Homogenization of Enhancing Thin Layers

TL;DR

This paper develops an explicit formula for the effective diffusion tensor in homogenized media with complex boundary conditions, extending the concept of Road effective boundary conditions to patterns and analyzing their properties.

Contribution

It introduces a new explicit formula for the effective diffusion tensor, extends Road EBCs to patterns including nodes, and proves the homogenization process commutes with deriving these boundary conditions.

Findings

Derived explicit formula for the effective diffusion tensor.

Extended Road EBCs to include patterns and nodes.

Provided rules for maximizing the trace of the diffusion tensor.

Abstract

This paper derives an explicit formula for the effective diffusion tensor by using the solutions to some effective cell problems after homogenizing Road effective boundary conditions (EBCs). The concept of Road EBCs was proposed recently by H. Li and X. Wang, and in this paper, we extend the effective conditions on closed curves to those on patterns, especially on the included nodes. We also prove that homogenization process commutes with the derivation of Road EBCs. By analyzing the effective diffusion tensor, we obtain several rules for maximizing its trace with given Road-effective-diffusivity/scale and length/scale in each cell and define a notion of balanced patterns. Moreover, we give an estimate of the trace of effective diffusion tensor of general patterns as Road-effective-diffusivity/scale goes to infinity.