×

FEM-BEM coupling methods for tokamak plasma axisymmetric free-boundary equilibrium computations in unbounded domains. (English) Zbl 1380.78003

Summary: Incorporating boundary conditions at infinity into simulations on bounded computational domains is a repeatedly occurring problem in scientific computing. The combination of finite element methods (FEM) and boundary element methods (BEM) is the obvious instrument, and we adapt here for the first time the two standard FEM-BEM coupling approaches to the free-boundary equilibrium problem: the Johnson-Nédélec coupling and the Bielak-MacCamy coupling. We recall also the classical approach for fusion applications, dubbed according to its first appearance von-Hagenow-Lackner coupling and present the less used alternative introduced by R. Albanese et al. [“On the solution of the magnetic flux equation in an infinite domain”, in: Proceedings of the 8th Europhysics Conference on Computational Physics, Computing in Plasma Physics. 41–44 (1986)]. We show that the von-Hagenow-Lackner coupling suffers from undesirable non-optimal convergence properties, that suggest that other coupling schemes, in particular Johnson-Nédélec or Albanese-Blum-de Barbieri are more appropriate for non-linear equilibrium problems. Moreover, we show that any of such coupling methods requires Newton-like iteration schemes for solving the corresponding non-linear discrete algebraic systems.

MSC:

78A30 Electro- and magnetostatics
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
78M15 Boundary element methods applied to problems in optics and electromagnetic theory
76W05 Magnetohydrodynamics and electrohydrodynamics
PDFBibTeX XMLCite
Full Text: DOI HAL

References:

