An efficient truncation scheme for Eulerian and total Lagrangian SPH methods

TL;DR

This paper introduces a truncation scheme for Wendland kernels in SPH methods, significantly improving computational efficiency while maintaining accuracy in fluid and solid dynamics simulations.

Contribution

The paper proposes a novel truncation approach for Wendland kernels in SPH, combined with kernel gradient correction and Laguerre-Gauss kernels, to enhance efficiency without sacrificing accuracy.

Findings

Truncated Wendland kernels achieve similar accuracy to standard kernels.

The truncation scheme significantly reduces computational cost.

Numerical examples confirm maintained accuracy and improved efficiency.

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 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.