A Delaunay Refinement Algorithm for the Particle Finite Element Method applied to Free Surface Flows

TL;DR

This work addresses robust simulation of free-surface flows with the particle finite element method (PFEM) by solving the incompressible Navier–Stokes equations on a dynamically updated mesh and accurately identifying the fluid domain. The authors introduce two key contributions: a Delaunay refinement-based mesh adaptation with quality guarantees and a size-field-driven approach to capture domain changes, and a multi-point constraint technique that enforces global incompressibility of empty bubbles to model bubbly flows within a single-fluid framework. Verifications on sloshing and rising-bubble tests, plus diverse 2D dam-break and falling-drop scenarios, demonstrate improved surface geometry accuracy, stability, and mass conservation, with results aligning well with analytical, experimental, and established numerical references. The method offers a robust, adaptable PFEM framework for complex, transient free-surface flows and bubbly phenomena, with potential extensions to three dimensions and more advanced size-field error estimation.

Abstract

This paper proposes two contributions to the calculation of free surface flows using the particle finite element method (PFEM). The PFEM is based on a Lagrangian approach: a set of particles defines the fluid. Then, unlike a pure Lagrangian method, all the particles are connected by a triangular mesh. The difficulty lies in locating the free surface from this mesh. It is a matter of deciding which of the elements in the mesh are part of the fluid domain, and to define a boundary - the free surface. Then, the incompressible Navier-Stokes equations are solved on the fluid domain and the particles' position is updated using the resulting velocity vector. Our first contribution is to propose an approach to adapt the mesh with theoretical guarantees of quality: the mesh generation community has acquired a lot of experience and understanding about mesh adaptation approaches with guarantees of quality on the final mesh. We use here a Delaunay refinement strategy, allowing to insert and remove nodes while gradually improving mesh quality. We show that this allows to create stable and smooth free surface geometries. Our PFEM approach models the topological evolution of one fluid. It is nevertheless necessary to apply conditions on the domain boundaries. When a boundary is a free surface, the flow on the other side is not modelled, it is represented by an external pressure. On the external free surface boundary, atmospheric pressure can be imposed. Nevertheless, there may be internal free surfaces: the fluid can fully encapsulate cavities to form bubbles. The pressure required to maintain the volume of those bubbles is a priori unknown. We propose a multi-point constraint approach to enforce global incompressibility of those empty bubbles. This approach allows to accurately model bubbly flows that involve two fluids with large density differences, while only modelling the heavier fluid.

Figures (25)

• Figure 1: A typical PFEM simulation of a water column collapsing. After a number of time steps, the initially well-distributed particles move according to the velocity field, resulting in a disordered state. Without any adaptation, the mesh quality suffers greatly from this motion of the particles, eventually leading to poor quality simulations.
• Figure 2: The main steps in the PFEM algorithm.
• Figure 3: Boundary conditions along the fluid.
• Figure 4: Left: $\alpha$-hull of a set of points. Center: the corresponding $\alpha$-shape. Right: The Delaunay triangulation of the $\alpha$-shape.
• Figure 5: To build the $\alpha$-shape of the set of fluid particles, we build the Delaunay triangulation. Only elements of the triangulation whose circumradii are smaller than a prescribed value are considered part of the fluid. The green element is added to the fluid, while the red element is not. Boundaries are naturally detected by the algorithm.