@article{22241, abstract = {We revisit the computation of 3D generalized winding numbers, a useful measure for inside-outside classification on triangle meshes with gaps, self-intersections, and open boundaries. At the core of our new method is an analytical reduction of the surface integral that defines the winding number, resulting in a single ray-mesh intersection test and an elementary sum over boundary edges per evaluation. This construction is orders of magnitude more efficient than the state of the art in practice, which we show in an extensive performance benchmark. Conveniently, the method also reduces to the best-available asymptotic complexity in the worst case, and it introduces no approximations apart from floating-point errors. Our algorithm is conceptually simple to understand, straightforward to implement and debug, and it works reliably even on extremely noisy and corrupt input geometry.}, author = {Xie, Peiyuan and Hafner, Christian and Wojtan, Christopher J}, issn = {1557-7368}, journal = {ACM Transactions on Graphics}, number = {4}, publisher = {Association for Computing Machinery}, title = {{Fast and exact winding numbers for triangle meshes}}, doi = {10.1145/3811339}, volume = {45}, year = {2026}, } @inproceedings{22129, abstract = {Inside–outside classification is widely used for geometry processing tasks such as surface reconstruction, geometry completion, and calculating signed distance fields. We introduce a new integral formulation of this problem, which assigns confidence scores that points are inside or outside, given incomplete boundary geometry. Even though our geometric construction does not appear in previous work, we show that it is unexpectedly linked to both the well-established generalized winding number (GWN) and pseudonormal methods for geometry completion, and it provably reduces to either one of them for specific values of a control parameter. The results obtained with our method frequently outperform screened Poisson surface reconstruction (PSR), GWN, and the pseudonormal method in terms of quality, and are at least on par with them on all of our examples. Unlike these methods, our algorithm naturally extends to the multi-label setting, in which regions with an arbitrary number of colors or physical materials can be reconstructed, and non-manifold features such as T-junctions may appear in the interface and boundary geometry}, author = {Wei, Ziyu and Hafner, Christian and Kalinov, Aleksei and Synak, Peter and Wojtan, Christopher J}, booktitle = {Computer Graphics Forum}, location = {Bern, Switzerland}, number = {5}, publisher = {Wiley}, title = {{Circles of confidence for multi-label geometry completion}}, doi = {10.1111/cgf.70516}, volume = {45}, year = {2026}, } @article{21923, abstract = {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.}, author = {Kalinov, Aleksei and Ly, Mickaël and Hafner, Christian and Wojtan, Christopher J}, issn = {0730-0301}, journal = {ACM Transactions on Graphics}, keywords = {Procedural animation}, location = {Los Angeles, CA, United States}, number = {4}, publisher = {Association for Computing Machinery}, title = {{Physics-inspired procedural texturing of extremely deformable surfaces}}, doi = {10.1145/3811353}, volume = {45}, year = {2026}, } @article{20628, abstract = {The realistic simulation of sand, soil, powders, rubble piles, and large collections of rigid bodies is a common and important problem in the fields of computer graphics, computational physics, and engineering. Direct simulation of these individual bodies quickly becomes expensive, so we often approximate the entire group as a continuum material that can be more easily computed using tools for solving partial differential equations, like the material point method (MPM). In this paper, we present a method for automatically extracting continuum material properties from a collection of rigid bodies. We use numerical homogenization with periodic boundary conditions to simulate an effectively infinite number of rigid bodies in contact. We then record the effective stress-strain relationships from these simulations and convert them into elastic properties and yield criteria for the continuum simulations. Our experiments validate existing theoretical models like the Mohr-Coulomb yield surface by extracting material behaviors from a collection of spheres in contact. We further generalize these existing models to more exotic materials derived from diverse and non-convex shapes. We observe complicated jamming behaviors from non-convex grains, and we introduce a new material model for materials with extremely high levels of internal friction and cohesion. We simulate these new continuum models using MPM with an improved return mapping technique. The end result is a complete system for turning an input rigid body simulation into an efficient continuum simulation with the same effective mechanical properties.