Simulating Fluid-Solid Interaction

Olivier Génevaux Arash Habibi Jean-Michel Dischler

LSIIT UMR CNRS-ULP 7005Boulevard Śebastien Brant, F67400 Illkirch, France{genevaux, habibi, dischler}@dpt-info.u-strasbg.fr

AbstractThough realistic eulerian fluid simulation systems now

provide believable movements, straightforward render-able surface representation, and affordable computationcosts, they are still unable to deal with non-static objectsin a realistic manner. Namely, objects can not have aninfluence on the fluid and be simultaneously affected bythe fluid’s motion. In this paper, a simulation scheme forfluids allowing automatic generation of physically plau-sible motions alongside realistic interactions with solidsis proposed. The method relies mainly on the definitionof a coupling force between the solids and the fluid, thusbridging the gap between commonly used eulerian fluidanimation models and lagrangian solid ones. This newmethod thus improves existing fluid simulations, makingthem capable of generating new kinds of motions, suchas a floating ball displaced by the wave created thanks toits own splash into the water.

Key words: Fluid simulation, interactions, Navier-Stokesequations, spring-masses simulation.

1 Introduction

Since realism is one of the major goals of computergraphics, numerous works attempt to provide so-called”realistic” methods to allow an easy creation of digitalequivalents for natural phenomena. Within these, the an-imation of fluids, and especially liquids, is a particularlydifficult task due to the underlying laws that govern flu-ids’ motions. Indeed, these laws, known as the Navier-Stokes, equations are highly unstable partial differentialequations that are quite difficult to solve in an efficientway, and even difficult to solve at all.

Multiple types of methods were proposed to simulatefluids. The first kind of method, which consists of whatcan be called ”empirical hand-crafted models”, does notinvolve the Navier-Stokes equations at all, or even phys-ical correctness, but offloads the task of creating a con-vincing velocity field to an animator, using velocity prim-itives [4]. Such an approach may be complemented us-ing a small scale turbulent velocity field that is added tothe large scale hand-designed flow to increase details and

give the fluid a more turbulent, thus better looking, be-havior [17, 18].

Figure 1: A ball falling inside a tank. — The ball ispushed upward due to the pressure wave it created.

The second kind of method, which encompasses parti-cle based simulations, does not make use of the Navier-Stokes equations either, but are nevertheless based onphysical considerations. Indeed, they are able to ex-hibit realistic fluid behaviors without resorting to an ex-plicit description of motions. These motions naturallyspring from the local forces in action between the par-ticles [13, 2, 9]. It should be noted that these models arealready able to deal with convincing interactions betweensolids and liquids. This arises from the fact that both en-tities are simulated using the same lagrangian represen-tation [10]. However this kind of simulation comes witha high computation cost stemming from the huge num-ber of forces that must be computed between the manyparticles needed to obtain realistically looking motion.

To avoid such a plethora of particles, physical macro-scopic modeling may be used. This is what is employedin a number of different works, from heavily simplifiedsimulation models to thorough ones. These simplifica-tions can be limitations to the simulable behaviors, suchas non turbulent fluids [20, 19], or fluids that can be rep-resented by a heightfield [11, 15]. In any case, it shouldbe noted that physical considerations are taken into ac-count, either explicitly by the solver or implicitly by theunderlying simulated equations.

The last class of simulators extends the former one.Indeed, the Navier-Stokes based simulators are undoubt-edly physically based, but what makes them specific isthat they use a comprehensive fluid model to its maxi-mum extent: they don’t put any restrictions on the prop-

erties of the simulated fluid, nor on the motions that canbe simulated starting from these equations. The only lim-itation arises from the allowed computational cost, whichcan be quite substantial when large scale but precise sim-ulations are required. Despite much less demanding thanlagrangian systems, these simulators can thus still becostly. The cost, along with the problem of liquid track-ing and surface determination, explains why these sim-ulators were first developed to run on two dimensionaldomains [1], before being expanded to full three dimen-sional domains [8]. Another interesting point of thesesimulation schemes is that movements can be easily con-trolled by animators without ruining the physical believ-ability of the whole animation [6].

Further development led to the solution of the twoaforementioned problems related to these methods: com-putation speed was improved using a more stable evolu-tion scheme [16], and a visually appealling surface rep-resentation was devised to allow realistic visualization ofliquids [7, 5]. This last point is not to be underestimateddue to the fact that current liquid simulations often relyon marker particles to discriminate between the fluid andits surrounding environment, thus not readily exhibitinga clear and good looking surface to be visualized.

However, if these schemes are able to handle a fluid in-side its static environment fairly well, they don’t provideany means to deal with true reciprocal interactions be-tween fluid and arbitrary solids, except for one work ap-plied to the explosion simulations [21], and another workrestricted to heightfield represented water [15]. The firstwork is not directly relevant to liquid simulation though,because it does not deal with free surface liquid while theheightfield nature of the second work is highly limitatingits use.

As a consequence, no animation system based on theNavier-Stokes equations is able to automatically dealwith a simple ball that splashes into water and then os-cillates due to the waves it just created without lots ofanimator’s work. This extraneous work therefore puts adamper on the usage of liquids in computer graphics pro-ductions.

