A Discontinuous Galerkin Material Point Method (Dgmpm) for the Simulation of Impact Problems in Solid Mechanics

Abstract

Abstract. The material point method is extended in this work to the Discontinuous Galerkin approximation framework for the simulation of impacts on elastic and hyperelastic solids. The formulation is based on the weak form of conservation laws on each cell of an eulerian grid in which volume integrals are discretized on a set of material points lying in that cell, and on the computation of Godunov fluxes at cells faces. The resulting method is first derived within the small strains framework and illustrated on a one-dimensional and a two-dimensional problem of impact on an elastic media. Then a one-dimensional hyperelastic problem of a solid undergoing large strains is presented, and a comparison is performed with an analytical solution.

Stability properties of the Discontinuous Galerkin Material Point Method for hyperbolic problems in one and two space dimensions

“…the cell size for the two aforementioned distributions of particles. First, it can be seen that both DGMPM results converge to the exact solution with a rate of approximately one, which is an already observed result [2]. Second, the error resulting from the use of the discrete operators perfectly fits that of the DGMPM computations, which validates the expressions derived in the previous section.…”

“…The same approach is now repeated with random distributions of particles. First, the number of particles lying in the cells is randomly selected in the range [1,2,3,4]. Then, the positions of the material points in the cells, which is assumed to be repeated in the whole mesh, are also randomly generated following a uniform distribution, that is: x p ∼ unif(x 2c(I)−1 , x 2c(I) ) ∀p.…”

“…∆xF (1) n + ∆yF (4) n ;F2 = − 1 2 ∆xF (1) n − ∆yF (2) n F 3 = 1 2 ∆xF (3) n + ∆yF (2) n ;F4 = 1 2 ∆xF (3) n − ∆yF (4) n A condensed way of writing those fluxes is adopted by means of the middle point of edge (j) with coordinates − → x (j) 1/2 , at which the shape functions are:…”

“…Note also that the numerical diffusion exhibited by the PIC can be limited by reducing the domain of influence of nodes rather than modifying the projections themselves. This approach is followed in the Discontinuous Galerkin Material Point Method (DGMPM) [1,2]. The introduction of the DG approximation within the MPM, combined with the use of the PIC projection, thus aims at providing non-oscillating discontinuous solutions with low numerical diffusion due to the support of the shape functions that reduces to one cell.…”

“…Furthermore, an interesting feature of the method consists in allowing the employment of mesh adaption strategies so that waves can be accurately captured in the current configuration. On the other hand, only linear shape functions leading to a first-order accuracy [2] have been considered, the extension of the method to higher-order approximations being the object of ongoing works. The low-order approximation is however not seen as a shortcoming for now since the development of the method has been focused so far on capturing discontinuous solutions for which the accuracy of any numerical scheme falls to one [3].…”