FreeGS 0.6.0

Free boundary Grad-Shafranov solver for tokamak plasma equilibria

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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](http://freegs.readthedocs.io/en/latest/).

Testing

Unit tests use [pytest](https://docs.pytest.org/en/latest/) so after installing, run with:

$ pytest

The tests are in the src/ subdirectory.

A convergence test is [described in the manual](https://freegs.readthedocs.io/en/latest/tests.html#convergence-test). To run:

$ python test-convergence.py

Examples

The Jupyter notebooks contain examples wuth 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

• plotting.py - Plotting routines using matplotlib

License

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

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](http://link.springer.com/article/10.3938/jkps.67.843) [arXiv:1503.03135](https://arxiv.org/abs/1503.03135)

• S.Jardin “Computational Methods in Plasma Physics” CRC Press

Versions

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