A POD-EIM reduced two-scale model for crystal growth

Abstract

Complex physical models depending on microstructures developing over time often result in simulation schemes that are very demanding concerning computational time. The two-scale model considered in the current presentation describes a phase transition of a binary mixture with the evolution of equiaxed dendritic microstructures. It consists of a macroscopic heat equation and a family of microscopic cell problems modeling the phase transition. Those phase transitions need to be resolved by very fine computational meshes leading to the demanding numerical complexity. The current study presents a reduced version of this two-scale model. The reduction aims at accelerating the microscopic model, which is parametrized by the macroscopic temperature, while maintaining the accuracy of the detailed system. Parameter dependency, non-linearity, time-dependency, coupled field-variables and high solution complexity are challenging difficulties. They are addressed by a combination of several approaches: Proper Orthogonal Decomposition (POD), Empirical Interpolation Method (EIM) and a partitioning approach generating sub-models for different solution regimes. A new partitioning criterion based on feature extraction is applied. The applicability of the reduction scheme is demonstrated experimentally: while the accuracy is largely maintained, the dimensionality of the detailed model and the computation time are reduced significantly.