Interactions Between Moderately Close Inclusions for the Laplace Equation

Abstract

The presence of small inclusions modifies the solution of the Laplace equation posed in a reference domain Ω 0 . This question has been deeply studied for a single inclusion or well separated inclusions. We investigate the case where the distance between the holes tends to zero but remains large with respect to their characteristic size. We first consider two perfectly insulated inclusions. In this configuration we give a complete multiscale asymptotic expansion of the solution to the Laplace equation. We also address the situation of a single inclusion close to a singular perturbation of the boundary ∂Ω 0 . We also present numerical experiments implementing a multiscale superposition method based on our first order expansion.

Artificial conditions for the linear elasticity equations

“…Among others, let us mention the following works using potential theory [38,5,3,4] and the reference monographs [30,31] for multiscale expansions. Following [10,36], the solution of (1.1) is given to first order by…”

“…When the distance between the x j cannot be assumed large with respect to ε, e.g. when x i − x j ≈ ε α for α ∈ (0, 1), the order of the error term in (1.2) is reduced (see [10]). The profiles v j 1 and v j 2 are obtained as solutions of a homogeneous Navier equation posed on the unbounded domain H j ∞ with Neumann conditions on the boundary of the normalized perturbation:…”

Multiscale expansion and numerical approximation for surface defects

“…This paper is a survey of articles [5,6,8,9,13,17,18]. We are interested in the influence of small geometrical perturbations on the solution of elliptic problems.…”

“…We provide a complete asymptotic description of the solution of the Laplace equation. We also present numerical simulations based on the multiscale superposition method derived from the first order expansion (cf [9]). We give an application of theses techniques in linear elasticity to predict the behavior till rupture of materials with microdefects (cf [6]).…”

“…Nous donnons un développement asymptotique multi-échelle complet de la solution de l'équation de Laplace dans la situation de deux inclusions. Nous présentonségalement quelques simulations numériques basées sur une méthode de superposition multi-échelle provenant du développement au premier ordre (cf [9]). Nousétendons ces techniques auxéquations de l'élasticité linéaire afin de prédire le comportementà rupture de certains matériaux présentant des micro-défauts (cf [6]).…”