有介面活性劑之曲率運動的數值方法

Abstract

在這篇論文裡，我們使用兩種數值方法去處理曲率運動的問題。第一種方法稱為介面追蹤法(front-tracking method)，主要是用有限差分法追蹤曲率運動後的曲線。為了瞭解非線性表面張力的影響，我們將介面活性劑加至曲線上，讓介面活性劑能夠在曲線上自由的擴散。我們所介紹的數值方法能夠保證隨著時間的流逝介面活性劑之濃度的總和不變。另一個方法稱為多重等位函數法(multiple level set method)，主要用此方法處理圖形結構變化的問題。圖形上的每條曲線的運動是由曲率運動所決定。在此問題中我們暫時不添加介面活性劑。

In this paper, we use two numerical methods to model a grain boundary evolution for motion by mean curvature. In the first part, we present a finite difference method to track a network of curves whose motion is determined by curvature. To study the effect of inhomogeneous surface tension on the evolution of the network of curves, we include surfactant which can diffuse along the curves. Our numerical method is based on a direct discretization of the governing equations which conserves the total surfactant mass in the curve network. In the second part, we present a multiple level set method to track the grain topological change. The motion of grain boundary is considered by mean curvature. In this part, there is no surfactant on each grain boundary. The governing equations consist of a level set equation for the grain boundary motion and reinitialization equation for reconstructing signed distance function. Thus, one important step is coupling of multiple level set functions. Numerical examples shows the topological change evolution of the grain structure.