The Generalized Interpolation Material Point Method

Abstract

The Material Point Method (MPM) discrete solution procedure for computational solid mechanics is generalized using a variational form and a Petrov- Galerkin discretization scheme, resulting in a family of methods named the Generalized Interpolation Material Point (GIMP) The generalization permits iden- tification with aspects of other point or node based dis- crete solution techniques which do not use a body-fixed grid, i.e. the methods. Similarities are noted and some practical advantages relative to some of these methods are identified. Examples are used to demon- strate and explain numerical artifact noise which can be expected in MPM calculations. This noise results in non- physical local variations at the material points, where constitutive response is evaluated. It is shown to destroy the explicit solution in one case, and seriously degrade it in another. History dependent, inelastic constitutive laws can be expected to evolve erroneously and report inac- curate stress states because of noisy input. The noise is due to the lack of smoothness of the interpolation func- tions, and occurs due to material points crossing compu- tational grid boundaries. The next degree of smoothness available in the GIMP methods is shown to be capable of eliminating cell crossing noise. keyword: MPM, PIC, meshless methods, Petrov- Galerkin discretization.