The Element-free Galerkin Meshless Method And Its Application

The element-free Galerkin meshless method (EFG) is one of the meshless methods. In this paper, the moving least-squares (MLS) approximation and EFG method based on MLS approximation and Galerkin weak form are introduced and applied to solve boundary value problems of a class of elliptic differential equations, interface problems and the quasistatic viscoelastic frictional contact problem with damage.The main contents in this paper are as follows:1. In chapter 2, the basic principles of the moving least-squares approximation are introduced. And the process to solve boundary value of problems of a class of elliptic differential equations by EFG is given and the ways to add the essential boundary conditions are also presented. Thus, the method is extended to solve problems of evolutionary variational partial differential equations. To verify the efficiency of EFG method, numerical examples are realized. Moreover, different treatment of boundary conditions and the choice of weight functions are discussed in numerical examples.2. Interface problems are introduced in chapter 3. The one and two dimensional interface problems and the immersed interface condition presented in detail elaborately. Combined with immersed interface method, a numerical scheme for interface problem is given by EFG, named immersed interface method with element-free Galerkin meshless method (IIM-EFG). In order to increase the accuracy, it is essential to choose the appropriate background grid and quadrature scheme. Two numerical examples are presented to verify the IIM-EFG method.3. The quasistatic viscoelastic frictional contact problem with damage is discussed in chapter 4 and its variational inequality (VI) is given. Use the FDM and MLS approximation discretize the time derivatives and displacement fields respectively and the full-discrete scheme is obtained. The convergence of EFG method is introduced. Furthermore, three numerical experiments are given. It validated the convergence order derived by theoretical analysis in the numerical example one with exact solution. And the other two examples discussed two different cases for the quasistatic viscoelastic frictional contact problems with damage. The numerical results demonstrate the effectiveness of EFG.