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"Minimal example:","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"using MPI\nusing PencilFFTs\nusing TimerOutputs\n\n# Enable timing of `PencilFFTs` functions\nTimerOutputs.enable_debug_timings(PencilFFTs)\nTimerOutputs.enable_debug_timings(PencilArrays)\nTimerOutputs.enable_debug_timings(Transpositions)\n\nMPI.Init()\n\nplan = PencilFFTPlan(#= args... =#)\n\n# [do stuff with `plan`...]\n\n# Retrieve and print timing data associated to `plan`\nto = timer(plan)\nprint_timer(to)","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"By default, each PencilFFTPlan has its own TimerOutput. If you already have a TimerOutput, you can pass it to the PencilFFTPlan constructor:","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"to = TimerOutput()\nplan = PencilFFTPlan(..., timer=to)\n\n# [do stuff with `plan`...]\n\nprint_timer(to)","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"#PencilFFTs","page":"Home","title":"PencilFFTs","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Fast Fourier transforms of MPI-distributed Julia arrays.","category":"page"},{"location":"#Introduction","page":"Home","title":"Introduction","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package provides multidimensional FFTs and related transforms on MPI-distributed Julia arrays via the PencilArrays package.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The name of this package originates from the decomposition of 3D domains along two out of three dimensions, sometimes called pencil decomposition. This is illustrated by the figure below,[1] where each coloured block is managed by a different MPI process. Typically, one wants to compute FFTs on a scalar or vector field along the three spatial dimensions. In the case of a pencil decomposition, 3D FFTs are performed one dimension at a time, along the non-decomposed direction. Transforms must then be interleaved with global data transpositions to switch between pencil configurations. In high-performance computing environments, such data transpositions are generally the most expensive part of a parallel FFT computation, due to the large cost of communications between computing nodes.","category":"page"},{"location":"","page":"Home","title":"Home","text":"
              \n \n
              ","category":"page"},{"location":"","page":"Home","title":"Home","text":"More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension N. The decompositions can be performed along an arbitrary number M N of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.","category":"page"},{"location":"#Features","page":"Home","title":"Features","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;\nFFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;\nin-place and out-of-place transforms;\nhigh scalability up to (at least) tens of thousands of MPI processes.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"PencilFFTs can be installed using the Julia package manager:","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> ] add PencilFFTs","category":"page"},{"location":"#Similar-projects","page":"Home","title":"Similar projects","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.\nPFFT is a very general parallel FFT library written in C.\nP3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.\n2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[1]: Figure adapted from this PhD thesis.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[2]: For the pencil decomposition represented in the figure, N = 3 and M = 2.","category":"page"},{"location":"PencilFFTs/#Distributed-FFT-plans","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"Distributed FFTs are implemented in the PencilFFTs module, and are built on top of the PencilArrays package.","category":"page"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"PencilFFTs/#Creating-plans","page":"Distributed FFT plans","title":"Creating plans","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"PencilFFTPlan","category":"page"},{"location":"PencilFFTs/#PencilFFTs.PencilFFTPlan","page":"Distributed FFT plans","title":"PencilFFTs.PencilFFTPlan","text":"PencilFFTPlan{T,N} <: AbstractFFTs.Plan{T}\n\nPlan for N-dimensional FFT-based transform on MPI-distributed data, where input data has type T.\n\n\n\nPencilFFTPlan(p::Pencil, transforms; kwargs...)\n\nCreate a PencilFFTPlan for distributed arrays following a given Pencil configuration. See variant below for details on the specification of transforms and on possible keyword arguments.\n\n\n\nPencilFFTPlan(\n A::PencilArray, transforms;\n fftw_flags = FFTW.ESTIMATE,\n fftw_timelimit = FFTW.NO_TIMELIMIT,\n permute_dims = Val(true),\n transpose_method = Transpositions.PointToPoint(),\n timer = timer(pencil(A)),\n)\n\nCreate plan for N-dimensional transform on MPI-distributed PencilArrays.\n\nExtended help\n\nThis creates a PencilFFTPlan for arrays sharing the same properties as A (dimensions, MPI decomposition, memory layout, ...), which describe data on an N-dimensional domain.\n\nTransforms\n\nThe transforms to be applied along each dimension are specified by the transforms argument. Possible transforms are defined as subtypes of Transforms.AbstractTransform, and are listed in Transform types. This argument may be either:\n\na tuple of N transforms to be applied along each dimension. For instance, transforms = (Transforms.R2R(FFTW.REDFT01), Transforms.RFFT(), Transforms.FFT());\na single transform to be applied along all dimensions. The input is automatically expanded into N equivalent transforms. For instance, for a three-dimensional array, transforms = Transforms.RFFT() specifies a 3D real-to-complex transform, and is equivalent to passing (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT()).\n\nNote that forward transforms are applied from left to right. In the last example, this means that a real-to-complex transform (RFFT) is first performed along the first dimension. This is followed by complex-to-complex transforms (FFT) along the second and third dimensions.\n\nInput data layout\n\nThe input PencilArray must satisfy the following constraints:\n\narray dimensions must not be permuted. This is the default when constructing PencilArrays.\nfor an M-dimensional domain decomposition (with M < N), the input array must be decomposed along the last M dimensions. For example, for a 2D decomposition of 3D data, the decomposed dimensions must be (2, 3). In particular, the first array dimension must not be distributed among different MPI processes.\nIn the PencilArrays package, the decomposed dimensions are specified at the moment of constructing a Pencil.\nthe element type must be compatible with the specified transform. For instance, real-to-complex transforms (Transforms.RFFT) require the input to be real floating point values. Other transforms, such as Transforms.R2R, accept both real and complex data.\n\nKeyword arguments\n\nThe keyword arguments fftw_flags and fftw_timelimit are passed to the FFTW plan creation functions (see AbstractFFTs docs).\npermute_dims determines whether the indices of the output data should be reversed. For instance, if the input data has global dimensions (Nx, Ny, Nz), then the output of a complex-to-complex FFT would have dimensions (Nz, Ny, Nx). This enables FFTs to always be performed along the first (i.e. fastest) array dimension, which could lead to performance gains. This option is enabled by default. For type inference reasons, it must be a value type (Val(true) or Val(false)).\ntranspose_method allows to select between implementations of the global data transpositions. See PencilArrays docs docs for details.\ntimer should be a TimerOutput object. See Measuring performance for details.\n\n\n\nPencilFFTPlan(\n dims_global::Dims{N}, transforms, proc_dims::Dims{M}, comm::MPI.Comm,\n [real_type = Float64]; extra_dims = (), kws...\n)\n\nCreate plan for N-dimensional transform.\n\nExtended help\n\nInstead of taking a PencilArray or a Pencil, this constructor requires the global dimensions of the input data, passed via the size_global argument.\n\nThe data is distributed over the MPI processes in the comm communicator. The distribution is performed over M dimensions (with M < N) according to the values in proc_dims, which specifies the number of MPI processes to put along each dimension.\n\nPencilArrays that may be transformed with the returned plan can be created using allocate_input.\n\nOptional arguments\n\nThe floating point precision can be selected by setting real_type parameter, which is Float64 by default.\nextra_dims may be used to specify the sizes of one or more extra dimensions that should not be transformed. These dimensions will be added to the rightmost (i.e. slowest) indices of the arrays. See Extra dimensions below for usage hints.\nsee the other constructor for more keyword arguments.\n\nExtra dimensions\n\nOne possible application of extra_dims is for describing the components of a vector or tensor field. However, this means that different PencilFFTPlans would need to be created for each kind of field (scalar, vector, ...). To avoid the creation of multiple plans, a possibly better alternative is to create tuples (or arrays) of PencilArrays using allocate_input and allocate_output.\n\nAnother more legitimate usage of extra_dims is to specify one or more Cartesian dimensions that should not be transformed nor split among MPI processes.\n\nExample\n\nSuppose we want to perform a 3D FFT of real data. The data is to be decomposed along two dimensions, over 8 MPI processes:\n\nsize_global = (64, 32, 128) # size of real input data\n\n# Perform real-to-complex transform along the first dimension, then\n# complex-to-complex transforms along the other dimensions.\ntransforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())\n# transforms = Transforms.RFFT() # this is equivalent to the above line\n\nproc_dims = (4, 2) # 2D decomposition\ncomm = MPI.COMM_WORLD\n\nplan = PencilFFTPlan(size_global, transforms, proc_dims, comm)\n\n\n\n\n\n","category":"type"},{"location":"PencilFFTs/#Allocating-data","page":"Distributed FFT plans","title":"Allocating data","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"allocate_input\nallocate_output","category":"page"},{"location":"PencilFFTs/#PencilFFTs.allocate_input","page":"Distributed FFT plans","title":"PencilFFTs.allocate_input","text":"allocate_input(p::PencilFFTPlan) -> PencilArray\nallocate_input(p::PencilFFTPlan, dims...) -> Array{PencilArray}\nallocate_input(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}\n\nAllocate uninitialised PencilArray that can hold input data for the given plan.\n\nThe second and third forms respectively allocate an array of PencilArrays of size dims, and a tuple of N PencilArrays.\n\nnote: In-place plans\nIf p is an in-place real-to-real or complex-to-complex plan, a ManyPencilArray is allocated. If p is an in-place real-to-complex plan, a ManyPencilArrayRFFT! is allocated. These types hold PencilArray wrappers for the input and output transforms (as well as for intermediate transforms) which share the same space in memory. The input and output PencilArrays should be respectively accessed by calling first(::ManyPencilArray) and last(::ManyPencilArray).ExampleSuppose p is an in-place PencilFFTPlan. Then,@assert is_inplace(p)\nA = allocate_input(p) :: ManyPencilArray\nv_in = first(A) :: PencilArray # input data view\nv_out = last(A) :: PencilArray # output data viewAlso note that in-place plans must be performed directly on the returned ManyPencilArray, and not on the contained PencilArray views:p * A # perform forward transform in-place\np \\ A # perform backward transform in-place\n# p * v_in # not allowed!!\n\n\n\n\n\n","category":"function"},{"location":"PencilFFTs/#PencilFFTs.allocate_output","page":"Distributed FFT plans","title":"PencilFFTs.allocate_output","text":"allocate_output(p::PencilFFTPlan) -> PencilArray\nallocate_output(p::PencilFFTPlan, dims...) -> Array{PencilArray}\nallocate_output(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}\n\nAllocate uninitialised PencilArray that can hold output data for the given plan.\n\nIf p is an in-place plan, a ManyPencilArray is allocated.\n\nSee allocate_input for details.\n\n\n\n\n\n","category":"function"},{"location":"PencilFFTs/#Methods","page":"Distributed FFT plans","title":"Methods","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"get_comm(::PencilFFTPlan)\nscale_factor(::PencilFFTPlan)\ntimer(::PencilFFTPlan)\nis_inplace(::PencilFFTPlan)","category":"page"},{"location":"PencilFFTs/#PencilArrays.Pencils.MPITopologies.get_comm-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilArrays.Pencils.MPITopologies.get_comm","text":"get_comm(p::PencilFFTPlan)\n\nGet MPI communicator associated to a PencilFFTPlan.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilFFTs.Transforms.scale_factor-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilFFTs.Transforms.scale_factor","text":"scale_factor(p::PencilFFTPlan)\n\nGet scale factor associated to a PencilFFTPlan.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilArrays.Pencils.timer-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilArrays.Pencils.timer","text":"timer(p::PencilFFTPlan)\n\nGet TimerOutput attached to a PencilFFTPlan.\n\nSee Measuring performance for details.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilFFTs.Transforms.is_inplace-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilFFTs.Transforms.is_inplace","text":"Transforms.is_inplace(p::PencilFFTPlan)\n\nReturns true if the given plan operates in-place on the input data, false otherwise.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#Internals","page":"Distributed FFT plans","title":"Internals","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"ManyPencilArrayRFFT!","category":"page"},{"location":"PencilFFTs/#PencilFFTs.ManyPencilArrayRFFT!","page":"Distributed FFT plans","title":"PencilFFTs.ManyPencilArrayRFFT!","text":"ManyPencilArrayRFFT!{T,N,M} <: AbstractManyPencilArray{N,M}\n\nContainer holding M different PencilArray views to the same underlying data buffer. All views share the same and dimensionality N. The element type T of the first view is real, that of subsequent views is Complex{T}. \n\nThis can be used to perform in-place real-to-complex plan, see alsoTransforms.RFFT!. It is used internally for such transforms by allocate_input and should not be constructed directly.\n\n\n\nManyPencilArrayRFFT!{T}(undef, pencils...; extra_dims=())\n\nCreate a ManyPencilArrayRFFT! container that can hold data of type T and Complex{T} associated to all the given Pencils.\n\nThe optional extra_dims argument is the same as for PencilArray.\n\nSee also ManyPencilArray\n\n\n\n\n\n","category":"type"}] +} diff --git a/previews/PR68/siteinfo.js b/previews/PR68/siteinfo.js new file mode 100644 index 00000000..ea011b52 --- /dev/null +++ b/previews/PR68/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "previews/PR68"; diff --git a/previews/PR39/tutorial/index.html b/previews/PR68/tutorial/index.html similarity index 62% rename from previews/PR39/tutorial/index.html rename to previews/PR68/tutorial/index.html index 84aa68e1..21f4eaf1 100644 --- a/previews/PR39/tutorial/index.html +++ b/previews/PR68/tutorial/index.html @@ -1,5 +1,5 @@ -Tutorial · PencilFFTs.jl
              Edit on GitHub

              Tutorial

              The following tutorial shows how to perform a 3D FFT of real periodic data defined on a grid of $N_x × N_y × N_z$ points.

              +Tutorial · PencilFFTs.jl
              GitHub

              Tutorial

              The following tutorial shows how to perform a 3D FFT of real periodic data defined on a grid of $N_x × N_y × N_z$ points.

              + Strong scaling of PencilFFTs +

              As seen above, PencilFFTs generally outperforms P3DFFT in its default setting. This is largely explained by the choice of using non-blocking point-to-point MPI communications (via MPI_Isend and MPI_Irecv), while P3DFFT uses collective MPI_Alltoallv calls. This enables PencilFFTs to perform data reordering operations on the partially received data while waiting for the incoming data, leading to better performance. Moreover, in contrast with P3DFFT, the high performance and scalability of PencilFFTs results from a highly generic code, handling decompositions in arbitrary dimensions and a relatively large (and extensible) variety of transformations.

              Note that PencilFFTs can optionally use collective communications (using MPI_Alltoallv) instead of point-to-point communications. For details, see the docs for PencilFFTPlan and for PencilArray transpositions. As seen above, collective communications generally perform worse than point-to-point ones, and runtimes are nearly indistinguishable from those of P3DFFT.

              Benchmark details

              The benchmarks were performed using Julia 1.7-beta3 and Intel MPI 2019. We used PencilFFTs v0.12.5 with FFTW.jl v1.4.3 and MPI.jl v0.19.0. We used the Fortran implementation of P3DFFT, version 2.7.6, which was built with Intel 2019 compilers and linked to FFTW 3.3.8. The cluster where the benchmarks were run has Intel Cascade Lake 6248 processors with 2×20 cores per node.

              The number of MPI processes along each decomposed dimension, $P_1$ and $P_2$, was automatically determined by a call to MPI_Dims_create, which tends to create a balanced decomposition with $P_1 ≈ P_2$. For instance, a total of 1024 processes is divided into $P_1 = P_2 = 32$. Different results may be obtained with other combinations, but this was not benchmarked.

              The source files used to generate this benchmark, as well as the raw benchmark results, are all available in the PencilFFTs repo.

              diff --git a/previews/PR76/generated/ffmpeg.err b/previews/PR76/generated/ffmpeg.err new file mode 100644 index 00000000..ab67442c --- /dev/null +++ b/previews/PR76/generated/ffmpeg.err @@ -0,0 +1,48 @@ +ffmpeg version 4.4.2-0ubuntu0.22.04.1 Copyright (c) 2000-2021 the FFmpeg developers + built with gcc 11 (Ubuntu 11.2.0-19ubuntu1) + configuration: --prefix=/usr --extra-version=0ubuntu0.22.04.1 --toolchain=hardened --libdir=/usr/lib/x86_64-linux-gnu --incdir=/usr/include/x86_64-linux-gnu --arch=amd64 --enable-gpl --disable-stripping --enable-gnutls --enable-ladspa --enable-libaom --enable-libass --enable-libbluray --enable-libbs2b --enable-libcaca --enable-libcdio --enable-libcodec2 --enable-libdav1d --enable-libflite --enable-libfontconfig --enable-libfreetype --enable-libfribidi --enable-libgme --enable-libgsm --enable-libjack --enable-libmp3lame --enable-libmysofa --enable-libopenjpeg --enable-libopenmpt --enable-libopus --enable-libpulse --enable-librabbitmq --enable-librubberband --enable-libshine --enable-libsnappy --enable-libsoxr --enable-libspeex --enable-libsrt --enable-libssh --enable-libtheora --enable-libtwolame --enable-libvidstab --enable-libvorbis --enable-libvpx --enable-libwebp --enable-libx265 --enable-libxml2 --enable-libxvid --enable-libzimg --enable-libzmq --enable-libzvbi --enable-lv2 --enable-omx --enable-openal --enable-opencl --enable-opengl --enable-sdl2 --enable-pocketsphinx --enable-librsvg --enable-libmfx --enable-libdc1394 --enable-libdrm --enable-libiec61883 --enable-chromaprint --enable-frei0r --enable-libx264 --enable-shared + libavutil 56. 70.100 / 56. 70.100 + libavcodec 58.134.100 / 58.134.100 + libavformat 58. 76.100 / 58. 76.100 + libavdevice 58. 13.100 / 58. 13.100 + libavfilter 7.110.100 / 7.110.100 + libswscale 5. 9.100 / 5. 9.100 + libswresample 3. 9.100 / 3. 9.100 + libpostproc 55. 9.100 / 55. 9.100 +Input #0, image2, from '/tmp/jl_1FHDYj/proc1_%04d.png': + Duration: 00:00:04.52, start: 0.000000, bitrate: N/A + Stream #0:0: Video: png, rgb24(pc), 1200x600, 25 fps, 25 tbr, 25 tbn, 25 tbc +Stream mapping: + Stream #0:0 -> #0:0 (png (native) -> h264 (libx264)) +Press [q] to stop, [?] for help +[libx264 @ 0x559c43cba700] using cpu capabilities: MMX2 SSE2Fast SSSE3 SSE4.2 AVX FMA3 BMI2 AVX2 +[libx264 @ 0x559c43cba700] profile High, level 3.1, 4:2:0, 8-bit +[libx264 @ 0x559c43cba700] 264 - core 163 r3060 5db6aa6 - H.264/MPEG-4 AVC codec - Copyleft 2003-2021 - http://www.videolan.org/x264.html - options: cabac=1 ref=3 deblock=1:0:0 analyse=0x3:0x113 me=hex subme=7 psy=1 psy_rd=1.00:0.00 mixed_ref=1 me_range=16 chroma_me=1 trellis=1 8x8dct=1 cqm=0 deadzone=21,11 fast_pskip=1 chroma_qp_offset=-2 threads=6 lookahead_threads=1 sliced_threads=0 nr=0 decimate=1 interlaced=0 bluray_compat=0 constrained_intra=0 bframes=3 b_pyramid=2 b_adapt=1 b_bias=0 direct=1 weightb=1 open_gop=0 weightp=2 keyint=250 keyint_min=25 scenecut=40 intra_refresh=0 rc_lookahead=40 rc=crf mbtree=1 crf=23.0 qcomp=0.60 qpmin=0 qpmax=69 qpstep=4 ip_ratio=1.40 aq=1:1.00 +Output #0, mp4, to 'vorticity_proc1.mp4': + Metadata: + encoder : Lavf58.76.100 + Stream #0:0: Video: h264 (avc1 / 0x31637661), yuv420p(tv, progressive), 1200x600, q=2-31, 25 fps, 12800 tbn + Metadata: + encoder : Lavc58.134.100 libx264 + Side data: + cpb: bitrate max/min/avg: 0/0/0 buffer size: 0 vbv_delay: N/A +frame= 3 fps=0.0 q=0.0 size= 0kB time=00:00:00.00 bitrate=N/A speed= 0x frame= 68 fps=0.0 q=28.0 size= 0kB time=00:00:00.60 bitrate= 0.6kbits/s speed=1.12x frame= 105 fps= 95 q=28.0 size= 0kB time=00:00:02.08 bitrate= 0.2kbits/s speed=1.89x frame= 143 fps= 88 q=28.0 size= 256kB time=00:00:03.60 bitrate= 582.6kbits/s speed=2.22x frame= 188 fps= 88 q=28.0 size= 256kB time=00:00:05.40 bitrate= 388.4kbits/s speed=2.54x frame= 230 fps= 88 q=28.0 size= 512kB time=00:00:07.08 bitrate= 592.5kbits/s speed= 2.7x frame= 270 fps= 86 q=28.0 size= 512kB time=00:00:08.68 bitrate= 483.3kbits/s speed=2.75x frame= 283 fps= 78 q=-1.0 Lsize= 841kB time=00:00:11.20 bitrate= 615.1kbits/s speed=3.07x +video:837kB audio:0kB subtitle:0kB other streams:0kB global headers:0kB muxing overhead: 0.496318% +[libx264 @ 0x559c43cba700] frame I:2 Avg QP:18.13 size: 37221 +[libx264 @ 0x559c43cba700] frame P:73 Avg QP:24.75 size: 8377 +[libx264 @ 0x559c43cba700] frame B:208 Avg QP:19.22 size: 819 +[libx264 @ 0x559c43cba700] consecutive B-frames: 0.7% 3.5% 1.1% 94.7% +[libx264 @ 0x559c43cba700] mb I I16..4: 26.9% 46.3% 26.8% +[libx264 @ 0x559c43cba700] mb P I16..4: 0.4% 1.6% 1.0% P16..4: 5.4% 6.7% 4.6% 0.0% 0.0% skip:80.3% +[libx264 @ 0x559c43cba700] mb B I16..4: 0.2% 0.1% 0.0% B16..8: 6.5% 1.1% 0.4% direct: 0.3% skip:91.4% L0:44.1% L1:47.0% BI: 8.9% +[libx264 @ 0x559c43cba700] 8x8 transform intra:48.0% inter:39.8% +[libx264 @ 0x559c43cba700] coded y,uvDC,uvAC intra: 40.0% 42.6% 35.8% inter: 2.8% 2.3% 0.7% +[libx264 @ 0x559c43cba700] i16 v,h,dc,p: 55% 35% 8% 3% +[libx264 @ 0x559c43cba700] i8 v,h,dc,ddl,ddr,vr,hd,vl,hu: 25% 11% 34% 5% 5% 4% 6% 3% 7% +[libx264 @ 0x559c43cba700] i4 v,h,dc,ddl,ddr,vr,hd,vl,hu: 29% 17% 17% 4% 7% 6% 9% 4% 7% +[libx264 @ 0x559c43cba700] i8c dc,h,v,p: 68% 14% 12% 5% +[libx264 @ 0x559c43cba700] Weighted P-Frames: Y:0.0% UV:0.0% +[libx264 @ 0x559c43cba700] ref P L0: 70.2% 15.0% 9.8% 5.0% +[libx264 @ 0x559c43cba700] ref B L0: 87.4% 10.2% 2.4% +[libx264 @ 0x559c43cba700] ref B L1: 98.3% 1.7% +[libx264 @ 0x559c43cba700] kb/s:605.10 diff --git a/previews/PR48/generated/ffmpeg.out b/previews/PR76/generated/ffmpeg.out similarity index 100% rename from previews/PR48/generated/ffmpeg.out rename to previews/PR76/generated/ffmpeg.out diff --git a/previews/PR76/generated/gradient/index.html b/previews/PR76/generated/gradient/index.html new file mode 100644 index 00000000..2fdaa46e --- /dev/null +++ b/previews/PR76/generated/gradient/index.html @@ -0,0 +1,110 @@ + +Gradient of a scalar field · PencilFFTs.jl
              GitHub

              Gradient of a scalar field

              This example shows different methods to compute the gradient of a real-valued 3D scalar field $θ(\bm{x})$ in Fourier space, where $\bm{x} = (x, y, z)$. It is assumed that the field is periodic with period $L = 2π$ along all dimensions.

              General procedure

              The discrete Fourier expansion of $θ$ writes

              \[θ(\bm{x}) = ∑_{\bm{k} ∈ \Z^3} \hat{θ}(\bm{k}) \, e^{i \bm{k} ⋅ \bm{x}},\]

              where $\bm{k} = (k_x, k_y, k_z)$ are the Fourier wave numbers and $\hat{θ}$ is the discrete Fourier transform of $θ$. Then, the spatial derivatives of $θ$ are given by

              \[\frac{∂ θ(\bm{x})}{∂ x_i} = +∑_{\bm{k} ∈ \Z^3} i k_i \hat{θ}(\bm{k}) \, e^{i \bm{k} ⋅ \bm{x}},\]

              where the subscript $i$ denotes one of the spatial components $x$, $y$ or $z$.

              In other words, to compute $\bm{∇} θ = (∂_x θ, ∂_y θ, ∂_z θ)$, one has to:

              1. transform $θ$ to Fourier space to obtain $\hat{θ}$,
              2. multiply $\hat{θ}$ by $i \bm{k}$,
              3. transform the result back to physical space to obtain $\bm{∇} θ$.

              Preparation

              In this section, we initialise a random real-valued scalar field $θ$ and compute its FFT. For more details see the Tutorial.

              using MPI
+using PencilFFTs
+using Random
+
+MPI.Init()
+
+# Input data dimensions (Nx × Ny × Nz)
+dims = (64, 32, 64)
+
+# Apply a 3D real-to-complex (r2c) FFT.
+transform = Transforms.RFFT()
+
+# Automatically create decomposition configuration
+comm = MPI.COMM_WORLD
+pen = Pencil(dims, comm)
+
+# Create plan
+plan = PencilFFTPlan(pen, transform)
+
+# Allocate data and initialise field
+θ = allocate_input(plan)
+randn!(θ)
+
+# Perform distributed FFT
+θ_hat = plan * θ

              Finally, we initialise the output that will hold ∇θ in Fourier space. Noting that ∇θ is a vector field, we choose to store it as a tuple of 3 PencilArrays.

              ∇θ_hat = allocate_output(plan, Val(3))
+
+# This is equivalent:
+# ∇θ_hat = ntuple(d -> similar(θ_hat), Val(3))
+
+summary(∇θ_hat)
              "Tuple{PencilArray{ComplexF64, 3, Array{ComplexF64, 3}, 3, 0, Pencil{3, 2, Permutation{(3, 2, 1), 3}, Vector{UInt8}}}, PencilArray{ComplexF64, 3, Array{ComplexF64, 3}, 3, 0, Pencil{3, 2, Permutation{(3, 2, 1), 3}, Vector{UInt8}}}, PencilArray{ComplexF64, 3, Array{ComplexF64, 3}, 3, 0, Pencil{3, 2, Permutation{(3, 2, 1), 3}, Vector{UInt8}}}}"

              Fourier wave numbers

              In general, the Fourier wave numbers are of the form $k_i = 0, ±\frac{2π}{L_i}, ±\frac{4π}{L_i}, ±\frac{6π}{L_i}, …$, where $L_i$ is the period along dimension $i$. When a real-to-complex Fourier transform is applied, roughly half of these wave numbers are redundant due to the Hermitian symmetry of the complex Fourier coefficients. In practice, this means that for the fastest dimension $x$ (along which a real-to-complex transform is performed), the negative wave numbers are dropped, i.e. $k_x = 0, \frac{2π}{L_x}, \frac{4π}{L_x}, …$.

              The AbstractFFTs package provides a convenient way to generate the Fourier wave numbers, using the functions fftfreq and rfftfreq. We can use these functions to initialise a "grid" of wave numbers associated to our 3D real-to-complex transform:

              using AbstractFFTs: fftfreq, rfftfreq
+
+box_size = (2π, 2π, 2π)  # Lx, Ly, Lz
+sample_rate = 2π .* dims ./ box_size
+
+# In our case (Lx = 2π and Nx even), this gives kx = [0, 1, 2, ..., Nx/2].
+kx = rfftfreq(dims[1], sample_rate[1])
+
+# In our case (Ly = 2π and Ny even), this gives
+# ky = [0, 1, 2, ..., Ny/2-1, -Ny/2, -Ny/2+1, ..., -1] (and similarly for kz).
+ky = fftfreq(dims[2], sample_rate[2])
+kz = fftfreq(dims[3], sample_rate[3])
+
+kvec = (kx, ky, kz)
              ([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0  …  23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, 32.0], [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0  …  -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0], [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0  …  -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0])

              Note that kvec now contains the wave numbers associated to the global domain. In the following, we will only need the wave numbers associated to the portion of the domain handled by the local MPI process.

              Method 1: global views

              PencilArrays, returned for instance by allocate_input and allocate_output, take indices that start at 1, regardless of the location of the subdomain associated to the local process on the global grid. (In other words, PencilArrays take local indices.) On the other hand, we have defined the wave number vector kvec which, for each MPI process, is defined over the global domain, and as such it takes global indices.

              One straightforward way of making data arrays compatible with wave numbers is to use global views, i.e. arrays that take global indices. These are generated from PencilArrays by calling the global_view function. Note that, in general, global indices do not start at 1 for a given MPI process. A given process will own a range of data given by indices in (i1:i2, j1:j2, k1:k2).

              θ_glob = global_view(θ_hat)
+∇θ_glob = global_view.(∇θ_hat)
+summary(θ_glob)
              "16×32×64 OffsetArray(::PencilArray{ComplexF64, 3}, 1:16, 1:32, 1:64) with eltype ComplexF64 with indices 1:16×1:32×1:64"

              Once we have global views, we can combine data and wave numbers using the portion of global indices owned by the local MPI process, as shown below. We can use CartesianIndices to iterate over the global indices associated to the local process.

              for I in CartesianIndices(θ_glob)
+    i, j, k = Tuple(I)  # unpack indices
+
+    # Wave number vector associated to current Cartesian index.
+    kx = kvec[1][i]
+    ky = kvec[2][j]
+    kz = kvec[3][k]
+
+    # Compute gradient in Fourier space.
+    # Note that modifying ∇θ_glob also modifies the original PencilArray ∇θ_hat.
+    ∇θ_glob[1][I] = im * kx * θ_glob[I]
+    ∇θ_glob[2][I] = im * ky * θ_glob[I]
+    ∇θ_glob[3][I] = im * kz * θ_glob[I]
+end

              The above loop can be written in a slightly more efficient manner by precomputing im * θ_glob[I]:

              @inbounds for I in CartesianIndices(θ_glob)
+    i, j, k = Tuple(I)
+
+    kx = kvec[1][i]
+    ky = kvec[2][j]
+    kz = kvec[3][k]
+
+    u = im * θ_glob[I]
+
+    ∇θ_glob[1][I] = kx * u
+    ∇θ_glob[2][I] = ky * u
+    ∇θ_glob[3][I] = kz * u
+end

              Also note that the above can be easily written in a more generic way, e.g. for arbitrary dimensions, thanks in part to the use of CartesianIndices. Moreover, in the above there is no notion of the dimension permutations discussed in the tutorial, as it is all hidden behind the implementation of PencilArrays. And as seen later in the benchmarks, these (hidden) permutations have zero cost, as the speed is identical to that of a function that explicitly takes into account these permutations.

              Finally, we can perform a backwards transform to obtain $\bm{∇} θ$ in physical space:

              ∇θ = plan \ ∇θ_hat;

              Note that the transform is automatically broadcast over the three fields of the ∇θ_hat vector, and the result ∇θ is also a tuple of three PencilArrays.

              Method 2: explicit global indexing

              Sometimes, one does not need to write generic code. In our case, one often knows the dimensionality of the problem and the memory layout of the data (i.e. the underlying index permutation).

              Below is a reimplementation of the above loop, using explicit indices instead of CartesianIndices, and assuming that the underlying index permutation is (3, 2, 1), that is, data is stored in $(z, y, x)$ order. As discussed in the tutorial, this is the default for transformed arrays. This example also serves as a more explicit explanation for what is going on in the first method.

              # Get local data range in the global grid.
+rng = axes(θ_glob)  # = (i1:i2, j1:j2, k1:k2)
              (OffsetArrays.IdOffsetRange(values=1:16, indices=1:16), OffsetArrays.IdOffsetRange(values=1:32, indices=1:32), OffsetArrays.IdOffsetRange(values=1:64, indices=1:64))

              For the loop below, we're assuming that the permutation is (3, 2, 1). In other words, the fastest index is the last one, and not the first one as it is usually in Julia. If the permutation is not (3, 2, 1), things will still work (well, except for the assertion below!), but the loop order will not be optimal.

              @assert permutation(θ_hat) === Permutation(3, 2, 1)
+
+@inbounds for i in rng[1], j in rng[2], k in rng[3]
+    kx = kvec[1][i]
+    ky = kvec[2][j]
+    kz = kvec[3][k]
+
+    # Note that we still access the arrays in (i, j, k) order.
+    # (The permutation happens behind the scenes!)
+    u = im * θ_glob[i, j, k]
+
+    ∇θ_glob[1][i, j, k] = kx * u
+    ∇θ_glob[2][i, j, k] = ky * u
+    ∇θ_glob[3][i, j, k] = kz * u
+end

              Method 3: using local indices

              Alternatively, we can avoid global views and work directly on PencilArrays using local indices that start at 1. In this case, part of the strategy is to construct a "local" grid of wave numbers that can also be accessed with local indices. This can be conveniently done using the localgrid function of the PencilArrays.jl package, which accepts a PencilArray (or its associated Pencil) and the global coordinates (here kvec):

              grid_fourier = localgrid(θ_hat, kvec)
              LocalRectilinearGrid{3} with Permutation(3, 2, 1) and coordinates:
+ (1) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0]
+ (2) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]
+ (3) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]

              Note that one can directly iterate on the returned grid object:

              @inbounds for I in CartesianIndices(grid_fourier)
+    # Wave number vector associated to current Cartesian index.
+    k⃗ = grid_fourier[I]
+    u = im * θ_hat[I]
+    ∇θ_hat[1][I] = k⃗[1] * u
+    ∇θ_hat[2][I] = k⃗[2] * u
+    ∇θ_hat[3][I] = k⃗[3] * u
+end

              This implementation is as efficient as the other examples, while being slightly shorter to write. Moreover, it is quite generic, and can be made independent of the number of dimensions with little effort.

              Method 4: using broadcasting

              Finally, note that the local grid object returned by localgrid makes it is possible to compute the gradient using broadcasting, thus fully avoiding scalar indexing. This can be quite convenient in some cases, and can also be very useful if one is working on GPUs (where scalar indexing is prohibitively expensive). Using broadcasting, the above examples simply become:

              @. ∇θ_hat[1] = im * grid_fourier[1] * θ_hat
+@. ∇θ_hat[2] = im * grid_fourier[2] * θ_hat
+@. ∇θ_hat[3] = im * grid_fourier[3] * θ_hat

              Once again, as shown in the benchmarks further below, this method performs quite similarly to the other ones.

              Summary

              The PencilArrays module provides different alternatives to deal with MPI-distributed data that may be subject to dimension permutations. In particular, one can choose to work with global indices (first two examples), with local indices (third example), or to avoid scalar indexing altogether (fourth example).

              If one wants to stay generic, making sure that the same code will work for arbitrary dimensions and will be efficient regardless of the underlying dimension permutation, methods 1, 3 or 4 should be preferred. These use CartesianIndices and make no assumptions on possible dimension permutations, which are by default enabled in the output of PencilFFTs transforms. In fact, such permutations are completely invisible in the implementations.

              The second method uses explicit (i, j, k) indices. It assumes that the underlying permutation is (3, 2, 1) to loop with i as the slowest index and k as the fastest, which is the optimal order in this case given the permutation. As such, the implementation is less generic than the others, and differences in performance are negligible with respect to more generic variants.

              Benchmark results

              The following are the benchmark results obtained from running examples/gradient.jl on a laptop, using 2 MPI processes and Julia 1.7.2, with an input array of global dimensions $64 × 32 × 64$. The different methods detailed above are marked on the right. The "lazy" marks indicate runs where the wave numbers were represented by lazy Frequencies objects (returned by rfftfreq and fftfreq). Otherwise, they were collected into Vectors. For some reason, plain Vectors are faster when working with grids generated by localgrid.

              In the script, additional implementations can be found which rely on a more advanced understanding of permutations and on the internals of the PencilArrays package. For instance, gradient_local_parent! directly works with the raw data stored in Julia Arrays, while gradient_local_linear! completely avoids CartesianIndices while staying generic and efficient. Nevertheless, these display roughly the same performance as the above examples.

                  gradient_global_view!...                  89.900 μs
+    gradient_global_view! (lazy)...           92.060 μs  [Method 1]
+    gradient_global_view_explicit!...         88.958 μs
+    gradient_global_view_explicit! (lazy)...  81.055 μs  [Method 2]
+    gradient_local!...                        92.305 μs
+    gradient_grid!...                         92.770 μs
+    gradient_grid! (lazy)...                  101.388 μs  [Method 3]
+    gradient_grid_broadcast!...               88.606 μs
+    gradient_grid_broadcast! (lazy)...        151.020 μs  [Method 4]
+    gradient_local_parent!...                 92.248 μs
+    gradient_local_linear!...                 91.212 μs
+    gradient_local_linear_explicit!...        90.992 μs

              This page was generated using Literate.jl.

              diff --git a/previews/PR76/generated/in-place/index.html b/previews/PR76/generated/in-place/index.html new file mode 100644 index 00000000..a128d5a2 --- /dev/null +++ b/previews/PR76/generated/in-place/index.html @@ -0,0 +1,31 @@ + +In-place transforms · PencilFFTs.jl
              GitHub

              In-place transforms

              Complex-to-complex and real-to-real transforms can be performed in-place, enabling important memory savings. The procedure is very similar to that of out-of-place transforms described in the tutorial. The differences are illustrated in the sections below.

              Creating a domain partition

              We start by partitioning a domain of dimensions $16×32×64$ along all available MPI processes.

              using PencilFFTs
+using MPI
+MPI.Init()
+
+dims_global = (16, 32, 64)  # global dimensions
              (16, 32, 64)

              Such a partitioning is described by a Pencil object. Here we choose to decompose the domain along the last two dimensions. In this case, the actual number of processes along each of these dimensions is chosen automatically.

              decomp_dims = (2, 3)
+comm = MPI.COMM_WORLD
+pen = Pencil(dims_global, decomp_dims, comm)
              Decomposition of 3D data
+    Data dimensions: (16, 32, 64)
+    Decomposed dimensions: (2, 3)
+    Data permutation: NoPermutation()
+    Array type: Array
              Allowed decompositions

              Distributed transforms using PencilFFTs.jl require that the first dimension is not decomposed. In other words, if one wants to perform transforms, then decomp_dims above must not contain 1.

              Creating in-place plans

              To create an in-place plan, pass an in-place transform such as Transforms.FFT! or Transforms.R2R! to PencilFFTPlan. For instance:

              # Perform a 3D in-place complex-to-complex FFT.
+transform = Transforms.FFT!()
+
+# Note that one can also combine different types of in-place transforms.
+# For instance:
+# transform = (
+#     Transforms.R2R!(FFTW.REDFT01),
+#     Transforms.FFT!(),
+#     Transforms.R2R!(FFTW.DHT),
+# )
              FFT!

              We can now create a distributed plan from the previously-created domain partition and the chosen transform.

              plan = PencilFFTPlan(pen, transform)
              Transforms: (FFT!, FFT!, FFT!)
+Input type: ComplexF64
+Global dimensions: (16, 32, 64) -> (16, 32, 64)
+MPI topology: 2D decomposition (2×1 processes)

              Note that in-place real-to-complex transforms are not currently supported. (In other words, the RFFT! transform type is not defined.)

              Allocating data

              As with out-of-place plans, data should be allocated using allocate_input. The difference is that, for in-place plans, this function returns a ManyPencilArray object, which is a container holding multiple PencilArray views sharing the same memory space.

              # Allocate data for the plan.
+# Since `plan` is in-place, this returns a `ManyPencilArray` container.
+A = allocate_input(plan)
+summary(A)
              "ManyPencilArray{ComplexF64, 3, 3, Tuple{PencilArray{ComplexF64, 3, Array{ComplexF64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{ComplexF64, 3, Array{ComplexF64, 3}, 3, 0, Pencil{3, 2, Permutation{(2, 1, 3), 3}, Vector{UInt8}}}, PencilArray{ComplexF64, 3, Array{ComplexF64, 3}, 3, 0, Pencil{3, 2, Permutation{(3, 2, 1), 3}, Vector{UInt8}}}}, Vector{ComplexF64}}"

              Note that allocate_output also works for in-place plans. In this case, it returns exactly the same thing as allocate_input.

              As shown in the next section, in-place plans must be applied on the returned ManyPencilArray. On the other hand, one usually wants to access and modify data, and for this one needs the PencilArray views contained in the ManyPencilArray. The input and output array views can be obtained by calling first(::ManyPencilArray) and last(::ManyPencilArray).

              For instance, we can initialise the input array with some data before transforming:

              using Random
+u_in = first(A)  # input data view
+randn!(u_in)
+summary(u_in)
              "16×16×64 PencilArray{ComplexF64, 3}(::Pencil{3, 2, NoPermutation, Array})"

              Applying plans

              Like in FFTW.jl, one can perform in-place transforms using the * and \ operators. As mentioned above, in-place plans must be applied on the ManyPencilArray containers returned by allocate_input.

              plan * A;  # performs in-place forward transform

              After performing an in-place transform, data contained in u_in has been overwritten and has no "physical" meaning. In other words, u_in should not be used at this point. To access the transformed data, one should retrieve the output data view using last(A).

              For instance, to compute the global sum of the transformed data:

              u_out = last(A)  # output data view
+sum(u_out)       # sum of transformed data (note that `sum` reduces over all processes)
              21690.495188969726 + 13651.674315154956im

              Finally, we can perform a backward transform and do stuff with the input view:

              plan \ A;  # perform in-place backward transform

              At this point, the data can be once again found in the input view u_in, while u_out should not be accessed.

              This page was generated using Literate.jl.

