A Shape Optimization Research For Quadrilateral Mesh Over The Triangular Surface

In recent years, with the fast development of computer technology, the Finite Element Method (EFM) has become one of the most powerful analysis tools available in solving complex engineering problems. As the first step, which is also one of the critical techniques,is the discretization of the object region,ie.the generation of the mesh. The result of the numerical simulation and speed of the analysis are strongly related to the quality of the mesh. Then it is very important to study the fast generation of high-quality mesh in the numerical simulation filed. A triangular mesh and a quadrilateral mesh are commonly used for 2D plane and over 3Dcurved surface in the FEM. Quadrilateral mesh in general has better precision and converge faster than the triangular mesh. For that reason,lots of engineers and scholars have attached great impotance on the quadrilateral mesh generation since the 1980s. A large number of approaches have been proposed in this field.In this paper,a simple survey of the development of the quadrilateral mesh generation will be given firstly. And then several typical quadrilateral mesh generation approaches will be presented. We mainly give a simple brief introduction to Morse theory and Morse function, and introduce a two-dimensional manifolds on the Morse-Smale complex. The main contribution of this paper is an optimization algorithm to the shape of the quadrangulation generated by the Morse-Smale complex. By the use of the traverse of the vertexes on the quadrangle on the surface, we can get better vertexes and boundaries after iterative approach, and then good quadrilateral mesh can be constructed.