Tied interface grid material point method for problems with localized extreme deformation

As a meshless method, the material point method (MPM) is capable of modeling problems with extreme deformation and material fragments. MPM uses a set of Lagrangian particles to discretize a material domain. The interaction between particles is carried out via an Eulerian background grid which is used as a finite element mesh to integrate momentum equations and to calculate spatial derivatives in each time step. Therefore, the accuracy of MPM is mainly dependent on the cell size of the background grid. But, a regular mesh with uniform cells is usually employed as the background grid, which results in poor efficiency for problems with localized extreme deformation. In this article, a tied interface grid material point method is proposed for such problems, in which the background grid with several cell sizes for different sub material domains can be used. The sub grid with refined cell size is used to cover the material domain undergoing extreme deformation, while the sub grid with coarse cell size used to cover the material domain elsewhere. The interaction between refined grid and coarse grid is implemented by a tied interface method. Several numerical examples including stress wave propagation, Taylor bar impact, and penetration problems, are studied to validate the accuracy and efficiency of the proposed method, which shows that the presented method possesses higher efficiency and lower memory requirement than MPM for problems with localized extreme deformation.