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Stable Fluids
Jos Stam
Alias wavefront
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 pro-
cedural based techniques, permit an animator to almost effortlessly
create interesting, swirling fluid-like behaviors. Also, the interac-
tion 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 phys-
ical 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 im-
plement. 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.
CR Categories: I.3.7 [Computer Graphics]: Three-Dimensional
Graphics and Realism—Animation
Keywords: animation of fluids, Navier-Stokes, stable solvers, im-
plicit elliptic PDE solvers, interactive modeling, gaseous phenom-
ena, advected textures
1 Introduction
One of the most intriguing problems in computer graphics is the
simulation of fluid-like behavior. A good fluid solver is of great
importance in many different areas. In the special effects industry
there is a high demand to convincingly mimic the appearance and
behavior of fluids such as smoke, water and fire. Paint programs
can also benefit from fluid solvers to emulate traditional techniques
such as watercolor and oil paint. Texture synthesis is another pos-
sible application. Indeed, many textures result from fluid-like pro-
cesses, such as erosion. The modeling and simulation of fluids is,
of course, also of prime importance in most scientific disciplines
and in engineering. Fluid mechanics is used as the standard math-
ematical framework on which these simulations are based. There
is a consensus among scientists that the Navier-Stokes equations
are a very good model for fluid flow. Thousands of books and
Alias wavefront, 1218 Third Ave, 8th Floor, Seattle, WA 98101, U.S.A.
jstam@aw.sgi.com
articles have been published in various areas on how to compute
these equations numerically. Which solver to use in practice de-
pends largely on the problem at hand and on the computing power
available. Most engineering tasks require that the simulation pro-
vide accurate bounds on the physical quantities involved to answer
questions related to safety, performance, etc. The visual appearance
(shape) of the flow is of secondary importance in these applications.
In computer graphics, on the other hand, the shape and the behav-
ior of the fluid are of primary interest, while physical accuracy is
secondary or in some cases irrelevant. Fluid solvers, for computer
graphics, should ideally provide a user with a tool that enables her
to achieve fluid-like effects in real-time. These factors are more im-
portant than strict physical accuracy, which would require too much
computational power.
In fact, most previous models in computer graphics 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 [15, 17]
or simple geometries such as leaves [23]. The complexity of the
flows was greatly improved with the introduction of random tur-
bulences [16, 20]. These turbulences are mass conserving and,
therefore, automatically exhibit rotational motion. Also the tur-
bulence is periodic in space and time, which is ideal for motion
“texture mapping” [19]. 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-Stokes equations were first imple-
mented in two-dimensions. Both Yaeger and Upson and Gamito
et al. used a vortex method coupled with a Poisson solver to cre-
ate two-dimensional animations of fluids [24, 8]. Later, Chen et
al. animated water surfaces from the pressure term given by a two-
dimensional 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-dimensional Navier-Stokes equations in creating
fluid-like animations [7]. Many effects which are hard to key frame
manually such as swirling motion and flows past objects are ob-
tained automatically. Their algorithm is based mainly on the work
of 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 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. [9]. The main prob-
lem with explicit solvers is that the numerical scheme can become
unstable for large time-steps. Instability leads to numerical sim-
ulations that “blow-up” and therefore have to be restarted with a
smaller time-step. The instability of these explicit algorithms sets
serious limits on speed and interactivity. Ideally, a user should be
able to interact in real-time with a fluid solver without having to
worry about possible “blow ups”.
In this paper, for the first time, we propose a stable algorithm
that solves the full Navier-Stokes equations. Our algorithm is very
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