A finite volume scheme suitable for nonlinear heat transfer in materials with anisotropic thermal conductivity is formulated, focussing on the difficulties arising from the discretization of complex domains which are typical in the simulation of industrially relevant processes. The discretization is based on unstructured constrained Delaunay triangulations of the domain. For simplicity, it is assumed that the thermal conductivity tensor has vanishing off-diagonal entries and that the anisotropy is independent of the temperature. Numerical simulations verify the accuracy of the method in two test 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.
Using a mathematical heat transfer model including anisotropic heat conduction, radiation, and radio frequency (RF) heating, we perform numerical computations of the temperature field in an axisymmetric growth apparatus during sublimation growth of silicon carbide (SiC) bulk single crystals by physical vapor transport (PVT) (modified Lely method). Because it is not unusual for the thermal insulation of a PVT growth apparatus to possess an anisotropic thermal conductivity, we numerically study the influence that this anisotropic thermal conductivity has on the temperature field in the growth chamber. Moreover, we also study the influence of the thickness of the insulation. Our results show that, depending on the insulation's orientation, even a moderate anisotropy in the insulation can result in temperature variations of more than 100 K at the growing crystal's surface, which should be taken into account for the simulation as well as for the design of a PVT growth apparatus.
The operator-splitting methods are based on splitting of the complex problem into a sequence of simpler tasks. A useful method is the iterative splitting method which ensures a consistent approximation in each step. In our paper, we suggest a new method which is based on the combination of the splitting time interval and the traditional iterative operator splitting. We analyse the local splitting error of the method. Numerical examples are given in order to demonstrate the method.