Scriptwriting has traditionally been text-centric, a modality that only partially conveys the produced audiovisual experience. A formative study with professional writers informed us that connecting textual and audiovisual modalities can aid ideation and iteration, especially for writing dialogues. In this work, we present Script2Screen, an AI-assisted tool that integrates scriptwriting with audiovisual scene creation in a unified, synchronized workflow. Focusing on dialogues in scripts, Script2Screen generates expressive scenes with emotional speeches and animated characters through a novel text-to-audiovisual-scene pipeline. The user interface provides fine-grained controls, allowing writers to fine-tune audiovisual elements such as character gestures, speech emotions, and camera angles. A user study with both novice and professional writers from various domains demonstrated that Script2Screen’s interactive audiovisual generation enhances the scriptwriting process, facilitating iterative refinement while complementing - rather than replacing - their creative efforts.

We present a low-rank Koopman operator formulation for accelerating deformable subspace simulation. Using a Dynamic Mode Decomposition (DMD) parameterization of the Koopman operator, our method learns the temporal evolution of deformable dynamics and predicts future states through efficient matrix evaluations instead of sequential time integration. This yields log-linear scaling in the number of time steps and allows large portions of the trajectory to be skipped while retaining accuracy. The resulting temporal efficiency is especially advantageous for optimization tasks such as control and initial-state estimation, where the objective often depends largely on the final configuration. To broaden the scope of Koopman-based reduced-order models in graphics, we introduce a discretization-agnostic extension that learns shared dynamic behavior across multiple shapes and mesh resolutions. Prior DMD-based approaches have been restricted to a single shape and discretization, which limits their usefulness for tasks involving geometry variation. Our formulation generalizes across both shape and discretization, which enables fast shape optimization that was previously impractical for DMD models. This expanded capability highlights the potential of Koopman operator learning as a practical tool for efficient deformable simulation and design.

Discontinuities in spatial derivatives appear in a wide range of physical systems, from creased thin sheets to materials with sharp stiffness transitions. Accurately modeling these features is essential for simulation but remains challenging for traditional mesh-based methods, which require discontinuity-aligned remeshing—entangling geometry with simulation and hindering generalization across shape families.

Designing subspaces for Reduced Order Modeling (ROM) is crucial for accelerating finite element simulations in graphics and engineering. Unfortunately, it's not always clear which subspace is optimal for arbitrary dynamic simulation. We propose to construct simulation subspaces from force distributions, allowing us to tailor such subspaces to common scene interactions involving constraint penalties, handles-based control, contact and musculoskeletal actuation. To achieve this we adopt a statistical perspective on Reduced Order Modelling, which allows us to push such user-designed force distributions through a linearized simulation to obtain a dual distribution on displacements. To construct our subspace, we then fit a low-rank Gaussian model to this displacement distribution, which we show generalizes Linear Modal Analysis subspaces for uncorrelated unit variance force distributions, as well as Green's Function subspaces for low rank force distributions. We show our framework allows for the construction of subspaces that are optimal both with respect to physical material properties, as well as arbitrary force distributions as observed in handle-based, contact, and musculoskeletal scene interactions.

Eigenanalysis of differential operators, such as the Laplace operator or elastic energy Hessian, is typically restricted to a single shape and its discretization, limiting reduced order modeling (ROM). We introduce the first eigenanalysis method for continuously parameterized shape families. Given a parametric shape, our method constructs spatial neural fields that represent eigen-functions across the entire shape space. It is agnostic to the specific shape representation, requiring only an inside/outside indicator function that depends on shape parameters. Eigenfunctions are computed by minimizing a variational principle over nested spaces with orthogonality constraints. Since eigenvalues may swap dominance at points of multiplicity, we jointly train multiple eigenfunctions while dynamically reordering them based on their eigenvalues at each step. Through causal gradient filtering , this reordering is reflected in backpropagation. Our method enables applications to operate over shape space, providing a single ROM that encapsulates vibration modes for all shapes, including previously unseen ones. Since our eigenanalysis is differentiable with respect to shape parameters, it facilitates eigenfunction-aware shape optimization. We evaluate our approach on shape optimization for sound synthesis and locomotion, as well as reduced-order modeling for elastodynamic simulation.

We present a novel reduced-order fluid simulation technique leveraging Dynamic Mode Decomposition (DMD) to achieve fast, memory-efficient, and user-controllable subspace simulation. We demonstrate that our approach combines the strengths of both spatial reduced order models (ROMs) as well as spectral decompositions. By optimizing for the operator that evolves a system state from one timestep to the next, rather than the system state itself, we gain both the compressive power of spatial ROMs as well as the intuitive physical dynamics of spectral methods. The latter property is of particular interest in graphics applications, where user control of fluid phenomena is of high demand. We demonstrate this in various applications including spatial and temporal modulation tools and fluid upscaling with added turbulence. We adapt DMD for graphics applications by reducing computational overhead, incorporating user-defined force inputs, and optimizing memory usage with randomized SVD. The integration of OptDMD and DMD with Control (DMDc) facilitates noise-robust reconstruction and real-time user interaction. We demonstrate the technique's robustness across diverse simulation scenarios, including artistic editing, time-reversal, and super-resolution. Through experimental validation on challenging scenarios, such as colliding vortex rings and boundary-interacting plumes, our method also exhibits superior performance and fidelity with significantly fewer basis functions compared to existing spatial ROMs. Leveraging the inherent linearity of the DMD formulation, we demonstrate a range of diverse applications. This work establishes another avenue for developing real-time, high-quality fluid simulations, enriching the space of fluid simulation techniques in interactive graphics and animation.

Architectural surface panelling often requires fabricating molds for panels, a process that can be cost-inefficient and material-wasteful when using traditional methods such as CNC milling. In this paper, we introduce a novel solution to generating molds for efficiently fabricating architectural panels. At the core of our method is a machine that utilizes a deflatable membrane as a flexible mold. By adjusting the deflation level and boundary element positions, the membrane can be reconfigured into various shapes, allowing for mass customization with significantly lower overhead costs. We devise an efficient algorithm that works in sync with our flexible mold machine that optimizes the placement of customizable boundary element positions, ensuring the fabricated panel matches the geometry of a given input shape: (1) Using a quadratic Weingarten surface arising from a natural assumption on the membrane's stress, we can approximate the initial placement of the boundary element from the input shape's geometry; (2) we solve the inverse problem with a simulator-in-the-loop optimizer by searching for the optimal placement of boundary curves with sensitivity analysis. We validate our approach by fabricating baseline panels and a facade with a wide range of curvature profiles, providing a detailed numerical analysis on simulation and fabrication, demonstrating significant advantages in cost and flexibility.