In this paper, a new method allowing to link a com-monly used eulerian fluid simulation to a lagrangian solidsimulation is proposed. While not new to fluid mechanics[3], such a method is new to the field of computer graph-ics. This way, arbitrary objects can be realistically in-corporated into fluid simulations without resorting to anyexternal intervention, as shown in Figure 1. Resultingmotions of fluid and solids automatically appear physi-cally correct since the strong coupling between the twodifferent entities are handled in a coherent way. This is adefinite improvement on previous work, where only uni-

directional actions between fluids and solids were con-sidered, so solids either responded to fluid’s motion butwithout any action on it, or vice-versa.

The new method presented in this paper relies essen-tially on the definition of a new force, computed by tak-ing into account both of the fluid and the solids, to capturethe interaction. This new force is then simultaneously in-troduced in both models, thus guaranteeing coherent be-haviors. Though this method may appear simple, it wasfound to be computationally effective while producingsatisfactory results, as can be seen in the included pic-tures and videos.

The remaining of the paper is organized as follows:section 2 describes both the separates fluids’ simulationmodel and the solids’ one. Section 3 details the couplingscheme between these two models. Section 4 presentslimitations of the method. Section 5 provides resultswhile section 6 analyses the proposed method as well asprovides some hints about improvements that could bemade.

2 Underlying simulations

Before describing how the interaction between liquidsand solids is handled, a description of the two distinctelementary simulation models is given.

2.1 Liquid simulationThe liquid related part of the simulator is based on theNavier-Stokes equations:

∂u∂t

= −(u · ∇)u− 1ρ∇p + ν∇2u + F, (1)

∇ · u = 0, (2)

where a fluid of densityρ and viscosityν is described:u stands for the velocity field,p for the pressure field,andF for the external forces, such as gravity. Equation1 expresses momentum conservation, while equation 2states that the liquid is incompressible, which means thatvolume must be conserved over time.

Since the simulator must be able to cope with a liquidthat does not fill all the simulation space, a way to dis-crimate between the liquid and its surrounding empty en-vironment must be devised. Moreover, full three dimen-sional motions must be achievable, so heightfield-like so-lutions, where waves are not able to overturn, are not rel-evant. The way to solve these two problems is to resortto Markers And Cellssimulation. In this kind of simu-lations, velocity and pressure are evaluated on cells of arectilinear static grid that defines the simulation domain,meanwhile liquid is tracked as a cloud of markers mov-ing inside this grid, as shown in Figure 2. These massless

markers are passively advected according to the under-lying velocity field, which is updated over time using theNavier-Stokes equations. Whether a given cell of the gridis considered full or empty of fluid by the simulator de-pends on the presence of at least one marker inside theconsidered cell.

Figure 2: Fluid Simulation. — Realized using a grid car-ryind fluid velocity and many markers to delineate thefluid presence.

It should be noted that the grid does not sample boththe pressure and the velocity fields at the same locationbut instead makes use of staggered sampling. The differ-ent components of the velocity field are not sampled onthe same position either. Focusing on the velocity sampletaken on a cell, each component is sampled on the cen-ter of the orthogonal faces to the considered componentorientation, whereas pressure is sampled on the center ofthe cell. Such a scheme is chosen due to better stabilityproperties compared with a scheme where the samplesare taken from the same location. A more detailed de-scription of the simulation grid can be found in articlesthat introduced the method in computer graphics [8].

Fluid evolution is achieved using the method pro-posed in [7]. The different parts of equation 1 are simu-lated using different strategies depending on their nature.The convection term−(u · ∇)u is handled by a semi-lagrangian integration process, the viscous termν∇2uis solved using direct differentiation while the externalforce termF contribution is based on simple Euler in-tegration. Mass conservation∇ · u = 0 is enforced bymeans of a projection step inside the target divergencefree space. This projection is computed by solving an ad-equately built equations system using a conjugate gradi-ent solver. This last step takes care of the pressure relatedterm−1/ρ ∇p of equation 1 too.

Apart from liquid simulation, liquid visualization mustalso be addressed. In order to get a surface, a marchingcube algorithm [12] is used. The underlying data field ofthe marching cube is built by counting the fluid markersinside each cell of a fine grid and smoothing these values.While not creating the best looking surfaces, this strategywas chosen among others because it is quite fast and easyto implement while it gives satisfactory smooth surfaces.However, a major drawback of such a surface extraction

system is that artificial fluid dissipation is introduced bythe smoothing step. Nonetheless attention must be drawnto the fact that this dissipation is only visual, and no fluidat all is lost at the simulation level using this technique.This dissipation can be seen on Figure 2 where some ofthe droplets visually disappear during the simulation.

2.2 Solid simulationAlthough fluid is simulated through a rather complex pro-cess, solids are much easier to handle. While solids canbe represented as rigid bodies delineated by a surfacemade of triangles, they can also be represented as a setof linked point masses [14]. This latter model, for whicha simple example is given in Figure 3, is especially wellsuited to animation. It is used in the presented method be-cause of its flexibility to handle both rigid and non-rigidbodies, its easy manipulation, and the way it fits well inthe interacting composite model.

