Ein Zweiskalenmodell für fest-flüssig Phasenübergänge mit dendritischer Mikrostruktur und Konvektion

在凝固过程中，例如金属的铸造，经常会出现固-液界面上特殊的树枝状(树枝状)组织。目前已有描述单个或少数枝晶演化的模型，但枝晶组织的小尺度(典型尺寸)阻碍了其在铸造过程中的数值模拟中的应用，例如整个铸件。在这个项目中，我们开发和分析了两组分混合物中具有树枝晶组织的凝固过程的双尺度模型。该模型由包含对流的相场模型导出，依赖于一个描述微观结构尺度的小参数e。利用齐次化理论中的技巧，构造了e&gt；0的一个极限问题。这个问题包括宏观的热和流体流动的微分方程式，以及模拟单个树枝状晶体演化的微观问题。数学分析包括解的存在唯一性的证明以及对两个模型解的差异的估计。通过对选定的模型问题进行数值模拟，研究双尺度模型的可靠性和适用性。

In solidification processes, as e.g. the casting of metals, there often occurs a specific dendritic (tree-like) microstructure of the solid-liquid interphase. Models describing the evolution of single or few dendrites are available, but the small scale (typical size) of the dendritic structures prevents their application in a numerical simulation of e.g. a whole workpiece in a casting process. In this project we develop and analyze two-scale models for solidification processes with dendritic microstructure in mixtures of two components. The model is derived from a phase field model including convection, depending on a small parameter e describing the scale of the microstructure. Using techniques from homogenization theory a limit problem for e > 0 is constructed. This problem consists of macroscopic differential equations for heat and fluid flow and of microscopic problems modeling the evolution of single dendritic crystals. A mathematical analysis includes a proof of the existence and uniqueness of solutions as well as an estimate for the difference of the solutions of both models. The reliability and applicability of the two-scale model shall be studied with the help of numerical simulations for selected model problems.