MFS-RBF Meshless Method For A Kind Of Fourth-order Problems

The problems described by the fourth-order elliptic variational inequalities or variational equations have been widely applied in the fields of physics, mechanics, engineering, for examples, the large deflection problem of thin plate, obstacle problem, contact problem. As the complexity of the problems, generally they are simplified to make the corresponding control equation to be easily calculated. Then, some numerical methods, such as the finite difference method, finite element method, boundary element method, are developed to solve these problems. In recent years, meshless method is a kind of new and efficient numerical method for solving partial differential equations. This study will focus on the special boundary-type meshless method by the coupling of the method of fundamental solutions and radial basis functions. And then this method will be applied in the large deflection problem of thin plate and obstacle problem.This thesis is composed of the following sections:1. In chapter 2, the method of fundamental solutions and radial basis functions are introduced. And then it presents the meshless collocation method based on the method of fundamental solutions and radial basis functions, namely MFS-RBF meshless method, and gives the construction of the virtual boundary nodes and the choice of radial basis functions. Numerical examples prove the validity of this method and reveal the effect of the choices of virtual boundary nodes and radial basis functions.2. In chapter 3, it introduces the obstacle problem described by a kind of fourth-order elliptic variational inequality. By the theory of duality method, it gives the MFS-RBF meshless method for obstacle problem. By numerical examples and compared with the finite element method, the results illustrate the effectiveness of the method.3. In chapter 4, it discusses the large deflection problem of thin plate depicted by Berger equation. An iterative algorithm has been put forward to solve the Berger equation, and then the MFS-RBF meshless method has been given. The numerical result reveals the efficiency of iterative meshless method. Compared with the local boundary integration method, this meshless method has the advantage of good accuracy, no need of any mesh and easy to implement. It is an efficient numerical method for Berger equation.4. In chapter 5, the conclusion and the prospect for future work are given.