We extend Progressive Dynamics [Zhang et al. 2024, 2025] from cloth and shells to volumetric finite elements, enabling an efficient level-of-detail (LOD) animation-design pipeline with predictive coarse-resolution previews for rapid iteration toward a final high-resolution volumetric elastodynamics animation. To achieve this, we introduce VelPro Splitter , a principled and substantially improved VelPro-type integrator [Zhang et al. 2025] that splits the current-level velocity and recombines its high-resolution component with prolonged coarse-level velocity. The resulting VelPro Splitter is general and applies to both shell and volumetric discretizations. This splitting strategy better generates the high-frequency dynamic details that motivate highresolution elastodynamics animation, going beyond the previously mostly geometric enrichment due to prolonged coarse-level velocity, while maintaining cross-level consistency in bulk deformation across LOD results. As a result, it largely decouples finest-level enrichment quality from the number of LOD levels, addressing a practical limitation of VelPro at small timestep sizes where many levels are required to obtain sufficient enrichment. To make this volumetric setting practical for Progressive Dynamics, we add two supporting contributions. First, we construct volumetric hierarchies and introduce a simple and effective topology-aware boundary-binding method that enables reliable prolongation between overlapping, but not-necessarily-conforming, meshes using a barycentric-like linear interpolant. Second, instead of applying the subspace approach of Zhang et al. [2024] for reducing coarse model locking, we show that a lightweight, resolution-based stiffness rescaling via a simple Young's-modulus adjustment [Chen et al. 2017] is both effective and well-suited for progressive volumetric simulation. Together, these contributions jointly realize Volumetric Progressive Dynamics. We demonstrate its high-fidelity LOD matching for volumetric elastodynamics across 1D, 2D, and 3D scenarios with high speeds, large deformations, and frictional contact.

The recently developed Progressive Dynamics framework [Zhang et al. 2024] addresses the long-standing challenge in enabling rapid iterative design for high-fidelity cloth and shell animation. In this work, we identify fundamental limitations of the original method in terms of stability and temporal continuity. For robust progressive dynamics simulation we seek methods that provide: (1) stability across all levels of detail (LOD) and timesteps, (2) temporally continuous animations without jumps or jittering, and (3) user-controlled balancing between geometric consistency and enrichment at each timestep, thereby making it a practical previewing tool with high-quality results at the finest level to be used as the final output. We propose a general framework, Progressive Dynamics++, for constructing a family of progressive dynamics integration methods that advance physical simulation states forward in both time and spatial resolution, which includes Zhang et al. [2024]'s method as one member. We analyze necessary stability conditions for Progressive Dynamics integrators and introduce a novel, stable method that significantly improves temporal continuity, supported by a new quantitative measure. Additionally, we present a quantitative analysis of the trade-off between geometric consistency and enrichment, along with strategies for balancing between these aspects in transitions across resolution and time.

We propose Progressive Dynamics, a coarse-to-fine, level-of-detail simulation method for the physics-based animation of complex frictionally contacting thin shell and cloth dynamics. Progressive Dynamics provides tight-matching consistency and progressive improvement across levels, with comparable quality and realism to high-fidelity, IPC-based shell simulations [Li et al. 2021] at finest resolutions. Together these features enable an efficient animation-design pipeline with predictive coarse-resolution previews providing rapid design iterations for a final, to-be-generated, high-resolution animation. In contrast, previously, to design such scenes with comparable dynamics would require prohibitively slow design iterations via repeated direct simulations on high-resolution meshes. We evaluate and demonstrate Progressive Dynamics's features over a wide range of challenging stress-tests, benchmarks, and animation design tasks. Here Progressive Dynamics efficiently computes consistent previews at costs comparable to coarsest-level direct simulations. Its matching progressive refinements across levels then generate rich, high-resolution animations with high-speed dynamics, impacts, and the complex detailing of the dynamic wrinkling, folding, and sliding of frictionally contacting thin shells and fabrics.

We propose a reduced-space elastodynamic solver that is well suited for augmenting rigged character animations with secondary motion. At the core of our method is a novel deformation subspace based on Linear Blend Skinning that overcomes many of the shortcomings prior subspace methods face. Our skinning subspace is parameterized entirely by a set of scalar weights, which we can obtain through a small, material-aware and rig-sensitive generalized eigenvalue problem. The resulting subspace can easily capture rotational motion and guarantees that the resulting simulation is rotation equivariant. We further propose a simple local-global solver for linear co-rotational elasticity and propose a clustering method to aggregate per-tetrahedra nonlinear energetic quantities. The result is a compact simulation that is fully decoupled from the complexity of the mesh.

