[Paper Review] Homogenization of quasiperiodic structures and two-scale cut-and-projection convergence

Wellander, Niklas, Sébastien Guenneau+1 more|arXiv (Cornell University)|Nov 8, 2019

Acoustic Wave Phenomena Research72 references157 citations

TL;DR

The paper develops and proves a two-scale cut-and-projection homogenization framework for quasiperiodic materials, establishing convergence, correctors, and applying it 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.

Motivation & Objective

• Motivate the study of effective properties for quasiperiodic composites and the need for homogenization tools beyond periodic settings.
• Describe and formalize the cut-and-projection approach for modeling quasiperiodic materials as higher-dimensional periodic structures.
• Establish two-scale convergence and corrector results in the quasiperiodic setting.
• Prove results that extend earlier work and illustrate with physical problems in electrostatics, elastostatics, and quasistatic magnetism.

Proposed method

• Introduce the cut-and-projection operator to model quasiperiodic materials as projections of higher-dimensional periodic structures.
• Develop distributional and weak two-scale convergence notions associated with a linear map R that satisfies an irrationality condition.
• Define R-dependent differential operators (grad_R, div_R, curl_R) and corresponding function spaces, and establish compactness and decomposition results.
• Provide corrector-type results for gradients in the quasiperiodic two-scale limit.
• Prove theorems that extend prior results (Wellander et al., 2018) and apply to electrostatic, elastostatic, and quasistatic magnetic PDEs.

Experimental results

Research questions

• RQ1 How can quasiperiodic material microstructures be analyzed using a two-scale homogenization framework via cut-and-projection?
• RQ2 What are the corrector results and limit models for PDEs with quasiperiodic coefficients under two-scale cut-and-projection convergence?
• RQ3 How do electrostatic, elastostatic, and quasistatic magnetic problems homogenize under quasiperiodic assumptions?
• RQ4 What role do the R-dependent differential operators and function spaces play in deriving the homogenized limits?
• RQ5 Do the homogenized properties depend on the specific cut, i.e., the matrix R, or are they invariant under the irrationality condition?

Key findings

• A two-scale cut-and-projection convergence framework is established for quasiperiodic media.
• Corrector results for gradients are derived in the quasiperiodic setting.
• The paper proves convergence results and provides the necessary functional-analytic tools for handling grad_R, div_R, and curl_R operators.
• The results extend previous statements and are illustrated through electrostatic, elastostatic, and quasistatic magnetic problems.
• The homogenization approach uses higher-dimensional periodic representations and a projection-based analysis that yields effective properties for quasiperiodic composites.