Stable FluidsJos Stam

Alias wavefront

AbstractBuilding animation tools for fluid-like motions is an important andchallenging problem with many applications in computer graphics.The use of physics-based models for fluid flow can greatly assistin creating such tools. Physical models, unlike key frame or pro-cedural based techniques, permit an animator to almost effortlesslycreate 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 tooexpensive to allow real-time interaction. This was largely due to thefact 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 producescomplex fluid-like flows. As well, our method is very easy to im-plement. The stability of our model allows us to take larger timesteps and therefore achieve faster simulations. We have used ourmodel in conjuction with advecting solid textures to create manyfluid-like animations interactively in two- and three-dimensions.CR Categories: I.3.7 [Computer Graphics]: Three-DimensionalGraphics and RealismAnimation

Keywords: animation of fluids, Navier-Stokes, stable solvers, im-plicit elliptic PDE solvers, interactive modeling, gaseous phenom-ena, advected textures

1 IntroductionOne of the most intriguing problems in computer graphics is thesimulation of fluid-like behavior. A good fluid solver is of greatimportance in many different areas. In the special effects industrythere is a high demand to convincingly mimic the appearance andbehavior of fluids such as smoke, water and fire. Paint programscan also benefit from fluid solvers to emulate traditional techniquessuch 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 disciplinesand in engineering. Fluid mechanics is used as the standard math-ematical framework on which these simulations are based. Thereis a consensus among scientists that the Navier-Stokes equationsare a very good model for fluid flow. Thousands of books and

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articles have been published in various areas on how to computethese equations numerically. Which solver to use in practice de-pends largely on the problem at hand and on the computing poweravailable. Most engineering tasks require that the simulation pro-vide accurate bounds on the physical quantities involved to answerquestions 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 issecondary or in some cases irrelevant. Fluid solvers, for computergraphics, should ideally provide a user with a tool that enables herto achieve fluid-like effects in real-time. These factors are more im-portant than strict physical accuracy, which would require too muchcomputational power.

In fact, most previous models in computer graphics were drivenby visual appearance and not by physical accuracy. Early flowmodels 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 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 motiontexture mapping [19]. Flows built up from a superposition offlow primitives all have the disadvantage that they do not responddynamically to user-applied external forces. Dynamical modelsof fluids based on the Navier-Stokes equations were first imple-mented in two-dimensions. Both Yaeger and Upson and Gamitoet al. used a vortex method coupled with a Poisson solver to cre-ate two-dimensional animations of fluids [24, 8]. Later, Chen etal. animated water surfaces from the pressure term given by a two-dimensional simulation of the Navier-Stokes equations [2]. Theirmethod unlike ours is both limited to two-dimensions and is un-stable. Kass and Miller linearize the shallow water equations tosimulate liquids [12]. The simplifications do not, however, cap-ture the interesting rotational motions characteristic of fluids. Morerecently, Foster and Metaxas clearly show the advantages of us-ing the full three-dimensional Navier-Stokes equations in creatingfluid-like animations [7]. Many effects which are hard to key framemanually such as swirling motion and flows past objects are ob-tained automatically. Their algorithm is based mainly on the workof Harlow and Welch in computational fluid dynamics, which datesback 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 differencing of the Navier-Stokes equations andan explicit time solver. Similar solvers and their source code arealso available from the book of Griebel et al. [9]. The main prob-lem with explicit solvers is that the numerical scheme can becomeunstable for large time-steps. Instability leads to numerical sim-ulations that blow-up and therefore have to be restarted with asmaller time-step. The instability of these explicit algorithms setsserious limits on speed and interactivity. Ideally, a user should beable to interact in real-time with a fluid solver without having toworry about possible blow ups.