Implicit Neural Spatial Representation (INSR) has emerged as an effective representation of spatially-dependent vector fields. This work explores solving time-dependent PDEs with INSR. Classical PDE solvers introduce both temporal and spatial discretizations. Common spatial discretizations include meshes and meshless point clouds, where each degree-of-freedom corresponds to a location in space. While these explicit spatial correspondences are intuitive to model and understand, these representations are not necessarily optimal for accuracy, memory usage, or adaptivity. Keeping the classical temporal discretization unchanged (e.g., explicit/implicit Euler), we explore INSR as an alternative spatial discretization, where spatial information is implicitly stored in the neural network weights. The network weights then evolve over time via time integration. Our approach does not require any training data generated by existing solvers because our approach is the solver itself. We validate our approach on various PDEs with examples involving large elastic deformations, turbulent fluids, and multi-scale phenomena. While slower to compute than traditional representations, our approach exhibits higher accuracy and lower memory consumption. Whereas classical solvers can dynamically adapt their spatial representation only by resorting to complex remeshing algorithms, our INSR approach is intrinsically adaptive. By tapping into the rich literature of classic time integrators, e.g., operator-splitting schemes, our method enables challenging simulations in contact mechanics and turbulent flows where previous neural-physics approaches struggle. Videos and codes are available on the project page: http://www.cs.columbia.edu/cg/INSR-PDE/

Linear reduced-order modeling (ROM) simplifies complex simulations by approximating the behavior of a system using a simplified kinematic representation. Typically, ROM is trained on input simulations created with a specific spatial discretization, and then serves to accelerate simulations with the same discretization. This discretization-dependence is restrictive.

Real-time elastodynamic solvers are well-suited for the rapid simulation of homogeneous elastic materials, with high-rates generally enabled by aggressive early termination of timestep solves. Unfortunately, the introduction of strong domain heterogeneities can make these solvers slow to converge. Stopping the solve short creates visible damping artifacts and rotational errors. To address these challenges we develop a reduced mixed finite element solver that preserves rich rotational motion, even at low-iteration regimes. Specifically, this solver augments time-step solve optimizations with auxillary stretch degrees of freedom at mesh elements, and maintains consistency with the primary positional degrees of freedoms at mesh nodes via explicit constraints. We make use of a Skinning Eigenmode subspace for our positional degrees of freedom. We accelerate integration of non-linear elastic energies with a cubature approximation, placing stretch degrees of freedom at cubature points. Across a wide range of examples we demonstrate that this subspace is particularly well suited for heterogeneous material simulation. Our resulting method is a subspace mixed finite element method completely decoupled from the resolution of the mesh that is well-suited for real-time simulation of heterogeneous domains.

We propose a hybrid neural network and physics framework for reduced-order modeling of elastoplasticity and fracture. State-of-the-art scientific computing models like the Material Point Method (MPM) faithfully simulate large-deformation elastoplasticity and fracture mechanics. However, their long runtime and large memory consumption render them unsuitable for applications constrained by computation time and memory usage, e.g., virtual reality. To overcome these barriers, we propose a reduced-order framework. Our key innovation is training a low-dimensional manifold for the Kirchhoff stress field via an implicit neural representation. This low-dimensional neural stress field (NSF) enables efficient evaluations of stress values and, correspondingly, internal forces at arbitrary spatial locations. In addition, we also train neural deformation and affine fields to build low-dimensional manifolds for the deformation and affine momentum fields. These neural stress, deformation, and affine fields share the same low-dimensional latent space, which uniquely embeds the high-dimensional simulation state. After training, we run new simulations by evolving in this single latent space, which drastically reduces the computation time and memory consumption. Our general continuum-mechanics-based reduced-order framework is applicable to any phenomena governed by the elastodynamics equation. To showcase the versatility of our framework, we simulate a wide range of material behaviors, including elastica, sand, metal, non-Newtonian fluids, fracture, contact, and collision. We demonstrate dimension reduction by up to 100,000 × and time savings by up to 10 ×.

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.

Standard liquid simulators apply operator splitting to independently solve for pressure and viscous stresses, a decoupling that induces incorrect free surface boundary conditions. Such methods are unable to simulate fluid phenomena reliant on the balance of pressure and viscous stresses, such as the liquid rope coil instability exhibited by honey. By contrast, unsteady Stokes solvers retain coupling between pressure and viscosity, thus resolving these phenomena, albeit using a much larger and thus more computationally expensive linear system compared to the decoupled approach. To accelerate solving the unsteady Stokes problem, we propose a reduced fluid model wherein interior regions are represented with incompressible polynomial vector fields. Sets of standard grid cells are consolidated into super-cells, each of which are modelled using a quadratic field of 26 degrees of freedom. We demonstrate that the reduced field must necessarily be at least quadratic, with the affine model being unable to correctly capture viscous forces. We reproduce the liquid rope coiling instability, as well as other simulated examples, to show that our reduced model is able to reproduce the same fluid phenomena at a smaller computational cost. Futhermore, we performed a crowdsourced user survey to verify that our method produces imperceptible differences compared to the full unsteady Stokes method.

We present an algorithmic approach to discover, study, and design multistable elastic knots. Elastic knots are physical realizations of closed curves embedded in 3-space. When endowed with the material thickness and bending resistance of a physical wire, these knots settle into equilibrium states that balance the forces induced by elastic deformation and self-contacts of the wire. In general, elastic knots can have many distinct equilibrium states, i.e. they are multistable mechanical systems. We propose a computational pipeline that combines randomized spatial sampling and physics simulation to efficiently find stable equilibrium states of elastic knots. Leveraging results from knot theory, we run our pipeline on thousands of different topological knot types to create an extensive data set of multistable knots. By applying a series of filters to this data, we discover new transformable knots with interesting geometric and physical properties. A further analysis across knot types reveals geometric and topological patterns, yielding constructive principles that generalize beyond the currently tabulated knot types. We show how multistable elastic knots can be used to design novel deployable structures and engaging recreational puzzles. Several physical prototypes at different scales highlight these applications and validate our simulation.

Abstract We present a simple and concise discretization of the covariant derivative vector Dirichlet energy for triangle meshes in 3D using Crouzeix‐Raviart finite elements. The discretization is based on linear discontinuous Galerkin elements, and is simple to implement, without compromising on quality: there are two degrees of freedom for each mesh edge, and the sparse Dirichlet energy matrix can be constructed in a single pass over all triangles using a short formula that only depends on the edge lengths, reminiscent of the scalar cotangent Laplacian. Our vector Dirichlet energy discretization can be used in a variety of applications, such as the calculation of Killing fields, parallel transport of vectors, and smooth vector field design. Experiments suggest convergence and suitability for applications similar to other discretizations of the vector Dirichlet energy.

Current quadratic smoothness energies for curved surfaces either exhibit distortions near the boundary due to zero Neumann boundary conditions or they do not correctly account for intrinsic curvature, which leads to unnatural-looking behavior away from the boundary. This leads to an unfortunate trade-off: One can either have natural behavior in the interior or a distortion-free result at the boundary, but not both. We introduce a generalized Hessian energy for curved surfaces, expressed in terms of the covariant one-form Dirichlet energy, the Gaussian curvature, and the exterior derivative. Energy minimizers solve the Laplace-Beltrami biharmonic equation, correctly accounting for intrinsic curvature, leading to natural-looking isolines. On the boundary, minimizers are as-linear-as-possible, which reduces the distortion of isolines at the boundary. We discretize the covariant one-form Dirichlet energy using Crouzeix-Raviart finite elements, arriving at a discrete formulation of the Hessian energy for applications on curved surfaces. We observe convergence of the discretization in our experiments.

