Abstract
The rest of this book is devoted to a presentation of mathematical modelling and numerical solution of some well known applications of low frequency electromagnetics (eddy current and magnetostatics) in industry. For each application, we present the set of equations that govern the process, give some mathematical results when they are known and present the numerical methods to solve the set of equations as well as a series of numerical experiments.
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References
Abeele, D.V., Degrez, G.: Efficient computational model for inductive plasma flows. AIAA J. 38(2), 234–242 (2000)
Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. U.S. Government Printing Office, Washington (1964)
Adams, R.: Sobolev Spaces. Academic, New York (1975)
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)
Albanese, R., Rubinacci, G.: Finite element methods for the solution of 2-D eddy current problems. Adv. Imaging Electron Phys. 102, 1–86 (1998)
Albanese, R., Rubinacci, G., Tamburrino, A., Ventre, S., Villone, F.: A fast 3-D eddy current integral formulation. COMPEL 20(2), 317–331 (2001)
Amiez, G., Gremaud, P.A.: On a numerical approach to stefan like problems. Numer. Math. 59, 71–89 (1991)
Amirat, Y., Touzani, R.: Self–inductance coefficient for toroidal thin conductors. Nonlinear Anal. B 131, 233–240 (2001)
Amirat, Y., Touzani, R.: Asymptotic behavior of the inductance coefficient for thin conductors. Math. Models Methods Appl. Sci. 12(2), 273–289 (2002)
Amirat, Y., Touzani, R.: A two-dimensional eddy current model using thin inductors. Asymptot. Anal. 58(3), 171–188 (2008)
Amirat, Y., Touzani, R.: A singular perturbation problem in eddy current models (2013, submitted)
Ammari, H., Nédélec, J.C.: Propagation d’ondes électromagnétiques à basses fréquences. J. Math. Pures Appl. 77, 839–849 (1998)
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)
Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three–dimensional non-smooth domains. Math. Methods Appl. Sci. 21, 823–864 (1998)
Arnold, D.N., Wendland, W.L.: On the asymptotic convergence of collocation methods. Math. Comput. 41(164), 349–381 (1983)
Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University, Cambridge (1967)
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)
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)
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)
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)
Bermúdez, A., Muñiz, M.C., Salgado, P.: Asymptotic approximation and numerical simulation of electromagnetic casting. Metall. Trans. B 34(1), 83–91 (2003)
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)
Bernardi, C., Dauge, M., Maday, Y.: Spectral Methods for Axisymmetric Domains. Series in Applied Mathematics. Gauthier-Villars, Paris (1999)
Bernardi, D., Colombo, V., Ghedini, E., Mentrelli, A.: Three-dimensional modeling of inductively coupled plasma torches. Pure Appl. Chem. 77(2), 359–372 (2005)
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)
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)
Biro, O.: Edge element formulations of eddy current problems. Comput. Methods Appl. Mech. Eng. 169, 391–405 (1999)
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)
Boccardo, L., Gallouët, T.: Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87(1), 149–169 (1989)
Bodart, O., Boureau, A.V., Touzani, R.: Numerical investigation of optimal control of induction heating processes. Appl. Math. Model. 25, 697–712 (2001)
Bonnans, J.F., Gilbert, J., Lemaréchal, C., Sagastizabal, C.: Numerical Optimization. Springer, New York (2006)
Bossavit, A.: On the numerical analysis of eddy current problems. Comput. Methods Appl. Mech. Eng. 27, 303–318 (1981)
Bossavit, A.: Two dual formulations of the 3-D eddy currents problem. COMPEL 4(2), 103–116 (1985)
Bossavit, A.: Computational Electromagnetism. Associated Press (1998)
Bossavit, A., Rodrigues, J.F.: On the electromagnetic induction heating problem in bounded domains. Adv. Math. Sci. Appl. 4(1), 79–92 (1994)
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)
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)
Braess, D.: Finite Elements, Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, Cambridge (2001)
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)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (2002)
