Available online 17 January 2025 (click here)

Background: 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.

Highlights:

Developed an improved explicit penalty formulation to impose Dirichlet boundary conditions and tie-contact constraints in both MPM and FEM-MPM coupling. The Dirichlet boundary conditions and contact constraints were discretized on material points, enhancing compatibility with complex geometrical shapes.

A dimensionless penalty factor was designed to minimize the decrease in the stable time step. The stability of this penalty factor has been discussed by eigenvalue analysis and did not need excessive tuning.

The contact force was formulated as a conservative force field, ensuring energy conservation during MPM-Rigid and FEM-MPM collisions.

The proposed method is easy to implement for both Updated Lagrangian and Total Lagrangian formulations, broadening its applicability to various engineering problems.