×

Numerical analysis of electric field formulations of the eddy current model. (English) Zbl 1084.78004

In this paper it is analyzed the eddy current problem with input current intensities as boundary data. The authors introduce and analyze a weak formulation in terms of the electric field and they propose a finite element method for its numerical solution. Next, the authors analyze the eddy current problem with electric potentials as boundary data. It is developed the weak formulation in terms of the electric field and it is analyzed a mixed finite element method for its numerical solution. In the last part of the paper there are justified the convergence properties, by comparing the numerical results obtained with a known analytical solution.

MSC:

78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Software:

TRIFOU
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alonso, A., Valli, A.: A domain decomposition approach for heterogeneous time-harmonic Maxwell equations. Comput. Meth. Appl. Mech. Engrg. 143, 97–112 (1997) · Zbl 0883.65096 · doi:10.1016/S0045-7825(96)01144-9
[2] Alonso, A., Valli, A.: An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations. Math. Comp. 68, 607–631 (1999) · Zbl 1043.78554 · doi:10.1090/S0025-5718-99-01013-3
[3] Alonso Rodríguez, A., Fernandes, P., Valli A.: The time-harmonic eddy current problem in general domains: solvability via scalar potentials. Lect. Notes Comput. Sci. Eng. 28, 143–163 (2003) · Zbl 1031.35140
[4] Alonso Rodríguez, A., Fernandes, P., Valli A.: Weak and strong formulations for the time-harmonic eddy-current problem in general multi-connected domains. Eur. J. Appl. Math. 14, 387–406 (2003) · Zbl 1051.78004 · doi:10.1017/S0956792503005151
[5] Alonso Rodríguez, A., Hiptmair, R., Valli A.: Mixed finite element approximation of eddy current problems. IMA J. Numer. Anal. 24, 255–271 (2004) · Zbl 1114.78012 · doi:10.1093/imanum/24.2.255
[6] Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional non-smooth domains. Math. Meth. Appl. Sci. 21, 823–864 (1998) · Zbl 0914.35094 · doi:10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B
[7] Bermúdez, A., Bullón, J., Pena, F., Salgado, P.: A numerical method for transient simulation of metallurgical compound electrodes. Finite Elem. Anal. Des. 39, 283–299 (2003) · Zbl 1213.74283 · doi:10.1016/S0168-874X(02)00069-0
[8] Bermúdez, A., Bullón, J., Pena, F.: A finite element method for the thermoelectrical modelling of electrodes. Commun. Numer. Meth. Engrg. 14, 581–593 (1998) · Zbl 0922.65089 · doi:10.1002/(SICI)1099-0887(199806)14:6<581::AID-CNM175>3.0.CO;2-S
[9] Bermúdez, A., Rodríguez, R., Salgado, P.: A finite element method with Lagrange multipliers for low-frequency harmonic Maxwell equations. SIAM J. Numer. Anal. 40, 1823–1849 (2002) · Zbl 1033.78009 · doi:10.1137/S0036142901390780
[10] Bermúdez, A., Rodríguez, R., Salgado, P.: Numerical analysis of the electric field formulation of an eddy currents problem. C. R. Acad. Sci. Paris, Serie I 337, 359–364 (2003) · Zbl 1038.78017
[11] Bermúdez, A., Rodríguez, R., Salgado, P.: Numerical treatment of realistic boundary conditions for the eddy current problem in an electrode via Lagrange multipliers. Math. Comp. 74, 123–151 (2005) · Zbl 1112.78019 · doi:10.1090/S0025-5718-04-01680-1
[12] Bermúdez, A., Rodríguez, R., Salgado, P.: Numerical solution of eddy current problems in bounded domains using realistic boundary conditions. Comput. Methods Appl. Mech. Engrg. 194, 411–426 (2005) · Zbl 1063.78015 · doi:10.1016/j.cma.2004.05.016
[13] Biro, O.: Edge element formulations of eddy current problems. Comput. Methods Appl. Mech. Engrg. 169, 391–405 (1999) · Zbl 0937.78002 · doi:10.1016/S0045-7825(98)00165-0
[14] Biro, O., Preis, K.: On the use of the magnetic vector potential in the finite element analysis of three-dimensional eddy current problems. IEEE Trans. Magn. 25, 3145–3159 (1989) · doi:10.1109/20.34388
[15] Bossavit, A.: Computational Electromagnetism. Variational Formulations, Complementarity, Edge Elements. Academic Press, San Diego, CA, 1998 · Zbl 0945.78001
[16] Bossavit, A.: Most general non-local boundary conditions for the Maxwell equation in a bounded region. COMPEL 19, 239–245 (2000) · Zbl 0966.78002
[17] Bossavit, A., Vérité, J.-C.: The “trifou” code: solving the 3D eddy currents problem by using H as state variable. IEEE Trans. Magn. MAG-19(6), 2465–2470 (1983)
[18] Buffa, A., Costabel, M., Sheen, D.: On traces for H(curl, {\(\Omega\)}) in Lipschitz domains. J. Math. Anal. Appl. 276, 845–876 (2002) · Zbl 1106.35304 · doi:10.1016/S0022-247X(02)00455-9
[19] Dular, P., Kuo-Peng, P., Geuzaine, C., Sadowski, N., Bastos, J. P. A.: Dual magnetodynamic formulations and their source fields associated with massive and stranded inductors. IEEE Trans. Magn. 36, 1293–1299 (2000) · doi:10.1109/20.877677
[20] Fernandes P. Gilardi, G.: Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Math. Models Methods Appl. Sci. 7, 957–991 (1997) · Zbl 0910.35123 · doi:10.1142/S0218202597000487
[21] Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer-Verlag, Berlin, 1986 · Zbl 0585.65077
[22] Golias, N.A., Anntonopoulus, C.S., Tsboukis, T.D., Kriezis, E.E.: 3D eddy current computation with edge elements in terms of the electric intensity. COMPEL 17, 667–673 (1998) · Zbl 0932.78013
[23] Hiptmair, R., Sterz, O.: Current and voltage excitation for the eddy current model. Internat. J. Numer. Model. 18, 1–21 (2005) · Zbl 1099.78021 · doi:10.1002/jnm.555
[24] Kanayama, H., Kikuchi, F.: 3-D eddy current computation using the Nédélec elements. Information 2, 37–45 (1999) · Zbl 1073.65561
[25] Meddahi, S., Selgas, V.: A mixed-FEM and BEM coupling for a three-dimensional eddy current problem. M2AN Math. Model. Numer. Anal. 37, 291–318 (2003) · Zbl 1031.78012
[26] Nédélec, J.-C.: Mixed finite elements in \(\mathbb{R}\)3. Numer. Math. 35, 315–341 (1980) · Zbl 0419.65069 · doi:10.1007/BF01396415
[27] Tu, H.T., Shao, K.R., Zhou, K.D.: H method for solving 3D eddy current problems. IEEE Trans. Magn. 31, 3518–3520 (1995) · doi:10.1109/20.489555
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.