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              \n \n \n
              ","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"By default, the domain is distributed on a 2D MPI topology of dimensions N_1 N_2. As an example, the above figure shows such a topology with N_1 = 4 and N_2 = 3, for a total of 12 MPI processes.","category":"page"},{"location":"tutorial/#tutorial:creating_plans","page":"Tutorial","title":"Creating plans","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The first thing to do is to create a domain decomposition configuration for the given dataset dimensions N_x N_y N_z. In the framework of PencilArrays, such a configuration is described by a Pencil object. As described in the PencilArrays docs, we can let the Pencil constructor automatically determine such a configuration. For this, only an MPI communicator and the dataset dimensions are needed:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using MPI\nusing PencilFFTs\n\nMPI.Init()\ncomm = MPI.COMM_WORLD\n\n# Input data dimensions (Nx × Ny × Nz)\ndims = (16, 32, 64)\npen = Pencil(dims, comm)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"By default this creates a 2D decomposition (for the case of a 3D dataset), but one can change this as detailed in the PencilArrays documentation linked above.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"We can now create a PencilFFTPlan, which requires information on decomposition configuration (the Pencil object) and on the transforms that will be applied:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# Apply a 3D real-to-complex (r2c) FFT.\ntransform = Transforms.RFFT()\n\n# Note that, for more control, one can instead separately specify the transforms along each dimension:\n# transform = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())\n\n# Create plan\nplan = PencilFFTPlan(pen, transform)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"See the PencilFFTPlan constructor for details on the accepted options, and the Transforms module for the possible transforms. It is also possible to enable fine-grained performance measurements via the TimerOutputs package, as described in Measuring performance.","category":"page"},{"location":"tutorial/#Allocating-data","page":"Tutorial","title":"Allocating data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Next, we want to apply the plan on some data. Transforms may only be applied on PencilArrays, which are array wrappers that include MPI decomposition information (in some sense, analogous to DistributedArrays in Julia's distributed computing approach). The helper function allocate_input can be used to allocate a PencilArray that is compatible with our plan:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# In our example, this returns a 3D PencilArray of real data (Float64).\nu = allocate_input(plan)\n\n# Fill the array with some (random) data\nusing Random\nrandn!(u)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"PencilArrays are a subtype of AbstractArray, and thus they support all common array operations.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Similarly, to preallocate output data, one can use allocate_output:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# In our example, this returns a 3D PencilArray of complex data (Complex{Float64}).\nv = allocate_output(plan)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"This is only required if one wants to apply the plans using a preallocated output (with mul!, see right below).","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The data types returned by allocate_input and allocate_output are slightly different when working with in-place transforms. See the in-place example for details.","category":"page"},{"location":"tutorial/#Applying-plans","page":"Tutorial","title":"Applying plans","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The interface to apply plans is consistent with that of AbstractFFTs. Namely, * and mul! are respectively used for forward transforms without and with preallocated output data. Similarly, \\ and ldiv! are used for backward transforms.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using LinearAlgebra # for mul!, ldiv!\n\n# Apply plan on `u` with `v` as an output\nmul!(v, plan, u)\n\n# Apply backward plan on `v` with `w` as an output\nw = similar(u)\nldiv!(w, plan, v) # now w ≈ u","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Note that, consistently with AbstractFFTs, normalisation is performed at the end of a backward transform, so that the original data is recovered when applying a forward followed by a backward transform.","category":"page"},{"location":"tutorial/#Accessing-and-modifying-data","page":"Tutorial","title":"Accessing and modifying data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For any given MPI process, a PencilArray holds the data associated to its local partition in the global geometry. PencilArrays are accessed using local indices that start at 1, regardless of the location of the local process in the MPI topology. Note that PencilArrays, being based on regular Arrays, support both linear and Cartesian indexing (see the Julia docs for details).","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For convenience, the global_view function can be used to generate an OffsetArray wrapper that takes global indices.","category":"page"},{"location":"tutorial/#tutorial:output_data_layout","page":"Tutorial","title":"Output data layout","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"In memory, the dimensions of the transform output are by default reversed with respect to the input. That is, if the order of indices in the input data is (x, y, z), then the output has order (z, y, x) in memory. This detail is hidden from the user, and output arrays are always accessed in the same order as the input data, regardless of the underlying output dimension permutation. This applies to PencilArrays and to OffsetArrays returned by global_view.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The reasoning behind dimension permutations, is that they allow to always perform FFTs along the fastest array dimension and to avoid a local data transposition, resulting in performance gains. A similar approach is followed by other parallel FFT libraries. FFTW itself, in its distributed-memory routines, includes a flag that enables a similar behaviour. In PencilFFTs, index permutation is the default, but it can be disabled via the permute_dims flag of PencilFFTPlan.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"A great deal of work has been spent in making generic index permutations as efficient as possible, both in intermediate and in the output state of the multidimensional transforms. This has been achieved, in part, by making sure that permutations such as (3, 2, 1) are compile-time constants.","category":"page"},{"location":"tutorial/#Further-reading","page":"Tutorial","title":"Further reading","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For details on working with PencilArrays see the PencilArrays docs.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The examples on the sidebar further illustrate the use of transforms and provide an introduction to working with MPI-distributed data in the form of PencilArrays. In particular, the gradient example illustrates different ways of computing things using Fourier-transformed distributed arrays. Then, the incompressible Navier–Stokes example is a more advanced and complete example of a possible application of the PencilFFTs package.","category":"page"},{"location":"benchmarks/#Benchmarks","page":"Benchmarks","title":"Benchmarks","text":"","category":"section"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The performance of PencilFFTs.jl is comparable to that of other open-source parallel FFT libraries implemented in lower-level languages. Below, we show comparisons with the Fortran implementation of P3DFFT, possibly the most popular of these libraries. The benchmarks were performed on the Jean–Zay cluster of the IDRIS French computing centre (CNRS).","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The figure below shows strong scaling benchmarks of 3D real-to-complex FFTs using 2D (\"pencil\") decomposition. The benchmarks were run for input arrays of dimensions N_x N_y N_z = 512^3, 1024^3 and 2048^3. Each timing is averaged over 100 repetitions.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"
              \n \n \n
              ","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"As seen above, PencilFFTs generally outperforms P3DFFT in its default setting. This is largely explained by the choice of using non-blocking point-to-point MPI communications (via MPI_Isend and MPI_Irecv), while P3DFFT uses collective MPI_Alltoallv calls. This enables PencilFFTs to perform data reordering operations on the partially received data while waiting for the incoming data, leading to better performance. Moreover, in contrast with P3DFFT, the high performance and scalability of PencilFFTs results from a highly generic code, handling decompositions in arbitrary dimensions and a relatively large (and extensible) variety of transformations.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"Note that PencilFFTs can optionally use collective communications (using MPI_Alltoallv) instead of point-to-point communications. For details, see the docs for PencilFFTPlan and for PencilArray transpositions. As seen above, collective communications generally perform worse than point-to-point ones, and runtimes are nearly indistinguishable from those of P3DFFT.","category":"page"},{"location":"benchmarks/#Benchmark-details","page":"Benchmarks","title":"Benchmark details","text":"","category":"section"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The benchmarks were performed using Julia 1.7-beta3 and Intel MPI 2019. We used PencilFFTs v0.12.5 with FFTW.jl v1.4.3 and MPI.jl v0.19.0. We used the Fortran implementation of P3DFFT, version 2.7.6, which was built with Intel 2019 compilers and linked to FFTW 3.3.8. The cluster where the benchmarks were run has Intel Cascade Lake 6248 processors with 2×20 cores per node.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The number of MPI processes along each decomposed dimension, P_1 and P_2, was automatically determined by a call to MPI_Dims_create, which tends to create a balanced decomposition with P_1 P_2. For instance, a total of 1024 processes is divided into P_1 = P_2 = 32. Different results may be obtained with other combinations, but this was not benchmarked.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The source files used to generate this benchmark, as well as the raw benchmark results, are all available in the PencilFFTs repo.","category":"page"},{"location":"GlobalFFTParams/#Global-FFT-parameters","page":"Global FFT parameters","title":"Global FFT parameters","text":"","category":"section"},{"location":"GlobalFFTParams/","page":"Global FFT parameters","title":"Global FFT parameters","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"GlobalFFTParams/","page":"Global FFT parameters","title":"Global FFT parameters","text":"GlobalFFTParams","category":"page"},{"location":"GlobalFFTParams/#PencilFFTs.GlobalFFTParams","page":"Global FFT parameters","title":"PencilFFTs.GlobalFFTParams","text":"GlobalFFTParams{T, N, inplace}\n\nSpecifies the global parameters for an N-dimensional distributed transform. These include the element type T and global data sizes of input and output data, as well as the transform types to be performed along each dimension.\n\n\n\nGlobalFFTParams(size_global, transforms, [real_type=Float64])\n\nDefine parameters for N-dimensional transform.\n\ntransforms must be a tuple of length N specifying the transforms to be applied along each dimension. Each element must be a subtype of Transforms.AbstractTransform. For all the possible transforms, see Transform types.\n\nThe element type must be a real type accepted by FFTW, i.e. either Float32 or Float64.\n\nNote that the transforms are applied one dimension at a time, with the leftmost dimension first for forward transforms.\n\nExample\n\nTo perform a 3D FFT of real data, first a real-to-complex FFT must be applied along the first dimension, followed by two complex-to-complex FFTs along the other dimensions:\n\njulia> size_global = (64, 32, 128); # size of real input data\n\njulia> transforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT());\n\njulia> fft_params = PencilFFTs.GlobalFFTParams(size_global, transforms)\nTransforms: (RFFT, FFT, FFT)\nInput type: Float64\nGlobal dimensions: (64, 32, 128) -> (33, 32, 128)\n\n\n\n\n\n","category":"type"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"EditURL = \"../../examples/in-place.jl\"","category":"page"},{"location":"generated/in-place/#In-place-transforms","page":"In-place transforms","title":"In-place transforms","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Complex-to-complex and real-to-real transforms can be performed in-place, enabling important memory savings. The procedure is very similar to that of out-of-place transforms described in the tutorial. The differences are illustrated in the sections below.","category":"page"},{"location":"generated/in-place/#Creating-a-domain-partition","page":"In-place transforms","title":"Creating a domain partition","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"We start by partitioning a domain of dimensions 163264 along all available MPI processes.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"using PencilFFTs\nusing MPI\nMPI.Init()\n\ndims_global = (16, 32, 64) # global dimensions","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Such a partitioning is described by a Pencil object. Here we choose to decompose the domain along the last two dimensions. In this case, the actual number of processes along each of these dimensions is chosen automatically.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"decomp_dims = (2, 3)\ncomm = MPI.COMM_WORLD\npen = Pencil(dims_global, decomp_dims, comm)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"warning: Allowed decompositions\nDistributed transforms using PencilFFTs.jl require that the first dimension is not decomposed. In other words, if one wants to perform transforms, then decomp_dims above must not contain 1.","category":"page"},{"location":"generated/in-place/#Creating-in-place-plans","page":"In-place transforms","title":"Creating in-place plans","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"To create an in-place plan, pass an in-place transform such as Transforms.FFT! or Transforms.R2R! to PencilFFTPlan. For instance:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"# Perform a 3D in-place complex-to-complex FFT.\ntransform = Transforms.FFT!()\n\n# Note that one can also combine different types of in-place transforms.\n# For instance:\n# transform = (\n# Transforms.R2R!(FFTW.REDFT01),\n# Transforms.FFT!(),\n# Transforms.R2R!(FFTW.DHT),\n# )","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"We can now create a distributed plan from the previously-created domain partition and the chosen transform.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan = PencilFFTPlan(pen, transform)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Note that in-place real-to-complex transforms are not currently supported. (In other words, the RFFT! transform type is not defined.)","category":"page"},{"location":"generated/in-place/#Allocating-data","page":"In-place transforms","title":"Allocating data","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"As with out-of-place plans, data should be allocated using allocate_input. The difference is that, for in-place plans, this function returns a ManyPencilArray object, which is a container holding multiple PencilArray views sharing the same memory space.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"# Allocate data for the plan.\n# Since `plan` is in-place, this returns a `ManyPencilArray` container.\nA = allocate_input(plan)\nsummary(A)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Note that allocate_output also works for in-place plans. In this case, it returns exactly the same thing as allocate_input.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"As shown in the next section, in-place plans must be applied on the returned ManyPencilArray. On the other hand, one usually wants to access and modify data, and for this one needs the PencilArray views contained in the ManyPencilArray. The input and output array views can be obtained by calling first(::ManyPencilArray) and last(::ManyPencilArray).","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"For instance, we can initialise the input array with some data before transforming:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"using Random\nu_in = first(A) # input data view\nrandn!(u_in)\nsummary(u_in)","category":"page"},{"location":"generated/in-place/#Applying-plans","page":"In-place transforms","title":"Applying plans","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Like in FFTW.jl, one can perform in-place transforms using the * and \\ operators. As mentioned above, in-place plans must be applied on the ManyPencilArray containers returned by allocate_input.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan * A; # performs in-place forward transform\nnothing #hide","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"After performing an in-place transform, data contained in u_in has been overwritten and has no \"physical\" meaning. In other words, u_in should not be used at this point. To access the transformed data, one should retrieve the output data view using last(A).","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"For instance, to compute the global sum of the transformed data:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"u_out = last(A) # output data view\nsum(u_out) # sum of transformed data (note that `sum` reduces over all processes)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Finally, we can perform a backward transform and do stuff with the input view:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan \\ A; # perform in-place backward transform\nnothing #hide","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"At this point, the data can be once again found in the input view u_in, while u_out should not be accessed.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"This page was generated using Literate.jl.","category":"page"},{"location":"Transforms/#Available-transforms","page":"Available transforms","title":"Available transforms","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"CurrentModule = PencilFFTs.Transforms","category":"page"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"Transforms","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms","page":"Available transforms","title":"PencilFFTs.Transforms","text":"Defines different one-dimensional FFT-based transforms.\n\nThe transforms are all subtypes of an AbstractTransform type.\n\nWhen possible, the names of the transforms are kept consistent with the functions exported by AbstractFFTs.jl and FFTW.jl.\n\n\n\n\n\n","category":"module"},{"location":"Transforms/#Transform-types","page":"Available transforms","title":"Transform types","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"FFT\nFFT!\nBFFT\nBFFT!\n\nRFFT\nRFFT!\nBRFFT\nBRFFT!\n\nR2R\nR2R!\n\nNoTransform\nNoTransform!","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.FFT","page":"Available transforms","title":"PencilFFTs.Transforms.FFT","text":"FFT()\n\nComplex-to-complex FFT.\n\nSee also AbstractFFTs.fft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.FFT!","page":"Available transforms","title":"PencilFFTs.Transforms.FFT!","text":"FFT!()\n\nIn-place version of FFT.\n\nSee also AbstractFFTs.fft!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BFFT","page":"Available transforms","title":"PencilFFTs.Transforms.BFFT","text":"BFFT()\n\nUnnormalised backward complex-to-complex FFT.\n\nLike AbstractFFTs.bfft, this transform is not normalised. To obtain the inverse transform, divide the output by the length of the transformed dimension.\n\nSee also AbstractFFTs.bfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.BFFT!","text":"BFFT()\n\nIn-place version of BFFT.\n\nSee also AbstractFFTs.bfft!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.RFFT","page":"Available transforms","title":"PencilFFTs.Transforms.RFFT","text":"RFFT()\n\nReal-to-complex FFT.\n\nSee also AbstractFFTs.rfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.RFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.RFFT!","text":"RFFT!()\n\nIn-place version of RFFT.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BRFFT","page":"Available transforms","title":"PencilFFTs.Transforms.BRFFT","text":"BRFFT(d::Integer)\nBRFFT((d1, d2, ..., dN))\n\nUnnormalised inverse of RFFT.\n\nTo obtain the inverse transform, divide the output by the length of the transformed dimension (of the real output array).\n\nAs described in the AbstractFFTs docs, the length of the output cannot be fully inferred from the input length. For this reason, the BRFFT constructor accepts an optional d argument indicating the output length.\n\nFor multidimensional datasets, a tuple of dimensions (d1, d2, ..., dN) may also be passed. This is equivalent to passing just dN. In this case, the last dimension (dN) is the one that changes size between the input and output. Note that this is the opposite of FFTW.brfft. The reason is that, in PencilFFTs, the last dimension is the one along which a complex-to-real transform is performed.\n\nSee also AbstractFFTs.brfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BRFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.BRFFT!","text":"BRFFT!(d::Integer)\nBRFFT!((d1, d2, ..., dN))\n\nIn-place version of BRFFT.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.R2R","page":"Available transforms","title":"PencilFFTs.Transforms.R2R","text":"R2R(kind)\n\nReal-to-real transform of type kind.\n\nThe possible values of kind are those described in the FFTW.r2r docs and the FFTW manual:\n\ndiscrete cosine transforms: FFTW.REDFT00, FFTW.REDFT01, FFTW.REDFFT10, FFTW.REDFFT11\ndiscrete sine transforms: FFTW.RODFT00, FFTW.RODFT01, FFTW.RODFFT10, FFTW.RODFFT11\ndiscrete Hartley transform: FFTW.DHT\n\nNote: half-complex format DFTs (FFTW.R2HC, FFTW.HC2R) are not currently supported.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.R2R!","page":"Available transforms","title":"PencilFFTs.Transforms.R2R!","text":"R2R!(kind)\n\nIn-place version of R2R.\n\nSee also FFTW.r2r!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.NoTransform","page":"Available transforms","title":"PencilFFTs.Transforms.NoTransform","text":"NoTransform()\n\nIdentity transform.\n\nSpecifies that no transformation should be applied.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.NoTransform!","page":"Available transforms","title":"PencilFFTs.Transforms.NoTransform!","text":"NoTransform!()\n\nIn-place version of NoTransform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#Internals","page":"Available transforms","title":"Internals","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"What follows is used internally in PencilFFTs.","category":"page"},{"location":"Transforms/#Types","page":"Available transforms","title":"Types","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"AbstractCustomPlan\nAbstractTransform\nIdentityPlan\nIdentityPlan!\nPlan","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.AbstractCustomPlan","page":"Available transforms","title":"PencilFFTs.Transforms.AbstractCustomPlan","text":"AbstractCustomPlan\n\nAbstract type defining a custom plan, to be used as an alternative to FFTW plans (FFTW.FFTWPlan).\n\nThe only custom plan defined in this module is IdentityPlan. The user can define other custom plans that are also subtypes of AbstractCustomPlan.\n\nNote that plan returns a subtype of either AbstractFFTs.Plan or AbstractCustomPlan.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.AbstractTransform","page":"Available transforms","title":"PencilFFTs.Transforms.AbstractTransform","text":"AbstractTransform\n\nSpecifies a one-dimensional FFT-based transform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.IdentityPlan","page":"Available transforms","title":"PencilFFTs.Transforms.IdentityPlan","text":"IdentityPlan\n\nType of plan associated to NoTransform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.IdentityPlan!","page":"Available transforms","title":"PencilFFTs.Transforms.IdentityPlan!","text":"IdentityPlan!\n\nType of plan associated to NoTransform!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.Plan","page":"Available transforms","title":"PencilFFTs.Transforms.Plan","text":"Plan = Union{AbstractFFTs.Plan, AbstractCustomPlan}\n\nUnion type representing any plan returned by plan.\n\nSee also AbstractCustomPlan.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#Functions","page":"Available transforms","title":"Functions","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"plan\n\nbinv\nscale_factor\n\neltype_input\neltype_output\nexpand_dims\nis_inplace\nkind\nlength_output","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.plan","page":"Available transforms","title":"PencilFFTs.Transforms.plan","text":"plan(transform::AbstractTransform, A, [dims];\n flags=FFTW.ESTIMATE, timelimit=Inf)\n\nCreate plan to transform array A along dimensions dims.\n\nIf dims is not specified, all dimensions of A are transformed.\n\nFor FFT plans, this function wraps the AbstractFFTs.jl and FFTW.jl plan creation functions. For more details on the function arguments, see AbstractFFTs.plan_fft.\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.binv","page":"Available transforms","title":"PencilFFTs.Transforms.binv","text":"binv(transform::AbstractTransform, d::Integer)\n\nReturns the backwards transform associated to the given transform.\n\nThe second argument must be the length of the first transformed dimension in the forward transform. It is used in particular when transform = RFFT(), to determine the length of the inverse (complex-to-real) transform. See the AbstractFFTs.irfft docs for details.\n\nThe backwards transform returned by this function is not normalised. The normalisation factor for a given array can be obtained by calling scale_factor.\n\nExample\n\njulia> binv(Transforms.FFT(), 42)\nBFFT\n\njulia> binv(Transforms.BRFFT(9), 42)\nRFFT\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.scale_factor","page":"Available transforms","title":"PencilFFTs.Transforms.scale_factor","text":"scale_factor(transform::AbstractTransform, A, [dims = 1:ndims(A)])\n\nGet factor required to normalise the given array after a transformation along dimensions dims (all dimensions by default).\n\nThe array A must have the dimensions of the transform input.\n\nImportant: the dimensions dims must be the same that were passed to plan.\n\nExamples\n\njulia> C = zeros(ComplexF32, 3, 4, 5);\n\njulia> scale_factor(Transforms.FFT(), C)\n60\n\njulia> scale_factor(Transforms.BFFT(), C)\n60\n\njulia> scale_factor(Transforms.BFFT(), C, 2:3)\n20\n\njulia> R = zeros(Float64, 3, 4, 5);\n\njulia> scale_factor(Transforms.RFFT(), R, 2)\n4\n\njulia> scale_factor(Transforms.RFFT(), R, 2:3)\n20\n\njulia> scale_factor(Transforms.BRFFT(8), C)\n96\n\njulia> scale_factor(Transforms.BRFFT(9), C)\n108\n\nThis will fail because the input of RFFT is real, and R is a complex array:\n\njulia> scale_factor(Transforms.RFFT(), C, 2:3)\nERROR: MethodError: no method matching scale_factor(::PencilFFTs.Transforms.RFFT, ::Array{ComplexF32, 3}, ::UnitRange{Int64})\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.eltype_input","page":"Available transforms","title":"PencilFFTs.Transforms.eltype_input","text":"eltype_input(transform::AbstractTransform, real_type<:AbstractFloat)\n\nDetermine input data type for a given transform given the floating point precision of the input data.\n\nSome transforms, such as R2R and NoTransform, can take both real and complex data. For those kinds of transforms, nothing is returned.\n\nExample\n\njulia> eltype_input(Transforms.FFT(), Float32)\nComplexF32 (alias for Complex{Float32})\n\njulia> eltype_input(Transforms.RFFT(), Float64)\nFloat64\n\njulia> eltype_input(Transforms.R2R(FFTW.REDFT01), Float64) # nothing\n\njulia> eltype_input(Transforms.NoTransform(), Float64) # nothing\n\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.eltype_output","page":"Available transforms","title":"PencilFFTs.Transforms.eltype_output","text":"eltype_output(transform::AbstractTransform, eltype_input)\n\nReturns the output data type for a given transform given the input type.\n\nThrows ArgumentError if the input data type is incompatible with the transform type.\n\nExample\n\njulia> eltype_output(Transforms.NoTransform(), Float32)\nFloat32\n\njulia> eltype_output(Transforms.RFFT(), Float64)\nComplexF64 (alias for Complex{Float64})\n\njulia> eltype_output(Transforms.BRFFT(4), ComplexF32)\nFloat32\n\njulia> eltype_output(Transforms.FFT(), Float64)\nERROR: ArgumentError: invalid input data type for PencilFFTs.Transforms.FFT: Float64\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.expand_dims","page":"Available transforms","title":"PencilFFTs.Transforms.expand_dims","text":"expand_dims(transform::AbstractTransform, Val(N))\n\nExpand a single multidimensional transform into one transform per dimension.\n\nExample\n\n# Expand a real-to-complex transform in 3 dimensions.\njulia> expand_dims(Transforms.RFFT(), Val(3))\n(RFFT, FFT, FFT)\n\njulia> expand_dims(Transforms.BRFFT(4), Val(3))\n(BFFT, BFFT, BRFFT{even})\n\njulia> expand_dims(Transforms.NoTransform(), Val(2))\n(NoTransform, NoTransform)\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.is_inplace","page":"Available transforms","title":"PencilFFTs.Transforms.is_inplace","text":"is_inplace(transform::AbstractTransform) -> Bool\nis_inplace(transforms::Vararg{AbtractTransform}) -> Union{Bool, Nothing}\n\nCheck whether a transform or a list of transforms is performed in-place.\n\nIf the list of transforms has a combination of in-place and out-of-place transforms, nothing is returned.\n\nExample\n\njulia> is_inplace(Transforms.RFFT())\nfalse\n\njulia> is_inplace(Transforms.NoTransform!())\ntrue\n\njulia> is_inplace(Transforms.FFT!(), Transforms.R2R!(FFTW.REDFT01))\ntrue\n\njulia> is_inplace(Transforms.FFT(), Transforms.R2R(FFTW.REDFT01))\nfalse\n\njulia> is_inplace(Transforms.FFT(), Transforms.R2R!(FFTW.REDFT01)) === nothing\ntrue\n\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.kind","page":"Available transforms","title":"PencilFFTs.Transforms.kind","text":"kind(transform::R2R)\n\nGet kind of real-to-real transform.\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.length_output","page":"Available transforms","title":"PencilFFTs.Transforms.length_output","text":"length_output(transform::AbstractTransform, length_in::Integer)\n\nReturns the length of the transform output, given the length of its input.\n\nThe input and output lengths are specified in terms of the respective input and output datatypes. For instance, for real-to-complex transforms, these are respectively the length of input real data and of output complex data.\n\n\n\n\n\n","category":"function"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"EditURL = \"../../examples/navier_stokes.jl\"","category":"page"},{"location":"generated/navier_stokes/#Navier–Stokes-equations","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"In this example, we numerically solve the incompressible Navier–Stokes equations","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"_t bmv + (bmv bm) bmv = -frac1ρ bm p + ν ^2 bmv\nquad bm bmv = 0","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where bmv(bmx t) and p(bmx t) are respectively the velocity and pressure fields, ν is the fluid kinematic viscosity and ρ is the fluid density.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.","category":"page"},{"location":"generated/navier_stokes/#First-steps","page":"Navier–Stokes equations","title":"First steps","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We start by loading the required packages, initialising MPI and setting the simulation parameters.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using MPI\nusing PencilFFTs\n\nMPI.Init()\ncomm = MPI.COMM_WORLD\nprocid = MPI.Comm_rank(comm) + 1\n\n# Simulation parameters\nNs = (64, 64, 64) # = (Nx, Ny, Nz)\nLs = (2π, 2π, 2π) # = (Lx, Ly, Lz)\n\n# Collocation points (\"global\" = over all processes).\n# We include the endpoint (length = N + 1) for convenience.\nxs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls) # = (x, y, z)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's check the number of MPI processes over which we're running our simulation:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"MPI.Comm_size(comm)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"pen = Pencil(Ns, comm)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"The subdomain associated to the local MPI process can be obtained using range_local:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"range_local(pen)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We now construct a distributed vector field that follows the decomposition configuration we just created:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v⃗₀ = (\n PencilArray{Float64}(undef, pen), # vx\n PencilArray{Float64}(undef, pen), # vy\n PencilArray{Float64}(undef, pen), # vz\n)\nsummary(v⃗₀[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We still need to fill this array with interesting values that represent a physical velocity field.","category":"page"},{"location":"generated/navier_stokes/#Initial-condition","page":"Navier–Stokes equations","title":"Initial condition","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"beginaligned\nv_x(x y z) = u₀ sin(k₀ x) cos(k₀ y) cos(k₀ z) \nv_y(x y z) = -u₀ cos(k₀ x) sin(k₀ y) cos(k₀ z) \nv_z(x y z) = 0\nendaligned","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where u₀ and k₀ are two parameters setting the amplitude and the period of the velocity field.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"grid = localgrid(pen, xs_global)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can use this to initialise the velocity field:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"u₀ = 1.0\nk₀ = 2π / Ls[1] # should be integer if L = 2π (to preserve periodicity)\n\n@. v⃗₀[1] = u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)\n@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)\n@. v⃗₀[3] = 0\nnothing # hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's plot a 2D slice of the velocity field managed by the local MPI process:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using GLMakie\n\n# Compute the norm of a vector field represented by a tuple of arrays.\nfunction vecnorm(v⃗::NTuple)\n vnorm = similar(v⃗[1])\n for n ∈ eachindex(v⃗[1])\n w = zero(eltype(vnorm))\n for v ∈ v⃗\n w += v[n]^2\n end\n vnorm[n] = sqrt(w)\n end\n vnorm\nend\n\n# This is useful for passing coordinates to Makie.contour!\nto_intervals(grid) = map(xs -> xs[begin]..xs[end], grid.coords)\n\nlet fig = Figure(size = (700, 600))\n ax = Axis3(fig[1, 1]; aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\")\n vnorm = parent(vecnorm(v⃗₀)) # use `parent` because Makie doesn't like custom array types...\n ct = contour!(\n ax, to_intervals(grid)..., vnorm;\n alpha = 0.2, levels = 4,\n colormap = :viridis,\n colorrange = (0.0, 1.0),\n highclip = (:red, 0.2), lowclip = (:green, 0.2),\n )\n cb = Colorbar(fig[1, 2], ct; label = \"Velocity magnitude\")\n fig\nend","category":"page"},{"location":"generated/navier_stokes/#Velocity-in-Fourier-space","page":"Navier–Stokes equations","title":"Velocity in Fourier space","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"bmv(bmx t) =\n_bmk hatbmv_bmk(t) e^i bmk bmx","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where bmk = (k_x k_y k_z) are the discrete wave numbers.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for k_x:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using AbstractFFTs: fftfreq, rfftfreq\n\nks_global = (\n rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]), # kx | real-to-complex\n fftfreq(Ns[2], 2π * Ns[2] / Ls[2]), # ky | complex-to-complex\n fftfreq(Ns[3], 2π * Ns[3] / Ls[3]), # kz | complex-to-complex\n)\n\nks_global[1]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ks_global[2]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ks_global[3]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"See PencilFFTPlan for details on creating plans and on optional keyword arguments.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients hatbmv_bmk:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v̂s = plan .* v⃗₀\nsummary(v̂s[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Note that, in Fourier space, the domain decomposition is performed along the directions x and y:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"pencil(v̂s[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"This is because the 3D FFTs are performed one dimension at a time, with the x direction first and the z direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers bmk associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"grid_fourier = localgrid(v̂s[1], ks_global)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, bmω = bm bmv. In Fourier space, this becomes hatbmω = i bmk hatbmv.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using StaticArrays: SVector\nusing LinearAlgebra: ×\n\nfunction curl_fourier!(\n ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,\n ) where {N}\n @inbounds for I ∈ eachindex(grid_fourier)\n # We use StaticArrays for the cross product between small vectors.\n ik⃗ = im * SVector(grid_fourier[I])\n v⃗ = SVector(getindex.(v̂s, Ref(I))) # = (v̂s[1][I], v̂s[2][I], ...)\n ω⃗ = ik⃗ × v⃗\n for n ∈ eachindex(ω⃗)\n ω̂s[n][I] = ω⃗[n]\n end\n end\n ω̂s\nend\n\nω̂s = similar.(v̂s)\ncurl_fourier!(ω̂s, v̂s, grid_fourier);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We finally transform back to physical space and plot the result:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ωs = plan .\\ ω̂s\n\nlet fig = Figure(size = (700, 600))\n ax = Axis3(fig[1, 1]; aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\")\n ω_norm = parent(vecnorm(ωs))\n ct = contour!(\n ax, to_intervals(grid)..., ω_norm;\n alpha = 0.1, levels = 0.8:0.2:2.0,\n colormap = :viridis, colorrange = (0.8, 2.0),\n highclip = (:red, 0.2), lowclip = (:green, 0.2),\n )\n cb = Colorbar(fig[1, 2], ct; label = \"Vorticity magnitude\")\n fig\nend","category":"page"},{"location":"generated/navier_stokes/#Computing-the-non-linear-term","page":"Navier–Stokes equations","title":"Computing the non-linear term","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"_t hatbmv_bmk =\n- mathcalP_bmk left widehat(bmv bm) bmv right\n- ν bmk^2 hatbmv_bmk\nquad text with quad\nmathcalP_bmk(hatbmF_bmk) = left( I - fracbmk \nbmkbmk^2 right) hatbmF_bmk","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where mathcalP_bmk is a projection operator allowing to preserve the incompressibility condition bm bmv = 0. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Now that we have the wave numbers bmk, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients hatbmv_bmk of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, hatbmF_bmk = left widehat(bmv bm) bmv right_bmk. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Below we implement a function that computes the non-linear term in Fourier space based on its convective form (bmv bm) bmv = bm (bmv bmv). Note that this equivalence uses the incompressibility condition bm bmv = 0.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using LinearAlgebra: mul!, ldiv! # for applying FFT plans in-place\n\n# Compute non-linear term in Fourier space from velocity field in physical\n# space. Optional keyword arguments may be passed to avoid allocations.\nfunction ns_nonlinear!(\n F̂s, vs, plan, grid_fourier;\n vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),\n )\n # Compute F_i = ∂_j (v_i v_j) for each i.\n # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)\n w, ŵ = vbuf, v̂buf\n @inbounds for (i, F̂i) ∈ enumerate(F̂s)\n F̂i .= 0\n vi = vs[i]\n for (j, vj) ∈ enumerate(vs)\n w .= vi .* vj # w = v_i * v_j in physical space\n mul!(ŵ, plan, w) # same in Fourier space\n # Add derivative in Fourier space\n for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n kj = k⃗[j]\n F̂i[I] += im * kj * ŵ[I]\n end\n end\n end\n F̂s\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"As an example, let's use this function on our initial velocity field:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"F̂s = similar.(v̂s)\nns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)\n ks_max = maximum.(abs, ks_global) # maximum stored wave numbers (kx_max, ky_max, kz_max)\n ks_lim = (2 / 3) .* ks_max\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I]\n if any(abs.(k⃗) .> ks_lim)\n for ŵ ∈ ŵs\n ŵ[I] = 0\n end\n end\n end\n ŵs\nend\n\n# We can apply this on the previously computed non-linear term:\ndealias_twothirds!(F̂s, grid_fourier, ks_global);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Finally, we implement the projection associated to the incompressibility condition:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function project_divergence_free!(ûs, grid_fourier)\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I]\n k² = sum(abs2, k⃗)\n iszero(k²) && continue # avoid division by zero\n û = getindex.(ûs, Ref(I)) # (ûs[1][I], ûs[2][I], ...)\n for i ∈ eachindex(û)\n ŵ = û[i]\n for j ∈ eachindex(û)\n ŵ -= k⃗[i] * k⃗[j] * û[j] / k²\n end\n ûs[i][I] = ŵ\n end\n end\n ûs\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)\nv̂s_proj .≈ v̂s # the last one may be false because v_z = 0 initially","category":"page"},{"location":"generated/navier_stokes/#Putting-it-all-together","page":"Navier–Stokes equations","title":"Putting it all together","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function ns_rhs!(\n dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,\n ) where {N}\n # 1. Compute non-linear term and dealias it\n (; plan, cache, ks_global, grid_fourier) = p\n F̂s = cache.F̂s\n ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])\n dealias_twothirds!(F̂s, grid_fourier, ks_global)\n\n # 2. Project onto divergence-free space\n project_divergence_free!(F̂s, grid_fourier)\n\n # 3. Transform velocity to Fourier space\n v̂s = cache.v̂s\n map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)\n\n # 4. Add viscous term (and multiply projected non-linear term by -1)\n ν = p.ν\n for n ∈ eachindex(v̂s)\n v̂ = v̂s[n]\n F̂ = F̂s[n]\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n k² = sum(abs2, k⃗)\n F̂[I] = -F̂[I] - ν * k² * v̂[I]\n end\n end\n\n # 5. Transform RHS back to physical space\n map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)\n\n nothing\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using OrdinaryDiffEqLowOrderRK # includes RK4\nusing RecursiveArrayTools: ArrayPartition\n\nns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)\nvs_init_ode = ArrayPartition(v⃗₀)\nsummary(vs_init_ode)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We now define solver parameters and temporary variables, and initialise the problem:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"params = (;\n ν = 5e-3, # kinematic viscosity\n plan, grid_fourier, ks_global,\n cache = (\n v̂s = similar.(v̂s),\n F̂s = similar.(v̂s),\n )\n)\n\ntspan = (0.0, 10.0)\nprob = ODEProblem{true}(ns_rhs!, vs_init_ode, tspan, params)\nintegrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum E(k), to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity ν must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large k), while high enough to allow the small-scale motions to be correctly resolved.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function energy_spectrum!(Ek, ks, v̂s, grid_fourier)\n Nk = length(Ek)\n @assert Nk == length(ks)\n Ek .= 0\n for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n knorm = sqrt(sum(abs2, k⃗))\n i = searchsortedfirst(ks, knorm)\n i > Nk && continue\n v⃗ = getindex.(v̂s, Ref(I)) # = (v̂s[1][I], v̂s[2][I], ...)\n factor = k⃗[1] == 0 ? 1 : 2 # account for Hermitian symmetry and r2c transform\n Ek[i] += factor * sum(abs2, v⃗) / 2\n end\n MPI.Allreduce!(Ek, +, get_comm(v̂s[1])) # sum across all processes\n Ek\nend\n\nks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])\nEk = similar(ks)\nv̂s = plan .* integrator.u.x\nenergy_spectrum!(Ek, ks, v̂s, grid_fourier)\nEk ./= scale_factor(plan)^2 # rescale energy\n\ncurl_fourier!(ω̂s, v̂s, grid_fourier)\nldiv!.(ωs, plan, ω̂s)\nω⃗_plot = Observable(ωs)\nk_plot = @view ks[2:end]\nE_plot = Observable(@view Ek[2:end])\nt_plot = Observable(integrator.t)\n\nfig = let\n fig = Figure(size = (1200, 600))\n ax = Axis3(\n fig[1, 1][1, 1]; title = @lift(\"t = $(round($t_plot, digits = 3))\"),\n aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\",\n )\n ω_mag = @lift parent(vecnorm($ω⃗_plot))\n ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)\n ct = contour!(\n ax, to_intervals(grid)..., ω_mag_norm;\n alpha = 0.3, levels = 3,\n colormap = :viridis, colorrange = (0.0, 1.0),\n highclip = (:red, 0.2), lowclip = (:green, 0.2),\n )\n cb = Colorbar(fig[1, 1][1, 2], ct; label = \"Normalised vorticity magnitude\")\n ax_sp = Axis(\n fig[1, 2];\n xlabel = \"k\", ylabel = \"E(k)\", xscale = log2, yscale = log10,\n title = \"Kinetic energy spectrum\",\n )\n ylims!(ax_sp, 1e-8, 1e0)\n scatterlines!(ax_sp, k_plot, E_plot)\n ks_slope = exp.(range(log(2.5), log(25.0), length = 3))\n E_fivethirds = @. 0.3 * ks_slope^(-5/3)\n @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)\n text!(ax_sp, L\"k^{-5/3}\"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))\n fig\nend\n\nusing Printf # hide\nwith_xvfb::Bool = ENV[\"DISPLAY\"] == \":99\" # hide\nnstep::Int = 0 # hide\ntmpdir::String = mktempdir() # hide\nfilename_frame(procid, nstep) = joinpath(tmpdir, @sprintf(\"proc%d_%04d.png\", procid, nstep)) # hide\nrecord(fig, \"vorticity_proc$procid.mp4\"; framerate = 10) do io\n with_xvfb && recordframe!(io) # hide\n while integrator.t < 20\n dt = 0.001\n step!(integrator, dt)\n t_plot[] = integrator.t\n mul!.(v̂s, plan, integrator.u.x) # current velocity in Fourier space\n curl_fourier!(ω̂s, v̂s, grid_fourier)\n ldiv!.(ω⃗_plot[], plan, ω̂s)\n ω⃗_plot[] = ω⃗_plot[] # to force updating the plot\n energy_spectrum!(Ek, ks, v̂s, grid_fourier)\n Ek ./= scale_factor(plan)^2 # rescale energy\n E_plot[] = E_plot[]\n global nstep += 1 # hide\n with_xvfb ? # hide\n save(filename_frame(procid, nstep), fig) : # hide\n recordframe!(io)\n end\nend;\n\nif with_xvfb # hide\n run(pipeline(`ffmpeg -y -r 10 -i $tmpdir/proc$(procid)_%04d.png -c:v libx264 -vf \"fps=25,format=yuv420p\" vorticity_proc$procid.mp4`; stdout = \"ffmpeg.out\", stderr = \"ffmpeg.err\")) # hide\nend # hide\nnothing # hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"
              \n \n
              ","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"This page was generated using Literate.jl.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"EditURL = \"../../examples/gradient.jl\"","category":"page"},{"location":"generated/gradient/#Gradient-of-a-scalar-field","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This example shows different methods to compute the gradient of a real-valued 3D scalar field θ(bmx) in Fourier space, where bmx = (x y z). It is assumed that the field is periodic with period L = 2π along all dimensions.","category":"page"},{"location":"generated/gradient/#General-procedure","page":"Gradient of a scalar field","title":"General procedure","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The discrete Fourier expansion of θ writes","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"θ(bmx) = _bmk Z^3 hatθ(bmk) e^i bmk bmx","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"where bmk = (k_x k_y k_z) are the Fourier wave numbers and hatθ is the discrete Fourier transform of θ. Then, the spatial derivatives of θ are given by","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"frac θ(bmx) x_i =\n_bmk Z^3 i k_i hatθ(bmk) e^i bmk bmx","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"where the subscript i denotes one of the spatial components x, y or z.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In other words, to compute bm θ = (_x θ _y θ _z θ), one has to:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"transform θ to Fourier space to obtain hatθ,\nmultiply hatθ by i bmk,\ntransform the result back to physical space to obtain bm θ.","category":"page"},{"location":"generated/gradient/#Preparation","page":"Gradient of a scalar field","title":"Preparation","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In this section, we initialise a random real-valued scalar field θ and compute its FFT. For more details see the Tutorial.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"using MPI\nusing PencilFFTs\nusing Random\n\nMPI.Init()\n\n# Input data dimensions (Nx × Ny × Nz)\ndims = (64, 32, 64)\n\n# Apply a 3D real-to-complex (r2c) FFT.\ntransform = Transforms.RFFT()\n\n# Automatically create decomposition configuration\ncomm = MPI.COMM_WORLD\npen = Pencil(dims, comm)\n\n# Create plan\nplan = PencilFFTPlan(pen, transform)\n\n# Allocate data and initialise field\nθ = allocate_input(plan)\nrandn!(θ)\n\n# Perform distributed FFT\nθ_hat = plan * θ\nnothing # hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, we initialise the output that will hold ∇θ in Fourier space. Noting that ∇θ is a vector field, we choose to store it as a tuple of 3 PencilArrays.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"∇θ_hat = allocate_output(plan, Val(3))\n\n# This is equivalent:\n# ∇θ_hat = ntuple(d -> similar(θ_hat), Val(3))\n\nsummary(∇θ_hat)","category":"page"},{"location":"generated/gradient/#Fourier-wave-numbers","page":"Gradient of a scalar field","title":"Fourier wave numbers","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In general, the Fourier wave numbers are of the form k_i = 0 frac2πL_i frac4πL_i frac6πL_i , where L_i is the period along dimension i. When a real-to-complex Fourier transform is applied, roughly half of these wave numbers are redundant due to the Hermitian symmetry of the complex Fourier coefficients. In practice, this means that for the fastest dimension x (along which a real-to-complex transform is performed), the negative wave numbers are dropped, i.e. k_x = 0 frac2πL_x frac4πL_x .","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The AbstractFFTs package provides a convenient way to generate the Fourier wave numbers, using the functions fftfreq and rfftfreq. We can use these functions to initialise a \"grid\" of wave numbers associated to our 3D real-to-complex transform:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"using AbstractFFTs: fftfreq, rfftfreq\n\nbox_size = (2π, 2π, 2π) # Lx, Ly, Lz\nsample_rate = 2π .* dims ./ box_size\n\n# In our case (Lx = 2π and Nx even), this gives kx = [0, 1, 2, ..., Nx/2].\nkx = rfftfreq(dims[1], sample_rate[1])\n\n# In our case (Ly = 2π and Ny even), this gives\n# ky = [0, 1, 2, ..., Ny/2-1, -Ny/2, -Ny/2+1, ..., -1] (and similarly for kz).\nky = fftfreq(dims[2], sample_rate[2])\nkz = fftfreq(dims[3], sample_rate[3])\n\nkvec = (kx, ky, kz)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that kvec now contains the wave numbers associated to the global domain. In the following, we will only need the wave numbers associated to the portion of the domain handled by the local MPI process.","category":"page"},{"location":"generated/gradient/#gradient_method_global","page":"Gradient of a scalar field","title":"Method 1: global views","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"PencilArrays, returned for instance by allocate_input and allocate_output, take indices that start at 1, regardless of the location of the subdomain associated to the local process on the global grid. (In other words, PencilArrays take local indices.) On the other hand, we have defined the wave number vector kvec which, for each MPI process, is defined over the global domain, and as such it takes global indices.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"One straightforward way of making data arrays compatible with wave numbers is to use global views, i.e. arrays that take global indices. These are generated from PencilArrays by calling the global_view function. Note that, in general, global indices do not start at 1 for a given MPI process. A given process will own a range of data given by indices in (i1:i2, j1:j2, k1:k2).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"θ_glob = global_view(θ_hat)\n∇θ_glob = global_view.(∇θ_hat)\nsummary(θ_glob)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Once we have global views, we can combine data and wave numbers using the portion of global indices owned by the local MPI process, as shown below. We can use CartesianIndices to iterate over the global indices associated to the local process.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"for I in CartesianIndices(θ_glob)\n i, j, k = Tuple(I) # unpack indices\n\n # Wave number vector associated to current Cartesian index.\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n # Compute gradient in Fourier space.\n # Note that modifying ∇θ_glob also modifies the original PencilArray ∇θ_hat.\n ∇θ_glob[1][I] = im * kx * θ_glob[I]\n ∇θ_glob[2][I] = im * ky * θ_glob[I]\n ∇θ_glob[3][I] = im * kz * θ_glob[I]\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The above loop can be written in a slightly more efficient manner by precomputing im * θ_glob[I]:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@inbounds for I in CartesianIndices(θ_glob)\n i, j, k = Tuple(I)\n\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n u = im * θ_glob[I]\n\n ∇θ_glob[1][I] = kx * u\n ∇θ_glob[2][I] = ky * u\n ∇θ_glob[3][I] = kz * u\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Also note that the above can be easily written in a more generic way, e.g. for arbitrary dimensions, thanks in part to the use of CartesianIndices. Moreover, in the above there is no notion of the dimension permutations discussed in the tutorial, as it is all hidden behind the implementation of PencilArrays. And as seen later in the benchmarks, these (hidden) permutations have zero cost, as the speed is identical to that of a function that explicitly takes into account these permutations.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, we can perform a backwards transform to obtain bm θ in physical space:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"∇θ = plan \\ ∇θ_hat;\nnothing #hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that the transform is automatically broadcast over the three fields of the ∇θ_hat vector, and the result ∇θ is also a tuple of three PencilArrays.","category":"page"},{"location":"generated/gradient/#gradient_method_global_explicit","page":"Gradient of a scalar field","title":"Method 2: explicit global indexing","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Sometimes, one does not need to write generic code. In our case, one often knows the dimensionality of the problem and the memory layout of the data (i.e. the underlying index permutation).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Below is a reimplementation of the above loop, using explicit indices instead of CartesianIndices, and assuming that the underlying index permutation is (3, 2, 1), that is, data is stored in (z y x) order. As discussed in the tutorial, this is the default for transformed arrays. This example also serves as a more explicit explanation for what is going on in the first method.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"# Get local data range in the global grid.\nrng = axes(θ_glob) # = (i1:i2, j1:j2, k1:k2)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"For the loop below, we're assuming that the permutation is (3, 2, 1). In other words, the fastest index is the last one, and not the first one as it is usually in Julia. If the permutation is not (3, 2, 1), things will still work (well, except for the assertion below!), but the loop order will not be optimal.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@assert permutation(θ_hat) === Permutation(3, 2, 1)\n\n@inbounds for i in rng[1], j in rng[2], k in rng[3]\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n # Note that we still access the arrays in (i, j, k) order.\n # (The permutation happens behind the scenes!)\n u = im * θ_glob[i, j, k]\n\n ∇θ_glob[1][i, j, k] = kx * u\n ∇θ_glob[2][i, j, k] = ky * u\n ∇θ_glob[3][i, j, k] = kz * u\nend","category":"page"},{"location":"generated/gradient/#gradient_method_local","page":"Gradient of a scalar field","title":"Method 3: using local indices","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Alternatively, we can avoid global views and work directly on PencilArrays using local indices that start at 1. In this case, part of the strategy is to construct a \"local\" grid of wave numbers that can also be accessed with local indices. This can be conveniently done using the localgrid function of the PencilArrays.jl package, which accepts a PencilArray (or its associated Pencil) and the global coordinates (here kvec):","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"grid_fourier = localgrid(θ_hat, kvec)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that one can directly iterate on the returned grid object:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@inbounds for I in CartesianIndices(grid_fourier)\n # Wave number vector associated to current Cartesian index.\n local k⃗ # hide\n k⃗ = grid_fourier[I]\n u = im * θ_hat[I]\n ∇θ_hat[1][I] = k⃗[1] * u\n ∇θ_hat[2][I] = k⃗[2] * u\n ∇θ_hat[3][I] = k⃗[3] * u\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This implementation is as efficient as the other examples, while being slightly shorter to write. Moreover, it is quite generic, and can be made independent of the number of dimensions with little effort.","category":"page"},{"location":"generated/gradient/#gradient_method_broadcast","page":"Gradient of a scalar field","title":"Method 4: using broadcasting","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, note that the local grid object returned by localgrid makes it is possible to compute the gradient using broadcasting, thus fully avoiding scalar indexing. This can be quite convenient in some cases, and can also be very useful if one is working on GPUs (where scalar indexing is prohibitively expensive). Using broadcasting, the above examples simply become:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@. ∇θ_hat[1] = im * grid_fourier[1] * θ_hat\n@. ∇θ_hat[2] = im * grid_fourier[2] * θ_hat\n@. ∇θ_hat[3] = im * grid_fourier[3] * θ_hat\nnothing # hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Once again, as shown in the benchmarks further below, this method performs quite similarly to the other ones.","category":"page"},{"location":"generated/gradient/#Summary","page":"Gradient of a scalar field","title":"Summary","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The PencilArrays module provides different alternatives to deal with MPI-distributed data that may be subject to dimension permutations. In particular, one can choose to work with global indices (first two examples), with local indices (third example), or to avoid scalar indexing altogether (fourth example).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"If one wants to stay generic, making sure that the same code will work for arbitrary dimensions and will be efficient regardless of the underlying dimension permutation, methods 1, 3 or 4 should be preferred. These use CartesianIndices and make no assumptions on possible dimension permutations, which are by default enabled in the output of PencilFFTs transforms. In fact, such permutations are completely invisible in the implementations.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The second method uses explicit (i, j, k) indices. It assumes that the underlying permutation is (3, 2, 1) to loop with i as the slowest index and k as the fastest, which is the optimal order in this case given the permutation. As such, the implementation is less generic than the others, and differences in performance are negligible with respect to more generic variants.","category":"page"},{"location":"generated/gradient/#gradient_benchmarks","page":"Gradient of a scalar field","title":"Benchmark results","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The following are the benchmark results obtained from running examples/gradient.jl on a laptop, using 2 MPI processes and Julia 1.7.2, with an input array of global dimensions 64 32 64. The different methods detailed above are marked on the right. The \"lazy\" marks indicate runs where the wave numbers were represented by lazy Frequencies objects (returned by rfftfreq and fftfreq). Otherwise, they were collected into Vectors. For some reason, plain Vectors are faster when working with grids generated by localgrid.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In the script, additional implementations can be found which rely on a more advanced understanding of permutations and on the internals of the PencilArrays package. For instance, gradient_local_parent! directly works with the raw data stored in Julia Arrays, while gradient_local_linear! completely avoids CartesianIndices while staying generic and efficient. Nevertheless, these display roughly the same performance as the above examples.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":" gradient_global_view!... 89.900 μs\n gradient_global_view! (lazy)... 92.060 μs [Method 1]\n gradient_global_view_explicit!... 88.958 μs\n gradient_global_view_explicit! (lazy)... 81.055 μs [Method 2]\n gradient_local!... 92.305 μs\n gradient_grid!... 92.770 μs\n gradient_grid! (lazy)... 101.388 μs [Method 3]\n gradient_grid_broadcast!... 88.606 μs\n gradient_grid_broadcast! (lazy)... 151.020 μs [Method 4]\n gradient_local_parent!... 92.248 μs\n gradient_local_linear!... 91.212 μs\n gradient_local_linear_explicit!... 90.992 μs","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This page was generated using Literate.jl.","category":"page"},{"location":"PencilFFTs_timers/#PencilFFTs.measuring_performance","page":"Measuring performance","title":"Measuring performance","text":"","category":"section"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"It is possible to measure the time spent in different sections of the distributed transforms using the TimerOutputs package. This has a (very small) performance overhead, so it is disabled by default. To enable time measurements, call TimerOutputs.enable_debug_timings after loading PencilFFTs (see below for an example). For more details see the TimerOutputs docs.","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"Minimal example:","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"using MPI\nusing PencilFFTs\nusing TimerOutputs\n\n# Enable timing of `PencilFFTs` functions\nTimerOutputs.enable_debug_timings(PencilFFTs)\nTimerOutputs.enable_debug_timings(PencilArrays)\nTimerOutputs.enable_debug_timings(Transpositions)\n\nMPI.Init()\n\nplan = PencilFFTPlan(#= args... =#)\n\n# [do stuff with `plan`...]\n\n# Retrieve and print timing data associated to `plan`\nto = timer(plan)\nprint_timer(to)","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"By default, each PencilFFTPlan has its own TimerOutput. If you already have a TimerOutput, you can pass it to the PencilFFTPlan constructor:","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"to = TimerOutput()\nplan = PencilFFTPlan(..., timer=to)\n\n# [do stuff with `plan`...]\n\nprint_timer(to)","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"#PencilFFTs","page":"Home","title":"PencilFFTs","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Fast Fourier transforms of MPI-distributed Julia arrays.","category":"page"},{"location":"#Introduction","page":"Home","title":"Introduction","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package provides multidimensional FFTs and related transforms on MPI-distributed Julia arrays via the PencilArrays package.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The name of this package originates from the decomposition of 3D domains along two out of three dimensions, sometimes called pencil decomposition. This is illustrated by the figure below,[1] where each coloured block is managed by a different MPI process. Typically, one wants to compute FFTs on a scalar or vector field along the three spatial dimensions. In the case of a pencil decomposition, 3D FFTs are performed one dimension at a time, along the non-decomposed direction. Transforms must then be interleaved with global data transpositions to switch between pencil configurations. In high-performance computing environments, such data transpositions are generally the most expensive part of a parallel FFT computation, due to the large cost of communications between computing nodes.","category":"page"},{"location":"","page":"Home","title":"Home","text":"
              \n \n
              ","category":"page"},{"location":"","page":"Home","title":"Home","text":"More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension N. The decompositions can be performed along an arbitrary number M N of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.","category":"page"},{"location":"#Features","page":"Home","title":"Features","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;\nFFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;\nin-place and out-of-place transforms;\nhigh scalability up to (at least) tens of thousands of MPI processes.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"PencilFFTs can be installed using the Julia package manager:","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> ] add PencilFFTs","category":"page"},{"location":"#Similar-projects","page":"Home","title":"Similar projects","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.\nPFFT is a very general parallel FFT library written in C.\nP3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.\n2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[1]: Figure adapted from this PhD thesis.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[2]: For the pencil decomposition represented in the figure, N = 3 and M = 2.","category":"page"},{"location":"PencilFFTs/#Distributed-FFT-plans","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"Distributed FFTs are implemented in the PencilFFTs module, and are built on top of the PencilArrays package.","category":"page"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"PencilFFTs/#Creating-plans","page":"Distributed FFT plans","title":"Creating plans","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"PencilFFTPlan","category":"page"},{"location":"PencilFFTs/#PencilFFTs.PencilFFTPlan","page":"Distributed FFT plans","title":"PencilFFTs.PencilFFTPlan","text":"PencilFFTPlan{T,N} <: AbstractFFTs.Plan{T}\n\nPlan for N-dimensional FFT-based transform on MPI-distributed data, where input data has type T.\n\n\n\nPencilFFTPlan(p::Pencil, transforms; kwargs...)\n\nCreate a PencilFFTPlan for distributed arrays following a given Pencil configuration. See variant below for details on the specification of transforms and on possible keyword arguments.\n\n\n\nPencilFFTPlan(\n A::PencilArray, transforms;\n fftw_flags = FFTW.ESTIMATE,\n fftw_timelimit = FFTW.NO_TIMELIMIT,\n permute_dims = Val(true),\n transpose_method = Transpositions.PointToPoint(),\n timer = timer(pencil(A)),\n)\n\nCreate plan for N-dimensional transform on MPI-distributed PencilArrays.\n\nExtended help\n\nThis creates a PencilFFTPlan for arrays sharing the same properties as A (dimensions, MPI decomposition, memory layout, ...), which describe data on an N-dimensional domain.\n\nTransforms\n\nThe transforms to be applied along each dimension are specified by the transforms argument. Possible transforms are defined as subtypes of Transforms.AbstractTransform, and are listed in Transform types. This argument may be either:\n\na tuple of N transforms to be applied along each dimension. For instance, transforms = (Transforms.R2R(FFTW.REDFT01), Transforms.RFFT(), Transforms.FFT());\na single transform to be applied along all dimensions. The input is automatically expanded into N equivalent transforms. For instance, for a three-dimensional array, transforms = Transforms.RFFT() specifies a 3D real-to-complex transform, and is equivalent to passing (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT()).\n\nNote that forward transforms are applied from left to right. In the last example, this means that a real-to-complex transform (RFFT) is first performed along the first dimension. This is followed by complex-to-complex transforms (FFT) along the second and third dimensions.\n\nInput data layout\n\nThe input PencilArray must satisfy the following constraints:\n\narray dimensions must not be permuted. This is the default when constructing PencilArrays.\nfor an M-dimensional domain decomposition (with M < N), the input array must be decomposed along the last M dimensions. For example, for a 2D decomposition of 3D data, the decomposed dimensions must be (2, 3). In particular, the first array dimension must not be distributed among different MPI processes.\nIn the PencilArrays package, the decomposed dimensions are specified at the moment of constructing a Pencil.\nthe element type must be compatible with the specified transform. For instance, real-to-complex transforms (Transforms.RFFT) require the input to be real floating point values. Other transforms, such as Transforms.R2R, accept both real and complex data.\n\nKeyword arguments\n\nThe keyword arguments fftw_flags and fftw_timelimit are passed to the FFTW plan creation functions (see AbstractFFTs docs).\npermute_dims determines whether the indices of the output data should be reversed. For instance, if the input data has global dimensions (Nx, Ny, Nz), then the output of a complex-to-complex FFT would have dimensions (Nz, Ny, Nx). This enables FFTs to always be performed along the first (i.e. fastest) array dimension, which could lead to performance gains. This option is enabled by default. For type inference reasons, it must be a value type (Val(true) or Val(false)).\ntranspose_method allows to select between implementations of the global data transpositions. See PencilArrays docs docs for details.\ntimer should be a TimerOutput object. See Measuring performance for details.\n\n\n\nPencilFFTPlan(\n dims_global::Dims{N}, transforms, proc_dims::Dims{M}, comm::MPI.Comm,\n [real_type = Float64]; extra_dims = (), kws...\n)\n\nCreate plan for N-dimensional transform.\n\nExtended help\n\nInstead of taking a PencilArray or a Pencil, this constructor requires the global dimensions of the input data, passed via the size_global argument.\n\nThe data is distributed over the MPI processes in the comm communicator. The distribution is performed over M dimensions (with M < N) according to the values in proc_dims, which specifies the number of MPI processes to put along each dimension.\n\nPencilArrays that may be transformed with the returned plan can be created using allocate_input.\n\nOptional arguments\n\nThe floating point precision can be selected by setting real_type parameter, which is Float64 by default.\nextra_dims may be used to specify the sizes of one or more extra dimensions that should not be transformed. These dimensions will be added to the rightmost (i.e. slowest) indices of the arrays. See Extra dimensions below for usage hints.\nsee the other constructor for more keyword arguments.\n\nExtra dimensions\n\nOne possible application of extra_dims is for describing the components of a vector or tensor field. However, this means that different PencilFFTPlans would need to be created for each kind of field (scalar, vector, ...). To avoid the creation of multiple plans, a possibly better alternative is to create tuples (or arrays) of PencilArrays using allocate_input and allocate_output.\n\nAnother more legitimate usage of extra_dims is to specify one or more Cartesian dimensions that should not be transformed nor split among MPI processes.\n\nExample\n\nSuppose we want to perform a 3D FFT of real data. The data is to be decomposed along two dimensions, over 8 MPI processes:\n\nsize_global = (64, 32, 128) # size of real input data\n\n# Perform real-to-complex transform along the first dimension, then\n# complex-to-complex transforms along the other dimensions.\ntransforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())\n# transforms = Transforms.RFFT() # this is equivalent to the above line\n\nproc_dims = (4, 2) # 2D decomposition\ncomm = MPI.COMM_WORLD\n\nplan = PencilFFTPlan(size_global, transforms, proc_dims, comm)\n\n\n\n\n\n","category":"type"},{"location":"PencilFFTs/#Allocating-data","page":"Distributed FFT plans","title":"Allocating data","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"allocate_input\nallocate_output","category":"page"},{"location":"PencilFFTs/#PencilFFTs.allocate_input","page":"Distributed FFT plans","title":"PencilFFTs.allocate_input","text":"allocate_input(p::PencilFFTPlan) -> PencilArray\nallocate_input(p::PencilFFTPlan, dims...) -> Array{PencilArray}\nallocate_input(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}\n\nAllocate uninitialised PencilArray that can hold input data for the given plan.\n\nThe second and third forms respectively allocate an array of PencilArrays of size dims, and a tuple of N PencilArrays.\n\nnote: In-place plans\nIf p is an in-place real-to-real or complex-to-complex plan, a ManyPencilArray is allocated. If p is an in-place real-to-complex plan, a ManyPencilArrayRFFT! is allocated. These types hold PencilArray wrappers for the input and output transforms (as well as for intermediate transforms) which share the same space in memory. The input and output PencilArrays should be respectively accessed by calling first(::ManyPencilArray) and last(::ManyPencilArray).ExampleSuppose p is an in-place PencilFFTPlan. Then,@assert is_inplace(p)\nA = allocate_input(p) :: ManyPencilArray\nv_in = first(A) :: PencilArray # input data view\nv_out = last(A) :: PencilArray # output data viewAlso note that in-place plans must be performed directly on the returned ManyPencilArray, and not on the contained PencilArray views:p * A # perform forward transform in-place\np \\ A # perform backward transform in-place\n# p * v_in # not allowed!!\n\n\n\n\n\n","category":"function"},{"location":"PencilFFTs/#PencilFFTs.allocate_output","page":"Distributed FFT plans","title":"PencilFFTs.allocate_output","text":"allocate_output(p::PencilFFTPlan) -> PencilArray\nallocate_output(p::PencilFFTPlan, dims...) -> Array{PencilArray}\nallocate_output(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}\n\nAllocate uninitialised PencilArray that can hold output data for the given plan.\n\nIf p is an in-place plan, a ManyPencilArray is allocated.\n\nSee allocate_input for details.\n\n\n\n\n\n","category":"function"},{"location":"PencilFFTs/#Methods","page":"Distributed FFT plans","title":"Methods","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"get_comm(::PencilFFTPlan)\nscale_factor(::PencilFFTPlan)\ntimer(::PencilFFTPlan)\nis_inplace(::PencilFFTPlan)","category":"page"},{"location":"PencilFFTs/#PencilArrays.Pencils.MPITopologies.get_comm-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilArrays.Pencils.MPITopologies.get_comm","text":"get_comm(p::PencilFFTPlan)\n\nGet MPI communicator associated to a PencilFFTPlan.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilFFTs.Transforms.scale_factor-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilFFTs.Transforms.scale_factor","text":"scale_factor(p::PencilFFTPlan)\n\nGet scale factor associated to a PencilFFTPlan.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilArrays.Pencils.timer-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilArrays.Pencils.timer","text":"timer(p::PencilFFTPlan)\n\nGet TimerOutput attached to a PencilFFTPlan.\n\nSee Measuring performance for details.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilFFTs.Transforms.is_inplace-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilFFTs.Transforms.is_inplace","text":"Transforms.is_inplace(p::PencilFFTPlan)\n\nReturns true if the given plan operates in-place on the input data, false otherwise.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#Internals","page":"Distributed FFT plans","title":"Internals","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"ManyPencilArrayRFFT!","category":"page"},{"location":"PencilFFTs/#PencilFFTs.ManyPencilArrayRFFT!","page":"Distributed FFT plans","title":"PencilFFTs.ManyPencilArrayRFFT!","text":"ManyPencilArrayRFFT!{T,N,M} <: AbstractManyPencilArray{N,M}\n\nContainer holding M different PencilArray views to the same underlying data buffer. All views share the same dimensionality N. The element type T of the first view is real, that of subsequent views is Complex{T}. \n\nThis can be used to perform in-place real-to-complex plan, see alsoTransforms.RFFT!. It is used internally for such transforms by allocate_input and should not be constructed directly.\n\n\n\nManyPencilArrayRFFT!{T}(undef, pencils...; extra_dims=())\n\nCreate a ManyPencilArrayRFFT! container that can hold data of type T and Complex{T} associated to all the given Pencils.\n\nThe optional extra_dims argument is the same as for PencilArray.\n\nSee also ManyPencilArray\n\n\n\n\n\n","category":"type"}] +} diff --git a/previews/PR76/siteinfo.js b/previews/PR76/siteinfo.js new file mode 100644 index 00000000..bfaaa0a6 --- /dev/null +++ b/previews/PR76/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "previews/PR76"; diff --git a/previews/PR40/tutorial/index.html b/previews/PR76/tutorial/index.html similarity index 59% rename from previews/PR40/tutorial/index.html rename to previews/PR76/tutorial/index.html index be9051d1..107d4390 100644 --- a/previews/PR40/tutorial/index.html +++ b/previews/PR76/tutorial/index.html @@ -1,5 +1,5 @@ -Tutorial · PencilFFTs.jl
              Edit on GitHub

              Tutorial

              The following tutorial shows how to perform a 3D FFT of real periodic data defined on a grid of $N_x × N_y × N_z$ points.

              +Tutorial · PencilFFTs.jl
              GitHub

              Tutorial

              The following tutorial shows how to perform a 3D FFT of real periodic data defined on a grid of $N_x × N_y × N_z$ points.

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z(Es?1@?vLdWdyX5+L``)>}-G{E|A*KzeX}zn%X-5qXXn_WeQ%g|7TGLwjHF5>`iUW zzX+wXuI>!Y3j}R!}H4Xsq8VD?*{ed$5@~*cy*ZxU=jpE0F0A33q7X)Gl1oqFrIcNy5qYCgj0BQh$IEaM+`5S;1Fjg3_1GWk1 z1yzASj4i-e2Y`cPs0M*ZC;$l_D+AyM2*eEdPYC!5DhB|JBfSNDK>&Cc04ETL?gMbv z0?y?+0KjLCa6lg>z&ikRfk2eFfCO>}k_8YCY~KSp*atY^7dS_R7Jvl%-UtBjo0hU^V*vEg1GbR101oH{9Rc>iv1|a)1OV7%90TG3``ZQJH;@m@fB1Jm>VQC0|egffkh=$uqJed-0RZv}0(`@m2KqAq9NZt6 z8yLV(NEm>F;{n%Uxwx7*01{jv0I&csb~AM~0;IN`* -

              More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension $N$. The decompositions can be performed along an arbitrary number $M < N$ of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.

              The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.

              Features

              • distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;

              • FFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;

              • in-place and out-of-place transforms;

              • high scalability up to (at least) tens of thousands of MPI processes.

              Installation

              PencilFFTs can be installed using the Julia package manager:

              julia> ] add PencilFFTs

              Similar projects

              • FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.

              • PFFT is a very general parallel FFT library written in C.

              • P3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.

              • 2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.

              • 1Figure adapted from this PhD thesis.
              • 2For the pencil decomposition represented in the figure, $N = 3$ and $M = 2$.
              +

              More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension $N$. The decompositions can be performed along an arbitrary number $M < N$ of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.

              The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.

              Features

              • distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;

              • FFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;

              • in-place and out-of-place transforms;

              • high scalability up to (at least) tens of thousands of MPI processes.

              Installation

              PencilFFTs can be installed using the Julia package manager:

              julia> ] add PencilFFTs

              Similar projects

              • FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.

              • PFFT is a very general parallel FFT library written in C.

              • P3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.

              • 2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.

              • 1Figure adapted from this PhD thesis.
              • 2For the pencil decomposition represented in the figure, $N = 3$ and $M = 2$.
              diff --git a/dev/search/index.html b/v0.14.4/search/index.html similarity index 94% rename from dev/search/index.html rename to v0.14.4/search/index.html index da67ae52..43a21eb8 100644 --- a/dev/search/index.html +++ b/v0.14.4/search/index.html @@ -1,2 +1,2 @@ -Search · PencilFFTs.jl

              Loading search...

                +Search · PencilFFTs.jl

                Loading search...

                  diff --git a/previews/PR59/search_index.js b/v0.14.4/search_index.js similarity index 100% rename from previews/PR59/search_index.js rename to v0.14.4/search_index.js diff --git a/v0.14.4/siteinfo.js b/v0.14.4/siteinfo.js new file mode 100644 index 00000000..68ad5b91 --- /dev/null +++ b/v0.14.4/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "v0.14.4"; diff --git a/previews/PR59/tutorial/index.html b/v0.14.4/tutorial/index.html similarity index 98% rename from previews/PR59/tutorial/index.html rename to v0.14.4/tutorial/index.html index a7255b6a..c23c7332 100644 --- a/previews/PR59/tutorial/index.html +++ b/v0.14.4/tutorial/index.html @@ -37,4 +37,4 @@ # Apply backward plan on `v` with `w` as an output w = similar(u) -ldiv!(w, plan, v) # now w ≈ u

                  Note that, consistently with AbstractFFTs, normalisation is performed at the end of a backward transform, so that the original data is recovered when applying a forward followed by a backward transform.

                  Accessing and modifying data

                  For any given MPI process, a PencilArray holds the data associated to its local partition in the global geometry. PencilArrays are accessed using local indices that start at 1, regardless of the location of the local process in the MPI topology. Note that PencilArrays, being based on regular Arrays, support both linear and Cartesian indexing (see the Julia docs for details).

                  For convenience, the global_view function can be used to generate an OffsetArray wrapper that takes global indices.

                  Output data layout

                  In memory, the dimensions of the transform output are by default reversed with respect to the input. That is, if the order of indices in the input data is (x, y, z), then the output has order (z, y, x) in memory. This detail is hidden from the user, and output arrays are always accessed in the same order as the input data, regardless of the underlying output dimension permutation. This applies to PencilArrays and to OffsetArrays returned by global_view.

                  The reasoning behind dimension permutations, is that they allow to always perform FFTs along the fastest array dimension and to avoid a local data transposition, resulting in performance gains. A similar approach is followed by other parallel FFT libraries. FFTW itself, in its distributed-memory routines, includes a flag that enables a similar behaviour. In PencilFFTs, index permutation is the default, but it can be disabled via the permute_dims flag of PencilFFTPlan.

                  A great deal of work has been spent in making generic index permutations as efficient as possible, both in intermediate and in the output state of the multidimensional transforms. This has been achieved, in part, by making sure that permutations such as (3, 2, 1) are compile-time constants.

                  Further reading

                  For details on working with PencilArrays see the PencilArrays docs.

                  The examples on the sidebar further illustrate the use of transforms and provide an introduction to working with MPI-distributed data in the form of PencilArrays. In particular, the gradient example illustrates different ways of computing things using Fourier-transformed distributed arrays. Then, the incompressible Navier–Stokes example is a more advanced and complete example of a possible application of the PencilFFTs package.

                  GitHub

                  PencilFFTs

                  Fast Fourier transforms of MPI-distributed Julia arrays.

                  Introduction

                  This package provides multidimensional FFTs and related transforms on MPI-distributed Julia arrays via the PencilArrays package.

                  The name of this package originates from the decomposition of 3D domains along two out of three dimensions, sometimes called pencil decomposition. This is illustrated by the figure below,[1] where each coloured block is managed by a different MPI process. Typically, one wants to compute FFTs on a scalar or vector field along the three spatial dimensions. In the case of a pencil decomposition, 3D FFTs are performed one dimension at a time, along the non-decomposed direction. Transforms must then be interleaved with global data transpositions to switch between pencil configurations. In high-performance computing environments, such data transpositions are generally the most expensive part of a parallel FFT computation, due to the large cost of communications between computing nodes.

                  + Pencil decomposition of 3D domains +

                  More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension $N$. The decompositions can be performed along an arbitrary number $M < N$ of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.

                  The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.

                  Features

                  • distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;

                  • FFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;

                  • in-place and out-of-place transforms;

                  • high scalability up to (at least) tens of thousands of MPI processes.

                  Installation

                  PencilFFTs can be installed using the Julia package manager:

                  julia> ] add PencilFFTs

                  Similar projects

                  • FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.

                  • PFFT is a very general parallel FFT library written in C.

                  • P3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.

                  • 2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.

                  • 1Figure adapted from this PhD thesis.
                  • 2For the pencil decomposition represented in the figure, $N = 3$ and $M = 2$.

                  GitHub

                  Navier–Stokes equations

                  In this example, we numerically solve the incompressible Navier–Stokes equations

                  \[∂_t \bm{v} + (\bm{v} ⋅ \bm{∇}) \bm{v} = -\frac{1}{ρ} \bm{∇} p + ν ∇^2 \bm{v}, +\quad \bm{∇} ⋅ \bm{v} = 0,\]

                  where $\bm{v}(\bm{x}, t)$ and $p(\bm{x}, t)$ are respectively the velocity and pressure fields, $ν$ is the fluid kinematic viscosity and $ρ$ is the fluid density.

                  We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.

                  First steps

                  We start by loading the required packages, initialising MPI and setting the simulation parameters.

                  using MPI
+using PencilFFTs
+
+MPI.Init()
+comm = MPI.COMM_WORLD
+procid = MPI.Comm_rank(comm) + 1
+
+# Simulation parameters
+Ns = (64, 64, 64)  # = (Nx, Ny, Nz)
+Ls = (2π, 2π, 2π)  # = (Lx, Ly, Lz)
+
+# Collocation points ("global" = over all processes).
+# We include the endpoint (length = N + 1) for convenience.
+xs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls)  # = (x, y, z)
                  (0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586)

                  Let's check the number of MPI processes over which we're running our simulation:

                  MPI.Comm_size(comm)
                  2

                  We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:

                  pen = Pencil(Ns, comm)
                  Decomposition of 3D data
+    Data dimensions: (64, 64, 64)
+    Decomposed dimensions: (2, 3)
+    Data permutation: NoPermutation()
+    Array type: Array

                  The subdomain associated to the local MPI process can be obtained using range_local:

                  range_local(pen)
                  (1:64, 1:32, 1:64)

                  We now construct a distributed vector field that follows the decomposition configuration we just created:

                  v⃗₀ = (
+    PencilArray{Float64}(undef, pen),  # vx
+    PencilArray{Float64}(undef, pen),  # vy
+    PencilArray{Float64}(undef, pen),  # vz
+)
+summary(v⃗₀[1])
                  "64×32×64 PencilArray{Float64, 3}(::Pencil{3, 2, NoPermutation, Array})"

                  We still need to fill this array with interesting values that represent a physical velocity field.

                  Initial condition

                  Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:

                  \[\begin{aligned} +v_x(x, y, z) &= u₀ \sin(k₀ x) \cos(k₀ y) \cos(k₀ z) \\ +v_y(x, y, z) &= -u₀ \cos(k₀ x) \sin(k₀ y) \cos(k₀ z) \\ +v_z(x, y, z) &= 0 +\end{aligned}\]

                  where $u₀$ and $k₀$ are two parameters setting the amplitude and the period of the velocity field.

                  To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:

                  grid = localgrid(pen, xs_global)
                  LocalRectilinearGrid{3} with coordinates:
+ (1) 0.0:0.09817477042468103:6.1850105367549055
+ (2) 0.0:0.09817477042468103:3.043417883165112
+ (3) 0.0:0.09817477042468103:6.1850105367549055

                  We can use this to initialise the velocity field:

                  u₀ = 1.0
+k₀ = 2π / Ls[1]  # should be integer if L = 2π (to preserve periodicity)
+
+@. v⃗₀[1] =  u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)
+@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)
+@. v⃗₀[3] =  0

                  Let's plot a 2D slice of the velocity field managed by the local MPI process:

                  using GLMakie
+
+# Compute the norm of a vector field represented by a tuple of arrays.
+function vecnorm(v⃗::NTuple)
+    vnorm = similar(v⃗[1])
+    for n ∈ eachindex(v⃗[1])
+        w = zero(eltype(vnorm))
+        for v ∈ v⃗
+            w += v[n]^2
+        end
+        vnorm[n] = sqrt(w)
+    end
+    vnorm
+end
+
+# This is useful for passing coordinates to Makie.contour!
+to_intervals(grid) = map(xs -> xs[begin]..xs[end], grid.coords)
+
+let fig = Figure(size = (700, 600))
+    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
+    vnorm = parent(vecnorm(v⃗₀))  # use `parent` because Makie doesn't like custom array types...
+    ct = contour!(
+        ax, to_intervals(grid)..., vnorm;
+        alpha = 0.2, levels = 4,
+        colormap = :viridis,
+        colorrange = (0.0, 1.0),
+        highclip = (:red, 0.2), lowclip = (:green, 0.2),
+    )
+    cb = Colorbar(fig[1, 2], ct; label = "Velocity magnitude")
+    fig
+end
                  Example block output

                  Velocity in Fourier space

                  In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series

                  \[\bm{v}(\bm{x}, t) = +∑_{\bm{k}} \hat{\bm{v}}_{\bm{k}}(t) \, e^{i \bm{k} ⋅ \bm{x}},\]

                  where $\bm{k} = (k_x, k_y, k_z)$ are the discrete wave numbers.

                  The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for $k_x$:

                  using AbstractFFTs: fftfreq, rfftfreq
+
+ks_global = (
+    rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]),  # kx | real-to-complex
+     fftfreq(Ns[2], 2π * Ns[2] / Ls[2]),  # ky | complex-to-complex
+     fftfreq(Ns[3], 2π * Ns[3] / Ls[3]),  # kz | complex-to-complex
+)
+
+ks_global[1]'
                  1×33 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
+ 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  27.0  28.0  29.0  30.0  31.0  32.0
                  ks_global[2]'
                  1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
+ 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0
                  ks_global[3]'
                  1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
+ 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0

                  To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:

                  plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())
                  Transforms: (RFFT, FFT, FFT)
+Input type: Float64
+Global dimensions: (64, 64, 64) -> (33, 64, 64)
+MPI topology: 2D decomposition (2×1 processes)

                  See PencilFFTPlan for details on creating plans and on optional keyword arguments.

