Numerical Simulation of Heat Transfer in Materials with Anisotropic Thermal Conductivity: a Finite Volume Scheme to Handle Complex Geometries

We devise a finite volume scheme for nonlinear heat transfer in materials with anisotropic thermal conductivity. We focus on the difficulties arising from the discretization of complex domains which are typical in the simulation of industrially relevant processes. For polyhedral domains in two dimensions, we consider Cartesian as well as cylindrical coordinates. Our finite volume scheme is based on unstructured constrained Delaunay triangulations of the domain. For simplicity, we assume that the thermal conductivity tensor has vanishing off-diagonal entries and that the anisotropy is independent of the temperature. We present numerical simulations, verifying our finite volume scheme in cases where a closed-form solution is available. Further results demonstrate the effectiveness of the method in computing the heat transfer in a complex growth apparatus used in crystal growth.