Author: christopherbatty

Jingrui Xing, Bin Wang, Mengyu Chu, Baoquan Chen

We present a grid-free fluid solver featuring a novel Gaussian representation. Drawing inspiration from the expressive capabilities of 3D Gaussian Splatting in multi-view image reconstruction, we model the continuous flow velocity as a weighted sum of multiple Gaussian functions. This representation is continuously differentiable, which enables us to derive spatial differentials directly and solve the time-dependent PDE via a custom first‑order optimization tailored to fluid dynamics. Compared to traditional discretizations, which typically adopt Eulerian, Lagrangian, or hybrid perspectives, our approach is inherently memory-efficient and spatially adaptive, enabling it to preserve fine-scale structures and vortices with high fidelity. While these advantages are also sought by implicit neural representations, GSR offers enhanced robustness, accuracy, and generality across diverse fluid phenomena, with improved computational efficiency during temporal evolution. Though our first‑order solver does not yet match the speed of fluid solvers using explicit representations, its continuous nature substantially reduces spatial discretization error and opens a new avenue for high‑fidelity simulation. We evaluate the proposed solver across a broad range of 2D and 3D fluid phenomena, demonstrating its ability to preserve intricate vortex dynamics, accurately capture boundary-induced effects such as Kármán vortex streets, and remain robust across long time horizons—all without additional parameter tuning. Our results suggest that GSR offers a compelling direction for future research in fluid simulation.

Gaussian Fluids: A Grid-Free Fluid Solver based on Gaussian Spatial Representation

Yijie Liu, Taiyuan Zhang, Xiaoxiao Yan, Han Yan, Nuoming Liu, Bo Ren

The freezing of thin films is a mesmerizing natural phenomenon, inspiring photographers to capture its beauty through their lenses and digital artists to recreate its allure using effects tools. In this paper, we present a novel method for physically simulating the intricate freezing dynamics on thin films. By accounting for the influence of phase and temperature changes on surface tension, our method reproduces Marangoni freezing and the “Snow-Globe Effect”, characterized by swirling ice dendrites on the film. We introduce a novel Phase Map method on top of the state-of-the-art Moving Eulerian-Lagrangian Particles (MELP) meshless framework, enabling dendritic crystal simulation on mobile particles and offering precise control over freezing patterns. We demonstrate that our method is able to capture a wide range of dynamic freezing processes of soap bubbles and is stable for complex boundaries in our experiments.

Controllable Complex Freezing Dynamics Simulation on Thin Films

Yuchen Sun, Junlin Li, Ruicheng Wang, Sinan Wang, Zhiqi Li, Bart G. van Bloemen Waanders, Bo Zhu

We propose Leapfrog Flow Maps (LFM) to simulate incompressible fluids with rich vortical flows in real time. Our key idea is to use a hybrid velocity-impulse scheme enhanced with leapfrog method to reduce the computational workload of impulse-based flow map methods, while possessing strong ability to preserve vortical structures and fluid details. In order to accelerate the impulse-to-velocity projection, we develop a fast matrix-free Algebraic Multigrid Preconditioned Conjugate Gradient (AMGPCG) solver with customized GPU optimization, which makes projection comparable with impulse evolution in terms of time cost. We demonstrate the performance of our method and its efficacy in a wide range of examples and experiments, such as real-time simulated burning fire ball and delta wingtip vortices.

Leapfrog Flow Maps for Real-Time Fluid Simulation

Duowen Chen^*, Zhiqi Li^*, Taiyuan Zhang, Jinjin He, Junwei Zhou, Bart G van Bloemen Waanders, Bo Zhu (* Joint First Authors)

This paper presents a unified compressible flow map framework designed to accommodate diverse compressible flow systems, including high-Mach-number flows (e.g., shock waves and supersonic aircraft), weakly compressible systems (e.g., smoke plumes and ink diffusion), and incompressible systems evolving through compressible acoustic quantities (e.g., free-surface shallow water). At the core of our approach is a theoretical foundation for compressible flow maps based on Lagrangian path integrals, a novel advection scheme for the conservative transport of density and energy, and a unified numerical framework for solving compressible flows with varying pressure treatments. We validate our method across three representative compressible flow systems, characterized by varying fluid morphologies, governing equations, and compressibility levels, demonstrating its ability to preserve and evolve spatiotemporal features such as vortical structures and wave interactions governed by different flow physics. Our results highlight a wide range of novel phenomena, from ink torus breakup to delta wing tail vortices and vortex shedding on free surfaces, significantly expanding the range of fluid systems that flow-map methods can handle.

Fluid Simulation on Compressible Flow Maps

Zhiqi Li, Candong Lin, Duowen Chen^, Xinyi Zhou, Shiying Xiong, Bo Zhu

We propose a novel gauge fluid solver that evolves Clebsch wave functions on particle flow maps (PFMs). The key insight underlying our work is that particle flow maps exhibit superior performance in transporting point elements—such as Clebsch components—compared to line and surface elements, which were the focus of previous methods relying on impulse and vortex gauge variables for flow maps. Our Clebsch PFM method incorporates three main contributions: a novel gauge transformation enabling accurate transport of wave functions on particle flow maps, an enhanced velocity reconstruction method for coarse grids, and a PFM-based simulation framework designed to better preserve fine-scale flow structures. We validate the Clebsch PFM method through a wide range of benchmark tests and simulation examples, ranging from leapfrogging vortex rings and vortex reconnections to Kelvin–Helmholtz instabilities, demonstrating that our method outperforms its impulse- or vortex-based counterparts on particle flow maps, particularly in preserving and evolving small-scale features.