                  We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$:

                  v̂s = plan .* v⃗₀
+summary(v̂s[1])
                  "16×64×64 PencilArray{ComplexF64, 3}(::Pencil{3, 2, Permutation{(3, 2, 1), 3}, Array})"

                  Note that, in Fourier space, the domain decomposition is performed along the directions $x$ and $y$:

                  pencil(v̂s[1])
                  Decomposition of 3D data
+    Data dimensions: (33, 64, 64)
+    Decomposed dimensions: (1, 2)
+    Data permutation: Permutation(3, 2, 1)
+    Array type: Array

                  This is because the 3D FFTs are performed one dimension at a time, with the $x$ direction first and the $z$ direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.

                  To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers $\bm{k}$ associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:

                  grid_fourier = localgrid(v̂s[1], ks_global)
                  LocalRectilinearGrid{3} with Permutation(3, 2, 1) and coordinates:
+ (1) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0]
+ (2) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]
+ (3) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]

                  As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, $\bm{ω} = \bm{∇} × \bm{v}$. In Fourier space, this becomes $\hat{\bm{ω}} = i \bm{k} × \hat{\bm{v}}$.

                  using StaticArrays: SVector
+using LinearAlgebra: ×
+
+function curl_fourier!(
+        ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,
+    ) where {N}
+    @inbounds for I ∈ eachindex(grid_fourier)
+        # We use StaticArrays for the cross product between small vectors.
+        ik⃗ = im * SVector(grid_fourier[I])
+        v⃗ = SVector(getindex.(v̂s, Ref(I)))  # = (v̂s[1][I], v̂s[2][I], ...)
+        ω⃗ = ik⃗ × v⃗
+        for n ∈ eachindex(ω⃗)
+            ω̂s[n][I] = ω⃗[n]
+        end
+    end
+    ω̂s
+end
+
+ω̂s = similar.(v̂s)
+curl_fourier!(ω̂s, v̂s, grid_fourier);

                  We finally transform back to physical space and plot the result:

                  ωs = plan .\ ω̂s
+
+let fig = Figure(size = (700, 600))
+    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
+    ω_norm = parent(vecnorm(ωs))
+    ct = contour!(
+        ax, to_intervals(grid)..., ω_norm;
+        alpha = 0.1, levels = 0.8:0.2:2.0,
+        colormap = :viridis, colorrange = (0.8, 2.0),
+        highclip = (:red, 0.2), lowclip = (:green, 0.2),
+    )
+    cb = Colorbar(fig[1, 2], ct; label = "Vorticity magnitude")
+    fig
+end
                  Example block output

                  Computing the non-linear term

                  One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as

                  \[∂_t \hat{\bm{v}}_{\bm{k}} = +- \mathcal{P}_{\bm{k}} \! \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right] +- ν |\bm{k}|^2 \hat{\bm{v}}_{\bm{k}} +\quad \text{ with } \quad +\mathcal{P}_{\bm{k}}(\hat{\bm{F}}_{\bm{k}}) = \left( I - \frac{\bm{k} ⊗ +\bm{k}}{|\bm{k}|^2} \right) \hat{\bm{F}}_{\bm{k}},\]

                  where $\mathcal{P}_{\bm{k}}$ is a projection operator allowing to preserve the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.

                  Now that we have the wave numbers $\bm{k}$, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$ of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, $\hat{\bm{F}}_{\bm{k}} = \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right]_{\bm{k}}$. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.

                  Below we implement a function that computes the non-linear term in Fourier space based on its convective form $(\bm{v} ⋅ \bm{∇}) \bm{v} = \bm{∇} ⋅ (\bm{v} ⊗ \bm{v})$. Note that this equivalence uses the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$.

                  using LinearAlgebra: mul!, ldiv!  # for applying FFT plans in-place
+
+# Compute non-linear term in Fourier space from velocity field in physical
+# space. Optional keyword arguments may be passed to avoid allocations.
+function ns_nonlinear!(
+        F̂s, vs, plan, grid_fourier;
+        vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),
+    )
+    # Compute F_i = ∂_j (v_i v_j) for each i.
+    # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)
+    w, ŵ = vbuf, v̂buf
+    @inbounds for (i, F̂i) ∈ enumerate(F̂s)
+        F̂i .= 0
+        vi = vs[i]
+        for (j, vj) ∈ enumerate(vs)
+            w .= vi .* vj     # w = v_i * v_j in physical space
+            mul!(ŵ, plan, w)  # same in Fourier space
+            # Add derivative in Fourier space
+            for I ∈ eachindex(grid_fourier)
+                k⃗ = grid_fourier[I]  # = (kx, ky, kz)
+                kj = k⃗[j]
+                F̂i[I] += im * kj * ŵ[I]
+            end
+        end
+    end
+    F̂s
+end
                  ns_nonlinear! (generic function with 1 method)

                  As an example, let's use this function on our initial velocity field:

                  F̂s = similar.(v̂s)
+ns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);

                  Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.

                  function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)
+    ks_max = maximum.(abs, ks_global)  # maximum stored wave numbers (kx_max, ky_max, kz_max)
+    ks_lim = (2 / 3) .* ks_max
+    @inbounds for I ∈ eachindex(grid_fourier)
+        k⃗ = grid_fourier[I]
+        if any(abs.(k⃗) .> ks_lim)
+            for ŵ ∈ ŵs
+                ŵ[I] = 0
+            end
+        end
+    end
+    ŵs
+end
+
+# We can apply this on the previously computed non-linear term:
+dealias_twothirds!(F̂s, grid_fourier, ks_global);

                  Finally, we implement the projection associated to the incompressibility condition:

                  function project_divergence_free!(ûs, grid_fourier)
+    @inbounds for I ∈ eachindex(grid_fourier)
+        k⃗ = grid_fourier[I]
+        k² = sum(abs2, k⃗)
+        iszero(k²) && continue  # avoid division by zero
+        û = getindex.(ûs, Ref(I))  # (ûs[1][I], ûs[2][I], ...)
+        for i ∈ eachindex(û)
+            ŵ = û[i]
+            for j ∈ eachindex(û)
+                ŵ -= k⃗[i] * k⃗[j] * û[j] / k²
+            end
+            ûs[i][I] = ŵ
+        end
+    end
+    ûs
+end
                  project_divergence_free! (generic function with 1 method)

                  We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:

                  v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)
+v̂s_proj .≈ v̂s  # the last one may be false because v_z = 0 initially
                  (true, true, false)

                  Putting it all together

                  To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:

                  function ns_rhs!(
+        dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,
+    ) where {N}
+    # 1. Compute non-linear term and dealias it
+    (; plan, cache, ks_global, grid_fourier) = p
+    F̂s = cache.F̂s
+    ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])
+    dealias_twothirds!(F̂s, grid_fourier, ks_global)
+
+    # 2. Project onto divergence-free space
+    project_divergence_free!(F̂s, grid_fourier)
+
+    # 3. Transform velocity to Fourier space
+    v̂s = cache.v̂s
+    map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)
+
+    # 4. Add viscous term (and multiply projected non-linear term by -1)
+    ν = p.ν
+    for n ∈ eachindex(v̂s)
+        v̂ = v̂s[n]
+        F̂ = F̂s[n]
+        @inbounds for I ∈ eachindex(grid_fourier)
+            k⃗ = grid_fourier[I]  # = (kx, ky, kz)
+            k² = sum(abs2, k⃗)
+            F̂[I] = -F̂[I] - ν * k² * v̂[I]
+        end
+    end
+
+    # 5. Transform RHS back to physical space
+    map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)
+
+    nothing
+end
                  ns_rhs! (generic function with 1 method)

                  For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.

                  using OrdinaryDiffEqLowOrderRK  # includes RK4
+using RecursiveArrayTools: ArrayPartition
+
+ns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)
+vs_init_ode = ArrayPartition(v⃗₀)
+summary(vs_init_ode)
                  "RecursiveArrayTools.ArrayPartition{Float64, Tuple{PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}}} with arrays:"

                  We now define solver parameters and temporary variables, and initialise the problem:

                  params = (;
+    ν = 5e-3,  # kinematic viscosity
+    plan, grid_fourier, ks_global,
+    cache = (
+        v̂s = similar.(v̂s),
+        F̂s = similar.(v̂s),
+    )
+)
+
+tspan = (0.0, 10.0)
+prob = ODEProblem{true}(ns_rhs!, vs_init_ode, tspan, params)
+integrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);

                  We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum $E(k)$, to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity $ν$ must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large $k$), while high enough to allow the small-scale motions to be correctly resolved.

                  function energy_spectrum!(Ek, ks, v̂s, grid_fourier)
+    Nk = length(Ek)
+    @assert Nk == length(ks)
+    Ek .= 0
+    for I ∈ eachindex(grid_fourier)
+        k⃗ = grid_fourier[I]  # = (kx, ky, kz)
+        knorm = sqrt(sum(abs2, k⃗))
+        i = searchsortedfirst(ks, knorm)
+        i > Nk && continue
+        v⃗ = getindex.(v̂s, Ref(I))  # = (v̂s[1][I], v̂s[2][I], ...)
+        factor = k⃗[1] == 0 ? 1 : 2  # account for Hermitian symmetry and r2c transform
+        Ek[i] += factor * sum(abs2, v⃗) / 2
+    end
+    MPI.Allreduce!(Ek, +, get_comm(v̂s[1]))  # sum across all processes
+    Ek
+end
+
+ks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])
+Ek = similar(ks)
+v̂s = plan .* integrator.u.x
+energy_spectrum!(Ek, ks, v̂s, grid_fourier)
+Ek ./= scale_factor(plan)^2  # rescale energy
+
+curl_fourier!(ω̂s, v̂s, grid_fourier)
+ldiv!.(ωs, plan, ω̂s)
+ω⃗_plot = Observable(ωs)
+k_plot = @view ks[2:end]
+E_plot = Observable(@view Ek[2:end])
+t_plot = Observable(integrator.t)
+
+fig = let
+    fig = Figure(size = (1200, 600))
+    ax = Axis3(
+        fig[1, 1][1, 1]; title = @lift("t = $(round($t_plot, digits = 3))"),
+        aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z",
+    )
+    ω_mag = @lift parent(vecnorm($ω⃗_plot))
+    ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)
+    ct = contour!(
+        ax, to_intervals(grid)..., ω_mag_norm;
+        alpha = 0.3, levels = 3,
+        colormap = :viridis, colorrange = (0.0, 1.0),
+        highclip = (:red, 0.2), lowclip = (:green, 0.2),
+    )
+    cb = Colorbar(fig[1, 1][1, 2], ct; label = "Normalised vorticity magnitude")
+    ax_sp = Axis(
+        fig[1, 2];
+        xlabel = "k", ylabel = "E(k)", xscale = log2, yscale = log10,
+        title = "Kinetic energy spectrum",
+    )
+    ylims!(ax_sp, 1e-8, 1e0)
+    scatterlines!(ax_sp, k_plot, E_plot)
+    ks_slope = exp.(range(log(2.5), log(25.0), length = 3))
+    E_fivethirds = @. 0.3 * ks_slope^(-5/3)
+    @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)
+    text!(ax_sp, L"k^{-5/3}"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))
+    fig
+end
+
+record(fig, "vorticity_proc$procid.mp4"; framerate = 10) do io
+    while integrator.t < 20
+        dt = 0.001
+        step!(integrator, dt)
+        t_plot[] = integrator.t
+        mul!.(v̂s, plan, integrator.u.x)  # current velocity in Fourier space
+        curl_fourier!(ω̂s, v̂s, grid_fourier)
+        ldiv!.(ω⃗_plot[], plan, ω̂s)
+        ω⃗_plot[] = ω⃗_plot[]  # to force updating the plot
+        energy_spectrum!(Ek, ks, v̂s, grid_fourier)
+        Ek ./= scale_factor(plan)^2  # rescale energy
+        E_plot[] = E_plot[]
+        recordframe!(io)
+    end
+end;
                  + +

                  This page was generated using Literate.jl.

                  • +ldiv!(w, plan, v) # now w ≈ u

                  Note that, consistently with AbstractFFTs, normalisation is performed at the end of a backward transform, so that the original data is recovered when applying a forward followed by a backward transform.

                  Accessing and modifying data

                  For any given MPI process, a PencilArray holds the data associated to its local partition in the global geometry. PencilArrays are accessed using local indices that start at 1, regardless of the location of the local process in the MPI topology. Note that PencilArrays, being based on regular Arrays, support both linear and Cartesian indexing (see the Julia docs for details).

                  For convenience, the global_view function can be used to generate an OffsetArray wrapper that takes global indices.

                  Output data layout

                  In memory, the dimensions of the transform output are by default reversed with respect to the input. That is, if the order of indices in the input data is (x, y, z), then the output has order (z, y, x) in memory. This detail is hidden from the user, and output arrays are always accessed in the same order as the input data, regardless of the underlying output dimension permutation. This applies to PencilArrays and to OffsetArrays returned by global_view.

                  The reasoning behind dimension permutations, is that they allow to always perform FFTs along the fastest array dimension and to avoid a local data transposition, resulting in performance gains. A similar approach is followed by other parallel FFT libraries. FFTW itself, in its distributed-memory routines, includes a flag that enables a similar behaviour. In PencilFFTs, index permutation is the default, but it can be disabled via the permute_dims flag of PencilFFTPlan.

                  A great deal of work has been spent in making generic index permutations as efficient as possible, both in intermediate and in the output state of the multidimensional transforms. This has been achieved, in part, by making sure that permutations such as (3, 2, 1) are compile-time constants.

                  Further reading

                  For details on working with PencilArrays see the PencilArrays docs.

                  The examples on the sidebar further illustrate the use of transforms and provide an introduction to working with MPI-distributed data in the form of PencilArrays. In particular, the gradient example illustrates different ways of computing things using Fourier-transformed distributed arrays. Then, the incompressible Navier–Stokes example is a more advanced and complete example of a possible application of the PencilFFTs package.

                  diff --git a/v0.15 b/v0.15 new file mode 120000 index 00000000..253bdcfe --- /dev/null +++ b/v0.15 @@ -0,0 +1 @@ +v0.15.2 \ No newline at end of file diff --git a/v0.15.0/GlobalFFTParams/index.html b/v0.15.0/GlobalFFTParams/index.html new file mode 100644 index 00000000..c9e557a7 --- /dev/null +++ b/v0.15.0/GlobalFFTParams/index.html @@ -0,0 +1,9 @@ + +Global FFT parameters · PencilFFTs.jl
                  Edit on GitHub

                  Global FFT parameters

                  PencilFFTs.GlobalFFTParams — Type
                  GlobalFFTParams{T, N, inplace}

                  Specifies the global parameters for an N-dimensional distributed transform. These include the element type T and global data sizes of input and output data, as well as the transform types to be performed along each dimension.

                  GlobalFFTParams(size_global, transforms, [real_type=Float64])

                  Define parameters for N-dimensional transform.

                  transforms must be a tuple of length N specifying the transforms to be applied along each dimension. Each element must be a subtype of Transforms.AbstractTransform. For all the possible transforms, see Transform types.

                  The element type must be a real type accepted by FFTW, i.e. either Float32 or Float64.

                  Note that the transforms are applied one dimension at a time, with the leftmost dimension first for forward transforms.

                  Example

                  To perform a 3D FFT of real data, first a real-to-complex FFT must be applied along the first dimension, followed by two complex-to-complex FFTs along the other dimensions:

                  julia> size_global = (64, 32, 128);  # size of real input data
+
+julia> transforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT());
+
+julia> fft_params = PencilFFTs.GlobalFFTParams(size_global, transforms)
+Transforms: (RFFT, FFT, FFT)
+Input type: Float64
+Global dimensions: (64, 32, 128) -> (33, 32, 128)
                  source
                  diff --git a/previews/PR55/PencilFFTs/index.html b/v0.15.0/PencilFFTs/index.html similarity index 54% rename from previews/PR55/PencilFFTs/index.html rename to v0.15.0/PencilFFTs/index.html index 9d18cc57..6753a6cc 100644 --- a/previews/PR55/PencilFFTs/index.html +++ b/v0.15.0/PencilFFTs/index.html @@ -1,5 +1,5 @@ -Distributed FFT plans · PencilFFTs.jl
                  Edit on GitHub

                  Distributed FFT plans

                  Distributed FFTs are implemented in the PencilFFTs module, and are built on top of the PencilArrays package.

                  Creating plans

                  PencilFFTs.PencilFFTPlan — Type
                  PencilFFTPlan{T,N} <: AbstractFFTs.Plan{T}

                  Plan for N-dimensional FFT-based transform on MPI-distributed data, where input data has type T.

                  PencilFFTPlan(p::Pencil, transforms; kwargs...)

                  Create a PencilFFTPlan for distributed arrays following a given Pencil configuration. See variant below for details on the specification of transforms and on possible keyword arguments.

                  PencilFFTPlan(
+Distributed FFT plans · PencilFFTs.jl
                  Edit on GitHub
                  Distributed FFT plans
                  Distributed FFTs are implemented in the PencilFFTs module, and are built on top of the PencilArrays package.
                  Creating plans
                  PencilFFTs.PencilFFTPlan — Type
                  PencilFFTPlan{T,N} <: AbstractFFTs.Plan{T}
                  Plan for N-dimensional FFT-based transform on MPI-distributed data, where input data has type T.
                  PencilFFTPlan(p::Pencil, transforms; kwargs...)
                  Create a PencilFFTPlan for distributed arrays following a given Pencil configuration. See variant below for details on the specification of transforms and on possible keyword arguments.
                  PencilFFTPlan(
     A::PencilArray, transforms;
     fftw_flags = FFTW.ESTIMATE,
     fftw_timelimit = FFTW.NO_TIMELIMIT,
@@ -19,13 +19,13 @@
 proc_dims = (4, 2)  # 2D decomposition
 comm = MPI.COMM_WORLD
 
-plan = PencilFFTPlan(size_global, transforms, proc_dims, comm)
                  source
                  Allocating data
                  PencilFFTs.allocate_input — Function
                  allocate_input(p::PencilFFTPlan)          -> PencilArray
+plan = PencilFFTPlan(size_global, transforms, proc_dims, comm)
                  source
                  Allocating data
                  PencilFFTs.allocate_input — Function
                  allocate_input(p::PencilFFTPlan)          -> PencilArray
 allocate_input(p::PencilFFTPlan, dims...) -> Array{PencilArray}
-allocate_input(p::PencilFFTPlan, Val(N))  -> NTuple{N, PencilArray}
                  Allocate uninitialised PencilArray that can hold input data for the given plan.
                  The second and third forms respectively allocate an array of PencilArrays of size dims, and a tuple of N PencilArrays.
                  In-place plans
                  If p is an in-place plan, a ManyPencilArray is allocated. This type holds PencilArray wrappers for the input and output transforms (as well as for intermediate transforms) which share the same space in memory. The input and output PencilArrays should be respectively accessed by calling first(::ManyPencilArray) and last(::ManyPencilArray).
                  Example
                  Suppose p is an in-place PencilFFTPlan. Then,
                  @assert is_inplace(p)
+allocate_input(p::PencilFFTPlan, Val(N))  -> NTuple{N, PencilArray}
                  Allocate uninitialised PencilArray that can hold input data for the given plan.
                  The second and third forms respectively allocate an array of PencilArrays of size dims, and a tuple of N PencilArrays.
                  In-place plans
                  If p is an in-place real-to-real or complex-to-complex plan, a ManyPencilArray is allocated. If p is an in-place real-to-complex plan, a ManyPencilArrayRFFT! is allocated. 
                  These types hold PencilArray wrappers for the input and output transforms (as well as for intermediate transforms) which share the same space in memory. The input and output PencilArrays should be respectively accessed by calling first(::ManyPencilArray) and last(::ManyPencilArray).
                  Example
                  Suppose p is an in-place PencilFFTPlan. Then,
                  @assert is_inplace(p)
 A = allocate_input(p) :: ManyPencilArray
 v_in = first(A)       :: PencilArray  # input data view
 v_out = last(A)       :: PencilArray  # output data view
                  Also note that in-place plans must be performed directly on the returned ManyPencilArray, and not on the contained PencilArray views:
                  p * A       # perform forward transform in-place
 p \ A       # perform backward transform in-place
-# p * v_in  # not allowed!!
                  source
                  source
                  PencilFFTs.allocate_output — Function
                  allocate_output(p::PencilFFTPlan)          -> PencilArray
 allocate_output(p::PencilFFTPlan, dims...) -> Array{PencilArray}
-allocate_output(p::PencilFFTPlan, Val(N))  -> NTuple{N, PencilArray}

                  Allocate uninitialised PencilArray that can hold output data for the given plan.

                  If p is an in-place plan, a ManyPencilArray is allocated.

                  See allocate_input for details.

                  source

                  Methods

                  +allocate_output(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}

                  Allocate uninitialised PencilArray that can hold output data for the given plan.

                  If p is an in-place plan, a ManyPencilArray is allocated.

                  See allocate_input for details.

                  source

                  Methods

                  Internals

                  PencilFFTs.ManyPencilArrayRFFT! — Type
                  ManyPencilArrayRFFT!{T,N,M} <: AbstractManyPencilArray{N,M}

                  Container holding M different PencilArray views to the same underlying data buffer. All views share the same and dimensionality N. The element type T of the first view is real, that of subsequent views is Complex{T}.

                  This can be used to perform in-place real-to-complex plan, see alsoTransforms.RFFT!. It is used internally for such transforms by allocate_input and should not be constructed directly.

                  ManyPencilArrayRFFT!{T}(undef, pencils...; extra_dims=())

                  Create a ManyPencilArrayRFFT! container that can hold data of type T and Complex{T} associated to all the given Pencils.

                  The optional extra_dims argument is the same as for PencilArray.

                  See also ManyPencilArray

                  source
                  diff --git a/v0.15.0/PencilFFTs_timers/index.html b/v0.15.0/PencilFFTs_timers/index.html new file mode 100644 index 00000000..dd848bc9 --- /dev/null +++ b/v0.15.0/PencilFFTs_timers/index.html @@ -0,0 +1,24 @@ + +Measuring performance · PencilFFTs.jl
                  Edit on GitHub

                  Measuring performance

                  It is possible to measure the time spent in different sections of the distributed transforms using the TimerOutputs package. This has a (very small) performance overhead, so it is disabled by default. To enable time measurements, call TimerOutputs.enable_debug_timings after loading PencilFFTs (see below for an example). For more details see the TimerOutputs docs.

                  Minimal example:

                  using MPI
+using PencilFFTs
+using TimerOutputs
+
+# Enable timing of `PencilFFTs` functions
+TimerOutputs.enable_debug_timings(PencilFFTs)
+TimerOutputs.enable_debug_timings(PencilArrays)
+TimerOutputs.enable_debug_timings(Transpositions)
+
+MPI.Init()
+
+plan = PencilFFTPlan(#= args... =#)
+
+# [do stuff with `plan`...]
+
+# Retrieve and print timing data associated to `plan`
+to = timer(plan)
+print_timer(to)

                  By default, each PencilFFTPlan has its own TimerOutput. If you already have a TimerOutput, you can pass it to the PencilFFTPlan constructor:

                  to = TimerOutput()
+plan = PencilFFTPlan(..., timer=to)
+
+# [do stuff with `plan`...]
+
+print_timer(to)
                  diff --git a/v0.15.0/Transforms/index.html b/v0.15.0/Transforms/index.html new file mode 100644 index 00000000..224f5678 --- /dev/null +++ b/v0.15.0/Transforms/index.html @@ -0,0 +1,75 @@ + +Available transforms · PencilFFTs.jl
                  Edit on GitHub

                  Available transforms

                  Transform types

                  PencilFFTs.Transforms.BFFT — Type
                  BFFT()

                  Unnormalised backward complex-to-complex FFT.

                  Like AbstractFFTs.bfft, this transform is not normalised. To obtain the inverse transform, divide the output by the length of the transformed dimension.

                  See also AbstractFFTs.bfft.

                  source
                  PencilFFTs.Transforms.BRFFT — Type
                  BRFFT(d::Integer)
+BRFFT((d1, d2, ..., dN))

                  Unnormalised inverse of RFFT.

                  To obtain the inverse transform, divide the output by the length of the transformed dimension (of the real output array).

                  As described in the AbstractFFTs docs, the length of the output cannot be fully inferred from the input length. For this reason, the BRFFT constructor accepts an optional d argument indicating the output length.

                  For multidimensional datasets, a tuple of dimensions (d1, d2, ..., dN) may also be passed. This is equivalent to passing just dN. In this case, the last dimension (dN) is the one that changes size between the input and output. Note that this is the opposite of FFTW.brfft. The reason is that, in PencilFFTs, the last dimension is the one along which a complex-to-real transform is performed.

                  See also AbstractFFTs.brfft.

                  source

                  Internals

                  What follows is used internally in PencilFFTs.

                  Types

                  PencilFFTs.Transforms.AbstractCustomPlan — Type
                  AbstractCustomPlan

                  Abstract type defining a custom plan, to be used as an alternative to FFTW plans (FFTW.FFTWPlan).

                  The only custom plan defined in this module is IdentityPlan. The user can define other custom plans that are also subtypes of AbstractCustomPlan.

                  Note that plan returns a subtype of either AbstractFFTs.Plan or AbstractCustomPlan.

                  source

                  Functions

                  PencilFFTs.Transforms.plan — Function
                  plan(transform::AbstractTransform, A, [dims];
+     flags=FFTW.ESTIMATE, timelimit=Inf)

                  Create plan to transform array A along dimensions dims.

                  If dims is not specified, all dimensions of A are transformed.

                  For FFT plans, this function wraps the AbstractFFTs.jl and FFTW.jl plan creation functions. For more details on the function arguments, see AbstractFFTs.plan_fft.

                  source
                  PencilFFTs.Transforms.binv — Function
                  binv(transform::AbstractTransform, d::Integer)

                  Returns the backwards transform associated to the given transform.

                  The second argument must be the length of the first transformed dimension in the forward transform. It is used in particular when transform = RFFT(), to determine the length of the inverse (complex-to-real) transform. See the AbstractFFTs.irfft docs for details.

                  The backwards transform returned by this function is not normalised. The normalisation factor for a given array can be obtained by calling scale_factor.

                  Example

                  julia> binv(Transforms.FFT(), 42)
+BFFT
+
+julia> binv(Transforms.BRFFT(9), 42)
+RFFT
                  source
                  PencilFFTs.Transforms.scale_factor — Function
                  scale_factor(transform::AbstractTransform, A, [dims = 1:ndims(A)])

                  Get factor required to normalise the given array after a transformation along dimensions dims (all dimensions by default).

                  The array A must have the dimensions of the transform input.

                  Important: the dimensions dims must be the same that were passed to plan.

                  Examples

                  julia> C = zeros(ComplexF32, 3, 4, 5);
+
+julia> scale_factor(Transforms.FFT(), C)
+60
+
+julia> scale_factor(Transforms.BFFT(), C)
+60
+
+julia> scale_factor(Transforms.BFFT(), C, 2:3)
+20
+
+julia> R = zeros(Float64, 3, 4, 5);
+
+julia> scale_factor(Transforms.RFFT(), R, 2)
+4
+
+julia> scale_factor(Transforms.RFFT(), R, 2:3)
+20
+
+julia> scale_factor(Transforms.BRFFT(8), C)
+96
+
+julia> scale_factor(Transforms.BRFFT(9), C)
+108

                  This will fail because the input of RFFT is real, and R is a complex array:

                  julia> scale_factor(Transforms.RFFT(), C, 2:3)
+ERROR: MethodError: no method matching scale_factor(::PencilFFTs.Transforms.RFFT, ::Array{ComplexF32, 3}, ::UnitRange{Int64})
                  source
                  PencilFFTs.Transforms.eltype_input — Function
                  eltype_input(transform::AbstractTransform, real_type<:AbstractFloat)

                  Determine input data type for a given transform given the floating point precision of the input data.

                  Some transforms, such as R2R and NoTransform, can take both real and complex data. For those kinds of transforms, nothing is returned.

                  Example

                  julia> eltype_input(Transforms.FFT(), Float32)
+ComplexF32 (alias for Complex{Float32})
+
+julia> eltype_input(Transforms.RFFT(), Float64)
+Float64
+
+julia> eltype_input(Transforms.R2R(FFTW.REDFT01), Float64)  # nothing
+
+julia> eltype_input(Transforms.NoTransform(), Float64)  # nothing
+
                  source
                  PencilFFTs.Transforms.eltype_output — Function
                  eltype_output(transform::AbstractTransform, eltype_input)

                  Returns the output data type for a given transform given the input type.

                  Throws ArgumentError if the input data type is incompatible with the transform type.

                  Example

                  julia> eltype_output(Transforms.NoTransform(), Float32)
+Float32
+
+julia> eltype_output(Transforms.RFFT(), Float64)
+ComplexF64 (alias for Complex{Float64})
+
+julia> eltype_output(Transforms.BRFFT(4), ComplexF32)
+Float32
+
+julia> eltype_output(Transforms.FFT(), Float64)
+ERROR: ArgumentError: invalid input data type for PencilFFTs.Transforms.FFT: Float64
                  source
                  PencilFFTs.Transforms.expand_dims — Function
                  expand_dims(transform::AbstractTransform, Val(N))

                  Expand a single multidimensional transform into one transform per dimension.

                  Example

                  # Expand a real-to-complex transform in 3 dimensions.
+julia> expand_dims(Transforms.RFFT(), Val(3))
+(RFFT, FFT, FFT)
+
+julia> expand_dims(Transforms.BRFFT(4), Val(3))
+(BFFT, BFFT, BRFFT{even})
+
+julia> expand_dims(Transforms.NoTransform(), Val(2))
+(NoTransform, NoTransform)
                  source
                  PencilFFTs.Transforms.is_inplace — Function
                  is_inplace(transform::AbstractTransform)         -> Bool
+is_inplace(transforms::Vararg{AbtractTransform}) -> Union{Bool, Nothing}

                  Check whether a transform or a list of transforms is performed in-place.

                  If the list of transforms has a combination of in-place and out-of-place transforms, nothing is returned.

                  Example

                  julia> is_inplace(Transforms.RFFT())
+false
+
+julia> is_inplace(Transforms.NoTransform!())
+true
+
+julia> is_inplace(Transforms.FFT!(), Transforms.R2R!(FFTW.REDFT01))
+true
+
+julia> is_inplace(Transforms.FFT(), Transforms.R2R(FFTW.REDFT01))
+false
+
+julia> is_inplace(Transforms.FFT(), Transforms.R2R!(FFTW.REDFT01)) === nothing
+true
+
                  source
                  PencilFFTs.Transforms.length_output — Function
                  length_output(transform::AbstractTransform, length_in::Integer)

                  Returns the length of the transform output, given the length of its input.

                  The input and output lengths are specified in terms of the respective input and output datatypes. For instance, for real-to-complex transforms, these are respectively the length of input real data and of output complex data.

                  source
                  diff --git a/previews/PR59/assets/custom.css b/v0.15.0/assets/custom.css similarity index 100% rename from previews/PR59/assets/custom.css rename to v0.15.0/assets/custom.css diff --git a/previews/PR48/assets/documenter.js b/v0.15.0/assets/documenter.js similarity index 98% rename from previews/PR48/assets/documenter.js rename to v0.15.0/assets/documenter.js index 4fb3baaf..6adfbbbf 100644 --- a/previews/PR48/assets/documenter.js +++ b/v0.15.0/assets/documenter.js @@ -1,15 +1,15 @@ // Generated by Documenter.jl requirejs.config({ paths: { - 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.0.1/languages/julia.min', + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/languages/julia.min', 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.12.1/jquery-ui.min', - 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.11/contrib/auto-render.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/contrib/auto-render.min', 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.6.0/jquery.min', 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', - 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.11/katex.min', - 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.0.1/highlight.min', - 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.0.1/languages/julia-repl.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/languages/julia-repl.min', }, shim: { "highlight-julia": { diff --git a/previews/PR59/assets/logo.svg b/v0.15.0/assets/logo.svg similarity index 100% rename from previews/PR59/assets/logo.svg rename to v0.15.0/assets/logo.svg diff --git a/previews/PR55/assets/search.js b/v0.15.0/assets/search.js similarity index 100% rename from previews/PR55/assets/search.js rename to v0.15.0/assets/search.js diff --git a/previews/PR59/assets/themes/documenter-dark.css b/v0.15.0/assets/themes/documenter-dark.css similarity index 100% rename from previews/PR59/assets/themes/documenter-dark.css rename to v0.15.0/assets/themes/documenter-dark.css diff --git a/previews/PR59/assets/themes/documenter-light.css b/v0.15.0/assets/themes/documenter-light.css similarity index 100% rename from previews/PR59/assets/themes/documenter-light.css rename to v0.15.0/assets/themes/documenter-light.css diff --git a/previews/PR48/assets/themeswap.js b/v0.15.0/assets/themeswap.js similarity index 100% rename from previews/PR48/assets/themeswap.js rename to v0.15.0/assets/themeswap.js diff --git a/previews/PR59/assets/tomate.js b/v0.15.0/assets/tomate.js similarity index 100% rename from previews/PR59/assets/tomate.js rename to v0.15.0/assets/tomate.js diff --git a/previews/PR59/assets/warner.js b/v0.15.0/assets/warner.js similarity index 100% rename from previews/PR59/assets/warner.js rename to v0.15.0/assets/warner.js diff --git a/v0.15.0/benchmarks/index.html b/v0.15.0/benchmarks/index.html new file mode 100644 index 00000000..1cb9fc97 --- /dev/null +++ b/v0.15.0/benchmarks/index.html @@ -0,0 +1,12 @@ + +Benchmarks · PencilFFTs.jl
                  Edit on GitHub

                  Benchmarks

                  The performance of PencilFFTs.jl is comparable to that of other open-source parallel FFT libraries implemented in lower-level languages. Below, we show comparisons with the Fortran implementation of P3DFFT, possibly the most popular of these libraries. The benchmarks were performed on the Jean–Zay cluster of the IDRIS French computing centre (CNRS).

                  The figure below shows strong scaling benchmarks of 3D real-to-complex FFTs using 2D ("pencil") decomposition. The benchmarks were run for input arrays of dimensions $N_x × N_y × N_z = 512^3$, $1024^3$ and $2048^3$. Each timing is averaged over 100 repetitions.

                  + + Strong scaling of PencilFFTs +

                  As seen above, PencilFFTs generally outperforms P3DFFT in its default setting. This is largely explained by the choice of using non-blocking point-to-point MPI communications (via MPI_Isend and MPI_Irecv), while P3DFFT uses collective MPI_Alltoallv calls. This enables PencilFFTs to perform data reordering operations on the partially received data while waiting for the incoming data, leading to better performance. Moreover, in contrast with P3DFFT, the high performance and scalability of PencilFFTs results from a highly generic code, handling decompositions in arbitrary dimensions and a relatively large (and extensible) variety of transformations.

                  Note that PencilFFTs can optionally use collective communications (using MPI_Alltoallv) instead of point-to-point communications. For details, see the docs for PencilFFTPlan and for PencilArray transpositions. As seen above, collective communications generally perform worse than point-to-point ones, and runtimes are nearly indistinguishable from those of P3DFFT.

                  Benchmark details

                  The benchmarks were performed using Julia 1.7-beta3 and Intel MPI 2019. We used PencilFFTs v0.12.5 with FFTW.jl v1.4.3 and MPI.jl v0.19.0. We used the Fortran implementation of P3DFFT, version 2.7.6, which was built with Intel 2019 compilers and linked to FFTW 3.3.8. The cluster where the benchmarks were run has Intel Cascade Lake 6248 processors with 2×20 cores per node.

                  The number of MPI processes along each decomposed dimension, $P_1$ and $P_2$, was automatically determined by a call to MPI_Dims_create, which tends to create a balanced decomposition with $P_1 ≈ P_2$. For instance, a total of 1024 processes is divided into $P_1 = P_2 = 32$. Different results may be obtained with other combinations, but this was not benchmarked.

                  The source files used to generate this benchmark, as well as the raw benchmark results, are all available in the PencilFFTs repo.

                  diff --git a/v0.15.0/generated/ffmpeg.err b/v0.15.0/generated/ffmpeg.err new file mode 100644 index 00000000..4570be9e --- /dev/null +++ b/v0.15.0/generated/ffmpeg.err @@ -0,0 +1,48 @@ +ffmpeg version 4.4.2-0ubuntu0.22.04.1 Copyright (c) 2000-2021 the FFmpeg developers + built with gcc 11 (Ubuntu 11.2.0-19ubuntu1) + configuration: --prefix=/usr --extra-version=0ubuntu0.22.04.1 --toolchain=hardened --libdir=/usr/lib/x86_64-linux-gnu --incdir=/usr/include/x86_64-linux-gnu --arch=amd64 --enable-gpl --disable-stripping --enable-gnutls --enable-ladspa --enable-libaom --enable-libass --enable-libbluray --enable-libbs2b --enable-libcaca --enable-libcdio --enable-libcodec2 --enable-libdav1d --enable-libflite --enable-libfontconfig --enable-libfreetype --enable-libfribidi --enable-libgme --enable-libgsm --enable-libjack --enable-libmp3lame --enable-libmysofa --enable-libopenjpeg --enable-libopenmpt --enable-libopus --enable-libpulse --enable-librabbitmq --enable-librubberband --enable-libshine --enable-libsnappy --enable-libsoxr --enable-libspeex --enable-libsrt --enable-libssh --enable-libtheora --enable-libtwolame --enable-libvidstab --enable-libvorbis --enable-libvpx --enable-libwebp --enable-libx265 --enable-libxml2 --enable-libxvid --enable-libzimg --enable-libzmq --enable-libzvbi --enable-lv2 --enable-omx --enable-openal --enable-opencl --enable-opengl --enable-sdl2 --enable-pocketsphinx --enable-librsvg --enable-libmfx --enable-libdc1394 --enable-libdrm --enable-libiec61883 --enable-chromaprint --enable-frei0r --enable-libx264 --enable-shared + libavutil 56. 70.100 / 56. 70.100 + libavcodec 58.134.100 / 58.134.100 + libavformat 58. 76.100 / 58. 76.100 + libavdevice 58. 13.100 / 58. 13.100 + libavfilter 7.110.100 / 7.110.100 + libswscale 5. 9.100 / 5. 9.100 + libswresample 3. 9.100 / 3. 9.100 + libpostproc 55. 9.100 / 55. 9.100 +Input #0, image2, from '/tmp/jl_y9ILVA/proc1_%04d.png': + Duration: 00:00:04.52, start: 0.000000, bitrate: N/A + Stream #0:0: Video: png, rgb24(pc), 1200x600, 25 fps, 25 tbr, 25 tbn, 25 tbc +Stream mapping: + Stream #0:0 -> #0:0 (png (native) -> h264 (libx264)) +Press [q] to stop, [?] for help +[libx264 @ 0x5611a40f0700] using cpu capabilities: MMX2 SSE2Fast SSSE3 SSE4.2 AVX FMA3 BMI2 AVX2 +[libx264 @ 0x5611a40f0700] profile High, level 3.1, 4:2:0, 8-bit +[libx264 @ 0x5611a40f0700] 264 - core 163 r3060 5db6aa6 - H.264/MPEG-4 AVC codec - Copyleft 2003-2021 - http://www.videolan.org/x264.html - options: cabac=1 ref=3 deblock=1:0:0 analyse=0x3:0x113 me=hex subme=7 psy=1 psy_rd=1.00:0.00 mixed_ref=1 me_range=16 chroma_me=1 trellis=1 8x8dct=1 cqm=0 deadzone=21,11 fast_pskip=1 chroma_qp_offset=-2 threads=3 lookahead_threads=1 sliced_threads=0 nr=0 decimate=1 interlaced=0 bluray_compat=0 constrained_intra=0 bframes=3 b_pyramid=2 b_adapt=1 b_bias=0 direct=1 weightb=1 open_gop=0 weightp=2 keyint=250 keyint_min=25 scenecut=40 intra_refresh=0 rc_lookahead=40 rc=crf mbtree=1 crf=23.0 qcomp=0.60 qpmin=0 qpmax=69 qpstep=4 ip_ratio=1.40 aq=1:1.00 +Output #0, mp4, to 'vorticity_proc1.mp4': + Metadata: + encoder : Lavf58.76.100 + Stream #0:0: Video: h264 (avc1 / 0x31637661), yuv420p(tv, progressive), 1200x600, q=2-31, 25 fps, 12800 tbn + Metadata: + encoder : Lavc58.134.100 libx264 + Side data: + cpb: bitrate max/min/avg: 0/0/0 buffer size: 0 vbv_delay: N/A +frame= 3 fps=0.0 q=0.0 size= 0kB time=00:00:00.00 bitrate=N/A speed= 0x frame= 35 fps=0.0 q=0.0 size= 0kB time=00:00:00.00 bitrate=N/A speed= 0x frame= 55 fps= 48 q=28.0 size= 0kB time=00:00:00.20 bitrate= 1.9kbits/s speed=0.176x frame= 73 fps= 44 q=28.0 size= 0kB time=00:00:00.92 bitrate= 0.4kbits/s speed=0.549x frame= 88 fps= 40 q=28.0 size= 0kB time=00:00:01.52 bitrate= 0.3kbits/s speed=0.688x frame= 103 fps= 36 q=28.0 size= 0kB time=00:00:02.12 bitrate= 0.2kbits/s speed=0.747x frame= 118 fps= 34 q=28.0 size= 0kB time=00:00:02.72 bitrate= 0.1kbits/s speed=0.795x frame= 133 fps= 34 q=28.0 size= 256kB time=00:00:03.32 bitrate= 631.8kbits/s speed=0.837x frame= 150 fps= 33 q=28.0 size= 256kB time=00:00:04.00 bitrate= 524.4kbits/s speed=0.881x frame= 165 fps= 32 q=28.0 size= 256kB time=00:00:04.60 bitrate= 456.0kbits/s speed=0.905x frame= 180 fps= 32 q=28.0 size= 256kB time=00:00:05.20 bitrate= 403.4kbits/s speed=0.928x frame= 195 fps= 32 q=28.0 size= 256kB time=00:00:05.80 bitrate= 361.6kbits/s speed=0.944x frame= 210 fps= 32 q=28.0 size= 256kB time=00:00:06.40 bitrate= 327.7kbits/s speed=0.961x frame= 225 fps= 31 q=28.0 size= 256kB time=00:00:07.00 bitrate= 299.6kbits/s speed=0.971x frame= 243 fps= 31 q=28.0 size= 512kB time=00:00:07.72 bitrate= 543.3kbits/s speed=0.997x frame= 255 fps= 31 q=28.0 size= 512kB time=00:00:08.20 bitrate= 511.5kbits/s speed=0.995x frame= 273 fps= 31 q=28.0 size= 512kB time=00:00:08.92 bitrate= 470.3kbits/s speed=1.02x frame= 283 fps= 28 q=-1.0 Lsize= 820kB time=00:00:11.20 bitrate= 599.7kbits/s speed=1.12x +video:816kB audio:0kB subtitle:0kB other streams:0kB global headers:0kB muxing overhead: 0.507219% +[libx264 @ 0x5611a40f0700] frame I:2 Avg QP:16.95 size: 37480 +[libx264 @ 0x5611a40f0700] frame P:74 Avg QP:25.18 size: 8146 +[libx264 @ 0x5611a40f0700] frame B:207 Avg QP:19.36 size: 758 +[libx264 @ 0x5611a40f0700] consecutive B-frames: 0.7% 4.2% 3.2% 91.9% +[libx264 @ 0x5611a40f0700] mb I I16..4: 30.2% 43.6% 26.1% +[libx264 @ 0x5611a40f0700] mb P I16..4: 0.3% 1.5% 1.0% P16..4: 5.8% 6.6% 4.5% 0.0% 0.0% skip:80.3% +[libx264 @ 0x5611a40f0700] mb B I16..4: 0.1% 0.1% 0.0% B16..8: 6.4% 0.9% 0.3% direct: 0.3% skip:91.9% L0:44.0% L1:47.5% BI: 8.5% +[libx264 @ 0x5611a40f0700] 8x8 transform intra:47.6% inter:37.8% +[libx264 @ 0x5611a40f0700] coded y,uvDC,uvAC intra: 40.6% 43.3% 34.9% inter: 2.7% 2.2% 0.6% +[libx264 @ 0x5611a40f0700] i16 v,h,dc,p: 66% 21% 10% 3% +[libx264 @ 0x5611a40f0700] i8 v,h,dc,ddl,ddr,vr,hd,vl,hu: 28% 12% 29% 5% 5% 4% 6% 4% 6% +[libx264 @ 0x5611a40f0700] i4 v,h,dc,ddl,ddr,vr,hd,vl,hu: 28% 18% 17% 5% 6% 5% 9% 4% 8% +[libx264 @ 0x5611a40f0700] i8c dc,h,v,p: 68% 15% 13% 5% +[libx264 @ 0x5611a40f0700] Weighted P-Frames: Y:0.0% UV:0.0% +[libx264 @ 0x5611a40f0700] ref P L0: 69.7% 15.3% 9.9% 5.1% +[libx264 @ 0x5611a40f0700] ref B L0: 88.2% 9.2% 2.6% +[libx264 @ 0x5611a40f0700] ref B L1: 98.4% 1.6% +[libx264 @ 0x5611a40f0700] kb/s:589.86 diff --git a/previews/PR59/generated/ffmpeg.out b/v0.15.0/generated/ffmpeg.out similarity index 100% rename from previews/PR59/generated/ffmpeg.out rename to v0.15.0/generated/ffmpeg.out diff --git a/previews/PR48/generated/gradient/index.html b/v0.15.0/generated/gradient/index.html similarity index 76% rename from previews/PR48/generated/gradient/index.html rename to v0.15.0/generated/gradient/index.html index 7279af35..9c0ada93 100644 --- a/previews/PR48/generated/gradient/index.html +++ b/v0.15.0/generated/gradient/index.html @@ -1,5 +1,5 @@ -Gradient of a scalar field · PencilFFTs.jl
                  Edit on GitHub

                  Gradient of a scalar field

                  This example shows different methods to compute the gradient of a real-valued 3D scalar field $θ(\bm{x})$ in Fourier space, where $\bm{x} = (x, y, z)$. It is assumed that the field is periodic with period $L = 2π$ along all dimensions.

                  General procedure

                  The discrete Fourier expansion of $θ$ writes

                  \[θ(\bm{x}) = ∑_{\bm{k} ∈ \Z^3} \hat{θ}(\bm{k}) \, e^{i \bm{k} ⋅ \bm{x}},\]

                  where $\bm{k} = (k_x, k_y, k_z)$ are the Fourier wave numbers and $\hat{θ}$ is the discrete Fourier transform of $θ$. Then, the spatial derivatives of $θ$ are given by

                  \[\frac{∂ θ(\bm{x})}{∂ x_i} = +Gradient of a scalar field · PencilFFTs.jl

                  Edit on GitHub

                  Gradient of a scalar field

                  This example shows different methods to compute the gradient of a real-valued 3D scalar field $θ(\bm{x})$ in Fourier space, where $\bm{x} = (x, y, z)$. It is assumed that the field is periodic with period $L = 2π$ along all dimensions.

                  General procedure

                  The discrete Fourier expansion of $θ$ writes

                  \[θ(\bm{x}) = ∑_{\bm{k} ∈ \Z^3} \hat{θ}(\bm{k}) \, e^{i \bm{k} ⋅ \bm{x}},\]

                  where $\bm{k} = (k_x, k_y, k_z)$ are the Fourier wave numbers and $\hat{θ}$ is the discrete Fourier transform of $θ$. Then, the spatial derivatives of $θ$ are given by

                  \[\frac{∂ θ(\bm{x})}{∂ x_i} = ∑_{\bm{k} ∈ \Z^3} i k_i \hat{θ}(\bm{k}) \, e^{i \bm{k} ⋅ \bm{x}},\]

                  where the subscript $i$ denotes one of the spatial components $x$, $y$ or $z$.

                  In other words, to compute $\bm{∇} θ = (∂_x θ, ∂_y θ, ∂_z θ)$, one has to:

                  1. transform $θ$ to Fourier space to obtain $\hat{θ}$,
                  2. multiply $\hat{θ}$ by $i \bm{k}$,
                  3. transform the result back to physical space to obtain $\bm{∇} θ$.

                  Preparation

                  In this section, we initialise a random real-valued scalar field $θ$ and compute its FFT. For more details see the Tutorial.

                  using MPI
 using PencilFFTs
 using Random
@@ -107,4 +107,4 @@
     gradient_grid_broadcast! (lazy)...        151.020 μs  [Method 4]
     gradient_local_parent!...                 92.248 μs
     gradient_local_linear!...                 91.212 μs
-    gradient_local_linear_explicit!...        90.992 μs

                  This page was generated using Literate.jl.

                  + gradient_local_linear_explicit!... 90.992 μs

                  This page was generated using Literate.jl.

                  diff --git a/previews/PR55/generated/in-place/index.html b/v0.15.0/generated/in-place/index.html similarity index 51% rename from previews/PR55/generated/in-place/index.html rename to v0.15.0/generated/in-place/index.html index ce5965b2..516fabb9 100644 --- a/previews/PR55/generated/in-place/index.html +++ b/v0.15.0/generated/in-place/index.html @@ -1,5 +1,5 @@ -In-place transforms · PencilFFTs.jl
                  Edit on GitHub

                  In-place transforms

                  Complex-to-complex and real-to-real transforms can be performed in-place, enabling important memory savings. The procedure is very similar to that of out-of-place transforms described in the tutorial. The differences are illustrated in the sections below.

                  Creating a domain partition

                  We start by partitioning a domain of dimensions $16×32×64$ along all available MPI processes.

                  using PencilFFTs
+In-place transforms · PencilFFTs.jl
                  Edit on GitHub
                  In-place transforms
                  Complex-to-complex and real-to-real transforms can be performed in-place, enabling important memory savings. The procedure is very similar to that of out-of-place transforms described in the tutorial. The differences are illustrated in the sections below.
                  Creating a domain partition
                  We start by partitioning a domain of dimensions $16×32×64$ along all available MPI processes.
                  using PencilFFTs
 using MPI
 MPI.Init()
 
@@ -28,4 +28,4 @@
 u_in = first(A)  # input data view
 randn!(u_in)
 summary(u_in)
                  "16×16×64 PencilArray{ComplexF64, 3}(::Pencil{3, 2, NoPermutation, Array})"
                  Applying plans
                  Like in FFTW.jl, one can perform in-place transforms using the * and \ operators. As mentioned above, in-place plans must be applied on the ManyPencilArray containers returned by allocate_input.
                  plan * A;  # performs in-place forward transform
                  After performing an in-place transform, data contained in u_in has been overwritten and has no "physical" meaning. In other words, u_in should not be used at this point. To access the transformed data, one should retrieve the output data view using last(A).
                  For instance, to compute the global sum of the transformed data:
                  u_out = last(A)  # output data view
-sum(u_out)       # sum of transformed data (note that `sum` reduces over all processes)
                  13604.928869874026 + 25414.584562229713im
                  Finally, we can perform a backward transform and do stuff with the input view:
                  plan \ A;  # perform in-place backward transform
                  At this point, the data can be once again found in the input view u_in, while u_out should not be accessed.
                  This page was generated using Literate.jl.
                  
+sum(u_out)       # sum of transformed data (note that `sum` reduces over all processes)
                  8496.556048339193 + 28723.810896390776im

                  Finally, we can perform a backward transform and do stuff with the input view:

                  plan \ A;  # perform in-place backward transform

                  At this point, the data can be once again found in the input view u_in, while u_out should not be accessed.

                  This page was generated using Literate.jl.

                  diff --git a/v0.15.0/generated/navier_stokes/index.html b/v0.15.0/generated/navier_stokes/index.html new file mode 100644 index 00000000..996d11c3 --- /dev/null +++ b/v0.15.0/generated/navier_stokes/index.html @@ -0,0 +1,303 @@ + +Navier–Stokes equations · PencilFFTs.jl
                  Edit on GitHub

                  Navier–Stokes equations

                  In this example, we numerically solve the incompressible Navier–Stokes equations

                  \[∂_t \bm{v} + (\bm{v} ⋅ \bm{∇}) \bm{v} = -\frac{1}{ρ} \bm{∇} p + ν ∇^2 \bm{v}, +\quad \bm{∇} ⋅ \bm{v} = 0,\]

                  where $\bm{v}(\bm{x}, t)$ and $p(\bm{x}, t)$ are respectively the velocity and pressure fields, $ν$ is the fluid kinematic viscosity and $ρ$ is the fluid density.

                  We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.

                  First steps

                  We start by loading the required packages, initialising MPI and setting the simulation parameters.

                  using MPI
+using PencilFFTs
+
+MPI.Init()
+comm = MPI.COMM_WORLD
+procid = MPI.Comm_rank(comm) + 1
+
+# Simulation parameters
+Ns = (64, 64, 64)  # = (Nx, Ny, Nz)
+Ls = (2π, 2π, 2π)  # = (Lx, Ly, Lz)
+
+# Collocation points ("global" = over all processes).
+# We include the endpoint (length = N + 1) for convenience.
+xs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls)  # = (x, y, z)
                  (0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586)

                  Let's check the number of MPI processes over which we're running our simulation:

                  MPI.Comm_size(comm)
                  2

                  We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:

                  pen = Pencil(Ns, comm)
                  Decomposition of 3D data
+    Data dimensions: (64, 64, 64)
+    Decomposed dimensions: (2, 3)
+    Data permutation: NoPermutation()
+    Array type: Array

                  The subdomain associated to the local MPI process can be obtained using range_local:

                  range_local(pen)
                  (1:64, 1:32, 1:64)

                  We now construct a distributed vector field that follows the decomposition configuration we just created:

                  v⃗₀ = (
+    PencilArray{Float64}(undef, pen),  # vx
+    PencilArray{Float64}(undef, pen),  # vy
+    PencilArray{Float64}(undef, pen),  # vz
+)
+summary(v⃗₀[1])
                  "64×32×64 PencilArray{Float64, 3}(::Pencil{3, 2, NoPermutation, Array})"

                  We still need to fill this array with interesting values that represent a physical velocity field.

                  Initial condition

                  Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:

                  \[\begin{aligned} +v_x(x, y, z) &= u₀ \sin(k₀ x) \cos(k₀ y) \cos(k₀ z) \\ +v_y(x, y, z) &= -u₀ \cos(k₀ x) \sin(k₀ y) \cos(k₀ z) \\ +v_z(x, y, z) &= 0 +\end{aligned}\]

                  where $u₀$ and $k₀$ are two parameters setting the amplitude and the period of the velocity field.

                  To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:

                  grid = localgrid(pen, xs_global)
                  LocalRectilinearGrid{3} with coordinates:
+ (1) 0.0:0.09817477042468103:6.1850105367549055
+ (2) 0.0:0.09817477042468103:3.043417883165112
+ (3) 0.0:0.09817477042468103:6.1850105367549055

                  We can use this to initialise the velocity field:

                  u₀ = 1.0
+k₀ = 2π / Ls[1]  # should be integer if L = 2π (to preserve periodicity)
+
+@. v⃗₀[1] =  u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)
+@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)
+@. v⃗₀[3] =  0

                  Let's plot a 2D slice of the velocity field managed by the local MPI process:

                  using GLMakie
+
+# Compute the norm of a vector field represented by a tuple of arrays.
+function vecnorm(v⃗::NTuple)
+    vnorm = similar(v⃗[1])
+    for n ∈ eachindex(v⃗[1])
+        w = zero(eltype(vnorm))
+        for v ∈ v⃗
+            w += v[n]^2
+        end
+        vnorm[n] = sqrt(w)
+    end
+    vnorm
+end
+
+let fig = Figure(resolution = (700, 600))
+    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
+    vnorm = vecnorm(v⃗₀)
+    ct = contour!(
+        ax, grid.x, grid.y, grid.z, vnorm;
+        alpha = 0.2, levels = 4,
+        colormap = :viridis, colorrange = (0.0, 1.0),
+    )
+    cb = Colorbar(fig[1, 2], ct; label = "Velocity magnitude")
+    fig
+end

                  Velocity in Fourier space

                  In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series

                  \[\bm{v}(\bm{x}, t) = +∑_{\bm{k}} \hat{\bm{v}}_{\bm{k}}(t) \, e^{i \bm{k} ⋅ \bm{x}},\]

                  where $\bm{k} = (k_x, k_y, k_z)$ are the discrete wave numbers.

                  The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for $k_x$:

                  using AbstractFFTs: fftfreq, rfftfreq
+
+ks_global = (
+    rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]),  # kx | real-to-complex
+     fftfreq(Ns[2], 2π * Ns[2] / Ls[2]),  # ky | complex-to-complex
+     fftfreq(Ns[3], 2π * Ns[3] / Ls[3]),  # kz | complex-to-complex
+)
+
+ks_global[1]'
                  1×33 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
+ 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  27.0  28.0  29.0  30.0  31.0  32.0
                  ks_global[2]'
                  1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
+ 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0
                  ks_global[3]'
                  1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
+ 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0

                  To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:

                  plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())
                  Transforms: (RFFT, FFT, FFT)
+Input type: Float64
+Global dimensions: (64, 64, 64) -> (33, 64, 64)
+MPI topology: 2D decomposition (2×1 processes)

                  See PencilFFTPlan for details on creating plans and on optional keyword arguments.

                  We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$:

                  v̂s = plan .* v⃗₀
+summary(v̂s[1])
                  "16×64×64 PencilArray{ComplexF64, 3}(::Pencil{3, 2, Permutation{(3, 2, 1), 3}, Array})"

                  Note that, in Fourier space, the domain decomposition is performed along the directions $x$ and $y$:

                  pencil(v̂s[1])
                  Decomposition of 3D data
+    Data dimensions: (33, 64, 64)
+    Decomposed dimensions: (1, 2)
+    Data permutation: Permutation(3, 2, 1)
+    Array type: Array

                  This is because the 3D FFTs are performed one dimension at a time, with the $x$ direction first and the $z$ direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.

                  To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers $\bm{k}$ associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:

                  grid_fourier = localgrid(v̂s[1], ks_global)
                  LocalRectilinearGrid{3} with Permutation(3, 2, 1) and coordinates:
+ (1) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0]
+ (2) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]
+ (3) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]

                  As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, $\bm{ω} = \bm{∇} × \bm{v}$. In Fourier space, this becomes $\hat{\bm{ω}} = i \bm{k} × \hat{\bm{v}}$.

                  using StaticArrays: SVector
+using LinearAlgebra: ×
+
+function curl_fourier!(
+        ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,
+    ) where {N}
+    @inbounds for I ∈ eachindex(grid_fourier)
+        # We use StaticArrays for the cross product between small vectors.
+        ik⃗ = im * SVector(grid_fourier[I])
+        v⃗ = SVector(getindex.(v̂s, Ref(I)))  # = (v̂s[1][I], v̂s[2][I], ...)
+        ω⃗ = ik⃗ × v⃗
+        for n ∈ eachindex(ω⃗)
+            ω̂s[n][I] = ω⃗[n]
+        end
+    end
+    ω̂s
+end
+
+ω̂s = similar.(v̂s)
+curl_fourier!(ω̂s, v̂s, grid_fourier);

                  We finally transform back to physical space and plot the result:

                  ωs = plan .\ ω̂s
+
+let fig = Figure(resolution = (700, 600))
+    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
+    ω_norm = vecnorm(ωs)
+    ct = contour!(
+        ax, grid.x, grid.y, grid.z, ω_norm;
+        alpha = 0.1, levels = 0.8:0.2:2.0,
+        colormap = :viridis, colorrange = (0.8, 2.0),
+    )
+    cb = Colorbar(fig[1, 2], ct; label = "Vorticity magnitude")
+    fig
+end

                  Computing the non-linear term

                  One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as

                  \[∂_t \hat{\bm{v}}_{\bm{k}} = +- \mathcal{P}_{\bm{k}} \! \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right] +- ν |\bm{k}|^2 \hat{\bm{v}}_{\bm{k}} +\quad \text{ with } \quad +\mathcal{P}_{\bm{k}}(\hat{\bm{F}}_{\bm{k}}) = \left( I - \frac{\bm{k} ⊗ +\bm{k}}{|\bm{k}|^2} \right) \hat{\bm{F}}_{\bm{k}},\]

                  where $\mathcal{P}_{\bm{k}}$ is a projection operator allowing to preserve the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.

                  Now that we have the wave numbers $\bm{k}$, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$ of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, $\hat{\bm{F}}_{\bm{k}} = \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right]_{\bm{k}}$. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.

