Energy-law Preserving Continuous Finite Element Methods for Simulation of Liquid Crystal and Multi-phase Flows

发布者：系统管理员发布时间：2012-12-12浏览次数：2255

报告题目：	Energy-law Preserving Continuous Finite Element Methods for Simulation of Liquid Crystal and Multi-phase Flows
报告人：	Ping Lin
	University of Dundee/University of Science & Technology Beijing
报告时间：	12月17日下午2：00-3：00
报告地点：	数学系第一报告厅
相关介绍：	The liquid crystal (LC) flow model is a coupling between orientation (director field) of LC molecules and a flow field. The model may probably be one of simplest complex fluids and is very similar to the Allen-Cahn phase field model for multiphase flows if the orientation variable is replaced by a phase function. There are a few large or small parameters involved in the model (e.g. the small penalty parameter for the unit length LC molecule or the small phase-change parameter, possibly large Reynolds number of the flow field, etc.). We propose a C^0 finite element formulation in space and a modified midpoint scheme in time which accurately preserves the inherent energy law of the model. We use C^0 elements because they are simpler than existing C^1 element and mixed element methods. We emphasise the energy law preservation because from the PDE analysis point of view the energy law is very important to correctly catch the evolution of singularities in the LC molecule orientation. In addition we will see numerical examples that the energy law preserving scheme performs better under some choices of parameters. We shall generalize the idea to the Cahn-Hilliard phase field model where the biharmonic operator is decomposed into two Laplacian operators. But we find that under our scheme non-physical oscillation near the interface occurs. We figure out the reason from the viewpoint of differential algebraic equations and then remove the non-physical oscillation by doing only one step of a modified backward Euler scheme at the initial time. A number of numerical examples demonstrate the good performance of the method. At the end of the talk we will show how to incorporate the temperature into the model and how to apply the method to a superconductivity model, especially at the regime of Hc2 or beyond. The talk is based on a few joint papers with Chun Liu, Qi Wang, Xingbin Pan and Roland Glowinski, etc.