A unified transport-velocity formulation for SPH simulation of cohesive granular materials

When simulating cohesive granular materials using smoothed particle hydrodynamics (SPH), tensile instability often arises, characterized by particle clustering and non-physical fractures. In two-dimensional scenarios, this issue is typically addressed by the artificial stress method, which introduces repulsive forces between particle pairs. However, extending this approach to three dimensions is considered complex due to the requirements of matrix diagonalization and coordinate system rotation. In this study, we introduce the transport-velocity formulation (TVF), a numerical technique widely used in SPH simulation of fluids to remove tensile instability, to address this issue. Furthermore, rather than being limited to inner particles alone as in the previous TVF, we develop a unified transport-velocity formulation (UTVF) that encompasses both free-surface and inner particles, by applying corrections to surface particles only in the tangential direction. This unified approach is tailored for large deformation and failure flow problems in cohesive granular materials, which often involve free surfaces. The proposed approach is first validated through benchmark cases of both fluids and elastic materials with known analytical solutions, demonstrating its convergence, stability, and accuracy. Comparisons with the artificial stress and particle shifting methods highlight the advantages of the UTVF in terms of momentum conservation and low dissipation. Subsequently, the developed UTVF is applied to the simulation of cohesive granular material failure and flows in both two-dimensional and three-dimensional settings. The results indicate that the proposed method effectively eliminates tensile instability, regardless of dimensionality. An open-source code is provided for further comparison and in-depth study. © 2025 The Authors