B-spline based boundary conditions in the material point method

Abstract

The material point method is an increasingly popular method for tackling solid mechanics problems involving large deformations. However, there are issues associated with applying boundary conditions in the method and, to date, no general approach for imposing both Neumann and Dirichlet boundary conditions has been proposed. In this paper, a new B-spline based boundary method is developed as a complete methodology for boundary representation, boundary tracking and boundary condition imposition in the standard material point method. The B-spline interpolation technique is employed to form continuous boundaries which are independent of the background mesh. Dirichlet boundary conditions are enforced by combining the B-spline boundaries with an implicit boundary method. Neumann boundary conditions are included by direct integration of surface tractions along the B-spline boundary. This general boundary method not only widens the problems that can be analysed by all variants of the material point method, when implemented using an implicit solver, but is also applicable to other embedded and non-matching mesh approaches. Although the Dirichlet boundary conditions are restricted to implicit methods, boundary representation, tracking and Neumann boundary condition enforcement can be applied to explicit and implicit methods.