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

Abstract

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

An explicit phase field material point method for modeling dynamic fracture problems

“…However, different from the analysis in Ni and Zhang 42 for the standard MPM, the mapping and reconstruction between particles and background grids have direct influence on the stability in ex‐PFMPM. The mapping and reconstruction process must be taken into consideration when calculating the critical time step.…”

“…The moving‐mesh MPM is actually the same as the standard FEM with the adoption of the lumped mass matrix except that the particles instead of the Gauss points serve as the quadrature points in the moving‐mesh MPM. As mentioned by Ni and Zhang, 42 the effect of particle position and neighboring cell interaction on the critical time step of the standard MPM need to be taken into consideration, when conducting the stability analysis. Therefore, to get an explicit critical time step formula based on the system eigenvalues in one dimension, as shown in Figure 3, we assume that the standard MPM has uniform mesh discretization in the 1D computational domain, and the mesh size is $$$ {x}_{I+1}-{x}_I={x}_I-{x}_{I-1}=l $$$.…”

“…In the Eulerian step, the mesh will be reset to its original position while the particles remain in their current positions. Ni et al 42 stated that the Eulerian step will not make the standard MPM scheme unstable, so that the stability is determined by whether the error will be amplified in Lagrangian part. Therefore, the aforementioned stability analysis of ex‐PFMPM is based on the moving mesh MPM.…”

“…Furthermore, when the parent coordinate of particle with hollow circle $$$ {r}_{\mathrm{h}}\to 1.0 $$$ in Figure 6A, $$$ {r}_{\mathrm{h}}\to 1.0 $$$ in Figure 6B, and $$$ {r}_{\mathrm{h}}\to -1.0\;\mathrm{or}\;1.0 $$$ in Figure 6C, the critical time step will severely decrease to 0. As discussed in Ni and Zhang, 42 the critical time step in MPM will considerably decrease because of the small nodal mass when particles are near the cell boundary. Similarly, when particles distribute near the cell boundary in ex‐PFMPM, a small nodal viscous dissipation value will be obtained at the opposite cell boundary, which causes a severely small critical time step.…”

“…Berzins 40 applied the stability analysis of Spigler and Vianello 41 to MPM and derived a stable time step bound for one dimension. Ni and Zhang 42 proved that the von Neumann stability analysis method is invalid in the explicit MPM and derived the critical time step formula by solving the system generalized eigenvalue problem. The corresponding formula illustrated the effect of particle distribution and neighboring cell interaction on the stability of the standard MPM.…”