基于局部Petrov-Galerkin离散方案的无网格法

基于局部Petrov-Galerkin离散方案,选用自然邻近插值构造试函数,用Shepard函数作为权函数,提出了一种无网格方法(MNNPG),这种方法充分发挥了局部Petrov-Galerkin法的优势,并且结合了自然邻近插值的特点,方便引入边界条件,由于以Shepard函数的圆形支集作为积分子域,用分片中点插值来完成区域积分,无需额外背景网格,是一种真正的无网格法。本文将该无网格方法用于求解二维弹性力学边值问题,算例结果很好地吻合了精确解,表明该方法具有良好的数值精度和稳定性。

Meshless method based on the Petrov-Galerkin discretization scheme

In this paper,a meshless method called as meshless natural neighbour Petrov-Galerkin method(MNNPG) is developed for elasto-statics based on the generalized meshless local Petrov-Galerkin method(MLPG).The natural neighbour interpolation is used to approximate the trial function and the Shepard function is chosen to be the test function over a circular local sub-domain.As this method joins the advantages of natural neighbour interpolation and meshless local Petrov-Galerkin method together,no background cells are needed for domain integration,no stiffness matrix assembly is required and no special treatment is needed to impose the essential boundary conditions.To improve the accuracy of presented method,the piecewise mid-point integration scheme is used to evaluate the domain integrals.In numerical examples,the accuracy and convergence of the presented method are studied and accurate results are obtained.