Bloch Wave Homogenization of Quasiperiodic Media

TL;DR

This paper develops a novel approach to homogenize quasiperiodic media by lifting almost periodic operators to higher-dimensional periodic operators, enabling the use of Bloch wave analysis for effective coefficient computation.

Contribution

It introduces a method to analyze quasiperiodic media via lifting to periodic operators and defines a quasiperiodic Bloch transform for homogenization.

Findings

Homogenized coefficients are expressed through the first Bloch eigenvalue of the lifted operator.

A quasiperiodic Bloch transform is defined for analyzing highly oscillating coefficients.

The approach successfully derives homogenization limits for quasiperiodic media.

Abstract

Quasiperiodic media is a class of almost periodic media which is generated from periodic media through a "cut and project" procedure. Bloch waves are typically defined through a direct integral decomposition of periodic operators. A suitable direct integral decomposition is not available for almost periodic operators. To remedy this, we lift an almost periodic operator to a degenerate periodic operator in higher dimensions. Approximate Bloch waves are obtained for a regularized version of the degenerate equation. Homogenized coefficients for quasiperiodic media are determined in terms of the first Bloch eigenvalue of the regularized lifted equation. A notion of quasiperiodic Bloch transform is defined and employed to obtain homogenization limit for an equation with highly oscillating quasiperiodic coefficients.