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Solution of Grad–Shafranov equation by the method of fundamental solutions

Published online by Cambridge University Press:  19 February 2014

D. Nath  and
M. S. Kalra
D. Nath*
Affiliation:
Nuclear Engineering and Technology Program, Indian Institute of Technology Kanpur, Kanpur 208 016, UP, India
M. S. Kalra
Affiliation:
Nuclear Engineering and Technology Program, Indian Institute of Technology Kanpur, Kanpur 208 016, UP, India
*
†Email address for correspondence: [email protected]
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Abstract

In this paper we have used the Method of Fundamental Solutions (MFS) to solve the Grad–Shafranov (GS) equation for the axisymmetric equilibria of tokamak plasmas with monomial sources. These monomials are the individual terms appearing on the right-hand side of the GS equation if one expands the nonlinear terms into polynomials. Unlike the Boundary Element Method (BEM), the MFS does not involve any singular integrals and is a meshless boundary-alone method. Its basic idea is to create a fictitious boundary around the actual physical boundary of the computational domain. This automatically removes the involvement of singular integrals. The results obtained by the MFS match well with the earlier results obtained using the BEM. The method is also applied to Solov'ev profiles and it is found that the results are in good agreement with analytical results.

Type
Papers
Information
Journal of Plasma Physics , Volume 80 , Issue 3 , June 2014 , pp. 477 - 494
Copyright
Copyright © Cambridge University Press 2014 

