The Material Point Method (MPM) is well suited to modelling dynamic solid mechanics problems undergoing large deformations with non-linear, history dependent material behaviour. However, the vast majority of existing material point method implementations do not inherit conservation properties (momenta and energy) from their continuum formulations. This paper provides, for the first time, a dynamic updated Lagrangian material point method for elasto-plastic materials undergoing large deformation that guarantees momenta and energy conservation. Sources of energy dissipation during point-to-grid and grid-to-point mappings for FLuid Implicit Particle (FLIP) and Particle In Cell (PIC) approaches are clarified and a novel time-stepping approach is proposed based on an efficient approximation of the Courant-Friedrich-Lewy (CFL) condition. The formulation provided in this paper provides a platform for understanding the energy conservation nature of future/existing features of material point methods, such as contact approaches.
Pretti, G., Coombs, W., Augarde, C., Sims, B., Puigvert, M., & Gutierrez, J. (2023). A conservation law consistent updated Lagrangian material point method for dynamic analysis. Journal of Computational Physics, 485, Article 112075. https://doi.org/10.1016/j.jcp.2023.112075