Applications of Variational Level Set Methods to Some Multiphase Problems

9706566赵宏凯 自由边界或界面的研究与数值计算 问题是相当具有挑战性的（特别是在三维）， 要么在自由边界中有拓扑变化， 有两个以上的相共享一个共同的边界（三重连接）。 这个项目将使用变分水平集方法，它可以 在一定程度上轻松自然地处理这些困难， 几个多相问题，如簇状皂的构型， 气泡、在天花板上或喷嘴末端的液滴的形成及其拓扑转变。表面张力的影响， 体积能量和外力将包括在变分方程中。 公式化。使用适当的抽象“表面”能量， 对一些最优图划分问题也将进行研究。在 与此同时，还将发展稳健有效的数值格式， 在二维和三维的数值计算这些问题。 由于分子的微观结构，表面能在许多物理现象中起着重要的作用，如肥皂泡，薄膜， 液滴的形成、润湿和去湿过程等。它们还 与数学文献中的极小曲面问题有关。是 众所周知，在封闭固定体积的封闭表面的所有可能构型中，球体具有最小的表面积。 这就解释了为什么肥皂泡或橘子是圆形的。的理解 它与其他物理效应的相互作用 并对这些自由边界问题进行了数值模拟 复杂的物理环境是这个项目的动机。 它将为许多实际应用提供非常有用的信息和工具，如最佳形状设计，蚀刻和沉积， 微芯片制造、电镀和喷墨打印机制造。 它也可以用于一些优化问题时，表面 能量被适当地解释，诸如用于高效并行计算的最优域分解切割、景观划分或金属切割。

9706566 Hongkai Zhao The study and numerical computation of free boundary or interface problems are quite challenging (especially in three dimensions) when either there are topological changes in the free boundaries or there are more than two phases that share a common boundary (triple junction). This project will use the variational level set approach, which can handle these difficulties easily and naturally to some extent, to model several multiphase problems such as the configurations of clustered soap bubbles, the formation of droplets on the ceiling or at the end of a nozzle and their topological transitions. The effects of surface tension, bulk energy and external forces will be included in the variational formulation. Using appropriate abstract 'surface' energy, applications to some optimal graph partitioning problems will also be made. At the same time, robust and efficient numerical schemes will be developed for numerical computation in 2D and 3D for these problems. Surface energy plays an important role in many physical phenomena due to the micro-structure of molecules, such as soap bubbles, thin films, droplets formation, wetting and dewetting process, etc. They are also related to minimal surface problems in mathematical literature. It is well known that among all possible configurations of closed surfaces which enclose a fixed volume, the sphere has the minimal surface area. This explain why soap bubbles or oranges are rounded. The understanding of the surface effects, its interactions with other physical effects and numerical simulation of these free boundary problems in quite complicated physical settings are the motivations for this project. It will provide extremely useful information and tools for many practical applications such as optimal shape design, etching and deposition in microchip fabrication, plating and manufacturing of ink jet printer. It can also be used in some optimization problems when the surface energy is properly interpreted, such as the optimal domain decomposition cut for efficient parallel computing, landscape dividing or metal cuttings.