Numerical experiments with the Bloch–Floquet approach in homogenization
Carlos Conca, Srinivasan Natesan, M. Vanninathan · International Journal for Numerical Methods in Engineering · 2005
Abstract This paper deals with a numerical study of classical homogenization of elliptic linear operators with periodic oscillating coefficients (period εY). 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 solutionuε. The novelty of their approach consists of using the spectral decomposition of the operator on ℝNto obtain a new approximation ofuε—the so‐called Bloch approximation θε—which provides an alternative to the classical two‐scale expansionuε(x)=u*(x)+Σεkuk(x,x/ε), and therefore, θεcontains implicitly at least the homogenized solutionu* and the first‐ and second‐order corrector terms. The Bloch approximation θεis obtained by computing, for every value of the Bloch variable η in the reciprocal cellY′ (Brillouin zone), the components ofu* 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 θεis closer touε, i.e. is a better approximation ofuεthan the first‐ and second‐order corrector terms, specifically in the case of high‐contrast materials. Copyright © 2005 John Wiley & Sons, Ltd.