Homogenization of a Phase Field Model for Binary Mixtures
Homogenization of a Phase Field Model for Binary Mixtures
复制标题
二元混合物相场模型的均质化
DOI：
10.1137/s1540345903425177
复制
发表时间：
2005
期刊：
影响因子：
--
通讯作者：
C. Eck
中科院分区：
文献类型：
--
作者：
C. Eck
本文的目的是推导一个双尺度模型，该模型描述二元混合物液固相变中等轴枝晶微观结构的演化。该方法基于 Caginalp 和 Xie Arch 提出的相场模型。理性机械。分析，142（1998），第293--329页]。假设固体核的周期性初始分布具有尺度周期 $\varepsilon > 0$ 并根据 $\varepsilon$ 缩放某些物理参数，则进行正式的渐近展开。结果是一个双尺度模型，由全局均质热方程和宏观域每个点的单固体晶体演化的局部单元问题组成。为了证明渐近展开的合理性，导出了对两尺度模型的解与尺度 $\varepsilon$ 的原始模型的解的差异的估计。此估计显示 $\varepsilon^{1/2}$ 阶误差。最后，说明了二尺度模型......
The aim of this article is the derivation of a two-scale model that describes the evolution of equiaxed dendritic microstructure in liquid-solid phase transitions of binary mixtures. The approach is based on a phase field model proposed by Caginalp and Xie Arch. Rational Mech. Anal., 142 (1998), pp. 293--329]. Assuming a periodic initial distribution of solid kernels with a period of scale $\varepsilon > 0$ and scaling certain physical parameters in the dependence of $\varepsilon$, a formal asymptotic expansion is carried out. The result is a two-scale model that consists of a global homogenized heat equation, and, at each point of the macroscopic domain, local cell problems for the evolution of single solid crystals. In order to justify the asymptotic expansion, an estimate for the difference of the solution of the two-scale model and the solution of the original model of scale $\varepsilon$ is derived. This estimate shows an error of order $\varepsilon^{1/2}$. Finally, the two-scale model is illustrated...