From 88b7744c740e1f74ece75b4ea34b77d246c343d7 Mon Sep 17 00:00:00 2001 From: Oriol Colomes Gene Date: Wed, 27 May 2020 12:41:04 +0200 Subject: [PATCH 01/33] Added 2D cylinder benchmark GMSH model --- models/cylinder_NSI.geo | 23 + models/cylinder_NSI.jl | 10 + models/cylinder_NSI.json | 1 + models/cylinder_NSI.msh | 8760 ++++++++++++++++++++++++++++++++++++++ 4 files changed, 8794 insertions(+) create mode 100644 models/cylinder_NSI.geo create mode 100644 models/cylinder_NSI.jl create mode 100644 models/cylinder_NSI.json create mode 100644 models/cylinder_NSI.msh diff --git a/models/cylinder_NSI.geo b/models/cylinder_NSI.geo new file mode 100644 index 00000000..2e419549 --- /dev/null +++ b/models/cylinder_NSI.geo @@ -0,0 +1,23 @@ +// Gmsh project created on Fri Mar 20 14:01:03 2020 +SetFactory("OpenCASCADE"); + +// Define Geometry +Rectangle(1) = {0, 0, 0, 2.2, 0.41, 0}; +Disk(2) = {0.2, 0.2, 0, 0.05, 0.05}; +BooleanDifference{ Surface{1}; Delete; }{ Surface{2}; Delete; } +Coherence; + +// Define mesh sizes +Transfinite Curve {7} = 40 Using Progression 1; +Transfinite Curve {5} = 40 Using Progression 1; +Transfinite Curve {8} = 20 Using Progression 1; +Transfinite Curve {6, 9} = 80 Using Progression 1; + +// Define Physical groups//+ +Physical Surface("fluid", 1) = {1}; +Physical Curve("inlet", 2) = {7}; +Physical Curve("outlet", 3) = {8}; +Physical Curve("noslip", 4) = {6, 9}; +Physical Curve("cylinder", 5) = {5}; +Physical Point("noslip", 4) = {6, 8, 9, 7}; +Physical Point("cylinder", 5) = {5}; diff --git a/models/cylinder_NSI.jl b/models/cylinder_NSI.jl new file mode 100644 index 00000000..5b00470f --- /dev/null +++ b/models/cylinder_NSI.jl @@ -0,0 +1,10 @@ +using Gridap +using Gridap.Io +using GridapGmsh + +model = GmshDiscreteModel("cylinder_NSI.msh") + +writevtk(model,"cilinder_NSI") + +fn = "cylinder_NSI.json" +to_json_file(model,fn) diff --git a/models/cylinder_NSI.json b/models/cylinder_NSI.json new file mode 100644 index 00000000..b66a0837 --- /dev/null +++ b/models/cylinder_NSI.json @@ -0,0 +1 @@ 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1322 +4318 1019 2097 1940 +4319 411 2172 413 +4320 1217 2108 1913 +4321 1204 2165 1206 +4322 2000 2090 1289 +4323 1872 2057 1183 +4324 1217 2145 2108 +4325 1943 2131 1267 +4326 1966 2102 1486 +4327 367 2118 1899 +4328 1847 2089 2006 +4329 1160 2150 1969 +4330 1252 2161 1931 +4331 124 1985 356 +4332 1626 2169 2048 +4333 334 2135 1982 +4334 1993 2090 1992 +4335 301 2133 1866 +4336 2006 2089 444 +4337 355 1992 157 +4338 1899 2118 1339 +4339 1948 2148 1146 +4340 351 2090 2000 +4341 357 2111 2109 +4342 1307 2103 1981 +4343 1040 2158 1977 +4344 2050 2166 1293 +4345 1289 2090 1993 +4346 356 1986 123 +4347 158 1994 355 +4348 2048 2169 916 +4349 1992 2090 351 +4350 2054 2164 1279 +4351 44 1994 2 +4352 2 1994 159 +4353 122 1986 4 +4354 4 1986 255 +4355 1931 1999 1246 +$EndElements