We propose a framework for simulating the complex dynamics of strands interacting with compressible, shear-dependent liquids, such as oil paint, mud, cream, melted chocolate, and pasta sauce. Our framework contains three main components: the strands modeled as discrete rods, the bulk liquid represented as a continuum (material point method), and a reduced-dimensional flow of liquid on the surface of the strands with detailed elastoviscoplastic behavior. These three components are tightly coupled together. To enable discrete strands interacting with continuum-based liquid, we develop models that account for the volume change of the liquid as it passes through strands and the momentum exchange between the strands and the liquid. We also develop an extended constraint-based collision handling method that supports cohesion between strands. Furthermore, we present a principled method to preserve the total momentum of a strand and its surface flow, as well as an analytic plastic flow approach for Herschel-Bulkley fluid that enables stable semi-implicit integration at larger time steps. We explore a series of challenging scenarios, involving splashing, shaking, and agitating the liquid which causes the strands to stick together and become entangled.

Robotic manipulation of deformable objects is a difficult problem especially because of the complexity of the many different ways an object can deform. Searching such a high-dimensional state space makes it difficult to recognize, track, and manipulate deformable objects. In this paper, we introduce a predictive, model-driven approach to address this challenge, using a precomputed, simulated database of deformable object models. Mesh models of common deformable garments are simulated with the garments picked up in multiple different poses under gravity, and stored in a database for fast and efficient retrieval. To validate this approach, we developed a comprehensive pipeline for manipulating clothing as in a typical laundry task. First, the database is used for category and the pose estimation is used for a garment in an arbitrary position. A fully featured 3-D model of the garment is constructed in real time, and volumetric features are then used to obtain the most similar model in the database to predict the object category and pose. Second, the database can significantly benefit the manipulation of deformable objects via nonrigid registration, providing accurate correspondences between the reconstructed object model and the database models. Third, the accurate model simulation can also be used to optimize the trajectories for the manipulation of deformable objects, such as the folding of garments. Extensive experimental results are shown for the above tasks using a variety of different clothings. Note to Practitioners-This paper provides an open source, extensible, 3-D database for dissemination to the robotics and graphics communities. Model-driven methods are proliferating, and they need to be applied, tested, and validated in real environments. A key idea we have exploited is to have an innovative and novel use of simulation. This database will serve as infrastructure for developing advanced robotic machine learning algorithms. We want to address this machine learning idea ourselves, but we expect the dissemination of the database to other researchers with different agendas and task applications, which will bring wide progress in this area. Our proposed methods, as mentioned earlier, can be easily applied to interrelated areas. One example is that the 3-D shape-based matching algorithm can be used for other objects, such as bottles, papers, and food. After integrating with other robotic systems, the use of the robot can be easily extended to other tasks, such as making food, cleaning room, and fetching objects, to assist our daily life.

Thin shells --- solids that are thin in one dimension compared to the other two --- often emit rich nonlinear sounds when struck. Strong excitations can even cause chaotic thin-shell vibrations, producing sounds whose energy spectrum diffuses from low to high frequencies over time --- a phenomenon known as wave turbulence. It is all these nonlinearities that grant shells such as cymbals and gongs their characteristic "glinting" sound. Yet, simulation models that efficiently capture these sound effects remain elusive. We propose a physically based, multi-scale reduced simulation method to synthesize nonlinear thin-shell sounds. We first split nonlinear vibrations into two scales, with a small low-frequency part simulated in a fully nonlinear way, and a high-frequency part containing many more modes approximated through time-varying linearization. This allows us to capture interesting nonlinearities in the shells' deformation, tens of times faster than previous approaches. Furthermore, we propose a method that enriches simulated sounds with wave turbulent sound details through a phenomenological diffusion model in the frequency domain, and thereby sidestep the expensive simulation of chaotic high-frequency dynamics. We show several examples of our simulations, illustrating the efficiency and realism of our model.

We propose a method for simulating the complex dynamics of partially and fully saturated woven and knit fabrics interacting with liquid, including the effects of buoyancy, nonlinear drag, pore (capillary) pressure, dripping, and convection-diffusion. Our model evolves the velocity fields of both the liquid and solid relying on mixture theory, as well as tracking a scalar saturation variable that affects the pore pressure forces in the fluid. We consider the porous microstructure implied by the fibers composing individual threads, and use it to derive homogenized drag and pore pressure models that faithfully reflect the anisotropy of fabrics. In addition to the bulk liquid and fabric motion, we derive a quasi-static flow model that accounts for liquid spreading within the fabric itself. Our implementation significantly extends standard numerical cloth and fluid models to support the diverse behaviors of wet fabric, and includes a numerical method tailored to cope with the challenging nonlinearities of the problem. We explore a range of fabric-water interactions to validate our model, including challenging animation scenarios involving splashing, wringing, and collisions with obstacles, along with qualitative comparisons against simple physical experiments.

While folds and pleats add interest to garments and cloth objects, incorporating them into an existing design manually or using existing software requires expertise and time. We present FoldSketch , a new system that supports simple and intuitive fold and pleat design. FoldSketch users specify the fold or pleat configuration they seek using a simple schematic sketching interface; the system then algorithmically generates both the fold-enhanced 3D garment geometry that conforms to user specifications, and the corresponding 2D patterns that reproduce this geometry within a simulation engine. While previous work aspired to compute the desired patterns for a given target 3D garment geometry, our main algorithmic challenge is that we do not have target geometry to start with. Real-life garment folds have complex profile shapes, and their exact geometry and location on a garment are intricately linked to a range of physical factors such as fabric properties and the garment's interaction with the wearer's body; it is therefore virtually impossible to predict the 3D shape of a fold-enhanced garment using purely geometric means. At the same time, using physical simulation to model folds requires appropriate 2D patterns and initial drape, neither of which can be easily provided by the user. We obtain both the 3D fold-enhanced garment and its corresponding patterns and initial drape via an alternating 2D-3D algorithm. We first expand the input patterns by allocating excess material for the expected fold formation; we then use these patterns to produce an estimated fold-enhanced drape geometry that balances designer expectations against physical reproducibility. We use the patterns and the estimated drape as input to a simulation generating an initial reproducible output. We improve the output's alignment with designer expectations by progressively refining the patterns and the estimated drape, converging to a final fully physically reproducible fold-enhanced garment. Our experiments confirm that FoldSketch reliably converges to a desired garment geometry and corresponding patterns and drape, and works well with different physical simulators. We demonstrate the versatility of our approach by showcasing a collection of garments augmented with diverse fold and pleat layouts specified via the FoldSketch interface, and further validate our approach via comparisons to alternative solutions and feedback from potential users.

Typical design for manufacturing applications requires simultaneous optimization of conflicting performance objectives: Design variations that improve one performance metric may decrease another performance metric. In these scenarios, there is no unique optimal design but rather a set of designs that are optimal for different trade-offs (called Pareto-optimal). In this work, we propose a novel approach to discover the Pareto front, allowing designers to navigate the landscape of compromises efficiently. Our approach is based on a first-order approximation of the Pareto front, which allows entire neighborhoods rather than individual points on the Pareto front to be captured. In addition to allowing for efficient discovery of the Pareto front and the corresponding mapping to the design space, this approach allows us to represent the entire trade-off manifold as a small collection of patches that comprise a high-quality and piecewise-smooth approximation. We illustrate how this technique can be used for navigating performance trade-offs in computer-aided design (CAD) models.

Developable surfaces are those that can be made by smoothly bending flat pieces without stretching or shearing. We introduce a definition of developability for triangle meshes which exactly captures two key properties of smooth developable surfaces, namely flattenability and presence of straight ruling lines. This definition provides a starting point for algorithms in developable surface modeling---we consider a variational approach that drives a given mesh toward developable pieces separated by regular seam curves. Computation amounts to gradient descent on an energy with support in the vertex star, without the need to explicitly cluster patches or identify seams. We briefly explore applications to developable design and manufacturing.

