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, that means if 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 analyse finally 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 realised 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 [34]. We analyse these ITCs in comparison with the often used perfect electric boundary condition, the conditions by Levi-Civita [26, 36], the shielding element by Nakata et.al. [29], the thin layer impedance boundary conditions by Mayergoyz et.al. [42, 27] and a family of ITCs derived by asymptotic expansions in the asymptotic regime of constant shielding by Schmidt and Tordeux [37]. 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.