The Hole-Filling Method and the Uniform Multiscale Computation of the Elastic Equations in Perforated Domains

Abstract

In this paper, we discuss the boundary value problem for the linear elastic equations in a perforated domain $\Omega^{\varepsilon}$. We fill all holes with a very compliant material, then we study the homogenization method and the multiscale analysis for the associated multiphase problem in a domain $\Omega$ without holes. We are interested in the asymptotic behavior of the solution for the multiphase problem as the material properties of one weak phase go to zero, which has a wide range of applications in shape optimization and in 3-D mesh generation. The main contribution obtained in this paper is to give a full mathematical justification for this limiting process in general senses. Finally, some numerical results are presented, which support strongly the theoretical results of this paper.