The Discontinuous Galerkin Material Point Method for variational hyperelastic–plastic solids

Abstract

The Discontinuous Galerkin Material Point Method (DGMPM) presented in Renaud et al. (2018)[14] is based on the discretization of a solid domain by means of particles in a background mesh. Owing to the employment of the discontinuous Galerkin approximation on the grid, the weak form of a hyperbolic system involves fluxes that are computed at cell interfaces by means of an approximate Riemann solver. Combining these fluxes with the projection of the updated solution from the nodes to the particles originally used in the Particle-In-Cell method allows a significative reduction of the numerical oscillations that pollute the classical MPM solutions. Although the DGMPM exhibits very promising aspects, such as the control of the time-stepping (Renaud et al., 2020 [43]) or the ability to locally increase the approximation order in an arbitrary grid, the method first needs to be tested in its early version on problems involving a more complex wave content. It is then proposed in this paper to couple the DGMPM with variational integrators of hyperelastic–plastic constitutive models. The genericity provided for dealing with rate-independent or rate-dependent plasticity, as well as the possibility to easily extend the DGMPM to thermomechanical problems, makes this class of integrators appealing. The approach is here illustrated on numerical examples for which comparisons are shown with the finite element and the material point methods, as well as a one-dimensional exact solution in the linearized geometrical limit.