Point Interpolation Meshless Method Based on Compactly Supported Radial Basis Functions

摘要

The local characteristic of compactly supported radial basis functions (RBFs), which makes the stiffness matrix of governing equations sparse and banded, is well adopted in meshless methods. However, its accuracy is not much high in interpolation. Interpolation functions in the point interpolation method (PIM) have delta function property which is convenient to implement essential boundary conditions. The limitation of the PIM is that the matrix may be singular. In order to improve the accuracy and avoid the limitation of PIM, this paper proposes a technique to modify the compactly supported RBFs based on the completeness. These modified compactly supported RBFs are used to construct interpolation functions, thus a compactly supported radial point interpolation method is obtained. With this modified PIM, the governing equations of two-dimensional elastic mechanics are discretized. This modification overcomes possible singularities associated with polynomial basis only. Furthermore, its shape functions have the property of delta function and the essential boundary conditions can be applied as easy as in conventional finite element method (FEM). In addition, the method is applied to two-dimensional static problems . A cantilever beam problem and an engineering problem are analyzed. The numerical results show that the proposed method is accurate, convenient and efficient.