The Weighted Least Squares Collocation Meshless Method Of RBF Local Interpolation And Preliminary Study Of Finite Point Difference Method

There are two parts in this paper. .The first part is the weighted least squares collocation meshless method using local radial basis functions (RBF) interpolation. And the second part is finite point difference method with the characteristics of the difference method and meshless method..Meshless method is a new effective numerical method on problem for determining solution of partial differential equations. The disadvantage of reconstructing the mesh-grid by FEM for non-continuous and large deformation problems, eta, can be overcomed by the meshless method. The weighted.least squares meshless method is one of the frequenly meshless method that has been used in numerical solution of differential equations. However, the past interpolation is often based on the nodes in all domain, so that the order of the shape function matrix is large. Thus computation time is increased. In the first part the weighted least squares collocation method of radial basis function local interpolation are derivated by the nodes in support domain. The validity of the method presented in this paper is tested through the two examples that are the Poisson equation and the bending of cantilever. The several key factors impacting on the convergence are discussed and some useful conclusions are obtained.A new method named finite point difference method based on idea of difference methods and characteristics of meshless method is presented in the second part. The domain is discreted by using of arbitrary distribution nodes of meshless method, then a new difference format, finite point difference method, is construced through the Taylor expansion of the nodes in the neighborhood of the center point. Because of the selection of nodes in finite point difference method can be arbitrary discreted, it adapt to the irregular boundary of any region. Thus, it can overcome the disadvantages of a lower accuracy by traditional difference in dealing with irregular boundary. On the other hand, due to the difference format has markedly strongpoint of simple iterative, good convergence rate, high-precision, so finite point difference method can overcome the disadvantages of meshless method, for example, the large amount of calculation, complicated shape function. Through the numerical solution for the Poisson and Hemholtz equation on three different regions, that is, rectangle, triangle and ellipse, the validity of this method is validated, and the results were compared with the finite element method and the finite difference method. The calculated results show that finite point difference method has high precision, especially for curve edge.In the first chapter, the weighted least squares collocation meshless method of RBF local interpolation is derived. In the second chapter, two numerical calculation results by using of this meshless method are obtained. In the third chapter, the finite point difference method is constructed and the numerical examples calucated.