ElectromagneticFields.jl

Common Interface for Electromagnetic Fields

ElectromagneticFields.jl provides a collection of analytically known electromagnetic fields — tokamak equilibria, Solov'ev solutions of the Grad-Shafranov equation, Penning traps, the Arnold-Beltrami-Childress field and several others — together with the machinery that turns any of them into fast, allocation-free evaluation code.

A field is described by very little: the covariant components of its vector potential, an electrostatic potential where there is one, and the chart it is written in. Everything a simulation actually needs — the magnetic field, its magnitude, the unit vector along it, an orthonormal frame adapted to it, the electric field, the metric, and the derivatives of all of these — is derived from that symbolically and compiled.

using ElectromagneticFields

field = FieldFunctions(AxisymmetricTokamakCylindricalEquilibrium(6.2, 5.3, 2.0))

B♭(field, 0.0, [6.5, 0.5, 0.25])

3-element StaticArraysCore.SVector{3, Float64} with indices SOneTo(3):
 -0.20384615384615384
  0.12230769230769223
 32.86

The result is a value. It can be built inside a function, stored, and passed to a vector field that depends on it — typically in the parameters of a GeometricEquations problem, from which the equation function evaluates the components it needs.

Where to go from here:

• Usage — constructing an equilibrium and generating its code.
• Interface — the complete list of quantities, their shapes, and how to hand a field to a solver.
• Code Generation — how the code is derived, what it costs, and how to add a field of your own.
• Coordinates and Fields — the differential geometry behind the naming and the derivation.
• Analytic Fields — the fields the package ships.

References

• Antoine J. Cerfon, Jeffrey P. Freidberg. "One size fits all" analytic solutions to the Grad–Shafranov equation. Physics of Plasmas 17, 032502, 2010.
• Patrick J. McCarthy. Analytical solutions to the Grad–Shafranov equation for tokamak equilibrium with dissimilar source functions. Physics of Plasmas 6, 3554, 1999.
• Yanyan Shi, Yajuan Sun, Yulei Wang, Jian Liu, Study of adaptive symplectic methods for simulating charged particle dynamics, Journal of Computational Dynamics 6, 429-448, 2019.

License

Copyright (c) Michael Kraus <michael.kraus@ipp.mpg.de>

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.