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We propose a numerical method for maintaining a dynamic rolling motion of animated gaseous phenomena, such as smoke, that avoids dissipation due to numerical error. We compensate for the errors induced by a semi-Lagrangian scheme using an error estimate for each time interval. We develop a new advection term and perform vortex advection based on a vorticity confinement force. Example simulations show that this method is able to keep smoke features alive, even near the center of a vortex. Copyright # 2005 John Wiley & Sons, Ltd.

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This chapter introduces the fundamental theory for understanding the entire contents of this book. A method to solve fluid equation for fire, water, and smoke simulations is explained. And also, background knowledge and research trends for modeling the natural environment such as ice formation are described. In addition to fluid dynamics, a coupling problem for solid–fluid collision detection and various methods of controlling fluid animation are dealt with. This chapter is organized as follows: underlying concepts for simulations of water, smoke, fire and ice are presented in Sects. 1.1–1.3. Collision response of fluid and target-driven fluid animation are discussed in Sect. 1.4.

This section proposes a numerical method for maintaining a dynamic rolling motion of animated gaseous phenomena, such as smoke, that avoids dissipation due to numerical error. It compensates for the errors induced by a semi-Lagrangian scheme using an error estimate for each time interval. And it develops a new advection term and perform vortex advection based on a vorticity confinement force. Example simulations show that this method is able to keep smoke features alive, even near the center of a vortex.

In the field of computer graphics, Navier-Stokes equations would be used for realistic simulations of smokes and currents. However, implementations derived from these equations are hard to achieve for real-time simulations, mainly due to its massive and complex calculations. Thus, there have been various attempts to approximate these equations for real-time simulation of smokes and others. When the advection terms of the equations are approximated by the Semi-Lagrange methods, the fluid density can be rapidly reduced and small-scale vorticity phenomena are easy to be missed, mainly due to the numerical losses over time. In this paper, we propose an improved numerical method to approximately calculate the advection terms, and thus eliminate these problems. To calculate the advection terms, our method starts to set critical regions around the target grid points. Then, among the grid points in a specific critical region, we search for a grid point which will be advected to the target grid point, and use the velocity of this grid point as its advection vector. This method would reduce the numerical losses in the calculation of densities and vorticity phenomena, and finally can implement more realistic smoke simulations. We also improve the overall efficiency of vector calculations and related operations through GPU-based implementation techniques, and thus finally achieve the real-time simulation.

Abstract We propose a fast and effective technique to improve sub-grid visual details of the grid based fluid simulation. Our method procedurally synthesizes the flow fields coming from the incompressible Navier-Stokes solver and the vorticity fields generated by vortex particle method for sub-grid turbulence. We are able to efficiently animate smoke which is highly turbulent and swirling with small scale details. Since this technique does not solve the linear system in high-resolution grids, it can perform fluid simulation more rapidly. We can easily estimate the influence of turbulent and swirling effect to the fluid flow.

We present a comprehensive methodology for realistically animating liquid phenomena. Our approach unifies existing computer graphics techniques for simulating fluids and extends them by incorporating more complex behavior. It is based on the Navier–Stokes equations which couple momentum and mass conservation to completely describe fluid motion. Our starting point is an environment containing an arbitrary distribution of fluid, and submerged or semisubmerged obstacles. Velocity and pressure are defined everywhere within this environment and updated using a set of finite difference expressions. The resulting vector and scalar fields are used to drive a height field equation representing the liquid surface. The nature of the coupling between obstacles in the environment and free variables allows for the simulation of a wide range of effects that were not possible with previous computer graphics fluid models. Wave effects such as reflection, refraction, and diffraction, as well as rotational effects such as eddies, vorticity, and splashing are a natural consequence of solving the system. In addition, the Lagrange equations of motion are used to place buoyant dynamic objects into a scene and track the position of spray and foam during the animation process. Typical disadvantages to dynamic simulations such as poor scalability and lack of control are addressed by assuming that stationary obstacles align with grid cells during the finite difference discretization, and by appending terms to the Navier–Stokes equations to include forcing functions. Free surfaces in our system are represented as either a collection of massless particles in 2D, or a height field which is suitable for many of the water rendering algorithms presented by researchers in recent years.