Brezis, H.: Analyse Fonctionnelle. Masson, Paris (1983)
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)
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)
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)
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)
Casas, E.: Pontryagin’s principle for state-constraint boundary control problems of seminlinear parabolic equations. SIAM J. Control Optim. 35, 1297–1327 (1997)
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)
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)
Chadebec, O., Colomb, J.L., Janet, F.: A review of magnetostatic moment method. IEEE Trans. Magn. 42(4), 515–520 (2006)
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)
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)
Clain, S.: Analyse mathématique et numérique d’un modèle de chauffage par induction. Ph.D. thesis, École Polytechnique Fédérale de Lausanne (1994)
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)
Clain, S., Touzani, R.: A two–dimensional stationary induction heating problems. Math. Methods Appl. Sci. 20, 759–766 (1997)
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)
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)
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)
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)
Coulaud, O., Henrot, A.: Numerical approximation of free boundary problem arising in electromagnetic shaping. Tech. Rep., INRIA (1992)
Crouzeix, M.: Variational approach of a magnetic shaping problem. Eur. J. Mech., B/Fluids 10(5), 527–536 (1991)
Dacorogna, B.: Direct Methods in the Calculus of Variations. Applied Mathematical Sciences. Springer, Berlin (1989)
Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology. Springer, Berlin (1990)
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)
Descloux, J.: A stability result for the magnetic shaping problem. Z. Angew. Math. Phys. 45, 544–555 (1994)
Descloux, J., Flück, M., Rappaz, J.: A problem of magnetostatics related to thin plates. Model. Math. Anal. Numer. 32, 859–876 (1998)
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)
Descloux, J., Flück, M., Rappaz, J.: Modelling and mathematical results arising from ferromagnetic problems. Sci. China Math. 55(5), 1053–1067 (2012)
Djaoua, M.: Équations intégrales pour un problème singulier dans le plan. Ph.D. thesis, École Polytechnique (1977)
Dreyfuss, P.: Analyse numérique d’une méthode intégrale sans singularité – application à l’électromagnétisme. Ph.D. thesis, École Polytechnique Fédérale de Lausanne (1999)
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)
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)
Egan, L.R., Furlani, E.P.: A computer simulation of an induction heating system. IEEE Trans. Magn. 27, 4343–4354 (1991)
Favennec, Y., Labbé, V., Bay, F.: Induction heating processes optimization: a general optimal control approach. J. Comput. Phys. 187, 68–94 (2003)
Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics. The Commemorative Ed. Addison Wesley, Redwood City (1989)
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)
Flück, M., Rumpf, M.: Bernoulli’s free-boundary problem, qualitative theory and numerical approximation. J. Reine Angew. Math. 486, 165–204 (1997)
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)
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)
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)
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)
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)
Friedman, M.J.: Mathematical study of the nonlinear singular integral magnetic field equation. I. SIAM J. Appl. Math. 39(1), 14–20 (1980)
Friedman, M.J.: Mathematical study of the nonlinear singular integral magnetic field equation. II. SIAM J. Appl. Math. 18(4), 644–653 (1981)
Friedman, M.J.: Mathematical study of the nonlinear singular integral magnetic field equation. III. SIAM J. Appl. Math. 12(4), 536–540 (1981)
Gagnoud, A., Etay, J., Garnier: Le problème de lévitation en frontière libre électromagnétique. J. Mec. Theor. Appl. 5(6), 911–925 (1986)
Gallouët, T., Herbin, R.: Existence of a solution to a coupled elliptic system. Appl. Math. Lett. 7(2), 49–55 (1994)
Gatica, G.: An alternative variational formulation for the Johnson & Nédélec’s coupling procedure. Rev. Math. Appl. 16, 17–41 (1995)
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)
