Implicit and Semi‐Implicit SPH Schemes for Large Deformation Problems in Geomechanics

Abstract

We present an implicit time integration scheme for smoothed particle hydrodynamics (SPH) employing multiplicative elastoplasticity in the finite deformation range. Time integration is based on the one‐step, unconditionally stable Newmark method formulated using an updated Lagrangian description. A novelty of the paper is an implicit evaluation of all variables including the neighbor lists of the SPH particles. The implicit scheme simplifies to a predictor–corrector (semi‐implicit) scheme when the zeroth iteration is taken as the predictor phase and the first iteration is taken as the corrector phase. We show that both schemes provide significant numerical stabilization of the SPH solution even with larger time steps and without the use of an artificial viscosity—a marked improvement over existing explicit SPH algorithms that rely on the satisfaction of the Courant–Friedrichs–Lewy condition and on artificial viscosity for stabilization. We demonstrate the robustness of the implicit and predictor–corrector schemes using benchmark examples that include sand column collapse and granular flow on an inclined flume.