Homogenization of quasiperiodic structures and two-scale cut-and-projection convergence

TL;DR

This paper develops a mathematical framework using cut-and-projection methods to analyze the effective properties of quasiperiodic composite materials, with applications in physics such as electromagnetism and elasticity.

Contribution

It introduces a rigorous characterization of convergence limits and correctors for quasiperiodic structures, extending previous results and providing new examples.

Findings

Established convergence limits for partial differential operators in quasiperiodic media

Provided proofs of previous theoretical results and extended them with new examples

Applied the framework to electrostatic, elastostatic, and quasistatic magnetic problems

Abstract

Quasiperiodic arrangements of the constitutive materials in composites result in effective properties with very unusual electromagnetic and elastic properties. The paper discusses the cut-and-projection method that is used to characterize effective properties of quasiperiodic materials. Characterization of cut-and-projection convergence limits of partial differential operators is presented, and correctors are established. We provide the proofs of the results announced in (Wellander et al., 2018) and give further examples. Applications to problems of interest in physics include electrostatic, elastostatic and quasistatic magnetic cases.