@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}, } @phdthesis{20551, abstract = {The space of codimension-2 shapes, such as curves in 3D and surfaces in 4D, is an infinite-dimensional manifold. This thesis explores geometric structures and dynamics on this space, with emphasis on their implications for physics, particularly hydrodynamics. Our investigation ranges from theoretical studies of infinite-dimensional symplectic and prequantum geometry to numerical computation of the time evolution of shapes. The thesis presents four main contributions. In the first part, we introduce implicit representations of codimension-2 shapes using a class of complex-valued functions, and prove that the space of these implicit representations forms a prequantum bundle over the codimension-2 shape space. This reveals a new geometric interpretation of the canonical symplectic structure on the codimension-2 shape space. In the second part, we use implicit representations to develop a simulation method for the dynamics of space curves. To handle chaotic systems such as vortex filaments in hydrodynamics, we exploit the infinite degrees of freedom, hidden in both the configuration and dynamics of implicit representations. In the third part, we introduce new symplectic structures on the space of space curves, which generalize the only previously known symplectic structure on this space, allowing for new Hamiltonian dynamics of space curves. In the fourth part, we apply a symplectic viewpoint to a differential geometric problem with practical applications. We derive a new area formula for spherical polygons via prequantization. }, author = {Ishida, Sadashige}, isbn = {978-3-99078-070-1}, issn = {2663-337X}, pages = {141}, publisher = {Institute of Science and Technology Austria}, title = {{Symplectic-prequantum structures and dynamics on the codimension-2 shape space}}, doi = {10.15479/AT-ISTA-20551}, year = {2025}, } @phdthesis{19630, abstract = {This thesis consists of three chapters, each corresponding to one publication. While each of these projects tackles a topic in a different area of research, they all share a common thread in the type of topological structure they handle - a partition of space into volumes separated by interfaces that meet in non-manifold junctions. In Chapter 2, we study clusters of soap bubbles from a simulation perspective. In particular, we develop a surface-only algorithm that couples large scale motion and shape deformation of soap bubble clusters with the small scale evolution of the thin film's thickness, which is responsible for visual phenomena like surface vortices, Newton's interference patterns, capillary waves, and deformation-dependent rupturing of films in a foam. We model film thickness as a reduced degree of freedom in the Navier-Stokes equations and from them derive three sets of equations governing normal and tangential motion of the soap film surface, as well as the evolution of the thin film thickness. We discretize these equations on a non-manifold triangle mesh, extending and adapting operators to handle complex topology. We also present an incompressible fluid solver for 2.5D films and an advection algorithm for convecting fields across non-manifold surface junctions. Our simulations enhance bubble solvers with additional effects caused by convection, rippling, draining, and evaporation of the thin film. In Chapter 3, we introduce a multi-material non-manifold mesh-based surface tracking algorithm that converts mesh defects, such as overlaps, self-intersections, and inversions 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, such as those presented in Chapter 2, but with an order of magnitude more interacting bubbles than what we could achieve before, and Boolean unions of non-manifold meshes consisting of millions of triangles. Lastly, in Chapter 4, we utilize developments in the theory of random geometric complexes facilitated by observations from Discrete Morse theory. We survey the methods and results obtained with this new approach, and discuss some of its shortcomings. We use simulations to illustrate the results and to form conjectures, getting numerical estimates for combinatorial, topological, and geometric properties of weighted and unweighted Delaunay mosaics, their dual Voronoi tessellations, and the Alpha and Wrap complexes contained in the mosaics.}, author = {Synak, Peter}, issn = {2663-337X}, pages = {106}, publisher = {Institute of Science and Technology Austria}, title = {{Methods for fluid simulation, surface tracking, and statistics of non-manifold structures}}, doi = {10.15479/AT-ISTA-19630}, year = {2025}, } @article{14703, abstract = {We present a discretization of the dynamic optimal transport problem for which we can obtain the convergence rate for the value of the transport cost to its continuous value when the temporal and spatial stepsize vanish. This convergence result does not require any regularity assumption on the measures, though experiments suggest that the rate is not sharp. Via an analysis of the duality gap we also obtain the convergence rates for the gradient of the optimal potentials and the velocity field under mild regularity assumptions. To obtain such rates we discretize the dual formulation of the dynamic optimal transport problem and use the mature literature related to the error due to discretizing the Hamilton-Jacobi equation.}, author = {Ishida, Sadashige and Lavenant, Hugo}, issn = {1615-3383}, journal = {Foundations of Computational Mathematics}, keywords = {Optimal transport, Hamilton-Jacobi equation, convex optimization}, publisher = {Springer Nature}, title = {{Quantitative convergence of a discretization of dynamic optimal transport using the dual formulation}}, doi = {10.1007/s10208-024-09686-3}, year = {2024}, } @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{12846, abstract = {We present a formula for the signed area of a spherical polygon via prequantization. In contrast to the traditional formula based on the Gauss-Bonnet theorem that requires measuring angles, the new formula mimics Green's theorem and is applicable to a wider range of degenerate spherical curves and polygons.}, author = {Chern, Albert and Ishida, Sadashige}, issn = {2470-6566}, journal = {SIAM Journal on Applied Algebra and Geometry}, number = {3}, pages = {782--796}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Area formula for spherical polygons via prequantization}}, doi = {10.1137/23M1565255}, volume = {8}, year = {2024}, } @unpublished{17361, abstract = {We present symplectic structures on the shape space of unparameterized space curves that generalize the classical Marsden-Weinstein structure. Our method integrates the Liouville 1-form of the Marsden-Weinstein structure with Riemannian structures that have been introduced in mathematical shape analysis. We also derive Hamiltonian vector fields for several classical Hamiltonian functions with respect to these new symplectic structures.}, author = {Bauer, Martin and Ishida, Sadashige and Michor, Peter W.}, booktitle = {arXiv}, keywords = {space of space curves, symplectic stuctures}, title = {{Symplectic structures on the space of space curves}}, doi = {10.48550/arXiv.2407.19908}, 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{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}, }