Fixed-point convergence of modular, steady-state heat transfer models coupling multiple scales and phenomena for melt-crystal growth

A block Gauss-Seidel iteration procedure is employed to couple two independent heat transfer codes representing different scales and phenomena in a melt-crystal growth system via shared domain boundaries. In general, such strategies are attractive because of their simplicity for solving problems that may be represented by linking, via shared boundary conditions, existing codes using a modular design. However, these approaches often meet with limited success due to convergence difficulties associated with the overall model. The mathematical framework of fixed-point iterations is employed to assess convergence behaviour for steady-state, non-linear heat transfer models. Analytical forms describe the convergence of one-dimensional problems and provide insight to the behaviour of more complicated, two-dimensional models. Certain implementations that are physically reasonable result in algorithms that will never converge, demonstrating that notions based on physical intuition may not be useful for predicting algorithm performance. A mixing of flux and temperature information along the shared model boundary leads to successful iterative strategies under many conditions.