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

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 dimension. The transmission conditions are derived asymptotically for vanishing sheet thickness ε where the skin depth is kept proportional to ε. The condition of order 1 turns out to be the perfect electric conductor boundary condition. The conditions of order 2 appear as generalised Poincaré-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 ε → 0 and the robustness of the equivalent transmission condition of order 2.

Computational helioseismology in the frequency domain: acoustic waves in axisymmetric solar models with flows

“…The framework can relatively easily be extended to the vectorial wave equation using existing Montjoie vectorial setups (e.g. Péron et al 2016, for Maxwell's equations). The scalar equation captures the propagation of acoustic waves through a solar-like medium and leads to an oscillation power spectrum that compares well with observations.…”

Approximate impedance for time-harmonic Maxwell's equations in a non planar domain with contrasted multi-thin layers

“…The concept of impedance boundary condition is widely used in the study of scattering of electromagnetic waves by obstacles covered with a thin coating. In the literature, one can find many applications in different fields: biology, elasticity systems, inverse problems, telecommunication, industry and others... (see, e.g., [1,2,7,8,9,10,12,13,15,16,18,19,20,25,26,27]). This boundary condition was introduced first for reasons linked to electrical engineering in the turn of the 18 th and 19 th century.…”

Asymptotic Models and Impedance Conditions for Highly Conductive Sheets in the Time-Harmonic Eddy Current Model

“…This work is concerned essentially with theoretical objectives. The numerical pertinence of ITCs have already been shown for the Maxwell equations in [27]. In [27], second order ITCs have been proposed for the time-harmonic Maxwell equations in three dimensions for vanishing sheet thickness ε where the skin depth is kept proportional to ε.…”

“…The numerical pertinence of ITCs have already been shown for the Maxwell equations in [27]. In [27], second order ITCs have been proposed for the time-harmonic Maxwell equations in three dimensions for vanishing sheet thickness ε where the skin depth is kept proportional to ε. These conditions appear as generalized Poincaré-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.…”

“…Here, we use techniques to derive impedance conditions for electromagnetic models in 3D including thin layers, see e.g. [10,27].…”