In this paper, for the first time, we propose a stable algorithmthat solves the full Navier-Stokes equations. Our algorithm is very

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Supplemental MaterialsSupplemental materials for this paper can be found in this directory.

easy to implement and allows a user to interact in real-time withthree-dimensional fluids on a graphics workstation. We achievethis by using time-steps much larger than the ones used by Fos-ter and Metaxas. To obtain a stable solver we depart from Fosterand Metaxas method of solution. Instead of their explicit Eulerianschemes, we use both Lagrangian and implicit methods to solve theNavier-Stokes equations. Our method cannot be found in the com-putational fluids literature, since it is custom made for computergraphics applications. The model would not be accurate enoughfor most engineering applications. Indeed, it suffers from too muchnumerical dissipation, i.e., the flow tends to dampen too rapidlyas compared to actual experiments. In a computer graphical appli-cation, on the other hand, this is not so bad, especially in an interac-tive system where the flow is kept alive by an animator applyingexternal forces. In fact, a flow which does not dampen at all mightbe too chaotic and difficult to control. As our results demonstratewe are able to produce nice swirling flows despite the numericaldissipation.

In this paper we apply our flows mainly to the simulation ofgaseous-like phenomena. We employ our solver to update boththe flow and the motion of densities within the flow. To furtherincrease the complexity of our animations we advect texture co-ordinates along with the density [13]. In this manner we are ableto synthesize highly detailed wispy gaseous flows even with lowresolution grids. We believe that the combination of physics-basedfluid solvers and solid textures is the most promising method ofachieving highly complex flows in computer graphics.

The next section presents the Navier-Stokes equations and thederivation which leads to our method of solution. That section con-tains all the fundamental ideas and techniques needed to obtain astable fluids solver. Since it relies on sophisticated mathematicaltechniques, it is written in a mathematical physics jargon whichshould be familiar to most computer graphics researchers workingin physics-based modeling. The application oriented reader whowishes only to implement our solver can skip Section 2 entirely andgo straight to Section 3. There we describe our implementation ofthe solver, providing sufficient information to code our technique.Section 4 is devoted to several applications that demonstrate thepower of our new solver. Finally, in Section 5 we conclude anddiscuss future research. To keep this within the confines of a shortpaper, we have decided not to include a tutorial-type section onfluid dynamics, since there are many excellent textbooks which pro-vide the necessary background and intuition. Readers who do nothave a background in fluid dynamics and who wish to fully under-stand the method in this paper should therefore consult such a text.Mathematically inclined readers may wish to start with the excel-lent book by Chorin and Marsden [3]. Readers with an engineeringbent on the other hand can consult the didactic book by Abbott [1].Also, Foster and Metaxas paper does a good job of introducing theconcepts from fluid dynamics to the computer graphics community.

2 Stable Navier-Stokes2.1 Basic EquationsIn this section we present the Navier-Stokes equations along withthe manipulations that lead to our stable solver. A fluid whose den-sity and temperature are nearly constant is described by a velocityfield u and a pressure field p. These quantities generally vary bothin space and in time and depend on the boundaries surrounding thefluid. We will denote the spatial coordinate by x, which for two-dimensional fluids is x = (x; y) and three-dimensional fluids isequal to (x; y; z). We have decided not to specialize our resultsfor a particular dimension. All results are thus valid for both two-dimensional and three-dimensional flows unless stated otherwise.Given that the velocity and the pressure are known for some initial

time t = 0, then the evolution of these quantities over time is givenby the Navier-Stokes equations [3]:

r u = 0 (1)@u

@t

= (u r)u

1

rp+ r

2

u+ f ; (2)

