Material point method is a variant of the finite element method (FEM) and is successfully applied in large deformation problems. Recently, material point method has been applied in a wide range of engineering applications including solid and solid-fluid interaction problems. However, describing the behavior of incompressible materials is a challenging problem in MPM. The explicit formulation and the linear elements used in the standard MPM exhibit numerical instabilities such as mesh locking and artificial pressure oscillations in material incompressibility. Further, the small time step used to obtain a reasonable numerical stability limits the application of MPM in problems particularly with long time durations. We present an implicit treatment of the pressure term in MPM to mitigate the numerical instabilities and small time steps in incompressible material problems. The set of velocity-pressure coupled governing equations resulted by the implicit formulation is solved using Choin's projection method. The numerical examples show that the present MPM implementation is capable of modeling incompressible materials without pressure oscillations using a significantly large time step.
Kularathna, S., & Soga, K. (2017). Projection Method in Material Point Method for Modeling Incompressible Materials. In Procedia Engineering (Vol. 175, pp. 57–64). Elsevier Ltd. https://doi.org/10.1016/j.proeng.2017.01.016