A fast algorithm for semi-analytically solving the homogenization boundary value problem for block locally-isotropic heterogeneous media

• Published in: Applied Mathematical Modelling 2021-04, Vol.92, p.23-43
• Main Authors: March, Nathan G., Carr, Elliot J., Turner, Ian W.
• Format: Article
• Language: English
• Subjects: Algorithms; Boundary value problems; Computational efficiency; Computer simulation; Computing costs; Computing time; Diffusion barriers; Diffusion rate; Diffusivity; Direct numerical simulation; Effective diffusivity; Finite volume method; Heterogeneous media; Homogenization; Mathematical analysis; Mathematical models; Semi-analytical solution; Steady-state diffusion equation
• ISSN: 0307-904X; 1088-8691; 0307-904X
• DOI: 10.1016/j.apm.2020.09.022

•New semi-analytical method for solving the homogenization boundary value problem.•Computes the effective diffusivity for block locally-isotropic heterogeneous media.•New method can capture highly complex heterogeneous geometries.•Faster than a standard finite volume method.•Potential to speed up coarse-scale simulations of diffusion in heterogeneous media. Direct numerical simulation of diffusion through heterogeneous media can be difficult due to the computational cost of resolving fine-scale heterogeneities. One method to overcome this difficulty is to homogenize the model by replacing the spatially-varying fine-scale diffusivity with an effective diffusivity calculated from the solution of an appropriate boundary value problem. In this paper, we present a new semi-analytical method for solving this boundary value problem and computing the effective diffusivity for pixellated, locally-isotropic, heterogeneous media. We compare our new solution method to a standard finite volume method and show that equivalent accuracy can be achieved in less computational time for several standard test cases. We also demonstrate how the new solution method can be applied to complex heterogeneous geometries represented by a two-dimensional grid of rectangular blocks. These results indicate that our new semi-analytical method has the potential to significantly speed up simulations of diffusion in heterogeneous media.