The virtual stress boundary method to impose nonconforming Neumann boundary conditions in the material point method

Abstract

The material point method (MPM) is a popular and powerful tool for simulating large deformation problems. The hybrid Eulerian–Lagrangian nature of the MPM means that the Lagrangian material points and the Eulerian background mesh are often nonconforming. Once the material and mesh boundaries become misaligned, imposing boundary conditions, such as Neumann boundary conditions (i.e., traction), becomes a challenge. The recently developed virtual stress boundary (VSB) method allows for imposing nonconforming Neumann boundary conditions without explicit knowledge of the boundary position. This is achieved through a problem transformation where the original boundary traction problem is replaced by an equivalent problem featuring a virtual stress field. This equivalent problem results in updated governing equations which are ultimately solved using a combination of particle-wise and cell-wise quadrature. In the current work, a modification to the VSB method is proposed to eliminate the need for cell-wise quadrature. Despite removing cell-wise quadrature, the modified VSB method maintains the accuracy observed in the original approach. Several numerical examples, including 1D and 2D benchmark problems, as well as a 3D demonstration problem, are presented to investigate the accuracy and illustrate the capability of the modified VSB method. Mesh refinement studies are included to show the method’s good convergence behavior.