Applications of Variational Level Set Methods to Some Multiphase Problems

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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.