                  Below we implement a function that computes the non-linear term in Fourier space based on its convective form $(\bm{v} ⋅ \bm{∇}) \bm{v} = \bm{∇} ⋅ (\bm{v} ⊗ \bm{v})$. Note that this equivalence uses the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$.

                  using LinearAlgebra: mul!, ldiv!  # for applying FFT plans in-place
+
+# Compute non-linear term in Fourier space from velocity field in physical
+# space. Optional keyword arguments may be passed to avoid allocations.
+function ns_nonlinear!(
+        F̂s, vs, plan, grid_fourier;
+        vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),
+    )
+    # Compute F_i = ∂_j (v_i v_j) for each i.
+    # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)
+    w, ŵ = vbuf, v̂buf
+    @inbounds for (i, F̂i) ∈ enumerate(F̂s)
+        F̂i .= 0
+        vi = vs[i]
+        for (j, vj) ∈ enumerate(vs)
+            w .= vi .* vj     # w = v_i * v_j in physical space
+            mul!(ŵ, plan, w)  # same in Fourier space
+            # Add derivative in Fourier space
+            for I ∈ eachindex(grid_fourier)
+                k⃗ = grid_fourier[I]  # = (kx, ky, kz)
+                kj = k⃗[j]
+                F̂i[I] += im * kj * ŵ[I]
+            end
+        end
+    end
+    F̂s
+end
                  ns_nonlinear! (generic function with 1 method)

                  As an example, let's use this function on our initial velocity field:

                  F̂s = similar.(v̂s)
+ns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);

                  Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.

                  function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)
+    ks_max = maximum.(abs, ks_global)  # maximum stored wave numbers (kx_max, ky_max, kz_max)
+    ks_lim = (2 / 3) .* ks_max
+    @inbounds for I ∈ eachindex(grid_fourier)
+        k⃗ = grid_fourier[I]
+        if any(abs.(k⃗) .> ks_lim)
+            for ŵ ∈ ŵs
+                ŵ[I] = 0
+            end
+        end
+    end
+    ŵs
+end
+
+# We can apply this on the previously computed non-linear term:
+dealias_twothirds!(F̂s, grid_fourier, ks_global);

                  Finally, we implement the projection associated to the incompressibility condition:

                  function project_divergence_free!(ûs, grid_fourier)
+    @inbounds for I ∈ eachindex(grid_fourier)
+        k⃗ = grid_fourier[I]
+        k² = sum(abs2, k⃗)
+        iszero(k²) && continue  # avoid division by zero
+        û = getindex.(ûs, Ref(I))  # (ûs[1][I], ûs[2][I], ...)
+        for i ∈ eachindex(û)
+            ŵ = û[i]
+            for j ∈ eachindex(û)
+                ŵ -= k⃗[i] * k⃗[j] * û[j] / k²
+            end
+            ûs[i][I] = ŵ
+        end
+    end
+    ûs
+end
                  project_divergence_free! (generic function with 1 method)

                  We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:

                  v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)
+v̂s_proj .≈ v̂s  # the last one may be false because v_z = 0 initially
                  (true, true, false)

                  Putting it all together

                  To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:

                  function ns_rhs!(
+        dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,
+    ) where {N}
+    # 1. Compute non-linear term and dealias it
+    (; plan, cache, ks_global, grid_fourier) = p
+    F̂s = cache.F̂s
+    ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])
+    dealias_twothirds!(F̂s, grid_fourier, ks_global)
+
+    # 2. Project onto divergence-free space
+    project_divergence_free!(F̂s, grid_fourier)
+
+    # 3. Transform velocity to Fourier space
+    v̂s = cache.v̂s
+    map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)
+
+    # 4. Add viscous term (and multiply projected non-linear term by -1)
+    ν = p.ν
+    for n ∈ eachindex(v̂s)
+        v̂ = v̂s[n]
+        F̂ = F̂s[n]
+        @inbounds for I ∈ eachindex(grid_fourier)
+            k⃗ = grid_fourier[I]  # = (kx, ky, kz)
+            k² = sum(abs2, k⃗)
+            F̂[I] = -F̂[I] - ν * k² * v̂[I]
+        end
+    end
+
+    # 5. Transform RHS back to physical space
+    map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)
+
+    nothing
+end
                  ns_rhs! (generic function with 1 method)

                  For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.

                  using OrdinaryDiffEq
+using RecursiveArrayTools: ArrayPartition
+
+ns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)
+vs_init_ode = ArrayPartition(v⃗₀)
+summary(vs_init_ode)
                  "RecursiveArrayTools.ArrayPartition{Float64, Tuple{PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}}} with arrays:"

                  We now define solver parameters and temporary variables, and initialise the problem:

                  params = (;
+    ν = 5e-3,  # kinematic viscosity
+    plan, grid_fourier, ks_global,
+    cache = (
+        v̂s = similar.(v̂s),
+        F̂s = similar.(v̂s),
+    )
+)
+
+tspan = (0.0, 10.0)
+prob = ODEProblem(ns_rhs!, vs_init_ode, tspan, params)
+integrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);

                  We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum $E(k)$, to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity $ν$ must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large $k$), while high enough to allow the small-scale motions to be correctly resolved.

                  function energy_spectrum!(Ek, ks, v̂s, grid_fourier)
+    Nk = length(Ek)
+    @assert Nk == length(ks)
+    Ek .= 0
+    for I ∈ eachindex(grid_fourier)
+        k⃗ = grid_fourier[I]  # = (kx, ky, kz)
+        knorm = sqrt(sum(abs2, k⃗))
+        i = searchsortedfirst(ks, knorm)
+        i > Nk && continue
+        v⃗ = getindex.(v̂s, Ref(I))  # = (v̂s[1][I], v̂s[2][I], ...)
+        factor = k⃗[1] == 0 ? 1 : 2  # account for Hermitian symmetry and r2c transform
+        Ek[i] += factor * sum(abs2, v⃗) / 2
+    end
+    MPI.Allreduce!(Ek, +, get_comm(v̂s[1]))  # sum across all processes
+    Ek
+end
+
+ks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])
+Ek = similar(ks)
+v̂s = plan .* integrator.u.x
+energy_spectrum!(Ek, ks, v̂s, grid_fourier)
+Ek ./= scale_factor(plan)^2  # rescale energy
+
+curl_fourier!(ω̂s, v̂s, grid_fourier)
+ldiv!.(ωs, plan, ω̂s)
+ω⃗_plot = Observable(ωs)
+k_plot = @view ks[2:end]
+E_plot = Observable(@view Ek[2:end])
+t_plot = Observable(integrator.t)
+
+fig = let
+    fig = Figure(resolution = (1200, 600))
+    ax = Axis3(
+        fig[1, 1][1, 1]; title = @lift("t = $(round($t_plot, digits = 3))"),
+        aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z",
+    )
+    ω_mag = @lift vecnorm($ω⃗_plot)
+    ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)
+    ct = contour!(
+        ax, grid.x, grid.y, grid.z, ω_mag_norm;
+        alpha = 0.3, levels = 3,
+        colormap = :viridis, colorrange = (0.0, 1.0),
+    )
+    cb = Colorbar(fig[1, 1][1, 2], ct; label = "Normalised vorticity magnitude")
+    ax_sp = Axis(
+        fig[1, 2];
+        xlabel = "k", ylabel = "E(k)", xscale = log2, yscale = log10,
+        title = "Kinetic energy spectrum",
+    )
+    ylims!(ax_sp, 1e-8, 1e0)
+    scatterlines!(ax_sp, k_plot, E_plot)
+    ks_slope = exp.(range(log(2.5), log(25.0), length = 3))
+    E_fivethirds = @. 0.3 * ks_slope^(-5/3)
+    @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)
+    text!(ax_sp, L"k^{-5/3}"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))
+    fig
+end
+
+record(fig, "vorticity_proc$procid.mp4"; framerate = 10) do io
+    while integrator.t < 20
+        dt = 0.001
+        step!(integrator, dt)
+        t_plot[] = integrator.t
+        mul!.(v̂s, plan, integrator.u.x)  # current velocity in Fourier space
+        curl_fourier!(ω̂s, v̂s, grid_fourier)
+        ldiv!.(ω⃗_plot[], plan, ω̂s)
+        ω⃗_plot[] = ω⃗_plot[]  # to force updating the plot
+        energy_spectrum!(Ek, ks, v̂s, grid_fourier)
+        Ek ./= scale_factor(plan)^2  # rescale energy
+        E_plot[] = E_plot[]
+        recordframe!(io)
+    end
+end;
                  + +

                  This page was generated using Literate.jl.

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b/v0.15.0/search/index.html similarity index 94% rename from previews/PR62/search/index.html rename to v0.15.0/search/index.html index 731fc44c..d2497780 100644 --- a/previews/PR62/search/index.html +++ b/v0.15.0/search/index.html @@ -1,2 +1,2 @@ -Search · PencilFFTs.jl

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                      diff --git a/v0.15.0/search_index.js b/v0.15.0/search_index.js new file mode 100644 index 00000000..85838151 --- /dev/null +++ b/v0.15.0/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"tutorial/#Tutorial","page":"Tutorial","title":"Tutorial","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The following tutorial shows how to perform a 3D FFT of real periodic data defined on a grid of N_x N_y N_z points.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"
                      \n \n \n
                      ","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"By default, the domain is distributed on a 2D MPI topology of dimensions N_1 N_2. As an example, the above figure shows such a topology with N_1 = 4 and N_2 = 3, for a total of 12 MPI processes.","category":"page"},{"location":"tutorial/#tutorial:creating_plans","page":"Tutorial","title":"Creating plans","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The first thing to do is to create a domain decomposition configuration for the given dataset dimensions N_x N_y N_z. In the framework of PencilArrays, such a configuration is described by a Pencil object. As described in the PencilArrays docs, we can let the Pencil constructor automatically determine such a configuration. For this, only an MPI communicator and the dataset dimensions are needed:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using MPI\nusing PencilFFTs\n\nMPI.Init()\ncomm = MPI.COMM_WORLD\n\n# Input data dimensions (Nx × Ny × Nz)\ndims = (16, 32, 64)\npen = Pencil(dims, comm)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"By default this creates a 2D decomposition (for the case of a 3D dataset), but one can change this as detailed in the PencilArrays documentation linked above.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"We can now create a PencilFFTPlan, which requires information on decomposition configuration (the Pencil object) and on the transforms that will be applied:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# Apply a 3D real-to-complex (r2c) FFT.\ntransform = Transforms.RFFT()\n\n# Note that, for more control, one can instead separately specify the transforms along each dimension:\n# transform = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())\n\n# Create plan\nplan = PencilFFTPlan(pen, transform)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"See the PencilFFTPlan constructor for details on the accepted options, and the Transforms module for the possible transforms. It is also possible to enable fine-grained performance measurements via the TimerOutputs package, as described in Measuring performance.","category":"page"},{"location":"tutorial/#Allocating-data","page":"Tutorial","title":"Allocating data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Next, we want to apply the plan on some data. Transforms may only be applied on PencilArrays, which are array wrappers that include MPI decomposition information (in some sense, analogous to DistributedArrays in Julia's distributed computing approach). The helper function allocate_input can be used to allocate a PencilArray that is compatible with our plan:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# In our example, this returns a 3D PencilArray of real data (Float64).\nu = allocate_input(plan)\n\n# Fill the array with some (random) data\nusing Random\nrandn!(u)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"PencilArrays are a subtype of AbstractArray, and thus they support all common array operations.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Similarly, to preallocate output data, one can use allocate_output:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# In our example, this returns a 3D PencilArray of complex data (Complex{Float64}).\nv = allocate_output(plan)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"This is only required if one wants to apply the plans using a preallocated output (with mul!, see right below).","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The data types returned by allocate_input and allocate_output are slightly different when working with in-place transforms. See the in-place example for details.","category":"page"},{"location":"tutorial/#Applying-plans","page":"Tutorial","title":"Applying plans","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The interface to apply plans is consistent with that of AbstractFFTs. Namely, * and mul! are respectively used for forward transforms without and with preallocated output data. Similarly, \\ and ldiv! are used for backward transforms.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using LinearAlgebra # for mul!, ldiv!\n\n# Apply plan on `u` with `v` as an output\nmul!(v, plan, u)\n\n# Apply backward plan on `v` with `w` as an output\nw = similar(u)\nldiv!(w, plan, v) # now w ≈ u","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Note that, consistently with AbstractFFTs, normalisation is performed at the end of a backward transform, so that the original data is recovered when applying a forward followed by a backward transform.","category":"page"},{"location":"tutorial/#Accessing-and-modifying-data","page":"Tutorial","title":"Accessing and modifying data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For any given MPI process, a PencilArray holds the data associated to its local partition in the global geometry. PencilArrays are accessed using local indices that start at 1, regardless of the location of the local process in the MPI topology. Note that PencilArrays, being based on regular Arrays, support both linear and Cartesian indexing (see the Julia docs for details).","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For convenience, the global_view function can be used to generate an OffsetArray wrapper that takes global indices.","category":"page"},{"location":"tutorial/#tutorial:output_data_layout","page":"Tutorial","title":"Output data layout","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"In memory, the dimensions of the transform output are by default reversed with respect to the input. That is, if the order of indices in the input data is (x, y, z), then the output has order (z, y, x) in memory. This detail is hidden from the user, and output arrays are always accessed in the same order as the input data, regardless of the underlying output dimension permutation. This applies to PencilArrays and to OffsetArrays returned by global_view.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The reasoning behind dimension permutations, is that they allow to always perform FFTs along the fastest array dimension and to avoid a local data transposition, resulting in performance gains. A similar approach is followed by other parallel FFT libraries. FFTW itself, in its distributed-memory routines, includes a flag that enables a similar behaviour. In PencilFFTs, index permutation is the default, but it can be disabled via the permute_dims flag of PencilFFTPlan.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"A great deal of work has been spent in making generic index permutations as efficient as possible, both in intermediate and in the output state of the multidimensional transforms. This has been achieved, in part, by making sure that permutations such as (3, 2, 1) are compile-time constants.","category":"page"},{"location":"tutorial/#Further-reading","page":"Tutorial","title":"Further reading","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For details on working with PencilArrays see the PencilArrays docs.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The examples on the sidebar further illustrate the use of transforms and provide an introduction to working with MPI-distributed data in the form of PencilArrays. In particular, the gradient example illustrates different ways of computing things using Fourier-transformed distributed arrays. Then, the incompressible Navier–Stokes example is a more advanced and complete example of a possible application of the PencilFFTs package.","category":"page"},{"location":"benchmarks/#Benchmarks","page":"Benchmarks","title":"Benchmarks","text":"","category":"section"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The performance of PencilFFTs.jl is comparable to that of other open-source parallel FFT libraries implemented in lower-level languages. Below, we show comparisons with the Fortran implementation of P3DFFT, possibly the most popular of these libraries. The benchmarks were performed on the Jean–Zay cluster of the IDRIS French computing centre (CNRS).","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The figure below shows strong scaling benchmarks of 3D real-to-complex FFTs using 2D (\"pencil\") decomposition. The benchmarks were run for input arrays of dimensions N_x N_y N_z = 512^3, 1024^3 and 2048^3. Each timing is averaged over 100 repetitions.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"
                      \n \n \n
                      ","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"As seen above, PencilFFTs generally outperforms P3DFFT in its default setting. This is largely explained by the choice of using non-blocking point-to-point MPI communications (via MPI_Isend and MPI_Irecv), while P3DFFT uses collective MPI_Alltoallv calls. This enables PencilFFTs to perform data reordering operations on the partially received data while waiting for the incoming data, leading to better performance. Moreover, in contrast with P3DFFT, the high performance and scalability of PencilFFTs results from a highly generic code, handling decompositions in arbitrary dimensions and a relatively large (and extensible) variety of transformations.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"Note that PencilFFTs can optionally use collective communications (using MPI_Alltoallv) instead of point-to-point communications. For details, see the docs for PencilFFTPlan and for PencilArray transpositions. As seen above, collective communications generally perform worse than point-to-point ones, and runtimes are nearly indistinguishable from those of P3DFFT.","category":"page"},{"location":"benchmarks/#Benchmark-details","page":"Benchmarks","title":"Benchmark details","text":"","category":"section"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The benchmarks were performed using Julia 1.7-beta3 and Intel MPI 2019. We used PencilFFTs v0.12.5 with FFTW.jl v1.4.3 and MPI.jl v0.19.0. We used the Fortran implementation of P3DFFT, version 2.7.6, which was built with Intel 2019 compilers and linked to FFTW 3.3.8. The cluster where the benchmarks were run has Intel Cascade Lake 6248 processors with 2×20 cores per node.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The number of MPI processes along each decomposed dimension, P_1 and P_2, was automatically determined by a call to MPI_Dims_create, which tends to create a balanced decomposition with P_1 P_2. For instance, a total of 1024 processes is divided into P_1 = P_2 = 32. Different results may be obtained with other combinations, but this was not benchmarked.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The source files used to generate this benchmark, as well as the raw benchmark results, are all available in the PencilFFTs repo.","category":"page"},{"location":"GlobalFFTParams/#Global-FFT-parameters","page":"Global FFT parameters","title":"Global FFT parameters","text":"","category":"section"},{"location":"GlobalFFTParams/","page":"Global FFT parameters","title":"Global FFT parameters","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"GlobalFFTParams/","page":"Global FFT parameters","title":"Global FFT parameters","text":"GlobalFFTParams","category":"page"},{"location":"GlobalFFTParams/#PencilFFTs.GlobalFFTParams","page":"Global FFT parameters","title":"PencilFFTs.GlobalFFTParams","text":"GlobalFFTParams{T, N, inplace}\n\nSpecifies the global parameters for an N-dimensional distributed transform. These include the element type T and global data sizes of input and output data, as well as the transform types to be performed along each dimension.\n\n\n\nGlobalFFTParams(size_global, transforms, [real_type=Float64])\n\nDefine parameters for N-dimensional transform.\n\ntransforms must be a tuple of length N specifying the transforms to be applied along each dimension. Each element must be a subtype of Transforms.AbstractTransform. For all the possible transforms, see Transform types.\n\nThe element type must be a real type accepted by FFTW, i.e. either Float32 or Float64.\n\nNote that the transforms are applied one dimension at a time, with the leftmost dimension first for forward transforms.\n\nExample\n\nTo perform a 3D FFT of real data, first a real-to-complex FFT must be applied along the first dimension, followed by two complex-to-complex FFTs along the other dimensions:\n\njulia> size_global = (64, 32, 128); # size of real input data\n\njulia> transforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT());\n\njulia> fft_params = PencilFFTs.GlobalFFTParams(size_global, transforms)\nTransforms: (RFFT, FFT, FFT)\nInput type: Float64\nGlobal dimensions: (64, 32, 128) -> (33, 32, 128)\n\n\n\n\n\n","category":"type"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"EditURL = \"https://github.com/jipolanco/PencilFFTs.jl/blob/master/docs/examples/in-place.jl\"","category":"page"},{"location":"generated/in-place/#In-place-transforms","page":"In-place transforms","title":"In-place transforms","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Complex-to-complex and real-to-real transforms can be performed in-place, enabling important memory savings. The procedure is very similar to that of out-of-place transforms described in the tutorial. The differences are illustrated in the sections below.","category":"page"},{"location":"generated/in-place/#Creating-a-domain-partition","page":"In-place transforms","title":"Creating a domain partition","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"We start by partitioning a domain of dimensions 163264 along all available MPI processes.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"using PencilFFTs\nusing MPI\nMPI.Init()\n\ndims_global = (16, 32, 64) # global dimensions","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Such a partitioning is described by a Pencil object. Here we choose to decompose the domain along the last two dimensions. In this case, the actual number of processes along each of these dimensions is chosen automatically.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"decomp_dims = (2, 3)\ncomm = MPI.COMM_WORLD\npen = Pencil(dims_global, decomp_dims, comm)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"warning: Allowed decompositions\nDistributed transforms using PencilFFTs.jl require that the first dimension is not decomposed. In other words, if one wants to perform transforms, then decomp_dims above must not contain 1.","category":"page"},{"location":"generated/in-place/#Creating-in-place-plans","page":"In-place transforms","title":"Creating in-place plans","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"To create an in-place plan, pass an in-place transform such as Transforms.FFT! or Transforms.R2R! to PencilFFTPlan. For instance:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"# Perform a 3D in-place complex-to-complex FFT.\ntransform = Transforms.FFT!()\n\n# Note that one can also combine different types of in-place transforms.\n# For instance:\n# transform = (\n# Transforms.R2R!(FFTW.REDFT01),\n# Transforms.FFT!(),\n# Transforms.R2R!(FFTW.DHT),\n# )","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"We can now create a distributed plan from the previously-created domain partition and the chosen transform.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan = PencilFFTPlan(pen, transform)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Note that in-place real-to-complex transforms are not currently supported. (In other words, the RFFT! transform type is not defined.)","category":"page"},{"location":"generated/in-place/#Allocating-data","page":"In-place transforms","title":"Allocating data","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"As with out-of-place plans, data should be allocated using allocate_input. The difference is that, for in-place plans, this function returns a ManyPencilArray object, which is a container holding multiple PencilArray views sharing the same memory space.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"# Allocate data for the plan.\n# Since `plan` is in-place, this returns a `ManyPencilArray` container.\nA = allocate_input(plan)\nsummary(A)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Note that allocate_output also works for in-place plans. In this case, it returns exactly the same thing as allocate_input.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"As shown in the next section, in-place plans must be applied on the returned ManyPencilArray. On the other hand, one usually wants to access and modify data, and for this one needs the PencilArray views contained in the ManyPencilArray. The input and output array views can be obtained by calling first(::ManyPencilArray) and last(::ManyPencilArray).","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"For instance, we can initialise the input array with some data before transforming:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"using Random\nu_in = first(A) # input data view\nrandn!(u_in)\nsummary(u_in)","category":"page"},{"location":"generated/in-place/#Applying-plans","page":"In-place transforms","title":"Applying plans","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Like in FFTW.jl, one can perform in-place transforms using the * and \\ operators. As mentioned above, in-place plans must be applied on the ManyPencilArray containers returned by allocate_input.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan * A; # performs in-place forward transform\nnothing #hide","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"After performing an in-place transform, data contained in u_in has been overwritten and has no \"physical\" meaning. In other words, u_in should not be used at this point. To access the transformed data, one should retrieve the output data view using last(A).","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"For instance, to compute the global sum of the transformed data:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"u_out = last(A) # output data view\nsum(u_out) # sum of transformed data (note that `sum` reduces over all processes)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Finally, we can perform a backward transform and do stuff with the input view:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan \\ A; # perform in-place backward transform\nnothing #hide","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"At this point, the data can be once again found in the input view u_in, while u_out should not be accessed.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"This page was generated using Literate.jl.","category":"page"},{"location":"Transforms/#Available-transforms","page":"Available transforms","title":"Available transforms","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"CurrentModule = PencilFFTs.Transforms","category":"page"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"Transforms","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms","page":"Available transforms","title":"PencilFFTs.Transforms","text":"Defines different one-dimensional FFT-based transforms.\n\nThe transforms are all subtypes of an AbstractTransform type.\n\nWhen possible, the names of the transforms are kept consistent with the functions exported by AbstractFFTs.jl and FFTW.jl.\n\n\n\n\n\n","category":"module"},{"location":"Transforms/#Transform-types","page":"Available transforms","title":"Transform types","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"FFT\nFFT!\nBFFT\nBFFT!\n\nRFFT\nRFFT!\nBRFFT\nBRFFT!\n\nR2R\nR2R!\n\nNoTransform\nNoTransform!","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.FFT","page":"Available transforms","title":"PencilFFTs.Transforms.FFT","text":"FFT()\n\nComplex-to-complex FFT.\n\nSee also AbstractFFTs.fft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.FFT!","page":"Available transforms","title":"PencilFFTs.Transforms.FFT!","text":"FFT!()\n\nIn-place version of FFT.\n\nSee also AbstractFFTs.fft!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BFFT","page":"Available transforms","title":"PencilFFTs.Transforms.BFFT","text":"BFFT()\n\nUnnormalised backward complex-to-complex FFT.\n\nLike AbstractFFTs.bfft, this transform is not normalised. To obtain the inverse transform, divide the output by the length of the transformed dimension.\n\nSee also AbstractFFTs.bfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.BFFT!","text":"BFFT()\n\nIn-place version of BFFT.\n\nSee also AbstractFFTs.bfft!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.RFFT","page":"Available transforms","title":"PencilFFTs.Transforms.RFFT","text":"RFFT()\n\nReal-to-complex FFT.\n\nSee also AbstractFFTs.rfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.RFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.RFFT!","text":"RFFT!()\n\nIn-place version of RFFT.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BRFFT","page":"Available transforms","title":"PencilFFTs.Transforms.BRFFT","text":"BRFFT(d::Integer)\nBRFFT((d1, d2, ..., dN))\n\nUnnormalised inverse of RFFT.\n\nTo obtain the inverse transform, divide the output by the length of the transformed dimension (of the real output array).\n\nAs described in the AbstractFFTs docs, the length of the output cannot be fully inferred from the input length. For this reason, the BRFFT constructor accepts an optional d argument indicating the output length.\n\nFor multidimensional datasets, a tuple of dimensions (d1, d2, ..., dN) may also be passed. This is equivalent to passing just dN. In this case, the last dimension (dN) is the one that changes size between the input and output. Note that this is the opposite of FFTW.brfft. The reason is that, in PencilFFTs, the last dimension is the one along which a complex-to-real transform is performed.\n\nSee also AbstractFFTs.brfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BRFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.BRFFT!","text":"BRFFT!(d::Integer)\nBRFFT!((d1, d2, ..., dN))\n\nIn-place version of BRFFT.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.R2R","page":"Available transforms","title":"PencilFFTs.Transforms.R2R","text":"R2R(kind)\n\nReal-to-real transform of type kind.\n\nThe possible values of kind are those described in the FFTW.r2r docs and the FFTW manual:\n\ndiscrete cosine transforms: FFTW.REDFT00, FFTW.REDFT01, FFTW.REDFFT10, FFTW.REDFFT11\ndiscrete sine transforms: FFTW.RODFT00, FFTW.RODFT01, FFTW.RODFFT10, FFTW.RODFFT11\ndiscrete Hartley transform: FFTW.DHT\n\nNote: half-complex format DFTs (FFTW.R2HC, FFTW.HC2R) are not currently supported.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.R2R!","page":"Available transforms","title":"PencilFFTs.Transforms.R2R!","text":"R2R!(kind)\n\nIn-place version of R2R.\n\nSee also FFTW.r2r!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.NoTransform","page":"Available transforms","title":"PencilFFTs.Transforms.NoTransform","text":"NoTransform()\n\nIdentity transform.\n\nSpecifies that no transformation should be applied.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.NoTransform!","page":"Available transforms","title":"PencilFFTs.Transforms.NoTransform!","text":"NoTransform!()\n\nIn-place version of NoTransform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#Internals","page":"Available transforms","title":"Internals","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"What follows is used internally in PencilFFTs.","category":"page"},{"location":"Transforms/#Types","page":"Available transforms","title":"Types","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"AbstractCustomPlan\nAbstractTransform\nIdentityPlan\nIdentityPlan!\nPlan","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.AbstractCustomPlan","page":"Available transforms","title":"PencilFFTs.Transforms.AbstractCustomPlan","text":"AbstractCustomPlan\n\nAbstract type defining a custom plan, to be used as an alternative to FFTW plans (FFTW.FFTWPlan).\n\nThe only custom plan defined in this module is IdentityPlan. The user can define other custom plans that are also subtypes of AbstractCustomPlan.\n\nNote that plan returns a subtype of either AbstractFFTs.Plan or AbstractCustomPlan.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.AbstractTransform","page":"Available transforms","title":"PencilFFTs.Transforms.AbstractTransform","text":"AbstractTransform\n\nSpecifies a one-dimensional FFT-based transform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.IdentityPlan","page":"Available transforms","title":"PencilFFTs.Transforms.IdentityPlan","text":"IdentityPlan\n\nType of plan associated to NoTransform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.IdentityPlan!","page":"Available transforms","title":"PencilFFTs.Transforms.IdentityPlan!","text":"IdentityPlan!\n\nType of plan associated to NoTransform!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.Plan","page":"Available transforms","title":"PencilFFTs.Transforms.Plan","text":"Plan = Union{AbstractFFTs.Plan, AbstractCustomPlan}\n\nUnion type representing any plan returned by plan.\n\nSee also AbstractCustomPlan.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#Functions","page":"Available transforms","title":"Functions","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"plan\n\nbinv\nscale_factor\n\neltype_input\neltype_output\nexpand_dims\nis_inplace\nkind\nlength_output","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.plan","page":"Available transforms","title":"PencilFFTs.Transforms.plan","text":"plan(transform::AbstractTransform, A, [dims];\n flags=FFTW.ESTIMATE, timelimit=Inf)\n\nCreate plan to transform array A along dimensions dims.\n\nIf dims is not specified, all dimensions of A are transformed.\n\nFor FFT plans, this function wraps the AbstractFFTs.jl and FFTW.jl plan creation functions. For more details on the function arguments, see AbstractFFTs.plan_fft.\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.binv","page":"Available transforms","title":"PencilFFTs.Transforms.binv","text":"binv(transform::AbstractTransform, d::Integer)\n\nReturns the backwards transform associated to the given transform.\n\nThe second argument must be the length of the first transformed dimension in the forward transform. It is used in particular when transform = RFFT(), to determine the length of the inverse (complex-to-real) transform. See the AbstractFFTs.irfft docs for details.\n\nThe backwards transform returned by this function is not normalised. The normalisation factor for a given array can be obtained by calling scale_factor.\n\nExample\n\njulia> binv(Transforms.FFT(), 42)\nBFFT\n\njulia> binv(Transforms.BRFFT(9), 42)\nRFFT\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.scale_factor","page":"Available transforms","title":"PencilFFTs.Transforms.scale_factor","text":"scale_factor(transform::AbstractTransform, A, [dims = 1:ndims(A)])\n\nGet factor required to normalise the given array after a transformation along dimensions dims (all dimensions by default).\n\nThe array A must have the dimensions of the transform input.\n\nImportant: the dimensions dims must be the same that were passed to plan.\n\nExamples\n\njulia> C = zeros(ComplexF32, 3, 4, 5);\n\njulia> scale_factor(Transforms.FFT(), C)\n60\n\njulia> scale_factor(Transforms.BFFT(), C)\n60\n\njulia> scale_factor(Transforms.BFFT(), C, 2:3)\n20\n\njulia> R = zeros(Float64, 3, 4, 5);\n\njulia> scale_factor(Transforms.RFFT(), R, 2)\n4\n\njulia> scale_factor(Transforms.RFFT(), R, 2:3)\n20\n\njulia> scale_factor(Transforms.BRFFT(8), C)\n96\n\njulia> scale_factor(Transforms.BRFFT(9), C)\n108\n\nThis will fail because the input of RFFT is real, and R is a complex array:\n\njulia> scale_factor(Transforms.RFFT(), C, 2:3)\nERROR: MethodError: no method matching scale_factor(::PencilFFTs.Transforms.RFFT, ::Array{ComplexF32, 3}, ::UnitRange{Int64})\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.eltype_input","page":"Available transforms","title":"PencilFFTs.Transforms.eltype_input","text":"eltype_input(transform::AbstractTransform, real_type<:AbstractFloat)\n\nDetermine input data type for a given transform given the floating point precision of the input data.\n\nSome transforms, such as R2R and NoTransform, can take both real and complex data. For those kinds of transforms, nothing is returned.\n\nExample\n\njulia> eltype_input(Transforms.FFT(), Float32)\nComplexF32 (alias for Complex{Float32})\n\njulia> eltype_input(Transforms.RFFT(), Float64)\nFloat64\n\njulia> eltype_input(Transforms.R2R(FFTW.REDFT01), Float64) # nothing\n\njulia> eltype_input(Transforms.NoTransform(), Float64) # nothing\n\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.eltype_output","page":"Available transforms","title":"PencilFFTs.Transforms.eltype_output","text":"eltype_output(transform::AbstractTransform, eltype_input)\n\nReturns the output data type for a given transform given the input type.\n\nThrows ArgumentError if the input data type is incompatible with the transform type.\n\nExample\n\njulia> eltype_output(Transforms.NoTransform(), Float32)\nFloat32\n\njulia> eltype_output(Transforms.RFFT(), Float64)\nComplexF64 (alias for Complex{Float64})\n\njulia> eltype_output(Transforms.BRFFT(4), ComplexF32)\nFloat32\n\njulia> eltype_output(Transforms.FFT(), Float64)\nERROR: ArgumentError: invalid input data type for PencilFFTs.Transforms.FFT: Float64\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.expand_dims","page":"Available transforms","title":"PencilFFTs.Transforms.expand_dims","text":"expand_dims(transform::AbstractTransform, Val(N))\n\nExpand a single multidimensional transform into one transform per dimension.\n\nExample\n\n# Expand a real-to-complex transform in 3 dimensions.\njulia> expand_dims(Transforms.RFFT(), Val(3))\n(RFFT, FFT, FFT)\n\njulia> expand_dims(Transforms.BRFFT(4), Val(3))\n(BFFT, BFFT, BRFFT{even})\n\njulia> expand_dims(Transforms.NoTransform(), Val(2))\n(NoTransform, NoTransform)\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.is_inplace","page":"Available transforms","title":"PencilFFTs.Transforms.is_inplace","text":"is_inplace(transform::AbstractTransform) -> Bool\nis_inplace(transforms::Vararg{AbtractTransform}) -> Union{Bool, Nothing}\n\nCheck whether a transform or a list of transforms is performed in-place.\n\nIf the list of transforms has a combination of in-place and out-of-place transforms, nothing is returned.\n\nExample\n\njulia> is_inplace(Transforms.RFFT())\nfalse\n\njulia> is_inplace(Transforms.NoTransform!())\ntrue\n\njulia> is_inplace(Transforms.FFT!(), Transforms.R2R!(FFTW.REDFT01))\ntrue\n\njulia> is_inplace(Transforms.FFT(), Transforms.R2R(FFTW.REDFT01))\nfalse\n\njulia> is_inplace(Transforms.FFT(), Transforms.R2R!(FFTW.REDFT01)) === nothing\ntrue\n\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.kind","page":"Available transforms","title":"PencilFFTs.Transforms.kind","text":"kind(transform::R2R)\n\nGet kind of real-to-real transform.\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.length_output","page":"Available transforms","title":"PencilFFTs.Transforms.length_output","text":"length_output(transform::AbstractTransform, length_in::Integer)\n\nReturns the length of the transform output, given the length of its input.\n\nThe input and output lengths are specified in terms of the respective input and output datatypes. For instance, for real-to-complex transforms, these are respectively the length of input real data and of output complex data.\n\n\n\n\n\n","category":"function"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"EditURL = \"https://github.com/jipolanco/PencilFFTs.jl/blob/master/docs/examples/navier_stokes.jl\"","category":"page"},{"location":"generated/navier_stokes/#Navier–Stokes-equations","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"In this example, we numerically solve the incompressible Navier–Stokes equations","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"_t bmv + (bmv bm) bmv = -frac1ρ bm p + ν ^2 bmv\nquad bm bmv = 0","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where bmv(bmx t) and p(bmx t) are respectively the velocity and pressure fields, ν is the fluid kinematic viscosity and ρ is the fluid density.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.","category":"page"},{"location":"generated/navier_stokes/#First-steps","page":"Navier–Stokes equations","title":"First steps","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We start by loading the required packages, initialising MPI and setting the simulation parameters.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using MPI\nusing PencilFFTs\n\nMPI.Init()\ncomm = MPI.COMM_WORLD\nprocid = MPI.Comm_rank(comm) + 1\n\n# Simulation parameters\nNs = (64, 64, 64) # = (Nx, Ny, Nz)\nLs = (2π, 2π, 2π) # = (Lx, Ly, Lz)\n\n# Collocation points (\"global\" = over all processes).\n# We include the endpoint (length = N + 1) for convenience.\nxs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls) # = (x, y, z)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's check the number of MPI processes over which we're running our simulation:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"MPI.Comm_size(comm)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"pen = Pencil(Ns, comm)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"The subdomain associated to the local MPI process can be obtained using range_local:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"range_local(pen)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We now construct a distributed vector field that follows the decomposition configuration we just created:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v⃗₀ = (\n PencilArray{Float64}(undef, pen), # vx\n PencilArray{Float64}(undef, pen), # vy\n PencilArray{Float64}(undef, pen), # vz\n)\nsummary(v⃗₀[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We still need to fill this array with interesting values that represent a physical velocity field.","category":"page"},{"location":"generated/navier_stokes/#Initial-condition","page":"Navier–Stokes equations","title":"Initial condition","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"beginaligned\nv_x(x y z) = u₀ sin(k₀ x) cos(k₀ y) cos(k₀ z) \nv_y(x y z) = -u₀ cos(k₀ x) sin(k₀ y) cos(k₀ z) \nv_z(x y z) = 0\nendaligned","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where u₀ and k₀ are two parameters setting the amplitude and the period of the velocity field.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"grid = localgrid(pen, xs_global)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can use this to initialise the velocity field:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"u₀ = 1.0\nk₀ = 2π / Ls[1] # should be integer if L = 2π (to preserve periodicity)\n\n@. v⃗₀[1] = u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)\n@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)\n@. v⃗₀[3] = 0\nnothing # hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's plot a 2D slice of the velocity field managed by the local MPI process:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using GLMakie\n\n# Compute the norm of a vector field represented by a tuple of arrays.\nfunction vecnorm(v⃗::NTuple)\n vnorm = similar(v⃗[1])\n for n ∈ eachindex(v⃗[1])\n w = zero(eltype(vnorm))\n for v ∈ v⃗\n w += v[n]^2\n end\n vnorm[n] = sqrt(w)\n end\n vnorm\nend\n\nlet fig = Figure(resolution = (700, 600))\n ax = Axis3(fig[1, 1]; aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\")\n vnorm = vecnorm(v⃗₀)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, vnorm;\n alpha = 0.2, levels = 4,\n colormap = :viridis, colorrange = (0.0, 1.0),\n )\n cb = Colorbar(fig[1, 2], ct; label = \"Velocity magnitude\")\n fig\nend","category":"page"},{"location":"generated/navier_stokes/#Velocity-in-Fourier-space","page":"Navier–Stokes equations","title":"Velocity in Fourier space","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"bmv(bmx t) =\n_bmk hatbmv_bmk(t) e^i bmk bmx","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where bmk = (k_x k_y k_z) are the discrete wave numbers.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for k_x:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using AbstractFFTs: fftfreq, rfftfreq\n\nks_global = (\n rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]), # kx | real-to-complex\n fftfreq(Ns[2], 2π * Ns[2] / Ls[2]), # ky | complex-to-complex\n fftfreq(Ns[3], 2π * Ns[3] / Ls[3]), # kz | complex-to-complex\n)\n\nks_global[1]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ks_global[2]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ks_global[3]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"See PencilFFTPlan for details on creating plans and on optional keyword arguments.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients hatbmv_bmk:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v̂s = plan .* v⃗₀\nsummary(v̂s[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Note that, in Fourier space, the domain decomposition is performed along the directions x and y:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"pencil(v̂s[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"This is because the 3D FFTs are performed one dimension at a time, with the x direction first and the z direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers bmk associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"grid_fourier = localgrid(v̂s[1], ks_global)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, bmω = bm bmv. In Fourier space, this becomes hatbmω = i bmk hatbmv.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using StaticArrays: SVector\nusing LinearAlgebra: ×\n\nfunction curl_fourier!(\n ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,\n ) where {N}\n @inbounds for I ∈ eachindex(grid_fourier)\n # We use StaticArrays for the cross product between small vectors.\n ik⃗ = im * SVector(grid_fourier[I])\n v⃗ = SVector(getindex.(v̂s, Ref(I))) # = (v̂s[1][I], v̂s[2][I], ...)\n ω⃗ = ik⃗ × v⃗\n for n ∈ eachindex(ω⃗)\n ω̂s[n][I] = ω⃗[n]\n end\n end\n ω̂s\nend\n\nω̂s = similar.(v̂s)\ncurl_fourier!(ω̂s, v̂s, grid_fourier);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We finally transform back to physical space and plot the result:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ωs = plan .\\ ω̂s\n\nlet fig = Figure(resolution = (700, 600))\n ax = Axis3(fig[1, 1]; aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\")\n ω_norm = vecnorm(ωs)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, ω_norm;\n alpha = 0.1, levels = 0.8:0.2:2.0,\n colormap = :viridis, colorrange = (0.8, 2.0),\n )\n cb = Colorbar(fig[1, 2], ct; label = \"Vorticity magnitude\")\n fig\nend","category":"page"},{"location":"generated/navier_stokes/#Computing-the-non-linear-term","page":"Navier–Stokes equations","title":"Computing the non-linear term","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"_t hatbmv_bmk =\n- mathcalP_bmk left widehat(bmv bm) bmv right\n- ν bmk^2 hatbmv_bmk\nquad text with quad\nmathcalP_bmk(hatbmF_bmk) = left( I - fracbmk \nbmkbmk^2 right) hatbmF_bmk","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where mathcalP_bmk is a projection operator allowing to preserve the incompressibility condition bm bmv = 0. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Now that we have the wave numbers bmk, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients hatbmv_bmk of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, hatbmF_bmk = left widehat(bmv bm) bmv right_bmk. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Below we implement a function that computes the non-linear term in Fourier space based on its convective form (bmv bm) bmv = bm (bmv bmv). Note that this equivalence uses the incompressibility condition bm bmv = 0.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using LinearAlgebra: mul!, ldiv! # for applying FFT plans in-place\n\n# Compute non-linear term in Fourier space from velocity field in physical\n# space. Optional keyword arguments may be passed to avoid allocations.\nfunction ns_nonlinear!(\n F̂s, vs, plan, grid_fourier;\n vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),\n )\n # Compute F_i = ∂_j (v_i v_j) for each i.\n # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)\n w, ŵ = vbuf, v̂buf\n @inbounds for (i, F̂i) ∈ enumerate(F̂s)\n F̂i .= 0\n vi = vs[i]\n for (j, vj) ∈ enumerate(vs)\n w .= vi .* vj # w = v_i * v_j in physical space\n mul!(ŵ, plan, w) # same in Fourier space\n # Add derivative in Fourier space\n for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n kj = k⃗[j]\n F̂i[I] += im * kj * ŵ[I]\n end\n end\n end\n F̂s\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"As an example, let's use this function on our initial velocity field:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"F̂s = similar.(v̂s)\nns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)\n ks_max = maximum.(abs, ks_global) # maximum stored wave numbers (kx_max, ky_max, kz_max)\n ks_lim = (2 / 3) .* ks_max\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I]\n if any(abs.(k⃗) .> ks_lim)\n for ŵ ∈ ŵs\n ŵ[I] = 0\n end\n end\n end\n ŵs\nend\n\n# We can apply this on the previously computed non-linear term:\ndealias_twothirds!(F̂s, grid_fourier, ks_global);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Finally, we implement the projection associated to the incompressibility condition:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function project_divergence_free!(ûs, grid_fourier)\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I]\n k² = sum(abs2, k⃗)\n iszero(k²) && continue # avoid division by zero\n û = getindex.(ûs, Ref(I)) # (ûs[1][I], ûs[2][I], ...)\n for i ∈ eachindex(û)\n ŵ = û[i]\n for j ∈ eachindex(û)\n ŵ -= k⃗[i] * k⃗[j] * û[j] / k²\n end\n ûs[i][I] = ŵ\n end\n end\n ûs\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)\nv̂s_proj .≈ v̂s # the last one may be false because v_z = 0 initially","category":"page"},{"location":"generated/navier_stokes/#Putting-it-all-together","page":"Navier–Stokes equations","title":"Putting it all together","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function ns_rhs!(\n dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,\n ) where {N}\n # 1. Compute non-linear term and dealias it\n (; plan, cache, ks_global, grid_fourier) = p\n F̂s = cache.F̂s\n ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])\n dealias_twothirds!(F̂s, grid_fourier, ks_global)\n\n # 2. Project onto divergence-free space\n project_divergence_free!(F̂s, grid_fourier)\n\n # 3. Transform velocity to Fourier space\n v̂s = cache.v̂s\n map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)\n\n # 4. Add viscous term (and multiply projected non-linear term by -1)\n ν = p.ν\n for n ∈ eachindex(v̂s)\n v̂ = v̂s[n]\n F̂ = F̂s[n]\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n k² = sum(abs2, k⃗)\n F̂[I] = -F̂[I] - ν * k² * v̂[I]\n end\n end\n\n # 5. Transform RHS back to physical space\n map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)\n\n nothing\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using OrdinaryDiffEq\nusing RecursiveArrayTools: ArrayPartition\n\nns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)\nvs_init_ode = ArrayPartition(v⃗₀)\nsummary(vs_init_ode)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We now define solver parameters and temporary variables, and initialise the problem:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"params = (;\n ν = 5e-3, # kinematic viscosity\n plan, grid_fourier, ks_global,\n cache = (\n v̂s = similar.(v̂s),\n F̂s = similar.(v̂s),\n )\n)\n\ntspan = (0.0, 10.0)\nprob = ODEProblem(ns_rhs!, vs_init_ode, tspan, params)\nintegrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum E(k), to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity ν must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large k), while high enough to allow the small-scale motions to be correctly resolved.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function energy_spectrum!(Ek, ks, v̂s, grid_fourier)\n Nk = length(Ek)\n @assert Nk == length(ks)\n Ek .= 0\n for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n knorm = sqrt(sum(abs2, k⃗))\n i = searchsortedfirst(ks, knorm)\n i > Nk && continue\n v⃗ = getindex.(v̂s, Ref(I)) # = (v̂s[1][I], v̂s[2][I], ...)\n factor = k⃗[1] == 0 ? 1 : 2 # account for Hermitian symmetry and r2c transform\n Ek[i] += factor * sum(abs2, v⃗) / 2\n end\n MPI.Allreduce!(Ek, +, get_comm(v̂s[1])) # sum across all processes\n Ek\nend\n\nks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])\nEk = similar(ks)\nv̂s = plan .* integrator.u.x\nenergy_spectrum!(Ek, ks, v̂s, grid_fourier)\nEk ./= scale_factor(plan)^2 # rescale energy\n\ncurl_fourier!(ω̂s, v̂s, grid_fourier)\nldiv!.(ωs, plan, ω̂s)\nω⃗_plot = Observable(ωs)\nk_plot = @view ks[2:end]\nE_plot = Observable(@view Ek[2:end])\nt_plot = Observable(integrator.t)\n\nfig = let\n fig = Figure(resolution = (1200, 600))\n ax = Axis3(\n fig[1, 1][1, 1]; title = @lift(\"t = $(round($t_plot, digits = 3))\"),\n aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\",\n )\n ω_mag = @lift vecnorm($ω⃗_plot)\n ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, ω_mag_norm;\n alpha = 0.3, levels = 3,\n colormap = :viridis, colorrange = (0.0, 1.0),\n )\n cb = Colorbar(fig[1, 1][1, 2], ct; label = \"Normalised vorticity magnitude\")\n ax_sp = Axis(\n fig[1, 2];\n xlabel = \"k\", ylabel = \"E(k)\", xscale = log2, yscale = log10,\n title = \"Kinetic energy spectrum\",\n )\n ylims!(ax_sp, 1e-8, 1e0)\n scatterlines!(ax_sp, k_plot, E_plot)\n ks_slope = exp.(range(log(2.5), log(25.0), length = 3))\n E_fivethirds = @. 0.3 * ks_slope^(-5/3)\n @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)\n text!(ax_sp, L\"k^{-5/3}\"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))\n fig\nend\n\nusing Printf # hide\nwith_xvfb = ENV[\"DISPLAY\"] == \":99\" # hide\nnstep = 0 # hide\nconst tmpdir = mktempdir() # hide\nfilename_frame(procid, nstep) = joinpath(tmpdir, @sprintf(\"proc%d_%04d.png\", procid, nstep)) # hide\nrecord(fig, \"vorticity_proc$procid.mp4\"; framerate = 10) do io\n with_xvfb && recordframe!(io) # hide\n while integrator.t < 20\n dt = 0.001\n step!(integrator, dt)\n t_plot[] = integrator.t\n mul!.(v̂s, plan, integrator.u.x) # current velocity in Fourier space\n curl_fourier!(ω̂s, v̂s, grid_fourier)\n ldiv!.(ω⃗_plot[], plan, ω̂s)\n ω⃗_plot[] = ω⃗_plot[] # to force updating the plot\n energy_spectrum!(Ek, ks, v̂s, grid_fourier)\n Ek ./= scale_factor(plan)^2 # rescale energy\n E_plot[] = E_plot[]\n global nstep += 1 # hide\n with_xvfb ? # hide\n save(filename_frame(procid, nstep), fig) : # hide\n recordframe!(io)\n end\nend;\n\nif with_xvfb # hide\n run(pipeline(`ffmpeg -y -r 10 -i $tmpdir/proc$(procid)_%04d.png -c:v libx264 -vf \"fps=25,format=yuv420p\" vorticity_proc$procid.mp4`; stdout = \"ffmpeg.out\", stderr = \"ffmpeg.err\")) # hide\nend # hide\nnothing # hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"
                      \n \n
                      ","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"This page was generated using Literate.jl.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"EditURL = \"https://github.com/jipolanco/PencilFFTs.jl/blob/master/docs/examples/gradient.jl\"","category":"page"},{"location":"generated/gradient/#Gradient-of-a-scalar-field","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This example shows different methods to compute the gradient of a real-valued 3D scalar field θ(bmx) in Fourier space, where bmx = (x y z). It is assumed that the field is periodic with period L = 2π along all dimensions.","category":"page"},{"location":"generated/gradient/#General-procedure","page":"Gradient of a scalar field","title":"General procedure","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The discrete Fourier expansion of θ writes","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"θ(bmx) = _bmk Z^3 hatθ(bmk) e^i bmk bmx","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"where bmk = (k_x k_y k_z) are the Fourier wave numbers and hatθ is the discrete Fourier transform of θ. Then, the spatial derivatives of θ are given by","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"frac θ(bmx) x_i =\n_bmk Z^3 i k_i hatθ(bmk) e^i bmk bmx","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"where the subscript i denotes one of the spatial components x, y or z.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In other words, to compute bm θ = (_x θ _y θ _z θ), one has to:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"transform θ to Fourier space to obtain hatθ,\nmultiply hatθ by i bmk,\ntransform the result back to physical space to obtain bm θ.","category":"page"},{"location":"generated/gradient/#Preparation","page":"Gradient of a scalar field","title":"Preparation","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In this section, we initialise a random real-valued scalar field θ and compute its FFT. For more details see the Tutorial.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"using MPI\nusing PencilFFTs\nusing Random\n\nMPI.Init()\n\n# Input data dimensions (Nx × Ny × Nz)\ndims = (64, 32, 64)\n\n# Apply a 3D real-to-complex (r2c) FFT.\ntransform = Transforms.RFFT()\n\n# Automatically create decomposition configuration\ncomm = MPI.COMM_WORLD\npen = Pencil(dims, comm)\n\n# Create plan\nplan = PencilFFTPlan(pen, transform)\n\n# Allocate data and initialise field\nθ = allocate_input(plan)\nrandn!(θ)\n\n# Perform distributed FFT\nθ_hat = plan * θ\nnothing # hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, we initialise the output that will hold ∇θ in Fourier space. Noting that ∇θ is a vector field, we choose to store it as a tuple of 3 PencilArrays.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"∇θ_hat = allocate_output(plan, Val(3))\n\n# This is equivalent:\n# ∇θ_hat = ntuple(d -> similar(θ_hat), Val(3))\n\nsummary(∇θ_hat)","category":"page"},{"location":"generated/gradient/#Fourier-wave-numbers","page":"Gradient of a scalar field","title":"Fourier wave numbers","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In general, the Fourier wave numbers are of the form k_i = 0 frac2πL_i frac4πL_i frac6πL_i , where L_i is the period along dimension i. When a real-to-complex Fourier transform is applied, roughly half of these wave numbers are redundant due to the Hermitian symmetry of the complex Fourier coefficients. In practice, this means that for the fastest dimension x (along which a real-to-complex transform is performed), the negative wave numbers are dropped, i.e. k_x = 0 frac2πL_x frac4πL_x .","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The AbstractFFTs package provides a convenient way to generate the Fourier wave numbers, using the functions fftfreq and rfftfreq. We can use these functions to initialise a \"grid\" of wave numbers associated to our 3D real-to-complex transform:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"using AbstractFFTs: fftfreq, rfftfreq\n\nbox_size = (2π, 2π, 2π) # Lx, Ly, Lz\nsample_rate = 2π .* dims ./ box_size\n\n# In our case (Lx = 2π and Nx even), this gives kx = [0, 1, 2, ..., Nx/2].\nkx = rfftfreq(dims[1], sample_rate[1])\n\n# In our case (Ly = 2π and Ny even), this gives\n# ky = [0, 1, 2, ..., Ny/2-1, -Ny/2, -Ny/2+1, ..., -1] (and similarly for kz).\nky = fftfreq(dims[2], sample_rate[2])\nkz = fftfreq(dims[3], sample_rate[3])\n\nkvec = (kx, ky, kz)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that kvec now contains the wave numbers associated to the global domain. In the following, we will only need the wave numbers associated to the portion of the domain handled by the local MPI process.","category":"page"},{"location":"generated/gradient/#gradient_method_global","page":"Gradient of a scalar field","title":"Method 1: global views","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"PencilArrays, returned for instance by allocate_input and allocate_output, take indices that start at 1, regardless of the location of the subdomain associated to the local process on the global grid. (In other words, PencilArrays take local indices.) On the other hand, we have defined the wave number vector kvec which, for each MPI process, is defined over the global domain, and as such it takes global indices.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"One straightforward way of making data arrays compatible with wave numbers is to use global views, i.e. arrays that take global indices. These are generated from PencilArrays by calling the global_view function. Note that, in general, global indices do not start at 1 for a given MPI process. A given process will own a range of data given by indices in (i1:i2, j1:j2, k1:k2).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"θ_glob = global_view(θ_hat)\n∇θ_glob = global_view.(∇θ_hat)\nsummary(θ_glob)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Once we have global views, we can combine data and wave numbers using the portion of global indices owned by the local MPI process, as shown below. We can use CartesianIndices to iterate over the global indices associated to the local process.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"for I in CartesianIndices(θ_glob)\n i, j, k = Tuple(I) # unpack indices\n\n # Wave number vector associated to current Cartesian index.\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n # Compute gradient in Fourier space.\n # Note that modifying ∇θ_glob also modifies the original PencilArray ∇θ_hat.\n ∇θ_glob[1][I] = im * kx * θ_glob[I]\n ∇θ_glob[2][I] = im * ky * θ_glob[I]\n ∇θ_glob[3][I] = im * kz * θ_glob[I]\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The above loop can be written in a slightly more efficient manner by precomputing im * θ_glob[I]:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@inbounds for I in CartesianIndices(θ_glob)\n i, j, k = Tuple(I)\n\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n u = im * θ_glob[I]\n\n ∇θ_glob[1][I] = kx * u\n ∇θ_glob[2][I] = ky * u\n ∇θ_glob[3][I] = kz * u\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Also note that the above can be easily written in a more generic way, e.g. for arbitrary dimensions, thanks in part to the use of CartesianIndices. Moreover, in the above there is no notion of the dimension permutations discussed in the tutorial, as it is all hidden behind the implementation of PencilArrays. And as seen later in the benchmarks, these (hidden) permutations have zero cost, as the speed is identical to that of a function that explicitly takes into account these permutations.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, we can perform a backwards transform to obtain bm θ in physical space:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"∇θ = plan \\ ∇θ_hat;\nnothing #hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that the transform is automatically broadcast over the three fields of the ∇θ_hat vector, and the result ∇θ is also a tuple of three PencilArrays.","category":"page"},{"location":"generated/gradient/#gradient_method_global_explicit","page":"Gradient of a scalar field","title":"Method 2: explicit global indexing","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Sometimes, one does not need to write generic code. In our case, one often knows the dimensionality of the problem and the memory layout of the data (i.e. the underlying index permutation).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Below is a reimplementation of the above loop, using explicit indices instead of CartesianIndices, and assuming that the underlying index permutation is (3, 2, 1), that is, data is stored in (z y x) order. As discussed in the tutorial, this is the default for transformed arrays. This example also serves as a more explicit explanation for what is going on in the first method.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"# Get local data range in the global grid.\nrng = axes(θ_glob) # = (i1:i2, j1:j2, k1:k2)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"For the loop below, we're assuming that the permutation is (3, 2, 1). In other words, the fastest index is the last one, and not the first one as it is usually in Julia. If the permutation is not (3, 2, 1), things will still work (well, except for the assertion below!), but the loop order will not be optimal.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@assert permutation(θ_hat) === Permutation(3, 2, 1)\n\n@inbounds for i in rng[1], j in rng[2], k in rng[3]\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n # Note that we still access the arrays in (i, j, k) order.\n # (The permutation happens behind the scenes!)\n u = im * θ_glob[i, j, k]\n\n ∇θ_glob[1][i, j, k] = kx * u\n ∇θ_glob[2][i, j, k] = ky * u\n ∇θ_glob[3][i, j, k] = kz * u\nend","category":"page"},{"location":"generated/gradient/#gradient_method_local","page":"Gradient of a scalar field","title":"Method 3: using local indices","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Alternatively, we can avoid global views and work directly on PencilArrays using local indices that start at 1. In this case, part of the strategy is to construct a \"local\" grid of wave numbers that can also be accessed with local indices. This can be conveniently done using the localgrid function of the PencilArrays.jl package, which accepts a PencilArray (or its associated Pencil) and the global coordinates (here kvec):","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"grid_fourier = localgrid(θ_hat, kvec)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that one can directly iterate on the returned grid object:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@inbounds for I in CartesianIndices(grid_fourier)\n # Wave number vector associated to current Cartesian index.\n local k⃗ # hide\n k⃗ = grid_fourier[I]\n u = im * θ_hat[I]\n ∇θ_hat[1][I] = k⃗[1] * u\n ∇θ_hat[2][I] = k⃗[2] * u\n ∇θ_hat[3][I] = k⃗[3] * u\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This implementation is as efficient as the other examples, while being slightly shorter to write. Moreover, it is quite generic, and can be made independent of the number of dimensions with little effort.","category":"page"},{"location":"generated/gradient/#gradient_method_broadcast","page":"Gradient of a scalar field","title":"Method 4: using broadcasting","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, note that the local grid object returned by localgrid makes it is possible to compute the gradient using broadcasting, thus fully avoiding scalar indexing. This can be quite convenient in some cases, and can also be very useful if one is working on GPUs (where scalar indexing is prohibitively expensive). Using broadcasting, the above examples simply become:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@. ∇θ_hat[1] = im * grid_fourier[1] * θ_hat\n@. ∇θ_hat[2] = im * grid_fourier[2] * θ_hat\n@. ∇θ_hat[3] = im * grid_fourier[3] * θ_hat\nnothing # hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Once again, as shown in the benchmarks further below, this method performs quite similarly to the other ones.","category":"page"},{"location":"generated/gradient/#Summary","page":"Gradient of a scalar field","title":"Summary","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The PencilArrays module provides different alternatives to deal with MPI-distributed data that may be subject to dimension permutations. In particular, one can choose to work with global indices (first two examples), with local indices (third example), or to avoid scalar indexing altogether (fourth example).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"If one wants to stay generic, making sure that the same code will work for arbitrary dimensions and will be efficient regardless of the underlying dimension permutation, methods 1, 3 or 4 should be preferred. These use CartesianIndices and make no assumptions on possible dimension permutations, which are by default enabled in the output of PencilFFTs transforms. In fact, such permutations are completely invisible in the implementations.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The second method uses explicit (i, j, k) indices. It assumes that the underlying permutation is (3, 2, 1) to loop with i as the slowest index and k as the fastest, which is the optimal order in this case given the permutation. As such, the implementation is less generic than the others, and differences in performance are negligible with respect to more generic variants.","category":"page"},{"location":"generated/gradient/#gradient_benchmarks","page":"Gradient of a scalar field","title":"Benchmark results","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The following are the benchmark results obtained from running examples/gradient.jl on a laptop, using 2 MPI processes and Julia 1.7.2, with an input array of global dimensions 64 32 64. The different methods detailed above are marked on the right. The \"lazy\" marks indicate runs where the wave numbers were represented by lazy Frequencies objects (returned by rfftfreq and fftfreq). Otherwise, they were collected into Vectors. For some reason, plain Vectors are faster when working with grids generated by localgrid.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In the script, additional implementations can be found which rely on a more advanced understanding of permutations and on the internals of the PencilArrays package. For instance, gradient_local_parent! directly works with the raw data stored in Julia Arrays, while gradient_local_linear! completely avoids CartesianIndices while staying generic and efficient. Nevertheless, these display roughly the same performance as the above examples.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":" gradient_global_view!... 89.900 μs\n gradient_global_view! (lazy)... 92.060 μs [Method 1]\n gradient_global_view_explicit!... 88.958 μs\n gradient_global_view_explicit! (lazy)... 81.055 μs [Method 2]\n gradient_local!... 92.305 μs\n gradient_grid!... 92.770 μs\n gradient_grid! (lazy)... 101.388 μs [Method 3]\n gradient_grid_broadcast!... 88.606 μs\n gradient_grid_broadcast! (lazy)... 151.020 μs [Method 4]\n gradient_local_parent!... 92.248 μs\n gradient_local_linear!... 91.212 μs\n gradient_local_linear_explicit!... 90.992 μs","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This page was generated using Literate.jl.","category":"page"},{"location":"PencilFFTs_timers/#PencilFFTs.measuring_performance","page":"Measuring performance","title":"Measuring performance","text":"","category":"section"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"It is possible to measure the time spent in different sections of the distributed transforms using the TimerOutputs package. This has a (very small) performance overhead, so it is disabled by default. To enable time measurements, call TimerOutputs.enable_debug_timings after loading PencilFFTs (see below for an example). For more details see the TimerOutputs docs.","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"Minimal example:","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"using MPI\nusing PencilFFTs\nusing TimerOutputs\n\n# Enable timing of `PencilFFTs` functions\nTimerOutputs.enable_debug_timings(PencilFFTs)\nTimerOutputs.enable_debug_timings(PencilArrays)\nTimerOutputs.enable_debug_timings(Transpositions)\n\nMPI.Init()\n\nplan = PencilFFTPlan(#= args... =#)\n\n# [do stuff with `plan`...]\n\n# Retrieve and print timing data associated to `plan`\nto = timer(plan)\nprint_timer(to)","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"By default, each PencilFFTPlan has its own TimerOutput. If you already have a TimerOutput, you can pass it to the PencilFFTPlan constructor:","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"to = TimerOutput()\nplan = PencilFFTPlan(..., timer=to)\n\n# [do stuff with `plan`...]\n\nprint_timer(to)","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"#PencilFFTs","page":"Home","title":"PencilFFTs","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Fast Fourier transforms of MPI-distributed Julia arrays.","category":"page"},{"location":"#Introduction","page":"Home","title":"Introduction","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package provides multidimensional FFTs and related transforms on MPI-distributed Julia arrays via the PencilArrays package.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The name of this package originates from the decomposition of 3D domains along two out of three dimensions, sometimes called pencil decomposition. This is illustrated by the figure below,[1] where each coloured block is managed by a different MPI process. Typically, one wants to compute FFTs on a scalar or vector field along the three spatial dimensions. In the case of a pencil decomposition, 3D FFTs are performed one dimension at a time, along the non-decomposed direction. Transforms must then be interleaved with global data transpositions to switch between pencil configurations. In high-performance computing environments, such data transpositions are generally the most expensive part of a parallel FFT computation, due to the large cost of communications between computing nodes.","category":"page"},{"location":"","page":"Home","title":"Home","text":"
                      \n \n
                      ","category":"page"},{"location":"","page":"Home","title":"Home","text":"More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension N. The decompositions can be performed along an arbitrary number M N of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.","category":"page"},{"location":"#Features","page":"Home","title":"Features","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;\nFFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;\nin-place and out-of-place transforms;\nhigh scalability up to (at least) tens of thousands of MPI processes.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"PencilFFTs can be installed using the Julia package manager:","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> ] add PencilFFTs","category":"page"},{"location":"#Similar-projects","page":"Home","title":"Similar projects","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.\nPFFT is a very general parallel FFT library written in C.\nP3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.\n2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[1]: Figure adapted from this PhD thesis.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[2]: For the pencil decomposition represented in the figure, N = 3 and M = 2.","category":"page"},{"location":"PencilFFTs/#Distributed-FFT-plans","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"Distributed FFTs are implemented in the PencilFFTs module, and are built on top of the PencilArrays package.","category":"page"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"PencilFFTs/#Creating-plans","page":"Distributed FFT plans","title":"Creating plans","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"PencilFFTPlan","category":"page"},{"location":"PencilFFTs/#PencilFFTs.PencilFFTPlan","page":"Distributed FFT plans","title":"PencilFFTs.PencilFFTPlan","text":"PencilFFTPlan{T,N} <: AbstractFFTs.Plan{T}\n\nPlan for N-dimensional FFT-based transform on MPI-distributed data, where input data has type T.\n\n\n\nPencilFFTPlan(p::Pencil, transforms; kwargs...)\n\nCreate a PencilFFTPlan for distributed arrays following a given Pencil configuration. See variant below for details on the specification of transforms and on possible keyword arguments.\n\n\n\nPencilFFTPlan(\n A::PencilArray, transforms;\n fftw_flags = FFTW.ESTIMATE,\n fftw_timelimit = FFTW.NO_TIMELIMIT,\n permute_dims = Val(true),\n transpose_method = Transpositions.PointToPoint(),\n timer = timer(pencil(A)),\n)\n\nCreate plan for N-dimensional transform on MPI-distributed PencilArrays.\n\nExtended help\n\nThis creates a PencilFFTPlan for arrays sharing the same properties as A (dimensions, MPI decomposition, memory layout, ...), which describe data on an N-dimensional domain.\n\nTransforms\n\nThe transforms to be applied along each dimension are specified by the transforms argument. Possible transforms are defined as subtypes of Transforms.AbstractTransform, and are listed in Transform types. This argument may be either:\n\na tuple of N transforms to be applied along each dimension. For instance, transforms = (Transforms.R2R(FFTW.REDFT01), Transforms.RFFT(), Transforms.FFT());\na single transform to be applied along all dimensions. The input is automatically expanded into N equivalent transforms. For instance, for a three-dimensional array, transforms = Transforms.RFFT() specifies a 3D real-to-complex transform, and is equivalent to passing (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT()).\n\nNote that forward transforms are applied from left to right. In the last example, this means that a real-to-complex transform (RFFT) is first performed along the first dimension. This is followed by complex-to-complex transforms (FFT) along the second and third dimensions.\n\nInput data layout\n\nThe input PencilArray must satisfy the following constraints:\n\narray dimensions must not be permuted. This is the default when constructing PencilArrays.\nfor an M-dimensional domain decomposition (with M < N), the input array must be decomposed along the last M dimensions. For example, for a 2D decomposition of 3D data, the decomposed dimensions must be (2, 3). In particular, the first array dimension must not be distributed among different MPI processes.\nIn the PencilArrays package, the decomposed dimensions are specified at the moment of constructing a Pencil.\nthe element type must be compatible with the specified transform. For instance, real-to-complex transforms (Transforms.RFFT) require the input to be real floating point values. Other transforms, such as Transforms.R2R, accept both real and complex data.\n\nKeyword arguments\n\nThe keyword arguments fftw_flags and fftw_timelimit are passed to the FFTW plan creation functions (see AbstractFFTs docs).\npermute_dims determines whether the indices of the output data should be reversed. For instance, if the input data has global dimensions (Nx, Ny, Nz), then the output of a complex-to-complex FFT would have dimensions (Nz, Ny, Nx). This enables FFTs to always be performed along the first (i.e. fastest) array dimension, which could lead to performance gains. This option is enabled by default. For type inference reasons, it must be a value type (Val(true) or Val(false)).\ntranspose_method allows to select between implementations of the global data transpositions. See PencilArrays docs docs for details.\ntimer should be a TimerOutput object. See Measuring performance for details.\n\n\n\nPencilFFTPlan(\n dims_global::Dims{N}, transforms, proc_dims::Dims{M}, comm::MPI.Comm,\n [real_type = Float64]; extra_dims = (), kws...\n)\n\nCreate plan for N-dimensional transform.\n\nExtended help\n\nInstead of taking a PencilArray or a Pencil, this constructor requires the global dimensions of the input data, passed via the size_global argument.\n\nThe data is distributed over the MPI processes in the comm communicator. The distribution is performed over M dimensions (with M < N) according to the values in proc_dims, which specifies the number of MPI processes to put along each dimension.\n\nPencilArrays that may be transformed with the returned plan can be created using allocate_input.\n\nOptional arguments\n\nThe floating point precision can be selected by setting real_type parameter, which is Float64 by default.\nextra_dims may be used to specify the sizes of one or more extra dimensions that should not be transformed. These dimensions will be added to the rightmost (i.e. slowest) indices of the arrays. See Extra dimensions below for usage hints.\nsee the other constructor for more keyword arguments.\n\nExtra dimensions\n\nOne possible application of extra_dims is for describing the components of a vector or tensor field. However, this means that different PencilFFTPlans would need to be created for each kind of field (scalar, vector, ...). To avoid the creation of multiple plans, a possibly better alternative is to create tuples (or arrays) of PencilArrays using allocate_input and allocate_output.\n\nAnother more legitimate usage of extra_dims is to specify one or more Cartesian dimensions that should not be transformed nor split among MPI processes.\n\nExample\n\nSuppose we want to perform a 3D FFT of real data. The data is to be decomposed along two dimensions, over 8 MPI processes:\n\nsize_global = (64, 32, 128) # size of real input data\n\n# Perform real-to-complex transform along the first dimension, then\n# complex-to-complex transforms along the other dimensions.\ntransforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())\n# transforms = Transforms.RFFT() # this is equivalent to the above line\n\nproc_dims = (4, 2) # 2D decomposition\ncomm = MPI.COMM_WORLD\n\nplan = PencilFFTPlan(size_global, transforms, proc_dims, comm)\n\n\n\n\n\n","category":"type"},{"location":"PencilFFTs/#Allocating-data","page":"Distributed FFT plans","title":"Allocating data","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"allocate_input\nallocate_output","category":"page"},{"location":"PencilFFTs/#PencilFFTs.allocate_input","page":"Distributed FFT plans","title":"PencilFFTs.allocate_input","text":"allocate_input(p::PencilFFTPlan) -> PencilArray\nallocate_input(p::PencilFFTPlan, dims...) -> Array{PencilArray}\nallocate_input(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}\n\nAllocate uninitialised PencilArray that can hold input data for the given plan.\n\nThe second and third forms respectively allocate an array of PencilArrays of size dims, and a tuple of N PencilArrays.\n\nnote: In-place plans\nIf p is an in-place real-to-real or complex-to-complex plan, a ManyPencilArray is allocated. If p is an in-place real-to-complex plan, a ManyPencilArrayRFFT! is allocated. These types hold PencilArray wrappers for the input and output transforms (as well as for intermediate transforms) which share the same space in memory. The input and output PencilArrays should be respectively accessed by calling first(::ManyPencilArray) and last(::ManyPencilArray).ExampleSuppose p is an in-place PencilFFTPlan. Then,@assert is_inplace(p)\nA = allocate_input(p) :: ManyPencilArray\nv_in = first(A) :: PencilArray # input data view\nv_out = last(A) :: PencilArray # output data viewAlso note that in-place plans must be performed directly on the returned ManyPencilArray, and not on the contained PencilArray views:p * A # perform forward transform in-place\np \\ A # perform backward transform in-place\n# p * v_in # not allowed!!\n\n\n\n\n\n","category":"function"},{"location":"PencilFFTs/#PencilFFTs.allocate_output","page":"Distributed FFT plans","title":"PencilFFTs.allocate_output","text":"allocate_output(p::PencilFFTPlan) -> PencilArray\nallocate_output(p::PencilFFTPlan, dims...) -> Array{PencilArray}\nallocate_output(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}\n\nAllocate uninitialised PencilArray that can hold output data for the given plan.\n\nIf p is an in-place plan, a ManyPencilArray is allocated.\n\nSee allocate_input for details.\n\n\n\n\n\n","category":"function"},{"location":"PencilFFTs/#Methods","page":"Distributed FFT plans","title":"Methods","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"get_comm(::PencilFFTPlan)\nscale_factor(::PencilFFTPlan)\ntimer(::PencilFFTPlan)\nis_inplace(::PencilFFTPlan)","category":"page"},{"location":"PencilFFTs/#PencilArrays.Pencils.MPITopologies.get_comm-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilArrays.Pencils.MPITopologies.get_comm","text":"get_comm(p::PencilFFTPlan)\n\nGet MPI communicator associated to a PencilFFTPlan.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilFFTs.Transforms.scale_factor-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilFFTs.Transforms.scale_factor","text":"scale_factor(p::PencilFFTPlan)\n\nGet scale factor associated to a PencilFFTPlan.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilArrays.Pencils.timer-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilArrays.Pencils.timer","text":"timer(p::PencilFFTPlan)\n\nGet TimerOutput attached to a PencilFFTPlan.\n\nSee Measuring performance for details.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilFFTs.Transforms.is_inplace-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilFFTs.Transforms.is_inplace","text":"Transforms.is_inplace(p::PencilFFTPlan)\n\nReturns true if the given plan operates in-place on the input data, false otherwise.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#Internals","page":"Distributed FFT plans","title":"Internals","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"ManyPencilArrayRFFT!","category":"page"},{"location":"PencilFFTs/#PencilFFTs.ManyPencilArrayRFFT!","page":"Distributed FFT plans","title":"PencilFFTs.ManyPencilArrayRFFT!","text":"ManyPencilArrayRFFT!{T,N,M} <: AbstractManyPencilArray{N,M}\n\nContainer holding M different PencilArray views to the same underlying data buffer. All views share the same and dimensionality N. The element type T of the first view is real, that of subsequent views is Complex{T}. \n\nThis can be used to perform in-place real-to-complex plan, see alsoTransforms.RFFT!. It is used internally for such transforms by allocate_input and should not be constructed directly.\n\n\n\nManyPencilArrayRFFT!{T}(undef, pencils...; extra_dims=())\n\nCreate a ManyPencilArrayRFFT! container that can hold data of type T and Complex{T} associated to all the given Pencils.\n\nThe optional extra_dims argument is the same as for PencilArray.\n\nSee also ManyPencilArray\n\n\n\n\n\n","category":"type"}] +} diff --git a/v0.15.0/siteinfo.js b/v0.15.0/siteinfo.js new file mode 100644 index 00000000..74e5169e --- /dev/null +++ b/v0.15.0/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "v0.15.0"; diff --git a/previews/PR55/tutorial/index.html b/v0.15.0/tutorial/index.html similarity index 70% rename from previews/PR55/tutorial/index.html rename to v0.15.0/tutorial/index.html index 7fac6b57..5954ad28 100644 --- a/previews/PR55/tutorial/index.html +++ b/v0.15.0/tutorial/index.html @@ -1,5 +1,5 @@ -Tutorial · PencilFFTs.jl
                      Edit on GitHub

