Multiscale expansion and numerical approximation for surface defects

Abstract

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. The cases of a single inclusion or several well-separated inclusions have been deeply studied. We recall here techniques to construct an asymptotic expansion. Then we consider moderately close inclusions, i.e. the distance between the inclusions tends to zero more slowly than their characteristic size. 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]). We explain how some mathematical questions about the loss of coercivity arise from the computation of the profiles appearing in the expansion (cf [8]).