where is the kinematic viscosity of the fluid, is its density andf is an external force. Some readers might be unfamiliar with thiscompact version of the Navier-Stokes equations. Eq. 2 is a vec-tor equation for the three (two in two-dimensions) components ofthe velocity field. The denotes a dot product between vec-tors, while the symbol r is the vector of spatial partial deriva-tives. More precisely, r = (@=@x; @=@y) in two-dimensions andr = (@=@x; @=@y; @=@z) in three-dimensions. We have also usedthe shorthand notation r2 = r r. The Navier-Stokes equationsare obtained by imposing that the fluid conserves both mass (Eq. 1)and momentum (Eq. 2). We refer the reader to any standard texton fluid mechanics for the actual derivation. These equations alsohave to be supplemented with boundary conditions. In this paperwe will consider two types of boundary conditions which are use-ful in practical applications: periodic boundary conditions and fixedboundary conditions. In the case of periodic boundaries the fluid isdefined on an n-dimensional torus (n = 2; 3). In this case thereare no walls, just a fluid which wraps around. Although such flu-ids are not encountered in practice, they are very useful in creatingevolving texture maps. Also, these boundary conditions lead to avery elegant implementation that uses the fast Fourier transform asshown below. The second type of boundary condition that we con-sider is when the fluid lies in some bounded domain D. In that case,the boundary conditions are given by a function u

D

defined on theboundary @D of the domain. See Foster and Metaxas work for agood discussion of these boundary conditions in the case of a hotfluid [7]. In any case, the boundary conditions should be such thatthe normal component of the velocity field is zero at the boundary;no matter should traverse walls.

The pressure and the velocity fields which appear in the Navier-Stokes equations are in fact related. A single equation for the ve-locity can be obtained by combining Eq. 1 and Eq. 2. We brieflyoutline the steps that lead to that equation, since it is fundamen-tal to our algorithm. We follow Chorin and Marsdens treatmentof the subject (p. 36ff, [3]). A mathematical result, known as theHelmholtz-Hodge Decomposition, states that any vector fieldw canuniquely be decomposed into the form:

w = u+rq; (3)where u has zero divergence: ru = 0 and q is a scalar field. Anyvector field is the sum of a mass conserving field and a gradientfield. This result allows us to define an operator P which projectsany vector field w onto its divergence free part u = Pw. Theoperator is in fact defined implicitly by multiplying both sides ofEq. 3 by r:

r w = r

2

q: (4)This is a Poisson equation for the scalar field q with the Neumannboundary condition @q

@n

= 0 on @D. A solution to this equation isused to compute the projection u:

u = Pw = wrq:

If we apply this projection operator on both sides of Eq. 2 we obtaina single equation for the velocity:

@u

@t

= P

(u r)u+ r

2

u+ f

; (5)

where we have used the fact that Pu = u and Prp = 0. This isour fundamental equation from which we will develop a stable fluidsolver.

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BBBBBBBBBBBBBBBBBBBBBBBBBBBBBB

BBBBBBBBBBBBBBBBBBBBBBBBBBBBBB

BBBBBBBBBBBBBBBBBBBBBBBBBBBBBB

BBBBBBBBBBBBBBBBBBBBBBBBBBBBBB

BBBBBBBBBBBBBBBBBBBBBBBBBBBBBB

BBBBBBBBBBBBBBBBBBBBBBBBBBBBBB

w0

1w2w 3w

4ww

u

q

u=0.

Figure 1: One simulation step of our solver is composed of steps.The first three steps may take the field out of the space of divergentfree fields. The last projection step ensures that the field is divergentfree after the entire simulation step.

x

p(x,s)p(x,t)

s0 t

Figure 2: To solve for the advection part, we trace each point ofthe field backward in time. The new velocity at x is thereforethe velocity that the particle had a time t ago at the old locationp(x;t).

2.2 Method of SolutionEq. 5 is solved from an initial state u

0

= u(x; 0) by marchingthrough time with a time step t. Let us assume that the field hasbeen resolved at a time t and that we wish to compute the field at alater time t+t. We resolve Eq. 5 over the time span t in foursteps. We start from the solution w

0

(x) = u(x; t) of the previoustime step and then sequentially resolve each term on the right handside of Eq. 5, followed by a projection onto the divergent free fields.The general procedure is illustrated in Figure 1. The steps are:

w

0

(x)

add force

z}|{

! w

1

(x)

advect

z}|{

! w

2

(x)

diuse

z}|{

! w

3

(x)

project

z}|{

! w

4

(x):

The solution at time t+t is then given by the last velocity field:u(x; t+t) = w

4

(x). A simulation is obtained by iterating thesesteps. We now explain how each step is computed in more detail.