In geometry processing, smoothness energies are commonly used to model scattered data interpolation, dense data denoising, and regularization during shape optimization. The squared Laplacian energy is a popular choice of energy and has a corresponding standard implementation: squaring the discrete Laplacian matrix. For compact domains, when values along the boundary are not known in advance, this construction bakes in low-order boundary conditions. This causes the geometric shape of the boundary to strongly bias the solution. For many applications, this is undesirable. Instead, we propose using the squared Frobenius norm of the Hessian as a smoothness energy. Unlike the squared Laplacian energy, this energy’s natural boundary conditions (those that best minimize the energy) correspond to meaningful high-order boundary conditions. These boundary conditions model free boundaries where the shape of the boundary should not bias the solution locally. Our analysis begins in the smooth setting and concludes with discretizations using finite-differences on 2D grids or mixed finite elements for triangle meshes. We demonstrate the core behavior of the squared Hessian as a smoothness energy for various tasks.

We propose a technique to simulate granular materials that exploits the dual strengths of discrete and continuum treatments. Discrete element simulations provide unmatched levels of detail and generality, but prove excessively costly when applied to large scale systems. Continuum approaches are computationally tractable, but limited in applicability due to built-in modeling assumptions; e.g., models suitable for granular flows typically fail to capture clogging, bouncing and ballistic motion. In our hybrid approach, an oracle dynamically partitions the domain into continuum regions where safe, and discrete regions where necessary. The domains overlap along transition zones, where a Lagrangian dynamics mass-splitting coupling principle enforces agreement between the two simulation states. Enrichment and homogenization operations allow the partitions to evolve over time. This approach accurately and efficiently simulates scenarios that previously required an entirely discrete treatment.

The diverse interactions between hair and liquid are complex and span multiple length scales, yet are central to the appearance of humans and animals in many situations. We therefore propose a novel multi-component simulation framework that treats many of the key physical mechanisms governing the dynamics of wet hair. The foundations of our approach are a discrete rod model for hair and a particle-in-cell model for fluids. To treat the thin layer of liquid that clings to the hair, we augment each hair strand with a height field representation. Our contribution is to develop the necessary physical and numerical models to evolve this new system and the interactions among its components. We develop a new reduced-dimensional liquid model to solve the motion of the liquid along the length of each hair, while accounting for its moving reference frame and influence on the hair dynamics. We derive a faithful model for surface tension-induced cohesion effects between adjacent hairs, based on the geometry of the liquid bridges that connect them. We adopt an empirically-validated drag model to treat the effects of coarse-scale interactions between hair and surrounding fluid, and propose new volume-conserving dripping and absorption strategies to transfer liquid between the reduced and particle-in-cell liquid representations. The synthesis of these techniques yields an effective wet hair simulator, which we use to animate hair flipping, an animal shaking itself dry, a spinning car wash roller brush dunked in liquid, and intricate hair coalescence effects, among several additional scenarios.

Computer Aided Design (CAD) is a multi-billion dollar industry used by almost every mechanical engineer in the world to create practically every existing manufactured shape. CAD models are not only widely available but also extremely useful in the growing field of fabrication-oriented design because they are parametric by construction and capture the engineer's design intent, including manufacturability. Harnessing this data, however, is challenging, because generating the geometry for a given parameter value requires time-consuming computations. Furthermore, the resulting meshes have different combinatorics, making the mesh data inherently discontinuous with respect to parameter adjustments. In our work, we address these challenges and develop tools that allow interactive exploration and optimization of parametric CAD data. To achieve interactive rates, we use precomputation on an adaptively sampled grid and propose a novel scheme for interpolating in this domain where each sample is a mesh with different combinatorics. Specifically, we extract partial correspondences from CAD representations for local mesh morphing and propose a novel interpolation method for adaptive grids that is both continuous/smooth and local (i.e., the influence of each sample is constrained to the local regions where mesh morphing can be computed). We show examples of how our method can be used to interactively visualize and optimize objects with a variety of physical properties.

Iterative algorithms are frequently used to resolve simultaneous impacts between rigid bodies in physical simulations. However, these algorithms lack formal guarantees of termination, which is sometimes viewed as potentially dangerous, so failsafes are used in practical codes to prevent infinite loops. We show such steps are unnecessary. In particular, we study the broad class of such algorithms that are conservative and satisfy a minimal set of physical correctness properties, and which encompasses recent methods like Generalized Reflections as well as pairwise schemes. We fully characterize finite termination of these algorithms. The only possible failure cases can be detected, and we describe a procedure for modifying the algorithms to provably ensure termination. We also describe modifications necessary to guarantee termination in the presence of numerical error due to the use of floating-point arithmetic. Finally, we discuss the challenges dissipation introduce for finite termination, and describe how dissipation models can be incorporated while retaining the termination guarantee.

Robotic manipulation of deformable objects remains a challenging task. One such task is to iron a piece of cloth autonomously. Given a roughly flattened cloth, the goal is to have an ironing plan that can iteratively apply a regular iron to remove all the major wrinkles by a robot. We present a novel solution to analyze the cloth surface by fusing two surface scan techniques: a curvature scan and a discontinuity scan. The curvature scan can estimate the height deviation of the cloth surface, while the discontinuity scan can effectively detect sharp surface features, such as wrinkles. We use this information to detect the regions that need to be pulled and extended before ironing, and the other regions where we want to detect wrinkles and apply ironing to remove the wrinkles. We demonstrate that our hybrid scan technique is able to capture and classify wrinkles over the surface robustly. Given detected wrinkles, we enable a robot to iron them using shape features. Experimental results show that using our wrinkle analysis algorithm, our robot is able to iron the cloth surface and effectively remove the wrinkles.

We propose a novel surface-only technique for simulating incompressible, inviscid and uniform-density liquids with surface tension in three dimensions. The liquid surface is captured by a triangle mesh on which a Lagrangian velocity field is stored. Because advection of the velocity field may violate the incompressibility condition, we devise an orthogonal projection technique to remove the divergence while requiring the evaluation of only two boundary integrals. The forces of surface tension, gravity, and solid contact are all treated by a boundary element solve, allowing us to perform detailed simulations of a wide range of liquid phenomena, including waterbells, droplet and jet collisions, fluid chains, and crown splashes.

Crumpling a thin sheet produces a characteristic sound, comprised of distinct clicking sounds corresponding to buckling events. We propose a physically based algorithm that automatically synthesizes crumpling sounds for a given thin shell animation. The resulting sound is a superposition of individually synthesized clicking sounds corresponding to visually significant and insignificant buckling events. We identify visually significant buckling events on the dynamically evolving thin surface mesh, and instantiate visually insignificant buckling events via a stochastic model that seeks to mimic the power-law distribution of buckling energies observed in many materials. In either case, the synthesis of a buckling sound employs linear modal analysis of the deformed thin shell. Because different buckling events in general occur at different deformed configurations, the question arises whether the calculation of linear modes can be reused. We amortize the cost of the linear modal analysis by dynamically partitioning the mesh into nearly rigid pieces: the modal analysis of a rigidly moving piece is retained over time, and the modal analysis of the assembly is obtained via Component Mode Synthesis (CMS). We illustrate our approach through a series of examples and a perceptual user study, demonstrating the utility of the sound synthesis method in producing realistic sounds at practical computation times.

