An improved explicit MPM formulation and its coupling scheme with FEM

Accurately imposing boundary conditions and contact constraints in the Material Point Method (MPM) and its coupling with the Finite Element Method (FEM-MPM) is challenging, especially when dealing with complex geometrical shapes and misalignment between material boundaries and the computational grid. To address these issues, an improved explicit penalty formulation based on particle positions is developed to effectively impose Dirichlet boundary conditions and tie-contact constraints in both MPM and FEM-MPM coupling. Specifically, the concepts of boundary reference points and tied reference points are introduced to discretize the penalty terms associated with Dirichlet boundary conditions and tied contact constraints, respectively. These methods are straightforward to implement and highly suitable for explicit computational frameworks. A dimensionless penalty factor selection scheme is designed to avoid excessive tunning and minimize the decrease in stable time step. Additionally, contact forces are formulated as a conservative force field, ensuring energy conservation during MPM-Rigid and FEM-MPM collisions, which enhance numerical performances. Moreover, Dirichlet boundary conditions and contact constraints are discretized on material points, improving compatibility with complex geometrical shapes. The proposed explicit computational framework is straightforward to implement in both Updated Lagrangian and Total Lagrangian formulations, broadening its applicability to various engineering problems. Finally, the robustness, accuracy, and efficiency of the proposed approach are demonstrated through a series of numerical experiments, showcasing precise implementation of irregular boundary conditions, accurate calculation of contact forces, and good energy conservation.