The easiest term to solve is the addition of the external force f .If we assume that the force does not vary considerably during thetime step, then

w

1

(x) = w

0

(x) + t f(x; t)

is a good approximation of the effect of the force on the field overthe time step t. In an interactive system this is a good approxi-mation, since forces are only applied at the beginning of each timestep.

The next step accounts for the effect of advection (or convec-tion) of the fluid on itself. A disturbance somewhere in the fluidpropagates according to the expression (u r)u. This termmakes the Navier-Stokes equations non-linear. Foster and Metaxasresolved this component using finite differencing. Their methodis stable only when the time step is sufficiently small such that

t < =juj, where is the spacing of their computationalgrid. Therefore, for small separations and/or large velocities, verysmall time steps have to be taken. On the other hand, we use a to-tally different approach which results in an unconditionally stablesolver. No matter how big the time step is, our simulations willnever blow up. Our method is based on a technique to solve par-tial differential equations known as the method of characteristics.Since this method is of crucial importance in obtaining our stablesolver, we provide all the mathematical details in Appendix A. Themethod, however, can be understood intuitively. At each time stepall the fluid particles are moved by the velocity of the fluid itself.Therefore, to obtain the velocity at a point x at the new time t+t,we backtrace the point x through the velocity field w

1

over a timet. This defines a path p(x; s) corresponding to a partial stream-line of the velocity field. The new velocity at the point x is thenset to the velocity that the particle, now at x, had at its previouslocation a time t ago:

w

2

(x) = w

1

(p(x;t)):

Figure 2 illustrates the above. This method has several advantages.Most importantly it is unconditionally stable. Indeed, from theabove equation we observe that the maximum value of the newfield is never larger than the largest value of the previous field.Secondly, the method is very easy to implement. All that is re-quired in practice is a particle tracer and a linear interpolator (seenext Section). This method is therefore both stable and simple toimplement, two highly desirable properties of any computer graph-ics fluid solver. We employed a similar scheme to move densitiesthrough user-defined velocity fields [19]. Versions of the method ofcharacteristics were also used by other researchers. The applicationwas either employed in visualizing flow fields [13, 18] or improv-ing the rendering of gas simulations [21, 5]. Our application ofthe technique is fundamentally different, since we use it to updatethe velocity field, which previous researchers did not dynamicallyanimate.

The third step solves for the effect of viscosity and is equivalentto a diffusion equation:

@w

2

@t

= r

2

w

2

:

This is a standard equation for which many numerical procedureshave been developed. The most straightforward way of solving thisequation is to discretize the diffusion operator r2 and then to doan explicit time step as Foster and Metaxas did [7]. However, thismethod is unstable when the viscosity is large. We prefer, therefore,to use an implicit method:

I tr

2

w

3

(x) = w

2

(x);

where I is the identity operator. When the diffusion operator isdiscretized, this leads to a sparse linear system for the unknownfield w

3

. Solving such a system can be done efficiently, however(see below).

The fourth step involves the projection step, which makes theresulting field divergence free. As pointed out in the previous sub-section this involves the resolution of the Poisson problem definedby Eq. 4:

r

2

q = r w

3

w

4

= w

3

rq:

The projection step, therefore, requires a good Poisson solver.Foster and Metaxas solved a similar equation using a relaxationscheme. Relaxation schemes, though, have poor convergence andusually require many iterations. Foster and Metaxas reported thatthey obtained good results even after a very small number of re-laxation steps. However, since we are using a different method toresolve for the advection step, we must use a more accurate method.

123

Indeed, the method of characteristics is more precise when the fieldis close to divergent free. More importantly from a visual point ofview, the projection step forces the fields to have vortices which re-sult in more swirling-like motions. For these reasons we have useda more accurate solver for the projection step.