Back and Forth Error Compensation and Correction (BFECC) was recently developed for interface computation by using the level set method. We show that it can be applied to reduce dissipation and diffusion encountered in various advection steps in fluid simulation such as velocity, smoke density and image advections. BFECC can be implemented easily on top of the first order upwinding or semi-Lagrangian integration of advection equations, while providing second order accuracy both in space and time. When applied to level set evolution, BFECC reduces volume loss significantly. We combine these techniques with variable density projection and show that they yield a realistic animations of two-phase flows. We demonstrate the benefits of this approach on the image advection and on the simulation of smoke, of bubbles in water, and of a highly dynamic interaction between water, a solid, and air.

We present a comprehensive methodology for realistically animating liquid phenomena. Physically accurate 3D motion is achieved by performing a two-stage calculation over an arbitrary environment of static obstacles surrounded by fluid. A finite difference approximation to the NavierStokes equations is first applied to a low resolution, voxelized representation of the scene. The resulting velocity and pressure fields describe the gross transport of liquid, including effects such as splashing, vorticity and overturning. Local fluid velocity is then used to drive a height field equation or to convect massless marker particles. The position of any free surface can thus be determined to a significantly higher resolution than that of the Navier-Stokes calculation. In addition, the pressure field, together with the Lagrange equations of motion, is used to simulate dynamic buoyant objects. Typical disadvantages to volumetric methods such as poor scalability and lack of control are addressed by...

This paper describes a method for animating suspended particle explosions. Rather than modeling the numerically troublesome, and largely invisible blast wave, the method uses a relatively stable incompressible fluid model to account for the motion of air and hot gases. The fluid's divergence field is adjusted directly to account for detonations and the generation and expansion of gaseous combustion products. Particles immersed in the fluid track the motion of particulate fuel and soot as they are advected by the fluid. Combustion is modeled using a simple but effective process governed by the particle and fluid systems. The method has enough flexibility to also approximate sprays of burning liquids. This paper includes several demonstrative examples showing air bursts, explosions near obstacles, confined explosions, and burning sprays. Because the method is based on components that allow large time integration steps, it only requires a few seconds of computation per frame for the examples shown.

This article presents a physically-based technique for simulating water. This work is motivated by the "stable fluids" method, developed by Stam [1999], to handle gaseous fluids. We extend this technique to water, which calls for the development of methods for modeling multiphase fluids and suppressing dissipation. We construct a multiphase fluid formulation by combining the Navier--Stokes equations with the level set method. By adopting constrained interpolation profile (CIP)-based advection, we reduce the numerical dissipation and diffusion significantly. We further reduce the dissipation by converting potentially dissipative cells into droplets or bubbles that undergo Lagrangian motion. Due to the multiphase formulation, the proposed method properly simulates the interaction of water with surrounding air, instead of simulating water in a void space. Moreover, the introduction of the nondissipative technique means that, in contrast to previous methods, the simulated water does not unnecessarily lose mass, and its motion is not damped to an unphysical extent. Experiments showed that the proposed method is stable and runs fast. It is demonstrated that two-dimensional simulation runs in real-time.

Vorticity confinement reintroduces the small scale detail lost when using efficient semi-Lagrangian schemes for simulating smoke and fire. However, it only amplifies the existing vorticity, and thus can be insufficient for highly turbulent effects such as explosions or rough water. We introduce a new hybrid technique that makes synergistic use of Lagrangian vortex particle methods and Eulerian grid based methods to overcome the weaknesses of both. Our approach uses vorticity confinement itself to couple these two methods together. We demonstrate that this approach can generate highly turbulent effects unachievable by standard grid based methods, and show applications to smoke, water and explosion simulations.

At interfaces between different fluids, properties such as density, viscosity, and molecular cohesion are discontinuous. To animate small-scale details of incompressible viscous multi-phase fluids realistically, we focus on the discontinuities in the state variables that express these properties. Surface tension of both free and bubble surfaces is modeled using the jump condition in the pressure field; and discontinuities in the velocity gradient field. driven by viscosity differences, are also considered. To obtain derivatives of the pressure and velocity fields with sub-grid accuracy, they are extrapolated across interfaces using continuous variables based on physical properties. The numerical methods that we present are easy to implement and do not impact the performance of existing solvers. Small-scale fluid motions, such as capillary instability, breakup of liquid sheets, and bubbly water can all be successfully animated.

