Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-28T14:52:11.218Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasi-static free-boundary equilibrium of toroidal plasma with CEDRES++: Computational methods and applications

Part of: Focus on Fusion

Published online by Cambridge University Press:  13 January 2015

H. Heumann ,
J. Blum ,
C. Boulbe ,
B. Faugeras ,
G. Selig ,
J.-M. Ané ,
S. Brémond ,
V. Grandgirard ,
P. Hertout  and
E. Nardon
H. Heumann*
Affiliation:
TEAM CASTOR, INRIA, 2004 Route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex, France and Laboratoire J.A. Dieudonné, UMR 7351, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France
J. Blum
Affiliation:
TEAM CASTOR, INRIA, 2004 Route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex, France and Laboratoire J.A. Dieudonné, UMR 7351, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France
C. Boulbe
Affiliation:
TEAM CASTOR, INRIA, 2004 Route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex, France and Laboratoire J.A. Dieudonné, UMR 7351, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France
B. Faugeras
Affiliation:
TEAM CASTOR, INRIA, 2004 Route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex, France and Laboratoire J.A. Dieudonné, UMR 7351, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France
G. Selig
Affiliation:
TEAM CASTOR, INRIA, 2004 Route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex, France and Laboratoire J.A. Dieudonné, UMR 7351, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France
J.-M. Ané
Affiliation:
CEA, IRFM, F-13108 Saint-Paul-lez-Durance, France
S. Brémond
Affiliation:
CEA, IRFM, F-13108 Saint-Paul-lez-Durance, France
V. Grandgirard
Affiliation:
CEA, IRFM, F-13108 Saint-Paul-lez-Durance, France
P. Hertout
Affiliation:
CEA, IRFM, F-13108 Saint-Paul-lez-Durance, France
E. Nardon
Affiliation:
CEA, IRFM, F-13108 Saint-Paul-lez-Durance, France
*
éEmail address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a comprehensive survey of various computational methods in CEDRES++ (Couplage Equilibre Diffusion Résistive pour l'Etude des Scénarios) for finding equilibria of toroidal plasma. Our focus is on free-boundary plasma equilibria, where either poloidal field coil currents or the temporal evolution of voltages in poloidal field circuit systems are given data. Centered around a piecewise linear finite element representation of the poloidal flux map, our approach allows in large parts the use of established numerical schemes. The coupling of a finite element method and a boundary element method gives consistent numerical solutions for equilibrium problems in unbounded domains. We formulate a new Newton method for the discretized nonlinear problem to tackle the various nonlinearities, including the free plasma boundary. The Newton method guarantees fast convergence and is the main building block for the inverse equilibrium problems that we can handle in CEDRES++ as well. The inverse problems aim at finding either poloidal field coil currents that ensure a desired shape and position of the plasma or at finding the evolution of the voltages in the poloidal field circuit systems that ensure a prescribed evolution of the plasma shape and position. We provide equilibrium simulations for the tokamaks ITER and WEST to illustrate the performance of CEDRES++ and its application areas.

