A Level Set Method on Particle Flow Maps

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.