An implicit high-order material point method.

The material point method (MPM) is a version of the particle-in-cell (PIC) which has substantial advantages over pure Lagrangian or Eulerian methods in numerical simulations of problems involving large deformations. The MPM helps to avoid mesh distortion and tangling problems related to Lagrangian methods and as well as the advection errors associated with Eulerian methods. Despite the MPM being promoted for its ability to solve large deformation problems the method suffers from instabilities when material points cross between elements. These instabilities are due to the lack of smoothness of the grid basis functions used for mapping information between the material points and the background grid. In this paper a novel high-order MPM is developed to eliminate the cell-crossing instability and improve the accuracy of the MPM method