Solving interface problems on three-dimensional microstructures is challenging due to the computational cost of resolving complex geometries and generating fitted meshes. Such problems arise, for example, in electrochemical systems, where microscopic transport processes occur in complex domains and are coupled across interfaces by reaction kinetics.
In this work, we derive a homogenized model using an asymptotic expansion approach. The resulting macroscopic model contains effective transport coefficients obtained from auxiliary cell problems, which capture the influence of the microscopic geometry. The cell problems are solved using a CutCell approach, while the fully resolved microscopic problem is computed with the Extended Finite Element Method (XFEM), allowing interface conditions to be imposed on linearized interfaces without requiring fitted meshes.
We demonstrate the approach for three-dimensional interface problems by comparing a fully resolved microscopic problem with the homogenized model. In particular, we investigate the convergence behavior and compare the XFEM discretization with a voxel-based approach.