Phase field methods for free boundary problems
Phase field methods for free boundary problems
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发表时间：
1982
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通讯作者：
G. Fix
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作者：
G. Fix
导出了存在过冷和表面张力影响的自由边界问题的相场模型。导出了一种数值逼近格式，并给出了实例数值结果。这项研究部分得到了ARO合同的支持。DAAG 29 - 80 - c - 0081。部分支持也根据NASA合同提供。NAS1-15810，而作者在汉普顿，VA 23665的科学与工程计算机应用研究所居住。大学图书馆卡耐基梅隆大学匹兹堡，宾夕法尼亚州15213。凝固过程的标准描述在经典的Stefan问题[1]中得到了体现。在这种情况下，T«T（x， T）表示温度场，T*表示相变温度。特别是，材料ft中的点x在T b> TA时处于液相，反之，在T 0， T(x， T) + T*时处于固相。过渡区域定义为（1.2）f(t) {x€R: t (x,t) t *}，对于该区域内的点（1.3）Xv + [D*grad t *vJ* 0, x£t (t)。这里X是潜热，v是F(t)的法向速度，\^ F(t)的法向速度，[•]_表示跨越F(t)的跳跃。为了完成问题v»的规范，指定初始条件，例如（1.4）T（x,0）»TQ(x) x€ft，对于给定的初始温度场TQ和边界条件。为简单起见，我们使用狄利克雷型条件，即（1-5）T(x， T) Tx(x), x£3ft, T > 0，其中T^是定义在ft的边界3ft上的给定温度场。
A phase field model is derived for free boundary problems where the effects of supercooling and surface tension are present. A scheme for obtaining numerical approximations is derived, and sample numerical results are presented. This research was supported in part by the ARO Contract No. DAAG 29-80-C-0081. Partial support was also provided under NASA Contract No. NAS1-15810 while the author was in residence at the Institute for Computer Applications in Science and Engineering, Hampton, VA 23665. UNIVERSITY LIBRARIES CARNEGIE-MELLON UNIVERSITY PITTSBURGH, PENNSYLVANIA 15213 1. The H-method for Stefan Problems The standard description of a solidification process is captured in the classical Stefan problem [1]. In this context T « T(x,t) denotes a temperature field with T* denoting the phase transition temperature. In particular, points x in the material ft are in the liquid phase when T > TA, and conversely, they are in the solid phase when T 0, T(x,t) + T*. The transition region is defined by (1.2) f(t) {x €R : T(x,t) T*}, and for points in this region (1.3) Xv + [D*grad T*vJ* 0, x £ T(t). Here X is the latent heat, v the normal velocity of F(t), \^ the normal to F(t), and [•]_ denotes the jump across F(t). To complete the specification of the problem v» specify initial conditions, e.g., (1.4) T(x,0) » TQ(x) x €ft, for a given initial temperature field TQ, and boundary conditions. For simplicity we use Dirichlet type conditions, namely (1-5) T(x,t) Tx(x), x £ 3ft, t > 0, where T^ is a given temperature field defined on the boundary 3ft of ft.