Novel analytical solution of a 2D heterogeneous diffusion problem with application to determining effective diffusion tensors

• New analytical solution method for locally-isotropic pixelated heterogeneous domains • New definition of effective diffusion tensor for pixelated heterogeneous media • Example application: homogenisation, including application to microstructure material • More accurate than semi-analytical technique and computationally efficient • Technique applicable to general problems defined on pixelated heterogeneous domains In this paper, we develop a novel analytical solution method for a steady-state diffusion equation on a locally-isotropic blocky or pixelated heterogeneous domain. Our approach involves decoupling the governing boundary-value problem through the introduction of unknown functions at the interfaces between individual homogeneous blocks. This approach allows exact solutions on each individual block to be obtained, expressed in terms of integrals involving the unknown interface functions. Applying appropriate continuity conditions at the interfaces then allows the construction of the solution over the full heterogeneous domain. Current practice for determining the integrals involving the unknown interface functions is to utilise numerical quadrature, yielding a semi-analytical solution. The novelty of our method involves determination of the integrals directly, yielding a fully analytical solution. Applications of the steady-state diffusion equation are numerous, however the specific example we demonstrate in this paper is homogenisation: a commonly utilised technique for upscaling flow through complex heterogeneous media. We demonstrate how our new analytical solution can be applied to compute effective diffusion tensors directly from a microCT image of the microstructure of a porous lignocellulosic material, and make comparisons against both a benchmark numerical method and a previously published semi-analytical method. In comparison, we find that our new analytical method yields a high degree of accuracy in comparable runtime.