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This article focuses on the intermediate length scales, where transport phenomena govern the spatial and temporal evolution of a structure. It presents the cellular automaton (CA) and phase field (PF) methods that represent the state of the art for modeling macrostructure and microstructure. The article describes the principles of the PF method and provides information on the applications of the PF method. The CA model is introduced as a computationally efficient method to predict grain structures in castings using the mesoscopic scale of individual grains. The article discusses the coupling of the CA to macroscopic calculation of heat, flow, and mass transfers in castings and applications to realistic casting conditions.

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Ch.-A. Gandin, I. Steinbach, Modeling of Microstructure Evolution during Solidification Processing, Fundamentals of Modeling for Metals Processing, Vol 22A, ASM Handbook, Edited By D.U. Furrer, S.L. Semiatin, ASM International, 2009, p 312–321,

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