Equivalent Transmission Conditions for the Time-Harmonic Maxwell Equations in 3D for a Medium with a Highly Conductive Thin Sheet

V Péron,M Duruflé,K Schmidt

doi:10.1137/15m1012116

Abstract

We propose equivalent transmission conditions of order 1 and 2 for thin and highly conducting sheets for the time-harmonic Maxwell's equation in three dimensions. The transmission conditions are derived asymptotically for vanishing sheet thickness $\varepsilon$ where the skin depth is kept proportional to $\varepsilon$. The condition of order 1 turns out to be the perfect electric conductor boundary condition. The conditions of order 2 appear as generalized Poincareź--Steklov maps between tangential components of the magnetic field and the electric field, and they are of Wentzell type involving second order surface differential operators. Numerical results with finite elements of higher order validate the asymptotic convergence for $\varepsilon\to0$ and the robustness of the equivalent transmission condition of order 2.

Highlights

Many electric and electronic devices feature thin conducting sheets providing efficient electromagnetic shielding. Due to their large aspect ratio and high conductivity of the flat or curved sheets the shielding properties are achieved with a minimum use of materials
Their large aspect ratio makes the numerical simulation of such devices more of a challenge, especially if standard methods like the finite element method (FEM) or the finite difference method shall be applied
The numerical modelling is much simplified if the thin conducting sheets are replaced by transmission conditions on an interface, which is usually its mid-surface

Summary

Introduction

Many electric and electronic devices feature thin conducting sheets providing efficient electromagnetic shielding. Already in 1902 Levi-Civita introduced equivalent transmission conditions [17] (see [1, 33]) for Maxwell’s equations He postulated that the electric field is continuous over the interface whereas the magnetic field has a discontinuity, which is proportional to the sheet thickness and conductivity. The electromagnetic properties in Ω are given by the piecewise-constant functions με, ε, and σε corresponding to the respective magnetic permeability, electric permittivity, and conductivity of the possibly different materials in the three subdomains They are given by μ−, με = μo, μ+, in Ωε−, in Ωεo, in Ωε+,.

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