PhD Defense - Jun (Blair) Geng, "Implicit and Explicit Smoothed Particle Hydrodynamics for Large Deformation Modeling in Geomechanics"

Large deformation problems in geomechanics often involve plastic yielding, strain localization, post-failure behavior, and moving free surfaces. These features pose major challenges for traditional mesh-based methods, which may suffer severe distortion and require costly remeshing. This dissertation develops an implicit updated Lagrangian smoothed particle hydrodynamics (SPH) framework for large deformation modeling in geomechanics. The conventional explicit hypoelastic-plastic SPH formulation is first reviewed as a baseline. Material quantities are evaluated at particles in their current positions, the constitutive response is expressed in rate form using the Jaumann rate of Cauchy stress, and the equations of motion are advanced by explicit time integration. This formulation is conditionally stable under the Courant–Friedrichs–Lewy (CFL) restriction and often requires artificial viscosity for numerical stabilization.

To address these limitations, the proposed framework combines an updated Lagrangian description with multiplicative elastoplasticity, Newmark time integration, and successive substitution. The fully implicit scheme evaluates particle states, interaction forces, and neighbor lists within the iterative solution, while a predictor–corrector formulation provides a more efficient semi-implicit alternative. Simple shear, granular column collapse, and inclined flume simulations demonstrate stable solutions at time steps beyond the conventional CFL limit without artificial viscosity. With only a few iterations per step, implicit updated Lagrangian SPH provides substantially improved stability while maintaining a favorable balance between accuracy and computational efficiency.