This paper addresses the problem of controlling the density and dynamics of smoke (a gas phenomenon) so that the synthetic appearance of the smoke (gas) resembles a still or moving object. Both the smoke region and the target object are represented as implicit functions. As a part of the target implicit function, a shape transformation is generated between an initial smoke region and the target object. In order to match the smoke surface with the target surface, we impose carefully designed velocity constraints on the smoke boundary during a dynamic fluid simulation. The velocity constraints are derived from an iterative functional minimization procedure for shape matching. The dynamics of the smoke is formulated using a novel compressible fluid model which can effectively absorb the discontinuities in the velocity field caused by imposed velocity constraints while reproducing realistic smoke appearances. As a result, a smoke region can evolve into a regular object and follow the motion of the object while maintaining its smoke appearance.

Surface parameterization is necessary for many graphics tasks: texture-preserving simplification, remeshing, surface painting, and precomputation of solid textures. The stretch caused by a given parameterization determines the sampling rate on the surface. In this article, we present an automatic parameterization method for segmenting a surface into patches that are then flattened with little stretch.
Many objects consist of regions of relatively simple shapes, each of which has a natural parameterization. Based on this observation, we describe a three-stage feature-based patch creation method for manifold surfaces. The first two stages, genus reduction and feature identification, are performed with the help of distance-based surface functions. In the last stage, we create one or two patches for each feature region based on a covariance matrix of the feature's surface points.
To reduce stretch during patch unfolding, we notice that stretch is a 2 × 2 tensor, which in ideal situations is the identity. Therefore, we use the Green-Lagrange tensor to measure and to guide the optimization process. Furthermore, we allow the boundary vertices of a patch to be optimized by adding scaffold triangles . We demonstrate our feature-based patch creation and patch unfolding methods for several textured models.
Finally, to evaluate the quality of a given parameterization, we describe an image-based error measure that takes into account stretch, seams, smoothness, packing efficiency, and surface visibility.

In this paper we present a new method for efficiently controlling animated smoke. Given a sequence of target smoke states, our method generates a smoke simulation in which the smoke is driven towards each of these targets in turn, while exhibiting natural-looking interesting smoke-like behavior. This control is made possible by two new terms that we add to the standard flow equations: (i) a driving force term that causes the fluid to carry the smoke towards a particular target, and (ii) a smoke gathering term that prevents the smoke from diffusing too much. These terms are explicitly defined by the instantaneous state of the system at each simulation timestep. Thus, no expensive optimization is required, allowing complex smoke animations to be generated with very little additional cost compared to ordinary flow simulations.

This paper describes a new animation technique for modeling the turbulent rotational motion that occurs when a hot gas interacts with solid objects and the surrounding medium. The method is especially useful for scenes involving swirling steam, rolling or billowing smoke, and gusting wind. It can also model gas motion due to fans and heat convection. The method combines specialized forms of the equations of motion of a hot gas with an efficient method for solving volumetric differential equations at low resolutions. Particular emphasis is given to issues of computational efficiency and ease-of-use of the method by an animator. We present the details of our model, together with examples illustrating its use. Keywords: Animation, Convection, Gaseous Phenomena, Gas Simulations, Physics-Based Modeling, Steam, Smoke, Turbulent Flow. 1 Introduction The turbulent motion of smoke and steam has always inspired interest amongst graphics researchers. The problem of modeling the complex inter-r...

Building animation tools for fluid-like motions is an important and challenging problem with many applications in computer graphics. The use of physics-based models for fluid flow can greatly assist in creating such tools. Physical models, unlike key frame or procedural based techniques, permit an animator to almost effortlessly create interesting, swirling fluid-like behaviors. Also, the interaction of flows with objects and virtual forces is handled elegantly. Until recently, it was believed that physical fluid models were too expensive to allow real-time interaction. This was largely due to the fact that previous models used unstable schemes to solve the physical equations governing a fluid. In this paper, for the first time, we propose an unconditionally stable model which still produces complex fluid-like flows. As well, our method is very easy to implement. The stability of our model allows us to take larger time steps and therefore achieve faster simulations. We have used our model in conjuction with advecting solid textures to create many fluid-like animations interactively in two- and three-dimensions.

