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Lagrangian Covector Fluid with Free Surface

Zhiqi Li, Barnabás Börcsök, Duowen Chen, Yutong Sun, Bo Zhu, Greg Turk

TL;DR

The paper presents a novel Lagrangian covector fluid framework that uses flow maps and path integrals to decouple the advection and projection steps for incompressible free surface flows. By introducing long range and short range mapping strategies and a unifying LMSP scheme, the authors transform challenging integral boundary Poisson problems into standard Dirichlet/Neumann solves, improving robustness and vorticity preservation. A Voronoi based meshfree implementation enables accurate gradient/divergence operators on moving particles, and results across 2D and 3D benchmarks demonstrate superior performance over purely particle based methods like the Power Particle Method. The approach opens new avenues for handling free surfaces in covector flow maps and offers a practical route toward robust, vortex preserving fluid simulations with complex boundaries.

Abstract

This paper introduces a novel Lagrangian fluid solver based on covector flow maps. We aim to address the challenges of establishing a robust flow-map solver for incompressible fluids under complex boundary conditions. Our key idea is to use particle trajectories to establish precise flow maps and tailor path integrals of physical quantities along these trajectories to reformulate the Poisson problem during the projection step. We devise a decoupling mechanism based on path-integral identities from flow-map theory. This mechanism integrates long-range flow maps for the main fluid body into a short-range projection framework, ensuring a robust treatment of free boundaries. We show that our method can effectively transform a long-range projection problem with integral boundaries into a Poisson problem with standard boundary conditions -- specifically, zero Dirichlet on the free surface and zero Neumann on solid boundaries. This transformation significantly enhances robustness and accuracy, extending the applicability of flow-map methods to complex free-surface problems.

Lagrangian Covector Fluid with Free Surface

TL;DR

The paper presents a novel Lagrangian covector fluid framework that uses flow maps and path integrals to decouple the advection and projection steps for incompressible free surface flows. By introducing long range and short range mapping strategies and a unifying LMSP scheme, the authors transform challenging integral boundary Poisson problems into standard Dirichlet/Neumann solves, improving robustness and vorticity preservation. A Voronoi based meshfree implementation enables accurate gradient/divergence operators on moving particles, and results across 2D and 3D benchmarks demonstrate superior performance over purely particle based methods like the Power Particle Method. The approach opens new avenues for handling free surfaces in covector flow maps and offers a practical route toward robust, vortex preserving fluid simulations with complex boundaries.

Abstract

This paper introduces a novel Lagrangian fluid solver based on covector flow maps. We aim to address the challenges of establishing a robust flow-map solver for incompressible fluids under complex boundary conditions. Our key idea is to use particle trajectories to establish precise flow maps and tailor path integrals of physical quantities along these trajectories to reformulate the Poisson problem during the projection step. We devise a decoupling mechanism based on path-integral identities from flow-map theory. This mechanism integrates long-range flow maps for the main fluid body into a short-range projection framework, ensuring a robust treatment of free boundaries. We show that our method can effectively transform a long-range projection problem with integral boundaries into a Poisson problem with standard boundary conditions -- specifically, zero Dirichlet on the free surface and zero Neumann on solid boundaries. This transformation significantly enhances robustness and accuracy, extending the applicability of flow-map methods to complex free-surface problems.
Paper Structure (33 sections, 1 theorem, 9 equations, 16 figures, 2 tables, 1 algorithm)

This paper contains 33 sections, 1 theorem, 9 equations, 16 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Particle-based fluid simulation
  4. Flow-map methods
  5. Gauge-based fluid
  6. Mathematical Foundation
  7. Flow Maps
  8. Covector Preliminaries
  9. Lie Advection
  10. Physical Model
  11. Covector Incompressible Fluid
  12. Covector Fluid on Flow Maps
  13. Path Integral on Lagrangian Flow Maps
  14. Incompressibility
  15. Long-Range Mapping, Long-Range Projection
  16. ...and 18 more sections

Key Result

proposition 1

Poission eq:possion_3 with initial guess $\hat{\lambda} =0$ is equivalent to Poission eq:possion_1 with initial guess $\Lambda_s^r=\Lambda_s^{s'}$

Figures (16)

  • Figure 1: Forward and backward flow maps defined on a particle.
  • Figure 2: Voronoi discretization
  • Figure 3: Particles represent either interior (blue) or boundary (gray-blue) fluid, boundaries (black), or air (orange).
  • Figure 4: A moving and rotating board in 2D traverses the domain from left to right, and then back.
  • Figure 5: Leapfrog vortices in 2D. PPM (top), our approach (bottom).
  • ...and 11 more figures

Theorems & Definitions (1)

  • proposition 1