Physics-Based Animation

Ahmed H. Mahmoud, Rahul Goel, Jonathan Ragan-Kelley, and Justin Solomon

We present a GPU-based system for automatic differentiation (AD) of functions defined on triangle meshes, designed to exploit the locality and sparsity in mesh-based computation. Our system evaluates derivatives using per-element forward-mode AD, confining all computation to registers and shared memory and assembling global gradients, sparse Jacobians, and sparse Hessians directly on the GPU. By avoiding global computation graphs, intermediate buffers, and device-host synchronization, our approach minimizes memory traffic and enables efficient differentiation under both static and dynamically changing sparsity. Our programming model lets users express energy terms over mesh neighborhoods, while our system automatically manages parallel execution, derivative propagation, sparse assembly, and matrix-free operations such as Hessian-vector products. Our system supports both scalar- and vector-valued objectives, dynamic interaction-driven sparsity updates, and seamless integration with external GPU sparse linear solvers. We evaluate our system on applications including elastic and cloth simulation, surface parameterization, mesh smoothing, frame field design, ARAP deformation, and spherical manifold optimization. Across these tasks, our system consistently outperforms state-of-the-art differentiation frameworks, including PyTorch, JAX, Warp, DrJIT, EnzymeAD, and Thallo. We demonstrate speedups across a range of solver types, from Newton and Gauss-Newton for nonlinear least squares to L-BFGS and gradient descent, and across different derivative usage modes, including Hessian-vector products as well as full sparse Hessian and Jacobian construction. Our system is available as open source at this https URL.

Locality-Aware Automatic Differentiation on the GPU for Mesh-Based Computations

Jiří Minarčík*, Michael Liu* (equal contribution), Keenan Crane, Minchen Li

Self-intersections are widespread in surface meshes and invalidate downstream simulation, fabrication, and learning pipelines. Existing approaches typically treat self-intersections as local collision events, but embeddedness (i.e., lack of self-intersections) is a global geometric property that cannot be enforced through local reasoning alone. We introduce an energy-based framework that enforces surface embeddedness simultaneously at the shape and mesh levels, based on the insight that successful untangling requires accounting for both global shape-level interactions and local mesh-level interactions. A shape-level energy captures global entanglement independent of discretization, while a mesh-level penalty regularizes local discrete interactions. Together, these energies enable reliable removal of self-intersections without changing mesh connectivity and apply to a broad class of geometries, including surfaces with boundary, non-manifold configurations, immersion failures, and multi-object scenes. Compared to prior state-of-the-art methods, our approach resolves self-intersections across challenging datasets, enabling reliable downstream processing of surface meshes.

Untangling Surfaces via Shape and Mesh Repulsion

Yue Chang, Peter Yichen Chen, Eitan Grinspun, Maurizio M. Chiaramonte

We present a low-rank Koopman operator formulation for accelerating deformable subspace simulation. Using a Dynamic Mode Decomposition (DMD) parameterization of the Koopman operator, our method learns the temporal evolution of deformable dynamics and predicts future states through efficient matrix evaluations instead of sequential time integration. This yields log-linear scaling in the number of time steps and allows large portions of the trajectory to be skipped while retaining accuracy. The resulting temporal efficiency is especially advantageous for optimization tasks such as control and initial-state estimation, where the objective often depends largely on the final configuration.
To broaden the scope of Koopman-based reduced-order models in graphics, we introduce a discretization-agnostic extension that learns shared dynamic behavior across multiple shapes and mesh resolutions. Prior DMD-based approaches have been restricted to a single shape and discretization, which limits their usefulness for tasks involving geometry variation. Our formulation generalizes across both shape and discretization, which enables fast shape optimization that was previously impractical for DMD models. This expanded capability highlights the potential of Koopman operator learning as a practical tool for efficient deformable simulation and design.

