DDFKs: Fluid Simulation with Dynamic Divergence-Free Kernels

2026-01-25

AI Summary

This work proposes DDFKs, a novel mesh-free solver that introduces dynamic divergence-free kernels (DFKs) into fluid simulation for the first time. Existing memory-efficient spatial representations struggle to enforce strict divergence-free constraints for incompressible flows while preserving high accuracy and fine-scale details. In contrast, DDFKs construct a continuous, adaptive velocity field using matrix-valued radial basis functions and vector-valued weights, inherently satisfying incompressibility without post-projection steps. This formulation significantly reduces numerical dissipation and naturally accommodates complex boundaries and intricate vortex structures. When combined with implicit neural or Gaussian spatial representations, the method achieves state-of-the-art performance in vortex preservation, temporal and memory efficiency, and generalizability across diverse incompressible flow scenarios, while enforcing the divergence-free condition more rigorously than prior approaches.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Computer Vision: Diffusion Models for Vision
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures

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.

Problem

• incompressible fluids
• divergence-free constraint
• fluid simulation
• spatial representations
• time-dependent PDEs

Innovation

• Divergence-Free Kernels
• Grid-Free Fluid Simulation
• Incompressible Flow
• Radial Basis Functions
• Implicit Neural Representations