A displacement-based material point method for weakly compressible free-surface flows

We introduce a novel displacement-based material point method for simulating weakly compressible free-surface flows and fluid–structure interaction. To address volumetric locking, we employ a B¯/F¯-inspired technique, previously developed for solid mechanics. This technique involves projecting the pressure and the dilatational part of the velocity gradient onto a lower-dimensional approximation space, eliminating complexities associated with two-field mixed formulations and operator splitting approaches. Additionally, to mitigate spurious pressure oscillations resulting from the use of a density-dependent equation of state, we enhance the framework with an artificial viscosity term. Finally, we employ higher-order spline background shape functions, resulting in a continuous representation of the velocity gradient and effectively preventing pressure jumps when material points cross element boundaries. Challenging numerical examples are provided to verify and validate our approach, demonstrating results that closely align with existing literature, exhibit reduced pressure oscillations, and are free of volumetric locking issues.