2020
2.
Qu, Z. S.; Pfefferlé, D.; Hudson, S. R.; Baillod, A.; Kumar, A.; Dewar, R. L.; Hole, M. J.
Coordinate Parameterisation and Spectral Method Optimisation for Beltrami Field Solver in Stellarator Geometry Journal Article
In: vol. 62, no. 12, pp. 124004, 2020, ISSN: 0741-3335.
Abstract | Links | BibTeX | Tags: curvilinear coordinates, MHD equilibrium, numerical method, SPEC
@article{quCoordinateParameterisationSpectral2020,
title = {Coordinate Parameterisation and Spectral Method Optimisation for Beltrami Field Solver in Stellarator Geometry},
author = {Z. S. Qu and D. Pfefferl\'{e} and S. R. Hudson and A. Baillod and A. Kumar and R. L. Dewar and M. J. Hole},
doi = {10.1088/1361-6587/abc08e},
issn = {0741-3335},
year = {2020},
date = {2020-11-01},
urldate = {2020-11-01},
volume = {62},
number = {12},
pages = {124004},
publisher = {IOP Publishing},
abstract = {The numerical solution of the stepped pressure equilibrium (Hudson et al 2012 Phys. Plasmas 19 112502) requires a fast and robust solver to obtain the Beltrami field in three-dimensional geometry such as stellarators. The spectral method implemented in the stepped pressure equilibrium code (SPEC) is efficient when the domain is a hollow torus, but ill-conditioning of the discretised linear equations occurs in the (solid) toroid due to the artificially singular coordinate parameterisation near the axis. In this work, we propose an improved choice for the reference axis to prevent coordinates surfaces from overlapping. Then, we examine the parity and asymptotics of the magnetic vector potential near the axis and suggest the use of recombined and rescaled Zernike radial basis functions. The maximum relative error in the magnetic field of the Wendelstein 7-X geometry is shown to reach 10-9 at high resolution in a series of convergence tests and benchmarks against the boundary integral equation solver for Taylor states. The new method is also reported to significantly improve the accuracy of multi-volume SPEC calculations. A comparison between free-boundary SPEC and the analytical Dommaschk potential is presented with higher-than-usual Fourier resolution. It is illustrated that we are able to resolve low amplitude current sheets when an interface is placed where there is no flux surface in the analytic solution. This was previously concealed because of insufficient numerical resolution.},
keywords = {curvilinear coordinates, MHD equilibrium, numerical method, SPEC},
pubstate = {published},
tppubtype = {article}
}
The numerical solution of the stepped pressure equilibrium (Hudson et al 2012 Phys. Plasmas 19 112502) requires a fast and robust solver to obtain the Beltrami field in three-dimensional geometry such as stellarators. The spectral method implemented in the stepped pressure equilibrium code (SPEC) is efficient when the domain is a hollow torus, but ill-conditioning of the discretised linear equations occurs in the (solid) toroid due to the artificially singular coordinate parameterisation near the axis. In this work, we propose an improved choice for the reference axis to prevent coordinates surfaces from overlapping. Then, we examine the parity and asymptotics of the magnetic vector potential near the axis and suggest the use of recombined and rescaled Zernike radial basis functions. The maximum relative error in the magnetic field of the Wendelstein 7-X geometry is shown to reach 10-9 at high resolution in a series of convergence tests and benchmarks against the boundary integral equation solver for Taylor states. The new method is also reported to significantly improve the accuracy of multi-volume SPEC calculations. A comparison between free-boundary SPEC and the analytical Dommaschk potential is presented with higher-than-usual Fourier resolution. It is illustrated that we are able to resolve low amplitude current sheets when an interface is placed where there is no flux surface in the analytic solution. This was previously concealed because of insufficient numerical resolution.
2014
1.
Pfefferlé, D; Cooper, W A; Graves, J P; Misev, C
In: Computer Physics Communications, vol. 185, no. 12, pp. 3127 - 3140, 2014, ISSN: 0010-4655.
Abstract | Links | BibTeX | Tags: cubic spline, curvilinear coordinates, fast particles, numerical method, VENUS-LEVIS
@article{pfefferle-levis,
title = {VENUS-LEVIS and its spline-Fourier interpolation of 3D toroidal magnetic field representation for guiding-centre and full-orbit simulations of charged energetic particles},
author = {D Pfefferl\'{e} and W A Cooper and J P Graves and C Misev},
doi = {10.1016/j.cpc.2014.08.007},
issn = {0010-4655},
year = {2014},
date = {2014-08-16},
journal = {Computer Physics Communications},
volume = {185},
number = {12},
pages = {3127 - 3140},
abstract = {Curvilinear guiding-centre drift and full-orbit equations of motion are presented as implemented in the VENUS-LEVIS code. A dedicated interpolation scheme based on Fourier reconstruction in the toroidal and poloidal directions and cubic spline in the radial direction of flux coordinate systems is detailed. This interpolation method exactly preserves the order of the RK4 integrating scheme which is crucial for the investigation of fast particle trajectories in 3D magnetic structures such as helical saturated tokamak plasma states, stellarator geometry and resonant magnetic perturbations (RMP). The initialisation of particles with respect to the guiding-centre is discussed. Two approaches to implement RMPs in orbit simulations are presented, one where the vacuum field is added to the 2D equilibrium, creating islands and stochastic regions, the other considering 3D nested flux-surfaces equilibrium including the RMPs.},
keywords = {cubic spline, curvilinear coordinates, fast particles, numerical method, VENUS-LEVIS},
pubstate = {published},
tppubtype = {article}
}
Curvilinear guiding-centre drift and full-orbit equations of motion are presented as implemented in the VENUS-LEVIS code. A dedicated interpolation scheme based on Fourier reconstruction in the toroidal and poloidal directions and cubic spline in the radial direction of flux coordinate systems is detailed. This interpolation method exactly preserves the order of the RK4 integrating scheme which is crucial for the investigation of fast particle trajectories in 3D magnetic structures such as helical saturated tokamak plasma states, stellarator geometry and resonant magnetic perturbations (RMP). The initialisation of particles with respect to the guiding-centre is discussed. Two approaches to implement RMPs in orbit simulations are presented, one where the vacuum field is added to the 2D equilibrium, creating islands and stochastic regions, the other considering 3D nested flux-surfaces equilibrium including the RMPs.