Outline of Nguetseng's approach to non-periodic homogenization

Abstract

Abstract. Classical homogenization theory developed for modeling heterogeneous media with regular fine structure is based on the assumption that the structure is periodic. Since the structure of real materials is not perfectly periodic, several approaches were proposed. The approach proposed by Gabriel Nguetseng in 2003 seems to be the most general deterministic approach, it covers periodic, almostperiodic and other structures. It is based on the notion of Banach algebra spectrum. The aim of this survey paper is to give a brief self-contained outline of this rather complicated approach described in many-page papers.

Convergence Along Mean Flows

“…Allaire [1]. In Section 2, we have formally derived the homogenized limit and obtained an explicit expression for the effective diffusion (20). As mentioned in Remark 6, there was an inherent assumption that the limits in the fast time variable exist and are finite.…”

“…As we do not aim to extend this theory beyond what already exists, we shall not give the theory in full generality and we refer the reader to existing literature (e.g. [13,31,32,33,38,7], see also [20] for an introductory exposition and [25] for a pedagogical exposition) for a more complete presentation and full proofs. In subsections 3.6-3.8 we introduce the new concept of Σ-convergence along flows and prove compactness results.…”

“…Remark 6. The expression (20) for the effective diffusion involves the averaging in the fast time variable. In this section dealing with formal derivation of the homogenized limit, we admit that the limits in the expression of the effective diffusion exist and are finite.…”

“…Remark 7. An interesting feature in the expression (20) is that the integrands are all in their flow representations. This suggests that the effective diffusion is the cumulative effect of the convection and diffusion effects averaged along the flows.…”