Bérangère Delourme

ON THE WELL-POSEDNESS, STABILITY AND ACCURACY OF AN ASYMPTOTIC MODEL FOR THIN PERIODIC INTERFACES IN ELECTROMAGNETIC SCATTERING PROBLEMS

We analyze the well-posedness and stability properties of a parameter dependent problem that models the reflection and transmission of electromagnetic waves at a thin and rapidly oscillating interface. The latter is modeled using approximate interface conditions that can be derived using asymptotic expansion of the exact solution with respect to the small parameter (proportional to the periodicity length of oscillations and the width of the interface). The obtained uniform stability results are then used to analyze the accuracy (with respect to the small parameter) of the proposed model.

High order asymptotics for wave propagation across thin periodic interfaces

This work deals with the scattering of acoustic waves by a thin ring that contains many regularly-spaced heterogeneities. We provide and justify a complete description of the solution with respect to the period and the thickness of the heterogeneities. Our approach mixes matched asymptotic expansions and homogenization theory.

On the homogenization of thin perforated walls of finite length

The present work deals with the resolution of the Poisson equation in a bounded domain made of a thin and periodic layer of finite length placed into a homogeneous medium. We provide and justify a high order asymptotic expansion which takes into account the boundary layer effect occurring in the vicinity of the periodic layer as well as the corner singularities appearing in the neighborhood of the extremities of the layer. Our approach combines the method of matched asymptotic expansions and the method of periodic surface homogenization.

Trapped modes in thin and infinite ladder like domains. Part 1 : existence results

The present paper deals with the wave propagation in a particular two dimensional structure, obtained from a localized perturbation of a reference periodic medium. This reference medium is a ladder like domain, namely a thin periodic structure (the thickness being characterized by a small parameter

Existence of guided waves due to a lineic perturbation of a 3D periodic medium

\epsilon > 0

Approximate Models for Wave Propagation Across Thin Periodic Interfaces

) whose limit (as

Rapidly convergent two-dimensional quasi-periodic Green function throughout the spectrum-including Wood anomalies

\epsilon

High-order asymptotics for the electromagnetic scattering by thin periodic layers

tends to 0) is a periodic graph. The localized perturbation consists in changing the geometry of the reference medium by modifying the thickness of one rung of the ladder. Considering the scalar Helmholtz equation with Neumann boundary conditions in this domain, we wonder whether such a geometrical perturbation is able to produce localized eigenmodes. To address this question, we use a standard approach of asymptotic analysis that consists of three main steps. We first find the formal limit of the eigenvalue problem as the

When a thin periodic layer meets corners: asymptotic analysis of a singular Poisson problem

\epsilon

Modèles et asymptotiques des interfaces fines et périodiques en électromagnétisme

tends to 0. In the present case, it corresponds to an eigenvalue problem for a second order differential operator defined along the periodic graph. Then, we proceed to an explicit calculation of the spectrum of the limit operator. Finally, we prove that the spectrum of the initial operator is close to the spectrum of the limit operator. In particular, we prove the existence of localized modes provided that the geometrical perturbation consists in diminishing the width of one rung of the periodic thin structure. Moreover, in that case, it is possible to create as many eigenvalues as one wants, provided that e is small enough. Numerical experiments illustrate the theoretical results.