An efficient truncation scheme for Eulerian and total Lagrangian smoothed particle hydrodynamics methods

Abstract

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 computational efficiency 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 to the SPH approximation. To decrease numerical errors from SPH approximation and the truncation method, we incorporate the Laguerre–Gauss kernel for particle relaxation to obtain the high-quality particle distribution with reduced residue [Wang et al., “A fourth-order kernel for improving numerical accuracy and stability in Eulerian and total Lagrangian SPH,” arXiv:2309.01581 (2023)], and the kernel gradient correction to rectify integration errors. 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 non-truncated Wendland kernels enable achieving the same level of accuracy but the former significantly increases the computational efficiency.