DDFKs: Fluid Simulation with Dynamic Divergence-Free Kernels

TL;DR

This paper tackles the challenge of simulating incompressible flows with memory-efficient spatial representations by introducing Dynamic Divergence-Free Kernels (DDFKs). By representing the velocity field as a sum of divergence-free kernels with matrix-valued radial basis components, the method enforces zero-divergence at all times, avoiding the need for pressure-projection or divergence losses common in INSR- or GSR-based solvers. The approach combines initialization, physics-based optimization, and periodic reinitialization to achieve accurate vorticity preservation, reduced numerical dissipation, and robust boundary handling across 2D and 3D scenarios, including complex geometries. The results show competitive accuracy against traditional Eulerian and memory-efficient methods, while delivering superior incompressibility guarantees and stability in long-horizon simulations, highlighting its potential for high-fidelity, grid-free fluid animation and engineering applications.

Abstract

Fluid simulations based on memory-efficient spatial representations like implicit neural spatial representations (INSRs) and Gaussian spatial representation (GSR), where the velocity fields are parameterized by neural networks or weighted Gaussian functions, has been an emerging research area. Though advantages over traditional discretizations like spatial adaptivity and continuous differentiability of these spatial representations are leveraged by fluid solvers, solving the time-dependent PDEs that governs the fluid dynamics remain challenging, especially in incompressible fluids where the divergence-free constraint is enforced. In this paper, we propose a grid-free solver Dynamic Divergence-Free Kernels (DDFKs) for incompressible flows based on divergence-free kernels (DFKs). Each DFK is incorporated with a matrix-valued radial basis function and a vector-valued weight, yielding a divergence-free vector field. We model the continuous flow velocity as the sum of multiple DFKs, thus enforcing incompressibility while being able to preserve different level of details. Quantitative and qualitative results show that our method achieves comparable accuracy, robustness, ability to preserve vortices, time and memory efficiency and generality across diverse phenomena to state-of-the-art methods using memory-efficient spatial representations, while excels at maintaining incompressibility. Though our first-order solver are slower than fluid solvers with traditional discretizations, our approach exhibits significantly lower numerical dissipation due to reduced discretization error. We demonstrate our method on diverse incompressible flow examples with rich vortices and various solid boundary conditions.

Figures (7)

• Figure 1: Compare between our method and Gaussian Fluids (GSR) xing2025gaussian-fluids. The images from left to right are simulation results of Leapfrog 2D at frames 0, 165, 456, 1050 and 1500, respectively.
• Figure 2: Karman vortex street example by Neural Monte Carlo (NMC), Gaussian fluids (GSR), and our method (DDFK). The sub-figures display the vorticity fields of frame 152, 199 and 199 of the simulation results, respectively.
• Figure 3: Ablation study on reinitialization. We show the simulation results and particle positions at frame 180.
• Figure 4: We compare our method with Gaussian Fluids (GSR) xing2025gaussian-fluids using vortices pass example of wide and narrow gaps. The wide setting The positions of the two spherical obstacles in the wide and narrow setting are $(0.5, 0.27) , (0.5, 0.73)$ and $(0.5, 0.285) , (0.5, 0.715)$. The radius of all the obstacles are $0.1174$.The figures from left to right are showing frames 200, 336, 520 and 879, respectively.
• Figure 5: We compare our method with traditional Eulerian grid and different memory-efficient spatial representations INSR and GSR on the Taylor vortex example. Enclosed by parenthesis on the first column is the average file size of the according representation of the velocity field. Images show the vorticity field at frame 0, 100, 200, 300, and 399, respectively.