The Poisson equation, when spatially discretized, becomes asparse linear system. Therefore, both the projection and the viscos-ity steps involve the solution of a large sparse system of equations.Multigrid methods, for example, can solve sparse linear systems inlinear time [10]. Since our advection solver is also linear in time,the complexity of our proposed algorithm is of complexity O(N).Foster and Metaxas solver has the same complexity. This perfor-mance is theoretically optimal since for a complicated fluid, anyalgorithm has to consult at least each cell of the computational grid.

2.3 Periodic Boundaries and the FFTWhen we consider a domain with periodic boundary conditions, ouralgorithm takes a particularly simple form. The periodicity allowsus to transform the velocity into the Fourier domain:

u(x; t) !

^

u(k; t):

In the Fourier domain the gradient operator r is equivalent tothe multiplication by ik, where i =

p

1. Consequently, both thediffusion step and the projection step are much simpler to solve.Indeed the diffusion operator and the projection operators in theFourier domain are

I tr

2

! 1 + tk

2

and

Pw !

^

P

^

w(k) =

^

w(k)

1

k

2

(k

^

w(k))k;

where k = jkj. The operator ^P projects the vector ^w(k) onto theplane which is normal to the wave number k. The Fourier transformof the velocity of a divergent free field is therefore always perpen-dicular to its wavenumbers. The diffusion can be interpreted as alow pass filter whose decay is proportional to both the time stepand the viscosity. These simple results demonstrate the power ofthe Fourier transform. Indeed, we are able to completely transcribeour solver in only a couple of lines. All that is required is a particletracer and a fast Fourier transform (FFT).

FourierStep(w0

,w4

,t):add force: w

1

= w

0

+t f

advect: w2

(x) = w

1

(p(x;t))

transform: ^w2

= FFTfw

2

g

diffuse: ^w3

(k) =

^

w

2

(k)=(1 + tk

2

)

project: ^w4

=

^

P

^

w

3

transform: w4

= FFT

1

f

^

w

4

g

Since the Fourier transform is of complexity O(N logN), thismethod is theoretically slightly more expensive than a method ofsolution relying on multi-grid solvers. However, this method isvery easy to implement. We have used this algorithm to generatethe liquid textures of Section 4.

2.4 Moving Substances through the FluidA non-reactive substance which is injected into the fluid will be ad-vected by it while diffusing at the same time. Common examples ofthis phenomenon include the patterns created by milk stirred in cof-fee or the smoke rising from a cigarette. Let a be any scalar quantitywhich is moved through the fluid. Examples of this quantity includethe density of dust, smoke or cloud droplets, the temperature of a

Figure 3: The values of the discretized fields are defined at the cen-ter of the grid cells.

fluid and a texture coordinate. The evolution of this scalar field isconveniently described by an advection diffusion type equation:

@a

@t

= u ra+

a

r

2

a

a+ S

a

;

where a

is a diffusion constant, a

is a dissipation rate and Sa

isa source term. This equation is very similar in form to the Navier-Stokes equation. Indeed, it includes an advection term, a diffusionterm and a force term S

a

. All these terms can be resolved exactlyin the same way as the velocity of the fluid. The dissipation termnot present in the Navier-Stokes equation is solved as follows overa time-step:

(1 + t

a

)a(x; t+t) = a(x; t):

Similar equations were used by Stam and Fiume to simulate fireand other gaseous phenomena [21]. However, their velocity fieldswere not computed dynamically.

We hope that the material in this section has convinced the readerthat our stable solver is indeed based on the full Navier-Stokesequations. Also, we have pointed to the numerical techniqueswhich should be used at each step of our solver. We now proceedto describe the implementation of our model in more detail.

