Maxwell and Eddy Current Equations

  • Rachid Touzani
  • Jacques Rappaz
Chapter
Part of the Scientific Computation book series (SCIENTCOMP)

Abstract

We start by presenting the full system of Maxwell equations stated in the whole space. This system describes propagation of electric and magnetic fields in the space. Using low frequency approximation, we present the system of eddy current equations that neglects displacement currents. We next define the vector potential associated to the magnetic induction and prove its uniqueness provided an appropriate gauge is defined. We then treat static cases and defined related problems. Finally, we state the equations for the quasi-harmonic regime.

Keywords

Maxwell Equation Magnetic Permeability Electric Current Density Green Formula Source Current Density 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Abeele, D.V., Degrez, G.: Efficient computational model for inductive plasma flows. AIAA J. 38(2), 234–242 (2000)Google Scholar
  2. 2.
    Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. U.S. Government Printing Office, Washington (1964)MATHGoogle Scholar
  3. 3.
    Adams, R.: Sobolev Spaces. Academic, New York (1975)MATHGoogle Scholar
  4. 4.
    Ainsworth, M., McLean, W., Tran, T.: The conditioning of boundary element equations on locally refined meshes and preconditioning by diagonal scaling. SIAM J. Numer. Anal. 36, 1901–1932 (1999)MathSciNetMATHGoogle Scholar
  5. 5.
    Albanese, R., Rubinacci, G.: Finite element methods for the solution of 2-D eddy current problems. Adv. Imaging Electron Phys. 102, 1–86 (1998)Google Scholar
  6. 6.
    Albanese, R., Rubinacci, G., Tamburrino, A., Ventre, S., Villone, F.: A fast 3-D eddy current integral formulation. COMPEL 20(2), 317–331 (2001)MATHGoogle Scholar
  7. 7.
    Amiez, G., Gremaud, P.A.: On a numerical approach to stefan like problems. Numer. Math. 59, 71–89 (1991)MathSciNetMATHGoogle Scholar
  8. 8.
    Amirat, Y., Touzani, R.: Self–inductance coefficient for toroidal thin conductors. Nonlinear Anal. B 131, 233–240 (2001)MathSciNetGoogle Scholar
  9. 9.
    Amirat, Y., Touzani, R.: Asymptotic behavior of the inductance coefficient for thin conductors. Math. Models Methods Appl. Sci. 12(2), 273–289 (2002)MathSciNetMATHGoogle Scholar
  10. 10.
    Amirat, Y., Touzani, R.: A two-dimensional eddy current model using thin inductors. Asymptot. Anal. 58(3), 171–188 (2008)MathSciNetMATHGoogle Scholar
  11. 11.
    Amirat, Y., Touzani, R.: A singular perturbation problem in eddy current models (2013, submitted)Google Scholar
  12. 12.
    Ammari, H., Nédélec, J.C.: Propagation d’ondes électromagnétiques à basses fréquences. J. Math. Pures Appl. 77, 839–849 (1998)Google Scholar
  13. 13.
    Ammari, H., Buffa, A., Nédélec, J.C.: A justification of eddy currents model for the Maxwell equations. SIAM J. Appl. Math. 60(5), 1805–1823 (2000)MathSciNetMATHGoogle Scholar
  14. 14.
    Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three–dimensional non-smooth domains. Math. Methods Appl. Sci. 21, 823–864 (1998)MathSciNetMATHGoogle Scholar
  15. 15.
    Arnold, D.N., Wendland, W.L.: On the asymptotic convergence of collocation methods. Math. Comput. 41(164), 349–381 (1983)MathSciNetMATHGoogle Scholar
  16. 16.
    Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University, Cambridge (1967)MATHGoogle Scholar
  17. 17.
    Bay, F., Labbé, V., Favennec, Y., Chenot, J.L.: A numerical model for induction heating processes coupling electromagnetism and thermomechanics. Int. J. Numer. Method Eng. 58(6), 839–867 (2003)MATHGoogle Scholar
  18. 18.
    Belgacem, F.B., Fournié, M., Gmati, N., Jelassi, F.: On the Schwarz algorithm for the elliptic exterior boundary value problems. Model. Math. Anal. Numer. 39(4), 693–714 (2005)MATHGoogle Scholar
  19. 19.
    Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vazquez, J.: An L 1–theory of existence and uniqueness of solutions of nonlinear elliptic equations. Annali della Scuola Normale Superiore di Pisa – Classe di Scienze, Ser. 4 22(2), 241–273 (1995)Google Scholar
  20. 20.
    Bermúdez, A., Rodríguez, R., Salgado, P.: A finite element method with Lagrange multipliers for low-frequency harmonic Maxwell equations. SIAM J. Numer. Anal. 40(5), 1823–1849 (2002)MathSciNetMATHGoogle Scholar
  21. 21.
