New Numerical Methods and Analysis for Navier Stokes Equations Involving Interfaces and Simulation of Electro-Migration of Voids

这项建议是为了继续PI的研究，开发、实施和应用有效的数值方法来解决界面问题，特别是具有移动界面和/或自由边界的问题。具体项目包括开发用于Navier-Stokes方程的浸没界面方法，模拟具有固定或移动界面的不可压缩粘性两相流。与此相关的一些重要项目包括发展新的投影和/或规范方法，以便即使在有界面的情况下也能保持二阶精度；能够满足二维和三维界面问题极大值原理的二阶椭圆求解器；使用笛卡尔网格和/或自适应笛卡尔网格来求解界面问题的有限元方法。另一个将被深入研究的应用是模拟存在晶界的集成电路中的空穴的电迁移。这项提议是为了开发有效的(快速、准确和实时的)方法来模拟一些重要的移动界面/自由边界问题。在石油工业中的应用包括模拟石油工业中的水和油的界面；在环境科学中模拟水中受污染的气泡；在材料科学中模拟晶体生长的图案式；在医学和生物科学中模拟细胞变形和生物膜的运动；在半导体工业中模拟集成电路中空穴的电迁移。多年来，我们开发了解决这些界面问题的先进方法，我们相信我们在这一领域处于领先地位。因此，我们对这一提议的成功非常有信心，并能够保持这一领域相对于其他国家的优势。在经济上，这一提议的成功可以节省进行真正实验所需的数百万美元。

This proposal is for the continuation of the PI's research toward thedevelopment, implementation, and application of efficient numerical methodsfor interface problems especially with moving interfaces and/or freeboundaries. Specific projects include development of the immersed interfacemethod for the Navier Stokes equations modeling incompressible viscous twophase flow with fixed or moving interfaces. Related to this topic, someimportant projects include development of new projection and/or gaugemethods so that the second order accuracy can be preserved even with thepresence of interfaces; second order elliptic solvers that can satisfy themaximum principle for interface problems in two and three dimensions; thefinite element methods using Cartesian and/or adaptive Cartesian grids forinterface problems. Another application which will be studied in depth isthe simulation of electro-migration of voids in integrated circuits withthe presence of grain boundaries.This proposal is about developing efficient (fast, accurate, and in realtime) methods to simulate some important moving interface/free boundary problems. Applications include the simulation of the interface between water and oil in petroleum industry; simulation of contaminated bubbles in water for environmental science; simulation of crystal growth of pattern formulation in material science; simulation of cell deformation and motion of biofilm in medical and biology sciences; and simulation of electro-migration of voids in an integrated circuit in semi-conductor industry. Over the years, we have developed advanced methods for solving these interface problems and we believe we are the leaders in this area. Therefore we are very confident in the success of the proposal and canmaintain the edge in this area over other countries. Economically, the success of this proposal can save millions of dollars that areneeded to carry out real experiments.