From 53e4ef2adcad4a00a9220b4137400cd0f2a03ae2 Mon Sep 17 00:00:00 2001 From: Oriol Colomes Gene Date: Wed, 27 May 2020 12:41:37 +0200 Subject: [PATCH 02/33] First draft of the transient incompressible Navier-Stokes tutorial --- src/transient_inc_navier_stokes.jl | 290 +++++++++++++++++++++++++++++ 1 file changed, 290 insertions(+) create mode 100644 src/transient_inc_navier_stokes.jl diff --git a/src/transient_inc_navier_stokes.jl b/src/transient_inc_navier_stokes.jl new file mode 100644 index 00000000..39849703 --- /dev/null +++ b/src/transient_inc_navier_stokes.jl @@ -0,0 +1,290 @@ +# In this tutorial, we will learn +# - How to solve nonlinear multi-field PDEs in Gridap +# - How to build FE spaces whose functions have zero mean value +# +# ## Problem statement +# +# The goal of this last tutorial is to solve a nonlinear multi-field PDE. As a model problem, we consider a well known benchmark in computational fluid dynamics, the lid-driven cavity for the incompressible Navier-Stokes equations. Formally, the PDE we want to solve is: find the velocity vector $u$ and the pressure $p$ such that +# +# ```math +# \left\lbrace +# \begin{aligned} +# -\Delta u + \mathit{Re}\ (u\cdot \nabla)\ u + \nabla p = 0 &\text{ in }\Omega,\\ +# \nabla\cdot u = 0 &\text{ in } \Omega,\\ +# u = g &\text{ on } \partial\Omega, +# \end{aligned} +# \right. +# ``` +# +# where the computational domain is the unit square $\Omega \doteq (0,1)^d$, $d=2$, $\mathit{Re}$ is the Reynolds number (here, we take $\mathit{Re}=10$), and $(w \cdot \nabla)\ u = (\nabla u)^t w$ is the well known convection operator. In this example, the driving force is the Dirichlet boundary velocity $g$, which is a non-zero horizontal velocity with a value of $g = (1,0)^t$ on the top side of the cavity, namely the boundary $(0,1)\times\{1\}$, and $g=0$ elsewhere on $\partial\Omega$. Since we impose Dirichlet boundary conditions on the entire boundary $\partial\Omega$, the mean value of the pressure is constrained to zero in order have a well posed problem, +# +# ```math +# \int_\Omega q \ {\rm d}\Omega = 0. +# ``` +# +# ## Numerical Scheme +# +# In order to approximate this problem we chose a formulation based on inf-sub stable $Q_k/P_{k-1}$ elements with continuous velocities and discontinuous pressures (see, e.g., [1] for specific details). The interpolation spaces are defined as follows. The velocity interpolation space is +# +# ```math +# V \doteq \{ v \in [C^0(\Omega)]^d:\ v|_T\in [Q_k(T)]^d \text{ for all } T\in\mathcal{T} \}, +# ``` +# where $T$ denotes an arbitrary cell of the FE mesh $\mathcal{T}$, and $Q_k(T)$ is the local polynomial space in cell $T$ defined as the multi-variate polynomials in $T$ of order less or equal to $k$ in each spatial coordinate. Note that, this is the usual continuous vector-valued Lagrangian FE space of order $k$ defined on a mesh of quadrilaterals