Physics-Based Animation

Zhen Chen, Etienne Vouga, Danny M. Kaufman

While thin shells have ubiquitous applications and have been studied inside and outside computer graphics for decades, there is little consensus on how to best discretize them. We systematically study models for simulating bending of Kirchhoff-Love shells, with the goal of making practical recommendations, backed by careful numerical experiments, of when and how these models should be used. We analyze ten of the most popular discrete bending models in computer graphics for thin-shell simulation, along with new variants that we propose ourselves. We first verify all models on an analytic test benchmark to probe convergence under refinement and mesh-dependence, and then stress-test with a second benchmark that considers behavior at sharp bends. Finally, we test benchmark leaders on a practical suite of challenging large-scale equilibrium and dynamic shell modeling problems, analyzing both full solution behavior and comparative compute costs. We identify leading existing models and their tradeoffs in terms of accuracy and performance. During this analysis we also identify some issues and modeling gaps in the best-performing discrete bending models. We construct new energy model variants to address some of these gaps, as well as formulas and algorithmic tools for their practical simulation, and finally recommend best practices for modeling thin-shell bending.

Better Bending: Analysis, Construction and Verification of Discrete
Bending Models for Kirchhoff-Love Shells

Shusen Liu, Yuzhong Guo, Lixin Ren, Ying Qiao, Xiaowei He

Simulating free-surface flows requires capturing the effects of incompress-
ibility, viscosity, and surface tension. Existing particle-based methods often
rely on operator splitting, which introduces coupling artifacts and limits sta-
bility. We propose a unified nonlinear optimization framework that achieves
a strong coupling of these three effects within a single solver. By leveraging
peridynamics, we formulate the discretization of distinct fluid mechanisms
under a consistent variational principle. Specifically, we recast fluid mo-
tion as a nonlinear variational optimization problem over particle positions,
which is solved via the semi-implicit successive substitution method. More-
over, the framework incorporates separate treatments for bulk and shear
viscosity, allowing for more refined control of different viscous fluid behav-
iors. To the best of our knowledge, this is the first particle-based unified
solver capable of fully resolving the interdependence of incompressibility,
viscosity, and surface tension, thereby significantly enhancing stability in
complex simulations of free-surface flows. The source code for the paper is
publicly available at https://github.com/peridyno/peridyno.

A Nonlocal Monolithic Variational Framework for Free Surface Flows

Yizao Tang, Yuechen Zhu, Xingyu Ni, Baoquan Chen

The simulation of sand–water mixtures requires capturing the stochastic behavior of individual sand particles within a uniform, continuous fluid medium, such as the characteristic of migration, deposition, and plugging across various scenarios. In this paper, we introduce a Granule-in-Cell (GIC) method for simulating such sand–water interaction. We leverage the Discrete Element Method (DEM) to capture the fine-scale details of individual granules and the Particle-in-Cell (PIC) method for its continuous spatial representation and particle-based structure for density projection. To combine these two frameworks, we treat granules as macroscopic transport flow rather than solid boundaries for the fluid. This bidirectional coupling allows our model to accommodate a range of interphase forces with different discretization schemes, resulting in a more realistic simulation with fully respect to the mass conservation equation. Experimental results demonstrate the effectiveness of our method in simulating complex sand–water interactions, while maintaining volume consistency. Notably, in the dam-breaking experiment, our simulation uniquely captures the distinct physical properties of sand under varying infiltration degree within a single scenario. Our work advances the state of the art in granule–fluid simulation, offering a unified framework that bridges mesoscopic and macroscopic dynamics.

The Granule-In-Cell Method for Simulating Sand–Water Mixtures

José Antonio Fernández-Fernández, Fabian Löschner, Jan Bender

Newton’s Method is widely used to find the solution of complex non-linear simulation problems. To guarantee a descent direction, it is common practice to clamp the negative eigenvalues of each element Hessian prior to assembly — a strategy known as Projected Newton (PN) — but this perturbation often hinders convergence. In this work, we observe that projecting only a small subset of element Hessians is sufficient to secure a descent direction. Building on this insight, we introduce Progressively Projected Newton (PPN), a novel variant of Newton’s Method that uses the current iterate’s residual to cheaply determine the subset of element Hessians to project. The benefit is twofold: most eigendecompositions are avoided and the global Hessian remains closer to its original form, reducing the number of Newton iterations. We compare PPN with PN and Project-on-Demand Newton (PDN) in a comprehensive set of experiments covering contact-free and contact-rich deformables, co-dimensional and rigid-body simulations, and a range of time step sizes, tolerances and resolutions. PPN reduces the amount of element projections in dynamic simulations by one order of magnitude while simultaneously improving convergence, consistently being the fastest solver in our benchmark.

Progressively Projected Newton’s Method

Alexandre Mercier-Aubin, Teseo Schneider, Paul G. Kry, Sheldon Andrews

We introduce affinification, a novel method for accelerating physics-based animation of elastic solids. During a time-dependent simulation, our method automatically partitions the space into affine and elastic regions depending on the deformation. As such, we capture localized deformations while significantly reducing computational costs with larger regions of model reduction. We design a new clustering method based on deformation rates to capture affinely deforming regions, and explore multiple heuristics for seeding, pattern generation, and the impact of physical parameters on coarsened regions. We compare our method with the ground truth, showing performance increasing with resolution and recorded simulations up to 17 times faster compared to elastic simulations, while retaining similar levels of visual fidelity.