3 Our Solver3.1 SetupOur implementation handles both the motion of fluids and the prop-agation by the fluid of any number of substances like mass-density,temperature or texture coordinates. Each quantity is defined on ei-ther a two-dimensional (NDIM=2) or three-dimensional (NDIM=3)grid, depending on the application. The grid is defined by its phys-ical dimensions: origin O[NDIM] and length L[NDIM] of eachside, and by its number of cells N[NDIM] in each coordinate. Thisin turn determines the size of each voxel D[i]=L[i]/N[i]. Thedefinition of the grid is an input to our program which is speci-fied by the animator. The velocity field is defined at the center ofeach cell as shown in Figure 3. Notice that previous researchers,e.g., [7], defined the velocity at the boundaries of the cells. Weprefer the cell-centered grid since it is more straightforward to im-plement. We allocate two grids for each component of the velocity:U0[NDIM] and U1[NDIM]. At each time step of our simulationone grid corresponds to the solution obtained in the previous step.We store the new solution in the second grid. After each step, thegrids are swapped. We also allocate two grids to hold a scalar fieldcorresponding to a substance transported by the flow. Although ourimplementation can handle any number of substances, for the sakeof clarity we present only the algorithm for one field in this section.This scalar quantity is stored in the grids S0 and S1. The speed ofinteractivity is controlled by a single time step dt, which can be aslarge as the animator wishes, since our algorithm is stable.

124

The physical properties of the fluid are a function of its viscosityvisc alone. By varying the viscosity, an animator can simulate awide range of substances ranging from glue-like matter to highlyturbulent flows. The properties of the substance are modeled by adiffusion constant kS and a dissipation rate aS. Along with theseparameters, the animator also must specify the values of these fieldson the boundary of the grid. There are basically two types: peri-odic or fixed. The boundary conditions can be of a different typefor each coordinate. When periodic boundary conditions are cho-sen, the fluid wraps around. This means that a piece of fluid whichleaves the grid on one side reenters the grid on the opposite side.In the case of fixed boundaries, the value of each physical quantitymust be specified at the boundary of the grid. The simplest methodis to set the field to zero at the boundary. We refer the reader toFoster and Metaxas paper for an excellent description of differentboundary conditions and their resulting effects [7]. In the resultssection we describe the boundary conditions chosen for each an-imation. For the special case when the boundary conditions areperiodic in each coordinate, a very elegant solver based on the fastFourier transform can be employed. This algorithm is described inSection 2.3. We do not repeat it here since the solver in this sectionis more general and can handle both types of boundary conditions.

The fluid is set into motion by applying external forces to it.We have written an animation system in which an animator witha mouse can apply directional forces to the fluid. The forces canalso be a function of other substances in the fluid. For example,a temperature field moving through the fluid can produce buoyantand turbulent forces. In our system we allow the user to createall sorts of dependencies between the various fields, some of whichare described in the results section of this paper. We do not describeour animation system in great detail since its functionality shouldbe evident from the examples of the next section. Instead we focuson our simulator, which takes the forces and parameters set by theanimator as an input.

3.2 The SimulatorOnce we worked out the mathematics underlying the Navier-Stokesequations in Section 2, our implementation became straightfor-ward. We wish to emphasize that the theoretical developments ofSection 2 are in no way gratuitous but are immensely useful in cod-ing compact solvers. In particular, casting the problem into a math-ematical setting has allowed us to take advantage of the large bodyof work done in the numerical analysis of partial differential equa-tions. We have written the solver as a separate library of routinesthat are called by the interactive animation system. The entire li-brary consists of only roughly 500 lines of C code. The two mainroutines of this library update either the velocity field Vstep ora scalar field Sstep over a given time step. We assume that theexternal force is given by an array of vectors F[NDIM] and thatthe source is given by an array Ssource for the scalar field. Thegeneral structure of our simulator looks like

while ( simulating ) f/* handle display and user interaction *//* get forces F and sources Ssource from the UI */Swap(U1,U0); Swap(S1,S0);Vstep ( U1, U0, visc, F, dt );Sstep ( S1, S0, kS, aS, U1, Ssource, dt );

g

The velocity solver is composed of four steps: the forces are addedto the field, the field is advected by itself, the field diffuses due toviscous friction within the fluid, and in the final step the velocity isforced to conserve mass. The general structure of this routine is:

