On the influence of the geometry on skin effect in electromagnetism

Abstract

International audienceWe consider the equations of electromagnetism set on a domain made of a dielectric and a conductor subdomain in a regime where the conductivity is large. Assuming smoothness for the dielectric--conductor interface, relying on recent works we prove that the solution of the Maxwell equations admits a multiscale asymptotic expansion with profile terms rapidly decaying inside the conductor. This <em> skin effect</em> is measured by introducing a skin depth function that turns out to depend on the mean curvature of the boundary of the conductor. We then confirm these asymptotic results by numerical experiments in various axisymmetric configurations. We also investigate numerically the case of a nonsmooth interface, namely a cylindrical conductor

Generalized impedance boundary conditions for strongly absorbing obstacle: The full wave equation

“…They then developed mathematical tools, based on compactness arguments, to establish error estimates up to order 3. Related works are the edgy current problem studied by MacCamy and Stephan in [16] and the study of the GIBCs for highly conducting obstacle for the Maxwell system in the time harmonic regime by Haddar, Joly, and Nguyen in [10] (see also the work of Caloz, Dauge, Faou, and Péron in [4]) and references therein.…”

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

“…Here, x ∈ R 3 are the cartesian coordinates, (y α , h) is a local normal coordinate system [4] to the surface Γ in the thin layer Ω ε o where y α , α = 1, 2 are tangential coordinates on Γ and h ∈ (− ε 2 , ε 2 ) is the normal coordinate to Γ. Moreover, the term E j is a "profile" defined on Γ × (− 1 2 , 1 2 ).…”

“…This derivation is order by order and for each order n it is in four steps: (i) Writing the operator curl curl in the domain Ω ε o in local coordinates (y α , h) [4] and performing the change of variable Y 3 = ε −1 h, i.e., rewriting it in in (y α , Y 3 )-coordinates, leads to an algebraic equation fixing the normal component e n of E n and a differential equation for the two tangential components E n , which are not completely defined yet. (ii) We expand the far field term E n at the two surfaces Γ ε ± of the conductor around its mid-surface Γ using the Taylor expansion.…”

“…Due to the small thickness of the conductor the derivatives in normal and the tangential directions scale differently in ε. Hence, it is convenient to use a local normal coordinate system in Ω ε o , see e.g., [4,App. A.1].…”

Thin Layer Models for Electromagnetism

“…Then, complex phenomenon known as electropermeabilization (or electroporation) may occur [21]: the cell membrane In order to avoid the meshing of the thin membrane, it is convenient to approximate the solution to problem (1), by replacing the thin layer by appropriate conditions across the surface Γ. The idea, as presented in [7], consists in rewriting the operator curl curl in the domain O ε m in local coordinates (x T , x 3 ) [1,7]. The variable x 3 ∈ (0, ε) is the Euclidean distance to Γ and x T denotes tangential coordinates on Γ.…”