Improved algorithm for analytical solution of the heat conduction problem in doubly periodic 2D composite materials

TL;DR

This paper introduces an improved algorithm for solving the heat conduction problem in doubly periodic 2D composite materials, enabling efficient computation of temperature and flux distributions with high accuracy.

Contribution

The paper presents a novel, more efficient algorithm for solving boundary value problems in doubly periodic composites using complex potentials and functional equations.

Findings

High computational efficiency demonstrated through examples

Algorithm accurately reconstructs temperature and flux at any point

Provides indirect estimates of solution accuracy

Abstract

We consider a boundary value problem in unbounded 2D doubly periodic composite with circular inclusions having arbitrary constant conductivities. By introducing complex potentials, the boundary value problem for the Laplace equation is transformed to a special R-linear BVP for doubly periodic analytic functions. This problem is solved with use of the method of functional equations. The R-linear BVP is transformed to a system of functional equations. A new improved algorithm for solution of the system is proposed. It allows one not only to compute the average property but to reconstruct the solution components (temperature and flux) at an arbitrary point of the composite. Several computational examples are discussed in details demonstrating high efficiency of the method. Indirect estimate of the algorithm accuracy has been also provided.