    Bermúdez, A., Muñiz, M.C., Salgado, P.: Asymptotic approximation and numerical simulation of electromagnetic casting. Metall. Trans. B 34(1), 83–91 (2003)Google Scholar
  22. 22.
    Bermúdez, A., Gómez, D., Muñiz, M.C., Salgado, P., Vázquez, R.: Numerical simulation of a thermo-electromagneto-hydrodynamic problem in an induction heating furnace. Appl. Numer. Math. 59(1), 2082–2104 (2009)MathSciNetMATHGoogle Scholar
  23. 23.
    Bernardi, C., Dauge, M., Maday, Y.: Spectral Methods for Axisymmetric Domains. Series in Applied Mathematics. Gauthier-Villars, Paris (1999)MATHGoogle Scholar
  24. 24.
    Bernardi, D., Colombo, V., Ghedini, E., Mentrelli, A.: Three-dimensional modeling of inductively coupled plasma torches. Pure Appl. Chem. 77(2), 359–372 (2005)Google Scholar
  25. 25.
    Bernardi, D., Colombo, V., Ghedini, E., Mentrelli, A., Trombetti, T.: 3-D numerical analysis of powder injection in inductively coupled plasma torches. IEEE Trans. Plasma Sci. 33(2), 424–425 (2005)Google Scholar
  26. 26.
    Besson, O., Bourgeois, J., Chevalier, P.A., Rappaz, J., Touzani, R.: Numerical modelling of electromagnetic casting processes. J. Comput. Phys. 92(2), 482–507 (1991)MathSciNetMATHGoogle Scholar
  27. 27.
    Biro, O.: Edge element formulations of eddy current problems. Comput. Methods Appl. Mech. Eng. 169, 391–405 (1999)MathSciNetMATHGoogle Scholar
  28. 28.
    Biro, O., Preis, K.: An edge finite element eddy current formulation using a reduced magnetic and a current vector potential. IEEE Trans. Magn. 36(5), 3128–3130 (2000)Google Scholar
  29. 29.
    Boccardo, L., Gallouët, T.: Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87(1), 149–169 (1989)MathSciNetMATHGoogle Scholar
  30. 30.
    Bodart, O., Boureau, A.V., Touzani, R.: Numerical investigation of optimal control of induction heating processes. Appl. Math. Model. 25, 697–712 (2001)MATHGoogle Scholar
  31. 31.
    Bonnans, J.F., Gilbert, J., Lemaréchal, C., Sagastizabal, C.: Numerical Optimization. Springer, New York (2006)MATHGoogle Scholar
  32. 32.
    Bossavit, A.: On the numerical analysis of eddy current problems. Comput. Methods Appl. Mech. Eng. 27, 303–318 (1981)MathSciNetMATHGoogle Scholar
  33. 33.
    Bossavit, A.: Two dual formulations of the 3-D eddy currents problem. COMPEL 4(2), 103–116 (1985)MathSciNetGoogle Scholar
  34. 34.
    Bossavit, A.: Computational Electromagnetism. Associated Press (1998)Google Scholar
  35. 35.
    Bossavit, A., Rodrigues, J.F.: On the electromagnetic induction heating problem in bounded domains. Adv. Math. Sci. Appl. 4(1), 79–92 (1994)MathSciNetMATHGoogle Scholar
  36. 36.
    Bossavit, A., Vérité, J.C.: The TRIFOU code: solving the 3-D eddy–current problem by using h as a state variable. IEEE Trans. Magn. (MAG–19) 6, 2465–2470 (1983)Google Scholar
  37. 37.
    Bouchon, F., Clain, S., Touzani, R.: Numerical solution of the free boundary Bernoulli problem using a level set formulation. Comput. Methods Appl. Mech. Eng. 194(36–38), 3934–3948 (2005)MathSciNetMATHGoogle Scholar
  38. 38.
    Braess, D.: Finite Elements, Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, Cambridge (2001)MATHGoogle Scholar
  39. 39.
    Brancher, J.P., Sero-Guilaume, O.: Sur l’équilibre des liquides magnétiques. Application à la magnétostatique. J. Mec. Theor. Appl. 2(2), 265–283 (1983)MATHGoogle Scholar
  40. 40.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (2002)MATHGoogle Scholar
  41. 41.
    Brezis, H.: Analyse Fonctionnelle. Masson, Paris (1983)MATHGoogle Scholar
  42. 42.
    Brooks, A.N., Hughes, T.J.R.: Streamline upwind/petrov-galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 32, 199–259 (1982)MathSciNetMATHGoogle Scholar
  43. 43.
    Buffa, A., Ciarlet, P. Jr.: On traces for functional spaces related to maxwell’s equations. Part I: an integration by parts formula in Lipschitz polyhedra. Math. Methods Appl. Sci. 24, 9–30 (2001)MathSciNetMATHGoogle Scholar
  44. 44.
