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Active Implicit Surface for Animation
This paper introduces a new model of deformable surfaces designed for animation, which we call active implicit surfaces. The underlying idea is to animate a potential field defined by discrete values stored in a grid, rather than directly animating a surface. This surface, defined as an iso-potential of the field, has the ability to follow a given object using a snake-like strategy. Surface tension and other characteristics such as constant surface area or constant volume may be added to this model. The implicit formulation allows the surface to easily experience topology changes during simulation. We present an optimized implementation: computations are restricted to a close neighborhood around the surface. Applications range from the coating of deformable materials simulated by particle systems (the surface hides the granularity of the underlying model) to the generation of metamorphosis between shapes that may not have the same topology.
Particle-based Viscoelastic Fluid Simulation
We present a new particle-based method for viscoelastic fluid simulation. We achieve realistic small-scale behavior of substances such as paint or mud as they splash on moving objects. Incompressibility and particle anti-clustering are enforced with a double density relaxation procedure which updates particle positions according to two opposing pressure terms. From this process surface tension effects emerge, enabling drop and filament formation. Elastic and non-linear plastic effects are obtained by adding springs with varying rest length between particles. We also extend the technique to handle interaction between fluid and dynamic objects. Various simulation scenarios are presented including rain drops, fountains, clay manipulation, and floating objects. The method is robust and stable, and can animate splashing behavior at interactive framerates.
Design of Tangent Vector Fields
Tangent vector fields are an essential ingredient in controlling surface appearance for applications ranging from anisotropic shading to texture synthesis and non-photorealistic rendering. To achieve a desired effect one is typically interested in smoothly varying fields that satisfy a sparse set of user-provided constraints. Using tools from Discrete Exterior Calculus, we present a simple and efficient algorithm for designing such fields over arbitrary triangle meshes. By representing the field as scalars over mesh edges (i.e., discrete 1-forms), we obtain an intrinsic, coordinate-free formulation in which field smoothness is enforced through discrete Laplace operators. Unlike previous methods, such a formulation leads to a linear system whose sparsity permits efficient pre-factorization. Constraints are incorporated through weighted least squares and can be updated rapidly enough to enable interactive design, as we demonstrate in the context of anisotropic texture synthesis.
Solid Texture Synthesis: A Survey
It's common belief that textures can simply and efficiently model 3D objects by separating appearance properties from their geometric description. Computer graphics has profusely used textures to model objects' external structure, through either photographs or procedural models. Whereas traditional 2D textures usually encode information about an object's external surface, researchers have proposed extensions for providing volumetric information, allowing encoding of objects' internal appearance. That is, these extensions provide appearance properties for each point in a predefined volumetric domain V C R3. Such textures are usually called solid textures. This survey illustrates the different algorithms for synthesizing and representing these textures.Surface texturing usually relies on a planar parameterization for associating texture attributes to a 3D object. A planar parameterization maps each 3D point on an object's surface to a 2D domain, which encodes texture attributes. This 3D-to-2D mapping might introduce a distortion, which generally depends on the complexity of the object's topology and shape. Finding a good planar parameterization-one that minimizes this distortion remains a challenge.
Solid Texture Synthesis from 2D Exemplars
We present a novel method for synthesizing solid textures from 2D texture exemplars. First, we extend optimization-based 2D texture synthesis to synthesize 3D texture solids. Next, the non-parametric texture optimization approach is integrated with histogram matching, which forces the global statistics of the synthesized solid to match those of the exemplar. This improves the convergence of the synthesis process and enables using smaller neighborhoods. In addition to producing compelling texture mapped surfaces, our method also effectively models the material in the interior of solid objects. We also demonstrate that our method is well-suited for synthesizing textures with a large number of channels per texel.
