Pixel-based boundary element method for computing effective thermal conductivity of heterogeneous materials

Abstract

In the present article, we propose an efficient 2D Pixel-based Boundary Element formulation to compute the effective thermal conductivity of heterogeneous materials. In the proposed formulation, each pixel of the digital image is represented by a subdomain with four boundary elements. We develop the formulation within the context of the Symmetric Galerkin Boundary Element Method and, apart from the Boundary Element Method itself, the other main ingredients are the Domain Decomposition technique, to treat each pixel as a subdomain, and the Element-by-Element technique used in conjunction with the Preconditioned Conjugate Gradient method to solve the resulting symmetric positive definite large linear system. In addition, Periodic Boundary Conditions are employed. We validate our formulation and implementation with the aid of analytical results for the case of a plate with periodic circular inclusions. Further, we illustrate the application of the proposed formulation by computing the effective thermal conductivity of a plate with periodic square inclusions, a woven-like material, and a cast iron sample, and compare the obtained results with those obtained via finite elements. Our results indicate that the Pixel-based Boundary Element formulation performs, at least, as good as the Pixel-based Finite Element formulation.