A generalized non-hourglass updated Lagrangian formulation for SPH solid dynamics

TL;DR

The paper addresses hourglass instabilities in updated Lagrangian SPH for solids by introducing a generalized non-hourglass formulation that adds a penalty force to correct misestimated shear in zero-energy deformation modes. The penalty relies on the velocity mismatch between linearly predicted and actual neighbor velocities, with plastic regions further modulating the force via a stress-return factor γ, ensuring reduced penalty during plastic flow. A dual-criteria time stepping scheme enhances efficiency, and the method is validated across elastic and plastic benchmarks, showing accuracy comparable to, or better than, existing essentially non-hourglass formulations, while preserving angular momentum in challenging cases. The approach is demonstrated on a range of elastic and plastically deforming problems, including high-speed impact and rotating/taylor-bar tests, and is released within the SPHinXsys open-source framework for broad adoption and industrial application.

Abstract

Hourglass modes, characterized by zigzag particle and stress distributions, are a common numerical instability encountered when simulating solid materials with updated Lagrangian smoother particle hydrodynamics (ULSPH). While recent solutions have effectively addressed this issue in elastic materials using an essentially non-hourglass formulation, extending these solutions to plastic materials with more complex constitutive equations has proven challenging due to the need to express shear forces in the form of a velocity Laplacian. To address this, a generalized non-hourglass formulation is proposed within the ULSPH framework, suitable for both elastic and plastic materials. Specifically, a penalty force is introduced into the momentum equation to resolve the disparity between the linearly predicted and actual velocities of neighboring particle pairs, thereby mitigating the hourglass issue. The stability, convergence, and accuracy of the proposed method are validated through a series of classical elastic and plastic cases, with a dual-criterion time-stepping scheme to improve computational efficiency. The results show that the present method not only matches or even surpasses the performance of the recent essentially non-hourglass formulation in elastic cases but also performs well in plastic scenarios.

Figures (31)

• Figure 1: 2D oscillating plate: model setup.
• Figure 2: 2D oscillating plate: evolution of particle configuration with time (t=0.05, 0.37 and 0.67) for (a) SPH-OG zhang2024essentially, (b) SPH-OAS gray2001sphzhang2024essentially, (c) SPH-ENOG zhang2024essentially, and (d) SPH-GNOG. Here, $v_f=0.05$, $L=0.2$, and $H=0.02$. The particles are colored by von Mises stress.
• Figure 3: 2D oscillating plate: temporal evolution of deflection at various resolutions. Here, $v_f=0.05$, $L=0.2$, and $H=0.02$.
• Figure 4: 2D oscillating plate: temporal evolution of elastic strain energy, kinetic energy, and total energy. Here, $v_f=0.05$, $L=0.2$, and $H=0.02$.
• Figure 5: 2D oscillating plate: test the long-time stability of the SPH-GNOG. The result is compared with those obtained by SPH-ENOG zhang2024essentially and SPH-OAS zhang2024essentially. Here, $L$=0.2, $H$=0.02, $H/dp=10$ and $\mathbf v_f$=0.05. The particles are colored by von Mises stress.