Type
Research Article
Information
Journal of Plasma Physics , Volume 81 , Issue 3 , June 2015 , 905810301
Check for updates
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Albanese, R., Blum, J. and De Barbieri, O. 1986 On the solution of the magnetic flux equation in an infinite domain. In: EPS. 8th Europhysics Conf. on Computing in Plasma Physics, European Physical Society, Mulhouse, France, pp. 41–44.Google Scholar
Albanese, R., Blum, J. and De Barbieri, O. 1987 Numerical studies of the Next European Torus via the PROTEUS code. In: 12th Conf. on Numerical Simulation of Plasmas, San Francisco.Google Scholar
Albanese, R. and Villone, F. 1998 The linearized CREATE-L plasma response model for the control of current, position and shape in tokamaks. Nucl. Fusion 38 (5), 723.CrossRefGoogle Scholar
Ané, J. M., Grandgirard, V., Albajar, F. and Johner, J. 2000 Design of next step tokamak: consistent analysis of plasma performance flux composition and poloidal field system. In: 18th IAEA Fusion Energy Conf., Sorrento, http://www.iaea.org/inis/collection/NCLCollectionStore/Public/33/029/33029055.pdf.Google Scholar
Ariola, M. and Pironti, A. 2008 Magnetic Control of Tokamak Plasmas. London: Springer.Google Scholar
Artaud, J. F.et al. 2010 The CRONOS suite of codes for integrated tokamak modelling. Nucl. Fusion 50 (4), 043 001.CrossRefGoogle Scholar
Bielak, J. and MacCamy, R. C. 1991 Symmetric finite element and boundary integral coupling methods for fluid-solid interaction. Q. Appl. Math. 49 (1), 107–119.CrossRefGoogle Scholar
Blum, J. 1989 Numerical Simulation and Optimal Control in Plasma Physics: With Applications to Tokamaks. Paris: Wiley/Gauthier-Villars.Google Scholar
Blum, J., Boulbe, C. and Faugeras, B. 2012 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.CrossRefGoogle Scholar
Blum, J. and Heumann, H. 2014 Optimal control for quasi-static evolution of plasma equilibrium in tokamaks. Technical Report, INRIA, Sophia Antipolis, HAL Id: hal-00988045, version 1.Google Scholar
Blum, J., Le Foll, J. and Thooris, B. 1981 The self-consistent equilibrium and diffusion code SCED. Comput. Phys. Commun. 24, 235–254.CrossRefGoogle Scholar
, Bucalossi, et al. 2011 Feasibility study of an actively cooled tungsten divertor in tore supra for ITER technology testing. Fusion Eng. Des. 86 (6–8), 684–688.CrossRefGoogle Scholar
Chen, G. and Zhou, J. 1992 Boundary Element Methods (Computational Mathematics and Applications). London: Academic Press, Ltd.Google Scholar
Ciarlet, P. G. 1978 The Finite Element Method for Elliptic Problems. Amsterdam: North-Holland Publishing Co.Google Scholar
Cooper, W. A., Hirshman, S. P., Merkel, P., Graves, J. P., Kisslinger, J., Wobig, H. F. G., Narushima, Y., Okamura, S. and Watanabe, K. Y. 2009 Three-dimensional anisotropic pressure free boundary equilibria. Comput. Phys. Commun. 180 (9), 1524–1533.CrossRefGoogle Scholar
Costabel, M. 1987 Principles of boundary element methods. In: Finite Elements in Physics (Lausanne, 1986), Amsterdam: North-Holland, pp. 243–274.Google Scholar
Costabel, M. and Stephan, E. P. 1990 Coupling of finite and boundary element methods for an elastoplastic interface problem. SIAM J. Numer. Anal. 27 (5), 1212–1226.CrossRefGoogle Scholar
Coster, D. P., Basiuk, V., Pereverzev, G., Kalupin, D., Zagórksi, R., Stankiewicz, R., Huynh, P. and Imbeaux, F. 2010 The european transport solver. IEEE Trans. Plasma Sci. 38 (9), 2085–2092.CrossRefGoogle Scholar
