FreeGS: Free boundary Grad-Shafranov solver

This Python module calculates plasma equilibria for tokamak fusion experiments, by solving the Grad-Shafranov equation with free boundaries. Given a set of coils, plasma profiles and shape, FreeGS finds the currents in the coils which produce a steady-state solution in force balance.

Note This is a work in progress, and probably contains bugs. There is a feature wishlist in issues, suggestions and contributions welcome!

Installing

FreeGS is available on PyPI

$ pip install --user freegs

or clone/download this repository and run setup:

$ git clone https://github.com/bendudson/freegs.git
$ cd freegs
$ python setup.py install --user

Documentation

The manual is in the docs subdirectory, and hosted here on readthedocs.

Testing

Unit tests use pytest so after installing, run with:

$ pytest

The tests are in the src/ subdirectory.

A convergence test is described in the manual. To run:

$ python test-convergence.py

Examples

The Jupyter notebooks contain examples with additional notes

• MAST-example.ipynb

There are also some Python scripts to run short tests and examples

$ python 01-freeboundary.py

This solves a free boundary problem, specifying the desired location of two X-points. Writes the equilibrium to a G-EQDSK file "lsn.geqdsk"

$ python 02-read-geqdsk.py

Reads in the file "lsn.geqdsk", inferring the coil currents from the plasma boundary and profiles in the G-EQDSK file.

$ python 03-mast.py

Calculates a double-null (CDND) equilibrium for MAST from scratch. Writes solution to G-EQDSK file "mast.geqdsk"

$ python 04-read-mast-geqdsk.py

Reads the file "mast.geqdsk", inferring the coil currents.

$ python 05-fixed-boundary.py 

This example solves a fixed boundary problem, in which the square edges of the domain are fixed. The plasma pressure on axis and plasma current are fixed.

$ python 06-xpoints.py

This demonstrates the coil current control code, finding X-points, and marking core region These routines are used inside the free boundary solver

Files

The "freegs" module consists of the following files:

• boundary.py - Operators for applying boundary conditions to plasma psi
• control.py - Routines for controlling coil currents based on constraints
• critical.py - Finds critical points (O- and X-points)
• equilibrium.py - Represents the plasma equilibrium state
• gradshafranov.py - Greens functions and operators for the Grad-Shafranov equation
• jtor.py - Routines for calculating toroidal current density (profiles)
• machine.py - Represents the coils and power supply circuits
• multigrid.py - The multigrid solver for the linear elliptic operator
• picard.py - Nonlinear solver, iterating the profiles and constraints
• newtonk.py - Same as above
• plotting.py - Plotting routines using matplotlib

License

Copyright 2016-2021 Ben Dudson, University of York, and other contributors.
Email: [email protected]

This program is free software: you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU Lesser General Public License for more details.

You should have received a copy of the GNU Lesser General Public License
along with this program.  If not, see <http://www.gnu.org/licenses/>.

References

• YoungMu Jeon, Development of a free boundary Tokamak Equlibrium Solver arXiv:1503.03135
• S.Jardin "Computational Methods in Plasma Physics" CRC Press

Versions

0.6.1 11th February 2021

• Fixes for HDF5 reading and writing
• Fix Numpy deprecations
• Catch errors in optimisation measures, making that more robust
• Fix core mask in geqdsk file reading
• Flake8 fixes
• Move to Github actions

0.6.0 29th January 2021

• Improve separatrix finding (Pablo Rodriguez-Fernandez)
• Update MAST-U machine description (James Harrison)
• Updated documentation
• Add tests for critical point finding
• Black format all library code
• Fix wall intersection calculation in optimisation
• Fix factor of pi in effectiveElongation (Chris Winnard)
• Miscellaneous fixes and tidying (Chris Winnard)
• Fix psi_bndry test in critical.core_mask
• Fix comments in Greens functions (James Morris)

0.5.0 25th March 2020

• More tests, thanks to @ZedThree.
• Added more flexible coil types, thanks to Chris Marsden. Includes support for shaped coils, multi-strand coils, and up-down mirrored coils.
• Basic support for reading and writing AEQDSK files
• Added h5py to requirements, fixes for latest NumPy

0.4.0 10th November 2019

• Add optimisation with Differential Evolution
• More unit testing, documentation

0.3.0 28th July 2019

• Add 4th-order solver for potential
• Add convergence test

0.2.0 12th March 2019

• Add field line tracer, freegs.fieldtracer
• Add Equilibrium.Btor toroidal field calculation
• Add Equilibrium.plasmaVolume
• Fix rlim, zlim saved into GEQDSK files