    Buffa, A., Ciarlet, P. Jr.: On traces for functional spaces related to Maxwell’s equations. Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications. Math. Methods Appl. Sci 24, 31–48 (2001)MathSciNetMATHGoogle Scholar
  45. 45.
    Casado-Díaz, J., Rebollo, T.C., Girault, V., Mármol, M.G., Murat, F.: Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in L 1. Numer. Math. 105(3), 337–374 (2006)Google Scholar
  46. 46.
    Casas, E.: Pontryagin’s principle for state-constraint boundary control problems of seminlinear parabolic equations. SIAM J. Control Optim. 35, 1297–1327 (1997)MathSciNetMATHGoogle Scholar
  47. 47.
    Chaboudez, C., Clain, S., Glardon, R., Rappaz, J., Swierkosz, M., Touzani, R.: Numerical modelling in induction heating of long workpieces. IEEE Trans. Magn. 30(6), 5028–5037 (1994)Google Scholar
  48. 48.
    Chaboudez, C., Clain, S., Mari, D., Glardon, R., Swierkosz, M., Rappaz, J.: Numerical modelling in induction heating for axisymmetric geometries. IEEE Trans. Magn. 33(1), 739–745 (1997)Google Scholar
  49. 49.
    Chadebec, O., Colomb, J.L., Janet, F.: A review of magnetostatic moment method. IEEE Trans. Magn. 42(4), 515–520 (2006)Google Scholar
  50. 50.
    Ciarlet, P.G.: Finite element methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. I, pp. 209–485. North-Holland, Amsterdam (1991)Google Scholar
  51. 51.
    Ciarlet, P.J., Sonnendrücker, E.: A decomposition of the electromagnetic field – application to the Darwin model. Math. Models Methods Appl. Sci. 7(8), 1085–1120 (1997)MathSciNetMATHGoogle Scholar
  52. 52.
    Clain, S.: Analyse mathématique et numérique d’un modèle de chauffage par induction. Ph.D. thesis, École Polytechnique Fédérale de Lausanne (1994)Google Scholar
  53. 53.
    Clain, S., Touzani, R.: Solution of a two–dimensional stationary induction heating problem without boundedness of the coefficients. Model. Math. Anal. Numer. 31(7), 845–870 (1997)MathSciNetMATHGoogle Scholar
  54. 54.
    Clain, S., Touzani, R.: A two–dimensional stationary induction heating problems. Math. Methods Appl. Sci. 20, 759–766 (1997)MathSciNetMATHGoogle Scholar
  55. 55.
    Clain, S., Rappaz, J., Swierkosz, M., Touzani, R.: Numerical modelling of induction heating for 2-D geometries. Math. Models Methods Appl. Sci. 3(6), 805–822 (1993)MathSciNetMATHGoogle Scholar
  56. 56.
    Clain, S., Rochette, D., Touzani, R.: A multislope MUSCL method on unstructured meshes applied to compressible euler equations for swirling flows. J. Comput. Phys. 229, 4884–4906 (2010)MathSciNetMATHGoogle Scholar
  57. 57.
    Clain, S., Touzani, R., Silva, M.L.D., Vacher, D., André, P.: A contribution on the numerical simulation of ICP torches. In: Fifth European Conference on Computational Fluid Dynamics, ECCOMAS CFD, Lisbon (2010)Google Scholar
  58. 58.
    Costabel, M.: Symmetric methods for the coupling of finite elements and boundary elements. In: Brebbia, C., Wendland, W., Kuhn, G. (eds.) Boundary Elements IX, pp. 411–420. Springer, Berlin (1987)Google Scholar
  59. 59.
    Coulaud, O., Henrot, A.: Numerical approximation of free boundary problem arising in electromagnetic shaping. Tech. Rep., INRIA (1992)Google Scholar
  60. 60.
    Crouzeix, M.: Variational approach of a magnetic shaping problem. Eur. J. Mech., B/Fluids 10(5), 527–536 (1991)Google Scholar
  61. 61.
    Dacorogna, B.: Direct Methods in the Calculus of Variations. Applied Mathematical Sciences. Springer, Berlin (1989)MATHGoogle Scholar
  62. 62.
    Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology. Springer, Berlin (1990)Google Scholar
  63. 63.
    Descloux, J.: Stability of the solutions of the bidimensional magnetic shaping problem in absence of surface tension. Eur. J. Mech. B/Fluids 10(5), 513–526 (1991)MathSciNetMATHGoogle Scholar
  64. 64.
    Descloux, J.: A stability result for the magnetic shaping problem. Z. Angew. Math. Phys. 45, 544–555 (1994)MathSciNetGoogle Scholar
  65. 65.
    Descloux, J., Flück, M., Rappaz, J.: A problem of magnetostatics related to thin plates. Model. Math. Anal. Numer. 32, 859–876 (1998)MATHGoogle Scholar
  66. 66.