Many high-level geometry processing tasks rely on low-level constructive solid geometry operations. Though trivial for implicit representations, boolean operations are notoriously difficult to execute robustly for explicit boundary representations. Existing methods for 3D triangle meshes fall short in one way or another. Some methods are fast but fail to produce closed, self-intersection free output. Other methods are robust but place prohibitively strict assumptions on the input, e.g., no hollow cavities, non-manifold edges or self-intersections. We propose a systematic recipe for conducting a family of exact constructive solid geometry operations. The two-stage method makes no general position assumptions and does not resort to numerical perturbation. The method is variadic , operating on any number of input meshes. This generalizes unary mesh-repair operations, classic binary boolean differencing, and n -ary operations such as finding all regions inside at least k out of n inputs. We demonstrate the superior effectiveness and robustness of our method on a dataset of 10,000 "real-world" meshes from a popular online repository. To encourage development, validation, and comparison, we release both our code and dataset to the public.

A reconfigurable is an object or collection of objects whose transformation between various states defines its functionality or aesthetic appeal. For example, consider a mechanical assembly composed of interlocking pieces, a transforming folding bicycle, or a space-saving arrangement of apartment furniture. Unlike traditional computer-aided design of static objects, specialized tools are required to address problems unique to the computational design and revision of objects undergoing rigid transformations. Collisions and interpenetrations as objects transition from one configuration to another prevent the physical realization of a design. We present a software environment intended to support fluid interactive design of reconfigurables, featuring tools that identify, visualize, monitor and resolve infeasible configurations. We demonstrate the versatility of the environment on a number of examples spanning mechanical systems, urban dwelling, and interlocking puzzles, some of which we then realize via additive manufacturing. Spatial-temporal information about collisions between objects is presented to the designer according to a cascading order of precedence. A designer may quickly determine when, and then where, and then how objects are colliding. This precedence guides the design and implementation of our four-dimensional spacetime bounding volume hierarchy for interactive-rate collision detection. On screen, the designer experiences a suite of interactive visualization and monitoring tools during editing: timeline notifications of new collisions, picture-in-picture windows for tracking collisions and suggestive hints for contact resolution. Contacts too tedious to remove manually can be eliminated automatically via our proposed constrained numerical optimization and swept-volume carving.

Mosaics are widely used for surface decoration to produce appealing visual effects. We present a method for synthesizing digital surface mosaics with irregularly shaped tiles, which are a type of tiles often used for mosaics design. Our method employs both continuous optimization and combinatorial optimization to improve tile arrangement. In the continuous optimization step, we iteratively partition the base surface into approximate Voronoi regions of the tiles and optimize the positions and orientations of the tiles to achieve a tight fit. Combination optimization performs tile permutation and replacement to further increase surface coverage and diversify tile selection. The alternative applications of these two optimization steps lead to rich combination of tiles and high surface coverage. We demonstrate the effectiveness of our solution with extensive experiments and comparisons.

Deformable objects such as garments are highly unstructured, making them difficult to recognize and manipulate. In this paper, we propose a novel method to teach a two-arm robot to efficiently track the states of a garment from an unknown state to a known state by iterative regrasping. The problem is formulated as a constrained weighted evaluation metric for evaluating the two desired grasping points during regrasping, which can also be used for a convergence criterion The result is then adopted as an estimation to initialize a regrasping, which is then considered as a new state for evaluation. The process stops when the predicted thin shell conclusively agrees with reconstruction. We show experimental results for regrasping a number of different garments including sweater, knitwear, pants, and leggings, etc.

Robotic manipulation of deformable objects remains a challenging task. One such task is folding a garment autonomously. Given start and end folding positions, what is an optimal trajectory to move the robotic arm to fold a garment? Certain trajectories will cause the garment to move, creating wrinkles, and gaps, other trajectories will fail altogether. We present a novel solution to find an optimal trajectory that avoids such problematic scenarios. The trajectory is optimized by minimizing a quadratic objective function in an off-line simulator, which includes material properties of the garment and frictional force on the table. The function measures the dissimilarity between a user folded shape and the folded garment in simulation, which is then used as an error measurement to create an optimal trajectory. We demonstrate that our two-arm robot can follow the optimized trajectories, achieving accurate and efficient manipulations of deformable objects.

Simulating the delightful dynamics of soap films, bubbles, and foams has traditionally required the use of a fully three-dimensional many-phase Navier-Stokes solver, even though their visual appearance is completely dominated by the thin liquid surface. We depart from earlier work on soap bubbles and foams by noting that their dynamics are naturally described by a Lagrangian vortex sheet model in whichcirculationis the primary variable. This leads us to derive a novel circulation-preserving surface-only discretization of foam dynamics driven by surface tension on a non-manifold triangle mesh. We represent the surface using a mesh-based multimaterial surface tracker which supports complex bubble topology changes, and evolve the surface according to the ambient air flow induced by a scalar circulation field stored on the mesh. Surface tension forces give rise to a simple update rule for circulation, even at non-manifold Plateau borders, based on a discrete measure of signed scalar mean curvature. We further incorporate vertex constraints to enable the interaction of soap films with wires. The result is a method that is at once simple, robust, and efficient, yet able to capture an array of soap films behaviors including foam rearrangement, catenoid collapse, blowing bubbles, and double bubbles being pulled apart.

We present an interactive tool for designing physical surfaces made from flexible interlocking quadrilateral elements of a single size and shape. With the element shape fixed, the design task becomes one of finding a discrete structure---i.e., element connectivity and binary orientations---that leads to a desired geometry. In order to address this challenging problem of combinatorial geometry, we propose a forward modeling tool that allows the user to interactively explore the space of feasible designs. Paralleling principles from conventional modeling software, our approach leverages a library of base shapes that can be instantiated, combined, and extended using two fundamental operations: merging and extrusion. In order to assist the user in building the designs, we furthermore propose a method to automatically generate assembly instructions. We demonstrate the versatility of our method by creating a diverse set of digital and physical examples that can serve as personalized lamps or decorative items.

We consider the simulation of dense foams composed of microscopic bubbles, such as shaving cream and whipped cream. We represent foam not as a collection of discrete bubbles, but instead as a continuum. We employ thematerial point method(MPM) to discretize a hyperelastic constitutive relation augmented with the Herschel-Bulkley model of non-Newtonian viscoplastic flow, which is known to closely approximate foam behavior. Since large shearing flows in foam can produce poor distributions of material points, a typical MPM implementation can produce non-physical internal holes in the continuum. To address these artifacts, we introduce a particle resampling method for MPM. In addition, we introduce an explicit tearing model to prevent regions from shearing into artificially thin, honey-like threads. We evaluate our method's efficacy by simulating a number of dense foams, and we validate our method by comparing to real-world footage of foam.

Abstract We present a user‐guided, semi‐automatic approach to completing large holes in a mesh. The reconstruction of the missing features in such holes is usually ambiguous. Thus, unsupervised methods may produce unsatisfactory results. To overcome this problem, we let the user indicate constraints by providing merely four points per important feature curve on the mesh. Our algorithm regards this input as an indication of an important broken feature curve. Our completion is formulated as a global energy minimization problem, with user‐defined spatial‐coherence constraints, allows for completion that adheres to the existing features. We demonstrate the method on example problems that are not handled satisfactorily by fully automatic methods.

