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.
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We present a computational framework for numerical homogenization of origami materials—thin sheets structured with periodic crease patterns that, once folded, exhibit diverse and often unusual mechanical properties. Whereas the in-plane stiffness of conventional sheet materials is typically orders of magnitude larger than their resistance to bending, origami-based folding introduces geometric structure that can drastically reshape both bending and stretching behavior. However, predicting how a particular crease pattern gives rise to effective macroscopic properties remains challenging due to the complex coupling of crease geometry, folding kinematics, and surface deformations. In this work, we introduce a computational framework that integrates simulation-based folding and numerical homogenization to explore the relationship between crease pattern and effective material behavior. To describe the macromechanical response of origami materials, we employ a quadratic energy model based on Classical Laminate Theory, together with a simplified treatment of crease plasticity. Our unified representation accommodates both straight- and curved-crease designs, revealing a rich space of origami materials with diverse behavior. In particular, we examine how pattern symmetry governs material symmetries, demonstrating examples that span the full spectrum from perfectly isotropic to highly anisotropic membrane and bending responses. Our framework further enables controlled exploration of parameter variations, illustrating how geometric features such as crease curvature shape macroscopic mechanical behavior. We showcase the potential of this approach through a broad set of examples, ranging from canonical straight-crease patterns such as Miura-ori to complex curved-crease tessellations. While a quantitative analysis is left for future work, we validate our homogenized descriptions against native-scale simulations and qualitatively compare deformation behaviors with real-world prototypes.
Constant mean curvature surfaces are æsthetically appealing geometric objects with applications in physics, differential geometry, and architecture. We present a straightforward discretization of constant mean curvature surfaces based on the classical observation that their Gauß maps are harmonic. Our construction is elementary—requiring only discrete Dirichlet energy minimization and a Poisson solve—yet it exactly mirrors this aspect of the smooth theory. A discrete analog of conjugation produces discrete constant Gauß curvature surfaces. Their unit offsets are discrete CMC surfaces with a conformal parameterization. Additionally, we introduce a novel Möbius-invariant discretization of the Dirichlet energy for sphere-valued maps on dual meshes that is derived from the discrete Willmore energy. It is more robust than standard formulations based on inverse cotangent weights. Our approach to the construction of constant mean curvature surfaces provides direct control over tangent planes along a boundary, if present, and naturally handles closed and periodic examples. We demonstrate the approach on a range of free-boundary, symmetric, and periodic CMC surfaces.
Incremental Potential Contact (IPC) has emerged as a robust and unifying formulation for contact-rich physical simulation by casting elasticity and collision handling as a single energy minimization problem. Achieving high performance, however, typically requires heavily specialized implementations that hard-code assumptions about energies, primitive types, and parameterizations, creating a major barrier to extensibility. Adding new energies or alternative parameterizations often requires re-deriving first and second-order derivatives, and implementing new assembly logic for the global Hessian and gradient. This challenge is further exacerbated by collision energies, where the same energy definition is often applied to mixed parameterizations, which can lead to a combinatorial explosion of parameterization-specific derivative and assembly cases. In this paper we introduce YASPS, a framework for physical simulation that resolves this limitation by making structural relationships explicit in a differentiable representation. YASPS introduces two relational operators, JOIN and UNION, which encode connectivity and heterogeneous parameterizations directly in the symbolic computation graph. Using symbolic differentiation over these operators, YASPS automatically derives local derivatives, and determines the induced sparsity and block structure of global gradients and Hessians from the same description while avoiding any code explosion induced by mixed-parameterizations. Targeting IPC workloads, YASPS compiles the resulting symbolic graphs into GPU kernels for local energy evaluation, derivative computation, and block-sparse matrix assembly, and solves the resulting Newton systems using a GPU-based iterative solver. This approach achieves performance competitive with state-of-the-art IPC implementations while enabling new energies and parameterizations to be added through localized symbolic definitions, without hand-written derivative or assembly code.