Davis, T. A. 2011 Suitesparse: a suite of sparse matrix software. http://faculty.cse.tamu.edu/davis/suitesparse.html.Google Scholar
Degtyarev, L. M. and Drozdov, V. V. 1985 An inverse variable technique in the MHD-equilibrium problem. Comput. Phys. Rep. 2 (7), 341–387.CrossRefGoogle Scholar
Degtyarev, L. M. and Drozdov, V. V. 1991 Adaptive mesh computation of magnetic hydrodynamic equilibrium. Int. J. Mod. Phys. C 02 (01), 30–38.CrossRefGoogle Scholar
Delfour, M. C. and Zolésio, J.-P. 2011 Shapes and Geometries, 2nd edn., Advances in Design and Control, Vol. 22. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM).CrossRefGoogle Scholar
DeLucia, J., Jardin, S. C. and Todd, A. M. M. 1980 An iterative metric method for solving the inverse tokamak equilibrium problem. J. Comput. Phys. 37 (2), 183–204.CrossRefGoogle Scholar
Falchetto, G. L.et al. 2014 The European Integrated Tokamak Modelling (ITM) effort: achievements and first physics results. Nucl. Fusion 54 (4), 043 018.CrossRefGoogle Scholar
Feistauer, M. and Sobotikova, V. 1990 Finite element approximation of nonlinear elliptic problems with discontinuous coefficients. ESAIM: Math. Modelling Numer. Anal. - Modlisation Mathématique et Analyse Numrique 24 (4), 457–500.CrossRefGoogle Scholar
Feneberg, W. and Lackner, K. 1973 Multipole tokamak equilibria. Nucl. Fusion 13 (4), 549.CrossRefGoogle Scholar
Fitzgerald, M., Appel, L. C. and Hole, M. J. 2013 EFIT tokamak equilibria with toroidal flow and anisotropic pressure using the two-temperature guiding-centre plasma. Nucl. Fusion 53 (11), 113 040.CrossRefGoogle Scholar
Freidberg, J. P. 1987 Ideal Magnetohydrodynamics. US: Plenum.CrossRefGoogle Scholar
Gatica, G. N. and Hsiao, G. C. 1995 The uncoupling of boundary integral and finite element methods for nonlinear boundary value problems. J. Math. Anal. Appl. 189 (2), 442–461.CrossRefGoogle Scholar
Glowinski, R. and Marrocco, A. 1974 Analyse numérique du champ magnétique d'un alternateur par éléments finis et sur-relaxation ponctuelle non linéaire. Comput. Methods Appl. Mech. Eng. 3 (1), 55–85.CrossRefGoogle Scholar
Goedbloed, J. P., Keppens, R. and Poedts, S. 2010 Advanced Magnetohydrodynamics: with Applications to Laboratory and Astrophysical Plasmas. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Goedbloed, J. P. and Lifschitz, A. 1997 Stationary symmetric magnetohydrodynamic flows. Phys. Plasmas (1994–present) 4 (10), 3544–3564.CrossRefGoogle Scholar
Goedbloed, J. P. and Poedts, S. 2004 Principles of Magnetohydrodynamics: with Applications to Laboratory and Astrophysical Plasmas. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Grad, H. 1967 Toroidal containment of a plasma. Phys. Fluids (1958–1988) 10 (1), 137–154.CrossRefGoogle Scholar
Grad, H. and Rubin, H. 1958 Hydromagnetic equilibria and force-free fields. In: Proc. 2nd UN Conf. on the Peaceful Uses of Atomic Energy, United Nations Publications, Geneva, Vol. 31, p. 190.Google Scholar
Grandgirard, V. 1999 Modélisation de l'équilibre d'un plasma de tokamak. PhD thesis, l'Université de Franche-Comté.Google Scholar
Gruber, R., Iacono, R. and Troyon, F. 1987 Computation of MHD equilibria by a quasi-inverse finite hybrid element approach. J. Comput. Phys. 73 (1), 168–182.CrossRefGoogle Scholar
Guazzotto, L., Betti, R., Manickam, J. and Kaye, S. 2004 Numerical study of tokamak equilibria with arbitrary flow. Phys. Plasmas (1994–present) 11 (2), 604–614.CrossRefGoogle Scholar
