Generalized Impedance Boundary Conditions for Scattering Problems From Strongly Absorbing Obstacles: The Case of Maxwell's Equations

Abstract

This paper is dedicated to the construction and analysis of so-called Generalized Impedance Boundary Conditions (GIBCs) for electromagnetic scattering problems from imperfect conductors with smooth boundaries. These boundary conditions can be seen as higher order approximations of a perfect conductor condition. We consider here the 3-D case with Maxwell equations in a harmonic regime. The construction of GIBCs is based on a scaled asymptotic expansion with respect to the skin depth. The asymptotic expansion is theoretically justified at any order and we give explicit expressions till the third order. These expressions are used to derive the GIBCs. The associated boundary value problem is analyzed and error estimates are obtained in terms of the skin depth.

Full Range Scattering Estimates and Their Application to Cloaking

“…Lemma 5 was proved and used in [5]. Similar inequalities related to the quantities div and curl were introduced in [6].…”

On the influence of the geometry on skin effect in electromagnetism

“…Several works are devoted to asymptotic expansions at high conductivity of the electromagnetic field in a domain made of two subdomains when the interface is smooth: see [20,11,12] for plane interface and eddy current approximation, and [8,3,4,17] for a three-dimensional model of skin effect in electromagnetism.…”

“…The skin effect reflects the flow of current near the surface of a conductor. After the early work [18], the mathematical analysis of the skin effect has been addressed more recently in several papers, [20,11,12,8].…”

“…The present work is motivated by recent studies [8,4,3,17] in which authors analyze the behavior of the electromagnetic fields solution of the Maxwell equations through an asymptotic expansion for large conductivity. In particular, uniform estimates for the electromagnetic field at high conductivity are proved in [3], whereas in the note [4] a suitable skin depth function is introduced on the interface between a conductor and an insulator to generalize the classical scalar quantity.…”

Corner asymptotics of the magnetic potential in the eddy‐current model

“…Impedance conditions were first proposed by Leontovich [13] and by Rytov [17] in the 1940's and then extended by Senior and Volakis [18]. These conditions can be derived up to the required precision for any conductor with a smooth surface: we refer to Haddar et al [9] for the mathematical justification of conditions of order 1, 2 and 3. Note however that such conditions are not valid near corners.…”