Bloch wave approach to almost periodic homogenization and approximations of effective coefficients

Abstract

Bloch wave homogenization is a spectral method for obtaining effective coefficients for periodically heterogeneous media. This method hinges on the direct integral decomposition of periodic operators, which is not available in a suitable form for almost periodic operators. In particular, the notion of Bloch eigenvalues and eigenvectors does not exist for almost periodic operators. However, we are able to recover the almost periodic homogenization result by employing a sequence of periodic approximations to almost periodic operators. We also establish a rate of convergence for approximations of homogenized tensors for a class of almost periodic media. The results are supported by a numerical study.

Combined effects of homogenization and singular perturbations: A bloch wave approach

“…Remark 4.6. In the papers [33,34], an additional artificial parameter is introduced in the Bloch eigenvalue problem to facilitate the homogenization method.…”

“…Unlike (1.1), these papers employ successive limits of the two parameters instead of simultaneous limits. Therefore, the non-dependence of the neighbourhood of analyticity on the second parameter is not required in [33,34].…”