We introduce a high-performance framework for solving nonconvex optimization problems arising in simulation and geometry processing applications. Our framework especially benefits high-dimensional problems whose unknowns are spatial coordinates in ℝd and whose objectives are expressed as a sum of element energies over small stencils. The Hessians of these problems have a block-sparse structure, where the nonzero entries are grouped into dense d × d blocks. Many works have exploited this structure to speed up the Hessian-vector products and nodal smoothers of iterative solvers. In contrast, accelerating direct sparse linear solvers, preferred for applications requiring high accuracy on unstructured grids, has been less explored. Our framework leverages block structure to accelerate all phases of a Newton-type minimization algorithm, from sparsity pattern construction to system assembly, matrix factorization, and solves. A fundamental piece of the framework is BlockCatamari, our highly tuned, block-accelerated multifrontal sparse Cholesky code that achieves significantly faster factorizations than existing direct solvers on shared-memory systems. We combine this solver with our fast parallel Hessian assembly routine, heuristics for conditionally enabling per-element Hessian projections, and strategies for minimizing symbolic factorization re-computations. We show that this leads to dramatic speedups on a large suite of benchmark problems involving injective surface parametrization and elasticity simulation with contact, and we further demonstrate the ease with which new energies can be defined at the different levels of abstraction offered by our framework.
Stochastic Monte Carlo solvers for partial differential equations (PDEs) recently gained popularity in computer graphics, finding applications in geometry processing, rendering, simulation, and visualization. At present, there exists no Monte Carlo solver for the rendering of biharmonic diffusion curves, an artist-friendly smooth vector graphics primitive. The fourth-order biharmonic equation of biharmonic diffusion curves can be split into two second-order PDEs, namely a Laplace and a Poisson equation. However, since biharmonic diffusion curves set Dirichlet and inhomogeneous Neumann conditions at the same time, these two second-order PDEs are tightly coupled and can hence not be solved directly. We propose to treat the rendering of biharmonic diffusion curves as an inverse problem, in which the Dirichlet data of the Laplace equation is unknown. We formulate a variational energy optimization, such that the user-defined boundary conditions are met. Thereby, the necessary gradients are estimated stochastically by solving two second-order problems with Dirichlet boundary conditions only.
While yarn-level garments offer rich dynamic details and compelling visual realism compared to triangle-based models, their wide adoption is hindered by the immense computational cost due to the presence of a large number of degrees of freedom (DOFs). This paper proposes a novel simulation framework designed to enhance the performance and stability for numerical simulation of nonlinear, non-convex, and high-resolution knitwear. Our method generalizes the Douglas-Rachford Splitting (DRS) scheme to resolve the non-convex coupling between stretching, shearing, bending, twisting, and contacting at each yarn thread. A key contribution is a nested decomposition strategy that decouples the non-convex variational energy into independent and convex sub-problems. Such convexification improves solver robustness and removes the necessity for frequent line searches. We provide a theoretically grounded strategy for metric selection for each sub-problem, derived from an analysis of the convergence guarantee of DRS. Consequently, our method achieves close-to-optimal convergence along the nonlinear iterations rather than relying on ad-hoc parameter tuning. The paper also clarifies a formal connection between our generalized DRS and the ADMM (Alternating Direction Method of Multipliers) framework, extending the applicability of our analysis to a broader set of constrained dynamics problems. Experimental results demonstrate that our method robustly handles complex knitwear simulation scenes with superior efficiency, stability, and physical fidelity compared to existing methods. With a matrix-free GPU parallelization, our method allows an interactive simulation rate of knitwear of multi-million DOFs.
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 its 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 in our tests. 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.
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.
Neutron transport simulation plays a central role in nuclear engineering, yet its inverse problems—such as shielding design and sensitivity analysis—remain extremely challenging. The governing transport equation shares a common mathematical foundation with the volume rendering equation used for participating media in computer graphics. Recent advances in differentiable rendering have demonstrated that inverse problems—such as recovering scene properties from observations—can be efficiently solved through gradient-based optimization, where the gradients are estimated through Monte Carlo methods. Inspired by these developments, we extend differentiable rendering techniques to neutron transport problems, enabling inverse design and sensitivity analysis in nuclear systems. To adapt these methods to the neutron transport problem, we generalize the automatic differentiation of Monte Carlo light transport simulation to multi-energy neutron transport; propose a new simplified way to handle moving geometric discontinuities; and propose new estimators for derivatives of volume integrals. We demonstrate the effectiveness of the proposed framework on several model inverse problems, including gradient-based optimization of multi-group neutron moderation, sensitivity analysis of material properties, and diagnostic sensor shape optimization. These results show that differentiable Monte Carlo neutron transport can support optimization and analysis tasks, opening new opportunities for inverse design in nuclear shielding and related applications.
