The Study of a Radial Point Interpolation Meshless Method with Polynomial Basis and It’s Application

Abstract: Meshless method rapidly developed in recent years is a numerical analysis method. It overcomes the concept of elements or meshes that used in conventional methods such as finite element method(FEM) and boundary element method(BEM). It is based on the node, thoroughly or partly cancelled mesh, avoided remeshing completely, so meshless method becomes a wide development and promising method for its flexibility and validity in solving partial differential equations.A radial point interpolation meshless method with polynomial basis is a new type meshless method. In this method, the shape function is constructed by the combination of radial and polynomial basis functions, the control equation is derived from the weak form of variational equation. The formulated shape functions in the radial point interpolation meshless method with polynomial basis has the property of delta function, which is covenient to implement essential boundary conditions. The method overcomes the main drawback in imposing essential boundary condition and solves the limitation of the PIM is that the matrix may be singular sometimes. The advantages of this meshless method are: the problem domain is represented by properly scattered points, and no need to make elements with nodes. In addition, high accuracy and efficient can be achieved and preprocess and post process is easy.This paper can be devided into two parts:The first part deals with some fundamental theory. Firstly, the paper systematically introduced persent development of meshless methods and the baisic theory of meshless method. Secondly, the principle of point interpolation method is described. At last, the basic principle of radial point interpolation meshless method with polynomial basis is mainly studied, then the method is applied to solve the two-dimensional planar elasticity problems in this paper and the discrete equtation from the variational principle(weak form) is obtained. Meanwhile, the process of realization is given concretely.In the second part of this paper, several examples are calculated and analyzed. First, one of the key factors which affects the precision of the method, and the distribution of nodes are discussed. A new nodal distribution method is firstly proposed based on combinating random and regular nodes. Moreover, numerical examples are carried out using this proposed method. The numerical results of all examples are compared with those of ANSYS and the theoretical ones. They show good agreement and verifies the reliability of the theory in the present paper. Finally, the meshless method is firstly applied to axial symmetry problem…
Key words: meshless method; point interpolation method; radial basis fuction; polynomial basis fuction; radial point interpolation method

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