We propose an interactive, optimization-in-the-loop tool for designing inflatable structures. Given a target shape, the user draws a network of seams defining desired segment boundaries in 3D. Our method computes optimally-shaped flat panels for the segments, such that the inflated structure is as close as possible to the target while satisfying the desired seam positions. Our approach is underpinned by physics-based pattern optimization, accurate coarse-scale simulation using tension field theory, and a specialized constraint-optimization method. Our system is fast enough to warrant interactive exploration of different seam layouts, including internal connections, and their effects on the inflated shape. We demonstrate the resulting design process on a varied set of simulation examples, some of which we have fabricated, demonstrating excellent agreement with the design intent.

We present a design system for linkage-based characters, combining form and function in an aesthetically-pleasing manner. Linkage-based character design exhibits a mix of discrete and continuous problems, making for a highly unintuitive design space that is difficult to navigate without assistance. Our system significantly simplifies this task by allowing users to interactively browse different topology options, thus guiding the discrete set of choices that need to be made. A subsequent continuous optimization step improves motion quality and, crucially, safeguards against singularities. We demonstrate the flexibility of our method on a diverse set of character designs, and then realize our designs by physically fabricating prototypes.

We present a computational approach for designing wire meshes , i.e., freeform surfaces composed of woven wires arranged in a regular grid. To facilitate shape exploration, we map material properties of wire meshes to the geometric model of Chebyshev nets . This abstraction is exploited to build an efficient optimization scheme. While the theory of Chebyshev nets suggests a highly constrained design space, we show that allowing controlled deviations from the underlying surface provides a rich shape space for design exploration. Our algorithm balances globally coupled material constraints with aesthetic and geometric design objectives that can be specified by the user in an interactive design session. In addition to sculptural art, wire meshes represent an innovative medium for industrial applications including composite materials and architectural façades. We demonstrate the effectiveness of our approach using a variety of digital and physical prototypes with a level of shape complexity unobtainable using previous methods.

We present a triangle mesh-based technique for tracking the evolution of three-dimensionalmultimaterial interfacesundergoing complex deformations. It is the first non-manifold triangle mesh tracking method to simultaneously maintain intersection-free meshes and support the proposed broad set of multimaterial remeshing and topological operations. We represent the interface as a non-manifold triangle mesh with material labels assigned to each half-face to distinguish volumetric regions. Starting from proposed application-dependent vertex velocities, we deform the mesh, seeking a non-intersecting, watertight solution. This goal necessitates development of various collision-safe, label-aware non-manifold mesh operations: multimaterial mesh improvement; T1 and T2 processes, topological transitions arising in foam dynamics and multiphase flows; and multimaterial merging, in which a new interface is created between colliding materials. We demonstrate the robustness and effectiveness of our approach on a range of scenarios including geometric flows and multiphase fluid animation.

We introduce an algorithm to synthesize missing geometry for a given triangle mesh that has “holes.” Similarly to previous work, the algorithm is context based in that it fills the hole by synthesizing geometry that is similar to the remainder of the input mesh. Our algorithm goes further to impose a coherence objective. A synthesis is coherent if every local neighborhood of the filled hole is similar to some local neighborhood of the input mesh. This requirement avoids undesired features such as can occur in context-based completion. We demonstrate the algorithm's ability to fill holes that were difficult or impossible to fill in a compelling manner by earlier approaches.

We develop an algorithm for the efficient and stable simulation of large-scale elastic rod assemblies. We observe that the time-integration step is severely restricted by a strong nonlinearity in the response of stretching modes to transversal impact, the degree of this nonlinearity varying greatly with the shape of the rod. Building on these observations, we propose a collision response algorithm that adapts its degree of nonlinearity. We illustrate the advantages of the resulting algorithm by analyzing simulations involving elastic rod assemblies of varying density and scale, with up to 1.7 million individual contacts per time step.

We propose a new method named compressive structured light for recovering inhomogeneous participating media. Whereas conventional structured light methods emit coded light patterns onto the surface of an opaque object to establish correspondence for triangulation, compressive structured light projects patterns into a volume of participating medium to produce images which are integral measurements of the volume density along the line of sight. For a typical participating medium encountered in the real world, the integral nature of the acquired images enables the use of compressive sensing techniques that can recover the entire volume density from only a few measurements. This makes the acquisition process more efficient and enables reconstruction of dynamic volumetric phenomena. Moreover, our method requires the projection of multiplexed coded illumination, which has the added advantage of increasing the signal-to-noise ratio of the acquisition. Finally, we propose an iterative algorithm to correct for the attenuation of the participating medium during the reconstruction process. We show the effectiveness of our method with simulations as well as experiments on the volumetric recovery of multiple translucent layers, 3D point clouds etched in glass, and the dynamic process of milk drops dissolving in water.

We present techniques for automatically parsing existing sewing patterns and converting them into 3D garment models. Our parser takes a sewing pattern in PDF format as input and starts by extracting the set of panels and styling elements (e.g. darts, pleats and hemlines) contained in the pattern. It then applies a combination of machine learning and integer programming to infer how the panels must be stitched together to form the garment. Our system includes an interactive garment simulator that takes the parsed result and generates the corresponding 3D model. Our fully automatic approach correctly parses 68% of the sewing patterns in our collection. Most of the remaining patterns contain only a few errors that can be quickly corrected within the garment simulator. Finally we present two applications that take advantage of our collection of parsed sewing patterns. Our garment hybrids application lets users smoothly interpolate multiple garments in the 2D space of patterns. Our sketch-based search application allows users to navigate the pattern collection by drawing the shape of panels.

Abstract We introduce a discrete paradigm for developable surface modeling. Unlike previous attempts at interactive developable surface modeling, our system is able to enforce exact developability at every step, ensuring that users do not inadvertently suggest configurations that leave the manifold of admissible folds of a flat two‐dimensional sheet. With methods for navigation of this highly nonlinear constraint space in place, we show how to formulate a discrete mean curvature bending energy measuring how far a given discrete developable surface is from being flat. This energy enables relaxation of user‐generated configurations and suggests a straightforward subdivision scheme that produces admissible smoothed versions of bent regions of our discrete developable surfaces.

We extend the Asynchronous Contact Mechanics algorithm [Harmon et al. 2009] and improve its performance by two orders of magnitude, using only optimizations that do not compromise ACM's three guarantees of safety, progress, and correctness. The key to this speedup is replacing ACM's timid, forward-looking mechanism for detecting collisions---locating and rescheduling separating plane kinetic data structures---with an optimistic speculative method inspired by Mirtich's rigid body Time Warp algorithm [2000]. Time warp allows us to perform collision detection over a window of time containing many of ACM's asynchronous trajectory changes; in this way we cull away large intervals as being collision free. Moreover, by replacing force processing intermingled with KDS rescheduling by windows of pure processing followed by collision detection, we transform an algorithm that is very difficult to parallelize into one that is embarrassingly parallel.