Discrete geometric mechanics for variational time integrators
Geometry at its most abstract is the study of symmetries and their associated invariants. Variational approaches based on such notions are commonly used in geometric modeling and discrete differential geometry. Here we will treat mechanics in a similar way. Indeed, the very essence of a mechanical system is characterized by its symmetries and invariants. Thus preserving these symmetries and invariants (e.g., certain momenta) into the discrete computational setting is of paramount importance if one wants discrete time integration to properly capture the underlying continuous motion. Motivated by the well-known variational and geometric nature of most dynamical systems, we review the use of discrete variational principles as a way to derive robust, and accurate time integrators.
The Jacobi-Maupertuis Principle in Variational Integrators
In this paper, we develop a hybrid variational integrator based on the Jacobi-Maupertuis Principle of Least Action. The Jacobi-Maupertuis principle states that for a mechanical system with total energy E and potential energy V(q), the curve traced out by the system on a constant energy surface minimizes the action given by ??[2(E-V(q))] ds where ds is the line element on the constant energy surface with respect to the kinetic energy of the system. The key feature is that the principle is a parametrization independent geodesic problem. We show that this principle can be combined with traditional variational integrators and can be used to efficiently handle high velocity regions where small time steps would otherwise be required. This is done by switching between the Hamilton principle and the Jacobi-Maupertuis principle depending upon the kinetic energy of the system. We demonstrate our technique for the Kepler problem and discuss some ongoing and future work in studying the energy and momentum behavior of the resulting integrator.
Advances in Water Resources
We present an overview of the most common numerical solution strategies for the incompressible NavierÔÇôStokes equations, including fully implicit formulations, artificial compressibility methods, penalty formulations, and operator splitting methods (pressure/velocity correction, projection methods). A unified framework that explains popular operator splitting methods as special cases of a fully implicit approach is also presented and can be used for constructing new and improved solution strategies. The exposition is mostly neutral to the spatial discretization technique, but we cover the need for staggered grids or mixed finite elements and outline some alternative stabilization techniques that allow using standard grids. Emphasis is put on showing the close relationship between (seemingly) different and competing solution approaches for incompressible viscous flow.
Tall Cell Fluids
We present a new Eulerian fluid simulation method, which allows real-time simulations of large scale three dimensional liquids. Such scenarios have hitherto been restricted to the domain of off-line computation. To reduce computation time we use a hybrid grid representation composed of regular cubic cells on top of a layer of tall cells. With this layout water above an arbitrary terrain can be represented without consuming an excessive amount of memory and compute power, while focusing effort on the area near the surface where it most matters. Additionally, we optimized the grid representation for a GPU implementation of the fluid solver. To further accelerate the simulation, we introduce a specialized multi-grid algorithm for solving the Poisson equation and propose solver modifications to keep the simulation stable for large time steps. We demonstrate the efficiency of our approach in several real-world scenarios, all running above 30 frames per second on a modern GPU. Some scenes include additional features such as two-way rigid body coupling as well as particle representations of sub-grid detail.
A Multigrid Fluid Pressure Solver Handling Separating Solid Boundary Conditions
We present a multigrid method for solving the linear complementarity problem (LCP) resulting from discretizing the Poisson equation subject to separating solid boundary conditions in an Eulerian liquid simulation's pressure projection step. The method requires only a few small changes to a multigrid solver for linear systems. Our generalized solver is fast enough to handle 3D liquid simulations with separating boundary conditions in practical domain sizes. Previous methods could only handle relatively small 2D domains in reasonable time because they used expensive quadratic programming (QP) solvers. We demonstrate our technique in several practical scenarios in which the omission of separating boundary conditions results in disturbing artifacts of liquid sticking to walls.
In this paper we introduce a discrete shell model describing the behavior of thin flexible structures, such as hats, leaves, and aluminum cans, which are characterized by a curved undeformed configuration. Previously such models required complex continuum mechanics formulations and correspondingly complex algorithms. We show that a simple shell model can be derived geometrically for triangle meshes and implemented quickly by modifying a standard cloth simulator. Our technique convincingly simulates a variety of curved objects with materials ranging from paper to metal, as we demonstrate with several examples including a comparison of a real and simulated falling hat.
Discrete Elastic Rods
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.