A sliding Mesh-Mortar method for a two dimensional Eddy currents model of electric engines
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 35 (2001) no. 2, pp. 191-228.

The paper deals with the application of a non-conforming domain decomposition method to the problem of the computation of induced currents in electric engines with moving conductors. The eddy currents model is considered as a quasi-static approximation of Maxwell equations and we study its two-dimensional formulation with either the modified magnetic vector potential or the magnetic field as primary variable. Two discretizations are proposed, the first one based on curved finite elements and the second one based on iso-parametric finite elements in both the static and moving parts. The coupling is obtained by means of the mortar element method (see [7]) and the approximation on the whole domain turns out to be non-conforming. In both cases optimal error estimates are provided. Numerical tests are then proposed in the case of standard first order finite elements to test the reliability and precision of the method. An application of the method to study the influence of the free part movement on the currents distribution is also provided.

Classification : 35Q60,  65N15,  65M55,  68U20,  78A30
Mots clés : Eddy currents problem, non-conforming finite element approximation, domain decomposition methods
     author = {Buffa, Annalisa and Maday, Yvon and Rapetti, Francesca},
     title = {A sliding Mesh-Mortar method for a two dimensional Eddy currents model of electric engines},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     pages = {191--228},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {2},
     year = {2001},
     zbl = {0986.35111},
     mrnumber = {1825696},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2001__35_2_191_0/}
Buffa, Annalisa; Maday, Yvon; Rapetti, Francesca. A sliding Mesh-Mortar method for a two dimensional Eddy currents model of electric engines. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 35 (2001) no. 2, pp. 191-228. http://www.numdam.org/item/M2AN_2001__35_2_191_0/

[1] R. Adams, Sobolev spaces. Academic Press, London (1976). | MR 450957 | Zbl 0314.46030

[2] R. Albanese and G. Rubinacci, Formulation of the eddy-current problem. IEEE proceedings 137 (1990).

[3] G. Anagnostou, A. Patera and Y. Maday, A sliding mesh for partial differential equations in nonstationary geometries: application to the incompressible Navier-Stockes equations. Tech. rep., Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie (1994).

[4] F. Ben Belgacem and Y. Maday, Non-conforming spectral element methodology tuned to parallel implementation. Comput. Meth. Appl. Mech. Engrg. 116 (1994) 59-67. | Zbl 0841.65096

[5] F. Ben Belgacem, Y. Maday, The mortar element method for three dimensional finite elements. RAIRO-Modél. Math. Anal. Numér. 2 (1997) 289-302. | Numdam | Zbl 0868.65082

[6] C. Bernardi, Optimal finite element interpolation of curved domains. SIAM J. Numer. Anal. 26 (1989) 1212-1240. | Zbl 0678.65003

[7] C. Bernardi, Y. Maday and A.T. Patera, A new nonconforming approach to domain decomposition: The mortar elements method, in Nonlinear partial differential equations and their applications, H. Brezis and J. Lions, Eds., Collège de France Seminar, Paris, Vol. XI (1994) 13-51. | Zbl 0797.65094

[8] A. Bossavit, Électromagnétisme en vue de la modélisation, Springer-Verlag, Paris (1986). | MR 1616583 | Zbl 0787.65090

[9] A. Bossavit, Calcul des courants induits et des forces électromagnétiques dans un système de conducteurs mobiles. RAIRO-Modél. Math. Anal. Numér. 23 (1989) 235-259. | Numdam | MR 1001329 | Zbl 0673.65084

[10] A. Bossavit, Le calcul des courants de Foucault en dimension 3, avec le champ électrique comme inconnue. I: Principes. Rev. Phys. Appl. 25 (1990) 189-197.

[11] F. Bouillault, Z. Ren and A. Razek, Modélisation tridimensionnelle des courants de Foucault à l'aide de méthodes mixtes avec différentes formulations. Rev. Phys. Appl. 25 (1990) 583-592.

[12] C.J. Carpenter, Comparison of alternative formulations of 3-dimensional magnetic-field and eddy-current problems at power frequencies. IEEE proceedings 124 (1977) 1026-1034.

[13] P. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978). | MR 520174 | Zbl 0383.65058

[14] R. Dautray and J.L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, 2nd edn. Masson, Paris (1987). | MR 918560 | Zbl 0642.35001

[15] B. Davat, Z. Ren and M. Lajoie-Mazenc, The movement in field modeling. IEEE, Trans. Magn. 21 (1985) 2296-2298.

[16] C.R.I. Emson, C.P. Riley, D.A. Walsh, K. Ueda and T. Kumano, Modeling eddy currents induced by rotating systems. IEEE, Trans. Magn. 34 (1998) 2593-2596.

[17] Y. Goldman, P. Joly and M. Kern, The electric field in the conductive half-space as a model in mining and petroleum prospection. Math. Meth. Appl. Sci. 11 (1989) 373-401. | Zbl 0693.65088

[18] J. Jackson, Classical electrodynamics. Wiley, New York (1952). | MR 436782 | Zbl 0997.78500

[19] S. Kurz, J. Fetzer, G. Lehenr, and W. Rucker, A novel formulation for 3d eddy current problems with moving bodies using a Lagrangian description and bem-fem coupling. IEEE, Trans. Magn. 34 (1998) 3068-3073.

[20] R. Leis, Initial Boundary value problems in mathematical physics. John Wiley and Sons (1986). | MR 841971 | Zbl 0599.35001

[21] Y. Marechal, G. Meunier, J. Coulomb and H. Magnin, A general purpose for restoring inter-element continuity. IEEE, Trans. Magn. 28 (1992) 1728-1731.

[22] A. Nicolet, F. Delincé, A. Genon and W. Legros, Finite elements-boundary elements coupling for the movement modeling in two dimensional structures. J. Phys. III 2 (1992) 2035-2044.

[23] A. Quarteroni and A. Valli, Numerical approximation of partial differential equations. Ser. Comput. Math. 23, Springer-Verlag (1993). | MR 1299729 | Zbl 0803.65088

[24] F. Rapetti, L. Santandrea, F. Bouillault and A. Razek, Simulating eddy currents distributions by a finite element method on moving non-matching grids. COMPEL 19 (2000) 10-29. | Zbl 0965.78013

[25] A. Razek, J. Coulomb, M. Felliachi and J. Sobonnadière, Conception of an air-gap element for dynamic analysis of the electromagnetic fields in electric machines. IEEE, Trans. Magn. 18 (1982) 655-659.

[26] D. Rodger, H. Lai and P. Leonard, Coupled elements for problems involving movement. IEEE, Trans. Magn. 26 (1990) 548-550.

[27] V. Thomeé, Galerkin finite element methods for parabolic problems. Ser. Comput. Math. 25, Springer (1997). | MR 1479170 | Zbl 0884.65097