A 3D face interpolated discretisation method for simulating anisotropic diffusive processes on meshes coming from wood morphology

Abstract

In this paper, a new 3D numerical discretisation method for solving the anisotropic, steady-state diffusion problem is developed and analysed. The scheme is constructed on hexahedral meshes using the geometrical properties of the cells. Indeed, each cell provides a local basis formed using the centres of its lateral faces. The discrete cell gradient approximation is then obtained by using three discrete directional derivatives resolved in terms of this basis, and by invoking the consistent relationships between opposite faces of the same cell. The face degrees of freedom are interpolated to reduce the complexity of the numerical scheme, resulting in the main unknowns being entirely nodal based. The scheme is unconditionally coercive and admits a unique solution. Various numerical experiments are performed to highlight the accuracy and the robustness of the method with respect to the mesh and anisotropy. An important outcome is that second order convergence is observed for all problems considered, even for highly deformed meshes. After this validation process, the method is applied to the prediction of the effective thermal conductivity of wood from its real 3D morphology. The property is estimated in the radial tangential and longitudinal directions. The solver is robust, efficient and yields to similar results compared to recent contributions.