    Descloux, J., Flück, M., Romerio, M.: A modelling of the stability of aluminium electrolytic cells. In: Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, vol. XIII (Paris, 1994/1996). Volume 391 of Pitman Research Notes in Mathematics Series, pp. 117–133. Longman, Harlow (1998)Google Scholar
  67. 67.
    Descloux, J., Flück, M., Rappaz, J.: Modelling and mathematical results arising from ferromagnetic problems. Sci. China Math. 55(5), 1053–1067 (2012)MathSciNetMATHGoogle Scholar
  68. 68.
    Djaoua, M.: Équations intégrales pour un problème singulier dans le plan. Ph.D. thesis, École Polytechnique (1977)Google Scholar
  69. 69.
    Dreyfuss, P.: Analyse numérique d’une méthode intégrale sans singularité – application à l’électromagnétisme. Ph.D. thesis, École Polytechnique Fédérale de Lausanne (1999)Google Scholar
  70. 70.
    Dreyfuss, P., Rappaz, J.: Numerical analysis of a non singular boundary integral method. Part I: the circular case. Math. Methods Appl. Sci. 24, 847–863 (2001)MathSciNetMATHGoogle Scholar
  71. 71.
    Dreyfuss, P., Rappaz, J.: Numerical analysis of a non singular boundary integral method. Part II: the general case. Math. Methods Appl. Sci. 25, 557–570 (2002)MathSciNetMATHGoogle Scholar
  72. 72.
    Egan, L.R., Furlani, E.P.: A computer simulation of an induction heating system. IEEE Trans. Magn. 27, 4343–4354 (1991)Google Scholar
  73. 73.
    Favennec, Y., Labbé, V., Bay, F.: Induction heating processes optimization: a general optimal control approach. J. Comput. Phys. 187, 68–94 (2003)MATHGoogle Scholar
  74. 74.
    Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics. The Commemorative Ed. Addison Wesley, Redwood City (1989)Google Scholar
  75. 75.
    Flotron, S.: Simulations numériques de phénomènes MHD–thermiques avec interface libre dans l’électrolyse de l’aluminium. Ph.D. thesis, École Polytechnique Fédérale de Lausanne (2013)Google Scholar
  76. 76.
    Flück, M., Rumpf, M.: Bernoulli’s free-boundary problem, qualitative theory and numerical approximation. J. Reine Angew. Math. 486, 165–204 (1997)MathSciNetGoogle Scholar
  77. 77.
    Flück, M., Hofer, T., Picasso, M., Rappaz, J., Steiner, G.: Scientific computing for aluminium production. Int. J. Numer. Anal. Model. 1(1), 1–20 (2008)Google Scholar
  78. 78.
    Flück, M., Janka, A., Laurent, C., Picasso, M., Rappaz, J., Steiner, G.: Some mathematical and numerical aspects in aluminium production. J. Sci. Comput. 1(1), 1–20 (2009)Google Scholar
  79. 79.
    Flück, M., Rappaz, J., Safa, Y.: Numerical simulation of thermal problems coupled with magnetohydrodynamic effects in aluminum cells. Appl. Math. Model. 33(3), 1479–1492 (2009)MathSciNetMATHGoogle Scholar
  80. 80.
    Flück, M., Hofer, T., Janka, A., Rappaz, J.: Numerical methods for ferromagnetic plates. Appl. Numer. Partial Differ. Equ. Comput. Methods Appl. Sci. 15, 169–182 (2010)Google Scholar
  81. 81.
    Franca, L.P., Muller, R.L., Hughes, T.J.R.: Convergence analyses of Galerkin least-squares methods for symmetric advective-diffusiv forms of the Stokes and incompressible Navier-stokes equations. Comput. Methods Appl. Mech. Eng. 105(2), 285–298 (1993)MATHGoogle Scholar
  82. 82.
    Friedman, M.J.: Mathematical study of the nonlinear singular integral magnetic field equation. I. SIAM J. Appl. Math. 39(1), 14–20 (1980)MATHGoogle Scholar
  83. 83.
    Friedman, M.J.: Mathematical study of the nonlinear singular integral magnetic field equation. II. SIAM J. Appl. Math. 18(4), 644–653 (1981)MATHGoogle Scholar
  84. 84.
    Friedman, M.J.: Mathematical study of the nonlinear singular integral magnetic field equation. III. SIAM J. Appl. Math. 12(4), 536–540 (1981)MATHGoogle Scholar
  85. 85.
    Gagnoud, A., Etay, J., Garnier: Le problème de lévitation en frontière libre électromagnétique. J. Mec. Theor. Appl. 5(6), 911–925 (1986)MATHGoogle Scholar
  86. 86.
    Gallouët, T., Herbin, R.: Existence of a solution to a coupled elliptic system. Appl. Math. Lett. 7(2), 49–55 (1994)MathSciNetMATHGoogle Scholar
  87. 87.
