A Discontinuous Galerkin Material Point Method for the solution of impact problems in solid dynamics

An extension of the Material Point Method [1] based on the Discontinuous Galerkin approximation (DG) [2] is presented here. A solid domain is represented by a collection of particles that can move and carry the fields of the problem inside an arbitrary computational grid in order to provide a Lagrangian description of the deformation without mesh tangling issues. The background mesh is then used as a support for the Discontinuous Galerkin approximation that leads to a weak form of conservation laws involving numerical fluxes defined at element faces. Those terms allow the introduction of the characteristic structure of hyperbolic problems within the numerical method by using an approximate Riemann solver [3]. The Discontinuous Galerkin Material Point Method, which can be viewed as a Discontinuous Galerkin Finite Element Method (DGFEM) with modified quadrature rule, aims at meeting advantages of both mesh-free and DG methods. The method is derived within the finite deformation framework for multidimensional problems by using a total Lagrangian formulation. A particular attention is paid to one specific discretization leading to a stability condition that allows to set the CFL number at one. The approach is illustrated and compared to existing or developed analytical solutions on one-dimensional problems and compared to the finite element method on two-dimensional simulations.