A space–time particle-position-based finite element formulation for free-surface flows

Numerical simulation of moving boundary flows with topological changes remains a significant challenge, particularly for free-surface incompressible flows. While the Particle Finite Element Method (PFEM) has proven effective for such problems, current formulations often rely on velocity-based approaches with time-marching schemes, which can limit accuracy, stability, and compatibility with monolithic fluid–structure coupling. This paper introduces a space–time, particle-position-based PFEM formulation that combines the strengths of PFEM (e.g., handling large deformations and topological changes) with the advantages of space–time finite element methods (e.g., improved stability and accuracy). The proposed method utilizes a mixed position-pressure framework in which the position is approximated using specially designed shape functions that are linear in the spatial directions and quadratic in the temporal direction. Meanwhile, the pressure is approximated using the same linear shape functions in the spatial directions but remains constant in the temporal direction. The space–time discretization is structured in time direction, allowing sequential solution slice by slice of the space–time domain, resulting in a nonlinear system of equations with the same size as that of standard time-marching methods. The results of numerical examples, including sloshing and dam-break scenarios, demonstrate the robustness and accuracy of the method, highlighting its effectiveness and showcasing its potential for more complex problems.