关于无网格方法中点插值形函数的研究

点插值法与其他无网格方法不同的是采用多项式近似来构造形函数，这种形函数具有Kronecker delta函数的特性，因此，易于施加本质边界条件．本文研究了点插值法中以单项式为基函数的形函数的建立及其性质，并通过矩阵三角化算法来克服形函数矩阵大奇异性．同时，本文所给出的数值算例验证了形函数具有Kronecker delta函数的特性，说明了点插值形函数具有精确的曲线拟合特性并能通过分片试验．

A Study on Shape Functions in the Meshless Point Interpolation Method

Unlike other meshless methods, the point interpolation method (PIM) adopts the polynomial approximation to construct the shape functions. The created shape functions possess the properties of the Kronecker delta function, which allows easy imposition of essential boundary conditions. This paper presents the construction of the shape functions using the monomial basic function and describes the properties of the shape functions. Matrix triangularization algorithm (MTA) was implemented to overcome the singularity of the matrix. The numerical examples showed that the shape functions and their derivatives of the point interpolation method possessed properties of the Kronecker delta function, and the shape functions had excellent accuracy in function fitting and could pass the patch test.