Material point method after 25 years: Theory, implementation, and applications

It has been 25 years since Sulsky and her coworkers developed the first version of the material point method (MPM): a quasi particle method to solve continuum mechanics problems. In the MPM, the continua are discretized by Lagrangian particles moving over a fixed Eulerian background grid. As a result, large deformation and contact can be treated effortlessly. Since then, many improved instances of the MPM have been developed and the MPM has found applications in many fields from geoengineering to movie industry. As the MPM has now been matured and a large body of literature on it exists, it is a good time to ponder and reflect on the developments of the method to date. To this end, this manuscript provides a concise introduction to the MPM, covering theory, implementation, and applications. All the algorithms required to have a working MPM implementation for the simulations of solids, fluids, and their interactions are provided. We have coded these algorithms in in-house open source programs and used them to study the performance of different MPM variants for large deformation solid mechanics problems. These problems exhibit large plastic deformation, fractures and contacts. Convergence of different MPMs (CPDI, GIMP, B-splines, total Lagrangian MPM, improved MPMs) are studied. Furthermore, MPM formulations for fluids/gases and heat conduction are also covered. Potential areas for improvement on the method have been identified. The paper is the first review of the MPM and presents a state of the art of the current MPM literature covering 339 references.