}, author = {Chen, Yi-Lu and Ly, Mickaël and Wojtan, Christopher J}, issn = {1557-7368}, journal = {ACM Transactions on Graphics}, location = {Hong Kong, China}, number = {6}, publisher = {Association for Computing Machinery}, title = {{Numerical homogenization of sand from grain-level simulations}}, doi = {10.1145/3763344}, volume = {44}, year = {2025}, } @article{17203, abstract = {The behavior of a rigid body primarily depends on its mass moments, which consist of the mass, center of mass, and moments of inertia. It is possible to manipulate these quantities without altering the geometric appearance of an object by introducing cavities in its interior. Algorithms that find cavities of suitable shapes and sizes have enabled the computational design of spinning tops, yo-yos, wheels, buoys, and statically balanced objects. Previous work is based, for example, on topology optimization on voxel grids, which introduces a large number of optimization variables and box constraints, or offset surface computation, which cannot guarantee that solutions to a feasible problem will always be found. In this work, we provide a mathematical analysis of constrained topology optimization problems that depend only on mass moments. This class of problems covers, among others, all applications mentioned above. Our main result is to show that no matter the outer shape of the rigid body to be optimized or the optimization objective and constraints considered, the optimal solution always features a quadric-shaped interface between material and cavities. This proves that optimal interfaces are always ellipsoids, hyperboloids, paraboloids, or one of a few degenerate cases, such as planes. This insight lets us replace a difficult topology optimization problem with a provably equivalent non-linear equation system in a small number (<10) of variables, which represent the coefficients of the quadric. This system can be solved in a few seconds for most examples, provides insights into the geometric structure of many specific applications, and lets us describe their solution properties. Finally, our method integrates seamlessly into modern fabrication workflows because our solutions are analytical surfaces that are native to the CAD domain.}, author = {Hafner, Christian and Ly, Mickaël and Wojtan, Christopher J}, issn = {1557-7368}, journal = {Transactions on Graphics}, keywords = {Topology Optimization, Mass Moments, Computational Geometry}, location = {Denver, Colorado}, number = {4}, publisher = {Association for Computing Machinery}, title = {{Spin-it faster: Quadrics solve all topology optimization problems that depend only on mass moments}}, doi = {10.1145/3658194}, volume = {43}, year = {2024}, } @inproceedings{17214, abstract = {Current numerical algorithms for simulating friction fall in one of two camps: smooth solvers sacrifice the stable treatment of static friction in exchange for fast convergence, and non-smooth solvers accurately compute friction at convergence rates that are often prohibitive for large graphics applications. We introduce a novel bridge between these two ideas that computes static and dynamic friction stably and efficiently. Our key idea is to convert the highly constrained non-smooth problem into an unconstrained smooth problem using logarithmic barriers that converges to the exact solution as accuracy increases. We phrase the problem as an interior point primal-dual problem that can be solved efficiently with Newton iteration. We observe quadratic convergence despite the non-smooth nature of the original problem, and our method is well-suited for large systems of tightly packed objects with many contact points. We demonstrate the efficacy of our method with stable piles of grains and stacks of objects, complex granular flows, and robust interlocking assemblies of rigid bodies.}, author = {Chen, Yi-Lu and Ly, Mickaël and Wojtan, Christopher J}, booktitle = {Special Interest Group on Computer Graphics and Interactive Techniques Conference Conference Papers '24}, isbn = {9798400705250}, keywords = {physical simulation, frictional contact, rigid body mechanics, non-smooth dynamics}, location = {Denver, United States}, publisher = {Association for Computing Machinery}, title = {{Primal-dual non-smooth friction for rigid body animation}}, doi = {10.1145/3641519.3657485}, year = {2024}, } @article{17219, abstract = {We introduce a multi-material non-manifold mesh-based surface tracking algorithm that converts self-intersections into topological changes. Our algorithm generalizes prior work on manifold surface tracking with topological changes: it preserves surface features like mesh-based methods, and it robustly handles topological changes like level set methods. Our method also offers improved efficiency and robustness over the state of the art. We demonstrate the effectiveness of the approach on a range of examples, including complex soap film simulations with thousands of interacting bubbles, and boolean unions of non-manifold meshes consisting of millions of triangles.