配点型广义节点无网格法的基本原理及其应用

Fundamental and application of generalized-node mesh-free collocation method

借鉴流形方法思想,引入广义节点的概念,对传统的无网格法进行了改进,建立了可具有任意高阶多项式插值函数的广义节点无网格方法。同时采用径向插值函数构造具有插值特性的逼近函数;采用配点法建立系统的离散方程。在阐述了这种方法基本原理的同时,针对线弹性力学问题给出了这种方法的数值计算列式。与传统无网格方法相比,这种方法更具有一般性;同时由于采用了配点法而不需要背景积分网格,所以可以认为这种方法是某种真正意义上的无网格法。当选取0阶广义节点位移插值函数时便可得到传统的无网格法;在不增加支持域内节点数目的条件下,通过选取高阶广义节点位移插值函数可以提高计算精度。最后通过算例分析,对0阶、1阶及2阶广义节点无网格法与现有的有关解答进行了对比,论证了其合理性。

In this paper, the concept of generalized node in manifold method is incorporated together with the mesh-free method to establish the interpolation functions with polynomials of arbitrary order. Based on the generalized-node interpolation functions, the generalized-node mesh-free method is developed to improve the conventional mesh-free method. At the same time, the radial basis function is used to construct the approximation functions with the feature of interpolation and the collocation method is employed to develop the discrete formulation of governing partial differential equations of the system. Therefore, the radial basis function （RBF） is applied to a direct collocation procedure and the generalized node is embedded in the RBF approximation. Then the mathematical formulation of the proposed method is developed with numerical implementation. The proposed generalized-node collocation meshfree method is a truly mesh-free method which combines the generalized node in manifold method with point collocation discretization of the governing partial differential equations. When the zero-order displacement interpolation function of the generalized node is chosen, the proposed method will be reduced to the conventional mesh-free point collocation method. In order to improve the accuracy with less supporting nodes, a higher-order displacement interpolation function is required. As numerical examples, a cantilever beam under end shear and an infinite plate with a hole subjected to uniform tensile load are respectively analyzed by the proposed method. It is shown that the numerical results computed by the proposed method can well agree with the solutions obtained by the theory of elasticity.