Notes on an intuitive approach to elliptic homogenization

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.

Figures (10)

• Figure 1: The heterogeneity of this fibrous material is apparent when viewed with an electron micrograph. The microstructure consists of a tangle of curved fibers with empty space between them. Image taken from kaviany_fluid_1991.
• Figure 2: Intuitively, it is not necessary to have fine-grained knowledge of the microstructure of the material of a heated body in order to calculate the temperature field $U(\mathbf X)$. At least, this is the case so long as the body is large compared to the length scale on which the material is heterogeneous. This is the notion of scale separation that is often encountered in the homogenization literature.
• Figure 3: As the frequency of the periodic oscillations of the material property increases, the corresponding oscillations in the solution die out. More concretely, in the presence of rapid fluctuations of the conductivity, the temperature responds as if the material were constant. The goal of elliptic homogenization is to determine this effective material property.
• Figure 4: A bar with periodic material heterogeneity. We think of one period of the fluctuation as the microstructure, and the domain as built from these cells of microstructure.
• Figure 5: Computing the homogenized conductivity with Eq. \ref{['kappahat']}, we solve the boundary value problem of Eq. \ref{['homogenized_1d']}. Even when the cells are large compared to the length of the bar, the homogenized solution provides a reasonable approximation of the true temperature field.