A Unified Analysis of Transmission Conditions for Thin Conducting Sheets in the Time-Harmonic Eddy Current Model

Abstract

We introduce tools for a unified analysis and a comparison of impedance transmission conditions (ITCs) for thin conducting sheets within the time-harmonic eddy current model in two dimensions. The first criterion is the robustness with respect to the frequency or skin depth, which means whether they give meaningful results for small and for large frequencies or conductivities. As a second tool we study the accuracy for a range of sheet thicknesses and frequencies for a relevant example and finally analyze their asymptotic order in different asymptotic regimes. For the latter we write all the ITCs in a common form and show how they can be realized within the finite element method. Two new conditions, which we call ITC-2-0 and ITC-2-1, are introduced in this article, which appear in a symmetric form. They are derived by asymptotic expansions in the asymptotic regime of constant ratio between skin depth and thickness like those in [K. Schmidt and A. Chernov, Technical report 1102, Institut for Numerical Simulation, University of Bonn, Bonn, Germany, 2011]. We analyze these ITCs in comparison with the often used perfect electric boundary condition, the conditions by Levi-Civita [T. Levi-Civita, Rend. Lincei (5), 11 (1902), pp. 75--81 (in Italian); K. Schmidt and S. Tordeux, Z. Angew. Math. Phys., 61 (2010), pp. 603--626] the shielding element by Nakata et al. [IEEE Trans Magn., 26 (1990), pp. 2379--2381], the thin layer impedance boundary conditions by Mayergoyz and others [O. Tozoni and I. Mayergoyz, Technika, Kiev, 1974 (in Russian); I. Mayergoyz and G. Bedrosian, IEEE Trans. Magn., 31 (1995), pp. 1319--1324] and a family of ITCs derived by asymptotic expansions in the asymptotic regime of constant shielding by Schmidt and [ESAIM: Math. Model. Numer. Anal., 45 (2011), pp. 1115--1140]. Our analysis shows the superiority of the transmission conditions derived by asymptotic expansions where especially the worst-case error level of the ITC-2-1 is remarkably lower than for all the other conditions.