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

The material point method (MPM) combines Eulerian method and Lagrangian method and thus both Lagrangian particle position and interaction between neighboring Eulerian grid cells will affect the simulation stability. However, the original critical time step formula in the standard MPM does not reflect the effect of particle position and neighboring cell interaction on stability and overestimates the critical time step so much that the CFL number has to be very small, even smaller than 0.1, to obtain a stable solution at extreme particle positions. Therefore, in many engineering applications, the standard MPM is very expensive due to the small CFL number. In this article, the effect of particle position and neighboring cell interaction on stability of the explicit MPM is studied. An explicit critical time step formula is obtained based on the system eigenvalues in one dimension, and is then extended to two and three dimensions. For extreme deformation problems, the geometric stiffness matrix is taken into consideration which modifies the sound speed of particles in the critical time step formula. Several tests are performed to verify our formula and show a decrease in amount of time steps used for simulation with our formula comparing with the original formula. K E Y W O R D S critical time step, explicit material point method, neighboring cell interaction, particle positions, stability analysis

Section: Critical Time Step Derivationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A precise critical time step formula for the explicit material point method

Zhang

2020

“…However, the local damping leads to energy dissipation, inaccuracies in the time‐dependent simulations, such as in the consolidation process and, in extreme cases, to an over‐damped system with qualitatively different characteristics. To avoid these severe limitations, for example, Lu et al and Renaud et al combined the time‐discontinuous Galerkin method to the MPM to control the spurious noises. Furthermore, Jiang et al have shown that the use of the PIC scheme can filter the spurious noises in the velocity but leads to an excessive dissipation.…”

Section: Introductionmentioning

confidence: 99%

Temporal and null‐space filter for the material point method

Tran

Sołowski

2019

Summary This paper presents algorithm improvements that reduce the numerical noise and increase the numerical stability of the material point method formulation. Because of the linear mapping required in each time step of the material point method algorithm, a possible mismatch between the number of material points and grid nodes leads to a loss of information. These null‐space related errors may accumulate and affect the numerical solution. To remove the null‐space errors, the presented algorithm utilizes a null‐space filter. The null‐space filter shown removes the null‐space errors, resulting from the rank deficient mapping and the difference between the number of material points and the number of nodes. The presented algorithm enhancements also include the use of the explicit generalized‐α integration method, which helps optimizing the numerical algorithmic dissipation. This paper demonstrates the performance of the improved algorithms in several numerical examples.

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Renaud

Heuzé

Stainier

2019

Self Cite

In this paper, stability conditions are derived for the Discontinuous Galerkin Material Point Method [1, 2] on the scalar linear advection equation for the sake of simplicity and without loss of generality for linear problems. The discrete systems resulting from the application of the DGMPM discretization in one and two space dimensions are first written. For these problems a second-order Runge-Kutta and the forward Euler time discretizations are respectively considered. Moreover, the numerical fluxes are computed at cell faces by means of either the Donor-Cell Upwind [3] or the Corner Transport Upwind [4] methods for multidimensional problems. Second, the discrete scheme equations are derived assuming that all cells of a background grid contain at least one particle. Although a Cartesian grid is considered in two space dimensions, the results can be extended to regular grids. The von Neumann linear stability analysis then allows the computation of the critical Courant number for a given space discretization. Though the DGMPM is equivalent to the first-order finite volume method if one particle lies in each element[1], so that the Courant number can be set to unity, other distributions of particles may restrict the stability region of the scheme. The study of several configurations is then proposed.