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.

翻译：本文针对采用粒子有限元法(PFEM)计算自由表面流问题做出两项贡献。PFEM基于拉格朗日方法：由一组粒子定义流体。与纯拉格朗日方法不同，所有粒子通过三角形网格连接。难点在于从该网格中定位自由表面，即判断网格中哪些单元属于流体域，并定义边界——自由表面。随后在流体域上求解不可压缩纳维-斯托克斯方程，并利用所得速度矢量更新粒子位置。我们的第一项贡献是提出一种具有理论质量保证的网格自适应方法：网格生成领域已在具有最终网格质量保证的自适应方法方面积累了丰富经验与深刻认识。此处采用Delaunay细化策略，允许在逐步提升网格质量的同时插入和移除节点。研究表明，该方法可生成稳定光滑的自由表面几何形状。我们的PFEM方法模拟单一流体的拓扑演化过程，但需对域边界施加条件。当边界为自由表面时，另一侧的流动无需建模，仅用外部压力表示。在外部自由表面边界上可施加大气压。然而可能存在内部自由表面：流体可完全包裹空腔形成气泡。维持这些气泡体积所需压力先验未知。我们提出一种多点约束方法来实现这些空泡的整体不可压缩性。该方法能够在仅对重流体建模的情况下，精确模拟包含大密度差两相流体的气泡流。