Three-dimensional mesh adaptation in PFEM

TL;DR

The paper tackles 3D chaotic free-surface flows with PFEM by addressing three geometric aspects: domain reconstruction, mesh adaptation, and boundary handling. It introduces an advected-boundary domain oracle tested with a winding-number criterion, a two-phase adaptive refinement (surface edge-splitting followed by volume circumcenter insertion guided by a size field), and a decoupled boundary management framework using simple geometric queries. Validation across vortex-in-a-box, viscous drop, and dam-break scenarios shows improved mass conservation, accurate topology changes, and robustness on complex or non-watertight geometries, highlighting PFEM's practical potential. The work lays the groundwork for scalable, high-fidelity 3D simulations and points toward future parallel and GPU implementations, with ongoing attention to slivers and mesh quality.

Abstract

Chaotic free surface flows are challenging problems to simulate numerically, mainly due to the significant changes in geometry and frequent topological changes. Methods that track the evolution of the fluid in a Lagrangian formulation are a natural choice. One such method is the Particle Finite Element Method (PFEM). As a hybrid particle-based and mesh-based method, PFEM leverages advantages from both approaches. The equations of motion are solved on a mesh using the finite element method and the obtained velocity field is used to displace the nodes of this mesh, considered as particles carrying all the relevant information across time steps. To avoid element distortion, the mesh is frequently re-generated. This introduces some challenges: How can the new shape of the domain be detected? How can the quality of the elements be kept acceptable? Can adaptive mesh refinement increase the accuracy and efficiency of the solver? Can PFEM simulations be performed in the presence of complex boundary geometries? In this work, three contributions to the geometry and mesh component of PFEM are introduced for three-dimensional free surface flow simulations. First, we propose a different domain reconstruction approach than the classically used alpha-shape procedure, namely through the use of the advected boundary from the previous time step as a predicate to represent the new shape of the domain. Second, an adaptive refinement procedure is proposed in two steps: refinement of the boundary surface followed by quality-based node insertion in the bulk. Third, an approach for managing boundaries in complex geometries is presented. A series of applications is shown to demonstrate the interest of the approach.

Figures (25)

• Figure 1: Illustration of the PFEM algorithm.
• Figure 2: Shape reconstruction of a dragon using the $\alpha$-shape, with the ground truth in grey. The $\alpha$-shape does not capture sharp angles. In this example, $\alpha=1.3$.
• Figure 3: Shape reconstruction of a dragon using the winding number, as proposed by windingNumber2, with the ground truth in grey. Sharp angles in the input are well recovered.
• Figure 4: The winding number $w(p)$ is equal to 1 inside an oriented closed curve, and 0 outside.
• Figure 5: Domain definition with winding number approach. The figures represent two time steps (left and right). From the previous time step, the boundary is advected using the velocity field. We then re-triangulate the nodes, and detect, for each triangle, whether its barycenter lies inside the closed volume defined by the advected boundary. We then recover the shape.