Low-Rank Koopman Deformables with Log-Linear Time Integration

Federico Sichetti, Zizhou Huang, Marco Attene, Denis Zorin, Enrico Puppo, Daniele Panozzo,

We propose a conservative algorithm to test the geometrical validity of simplicial (triangles, tetrahedra), tensor product (quadrilaterals, hexahedra), and mixed (prisms) elements of arbitrary polynomial order as they deform linearly within a time interval. Our algorithm uses a combination of adaptive Bézier refinement and bisection search to determine if, when, and where the Jacobian determinant of an element’s polynomial geometric map becomes negative in the transition from one configuration to another. In elastodynamic simulation, our algorithm guarantees that the system remains physically valid during the entire trajectory, not only at discrete time steps. Unlike previous approaches, physical validity is preserved even when our method is implemented using floating point arithmetic. Hence, our algorithm is only slightly slower than existing non-conservative methods while providing guarantees and while being an easy drop-in replacement for current validity tests. To prove the practical effectiveness of our algorithm, we demonstrate its use in a high-order Incremental Potential Contact (IPC) elastodynamic simulator and experimentally show that it prevents invalid, simulation-breaking configurations that would otherwise occur using non-conservative methods.

High-Order Continuous Geometrical Validity

Anka He Chen, Jerry Hsu, Youssef Ayman, Miles Macklin

We present Divide and Truncate (DAT), a unified framework for coupling multi-physics systems through penetration-free collision handling, including rigid bodies, volumetric soft bodies, thin shells, rods, and animated objects. By partitioning the ambient space into exclusive regions and truncating displacements to remain within them, DAT guarantees penetration-free contact resolution. Our \emph{Planar-DAT} variant further refines this by restricting only motion toward nearby surfaces, leaving tangential movement unconstrained, which addresses the artificial damping and deadlock problems of previous works. The framework is material-agnostic: each object responds to contact without knowledge of the opposing body’s physics. Our method is also solver-agnostic; it can be integrated seamlessly with any iterative optimizer as a post-processing step, enabling robust simulation of complex multi-body interactions.

Divide and Truncate: A Penetration and Inversion Free Framework for Coupled Multi-physics Systems

Daria Nogina, Silvia Sellán

Smoothed Particle Hydrodynamics (SPH) simulations rely on accurately and efficiently modeling fluid-solid interactions. However, particle-based coupling strategies introduce non-deterministic discretization errors, and implicit methods achieve high accuracy at the cost of expensive numerical integration. We introduce Tube Maps, a drop-in replacement for SPH boundary density computation that achieves accuracy comparable to implicit methods while dramatically reducing their computational cost. Our key observation is that the boundary density integral is fully determined by the local surface geometry near a fluid particle’s closest point. By expressing this geometry in tubular coordinates, we reduce the original three-dimensional integral to a one-dimensional closed-form expression that can be evaluated in constant time. We thus eliminate numerical quadrature and reduce boundary handling costs by one to three orders of magnitude, enabling fast and accurate SPH simulations with time-varying curved solids.

Tube Maps: Fast SPH Boundary Handling in Tubular Coordinates

José Antonio Fernández-Fernández, Fabian Löschner, Lukas Westhofen, Andreas Longva, Jan Bender

Optimization time integrators are effective at solving complex multi-physics problems including deformable solids with non-linear material models, contact with friction, strain limiting, etc. For challenging problems, Newton-type optimizers are often used, which necessitates first- and second-order derivatives of the global non-linear objective function. Manually differentiating, implementing, testing, optimizing, and maintaining the resulting code is extremely time-consuming, error-prone, and precludes quick changes to the model, even when using tools that assist with parts of such pipeline. We present SymX, an open source framework that computes the required derivatives of the different energy contributions by symbolic differentiation, generates optimized code, compiles it on-the-fly, and performs the global assembly. The user only has to provide the symbolic expression of each energy for a single representative element in its corresponding discretization and our system will determine the assembled derivatives for the whole simulation. We demonstrate the versatility of SymX in complex simulations featuring different non-linear materials, high-order finite elements, rigid body systems, adaptive discretizations, frictional contact, and coupling of multiple interacting physical systems. SymX’s derivatives offer performance on par with SymPy, an established off-the-shelf symbolic engine, and produces simulations at least one order of magnitude faster than TinyAD, an alternative state-of-the-art integral solution.

