Homogenization Of Eigenvalue Problem Research Articles

English

The goal of this paper is to establish a general homogenization result for linearized elasticity of an eigenvalue problem defined over perforated domains, beyond the periodic setting, within the framework of the H0-convergence theory. Our main homogenization result states that the knowledge of the fourth-order tensor A0, the H0-limit of Ae, is sufficient to determine the homogenized eigenvalue problem and preserve the structure of the spectrum. This theorem is proved essentially by using Tartar’s method of test functions, and some general arguments of spectral analysis used in the literature on the homogenization of eigenvalue problems. Moreover, we give a result on a particular case of a simple eigenvalue of the homogenized problem. We conclude our work by some comments and perspectives.

Homogenization of eigenvalue problems of elasticity in perforated domains

Nandakumar, AK., Homogenization of eigenvalue problems of elasticity in perforated domains, Asymptotic Analysis 9 (1994) 337-358. In this paper, we study the homogenization of eigenvalue problem associated with the elasticity system in a periodically (with period e > 0, a small parameter) perforated domain with tiny holes. The critical size of the holes a. is given by a. = Coe N IN -2 if N~3 and a. = exp( -Col e2) if N = 2, where Co is a constant and N is the dimension. We will study the above eigenvalue problem as e -> ° and will obtain the homogenized system. We also study the correctors for the eigenvalues and eigenvectors.

Homogenization of eigenvalue problems in perforated domains

In this paper, we treat some eigenvalue problems in periodically perforated domains and study the asymptotic behaviour of the eigenvalues and the eigenvectors when the number of holes in the domain increases to infinity Using the method of asymptotic expansion, we give explicit formula for the homogenized coefficients and expansion for eigenvalues and eigenvectors. If we denote by ε the size of each hole in the domain, then we obtain the following aysmptotic expansion for the eigenvalues: Dirichlet: λε = ε−2 λ + λ0 +O (ε), Stekloff: λε = ελ1 +O (ε2), Neumann: λε = λ0 + ελ1 +O (ε2). Using the method of energy, we prove a theorem of convergence in each case considered here. We briefly study correctors in the case of Neumann eigenvalue problem.