or hexahedra. On the other hand, the space for the pressure is +# +# ```math +# \begin{aligned} +# Q_0 &\doteq \{ q \in Q: \ \int_\Omega q \ {\rm d}\Omega = 0\}, \text{ with}\\ +# Q &\doteq \{ q \in L^2(\Omega):\ q|_T\in P_{k-1}(T) \text{ for all } T\in\mathcal{T}\}, +# \end{aligned} +# ``` +# where $P_{k-1}(T)$ is the polynomial space of multi-variate polynomials in $T$ of degree less or equal to $k-1$. Note that functions in $Q_0$ are strongly constrained to have zero mean value. This is achieved in the code by removing one degree of freedom from the (unconstrained) interpolation space $Q$ and adding a constant to the computed pressure so that the resulting function has zero mean value. +# +# The weak form associated to these interpolation spaces reads: find $(u,p)\in U_g \times Q_0$ such that $[r(u,p)](v,q)=0$ for all $(v,q)\in V_0 \times Q_0$ +# where $U_g$ and $V_0$ are the set of functions in $V$ fulfilling the Dirichlet boundary condition $g$ and $0$ on $\partial\Omega$ respectively. The weak residual $r$ evaluated at a given pair $(u,p)$ is the linear form defined as +# +# ```math +# [r(u,p)](v,q) \doteq a((u,p),(v,q))+ [c(u)](v), +# ``` +# with +# ```math +# \begin{aligned} +# a((u,p),(v,q)) &\doteq \int_{\Omega} \nabla v \cdot \nabla u \ {\rm d}\Omega - \int_{\Omega} (\nabla\cdot v) \ p \ {\rm d}\Omega + \int_{\Omega} q \ (\nabla \cdot u) \ {\rm d}\Omega,\\ +# [c(u)](v) &\doteq \int_{\Omega} v \cdot \left( (u\cdot\nabla)\ u \right)\ {\rm d}\Omega.\\ +# \end{aligned} +# ``` +# Note that the bilinear form $a$ is associated with the linear part of the PDE, whereas $c$ is the contribution to the residual resulting from the convective term. +# +# In order to solve this nonlinear weak equation with a Newton-Raphson method, one needs to compute the Jacobian associated with the residual $r$. In this case, the Jacobian $j$ evaluated at a pair $(u,p)$ is the bilinear form defined as +# +# ```math +# [j(u,p)]((\delta u, \delta p),(v,q)) \doteq a((\delta u,\delta p),(v,q)) + [{\rm d}c(u)](\delta u,v), +# ``` +# where ${\rm d}c$ results from the linearization of the convective term, namely +# ```math +# [{\rm d}c(u)](\delta u,v) \doteq \int_{\Omega} v \cdot \left( (u\cdot\nabla)\ \delta u \right) \ {\rm d}\Omega + \int_{\Omega} v \cdot \left( (\delta u\cdot\nabla)\ u \right) \ {\rm d}\Omega. +# ``` +# The implementation of this numerical scheme is done in Gridap by combining the concepts previously seen for single-field nonlinear PDEs and linear multi-field problems. +# +# ## Discrete model +# +# We start with the discretization of the computational domain. We consider a $100\times100$ Cartesian mesh of the unit square. + +using Gridap +using ForwardDiff +using LinearAlgebra +using Test +using GridapODEs.ODETools +using GridapODEs.TransientFETools +using Gridap.FESpaces: get_algebraic_operator +using Printf + +import