Eulerian-Lagrangian Fluid Simulation on Particle Flow Maps

TL;DR

PFM introduces an Eulerian-Lagrangian framework that uses forward particle trajectories as samples of a perfect flow map and transports impulse and its gradients via a dual-scale flow map on each particle. It eliminates the neural buffer and backtracking of NFM by forward-evolving the backward map Jacobian $m T$ and performing a particle-to-grid transfer of $ m m$ and $ abla m m$, coupled with a MAC-grid Poisson solve for incompressibility. The approach achieves up to 49x faster speed and up to 41% memory reduction than NFM while preserving vortical structures in 2D and 3D tests, including leapfrogging vortices, vortex-tube reconnections, and turbulent flows. It introduces a long-short flow map with temporal samples, adaptive transport of gradients via $m T$, and impulse-based transfer with reinitialization, offering a scalable, accurate alternative for high-fidelity fluid simulation in graphics and physics domains.

Abstract

We propose a novel Particle Flow Map (PFM) method to enable accurate long-range advection for incompressible fluid simulation. The foundation of our method is the observation that a particle trajectory generated in a forward simulation naturally embodies a perfect flow map. Centered on this concept, we have developed an Eulerian-Lagrangian framework comprising four essential components: Lagrangian particles for a natural and precise representation of bidirectional flow maps; a dual-scale map representation to accommodate the mapping of various flow quantities; a particle-to-grid interpolation scheme for accurate quantity transfer from particles to grid nodes; and a hybrid impulse-based solver to enforce incompressibility on the grid. The efficacy of PFM has been demonstrated through various simulation scenarios, highlighting the evolution of complex vortical structures and the details of turbulent flows. Notably, compared to NFM, PFM reduces computing time by up to 49 times and memory consumption by up to 41%, while enhancing vorticity preservation as evidenced in various tests like leapfrog, vortex tube, and turbulent flow.

Figures (24)

• Figure 1: Particle is a flow map.
• Figure 2: (a) Velocity field streamlines of a single vortex. (b) Angular velocity along $y$ = 0.2 of the velocity field in (a). (c) Deviation of $\mathcal{F}\mathcal{T}$ relative to the identity matrix. (d) Discrepancy between the forward-evolved $\mathcal{T}$ and the backward-evolved $\mathcal{T}$. In both (c) and (d), the errors correspond to the average of the individual errors across all particles, where the error on each particle is quantified by the Frobenius norm of the disparities between the respective matrices of each particle. (e) Disparity between the analytical velocity field and the velocity field reconstructed using our method. It demonstrates that incorporating impulse gradients in particle-to-grid process plays a crucial role in reducing errors.
• Figure 3: These images depict the vortices created by a fish's tail as it swims through water. The fish's periodic tail movements generate cyclical vortices, and the nesting of these vortex tubes creates a layered and complex structure.
• Figure 4: These images shows vortex formation process initiated by a propeller as it rotates under wind influence. These visuals provide a clear view of the intricate, spiral-shaped patterns formed by the interconnected vortex tubes.
• Figure 5: Adaptive flow map from time $a$ to time $c$ with temporal samples $[b_{n}, ..., b_2, b_1]$. Impulse is stored on each time sample and backward map Jacobian is stored between adjacent time samples.

Theorems & Definitions (9)

• Remark 1
• Remark 2
• Theorem B.1
• proof
• Theorem B.2
• proof
• Theorem C.1
• proof
• Theorem C.2