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References

REFERENCES

Atanasiu, C. V., Gunter, S. and Miron, I. G. 2004 Analytical solutions of Grad–Shafranov equation. Phys. Plasmas 11 (7), 3510–3518.Google Scholar
Aydin, S. H. and Tezer-Sezgin, M. 2008 Numerical solution of Grad–Shafranov equation for the distribution of magnetic flux in nuclear fusion devices. Turkish J. Eng. Env. Sci. 32 (5), 265–275.Google Scholar
Bandyopadhyay, I., Deshpande, S. P. and Chaturvedi, S. 2001 Design analysis of plasma position control in SST1. Fusion Eng. Des. 54 (2), 151–166.Google Scholar
Batchelor, D. A.et al. 2007 Ch. III.2: simulation of fusion plasmas: current status and future direction. Plasma Sci. Technol. 9 (3), 312–387.Google Scholar
Cerfon, A. J. and Freidberg, J. P. 2010 One size fits all analytic solutions to the Grad–Shafranov equation. Phys. Plasmas 17 (3), 032502.Google Scholar
Chen, C. S., Karageorghis, A. and Smyrlis, Y. S. 2008 (Eds.) The Method of Fundamental Solutions – A Meshless Method. Atlanta, GA: Dynamic.Google Scholar
Chen, W., Shen, L. J., Shen, Z. J. and Yuan, G. W. 2005 Boundary knot method for poisson equations. Eng. Anal. Bound. Elem. 29 (8), 756–760.Google Scholar
Chen, J. T., Wu, C. S., Lee, Y. T. and Chen, K. H. 2007 On the equivalence of the Trefftz method and method of fundamental solutions for Laplace and biharmonic equations. Compt. Math. Appl. 53 (6), 851–879.Google Scholar
Freidberg, J. P. 1982 Ideal Magnetohydrodynamics. New York, NY: Plenum Press.Google Scholar
Goedbloed, J. P., Keppens, R. and Poedts, S. 2010 Principles of Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas. Cambridge, UK: Cambridge University Press.Google Scholar
Grad, H. and Rubin, H. 1958 Hydromagnetic equilibria and force-Ffree fields. In:Proceedings of the Second United Nations Conference on the Peaceful Uses of Atomic Energy, United Nations, Geneva, Vol. 31, pp. 190–197.Google Scholar
Hender, T. C.et al. 2007 Chapter 3: MHD stability, Operational limits and disruptions. Nucl. Fusion 47 (6), S128–202.Google Scholar
Itagaki, M. and Fukunaga, T. 2006 Boundary element modeling to solve the Grad–Shafranov equation as an axisymmetric problem. Eng. Anal. Bound. Elem. 30 (9), 746–757.Google Scholar
Itagaki, M., Kamisawada, J. and Oikawa, S. 2004 Boundary-only integral equation approach based on polynomial expansion of plasma current profile to solve the Grad-Shafranov equation. Nucl. Fusion 44 (7), 427–437.Google Scholar
ITER Physics Expert Group on Confinement and Transport, ITER Physics Expert Group on Confinement Modeling and Database, ITER Physics Basis Editors. 1999 Chapter 2: plasma confinement and transport. Nucl. Fusion 39 (12), 2175–2249.Google Scholar
ITER Physics Basis Editors, ITER Physics Expert Group Chairs and Co-Chairs, ITER Joint Central Team and Physics Integration Unit. 1999 Chapter 1: overview and summary. Nucl. Fusion 39 (12), 2137–2174.Google Scholar
Jardin, S. C. 2010 Computational Methods in Plasma Physics. Boca Raton, FL: CRC Press.Google Scholar
Karageorghis, A. and Fairweather, G. 1989 The method of fundamental solutions for the solution of nonlinear plane potential problems. IMA J. Numer. Anal. 9 (2), 231–242.CrossRefGoogle Scholar
Kupradze, V. D. 1964 A method for the approximate solution of limiting problems in mathematical physics. USSR Comput. Math. Math. Phys 4 (6), 199–205.Google Scholar
Kurihara, K. 1993 Tokamak plasma shape identification based on boundary integral equations and real-time shape visualization system. Nucl. Fusion 33 (3), 399–412.Google Scholar
Kurihara, K. 2000 A new shape reproduction method based on the Cauchy-condition surface for real-time tokamak reactor control. Fusion Eng. Des. 51–52, 1049–1057. doi:10.1016/S0920-3796(00)00174-5.Google Scholar
Leuer, J. A., Schaffer, M. J., Parks, P. B. and Brown, M. R. 2001 Calculation of free boundary SSX doublet equilibria using the finite element method. In Proceedings of the 43rd Annual Meeting of the APS, Long Beach, CA, GP1.079, RP1.008.Google Scholar
Ling, K. M and Jardin, S. C. 1985 The Princeton spectral equilibrium code – PSEC. J. Comput. Phys. 58 (3), 300–335.Google Scholar
Partridge, P. W. and Sensale, B. 2000 The method of fundamental solutions with dual reciprocity for diffusion and diffusion-convection using sub-domains. Eng. Anal. Bound. Elem. 24 (9), 633–641.Google Scholar
Pataki, A., Cerfon, A. J., Freidberg, J. P., Greengard, L. and ONeil, M. 2013 A fast, high-order solver for the Grad–Shafranov equation. J. Comput. Phys. 243, 28–45.Google Scholar
Press, W. H., Teukolsky, S. A., Vellerling, W. T. and Flannery, B. P. 1992 Numerical Recipes in Fortran 77. Cambridge, UK: Cambridge University Press.Google Scholar
Rampp, M., Preuss, R., Fischer, R., Hallatschek, K. and Giannone, L. 2012 A parallel Grad–Shafranov solver for real-time control of tokamak plasmas. Fusion Sci. Technol. 62 (3), 409–418.Google Scholar
Salgado-Ibarra, E. A. 2011 Boundary Element Method (BEM) and Method of Fundamental Solutions (MFS) for the boundary value problems of the 2-D Laplace's equation. Master's thesis, University of Nevada, Las Vegas, NV.Google Scholar
Shafranov, V. D. 1958 On magnetohydrodynamical equilibrium configurations. Soviet J. Exp. Theor. Phys. 6, 545–554.Google Scholar
Smyrlis, Y. S. and Karageorghis, A. 2001 Some aspects of the method of fundamental solutions for certain harmonic problems. J. Sci. Comput. 16 (3)341–371.Google Scholar
Smyrlis, Y. S. and Karageorghis, A. 2004 Numerical analysis of the MFS for certain harmonic problems. M2AN Math. Model. Numer. Anal. 38 (3), 495–517.Google Scholar
Solov'ev, L. S. 1968 The theory of hydromagnetic stability of toroidal plasma configurations. Soviet J. Exp. Theor. Phys. 26 (2), 400–407.Google Scholar
Suárez, P. U. 2010 Galerkin Boundary Integral Analysis of the Grad–Shafranov Equation. Saarbrücken, Germany: Lap Lambert.Google Scholar
Takeda, T. and Tokuda, S. 1991 Computation of MHD equilibrium of tokamak plasma. J. Comput. Phys. 93 (1), 1–107.Google Scholar
Thompson, J. F., Soni, B. K. and Weatherill, N. P. 1998 Handbook of Grid Generation. Boca Raton, FL: CRC Press.Google Scholar
Troyon, F., Gruber, R., Saurenmann, H., Semenzato, S. and Succi, S. 1984 MHD-limits to plasma confinement. Plasma Phys. Control. Fusion 26(1A), 209–215.Google Scholar
Tsangaris, Th., Smyrlis, Y. S. and Karageorghis, A. 2006 Numerical analysis of the method of fundamental solutions for harmonic problems in annular domains. Numer. Methods Partial Differ. Equ. 22 (3), 507–539.Google Scholar
Tsuchimoto, M., Honma, T., Yatsu, S. and Kaji, I. 1987 Nonlinear analysis of MHD equilibria of toroidal plasmas using boundary element method. Eng. Anal. 4 (4), 221–227.Google Scholar
Tsuchimoto, M., Honma, T., Yatsu, S., Kasahara, T. and Kaji, I. 1988 An analysis of nonlinear MHD equilibria of compact tori by using boundary element method. IEEE Trans. Magn. 24 (1), 248–251.Google Scholar
Wei, T., Hon, Y. C. and Ling, L. 2007 Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators. Eng. Anal. Bound. Elem. 31 (4), 373–385.Google Scholar
Zhou, D. and Wei, T. 2008 The method of fundamental solutions for solving a Cauchy problem of Laplaces equation in a multi-connected domain. Inverse Probl. Sci. Eng. 16 (3), 389–411.Google Scholar