On the implicit implementation of the Generalised Interpolation Material Point method.

The modelling of solid mechanics problems often requires a method which allows large deformations and can account for non-linear behaviour. The Material Point Method (MPM) allows this by modelling a problem domain using particles (or Material Points, MPs) at which state variables are stored and tracked. Calculations are then carried out on a regular background mesh to which state variables are mapped from the particles. This background mesh is reset after each load (or time) step whereas the positions of the MPs are updated by the transfer of mesh displacements to the MP locations. In the standard MPM a problem can arise when a particle crosses the boundary between one background element and another. This is due to the fact that the shape functions are not continuous between elements. A significant advance in the MPM was made in [1] where this cell crossing instability was addressed using the Generalised Interpolation Material Point (GIMP) method. The GIMP method attempts to alleviate the problem associated with particles crossing between elements by modifying the particle characteristic functions so that particles influence nodes other than those associated with the element they are inside. In the current literature the GIMP method has been implemented using an explicit time stepping scheme. In this paper the GIMP method is recast in an implicit form. In particular, the formulation presented in this paper provides a method to determine the element stiness matrix, giving a truly implicit approach. The method is demonstrated first on 1D stress analysis problems for clarity and then extended to 2D