We present an interactive animation editor for complex deformable object animations. Given an existing animation, the artist directly manipulates the deformable body at any time frame, and the surrounding animation immediately adjusts in response. The automatic adjustments are designed to respect physics, preserve detail in both the input motion and geometry, respect prescribed bilateral contact constraints, and controllably and smoothly decay in space-time. While the utility of interactive editing for rigid body and articulated figure animations is widely recognized, a corresponding approach to deformable bodies has not been technically feasible before. We achieve interactive rates by combining spacetime model reduction, rotation-strain coordinate warping, linearized elasticity, and direct manipulation. This direct editing tool can serve the final stages of animation production, which often call for detailed, direct adjustments that are otherwise tedious to realize by re-simulation or frame-by-frame editing.

We present the first reduced-dimensional technique to simulate the dynamics of thin sheets of viscous incompressible liquid in three dimensions. Beginning from a discrete Lagrangian model for elastic thin shells, we apply the Stokes-Rayleigh analogy to derive a simple yet consistent model for viscous forces. We incorporate nonlinear surface tension forces with a formulation based on minimizing discrete surface area, and preserve the quality of triangular mesh elements through local remeshing operations. Simultaneously, we track and evolve the thickness of each triangle to exactly conserve liquid volume. This approach enables the simulation of extremely thin sheets of viscous liquids, which are difficult to animate with existing volumetric approaches. We demonstrate our method with examples of several characteristic viscous sheet behaviors, including stretching, buckling, sagging, and wrinkling.

Resolving simultaneous impacts is an open and significant problem in collision response modeling. Existing algorithms in this domain fail to fulfill at least one of five physical desiderata. To address this we present a simple generalized impact model motivated by both the successes and pitfalls of two popular approaches: pair-wise propagation and linear complementarity models. Our algorithm is the first to satisfy all identified desiderata, including simultaneously guaranteeing symmetry preservation, kinetic energy conservation, and allowing break-away. Furthermore, we address the associated problem of inelastic collapse, proposing a complementary generalized restitution model that eliminates this source of nontermination. We then consider the application of our models to the synchronous time-integration of large-scale assemblies of impacting rigid bodies. To enable such simulations we formulate a consistent frictional impact model that continues to satisfy the desiderata. Finally, we validate our proposed algorithm by correctly capturing the observed characteristics of physical experiments including the phenomenon of extended patterns in vertically oscillated granular materials.

We propose an example-based approach for simulating complex elastic material behavior. Supplied with a few poses that characterize a given object, our system starts by constructing a space of prefered deformations by means of interpolation. During simulation, this example manifold then acts as an additional elastic attractor that guides the object towards its space of prefered shapes. Added on top of existing solid simulation codes, this example potential effectively allows us to implement inhomogeneous and anisotropic materials in a direct and intuitive way. Due to its example-based interface, our method promotes an art-directed approach to solid simulation, which we exemplify on a set of practical examples.

We present a novel interactive tool for garment design that enables, for the first time, interactive bidirectional editing between 2D patterns and 3D high-fidelity simulated draped forms. This provides a continuous, interactive, and natural design modality in which 2D and 3D representations are simultaneously visible and seamlessly maintain correspondence. Artists can now interactively edit 2D pattern designs and immediately obtain stable accurate feedback online, thus enabling rapid prototyping and an intuitive understanding of complex drape form.

We present a continuum-based discrete model for thin threads of viscous fluid by drawing upon the Rayleigh analogy to elastic rods, demonstrating canonical coiling, folding, and breakup in dynamic simulations. Our derivation emphasizes space-time symmetry, which sheds light on the role of time-parallel transport in eliminating---without approximation---all but an O ( n ) band of entries of the physical system's energy Hessian. The result is a fast, unified, implicit treatment of viscous threads and elastic rods that closely reproduces a variety of fascinating physical phenomena, including hysteretic transitions between coiling regimes, competition between surface tension and gravity, and the first numerical fluid-mechanical sewing machine. The novel implicit treatment also yields an order of magnitude speedup in our elastic rod dynamics.

We develop an accurate, unified treatment of elastica. Following the method of resultant-based formulation to its logical extreme, we derive a higher-order integration rule, or elaston , measuring stretching, shearing, bending, and twisting along any axis. The theory and accompanying implementation do not distinguish between forms of different dimension (solids, shells, rods), nor between manifold regions and non-manifold junctions. Consequently, a single code accurately models a diverse range of elastoplastic behaviors, including buckling, writhing, cutting and merging. Emphasis on convergence to the continuum sets us apart from early unification efforts.

Example-based texture synthesis algorithms generate novel texture images from example data. A popular hierarchical pixel-based approach uses spatial jitter to introduce diversity, at the risk of breaking coarse structure beyond repair. We propose a multiscale descriptor that enables appearance-space jitter, which retains structure. This idea enables repurposing of existing texture synthesis implementations for a qualitatively different problem statement and class of inputs: generating hybrids of structured images.

Transferring existing mesh deformation from one character to another is a simple way to accelerate the laborious process of mesh animation. In many cases, it is useful to preserve the semantic characteristics of the motion instead of its literal deformation. For example, when applying the walking motion of a human to a flamingo, the knees should bend in the opposite direction. Semantic deformation transfer accomplishes this task with a shape space that enables interpolation and projection with standard linear algebra. Given several example mesh pairs, semantic deformation transfer infers a correspondence between the shape spaces of the two characters. This enables automatic transfer of new poses and animations.

We develop a method for reliable simulation of elastica in complex contact scenarios. Our focus is on firmly establishing three parameter-independent guarantees: that simulations of well-posed problems (a) have no interpenetrations, (b) obey causality, momentum- and energy-conservation laws, and (c) complete in finite time. We achieve these guarantees through a novel synthesis of asynchronous variational integrators, kinetic data structures, and a discretization of the contact barrier potential by an infinite sum of nested quadratic potentials. In a series of two- and three-dimensional examples, we illustrate that this method more easily handles challenging problems involving complex contact geometries, sharp features, and sliding during extremely tight contact.

We present a method for simulating highly detailed cutting and fracturing of thin shells using low-resolution simulation meshes. Instead of refining or remeshing the underlying simulation domain to resolve complex cut paths, we adapt the extended finite element method (XFEM) and enrich our approximation by customdesigned basis functions, while keeping the simulation mesh unchanged. The enrichment functions are stored in enrichment textures , which allows for fracture and cutting discontinuities at a resolution much finer than the underlying mesh, similar to image textures for increased visual resolution. Furthermore, we propose harmonic enrichment functions to handle multiple, intersecting, arbitrarily shaped, progressive cuts per element in a simple and unified framework. Our underlying shell simulation is based on discontinuous Galerkin (DG) FEM, which relaxes the restrictive requirement of C 1 continuous basis functions and thus allows for simpler, C 0 continuous XFEM enrichment functions.

The nascent field of discrete differential geometry deals with discrete geometric objects (such as polygons) which act as analogues to continuous geometric objects (such as curves). The discrete objects can be measured (length, area) and can interact with other discrete objects (collision/response). From a computational standpoint, the discrete objects are attractive, because they have been designed from the ground up with data-structures and algorithms in mind. From a mathematical standpoint, they present a great challenge: the discrete objects should have properties which are analogues of the properties of continuous objects. One important property of curves and surfaces is their curvature, which plays a significant role in many application areas (see, e.g., Chapters 4 and 5). In the continuous domain there are remarkable theorems dealing with curvature; a key requirement for a discrete curve with discrete curvature is that it satisfies analogous theorems. In this chapter we examine the curvature of continuous and discrete curves on the plane.

