Based On Sliding Kriging Interpolation Meshless Local Petrov-galerkin Method

Meshless method is a new numerical method for solving the partial differential equations. It is different from the finite element method, for the meshless method uses a set of nodes to discretize the problem domain, and constructs the approximate function by the discrete nodes. As the meshless method does not need remesh, and can be completely or partially eliminate the impact of the meshes, it becomes a hot point of the field of computational mechanics.The meshless local Petrov-Galerkin method (MLPG) is a truly meshless method, which is obtained form the local weak form of the weighted residual method and the moving least squares method, whether it is constructed the trial function, or the numerical integration is not dependent on meshes. However, the shape function constructed by the moving least square method has not the Kronecker function property. Hence, it is not easy to impose the essential boundary conditions and may be easy to form ill-conditioned simultaneous equations, thereby reducing the computational efficiency of the method. In order to overcome these shortcomings, an improved meshless local Petrov- Galerkin method is proposed in this paper, in which the moving Kriging interpolation method is used to construct the trial function and the Heaviside step function is used as the test function. Furthermore, it is applied to solve potential problems, transient heat conduction problems, elasticity problems and elastodynamics problems respectively. The main researches of this thesis are as follows.The improved meshless local Petrov-Galerkin method is applied to two-dimensional potential problems, and the improved meshless local Petrov-Galerkin method based on moving Kriging interpolation method for potential problem is presented, and the corresponding formulae are obtained. Compared with the traditional meshless local Petrov-Galerkin method, the present method has greater computational efficiency and greater precision, and the essential boundary conditions can be imposed more easily.Based on the study of the heat conduction problems, the improved meshless local Petrov-Galerkin method is applied to two-dimensional transient heat conduction problems. Combining the Galerkin weak form of transient heat conduction problems, the improved meshless local Petrov-Galerkin method for transient heat conduction problems is investigated. And the corresponding formulae are obtained.The improved meshless local Petrov-Galerkin method is applied to two-dimensional elasticity. The discrete equation is produced from the weak form of weighted residual method, and then the improved meshless local Petrov-Galerkin method for elasticity is presented. And the corresponding formulae are obtained.The improved meshless local Petrov-Galerkin method is applied to two-dimensional elastodynamics. The Galerkin weak form of elasto-dynamics problems is employed to obtain the discretized system equations, and the Newmark time integration method is used for time history analyses. Then the improved meshless local Petrov-Galerkin method for elastiodynamics is presented. And the corresponding formulae are obtained.In order to show the efficiency of the improved meshless local Petrov- Galerkin method in the dissertation, the MATLAB codes of the methods above have been written. Some numerical examples are provided, and the validity and efficiency of these methods are demonstrated.