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

在凝固过程中，例如金属的铸造，通常会出现固-液界面的特定树枝状（树状）微观结构。描述单个或几个枝晶演变的模型是可用的，但是枝晶结构的小尺度（典型尺寸）阻止了它们在例如铸造过程中的整个工件的数值模拟中的应用。在这个项目中，我们开发和分析两种成分的混合物中的枝晶显微组织的凝固过程的双尺度模型。该模型是来自相场模型，包括对流，取决于一个小参数e描述的微观结构的规模。利用均匀化理论的技巧，构造了一个e > 0的极限问题.这个问题包括宏观微分方程的热量和流体流动和微观问题建模的演变单枝晶。数学分析包括解的存在性和唯一性的证明以及两个模型的解的差异的估计。应借助选定模型问题的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.