Mesh adaption for two-dimensional bounded and free-surface flows with the particle finite element method

The particle finite element method (PFEM) is a Lagrangian method that avoids large mesh distortion through automatic remeshing when the computational grid becomes too distorted. The method is well adapted for flows with deforming interfaces and moving boundaries. However, the α-shape technique used to identify these boundaries presupposes a mesh of approximately uniform size. Moreover, the α-shape criterion is purely geometric and, thus, leads to violations of mass conservation at boundaries. We propose a new algorithm for mesh refinement and adaptation in two dimensions to improve the ratio accuracy to computational cost of the PFEM. A local target mesh size is prescribed according to geometric and/or physics-based criteria, and particles are added or removed to approximately enforce this target mesh size. Additionally, the new boundary recognition algorithm relies on the tagging of boundary nodes and a local α-shape criterion that depends on the target mesh size. The method allows thereby reducing mass conservation errors at free surfaces and improving the local accuracy through mesh refinement and simultaneously offers a new boundary tracking algorithm. The new algorithm is tested on four two-dimensional validation cases. The first two cases, i.e., the lid-driven cavity flow at Reynolds number 400 and the flow around a static cylinder at Reynolds numbers below 200, do not feature a free surface and mainly illustrate the mesh refinement capability. The last two test cases consist in the sloshing problem in a reservoir subjected to forced oscillations and the fall of a 2D liquid drop into a tank filled with the same viscous fluid. These last two cases demonstrate the more accurate representation of the free surface and a corresponding reduction of the error in mass conservation.