Helton, F. J. and Wang, T. S. 1978 MHD equilibrium in non-circular tokamaks with field-shaping coil systems. Nucl. Fusion 18 (11), 1523.CrossRefGoogle Scholar
Hertout, P., Boulbe, C., Nardon, E., Blum, J., Bremond, S., Bucalossi, J., Faugeras, B., Grandgirard, V. and Moreau, P. 2011 The cedres++ equilibrium code and its application to ITER, JT-60SA and Tore Supra. Fusion Eng. Des. 86, 1045–1048.CrossRefGoogle Scholar
Hinton, F. L. and Hazeltine, R. D. 1976 Theory of plasma transport in toroidal confinement systems. Rev. Mod. Phys. 48, 239–308.CrossRefGoogle Scholar
Hiptmair, R. 2003 Coupling of finite elements and boundary elements in electromagnetic scattering. SIAM J. Numer. Anal. 41 (3), 919–944.CrossRefGoogle Scholar
Hirshman, S. P. and Betancourt, O. 1991 Preconditioned descent algorithm for rapid calculations of magnetohydrodynamic equilibria. J. Comput. Phys. 96 (1), 99–109.CrossRefGoogle Scholar
Hirshman, S. P. and Jardin, S. C. 1979 Two-dimensional transport of tokamak plasmas. Phys. Fluids 22 (4), 731–742.CrossRefGoogle Scholar
Hofmann, F. and Tonetti, G. 1988 Tokamak equilibrium reconstruction using Faraday rotation measurements. Nucl. Fusion 28 (10), 1871.CrossRefGoogle Scholar
Huysmans, G. T. A. and Czarny, O. 2007 MHD stability in X-point geometry: simulation of ELMs. Nucl. Fusion 47 (7), 659.CrossRefGoogle Scholar
Huysmans, G. T. A., Goedbloed, J. P. and Kerner, W. 1991 Isoparametric bicubic Hermite elements for solution of the Grad–Shafranov equation. In: Proc. CP90 Conf. on Comp. Phys., Vol. 2, pp. 371–376.Google Scholar
ITM 2013 Integrated tokamak modelling. http://portal.efda-itm.eu/, integrated Tokamak Modelling.Google Scholar
Jardin, S. C. 2010 Computational Methods in Plasma Physics. Boca Raton, Florida: CRC Press/Taylor and Francis.CrossRefGoogle Scholar
Jardin, S. C., Pomphrey, N. and DeLucia, J. 1986 Dynamic modeling of transport and positional control of tokamaks. J. Comput. Phys. 66, 481–507.CrossRefGoogle Scholar
Johnson, J. L.et al. 1979 Numerical determination of axisymmetric toroidal magnetohydrodynamic equilibria. J. Comput. Phys. 32 (2), 212–234.CrossRefGoogle Scholar
Lackner, K. 1976 Computation of ideal MHD equilibria. Comput. Phys. Commun. 12 (1), 33–44.CrossRefGoogle Scholar
Lao, L. L., Ferron, J. R., Geoebner, R. J., Howl, W., St. John, H. E., Strait, E. J. and Taylor, T. S. 1990 Equilibrium analysis of current profiles in Tokamaks. Nucl. Fusion 30 (6), 1035.CrossRefGoogle Scholar
Lao, L. L., Hirshman, S. P. and Wieland, R. M. 1981 Variational moment solutions to the Grad–Shafranov equation. Phys. Fluids 24 (8), 1431–1440.CrossRefGoogle Scholar
Lao, L. L., John, H. St., Stambaugh, R. D., Kellman, A. G. and Pfeiffer, W. 1985 Reconstruction of current profile parameters and plasma shapes in tokamaks. Nucl. Fusion 25 (11), 1611.CrossRefGoogle Scholar
Li, J., Melenk, J. M., Wohlmuth, B. and Zou, J. 2010 Optimal a priori estimates for higher order finite elements for elliptic interface problems. Appl. Numer. Math. 60 (1–2), 19–37.CrossRefGoogle Scholar
Ling, K. M. and Jardin, S. C. 1985 The Princeton spectral equilibrium code: PSEC. J. Comput. Phys. 58 (3), 300–335.CrossRefGoogle Scholar
Lüst, R. and Schlüter, A. 1957 Axialsymmetrische magnetohydrodynamische Gleichgewichtskonfigurationen. Z. Naturforsch. A12, 850–854.CrossRefGoogle Scholar
