Numerical experiments with the Bloch–Floquet approach in homogenization
Author
dc.contributor.author
Conca Rosende, Carlos
Author
dc.contributor.author
Natesan, S.
es_CL
Author
dc.contributor.author
Vanninathan, Muthusamy
es_CL
Admission date
dc.date.accessioned
2013-12-30T18:37:41Z
Available date
dc.date.available
2013-12-30T18:37:41Z
Publication date
dc.date.issued
2006
Cita de ítem
dc.identifier.citation
Int. J. Numer. Meth. Engng 2006; 65:1444–1471
en_US
Identifier
dc.identifier.other
DOI: 10.1002/nme.1502
Identifier
dc.identifier.uri
https://repositorio.uchile.cl/handle/2250/125909
Abstract
dc.description.abstract
This paper deals with a numerical study of classical homogenization of elliptic linear operators
with periodic oscillating coefficients (period 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 . The novelty of their approach consists of using
the spectral decomposition of the operator on RN to obtain a new approximation of u —the socalled
Bloch approximation —which provides an alternative to the classical two-scale expansion
u (x)=u
∗
(x) + kuk(x, x/ ), and therefore, contains implicitly at least the homogenized solution
u
∗ and the first- and second-order corrector terms.
The Bloch approximation is obtained by computing, for every value of the Bloch variable
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 is closer to u , i.e. is a
better approximation of u than the first- and second-order corrector terms, specifically in the case of
high-contrast materials.