Clebsch Gauge Fluid on Particle Flow Maps

Zhiqi Li*, Ruicheng Wang*, Junlin Li*, Duowen Chen, Sinan Wang, Bo Zhu (* Co-first Authors)

We propose the Epsilon Difference Gradient Evolution (EDGE) method for accurate flow-map calculation on grids via Hermite interpolation without using velocity buffers. Our key idea is to integrate Gradient Evolution for accurate first-order derivatives and a tetrahedron-based Epsilon Difference scheme to compute higher-order derivatives with reduced memory consumption. EDGE achieves O(1) memory usage, independent of flow map length, while maintaining vorticity preservation comparable to buffer-based methods. We validate our methods across diverse vortical flow scenarios, demonstrating up to 90% backward map memory reduction and significant computational efficiency, broadening the applicability of flow-map methods to large-scale and complex fluid simulations.

EDGE: Epsilon-Difference Gradient Evolution for Buffer-Free Flow Maps

Mengdi Wang, Fan Feng, Junlin Li, Bo Zhu

We propose the adaptive hybrid particle-grid flow map method, a novel flow-map approach that leverages Lagrangian particles to simultaneously transport impulse and guide grid adaptation, introducing a fully adaptive flow map-based fluid simulation framework. The core idea of our method is to maintain flow-map trajectories separately on grid nodes and particles: the grid-based representation tracks long-range flow maps at a coarse spatial resolution, while the particle-based representation tracks both long and short-range flow maps, enhanced by their gradients, at a fine resolution. This hybrid Eulerian-Lagrangian flow-map representation naturally enables adaptivity for both advection and projection steps. We implement this method in Cirrus, a GPU-based fluid simulation framework designed for octree-like adaptive grids enhanced with particle trackers. The efficacy of our system is demonstrated through numerical tests and various simulation examples, achieving up to 512x512x2048 effective resolution on an RTX 4090 GPU. We achieve a 1.5 to 2x speedup with our GPU optimization over the Particle Flow Map method on the same hardware, while the adaptive grid implementation offers efficiency gains of one to two orders of magnitude by reducing computational resource requirements. The source code has been made publicly available at: https://wang-mengdi.github.io/proj/25-cirrus/.

Cirrus: Adaptive Hybrid Particle-Grid Flow Maps on GPU

Sinan Wang, Junwei Zhou, Fan Feng, Zhiqi Li, Yuchen Sun, Duowen Chen, Greg Turk, Bo Zhu

We propose the Vortex Particle Flow Map (VPFM) method to simulate incompressible flow with complex vortical evolution in the presence of dynamic solid boundaries. The core insight of our approach is that vorticity is an ideal quantity for evolution on particle flow maps, enabling significantly longer flow map distances compared to other fluid quantities like velocity or impulse. To achieve this goal, we developed a hybrid Eulerian-Lagrangian representation that evolves vorticity and flow map quantities on vortex particles, while reconstructing velocity on a background grid. The method integrates three key components: (1) a vorticity-based particle flow map framework, (2) an accurate Hessian evolution scheme on particles, and (3) a solid boundary treatment for no-through and no-slip conditions in VPFM. These components collectively allow a substantially longer flow map length (3-12 times longer) than the state-of-the-art, enhancing vorticity preservation over extended spatiotemporal domains. We validated the performance of VPFM through diverse simulations, demonstrating its effectiveness in capturing complex vortex dynamics and turbulence phenomena.

Fluid Simulation on Vortex Particle Flow Maps

Leticia Mattos Da Silva, Silvia Sellán, Natalia Pacheco-Tallaj, Justin Solomon

Numerical schemes for time integration are the cornerstone of dynamical simulations for deformable solids. The most popular time integrators for isotropic distortion energies rely on nonlinear root-finding solvers, most prominently, Newton’s method. These solvers are computationally expensive and sensitive to ill-conditioned Hessians and poor initial guesses; these difficulties can particularly hamper the effectiveness of variational integrators, whose momentum conservation properties require reliable root-finding. To tackle these difficulties, this paper shows how to express variational time integration for a large class of elastic energies as an optimization problem with a “hidden” convex substructure. This hidden convexity suggests uses of optimization techniques with rigorous convergence analysis, guaranteed inversion-free elements, and conservation of physical invariants up to tolerance/numerical precision. In particular, we propose an Alternating Direction Method of Multipliers (ADMM) algorithm combined with a proximal operator step to solve our formulation. Empirically, our integrator improves the performance of elastic simulation tasks, as we demonstrate in a number of examples.

Variational Elastodynamic Simulation

Mengdi Wang, Fan Feng, Junlin Li, Bo Zhu

We propose the adaptive hybrid particle-grid flow map method, a novel flow-map approach that leverages Lagrangian particles to simultaneously transport impulse and guide grid adaptation, introducing a fully adaptive flow map-based fluid simulation framework. The core idea of our method is to maintain flow-map trajectories separately on grid nodes and particles: the grid-based representation tracks long-range flow maps at a coarse spatial resolution, while the particle-based representation tracks both long and short-range flow maps, enhanced by their gradients, at a fine resolution. This hybrid Eulerian-Lagrangian flow-map representation naturally enables adaptivity for both advection and projection steps. We implement this method in Cirrus, a GPU-based fluid simulation framework designed for octree-like adaptive grids enhanced with particle trackers. The efficacy of our system is demonstrated through numerical tests and various simulation examples, achieving up to 512x512x2048 effective resolution on an RTX 4090 GPU. We achieve a 1.5 to 2x speedup with our GPU optimization over the Particle Flow Map method on the same hardware, while the adaptive grid implementation offers efficiency gains of one to two orders of magnitude by reducing computational resource requirements. The source code has been made publicly available at: https://wang-mengdi.github.io/proj/25-cirrus/.

Cirrus: Adaptive Hybrid Particle-Grid Flow Maps on GPU