                      Tutorial

                      The following tutorial shows how to perform a 3D FFT of real periodic data defined on a grid of $N_x × N_y × N_z$ points.

                      +Tutorial · PencilFFTs.jl
                      Edit on GitHub

                      Tutorial

                      The following tutorial shows how to perform a 3D FFT of real periodic data defined on a grid of $N_x × N_y × N_z$ points.

                      + Strong scaling of PencilFFTs +

                      As seen above, PencilFFTs generally outperforms P3DFFT in its default setting. This is largely explained by the choice of using non-blocking point-to-point MPI communications (via MPI_Isend and MPI_Irecv), while P3DFFT uses collective MPI_Alltoallv calls. This enables PencilFFTs to perform data reordering operations on the partially received data while waiting for the incoming data, leading to better performance. Moreover, in contrast with P3DFFT, the high performance and scalability of PencilFFTs results from a highly generic code, handling decompositions in arbitrary dimensions and a relatively large (and extensible) variety of transformations.

                      Note that PencilFFTs can optionally use collective communications (using MPI_Alltoallv) instead of point-to-point communications. For details, see the docs for PencilFFTPlan and for PencilArray transpositions. As seen above, collective communications generally perform worse than point-to-point ones, and runtimes are nearly indistinguishable from those of P3DFFT.

                      Benchmark details

                      The benchmarks were performed using Julia 1.7-beta3 and Intel MPI 2019. We used PencilFFTs v0.12.5 with FFTW.jl v1.4.3 and MPI.jl v0.19.0. We used the Fortran implementation of P3DFFT, version 2.7.6, which was built with Intel 2019 compilers and linked to FFTW 3.3.8. The cluster where the benchmarks were run has Intel Cascade Lake 6248 processors with 2×20 cores per node.

                      The number of MPI processes along each decomposed dimension, $P_1$ and $P_2$, was automatically determined by a call to MPI_Dims_create, which tends to create a balanced decomposition with $P_1 ≈ P_2$. For instance, a total of 1024 processes is divided into $P_1 = P_2 = 32$. Different results may be obtained with other combinations, but this was not benchmarked.

                      The source files used to generate this benchmark, as well as the raw benchmark results, are all available in the PencilFFTs repo.

                      diff --git a/previews/PR55/generated/gradient/index.html b/v0.15.1/generated/gradient/index.html similarity index 76% rename from previews/PR55/generated/gradient/index.html rename to v0.15.1/generated/gradient/index.html index 0fb1d6d2..33465974 100644 --- a/previews/PR55/generated/gradient/index.html +++ b/v0.15.1/generated/gradient/index.html @@ -1,5 +1,5 @@ -Gradient of a scalar field · PencilFFTs.jl
                      Edit on GitHub

                      Gradient of a scalar field

                      This example shows different methods to compute the gradient of a real-valued 3D scalar field $θ(\bm{x})$ in Fourier space, where $\bm{x} = (x, y, z)$. It is assumed that the field is periodic with period $L = 2π$ along all dimensions.

                      General procedure

                      The discrete Fourier expansion of $θ$ writes

                      \[θ(\bm{x}) = ∑_{\bm{k} ∈ \Z^3} \hat{θ}(\bm{k}) \, e^{i \bm{k} ⋅ \bm{x}},\]

                      where $\bm{k} = (k_x, k_y, k_z)$ are the Fourier wave numbers and $\hat{θ}$ is the discrete Fourier transform of $θ$. Then, the spatial derivatives of $θ$ are given by

                      \[\frac{∂ θ(\bm{x})}{∂ x_i} = +Gradient of a scalar field · PencilFFTs.jl

                      Edit on GitHub

                      Gradient of a scalar field

                      This example shows different methods to compute the gradient of a real-valued 3D scalar field $θ(\bm{x})$ in Fourier space, where $\bm{x} = (x, y, z)$. It is assumed that the field is periodic with period $L = 2π$ along all dimensions.

                      General procedure

                      The discrete Fourier expansion of $θ$ writes

                      \[θ(\bm{x}) = ∑_{\bm{k} ∈ \Z^3} \hat{θ}(\bm{k}) \, e^{i \bm{k} ⋅ \bm{x}},\]

                      where $\bm{k} = (k_x, k_y, k_z)$ are the Fourier wave numbers and $\hat{θ}$ is the discrete Fourier transform of $θ$. Then, the spatial derivatives of $θ$ are given by

                      \[\frac{∂ θ(\bm{x})}{∂ x_i} = ∑_{\bm{k} ∈ \Z^3} i k_i \hat{θ}(\bm{k}) \, e^{i \bm{k} ⋅ \bm{x}},\]

                      where the subscript $i$ denotes one of the spatial components $x$, $y$ or $z$.

                      In other words, to compute $\bm{∇} θ = (∂_x θ, ∂_y θ, ∂_z θ)$, one has to:

                      1. transform $θ$ to Fourier space to obtain $\hat{θ}$,
                      2. multiply $\hat{θ}$ by $i \bm{k}$,
                      3. transform the result back to physical space to obtain $\bm{∇} θ$.

                      Preparation

                      In this section, we initialise a random real-valued scalar field $θ$ and compute its FFT. For more details see the Tutorial.

                      using MPI
 using PencilFFTs
 using Random
@@ -107,4 +107,4 @@
     gradient_grid_broadcast! (lazy)...        151.020 μs  [Method 4]
     gradient_local_parent!...                 92.248 μs
     gradient_local_linear!...                 91.212 μs
-    gradient_local_linear_explicit!...        90.992 μs

                      This page was generated using Literate.jl.

                      + gradient_local_linear_explicit!... 90.992 μs

                      This page was generated using Literate.jl.

                      diff --git a/v0.15.1/generated/in-place/index.html b/v0.15.1/generated/in-place/index.html new file mode 100644 index 00000000..7b95caae --- /dev/null +++ b/v0.15.1/generated/in-place/index.html @@ -0,0 +1,31 @@ + +In-place transforms · PencilFFTs.jl
                      Edit on GitHub

                      In-place transforms

                      Complex-to-complex and real-to-real transforms can be performed in-place, enabling important memory savings. The procedure is very similar to that of out-of-place transforms described in the tutorial. The differences are illustrated in the sections below.

                      Creating a domain partition

                      We start by partitioning a domain of dimensions $16×32×64$ along all available MPI processes.

                      using PencilFFTs
+using MPI
+MPI.Init()
+
+dims_global = (16, 32, 64)  # global dimensions
                      (16, 32, 64)

                      Such a partitioning is described by a Pencil object. Here we choose to decompose the domain along the last two dimensions. In this case, the actual number of processes along each of these dimensions is chosen automatically.

                      decomp_dims = (2, 3)
+comm = MPI.COMM_WORLD
+pen = Pencil(dims_global, decomp_dims, comm)
                      Decomposition of 3D data
+    Data dimensions: (16, 32, 64)
+    Decomposed dimensions: (2, 3)
+    Data permutation: NoPermutation()
+    Array type: Array
                      Allowed decompositions

                      Distributed transforms using PencilFFTs.jl require that the first dimension is not decomposed. In other words, if one wants to perform transforms, then decomp_dims above must not contain 1.

                      Creating in-place plans

                      To create an in-place plan, pass an in-place transform such as Transforms.FFT! or Transforms.R2R! to PencilFFTPlan. For instance:

                      # Perform a 3D in-place complex-to-complex FFT.
+transform = Transforms.FFT!()
+
+# Note that one can also combine different types of in-place transforms.
+# For instance:
+# transform = (
+#     Transforms.R2R!(FFTW.REDFT01),
+#     Transforms.FFT!(),
+#     Transforms.R2R!(FFTW.DHT),
+# )
                      FFT!

                      We can now create a distributed plan from the previously-created domain partition and the chosen transform.

                      plan = PencilFFTPlan(pen, transform)
                      Transforms: (FFT!, FFT!, FFT!)
+Input type: ComplexF64
+Global dimensions: (16, 32, 64) -> (16, 32, 64)
+MPI topology: 2D decomposition (2×1 processes)

                      Note that in-place real-to-complex transforms are not currently supported. (In other words, the RFFT! transform type is not defined.)

                      Allocating data

                      As with out-of-place plans, data should be allocated using allocate_input. The difference is that, for in-place plans, this function returns a ManyPencilArray object, which is a container holding multiple PencilArray views sharing the same memory space.

                      # Allocate data for the plan.
+# Since `plan` is in-place, this returns a `ManyPencilArray` container.
+A = allocate_input(plan)
+summary(A)
                      "ManyPencilArray{ComplexF64, 3, 3, Tuple{PencilArray{ComplexF64, 3, Base.ReshapedArray{ComplexF64, 3, SubArray{ComplexF64, 1, Vector{ComplexF64}, Tuple{Base.OneTo{Int64}}, true}, Tuple{}}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{ComplexF64, 3, Base." ⋯ 140 bytes ⋯ "tation{(2, 1, 3), 3}, Vector{UInt8}}}, PencilArray{ComplexF64, 3, Base.ReshapedArray{ComplexF64, 3, SubArray{ComplexF64, 1, Vector{ComplexF64}, Tuple{Base.OneTo{Int64}}, true}, Tuple{}}, 3, 0, Pencil{3, 2, Permutation{(3, 2, 1), 3}, Vector{UInt8}}}}, Vector{ComplexF64}}"

                      Note that allocate_output also works for in-place plans. In this case, it returns exactly the same thing as allocate_input.

                      As shown in the next section, in-place plans must be applied on the returned ManyPencilArray. On the other hand, one usually wants to access and modify data, and for this one needs the PencilArray views contained in the ManyPencilArray. The input and output array views can be obtained by calling first(::ManyPencilArray) and last(::ManyPencilArray).

                      For instance, we can initialise the input array with some data before transforming:

                      using Random
+u_in = first(A)  # input data view
+randn!(u_in)
+summary(u_in)
                      "16×16×64 PencilArray{ComplexF64, 3}(::Pencil{3, 2, NoPermutation, Array})"

                      Applying plans

                      Like in FFTW.jl, one can perform in-place transforms using the * and \ operators. As mentioned above, in-place plans must be applied on the ManyPencilArray containers returned by allocate_input.

                      plan * A;  # performs in-place forward transform

                      After performing an in-place transform, data contained in u_in has been overwritten and has no "physical" meaning. In other words, u_in should not be used at this point. To access the transformed data, one should retrieve the output data view using last(A).

                      For instance, to compute the global sum of the transformed data:

                      u_out = last(A)  # output data view
+sum(u_out)       # sum of transformed data (note that `sum` reduces over all processes)
                      -14040.358823494502 + 2033.9291021805539im

                      Finally, we can perform a backward transform and do stuff with the input view:

                      plan \ A;  # perform in-place backward transform

                      At this point, the data can be once again found in the input view u_in, while u_out should not be accessed.

                      This page was generated using Literate.jl.

                      diff --git a/v0.15.1/generated/navier_stokes/index.html b/v0.15.1/generated/navier_stokes/index.html new file mode 100644 index 00000000..f289aa6d --- /dev/null +++ b/v0.15.1/generated/navier_stokes/index.html @@ -0,0 +1,317 @@ + +Navier–Stokes equations · PencilFFTs.jl
                      Edit on GitHub

                      Navier–Stokes equations

                      In this example, we numerically solve the incompressible Navier–Stokes equations

                      \[∂_t \bm{v} + (\bm{v} ⋅ \bm{∇}) \bm{v} = -\frac{1}{ρ} \bm{∇} p + ν ∇^2 \bm{v}, +\quad \bm{∇} ⋅ \bm{v} = 0,\]

                      where $\bm{v}(\bm{x}, t)$ and $p(\bm{x}, t)$ are respectively the velocity and pressure fields, $ν$ is the fluid kinematic viscosity and $ρ$ is the fluid density.

                      We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.

                      First steps

                      We start by loading the required packages, initialising MPI and setting the simulation parameters.

                      using MPI
+using PencilFFTs
+
+MPI.Init()
+comm = MPI.COMM_WORLD
+procid = MPI.Comm_rank(comm) + 1
+
+# Simulation parameters
+Ns = (64, 64, 64)  # = (Nx, Ny, Nz)
+Ls = (2π, 2π, 2π)  # = (Lx, Ly, Lz)
+
+# Collocation points ("global" = over all processes).
+# We include the endpoint (length = N + 1) for convenience.
+xs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls)  # = (x, y, z)
                      (0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586)

                      Let's check the number of MPI processes over which we're running our simulation:

                      MPI.Comm_size(comm)
                      2

                      We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:

                      pen = Pencil(Ns, comm)
                      Decomposition of 3D data
+    Data dimensions: (64, 64, 64)
+    Decomposed dimensions: (2, 3)
+    Data permutation: NoPermutation()
+    Array type: Array

                      The subdomain associated to the local MPI process can be obtained using range_local:

                      range_local(pen)
                      (1:64, 1:32, 1:64)

                      We now construct a distributed vector field that follows the decomposition configuration we just created:

                      v⃗₀ = (
+    PencilArray{Float64}(undef, pen),  # vx
+    PencilArray{Float64}(undef, pen),  # vy
+    PencilArray{Float64}(undef, pen),  # vz
+)
+summary(v⃗₀[1])
                      "64×32×64 PencilArray{Float64, 3}(::Pencil{3, 2, NoPermutation, Array})"

                      We still need to fill this array with interesting values that represent a physical velocity field.

                      Initial condition

                      Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:

                      \[\begin{aligned} +v_x(x, y, z) &= u₀ \sin(k₀ x) \cos(k₀ y) \cos(k₀ z) \\ +v_y(x, y, z) &= -u₀ \cos(k₀ x) \sin(k₀ y) \cos(k₀ z) \\ +v_z(x, y, z) &= 0 +\end{aligned}\]

                      where $u₀$ and $k₀$ are two parameters setting the amplitude and the period of the velocity field.

                      To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:

                      grid = localgrid(pen, xs_global)
                      LocalRectilinearGrid{3} with coordinates:
+ (1) 0.0:0.09817477042468103:6.1850105367549055
+ (2) 0.0:0.09817477042468103:3.043417883165112
+ (3) 0.0:0.09817477042468103:6.1850105367549055

                      We can use this to initialise the velocity field:

                      u₀ = 1.0
+k₀ = 2π / Ls[1]  # should be integer if L = 2π (to preserve periodicity)
+
+@. v⃗₀[1] =  u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)
+@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)
+@. v⃗₀[3] =  0

                      Let's plot a 2D slice of the velocity field managed by the local MPI process:

                      using GLMakie
+
+# Compute the norm of a vector field represented by a tuple of arrays.
+function vecnorm(v⃗::NTuple)
+    vnorm = similar(v⃗[1])
+    for n ∈ eachindex(v⃗[1])
+        w = zero(eltype(vnorm))
+        for v ∈ v⃗
+            w += v[n]^2
+        end
+        vnorm[n] = sqrt(w)
+    end
+    vnorm
+end
+
+let fig = Figure(resolution = (700, 600))
+    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
+    vnorm = vecnorm(v⃗₀)
+    ct = contour!(
+        ax, grid.x, grid.y, grid.z, vnorm;
+        alpha = 0.2, levels = 4,
+        colormap = :viridis, colorrange = (0.0, 1.0),
+    )
+    cb = Colorbar(fig[1, 2], ct; label = "Velocity magnitude")
+    fig
+end

                      Velocity in Fourier space

                      In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series

                      \[\bm{v}(\bm{x}, t) = +∑_{\bm{k}} \hat{\bm{v}}_{\bm{k}}(t) \, e^{i \bm{k} ⋅ \bm{x}},\]

                      where $\bm{k} = (k_x, k_y, k_z)$ are the discrete wave numbers.

                      The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for $k_x$:

                      using AbstractFFTs: fftfreq, rfftfreq
+
+ks_global = (
+    rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]),  # kx | real-to-complex
+     fftfreq(Ns[2], 2π * Ns[2] / Ls[2]),  # ky | complex-to-complex
+     fftfreq(Ns[3], 2π * Ns[3] / Ls[3]),  # kz | complex-to-complex
+)
+
+ks_global[1]'
                      1×33 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
+ 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  27.0  28.0  29.0  30.0  31.0  32.0
                      ks_global[2]'
                      1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
+ 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0
                      ks_global[3]'
                      1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
+ 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0

                      To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:

                      plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())
                      Transforms: (RFFT, FFT, FFT)
+Input type: Float64
+Global dimensions: (64, 64, 64) -> (33, 64, 64)
+MPI topology: 2D decomposition (2×1 processes)

                      See PencilFFTPlan for details on creating plans and on optional keyword arguments.

                      We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$:

                      v̂s = plan .* v⃗₀
+summary(v̂s[1])
                      "16×64×64 PencilArray{ComplexF64, 3}(::Pencil{3, 2, Permutation{(3, 2, 1), 3}, Array})"

                      Note that, in Fourier space, the domain decomposition is performed along the directions $x$ and $y$:

                      pencil(v̂s[1])
                      Decomposition of 3D data
+    Data dimensions: (33, 64, 64)
+    Decomposed dimensions: (1, 2)
+    Data permutation: Permutation(3, 2, 1)
+    Array type: Array

                      This is because the 3D FFTs are performed one dimension at a time, with the $x$ direction first and the $z$ direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.

                      To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers $\bm{k}$ associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:

                      grid_fourier = localgrid(v̂s[1], ks_global)
                      LocalRectilinearGrid{3} with Permutation(3, 2, 1) and coordinates:
+ (1) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0]
+ (2) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]
+ (3) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]

                      As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, $\bm{ω} = \bm{∇} × \bm{v}$. In Fourier space, this becomes $\hat{\bm{ω}} = i \bm{k} × \hat{\bm{v}}$.

                      using StaticArrays: SVector
+using LinearAlgebra: ×
+
+function curl_fourier!(
+        ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,
+    ) where {N}
+    @inbounds for I ∈ eachindex(grid_fourier)
+        # We use StaticArrays for the cross product between small vectors.
+        ik⃗ = im * SVector(grid_fourier[I])
+        v⃗ = SVector(getindex.(v̂s, Ref(I)))  # = (v̂s[1][I], v̂s[2][I], ...)
+        ω⃗ = ik⃗ × v⃗
+        for n ∈ eachindex(ω⃗)
+            ω̂s[n][I] = ω⃗[n]
+        end
+    end
+    ω̂s
+end
+
+ω̂s = similar.(v̂s)
+curl_fourier!(ω̂s, v̂s, grid_fourier);

                      We finally transform back to physical space and plot the result:

                      ωs = plan .\ ω̂s
+
+let fig = Figure(resolution = (700, 600))
+    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
+    ω_norm = vecnorm(ωs)
+    ct = contour!(
+        ax, grid.x, grid.y, grid.z, ω_norm;
+        alpha = 0.1, levels = 0.8:0.2:2.0,
+        colormap = :viridis, colorrange = (0.8, 2.0),
+    )
+    cb = Colorbar(fig[1, 2], ct; label = "Vorticity magnitude")
+    fig
+end

                      Computing the non-linear term

                      One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as

                      \[∂_t \hat{\bm{v}}_{\bm{k}} = +- \mathcal{P}_{\bm{k}} \! \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right] +- ν |\bm{k}|^2 \hat{\bm{v}}_{\bm{k}} +\quad \text{ with } \quad +\mathcal{P}_{\bm{k}}(\hat{\bm{F}}_{\bm{k}}) = \left( I - \frac{\bm{k} ⊗ +\bm{k}}{|\bm{k}|^2} \right) \hat{\bm{F}}_{\bm{k}},\]

                      where $\mathcal{P}_{\bm{k}}$ is a projection operator allowing to preserve the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.

                      Now that we have the wave numbers $\bm{k}$, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$ of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, $\hat{\bm{F}}_{\bm{k}} = \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right]_{\bm{k}}$. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.

                      Below we implement a function that computes the non-linear term in Fourier space based on its convective form $(\bm{v} ⋅ \bm{∇}) \bm{v} = \bm{∇} ⋅ (\bm{v} ⊗ \bm{v})$. Note that this equivalence uses the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$.

                      using LinearAlgebra: mul!, ldiv!  # for applying FFT plans in-place
+
+# Compute non-linear term in Fourier space from velocity field in physical
+# space. Optional keyword arguments may be passed to avoid allocations.
+function ns_nonlinear!(
+        F̂s, vs, plan, grid_fourier;
+        vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),
+    )
+    # Compute F_i = ∂_j (v_i v_j) for each i.
+    # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)
+    w, ŵ = vbuf, v̂buf
+    @inbounds for (i, F̂i) ∈ enumerate(F̂s)
+        F̂i .= 0
+        vi = vs[i]
+        for (j, vj) ∈ enumerate(vs)
+            w .= vi .* vj     # w = v_i * v_j in physical space
+            mul!(ŵ, plan, w)  # same in Fourier space
+            # Add derivative in Fourier space
+            for I ∈ eachindex(grid_fourier)
+                k⃗ = grid_fourier[I]  # = (kx, ky, kz)
+                kj = k⃗[j]
+                F̂i[I] += im * kj * ŵ[I]
+            end
+        end
+    end
+    F̂s
+end
                      ns_nonlinear! (generic function with 1 method)

                      As an example, let's use this function on our initial velocity field:

                      F̂s = similar.(v̂s)
+ns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);

                      Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.

                      function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)
+    ks_max = maximum.(abs, ks_global)  # maximum stored wave numbers (kx_max, ky_max, kz_max)
+    ks_lim = (2 / 3) .* ks_max
+    @inbounds for I ∈ eachindex(grid_fourier)
+        k⃗ = grid_fourier[I]
+        if any(abs.(k⃗) .> ks_lim)
+            for ŵ ∈ ŵs
+                ŵ[I] = 0
+            end
+        end
+    end
+    ŵs
+end
+
+# We can apply this on the previously computed non-linear term:
+dealias_twothirds!(F̂s, grid_fourier, ks_global);

                      Finally, we implement the projection associated to the incompressibility condition:

                      function project_divergence_free!(ûs, grid_fourier)
+    @inbounds for I ∈ eachindex(grid_fourier)
+        k⃗ = grid_fourier[I]
+        k² = sum(abs2, k⃗)
+        iszero(k²) && continue  # avoid division by zero
+        û = getindex.(ûs, Ref(I))  # (ûs[1][I], ûs[2][I], ...)
+        for i ∈ eachindex(û)
+            ŵ = û[i]
+            for j ∈ eachindex(û)
+                ŵ -= k⃗[i] * k⃗[j] * û[j] / k²
+            end
+            ûs[i][I] = ŵ
+        end
+    end
+    ûs
+end
                      project_divergence_free! (generic function with 1 method)

                      We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:

                      v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)
+v̂s_proj .≈ v̂s  # the last one may be false because v_z = 0 initially
                      (true, true, false)

                      Putting it all together

                      To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:

                      function ns_rhs!(
+        dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,
+    ) where {N}
+    # 1. Compute non-linear term and dealias it
+    (; plan, cache, ks_global, grid_fourier) = p
+    F̂s = cache.F̂s
+    ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])
+    dealias_twothirds!(F̂s, grid_fourier, ks_global)
+
+    # 2. Project onto divergence-free space
+    project_divergence_free!(F̂s, grid_fourier)
+
+    # 3. Transform velocity to Fourier space
+    v̂s = cache.v̂s
+    map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)
+
+    # 4. Add viscous term (and multiply projected non-linear term by -1)
+    ν = p.ν
+    for n ∈ eachindex(v̂s)
+        v̂ = v̂s[n]
+        F̂ = F̂s[n]
+        @inbounds for I ∈ eachindex(grid_fourier)
+            k⃗ = grid_fourier[I]  # = (kx, ky, kz)
+            k² = sum(abs2, k⃗)
+            F̂[I] = -F̂[I] - ν * k² * v̂[I]
+        end
+    end
+
+    # 5. Transform RHS back to physical space
+    map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)
+
+    nothing
+end
                      ns_rhs! (generic function with 1 method)

                      For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.

                      using OrdinaryDiffEq
+using RecursiveArrayTools: ArrayPartition
+
+ns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)
+vs_init_ode = ArrayPartition(v⃗₀)
+summary(vs_init_ode)
                      "RecursiveArrayTools.ArrayPartition{Float64, Tuple{PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}}} with arrays:"

                      We now define solver parameters and temporary variables, and initialise the problem:

                      params = (;
+    ν = 5e-3,  # kinematic viscosity
+    plan, grid_fourier, ks_global,
+    cache = (
+        v̂s = similar.(v̂s),
+        F̂s = similar.(v̂s),
+    )
+)
+
+tspan = (0.0, 10.0)
+prob = ODEProblem(ns_rhs!, vs_init_ode, tspan, params)
+integrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);

                      We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum $E(k)$, to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity $ν$ must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large $k$), while high enough to allow the small-scale motions to be correctly resolved.

                      function energy_spectrum!(Ek, ks, v̂s, grid_fourier)
+    Nk = length(Ek)
+    @assert Nk == length(ks)
+    Ek .= 0
+    for I ∈ eachindex(grid_fourier)
+        k⃗ = grid_fourier[I]  # = (kx, ky, kz)
+        knorm = sqrt(sum(abs2, k⃗))
+        i = searchsortedfirst(ks, knorm)
+        i > Nk && continue
+        v⃗ = getindex.(v̂s, Ref(I))  # = (v̂s[1][I], v̂s[2][I], ...)
+        factor = k⃗[1] == 0 ? 1 : 2  # account for Hermitian symmetry and r2c transform
+        Ek[i] += factor * sum(abs2, v⃗) / 2
+    end
+    MPI.Allreduce!(Ek, +, get_comm(v̂s[1]))  # sum across all processes
+    Ek
+end
+
+ks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])
+Ek = similar(ks)
+v̂s = plan .* integrator.u.x
+energy_spectrum!(Ek, ks, v̂s, grid_fourier)
+Ek ./= scale_factor(plan)^2  # rescale energy
+
+curl_fourier!(ω̂s, v̂s, grid_fourier)
+ldiv!.(ωs, plan, ω̂s)
+ω⃗_plot = Observable(ωs)
+k_plot = @view ks[2:end]
+E_plot = Observable(@view Ek[2:end])
+t_plot = Observable(integrator.t)
+
+fig = let
+    fig = Figure(resolution = (1200, 600))
+    ax = Axis3(
+        fig[1, 1][1, 1]; title = @lift("t = $(round($t_plot, digits = 3))"),
+        aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z",
+    )
+    ω_mag = @lift vecnorm($ω⃗_plot)
+    ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)
+    ct = contour!(
+        ax, grid.x, grid.y, grid.z, ω_mag_norm;
+        alpha = 0.3, levels = 3,
+        colormap = :viridis, colorrange = (0.0, 1.0),
+    )
+    cb = Colorbar(fig[1, 1][1, 2], ct; label = "Normalised vorticity magnitude")
+    ax_sp = Axis(
+        fig[1, 2];
+        xlabel = "k", ylabel = "E(k)", xscale = log2, yscale = log10,
+        title = "Kinetic energy spectrum",
+    )
+    ylims!(ax_sp, 1e-8, 1e0)
+    scatterlines!(ax_sp, k_plot, E_plot)
+    ks_slope = exp.(range(log(2.5), log(25.0), length = 3))
+    E_fivethirds = @. 0.3 * ks_slope^(-5/3)
+    @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)
+    text!(ax_sp, L"k^{-5/3}"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))
+    fig
+end
+
+using Printf # hide
+with_xvfb = ENV["DISPLAY"] == ":99" # hide
+nstep = 0  # hide
+const tmpdir = mktempdir()  # hide
+filename_frame(procid, nstep) = joinpath(tmpdir, @sprintf("proc%d_%04d.png", procid, nstep)) # hide
+record(fig, "vorticity_proc$procid.mp4"; framerate = 10) do io
+    with_xvfb && recordframe!(io) # hide
+    while integrator.t < 20
+        dt = 0.001
+        step!(integrator, dt)
+        t_plot[] = integrator.t
+        mul!.(v̂s, plan, integrator.u.x)  # current velocity in Fourier space
+        curl_fourier!(ω̂s, v̂s, grid_fourier)
+        ldiv!.(ω⃗_plot[], plan, ω̂s)
+        ω⃗_plot[] = ω⃗_plot[]  # to force updating the plot
+        energy_spectrum!(Ek, ks, v̂s, grid_fourier)
+        Ek ./= scale_factor(plan)^2  # rescale energy
+        E_plot[] = E_plot[]
+        global nstep += 1  # hide
+        with_xvfb ?  # hide
+        save(filename_frame(procid, nstep), fig) :  # hide
+        recordframe!(io)
+    end
+end;
+
+if with_xvfb  # hide
+    run(pipeline(`ffmpeg -y -r 10 -i $tmpdir/proc$(procid)_%04d.png -c:v libx264 -vf "fps=25,format=yuv420p" vorticity_proc$procid.mp4`; stdout = "ffmpeg.out", stderr = "ffmpeg.err"))   # hide
+end              # hide
+nothing          # hide
                      + +

                      This page was generated using Literate.jl.

                      diff --git a/previews/PR62/img/benchmark_idris.svg b/v0.15.1/img/benchmark_idris.svg similarity index 100% rename from previews/PR62/img/benchmark_idris.svg rename to v0.15.1/img/benchmark_idris.svg diff --git a/v0.15.1/img/pencils.svg b/v0.15.1/img/pencils.svg new file mode 100644 index 00000000..3e110436 --- /dev/null +++ b/v0.15.1/img/pencils.svg @@ -0,0 +1,1067 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/v0.15.1/index.html b/v0.15.1/index.html new file mode 100644 index 00000000..a345a637 --- /dev/null +++ b/v0.15.1/index.html @@ -0,0 +1,7 @@ + +Home · PencilFFTs.jl
                      Edit on GitHub

                      PencilFFTs

                      Fast Fourier transforms of MPI-distributed Julia arrays.

                      Introduction

                      This package provides multidimensional FFTs and related transforms on MPI-distributed Julia arrays via the PencilArrays package.

                      The name of this package originates from the decomposition of 3D domains along two out of three dimensions, sometimes called pencil decomposition. This is illustrated by the figure below,[1] where each coloured block is managed by a different MPI process. Typically, one wants to compute FFTs on a scalar or vector field along the three spatial dimensions. In the case of a pencil decomposition, 3D FFTs are performed one dimension at a time, along the non-decomposed direction. Transforms must then be interleaved with global data transpositions to switch between pencil configurations. In high-performance computing environments, such data transpositions are generally the most expensive part of a parallel FFT computation, due to the large cost of communications between computing nodes.

                      + Pencil decomposition of 3D domains +

                      More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension $N$. The decompositions can be performed along an arbitrary number $M < N$ of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.

                      The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.

                      Features

                      • distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;

                      • FFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;

                      • in-place and out-of-place transforms;

                      • high scalability up to (at least) tens of thousands of MPI processes.

                      Installation

                      PencilFFTs can be installed using the Julia package manager:

                      julia> ] add PencilFFTs

                      Similar projects

                      • FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.

                      • PFFT is a very general parallel FFT library written in C.

                      • P3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.

                      • 2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.

                      • 1Figure adapted from this PhD thesis.
                      • 2For the pencil decomposition represented in the figure, $N = 3$ and $M = 2$.
                      diff --git a/v0.15.1/search/index.html b/v0.15.1/search/index.html new file mode 100644 index 00000000..0175e1da --- /dev/null +++ b/v0.15.1/search/index.html @@ -0,0 +1,2 @@ + +Search · PencilFFTs.jl

                      Loading search...

