Numerical experiments with the Bloch-Floquet approach in homogenization

Abstract

This paper deals with a numerical study of classical homogenization of elliptic linear operators with periodic oscillating coefficients (period epsilon Y). The importance of such problems in engineering applications is quite well-known. A method introduced by Conca and Vanninathan [SIAM J. Appl. Math. 1997; 57:1639-1659] based on Bloch waves that homogenize this kind of operators is used for the numerical approximation of their solution u(epsilon). The novelty of their approach consists of using the spectral decomposition of the operator on R-N to obtain a new approximation of u(epsilon)-the so-called Bloch approximation theta(epsilon)-which provides an alternative to the classical two-scale expansion u(epsilon)(x) = u*(x) + Sigma epsilon(k) u(k)(x,x/epsilon), and therefore, theta(epsilon) contains implicitly at least the homogenized solution u* and the first- and second-order corrector terms. The Bloch approximation theta(epsilon) is obtained by computing, for every value of the Bloch variable eta in the reciprocal cell Y' (Brillouin zone), the components of u* on the first Bloch mode associated with the periodic structure of the medium. Though theoretical basis of the method already exists, there is no evidence of its numerical performance. The main goal of this paper is to report on some numerical experiments including a comparative study between both the classical and Bloch approaches. The important conclusion emerging from the numerical results states that theta(epsilon) is closer to u(epsilon), i.e. is a better approximation of u(epsilon) than the first- and second-order corrector terms, specifically in the case of high-contrast materials.