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38
Stable Fluids
, 1999
"... Building animation tools for fluidlike motions is an important and challenging problem with many applications in computer graphics. The use of physicsbased models for fluid flow can greatly assist in creating such tools. Physical models, unlike key frame or procedural based techniques, permit an a ..."
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Cited by 418 (8 self)
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Building animation tools for fluidlike motions is an important and challenging problem with many applications in computer graphics. The use of physicsbased 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 fluidlike 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 realtime 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 fluidlike 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 fluidlike animations interactively in two and threedimensions.
Practical animation of liquids
 Graphical Models and Image Processing
, 1996
"... 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 NavierStokes equations which couple momentum and mass con ..."
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Cited by 343 (23 self)
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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 NavierStokes 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 computergraphics 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 NavierStokes 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.
Turbulent Wind Fields for Gaseous Phenomena
, 1993
"... The realistic depiction of smoke, steam, mist and water reacting to a turbulent field such as wind is an attractive and challenging problem. Its solution requires interlocking models for turbulent fields, gaseous flow, and realistic illumination. We present a model for turbulent wind flow having a d ..."
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Cited by 106 (9 self)
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The realistic depiction of smoke, steam, mist and water reacting to a turbulent field such as wind is an attractive and challenging problem. Its solution requires interlocking models for turbulent fields, gaseous flow, and realistic illumination. We present a model for turbulent wind flow having a deterministic component to specify largescale behaviour, and a stochastic component to model turbulent smallscale behaviour. The smallscale component is generated using spacetime Fourier synthesis. Turbulent wind fields can be superposed interactively to create subtle behaviour. An advectiondiffusion model is used to animate particlebased gaseous phenomena embedded in a wind field, and we derive an efficient physicallybased illumination model for rendering the system. Because the number of particles can be quite large, we present a clustering algorithm for efficient animation and rendering. CR Categories and Subject Descriptors: I.3.7 [Com puter Graphics]: ThreeDimensional Graphics...
Melting and Flowing
, 2002
"... We present a fast and stable system for animating materials that melt, flow, and solidify. Examples of realworld materials that exhibit these phenomena include melting candles, lava flow, the hardening of cement, icicle formation, and limestone deposition. We animate such phenomena by physical simu ..."
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Cited by 60 (4 self)
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We present a fast and stable system for animating materials that melt, flow, and solidify. Examples of realworld materials that exhibit these phenomena include melting candles, lava flow, the hardening of cement, icicle formation, and limestone deposition. We animate such phenomena by physical simulation of fluids  in particular the incompressible viscous NavierStokes equations with free surfaces, treating solid and nearlysolid materials as very high viscosity fluids. The computational method is a modification of the MarkerandCell (MAC) algorithm in order to rapidly simulate fluids with variable and arbitrarily high viscosity. This allows the viscosity of the material to change in space and time according to variation in temperature, water content, or any other spatial variable, allowing different locations in the same continuous material to exhibit states ranging from the absolute rigidity or slight bending of hardened wax to the splashing and sloshing of water. We create detailed polygonal models of the fluid by splatting particles into a volumetric grid and we render these models using ray tracing with subsurface scattering. We demonstrate the method with examples of several viscous materials including melting wax and sand drip castles.
Subdivision Schemes for Fluid Flow
 PROCEEDINGS OF SIGGRAPH 99
, 1999
"... The motion of fluids has been a topic of study for hundredsof years. In its most general setting, fluid flow is governed by a system of nonlinear partial differential equations known as the NavierStokes equations. However, in several important settings, these equations degenerate into simpler syst ..."
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Cited by 26 (2 self)
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The motion of fluids has been a topic of study for hundredsof years. In its most general setting, fluid flow is governed by a system of nonlinear partial differential equations known as the NavierStokes equations. However, in several important settings, these equations degenerate into simpler systems of linear partial differential equations. This paper will show that flows corresponding to these linear equations can be modeled using subdivision schemes for vector fields. Given an initial, coarse vector field, these schemes generate an increasingly dense sequence of vector fields. The limit of this sequence is a continuous vector field defining a flow that follows the initial vector field. The beauty of this approach is that realistic flows can now be modeled and manipulated in real time using their associated subdivision schemes.
The Computer Modelling of Fallen Snow
, 2000
"... One of nature's greatest beauties is the way fresh snowcovers the world in a perfect blanket of crystalline white. Snow replaces sharp angles with gentle curves, and clings to surfaces to form ghostly silhouettes. It is said the Inuit have 50 di#erentwords for snow, yet even they can be left speechl ..."