SymX: Energy-based Simulation from Symbolic Expressions

Yuqi Meng, Yihao Shi, Kemeng Huang, Zixuan Lu, Ning Guo, Taku Komura, Yin Yang, Minchen Li

We present an efficient B-spline finite element method (FEM) for cloth simulation. While higher-order FEM has long promised higher accuracy, its adoption in cloth simulators has been limited by larger computational costs while generating results with similar visual quality. Our contribution is a full algorithmic pipeline that makes cloth simulation using quadratic B-spline surfaces faster than standard linear FEM in practice while consistently improving accuracy and visual fidelity. Using quadratic B-spline basis functions, we obtain a globally C¹-continuous displacement field that supports consistent discretization of both membrane and bending energies, effectively reducing locking artifacts and mesh dependence common to linear elements. To close the performance gap, we introduce a reduced integration scheme that separately optimizes quadrature rules for membrane and bending energies, an accelerated Hessian assembly procedure tailored to the spline structure, and an optimized linear solver based on partial factorization. Together, these optimizations make high-order, smooth cloth simulation competitive at scale, yielding an average 2× speedup over linear FEM. Extensive experiments demonstrate improved accuracy, wrinkle detail, and robustness, including contact-rich scenarios, relative to linear FEM and recent higher-order approaches. Our method enables realistic wrinkling dynamics across a wide range of material parameters and supports practical garment animation, providing a new promising spatial discretization for high-quality cloth simulation.

Efficient B-Spline Finite Elements for Cloth Simulation

Zhiyong He, Dewen Guo, Minghao Guo, Yili Zhao, Wojciech Matusik, Hao Su, Chenfanfu Jiang, Peter Yichen Chen, Yin Yang

Simulating large-scale articulated assemblies poses a significant challenge due to the numerical stiffness and geometric complexity of jointed structures. Conventional rigid body solvers struggle with the high nonlinearity induced by rotation parameterization. This difficulty becomes more pronounced for multiple two-way-coupled bodies. This paper introduces a novel framework that leverages the linear kinematic mapping of Affine Body Dynamics (ABD). As ABD targets near-rigid objects, the constitutive variations of different materials become negligible, which justifies a co-rotational approach to isolate geometric nonlinearities of the system. This insight enables the use of constant system matrices that can be pre-factorized throughout the simulation, even with fully implicit integration schemes. To manage the high DOF counts of large-scale systems, we map primal body coordinates onto a compact dual space defined by minimal joint degrees of freedom. By solving the resulting KKT systems, our method ensures exact constraint enforcement and physically accurate motion propagation. We provide a suite of specialized solvers tailored for diverse joint topologies, including chains, trees, closed loops, and irregular networks. Experimental results show that our approach achieves interactive rates for systems with hundreds of thousands of bodies on a single CPU core, while maintaining excellent stability at large time steps.

M-ABD: Scalable, Efficient, and Robust Multi-Affine-Body Dynamics

Nicholas Mcdonald, Guillaume Cordonnier

Mountainous terrains evolve over geological timescales through erosion processes driven by the complex interplay of transported quantities such as water, sediment, and rockfall. A key challenge in erosion modeling is the simultaneous simulation of transport and erosive processes, which differ in temporal scales by several orders of magnitude. We address this challenge with a novel, parallel, stochastic particle-based method capable of simulating transport over geological timescales. Our approach relaxes the strong assumptions on velocity required by prior works (e.g., based on the Stream Power Law), enabling a new erosion model grounded in a more general form of momentum conservation. We demonstrate that our scheme accurately solves the underlying conservation laws and avoids artifacts common in previous works. Furthermore, we show that our new erosion model captures multiscale geomorphological features, producing coherent basin structures and dynamic phenomena such as braided rivers, meanders, and deltas.

Stochastic geomorphological transport for terrain erosion simulation