Homogenization of the time-dependent heat equation on planar one-dimensional periodic structures

TL;DR

This paper develops a homogenized model for the time-dependent heat equation on a periodic graph structure, deriving an effective two-dimensional heat equation with a conductivity tensor influenced by the graph's topology and edge lengths.

Contribution

It introduces a novel homogenization approach for the heat equation on periodic graphs, providing an explicit algebraic formula for the effective conductivity tensor.

Findings

Homogenized model is well-posed and derived explicitly.

Numerical experiments confirm convergence to the homogenized limit.

Convergence order in mesh period δ is demonstrated.

Abstract

In this paper we consider the homogenization of a time-dependent heat conduction problem on a planar one-dimensional periodic structure. On the edges of a graph the one-dimensional heat equation is posed, while the Kirchhoff junction condition is applied at all (inner) vertices. Using the two-scale convergence adapted to homogenization of lower-dimensional problems we obtain the limit homogenized problem defined on a two-dimensional domain that is occupied by the mesh when the mesh period $\delta$ tends to $0$. The homogenized model is given by the classical heat equation with the conductivity tensor depending on the unit cell graph only through the topology of the graph and lengthes of its edges. We show the well-posedness of the limit problem and give a purely algebraic formula for the computation of the homogenized conductivity tensor. The analysis is completed by numerical experiments showing a convergence to the limit problem where the convergence order in $\delta$.