}, author = {Synak, Peter and Kalinov, Aleksei and Strugaru, Irina-Malina and Etemadihaghighi, Arian and Yang, Huidong and Wojtan, Christopher J}, issn = {1557-7368}, journal = {ACM Transactions on Graphics}, keywords = {surface tracking, topology change, non- manifold meshes, multi-material flows, solid modeling}, number = {4}, publisher = {Association for Computing Machinery}, title = {{Multi-material mesh-based surface tracking with implicit topology changes}}, doi = {10.1145/3658223}, volume = {43}, year = {2024}, } @article{14240, abstract = {This paper introduces a novel method for simulating large bodies of water as a height field. At the start of each time step, we partition the waves into a bulk flow (which approximately satisfies the assumptions of the shallow water equations) and surface waves (which approximately satisfy the assumptions of Airy wave theory). We then solve the two wave regimes separately using appropriate state-of-the-art techniques, and re-combine the resulting wave velocities at the end of each step. This strategy leads to the first heightfield wave model capable of simulating complex interactions between both deep and shallow water effects, like the waves from a boat wake sloshing up onto a beach, or a dam break producing wave interference patterns and eddies. We also analyze the numerical dispersion created by our method and derive an exact correction factor for waves at a constant water depth, giving us a numerically perfect re-creation of theoretical water wave dispersion patterns.}, author = {Jeschke, Stefan and Wojtan, Christopher J}, issn = {1557-7368}, journal = {ACM Transactions on Graphics}, number = {4}, publisher = {Association for Computing Machinery}, title = {{Generalizing shallow water simulations with dispersive surface waves}}, doi = {10.1145/3592098}, volume = {42}, year = {2023}, } @article{14628, abstract = {We introduce a compact, intuitive procedural graph representation for cellular metamaterials, which are small-scale, tileable structures that can be architected to exhibit many useful material properties. Because the structures’ “architectures” vary widely—with elements such as beams, thin shells, and solid bulks—it is difficult to explore them using existing representations. Generic approaches like voxel grids are versatile, but it is cumbersome to represent and edit individual structures; architecture-specific approaches address these issues, but are incompatible with one another. By contrast, our procedural graph succinctly represents the construction process for any structure using a simple skeleton annotated with spatially varying thickness. To express the highly constrained triply periodic minimal surfaces (TPMS) in this manner, we present the first fully automated version of the conjugate surface construction method, which allows novices to create complex TPMS from intuitive input. We demonstrate our representation’s expressiveness, accuracy, and compactness by constructing a wide range of established structures and hundreds of novel structures with diverse architectures and material properties. We also conduct a user study to verify our representation’s ease-of-use and ability to expand engineers’ capacity for exploration.}, author = {Makatura, Liane and Wang, Bohan and Chen, Yi-Lu and Deng, Bolei and Wojtan, Christopher J and Bickel, Bernd and Matusik, Wojciech}, issn = {1557-7368}, journal = {ACM Transactions on Graphics}, keywords = {Computer Graphics and Computer-Aided Design}, number = {5}, publisher = {Association for Computing Machinery}, title = {{Procedural metamaterials: A unified procedural graph for metamaterial design}}, doi = {10.1145/3605389}, volume = {42}, year = {2023}, } @inproceedings{14748, author = {Chen, Yi-Lu and Ly, Mickaël and Wojtan, Christopher J}, booktitle = {Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation}, isbn = {9798400702686}, location = {Los Angeles, CA, United States}, publisher = {Association for Computing Machinery}, title = {{Unified treatment of contact, friction and shock-propagation in rigid body animation}}, doi = {10.1145/3606037.3606836}, year = {2023}, } @misc{15292, abstract = {We present a rigid body animation technique which prevents solids from interpenetrating, dissipates energy through friction, and propagates shocks through contacts. We employ the Alternating Direction Method of Multipliers (ADMM) to couple non-smooth Coulomb friction with impact propagation, allowing efficient and accurate non-smooth dynamics along with a correct transmission of impacts through assemblies of rigid bodies. We further extend our method to model adhesion, dynamic friction and lubricated contact.}, author = {Chen, Yi-Lu and Ly, Mickaël and Wojtan, Christopher J}, booktitle = {Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation}, location = {Los Angeles, CA, United States}, publisher = {ACM}, title = {{Unified treatment of contact, friction and shock-propagation in rigid body animation}}, doi = {10.1145/3606037.3606836}, year = {2023}, } @article{11432, abstract = {This paper proposes a method for simulating liquids in large bodies of water by coupling together a water surface wave simulator with a 3D Navier-Stokes simulator. The surface wave simulation uses the equivalent sources method (ESM) to efficiently animate large bodies of water with precisely controllable wave propagation behavior. The 3D liquid simulator animates complex non-linear fluid behaviors like splashes and breaking waves using off-the-shelf simulators using FLIP or the level set method with semi-Lagrangian advection. We combine the two approaches by using the 3D solver to animate localized non-linear behaviors, and the 2D wave solver to animate larger regions with linear surface physics. We use the surface motion from the 3D solver as boundary conditions for 2D surface wave simulator, and we use the velocity and surface heights from the 2D surface wave simulator as boundary conditions for the 3D fluid simulation. We also introduce a novel technique for removing visual artifacts caused by numerical errors in 3D fluid solvers: we use experimental data to estimate the artificial dispersion caused by the 3D solver and we then carefully tune the wave speeds of the 2D solver to match it, effectively eliminating any differences in wave behavior across the boundary. To the best of our knowledge, this is the first time such a empirically driven error compensation approach has been used to remove coupling errors from a physics simulator. Our coupled simulation approach leverages the strengths of each simulation technique, animating large environments with seamless transitions between 2D and 3D physics.