×

On the influence of the geometry on skin effect in electromagnetism. (English) Zbl 1225.78003

Summary: We consider the equations of electromagnetism set on a domain made of a dielectric and a conductor subdomain in a regime where the conductivity is large. Assuming smoothness for the dielectric – conductor interface, relying on recent works we prove that the solution of the Maxwell equations admits a multiscale asymptotic expansion with profile terms rapidly decaying inside the conductor. This skin effect is measured by introducing a skin depth function that turns out to depend on the mean curvature of the boundary of the conductor. We then confirm these asymptotic results by numerical experiments in various axisymmetric configurations. We also investigate numerically the case of a nonsmooth interface, namely a cylindrical conductor.

MSC:

78A25 Electromagnetic theory (general)
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
PDFBibTeX XMLCite
Full Text: DOI HAL

References:

[1] C. Bernardi, M. Dauge, Y. Maday, Spectral methods for axisymmetric domains, Series in Applied Mathematics (Paris), vol. 3, Gauthier-Villars, Éditions Scientifiques et Médicales, Elsevier, Paris, 1999. Numerical algorithms and tests due to Mejdi Azaı¨ez.; C. Bernardi, M. Dauge, Y. Maday, Spectral methods for axisymmetric domains, Series in Applied Mathematics (Paris), vol. 3, Gauthier-Villars, Éditions Scientifiques et Médicales, Elsevier, Paris, 1999. Numerical algorithms and tests due to Mejdi Azaı¨ez. · Zbl 0929.35001
[2] P. Boissoles, Problèmes mathématiques et numériques issus de l’imagerie par résonance magnétique nucléaire, Ph.D. Thesis, Université Rennes 1, 2005.; P. Boissoles, Problèmes mathématiques et numériques issus de l’imagerie par résonance magnétique nucléaire, Ph.D. Thesis, Université Rennes 1, 2005.
[3] Caloz, G.; Dauge, M.; Péron, V., Uniform estimates for transmission problems with high contrast in heat conduction and electromagnetism, J. Math. Anal. Appl., 370, 2, 555-572 (2010) · Zbl 1194.35452
[4] Dauge, M.; Faou, E.; Péron, V., Comportement asymptotique à haute conductivité de l’épaisseur de peau en électromagnétisme, C.R. Acad. Sci. Paris Sér. I Math., 348, 7-8, 385-390 (2010) · Zbl 1188.35022
[5] Faou, E., Elasticity on a thin shell: formal series solution, Asymptot. Anal., 31, 3-4, 317-361 (2002) · Zbl 1046.74031
[6] Goldstein, H., Classical Mechanics. Classical Mechanics, Addison-Wesley Series in Physics (1980), Addison-Wesley Publishing Co.: Addison-Wesley Publishing Co. Reading, Mass
[7] Haddar, H.; Joly, P.; Nguyen, H.-M., Generalized impedance boundary conditions for scattering problems from strongly absorbing obstacles: the case of Maxwell’s equations, Math. Models Methods Appl. Sci., 18, 10, 1787-1827 (2008) · Zbl 1170.35094
[8] Hiptmair, R.; Ledger, P. D., Computation of resonant modes for axisymmetric Maxwell cavities using hp-version edge finite elements, Int. J. Numer. Methods Engrg., 62, 12, 1652-1676 (2005) · Zbl 1121.78013
[9] T. Levi-Civita, The absolute differential calculus, Dover Phoenix Editions, Dover Publications Inc., Mineola, NY, 2005 (Calculus of tensors, Translated from the Italian by Marjorie Long, Edited by Enrico Persico, Reprint of the 1926 translation).; T. Levi-Civita, The absolute differential calculus, Dover Phoenix Editions, Dover Publications Inc., Mineola, NY, 2005 (Calculus of tensors, Translated from the Italian by Marjorie Long, Edited by Enrico Persico, Reprint of the 1926 translation). · Zbl 1206.53013
[10] MacCamy, R. C.; Stephan, E., Solution procedures for three-dimensional eddy current problems, J. Math. Anal. Appl., 101, 2, 348-379 (1984) · Zbl 0563.35054
[11] MacCamy, R. C.; Stephan, E., A skin effect approximation for eddy current problems, Arch. Ratio. Mech. Anal., 90, 1, 87-98 (1985) · Zbl 0595.35096
[12] D. Martin, Mélina, bibliothèque de calculs éléments finis, 1990-2010. Source code: <http://anum-maths.univ-rennes1.fr/melina>; D. Martin, Mélina, bibliothèque de calculs éléments finis, 1990-2010. Source code: <http://anum-maths.univ-rennes1.fr/melina>
[13] Naghdi, P., Foundations of elastic shell theory, (Progress in Solid Mechanics, vol. IV (1963), North-Holland: North-Holland Amsterdam), 1-90
[14] Nkemzi, B., On the solution of Maxwell’s equations in axisymmetric domains with edges, ZAMM Z. Angew. Math. Mech., 85, 8, 571-592 (2005) · Zbl 1088.35070
[15] Pardo, D.; Demkowicz, L.; Torres-Verdı´n, C.; Paszynski, M., Two-dimensional high-accuracy simulation of resistivity logging-while-drilling (LWD) measurements using a self-adaptive goal-oriented hp finite element method, SIAM J. Appl. Math., 66, 6, 2085-2106 (2006) · Zbl 1116.78014
[16] V. Péron, Modélisation mathématique de phénomènes électromagnétiques dans des matériaux à fort contraste, Ph.D. Thesis, Université Rennes 1, 2009. <http://tel.archives-ouvertes.fr/tel-00421736/fr/>; V. Péron, Modélisation mathématique de phénomènes électromagnétiques dans des matériaux à fort contraste, Ph.D. Thesis, Université Rennes 1, 2009. <http://tel.archives-ouvertes.fr/tel-00421736/fr/>
[17] Rytov, S. M., Calcul du skin effect par la méthode des perturbations, J. Phys., 11, 3, 233-242 (1940) · JFM 66.1129.02
[18] Schwab, C.; Suri, M., The \(p\) and hp versions of the finite element method for problems with boundary layers, Math. Comput., 65, 216, 1403-1429 (1996) · Zbl 0853.65115
[19] Stephan, E., Solution procedures for interface problems in acoustics and electromagnetics, (Theoretical Acoustics and Numerical Techniques. Theoretical Acoustics and Numerical Techniques, CISM Courses and Lectures, vol. 277 (1983), Springer: Springer Vienna), 291-348 · Zbl 0578.76078
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.