    Gatica, G.: An alternative variational formulation for the Johnson & Nédélec’s coupling procedure. Rev. Math. Appl. 16, 17–41 (1995)MathSciNetMATHGoogle Scholar
  88. 88.
    Gauthier-Béchonnet, S.: Résolution et mise en œuvre d’un modèle tridimensionnal des courants de foucault. Ph.D. thesis, Université Blaise Pascal, Clermont-Ferrand (1998)Google Scholar
  89. 89.
    Gerbeau, J.F., Le Bris, C., Bercovier, M.: Existence of solution for a density-dependent magnetohydrodynamic equation. Adv. Differ. Equ. 2(3), 427–452 (1997)MATHGoogle Scholar
  90. 90.
    Gerbeau, J.F., Le Bris, C., Bercovier, M.: Spurious velocities in the steady flow of an incompressible fluid subjected to external forces. Int. J. Numer. Method Fluids 25, 679–695 (1997)MATHGoogle Scholar
  91. 91.
    Gerbeau, J.F., Le Bris, C., Le Lièvre, T.: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Oxford University Press, Oxford (2006)MATHGoogle Scholar
  92. 92.
    Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1970)Google Scholar
  93. 93.
    Girault, V., Raviart, P.A.: Finite Element Approximation of the Navier-Stokes Equations. Springer, Berlin (1985)Google Scholar
  94. 94.
    Gleize, A., Gonzales, J., Freton, P.: Thermal plasma modelling. J. Phys. D Appl. Phys. 38, R153–R183 (2005)Google Scholar
  95. 95.
    Haddar, H., Joly, P.: Effective boundary conditions for thin ferromagnetic layers: the one dimensional model. SIAM J. Appl. Math. 6(4), 1386–1417 (2001)MathSciNetGoogle Scholar
  96. 96.
    Henneron, T.: Contribution à la prise en compte des grandeurs globales dans les problèmes d’électromagnétisme résolus avec la méthode des éléments finis. Ph.D. thesis, Université Lille I (2004)Google Scholar
  97. 97.
    Henrot, A., Pierre, M.: Un problème inverse en formage des métaux liquides. Model. Math. Anal. Numer. 23(1), 155–177 (1989)MathSciNetMATHGoogle Scholar
  98. 98.
    Henrot, A., Pierre, M.: Variation et optimisation de forme: une analyse géométrique. Springer, Berlin (2005)Google Scholar
  99. 99.
    Hernández, R.V.: Contributions to the mathematical study of some problems in magnetohydrodynamics and induction heating. Ph.D. thesis, Universidade de Santiago de Compostela (2008)Google Scholar
  100. 100.
    Hiptmair, R.: Symmetric coupling for eddy current problems. SIAM J. Numer. Anal. 40(1), 41–65 (2002)MathSciNetMATHGoogle Scholar
  101. 101.
    Hiptmair, R., Sterz, O.: Current and voltage excitations for the eddy current model. Int. J. Numer. Model. 18, 1–21 (2005)MATHGoogle Scholar
  102. 102.
    Hofer, T.: Numerical simulation and optimization of the alumina distribution in an aluminium electrolysis pot. Ph.D. thesis, École Polytechnique Fédérale de Lausanne (2011)Google Scholar
  103. 103.
    Hsiao, G.: On boundary integral equations of the first kind. J. Comput. Math. 7(2), 121–131 (1989)MathSciNetMATHGoogle Scholar
  104. 104.
    Hsiao, G.: Boundary element methods – an overview. Appl. Numer. Math. 56, 1356–1369 (2006)MathSciNetMATHGoogle Scholar
  105. 105.
    Hsiao, G., Wendland, W.: Boundary element methods: foundation and error analysis. In: Encyclopedia of Computational Mechanics, vol. 1, chap.  12, pp. 339–373. Wiley (2005)
  106. 106.
    Hughes, T.J.R.: The Finite Element Method, Linear Static and Dynamic Finite Element Analysis. Dover, Mineola (2000)MATHGoogle Scholar
  107. 107.
    Jackson, J.: Classical Electrodynamics. Wiley, London (1965)Google Scholar
  108. 108.
    Johnson, C., Nédélec, J.C.: On the coupling of boundary integral and finite element methods. Math. Comput. 35, 1063–1079 (1980)MATHGoogle Scholar
  109. 109.
    Joly, P., Vacus, O.: Mathematical and numerical studies of nonlinear ferromagnetic material. Model. Math. Anal. Numer. 33(3), 593–626 (1999)MathSciNetMATHGoogle Scholar
  110. 110.
    Kanayama, H., Tagami, D., Saito, M., Kikuchi, F.: A numerical method for 3-D eddy current problems. Jpn. J. Ind. Appl. Math. 18(2), 603–612 (2001)MathSciNetMATHGoogle Scholar
  111. 111.