                        diff --git a/v0.15.1/search_index.js b/v0.15.1/search_index.js new file mode 100644 index 00000000..85838151 --- /dev/null +++ b/v0.15.1/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"tutorial/#Tutorial","page":"Tutorial","title":"Tutorial","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The following tutorial shows how to perform a 3D FFT of real periodic data defined on a grid of N_x N_y N_z points.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"
                        \n \n \n
                        ","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"By default, the domain is distributed on a 2D MPI topology of dimensions N_1 N_2. As an example, the above figure shows such a topology with N_1 = 4 and N_2 = 3, for a total of 12 MPI processes.","category":"page"},{"location":"tutorial/#tutorial:creating_plans","page":"Tutorial","title":"Creating plans","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The first thing to do is to create a domain decomposition configuration for the given dataset dimensions N_x N_y N_z. In the framework of PencilArrays, such a configuration is described by a Pencil object. As described in the PencilArrays docs, we can let the Pencil constructor automatically determine such a configuration. For this, only an MPI communicator and the dataset dimensions are needed:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using MPI\nusing PencilFFTs\n\nMPI.Init()\ncomm = MPI.COMM_WORLD\n\n# Input data dimensions (Nx × Ny × Nz)\ndims = (16, 32, 64)\npen = Pencil(dims, comm)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"By default this creates a 2D decomposition (for the case of a 3D dataset), but one can change this as detailed in the PencilArrays documentation linked above.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"We can now create a PencilFFTPlan, which requires information on decomposition configuration (the Pencil object) and on the transforms that will be applied:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# Apply a 3D real-to-complex (r2c) FFT.\ntransform = Transforms.RFFT()\n\n# Note that, for more control, one can instead separately specify the transforms along each dimension:\n# transform = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())\n\n# Create plan\nplan = PencilFFTPlan(pen, transform)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"See the PencilFFTPlan constructor for details on the accepted options, and the Transforms module for the possible transforms. It is also possible to enable fine-grained performance measurements via the TimerOutputs package, as described in Measuring performance.","category":"page"},{"location":"tutorial/#Allocating-data","page":"Tutorial","title":"Allocating data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Next, we want to apply the plan on some data. Transforms may only be applied on PencilArrays, which are array wrappers that include MPI decomposition information (in some sense, analogous to DistributedArrays in Julia's distributed computing approach). The helper function allocate_input can be used to allocate a PencilArray that is compatible with our plan:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# In our example, this returns a 3D PencilArray of real data (Float64).\nu = allocate_input(plan)\n\n# Fill the array with some (random) data\nusing Random\nrandn!(u)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"PencilArrays are a subtype of AbstractArray, and thus they support all common array operations.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Similarly, to preallocate output data, one can use allocate_output:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# In our example, this returns a 3D PencilArray of complex data (Complex{Float64}).\nv = allocate_output(plan)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"This is only required if one wants to apply the plans using a preallocated output (with mul!, see right below).","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The data types returned by allocate_input and allocate_output are slightly different when working with in-place transforms. See the in-place example for details.","category":"page"},{"location":"tutorial/#Applying-plans","page":"Tutorial","title":"Applying plans","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The interface to apply plans is consistent with that of AbstractFFTs. Namely, * and mul! are respectively used for forward transforms without and with preallocated output data. Similarly, \\ and ldiv! are used for backward transforms.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using LinearAlgebra # for mul!, ldiv!\n\n# Apply plan on `u` with `v` as an output\nmul!(v, plan, u)\n\n# Apply backward plan on `v` with `w` as an output\nw = similar(u)\nldiv!(w, plan, v) # now w ≈ u","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Note that, consistently with AbstractFFTs, normalisation is performed at the end of a backward transform, so that the original data is recovered when applying a forward followed by a backward transform.","category":"page"},{"location":"tutorial/#Accessing-and-modifying-data","page":"Tutorial","title":"Accessing and modifying data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For any given MPI process, a PencilArray holds the data associated to its local partition in the global geometry. PencilArrays are accessed using local indices that start at 1, regardless of the location of the local process in the MPI topology. Note that PencilArrays, being based on regular Arrays, support both linear and Cartesian indexing (see the Julia docs for details).","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For convenience, the global_view function can be used to generate an OffsetArray wrapper that takes global indices.","category":"page"},{"location":"tutorial/#tutorial:output_data_layout","page":"Tutorial","title":"Output data layout","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"In memory, the dimensions of the transform output are by default reversed with respect to the input. That is, if the order of indices in the input data is (x, y, z), then the output has order (z, y, x) in memory. This detail is hidden from the user, and output arrays are always accessed in the same order as the input data, regardless of the underlying output dimension permutation. This applies to PencilArrays and to OffsetArrays returned by global_view.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The reasoning behind dimension permutations, is that they allow to always perform FFTs along the fastest array dimension and to avoid a local data transposition, resulting in performance gains. A similar approach is followed by other parallel FFT libraries. FFTW itself, in its distributed-memory routines, includes a flag that enables a similar behaviour. In PencilFFTs, index permutation is the default, but it can be disabled via the permute_dims flag of PencilFFTPlan.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"A great deal of work has been spent in making generic index permutations as efficient as possible, both in intermediate and in the output state of the multidimensional transforms. This has been achieved, in part, by making sure that permutations such as (3, 2, 1) are compile-time constants.","category":"page"},{"location":"tutorial/#Further-reading","page":"Tutorial","title":"Further reading","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For details on working with PencilArrays see the PencilArrays docs.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The examples on the sidebar further illustrate the use of transforms and provide an introduction to working with MPI-distributed data in the form of PencilArrays. In particular, the gradient example illustrates different ways of computing things using Fourier-transformed distributed arrays. Then, the incompressible Navier–Stokes example is a more advanced and complete example of a possible application of the PencilFFTs package.","category":"page"},{"location":"benchmarks/#Benchmarks","page":"Benchmarks","title":"Benchmarks","text":"","category":"section"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The performance of PencilFFTs.jl is comparable to that of other open-source parallel FFT libraries implemented in lower-level languages. Below, we show comparisons with the Fortran implementation of P3DFFT, possibly the most popular of these libraries. The benchmarks were performed on the Jean–Zay cluster of the IDRIS French computing centre (CNRS).","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The figure below shows strong scaling benchmarks of 3D real-to-complex FFTs using 2D (\"pencil\") decomposition. The benchmarks were run for input arrays of dimensions N_x N_y N_z = 512^3, 1024^3 and 2048^3. Each timing is averaged over 100 repetitions.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"
                        \n \n \n
                        ","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"As seen above, PencilFFTs generally outperforms P3DFFT in its default setting. This is largely explained by the choice of using non-blocking point-to-point MPI communications (via MPI_Isend and MPI_Irecv), while P3DFFT uses collective MPI_Alltoallv calls. This enables PencilFFTs to perform data reordering operations on the partially received data while waiting for the incoming data, leading to better performance. Moreover, in contrast with P3DFFT, the high performance and scalability of PencilFFTs results from a highly generic code, handling decompositions in arbitrary dimensions and a relatively large (and extensible) variety of transformations.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"Note that PencilFFTs can optionally use collective communications (using MPI_Alltoallv) instead of point-to-point communications. For details, see the docs for PencilFFTPlan and for PencilArray transpositions. As seen above, collective communications generally perform worse than point-to-point ones, and runtimes are nearly indistinguishable from those of P3DFFT.","category":"page"},{"location":"benchmarks/#Benchmark-details","page":"Benchmarks","title":"Benchmark details","text":"","category":"section"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The benchmarks were performed using Julia 1.7-beta3 and Intel MPI 2019. We used PencilFFTs v0.12.5 with FFTW.jl v1.4.3 and MPI.jl v0.19.0. We used the Fortran implementation of P3DFFT, version 2.7.6, which was built with Intel 2019 compilers and linked to FFTW 3.3.8. The cluster where the benchmarks were run has Intel Cascade Lake 6248 processors with 2×20 cores per node.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The number of MPI processes along each decomposed dimension, P_1 and P_2, was automatically determined by a call to MPI_Dims_create, which tends to create a balanced decomposition with P_1 P_2. For instance, a total of 1024 processes is divided into P_1 = P_2 = 32. Different results may be obtained with other combinations, but this was not benchmarked.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The source files used to generate this benchmark, as well as the raw benchmark results, are all available in the PencilFFTs repo.","category":"page"},{"location":"GlobalFFTParams/#Global-FFT-parameters","page":"Global FFT parameters","title":"Global FFT parameters","text":"","category":"section"},{"location":"GlobalFFTParams/","page":"Global FFT parameters","title":"Global FFT parameters","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"GlobalFFTParams/","page":"Global FFT parameters","title":"Global FFT parameters","text":"GlobalFFTParams","category":"page"},{"location":"GlobalFFTParams/#PencilFFTs.GlobalFFTParams","page":"Global FFT parameters","title":"PencilFFTs.GlobalFFTParams","text":"GlobalFFTParams{T, N, inplace}\n\nSpecifies the global parameters for an N-dimensional distributed transform. These include the element type T and global data sizes of input and output data, as well as the transform types to be performed along each dimension.\n\n\n\nGlobalFFTParams(size_global, transforms, [real_type=Float64])\n\nDefine parameters for N-dimensional transform.\n\ntransforms must be a tuple of length N specifying the transforms to be applied along each dimension. Each element must be a subtype of Transforms.AbstractTransform. For all the possible transforms, see Transform types.\n\nThe element type must be a real type accepted by FFTW, i.e. either Float32 or Float64.\n\nNote that the transforms are applied one dimension at a time, with the leftmost dimension first for forward transforms.\n\nExample\n\nTo perform a 3D FFT of real data, first a real-to-complex FFT must be applied along the first dimension, followed by two complex-to-complex FFTs along the other dimensions:\n\njulia> size_global = (64, 32, 128); # size of real input data\n\njulia> transforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT());\n\njulia> fft_params = PencilFFTs.GlobalFFTParams(size_global, transforms)\nTransforms: (RFFT, FFT, FFT)\nInput type: Float64\nGlobal dimensions: (64, 32, 128) -> (33, 32, 128)\n\n\n\n\n\n","category":"type"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"EditURL = \"https://github.com/jipolanco/PencilFFTs.jl/blob/master/docs/examples/in-place.jl\"","category":"page"},{"location":"generated/in-place/#In-place-transforms","page":"In-place transforms","title":"In-place transforms","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Complex-to-complex and real-to-real transforms can be performed in-place, enabling important memory savings. The procedure is very similar to that of out-of-place transforms described in the tutorial. The differences are illustrated in the sections below.","category":"page"},{"location":"generated/in-place/#Creating-a-domain-partition","page":"In-place transforms","title":"Creating a domain partition","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"We start by partitioning a domain of dimensions 163264 along all available MPI processes.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"using PencilFFTs\nusing MPI\nMPI.Init()\n\ndims_global = (16, 32, 64) # global dimensions","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Such a partitioning is described by a Pencil object. Here we choose to decompose the domain along the last two dimensions. In this case, the actual number of processes along each of these dimensions is chosen automatically.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"decomp_dims = (2, 3)\ncomm = MPI.COMM_WORLD\npen = Pencil(dims_global, decomp_dims, comm)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"warning: Allowed decompositions\nDistributed transforms using PencilFFTs.jl require that the first dimension is not decomposed. In other words, if one wants to perform transforms, then decomp_dims above must not contain 1.","category":"page"},{"location":"generated/in-place/#Creating-in-place-plans","page":"In-place transforms","title":"Creating in-place plans","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"To create an in-place plan, pass an in-place transform such as Transforms.FFT! or Transforms.R2R! to PencilFFTPlan. For instance:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"# Perform a 3D in-place complex-to-complex FFT.\ntransform = Transforms.FFT!()\n\n# Note that one can also combine different types of in-place transforms.\n# For instance:\n# transform = (\n# Transforms.R2R!(FFTW.REDFT01),\n# Transforms.FFT!(),\n# Transforms.R2R!(FFTW.DHT),\n# )","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"We can now create a distributed plan from the previously-created domain partition and the chosen transform.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan = PencilFFTPlan(pen, transform)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Note that in-place real-to-complex transforms are not currently supported. (In other words, the RFFT! transform type is not defined.)","category":"page"},{"location":"generated/in-place/#Allocating-data","page":"In-place transforms","title":"Allocating data","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"As with out-of-place plans, data should be allocated using allocate_input. The difference is that, for in-place plans, this function returns a ManyPencilArray object, which is a container holding multiple PencilArray views sharing the same memory space.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"# Allocate data for the plan.\n# Since `plan` is in-place, this returns a `ManyPencilArray` container.\nA = allocate_input(plan)\nsummary(A)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Note that allocate_output also works for in-place plans. In this case, it returns exactly the same thing as allocate_input.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"As shown in the next section, in-place plans must be applied on the returned ManyPencilArray. On the other hand, one usually wants to access and modify data, and for this one needs the PencilArray views contained in the ManyPencilArray. The input and output array views can be obtained by calling first(::ManyPencilArray) and last(::ManyPencilArray).","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"For instance, we can initialise the input array with some data before transforming:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"using Random\nu_in = first(A) # input data view\nrandn!(u_in)\nsummary(u_in)","category":"page"},{"location":"generated/in-place/#Applying-plans","page":"In-place transforms","title":"Applying plans","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Like in FFTW.jl, one can perform in-place transforms using the * and \\ operators. As mentioned above, in-place plans must be applied on the ManyPencilArray containers returned by allocate_input.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan * A; # performs in-place forward transform\nnothing #hide","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"After performing an in-place transform, data contained in u_in has been overwritten and has no \"physical\" meaning. In other words, u_in should not be used at this point. To access the transformed data, one should retrieve the output data view using last(A).","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"For instance, to compute the global sum of the transformed data:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"u_out = last(A) # output data view\nsum(u_out) # sum of transformed data (note that `sum` reduces over all processes)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Finally, we can perform a backward transform and do stuff with the input view:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan \\ A; # perform in-place backward transform\nnothing #hide","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"At this point, the data can be once again found in the input view u_in, while u_out should not be accessed.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"This page was generated using Literate.jl.","category":"page"},{"location":"Transforms/#Available-transforms","page":"Available transforms","title":"Available transforms","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"CurrentModule = PencilFFTs.Transforms","category":"page"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"Transforms","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms","page":"Available transforms","title":"PencilFFTs.Transforms","text":"Defines different one-dimensional FFT-based transforms.\n\nThe transforms are all subtypes of an AbstractTransform type.\n\nWhen possible, the names of the transforms are kept consistent with the functions exported by AbstractFFTs.jl and FFTW.jl.\n\n\n\n\n\n","category":"module"},{"location":"Transforms/#Transform-types","page":"Available transforms","title":"Transform types","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"FFT\nFFT!\nBFFT\nBFFT!\n\nRFFT\nRFFT!\nBRFFT\nBRFFT!\n\nR2R\nR2R!\n\nNoTransform\nNoTransform!","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.FFT","page":"Available transforms","title":"PencilFFTs.Transforms.FFT","text":"FFT()\n\nComplex-to-complex FFT.\n\nSee also AbstractFFTs.fft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.FFT!","page":"Available transforms","title":"PencilFFTs.Transforms.FFT!","text":"FFT!()\n\nIn-place version of FFT.\n\nSee also AbstractFFTs.fft!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BFFT","page":"Available transforms","title":"PencilFFTs.Transforms.BFFT","text":"BFFT()\n\nUnnormalised backward complex-to-complex FFT.\n\nLike AbstractFFTs.bfft, this transform is not normalised. To obtain the inverse transform, divide the output by the length of the transformed dimension.\n\nSee also AbstractFFTs.bfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.BFFT!","text":"BFFT()\n\nIn-place version of BFFT.\n\nSee also AbstractFFTs.bfft!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.RFFT","page":"Available transforms","title":"PencilFFTs.Transforms.RFFT","text":"RFFT()\n\nReal-to-complex FFT.\n\nSee also AbstractFFTs.rfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.RFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.RFFT!","text":"RFFT!()\n\nIn-place version of RFFT.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BRFFT","page":"Available transforms","title":"PencilFFTs.Transforms.BRFFT","text":"BRFFT(d::Integer)\nBRFFT((d1, d2, ..., dN))\n\nUnnormalised inverse of RFFT.\n\nTo obtain the inverse transform, divide the output by the length of the transformed dimension (of the real output array).\n\nAs described in the AbstractFFTs docs, the length of the output cannot be fully inferred from the input length. For this reason, the BRFFT constructor accepts an optional d argument indicating the output length.\n\nFor multidimensional datasets, a tuple of dimensions (d1, d2, ..., dN) may also be passed. This is equivalent to passing just dN. In this case, the last dimension (dN) is the one that changes size between the input and output. Note that this is the opposite of FFTW.brfft. The reason is that, in PencilFFTs, the last dimension is the one along which a complex-to-real transform is performed.\n\nSee also AbstractFFTs.brfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BRFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.BRFFT!","text":"BRFFT!(d::Integer)\nBRFFT!((d1, d2, ..., dN))\n\nIn-place version of BRFFT.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.R2R","page":"Available transforms","title":"PencilFFTs.Transforms.R2R","text":"R2R(kind)\n\nReal-to-real transform of type kind.\n\nThe possible values of kind are those described in the FFTW.r2r docs and the FFTW manual:\n\ndiscrete cosine transforms: FFTW.REDFT00, FFTW.REDFT01, FFTW.REDFFT10, FFTW.REDFFT11\ndiscrete sine transforms: FFTW.RODFT00, FFTW.RODFT01, FFTW.RODFFT10, FFTW.RODFFT11\ndiscrete Hartley transform: FFTW.DHT\n\nNote: half-complex format DFTs (FFTW.R2HC, FFTW.HC2R) are not currently supported.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.R2R!","page":"Available transforms","title":"PencilFFTs.Transforms.R2R!","text":"R2R!(kind)\n\nIn-place version of R2R.\n\nSee also FFTW.r2r!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.NoTransform","page":"Available transforms","title":"PencilFFTs.Transforms.NoTransform","text":"NoTransform()\n\nIdentity transform.\n\nSpecifies that no transformation should be applied.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.NoTransform!","page":"Available transforms","title":"PencilFFTs.Transforms.NoTransform!","text":"NoTransform!()\n\nIn-place version of NoTransform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#Internals","page":"Available transforms","title":"Internals","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"What follows is used internally in PencilFFTs.","category":"page"},{"location":"Transforms/#Types","page":"Available transforms","title":"Types","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"AbstractCustomPlan\nAbstractTransform\nIdentityPlan\nIdentityPlan!\nPlan","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.AbstractCustomPlan","page":"Available transforms","title":"PencilFFTs.Transforms.AbstractCustomPlan","text":"AbstractCustomPlan\n\nAbstract type defining a custom plan, to be used as an alternative to FFTW plans (FFTW.FFTWPlan).\n\nThe only custom plan defined in this module is IdentityPlan. The user can define other custom plans that are also subtypes of AbstractCustomPlan.\n\nNote that plan returns a subtype of either AbstractFFTs.Plan or AbstractCustomPlan.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.AbstractTransform","page":"Available transforms","title":"PencilFFTs.Transforms.AbstractTransform","text":"AbstractTransform\n\nSpecifies a one-dimensional FFT-based transform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.IdentityPlan","page":"Available transforms","title":"PencilFFTs.Transforms.IdentityPlan","text":"IdentityPlan\n\nType of plan associated to NoTransform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.IdentityPlan!","page":"Available transforms","title":"PencilFFTs.Transforms.IdentityPlan!","text":"IdentityPlan!\n\nType of plan associated to NoTransform!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.Plan","page":"Available transforms","title":"PencilFFTs.Transforms.Plan","text":"Plan = Union{AbstractFFTs.Plan, AbstractCustomPlan}\n\nUnion type representing any plan returned by plan.\n\nSee also AbstractCustomPlan.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#Functions","page":"Available transforms","title":"Functions","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"plan\n\nbinv\nscale_factor\n\neltype_input\neltype_output\nexpand_dims\nis_inplace\nkind\nlength_output","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.plan","page":"Available transforms","title":"PencilFFTs.Transforms.plan","text":"plan(transform::AbstractTransform, A, [dims];\n flags=FFTW.ESTIMATE, timelimit=Inf)\n\nCreate plan to transform array A along dimensions dims.\n\nIf dims is not specified, all dimensions of A are transformed.\n\nFor FFT plans, this function wraps the AbstractFFTs.jl and FFTW.jl plan creation functions. For more details on the function arguments, see AbstractFFTs.plan_fft.\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.binv","page":"Available transforms","title":"PencilFFTs.Transforms.binv","text":"binv(transform::AbstractTransform, d::Integer)\n\nReturns the backwards transform associated to the given transform.\n\nThe second argument must be the length of the first transformed dimension in the forward transform. It is used in particular when transform = RFFT(), to determine the length of the inverse (complex-to-real) transform. See the AbstractFFTs.irfft docs for details.\n\nThe backwards transform returned by this function is not normalised. The normalisation factor for a given array can be obtained by calling scale_factor.\n\nExample\n\njulia> binv(Transforms.FFT(), 42)\nBFFT\n\njulia> binv(Transforms.BRFFT(9), 42)\nRFFT\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.scale_factor","page":"Available transforms","title":"PencilFFTs.Transforms.scale_factor","text":"scale_factor(transform::AbstractTransform, A, [dims = 1:ndims(A)])\n\nGet factor required to normalise the given array after a transformation along dimensions dims (all dimensions by default).\n\nThe array A must have the dimensions of the transform input.\n\nImportant: the dimensions dims must be the same that were passed to plan.\n\nExamples\n\njulia> C = zeros(ComplexF32, 3, 4, 5);\n\njulia> scale_factor(Transforms.FFT(), C)\n60\n\njulia> scale_factor(Transforms.BFFT(), C)\n60\n\njulia> scale_factor(Transforms.BFFT(), C, 2:3)\n20\n\njulia> R = zeros(Float64, 3, 4, 5);\n\njulia> scale_factor(Transforms.RFFT(), R, 2)\n4\n\njulia> scale_factor(Transforms.RFFT(), R, 2:3)\n20\n\njulia> scale_factor(Transforms.BRFFT(8), C)\n96\n\njulia> scale_factor(Transforms.BRFFT(9), C)\n108\n\nThis will fail because the input of RFFT is real, and R is a complex array:\n\njulia> scale_factor(Transforms.RFFT(), C, 2:3)\nERROR: MethodError: no method matching scale_factor(::PencilFFTs.Transforms.RFFT, ::Array{ComplexF32, 3}, ::UnitRange{Int64})\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.eltype_input","page":"Available transforms","title":"PencilFFTs.Transforms.eltype_input","text":"eltype_input(transform::AbstractTransform, real_type<:AbstractFloat)\n\nDetermine input data type for a given transform given the floating point precision of the input data.\n\nSome transforms, such as R2R and NoTransform, can take both real and complex data. For those kinds of transforms, nothing is returned.\n\nExample\n\njulia> eltype_input(Transforms.FFT(), Float32)\nComplexF32 (alias for Complex{Float32})\n\njulia> eltype_input(Transforms.RFFT(), Float64)\nFloat64\n\njulia> eltype_input(Transforms.R2R(FFTW.REDFT01), Float64) # nothing\n\njulia> eltype_input(Transforms.NoTransform(), Float64) # nothing\n\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.eltype_output","page":"Available transforms","title":"PencilFFTs.Transforms.eltype_output","text":"eltype_output(transform::AbstractTransform, eltype_input)\n\nReturns the output data type for a given transform given the input type.\n\nThrows ArgumentError if the input data type is incompatible with the transform type.\n\nExample\n\njulia> eltype_output(Transforms.NoTransform(), Float32)\nFloat32\n\njulia> eltype_output(Transforms.RFFT(), Float64)\nComplexF64 (alias for Complex{Float64})\n\njulia> eltype_output(Transforms.BRFFT(4), ComplexF32)\nFloat32\n\njulia> eltype_output(Transforms.FFT(), Float64)\nERROR: ArgumentError: invalid input data type for PencilFFTs.Transforms.FFT: Float64\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.expand_dims","page":"Available transforms","title":"PencilFFTs.Transforms.expand_dims","text":"expand_dims(transform::AbstractTransform, Val(N))\n\nExpand a single multidimensional transform into one transform per dimension.\n\nExample\n\n# Expand a real-to-complex transform in 3 dimensions.\njulia> expand_dims(Transforms.RFFT(), Val(3))\n(RFFT, FFT, FFT)\n\njulia> expand_dims(Transforms.BRFFT(4), Val(3))\n(BFFT, BFFT, BRFFT{even})\n\njulia> expand_dims(Transforms.NoTransform(), Val(2))\n(NoTransform, NoTransform)\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.is_inplace","page":"Available transforms","title":"PencilFFTs.Transforms.is_inplace","text":"is_inplace(transform::AbstractTransform) -> Bool\nis_inplace(transforms::Vararg{AbtractTransform}) -> Union{Bool, Nothing}\n\nCheck whether a transform or a list of transforms is performed in-place.\n\nIf the list of transforms has a combination of in-place and out-of-place transforms, nothing is returned.\n\nExample\n\njulia> is_inplace(Transforms.RFFT())\nfalse\n\njulia> is_inplace(Transforms.NoTransform!())\ntrue\n\njulia> is_inplace(Transforms.FFT!(), Transforms.R2R!(FFTW.REDFT01))\ntrue\n\njulia> is_inplace(Transforms.FFT(), Transforms.R2R(FFTW.REDFT01))\nfalse\n\njulia> is_inplace(Transforms.FFT(), Transforms.R2R!(FFTW.REDFT01)) === nothing\ntrue\n\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.kind","page":"Available transforms","title":"PencilFFTs.Transforms.kind","text":"kind(transform::R2R)\n\nGet kind of real-to-real transform.\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.length_output","page":"Available transforms","title":"PencilFFTs.Transforms.length_output","text":"length_output(transform::AbstractTransform, length_in::Integer)\n\nReturns the length of the transform output, given the length of its input.\n\nThe input and output lengths are specified in terms of the respective input and output datatypes. For instance, for real-to-complex transforms, these are respectively the length of input real data and of output complex data.\n\n\n\n\n\n","category":"function"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"EditURL = \"https://github.com/jipolanco/PencilFFTs.jl/blob/master/docs/examples/navier_stokes.jl\"","category":"page"},{"location":"generated/navier_stokes/#Navier–Stokes-equations","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"In this example, we numerically solve the incompressible Navier–Stokes equations","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"_t bmv + (bmv bm) bmv = -frac1ρ bm p + ν ^2 bmv\nquad bm bmv = 0","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where bmv(bmx t) and p(bmx t) are respectively the velocity and pressure fields, ν is the fluid kinematic viscosity and ρ is the fluid density.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.","category":"page"},{"location":"generated/navier_stokes/#First-steps","page":"Navier–Stokes equations","title":"First steps","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We start by loading the required packages, initialising MPI and setting the simulation parameters.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using MPI\nusing PencilFFTs\n\nMPI.Init()\ncomm = MPI.COMM_WORLD\nprocid = MPI.Comm_rank(comm) + 1\n\n# Simulation parameters\nNs = (64, 64, 64) # = (Nx, Ny, Nz)\nLs = (2π, 2π, 2π) # = (Lx, Ly, Lz)\n\n# Collocation points (\"global\" = over all processes).\n# We include the endpoint (length = N + 1) for convenience.\nxs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls) # = (x, y, z)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's check the number of MPI processes over which we're running our simulation:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"MPI.Comm_size(comm)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"pen = Pencil(Ns, comm)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"The subdomain associated to the local MPI process can be obtained using range_local:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"range_local(pen)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We now construct a distributed vector field that follows the decomposition configuration we just created:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v⃗₀ = (\n PencilArray{Float64}(undef, pen), # vx\n PencilArray{Float64}(undef, pen), # vy\n PencilArray{Float64}(undef, pen), # vz\n)\nsummary(v⃗₀[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We still need to fill this array with interesting values that represent a physical velocity field.","category":"page"},{"location":"generated/navier_stokes/#Initial-condition","page":"Navier–Stokes equations","title":"Initial condition","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"beginaligned\nv_x(x y z) = u₀ sin(k₀ x) cos(k₀ y) cos(k₀ z) \nv_y(x y z) = -u₀ cos(k₀ x) sin(k₀ y) cos(k₀ z) \nv_z(x y z) = 0\nendaligned","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where u₀ and k₀ are two parameters setting the amplitude and the period of the velocity field.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"grid = localgrid(pen, xs_global)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can use this to initialise the velocity field:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"u₀ = 1.0\nk₀ = 2π / Ls[1] # should be integer if L = 2π (to preserve periodicity)\n\n@. v⃗₀[1] = u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)\n@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)\n@. v⃗₀[3] = 0\nnothing # hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's plot a 2D slice of the velocity field managed by the local MPI process:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using GLMakie\n\n# Compute the norm of a vector field represented by a tuple of arrays.\nfunction vecnorm(v⃗::NTuple)\n vnorm = similar(v⃗[1])\n for n ∈ eachindex(v⃗[1])\n w = zero(eltype(vnorm))\n for v ∈ v⃗\n w += v[n]^2\n end\n vnorm[n] = sqrt(w)\n end\n vnorm\nend\n\nlet fig = Figure(resolution = (700, 600))\n ax = Axis3(fig[1, 1]; aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\")\n vnorm = vecnorm(v⃗₀)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, vnorm;\n alpha = 0.2, levels = 4,\n colormap = :viridis, colorrange = (0.0, 1.0),\n )\n cb = Colorbar(fig[1, 2], ct; label = \"Velocity magnitude\")\n fig\nend","category":"page"},{"location":"generated/navier_stokes/#Velocity-in-Fourier-space","page":"Navier–Stokes equations","title":"Velocity in Fourier space","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"bmv(bmx t) =\n_bmk hatbmv_bmk(t) e^i bmk bmx","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where bmk = (k_x k_y k_z) are the discrete wave numbers.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for k_x:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using AbstractFFTs: fftfreq, rfftfreq\n\nks_global = (\n rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]), # kx | real-to-complex\n fftfreq(Ns[2], 2π * Ns[2] / Ls[2]), # ky | complex-to-complex\n fftfreq(Ns[3], 2π * Ns[3] / Ls[3]), # kz | complex-to-complex\n)\n\nks_global[1]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ks_global[2]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ks_global[3]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"See PencilFFTPlan for details on creating plans and on optional keyword arguments.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients hatbmv_bmk:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v̂s = plan .* v⃗₀\nsummary(v̂s[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Note that, in Fourier space, the domain decomposition is performed along the directions x and y:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"pencil(v̂s[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"This is because the 3D FFTs are performed one dimension at a time, with the x direction first and the z direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers bmk associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"grid_fourier = localgrid(v̂s[1], ks_global)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, bmω = bm bmv. In Fourier space, this becomes hatbmω = i bmk hatbmv.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using StaticArrays: SVector\nusing LinearAlgebra: ×\n\nfunction curl_fourier!(\n ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,\n ) where {N}\n @inbounds for I ∈ eachindex(grid_fourier)\n # We use StaticArrays for the cross product between small vectors.\n ik⃗ = im * SVector(grid_fourier[I])\n v⃗ = SVector(getindex.(v̂s, Ref(I))) # = (v̂s[1][I], v̂s[2][I], ...)\n ω⃗ = ik⃗ × v⃗\n for n ∈ eachindex(ω⃗)\n ω̂s[n][I] = ω⃗[n]\n end\n end\n ω̂s\nend\n\nω̂s = similar.(v̂s)\ncurl_fourier!(ω̂s, v̂s, grid_fourier);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We finally transform back to physical space and plot the result:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ωs = plan .\\ ω̂s\n\nlet fig = Figure(resolution = (700, 600))\n ax = Axis3(fig[1, 1]; aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\")\n ω_norm = vecnorm(ωs)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, ω_norm;\n alpha = 0.1, levels = 0.8:0.2:2.0,\n colormap = :viridis, colorrange = (0.8, 2.0),\n )\n cb = Colorbar(fig[1, 2], ct; label = \"Vorticity magnitude\")\n fig\nend","category":"page"},{"location":"generated/navier_stokes/#Computing-the-non-linear-term","page":"Navier–Stokes equations","title":"Computing the non-linear term","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"_t hatbmv_bmk =\n- mathcalP_bmk left widehat(bmv bm) bmv right\n- ν bmk^2 hatbmv_bmk\nquad text with quad\nmathcalP_bmk(hatbmF_bmk) = left( I - fracbmk \nbmkbmk^2 right) hatbmF_bmk","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where mathcalP_bmk is a projection operator allowing to preserve the incompressibility condition bm bmv = 0. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Now that we have the wave numbers bmk, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients hatbmv_bmk of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, hatbmF_bmk = left widehat(bmv bm) bmv right_bmk. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Below we implement a function that computes the non-linear term in Fourier space based on its convective form (bmv bm) bmv = bm (bmv bmv). Note that this equivalence uses the incompressibility condition bm bmv = 0.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using LinearAlgebra: mul!, ldiv! # for applying FFT plans in-place\n\n# Compute non-linear term in Fourier space from velocity field in physical\n# space. Optional keyword arguments may be passed to avoid allocations.\nfunction ns_nonlinear!(\n F̂s, vs, plan, grid_fourier;\n vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),\n )\n # Compute F_i = ∂_j (v_i v_j) for each i.\n # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)\n w, ŵ = vbuf, v̂buf\n @inbounds for (i, F̂i) ∈ enumerate(F̂s)\n F̂i .= 0\n vi = vs[i]\n for (j, vj) ∈ enumerate(vs)\n w .= vi .* vj # w = v_i * v_j in physical space\n mul!(ŵ, plan, w) # same in Fourier space\n # Add derivative in Fourier space\n for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n kj = k⃗[j]\n F̂i[I] += im * kj * ŵ[I]\n end\n end\n end\n F̂s\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"As an example, let's use this function on our initial velocity field:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"F̂s = similar.(v̂s)\nns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)\n ks_max = maximum.(abs, ks_global) # maximum stored wave numbers (kx_max, ky_max, kz_max)\n ks_lim = (2 / 3) .* ks_max\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I]\n if any(abs.(k⃗) .> ks_lim)\n for ŵ ∈ ŵs\n ŵ[I] = 0\n end\n end\n end\n ŵs\nend\n\n# We can apply this on the previously computed non-linear term:\ndealias_twothirds!(F̂s, grid_fourier, ks_global);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Finally, we implement the projection associated to the incompressibility condition:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function project_divergence_free!(ûs, grid_fourier)\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I]\n k² = sum(abs2, k⃗)\n iszero(k²) && continue # avoid division by zero\n û = getindex.(ûs, Ref(I)) # (ûs[1][I], ûs[2][I], ...)\n for i ∈ eachindex(û)\n ŵ = û[i]\n for j ∈ eachindex(û)\n ŵ -= k⃗[i] * k⃗[j] * û[j] / k²\n end\n ûs[i][I] = ŵ\n end\n end\n ûs\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)\nv̂s_proj .≈ v̂s # the last one may be false because v_z = 0 initially","category":"page"},{"location":"generated/navier_stokes/#Putting-it-all-together","page":"Navier–Stokes equations","title":"Putting it all together","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function ns_rhs!(\n dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,\n ) where {N}\n # 1. Compute non-linear term and dealias it\n (; plan, cache, ks_global, grid_fourier) = p\n F̂s = cache.F̂s\n ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])\n dealias_twothirds!(F̂s, grid_fourier, ks_global)\n\n # 2. Project onto divergence-free space\n project_divergence_free!(F̂s, grid_fourier)\n\n # 3. Transform velocity to Fourier space\n v̂s = cache.v̂s\n map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)\n\n # 4. Add viscous term (and multiply projected non-linear term by -1)\n ν = p.ν\n for n ∈ eachindex(v̂s)\n v̂ = v̂s[n]\n F̂ = F̂s[n]\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n k² = sum(abs2, k⃗)\n F̂[I] = -F̂[I] - ν * k² * v̂[I]\n end\n end\n\n # 5. Transform RHS back to physical space\n map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)\n\n nothing\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using OrdinaryDiffEq\nusing RecursiveArrayTools: ArrayPartition\n\nns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)\nvs_init_ode = ArrayPartition(v⃗₀)\nsummary(vs_init_ode)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We now define solver parameters and temporary variables, and initialise the problem:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"params = (;\n ν = 5e-3, # kinematic viscosity\n plan, grid_fourier, ks_global,\n cache = (\n v̂s = similar.(v̂s),\n F̂s = similar.(v̂s),\n )\n)\n\ntspan = (0.0, 10.0)\nprob = ODEProblem(ns_rhs!, vs_init_ode, tspan, params)\nintegrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum E(k), to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity ν must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large k), while high enough to allow the small-scale motions to be correctly resolved.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function energy_spectrum!(Ek, ks, v̂s, grid_fourier)\n Nk = length(Ek)\n @assert Nk == length(ks)\n Ek .= 0\n for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n knorm = sqrt(sum(abs2, k⃗))\n i = searchsortedfirst(ks, knorm)\n i > Nk && continue\n v⃗ = getindex.(v̂s, Ref(I)) # = (v̂s[1][I], v̂s[2][I], ...)\n factor = k⃗[1] == 0 ? 1 : 2 # account for Hermitian symmetry and r2c transform\n Ek[i] += factor * sum(abs2, v⃗) / 2\n end\n MPI.Allreduce!(Ek, +, get_comm(v̂s[1])) # sum across all processes\n Ek\nend\n\nks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])\nEk = similar(ks)\nv̂s = plan .* integrator.u.x\nenergy_spectrum!(Ek, ks, v̂s, grid_fourier)\nEk ./= scale_factor(plan)^2 # rescale energy\n\ncurl_fourier!(ω̂s, v̂s, grid_fourier)\nldiv!.(ωs, plan, ω̂s)\nω⃗_plot = Observable(ωs)\nk_plot = @view ks[2:end]\nE_plot = Observable(@view Ek[2:end])\nt_plot = Observable(integrator.t)\n\nfig = let\n fig = Figure(resolution = (1200, 600))\n ax = Axis3(\n fig[1, 1][1, 1]; title = @lift(\"t = $(round($t_plot, digits = 3))\"),\n aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\",\n )\n ω_mag = @lift vecnorm($ω⃗_plot)\n ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, ω_mag_norm;\n alpha = 0.3, levels = 3,\n colormap = :viridis, colorrange = (0.0, 1.0),\n )\n cb = Colorbar(fig[1, 1][1, 2], ct; label = \"Normalised vorticity magnitude\")\n ax_sp = Axis(\n fig[1, 2];\n xlabel = \"k\", ylabel = \"E(k)\", xscale = log2, yscale = log10,\n title = \"Kinetic energy spectrum\",\n )\n ylims!(ax_sp, 1e-8, 1e0)\n scatterlines!(ax_sp, k_plot, E_plot)\n ks_slope = exp.(range(log(2.5), log(25.0), length = 3))\n E_fivethirds = @. 0.3 * ks_slope^(-5/3)\n @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)\n text!(ax_sp, L\"k^{-5/3}\"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))\n fig\nend\n\nusing Printf # hide\nwith_xvfb = ENV[\"DISPLAY\"] == \":99\" # hide\nnstep = 0 # hide\nconst tmpdir = mktempdir() # hide\nfilename_frame(procid, nstep) = joinpath(tmpdir, @sprintf(\"proc%d_%04d.png\", procid, nstep)) # hide\nrecord(fig, \"vorticity_proc$procid.mp4\"; framerate = 10) do io\n with_xvfb && recordframe!(io) # hide\n while integrator.t < 20\n dt = 0.001\n step!(integrator, dt)\n t_plot[] = integrator.t\n mul!.(v̂s, plan, integrator.u.x) # current velocity in Fourier space\n curl_fourier!(ω̂s, v̂s, grid_fourier)\n ldiv!.(ω⃗_plot[], plan, ω̂s)\n ω⃗_plot[] = ω⃗_plot[] # to force updating the plot\n energy_spectrum!(Ek, ks, v̂s, grid_fourier)\n Ek ./= scale_factor(plan)^2 # rescale energy\n E_plot[] = E_plot[]\n global nstep += 1 # hide\n with_xvfb ? # hide\n save(filename_frame(procid, nstep), fig) : # hide\n recordframe!(io)\n end\nend;\n\nif with_xvfb # hide\n run(pipeline(`ffmpeg -y -r 10 -i $tmpdir/proc$(procid)_%04d.png -c:v libx264 -vf \"fps=25,format=yuv420p\" vorticity_proc$procid.mp4`; stdout = \"ffmpeg.out\", stderr = \"ffmpeg.err\")) # hide\nend # hide\nnothing # hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"
                        \n \n
                        ","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"This page was generated using Literate.jl.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"EditURL = \"https://github.com/jipolanco/PencilFFTs.jl/blob/master/docs/examples/gradient.jl\"","category":"page"},{"location":"generated/gradient/#Gradient-of-a-scalar-field","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This example shows different methods to compute the gradient of a real-valued 3D scalar field θ(bmx) in Fourier space, where bmx = (x y z). It is assumed that the field is periodic with period L = 2π along all dimensions.","category":"page"},{"location":"generated/gradient/#General-procedure","page":"Gradient of a scalar field","title":"General procedure","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The discrete Fourier expansion of θ writes","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"θ(bmx) = _bmk Z^3 hatθ(bmk) e^i bmk bmx","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"where bmk = (k_x k_y k_z) are the Fourier wave numbers and hatθ is the discrete Fourier transform of θ. Then, the spatial derivatives of θ are given by","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"frac θ(bmx) x_i =\n_bmk Z^3 i k_i hatθ(bmk) e^i bmk bmx","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"where the subscript i denotes one of the spatial components x, y or z.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In other words, to compute bm θ = (_x θ _y θ _z θ), one has to:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"transform θ to Fourier space to obtain hatθ,\nmultiply hatθ by i bmk,\ntransform the result back to physical space to obtain bm θ.","category":"page"},{"location":"generated/gradient/#Preparation","page":"Gradient of a scalar field","title":"Preparation","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In this section, we initialise a random real-valued scalar field θ and compute its FFT. For more details see the Tutorial.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"using MPI\nusing PencilFFTs\nusing Random\n\nMPI.Init()\n\n# Input data dimensions (Nx × Ny × Nz)\ndims = (64, 32, 64)\n\n# Apply a 3D real-to-complex (r2c) FFT.\ntransform = Transforms.RFFT()\n\n# Automatically create decomposition configuration\ncomm = MPI.COMM_WORLD\npen = Pencil(dims, comm)\n\n# Create plan\nplan = PencilFFTPlan(pen, transform)\n\n# Allocate data and initialise field\nθ = allocate_input(plan)\nrandn!(θ)\n\n# Perform distributed FFT\nθ_hat = plan * θ\nnothing # hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, we initialise the output that will hold ∇θ in Fourier space. Noting that ∇θ is a vector field, we choose to store it as a tuple of 3 PencilArrays.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"∇θ_hat = allocate_output(plan, Val(3))\n\n# This is equivalent:\n# ∇θ_hat = ntuple(d -> similar(θ_hat), Val(3))\n\nsummary(∇θ_hat)","category":"page"},{"location":"generated/gradient/#Fourier-wave-numbers","page":"Gradient of a scalar field","title":"Fourier wave numbers","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In general, the Fourier wave numbers are of the form k_i = 0 frac2πL_i frac4πL_i frac6πL_i , where L_i is the period along dimension i. When a real-to-complex Fourier transform is applied, roughly half of these wave numbers are redundant due to the Hermitian symmetry of the complex Fourier coefficients. In practice, this means that for the fastest dimension x (along which a real-to-complex transform is performed), the negative wave numbers are dropped, i.e. k_x = 0 frac2πL_x frac4πL_x .","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The AbstractFFTs package provides a convenient way to generate the Fourier wave numbers, using the functions fftfreq and rfftfreq. We can use these functions to initialise a \"grid\" of wave numbers associated to our 3D real-to-complex transform:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"using AbstractFFTs: fftfreq, rfftfreq\n\nbox_size = (2π, 2π, 2π) # Lx, Ly, Lz\nsample_rate = 2π .* dims ./ box_size\n\n# In our case (Lx = 2π and Nx even), this gives kx = [0, 1, 2, ..., Nx/2].\nkx = rfftfreq(dims[1], sample_rate[1])\n\n# In our case (Ly = 2π and Ny even), this gives\n# ky = [0, 1, 2, ..., Ny/2-1, -Ny/2, -Ny/2+1, ..., -1] (and similarly for kz).\nky = fftfreq(dims[2], sample_rate[2])\nkz = fftfreq(dims[3], sample_rate[3])\n\nkvec = (kx, ky, kz)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that kvec now contains the wave numbers associated to the global domain. In the following, we will only need the wave numbers associated to the portion of the domain handled by the local MPI process.","category":"page"},{"location":"generated/gradient/#gradient_method_global","page":"Gradient of a scalar field","title":"Method 1: global views","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"PencilArrays, returned for instance by allocate_input and allocate_output, take indices that start at 1, regardless of the location of the subdomain associated to the local process on the global grid. (In other words, PencilArrays take local indices.) On the other hand, we have defined the wave number vector kvec which, for each MPI process, is defined over the global domain, and as such it takes global indices.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"One straightforward way of making data arrays compatible with wave numbers is to use global views, i.e. arrays that take global indices. These are generated from PencilArrays by calling the global_view function. Note that, in general, global indices do not start at 1 for a given MPI process. A given process will own a range of data given by indices in (i1:i2, j1:j2, k1:k2).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"θ_glob = global_view(θ_hat)\n∇θ_glob = global_view.(∇θ_hat)\nsummary(θ_glob)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Once we have global views, we can combine data and wave numbers using the portion of global indices owned by the local MPI process, as shown below. We can use CartesianIndices to iterate over the global indices associated to the local process.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"for I in CartesianIndices(θ_glob)\n i, j, k = Tuple(I) # unpack indices\n\n # Wave number vector associated to current Cartesian index.\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n # Compute gradient in Fourier space.\n # Note that modifying ∇θ_glob also modifies the original PencilArray ∇θ_hat.\n ∇θ_glob[1][I] = im * kx * θ_glob[I]\n ∇θ_glob[2][I] = im * ky * θ_glob[I]\n ∇θ_glob[3][I] = im * kz * θ_glob[I]\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The above loop can be written in a slightly more efficient manner by precomputing im * θ_glob[I]:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@inbounds for I in CartesianIndices(θ_glob)\n i, j, k = Tuple(I)\n\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n u = im * θ_glob[I]\n\n ∇θ_glob[1][I] = kx * u\n ∇θ_glob[2][I] = ky * u\n ∇θ_glob[3][I] = kz * u\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Also note that the above can be easily written in a more generic way, e.g. for arbitrary dimensions, thanks in part to the use of CartesianIndices. Moreover, in the above there is no notion of the dimension permutations discussed in the tutorial, as it is all hidden behind the implementation of PencilArrays. And as seen later in the benchmarks, these (hidden) permutations have zero cost, as the speed is identical to that of a function that explicitly takes into account these permutations.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, we can perform a backwards transform to obtain bm θ in physical space:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"∇θ = plan \\ ∇θ_hat;\nnothing #hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that the transform is automatically broadcast over the three fields of the ∇θ_hat vector, and the result ∇θ is also a tuple of three PencilArrays.","category":"page"},{"location":"generated/gradient/#gradient_method_global_explicit","page":"Gradient of a scalar field","title":"Method 2: explicit global indexing","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Sometimes, one does not need to write generic code. In our case, one often knows the dimensionality of the problem and the memory layout of the data (i.e. the underlying index permutation).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Below is a reimplementation of the above loop, using explicit indices instead of CartesianIndices, and assuming that the underlying index permutation is (3, 2, 1), that is, data is stored in (z y x) order. As discussed in the tutorial, this is the default for transformed arrays. This example also serves as a more explicit explanation for what is going on in the first method.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"# Get local data range in the global grid.\nrng = axes(θ_glob) # = (i1:i2, j1:j2, k1:k2)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"For the loop below, we're assuming that the permutation is (3, 2, 1). In other words, the fastest index is the last one, and not the first one as it is usually in Julia. If the permutation is not (3, 2, 1), things will still work (well, except for the assertion below!), but the loop order will not be optimal.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@assert permutation(θ_hat) === Permutation(3, 2, 1)\n\n@inbounds for i in rng[1], j in rng[2], k in rng[3]\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n # Note that we still access the arrays in (i, j, k) order.\n # (The permutation happens behind the scenes!)\n u = im * θ_glob[i, j, k]\n\n ∇θ_glob[1][i, j, k] = kx * u\n ∇θ_glob[2][i, j, k] = ky * u\n ∇θ_glob[3][i, j, k] = kz * u\nend","category":"page"},{"location":"generated/gradient/#gradient_method_local","page":"Gradient of a scalar field","title":"Method 3: using local indices","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Alternatively, we can avoid global views and work directly on PencilArrays using local indices that start at 1. In this case, part of the strategy is to construct a \"local\" grid of wave numbers that can also be accessed with local indices. This can be conveniently done using the localgrid function of the PencilArrays.jl package, which accepts a PencilArray (or its associated Pencil) and the global coordinates (here kvec):","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"grid_fourier = localgrid(θ_hat, kvec)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that one can directly iterate on the returned grid object:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@inbounds for I in CartesianIndices(grid_fourier)\n # Wave number vector associated to current Cartesian index.\n local k⃗ # hide\n k⃗ = grid_fourier[I]\n u = im * θ_hat[I]\n ∇θ_hat[1][I] = k⃗[1] * u\n ∇θ_hat[2][I] = k⃗[2] * u\n ∇θ_hat[3][I] = k⃗[3] * u\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This implementation is as efficient as the other examples, while being slightly shorter to write. Moreover, it is quite generic, and can be made independent of the number of dimensions with little effort.","category":"page"},{"location":"generated/gradient/#gradient_method_broadcast","page":"Gradient of a scalar field","title":"Method 4: using broadcasting","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, note that the local grid object returned by localgrid makes it is possible to compute the gradient using broadcasting, thus fully avoiding scalar indexing. This can be quite convenient in some cases, and can also be very useful if one is working on GPUs (where scalar indexing is prohibitively expensive). Using broadcasting, the above examples simply become:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@. ∇θ_hat[1] = im * grid_fourier[1] * θ_hat\n@. ∇θ_hat[2] = im * grid_fourier[2] * θ_hat\n@. ∇θ_hat[3] = im * grid_fourier[3] * θ_hat\nnothing # hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Once again, as shown in the benchmarks further below, this method performs quite similarly to the other ones.","category":"page"},{"location":"generated/gradient/#Summary","page":"Gradient of a scalar field","title":"Summary","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The PencilArrays module provides different alternatives to deal with MPI-distributed data that may be subject to dimension permutations. In particular, one can choose to work with global indices (first two examples), with local indices (third example), or to avoid scalar indexing altogether (fourth example).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"If one wants to stay generic, making sure that the same code will work for arbitrary dimensions and will be efficient regardless of the underlying dimension permutation, methods 1, 3 or 4 should be preferred. These use CartesianIndices and make no assumptions on possible dimension permutations, which are by default enabled in the output of PencilFFTs transforms. In fact, such permutations are completely invisible in the implementations.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The second method uses explicit (i, j, k) indices. It assumes that the underlying permutation is (3, 2, 1) to loop with i as the slowest index and k as the fastest, which is the optimal order in this case given the permutation. As such, the implementation is less generic than the others, and differences in performance are negligible with respect to more generic variants.","category":"page"},{"location":"generated/gradient/#gradient_benchmarks","page":"Gradient of a scalar field","title":"Benchmark results","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The following are the benchmark results obtained from running examples/gradient.jl on a laptop, using 2 MPI processes and Julia 1.7.2, with an input array of global dimensions 64 32 64. The different methods detailed above are marked on the right. The \"lazy\" marks indicate runs where the wave numbers were represented by lazy Frequencies objects (returned by rfftfreq and fftfreq). Otherwise, they were collected into Vectors. For some reason, plain Vectors are faster when working with grids generated by localgrid.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In the script, additional implementations can be found which rely on a more advanced understanding of permutations and on the internals of the PencilArrays package. For instance, gradient_local_parent! directly works with the raw data stored in Julia Arrays, while gradient_local_linear! completely avoids CartesianIndices while staying generic and efficient. Nevertheless, these display roughly the same performance as the above examples.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":" gradient_global_view!... 89.900 μs\n gradient_global_view! (lazy)... 92.060 μs [Method 1]\n gradient_global_view_explicit!... 88.958 μs\n gradient_global_view_explicit! (lazy)... 81.055 μs [Method 2]\n gradient_local!... 92.305 μs\n gradient_grid!... 92.770 μs\n gradient_grid! (lazy)... 101.388 μs [Method 3]\n gradient_grid_broadcast!... 88.606 μs\n gradient_grid_broadcast! (lazy)... 151.020 μs [Method 4]\n gradient_local_parent!... 92.248 μs\n gradient_local_linear!... 91.212 μs\n gradient_local_linear_explicit!... 90.992 μs","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This page was generated using Literate.jl.","category":"page"},{"location":"PencilFFTs_timers/#PencilFFTs.measuring_performance","page":"Measuring performance","title":"Measuring performance","text":"","category":"section"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"It is possible to measure the time spent in different sections of the distributed transforms using the TimerOutputs package. This has a (very small) performance overhead, so it is disabled by default. To enable time measurements, call TimerOutputs.enable_debug_timings after loading PencilFFTs (see below for an example). For more details see the TimerOutputs docs.","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"Minimal example:","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"using MPI\nusing PencilFFTs\nusing TimerOutputs\n\n# Enable timing of `PencilFFTs` functions\nTimerOutputs.enable_debug_timings(PencilFFTs)\nTimerOutputs.enable_debug_timings(PencilArrays)\nTimerOutputs.enable_debug_timings(Transpositions)\n\nMPI.Init()\n\nplan = PencilFFTPlan(#= args... =#)\n\n# [do stuff with `plan`...]\n\n# Retrieve and print timing data associated to `plan`\nto = timer(plan)\nprint_timer(to)","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"By default, each PencilFFTPlan has its own TimerOutput. If you already have a TimerOutput, you can pass it to the PencilFFTPlan constructor:","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"to = TimerOutput()\nplan = PencilFFTPlan(..., timer=to)\n\n# [do stuff with `plan`...]\n\nprint_timer(to)","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"#PencilFFTs","page":"Home","title":"PencilFFTs","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Fast Fourier transforms of MPI-distributed Julia arrays.","category":"page"},{"location":"#Introduction","page":"Home","title":"Introduction","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package provides multidimensional FFTs and related transforms on MPI-distributed Julia arrays via the PencilArrays package.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The name of this package originates from the decomposition of 3D domains along two out of three dimensions, sometimes called pencil decomposition. This is illustrated by the figure below,[1] where each coloured block is managed by a different MPI process. Typically, one wants to compute FFTs on a scalar or vector field along the three spatial dimensions. In the case of a pencil decomposition, 3D FFTs are performed one dimension at a time, along the non-decomposed direction. Transforms must then be interleaved with global data transpositions to switch between pencil configurations. In high-performance computing environments, such data transpositions are generally the most expensive part of a parallel FFT computation, due to the large cost of communications between computing nodes.","category":"page"},{"location":"","page":"Home","title":"Home","text":"
                        \n \n
                        ","category":"page"},{"location":"","page":"Home","title":"Home","text":"More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension N. The decompositions can be performed along an arbitrary number M N of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.","category":"page"},{"location":"#Features","page":"Home","title":"Features","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;\nFFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;\nin-place and out-of-place transforms;\nhigh scalability up to (at least) tens of thousands of MPI processes.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"PencilFFTs can be installed using the Julia package manager:","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> ] add PencilFFTs","category":"page"},{"location":"#Similar-projects","page":"Home","title":"Similar projects","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.\nPFFT is a very general parallel FFT library written in C.\nP3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.\n2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[1]: Figure adapted from this PhD thesis.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[2]: For the pencil decomposition represented in the figure, N = 3 and M = 2.","category":"page"},{"location":"PencilFFTs/#Distributed-FFT-plans","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"Distributed FFTs are implemented in the PencilFFTs module, and are built on top of the PencilArrays package.","category":"page"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"PencilFFTs/#Creating-plans","page":"Distributed FFT plans","title":"Creating plans","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"PencilFFTPlan","category":"page"},{"location":"PencilFFTs/#PencilFFTs.PencilFFTPlan","page":"Distributed FFT plans","title":"PencilFFTs.PencilFFTPlan","text":"PencilFFTPlan{T,N} <: AbstractFFTs.Plan{T}\n\nPlan for N-dimensional FFT-based transform on MPI-distributed data, where input data has type T.\n\n\n\nPencilFFTPlan(p::Pencil, transforms; kwargs...)\n\nCreate a PencilFFTPlan for distributed arrays following a given Pencil configuration. See variant below for details on the specification of transforms and on possible keyword arguments.\n\n\n\nPencilFFTPlan(\n A::PencilArray, transforms;\n fftw_flags = FFTW.ESTIMATE,\n fftw_timelimit = FFTW.NO_TIMELIMIT,\n permute_dims = Val(true),\n transpose_method = Transpositions.PointToPoint(),\n timer = timer(pencil(A)),\n)\n\nCreate plan for N-dimensional transform on MPI-distributed PencilArrays.\n\nExtended help\n\nThis creates a PencilFFTPlan for arrays sharing the same properties as A (dimensions, MPI decomposition, memory layout, ...), which describe data on an N-dimensional domain.\n\nTransforms\n\nThe transforms to be applied along each dimension are specified by the transforms argument. Possible transforms are defined as subtypes of Transforms.AbstractTransform, and are listed in Transform types. This argument may be either:\n\na tuple of N transforms to be applied along each dimension. For instance, transforms = (Transforms.R2R(FFTW.REDFT01), Transforms.RFFT(), Transforms.FFT());\na single transform to be applied along all dimensions. The input is automatically expanded into N equivalent transforms. For instance, for a three-dimensional array, transforms = Transforms.RFFT() specifies a 3D real-to-complex transform, and is equivalent to passing (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT()).\n\nNote that forward transforms are applied from left to right. In the last example, this means that a real-to-complex transform (RFFT) is first performed along the first dimension. This is followed by complex-to-complex transforms (FFT) along the second and third dimensions.\n\nInput data layout\n\nThe input PencilArray must satisfy the following constraints:\n\narray dimensions must not be permuted. This is the default when constructing PencilArrays.\nfor an M-dimensional domain decomposition (with M < N), the input array must be decomposed along the last M dimensions. For example, for a 2D decomposition of 3D data, the decomposed dimensions must be (2, 3). In particular, the first array dimension must not be distributed among different MPI processes.\nIn the PencilArrays package, the decomposed dimensions are specified at the moment of constructing a Pencil.\nthe element type must be compatible with the specified transform. For instance, real-to-complex transforms (Transforms.RFFT) require the input to be real floating point values. Other transforms, such as Transforms.R2R, accept both real and complex data.\n\nKeyword arguments\n\nThe keyword arguments fftw_flags and fftw_timelimit are passed to the FFTW plan creation functions (see AbstractFFTs docs).\npermute_dims determines whether the indices of the output data should be reversed. For instance, if the input data has global dimensions (Nx, Ny, Nz), then the output of a complex-to-complex FFT would have dimensions (Nz, Ny, Nx). This enables FFTs to always be performed along the first (i.e. fastest) array dimension, which could lead to performance gains. This option is enabled by default. For type inference reasons, it must be a value type (Val(true) or Val(false)).\ntranspose_method allows to select between implementations of the global data transpositions. See PencilArrays docs docs for details.\ntimer should be a TimerOutput object. See Measuring performance for details.\n\n\n\nPencilFFTPlan(\n dims_global::Dims{N}, transforms, proc_dims::Dims{M}, comm::MPI.Comm,\n [real_type = Float64]; extra_dims = (), kws...\n)\n\nCreate plan for N-dimensional transform.\n\nExtended help\n\nInstead of taking a PencilArray or a Pencil, this constructor requires the global dimensions of the input data, passed via the size_global argument.\n\nThe data is distributed over the MPI processes in the comm communicator. The distribution is performed over M dimensions (with M < N) according to the values in proc_dims, which specifies the number of MPI processes to put along each dimension.\n\nPencilArrays that may be transformed with the returned plan can be created using allocate_input.\n\nOptional arguments\n\nThe floating point precision can be selected by setting real_type parameter, which is Float64 by default.\nextra_dims may be used to specify the sizes of one or more extra dimensions that should not be transformed. These dimensions will be added to the rightmost (i.e. slowest) indices of the arrays. See Extra dimensions below for usage hints.\nsee the other constructor for more keyword arguments.\n\nExtra dimensions\n\nOne possible application of extra_dims is for describing the components of a vector or tensor field. However, this means that different PencilFFTPlans would need to be created for each kind of field (scalar, vector, ...). To avoid the creation of multiple plans, a possibly better alternative is to create tuples (or arrays) of PencilArrays using allocate_input and allocate_output.\n\nAnother more legitimate usage of extra_dims is to specify one or more Cartesian dimensions that should not be transformed nor split among MPI processes.\n\nExample\n\nSuppose we want to perform a 3D FFT of real data. The data is to be decomposed along two dimensions, over 8 MPI processes:\n\nsize_global = (64, 32, 128) # size of real input data\n\n# Perform real-to-complex transform along the first dimension, then\n# complex-to-complex transforms along the other dimensions.\ntransforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())\n# transforms = Transforms.RFFT() # this is equivalent to the above line\n\nproc_dims = (4, 2) # 2D decomposition\ncomm = MPI.COMM_WORLD\n\nplan = PencilFFTPlan(size_global, transforms, proc_dims, comm)\n\n\n\n\n\n","category":"type"},{"location":"PencilFFTs/#Allocating-data","page":"Distributed FFT plans","title":"Allocating data","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"allocate_input\nallocate_output","category":"page"},{"location":"PencilFFTs/#PencilFFTs.allocate_input","page":"Distributed FFT plans","title":"PencilFFTs.allocate_input","text":"allocate_input(p::PencilFFTPlan) -> PencilArray\nallocate_input(p::PencilFFTPlan, dims...) -> Array{PencilArray}\nallocate_input(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}\n\nAllocate uninitialised PencilArray that can hold input data for the given plan.\n\nThe second and third forms respectively allocate an array of PencilArrays of size dims, and a tuple of N PencilArrays.\n\nnote: In-place plans\nIf p is an in-place real-to-real or complex-to-complex plan, a ManyPencilArray is allocated. If p is an in-place real-to-complex plan, a ManyPencilArrayRFFT! is allocated. These types hold PencilArray wrappers for the input and output transforms (as well as for intermediate transforms) which share the same space in memory. The input and output PencilArrays should be respectively accessed by calling first(::ManyPencilArray) and last(::ManyPencilArray).ExampleSuppose p is an in-place PencilFFTPlan. Then,@assert is_inplace(p)\nA = allocate_input(p) :: ManyPencilArray\nv_in = first(A) :: PencilArray # input data view\nv_out = last(A) :: PencilArray # output data viewAlso note that in-place plans must be performed directly on the returned ManyPencilArray, and not on the contained PencilArray views:p * A # perform forward transform in-place\np \\ A # perform backward transform in-place\n# p * v_in # not allowed!!\n\n\n\n\n\n","category":"function"},{"location":"PencilFFTs/#PencilFFTs.allocate_output","page":"Distributed FFT plans","title":"PencilFFTs.allocate_output","text":"allocate_output(p::PencilFFTPlan) -> PencilArray\nallocate_output(p::PencilFFTPlan, dims...) -> Array{PencilArray}\nallocate_output(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}\n\nAllocate uninitialised PencilArray that can hold output data for the given plan.\n\nIf p is an in-place plan, a ManyPencilArray is allocated.\n\nSee allocate_input for details.\n\n\n\n\n\n","category":"function"},{"location":"PencilFFTs/#Methods","page":"Distributed FFT plans","title":"Methods","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"get_comm(::PencilFFTPlan)\nscale_factor(::PencilFFTPlan)\ntimer(::PencilFFTPlan)\nis_inplace(::PencilFFTPlan)","category":"page"},{"location":"PencilFFTs/#PencilArrays.Pencils.MPITopologies.get_comm-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilArrays.Pencils.MPITopologies.get_comm","text":"get_comm(p::PencilFFTPlan)\n\nGet MPI communicator associated to a PencilFFTPlan.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilFFTs.Transforms.scale_factor-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilFFTs.Transforms.scale_factor","text":"scale_factor(p::PencilFFTPlan)\n\nGet scale factor associated to a PencilFFTPlan.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilArrays.Pencils.timer-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilArrays.Pencils.timer","text":"timer(p::PencilFFTPlan)\n\nGet TimerOutput attached to a PencilFFTPlan.\n\nSee Measuring performance for details.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilFFTs.Transforms.is_inplace-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilFFTs.Transforms.is_inplace","text":"Transforms.is_inplace(p::PencilFFTPlan)\n\nReturns true if the given plan operates in-place on the input data, false otherwise.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#Internals","page":"Distributed FFT plans","title":"Internals","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"ManyPencilArrayRFFT!","category":"page"},{"location":"PencilFFTs/#PencilFFTs.ManyPencilArrayRFFT!","page":"Distributed FFT plans","title":"PencilFFTs.ManyPencilArrayRFFT!","text":"ManyPencilArrayRFFT!{T,N,M} <: AbstractManyPencilArray{N,M}\n\nContainer holding M different PencilArray views to the same underlying data buffer. All views share the same and dimensionality N. The element type T of the first view is real, that of subsequent views is Complex{T}. \n\nThis can be used to perform in-place real-to-complex plan, see alsoTransforms.RFFT!. It is used internally for such transforms by allocate_input and should not be constructed directly.\n\n\n\nManyPencilArrayRFFT!{T}(undef, pencils...; extra_dims=())\n\nCreate a ManyPencilArrayRFFT! container that can hold data of type T and Complex{T} associated to all the given Pencils.\n\nThe optional extra_dims argument is the same as for PencilArray.\n\nSee also ManyPencilArray\n\n\n\n\n\n","category":"type"}] +} diff --git a/v0.15.1/siteinfo.js b/v0.15.1/siteinfo.js new file mode 100644 index 00000000..af5dc7cd --- /dev/null +++ b/v0.15.1/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "v0.15.1"; diff --git a/previews/PR48/tutorial/index.html b/v0.15.1/tutorial/index.html similarity index 69% rename from previews/PR48/tutorial/index.html rename to v0.15.1/tutorial/index.html index fbf7b386..c8651d7b 100644 --- a/previews/PR48/tutorial/index.html +++ b/v0.15.1/tutorial/index.html @@ -1,5 +1,5 @@ -Tutorial · PencilFFTs.jl
                        Edit on GitHub

                        Tutorial

                        The following tutorial shows how to perform a 3D FFT of real periodic data defined on a grid of $N_x × N_y × N_z$ points.

                        +Tutorial · PencilFFTs.jl
                        Edit on GitHub

                        Tutorial

                        The following tutorial shows how to perform a 3D FFT of real periodic data defined on a grid of $N_x × N_y × N_z$ points.

                        + Strong scaling of PencilFFTs +

                        As seen above, PencilFFTs generally outperforms P3DFFT in its default setting. This is largely explained by the choice of using non-blocking point-to-point MPI communications (via MPI_Isend and MPI_Irecv), while P3DFFT uses collective MPI_Alltoallv calls. This enables PencilFFTs to perform data reordering operations on the partially received data while waiting for the incoming data, leading to better performance. Moreover, in contrast with P3DFFT, the high performance and scalability of PencilFFTs results from a highly generic code, handling decompositions in arbitrary dimensions and a relatively large (and extensible) variety of transformations.

                        Note that PencilFFTs can optionally use collective communications (using MPI_Alltoallv) instead of point-to-point communications. For details, see the docs for PencilFFTPlan and for PencilArray transpositions. As seen above, collective communications generally perform worse than point-to-point ones, and runtimes are nearly indistinguishable from those of P3DFFT.

                        Benchmark details

                        The benchmarks were performed using Julia 1.7-beta3 and Intel MPI 2019. We used PencilFFTs v0.12.5 with FFTW.jl v1.4.3 and MPI.jl v0.19.0. We used the Fortran implementation of P3DFFT, version 2.7.6, which was built with Intel 2019 compilers and linked to FFTW 3.3.8. The cluster where the benchmarks were run has Intel Cascade Lake 6248 processors with 2×20 cores per node.

                        The number of MPI processes along each decomposed dimension, $P_1$ and $P_2$, was automatically determined by a call to MPI_Dims_create, which tends to create a balanced decomposition with $P_1 ≈ P_2$. For instance, a total of 1024 processes is divided into $P_1 = P_2 = 32$. Different results may be obtained with other combinations, but this was not benchmarked.

                        The source files used to generate this benchmark, as well as the raw benchmark results, are all available in the PencilFFTs repo.

                        diff --git a/v0.15.2/generated/ffmpeg.err b/v0.15.2/generated/ffmpeg.err new file mode 100644 index 00000000..7a323ed2 --- /dev/null +++ b/v0.15.2/generated/ffmpeg.err @@ -0,0 +1,48 @@ +ffmpeg version 4.4.2-0ubuntu0.22.04.1 Copyright (c) 2000-2021 the FFmpeg developers + built with gcc 11 (Ubuntu 11.2.0-19ubuntu1) + configuration: --prefix=/usr --extra-version=0ubuntu0.22.04.1 --toolchain=hardened --libdir=/usr/lib/x86_64-linux-gnu --incdir=/usr/include/x86_64-linux-gnu --arch=amd64 --enable-gpl --disable-stripping --enable-gnutls --enable-ladspa --enable-libaom --enable-libass --enable-libbluray --enable-libbs2b --enable-libcaca --enable-libcdio --enable-libcodec2 --enable-libdav1d --enable-libflite --enable-libfontconfig --enable-libfreetype --enable-libfribidi --enable-libgme --enable-libgsm --enable-libjack --enable-libmp3lame --enable-libmysofa --enable-libopenjpeg --enable-libopenmpt --enable-libopus --enable-libpulse --enable-librabbitmq --enable-librubberband --enable-libshine --enable-libsnappy --enable-libsoxr --enable-libspeex --enable-libsrt --enable-libssh --enable-libtheora --enable-libtwolame --enable-libvidstab --enable-libvorbis --enable-libvpx --enable-libwebp --enable-libx265 --enable-libxml2 --enable-libxvid --enable-libzimg --enable-libzmq --enable-libzvbi --enable-lv2 --enable-omx --enable-openal --enable-opencl --enable-opengl --enable-sdl2 --enable-pocketsphinx --enable-librsvg --enable-libmfx --enable-libdc1394 --enable-libdrm --enable-libiec61883 --enable-chromaprint --enable-frei0r --enable-libx264 --enable-shared + libavutil 56. 70.100 / 56. 70.100 + libavcodec 58.134.100 / 58.134.100 + libavformat 58. 76.100 / 58. 76.100 + libavdevice 58. 13.100 / 58. 13.100 + libavfilter 7.110.100 / 7.110.100 + libswscale 5. 9.100 / 5. 9.100 + libswresample 3. 9.100 / 3. 9.100 + libpostproc 55. 9.100 / 55. 9.100 +Input #0, image2, from '/tmp/jl_vjIWbl/proc1_%04d.png': + Duration: 00:00:04.52, start: 0.000000, bitrate: N/A + Stream #0:0: Video: png, rgb24(pc), 1200x600, 25 fps, 25 tbr, 25 tbn, 25 tbc +Stream mapping: + Stream #0:0 -> #0:0 (png (native) -> h264 (libx264)) +Press [q] to stop, [?] for help +[libx264 @ 0x55eed4d03700] using cpu capabilities: MMX2 SSE2Fast SSSE3 SSE4.2 AVX FMA3 BMI2 AVX2 +[libx264 @ 0x55eed4d03700] profile High, level 3.1, 4:2:0, 8-bit +[libx264 @ 0x55eed4d03700] 264 - core 163 r3060 5db6aa6 - H.264/MPEG-4 AVC codec - Copyleft 2003-2021 - http://www.videolan.org/x264.html - options: cabac=1 ref=3 deblock=1:0:0 analyse=0x3:0x113 me=hex subme=7 psy=1 psy_rd=1.00:0.00 mixed_ref=1 me_range=16 chroma_me=1 trellis=1 8x8dct=1 cqm=0 deadzone=21,11 fast_pskip=1 chroma_qp_offset=-2 threads=6 lookahead_threads=1 sliced_threads=0 nr=0 decimate=1 interlaced=0 bluray_compat=0 constrained_intra=0 bframes=3 b_pyramid=2 b_adapt=1 b_bias=0 direct=1 weightb=1 open_gop=0 weightp=2 keyint=250 keyint_min=25 scenecut=40 intra_refresh=0 rc_lookahead=40 rc=crf mbtree=1 crf=23.0 qcomp=0.60 qpmin=0 qpmax=69 qpstep=4 ip_ratio=1.40 aq=1:1.00 +Output #0, mp4, to 'vorticity_proc1.mp4': + Metadata: + encoder : Lavf58.76.100 + Stream #0:0: Video: h264 (avc1 / 0x31637661), yuv420p(tv, progressive), 1200x600, q=2-31, 25 fps, 12800 tbn + Metadata: + encoder : Lavc58.134.100 libx264 + Side data: + cpb: bitrate max/min/avg: 0/0/0 buffer size: 0 vbv_delay: N/A +frame= 3 fps=0.0 q=0.0 size= 0kB time=00:00:00.00 bitrate=N/A speed= 0x frame= 70 fps=0.0 q=28.0 size= 0kB time=00:00:00.68 bitrate= 0.6kbits/s speed=1.21x frame= 108 fps= 98 q=28.0 size= 0kB time=00:00:02.20 bitrate= 0.2kbits/s speed= 2x frame= 145 fps= 90 q=28.0 size= 256kB time=00:00:03.68 bitrate= 570.0kbits/s speed=2.27x frame= 185 fps= 87 q=28.0 size= 256kB time=00:00:05.28 bitrate= 397.3kbits/s speed=2.49x frame= 225 fps= 85 q=28.0 size= 256kB time=00:00:06.88 bitrate= 304.9kbits/s speed=2.61x frame= 260 fps= 82 q=28.0 size= 512kB time=00:00:08.28 bitrate= 506.6kbits/s speed=2.61x frame= 283 fps= 74 q=-1.0 Lsize= 821kB time=00:00:11.20 bitrate= 600.9kbits/s speed=2.95x +video:817kB audio:0kB subtitle:0kB other streams:0kB global headers:0kB muxing overhead: 0.506232% +[libx264 @ 0x55eed4d03700] frame I:2 Avg QP:16.56 size: 37727 +[libx264 @ 0x55eed4d03700] frame P:74 Avg QP:25.32 size: 8153 +[libx264 @ 0x55eed4d03700] frame B:207 Avg QP:18.80 size: 761 +[libx264 @ 0x55eed4d03700] consecutive B-frames: 0.7% 4.2% 3.2% 91.9% +[libx264 @ 0x55eed4d03700] mb I I16..4: 32.8% 41.2% 26.0% +[libx264 @ 0x55eed4d03700] mb P I16..4: 0.3% 1.5% 0.9% P16..4: 5.7% 6.7% 4.5% 0.0% 0.0% skip:80.4% +[libx264 @ 0x55eed4d03700] mb B I16..4: 0.1% 0.1% 0.0% B16..8: 6.6% 0.9% 0.4% direct: 0.3% skip:91.7% L0:44.2% L1:47.5% BI: 8.2% +[libx264 @ 0x55eed4d03700] 8x8 transform intra:47.5% inter:37.5% +[libx264 @ 0x55eed4d03700] coded y,uvDC,uvAC intra: 40.9% 43.5% 35.3% inter: 2.7% 2.2% 0.7% +[libx264 @ 0x55eed4d03700] i16 v,h,dc,p: 69% 20% 9% 3% +[libx264 @ 0x55eed4d03700] i8 v,h,dc,ddl,ddr,vr,hd,vl,hu: 28% 11% 30% 5% 5% 4% 6% 4% 7% +[libx264 @ 0x55eed4d03700] i4 v,h,dc,ddl,ddr,vr,hd,vl,hu: 29% 18% 17% 5% 6% 6% 9% 4% 7% +[libx264 @ 0x55eed4d03700] i8c dc,h,v,p: 67% 15% 13% 5% +[libx264 @ 0x55eed4d03700] Weighted P-Frames: Y:0.0% UV:0.0% +[libx264 @ 0x55eed4d03700] ref P L0: 69.5% 15.3% 10.0% 5.1% +[libx264 @ 0x55eed4d03700] ref B L0: 87.1% 10.1% 2.8% +[libx264 @ 0x55eed4d03700] ref B L1: 98.4% 1.6% +[libx264 @ 0x55eed4d03700] kb/s:591.01 diff --git a/previews/PR62/generated/ffmpeg.out b/v0.15.2/generated/ffmpeg.out similarity index 100% rename from previews/PR62/generated/ffmpeg.out rename to v0.15.2/generated/ffmpeg.out diff --git a/v0.15.2/generated/gradient/index.html b/v0.15.2/generated/gradient/index.html new file mode 100644 index 00000000..9746badf --- /dev/null +++ b/v0.15.2/generated/gradient/index.html @@ -0,0 +1,110 @@ + +Gradient of a scalar field · PencilFFTs.jl
                        GitHub

                        Gradient of a scalar field

                        This example shows different methods to compute the gradient of a real-valued 3D scalar field $θ(\bm{x})$ in Fourier space, where $\bm{x} = (x, y, z)$. It is assumed that the field is periodic with period $L = 2π$ along all dimensions.