}, author = {Schreck, Camille and Wojtan, Christopher J}, issn = {1467-8659}, journal = {Computer Graphics Forum}, number = {2}, pages = {343--353}, publisher = {Wiley}, title = {{Coupling 3D liquid simulation with 2D wave propagation for large scale water surface animation using the equivalent sources method}}, doi = {10.1111/cgf.14478}, volume = {41}, year = {2022}, } @article{11736, abstract = {This paper introduces a methodology for inverse-modeling of yarn-level mechanics of cloth, based on the mechanical response of fabrics in the real world. We compiled a database from physical tests of several different knitted fabrics used in the textile industry. These data span different types of complex knit patterns, yarn compositions, and fabric finishes, and the results demonstrate diverse physical properties like stiffness, nonlinearity, and anisotropy. We then develop a system for approximating these mechanical responses with yarn-level cloth simulation. To do so, we introduce an efficient pipeline for converting between fabric-level data and yarn-level simulation, including a novel swatch-level approximation for speeding up computation, and some small-but-necessary extensions to yarn-level models used in computer graphics. The dataset used for this paper can be found at http://mslab.es/projects/YarnLevelFabrics.}, author = {Sperl, Georg and Sánchez-Banderas, Rosa M. and Li, Manwen and Wojtan, Christopher J and Otaduy, Miguel A.}, issn = {1557-7368}, journal = {ACM Transactions on Graphics}, number = {4}, publisher = {Association for Computing Machinery}, title = {{Estimation of yarn-level simulation models for production fabrics}}, doi = {10.1145/3528223.3530167}, volume = {41}, year = {2022}, } @article{12431, abstract = {This paper presents a new representation of curve dynamics, with applications to vortex filaments in fluid dynamics. Instead of representing these filaments with explicit curve geometry and Lagrangian equations of motion, we represent curves implicitly with a new co-dimensional 2 level set description. Our implicit representation admits several redundant mathematical degrees of freedom in both the configuration and the dynamics of the curves, which can be tailored specifically to improve numerical robustness, in contrast to naive approaches for implicit curve dynamics that suffer from overwhelming numerical stability problems. Furthermore, we note how these hidden degrees of freedom perfectly map to a Clebsch representation in fluid dynamics. Motivated by these observations, we introduce untwisted level set functions and non-swirling dynamics which successfully regularize sources of numerical instability, particularly in the twisting modes around curve filaments. A consequence is a novel simulation method which produces stable dynamics for large numbers of interacting vortex filaments and effortlessly handles topological changes and re-connection events.}, author = {Ishida, Sadashige and Wojtan, Christopher J and Chern, Albert}, issn = {1557-7368}, journal = {ACM Transactions on Graphics}, number = {6}, publisher = {Association for Computing Machinery}, title = {{Hidden degrees of freedom in implicit vortex filaments}}, doi = {10.1145/3550454.3555459}, volume = {41}, year = {2022}, } @misc{9327, abstract = {This archive contains the missing sweater mesh animations and displacement models for the code of "Mechanics-Aware Deformation of Yarn Pattern Geometry" Code Repository: https://git.ist.ac.at/gsperl/MADYPG}, author = {Sperl, Georg and Narain, Rahul and Wojtan, Christopher J}, publisher = {IST Austria}, title = {{Mechanics-Aware Deformation of Yarn Pattern Geometry (Additional Animation/Model Data)}}, doi = {10.15479/AT:ISTA:9327}, year = {2021}, } @article{9818, abstract = {Triangle mesh-based simulations are able to produce satisfying animations of knitted and woven cloth; however, they lack the rich geometric detail of yarn-level simulations. Naive texturing approaches do not consider yarn-level physics, while full yarn-level simulations may become prohibitively expensive for large garments. We propose a method to animate yarn-level cloth geometry on top of an underlying deforming mesh in a mechanics-aware fashion. Using triangle strains to interpolate precomputed yarn geometry, we are able to reproduce effects such as knit loops tightening under stretching. In combination with precomputed mesh animation or real-time mesh simulation, our method is able to animate yarn-level cloth in real-time at large scales.