The modeling of melt crystal growth processes is an inherently multi-scale challenge, with relevant length scales ranging from furnace dimensions to atomic-sized features in the grown crystal. A faithful depiction of the interaction of these scales is needed to advance our scientific understanding of crystal growth processes, as well as to make our process models quantitatively predictive and technologically useful. However, efficient and robust techniques are needed to couple strongly nonlinear conjugate heat transfer with models that depict phase change and crystal growth.

A strategy that simultaneously represents all chosen phenomena at all scales in a single, large mathematical model is referred to as a monolithic, analytic, or direct-coupling approach. From the points of view of mathematical self- consistency and algorithmic robustness, this course of action is preferred and is the chosen strategy for many global- scale models for melt crystal growth processes. However, such approaches require intensive and coordinated programming efforts, are typically system-specific, and are often difficult to maintain and modify. Due to these challenges, there is an increasing desire for innovative ways to couple existing software that have been developed to solve specific problems, especially for modeling multi-scale and multi-physics problems. Such alternative approaches are modular, partitioned, or synthetic, in which different models and solvers are used together. In this manner, software tools that have desired features can be exploited to enhance existing capabilities.

We discuss the mathematical and algorithmic challenges for the modular coupling of global-scale furnace heat transfer models and local-scale models for melt crystal growth based on an innovative Block-Newton approach implemented using a Jacobian-free Newton-Krylov algorithm. To clarify some of the underlying issues, we present initial studies of this and other approaches using simple models. Then we present our experience with the coupling of the global model, CrysMAS developed by the Crystal Growth Laboratory, Erlangen, Germany, that computes high-temperature, furnace heat transfer, with the local model, Cats2D and Cats3D, developed by the Derby group to solve for heat transfer, incompressible melt flow, and melt-crystal interface shape.