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)
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)
Gerbeau, J.F., Le Bris, C., Le Lièvre, T.: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Oxford University Press, Oxford (2006)
Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1970)
Girault, V., Raviart, P.A.: Finite Element Approximation of the Navier-Stokes Equations. Springer, Berlin (1985)
Gleize, A., Gonzales, J., Freton, P.: Thermal plasma modelling. J. Phys. D Appl. Phys. 38, R153–R183 (2005)
Haddar, H., Joly, P.: Effective boundary conditions for thin ferromagnetic layers: the one dimensional model. SIAM J. Appl. Math. 6(4), 1386–1417 (2001)
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)
Henrot, A., Pierre, M.: Un problème inverse en formage des métaux liquides. Model. Math. Anal. Numer. 23(1), 155–177 (1989)
Henrot, A., Pierre, M.: Variation et optimisation de forme: une analyse géométrique. Springer, Berlin (2005)
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)
Hiptmair, R.: Symmetric coupling for eddy current problems. SIAM J. Numer. Anal. 40(1), 41–65 (2002)
Hiptmair, R., Sterz, O.: Current and voltage excitations for the eddy current model. Int. J. Numer. Model. 18, 1–21 (2005)
Hofer, T.: Numerical simulation and optimization of the alumina distribution in an aluminium electrolysis pot. Ph.D. thesis, École Polytechnique Fédérale de Lausanne (2011)
Hsiao, G.: On boundary integral equations of the first kind. J. Comput. Math. 7(2), 121–131 (1989)
Hsiao, G.: Boundary element methods – an overview. Appl. Numer. Math. 56, 1356–1369 (2006)
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)
Hughes, T.J.R.: The Finite Element Method, Linear Static and Dynamic Finite Element Analysis. Dover, Mineola (2000)
Jackson, J.: Classical Electrodynamics. Wiley, London (1965)
Johnson, C., Nédélec, J.C.: On the coupling of boundary integral and finite element methods. Math. Comput. 35, 1063–1079 (1980)
Joly, P., Vacus, O.: Mathematical and numerical studies of nonlinear ferromagnetic material. Model. Math. Anal. Numer. 33(3), 593–626 (1999)
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)
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)
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)
Kuhn, M., Steinbach, O.: Symmetric coupling of finite and boundary elements for exterior magnetic field problems. Math. Methods Appl. Sci. 25, 357–371 (2002)
Kuster, C., Gremaud, P., Touzani, R.: Fast numerical methods for Bernoulli free boundary problems. SIAM J. Sci. Comput. 29(2), 622–634 (2007)
Labridis, D., Dokopoulos, P.: Calculation of eddy current losses in nonlinear ferromagnetic materials. IEEE Trans. Magn. 25, 2665–2669 (1989)
Landau, L., Lifshitz, E.: Electrodynamics of Continuous Media. Pergamon, London (1960)
Landau, L., Lifshitz, E.: Fluid Mechanics. Pergamon, London (1960)
Leray, J., Schauder, J.: Topologie et équations fonctionnelles. Ann. Sci. Ecole Norm. Sup. 51, 45–78 (1934)
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)
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)
Li, B.Q.: The fluid flow aspects of electromagnetic levitation processes. Int. J. Eng. Sci. 32(1), 45–67 (1989)
Li, H.: Finite element analysis for the axisymmetric Laplace operator on polygonal domains. J. Comput. Appl. Math. 235, 5155–5176 (2011)
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)
Lions, J.L., Magenes, E.: Problèmes aux limites non homogènes et applications, Tome I. Dunod, Paris (1968)
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)
Masserey, A.: Optimisation et simulation numérique du chauffage par induction pour le procédé de thixoformage. Ph.D. thesis, École Polytechnique Fédérale de Lausanne (2002)
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)
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)
Meir, A.: Thermally coupled, stationary, incompressible MHD flow; existence, uniqueness, and finite element approximation. Numer. Methods PDE 11, 311–337 (1995)
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)
Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford (2003)
Monk, P., Vacus, O.: Error estimates for a numerical scheme for ferromagnetic problems. SIAM J. Numer. Anal. 36(3), 696–718 (1999)
Montaser, A., Golightly, D.W. (eds.): Inductively Coupled Plasmas in Analytical Atomic Spectrometry. VCH Publishers, Inc., New York (1992)
