Essentially non-hourglass and non-tensile-instability SPH elastic dynamics

TL;DR

This work addresses tensile instability in updated Lagrangian SPH by revealing hourglass modes as the true instability source and introducing an essentially non-hourglass formulation based on an angular-momentum-conservative Laplacian for shear acceleration, yielding $\dot{\mathbf v}^{s} \approx \frac{G}{\rho} \int_{0}^{t} \nabla^{2} \mathbf v \, dt$ with $\nabla^{2} \mathbf v = 2 \zeta \sum_j \frac{\mathbf e_{ij} \cdot \mathbf v_{ij}}{r_{ij}} \nabla_i W_{ij} V_j$. This approach eliminates hourglass modes and tensile instability in 2D and 3D elastic dynamics across varied benchmarks, challenging the conventional view that tension itself drives instability. The study further enhances efficiency with a dual-criteria time stepping scheme, using $\Delta t_{ad} = CFL_{ad}\frac{h}{|\mathbf v|_{max}}$ and $\Delta t_{ac} = CFL_{ac}\frac{h}{c_0+|\mathbf v|_{max}}$, and updating particle configurations only during the advection step. The authors validate convergence, accuracy, and robustness on plates, rings, and ball-plate interactions, and provide open-source code in SPHinXsys for reproducibility and extension to plastic dynamics.

Abstract

Since the tension instability was discovered in updated Lagrangian smoothed particle hydrodynamics (ULSPH) at the end of the 20th century, researchers have made considerable efforts to suppress its occurrence. However, up to the present day, this problem has not been fundamentally resolved. In this paper, the concept of hourglass modes is firstly introduced into ULSPH, and the inherent causes of tension instability in elastic dynamics are clarified based on this brand-new perspective. Specifically, we present an essentially non-hourglass formulation by decomposing the shear acceleration with the Laplacian operator, and a comprehensive set of challenging benchmark cases for elastic dynamics is used to showcase that our method can completely eliminate tensile instability by resolving hourglass modes. The present results reveal the true origin of tension instability and challenge the traditional understanding of its sources, i.e., hourglass modes are the real culprit behind inducing this instability in tension zones rather that the tension itself. Furthermore, a time integration scheme known as dual-criteria time stepping is adopted into the simulation of solids for the first time, to significantly enhance computational efficiency.

Figures (21)

• Figure 1: Illustration for (a) non-tensile-instability and non-hourglass modes, (b) tensile instability and (c) hourglass modes in ULSPH simulations of 2D oscillating plates. The particles are colored with von Mises stress.
• Figure 2: Model setup for 2D oscillating plate
• Figure 3: Evolution of particle configuration with time (t=0.05, 0.37 and 0.67) for (a) SPH-OG, (b) SPH-OAS, and (c) SPH-ENOG. The particles are colored by von Mises stress.
• Figure 4: Convergence verification for the 2D oscillating plate with the present SPH-ENOG. The curves show the changes of amplitude with time. Here, $L$=0.2, $H$=0.02, and $\mathbf v_f$=0.05.
• Figure 5: Test the long-term stability for the SPH-ENOG, the result is compared with the SPH-OAS. Here, $L$=0.2, $H$=0.02, $H/dp=10$ and $\mathbf v_f$=0.05. The particles are colored by von Mises stress.