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Multiscale Problems in Solidification Processes

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Analysis, Modeling and Simulation of Multiscale Problems

Summary

Our objective is to describe solidification phenomena in alloy systems. In the classical approach, balance equations in the phases are coupled to conditions on the phase boundaries which are modelled as moving hypersurfaces. The Gibbs-Thomson condition ensures that the evolution is consistent with thermodynamics. We present a derivation of that condition by defining the motion via a localized gradient flow of the entropy. Another general framework for modelling solidification of alloys with multiple phases and components is based on the phase field approach. The phase boundary motion is then given by a system of Allen-Cahn type equations for order parameters. In the sharp interface limit, i.e., if the smallest length scale β related to the thickness of the diffuse phase boundaries converges to zero, a model with moving boundaries is recovered. In the case of two phases it can even be shown that the approximation of the sharp interface model by the phase field model is of second order in β. Nowadays it is not possible to simulate the microstructure evolution in a whole workpiece. We present a two-scale model derived by homogenization methods including a mathematical justification by an estimate of the model error.

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Author information

Authors and Affiliations

  1. Institute for Applied Mathematics, University Erlangen-Nürnberg, Martensstraße 3, 91058, Erlangen

    Christof Eck

  2. NWF I — Mathematik, University of Regensburg, 93040, Regensburg

    Harald Garcke & Björn Stinner

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  1. Christof Eck
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  2. Harald Garcke
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  3. Björn Stinner
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Editors and Affiliations

  1. Weierstraß-Institut für Angewandte, Analysis und Stochastik, Mohrenstraße 39, 10117, Berlin, Germany

    Alexander Mielke

  2. Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489, Berlin, Germany

    Alexander Mielke

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Eck, C., Garcke, H., Stinner, B. (2006). Multiscale Problems in Solidification Processes. In: Mielke, A. (eds) Analysis, Modeling and Simulation of Multiscale Problems. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-35657-6_2

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