Journal of Crystal Growth 303 (2007) 114–123 Strategies for the coupling of global and local crystal growth models Jeffrey J. Derby , Lisa Lun, Andrew Yeckel Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455-0132, USA Available online 23 January 2007 Abstract The modular coupling of existing numerical codes to model crystal growth processes will provide for maximum effectiveness, capability, and flexibility. However, significant challenges are posed to make these coupled models mathematically self-consistent and algorithmically robust. This paper presents sample results from a coupling of the CrysVUn code, used here to compute furnace-scale heat transfer, and Cats2D, used to calculate melt fluid dynamics and phase-change phenomena, to form a global model for a Bridgman crystal growth system. However, the strategy used to implement the CrysVUn–Cats2D coupling is unreliable and inefficient. The implementation of under-relaxation within a block Gauss–Seidel iteration is shown to be ineffective for improving the coupling performance in a model one-dimensional problem representative of a melt crystal growth model. Ideas to overcome current convergence limitations using approximations to a full Newton iteration method are discussed. r 2007 Elsevier B.V. All rights reserved. PACS: 02.60.Cb; 02.70.c; 81.10.h; 81.10.Aj; 81.10.Fq Keywords: A1. Computer simulation; A1. Directional solidification; A1. Heat transfer; A2. Bridgman technique; A2. Growth from melt 1. Introduction 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 [1]. 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. There are also mathematical and algorithmic challenges for the coupling of different models that describe these phenomena at different scales. In the following study, we desire to address several modeling and mathematical issues associated with the coupling of global-scale furnace heat transfer models and local-scale models for melt crystal growth. Our goal from a software or model development perspective is to make use of existing codes that are appropriate for the tasks at hand and couple them together in a modular manner that is mathematically self-consistent and algorithmically robust. Furthermore, we desire to maintain maximum flexibility in the use of these code modules. We desire to treat each code as if it were a ‘‘black box,’’ meaning that we may not have access to its internal mathematical workings. We also demand that the codes exchange a minimum amount of information in the overall model formulation. Specifically, we propose that informa- tion be passed only along predefined domain boundaries. For the solution of the governing partial differential equations of interest here, this strategy is equivalent to setting and exchanging boundary conditions for different models. A strategy that simultaneously represents all chosen phenomena in all domains in a single, large mathematical model is referred to as a monolithic, analytic, or direct- coupling approach [2]. 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 [3–7]. However, such approaches require inten- sive and coordinated programming efforts, are typically system-specific, and are often difficult to maintain and modify. ARTICLE IN PRESS www.elsevier.com/locate/jcrysgro 0022-0248/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2006.11.343 Corresponding author. Tel.: +1 612 625 8881; fax: +1 612 626 7246. E-mail address: derby@umn.edu (J.J. Derby).