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Preprint 36-2012

A unified analysis of transmission conditions for thin conducting sheets in the time-harmonic eddy current model

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Author(s) : Kersten Schmidt , Alexey Chernov

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 36-2012

MSC 2000

65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
35C20 Asymptotic expansions
35J25 Boundary value problems for second-order, elliptic equations
41A60 Asymptotic approximations, asymptotic expansions
35B40 Asymptotic behavior of solutions
78M30 Variational methods

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 [26]. We analyse these ITCs in comparison with the often used perfect electric boundary condition, the shielding element by Nakata et.al. [23], the thin layer impedance boundary conditions by Mayergoyz and Bedrosian [22] and a family of ITCs derived by asymptotic expansions in the asymptotic regime of constant shielding by Schmidt and Tordeux [28]. 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.

Keywords : Impedance Transmission Conditions, Thin Conducting Sheets, Asymptotic Expansions

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