Affinification: A Fine Approximation of Deformations

Liangwang Ruan, Bin Wang, Tiantian Liu, Baoquan Chen

We present an efficient and scalable method for the inverse shape design problem of elastic objects, with broad applicability to diverse materials and interactive editing. The core idea is to decouple material nonlinearity from geometry optimization by introducing the Cauchy stress tensor as an auxiliary variable. We design a three-stage scheme that iteratively optimizes the stress tensors and the rest shape, with each stage being well-posed and efficiently-solvable. To address the lack of a theoretical convergence guarantee arising from the decoupled energy formulation, we incorporate a relaxation method that ensures robust stability in practice. As a result, our method achieves a 3× speedup over the state-of-the-art asymptotic method [Jia21] on a model with 40k vertices and 112k elements (Fig. 2), and exhibits near-linear scalability to large systems (Fig. 8). We demonstrate applications including rest shape design for various materials (ranging from standard models to complex spline-based materials [XSZB15]), interactive material and force editing, and elastic object reconstruction from images.

STAGED: Stress-Tensor Assisted Global-local-global solver for interactive Elastic shape Design

Duowen Chen, Zhiqiang Lao, Yu Guo, Heather Yu

We introduce a hybrid pipeline that combines classical fluid simulation with modern generative video models to produce high- quality, controllable fluid effects without implementationally difficult solvers or costly ray-tracing. First, a lightweight physics- based simulator enforces core properties like incompressibility and lets artists specify layout, boundary conditions, and source positions. Second, we render a simple ‘control video’ via real-time rasterisation (diffuse shading, masks, depth) to capture scene structure and material regions. Third, a text-guided diffusion transformer (e.g., VACE) treats this control video as a canvas, refining it by adding foam, bubbles, splashes, and realistic colour blending for multiple materials. Our method leverages pre- trained video generators’ implicit physical priors, while masking and noise-warping ensure precise, per-material control and seamless mixtures in latent space. Compared to purely simulation-based or generative model based text-only approaches, we avoid implementing specialised multiphase algorithms and expensive rendering passes, yet retain full artistic control over fluid behaviour and appearance. We demonstrate that this training-free strategy delivers photorealistic fluid videos, supports diverse effects (multiphase flows, transparent media and wet foams), and simplifies the artist’s workflow by unifying simulation, shading, and generative rendering in a single, extensible framework.

Fluid Composer: Fluid Detail Composition and Rendering Using Video Diffusion Models

Zoé Herson, Axel Paris, Élie Michel

We present a real-time method to capture and simulate the dynamic behavior of watercolor painting. We develop a physically accurate, grid-based, real-time fluid simulation based on a reparameterized Thin Film model. The equations are rewritten so as to create principled parameters that finely control the length, thickness, and frequency of dripping. Our close connection with physics allows both theoretical and experimental validation of our method. The resulting system can reproduce dripping, fluid-air interface, and pigment advection and diffusion, all controllable by the user in real-time. Our experiments show that artists can use our system to create interesting and varied digital paintings.

Dripping Thin Films for Real-time Digital Painting

Junyuan Liu, Shusen Liu, Yuzhong Guo, Ruikai Liang, Yin Li, Xiaowei He

While semi-analytical boundary handling techniques have proven effective for modeling particle-based fluid-solid interactions, they can become unstable when applied to mesh boundaries undergoing dynamic motion or featuring complex, sharp geometries. We propose a novel semi-analytical energy model for boundary handling that unifies fluid simulation and boundary interactions within a variational framework. The model comprises two key components: a semi-analytical bulk energy formulation that mitigates particle deficiency issues in the evaluation of bulk energy, and a nonlocal contact potential that effectively prevents particle penetration into boundaries. Both energy terms are naturally compatible with the Semi-Implicit SPH (SISPH), and a unified Hessian-free solver combined with reduced-order collision detection enables an efficient and stable GPU-based implementation for both fluid dynamics and nonlinear fluid-solid interactions. Furthermore, the unified treatment of fluid bulk energy and boundary energy via the semi-analytical formulation robustly corrects penetrations in practice, even under severe compression scenarios involving complex moving boundaries. Compared with existing semi-analytical boundary treatments, our method is more robust under fast boundary motion and strong compression. Across challenging benchmarks with sharp features, narrow gaps, and moving meshes, it remains stable and penetration-free where prior methods often fail.

A Semi-Analytical Energy Model for Particle-Based Fluid Simulation Involving Complex Moving Boundaries

Fumiya Narita, Takashi Kanai

Tall cell grids have been proposed as an efficient approach to accelerate large-scale liquid simulation. In this framework, regions near the liquid surface are discretized with regular grids, while regions farther away are represented by elongated rectangular cells. The regular grid region close to the surface is referred to as the optical layer. In previous work, the thickness of this optical layer was uniformly fixed across the entire liquid domain. In this paper, we propose a novel tall cell grid structure in which the thickness of the optical layer is dynamically adjusted according to the motion of the liquid. This adaptive strategy reduces the number of grid cells required in the projection step without compromising visual quality, thereby accelerating the overall simulation. Furthermore, we introduce a two-way coupling scheme between rigid bodies and liquids in regions where the optical layer remains thin. Our algorithm is simple and can be easily integrated into existing tall cell grid frameworks.

Adaptive Optical Layers: Efficient Tall Cell Grids for Liquid Simulation