In smoothed particle hydrodynamics (SPH) method, the particle-based approximations are implemented via kernel functions, and the evaluation of performance involves two key criteria: numerical accuracy and computational efficiency. In the SPH community, the Wendland kernel reigns as the prevailing choice due to its commendable accuracy and reasonable computational efficiency. Nevertheless, there exists an urgent need to enhance the computational efficiency of numerical methods while upholding accuracy. In this paper, we employ a truncation approach to limit the compact support of the Wendland kernel to 1.6h. This decision is based on the observation that particles within the range of 1.6h to 2h make negligible contributions, practically approaching zero, to the SPH approximation. To address integration errors stemming from the truncation, we incorporate the Laguerre-Gauss kernel for particle relaxation due to the fact that this kernel has been demonstrated to enable the attainment of particle distributions with reduced residue and integration errors \cite{wang2023fourth}. Furthermore, we introduce the kernel gradient correction to rectify numerical errors from the SPH approximation of kernel gradient and the truncated compact support. A comprehensive set of numerical examples including fluid dynamics in Eulerian formulation and solid dynamics in total Lagrangian formulation are tested and have demonstrated that truncated and standard Wendland kernels enable achieve the same level accuracy but the former significantly increase the computational efficiency.

翻译：在光滑粒子流体动力学（SPH）方法中，基于粒子的近似通过核函数实现，其性能评估涉及两个关键标准：数值精度和计算效率。在SPH领域，Wendland核因其良好的精度和合理的计算效率成为主流选择。然而，在保证精度的同时提高数值方法的计算效率存在迫切需求。本文采用截断方法将Wendland核的紧支撑范围限制为1.6h。这一决策基于以下观察：在1.6h至2h范围内的粒子对SPH近似的贡献可忽略不计（实际趋近于零）。为解决截断导致的积分误差，我们引入Laguerre-Gauss核进行粒子松弛，因为该核已被证明能够实现残余积分误差更小的粒子分布\cite{wang2023fourth}。此外，我们引入核梯度校正以修正SPH近似核梯度及截断紧支撑带来的数值误差。通过涵盖欧拉框架下流体动力学和完全拉格朗日框架下固体动力学的综合数值算例验证，结果表明：截断核与标准Wendland核可实现相同精度，但前者显著提升了计算效率。