High order transmission conditions for thin conductive sheets in magneto-quasistatics

Schmidt, Kersten; Tordeux, Sébastien

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 6, pp. 1115-1140.

We propose transmission conditions of order 1, 2 and 3 approximating the shielding behaviour of thin conducting curved sheets for the magneto-quasistatic eddy current model in 2D. This model reduction applies to sheets whose thicknesses ε are at the order of the skin depth or essentially smaller. The sheet has itself not to be resolved, only its midline is represented by an interface. The computation is directly in one step with almost no additional cost. We prove the well-posedness w.r.t. to the small parameter ε and obtain optimal bound for the modelling error outside the sheet of order ${\epsilon }^{N+1}$ for the condition of order N. We end the paper with numerical experiments involving high order finite elements for sheets with varying curvature.

Classification : 65N30, 35C20, 35J25, 41A60, 35B40, 78M30, 78M35
Mots clés : asymptotic expansions, transmission condition, thin conducting sheets

[1] X. Antoine, H. Barucq and L. Vernhet, High-frequency asymptotic analysis of a dissipative transmission problem resulting in generalized impedance boundary conditions. Asymptot. Anal. 26 (2001) 257-283. | MR | Zbl

[2] N. Bartoli and A. Bendali, Robust and high-order effective boundary conditions for perfectly conducting scatterers coated by a thin dielectric layer. IMA J. Appl. Math. 67 (2002) 479-508. | MR | Zbl

[3] A. Bendali and K. Lemrabet, The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation. SIAM J. Appl. Math. 6 (1996) 1664-1693. | MR | Zbl

[4] A. Bendali and K. Lemrabet, Asymptotic analysis of the scattering of a time-harmonic electromagnetic wave by a perfectly conducting metal coated with a thin dielectric shell. Asymptot. Anal. 57 (2008) 199-227. | MR | Zbl

[5] A. Bossavit, Computational Electromagnetism. Variational Formulation, Complementarity, Edge Elements. No. 2 in Academic Press Electromagnetism Series. Academic Press, San Diego (1998). | MR | Zbl

[6] D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 3th edition. Cambridge University Press (2007). | Zbl

[7] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer-Verlag, New York (2010). | MR | Zbl

[8] G. Caloz, M. Costabel, M. Dauge and G. Vial, Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer. Asymptot. Anal. 50 (2006) 121-173. | MR | Zbl

[9] Concepts Development Team. Webpage of Numerical C++ Library Concepts 2. http://www.concepts.math.ethz.ch (2011).

[10] M. Duruflé, H. Haddar and P. Joly, Higher order generalized impedance boundary conditions in electromagnetic scattering problems. C.R. Phys. 7 (2006) 533-542.

[11] P. Frauenfelder and C. Lage, Concepts - an object-oriented software package for partial differential equations. ESAIM: M2AN 36 (2002) 937-951. | Numdam | MR | Zbl

[12] H. Haddar, P. Joly and H.M. Nguyen, Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: the scalar case. Math. Models Methods Appl. Sci. 15 (2005) 1273-1300. | MR | Zbl

[13] H. Haddar, P. Joly and H.M. Nguyen, Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: the case of Maxwell s equations. Math. Models Methods Appl. Sci. 18 (2008) 1787-1827. | MR | Zbl

[14] H. Igarashi, A. Kost and T. Honma, A boundary element analysis of magnetic shielding for electron microscopes. Compel 17 (1998) 585-594. | Zbl

[15] P. Joly and S. Tordeux, Asymptotic analysis of an approximate model for time harmonic waves in media with thin slots. ESAIM: M2AN 40 (2006) 63-97. | Numdam | MR | Zbl

[16] L. Krähenbühl and D. Muller, Thin layers in electrial engineering. Example of shell models in analysing eddy-currents by boundary and finite element methods. IEEE Trans. Magn. 29 (1993) 1450-1455.

[17] M.A. Leontovich, On approximate boundary conditions for electromagnetic fields on the surface of highly conducting bodies (in russian). Research in the propagation of radio waves. Moscow, Academy of Sciences (1948) 5-12.

[18] W. Mclean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press (2000). | MR | Zbl

[19] A.M. Miri, N.A. Riegel and C. Meinecke, FE calculation of transient eddy currents in thin conductive sheets using dynamic boundary conditions. Int. J. Numer. Model. 11 (1998) 307-316. | Zbl

[20] T. Nakata, N. Takahashi, K. Fujiwara and Y. Shiraki, 3D magnetic field analysis using special elements. IEEE Trans. Magn. 26 (1990) 2379-2381.

[21] V. Péron and C. Poignard, Approximate transmission conditions for time-harmonic Maxwell equations in a domain with thin layer. Research Report RR-6775, INRIA (2008).

[22] R. Perrussel and C. Poignard, Asymptotic Transmission Conditions for Steady-State Potential in a High Contrast Medium. A Uniform Variational Formulation for Resistive Thin Layers. Research Report RR-7163, INRIA (2010).

[23] K. Schmidt, High-order numerical modeling of highly conductive thin sheets. Ph.D. thesis, ETH Zürich (2008).

[24] K. Schmidt and S. Tordeux, Asymptotic modelling of conductive thin sheets. Z. Angew. Math. Phys. 61 (2010) 603-626. | MR | Zbl

[25] K. Schmidt, O. Sterz and R. Hiptmair, Estimating the eddy-current modelling error. IEEE Trans. Magn. 44 (2008) 686-689.

[26] T. Senior and J. Volakis, Approximate Boundary Conditions in Electromagnetics. Institution of Electrical Engineers (1995). | Zbl

[27] A.N. Shchukin, Propagation of Radio Waves (in Russian). Svyazizdat, Moscow (1940).