### BibTeX

@INPROCEEDINGS{Stam99stablefluids,

author = {Jos Stam},

title = {Stable Fluids},

booktitle = {},

year = {1999},

pages = {121--128}

}

### Years of Citing Articles

### OpenURL

### Abstract

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.

### Citations

2009 |
Numerical Recipes in C: The Art of Scientific Computing
- Press, Teukolsky, et al.
- 1997
(Show Context)
Citation Context ...article traces a path starting at X throughthe field U over a time -dt. The endpoint of this path is the new point X0. We use both a simple second order Runge-Kutta (RK2)method for the particle trace =-=[14]-=- and an adaptive particle tracer, which subsamples the time step only in regions of high velocity gra-dients, such as near object boundaries. The routine LinInterp linearly interpolates the value of t... |

803 |
Multi-Grid Methods and Applications
- Hackbusch
- 1985
(Show Context)
Citation Context ...stem. Therefore, both the projection and the viscosity steps involve the solution of a large sparse system of equations. Multigrid methods, for example, can solve sparse linear systems in linear time =-=[12]. Since our adv-=-ection solver is also linear in time, the complexity of our proposed algorithm is of complexity ��¥���� . Foster and Metaxas’ solver has the same complexity. This performance is theoret... |

356 | Practical animation of liquids
- Foster, Fedkiw
- 2001
(Show Context)
Citation Context ...te the overall motion, while the solid texture can add additional detail for higher quality animations. Also we have not addressed the problem of simulating fluids with free boundaries, such as water =-=[8]-=-. This problem is considerably more difficult, since the geometry of the boundary evolves dynamically over time. We hope, however, that our stable solvers may be applied to this problem as well. Also,... |

354 |
Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8
- Harlow, Welsh
- 1965
(Show Context)
Citation Context ...ly such as swirling motion and flows past objects are obtained automatically. Their algorithm is based mainly on the work of Harlow and Welch in computational fluid dynamics, which dates back to 1965 =-=[13]-=-. Since then many other techniques which Foster and Metaxas could have used have been developed. However, their model has the advantage of being simple to code, since it is based on a finite differenc... |

331 | Particle systems - a technique for modeling a class of fuzzy objects
- Reeves, W
- 1983
(Show Context)
Citation Context ...were driven by visual appearance and not by physical accuracy. Early flow models were built from simple primitives. Various combinations of these primitives allowed the animation of particles systems =-=[17, 19]-=- or simple geometries such as leaves [25]. The complexity of the flows was greatly improved with the introduction of random turbulences [18, 22]. These turbulences are mass conserving and, therefore, ... |

208 |
Texturing & Modeling: A Procedural Approach
- Ebert, Musgrave, et al.
- 2003
(Show Context)
Citation Context ...er, discredit previous, more visually oriented models. In particular, we believe that the combination of our fluid solvers with solid textures, for example, may be a promising area of future research =-=[6]-=-. Our fluid solvers can be used to generate the overall motion, while the solid texture can add additional detail for higher quality animations. Also we have not addressed the problem of simulating fl... |

156 | Modeling the motion of a hot turbulent gas
- Foster, Metaxas
- 1997
(Show Context)
Citation Context ...rotational motions characteristic of fluids. More recently, Foster and Metaxas clearly show the advantages of using the full three-dimensional Navier-Stokes equations in creating fluidlike animations =-=[9]-=-. Many effects which are hard to key frame manually such as swirling motion and flows past objects are obtained automatically. Their algorithm is based mainly on the work of Harlow and Welch in comput... |

129 | Depicting fire and other gaseous phenomena using diffusion processes
- Stam, Fiume
- 1995
(Show Context)
Citation Context ...]. Versions of the method of characteristics were also used by other researchers. The application was either employed in visualizing flow fields [15, 20] or improving the rendering of gas simulations =-=[23, 7]-=-. Our application of the technique is fundamentally different, since we use it to update the velocity field, which previous researchers did not dynamically animate. The third step solves for the effec... |

111 | Turbulent wind fields for gaseous phenomena
- Stam, Fiume
- 1993
(Show Context)
Citation Context ...e primitives allowed the animation of particles systems [17, 19] or simple geometries such as leaves [25]. The complexity of the flows was greatly improved with the introduction of random turbulences =-=[18, 22]. Th-=-ese turbulences are mass conserving and, therefore, automatically exhibit rotational motion. Also the turbulence is periodic in space and time, which is ideal for motion “texture mapping” [21]. Fl... |