    Kim, D.H., Hahn, S.Y., Park, I.H., Cha, G.: Computation of three–dimensional electromagnetic field including moving media by indirect boundary integral equation method. IEEE Trans. Magn. 35(3), 1932–1938 (1999)Google Scholar
  112. 112.
    Klein, O., Philip, P.: Correct voltage distribution for axisymmetric sinusoidal modelling of induction heating with prescribing current, voltage, or power. IEEE Trans. Magn. 38(3), 1519–1523 (2002)Google Scholar
  113. 113.
    Kuhn, M., Steinbach, O.: Symmetric coupling of finite and boundary elements for exterior magnetic field problems. Math. Methods Appl. Sci. 25, 357–371 (2002)MathSciNetMATHGoogle Scholar
  114. 114.
    Kuster, C., Gremaud, P., Touzani, R.: Fast numerical methods for Bernoulli free boundary problems. SIAM J. Sci. Comput. 29(2), 622–634 (2007)MathSciNetMATHGoogle Scholar
  115. 115.
    Labridis, D., Dokopoulos, P.: Calculation of eddy current losses in nonlinear ferromagnetic materials. IEEE Trans. Magn. 25, 2665–2669 (1989)Google Scholar
  116. 116.
    Landau, L., Lifshitz, E.: Electrodynamics of Continuous Media. Pergamon, London (1960)MATHGoogle Scholar
  117. 117.
    Landau, L., Lifshitz, E.: Fluid Mechanics. Pergamon, London (1960)Google Scholar
  118. 118.
    Leray, J., Schauder, J.: Topologie et équations fonctionnelles. Ann. Sci. Ecole Norm. Sup. 51, 45–78 (1934)MathSciNetGoogle Scholar
  119. 119.
    Leroux, M.N.: Résolution numérique du problème du potentiel dans le plan par une méthode variationnelle d’éléments finis. Ph.D. thesis, Université de Rennes (1974)Google Scholar
  120. 120.
    Leroux, M.N.: Méthode d’éléments finis pour la résolution numérique de problèmes extérieurs en dimension 2. R.A.I.R.O. Analyse Numérique 11(1), 27–60 (1977)Google Scholar
  121. 121.
    Li, B.Q.: The fluid flow aspects of electromagnetic levitation processes. Int. J. Eng. Sci. 32(1), 45–67 (1989)Google Scholar
  122. 122.
    Li, H.: Finite element analysis for the axisymmetric Laplace operator on polygonal domains. J. Comput. Appl. Math. 235, 5155–5176 (2011)MathSciNetMATHGoogle Scholar
  123. 123.
    Li, B.Q., Evans, J.W.: Computation of shapes of electromagnetically supported menisci in electromagnetic casters. Part I: calculations in two dimensions. IEEE Trans. Magn. 25(6), 4443–4448 (1989)Google Scholar
  124. 124.
    Lions, J.L., Magenes, E.: Problèmes aux limites non homogènes et applications, Tome I. Dunod, Paris (1968)MATHGoogle Scholar
  125. 125.
    Massé, P., Morel, B., Breville, T.: A finite element prediction correction scheme for magneto-thermal coupled problem during Curie transition. IEEE Trans. Magn. 25, 181–183 (1989)Google Scholar
  126. 126.
    Masserey, A.: Optimisation et simulation numérique du chauffage par induction pour le procédé de thixoformage. Ph.D. thesis, École Polytechnique Fédérale de Lausanne (2002)Google Scholar
  127. 127.
    Masserey, A., Rappaz, J., Rozsnyo, R., Touzani, R.: Optimal control of an induction heating process for thixoforming. IEEE Trans. Magn. 40(3), 1657–1663 (2004)Google Scholar
  128. 128.
    Masserey, A., Rappaz, J., Rozsnyo, R., Touzani, R.: Power formulation for the optimal control of an industrial induction heating process for thixoforming. Int. J. Appl. Electromagn. Mech. 19, 51–56 (2004)Google Scholar
  129. 129.
    Meir, A.: Thermally coupled, stationary, incompressible MHD flow; existence, uniqueness, and finite element approximation. Numer. Methods PDE 11, 311–337 (1995)MathSciNetMATHGoogle Scholar
  130. 130.
    Meir, A., Schmidt, P.G.: Variational methods for stationary MHD flow under natural interface conditions. Nonlinear Anal. Theory Methods Appl. 24(4), 659–689 (1996)MathSciNetGoogle Scholar
  131. 131.
    Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford (2003)MATHGoogle Scholar
  132. 132.
    Monk, P., Vacus, O.: Error estimates for a numerical scheme for ferromagnetic problems. SIAM J. Numer. Anal. 36(3), 696–718 (1999)MathSciNetGoogle Scholar
  133. 133.
    Montaser, A., Golightly, D.W. (eds.): Inductively Coupled Plasmas in Analytical Atomic Spectrometry. VCH Publishers, Inc., New York (1992)Google Scholar
  134. 134.