[1] Albanese, R.; Ambrosino, R.; Mattei, M., CREATE-NL+: a robust control-oriented free boundary dynamic plasma equilibrium solver, Proceedings of the 28th Symposium on Fusion Technology (SOFT-28). Proceedings of the 28th Symposium on Fusion Technology (SOFT-28), Fusion Eng. Des., 96-97, 664-667 (2015)
[2] Albanese, R.; Blum, J.; De Barbieri, O., On the solution of the magnetic flux equation in an infinite domain, (EPS. 8th Europhysics Conference on Computing in Plasma Physics (1986) (1986)), 41-44
[3] Albanese, R.; Blum, J.; De Barbieri, O., Numerical studies of the Next European Torus via the PROTEUS code, (12th Conf. on Numerical Simulation of Plasmas. 12th Conf. on Numerical Simulation of Plasmas, San Francisco (1987))
[4] Bermudez, A.; Gomez, D.; Muniz, M. C.; Salgado, P., A FEM/BEM for axisymmetric electromagnetic and thermal modelling of induction furnaces, Int. J. Numer. Methods Eng., 71, 7, 856-878 (2007) · Zbl 1194.78050
[5] Bielak, J.; MacCamy, R. C., An exterior interface problem in two-dimensional elastodynamics, Q. Appl. Math., 41, 1, 143-159 (1983/84) · Zbl 0519.73021
[6] Blum, J., Numerical Simulation and Optimal Control in Plasma Physics (1989), Wiley/Gauthier-Villars
[7] Blum, J.; Boulbe, C.; Faugeras, B., Real-time plasma equilibrium reconstruction in a tokamak, Proceedings of the 6th International Conference on Inverse Problems in Engineering: Theory and Practice. Proceedings of the 6th International Conference on Inverse Problems in Engineering: Theory and Practice, J. Phys. Conf. Ser., 135, Article 012019 pp. (2008) · Zbl 1382.76295
[8] Blum, J.; Boulbe, C.; Faugeras, B., Reconstruction of the equilibrium of the plasma in a tokamak and identification of the current density profile in real time, J. Comput. Phys., 231, 3, 960-980 (2012) · Zbl 1382.76295
[9] Blum, J.; Le Foll, J.; Thooris, B., The self-consistent equilibrium and diffusion code SCED, Comput. Phys. Commun., 24, 235-254 (1981)
[11] Costabel, M.; Ervin, V. J.; Stephan, E. P., Experimental convergence rates for various couplings of boundary and finite elements, Math. Comput. Model., 15, 3, 93-102 (1991) · Zbl 0727.65096
[12] Costabel, M.; Stephan, E. P., Coupling of finite and boundary element methods for an elastoplastic interface problem, SIAM J. Numer. Anal., 27, 5, 1212-1226 (1990) · Zbl 0725.73090
[13] Cuvelier, F.; Japhet, C.; Scarella, G., An efficient way to assemble finite element matrices in vector languages, BIT Numer. Math., 56, 3, 833-864 (2016) · Zbl 1351.65088
[14] Davis, T. A., Umfpack (2011)
[15] Fairweather, G., Finite Element Galerkin Methods for Differential Equations, Lect. Notes Pure Appl. Math., vol. 34 (1978), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York, Basel · Zbl 0372.65044
[16] Freidberg, J. P., Ideal Magnetohydrodynamics (1987), Plenum US
[17] Gatica, G. N.; Hsiao, G. C., The uncoupling of boundary integral and finite element methods for nonlinear boundary value problems, J. Math. Anal. Appl., 189, 2, 442-461 (1995) · Zbl 0821.65073
[18] Goedbloed, J. P.; Poedts, S., Principles of Magnetohydrodynamics: with Applications to Laboratory and Astrophysical Plasmas (2004), Cambridge University Press
[19] Grandgirard, V., Modélisation de l’équilibre d’un plasma de tokamak (1999), Université de Franche-Comté, PhD thesis
[20] Heumann, H.; Blum, J.; Boulbe, C.; Faugeras, B.; Selig, G.; Ané, J.-M.; Brémond, S.; Grangirard, V.; Hertout, P.; Nardon, E., Quasi-static free-boundary equilibrium of toroidal plasma with CEDRES++: Computational methods and applications, J. Plasma Phys., 81, 3 (2015)
[21] Hinton, F. L.; Hazeltine, R. D., Theory of plasma transport in toroidal confinement systems, Rev. Mod. Phys., 48, 239-308 (April 1976)
[22] Hiptmair, R., Coupling of finite elements and boundary elements in electromagnetic scattering, SIAM J. Numer. Anal., 41, 3, 919-944 (2003) · Zbl 1049.78021
[23] Hsiao, G. C.; Zhang, S., Optimal order multigrid methods for solving exterior boundary value problems, SIAM J. Numer. Anal., 31, 3, 680-694 (1994) · Zbl 0805.65113
[24] Itagaki, M.; Fukunaga, T., Boundary element modelling to solve the Grad-Shafranov equation as an axisymmetric problem, Eng. Anal. Bound. Elem., 30, 9, 746-757 (2006) · Zbl 1195.76284
[25] Itagaki, M.; Kamisawada, J.; Oikawa, S., Boundary-only integral equation approach based on polynomial expansion of plasma current profile to solve the Grad-Shafranov equation, Nucl. Fusion, 44, 3, 427 (2004)
[26] Jackson, J. D., Classical Electrodynamics (1975), Wiley · Zbl 0997.78500
[27] Jardin, S. C., Computational Methods in Plasma Physics (2010), CRC Press/Taylor & Francis: CRC Press/Taylor & Francis Boca Raton, FL · Zbl 1198.76002
[28] Johnson, C.; Nedelec, J. C., On the coupling of boundary integral and finite element methods, Math. Comput., 35, 152, 1063-1079 (1980) · Zbl 0451.65083
[29] Koko, J., Vectorized Matlab codes for linear two-dimensional elasticity, Sci. Program., 15, 3, 157-172 (August 2007)
[30] Lackner, K., Computation of ideal MHD equilibria, Comput. Phys. Commun., 12, 1, 33-44 (1976)
[31] Lao, L. L.; Ferron, J. R.; Geoebner, R. J.; Howl, W.; John, H. E.St.; Strait, E. J.; Taylor, T. S., Equilibrium analysis of current profiles in Tokamaks, Nucl. Fusion, 30, 6, 1035 (1990)
[32] Mc Carthy, P. J.; Martin, P.; Schneider, W., The CLISTE Interpretive Equilibrium Code (1999), Max-Planck-Institut fur Plasmaphysik, Technical IPP report 5/85
[33] Moret, J.-M.; Duval, B. P.; Le, H. B.; Coda, S.; Felici, F.; Reimerdes, H., Tokamak equilibrium reconstruction code LIUQE and its real time implementation, Fusion Eng. Des., 91, 1-15 (2015)
[34] Sauter, Stefan A.; Schwab, Christoph, Boundary Element Methods, Springer Ser. Comput. Math., vol. 39 (2011), Springer-Verlag: Springer-Verlag Berlin, Translated and expanded from the 2004 German original · Zbl 1215.65183
[35] Sayas, F.-J., The validity of Johnson-Nedelec BEM-FEM coupling on polygonal interfaces, SIAM Rev., 55, 1, 131-146 (2013) · Zbl 1270.65070
[36] Steinbach, O., Numerical Approximation Methods for Elliptic Boundary Value ProblemsFinite and Boundary Elements (2008), Springer: Springer New York, Translated from the 2003 German original
[37] Stephan, E. P., Coupling of finite elements and boundary elements for some nonlinear interface problems, Comput. Methods Appl. Mech. Eng., 101, 1-3, 61-72 (1992) · Zbl 0778.73076
[38] Takeda, T.; Tokuda, S., Computation of MHD equilibrium of tokamak plasma, J. Comput. Phys., 93, 1, 1-107 (1991) · Zbl 0716.76086
[39] von Hagenow, K.; Lackner, K., Computation of axisymmetric MHD equilibria, (7th Conf. on Numerical Simulation of Plasmas. 7th Conf. on Numerical Simulation of Plasmas, New York (1975)), 140
[40] Wesson, J., Tokamaks, Int. Ser. Monogr. Phys. (2004), Oxford University Press · Zbl 1111.82054
[41] Zhao, K.; Vouvakis, M. N.; Lee, J.-F., Solving electromagnetic problems using a novel symmetric FEM-BEM approach, IEEE Trans. Magn., 42, 4, 583-586 (April 2006)
[42] Zienkiewicz, O. C.; Kelly, D. W.; Bettess, P., Marriage à la mode - the best of both worlds (finite elements and boundary integrals), (Glowinski, R.; Rodin, E. Y.; Zienkiewicz, O. C., Energy Methods in Finite Element Analysis (1979), Wiley: Wiley Chichester, UK), 81-107 · Zbl 0418.73065
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.