Vstep ( U1, U0, visc, F, dt )

for(i=0;i

4 ResultsOur Navier-Stokes solver can be used in many applications requir-ing fluid-like motions. We have implemented both the two- and thethree-dimensional solvers in an interactive modeler that allows auser to interact with the fluids in real-time. The motion is modeledby either adding density into the fluid or by applying forces. Theevolution of the velocity and the density is then computed using oursolver. To further increase the visual complexity of the flows, weadd textural detail to the density. By moving the texture coordinatesusing the scalar solver as well, we achieve highly detailed flows. Tocompensate for the high distortions that the texture maps undergo,we use three sets of texture coordinates which are periodically resetto their initial (unperturbed) values. At every moment the resultingtexture map is the superposition of these three texture maps. Thisidea was first suggested by Max et al. [13].

Figure 4.(a) shows a sequence of frames from an animationwhere the user interacts with one of our liquid textures. The figureon the backcover of the SIGGRAPH proceedings is another frameof a similar sequence with a larger grid size (1002).

Figures 4.(b) through 4.(g) show frames from various animationsthat we generated using our three-dimensional solver. In each casethe animations were created by allowing the animator to place den-sity and apply forces in real-time. The gases are volume renderedusing the three-dimensional hardware texture mapping capabilitiesof our SGI Octane workstation. We also added a single pass thatcomputes self-shadowing effects from a directional light source ina fixed position. It should be evident that the quality of the render-ings could be further improved using more sophisticated renderinghardware or software. Our grid sizes ranged from 163 to 303 withframe rates fast enough to monitor the animations while being ableto control their behavior. In most of these animations we added anoise term which is proportional to the amount of density (thefactor of proportionality being a user defined parameter). This pro-duced nice billowing motions in some of our animations. In Fig-ures 4.(d)-(e) we used a fractal texture map, while in Figure 4.(g)we used a texture map consisting of evenly spaced lines.

All of our animations were created on an SGI Octane worksta-tion with a R10K processor and 192 Mbytes of memory.

5 ConclusionsThe motivation of this paper was to create a general software sys-tem that allows an animator to design fluid-like motions in real time.Our initial intention was to base our system on Foster and Metaxaswork. However, the instabilities inherent in their method forced usto develop a new algorithm. Our solver has the property of beingunconditionally stable and it can handle a wide variety of fluids inboth two- and three-dimensions. The results that accompany thispaper clearly demonstrate that our solver is powerful enough to al-low an animator to achieve many fluid-like effects. We thereforebelieve that our solver is a substantial improvement over previouswork in this area. The work presented here does not, however, dis-credit previous, more visually oriented models. In particular, webelieve that the combination of our fluid solvers with solid textures,for example, may be a promising area of future research [4]. Ourfluid solvers can be used to generate the overall motion, while thesolid texture can add additional detail for higher quality animations.

Also we have not addressed the problem of simulating fluids withfree boundaries, such as water [6]. This problem is considerablymore difficult, since the geometry of the boundary evolves dynam-ically over time. We hope, however, that our stable solvers maybe applied to this problem as well. Also, we wish to extend oursolver to finite element boundary-fitted meshes. We are currentlyinvestigating such extensions.

AcknowledgmentsI would like to thank Marcus Grote for his informed input on fluiddynamics and for pointing me to reference [3]. Thanks to DuncanBrinsmead for his constructive criticisms all along. Thanks also toPamela Jackson for carefully proofreading the paper and to BradClarkson for his help with creating the videos.

A Method of CharacteristicsThe method of characteristics can be used to solve advection equa-tions of the type

@a(x; t)

@t

= v(x) ra(x; t) and a(x; 0) = a

0

(x);

where a is a scalar field, v is a steady vector field and a0

is the fieldat time t = 0. Let p(x

0

; t) denote the characteristics of the vectorfield v which flow through the point x

0

at t = 0:

d

dt

p(x

0

; t) = v(p(x

0

; t)) and p(x

0

; 0) = x

0

:

Now let a(x0

; t) = a(p(x

0

; t); t) be the value of the field along thecharacteristic passing through the point x

0

at t = 0. The variationof this quantity over time can be computed using the chain rule ofdifferentiation:

da

dt

=

@a

@t

+ v ra = 0:

This shows that the value of the scalar does not vary along thestreamlines. In particular, we have a(x

0

; t) = a(x

0

; 0) = a

0

(x

0

).

