TY - GEN AB - 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. AU - Chern, Albert AU - Ishida, Sadashige ID - 12846 T2 - arXiv TI - Area formula for spherical polygons via prequantization ER - TY - JOUR AB - 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. AU - Makatura, Liane AU - Wang, Bohan AU - Chen, Yi-Lu AU - Deng, Bolei AU - Wojtan, Christopher J AU - Bickel, Bernd AU - Matusik, Wojciech ID - 14628 IS - 5 JF - ACM Transactions on Graphics KW - Computer Graphics and Computer-Aided Design SN - 0730-0301 TI - Procedural metamaterials: A unified procedural graph for metamaterial design VL - 42 ER - TY - GEN AB - 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. AU - Ishida, Sadashige AU - Lavenant, Hugo ID - 14703 KW - Optimal transport KW - Hamilton-Jacobi equation KW - convex optimization T2 - arXiv TI - Quantitative convergence of a discretization of dynamic optimal transport using the dual formulation ER - TY - JOUR AB - 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. AU - Jeschke, Stefan AU - Wojtan, Christopher J ID - 14240 IS - 4 JF - ACM Transactions on Graphics SN - 0730-0301 TI - Generalizing shallow water simulations with dispersive surface waves VL - 42 ER - TY - GEN AU - Chen, Yi-Lu AU - Ly, Mickaël AU - Wojtan, Christopher J ID - 14748 SN - 9798400702686 T2 - Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation TI - Unified treatment of contact, friction and shock-propagation in rigid body animation ER - TY - JOUR AB - 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. AU - Schreck, Camille AU - Wojtan, Christopher J ID - 11432 IS - 2 JF - Computer Graphics Forum SN - 0167-7055 TI - Coupling 3D liquid simulation with 2D wave propagation for large scale water surface animation using the equivalent sources method VL - 41 ER - TY - JOUR AB - We revisit two basic Direct Simulation Monte Carlo Methods to model aggregation kinetics and extend them for aggregation processes with collisional fragmentation (shattering). We test the performance and accuracy of the extended methods and compare their performance with efficient deterministic finite-difference method applied to the same model. We validate the stochastic methods on the test problems and apply them to verify the existence of oscillating regimes in the aggregation-fragmentation kinetics recently detected in deterministic simulations. We confirm the emergence of steady oscillations of densities in such systems and prove the stability of the oscillations with respect to fluctuations and noise. AU - Kalinov, Aleksei AU - Osinskiy, A.I. AU - Matveev, S.A. AU - Otieno, W. AU - Brilliantov, N.V. ID - 11556 JF - Journal of Computational Physics KW - Computer Science Applications KW - Physics and Astronomy (miscellaneous) KW - Applied Mathematics KW - Computational Mathematics KW - Modeling and Simulation KW - Numerical Analysis SN - 0021-9991 TI - Direct simulation Monte Carlo for new regimes in aggregation-fragmentation kinetics VL - 467 ER - TY - JOUR AB - 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. AU - Sperl, Georg AU - Sánchez-Banderas, Rosa M. AU - Li, Manwen AU - Wojtan, Christopher J AU - Otaduy, Miguel A. ID - 11736 IS - 4 JF - ACM Transactions on Graphics SN - 0730-0301 TI - Estimation of yarn-level simulation models for production fabrics VL - 41 ER - TY - JOUR AB - 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. AU - Ishida, Sadashige AU - Wojtan, Christopher J AU - Chern, Albert ID - 12431 IS - 6 JF - ACM Transactions on Graphics SN - 0730-0301 TI - Hidden degrees of freedom in implicit vortex filaments VL - 41 ER - TY - THES AB - The complex yarn structure of knitted and woven fabrics gives rise to both a mechanical and visual complexity. The small-scale interactions of yarns colliding with and pulling on each other result in drastically different large-scale stretching and bending behavior, introducing anisotropy, curling, and more. While simulating cloth as individual yarns can reproduce this complexity and match the quality of real fabric, it may be too computationally expensive for large fabrics. On the other hand, continuum-based approaches do not need to discretize the cloth at a stitch-level, but it is non-trivial to find a material model that would replicate the large-scale behavior of yarn fabrics, and they discard the intricate visual detail. In this thesis, we discuss three methods to try and bridge the gap between small-scale and large-scale yarn mechanics using numerical homogenization: fitting a continuum model to periodic yarn simulations, adding mechanics-aware yarn detail onto thin-shell simulations, and quantitatively fitting yarn parameters to physical measurements of real fabric. To start, we present a method for animating yarn-level cloth effects using a thin-shell solver. We first use a large number of periodic yarn-level simulations to build a model of the potential energy density of the cloth, and then use it to compute forces in a thin-shell simulator. The resulting simulations faithfully reproduce expected effects like the stiffening of woven fabrics and the highly deformable nature and anisotropy of knitted fabrics at a fraction of the cost of full yarn-level simulation. While our thin-shell simulations are able to capture large-scale yarn mechanics, they lack the rich visual detail of yarn-level simulations. Therefore, we propose a method to animate yarn-level cloth geometry on top of an underlying deforming mesh in a mechanics-aware fashion in real time. Using triangle strains to interpolate precomputed yarn geometry, we are able to reproduce effects such as knit loops tightening under stretching at negligible cost. Finally, we introduce a methodology for inverse-modeling of yarn-level mechanics of cloth, based on the mechanical response of fabrics in the real world. We compile a database from physical tests of several knitted fabrics used in the textile industry spanning diverse physical properties like stiffness, nonlinearity, and anisotropy. We then develop a system for approximating these mechanical responses with yarn-level cloth simulation, using homogenized shell models to speed up computation and adding some small-but-necessary extensions to yarn-level models used in computer graphics. AU - Sperl, Georg ID - 12358 SN - 2663-337X TI - Homogenizing yarn simulations: Large-scale mechanics, small-scale detail, and quantitative fitting ER -