(291a) Efficient Strategies for Coupling Multi-Scale Models for Bulk Crystal Growth

The modeling of bulk 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.

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.  Such methods can be used to link together existing best-in-class tools to tackle complex multi-physics and multiscale problems, without requiring extraordinary programming effort.

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. The most promising implementation is 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, CrysVUn developed by the Crystal Growth Laboratory, Erlangen, Germany, that computes high-temperature, furnace heat transfer, with the local model, Cats2D developed by the Derby group, which solves for heat transfer, incompressible melt flow, and melt-crystal interface shape.  Prospects for quantitative process modeling and the ability to represent three-dimensional and transient phenomena in bulk crystal growth are discussed.

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This work has been supported in part by the Department of Energy, National Nuclear Security Administration, under Award Numbers DE-FG52-06NA27498 and DE-FG52-08NA28768, the content of which does not necessarily reflect the position or policy of the United States Government, and no official endorsement should be inferred.