Numerical investigation into coarse-scale models of diffusion in complex heterogeneous media

TL;DR

This paper explores how different factors affect the accuracy of coarse-scale diffusion models in complex heterogeneous media, emphasizing the importance of homogenization cell size and boundary conditions for computational efficiency and precision.

Contribution

It provides new insights into the error sources in homogenization and demonstrates optimal choices for boundary conditions and cell size in coarse-scale diffusion modeling.

Findings

Periodic boundary conditions are optimal for homogenization cells.

Using the smallest feasible homogenization cell improves accuracy.

Error analysis guides better coarse-scale model design.

Abstract

Computational modelling of diffusion in heterogeneous media is prohibitively expensive for problems with fine-scale heterogeneities. A common strategy for resolving this issue is to decompose the domain into a number of non-overlapping sub-domains and homogenize the spatially-dependent diffusivity within each sub-domain (homogenization cell). This process yields a coarse-scale model for approximating the solution behaviour of the original fine-scale model at a reduced computational cost. In this paper, we study coarse-scale diffusion models in block heterogeneous media and investigate, for the first time, the effect that various factors have on the accuracy of resulting coarse-scale solutions. We present new findings on the error associated with homogenization as well as confirm via numerical experimentation that periodic boundary conditions are the best choice for the homogenization cell and demonstrate that the smallest homogenization cell that is computationally feasible should be used in numerical simulations.