                        General procedure

                        The discrete Fourier expansion of $θ$ writes

                        \[θ(\bm{x}) = ∑_{\bm{k} ∈ \Z^3} \hat{θ}(\bm{k}) \, e^{i \bm{k} ⋅ \bm{x}},\]

                        where $\bm{k} = (k_x, k_y, k_z)$ are the Fourier wave numbers and $\hat{θ}$ is the discrete Fourier transform of $θ$. Then, the spatial derivatives of $θ$ are given by

                        \[\frac{∂ θ(\bm{x})}{∂ x_i} = +∑_{\bm{k} ∈ \Z^3} i k_i \hat{θ}(\bm{k}) \, e^{i \bm{k} ⋅ \bm{x}},\]

                        where the subscript $i$ denotes one of the spatial components $x$, $y$ or $z$.

                        In other words, to compute $\bm{∇} θ = (∂_x θ, ∂_y θ, ∂_z θ)$, one has to:

                        1. transform $θ$ to Fourier space to obtain $\hat{θ}$,
                        2. multiply $\hat{θ}$ by $i \bm{k}$,
                        3. transform the result back to physical space to obtain $\bm{∇} θ$.

                        Preparation

                        In this section, we initialise a random real-valued scalar field $θ$ and compute its FFT. For more details see the Tutorial.

                        using MPI
+using PencilFFTs
+using Random
+
+MPI.Init()
+
+# Input data dimensions (Nx × Ny × Nz)
+dims = (64, 32, 64)
+
+# Apply a 3D real-to-complex (r2c) FFT.
+transform = Transforms.RFFT()
+
+# Automatically create decomposition configuration
+comm = MPI.COMM_WORLD
+pen = Pencil(dims, comm)
+
+# Create plan
+plan = PencilFFTPlan(pen, transform)
+
+# Allocate data and initialise field
+θ = allocate_input(plan)
+randn!(θ)
+
+# Perform distributed FFT
+θ_hat = plan * θ

                        Finally, we initialise the output that will hold ∇θ in Fourier space. Noting that ∇θ is a vector field, we choose to store it as a tuple of 3 PencilArrays.

                        ∇θ_hat = allocate_output(plan, Val(3))
+
+# This is equivalent:
+# ∇θ_hat = ntuple(d -> similar(θ_hat), Val(3))
+
+summary(∇θ_hat)
                        "Tuple{PencilArray{ComplexF64, 3, Array{ComplexF64, 3}, 3, 0, Pencil{3, 2, Permutation{(3, 2, 1), 3}, Vector{UInt8}}}, PencilArray{ComplexF64, 3, Array{ComplexF64, 3}, 3, 0, Pencil{3, 2, Permutation{(3, 2, 1), 3}, Vector{UInt8}}}, PencilArray{ComplexF64, 3, Array{ComplexF64, 3}, 3, 0, Pencil{3, 2, Permutation{(3, 2, 1), 3}, Vector{UInt8}}}}"

                        Fourier wave numbers

                        In general, the Fourier wave numbers are of the form $k_i = 0, ±\frac{2π}{L_i}, ±\frac{4π}{L_i}, ±\frac{6π}{L_i}, …$, where $L_i$ is the period along dimension $i$. When a real-to-complex Fourier transform is applied, roughly half of these wave numbers are redundant due to the Hermitian symmetry of the complex Fourier coefficients. In practice, this means that for the fastest dimension $x$ (along which a real-to-complex transform is performed), the negative wave numbers are dropped, i.e. $k_x = 0, \frac{2π}{L_x}, \frac{4π}{L_x}, …$.

                        The AbstractFFTs package provides a convenient way to generate the Fourier wave numbers, using the functions fftfreq and rfftfreq. We can use these functions to initialise a "grid" of wave numbers associated to our 3D real-to-complex transform:

                        using AbstractFFTs: fftfreq, rfftfreq
+
+box_size = (2π, 2π, 2π)  # Lx, Ly, Lz
+sample_rate = 2π .* dims ./ box_size
+
+# In our case (Lx = 2π and Nx even), this gives kx = [0, 1, 2, ..., Nx/2].
+kx = rfftfreq(dims[1], sample_rate[1])
+
+# In our case (Ly = 2π and Ny even), this gives
+# ky = [0, 1, 2, ..., Ny/2-1, -Ny/2, -Ny/2+1, ..., -1] (and similarly for kz).
+ky = fftfreq(dims[2], sample_rate[2])
+kz = fftfreq(dims[3], sample_rate[3])
+
+kvec = (kx, ky, kz)
                        ([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0  …  23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, 32.0], [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0  …  -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0], [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0  …  -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0])

                        Note that kvec now contains the wave numbers associated to the global domain. In the following, we will only need the wave numbers associated to the portion of the domain handled by the local MPI process.

                        Method 1: global views

                        PencilArrays, returned for instance by allocate_input and allocate_output, take indices that start at 1, regardless of the location of the subdomain associated to the local process on the global grid. (In other words, PencilArrays take local indices.) On the other hand, we have defined the wave number vector kvec which, for each MPI process, is defined over the global domain, and as such it takes global indices.

                        One straightforward way of making data arrays compatible with wave numbers is to use global views, i.e. arrays that take global indices. These are generated from PencilArrays by calling the global_view function. Note that, in general, global indices do not start at 1 for a given MPI process. A given process will own a range of data given by indices in (i1:i2, j1:j2, k1:k2).

                        θ_glob = global_view(θ_hat)
+∇θ_glob = global_view.(∇θ_hat)
+summary(θ_glob)
                        "16×32×64 OffsetArray(::PencilArray{ComplexF64, 3}, 1:16, 1:32, 1:64) with eltype ComplexF64 with indices 1:16×1:32×1:64"

                        Once we have global views, we can combine data and wave numbers using the portion of global indices owned by the local MPI process, as shown below. We can use CartesianIndices to iterate over the global indices associated to the local process.

                        for I in CartesianIndices(θ_glob)
+    i, j, k = Tuple(I)  # unpack indices
+
+    # Wave number vector associated to current Cartesian index.
+    kx = kvec[1][i]
+    ky = kvec[2][j]
+    kz = kvec[3][k]
+
+    # Compute gradient in Fourier space.
+    # Note that modifying ∇θ_glob also modifies the original PencilArray ∇θ_hat.
+    ∇θ_glob[1][I] = im * kx * θ_glob[I]
+    ∇θ_glob[2][I] = im * ky * θ_glob[I]
+    ∇θ_glob[3][I] = im * kz * θ_glob[I]
+end

                        The above loop can be written in a slightly more efficient manner by precomputing im * θ_glob[I]:

                        @inbounds for I in CartesianIndices(θ_glob)
+    i, j, k = Tuple(I)
+
+    kx = kvec[1][i]
+    ky = kvec[2][j]
+    kz = kvec[3][k]
+
+    u = im * θ_glob[I]
+
+    ∇θ_glob[1][I] = kx * u
+    ∇θ_glob[2][I] = ky * u
+    ∇θ_glob[3][I] = kz * u
+end

                        Also note that the above can be easily written in a more generic way, e.g. for arbitrary dimensions, thanks in part to the use of CartesianIndices. Moreover, in the above there is no notion of the dimension permutations discussed in the tutorial, as it is all hidden behind the implementation of PencilArrays. And as seen later in the benchmarks, these (hidden) permutations have zero cost, as the speed is identical to that of a function that explicitly takes into account these permutations.

                        Finally, we can perform a backwards transform to obtain $\bm{∇} θ$ in physical space:

                        ∇θ = plan \ ∇θ_hat;

                        Note that the transform is automatically broadcast over the three fields of the ∇θ_hat vector, and the result ∇θ is also a tuple of three PencilArrays.

                        Method 2: explicit global indexing

                        Sometimes, one does not need to write generic code. In our case, one often knows the dimensionality of the problem and the memory layout of the data (i.e. the underlying index permutation).

                        Below is a reimplementation of the above loop, using explicit indices instead of CartesianIndices, and assuming that the underlying index permutation is (3, 2, 1), that is, data is stored in $(z, y, x)$ order. As discussed in the tutorial, this is the default for transformed arrays. This example also serves as a more explicit explanation for what is going on in the first method.

                        # Get local data range in the global grid.
+rng = axes(θ_glob)  # = (i1:i2, j1:j2, k1:k2)
                        (OffsetArrays.IdOffsetRange(values=1:16, indices=1:16), OffsetArrays.IdOffsetRange(values=1:32, indices=1:32), OffsetArrays.IdOffsetRange(values=1:64, indices=1:64))

                        For the loop below, we're assuming that the permutation is (3, 2, 1). In other words, the fastest index is the last one, and not the first one as it is usually in Julia. If the permutation is not (3, 2, 1), things will still work (well, except for the assertion below!), but the loop order will not be optimal.

                        @assert permutation(θ_hat) === Permutation(3, 2, 1)
+
+@inbounds for i in rng[1], j in rng[2], k in rng[3]
+    kx = kvec[1][i]
+    ky = kvec[2][j]
+    kz = kvec[3][k]
+
+    # Note that we still access the arrays in (i, j, k) order.
+    # (The permutation happens behind the scenes!)
+    u = im * θ_glob[i, j, k]
+
+    ∇θ_glob[1][i, j, k] = kx * u
+    ∇θ_glob[2][i, j, k] = ky * u
+    ∇θ_glob[3][i, j, k] = kz * u
+end

                        Method 3: using local indices

                        Alternatively, we can avoid global views and work directly on PencilArrays using local indices that start at 1. In this case, part of the strategy is to construct a "local" grid of wave numbers that can also be accessed with local indices. This can be conveniently done using the localgrid function of the PencilArrays.jl package, which accepts a PencilArray (or its associated Pencil) and the global coordinates (here kvec):

                        grid_fourier = localgrid(θ_hat, kvec)
                        LocalRectilinearGrid{3} with Permutation(3, 2, 1) and coordinates:
+ (1) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0]
+ (2) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]
+ (3) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]

                        Note that one can directly iterate on the returned grid object:

                        @inbounds for I in CartesianIndices(grid_fourier)
+    # Wave number vector associated to current Cartesian index.
+    k⃗ = grid_fourier[I]
+    u = im * θ_hat[I]
+    ∇θ_hat[1][I] = k⃗[1] * u
+    ∇θ_hat[2][I] = k⃗[2] * u
+    ∇θ_hat[3][I] = k⃗[3] * u
+end

                        This implementation is as efficient as the other examples, while being slightly shorter to write. Moreover, it is quite generic, and can be made independent of the number of dimensions with little effort.

                        Method 4: using broadcasting

                        Finally, note that the local grid object returned by localgrid makes it is possible to compute the gradient using broadcasting, thus fully avoiding scalar indexing. This can be quite convenient in some cases, and can also be very useful if one is working on GPUs (where scalar indexing is prohibitively expensive). Using broadcasting, the above examples simply become:

                        @. ∇θ_hat[1] = im * grid_fourier[1] * θ_hat
+@. ∇θ_hat[2] = im * grid_fourier[2] * θ_hat
+@. ∇θ_hat[3] = im * grid_fourier[3] * θ_hat

                        Once again, as shown in the benchmarks further below, this method performs quite similarly to the other ones.

                        Summary

                        The PencilArrays module provides different alternatives to deal with MPI-distributed data that may be subject to dimension permutations. In particular, one can choose to work with global indices (first two examples), with local indices (third example), or to avoid scalar indexing altogether (fourth example).

                        If one wants to stay generic, making sure that the same code will work for arbitrary dimensions and will be efficient regardless of the underlying dimension permutation, methods 1, 3 or 4 should be preferred. These use CartesianIndices and make no assumptions on possible dimension permutations, which are by default enabled in the output of PencilFFTs transforms. In fact, such permutations are completely invisible in the implementations.

                        The second method uses explicit (i, j, k) indices. It assumes that the underlying permutation is (3, 2, 1) to loop with i as the slowest index and k as the fastest, which is the optimal order in this case given the permutation. As such, the implementation is less generic than the others, and differences in performance are negligible with respect to more generic variants.

                        Benchmark results

                        The following are the benchmark results obtained from running examples/gradient.jl on a laptop, using 2 MPI processes and Julia 1.7.2, with an input array of global dimensions $64 × 32 × 64$. The different methods detailed above are marked on the right. The "lazy" marks indicate runs where the wave numbers were represented by lazy Frequencies objects (returned by rfftfreq and fftfreq). Otherwise, they were collected into Vectors. For some reason, plain Vectors are faster when working with grids generated by localgrid.

                        In the script, additional implementations can be found which rely on a more advanced understanding of permutations and on the internals of the PencilArrays package. For instance, gradient_local_parent! directly works with the raw data stored in Julia Arrays, while gradient_local_linear! completely avoids CartesianIndices while staying generic and efficient. Nevertheless, these display roughly the same performance as the above examples.

                            gradient_global_view!...                  89.900 μs
+    gradient_global_view! (lazy)...           92.060 μs  [Method 1]
+    gradient_global_view_explicit!...         88.958 μs
+    gradient_global_view_explicit! (lazy)...  81.055 μs  [Method 2]
+    gradient_local!...                        92.305 μs
+    gradient_grid!...                         92.770 μs
+    gradient_grid! (lazy)...                  101.388 μs  [Method 3]
+    gradient_grid_broadcast!...               88.606 μs
+    gradient_grid_broadcast! (lazy)...        151.020 μs  [Method 4]
+    gradient_local_parent!...                 92.248 μs
+    gradient_local_linear!...                 91.212 μs
+    gradient_local_linear_explicit!...        90.992 μs

                        This page was generated using Literate.jl.

                        diff --git a/v0.15.2/generated/in-place/index.html b/v0.15.2/generated/in-place/index.html new file mode 100644 index 00000000..7b5e54cf --- /dev/null +++ b/v0.15.2/generated/in-place/index.html @@ -0,0 +1,31 @@ + +In-place transforms · PencilFFTs.jl
                        GitHub

                        In-place transforms

                        Complex-to-complex and real-to-real transforms can be performed in-place, enabling important memory savings. The procedure is very similar to that of out-of-place transforms described in the tutorial. The differences are illustrated in the sections below.

                        Creating a domain partition

                        We start by partitioning a domain of dimensions $16×32×64$ along all available MPI processes.

                        using PencilFFTs
+using MPI
+MPI.Init()
+
+dims_global = (16, 32, 64)  # global dimensions
                        (16, 32, 64)

                        Such a partitioning is described by a Pencil object. Here we choose to decompose the domain along the last two dimensions. In this case, the actual number of processes along each of these dimensions is chosen automatically.

                        decomp_dims = (2, 3)
+comm = MPI.COMM_WORLD
+pen = Pencil(dims_global, decomp_dims, comm)
                        Decomposition of 3D data
+    Data dimensions: (16, 32, 64)
+    Decomposed dimensions: (2, 3)
+    Data permutation: NoPermutation()
+    Array type: Array
                        Allowed decompositions

                        Distributed transforms using PencilFFTs.jl require that the first dimension is not decomposed. In other words, if one wants to perform transforms, then decomp_dims above must not contain 1.

                        Creating in-place plans

                        To create an in-place plan, pass an in-place transform such as Transforms.FFT! or Transforms.R2R! to PencilFFTPlan. For instance:

                        # Perform a 3D in-place complex-to-complex FFT.
+transform = Transforms.FFT!()
+
+# Note that one can also combine different types of in-place transforms.
+# For instance:
+# transform = (
+#     Transforms.R2R!(FFTW.REDFT01),
+#     Transforms.FFT!(),
+#     Transforms.R2R!(FFTW.DHT),
+# )
                        FFT!

                        We can now create a distributed plan from the previously-created domain partition and the chosen transform.

                        plan = PencilFFTPlan(pen, transform)
                        Transforms: (FFT!, FFT!, FFT!)
+Input type: ComplexF64
+Global dimensions: (16, 32, 64) -> (16, 32, 64)
+MPI topology: 2D decomposition (2×1 processes)

                        Note that in-place real-to-complex transforms are not currently supported. (In other words, the RFFT! transform type is not defined.)

                        Allocating data

                        As with out-of-place plans, data should be allocated using allocate_input. The difference is that, for in-place plans, this function returns a ManyPencilArray object, which is a container holding multiple PencilArray views sharing the same memory space.

                        # Allocate data for the plan.
+# Since `plan` is in-place, this returns a `ManyPencilArray` container.
+A = allocate_input(plan)
+summary(A)
                        "ManyPencilArray{ComplexF64, 3, 3, Tuple{PencilArray{ComplexF64, 3, Base.ReshapedArray{ComplexF64, 3, SubArray{ComplexF64, 1, Vector{ComplexF64}, Tuple{Base.OneTo{Int64}}, true}, Tuple{}}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{ComplexF64, 3, Base." ⋯ 140 bytes ⋯ "tation{(2, 1, 3), 3}, Vector{UInt8}}}, PencilArray{ComplexF64, 3, Base.ReshapedArray{ComplexF64, 3, SubArray{ComplexF64, 1, Vector{ComplexF64}, Tuple{Base.OneTo{Int64}}, true}, Tuple{}}, 3, 0, Pencil{3, 2, Permutation{(3, 2, 1), 3}, Vector{UInt8}}}}, Vector{ComplexF64}}"

                        Note that allocate_output also works for in-place plans. In this case, it returns exactly the same thing as allocate_input.

                        As shown in the next section, in-place plans must be applied on the returned ManyPencilArray. On the other hand, one usually wants to access and modify data, and for this one needs the PencilArray views contained in the ManyPencilArray. The input and output array views can be obtained by calling first(::ManyPencilArray) and last(::ManyPencilArray).

                        For instance, we can initialise the input array with some data before transforming:

                        using Random
+u_in = first(A)  # input data view
+randn!(u_in)
+summary(u_in)
                        "16×16×64 PencilArray{ComplexF64, 3}(::Pencil{3, 2, NoPermutation, Array})"

                        Applying plans

                        Like in FFTW.jl, one can perform in-place transforms using the * and \ operators. As mentioned above, in-place plans must be applied on the ManyPencilArray containers returned by allocate_input.

                        plan * A;  # performs in-place forward transform

                        After performing an in-place transform, data contained in u_in has been overwritten and has no "physical" meaning. In other words, u_in should not be used at this point. To access the transformed data, one should retrieve the output data view using last(A).

                        For instance, to compute the global sum of the transformed data:

                        u_out = last(A)  # output data view
+sum(u_out)       # sum of transformed data (note that `sum` reduces over all processes)
                        -406.14918822270556 - 34413.04836644857im

                        Finally, we can perform a backward transform and do stuff with the input view:

                        plan \ A;  # perform in-place backward transform

                        At this point, the data can be once again found in the input view u_in, while u_out should not be accessed.

                        This page was generated using Literate.jl.

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image/svg+xml + + + Matplotlib v3.4.3, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 128 + + + + + + + + + + + + + + + 256 + + + + + + + + + + + + + + + 512 + + + + + + + + + + + + + + + 1024 + + + + + + + + + + + + + + + 2048 + + + + + + + + + + + + + + + 4096 + + + + + + + + + + + + + + + 8192 + + + + + + + + + + + + + + + 16384 + + + + + + MPI processes + + + + + + + + + + + + + + + + + 10 + + + + + + + + + + + + + + + 100 + + + + + + + + + + + + + + + 1000 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Time (milliseconds) + + + + + + + + + + + + + + + + + + + + + + + + + + + + 512 + 3 + + + + + + + + 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00000000..a5cef848 --- /dev/null +++ b/v0.15.2/index.html @@ -0,0 +1,7 @@ + +Home · PencilFFTs.jl
                        GitHub

                        PencilFFTs

                        Fast Fourier transforms of MPI-distributed Julia arrays.

                        Introduction

                        This package provides multidimensional FFTs and related transforms on MPI-distributed Julia arrays via the PencilArrays package.

                        The name of this package originates from the decomposition of 3D domains along two out of three dimensions, sometimes called pencil decomposition. This is illustrated by the figure below,[1] where each coloured block is managed by a different MPI process. Typically, one wants to compute FFTs on a scalar or vector field along the three spatial dimensions. In the case of a pencil decomposition, 3D FFTs are performed one dimension at a time, along the non-decomposed direction. Transforms must then be interleaved with global data transpositions to switch between pencil configurations. In high-performance computing environments, such data transpositions are generally the most expensive part of a parallel FFT computation, due to the large cost of communications between computing nodes.

                        + Pencil decomposition of 3D domains +

                        More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension $N$. The decompositions can be performed along an arbitrary number $M < N$ of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.

                        The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.

                        Features

                        • distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;

                        • FFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;

                        • in-place and out-of-place transforms;

                        • high scalability up to (at least) tens of thousands of MPI processes.

                        Installation

                        PencilFFTs can be installed using the Julia package manager:

                        julia> ] add PencilFFTs

                        Similar projects

                        • FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.

                        • PFFT is a very general parallel FFT library written in C.

                        • P3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.

                        • 2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.

                        • 1Figure adapted from this PhD thesis.
                        • 2For the pencil decomposition represented in the figure, $N = 3$ and $M = 2$.
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                        ","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"By default, the domain is distributed on a 2D MPI topology of dimensions N_1 N_2. As an example, the above figure shows such a topology with N_1 = 4 and N_2 = 3, for a total of 12 MPI processes.","category":"page"},{"location":"tutorial/#tutorial:creating_plans","page":"Tutorial","title":"Creating plans","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The first thing to do is to create a domain decomposition configuration for the given dataset dimensions N_x N_y N_z. In the framework of PencilArrays, such a configuration is described by a Pencil object. As described in the PencilArrays docs, we can let the Pencil constructor automatically determine such a configuration. For this, only an MPI communicator and the dataset dimensions are needed:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using MPI\nusing PencilFFTs\n\nMPI.Init()\ncomm = MPI.COMM_WORLD\n\n# Input data dimensions (Nx × Ny × Nz)\ndims = (16, 32, 64)\npen = Pencil(dims, comm)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"By default this creates a 2D decomposition (for the case of a 3D dataset), but one can change this as detailed in the PencilArrays documentation linked above.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"We can now create a PencilFFTPlan, which requires information on decomposition configuration (the Pencil object) and on the transforms that will be applied:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# Apply a 3D real-to-complex (r2c) FFT.\ntransform = Transforms.RFFT()\n\n# Note that, for more control, one can instead separately specify the transforms along each dimension:\n# transform = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())\n\n# Create plan\nplan = PencilFFTPlan(pen, transform)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"See the PencilFFTPlan constructor for details on the accepted options, and the Transforms module for the possible transforms. It is also possible to enable fine-grained performance measurements via the TimerOutputs package, as described in Measuring performance.","category":"page"},{"location":"tutorial/#Allocating-data","page":"Tutorial","title":"Allocating data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Next, we want to apply the plan on some data. Transforms may only be applied on PencilArrays, which are array wrappers that include MPI decomposition information (in some sense, analogous to DistributedArrays in Julia's distributed computing approach). The helper function allocate_input can be used to allocate a PencilArray that is compatible with our plan:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# In our example, this returns a 3D PencilArray of real data (Float64).\nu = allocate_input(plan)\n\n# Fill the array with some (random) data\nusing Random\nrandn!(u)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"PencilArrays are a subtype of AbstractArray, and thus they support all common array operations.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Similarly, to preallocate output data, one can use allocate_output:","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"# In our example, this returns a 3D PencilArray of complex data (Complex{Float64}).\nv = allocate_output(plan)","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"This is only required if one wants to apply the plans using a preallocated output (with mul!, see right below).","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The data types returned by allocate_input and allocate_output are slightly different when working with in-place transforms. 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Similarly, \\ and ldiv! are used for backward transforms.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"using LinearAlgebra # for mul!, ldiv!\n\n# Apply plan on `u` with `v` as an output\nmul!(v, plan, u)\n\n# Apply backward plan on `v` with `w` as an output\nw = similar(u)\nldiv!(w, plan, v) # now w ≈ u","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"Note that, consistently with AbstractFFTs, normalisation is performed at the end of a backward transform, so that the original data is recovered when applying a forward followed by a backward transform.","category":"page"},{"location":"tutorial/#Accessing-and-modifying-data","page":"Tutorial","title":"Accessing and modifying data","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For any given MPI process, a PencilArray holds the data associated to its local partition in the global geometry. 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This detail is hidden from the user, and output arrays are always accessed in the same order as the input data, regardless of the underlying output dimension permutation. This applies to PencilArrays and to OffsetArrays returned by global_view.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The reasoning behind dimension permutations, is that they allow to always perform FFTs along the fastest array dimension and to avoid a local data transposition, resulting in performance gains. A similar approach is followed by other parallel FFT libraries. FFTW itself, in its distributed-memory routines, includes a flag that enables a similar behaviour. 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This has been achieved, in part, by making sure that permutations such as (3, 2, 1) are compile-time constants.","category":"page"},{"location":"tutorial/#Further-reading","page":"Tutorial","title":"Further reading","text":"","category":"section"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"For details on working with PencilArrays see the PencilArrays docs.","category":"page"},{"location":"tutorial/","page":"Tutorial","title":"Tutorial","text":"The examples on the sidebar further illustrate the use of transforms and provide an introduction to working with MPI-distributed data in the form of PencilArrays. In particular, the gradient example illustrates different ways of computing things using Fourier-transformed distributed arrays. Then, the incompressible Navier–Stokes example is a more advanced and complete example of a possible application of the PencilFFTs package.","category":"page"},{"location":"benchmarks/#Benchmarks","page":"Benchmarks","title":"Benchmarks","text":"","category":"section"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The performance of PencilFFTs.jl is comparable to that of other open-source parallel FFT libraries implemented in lower-level languages. Below, we show comparisons with the Fortran implementation of P3DFFT, possibly the most popular of these libraries. The benchmarks were performed on the Jean–Zay cluster of the IDRIS French computing centre (CNRS).","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The figure below shows strong scaling benchmarks of 3D real-to-complex FFTs using 2D (\"pencil\") decomposition. The benchmarks were run for input arrays of dimensions N_x N_y N_z = 512^3, 1024^3 and 2048^3. Each timing is averaged over 100 repetitions.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"
                        \n \n \n
                        ","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"As seen above, PencilFFTs generally outperforms P3DFFT in its default setting. This is largely explained by the choice of using non-blocking point-to-point MPI communications (via MPI_Isend and MPI_Irecv), while P3DFFT uses collective MPI_Alltoallv calls. This enables PencilFFTs to perform data reordering operations on the partially received data while waiting for the incoming data, leading to better performance. Moreover, in contrast with P3DFFT, the high performance and scalability of PencilFFTs results from a highly generic code, handling decompositions in arbitrary dimensions and a relatively large (and extensible) variety of transformations.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"Note that PencilFFTs can optionally use collective communications (using MPI_Alltoallv) instead of point-to-point communications. For details, see the docs for PencilFFTPlan and for PencilArray transpositions. As seen above, collective communications generally perform worse than point-to-point ones, and runtimes are nearly indistinguishable from those of P3DFFT.","category":"page"},{"location":"benchmarks/#Benchmark-details","page":"Benchmarks","title":"Benchmark details","text":"","category":"section"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The benchmarks were performed using Julia 1.7-beta3 and Intel MPI 2019. We used PencilFFTs v0.12.5 with FFTW.jl v1.4.3 and MPI.jl v0.19.0. We used the Fortran implementation of P3DFFT, version 2.7.6, which was built with Intel 2019 compilers and linked to FFTW 3.3.8. The cluster where the benchmarks were run has Intel Cascade Lake 6248 processors with 2×20 cores per node.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The number of MPI processes along each decomposed dimension, P_1 and P_2, was automatically determined by a call to MPI_Dims_create, which tends to create a balanced decomposition with P_1 P_2. For instance, a total of 1024 processes is divided into P_1 = P_2 = 32. Different results may be obtained with other combinations, but this was not benchmarked.","category":"page"},{"location":"benchmarks/","page":"Benchmarks","title":"Benchmarks","text":"The source files used to generate this benchmark, as well as the raw benchmark results, are all available in the PencilFFTs repo.","category":"page"},{"location":"GlobalFFTParams/#Global-FFT-parameters","page":"Global FFT parameters","title":"Global FFT parameters","text":"","category":"section"},{"location":"GlobalFFTParams/","page":"Global FFT parameters","title":"Global FFT parameters","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"GlobalFFTParams/","page":"Global FFT parameters","title":"Global FFT parameters","text":"GlobalFFTParams","category":"page"},{"location":"GlobalFFTParams/#PencilFFTs.GlobalFFTParams","page":"Global FFT parameters","title":"PencilFFTs.GlobalFFTParams","text":"GlobalFFTParams{T, N, inplace}\n\nSpecifies the global parameters for an N-dimensional distributed transform. These include the element type T and global data sizes of input and output data, as well as the transform types to be performed along each dimension.\n\n\n\nGlobalFFTParams(size_global, transforms, [real_type=Float64])\n\nDefine parameters for N-dimensional transform.\n\ntransforms must be a tuple of length N specifying the transforms to be applied along each dimension. Each element must be a subtype of Transforms.AbstractTransform. For all the possible transforms, see Transform types.\n\nThe element type must be a real type accepted by FFTW, i.e. either Float32 or Float64.\n\nNote that the transforms are applied one dimension at a time, with the leftmost dimension first for forward transforms.\n\nExample\n\nTo perform a 3D FFT of real data, first a real-to-complex FFT must be applied along the first dimension, followed by two complex-to-complex FFTs along the other dimensions:\n\njulia> size_global = (64, 32, 128); # size of real input data\n\njulia> transforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT());\n\njulia> fft_params = PencilFFTs.GlobalFFTParams(size_global, transforms)\nTransforms: (RFFT, FFT, FFT)\nInput type: Float64\nGlobal dimensions: (64, 32, 128) -> (33, 32, 128)\n\n\n\n\n\n","category":"type"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"EditURL = \"../../examples/in-place.jl\"","category":"page"},{"location":"generated/in-place/#In-place-transforms","page":"In-place transforms","title":"In-place transforms","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Complex-to-complex and real-to-real transforms can be performed in-place, enabling important memory savings. The procedure is very similar to that of out-of-place transforms described in the tutorial. The differences are illustrated in the sections below.","category":"page"},{"location":"generated/in-place/#Creating-a-domain-partition","page":"In-place transforms","title":"Creating a domain partition","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"We start by partitioning a domain of dimensions 163264 along all available MPI processes.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"using PencilFFTs\nusing MPI\nMPI.Init()\n\ndims_global = (16, 32, 64) # global dimensions","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Such a partitioning is described by a Pencil object. Here we choose to decompose the domain along the last two dimensions. In this case, the actual number of processes along each of these dimensions is chosen automatically.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"decomp_dims = (2, 3)\ncomm = MPI.COMM_WORLD\npen = Pencil(dims_global, decomp_dims, comm)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"warning: Allowed decompositions\nDistributed transforms using PencilFFTs.jl require that the first dimension is not decomposed. In other words, if one wants to perform transforms, then decomp_dims above must not contain 1.","category":"page"},{"location":"generated/in-place/#Creating-in-place-plans","page":"In-place transforms","title":"Creating in-place plans","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"To create an in-place plan, pass an in-place transform such as Transforms.FFT! or Transforms.R2R! to PencilFFTPlan. For instance:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"# Perform a 3D in-place complex-to-complex FFT.\ntransform = Transforms.FFT!()\n\n# Note that one can also combine different types of in-place transforms.\n# For instance:\n# transform = (\n# Transforms.R2R!(FFTW.REDFT01),\n# Transforms.FFT!(),\n# Transforms.R2R!(FFTW.DHT),\n# )","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"We can now create a distributed plan from the previously-created domain partition and the chosen transform.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan = PencilFFTPlan(pen, transform)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Note that in-place real-to-complex transforms are not currently supported. (In other words, the RFFT! transform type is not defined.)","category":"page"},{"location":"generated/in-place/#Allocating-data","page":"In-place transforms","title":"Allocating data","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"As with out-of-place plans, data should be allocated using allocate_input. The difference is that, for in-place plans, this function returns a ManyPencilArray object, which is a container holding multiple PencilArray views sharing the same memory space.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"# Allocate data for the plan.\n# Since `plan` is in-place, this returns a `ManyPencilArray` container.\nA = allocate_input(plan)\nsummary(A)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Note that allocate_output also works for in-place plans. In this case, it returns exactly the same thing as allocate_input.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"As shown in the next section, in-place plans must be applied on the returned ManyPencilArray. On the other hand, one usually wants to access and modify data, and for this one needs the PencilArray views contained in the ManyPencilArray. The input and output array views can be obtained by calling first(::ManyPencilArray) and last(::ManyPencilArray).","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"For instance, we can initialise the input array with some data before transforming:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"using Random\nu_in = first(A) # input data view\nrandn!(u_in)\nsummary(u_in)","category":"page"},{"location":"generated/in-place/#Applying-plans","page":"In-place transforms","title":"Applying plans","text":"","category":"section"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Like in FFTW.jl, one can perform in-place transforms using the * and \\ operators. As mentioned above, in-place plans must be applied on the ManyPencilArray containers returned by allocate_input.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan * A; # performs in-place forward transform\nnothing #hide","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"After performing an in-place transform, data contained in u_in has been overwritten and has no \"physical\" meaning. In other words, u_in should not be used at this point. To access the transformed data, one should retrieve the output data view using last(A).","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"For instance, to compute the global sum of the transformed data:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"u_out = last(A) # output data view\nsum(u_out) # sum of transformed data (note that `sum` reduces over all processes)","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"Finally, we can perform a backward transform and do stuff with the input view:","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"plan \\ A; # perform in-place backward transform\nnothing #hide","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"At this point, the data can be once again found in the input view u_in, while u_out should not be accessed.","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"","category":"page"},{"location":"generated/in-place/","page":"In-place transforms","title":"In-place transforms","text":"This page was generated using Literate.jl.","category":"page"},{"location":"Transforms/#Available-transforms","page":"Available transforms","title":"Available transforms","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"CurrentModule = PencilFFTs.Transforms","category":"page"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"Transforms","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms","page":"Available transforms","title":"PencilFFTs.Transforms","text":"Defines different one-dimensional FFT-based transforms.\n\nThe transforms are all subtypes of an AbstractTransform type.\n\nWhen possible, the names of the transforms are kept consistent with the functions exported by AbstractFFTs.jl and FFTW.jl.\n\n\n\n\n\n","category":"module"},{"location":"Transforms/#Transform-types","page":"Available transforms","title":"Transform types","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"FFT\nFFT!\nBFFT\nBFFT!\n\nRFFT\nRFFT!\nBRFFT\nBRFFT!\n\nR2R\nR2R!\n\nNoTransform\nNoTransform!","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.FFT","page":"Available transforms","title":"PencilFFTs.Transforms.FFT","text":"FFT()\n\nComplex-to-complex FFT.\n\nSee also AbstractFFTs.fft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.FFT!","page":"Available transforms","title":"PencilFFTs.Transforms.FFT!","text":"FFT!()\n\nIn-place version of FFT.\n\nSee also AbstractFFTs.fft!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BFFT","page":"Available transforms","title":"PencilFFTs.Transforms.BFFT","text":"BFFT()\n\nUnnormalised backward complex-to-complex FFT.\n\nLike AbstractFFTs.bfft, this transform is not normalised. To obtain the inverse transform, divide the output by the length of the transformed dimension.\n\nSee also AbstractFFTs.bfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.BFFT!","text":"BFFT()\n\nIn-place version of BFFT.\n\nSee also AbstractFFTs.bfft!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.RFFT","page":"Available transforms","title":"PencilFFTs.Transforms.RFFT","text":"RFFT()\n\nReal-to-complex FFT.\n\nSee also AbstractFFTs.rfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.RFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.RFFT!","text":"RFFT!()\n\nIn-place version of RFFT.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BRFFT","page":"Available transforms","title":"PencilFFTs.Transforms.BRFFT","text":"BRFFT(d::Integer)\nBRFFT((d1, d2, ..., dN))\n\nUnnormalised inverse of RFFT.\n\nTo obtain the inverse transform, divide the output by the length of the transformed dimension (of the real output array).\n\nAs described in the AbstractFFTs docs, the length of the output cannot be fully inferred from the input length. For this reason, the BRFFT constructor accepts an optional d argument indicating the output length.\n\nFor multidimensional datasets, a tuple of dimensions (d1, d2, ..., dN) may also be passed. This is equivalent to passing just dN. In this case, the last dimension (dN) is the one that changes size between the input and output. Note that this is the opposite of FFTW.brfft. The reason is that, in PencilFFTs, the last dimension is the one along which a complex-to-real transform is performed.\n\nSee also AbstractFFTs.brfft.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.BRFFT!","page":"Available transforms","title":"PencilFFTs.Transforms.BRFFT!","text":"BRFFT!(d::Integer)\nBRFFT!((d1, d2, ..., dN))\n\nIn-place version of BRFFT.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.R2R","page":"Available transforms","title":"PencilFFTs.Transforms.R2R","text":"R2R(kind)\n\nReal-to-real transform of type kind.\n\nThe possible values of kind are those described in the FFTW.r2r docs and the FFTW manual:\n\ndiscrete cosine transforms: FFTW.REDFT00, FFTW.REDFT01, FFTW.REDFFT10, FFTW.REDFFT11\ndiscrete sine transforms: FFTW.RODFT00, FFTW.RODFT01, FFTW.RODFFT10, FFTW.RODFFT11\ndiscrete Hartley transform: FFTW.DHT\n\nNote: half-complex format DFTs (FFTW.R2HC, FFTW.HC2R) are not currently supported.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.R2R!","page":"Available transforms","title":"PencilFFTs.Transforms.R2R!","text":"R2R!(kind)\n\nIn-place version of R2R.\n\nSee also FFTW.r2r!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.NoTransform","page":"Available transforms","title":"PencilFFTs.Transforms.NoTransform","text":"NoTransform()\n\nIdentity transform.\n\nSpecifies that no transformation should be applied.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.NoTransform!","page":"Available transforms","title":"PencilFFTs.Transforms.NoTransform!","text":"NoTransform!()\n\nIn-place version of NoTransform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#Internals","page":"Available transforms","title":"Internals","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"What follows is used internally in PencilFFTs.","category":"page"},{"location":"Transforms/#Types","page":"Available transforms","title":"Types","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"AbstractCustomPlan\nAbstractTransform\nIdentityPlan\nIdentityPlan!\nPlan","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.AbstractCustomPlan","page":"Available transforms","title":"PencilFFTs.Transforms.AbstractCustomPlan","text":"AbstractCustomPlan\n\nAbstract type defining a custom plan, to be used as an alternative to FFTW plans (FFTW.FFTWPlan).\n\nThe only custom plan defined in this module is IdentityPlan. The user can define other custom plans that are also subtypes of AbstractCustomPlan.\n\nNote that plan returns a subtype of either AbstractFFTs.Plan or AbstractCustomPlan.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.AbstractTransform","page":"Available transforms","title":"PencilFFTs.Transforms.AbstractTransform","text":"AbstractTransform\n\nSpecifies a one-dimensional FFT-based transform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.IdentityPlan","page":"Available transforms","title":"PencilFFTs.Transforms.IdentityPlan","text":"IdentityPlan\n\nType of plan associated to NoTransform.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.IdentityPlan!","page":"Available transforms","title":"PencilFFTs.Transforms.IdentityPlan!","text":"IdentityPlan!\n\nType of plan associated to NoTransform!.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#PencilFFTs.Transforms.Plan","page":"Available transforms","title":"PencilFFTs.Transforms.Plan","text":"Plan = Union{AbstractFFTs.Plan, AbstractCustomPlan}\n\nUnion type representing any plan returned by plan.\n\nSee also AbstractCustomPlan.\n\n\n\n\n\n","category":"type"},{"location":"Transforms/#Functions","page":"Available transforms","title":"Functions","text":"","category":"section"},{"location":"Transforms/","page":"Available transforms","title":"Available transforms","text":"plan\n\nbinv\nscale_factor\n\neltype_input\neltype_output\nexpand_dims\nis_inplace\nkind\nlength_output","category":"page"},{"location":"Transforms/#PencilFFTs.Transforms.plan","page":"Available transforms","title":"PencilFFTs.Transforms.plan","text":"plan(transform::AbstractTransform, A, [dims];\n flags=FFTW.ESTIMATE, timelimit=Inf)\n\nCreate plan to transform array A along dimensions dims.\n\nIf dims is not specified, all dimensions of A are transformed.\n\nFor FFT plans, this function wraps the AbstractFFTs.jl and FFTW.jl plan creation functions. For more details on the function arguments, see AbstractFFTs.plan_fft.\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.binv","page":"Available transforms","title":"PencilFFTs.Transforms.binv","text":"binv(transform::AbstractTransform, d::Integer)\n\nReturns the backwards transform associated to the given transform.\n\nThe second argument must be the length of the first transformed dimension in the forward transform. It is used in particular when transform = RFFT(), to determine the length of the inverse (complex-to-real) transform. See the AbstractFFTs.irfft docs for details.\n\nThe backwards transform returned by this function is not normalised. The normalisation factor for a given array can be obtained by calling scale_factor.\n\nExample\n\njulia> binv(Transforms.FFT(), 42)\nBFFT\n\njulia> binv(Transforms.BRFFT(9), 42)\nRFFT\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.scale_factor","page":"Available transforms","title":"PencilFFTs.Transforms.scale_factor","text":"scale_factor(transform::AbstractTransform, A, [dims = 1:ndims(A)])\n\nGet factor required to normalise the given array after a transformation along dimensions dims (all dimensions by default).\n\nThe array A must have the dimensions of the transform input.\n\nImportant: the dimensions dims must be the same that were passed to plan.\n\nExamples\n\njulia> C = zeros(ComplexF32, 3, 4, 5);\n\njulia> scale_factor(Transforms.FFT(), C)\n60\n\njulia> scale_factor(Transforms.BFFT(), C)\n60\n\njulia> scale_factor(Transforms.BFFT(), C, 2:3)\n20\n\njulia> R = zeros(Float64, 3, 4, 5);\n\njulia> scale_factor(Transforms.RFFT(), R, 2)\n4\n\njulia> scale_factor(Transforms.RFFT(), R, 2:3)\n20\n\njulia> scale_factor(Transforms.BRFFT(8), C)\n96\n\njulia> scale_factor(Transforms.BRFFT(9), C)\n108\n\nThis will fail because the input of RFFT is real, and R is a complex array:\n\njulia> scale_factor(Transforms.RFFT(), C, 2:3)\nERROR: MethodError: no method matching scale_factor(::PencilFFTs.Transforms.RFFT, ::Array{ComplexF32, 3}, ::UnitRange{Int64})\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.eltype_input","page":"Available transforms","title":"PencilFFTs.Transforms.eltype_input","text":"eltype_input(transform::AbstractTransform, real_type<:AbstractFloat)\n\nDetermine input data type for a given transform given the floating point precision of the input data.\n\nSome transforms, such as R2R and NoTransform, can take both real and complex data. For those kinds of transforms, nothing is returned.\n\nExample\n\njulia> eltype_input(Transforms.FFT(), Float32)\nComplexF32 (alias for Complex{Float32})\n\njulia> eltype_input(Transforms.RFFT(), Float64)\nFloat64\n\njulia> eltype_input(Transforms.R2R(FFTW.REDFT01), Float64) # nothing\n\njulia> eltype_input(Transforms.NoTransform(), Float64) # nothing\n\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.eltype_output","page":"Available transforms","title":"PencilFFTs.Transforms.eltype_output","text":"eltype_output(transform::AbstractTransform, eltype_input)\n\nReturns the output data type for a given transform given the input type.\n\nThrows ArgumentError if the input data type is incompatible with the transform type.\n\nExample\n\njulia> eltype_output(Transforms.NoTransform(), Float32)\nFloat32\n\njulia> eltype_output(Transforms.RFFT(), Float64)\nComplexF64 (alias for Complex{Float64})\n\njulia> eltype_output(Transforms.BRFFT(4), ComplexF32)\nFloat32\n\njulia> eltype_output(Transforms.FFT(), Float64)\nERROR: ArgumentError: invalid input data type for PencilFFTs.Transforms.FFT: Float64\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.expand_dims","page":"Available transforms","title":"PencilFFTs.Transforms.expand_dims","text":"expand_dims(transform::AbstractTransform, Val(N))\n\nExpand a single multidimensional transform into one transform per dimension.\n\nExample\n\n# Expand a real-to-complex transform in 3 dimensions.\njulia> expand_dims(Transforms.RFFT(), Val(3))\n(RFFT, FFT, FFT)\n\njulia> expand_dims(Transforms.BRFFT(4), Val(3))\n(BFFT, BFFT, BRFFT{even})\n\njulia> expand_dims(Transforms.NoTransform(), Val(2))\n(NoTransform, NoTransform)\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.is_inplace","page":"Available transforms","title":"PencilFFTs.Transforms.is_inplace","text":"is_inplace(transform::AbstractTransform) -> Bool\nis_inplace(transforms::Vararg{AbtractTransform}) -> Union{Bool, Nothing}\n\nCheck whether a transform or a list of transforms is performed in-place.\n\nIf the list of transforms has a combination of in-place and out-of-place transforms, nothing is returned.\n\nExample\n\njulia> is_inplace(Transforms.RFFT())\nfalse\n\njulia> is_inplace(Transforms.NoTransform!())\ntrue\n\njulia> is_inplace(Transforms.FFT!(), Transforms.R2R!(FFTW.REDFT01))\ntrue\n\njulia> is_inplace(Transforms.FFT(), Transforms.R2R(FFTW.REDFT01))\nfalse\n\njulia> is_inplace(Transforms.FFT(), Transforms.R2R!(FFTW.REDFT01)) === nothing\ntrue\n\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.kind","page":"Available transforms","title":"PencilFFTs.Transforms.kind","text":"kind(transform::R2R)\n\nGet kind of real-to-real transform.\n\n\n\n\n\n","category":"function"},{"location":"Transforms/#PencilFFTs.Transforms.length_output","page":"Available transforms","title":"PencilFFTs.Transforms.length_output","text":"length_output(transform::AbstractTransform, length_in::Integer)\n\nReturns the length of the transform output, given the length of its input.\n\nThe input and output lengths are specified in terms of the respective input and output datatypes. For instance, for real-to-complex transforms, these are respectively the length of input real data and of output complex data.\n\n\n\n\n\n","category":"function"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"EditURL = \"../../examples/navier_stokes.jl\"","category":"page"},{"location":"generated/navier_stokes/#Navier–Stokes-equations","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"In this example, we numerically solve the incompressible Navier–Stokes equations","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"_t bmv + (bmv bm) bmv = -frac1ρ bm p + ν ^2 bmv\nquad bm bmv = 0","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where bmv(bmx t) and p(bmx t) are respectively the velocity and pressure fields, ν is the fluid kinematic viscosity and ρ is the fluid density.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.","category":"page"},{"location":"generated/navier_stokes/#First-steps","page":"Navier–Stokes equations","title":"First steps","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We start by loading the required packages, initialising MPI and setting the simulation parameters.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using MPI\nusing PencilFFTs\n\nMPI.Init()\ncomm = MPI.COMM_WORLD\nprocid = MPI.Comm_rank(comm) + 1\n\n# Simulation parameters\nNs = (64, 64, 64) # = (Nx, Ny, Nz)\nLs = (2π, 2π, 2π) # = (Lx, Ly, Lz)\n\n# Collocation points (\"global\" = over all processes).\n# We include the endpoint (length = N + 1) for convenience.\nxs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls) # = (x, y, z)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's check the number of MPI processes over which we're running our simulation:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"MPI.Comm_size(comm)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"pen = Pencil(Ns, comm)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"The subdomain associated to the local MPI process can be obtained using range_local:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"range_local(pen)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We now construct a distributed vector field that follows the decomposition configuration we just created:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v⃗₀ = (\n PencilArray{Float64}(undef, pen), # vx\n PencilArray{Float64}(undef, pen), # vy\n PencilArray{Float64}(undef, pen), # vz\n)\nsummary(v⃗₀[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We still need to fill this array with interesting values that represent a physical velocity field.","category":"page"},{"location":"generated/navier_stokes/#Initial-condition","page":"Navier–Stokes equations","title":"Initial condition","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"beginaligned\nv_x(x y z) = u₀ sin(k₀ x) cos(k₀ y) cos(k₀ z) \nv_y(x y z) = -u₀ cos(k₀ x) sin(k₀ y) cos(k₀ z) \nv_z(x y z) = 0\nendaligned","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where u₀ and k₀ are two parameters setting the amplitude and the period of the velocity field.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"grid = localgrid(pen, xs_global)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can use this to initialise the velocity field:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"u₀ = 1.0\nk₀ = 2π / Ls[1] # should be integer if L = 2π (to preserve periodicity)\n\n@. v⃗₀[1] = u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)\n@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)\n@. v⃗₀[3] = 0\nnothing # hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Let's plot a 2D slice of the velocity field managed by the local MPI process:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using GLMakie\n\n# Compute the norm of a vector field represented by a tuple of arrays.\nfunction vecnorm(v⃗::NTuple)\n vnorm = similar(v⃗[1])\n for n ∈ eachindex(v⃗[1])\n w = zero(eltype(vnorm))\n for v ∈ v⃗\n w += v[n]^2\n end\n vnorm[n] = sqrt(w)\n end\n vnorm\nend\n\nlet fig = Figure(resolution = (700, 600))\n ax = Axis3(fig[1, 1]; aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\")\n vnorm = parent(vecnorm(v⃗₀)) # use `parent` because Makie doesn't like custom array types...\n ct = contour!(\n ax, grid.x, grid.y, grid.z, vnorm;\n alpha = 0.2, levels = 4,\n colormap = :viridis,\n colorrange = (0.0, 1.0),\n highclip = (:red, 0.2), lowclip = (:green, 0.2),\n )\n cb = Colorbar(fig[1, 2], ct; label = \"Velocity magnitude\")\n fig\nend","category":"page"},{"location":"generated/navier_stokes/#Velocity-in-Fourier-space","page":"Navier–Stokes equations","title":"Velocity in Fourier space","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"bmv(bmx t) =\n_bmk hatbmv_bmk(t) e^i bmk bmx","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where bmk = (k_x k_y k_z) are the discrete wave numbers.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for k_x:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using AbstractFFTs: fftfreq, rfftfreq\n\nks_global = (\n rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]), # kx | real-to-complex\n fftfreq(Ns[2], 2π * Ns[2] / Ls[2]), # ky | complex-to-complex\n fftfreq(Ns[3], 2π * Ns[3] / Ls[3]), # kz | complex-to-complex\n)\n\nks_global[1]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ks_global[2]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ks_global[3]'","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"See PencilFFTPlan for details on creating plans and on optional keyword arguments.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients hatbmv_bmk:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v̂s = plan .* v⃗₀\nsummary(v̂s[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Note that, in Fourier space, the domain decomposition is performed along the directions x and y:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"pencil(v̂s[1])","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"This is because the 3D FFTs are performed one dimension at a time, with the x direction first and the z direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers bmk associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"grid_fourier = localgrid(v̂s[1], ks_global)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, bmω = bm bmv. In Fourier space, this becomes hatbmω = i bmk hatbmv.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using StaticArrays: SVector\nusing LinearAlgebra: ×\n\nfunction curl_fourier!(\n ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,\n ) where {N}\n @inbounds for I ∈ eachindex(grid_fourier)\n # We use StaticArrays for the cross product between small vectors.\n ik⃗ = im * SVector(grid_fourier[I])\n v⃗ = SVector(getindex.(v̂s, Ref(I))) # = (v̂s[1][I], v̂s[2][I], ...)\n ω⃗ = ik⃗ × v⃗\n for n ∈ eachindex(ω⃗)\n ω̂s[n][I] = ω⃗[n]\n end\n end\n ω̂s\nend\n\nω̂s = similar.(v̂s)\ncurl_fourier!(ω̂s, v̂s, grid_fourier);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We finally transform back to physical space and plot the result:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"ωs = plan .\\ ω̂s\n\nlet fig = Figure(resolution = (700, 600))\n ax = Axis3(fig[1, 1]; aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\")\n ω_norm = parent(vecnorm(ωs))\n ct = contour!(\n ax, grid.x, grid.y, grid.z, ω_norm;\n alpha = 0.1, levels = 0.8:0.2:2.0,\n colormap = :viridis, colorrange = (0.8, 2.0),\n highclip = (:red, 0.2), lowclip = (:green, 0.2),\n )\n cb = Colorbar(fig[1, 2], ct; label = \"Vorticity magnitude\")\n fig\nend","category":"page"},{"location":"generated/navier_stokes/#Computing-the-non-linear-term","page":"Navier–Stokes equations","title":"Computing the non-linear term","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"_t hatbmv_bmk =\n- mathcalP_bmk left widehat(bmv bm) bmv right\n- ν bmk^2 hatbmv_bmk\nquad text with quad\nmathcalP_bmk(hatbmF_bmk) = left( I - fracbmk \nbmkbmk^2 right) hatbmF_bmk","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"where mathcalP_bmk is a projection operator allowing to preserve the incompressibility condition bm bmv = 0. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Now that we have the wave numbers bmk, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients hatbmv_bmk of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, hatbmF_bmk = left widehat(bmv bm) bmv right_bmk. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Below we implement a function that computes the non-linear term in Fourier space based on its convective form (bmv bm) bmv = bm (bmv bmv). Note that this equivalence uses the incompressibility condition bm bmv = 0.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using LinearAlgebra: mul!, ldiv! # for applying FFT plans in-place\n\n# Compute non-linear term in Fourier space from velocity field in physical\n# space. Optional keyword arguments may be passed to avoid allocations.\nfunction ns_nonlinear!(\n F̂s, vs, plan, grid_fourier;\n vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),\n )\n # Compute F_i = ∂_j (v_i v_j) for each i.\n # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)\n w, ŵ = vbuf, v̂buf\n @inbounds for (i, F̂i) ∈ enumerate(F̂s)\n F̂i .= 0\n vi = vs[i]\n for (j, vj) ∈ enumerate(vs)\n w .= vi .* vj # w = v_i * v_j in physical space\n mul!(ŵ, plan, w) # same in Fourier space\n # Add derivative in Fourier space\n for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n kj = k⃗[j]\n F̂i[I] += im * kj * ŵ[I]\n end\n end\n end\n F̂s\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"As an example, let's use this function on our initial velocity field:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"F̂s = similar.(v̂s)\nns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)\n ks_max = maximum.(abs, ks_global) # maximum stored wave numbers (kx_max, ky_max, kz_max)\n ks_lim = (2 / 3) .* ks_max\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I]\n if any(abs.(k⃗) .> ks_lim)\n for ŵ ∈ ŵs\n ŵ[I] = 0\n end\n end\n end\n ŵs\nend\n\n# We can apply this on the previously computed non-linear term:\ndealias_twothirds!(F̂s, grid_fourier, ks_global);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"Finally, we implement the projection associated to the incompressibility condition:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function project_divergence_free!(ûs, grid_fourier)\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I]\n k² = sum(abs2, k⃗)\n iszero(k²) && continue # avoid division by zero\n û = getindex.(ûs, Ref(I)) # (ûs[1][I], ûs[2][I], ...)\n for i ∈ eachindex(û)\n ŵ = û[i]\n for j ∈ eachindex(û)\n ŵ -= k⃗[i] * k⃗[j] * û[j] / k²\n end\n ûs[i][I] = ŵ\n end\n end\n ûs\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)\nv̂s_proj .≈ v̂s # the last one may be false because v_z = 0 initially","category":"page"},{"location":"generated/navier_stokes/#Putting-it-all-together","page":"Navier–Stokes equations","title":"Putting it all together","text":"","category":"section"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function ns_rhs!(\n dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,\n ) where {N}\n # 1. Compute non-linear term and dealias it\n (; plan, cache, ks_global, grid_fourier) = p\n F̂s = cache.F̂s\n ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])\n dealias_twothirds!(F̂s, grid_fourier, ks_global)\n\n # 2. Project onto divergence-free space\n project_divergence_free!(F̂s, grid_fourier)\n\n # 3. Transform velocity to Fourier space\n v̂s = cache.v̂s\n map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)\n\n # 4. Add viscous term (and multiply projected non-linear term by -1)\n ν = p.ν\n for n ∈ eachindex(v̂s)\n v̂ = v̂s[n]\n F̂ = F̂s[n]\n @inbounds for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n k² = sum(abs2, k⃗)\n F̂[I] = -F̂[I] - ν * k² * v̂[I]\n end\n end\n\n # 5. Transform RHS back to physical space\n map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)\n\n nothing\nend","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"using OrdinaryDiffEq\nusing RecursiveArrayTools: ArrayPartition\n\nns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)\nvs_init_ode = ArrayPartition(v⃗₀)\nsummary(vs_init_ode)","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We now define solver parameters and temporary variables, and initialise the problem:","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"params = (;\n ν = 5e-3, # kinematic viscosity\n plan, grid_fourier, ks_global,\n cache = (\n v̂s = similar.(v̂s),\n F̂s = similar.(v̂s),\n )\n)\n\ntspan = (0.0, 10.0)\nprob = ODEProblem{true}(ns_rhs!, vs_init_ode, tspan, params)\nintegrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);\nnothing #hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum E(k), to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity ν must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large k), while high enough to allow the small-scale motions to be correctly resolved.","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"function energy_spectrum!(Ek, ks, v̂s, grid_fourier)\n Nk = length(Ek)\n @assert Nk == length(ks)\n Ek .= 0\n for I ∈ eachindex(grid_fourier)\n k⃗ = grid_fourier[I] # = (kx, ky, kz)\n knorm = sqrt(sum(abs2, k⃗))\n i = searchsortedfirst(ks, knorm)\n i > Nk && continue\n v⃗ = getindex.(v̂s, Ref(I)) # = (v̂s[1][I], v̂s[2][I], ...)\n factor = k⃗[1] == 0 ? 1 : 2 # account for Hermitian symmetry and r2c transform\n Ek[i] += factor * sum(abs2, v⃗) / 2\n end\n MPI.Allreduce!(Ek, +, get_comm(v̂s[1])) # sum across all processes\n Ek\nend\n\nks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])\nEk = similar(ks)\nv̂s = plan .* integrator.u.x\nenergy_spectrum!(Ek, ks, v̂s, grid_fourier)\nEk ./= scale_factor(plan)^2 # rescale energy\n\ncurl_fourier!(ω̂s, v̂s, grid_fourier)\nldiv!.(ωs, plan, ω̂s)\nω⃗_plot = Observable(ωs)\nk_plot = @view ks[2:end]\nE_plot = Observable(@view Ek[2:end])\nt_plot = Observable(integrator.t)\n\nfig = let\n fig = Figure(resolution = (1200, 600))\n ax = Axis3(\n fig[1, 1][1, 1]; title = @lift(\"t = $(round($t_plot, digits = 3))\"),\n aspect = :data, xlabel = \"x\", ylabel = \"y\", zlabel = \"z\",\n )\n ω_mag = @lift parent(vecnorm($ω⃗_plot))\n ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)\n ct = contour!(\n ax, grid.x, grid.y, grid.z, ω_mag_norm;\n alpha = 0.3, levels = 3,\n colormap = :viridis, colorrange = (0.0, 1.0),\n highclip = (:red, 0.2), lowclip = (:green, 0.2),\n )\n cb = Colorbar(fig[1, 1][1, 2], ct; label = \"Normalised vorticity magnitude\")\n ax_sp = Axis(\n fig[1, 2];\n xlabel = \"k\", ylabel = \"E(k)\", xscale = log2, yscale = log10,\n title = \"Kinetic energy spectrum\",\n )\n ylims!(ax_sp, 1e-8, 1e0)\n scatterlines!(ax_sp, k_plot, E_plot)\n ks_slope = exp.(range(log(2.5), log(25.0), length = 3))\n E_fivethirds = @. 0.3 * ks_slope^(-5/3)\n @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)\n text!(ax_sp, L\"k^{-5/3}\"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))\n fig\nend\n\nusing Printf # hide\nwith_xvfb = ENV[\"DISPLAY\"] == \":99\" # hide\nnstep = 0 # hide\nconst tmpdir = mktempdir() # hide\nfilename_frame(procid, nstep) = joinpath(tmpdir, @sprintf(\"proc%d_%04d.png\", procid, nstep)) # hide\nrecord(fig, \"vorticity_proc$procid.mp4\"; framerate = 10) do io\n with_xvfb && recordframe!(io) # hide\n while integrator.t < 20\n dt = 0.001\n step!(integrator, dt)\n t_plot[] = integrator.t\n mul!.(v̂s, plan, integrator.u.x) # current velocity in Fourier space\n curl_fourier!(ω̂s, v̂s, grid_fourier)\n ldiv!.(ω⃗_plot[], plan, ω̂s)\n ω⃗_plot[] = ω⃗_plot[] # to force updating the plot\n energy_spectrum!(Ek, ks, v̂s, grid_fourier)\n Ek ./= scale_factor(plan)^2 # rescale energy\n E_plot[] = E_plot[]\n global nstep += 1 # hide\n with_xvfb ? # hide\n save(filename_frame(procid, nstep), fig) : # hide\n recordframe!(io)\n end\nend;\n\nif with_xvfb # hide\n run(pipeline(`ffmpeg -y -r 10 -i $tmpdir/proc$(procid)_%04d.png -c:v libx264 -vf \"fps=25,format=yuv420p\" vorticity_proc$procid.mp4`; stdout = \"ffmpeg.out\", stderr = \"ffmpeg.err\")) # hide\nend # hide\nnothing # hide","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"
                        \n \n
                        ","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"","category":"page"},{"location":"generated/navier_stokes/","page":"Navier–Stokes equations","title":"Navier–Stokes equations","text":"This page was generated using Literate.jl.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"EditURL = \"../../examples/gradient.jl\"","category":"page"},{"location":"generated/gradient/#Gradient-of-a-scalar-field","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This example shows different methods to compute the gradient of a real-valued 3D scalar field θ(bmx) in Fourier space, where bmx = (x y z). It is assumed that the field is periodic with period L = 2π along all dimensions.","category":"page"},{"location":"generated/gradient/#General-procedure","page":"Gradient of a scalar field","title":"General procedure","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The discrete Fourier expansion of θ writes","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"θ(bmx) = _bmk Z^3 hatθ(bmk) e^i bmk bmx","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"where bmk = (k_x k_y k_z) are the Fourier wave numbers and hatθ is the discrete Fourier transform of θ. Then, the spatial derivatives of θ are given by","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"frac θ(bmx) x_i =\n_bmk Z^3 i k_i hatθ(bmk) e^i bmk bmx","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"where the subscript i denotes one of the spatial components x, y or z.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In other words, to compute bm θ = (_x θ _y θ _z θ), one has to:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"transform θ to Fourier space to obtain hatθ,\nmultiply hatθ by i bmk,\ntransform the result back to physical space to obtain bm θ.","category":"page"},{"location":"generated/gradient/#Preparation","page":"Gradient of a scalar field","title":"Preparation","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In this section, we initialise a random real-valued scalar field θ and compute its FFT. For more details see the Tutorial.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"using MPI\nusing PencilFFTs\nusing Random\n\nMPI.Init()\n\n# Input data dimensions (Nx × Ny × Nz)\ndims = (64, 32, 64)\n\n# Apply a 3D real-to-complex (r2c) FFT.\ntransform = Transforms.RFFT()\n\n# Automatically create decomposition configuration\ncomm = MPI.COMM_WORLD\npen = Pencil(dims, comm)\n\n# Create plan\nplan = PencilFFTPlan(pen, transform)\n\n# Allocate data and initialise field\nθ = allocate_input(plan)\nrandn!(θ)\n\n# Perform distributed FFT\nθ_hat = plan * θ\nnothing # hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, we initialise the output that will hold ∇θ in Fourier space. Noting that ∇θ is a vector field, we choose to store it as a tuple of 3 PencilArrays.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"∇θ_hat = allocate_output(plan, Val(3))\n\n# This is equivalent:\n# ∇θ_hat = ntuple(d -> similar(θ_hat), Val(3))\n\nsummary(∇θ_hat)","category":"page"},{"location":"generated/gradient/#Fourier-wave-numbers","page":"Gradient of a scalar field","title":"Fourier wave numbers","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In general, the Fourier wave numbers are of the form k_i = 0 frac2πL_i frac4πL_i frac6πL_i , where L_i is the period along dimension i. When a real-to-complex Fourier transform is applied, roughly half of these wave numbers are redundant due to the Hermitian symmetry of the complex Fourier coefficients. In practice, this means that for the fastest dimension x (along which a real-to-complex transform is performed), the negative wave numbers are dropped, i.e. k_x = 0 frac2πL_x frac4πL_x .","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The AbstractFFTs package provides a convenient way to generate the Fourier wave numbers, using the functions fftfreq and rfftfreq. We can use these functions to initialise a \"grid\" of wave numbers associated to our 3D real-to-complex transform:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"using AbstractFFTs: fftfreq, rfftfreq\n\nbox_size = (2π, 2π, 2π) # Lx, Ly, Lz\nsample_rate = 2π .* dims ./ box_size\n\n# In our case (Lx = 2π and Nx even), this gives kx = [0, 1, 2, ..., Nx/2].\nkx = rfftfreq(dims[1], sample_rate[1])\n\n# In our case (Ly = 2π and Ny even), this gives\n# ky = [0, 1, 2, ..., Ny/2-1, -Ny/2, -Ny/2+1, ..., -1] (and similarly for kz).\nky = fftfreq(dims[2], sample_rate[2])\nkz = fftfreq(dims[3], sample_rate[3])\n\nkvec = (kx, ky, kz)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that kvec now contains the wave numbers associated to the global domain. In the following, we will only need the wave numbers associated to the portion of the domain handled by the local MPI process.","category":"page"},{"location":"generated/gradient/#gradient_method_global","page":"Gradient of a scalar field","title":"Method 1: global views","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"PencilArrays, returned for instance by allocate_input and allocate_output, take indices that start at 1, regardless of the location of the subdomain associated to the local process on the global grid. (In other words, PencilArrays take local indices.) On the other hand, we have defined the wave number vector kvec which, for each MPI process, is defined over the global domain, and as such it takes global indices.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"One straightforward way of making data arrays compatible with wave numbers is to use global views, i.e. arrays that take global indices. These are generated from PencilArrays by calling the global_view function. Note that, in general, global indices do not start at 1 for a given MPI process. A given process will own a range of data given by indices in (i1:i2, j1:j2, k1:k2).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"θ_glob = global_view(θ_hat)\n∇θ_glob = global_view.(∇θ_hat)\nsummary(θ_glob)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Once we have global views, we can combine data and wave numbers using the portion of global indices owned by the local MPI process, as shown below. We can use CartesianIndices to iterate over the global indices associated to the local process.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"for I in CartesianIndices(θ_glob)\n i, j, k = Tuple(I) # unpack indices\n\n # Wave number vector associated to current Cartesian index.\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n # Compute gradient in Fourier space.\n # Note that modifying ∇θ_glob also modifies the original PencilArray ∇θ_hat.\n ∇θ_glob[1][I] = im * kx * θ_glob[I]\n ∇θ_glob[2][I] = im * ky * θ_glob[I]\n ∇θ_glob[3][I] = im * kz * θ_glob[I]\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The above loop can be written in a slightly more efficient manner by precomputing im * θ_glob[I]:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@inbounds for I in CartesianIndices(θ_glob)\n i, j, k = Tuple(I)\n\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n u = im * θ_glob[I]\n\n ∇θ_glob[1][I] = kx * u\n ∇θ_glob[2][I] = ky * u\n ∇θ_glob[3][I] = kz * u\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Also note that the above can be easily written in a more generic way, e.g. for arbitrary dimensions, thanks in part to the use of CartesianIndices. Moreover, in the above there is no notion of the dimension permutations discussed in the tutorial, as it is all hidden behind the implementation of PencilArrays. And as seen later in the benchmarks, these (hidden) permutations have zero cost, as the speed is identical to that of a function that explicitly takes into account these permutations.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, we can perform a backwards transform to obtain bm θ in physical space:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"∇θ = plan \\ ∇θ_hat;\nnothing #hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that the transform is automatically broadcast over the three fields of the ∇θ_hat vector, and the result ∇θ is also a tuple of three PencilArrays.","category":"page"},{"location":"generated/gradient/#gradient_method_global_explicit","page":"Gradient of a scalar field","title":"Method 2: explicit global indexing","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Sometimes, one does not need to write generic code. In our case, one often knows the dimensionality of the problem and the memory layout of the data (i.e. the underlying index permutation).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Below is a reimplementation of the above loop, using explicit indices instead of CartesianIndices, and assuming that the underlying index permutation is (3, 2, 1), that is, data is stored in (z y x) order. As discussed in the tutorial, this is the default for transformed arrays. This example also serves as a more explicit explanation for what is going on in the first method.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"# Get local data range in the global grid.\nrng = axes(θ_glob) # = (i1:i2, j1:j2, k1:k2)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"For the loop below, we're assuming that the permutation is (3, 2, 1). In other words, the fastest index is the last one, and not the first one as it is usually in Julia. If the permutation is not (3, 2, 1), things will still work (well, except for the assertion below!), but the loop order will not be optimal.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@assert permutation(θ_hat) === Permutation(3, 2, 1)\n\n@inbounds for i in rng[1], j in rng[2], k in rng[3]\n local kx, ky, kz # hide\n kx = kvec[1][i]\n ky = kvec[2][j]\n kz = kvec[3][k]\n\n # Note that we still access the arrays in (i, j, k) order.\n # (The permutation happens behind the scenes!)\n u = im * θ_glob[i, j, k]\n\n ∇θ_glob[1][i, j, k] = kx * u\n ∇θ_glob[2][i, j, k] = ky * u\n ∇θ_glob[3][i, j, k] = kz * u\nend","category":"page"},{"location":"generated/gradient/#gradient_method_local","page":"Gradient of a scalar field","title":"Method 3: using local indices","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Alternatively, we can avoid global views and work directly on PencilArrays using local indices that start at 1. In this case, part of the strategy is to construct a \"local\" grid of wave numbers that can also be accessed with local indices. This can be conveniently done using the localgrid function of the PencilArrays.jl package, which accepts a PencilArray (or its associated Pencil) and the global coordinates (here kvec):","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"grid_fourier = localgrid(θ_hat, kvec)","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Note that one can directly iterate on the returned grid object:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@inbounds for I in CartesianIndices(grid_fourier)\n # Wave number vector associated to current Cartesian index.\n local k⃗ # hide\n k⃗ = grid_fourier[I]\n u = im * θ_hat[I]\n ∇θ_hat[1][I] = k⃗[1] * u\n ∇θ_hat[2][I] = k⃗[2] * u\n ∇θ_hat[3][I] = k⃗[3] * u\nend","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This implementation is as efficient as the other examples, while being slightly shorter to write. Moreover, it is quite generic, and can be made independent of the number of dimensions with little effort.","category":"page"},{"location":"generated/gradient/#gradient_method_broadcast","page":"Gradient of a scalar field","title":"Method 4: using broadcasting","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Finally, note that the local grid object returned by localgrid makes it is possible to compute the gradient using broadcasting, thus fully avoiding scalar indexing. This can be quite convenient in some cases, and can also be very useful if one is working on GPUs (where scalar indexing is prohibitively expensive). Using broadcasting, the above examples simply become:","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"@. ∇θ_hat[1] = im * grid_fourier[1] * θ_hat\n@. ∇θ_hat[2] = im * grid_fourier[2] * θ_hat\n@. ∇θ_hat[3] = im * grid_fourier[3] * θ_hat\nnothing # hide","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"Once again, as shown in the benchmarks further below, this method performs quite similarly to the other ones.","category":"page"},{"location":"generated/gradient/#Summary","page":"Gradient of a scalar field","title":"Summary","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The PencilArrays module provides different alternatives to deal with MPI-distributed data that may be subject to dimension permutations. In particular, one can choose to work with global indices (first two examples), with local indices (third example), or to avoid scalar indexing altogether (fourth example).","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"If one wants to stay generic, making sure that the same code will work for arbitrary dimensions and will be efficient regardless of the underlying dimension permutation, methods 1, 3 or 4 should be preferred. These use CartesianIndices and make no assumptions on possible dimension permutations, which are by default enabled in the output of PencilFFTs transforms. In fact, such permutations are completely invisible in the implementations.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The second method uses explicit (i, j, k) indices. It assumes that the underlying permutation is (3, 2, 1) to loop with i as the slowest index and k as the fastest, which is the optimal order in this case given the permutation. As such, the implementation is less generic than the others, and differences in performance are negligible with respect to more generic variants.","category":"page"},{"location":"generated/gradient/#gradient_benchmarks","page":"Gradient of a scalar field","title":"Benchmark results","text":"","category":"section"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"The following are the benchmark results obtained from running examples/gradient.jl on a laptop, using 2 MPI processes and Julia 1.7.2, with an input array of global dimensions 64 32 64. The different methods detailed above are marked on the right. The \"lazy\" marks indicate runs where the wave numbers were represented by lazy Frequencies objects (returned by rfftfreq and fftfreq). Otherwise, they were collected into Vectors. For some reason, plain Vectors are faster when working with grids generated by localgrid.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"In the script, additional implementations can be found which rely on a more advanced understanding of permutations and on the internals of the PencilArrays package. For instance, gradient_local_parent! directly works with the raw data stored in Julia Arrays, while gradient_local_linear! completely avoids CartesianIndices while staying generic and efficient. Nevertheless, these display roughly the same performance as the above examples.","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":" gradient_global_view!... 89.900 μs\n gradient_global_view! (lazy)... 92.060 μs [Method 1]\n gradient_global_view_explicit!... 88.958 μs\n gradient_global_view_explicit! (lazy)... 81.055 μs [Method 2]\n gradient_local!... 92.305 μs\n gradient_grid!... 92.770 μs\n gradient_grid! (lazy)... 101.388 μs [Method 3]\n gradient_grid_broadcast!... 88.606 μs\n gradient_grid_broadcast! (lazy)... 151.020 μs [Method 4]\n gradient_local_parent!... 92.248 μs\n gradient_local_linear!... 91.212 μs\n gradient_local_linear_explicit!... 90.992 μs","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"","category":"page"},{"location":"generated/gradient/","page":"Gradient of a scalar field","title":"Gradient of a scalar field","text":"This page was generated using Literate.jl.","category":"page"},{"location":"PencilFFTs_timers/#PencilFFTs.measuring_performance","page":"Measuring performance","title":"Measuring performance","text":"","category":"section"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"It is possible to measure the time spent in different sections of the distributed transforms using the TimerOutputs package. This has a (very small) performance overhead, so it is disabled by default. To enable time measurements, call TimerOutputs.enable_debug_timings after loading PencilFFTs (see below for an example). For more details see the TimerOutputs docs.","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"Minimal example:","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"using MPI\nusing PencilFFTs\nusing TimerOutputs\n\n# Enable timing of `PencilFFTs` functions\nTimerOutputs.enable_debug_timings(PencilFFTs)\nTimerOutputs.enable_debug_timings(PencilArrays)\nTimerOutputs.enable_debug_timings(Transpositions)\n\nMPI.Init()\n\nplan = PencilFFTPlan(#= args... =#)\n\n# [do stuff with `plan`...]\n\n# Retrieve and print timing data associated to `plan`\nto = timer(plan)\nprint_timer(to)","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"By default, each PencilFFTPlan has its own TimerOutput. If you already have a TimerOutput, you can pass it to the PencilFFTPlan constructor:","category":"page"},{"location":"PencilFFTs_timers/","page":"Measuring performance","title":"Measuring performance","text":"to = TimerOutput()\nplan = PencilFFTPlan(..., timer=to)\n\n# [do stuff with `plan`...]\n\nprint_timer(to)","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"#PencilFFTs","page":"Home","title":"PencilFFTs","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Fast Fourier transforms of MPI-distributed Julia arrays.","category":"page"},{"location":"#Introduction","page":"Home","title":"Introduction","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package provides multidimensional FFTs and related transforms on MPI-distributed Julia arrays via the PencilArrays package.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The name of this package originates from the decomposition of 3D domains along two out of three dimensions, sometimes called pencil decomposition. This is illustrated by the figure below,[1] where each coloured block is managed by a different MPI process. Typically, one wants to compute FFTs on a scalar or vector field along the three spatial dimensions. In the case of a pencil decomposition, 3D FFTs are performed one dimension at a time, along the non-decomposed direction. Transforms must then be interleaved with global data transpositions to switch between pencil configurations. In high-performance computing environments, such data transpositions are generally the most expensive part of a parallel FFT computation, due to the large cost of communications between computing nodes.","category":"page"},{"location":"","page":"Home","title":"Home","text":"
                        \n \n
                        ","category":"page"},{"location":"","page":"Home","title":"Home","text":"More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension N. The decompositions can be performed along an arbitrary number M N of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.","category":"page"},{"location":"#Features","page":"Home","title":"Features","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;\nFFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;\nin-place and out-of-place transforms;\nhigh scalability up to (at least) tens of thousands of MPI processes.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"PencilFFTs can be installed using the Julia package manager:","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> ] add PencilFFTs","category":"page"},{"location":"#Similar-projects","page":"Home","title":"Similar projects","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.\nPFFT is a very general parallel FFT library written in C.\nP3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.\n2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[1]: Figure adapted from this PhD thesis.","category":"page"},{"location":"","page":"Home","title":"Home","text":"[2]: For the pencil decomposition represented in the figure, N = 3 and M = 2.","category":"page"},{"location":"PencilFFTs/#Distributed-FFT-plans","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"Distributed FFTs are implemented in the PencilFFTs module, and are built on top of the PencilArrays package.","category":"page"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"CurrentModule = PencilFFTs","category":"page"},{"location":"PencilFFTs/#Creating-plans","page":"Distributed FFT plans","title":"Creating plans","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"PencilFFTPlan","category":"page"},{"location":"PencilFFTs/#PencilFFTs.PencilFFTPlan","page":"Distributed FFT plans","title":"PencilFFTs.PencilFFTPlan","text":"PencilFFTPlan{T,N} <: AbstractFFTs.Plan{T}\n\nPlan for N-dimensional FFT-based transform on MPI-distributed data, where input data has type T.\n\n\n\nPencilFFTPlan(p::Pencil, transforms; kwargs...)\n\nCreate a PencilFFTPlan for distributed arrays following a given Pencil configuration. See variant below for details on the specification of transforms and on possible keyword arguments.\n\n\n\nPencilFFTPlan(\n A::PencilArray, transforms;\n fftw_flags = FFTW.ESTIMATE,\n fftw_timelimit = FFTW.NO_TIMELIMIT,\n permute_dims = Val(true),\n transpose_method = Transpositions.PointToPoint(),\n timer = timer(pencil(A)),\n)\n\nCreate plan for N-dimensional transform on MPI-distributed PencilArrays.\n\nExtended help\n\nThis creates a PencilFFTPlan for arrays sharing the same properties as A (dimensions, MPI decomposition, memory layout, ...), which describe data on an N-dimensional domain.\n\nTransforms\n\nThe transforms to be applied along each dimension are specified by the transforms argument. Possible transforms are defined as subtypes of Transforms.AbstractTransform, and are listed in Transform types. This argument may be either:\n\na tuple of N transforms to be applied along each dimension. For instance, transforms = (Transforms.R2R(FFTW.REDFT01), Transforms.RFFT(), Transforms.FFT());\na single transform to be applied along all dimensions. The input is automatically expanded into N equivalent transforms. For instance, for a three-dimensional array, transforms = Transforms.RFFT() specifies a 3D real-to-complex transform, and is equivalent to passing (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT()).\n\nNote that forward transforms are applied from left to right. In the last example, this means that a real-to-complex transform (RFFT) is first performed along the first dimension. This is followed by complex-to-complex transforms (FFT) along the second and third dimensions.\n\nInput data layout\n\nThe input PencilArray must satisfy the following constraints:\n\narray dimensions must not be permuted. This is the default when constructing PencilArrays.\nfor an M-dimensional domain decomposition (with M < N), the input array must be decomposed along the last M dimensions. For example, for a 2D decomposition of 3D data, the decomposed dimensions must be (2, 3). In particular, the first array dimension must not be distributed among different MPI processes.\nIn the PencilArrays package, the decomposed dimensions are specified at the moment of constructing a Pencil.\nthe element type must be compatible with the specified transform. For instance, real-to-complex transforms (Transforms.RFFT) require the input to be real floating point values. Other transforms, such as Transforms.R2R, accept both real and complex data.\n\nKeyword arguments\n\nThe keyword arguments fftw_flags and fftw_timelimit are passed to the FFTW plan creation functions (see AbstractFFTs docs).\npermute_dims determines whether the indices of the output data should be reversed. For instance, if the input data has global dimensions (Nx, Ny, Nz), then the output of a complex-to-complex FFT would have dimensions (Nz, Ny, Nx). This enables FFTs to always be performed along the first (i.e. fastest) array dimension, which could lead to performance gains. This option is enabled by default. For type inference reasons, it must be a value type (Val(true) or Val(false)).\ntranspose_method allows to select between implementations of the global data transpositions. See PencilArrays docs docs for details.\ntimer should be a TimerOutput object. See Measuring performance for details.\n\n\n\nPencilFFTPlan(\n dims_global::Dims{N}, transforms, proc_dims::Dims{M}, comm::MPI.Comm,\n [real_type = Float64]; extra_dims = (), kws...\n)\n\nCreate plan for N-dimensional transform.\n\nExtended help\n\nInstead of taking a PencilArray or a Pencil, this constructor requires the global dimensions of the input data, passed via the size_global argument.\n\nThe data is distributed over the MPI processes in the comm communicator. The distribution is performed over M dimensions (with M < N) according to the values in proc_dims, which specifies the number of MPI processes to put along each dimension.\n\nPencilArrays that may be transformed with the returned plan can be created using allocate_input.\n\nOptional arguments\n\nThe floating point precision can be selected by setting real_type parameter, which is Float64 by default.\nextra_dims may be used to specify the sizes of one or more extra dimensions that should not be transformed. These dimensions will be added to the rightmost (i.e. slowest) indices of the arrays. See Extra dimensions below for usage hints.\nsee the other constructor for more keyword arguments.\n\nExtra dimensions\n\nOne possible application of extra_dims is for describing the components of a vector or tensor field. However, this means that different PencilFFTPlans would need to be created for each kind of field (scalar, vector, ...). To avoid the creation of multiple plans, a possibly better alternative is to create tuples (or arrays) of PencilArrays using allocate_input and allocate_output.\n\nAnother more legitimate usage of extra_dims is to specify one or more Cartesian dimensions that should not be transformed nor split among MPI processes.\n\nExample\n\nSuppose we want to perform a 3D FFT of real data. The data is to be decomposed along two dimensions, over 8 MPI processes:\n\nsize_global = (64, 32, 128) # size of real input data\n\n# Perform real-to-complex transform along the first dimension, then\n# complex-to-complex transforms along the other dimensions.\ntransforms = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())\n# transforms = Transforms.RFFT() # this is equivalent to the above line\n\nproc_dims = (4, 2) # 2D decomposition\ncomm = MPI.COMM_WORLD\n\nplan = PencilFFTPlan(size_global, transforms, proc_dims, comm)\n\n\n\n\n\n","category":"type"},{"location":"PencilFFTs/#Allocating-data","page":"Distributed FFT plans","title":"Allocating data","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"allocate_input\nallocate_output","category":"page"},{"location":"PencilFFTs/#PencilFFTs.allocate_input","page":"Distributed FFT plans","title":"PencilFFTs.allocate_input","text":"allocate_input(p::PencilFFTPlan) -> PencilArray\nallocate_input(p::PencilFFTPlan, dims...) -> Array{PencilArray}\nallocate_input(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}\n\nAllocate uninitialised PencilArray that can hold input data for the given plan.\n\nThe second and third forms respectively allocate an array of PencilArrays of size dims, and a tuple of N PencilArrays.\n\nnote: In-place plans\nIf p is an in-place real-to-real or complex-to-complex plan, a ManyPencilArray is allocated. If p is an in-place real-to-complex plan, a ManyPencilArrayRFFT! is allocated. These types hold PencilArray wrappers for the input and output transforms (as well as for intermediate transforms) which share the same space in memory. The input and output PencilArrays should be respectively accessed by calling first(::ManyPencilArray) and last(::ManyPencilArray).ExampleSuppose p is an in-place PencilFFTPlan. Then,@assert is_inplace(p)\nA = allocate_input(p) :: ManyPencilArray\nv_in = first(A) :: PencilArray # input data view\nv_out = last(A) :: PencilArray # output data viewAlso note that in-place plans must be performed directly on the returned ManyPencilArray, and not on the contained PencilArray views:p * A # perform forward transform in-place\np \\ A # perform backward transform in-place\n# p * v_in # not allowed!!\n\n\n\n\n\n","category":"function"},{"location":"PencilFFTs/#PencilFFTs.allocate_output","page":"Distributed FFT plans","title":"PencilFFTs.allocate_output","text":"allocate_output(p::PencilFFTPlan) -> PencilArray\nallocate_output(p::PencilFFTPlan, dims...) -> Array{PencilArray}\nallocate_output(p::PencilFFTPlan, Val(N)) -> NTuple{N, PencilArray}\n\nAllocate uninitialised PencilArray that can hold output data for the given plan.\n\nIf p is an in-place plan, a ManyPencilArray is allocated.\n\nSee allocate_input for details.\n\n\n\n\n\n","category":"function"},{"location":"PencilFFTs/#Methods","page":"Distributed FFT plans","title":"Methods","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"get_comm(::PencilFFTPlan)\nscale_factor(::PencilFFTPlan)\ntimer(::PencilFFTPlan)\nis_inplace(::PencilFFTPlan)","category":"page"},{"location":"PencilFFTs/#PencilArrays.Pencils.MPITopologies.get_comm-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilArrays.Pencils.MPITopologies.get_comm","text":"get_comm(p::PencilFFTPlan)\n\nGet MPI communicator associated to a PencilFFTPlan.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilFFTs.Transforms.scale_factor-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilFFTs.Transforms.scale_factor","text":"scale_factor(p::PencilFFTPlan)\n\nGet scale factor associated to a PencilFFTPlan.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilArrays.Pencils.timer-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilArrays.Pencils.timer","text":"timer(p::PencilFFTPlan)\n\nGet TimerOutput attached to a PencilFFTPlan.\n\nSee Measuring performance for details.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#PencilFFTs.Transforms.is_inplace-Tuple{PencilFFTPlan}","page":"Distributed FFT plans","title":"PencilFFTs.Transforms.is_inplace","text":"Transforms.is_inplace(p::PencilFFTPlan)\n\nReturns true if the given plan operates in-place on the input data, false otherwise.\n\n\n\n\n\n","category":"method"},{"location":"PencilFFTs/#Internals","page":"Distributed FFT plans","title":"Internals","text":"","category":"section"},{"location":"PencilFFTs/","page":"Distributed FFT plans","title":"Distributed FFT plans","text":"ManyPencilArrayRFFT!","category":"page"},{"location":"PencilFFTs/#PencilFFTs.ManyPencilArrayRFFT!","page":"Distributed FFT plans","title":"PencilFFTs.ManyPencilArrayRFFT!","text":"ManyPencilArrayRFFT!{T,N,M} <: AbstractManyPencilArray{N,M}\n\nContainer holding M different PencilArray views to the same underlying data buffer. All views share the same dimensionality N. The element type T of the first view is real, that of subsequent views is Complex{T}. \n\nThis can be used to perform in-place real-to-complex plan, see alsoTransforms.RFFT!. It is used internally for such transforms by allocate_input and should not be constructed directly.\n\n\n\nManyPencilArrayRFFT!{T}(undef, pencils...; extra_dims=())\n\nCreate a ManyPencilArrayRFFT! container that can hold data of type T and Complex{T} associated to all the given Pencils.\n\nThe optional extra_dims argument is the same as for PencilArray.\n\nSee also ManyPencilArray\n\n\n\n\n\n","category":"type"}] +} diff --git a/v0.15.2/siteinfo.js b/v0.15.2/siteinfo.js new file mode 100644 index 00000000..ea8a89be --- /dev/null +++ b/v0.15.2/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "v0.15.2"; diff --git a/v0.15.2/tutorial/index.html b/v0.15.2/tutorial/index.html new file mode 100644 index 00000000..ae448bab --- /dev/null +++ b/v0.15.2/tutorial/index.html @@ -0,0 +1,40 @@ + +Tutorial · PencilFFTs.jl
                        GitHub

