An efficient truncation scheme for Eulerian and total Lagrangian SPH methods

TL;DR

This work tackles the computational inefficiency of SPH by truncating the Wendland kernel's compact support to $1.6h$ and compensating the resultant errors with a kernel-gradient correction and Laguerre-Gauss particle relaxation. Through quantitative error analysis and extensive simulations in both Eulerian SPH for fluids and total Lagrangian SPH for solids, the truncated scheme achieves comparable accuracy to the standard Wendland kernel while significantly reducing neighbor counts and CPU time. The approach demonstrates robust performance across challenging test cases, including shock-dominated, shear, and complex solid deformation problems, suggesting broad practical impact for speeding up SPH without sacrificing fidelity. The findings indicate substantial efficiency gains in a range of geometries and flow regimes, making the truncated kernel approach attractive for large-scale or real-time SPH applications.

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.

Figures (14)

• Figure 1: Wendland kernel: function (red) and its derivative (blue) in the compact support and two different compact support size with $\kappa=2$ and $\kappa=1.6$ are plotted as black and green circles, respectively.
• Figure 2: The convergence study of error for kernel gradient on lattice distribution (upper panel) and the relaxed distributions using Wendland (middle panel) and Laguerre-Gauss (bottom panel) kernels, respectively.
• Figure 3: Double Mach reflection of a strong shock: Density contour and its zoom-in view ranging from $1.3$ to $24.0$ obtained by SW (top panel) and TW (bottom panel) with the resolution $dp=1/125$.
• Figure 4: Double shear layer flow : The vorticity contour ranging from $-25$ to $25$ obtained by SW (left panel) and TW (right panel) with the spatial resolution $dp=1/128$ at different times $t=0.0$, $0.8$ and $1.2$.
• Figure 5: Lid-driven square cavity flow: The square geometry and boundary conditions. Besides, the velocity contour ranging from $2.2 \times 10^{-5}$ to $0.96$ obtained by TW with the resolution as $dp=1/1/129$ under the Reynolds number $Re=400$.