    Natarajan, T., El-Kaddah, N.: A methodology for two-dimensional finite element analysis of electromagnetically driven flow in induction stirring systems. IEEE Trans. Magn. 35(3), 1773–1776 (1999)Google Scholar
  135. 135.
    Nédélec, J.C.: Notions sur les équations intégrales de la physique. Centre de Mathématiques Appliquées, École Polytechnique, Palaiseau (1977)Google Scholar
  136. 136.
    Nédélec, J.C.: Mixed finite elements in 3. Numer. Math. 35(3), 315–341 (1980)MathSciNetMATHGoogle Scholar
  137. 137.
    Nédélec, J.C.: A new family of mixed finite elements in 3. Numer. Math. 50, 57–81 (1986)MathSciNetMATHGoogle Scholar
  138. 138.
    Nédélec, J.C.: Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems. Springer, New York (2001)MATHGoogle Scholar
  139. 139.
    Neff, H.: Introductory Electromagnetics. Wiley, New York (1991)Google Scholar
  140. 140.
    Parietti, C.: Modélisation mathématique et analyse numérique d’un problème de chauffage électromagnétique. Ph.D. thesis, École Polytechnique Fédérale de Lausanne (1998)Google Scholar
  141. 141.
    Parietti, C., Rappaz, J.: A quasi–static two–dimensional induction heating problem. Part I: modelling and analysis. Math. Models Methods Appl. Sci. 8(6), 1003–1021 (1998)MathSciNetMATHGoogle Scholar
  142. 142.
    Parietti, C., Rappaz, J.: A quasi–static two–dimensional induction heating problem. Part II: numerical analysis. Math. Models Methods Appl. Sci. 9(9), 1333–1350 (1999)MathSciNetMATHGoogle Scholar
  143. 143.
    Pierre, M., Roche, J.R.: Computation of free surfaces in the electromagnetic shaping of liquid metals by optimization algorithms. Eur. J. Mech. B/Fluids 10(5), 489–500 (1991)MathSciNetMATHGoogle Scholar
  144. 144.
    Pierre, M., Roche, J.R.: Numerical simulation of electromagnetic shaping of liquid metals. Tech. Rep., INRIA (1992)Google Scholar
  145. 145.
    Rapetti, F., Bouillaut, F., Santandrea, L., Buffa, A., Maday, Y., Razek, A.: Calculation of eddy currents with edge elements on non-matching grids in moving structures. IEEE Trans. Magn. 10(5), 482–507 (1991)Google Scholar
  146. 146.
    Rappaz, J., Swierkosz, M.: Mathematical modeling and numerical simulation of induction heating process. Appl. Math. Comput. Sci. 6(2), 207–221 (1996)MathSciNetMATHGoogle Scholar
  147. 147.
    Rappaz, J., Swierkosz, M.: Boundary-element method yields external vector potentials in complex industrial applications. Comput. Phys. 11(2), 145–150 (1997)Google Scholar
  148. 148.
    Rappaz, J., Touzani, R.: Modelling of a two–dimensional magnetohydrodynamic problem. Eur. J. Mech. B/Fluids 10(5), 482–507 (1991)MathSciNetGoogle Scholar
  149. 149.
    Rappaz, J., Touzani, R.: On a two–dimensional Magnetohydrodynamic problem, I: modelling and analysis. Model. Math. Anal. Numer. 26(2), 347–364 (1992)MathSciNetMATHGoogle Scholar
  150. 150.
    Rappaz, J., Touzani, R.: On a two–dimensional Magnetohydrodynamic problem, II: numerical analysis. Model. Math. Anal. Numer. 30(2), 215–235 (1996)MathSciNetMATHGoogle Scholar
  151. 151.
    Rappaz, M., Bellet, M., Deville, M.: Modélisation numérique en science des matériaux. Presses Polytechniques et Universitaires Romandes, Lausanne (1998)MATHGoogle Scholar
  152. 152.
    Rappaz, J., Swierkosz, M., Trophime, C.: Un modèle mathématique et numérique pour un logiciel de simulation tridumensionnelle d’induction électromagnétique. Tech. Rep., École Polytechnique Fédérale de Lausanne (1999)Google Scholar
  153. 153.
    Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Bänsch, E., Dold, A. (eds.) Mathematical Aspects of Finite Element Methods. Lecture Notes in Mathematics, vol. 606, p. 503. Springer, New York/Rome (1977)Google Scholar
  154. 154.
    Reitz, J., Milford, F.: Foundations of Electromagnetic Theory. Addison–Wesley, Reading (1975)Google Scholar
  155. 155.
    Robinson, N.: Electromagnetism. Oxford Physics Series. Clarendon, Oxford (1973)Google Scholar
  156. 156.
    Rochette, D.: Contributions à la simulation d’écoulements de plasma haute pression appliquée aux appareillages de coupure et torches à plasma. Ph.D. thesis, Université Blaise Pascal (2012). Habilitation ThesisGoogle Scholar
  157. 157.
