The Virtual Stress Boundary Method to Impose Nonconforming Neumann Boundary Conditions in the High‐Order Material Point Method

ABSTRACT Previous studies have demonstrated that the virtual stress boundary (VSB) method accurately and efficiently imposes nonconforming Neumann boundary conditions in the material point method (MPM). A key advantage of the VSB method is its ability to impose boundary conditions without requiring explicit knowledge of the boundary's position. This is achieved through a transformation that replaces the original boundary traction with an equivalent virtual stress field. The inclusion of this virtual stress field allows the governing equations to be solved using only volume integrals, which are easily computed in the MPM using a particle‐wise quadrature scheme. In this study, the VSB method is extended to accommodate various high‐order MPM variations. The updates mainly involve addressing the non‐local support domain of high‐order shape functions and enhancing quadrature accuracy in filled cells. The accuracy and versatility of the combined high‐order MPM and VSB method are demonstrated through numerical examples in 1D, 2D, and 3D.