Applications of Variational Level Set Methods to Some Multiphase Problems

变分水平集方法在一些多相问题中的应用

• 批准号: 9706566
• 负责人: Hong-Kai Zhao
• 金额: $ 6.56万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706566&HistoricalAwards=false
• 关键词: Applications; Variational; Level; Set; Methods

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.

当自由边界存在拓扑变化或有两个以上的相共享一个共同的边界（三重结）时，自由边界或界面问题的研究和数值计算是相当具有挑战性的（特别是在三维空间中）。本项目将使用变分水平集方法，该方法可以在一定程度上轻松自然地处理这些困难，以模拟几个多相问题，例如群集肥皂泡的配置，顶部或喷嘴末端液滴的形成及其拓扑转换。变分公式将包括表面张力、体积能和外力的影响。利用适当的抽象“表面”能量，还将应用于一些最优图划分问题。同时，将为这些问题的二维和三维数值计算开发鲁棒和高效的数值格式。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、润湿和脱湿过程等。它们也与数学文献中的最小曲面问题有关。众所周知，在包围一个固定体积的所有可能的封闭表面构型中，球面具有最小的表面积。这就解释了为什么肥皂泡或橘子是圆形的。对表面效应的理解，它与其他物理效应的相互作用，以及在相当复杂的物理环境中对这些自由边界问题的数值模拟是本项目的动机。它将为许多实际应用提供非常有用的信息和工具，如微芯片制造中的最佳形状设计，蚀刻和沉积，电镀和喷墨打印机的制造。当表面能被正确解释时，它也可以用于一些优化问题，如高效并行计算的最优域分解切割，景观划分或金属切割。