This paper presents a high-order node-based smoothed radial point interpolation method (NS-RPIM) with linear strain fields in smoothing domains. The linear smoothed strains are constructed by complete order of polynomial functions and normalized with reference to the central point of the smoothing region. The new NS-RPIM is one order higher than those of the existing methods which use piecewise constant strains. This high-order method still uses linear displacements within each triangular background cell, but linear strains are created over smoothing domains using the pick-out theory. Because the smoothed strain and the compatible strain within a local region are equal in an integral sense, the unknown parameters in the reconstructed strain functions can be determined uniquely by using three linearly independent weight functions. The numerical verification and computational efficiency are investigated and compared with the standard node-based smoothing models. It is found that the linear strain NS-RPIM shows improvement on both the convergence and computational efficiency in terms of error norm in displacement and is temporally stable. Since no spurious non-zero energy mode appears, this approach can be employed directly for the vibration analysis of solids.

本文提出了一种基于高阶节点的平滑径向点插值方法（NS-RPIM），该方法在平滑域中具有线性应变场。线性平滑应变是由多项式函数的完全阶数构成的，并参考平滑区域的中心点进行归一化。新的NS-RPIM比使用分段恒定应变的现有方法高一个数量级。这种高阶方法仍然使用每个三角形背景单元内的线性位移，但是使用拾取理论在平滑域上创建了线性应变。由于局部区域内的平滑应变和兼容应变在积分意义上相等，因此可以使用三个线性独立的权重函数唯一地确定重构应变函数中的未知参数。研究了数值验证和计算效率，并将其与基于标准节点的平滑模型进行了比较。发现线性应变NS-RPIM在位移误差范式方面显示出在收敛性和计算效率上的改进，并且在时间上是稳定的。由于没有出现虚假的非零能量模式，因此可以将这种方法直接用于固体的振动分析。