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 perelement 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, Dr.JIT, 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 https://github.com/owensgroup/RXMesh.
We introduce Kinematic Kitbashing, an optimization framework that synthesizes articulated 3D objects by assembling reusable parts conditioned on an abstract kinematic graph. Given the graph and a library of articulated parts, our method optimizes per-part similarity transformations that place, orient, and scale each component into a coherent articulated object; optional graph edits further enable novel assemblies beyond the prescribed connectivity. Central to our method is an exemplar-based analogy for part placement: each reused component is paired with a single source asset that exemplifies how it attaches to its parent. We capture this attachment context using vector distance fields and measure consistency by integrating the matching error over the joint’s full motion range. This yields a kinematics-aware attachment energy that favors placements that preserve the exemplar’s local attachment neighborhood throughout articulation. To incorporate task-level functionality, we use this attachment energy as a prior in an annealed Langevin sampling framework, enabling gradient-free optimization of black-box functionality objectives. We demonstrate the versatility of kinematic kitbashing across diverse applications, including instantiating kinematic graphs from user-selected or automatically retrieved parts, synthesizing assemblies with user-defined functionality, and re-targeting articulations via graph edits.
Computing injective, flip-free embeddings of simplicial meshes is a fundamental challenge in geometry processing. While the Total Lifted Content (TLC) energy provides a smooth functional with theoretically guaranteed injective global minima, its non-convex nature presents significant optimization hurdles. In this work, we propose a Nonlinear Multigrid (MG/Opt) framework specifically tailored for efficient TLC energy minimization. Our approach leverages a topology-preserving mesh hierarchy to rapidly eliminate low-frequency errors on coarse scales while maintaining rigorous gradient consistency between hierarchical levels through linear correction terms. To address the persistent challenge of local inversions, we introduce a targeted smoothing strategy and an adaptive parameter scheduling mechanism that balances global convergence speed with local injectivity precision. Experimental evaluations across extensive 2D and 3D benchmark datasets demonstrate the robustness of our approach. Our method achieves a 100% convergence success rate across all tested 2D cases and all but one 3D dataset (the VOLMAP benchmark). Furthermore, our MG/Opt framework delivers substantial performance gains, achieving up to 63 × speedup on high-resolution subdivided datasets compared to traditional solvers. These results confirm that our multigrid strategy provides a scalable and reliable solution for large-scale injective mapping problems. Code and data for this paper are at https://github.com/distmeshM/MGOpt_TLC
Text-to-image diffusion models generate images by gradually converting white Gaussian noise into a natural image. White Gaussian noise is well suited for producing diverse outputs from a single text prompt due to its absence of structure. However, this very property limits control over, and predictability of, specific visual attributes, as the noise is not human-interpretable. In this work, we investigate the characteristics of the input noise in diffusion models. We show that, although all frequencies in white Gaussian noise have comparable statistical energy, low-frequency components primarily determine the image’s global structure and color composition, while high-frequency components control finer details. Building on this observation, we demonstrate that simple manipulations of the low-frequency noise using low-frequency image priors can effectively condition the generation process to reconstruct these low-frequency visual cues. This allows us to define a simple, training-free method with minimal overhead that steers overall image structure and color, while letting high-frequency components freely emerge as fine details, enabling variability across generated outputs.
Smith microfacet models are widely used to describe the interaction of light with surfaces. However, conventional microfacet models account for only a single bounce on microsurfaces, which leads to energy loss. To address the problem of missing energy, several unbiased methods have been proposed to compute multiple bounces on Smith microsurfaces. In addition, a position-free multiple-bounce microfacet model has been introduced by assuming that each bounce of a light path is height-independent, leading to lower variance but biased results. In this paper, we propose a novel derivation of multiple-bounce Smith microfacet bidirectional reflectance distribution functions (BRDFs) based on a partial adoption of the independent-bounce assumption. Our model treats most bounces as height-interdependent, leading to a height-interdependent path formulation. We further propose a successive height preintegration to make this formulation position-free. Although our model still exhibits an extremely small bias, it achieves lower variance than prior unbiased methods. Furthermore, we introduce a path type-aware BRDF evaluation that further reduces variance.