107 | Particle animation and rendering using data parallel computation
- Sims
- 1990
(Show Context)
Citation Context ...were driven by visual appearance and not by physical accuracy. Early flow models were built from simple primitives. Various combinations of these primitives allowed the animation of particles systems =-=[17, 19]-=- or simple geometries such as leaves [25]. The complexity of the flows was greatly improved with the introduction of random turbulences [18, 22]. These turbulences are mass conserving and, therefore, ... |

92 | Rapid, stable fluid dynamics for computer graphics - Kass, Miller - 1990 |

89 |
Animation aerodynamics
- Wejchert, Haumann
- 1991
(Show Context)
Citation Context ...hysical accuracy. Early flow models were built from simple primitives. Various combinations of these primitives allowed the animation of particles systems [17, 19] or simple geometries such as leaves =-=[25]-=-. The complexity of the flows was greatly improved with the introduction of random turbulences [18, 22]. These turbulences are mass conserving and, therefore, automatically exhibit rotational motion. ... |

67 |
On the solution of nonlinear hyperbolic differential equations by finite differences
- Courant, Isaacson, et al.
- 1952
(Show Context)
Citation Context ...we use both Lagrangian and implicit methods to solve the Navier-Stokes equations. Our approach falls into the class of so-called SemiLagrangian schemes that were first introduced in the early fifties =-=[5]-=-. These schemes are rarely used in engineering applications because they suffer from too much numerical dissipation: the simulated fluid tends to dampen more rapidly than an actual fluid. This shortco... |

55 |
Stable Fluid Dynamics for Computer Graphics
- Kass, Miller, et al.
- 1990
(Show Context)
Citation Context ...al simulation of the Navier-Stokes equations [3]. Their method unlike ours is both limited to two-dimensions and is unstable. Kass and Miller linearize the shallow water equations to simulate liquids =-=[14]-=-. The simplifications do not, however, capture the interesting rotational motions characteristic of fluids. More recently, Foster and Metaxas clearly show the advantages of using the full three-dimens... |

50 |
Numerical Simulation in Fluid Dynamics, a Practical Introduction
- Dornseifer, Neunhoeffer
- 1998
(Show Context)
Citation Context ...o code, since it is based on a finite differencing of the Navier-Stokes equations and an explicit time solver. Similar solvers and their source code are also available from the book of Griebel et al. =-=[11]. Th-=-e main problem with explicit solvers is that the numerical scheme can become unstable for large time-steps. Instability leads to numerical simulations that “blow-up” and therefore have to be resta... |

44 |
Visualizing Wind Velocities by Advecting Cloud Textures
- Max, Crawfis, et al.
- 1992
(Show Context)
Citation Context ...na. We employ our solver to update both the flow and the motion of densities within the flow. To further increase the complexity of our animations we advect texture coordinates along with the density =-=[15]. In-=- this manner we are able to synthesize highly detailed “wispy” gaseous flows even with low resolution grids. We believe that the combination of physics-based fluid solvers and solid textures is th... |

37 |
Stochastic motion – motion under the influence of wind, EUROGRAPHICS’92
- Shinya, Fournier
- 1992
(Show Context)
Citation Context ...e primitives allowed the animation of particles systems [17, 19] or simple geometries such as leaves [25]. The complexity of the flows was greatly improved with the introduction of random turbulences =-=[18, 22]. Th-=-ese turbulences are mass conserving and, therefore, automatically exhibit rotational motion. Also the turbulence is periodic in space and time, which is ideal for motion “texture mapping” [21]. Fl... |

30 | Real-time fluid simulation in a dynamic virtual environment
- Chen, Lobo, et al.
- 1997
(Show Context)
Citation Context ... solver to create two-dimensional animations of fluids [26, 10]. Later, Chen et al. animated water surfaces from the pressure term given by a two-dimensional simulation of the Navier-Stokes equations =-=[3]-=-. Their method unlike ours is both limited to two-dimensions and is unstable. Kass and Miller linearize the shallow water equations to simulate liquids [14]. The simplifications do not, however, captu... |

30 |
Combining Physical and Visual Simulation - Creation of the Planet Jupiter for the Film “2010
- Yaeger, Upson, et al.
- 1986
(Show Context)
Citation Context ...er-Stokes equations were first implemented in two-dimensions. Both Yaeger and Upson and Gamito et al. used a vortex method coupled with a Poisson solver to create two-dimensional animations of fluids =-=[26, 10]-=-. Later, Chen et al. animated water surfaces from the pressure term given by a two-dimensional simulation of the Navier-Stokes equations [3]. Their method unlike ours is both limited to two-dimensions... |