    Rodríguez, A.A., Valli, A.: Eddy Current Approximation of Maxwell Equations. Springer, Milan (2010)MATHGoogle Scholar
  158. 158.
    Rodríguez, A.A., Valli, A., Hernández, R.V.: A formulation of the eddy current problem in the presence of electric ports. Numer. Math. 113, 643–672 (2009)MathSciNetMATHGoogle Scholar
  159. 159.
    Rogier, F.: Problèmes mathématiques et numériques liés à l’approximation de la géométrie d’un corps diffractant dans les équations de l’électromagnétisme. Ph.D. thesis, École Polytechnique (1989)Google Scholar
  160. 160.
    Roy, S.S., Cramb, A.W., Hoburg, J.F.: Magnetic shaping of columns of liquid sodium. Metall. Trans. B 26(1), 1191–1197 (1995)Google Scholar
  161. 161.
    Sakane, J., Li, B., Evans, J.: Mathematical modeling of meniscus profile and melt flow in electromagnetic casters. Metall. Trans. B 19(2), 397–408 (1988)Google Scholar
  162. 162.
    Schercliff, J.A.: Magnetic shaping of molten metal columns. Proc. R. Soc. Lond. A Math. Phys. Sci. 275(1763), 455–473 (1981)Google Scholar
  163. 163.
    Schmidlin, G., Fischer, U., Andjelic, Z., Schwab, C.: Preconditioning the second-kind boundary integral equations for 3-D eddy current problems. Int. J. Numer. Meth. Eng. 10(5), 482–507 (1991)Google Scholar
  164. 164.
    Sethian, J.: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Materials Science. Cambridge University Press, Cambridge (1999)MATHGoogle Scholar
  165. 165.
    Sneyd, A., Moffat, H.: Fluid dynamical aspects of the levitation melting process. J. Fluid Mech. 117, 45–70 (1982)MathSciNetMATHGoogle Scholar
  166. 166.
    Steiner, G.: Simulation numérique de phénomènes MHD: application à l’électrolyse de l’aluminium. Ph.D. thesis, École Polytechnique Fédérale de Lausanne (2009)Google Scholar
  167. 167.
    Stephan, E.: Coupling of boundary element methods and finite element methods. In: Encyclopedia of Computational Mechanics, vol. 1, chap.  13, pp. 375–412. Wiley, Chichester (2005)
  168. 168.
    Szabó, B., Babuška, I.: Finite Element Analysis. Wiley-Interscience, New York (1991)MATHGoogle Scholar
  169. 169.
    Touzani, R.: Un problème de courant de Foucault avec inducteur filiforme. C. R. Acad. Sci. tome 319, Série I, 771–776 (1994)Google Scholar
  170. 170.
    Touzani, R.: Analysis of an eddy current problem involving a thin inductor. Comput. Methods Appl. Mech. Eng. 131, 233–240 (1996)MathSciNetMATHGoogle Scholar
  171. 171.
    Vérité, J.C.: Trifou: un code de calcul tridimensionnel des courants de foucault. EDF Bulletin de la direction des études et recherches, Série C, Mathématiques et Informatique 2, 79–92 (1983)Google Scholar
  172. 172.
    Vérité, J.C.: Traitement du potentiel scalaire magnétique extérieur dans le cas d’un domaine multiplement connexe. application au code TRIFOU. EDF Bulletin de la direction des études et recherches, Série C, Mathématiques et Informatique 1, 61–75 (1986)Google Scholar
  173. 173.
    Wang, J., Xie, D., Yao, Y., Mohammed, O.: A modified solution for large sparse symmetric linear systems in electromagnetic field analysis. IEEE Trans. Magn. 37(5), 3494–3497 (2001)Google Scholar
  174. 174.
    Wanser, S., Krähenbühl, L., Nicolas, A.: A computation of 3D induction hardening problems by combined finite and boundary element methods. IEEE Trans. Magn. 30(5), 3320–3323 (1994)Google Scholar
  175. 175.
    Wendland, W.L.: On the asymptotic convergence of boundary integral methods. In: Brebbia, C.A. (ed.) Boundary Element Methods, pp. 412–430. Springer, Berlin (1981)Google Scholar
  176. 176.
    Xue, S., Proulx, P., Boulos, M.I.: Extended-field electromagnetic model for inductively coupled plasma. J. Phys. D Appl. Phys. 34(4), 1897–1906 (2001)Google Scholar
  177. 177.
    Yamazaki, K.: Transient eddy current analysis for moving conductors using adaptive moving coordinate systems. IEEE Trans. Magn. 36(4), 785–789 (2000)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Rachid Touzani
    • 1
  • Jacques Rappaz
    • 2
  1. 1.Université Blaise PascalClermont-FerrandFrance
  2. 2.Ecole Polytechnique Fédérale de LausanneLausanneSwitzerland

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