In this paper, we propose a new approach to numerical smoke simulation for computer graphics applications. The method proposed here exploits physics unique to smoke in order to design a numerical method that is both fast and efficient on the relatively coarse grids traditionally used in computer graphics applications (as compared to the much finer grids used in the computational fluid dynamics literature). We use the inviscid Euler equations in our model, since they are usually more appropriate for gas modeling and less computationally intensive than the viscous NavierStokes equations used by others. In addition, we introduce a physically consistent vorticity confinement term to model the small scale rolling features characteristic of smoke that are absent on most coarse grid simulations. Our model also correctly handles the interaction of smoke with moving objects. Keywords: Smoke, computational fluid dynamics, Navier-Stokes equations, Euler equations, Semi-Lagrangian methods, stable fluids, vorticity confinement, participating media 1

Notation 1. Numerical simulation - a key technology of the future 2. The mathematical description of flows 3. The numerical treatment of the Navier-Stokes equations 4. Visualization techniques 5. Example applications 6. Free boundary value problems 7. Example applications for free boundary value problems 8. Parallelization 9. Energy transport 10. Turbulence 11. Extension to Three dimensions 12. Concluding remarks Appendix A Appendix B Bibliography Index.

We present numerical schemes for the incompressible Navier–Stokes equations based on a primitive variable for-mulation in which the incompressibility constraint has been replaced by a pressure Poisson equation. The pressure is treated explicitly in time, completely decoupling the computation of the momentum and kinematic equations. The result is a class of extremely efficient Navier–Stokes solvers. Full time accuracy is achieved for all flow variables. The key to the schemes is a Neumann boundary condition for the pressure Poisson equation which enforces the incompressibility condition for the velocity field. Irrespective of explicit or implicit time discretization of the viscous term in the mo-mentum equation the explicit time discretization of the pressure term does not affect the time step constraint. Indeed, we prove unconditional stability of the new formulation for the Stokes equation with explicit treatment of the pressure term and first or second order implicit treatment of the viscous term. Systematic numerical experiments for the full Navier–Stokes equations indicate that a second order implicit time discretization of the viscous term, with the pressure and convective terms treated explicitly, is stable under the standard CFL condition. Additionally, various numerical examples are presented, including both implicit and explicit time discretizations, using spectral and finite difference spatial discretizations, demonstrating the accuracy, flexibility and efficiency of this class of schemes. In particular, a Galerkin formulation is presented requiring only C 0 elements to implement.

High order methods are of great interest in the study of turbulent flows in complex geometries by means of direct simulation. With this goal in mind, the incompressible Navier–Stokes equations are discretized in space by a compact fourth order finite difference method on a staggered grid. The equations are integrated in time by a second order semi-implicit method. Stable boundary conditions are implemented and the grid is allowed to be curvilinear in two space dimensions. The method is extended to three dimensions by a Fourier expansion. In every time step, a system of linear equations is solved for the velocity and the pressure by an outer and an inner iteration with preconditioning. The convergence properties of the iterative method are analyzed. The order of accuracy of the method is demonstrated in numerical experiments. The method is used to compute the flow in a channel, the driven cavity and a constricted channel.

We propose a method that significantly improves the accuracy of the level set method and could be of value for numerical solutions of differential equations in general. Level set methods use a level set function, usually an approximate signed distance function, Φ, to represent the interface as the zero set of Φ. When Φ is advanced to the next time level by an advection equation, its new zero level set will represent the new interface position. But the non-zero curvature of the interface will result in uneven gradients of the level set function which induces extra numerical error. Instead of attempting to reduce this error directly, we update the level set function Φ forward in time and then backward to get another copy of the level set function, say Φ1. Φ1 and Φ should have been equal if there were no numerical error. Therefore Φ−Φ1 provides us the information of error induced by uneven gradients and this information can be used to compensate Φ before updating Φ forward again in time.

In this paper, we present a method for visual simulation of gaseous phenomena based on the vortex method. This method uses a localized vortex flow as a basic building block and combines those blocks to describe a whole flow field. As a result, we achieve computational efficiency by concentrating only on a localized vorticity region while generating dynamic swirling fluid flows. Based on the Lagrangian framework, we resolve various boundary conditions. By exploiting the panel method, we satisfy the no-through boundary condition in a Lagrangian way. A simple and effective way of handling the no-slip boundary condition is also presented. In treating the no-slip boundary condition, we allow a user to control the roughness of the boundary surface, which further improves visual realism.

