iGIMP: An implicit generalised interpolation material point method for large deformations

The Material Point Method (MPM) uses a combined Eulerian-Lagrangian approach to solve problems involving large deformations. A problem domain is discretised as material points which are advected on a background grid. Problems are encountered with the original MPM when material points cross between grid cells, and this has been tackled by the development of the Generalised Interpolation MPM, where material points’ domains of influence extend beyond the currently occupied grid cell. In this paper, the Generalised Interpolation Material Point (GIMP) Method has been implemented implicitly in a manner that allows a global stiffness matrix to be constructed similar to that in the Finite Element Method (FEM) by combining contributions from individual elements on the background grid. An updated Lagrangian finite deformation framework has been used to ensure non-linear behaviour within each of the loadsteps. The weighting functions used for this which make the GIMP method different to standard MPM are presented and the implementation is explained. Specific details on computing the deformation gradient to be consistent with the updated Lagrangian framework and the updating of the material point influence domains are outlined, both of which are currently unclear in the published literature. It is then shown through numerical examples that for both small and large deformation elastic and elasto-plastic problems, the implicit GIMP method agrees well with analytical solutions and exhibits convergence properties between that of linear and quadratic FEM.