Imposition of essential boundary conditions in the material point method.

There is increasing interest in the Material Point Method (MPM) as a means of modelling solid mechanics problems in which very large deformations occur, e.g. in the study of landslides and metal forming, however some aspects vital to wider use of the method have to date been ignored, in particular methods for imposing essential boundary conditions in the case where the problem domain boundary does not coincide with the background grid element edges. In this paper we develop a simple procedure originally devised for standard finite elements for the imposition of essential boundary conditions, to the MPM, expanding its capabilities to boundaries of any inclination. To the authors' knowledge this is the first time that a method has been proposed that allows arbitrary Dirichlet boundary conditions (zero and non-zero values at any inclination) to be imposed in the MPM. The method presented in this paper is different from other MPM boundary approximation approaches, in that: (i) the boundaries are independent of the background mesh, (ii) artificially stiff regions of material points are avoided and (iii) the method does not rely on spurious mirroring of the problem domain to imposed symmetry. The main contribution of this work is equally applicable to standard finite elements and the MPM