We present a new fluid animation technique in which liquid and gas interact with each other, using the exampleof bubbles rising in water. In contrast to previous studies which only focused on one fluid, our system considersboth the liquid and the gas simultaneously. In addition to the flowing motion, the interactions between liquid andgas cause buoyancy, surface tension, deformation and movement of the bubbles. For the natural manipulationof topological changes and the removal of the numerical diffusion, we combine the volume-of-fluid method andthe front-tracking method developed in the field of computational fluid dynamics. Our minimum-stress surfacetension method enables this complementary combination. The interfaces are constructed using the marching cubesalgorithm. Optical effects are rendered using vertex shader techniques.
Categories and Subject Descriptors (according to ACM CCS): I.3.7 [Computer Graphics]: Animation

We propose a new fluid control technique that uses a geometrically induced potential field. Instead of optimizing the control forces exerted at each frame, as was done in previous work, a potential is added as an extra dimension to the simulation space which coerces the fluid inside this space to form the target shape. This type of shape control requires practically no additional computing by the Navier-Stokes solver at run-time, and adds little overhead to implementation. The confinement potentials are induced from geometric information given by animators, and so the control forces that take fluids to a lower potential can be decided in a preprocessing step. We show that a slightly generalized Navier-Stokes equation for fluids in potential fields can be simulated without changing the solver itself. A harmonic potential function can be quickly found with the Poisson solver which is already implemented as a part of the Navier-Stokes solver. 2 and 3 dimensional flows designed by common methods such as hand drawing, traditional shape modeling and key-framing, can be animated efficiently with our control technique. Copyright © 2004 John Wiley & Sons, Ltd.

We present a method for simulating water and smoke on an unrestricted octree data structure exploiting mesh refinement techniques to capture the small scale visual detail. We propose a new technique for discretizing the Poisson equation on this octree grid. The resulting linear system is symmetric positive definite enabling the use of fast solution methods such as preconditioned conjugate gradients, whereas the standard approximation to the Poisson equation on an octree grid results in a non-symmetric linear system which is more computationally challenging to invert. The semi-Lagrangian characteristic tracing technique is used to advect the velocity, smoke density, and even the level set making implementation on an octree straightforward. In the case of smoke, we have multiple refinement criteria including object boundaries, optical depth, and vorticity concentration. In the case of water, we refine near the interface as determined by the zero isocontour of the level set function.

The discovery of coherent structures in turbulence has fostered the hope
that the study of vortices will lead to models and an understanding of
turbulent flow, thereby solving or at least making less mysterious one
of the great unresolved problems of classical physics. Vortex dynamics
is a natural paradigm for the field of chaotic motion and modern
dynamical system theory. The emphasis in this monograph is on the
classical theory of inviscid incompressible fluids containing finite
regions of vorticity. The effects of viscosity, compressiblity,
inhomogeneity, and stratification are enormously important in many
fields of application, from hypersonic flight to global environmental
fluid mechanics. However, this volume focuses on those aspects of fluid
motion that are primarily controlled by the vorticity and are such that
the effects of the other fluid properties are secondary. This book will
be of interest to students of fluid mechanics, turbulence, and vortex
methods as well as to applied mathematicians and engineers.

We describe a method that permits the high performance simulation of fluid phenomena such as smoke, with high-level control for the artist. Our key primitives are vortex filament and vortex ring: vorticity defines a flow as well as velocity does, and for numerous interesting flows such as smoke or explosions this information is very compact and tightly linked to the visual features of the fluid. We treat these vortices as 1D Lagrangian primitives (i.e. connected particles), which permit unbounded fluids and very accurate positioning of the features. The simulation of passive density particles for rendering is totally independent of the fluid animation itself. Thus, the animation can be efficiently simulated, edited and even stored, while the fluid resolution used for rendering can be arbitrarily high. We aim at plausible fluids rather than physical accuracy. For efficiency and stability, we introduce a new formalization of the Biot-Savart law and a modified Biot-Savart kernel. Our model also introduces a hierarchical filament structure for animation LOD, turbulent noise, and an original scheme for density particles.

A collection is presented of black-and-white photographs illustrating
the variety of fluid phenomena which can be studied by means of flow
visualization techniques. The order of presentation is in general from
low speeds to high: creeping and laminar flow; boundary layer
separation; vortices, instability and turbulence; free surface flow and
natural convection; subsonic flow, shock waves, and supersonic flow
(including the hypersonic regime). Reynolds numbers are based on
diameter unless otherwise specified. The flow visualization techniques
employed include smoke filaments in air, bubbles in fluids, metal
particles in oils, gases having different optical characteristics such
as CO2 in air, differently colored fluids, and light sheet illumination.