An implicit material point method using a cell-based integration scheme for large deformation static problems

A novel implicit material point method (MPM) using a cell-based integration scheme is proposed to solve large deformation static problems. An incremental weak form based on the updated Lagrangian approach is formulated for the implicit MPM. The volume integrals of the incremental weak form are evaluated at the integration points of grid cells instead of material points, which eliminates the cell-crossing error and reduces the integration error in MPM computations. Grid cells are equally sub-divided into grid cell sub-domains. The centers and the particle volumes of the grid cell sub-domains are, respectively, taken as the integration points and corresponding weights for the numerical integration of the incremental weak form. Particle information is transferred through grid nodes to the integration points of grid cells by using grid shape functions. A volume-weighted nodal averaging scheme is used for transferring the deformation gradient from material particles to grid nodes to correctly consider the particle volumes associated with the deformation gradient. Numerical results show that the present implicit MPM can effectively and efficiently solve large deformation static problems.