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)
Nédélec, J.C.: Notions sur les équations intégrales de la physique. Centre de Mathématiques Appliquées, École Polytechnique, Palaiseau (1977)
Nédélec, J.C.: Mixed finite elements in ℝ 3. Numer. Math. 35(3), 315–341 (1980)
Nédélec, J.C.: A new family of mixed finite elements in ℝ 3. Numer. Math. 50, 57–81 (1986)
Nédélec, J.C.: Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems. Springer, New York (2001)
Neff, H.: Introductory Electromagnetics. Wiley, New York (1991)
Parietti, C.: Modélisation mathématique et analyse numérique d’un problème de chauffage électromagnétique. Ph.D. thesis, École Polytechnique Fédérale de Lausanne (1998)
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)
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)
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)
Pierre, M., Roche, J.R.: Numerical simulation of electromagnetic shaping of liquid metals. Tech. Rep., INRIA (1992)
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)
Rappaz, J., Swierkosz, M.: Mathematical modeling and numerical simulation of induction heating process. Appl. Math. Comput. Sci. 6(2), 207–221 (1996)
Rappaz, J., Swierkosz, M.: Boundary-element method yields external vector potentials in complex industrial applications. Comput. Phys. 11(2), 145–150 (1997)
Rappaz, J., Touzani, R.: Modelling of a two–dimensional magnetohydrodynamic problem. Eur. J. Mech. B/Fluids 10(5), 482–507 (1991)
Rappaz, J., Touzani, R.: On a two–dimensional Magnetohydrodynamic problem, I: modelling and analysis. Model. Math. Anal. Numer. 26(2), 347–364 (1992)
Rappaz, J., Touzani, R.: On a two–dimensional Magnetohydrodynamic problem, II: numerical analysis. Model. Math. Anal. Numer. 30(2), 215–235 (1996)
Rappaz, M., Bellet, M., Deville, M.: Modélisation numérique en science des matériaux. Presses Polytechniques et Universitaires Romandes, Lausanne (1998)
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)
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)
Reitz, J., Milford, F.: Foundations of Electromagnetic Theory. Addison–Wesley, Reading (1975)
Robinson, N.: Electromagnetism. Oxford Physics Series. Clarendon, Oxford (1973)
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 Thesis
Rodríguez, A.A., Valli, A.: Eddy Current Approximation of Maxwell Equations. Springer, Milan (2010)
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)
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)
Roy, S.S., Cramb, A.W., Hoburg, J.F.: Magnetic shaping of columns of liquid sodium. Metall. Trans. B 26(1), 1191–1197 (1995)
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)
Schercliff, J.A.: Magnetic shaping of molten metal columns. Proc. R. Soc. Lond. A Math. Phys. Sci. 275(1763), 455–473 (1981)
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)
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)
Sneyd, A., Moffat, H.: Fluid dynamical aspects of the levitation melting process. J. Fluid Mech. 117, 45–70 (1982)
Steiner, G.: Simulation numérique de phénomènes MHD: application à l’électrolyse de l’aluminium. Ph.D. thesis, École Polytechnique Fédérale de Lausanne (2009)
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)
Szabó, B., Babuška, I.: Finite Element Analysis. Wiley-Interscience, New York (1991)
Touzani, R.: Un problème de courant de Foucault avec inducteur filiforme. C. R. Acad. Sci. tome 319, Série I, 771–776 (1994)
Touzani, R.: Analysis of an eddy current problem involving a thin inductor. Comput. Methods Appl. Mech. Eng. 131, 233–240 (1996)
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)
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)
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)
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)
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)
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)
Yamazaki, K.: Transient eddy current analysis for moving conductors using adaptive moving coordinate systems. IEEE Trans. Magn. 36(4), 785–789 (2000)
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Touzani, R., Rappaz, J. (2014). Induction Heating Processes. In: Mathematical Models for Eddy Currents and Magnetostatics. Scientific Computation. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0202-8_8
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