The imposition of nonconforming Neumann boundary condition in the material point method without boundary representation

Since its original formulation, the material point method (MPM) has been repeatedly further developed to become a powerful tool for simulating large deformation problems. However, the imposition of boundary condition in the MPM remains a challenge since most material points are not located at the boundary and regular background mesh is most commonly used. Therefore, the existing methods to impose boundary condition always require a boundary tracking or reconstruction algorithm, which is not always efficient or accurate. In the current work, a novel strategy to apply nonconforming Neumann boundary condition is proposed that does not require the exact boundary position. The original problem with a nonconforming traction boundary condition is transformed into an equivalent problem with a prescribed virtual stress field. Importantly, both the original and the transformed problems produce the exact same response within the material domain. To solve the equivalent problem, a modified governing equation without boundary representation is subsequently constructed, where the volume integral terms are computed by both particle quadrature and cell quadrature. Several numerical examples are investigated to assess the accuracy and demonstrate the capability of the proposed approach in simulating different engineering problems. These numerical examples include mesh refinement to illustrate the method’s good convergence and a 3D complex surface to illustrate the method’s capability.