We present a discrete treatment of adapted framed curves, parallel transport, and holonomy, thus establishing the language for a discrete geometric model of thin flexible rods with arbitrary cross section and undeformed configuration. Our approach differs from existing simulation techniques in the graphics and mechanics literature both in the kinematic description---we represent the material frame by its angular deviation from the natural Bishop frame---as well as in the dynamical treatment---we treat the centerline as dynamic and the material frame as quasistatic. Additionally, we describe a manifold projection method for coupling rods to rigid-bodies and simultaneously enforcing rod inextensibility. The use of quasistatics and constraints provides an efficient treatment for stiff twisting and stretching modes; at the same time, we retain the dynamic bending of the centerline and accurately reproduce the coupling between bending and twisting modes. We validate the discrete rod model via quantitative buckling, stability, and coupled-mode experiments, and via qualitative knot-tying comparisons.

Example-based texture synthesis algorithms have gained widespread popularity for their ability to take a single input image and create a perceptually similar non-periodic texture. However, previous methods rely on single input exemplars that can capture only a limited band of spatial scales. For example, synthesizing a continent-like appearance at a variety of zoom levels would require an impractically high input resolution. In this paper, we develop a multiscale texture synthesis algorithm. We propose a novel example-based representation, which we call an exemplar graph, that simply requires a few low-resolution input exemplars at different scales. Moreover, by allowing loops in the graph, we can create infinite zooms and infinitely detailed textures that are impossible with current example-based methods. We also introduce a technique that ameliorates inconsistencies in the user's input, and show that the application of this method yields improved interscale coherence and higher visual quality. We demonstrate optimizations for both CPU and GPU implementations of our method, and use them to produce animations with zooming and panning at multiple scales, as well as static gigapixel-sized images with features spanning many spatial scales.

Robust treatment of complex collisions is a challenging problem in cloth simulation. Some state of the art methods resolve collisions iteratively, invoking a fail-safe when a bound on iteration count is exceeded. The best-known fail-safe rigidifies the contact region, causing simulation artifacts. We present a fail-safe that cancels impact but not sliding motion, considerably reducing artificial dissipation. We equip the proposed fail-safe with an approximation of Coulomb friction, allowing finer control of sliding dissipation.

This paper describes the ongoing implementation of the SimX system for multi-experiment computational studies within the SCIRun problem solving environment. The modular, component-based nature of SCIRun enables a natural integration of the SimX runtime modules with the simulation codes that constitute the experiments underlying the study, and provides a rich steering and visualization environment for study interactions. Experience with a computational study involving a SCIRun defibrillator device simulation code (DefibSim) highlights these advantages, and identifies several avenues for future work.

We combine the often opposing forces of artistic freedom and mathematical determinism to enrich a given animation or simulation of a surface with physically based detail. We present a process called tracking , which takes as input a rough animation or simulation and enhances it with physically simulated detail. Building on the foundation of constrained Lagrangian mechanics, we propose weak-form constraints for tracking the input motion. This method allows the artist to choose where to add details such as characteristic wrinkles and folds of various thin shell materials and dynamical effects of physical forces. We demonstrate multiple applications ranging from enhancing an artist's animated character to guiding a simulated inanimate object.

Many textiles do not noticeably stretch under their own weight. Unfortunately, for better performance many cloth solvers disregard this fact. We propose a method to obtain very low strain along the warp and weft direction using Constrained Lagrangian Mechanics and a novel fast projection method. The resulting algorithm acts as a velocity filter that easily integrates into existing simulation code.

Filtering is critical for representing detail, such as color textures or normal maps, across a variety of scales. While MIP-mapping texture maps is commonplace, accurate normal map filtering remains a challenging problem because of nonlinearities in shading---we cannot simply average nearby surface normals. In this paper, we show analytically that normal map filtering can be formalized as a spherical convolution of the normal distribution function (NDF) and the BRDF, for a large class of common BRDFs such as Lambertian, microfacet and factored measurements. This theoretical result explains many previous filtering techniques as special cases, and leads to a generalization to a broader class of measured and analytic BRDFs. Our practical algorithms leverage a significant body of work that has studied lighting-BRDF convolution. We show how spherical harmonics can be used to filter the NDF for Lambertian and low-frequency specular BRDFs, while spherical von Mises-Fisher distributions can be used for high-frequency materials.

Advances in high-performance computing have led to the broad use of computational studies in everyday engineering and scientific applications. A single study may require thousands of computational experiments, each corresponding to individual runs of simulation software with different parameter settings; in complex studies, the pattern of parameter changes is complex and may have to be adjusted by the user based on partial simulation results. Unfortunately, existing tools have limited high-level support for managing large ensembles of simultaneous computational experiments. In this paper, we present a system architecture for interactive computational studies targeting two goals. The first is to provide a framework for high-level user interaction with computational studies, rather than individual experiments; the second is to maximize the size of the studies that can be performed at close to interactive rates. We describe a prototype implementation of the system and demonstrate performance improvements obtained using our approach for a simple model problem.

Abstract Discrete curvature and shape operators, which capture complete information about directional curvatures at a point, are essential in a variety of applications: simulation of deformable two‐dimensional objects, variational modeling and geometric data processing. In many of these applications, objects are represented by meshes. Currently, a spectrum of approaches for formulating curvature operators for meshes exists, ranging from highly accurate but computationally expensive methods used in engineering applications to efficient but less accurate techniques popular in simulation for computer graphics. We propose a simple and efficient formulation for the shape operator for variational problems on general meshes, using degrees of freedom associated with normals. On the one hand, it is similar in its simplicity to some of the discrete curvature operators commonly used in graphics; on the other hand, it passes a number of important convergence tests and produces consistent results for different types of meshes and mesh refinement.

Many computer graphics applications require high-intensity numerical simulation. We show that such computations can be performed efficiently on the GPU, which we regard as a full function streaming processor with high floating-point performance. We implemented two basic, broadly useful, computational kernels: a sparse matrix conjugate gradient solver and a regular-grid multigrid solver . Real time applications ranging from mesh smoothing and parameterization to fluid solvers and solid mechanics can greatly benefit from these, evidence our example applications of geometric flow and fluid simulation running on NVIDIA's GeForce FX.

Finite element solvers are a basic component of simulation applications; they are common in computer graphics, engineering, and medical simulations. Although adaptive solvers can be of great value in reducing the often high computational cost of simulations they are not employed broadly. Indeed, building adaptive solvers can be a daunting task especially for 3D finite elements. In this paper we are introducing a new approach to produce conforming, hierarchical, adaptive refinement methods (CHARMS). The basic principle of our approach is to refine basis functions, not elements. This removes a number of implementation headaches associated with other approaches and is a general technique independent of domain dimension (here 2D and 3D), element type (e.g., triangle, quad, tetrahedron, hexahedron), and basis function order (piece-wise linear, higher order B-splines, Loop subdivision, etc.). The (un-)refinement algorithms are simple and require little in terms of data structure support. We demonstrate the versatility of our new approach through 2D and 3D examples, including medical applications and thin-shell animations.

Subdivision surfaces are an attractive representation when modeling arbitrary-topology free-form surfaces and show great promise for applications in engineering design and computer animation. Interference detection is a critical tool in many of these applications. In this paper, we derive normal bounds for subdivision surfaces and use these to develop an efficient algorithm for (self-) interference detection.