A Level Set Method on Particle Flow Maps

TL;DR

benchmark comparisons in 2D and 3D demonstrate that PFM-LS achieves state-of-the-art volume preservation and shape fidelity against a broad range of existing level-set methods.

Abstract

This paper introduces a Particle Flow Map Level Set (PFM-LS) method for high-fidelity interface tracking. We store level-set values, gradients, and Hessians on particles concentrated in a narrow band around the interface, advecting them via bidirectional flow maps while using a conventional grid-based representation elsewhere. By interpreting the level set value as a 3-form and its gradient as a 1-form, PFM-LS achieves exceptional geometric fidelity during complex deformations and preserves sub-grid features that traditional methods cannot capture. Our dual-timescale approach utilizes long-range maps for values and gradients, with frequent reinitialization of short-range maps for the distortion-sensitive Hessian, alongside adaptive particle control that maintains sufficient density within the narrow band. We also develop a hybrid particle-grid quasi-Newton redistancing scheme that preserves fine-scale features while enforcing the signed-distance property. Benchmark comparisons in 2D and 3D demonstrate that PFM-LS achieves state-of-the-art volume preservation and shape fidelity against a broad range of existing level-set methods.

Figures (18)

• Figure 1: Particle trajectory from time $a$ to $c$ naturally defines both forward map $\boldsymbol{\phi}_{[a,c]}$ and backward map $\boldsymbol{\psi}_{[c,a]}$, forming a perfect bidirectional flow map.
• Figure 2: Transport of differential forms under the flow map. Point elements (3-forms, e.g., $\rho$) are preserved; line elements (2-forms, e.g., vorticity) transform via $F = \nabla\boldsymbol{\phi}$; surface elements (1-forms, e.g., impulse) transform via $T^\top$.
• Figure 3: Illustration of gradient-augmented level sets with particles. Red and blue particles carry level set values and gradient information (arrows) near the interface $\varphi = 0$. This higher-order representation enables capturing sub-grid features such as small droplets (left), thin films (middle), and annular structures (right) that traditional methods cannot resolve.
• Figure 4: Narrow band particle flow map level set representation. The octagonal interface (orange) is tracked by particles (blue dots) concentrated only in a narrow band around the interface. The zoomed regions show particles carrying gradient information (arrows) at two time points, with the backward map $\boldsymbol{\psi}_{[c, a]}$ connecting their positions. This focused approach maintains high accuracy near the interface while reducing computational cost elsewhere.
• Figure 5: Volume preservation comparison during $200\times200$2D Zalesak's disk rotation test. Our method (orange lines) maintains nearly perfect volume conservation throughout the entire revolution. The inset shows detailed performance during the final moments, where our method preserves over 99.9999% of the original volume, significantly outperforming all other methods. Most reference map-augmented methods (RM suffix) achieve good preservation, while traditional methods like Linear show substantial volume loss. Note that NBFLIP struggles in this test despite its particle-based nature.