}, author = {Sperl, Georg and Narain, Rahul and Wojtan, Christopher J}, issn = {1557-7368}, journal = {ACM Transactions on Graphics}, number = {4}, publisher = {Association for Computing Machinery}, title = {{Mechanics-aware deformation of yarn pattern geometry}}, doi = {10.1145/3450626.3459816}, volume = {40}, year = {2021}, } @article{5681, abstract = {We introduce dynamically warping grids for adaptive liquid simulation. Our primary contributions are a strategy for dynamically deforming regular grids over the course of a simulation and a method for efficiently utilizing these deforming grids for liquid simulation. Prior work has shown that unstructured grids are very effective for adaptive fluid simulations. However, unstructured grids often lead to complicated implementations and a poor cache hit rate due to inconsistent memory access. Regular grids, on the other hand, provide a fast, fixed memory access pattern and straightforward implementation. Our method combines the advantages of both: we leverage the simplicity of regular grids while still achieving practical and controllable spatial adaptivity. We demonstrate that our method enables adaptive simulations that are fast, flexible, and robust to null-space issues. At the same time, our method is simple to implement and takes advantage of existing highly-tuned algorithms.}, author = {Hikaru, Ibayashi and Wojtan, Christopher J and Thuerey, Nils and Igarashi, Takeo and Ando, Ryoichi}, issn = {1941-0506}, journal = {IEEE Transactions on Visualization and Computer Graphics}, number = {6}, pages = {2288--2302}, publisher = {IEEE}, title = {{Simulating liquids on dynamically warping grids}}, doi = {10.1109/TVCG.2018.2883628}, volume = {26}, year = {2020}, } @article{8384, abstract = {Previous research on animations of soap bubbles, films, and foams largely focuses on the motion and geometric shape of the bubble surface. These works neglect the evolution of the bubble’s thickness, which is normally responsible for visual phenomena like surface vortices, Newton’s interference patterns, capillary waves, and deformation-dependent rupturing of films in a foam. In this paper, we model these natural phenomena by introducing the film thickness as a reduced degree of freedom in the Navier-Stokes equations and deriving their equations of motion. We discretize the equations on a nonmanifold triangle mesh surface and couple it to an existing bubble solver. In doing so, we also introduce an incompressible fluid solver for 2.5D films and a novel advection algorithm for convecting fields across non-manifold surface junctions. Our simulations enhance state-of-the-art bubble solvers with additional effects caused by convection, rippling, draining, and evaporation of the thin film.}, author = {Ishida, Sadashige and Synak, Peter and Narita, Fumiya and Hachisuka, Toshiya and Wojtan, Christopher J}, issn = {1557-7368}, journal = {ACM Transactions on Graphics}, number = {4}, publisher = {Association for Computing Machinery}, title = {{A model for soap film dynamics with evolving thickness}}, doi = {10.1145/3386569.3392405}, volume = {39}, year = {2020}, } @article{8385, abstract = {We present a method for animating yarn-level cloth effects using a thin-shell solver. We accomplish this through numerical homogenization: we first use a large number of yarn-level simulations to build a model of the potential energy density of the cloth, and then use this energy density function to compute forces in a thin shell simulator. We model several yarn-based materials, including both woven and knitted fabrics. Our model faithfully reproduces expected effects like the stiffness of woven fabrics, and the highly deformable nature and anisotropy of knitted fabrics. Our approach does not require any real-world experiments nor measurements; because the method is based entirely on simulations, it can generate entirely new material models quickly, without the need for testing apparatuses or human intervention. We provide data-driven models of several woven and knitted fabrics, which can be used for efficient simulation with an off-the-shelf cloth solver.}, author = {Sperl, Georg and Narain, Rahul and Wojtan, Christopher J}, issn = {1557-7368}, journal = {ACM Transactions on Graphics}, number = {4}, publisher = {Association for Computing Machinery}, title = {{Homogenized yarn-level cloth}}, doi = {10.1145/3386569.3392412}, volume = {39}, year = {2020}, } @article{8535, abstract = {We propose a method to enhance the visual detail of a water surface simulation. Our method works as a post-processing step which takes a simulation as input and increases its apparent resolution by simulating many detailed Lagrangian water waves on top of it. We extend linear water wave theory to work in non-planar domains which deform over time, and we discretize the theory using Lagrangian wave packets attached to spline curves. The method is numerically stable and trivially parallelizable, and it produces high frequency ripples with dispersive wave-like behaviors customized to the underlying fluid simulation.}, author = {Skrivan, Tomas and Soderstrom, Andreas and Johansson, John and Sprenger, Christoph and Museth, Ken and Wojtan, Christopher J}, issn = {1557-7368}, journal = {ACM Transactions on Graphics}, number = {4}, publisher = {Association for Computing Machinery}, title = {{Wave curves: Simulating Lagrangian water waves on dynamically deforming surfaces}}, doi = {10.1145/3386569.3392466}, volume = {39}, year = {2020}, }