The two-scale Fourier transform approach to homogenization; periodic homogenization in Fourier space

Article type: Research Article

Authors: Wellander, Niklas

Affiliations: Swedish Defence Research Agency, FOI, SE-581 11 Linköping, Sweden

Note: [] Address for correspondence: Niklas Wellander, Swedish Defence Research Agency, FOI, P.O. Box 1165, SE-581 11 Linköping, Sweden. Tel.: +46 13 37 80 00; Fax: +46 13 37 81 70; E-mail: niklas.wellander@foi.se

Abstract: A two-scale Fourier transform for periodic homogenization in Fourier space is introduced. The transform connects the various existing techniques for periodic homogenization, i.e., two-scale convergence, periodic unfolding and the Floquet–Bloch expansion approach to homogenization. It turns out that the two-scale compactness results are easily obtained by the use of the two-scale Fourier transform. Moreover, the Floquet–Bloch eigenvalue problems for differential operators is recovered in a natural and straight forward way by the use of this transform. The transform is generalized to the (N+1)-scale case.

Keywords: two-scale Fourier transform, two-scale convergence, weak convergence, homogenization, two-scale transform, periodic unfolding, periodic extension, Floquet–Bloch homogenization

DOI: 10.3233/ASY-2008-0914

Journal: Asymptotic Analysis, vol. 62, no. 1-2, pp. 1-40, 2009