Scattering of a scalar time-harmonic wave by a penetrable obstacle with a thin layer

Abstract

This work looks at the asymptotic behaviour of the solution to the Helmholtz equation in a penetrable domain of $\mathbb{R}$3 with a thin layer of thickness δ which tends to 0. We use the method of multi-scale expansion to derive and justify an asymptotic expansion of the solution with respect to the thickness δ up to any order. We then provide approximate transmission conditions of order two defined on an interface located inside the thin layer, with accuracy up to O(δ2), which allow one to take into account the influence of the thin layer.

On Ventcel-type transmission conditions for a Helmholtz problem with a non-uniform thin layer

“…Proof. The existence and uniqueness of the series u n − n , u n + n and U n m,β n as well as the error estimate follow the same lines as Theorem 13 in [9] and Theorem 5.1 in [7].…”

“…We start by the following theorem in which we state the existence and the uniqueness result and give a uniform estimate of the solution u ε with respect to ε, for which a proof can be found in [9,7].…”

“…A simple calculation shows that the terms w n − , w n + coincide with u n − , u n + , for n ∈ {0, 1}. Furthermore, each term in (64) is bounded in H 1 (Ω) (cf., e.g., [9,Theorem 4.1]). Let R w be the remainder obtained by truncating the asymptotic expansions (64) at order 3…”

“…This work is a continuation of a series of two papers [9,10] in which the authors studied a similar problem defined in a domain with a uniform thin layer.…”

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.…”