Therefore, the initial field and the characteristics entirely define thesolution to the advection problem. The field for a given time t andlocation x is computed by first tracing the location x back in timealong the characteristic to get the point x

0

, and then evaluating theinitial field at that point:

a(p(x

0

; t); t) = a

0

(x

0

):

We use this method to solve the advection equation over a timeinterval [t; t+t] for the fluid. In this case, v = u(x; t) and a

0

isany of the components of the fluids velocity at time t.

B FISHPAK RoutinesThe linear solver POIS3D from FISHPAK is designed to solve ageneral system of finite difference equations of the type:

K1*(S[i-1,j,k]-2*S[i,j,k]+S[i+1,j,k]) +K2*(S[i,j-1,k]-2*S[i,j,k]+S[i,j+1,k]) +A[k]*S[i,j,k-1]+B[k]*S[i,j,k]+ .

For the diffusion solver, the values of the constants on the left handside are:

K1 = -dt*k/(D[0]*D[0]),K2 = -dt*k/(D[1]*D[1]),A[k] = C[k] = -dt*k/(D[2]*D[2]) andB[k] = 1+2*dt*k/(D[2]*D[2]),

while the right hand side is equal to the grid containing the previoussolution: F=S0. In the projection step these constants are equal to

K1 = 1/(D[0]*D[0]), K2 = 1/(D[1]*D[1]),A[k] = C[k] = 1/(D[2]*D[2]) andB[k] = -2/(D[2]*D[2]),

126

while the right hand side is equal to the divergence of the velocityfield:

F[i,j,k] = 0.5*((U[i+1,j,k]-U[i-1,j,k])/D[0]+(U[i,j+1,k]-U[i,j-1,k])/D[1]+(U[i,j,k+1]-U[i,j,k-1])/D[2]).

The gradient of the solution is then subtracted from the previoussolution:

U1[0][i,j,k] = U0[0][i,j,k] -0.5*(S[i+1,j,k]-S[i-1,j,k])/D[0],

U1[1][i,j,k] = U0[1][i,j,k] -0.5*(S[i,j+1,k]-S[i,j-1,k])/D[1],

U1[2][i,j,k] = U0[2][i,j,k] -0.5*(S[i,j,k+1]-S[i,j,k-1])/D[2].

The FISHPAK routine is also able to handle different types ofboundary conditions, both periodic and fixed.

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[14] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetter-ling. Numerical Recipes in C. The Art of Scientific Computing.Cambridge University Press, Cambridge, 1988.

[15] W. T. Reeves. Particle Systems. A Technique for Modelinga Class of Fuzzy Objects. ACM Computer Graphics (SIG-GRAPH 83), 17(3):359376, July 1983.

[16] M. Shinya and A. Fournier. Stochastic Motion - Motion Underthe Influence of Wind. In Proceedings of Eurographics 92,pages 119128, September 1992.

[17] K. Sims. Particle Animation and Rendering Using Data Paral-lel Computation. ACM Computer Graphics (SIGGRAPH 90),24(4):405413, August 1990.

[18] K. Sims. Choreographed Image Flow. The Journal Of Visual-ization And Computer Animation, 3:3143, 1992.

[19] J. Stam. A General Animation Framework for Gaseous Phe-nomena. ERCIM Research Report, R047, January 1997.http://www.ercim.org/publications/technical reports/047-abstract.html.

[20] J. Stam and E. Fiume. Turbulent Wind Fields for GaseousPhenomena. In Proceedings of SIGGRAPH 93, pages 369376. Addison-Wesley Publishing Company, August 1993.

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(a)

(b) (c)

(e)

(g)

(b)

(d)

(f)

Figure 4: Snapshots from our interactive fluid solver.

128