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Cited by 23 (0 self)
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One of nature's greatest beauties is the way fresh snowcovers the world in a perfect blanket of crystalline white. Snow replaces sharp angles with gentle curves, and clings to surfaces to form ghostly silhouettes. It is said the Inuit have 50 di#erentwords for snow, yet even they can be left speechless, as snow is one of the most complex natural materials in existence.
Interactive animation of ocean waves
 In Symposium on Computer Animation
, 2002
"... We present an adaptive scheme for the interactive animation and display of ocean waves far from the coast. Relying on a procedural wave model, the method restricts computations to the visible part of the ocean surface, adapts the geometric resolution to the viewing distance and only considers the vi ..."
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Cited by 23 (1 self)
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We present an adaptive scheme for the interactive animation and display of ocean waves far from the coast. Relying on a procedural wave model, the method restricts computations to the visible part of the ocean surface, adapts the geometric resolution to the viewing distance and only considers the visible waves wavelengths. This yields realtime performances, even when the camera moves. The method allows the user to interactively fly over an unbounded animated ocean, which was not possible using previous approaches.
PositionBased Physics: Simulating the Motion of Many Highly Interacting Spheres and Polyhedra
 In Computer Graphics
, 1996
"... This paper proposes a simplified positionbased physics that allows us to rapidly generate "piles" or "clumps" of many objects: local energy minima under a variety of potential energy functions. We can also generate plausible motions for many highly interacting objects from arbitrary starting positi ..."
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Cited by 16 (4 self)
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This paper proposes a simplified positionbased physics that allows us to rapidly generate "piles" or "clumps" of many objects: local energy minima under a variety of potential energy functions. We can also generate plausible motions for many highly interacting objects from arbitrary starting positions to a local energy minimum. We present an efficient and numerically stable algorithm for carrying out positionbased physics on spheres and nonrotating polyhedra through the use of linear programming. This algorithm is a generalization of an algorithm for finding tight packings of (nonrotating) polygons in two dimensions. This work introduces linear programming as a useful tool for graphics animation. As its name implies, positionbased physics does not contain a notion of velocity, and thus it is not suitable for simulating the motion of freeflying, unencumbered objects. However, it generates realistic motions of "crowded" sets of objects in confined spaces, and it does so at least two...
Simulation of Bubbles in Foam With The Volume Control Method
"... Figure 1: When the level set is advected by the BFECC [Dupont and Liu 2003] method, the simulation of a rising bubble produces volume loss (top). When the proposed volume control method is used, the volume of bubble is preserved regardless of the length of the simulation (bottom). From left to right ..."
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Cited by 12 (0 self)
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Figure 1: When the level set is advected by the BFECC [Dupont and Liu 2003] method, the simulation of a rising bubble produces volume loss (top). When the proposed volume control method is used, the volume of bubble is preserved regardless of the length of the simulation (bottom). From left to right, each column shows the bubble at t = 0, 0.0625, 0.125, 0.25, 0.5, and 10.0 second. The image on the far right shows a foam structure obtained after raising more than 400 bubbles. Liquid and gas interactions often produce bubbles that stay for a long time without bursting on the surface, making a dry foam structure. Such long lasting bubbles simulated by the level set method can suffer from a small but steady volume error that accumulates to a visible amount of volume change. We propose to address this problem by using the volume control method. We track the volume change of each connected region, and apply a carefully computed divergence that compensates undesired volume changes. To compute the divergence, we construct a mathematical model of the volume change, choose control strategies that regulate the modeled volume error, and establish methods to compute the control gains that provide robust and fast reduction of the volume error, and (if desired) the control of how the volume changes over time. 1
An accurate model of wave refraction over shallow water
 Computers & Graphics
, 2002
"... Abstract. A computer model of wave refraction is desirable, in the context of landscape modeling, to generate the familiar wave patterns seen near coastlines. In this article, we present a new method for the calculation of shallow water wave refraction. The method is more accurate than previously ex ..."
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Cited by 11 (1 self)
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Abstract. A computer model of wave refraction is desirable, in the context of landscape modeling, to generate the familiar wave patterns seen near coastlines. In this article, we present a new method for the calculation of shallow water wave refraction. The method is more accurate than previously existing methods and provides realistic wave refraction effects. We resort to Fermat’s principle of the shortest path and compute the propagation of wavefronts over an arbitrary inhomogeneous medium. The propagation of wavefronts produces a phase map for each terrain. This phase map is then coupled with a geometric model of waves to generate a heightfield representation of the sea surface. 1