Physics-Based Animation

Lorenzo Diazzi, Jiacheng Dai, Daniele Panozzo, Marco Attene

We present an algorithm that produces high quality tetrahedral meshes conforming with input polyhedra. Our meshing algorithm is based on Ruppert’s Delaunay refinement where convergence is guaranteed thanks to a novel chamfering approach that removes all acute angles from the input. On such a modified input Delaunay refinement produces a Delaunay tetrahedrization where all the faces have bounded angles. The input portions that were removed by the chamfering are re-inserted in this tetrahedrization to achieve exact conformance at the cost of a small number of bad-shaped tetrahedra near the formerly acute input angles. Numerical robustness is guaranteed along all the phases thanks to a clever use of modern indirect geometric predicates and the definition of a new type of implicit point to represent Steiner vertices on the input faces. In practice, our prototype implementation produces meshes having a quality comparable to the state-of-the-art tetgen software: while tetgen fails on 37% of the 3942 valid models in the Thingi10k dataset, our method succeeds on all of them.

Surface chamfering for robust tetrahedral meshing

Aleksei Kalinov, Mickaël Ly, Christian Hafner, Chris Wojtan

The appearance of simulated natural phenomena heavily depends on the way surfaces are textured. However, applying texture maps to dynamic deformable surfaces presents a significant challenge, due to ever-shifting differences in length scales involved. When these surfaces move and advect the texture along with them, their final appearance degrades as deformed regions dramatically distort their texture map. Modifications to the texture directly at the pixel level in response to the deformation may introduce ghosting artifacts and look unnatural. In the real world, the appearance of surface details on a deforming material changes through the interplay of physical processes such as rupturing, exposure of internal structure, or wrinkling. Motivated by these behaviors, in this work we explore how physical principles can guide the texturing methods based on the measure of surface deformation. We present two novel wave-based procedural texturing algorithms which reproduce common physical properties like advection and self-similarity, enabling the plausible animation of deforming objects with extreme texture map distortions. Our algorithms are fully procedural, require no actual physics simulation, and store no state or history of deformation besides the input UV map, making them highly parallelizable on the GPU and efficient enough for real-time applications. We show the versatility of the method by animating physical phenomena with extreme deformations such as flowing lava, stretching putty and outpouring sludge.

Physics-inspired procedural texturing of extremely deformable surfaces

Xuwen Chen, Junyu Wang, Cheng Yu, Xingyu Ni, Meng Zhang, Bin Wang, Mengyu Chu, Baoquan Chen

Continuous Collision Detection is essential in simulation and modeling for accurately identifying object collisions. While robust CCD techniques have matured for triangle meshes, ensuring floating-point robustness for parametric surfaces remains an open challenge due to their representational complexity and heightened algorithmic sensitivity. In this paper, we present the first floating-point-robust CCD framework for parametric surfaces. Built on the Time-Dependent Inclusion-Based Method (TDIBM), our approach introduces a novel error decomposition strategy that separates coefficient and arithmetic errors, enabling structured analysis and safety guarantees. To rigorously benchmark robustness, we develop a rational arithmetic-based dataset by inverting the CCD process — we generate exact ground-truth datasets from prescribed collision outcomes. Our construction captures both typical scenarios and near-degenerate cases. We evaluate several CCD algorithms using this benchmark to provide an in-depth analysis. Together, our method and dataset establish a comprehensive foundation for analyzing, benchmarking, and improving floating-point robustness in parametric surface CCD. Code and dataset will be published upon acceptance.

Floating-Point Robustness in Parametric Surface Continuous Collision Detection: From Algorithm to Benchmarking

Behrooz Zarebavani, Ahmed H. Mahmoud, Ana Dodik, Changcheng Yuan, Serban D. Porumbescu, John D. Owens, Maryam Mehri Dehnavi, Justin Solomon

We present a fast sparse matrix permutation algorithm tailored to linear systems arising from triangle meshes. Our approach produces nested-dissection-style permutations while significantly reducing permutation runtime overhead. Rather than enforcing strict balance and separator optimality, the algorithm deliberately relaxes these design decisions to favor fast partitioning and efficient elimination-tree construction. Our method decomposes permutation into patch-level local orderings and a compact quotient-graph ordering of separators, preserving the essential structure required by sparse Cholesky factorization while avoiding its most expensive components. We integrate our algorithm into vendor-maintained sparse Cholesky solvers on both CPUs and GPUs. Across a range of graphics applications, including single factorizations and repeated factorizations, our method reduces permutation time and improves the sparse Cholesky solve performance by up to 6.27x. Our code is available at this https URL.

Fast Sparse Matrix Permutation for Mesh-Based Direct Solvers

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