Luxon, J. L. and Brown, B. B. 1982 Magnetic analysis of non-circular cross-section tokamaks. Nucl. Fusion 22 (6), 813.CrossRefGoogle Scholar
Maschke, E. K. and Perrin, H. J. 1984 An analytic solution of the stationary MHD equations for a rotating toroidal plasma. Phys. Lett. A 102 (3), 106–108.CrossRefGoogle Scholar
Mc Carthy, P. J., Martin, P. and Schneider, W. 1999 The CLISTE interpretive equilibrium code. Technical Report IPP Report 5/85. Max-Planck-Institut fur Plasmaphysik.Google Scholar
Murat, F. and Simon, J. 1976 Sur le contrôle par un domaine géométrique. Technical Report 76015. Laboratoire d'Analyse Numérique, Université de Paris 6.Google Scholar
Nédélec, J.-C. 2001 Acoustic and Electromagnetic Equations Applied Mathematical Sciences, 144). New York: Springer-Verlag.CrossRefGoogle Scholar
Nocedal, J. and Wright, S. J. 2006 Numerical Optimization, 2nd edn.Springer Series in Operations Research and Financial Engineering. New York: Springer.Google Scholar
Parail, V.et al. 2013 Self-consistent simulation of plasma scenarios for iter using a combination of 1.5d transport codes and free-boundary equilibrium codes. Nucl. Fusion 53 (11), 113 002.CrossRefGoogle Scholar
Park, W., Belova, E. V., Fu, G. Y., Tang, X. Z., Strauss, H. R. and Sugiyama, L. E. 1999 Plasma simulation studies using multilevel physics models. Phys. Plasmas (1994-present) 6 (5), 1796–1803.CrossRefGoogle Scholar
Pechstein, C. and Jüttler, B. 2006 Monotonicity-preserving interproximation of B-H-curves. J. Comput. Appl. Math. 196 (1), 45–57.CrossRefGoogle Scholar
Pustovitov, V. D. 2010 Anisotropic pressure effects on plasma equilibrium in toroidal systems. Plasma Phys. Control. Fusion 52 (6), 065 001.CrossRefGoogle Scholar
Renard, Y. and Pommier, J. 2014 GetFEM++, an open-source finite element library. http://download.gna.org/getfem/html/homepage/index.html.Google Scholar
Schwab, C. 2004 p- and hp- Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. Oxford: Clarendon Press.Google Scholar
Shafranov, V. D. 1958 On magnetohydrodynamical equilibrium configurations. Sov. J. Exp. Theor. Phys. 6, 545.Google Scholar
Shewchuk, J. R. 1996 Triangle: Engineering a 2D quality mesh generator and delaunay triangulator. In: Applied Computational Geometry: Towards Geometric Engineering, Vol. 1148 (ed. Lin, M. C. and Manocha, D.), Lecture Notes in Computer Science, Springer-Verlag, from the First ACM Workshop on Applied Computational Geometry, Springer, Berlin, pp. 203–222.CrossRefGoogle Scholar
Stephan, E. P. 1992 Coupling of finite elements and boundary elements for some nonlinear interface problems. Comput. Methods Appl. Mech. Eng. 101 (1–3), 61–72.CrossRefGoogle Scholar
Turkington, B., Lifschitz, A., Eydeland, A. and Spruck, J. 1993 Multiconstrained variational problems in magnetohydrodynamics: equilibrium and slow evolution. J. Comput. Phys. 106 (2), 269–285.CrossRefGoogle Scholar
Wesson, J. and Campbell, D. J. 2004 Tokamaks. The International Series of Monographs in Physics, 3rd edn. Vol. 118, Oxford: Clarendon Press.Google Scholar
Zhao, K., Vouvakis, M. N. and Lee, J.-F. 2006 Solving electromagnetic problems using a novel symmetric fem-bem approach. IEEE Trans. Magn. 42 (4), 583–586.CrossRefGoogle Scholar
Zwingmann, W., Eriksson, L.-G. and Stubberfield, P. 2001 Equilibrium analysis of tokamak discharges with anisotropic pressure. Plasma Phys. Control. Fusion 43 (11), 1441.CrossRefGoogle Scholar