Researches On The Interpolating Meshless Methods

The meshless method is a new numerical method which is developed after the finite element method. In this method, only nodal information is needed to form the approximation or interpolating function of a variable without considering the mesh, then it has simple pre-processing and high computing accuracy,and thus became one of popular research topics in computational science.The existing meshless methods, such as the element-free Galerkin method etc., which are based on the moving least-squares (MLS) approximation, are of low efficiency, and cannot apply boundary conditions directly. In view of these, we present an improvement of the interpolating moving least-squares (IMLS) method and study the corresponding meshless method including the interpolating element-free Galerkin (IEFG) method and interpolating boundary element-free method (IBEFM). Afterwards, the method is applied to potential problems and elasticity problems. The proposed method successfully overcomes the difficulty of the application of the boundary conditions in the meshless methods based on the MLS approximation.In this thesis, the IMLS method proposed by Lancaster is improved, and then the improved IMLS method is presented, and the interpolating property of the corresponding shape function is proved. Compared with the IMLS method proposed by Lancaster, there is a simpler formula of the shape function in the improved IMLS method so that the new method has higher computing efficiency of the shape function which, as we know, is an important aspect in the meshless method, and has a direct impact on the computing efficiency of the meshless method based on the shape function.Considering the unclear physics meaning of the functional in the existing complex variable moving least-squares (CVMLS) approximation, by constructing a new functional which has an explicit physics meaning,a new CVMLS approximation is presented in this thesis. Based on the new CVMLS approximation, the complex variable interpolating moving least-squares (CVIMLS) method is presented, and the interpolating property of the shape function of the CVIMLS method is proved.Combining the shape function constructed by the improved IMLS method in this thesis and Galerkin weak form for the potential problem, the interpolating element-free Galerkin (IEFG) method for the potential problem is put forward, and the corresponding formulae are obtained. Compared with the conventional EFG method, the boundary conditions can be directly applied in the IEFG method which makes the computing efficiency of the method higher. Some numerical examples are developed to demonstrate the validity of the method.Combining the shape function constructed by the improved IMLS method and Galerkin weak form for the elasticity problem, the IEFG method for elasticity problem is presented, and the corresponding formulae are obtained. Some numerical examples are presented to demonstrate the validity of the method.Combining the shape function constructed by the improved IMLS method and the boundary integral equation method for the potential problem, the interpolating boundary element-free method (IBEFM) for potential problem is presented, and the corresponding formulae are obtained. Compared with the conventional BEFM, the boundary conditions can be applied directly in the IBEFM, then the IBEFM has higher computing efficiency. Some numerical examples are presented to demonstrate the validity of the method.Combining the shape function constructed by the improved IMLS method in this thesis and the boundary integral equation method for the elasticity problem, the interpolating boundary element-free method (IBEFM) for elasticity problem is proposed, and the corresponding formulae are obtained. Some numerical examples are developed to demonstrate the validity of the method.For the IEFG method and the IBEFM for potential and elasticity problems, the corresponding algorithm flows are presented, and the corresponding computer programs are written. Numerical examples show that the proposed method is valid and the relevant results are correct.