Thin shell structures exhibit complex behaviors critical for modeling and design across wide-ranging applications. Capturing their mechanical response requires finely detailed, high-resolution meshes. Corresponding simulations for predicting equilibria with these meshes are expensive, whereas coarse-mesh simulations can be fast but generate unacceptable artifacts and inaccuracies. The recently proposed progressive simulation framework [Zhang et al. 2022] offers a promising avenue to address these limitations with consistent and progressively improving simulation over a hierarchy of increasingly higher-resolution models. Unfortunately, it is currently severely limited in application to meshes and shapes generated via Loop subdivision. We propose Progressive Shells Quasistatics to extend progressive simulation to the high-fidelity modeling and design of all input shell (and plate) geometries with unstructured (as well as structured) triangle meshes. To do so, we construct a fine-to-coarse hierarchy with a novel nonlinear prolongation operator custom-suited for curved-surface simulation that is rest-shape preserving, supports complex curved boundaries, and enables the reconstruction of detailed geometries from coarse-level meshes. Then, to enable convergent, high-quality solutions with robust contact handling, we propose a new, safe, and efficient shape-preserving upsampling method that ensures non-intersection and strain limits during refinement. With these core contributions, Progressive Shell Quasistatics enables, for the first time, wide generality for progressive simulation, including support for arbitrary curved-shell geometries, progressive collision objects, curved boundaries, and unstructured triangle meshes - all while ensuring that preview and final solutions remain free of intersections. We demonstrate these features across a wide range of stress-tests where progressive simulation captures the wrinkling, folding, twisting, and buckling behaviors of frictionally contacting thin shells with orders-of-magnitude speed-up in examples over direct fine-resolution simulation.

The trade-off between speed and fidelity in cloth simulation is a fundamental computational problem in computer graphics and computational design. Coarse cloth models provide the interactive performance required by designers, but they can not be simulated at higher resolutions ("up-resed") without introducing simulation artifacts and/or unpredicted outcomes, such as different folds, wrinkles and drapes. But how can a coarse simulation predict the result of an unconstrained, high-resolution simulation that has not yet been run? We propose Progressive Cloth Simulation (PCS), a new forward simulation method for efficient preview of cloth quasistatics on exceedingly coarse triangle meshes with consistent and progressive improvement over a hierarchy of increasingly higher-resolution models. PCS provides an efficient coarse previewing simulation method that predicts the coarse-scale folds and wrinkles that will be generated by a corresponding converged, high-fidelity C-IPC simulation of the cloth drape's equilibrium. For each preview PCS can generate an increasing-resolution sequence of consistent models that progress towards this converged solution. This successive improvement can then be interrupted at any point, for example, whenever design parameters are updated. PCS then ensures feasibility at all resolutions, so that predicted solutions remain intersection-free and capture the complex folding and buckling behaviors of frictionally contacting cloth.

Abstract Across computer graphics, vision, robotics and simulation, many applications rely on determining the 3D rotation that aligns two objects or sets of points. The standard solution is to use singular value decomposition (SVD), where the optimal rotation is recovered as the product of the singular vectors. Faster computation of only the rotation is possible using suitable parameterizations of the rotations and iterative optimization. We propose such a method based on the Cayley transformations. The resulting optimization problem allows better local quadratic approximation compared to the Taylor approximation of the exponential map. This results in both faster convergence as well as more stable approximation compared to other iterative approaches. It also maps well to AVX vectorization. We compare our implementation with a wide range of alternatives on real and synthetic data. The results demonstrate up to two orders of magnitude of speedup compared to a straightforward SVD implementation and a 1.5‐6 times speedup over popular optimized code.

This paper introduces a novel geometric multigrid solver for unstructured curved surfaces. Multigrid methods are highly efficient iterative methods for solving systems of linear equations. Despite the success in solving problems defined on structured domains, generalizing multigrid to unstructured curved domains remains a challenging problem. The critical missing ingredient is a prolongation operator to transfer functions across different multigrid levels. We propose a novel method for computing the prolongation for triangulated surfaces based on intrinsic geometry, enabling an efficient geometric multigrid solver for curved surfaces. Our surface multigrid solver achieves better convergence than existing multigrid methods. Compared to direct solvers, our solver is orders of magnitude faster. We evaluate our method on many geometry processing applications and a wide variety of complex shapes with and without boundaries. By simply replacing the direct solver, we upgrade existing algorithms to interactive frame rates, and shift the computational bottleneck away from solving linear systems.

Recent years have seen an increasing interest in the authoring and crafting of personal visualizations. Mainstream data analysis and authoring tools lack the flexibility for customization and personalization, whereas tools from the research community either require creativity and drawing skills, or are limited to simple vector graphics. We present DataQuilt, a novel system that enables visualization authors to iteratively design pictorial visualizations as collages. Real images (e.g., paintings, photographs, sketches) act as both inspiration and as a resource of visual elements that can be mapped to data. The creative pipeline involves the semi-guided extraction of relevant elements of an image (arbitrary regions, regular shapes, color palettes, textures) aided by computer vision techniques; the binding of these graphical elements and their features to data in order to create meaningful visualizations; and the iterative refinement of both features and visualizations through direct manipulation. We demonstrate the usability of DataQuilt in a controlled study and its expressiveness through a collection of authored visualizations from a second open-ended study.

We present a novel approach to enrich arbitrary rig animations with elastodynamic secondary effects. Unlike previous methods which pit rig displacements and physical forces as adversaries against each other, we advocate that physics should complement artists' intentions. We propose optimizing for elastodynamic displacements in the subspace orthogonal to displacements that can be created by the rig. This ensures that the additional dynamic motions do not undo the rig animation. The complementary space is high-dimensional, algebraically constructed without manual oversight, and capable of rich high-frequency dynamics. Unlike prior tracking methods, we do not require extra painted weights, segmentation into fixed and free regions or tracking clusters. Our method is agnostic to the physical model and plugs into non-linear FEM simulations, geometric as-rigid-as-possible energies, or mass-spring models. Our method does not require a particular type of rig and adds secondary effects to skeletal animations, cage-based deformations, wire deformers, motion capture data, and rigid-body simulations.