GridapODEs.TransientFETools: ∂t + +# Problem parameters +const Um = 1.0 +const H = 0.41 +const ⌀ = 0.1 +const ρ = 1.0 + +# Boundary conditions +u_in(x,t) = VectorValue( 1.5 * Um * x[2] * ( H - x[2] ) / ( (H/2)^2 ), 0.0 ) +u_noSlip(x,t) = VectorValue( 0.0, 0.0 ) +u_in(t::Real) = x -> u_in(x,t) +u_noSlip(t::Real) = x -> u_noSlip(x,t) +∂tu_in(t) = x -> VectorValue(0.0,0.0) +∂tu_in(x,t) = ∂tu_in(t)(x) +∂t(::typeof(u_in)) = ∂tu_in +∂t(::typeof(u_noSlip)) = ∂tu_in + +# Domain +#n = 4 +#domain = (0,1,0,1) +#partition = (n,n) +#model = CartesianDiscreteModel(domain,partition) +#labels = get_face_labeling(model) +#add_tag_from_tags!(labels,"diri1",[6,]) +#add_tag_from_tags!(labels,"diri0",[1,2,3,4,5,7,8]) + +model = DiscreteModelFromFile("../models/cylinder_NSI.json") +writevtk(model,"model") + +# Time stepping settings +t0 = 0.0 +tF = 0.1 +dt = 0.05 +θ = 0.5 + +# ## FE spaces +D = 2 +order = 2 +V = FESpace( + reffe=:Lagrangian, conformity=:H1, valuetype=VectorValue{D,Float64}, + model=model, order=order, dirichlet_tags=["inlet","noslip","cylinder"]) +Q = TestFESpace( + reffe=:Lagrangian, conformity=:H1, valuetype=Float64, + model=model, order=order-1, constraint=:zeromean) + + + +p(x,t) = 0.0 +p(t::Real) = x -> p(x,t) + +function u_t0(x,t) + if(x[2]==1.0 && x[1]>0.0 && x[1]<1.0) + return VectorValue(1.0,0.0) + else + return VectorValue(0.0,0.0) + end +end +u_t0(t::Real) = x->u_t0(x,t) + +U = TransientTrialFESpace(V,[u_in,u_noSlip,u_noSlip]) +U0 = TrialFESpace(V,[u_in(t0),u_noSlip(t0),u_noSlip(t0)]) +P = TrialFESpace(Q) + +Y = MultiFieldFESpace([V, Q]) +X = MultiFieldFESpace([U, P]) +X0 = MultiFieldFESpace([U0, P]) + +# Initial FE spaces +#U0 = U(t0) +#P0 = P(t0) +#X0 = X(t0) + +# ## Navier-Stokes weak form +@law conv(u,∇u) = (∇u')*u +@law dconv(du,∇du,u,∇u) = conv(u,∇du)+conv(du,∇u) + +const Re = 100.0 +const ν = Um*⌀ / Re +@law σ_dev(ε) = 2*ν*ε +function a(x,y) + u, p = x + v, q = y + inner(ε(v),σ_dev(ε(u))) - (∇*v)*p + q*(∇*u) +end + +c(u,v) = v*conv(u,∇(u)) +dc(u,du,v) = v*dconv(du,∇(du),u,∇(u)) + +function res(t,x,xt,y) + u, p = x + ut, pt = xt + v, q = y + inner(ut,v) + a(x,y) + c(u,v) +end + +function jac(t,x,xt,dx,y) + u, p = x + v, q = y + du, dp = dx + a(dx,y)+ dc(u,du,v) +end + +function jac_t(t,x,xt,dxt,y) + dut, dpt = dxt + v, q = y + inner(dut,v) +end + +# Triangulation and CellQuadrature +trian = Triangulation(model) +degree = (order-1)*2 +quad = CellQuadrature(trian,degree) + +# Navier-Stokes FE operator +t_Ω = FETerm(res,jac,jac_t,trian,quad) +op = TransientFEOperator(X,Y,t_Ω) + +# Stokes FE operator +t_Stokes_Ω = LinearFETerm(a,trian,quad) +op_Stokes = FEOperator(X0,Y,t_Stokes_Ω) + +# Stokes solution +ls = LUSolver() +solver_Stokes = FESolver(ls) +xh0 = solve(solver_Stokes,op_Stokes) +uh0 = xh0[1] +ph0 = xh0[2] + +l2(w) = w*w +#println("L2-norm: ", sqrt(sum( integrate(l2(uh0),trian,quad) ))) + +# Transient Navier-Stokes solution +using LineSearches: BackTracking +nls = NLSolver(show_trace=true, method=:newton, linesearch=BackTracking()) +odes = ThetaMethod(nls,dt,θ) +solver = TransientFESolver(odes) +sol_t = solve(solver,op,xh0,t0,tF) + +# Post-processing + +# Cylinder triangulation +trian_Γc = BoundaryTriangulation(model,"cylinder") +quad_Γc = CellQuadrature(trian_Γc,degree) +n_Γc = get_normal_vector(trian_Γc) + +# Drag & Lift coefficients +coeff(F) = 2*F / (ρ*Um^2*⌀) +using Plots +tpl = [0.0] +CLpl = [0.0] +pl = plot(tpl,CLpl,leg=false) +anim = Animation() + +_t_n = t0 +_phn = ph0 +_uhn = uh0 + +# Paraview files +outputName = "ins-results" +using WriteVTK +pvd = paraview_collection(outputName) +pvd[0.0] = writevtk(trian,outputName,cellfields=["uh_tn"=>uh0,"ph_tn"=>ph0])[1] + +# import Base.push! +# function push!(output::vtkOutput, vtkFile::Array{String,1}, time::Float64) +# append!(output.vtkFiles, vtkFile) +# push!(output.times, time) +# end +# function writeFiles(output::vtkOutput) +# if !isdir(output.outputName) +# mkdir(output.outputName) +# end +# fileName = output.outputName * "/" * output.outputName * ".pvd" +# +# end + +#push!(vtkFiles,writevtk(trian,"ins-result0.0",cellfields=["uh_tn"=>uh0,"ph_tn"=>ph0])) +#mkdir(outputName) +#vtkFiles = writevtk(trian,"ins-result0.0",cellfields=["uh_tn"=>uh0,"ph_tn"=>ph0])) + +for (xh_tn, tn) in sol_t + global _t_n + global _phn + global _uhn + _t_n += dt + uh_tn = xh_tn.blocks[1] + ph_tn = xh_tn.blocks[2] + uhθ = θ*uh_tn + (1.0 - θ)*_uhn + phθ = θ*ph_tn + (1.0 - θ)*_phn + _uhn = uh_tn + _phn = ph_tn + pvd[_t_n] = writevtk(trian,outputName,cellfields=["uh_tn"=>uh_tn,"ph_tn"=>ph_tn]) + #println("L2-norm: ", sqrt(sum( integrate(l2(uh_tn),trian,quad) ))) + uh_Γc = restrict(uh_tn,trian_Γc) + ph_Γc = restrict(phθ,trian_Γc) + FD, FL = sum( integrate( (σ_dev(ε(uh_Γc))*n_Γc - ph_Γc*n_Γc), trian_Γc, quad_Γc ) ) + println("CD: ", coeff(FD)) + println("CL: ", coeff(FL)) + push!(tpl,_t_n) + push!(CLpl,coeff(FL)) + push!(pl, tpl, CLpl) + frame(anim) +end +anim = @animate for i in 2:length(tpl) + plot(tpl[1:i],CLpl[1:i]) +end +vtk_save(pvd) +gif(anim,fps=5) +#plot(tpl,CLpl) From cdff0f991d6ac84579e10249b9095a495d7e7772 Mon Sep 17 00:00:00 2001 From: Oriol Colomes Date: Thu, 4 Jun 2020 22:12:53 +0200 Subject: [PATCH 03/33] - Added cylinder.gif - Further developments of the transient_inc_navier_stokes tutorial --- assets/cylinder_ins/cylinder.gif | Bin 0 -> 13501143 bytes src/transient_inc_navier_stokes.jl | 489 ++++++++++++++--------------- 2 files changed, 242 insertions(+), 247 deletions(-) create mode 100644 assets/cylinder_ins/cylinder.gif diff --git a/assets/cylinder_ins/cylinder.gif b/assets/cylinder_ins/cylinder.gif new file mode 100644 index 0000000000000000000000000000000000000000..cb91f14c76d8f308a5ba920d5d72f123047713c4 GIT binary patch literal 13501143 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