A POD-EIM reduced two-scale model for crystal growth
- PDF / 2,696,839 Bytes
- 27 Pages / 439.642 x 666.49 pts Page_size
- 81 Downloads / 173 Views
A POD-EIM reduced two-scale model for crystal growth Magnus Redeker · Bernard Haasdonk
Received: 31 October 2013 / Accepted: 4 June 2014 / Published online: 9 July 2014 © Springer Science+Business Media New York 2014
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.
Communicated by: Karsten Urban This presentation is dedicated to Prof. Christof Eck. M. Redeker () · B. Haasdonk Institute for Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany e-mail: [email protected] B. Haasdonk e-mail: [email protected]
988
M. Redeker, B. Haasdonk
Keywords Model reduction · Proper orthogonal decomposition · Empirical interpolation · Parametrized two-scale model Mathematics Subject Classification (2010) 78M34
1 Introduction Many technically relevant processes exhibit a priori unknown microstructures that evolve in time. Important examples are solidification processes with dendritic and eutectic microstructures [7, 27], flow in porous media with changing pore geometry as a consequence of elastoplastic deformations and deposition or desorption of matter, and microstructures in epitaxial growth of thin solid layers. Due to huge differences in relevant length and possibly also time scales, it is usually not feasible to simulate such processes by a direct numerical discretization of a full model. A much more promising approach is the usage of homogenization or averaging techniques, that were originally developed with the aim to find purely macroscopic models with suitable constitutive laws that model the properties of the microstructure [6, 20, 22]. The applicat
Data Loading...
