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

TL;DR

This work targets robust, volume-conserving free-surface fluid simulation using partial optimal transport. It replaces heavy cell clipping with an analytic construction: restricting Laguerre cells to spheres and computing volumes and facet areas via pyramid decomposition and spherical patches, all tied to a convex Kantorovich objective $K(\mathbf{\Psi})$ and its gradient $\partial K/\partial \psi_i=\nu_i-|V_i|$. The contributions include an explicit analytic geometry for restricted Laguerre cells, a complete rendering pipeline based on this structure, and demonstrations of strong performance on large-scale simulations. The approach enables precise surface representation, scalable computation, and a rendering framework closely integrated with the OT-based fluid solver, with potential for GPU acceleration and broader deformation-mechanics applications.

Abstract

We propose an analytic construction of the geometry required for free-surface fluid simulations and deformation mechanics based on partial optimal transport such as the Gallouët-Mérigot's scheme or the Power Particles method. 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, our algorithm efficiently computes the generalized Laguerres cells, that is, intersections between Laguerre cells and spheres. This makes it possible to more precisely compute the volume and the area of the facets as well as strongly reducing the number of operations required to obtain the geometry. Additionally, we provide a dedicated rendering framework solely based on the computed volumetric structure.

Figures (13)

• Figure 1: Illustration of Eulerian and Lagrangian representations.
• Figure 2: Illustrations of the two methods adding free surface to the optimal transport-based fluid simulation. While air particles can be used to discretize the second phase Goes2015, it also involves a higher cost. In contrast, the analytic characterization Levy2022 offers an explicit geometry of the free surface without requiring additional cells.
• Figure 3: Illustrations of the restriction geometry inherent to the free surface scheme. (a) Laguerre diagram (white), restricted diagram (blue) and free surface (red). (b) Restricted Laguerre facets (blue) and free surface (red). (c) Discretization performed in the previous method Levy2022.
• Figure 4: Pyramid decomposition used for the integration of quantities over the restricted cells. Pyramids are defined by an apex located on the site (white) and a base which is a restricted Laguerre facet (blue) or a part of the spherical shell (red).
• Figure 5: Illustration of the area computation with spherical patches. (a) A spherical patch is bounded by arcs and vertices derived from the Laguerre cell and the sphere which may require exact predicates. (b) The area of projection of the cell facets onto the sphere can be computed without dedicated exact predicates.