Notes on an intuitive approach to elliptic homogenization

TL;DR

This paper presents an intuitive, physically motivated derivation of homogenized coefficients for elliptic problems, avoiding perturbation theory, and extends the discussion to heat conduction on curved surfaces with multiscale features.

Contribution

It introduces a non-perturbative, physically motivated approach to elliptic homogenization and explores homogenization of the Laplace-Beltrami operator on curved surfaces.

Findings

Derived homogenized coefficients without perturbation theory

Extended homogenization concepts to curved surfaces with multiscale curvature

Provided insights into heat transfer in complex geometries

Abstract

Elliptic homogenization is used to determine coarse-grained properties of materials with features on small scales for heat transfer and elasticity. When microstructural features of a material have rapid, periodic fluctuations, the solution corresponding to a "homogenized" coefficient field closely resembles the true solution based on the heterogeneous material. Most presentations of elliptic homogenization rely on methods from perturbation theory, which can make an intuitive, physical understanding of the homogenized coefficients elusive. In this set of notes, we derive the homogenized coefficients for one- and two-dimensional elliptic boundary value problems based on arguments which are physically motivated, and with no recourse to perturbation theory. Then, we discuss homogenization of the Laplace-Beltrami operator for heat conduction on thin surfaces with multiscale curvature, an example which has seen minimal treatment in existing literature.