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Numerical experiments with the Bloch–Floquet approach in homogenization
Author(s) -
Conca C.,
Natesan S.,
Vanninathan M.
Publication year - 2005
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1502
Subject(s) - bloch wave , homogenization (climate) , mathematics , operator (biology) , mathematical analysis , numerical analysis , floquet theory , brillouin zone , mathematical physics , quantum mechanics , physics , nonlinear system , biodiversity , ecology , biochemistry , chemistry , repressor , gene , transcription factor , biology
Abstract This paper deals with a numerical study of classical homogenization of elliptic linear operators with periodic oscillating coefficients (period ε 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 ε . The novelty of their approach consists of using the spectral decomposition of the operator on ℝ N to obtain a new approximation of u ε —the so‐called Bloch approximation θ ε —which provides an alternative to the classical two‐scale expansion u ε ( x )= u *( x )+Σε k u k ( x , x /ε), and therefore, θ ε contains implicitly at least the homogenized solution u * and the first‐ and second‐order corrector terms. The Bloch approximation θ ε is obtained by computing, for every value of the Bloch variable η 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 θ ε is closer to u ε , i.e. is a better approximation of u ε than the first‐ and second‐order corrector terms, specifically in the case of high‐contrast materials. Copyright © 2005 John Wiley & Sons, Ltd.

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