More Power to the Particles: Analytic Geometry for Partial Optimal Transport-based Fluid simulation

Abstract

We propose unified data structures and algorithms for free-surface fluid simulations based on partial optimal transport, such as the Power Particles method or Gallou\"et-M\'erigot's scheme. Such methods previously relied on a discretization of the cells by leveraging a classical convex cell clipping algorithm. However, this results in a heavy computational cost and a coarse approximation of the evaluated quantities. In contrast, we propose to analytically construct the generalized Laguerre cells characterized by intersections between Laguerre cells and spheres. This makes it possible to accurately compute the differential quantities used by the Newton algorithm, that is, the areas of the (curved) facets and the volumes of the (generalized) Laguerre cells. This significantly improves the convergence of the Newton algorithm, hence the robustness of the simulations, even in challenging scenarios with high velocities and chocs. Moreover, this drastically reduces the computational cost as compared to previous works. Based on our data structure, we propose a framework that combines (1) the numerical solution mechanism for partial optimal transport, (2) the fluid simulation scheme and (3) the rendering. The aforementioned three components are implemented on the GPU, providing further speedup and avoiding data transfers. This is made possible by the compactness of our data structure combined with a massively parallel implementation. We report the result of numerical experiments featuring highly detailed, large-scale simulations and high variations of physical properties within the same simulation.