Eigenanalysis of partial differential operators is essential in reduced-order modeling for physical simulation, providing eigenmode representations in elasticity, acoustics, and transient heat transport. While recent neural, mesh-free, and geometry-agnostic approaches enable differentiable eigenanalysis over continuously parameterized shape spaces, boundary conditions of eigenfunctions are limited to Neumann-type (natural) conditions, hindering their applicability in scenarios where boundary conditions must be precisely controlled or optimized. In this work, we focus on Laplace-type operators and extend shape space neural eigenanalysis to handle boundary conditions beyond natural Neumann settings. Building on the same variational, energy-based formulation, we show that Dirichlet, Robin, and mixed boundary conditions can be incorporated without altering the underlying eigenvalue optimization principle. In our formulation, boundary configurations—including boundary placement and Robin coefficients modeling boundary exchange processes—are treated as first-class parameters rather than fixed constraints. When combined with shape-parameterized domains, this leads to a joint shape–boundary space formulation, allowing eigenfunctions and spectra to be evaluated consistently across variations in both geometry and boundary configuration. We conduct experiments on representative applications such as boundary-driven spectral optimization for rigid-walled cavity resonance tuning, reduced-order simulation with changing supports, and analysis of transient thermal behavior under varying boundary exchange conditions. By elevating boundary conditions from fixed constraints to operator-wise parameterization, our approach broadens the applicability of Laplace-type neural eigenanalysis to physical systems where boundary constraints serve as critical design and control variables.
Misinformation on the web increasingly appears in multimodal forms, combining text, images, and OCR-rendered content in ways that amplify harm to public trust and vulnerable communities. While prior fact-checking systems often rely on unimodal signals or shallow fusion strategies, modern misinformation campaigns operate across modalities and require models that can reason over subtle cross-modal inconsistencies in a transparent and responsible manner. We introduce MultiCheck, a lightweight and interpretable framework for multimodal fact verification that jointly analyzes textual, visual, and OCR evidence. At its core, MultiCheck employs a relational fusion module based on element-wise difference and product operations, allowing for explicit cross-modal interaction modeling with minimal computational overhead. A contrastive alignment objective further helps the model distinguish between supporting and refuting evidence while maintaining a small memory and energy footprint, making it suitable for low-resource deployment. Evaluated on the Factify-2 (5-class) and MOCHEG (3-class) benchmarks, MultiCheck achieves substantial performance improvement and remains robust under noisy OCR and missing modality conditions. Overall, MultiCheck is efficient, interpretable, and robust for multimodal verification. Our code is available at the following: https://github.com/Adityakishore09/MultiCheck_WWW-2026 GitHub repository.
The escalating energy consumption and carbon footprint of training large-scale Web AI models pose urgent challenges for sustainable development. Dataset Distillation (DD) offers a promising avenue for green AI by compressing large datasets into small synthetic ones for efficient training. However, most existing DD methods overfit to the inductive biases of specific source architectures (e.g., CNNs or ViTs), resulting in poor cross-model generalization. This limitation necessitates redundant re-distillation processes for different architectures, severely undermining the energy-saving potential of DD. To address this, we introduce Adversarial Mutual Information Distillation (AMID), a rigorous framework designed to create highly reusable and robust synthetic datasets. From an information-theoretic perspective, we cast model-agnosticism as minimizing the mutual information (MI) between the synthetic data and the specific identity of the distillation model. We convert this intractable objective into a tractable two-player adversarial game, which unifies knowledge preservation with adversarial unlearning of architectural bias. Extensive experiments on CIFAR-10 and Tiny ImageNet demonstrate that AMID achieves state-of-the-art cross-architecture generalization across diverse CNNs and ViTs. Crucially, our analysis confirms that AMID significantly reduces the computational overhead and CO2 emissions of downstream training while maintaining robust performance, paving the way for energy-efficient, transferable, and sustainable Web AI ecosystems.