Assessing and improving strong-shock accuracy in the material point method

Abstract

The Material Point Method (MPM) has appealing attributes for simulations involving large deformation of materials with history-dependent constitutive laws. It avoids the mesh tangling errors of Lagrangian finite element methods, as well as the advection errors typical of Eulerian or ALE methods. Recent developments in the MPM have led to a better understanding of the error and dissipation that arise in the particle-to-grid and grid-to-particle mapping, but significant error and high-frequency noise can still occur in simulations involving strong shock waves. This error appears to be associated with the kinematics on the background computational grid common to all MPM implementations. An investigation of forward- and reverse-ballistic impact simulations has revealed that sub grid-scale variations in the material state arise as the shock front enters a grid cell, and that this non-equilibrium state can persist unless the background grid has some means to relax the resultant particle-scale noise. Relaxation can occur when (i) the background grid shape functions have non-constant gradient, (ii) the shocked material has a sufficiently high velocity relative to the background grid, or (iii) the velocity field is enriched with additional degrees of freedom for treating weak discontinuities between particles.