A particle-position-based finite element formulation for free-surface flows with topological changes

Numerical simulation of moving boundary flows with topological changes in the fluid domain presents a significant challenge, as traditional mesh-based formulations cannot directly deal with these changes. In this work, we propose a novel position-based Particle Finite Element Method derived from the stationary total mechanical energy principle applied to incompressible Newtonian flows. The use of positions as the main variables instead of velocities might have some advantages in certain scenarios: i. it eliminates the need for an additional step to compute displacements based on particles velocities; ii. it naturally accounts for the geometric non-linearities through the deformation gradient and iii. the resulting approach seems ideal to simulate FSI problems in a monolithic way as hyperelastic materials are usually written in terms of position or displacement. The solution process follows the well-established PFEM, where the entire domain is discretized with particles and a finite element mesh is generated over them to solve the motion equations using the proposed approach. The reference is updated whenever the mesh needs to be reconstructed, resulting in a partially updated Lagrangian description. The incompressibility constraint is enforced using a mixed approach, where pressure is interpolated in the same polynomial basis as the positions. LBB condition is circumvented by applying a Lagrangian version of the pressure stabilization/Petrov–Galerkin (PSPG) technique. The proposed formulation is verified through numerical examples, demonstrating its robustness, accuracy, and applicability.