Semi-Lagrangian reproducing kernel formulation and application to modeling earth moving operations

Lagrangian formulation has traditionally been introduced for solids under large deformation. Earth-moving operations on the other hand, involve excessive material deformation and damage that cannot be modeled by the Lagrangian formulation. In this paper, a Semi-Lagrangian reproducing kernel (RK) formulation is introduced for modeling earth-moving operations. von Neumann stability analyses of Lagrangian and Semi-Lagrangian Reproducing Kernel (RK) formulations are first performed. The analysis results show that Semi-Lagrangian weak form integrated by a direct nodal integration (DNI), resembling Smoothed Particle Hydrodynamics (SPH), leads to a tensile instability that is inconsistent with the material stability. On the contrary, Semi-Lagrangian RK weak form integrated using the stabilized conforming nodal integration (SCNI) exhibits consistent numerical stability and material stability. The stable time steps for Lagrangian and Semi-Lagrangian RK formulations using central difference are also estimated and an enhanced stability is observed when the weak forms are integrated by SCNI compared to that using DNI or one-point Gauss quadrature. Earth moving simulation is performed to demonstrate the applicability of the proposed Semi-Lagrangian RK formulations to excessive material deformation and fragment problems.