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

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 δ 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 δ depends on the unit cell pattern.

2020 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 80M40
• 35R02

Acknowledgments

The authors would like to thank Luka Grubišić (University of Zagreb), Herbert Egger (TU Darmstadt) and Andro Mikelić (University of Lyon 1) for fruitful discussions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors gratefully acknowledge the financial support of their research cooperation ‘Asymptotic and algebraic analysis of nonlinear eigenvalue problems for biomechanical and photonic devices’ through the bilateral program ‘Procope’ between the German Academic Exchange Service (DAAD) based on funding of the German Federal Ministry of Education and Research [project ID 57334847] and the Croatian Ministry of Science (contract number 910-08/16-01/00209). The research was partly conducted during the stay of the first author at the Graduate School Computational Engineering (CE) at the Technical University (TU) Darmstadt and he is grateful to the support and hospitality. Work of the first and the last author has been supported by the [grant number HRZZ 2735] of the Croatian Science Foundation and the COST Action: CA18232 - Mathematical models for interacting dynamics on networks.