In this paper, a meshless generalized finite difference (FD) method is developed and presented for solving 2D elasticity problems. Different with other types of generalized FD method (GFDM) commonly constructed with moving least square (MLS) or radial basis function (RBF) shape functions, the present method is developed based upon B-spline based shape function. The method is a truly meshless approach. Key aspects attributed to the method are: B-spline basis functions augmented with polynomials are employed to construct its shape function. This allows B-splines with lower order to be chosen for the approximation and keeping the efficiency of computation related to tensor product operation of B-spline basis functions. In addition, as distribution of stencil nodes affects numerical performance of generalized FD method, neighboring nodes from triangle cells surrounding a center node are selected for building the supporting domains. While meeting compact stencil requirement, the selection eliminates necessity for determining appropriate number of supporting nodes or size of supporting domains. As a result, the proposed method shows good numerical approximation and accuracy for 2D elasticity problems. Numerical examples are presented to show the effectiveness of the proposed method for solving several 2D elasticity problems in various geometries.

本文提出并提出了一种无网格的广义有限差分（FD）方法来解决二维弹性问题。与通常由移动最小二乘（MLS）或径向基函数（RBF）形状函数构造的其他类型的广义FD方法（GFDM）不同，本方法是基于基于B样条的形状函数开发的。该方法是真正的无网格方法。归因于该方法的关键方面是：用多项式扩展的B样条基函数来构造其形状函数。这允许选择较低阶的B样条进行逼近，并保持与B样条基函数的张量积运算有关的计算效率。此外，由于模具节点的分布会影响广义FD方法的数值性能，从围绕中心节点的三角形单元中选择相邻节点以构建支持域。在满足紧凑的模板要求的同时，该选择消除了确定适当数量的支持节点或支持域大小的必要性。结果，所提出的方法对于二维弹性问题显示出良好的数值逼近和精度。数值例子表明了所提出的方法在解决各种几何形状中的几个二维弹性问题上的有效性。所提出的方法对于二维弹性问题显示出良好的数值逼近和精度。数值例子表明了所提出的方法在解决各种几何形状中的几个二维弹性问题上的有效性。所提出的方法对于二维弹性问题显示出良好的数值逼近和精度。数值例子表明了所提出的方法在解决各种几何形状中的几个二维弹性问题上的有效性。