VEMPIC: Particle-in-Polyhedron Fluid Simulation for Intricate Solid Boundaries

The comprehensive visual modeling of fluid motion has historically been a challenging task, due in no small part to the difficulties inherent in geometries that are non-manifold, open, or thin. Modern geometric cut-cell mesh generators have been shown to produce, both robustly and quickly, workable volumetric elements in the presence of these problematic geometries, and the resulting volumetric representation would seem to offer an ideal infrastructure with which to perform fluid simulations. However, cut-cell mesh elements are general polyhedra that often contain holes and are non-convex; it is therefore difficult to construct the explicit function spaces required to employ standard functional discretizations, such as the Finite Element Method. The Virtual Element Method (VEM) has recently emerged as a functional discretization that successfully operates with complex polyhedral elements through a weak formulation of its function spaces. We present a novel cut-cell fluid simulation framework that exactly represents boundary geometry during the simulation. Our approach enables, for the first time, detailed fluid simulation with "in-the-wild" obstacles, including ones that contain non-manifold parts, self-intersections, and extremely thin features. Our key technical contribution is the generalization of the Particle-In-Cell fluid simulation methodology to arbitrary polyhedra using VEM. Coupled with a robust cut-cell generation scheme, this produces a fluid simulation algorithm that can operate on previously infeasible geometries without requiring any additional mesh modification or repair.