Examination and Analysis of Implementation Choices within the Material Point Method (MPM)

The Material Point Method (MPM) has shown itself to be a powerful tool in the simulation of large deformation problems, especially those involving complex geometries and contact where typical finite element type methods frequently fail. While these large complex problems lead to some impressive simulations and solutions, there has been a lack of basic analysis characterizing the errors present in the method, even on the simplest of problems. The large number of choices one has when implementing the method, such as the choice of basis functions and boundary treatments, further complicates this error analysis. In this paper we explore some of the many choices one can make when implementing an MPM algorithm and the numerical ramifications of these choices. Specifically, we analyze and demonstrate how the smoothing length within the Generalized Interpolation Material Point Method (GIMP) can affect the error and stability properties of the method. We also demonstrate how various choices of basis functions and boundary treatments affect the spatial convergence properties of MPM. Keyword: Material Point Method, GIMP, Meshfree Methods, Meshless Methods, Particle Methods, Smoothed Particle Hydrodynamics, Quadrature