Analysis and mitigation of spatial integration errors for the material point method

Abstract

The material point method (MPM) is a robust numerical simulation approach for continuum mechanics problems involving large material deformations coupled to changing surface topographies. These types of problems are present in many different engineering contexts, from understanding the failure processes of earthen slopes to predicting the strengths and failure mechanisms of body armor to modeling the impact forces of waves in fluid tanks. By using a set of persistent material point tracers to follow the motion and deformation of the continuum material while solving the equations of motion on a static simulation grid, the MPM avoids several shortcomings of more traditional numerical approaches including blurring of material surfaces -as in Eulerian finite element or finite volume methods (FEMs or FVMs) -and mesh tangling -as in Lagrangian FEMs. Despite its robustness, MPM is known to develop significant numerical errors: namely, (i) the particle ringing instability and (ii) solution dependent discretization and integration errors. In this work, we present an analysis of local-in-time, spatial integration errors in the MPM and several techniques designed to mitigate these errors. Error mitigation approaches previously described in the literature are compared to a new method we propose for problems involving very large material deformations. The proposed method is shown to offer substantial improvement over standard MPM for simulations of fluid-like materials without requiring significant augmentation of existing MPM frameworks.

Circumventing volumetric locking in explicit material point methods: A simple, efficient, and general approach

“…Further, for a quantitative comparison, in Figure 13 we plot the time evolutions of the center-of-mass in the flow direction calculated from the standard and F MPM solutions, similar to an analysis performed in Baumgarten and Kamrin. 56 It can be seen that the center-of-mass evolves nearly identically in the F MPM formulations obtained with GIMP and B-splines basis functions, while the standard MPM solutions exhibit significant differences. This comparison confirms the proposed method performs well and consistently irrespective of the basis functions.…”

“…Further, for a quantitative comparison, in Figure 13 we plot the time evolutions of the center‐of‐mass in the flow direction calculated from the standard and $$$ \overline{\boldsymbol{F}} $$$ MPM solutions, similar to an analysis performed in Baumgarten and Kamrin 56 . It can be seen that the center‐of‐mass evolves nearly identically in the $$$ \overline{\boldsymbol{F}} $$$ MPM formulations obtained with GIMP and B‐splines basis functions, while the standard MPM solutions exhibit significant differences.…”