25 |
Solid Spaces and Inverse Particle Systems for Controlling the Animation of Gases and Fluids,” The Visual
- Ebert, Carlson, et al.
- 1990
(Show Context)
Citation Context ...]. Versions of the method of characteristics were also used by other researchers. The application was either employed in visualizing flow fields [15, 20] or improving the rendering of gas simulations =-=[23, 7]-=-. Our application of the technique is fundamentally different, since we use it to update the velocity field, which previous researchers did not dynamically animate. The third step solves for the effec... |

23 | Two-Dimensional Simulation of Gaseous Phenomena using Vortex Particles - Gamito, Lopes, et al. - 1995 |

21 |
Efficient fortran subprograms for the solution of separable elliptic partial differential equations
- Swarztrauber, P, et al.
- 1979
(Show Context)
Citation Context ...er SolveLin. The best theoretical choice is the multi-grid algorithm [12]. However, we used a solver from the FISHPAK library since it was very easy to incorporate into our code and gave good results =-=[24] �-=-�� . In practice, it turned out to be faster than our implementation of the multi-grid algorithm. In Appendix B, we show exactly how these routines are used to perform both the Diffuse step and the Pr... |

20 |
Computational Fluid Dynamics An Introduction for Engineers
- Abbott
- 1989
(Show Context)
Citation Context ...a text. Mathematically inclined readers may wish to start with the excellent book by Chorin and Marsden [4]. Readers with an engineering bent on the other hand can consult the didactic book by Abbott =-=[2]. -=-Also, Foster and Metaxas’ paper does a good job of introducing the concepts from fluid dynamics to the computer graphics community. 2 Stable Navier-Stokes 2.1 Basic Equations In this section we pres... |

14 |
A Mathematical Introduction to Fluid Mechanics (SpringerVerlag
- Chorin, Marsden
- 1993
(Show Context)
Citation Context ...ynamics and who wish to fully understand the method in this paper should therefore consult such a text. Mathematically inclined readers may wish to start with the excellent book by Chorin and Marsden =-=[4]. -=-Readers with an engineering bent on the other hand can consult the didactic book by Abbott [2]. Also, Foster and Metaxas’ paper does a good job of introducing the concepts from fluid dynamics to the... |

10 | A general animation framework for gaseous phenomena
- STAM
- 1997
(Show Context)
Citation Context ...[18, 22]. These turbulences are mass conserving and, therefore, automatically exhibit rotational motion. Also the turbulence is periodic in space and time, which is ideal for motion “texture mapping=-=” [21]-=-. Flows built up from a superposition of flow primitives all have the disadvantage that they do not respond dynamically to user-applied external forces. Dynamical models of fluids based on the Navier-... |

9 |
Choreographed image flow
- SIMS
- 1992
(Show Context)
Citation Context ...move densities through user-defined velocity fields [21]. Versions of the method of characteristics were also used by other researchers. The application was either employed in visualizing flow fields =-=[15, 20]-=- or improving the rendering of gas simulations [23, 7]. Our application of the technique is fundamentally different, since we use it to update the velocity field, which previous researchers did not dy... |

1 |
Numerical Calculation ofTime-Dependent Viscous Incompressible Flow of Fluid with Free Surface. The Physics of Fluids
- Harlow, Welch
(Show Context)
Citation Context ...lly such as swirling motion and flows past objects are obtained automatically. Their algorithm is based mainly on the workof Harlow and Welch in computational fluid dynamics, which dates back to 1965 =-=[11]-=-. Since then many other techniques which Fos-ter and Metaxas could have used have been developed. However, their model has the advantage of being simple to code, since it isbased on a finite differenc... |

1 |
Stable Fluid Dynamics forComputer Graphics
- Rapid
- 1990
(Show Context)
Citation Context ...l simulation of the Navier-Stokes equations [2]. Their method unlike ours is both limited to two-dimensions and is un-stable. Kass and Miller linearize the shallow water equations to simulate liquids =-=[12]-=-. The simplifications do not, however, cap-ture the interesting rotational motions characteristic of fluids. More recently, Foster and Metaxas clearly show the advantages of us-ing the full three-dime... |

1 |
Animation Aerodynamics.ACM Computer Graphics (SIGGRAPH '91
- Wejchert, Haumann
- 1991
(Show Context)
Citation Context ...physical accuracy. Early flow models were built from simple primitives. Various combinations ofthese primitives allowed the animation of particles systems [15, 17] or simple geometries such as leaves =-=[23]-=-. The complexity of theflows was greatly improved with the introduction of random turbulences [16, 20]. These turbulences are mass conserving and,therefore, automatically exhibit rotational motion. Al... |