                        Tutorial

                        The following tutorial shows how to perform a 3D FFT of real periodic data defined on a grid of $N_x × N_y × N_z$ points.

                        + + Pencil decomposition of 3D domains +

                        By default, the domain is distributed on a 2D MPI topology of dimensions $N_1 × N_2$. As an example, the above figure shows such a topology with $N_1 = 4$ and $N_2 = 3$, for a total of 12 MPI processes.

                        Creating plans

                        The first thing to do is to create a domain decomposition configuration for the given dataset dimensions $N_x × N_y × N_z$. In the framework of PencilArrays, such a configuration is described by a Pencil object. As described in the PencilArrays docs, we can let the Pencil constructor automatically determine such a configuration. For this, only an MPI communicator and the dataset dimensions are needed:

                        using MPI
+using PencilFFTs
+
+MPI.Init()
+comm = MPI.COMM_WORLD
+
+# Input data dimensions (Nx × Ny × Nz)
+dims = (16, 32, 64)
+pen = Pencil(dims, comm)

                        By default this creates a 2D decomposition (for the case of a 3D dataset), but one can change this as detailed in the PencilArrays documentation linked above.

                        We can now create a PencilFFTPlan, which requires information on decomposition configuration (the Pencil object) and on the transforms that will be applied:

                        # Apply a 3D real-to-complex (r2c) FFT.
+transform = Transforms.RFFT()
+
+# Note that, for more control, one can instead separately specify the transforms along each dimension:
+# transform = (Transforms.RFFT(), Transforms.FFT(), Transforms.FFT())
+
+# Create plan
+plan = PencilFFTPlan(pen, transform)

                        See the PencilFFTPlan constructor for details on the accepted options, and the Transforms module for the possible transforms. It is also possible to enable fine-grained performance measurements via the TimerOutputs package, as described in Measuring performance.

                        Allocating data

                        Next, we want to apply the plan on some data. Transforms may only be applied on PencilArrays, which are array wrappers that include MPI decomposition information (in some sense, analogous to DistributedArrays in Julia's distributed computing approach). The helper function allocate_input can be used to allocate a PencilArray that is compatible with our plan:

                        # In our example, this returns a 3D PencilArray of real data (Float64).
+u = allocate_input(plan)
+
+# Fill the array with some (random) data
+using Random
+randn!(u)

                        PencilArrays are a subtype of AbstractArray, and thus they support all common array operations.

                        Similarly, to preallocate output data, one can use allocate_output:

                        # In our example, this returns a 3D PencilArray of complex data (Complex{Float64}).
+v = allocate_output(plan)

                        This is only required if one wants to apply the plans using a preallocated output (with mul!, see right below).

                        The data types returned by allocate_input and allocate_output are slightly different when working with in-place transforms. See the in-place example for details.

                        Applying plans

                        The interface to apply plans is consistent with that of AbstractFFTs. Namely, * and mul! are respectively used for forward transforms without and with preallocated output data. Similarly, \ and ldiv! are used for backward transforms.

                        using LinearAlgebra  # for mul!, ldiv!
+
+# Apply plan on `u` with `v` as an output
+mul!(v, plan, u)
+
+# Apply backward plan on `v` with `w` as an output
+w = similar(u)
+ldiv!(w, plan, v)  # now w ≈ u

                        Note that, consistently with AbstractFFTs, normalisation is performed at the end of a backward transform, so that the original data is recovered when applying a forward followed by a backward transform.

                        Accessing and modifying data

                        For any given MPI process, a PencilArray holds the data associated to its local partition in the global geometry. PencilArrays are accessed using local indices that start at 1, regardless of the location of the local process in the MPI topology. Note that PencilArrays, being based on regular Arrays, support both linear and Cartesian indexing (see the Julia docs for details).

                        For convenience, the global_view function can be used to generate an OffsetArray wrapper that takes global indices.

                        Output data layout

                        In memory, the dimensions of the transform output are by default reversed with respect to the input. That is, if the order of indices in the input data is (x, y, z), then the output has order (z, y, x) in memory. This detail is hidden from the user, and output arrays are always accessed in the same order as the input data, regardless of the underlying output dimension permutation. This applies to PencilArrays and to OffsetArrays returned by global_view.

                        The reasoning behind dimension permutations, is that they allow to always perform FFTs along the fastest array dimension and to avoid a local data transposition, resulting in performance gains. A similar approach is followed by other parallel FFT libraries. FFTW itself, in its distributed-memory routines, includes a flag that enables a similar behaviour. In PencilFFTs, index permutation is the default, but it can be disabled via the permute_dims flag of PencilFFTPlan.

                        A great deal of work has been spent in making generic index permutations as efficient as possible, both in intermediate and in the output state of the multidimensional transforms. This has been achieved, in part, by making sure that permutations such as (3, 2, 1) are compile-time constants.

                        Further reading

                        For details on working with PencilArrays see the PencilArrays docs.

                        The examples on the sidebar further illustrate the use of transforms and provide an introduction to working with MPI-distributed data in the form of PencilArrays. In particular, the gradient example illustrates different ways of computing things using Fourier-transformed distributed arrays. Then, the incompressible Navier–Stokes example is a more advanced and complete example of a possible application of the PencilFFTs package.

                        diff --git a/versions.js b/versions.js index e1679a69..a0dc18ad 100644 --- a/versions.js +++ b/versions.js @@ -1,5 +1,6 @@ var DOC_VERSIONS = [ "stable", + "v0.15", "v0.14", "v0.13", "v0.12", @@ -16,5 +17,5 @@ var DOC_VERSIONS = [ "v0.1", "dev", ]; -var DOCUMENTER_NEWEST = "v0.14.3"; +var DOCUMENTER_NEWEST = "v0.15.2"; var DOCUMENTER_STABLE = "stable";
                        GitHub

                        Navier–Stokes equations

                        In this example, we numerically solve the incompressible Navier–Stokes equations

                        \[∂_t \bm{v} + (\bm{v} ⋅ \bm{∇}) \bm{v} = -\frac{1}{ρ} \bm{∇} p + ν ∇^2 \bm{v}, +\quad \bm{∇} ⋅ \bm{v} = 0,\]

                        where $\bm{v}(\bm{x}, t)$ and $p(\bm{x}, t)$ are respectively the velocity and pressure fields, $ν$ is the fluid kinematic viscosity and $ρ$ is the fluid density.

                        We solve the above equations a 3D periodic domain using a standard Fourier pseudo-spectral method.

                        First steps

                        We start by loading the required packages, initialising MPI and setting the simulation parameters.

                        using MPI
+using PencilFFTs
+
+MPI.Init()
+comm = MPI.COMM_WORLD
+procid = MPI.Comm_rank(comm) + 1
+
+# Simulation parameters
+Ns = (64, 64, 64)  # = (Nx, Ny, Nz)
+Ls = (2π, 2π, 2π)  # = (Lx, Ly, Lz)
+
+# Collocation points ("global" = over all processes).
+# We include the endpoint (length = N + 1) for convenience.
+xs_global = map((N, L) -> range(0, L; length = N + 1), Ns, Ls)  # = (x, y, z)
                        (0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586, 0.0:0.09817477042468103:6.283185307179586)

                        Let's check the number of MPI processes over which we're running our simulation:

                        MPI.Comm_size(comm)
                        2

                        We can now create a partitioning of the domain based on the number of grid points (Ns) and on the number of MPI processes. There are different ways to do this. For simplicity, here we do it automatically following the PencilArrays.jl docs:

                        pen = Pencil(Ns, comm)
                        Decomposition of 3D data
+    Data dimensions: (64, 64, 64)
+    Decomposed dimensions: (2, 3)
+    Data permutation: NoPermutation()
+    Array type: Array

                        The subdomain associated to the local MPI process can be obtained using range_local:

                        range_local(pen)
                        (1:64, 1:32, 1:64)

                        We now construct a distributed vector field that follows the decomposition configuration we just created:

                        v⃗₀ = (
+    PencilArray{Float64}(undef, pen),  # vx
+    PencilArray{Float64}(undef, pen),  # vy
+    PencilArray{Float64}(undef, pen),  # vz
+)
+summary(v⃗₀[1])
                        "64×32×64 PencilArray{Float64, 3}(::Pencil{3, 2, NoPermutation, Array})"

                        We still need to fill this array with interesting values that represent a physical velocity field.

                        Initial condition

                        Let's set the initial condition in physical space. In this example, we choose the Taylor–Green vortex configuration as an initial condition:

                        \[\begin{aligned} +v_x(x, y, z) &= u₀ \sin(k₀ x) \cos(k₀ y) \cos(k₀ z) \\ +v_y(x, y, z) &= -u₀ \cos(k₀ x) \sin(k₀ y) \cos(k₀ z) \\ +v_z(x, y, z) &= 0 +\end{aligned}\]

                        where $u₀$ and $k₀$ are two parameters setting the amplitude and the period of the velocity field.

                        To set the initial condition, each MPI process needs to know which portion of the physical grid it has been attributed. For this, PencilArrays.jl includes a localgrid helper function:

                        grid = localgrid(pen, xs_global)
                        LocalRectilinearGrid{3} with coordinates:
+ (1) 0.0:0.09817477042468103:6.1850105367549055
+ (2) 0.0:0.09817477042468103:3.043417883165112
+ (3) 0.0:0.09817477042468103:6.1850105367549055

                        We can use this to initialise the velocity field:

                        u₀ = 1.0
+k₀ = 2π / Ls[1]  # should be integer if L = 2π (to preserve periodicity)
+
+@. v⃗₀[1] =  u₀ * sin(k₀ * grid.x) * cos(k₀ * grid.y) * cos(k₀ * grid.z)
+@. v⃗₀[2] = -u₀ * cos(k₀ * grid.x) * sin(k₀ * grid.y) * cos(k₀ * grid.z)
+@. v⃗₀[3] =  0

                        Let's plot a 2D slice of the velocity field managed by the local MPI process:

                        using GLMakie
+
+# Compute the norm of a vector field represented by a tuple of arrays.
+function vecnorm(v⃗::NTuple)
+    vnorm = similar(v⃗[1])
+    for n ∈ eachindex(v⃗[1])
+        w = zero(eltype(vnorm))
+        for v ∈ v⃗
+            w += v[n]^2
+        end
+        vnorm[n] = sqrt(w)
+    end
+    vnorm
+end
+
+let fig = Figure(resolution = (700, 600))
+    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
+    vnorm = parent(vecnorm(v⃗₀))  # use `parent` because Makie doesn't like custom array types...
+    ct = contour!(
+        ax, grid.x, grid.y, grid.z, vnorm;
+        alpha = 0.2, levels = 4,
+        colormap = :viridis,
+        colorrange = (0.0, 1.0),
+        highclip = (:red, 0.2), lowclip = (:green, 0.2),
+    )
+    cb = Colorbar(fig[1, 2], ct; label = "Velocity magnitude")
+    fig
+end
                        Example block output

                        Velocity in Fourier space

                        In the Fourier pseudo-spectral method, the periodic velocity field is discretised in space as a truncated Fourier series

                        \[\bm{v}(\bm{x}, t) = +∑_{\bm{k}} \hat{\bm{v}}_{\bm{k}}(t) \, e^{i \bm{k} ⋅ \bm{x}},\]

                        where $\bm{k} = (k_x, k_y, k_z)$ are the discrete wave numbers.

                        The wave numbers can be obtained using the fftfreq function. Since we perform a real-to-complex transform along the first dimension, we use rfftfreq instead for $k_x$:

                        using AbstractFFTs: fftfreq, rfftfreq
+
+ks_global = (
+    rfftfreq(Ns[1], 2π * Ns[1] / Ls[1]),  # kx | real-to-complex
+     fftfreq(Ns[2], 2π * Ns[2] / Ls[2]),  # ky | complex-to-complex
+     fftfreq(Ns[3], 2π * Ns[3] / Ls[3]),  # kz | complex-to-complex
+)
+
+ks_global[1]'
                        1×33 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
+ 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  27.0  28.0  29.0  30.0  31.0  32.0
                        ks_global[2]'
                        1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
+ 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0
                        ks_global[3]'
                        1×64 adjoint(::AbstractFFTs.Frequencies{Float64}) with eltype Float64:
+ 0.0  1.0  2.0  3.0  4.0  5.0  6.0  …  -6.0  -5.0  -4.0  -3.0  -2.0  -1.0

                        To transform the velocity field to Fourier space, we first create a real-to-complex FFT plan to be applied to one of the velocity components:

                        plan = PencilFFTPlan(v⃗₀[1], Transforms.RFFT())
                        Transforms: (RFFT, FFT, FFT)
+Input type: Float64
+Global dimensions: (64, 64, 64) -> (33, 64, 64)
+MPI topology: 2D decomposition (2×1 processes)

                        See PencilFFTPlan for details on creating plans and on optional keyword arguments.

                        We can now apply this plan to the three velocity components to obtain the respective Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$:

                        v̂s = plan .* v⃗₀
+summary(v̂s[1])
                        "16×64×64 PencilArray{ComplexF64, 3}(::Pencil{3, 2, Permutation{(3, 2, 1), 3}, Array})"

                        Note that, in Fourier space, the domain decomposition is performed along the directions $x$ and $y$:

                        pencil(v̂s[1])
                        Decomposition of 3D data
+    Data dimensions: (33, 64, 64)
+    Decomposed dimensions: (1, 2)
+    Data permutation: Permutation(3, 2, 1)
+    Array type: Array

                        This is because the 3D FFTs are performed one dimension at a time, with the $x$ direction first and the $z$ direction last. To efficiently perform an FFT along a given direction (taking advantage of serial FFT implementations like FFTW), all the data along that direction must be contained locally within a single MPI process. For that reason, data redistributions (or transpositions) among MPI processes are performed behind the scenes during each FFT computation. Such transpositions require important communications between MPI processes, and are usually the most time-consuming aspect of massively-parallel simulations using this kind of methods.

                        To solve the Navier–Stokes equations in Fourier space, we will also need the respective wave numbers $\bm{k}$ associated to the local MPI process. Similarly to the local grid points, these are obtained using the localgrid function:

                        grid_fourier = localgrid(v̂s[1], ks_global)
                        LocalRectilinearGrid{3} with Permutation(3, 2, 1) and coordinates:
+ (1) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0]
+ (2) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]
+ (3) [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, -32.0, -31.0, -30.0, -29.0, -28.0, -27.0, -26.0, -25.0, -24.0, -23.0, -22.0, -21.0, -20.0, -19.0, -18.0, -17.0, -16.0, -15.0, -14.0, -13.0, -12.0, -11.0, -10.0, -9.0, -8.0, -7.0, -6.0, -5.0, -4.0, -3.0, -2.0, -1.0]

                        As an example, let's first use this to compute and plot the vorticity associated to the initial condition. The vorticity is defined as the curl of the velocity, $\bm{ω} = \bm{∇} × \bm{v}$. In Fourier space, this becomes $\hat{\bm{ω}} = i \bm{k} × \hat{\bm{v}}$.

                        using StaticArrays: SVector
+using LinearAlgebra: ×
+
+function curl_fourier!(
+        ω̂s::NTuple{N, <:PencilArray}, v̂s::NTuple{N, <:PencilArray}, grid_fourier,
+    ) where {N}
+    @inbounds for I ∈ eachindex(grid_fourier)
+        # We use StaticArrays for the cross product between small vectors.
+        ik⃗ = im * SVector(grid_fourier[I])
+        v⃗ = SVector(getindex.(v̂s, Ref(I)))  # = (v̂s[1][I], v̂s[2][I], ...)
+        ω⃗ = ik⃗ × v⃗
+        for n ∈ eachindex(ω⃗)
+            ω̂s[n][I] = ω⃗[n]
+        end
+    end
+    ω̂s
+end
+
+ω̂s = similar.(v̂s)
+curl_fourier!(ω̂s, v̂s, grid_fourier);

                        We finally transform back to physical space and plot the result:

                        ωs = plan .\ ω̂s
+
+let fig = Figure(resolution = (700, 600))
+    ax = Axis3(fig[1, 1]; aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z")
+    ω_norm = parent(vecnorm(ωs))
+    ct = contour!(
+        ax, grid.x, grid.y, grid.z, ω_norm;
+        alpha = 0.1, levels = 0.8:0.2:2.0,
+        colormap = :viridis, colorrange = (0.8, 2.0),
+        highclip = (:red, 0.2), lowclip = (:green, 0.2),
+    )
+    cb = Colorbar(fig[1, 2], ct; label = "Vorticity magnitude")
+    fig
+end
                        Example block output

                        Computing the non-linear term

                        One can show that, in Fourier space, the incompressible Navier–Stokes equations can be written as

                        \[∂_t \hat{\bm{v}}_{\bm{k}} = +- \mathcal{P}_{\bm{k}} \! \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right] +- ν |\bm{k}|^2 \hat{\bm{v}}_{\bm{k}} +\quad \text{ with } \quad +\mathcal{P}_{\bm{k}}(\hat{\bm{F}}_{\bm{k}}) = \left( I - \frac{\bm{k} ⊗ +\bm{k}}{|\bm{k}|^2} \right) \hat{\bm{F}}_{\bm{k}},\]

                        where $\mathcal{P}_{\bm{k}}$ is a projection operator allowing to preserve the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$. This operator encodes the action of the pressure gradient term, which serves precisely to enforce incompressibility. Note that, because of this, the pressure gradient dissapears from the equations.

                        Now that we have the wave numbers $\bm{k}$, computing the linear viscous term in Fourier space is straighforward once we have the Fourier coefficients $\hat{\bm{v}}_{\bm{k}}$ of the velocity field. What is slightly more challenging (and much more costly) is the computation of the non-linear term in Fourier space, $\hat{\bm{F}}_{\bm{k}} = \left[ \widehat{(\bm{v} ⋅ \bm{∇}) \bm{v}} \right]_{\bm{k}}$. In the pseudo-spectral method, the quadratic nonlinearity is computed by collocation in physical space (i.e. this term is evaluated at grid points), while derivatives are computed in Fourier space. This requires transforming fields back and forth between both spaces.

                        Below we implement a function that computes the non-linear term in Fourier space based on its convective form $(\bm{v} ⋅ \bm{∇}) \bm{v} = \bm{∇} ⋅ (\bm{v} ⊗ \bm{v})$. Note that this equivalence uses the incompressibility condition $\bm{∇} ⋅ \bm{v} = 0$.

                        using LinearAlgebra: mul!, ldiv!  # for applying FFT plans in-place
+
+# Compute non-linear term in Fourier space from velocity field in physical
+# space. Optional keyword arguments may be passed to avoid allocations.
+function ns_nonlinear!(
+        F̂s, vs, plan, grid_fourier;
+        vbuf = similar(vs[1]), v̂buf = similar(F̂s[1]),
+    )
+    # Compute F_i = ∂_j (v_i v_j) for each i.
+    # In Fourier space: F̂_i = im * k_j * FFT(v_i * v_j)
+    w, ŵ = vbuf, v̂buf
+    @inbounds for (i, F̂i) ∈ enumerate(F̂s)
+        F̂i .= 0
+        vi = vs[i]
+        for (j, vj) ∈ enumerate(vs)
+            w .= vi .* vj     # w = v_i * v_j in physical space
+            mul!(ŵ, plan, w)  # same in Fourier space
+            # Add derivative in Fourier space
+            for I ∈ eachindex(grid_fourier)
+                k⃗ = grid_fourier[I]  # = (kx, ky, kz)
+                kj = k⃗[j]
+                F̂i[I] += im * kj * ŵ[I]
+            end
+        end
+    end
+    F̂s
+end
                        ns_nonlinear! (generic function with 1 method)

                        As an example, let's use this function on our initial velocity field:

                        F̂s = similar.(v̂s)
+ns_nonlinear!(F̂s, v⃗₀, plan, grid_fourier);

                        Strictly speaking, computing the non-linear term by collocation can lead to aliasing errors, as the quadratic term excites Fourier modes that fall beyond the range of resolved wave numbers. The typical solution is to apply Orzsag's 2/3 rule to zero-out the Fourier coefficients associated to the highest wave numbers. We define a function that applies this procedure below.

                        function dealias_twothirds!(ŵs::Tuple, grid_fourier, ks_global)
+    ks_max = maximum.(abs, ks_global)  # maximum stored wave numbers (kx_max, ky_max, kz_max)
+    ks_lim = (2 / 3) .* ks_max
+    @inbounds for I ∈ eachindex(grid_fourier)
+        k⃗ = grid_fourier[I]
+        if any(abs.(k⃗) .> ks_lim)
+            for ŵ ∈ ŵs
+                ŵ[I] = 0
+            end
+        end
+    end
+    ŵs
+end
+
+# We can apply this on the previously computed non-linear term:
+dealias_twothirds!(F̂s, grid_fourier, ks_global);

                        Finally, we implement the projection associated to the incompressibility condition:

                        function project_divergence_free!(ûs, grid_fourier)
+    @inbounds for I ∈ eachindex(grid_fourier)
+        k⃗ = grid_fourier[I]
+        k² = sum(abs2, k⃗)
+        iszero(k²) && continue  # avoid division by zero
+        û = getindex.(ûs, Ref(I))  # (ûs[1][I], ûs[2][I], ...)
+        for i ∈ eachindex(û)
+            ŵ = û[i]
+            for j ∈ eachindex(û)
+                ŵ -= k⃗[i] * k⃗[j] * û[j] / k²
+            end
+            ûs[i][I] = ŵ
+        end
+    end
+    ûs
+end
                        project_divergence_free! (generic function with 1 method)

                        We can verify the correctness of the projection operator by checking that the initial velocity field is not modified by it, since it is already incompressible:

                        v̂s_proj = project_divergence_free!(copy.(v̂s), grid_fourier)
+v̂s_proj .≈ v̂s  # the last one may be false because v_z = 0 initially
                        (true, true, false)

                        Putting it all together

                        To perform the time integration of the Navier–Stokes equations, we will use the timestepping routines implemented in the DifferentialEquations.jl suite. For simplicity, we use here an explicit Runge–Kutta scheme. In this case, we just need to write a function that computes the right-hand side of the Navier–Stokes equations in Fourier space:

                        function ns_rhs!(
+        dvs::NTuple{N, <:PencilArray}, vs::NTuple{N, <:PencilArray}, p, t,
+    ) where {N}
+    # 1. Compute non-linear term and dealias it
+    (; plan, cache, ks_global, grid_fourier) = p
+    F̂s = cache.F̂s
+    ns_nonlinear!(F̂s, vs, plan, grid_fourier; vbuf = dvs[1], v̂buf = cache.v̂s[1])
+    dealias_twothirds!(F̂s, grid_fourier, ks_global)
+
+    # 2. Project onto divergence-free space
+    project_divergence_free!(F̂s, grid_fourier)
+
+    # 3. Transform velocity to Fourier space
+    v̂s = cache.v̂s
+    map((v, v̂) -> mul!(v̂, plan, v), vs, v̂s)
+
+    # 4. Add viscous term (and multiply projected non-linear term by -1)
+    ν = p.ν
+    for n ∈ eachindex(v̂s)
+        v̂ = v̂s[n]
+        F̂ = F̂s[n]
+        @inbounds for I ∈ eachindex(grid_fourier)
+            k⃗ = grid_fourier[I]  # = (kx, ky, kz)
+            k² = sum(abs2, k⃗)
+            F̂[I] = -F̂[I] - ν * k² * v̂[I]
+        end
+    end
+
+    # 5. Transform RHS back to physical space
+    map((dv, dv̂) -> ldiv!(dv, plan, dv̂), dvs, F̂s)
+
+    nothing
+end
                        ns_rhs! (generic function with 1 method)

                        For the time-stepping, we load OrdinaryDiffEq.jl from the DifferentialEquations.jl suite and set-up the simulation. Since DifferentialEquations.jl can't directly deal with tuples of arrays, we convert the input data to the ArrayPartition type and write an interface function to make things work with our functions defined above.

                        using OrdinaryDiffEq
+using RecursiveArrayTools: ArrayPartition
+
+ns_rhs!(dv::ArrayPartition, v::ArrayPartition, args...) = ns_rhs!(dv.x, v.x, args...)
+vs_init_ode = ArrayPartition(v⃗₀)
+summary(vs_init_ode)
                        "RecursiveArrayTools.ArrayPartition{Float64, Tuple{PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}, PencilArray{Float64, 3, Array{Float64, 3}, 3, 0, Pencil{3, 2, NoPermutation, Vector{UInt8}}}}} with arrays:"

                        We now define solver parameters and temporary variables, and initialise the problem:

                        params = (;
+    ν = 5e-3,  # kinematic viscosity
+    plan, grid_fourier, ks_global,
+    cache = (
+        v̂s = similar.(v̂s),
+        F̂s = similar.(v̂s),
+    )
+)
+
+tspan = (0.0, 10.0)
+prob = ODEProblem{true}(ns_rhs!, vs_init_ode, tspan, params)
+integrator = init(prob, RK4(); dt = 1e-3, save_everystep = false);

                        We finally solve the problem over time and plot the vorticity associated to the solution. It is also useful to look at the energy spectrum $E(k)$, to see if the small scales are correctly resolved. To obtain a turbulent flow, the viscosity $ν$ must be small enough to allow the transient appearance of an energy cascade towards the small scales (i.e. from small to large $k$), while high enough to allow the small-scale motions to be correctly resolved.

                        function energy_spectrum!(Ek, ks, v̂s, grid_fourier)
+    Nk = length(Ek)
+    @assert Nk == length(ks)
+    Ek .= 0
+    for I ∈ eachindex(grid_fourier)
+        k⃗ = grid_fourier[I]  # = (kx, ky, kz)
+        knorm = sqrt(sum(abs2, k⃗))
+        i = searchsortedfirst(ks, knorm)
+        i > Nk && continue
+        v⃗ = getindex.(v̂s, Ref(I))  # = (v̂s[1][I], v̂s[2][I], ...)
+        factor = k⃗[1] == 0 ? 1 : 2  # account for Hermitian symmetry and r2c transform
+        Ek[i] += factor * sum(abs2, v⃗) / 2
+    end
+    MPI.Allreduce!(Ek, +, get_comm(v̂s[1]))  # sum across all processes
+    Ek
+end
+
+ks = rfftfreq(Ns[1], 2π * Ns[1] / Ls[1])
+Ek = similar(ks)
+v̂s = plan .* integrator.u.x
+energy_spectrum!(Ek, ks, v̂s, grid_fourier)
+Ek ./= scale_factor(plan)^2  # rescale energy
+
+curl_fourier!(ω̂s, v̂s, grid_fourier)
+ldiv!.(ωs, plan, ω̂s)
+ω⃗_plot = Observable(ωs)
+k_plot = @view ks[2:end]
+E_plot = Observable(@view Ek[2:end])
+t_plot = Observable(integrator.t)
+
+fig = let
+    fig = Figure(resolution = (1200, 600))
+    ax = Axis3(
+        fig[1, 1][1, 1]; title = @lift("t = $(round($t_plot, digits = 3))"),
+        aspect = :data, xlabel = "x", ylabel = "y", zlabel = "z",
+    )
+    ω_mag = @lift parent(vecnorm($ω⃗_plot))
+    ω_mag_norm = @lift $ω_mag ./ maximum($ω_mag)
+    ct = contour!(
+        ax, grid.x, grid.y, grid.z, ω_mag_norm;
+        alpha = 0.3, levels = 3,
+        colormap = :viridis, colorrange = (0.0, 1.0),
+        highclip = (:red, 0.2), lowclip = (:green, 0.2),
+    )
+    cb = Colorbar(fig[1, 1][1, 2], ct; label = "Normalised vorticity magnitude")
+    ax_sp = Axis(
+        fig[1, 2];
+        xlabel = "k", ylabel = "E(k)", xscale = log2, yscale = log10,
+        title = "Kinetic energy spectrum",
+    )
+    ylims!(ax_sp, 1e-8, 1e0)
+    scatterlines!(ax_sp, k_plot, E_plot)
+    ks_slope = exp.(range(log(2.5), log(25.0), length = 3))
+    E_fivethirds = @. 0.3 * ks_slope^(-5/3)
+    @views lines!(ax_sp, ks_slope, E_fivethirds; color = :black, linestyle = :dot)
+    text!(ax_sp, L"k^{-5/3}"; position = (ks_slope[2], E_fivethirds[2]), align = (:left, :bottom))
+    fig
+end
+
+record(fig, "vorticity_proc$procid.mp4"; framerate = 10) do io
+    while integrator.t < 20
+        dt = 0.001
+        step!(integrator, dt)
+        t_plot[] = integrator.t
+        mul!.(v̂s, plan, integrator.u.x)  # current velocity in Fourier space
+        curl_fourier!(ω̂s, v̂s, grid_fourier)
+        ldiv!.(ω⃗_plot[], plan, ω̂s)
+        ω⃗_plot[] = ω⃗_plot[]  # to force updating the plot
+        energy_spectrum!(Ek, ks, v̂s, grid_fourier)
+        Ek ./= scale_factor(plan)^2  # rescale energy
+        E_plot[] = E_plot[]
+        recordframe!(io)
+    end
+end;
                        + +

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                        PencilFFTs

                        Fast Fourier transforms of MPI-distributed Julia arrays.

                        Introduction

                        This package provides multidimensional FFTs and related transforms on MPI-distributed Julia arrays via the PencilArrays package.

                        The name of this package originates from the decomposition of 3D domains along two out of three dimensions, sometimes called pencil decomposition. This is illustrated by the figure below,[1] where each coloured block is managed by a different MPI process. Typically, one wants to compute FFTs on a scalar or vector field along the three spatial dimensions. In the case of a pencil decomposition, 3D FFTs are performed one dimension at a time, along the non-decomposed direction. Transforms must then be interleaved with global data transpositions to switch between pencil configurations. In high-performance computing environments, such data transpositions are generally the most expensive part of a parallel FFT computation, due to the large cost of communications between computing nodes.

                        + Pencil decomposition of 3D domains +

                        More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension $N$. The decompositions can be performed along an arbitrary number $M < N$ of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.

                        The generic and efficient implementation of this package is greatly enabled by the use of zero-cost abstractions in Julia. As shown in the Benchmarks section, PencilFFTs scales well to large numbers of processes, and performs similarly to the Fortran implementation of P3DFFT, possibly the most popular library for computing parallel FFTs using 2D domain decomposition.

                        Features

                        • distributed N-dimensional FFTs of MPI-distributed Julia arrays, using the PencilArrays package;

                        • FFTs and related transforms (e.g. DCTs / Chebyshev transforms) may be arbitrarily combined along different dimensions;

                        • in-place and out-of-place transforms;

                        • high scalability up to (at least) tens of thousands of MPI processes.

                        Installation

                        PencilFFTs can be installed using the Julia package manager:

                        julia> ] add PencilFFTs

                        Similar projects

                        • FFTW3 implements distributed-memory transforms using MPI, but these are limited to 1D decompositions. Also, this functionality is not currently included in the FFTW.jl wrappers.

                        • PFFT is a very general parallel FFT library written in C.

                        • P3DFFT implements parallel 3D FFTs using pencil decomposition in Fortran and C++.

                        • 2DECOMP&FFT is another parallel 3D FFT library using pencil decomposition written in Fortran.

                        • 1Figure adapted from this PhD thesis.
                        • 2For the pencil decomposition represented in the figure, $N = 3$ and $M = 2$.

                        • More generally, PencilFFTs allows to decompose and perform FFTs on geometries of arbitrary dimension $N$. The decompositions can be performed along an arbitrary number $M < N$ of dimensions.[2] Moreover, the transforms applied along each dimension can be arbitrarily chosen (and combined) among those supported by FFTW.jl, including complex-to-complex, real-to-complex and real-to-real transforms.