Figure 3: Solid Simulation. — Example of a 2D square:4 nodes are linked using viscous springs.

A spring-mass solid, as one can infer from its name, ismade of a set of masses linked by interactions. Thesecohesion interactions are chosen to be not only elasticbut visco-elastic, which means that the intensityf of theforce between two given particlesi andj is not only de-pendant on their distance but on their closing speed too:

f = −kcoh · (|xi − xj | − dcoh)

−zcoh ·d

dt|xi − xj |.

The forceF is obviously aligned with the direction im-plied by the positions of the two particles:

F = f · xi − xj|xi − xj |

.

To evolve the solid over time, the position of each massis updated independently of the others according to New-ton’s second law taking account of all the applied forces:

x =1m

∫∫ ∑i

Fi dt2.

TheFi forces encompass all the applied forces to themass, coming either from the internal or external interac-tions, such as gravity or obstacle collision forces. The

aforementioned flexibility of spring-mass solids arisesfrom this separate time evolution process of every mass:since their relative position is allowed to change relatedto the stiffness of interactions, all kind of solids can bebuilt, from extremely plastic to the most sturdy ones.

3 Interfacing the two models

To allow a true reciprocal interaction between the twodifferent eulerian and lagrangian models prevously de-scribed in sections 2.1 and 2.2, an interface to bind themis needed.

This interface is chosen to stand as a representation ofthe fluid for the solids, that is, to build a lagrangian de-scription of the eulerian world. In fact, looking at thefluid simulator, this very representation is already avail-able through the use of the massless markers whose roleis to delineate the presence of fluid. Thus, the interface ischosen to be built out of these markers. This way, no newentity is introduced inside the model, helping to avoid un-necessary overcomplication of the whole composite sim-ulator.

Indeed, these markers are already able to easily, yetprecisely, discriminate where fluid is actually located in-side the environment. This is an extremely important factsince the position of the interface is part of the way it af-fects both models. This position dependance is undoubtlyneeded due to the free-surface nature of the simulatedliquids. Indeed, since solids can be half-submerged, itwould be a rough oversimplification to assume that all oftheir surrounding environment is composed only of fluid.Moreover, the point cloud nature of this interface will al-low linkage of the interface with the two different models.

Another interesting property of this interface is thatsince it is defined as a set of points, it is trivial to de-fine a restricted interface as a subset of all the markers.This property is of the utmost interest since it allows oneto easily manipulate an almost steady interface made ofa constant number of points even if the number of fluidmarkers is increased to improve fluid representation. Thisallows an animator to avoid altering dynamic propertiesof the interface between a low resolution prototyping an-imation and a full resolution production one.

The composite simulator is thus formed of three logicalentities:

• The eulerian Navier-Stokes based fluid simulator;

• The lagrangian spring-mass solids simulator;

• The interface that must be able to flow data betweenthem, translating these data when they are not read-ily interpretable by the destination model.

The final coupling is asymmetric at a coarse level, ascan be seen on the synthetic decription depicted in Figure

4 and is comprised of two parts. On the one hand, theconnection between the interface and the solid model issymetric through the definition of a force equally reintro-duced in the two entities. On the other hand, the bindingbetween the fluid model and the interface is an assymet-ric one, where velocity information are transferred fromliquid toward the interface while forces information flowthe reverse way. This scheme has been chosen instead ofthe converse one, where velocity would flow from solidsto liquid and force from liquid to solids, since it allowsthe aforementioned interface definition which is experi-mentally found to be effective and efficient.

Figure 4: Overview of the coupling scheme.

3.1 Interface / Solids relationshipThe relation between the interface and the solids is han-dled in an efficient way by the use of a force exerted be-tween each markeri of the interface and each nodal massj of the solids. This elementary forcefelemij is visco-elastic, such as the internal cohesion force of solids, butwith a limited radius of influence:

f̄elemij = −kint · (|xi − xj | − dint)

−zint ·d

dt|xi − xj |,

felemij ={

f̄elemij if |xi − xj | ≤ R0 if |xi − xj | > R

.

This way, solids are naturally affected by the nearby in-terface, which is itself determined by liquid movements.The limited interaction distance between solid and in-terface guarantees that objects are only influenced byclosely located liquid motions, and not by far, thus phys-ically irrelevant, liquid movements.

The radiusR is closely related to the way solids aremodeled: its size must be big enough to avoid holes inthe solids while correctly approximating their surfaces.

No general guideline can be given to setup the stiffnesskint and the viscosityzint, except thatkint must be stiffenough to repel water markers from the interior of thesolid. Unless interior is preserved from fluid, objectswill be filled with fluid, preventing a buoyancy force todevelop. This buoyancy force automatically stems frompressure difference in the fluid around the object due togravity.

Given this force definition, the complete interactionforce can be computed for each node belonging to a solidFsolid, as well as for all of the interface markersFmark,as the sum of the forces in which the entity is involved:

Fsolidi =∑

j

Felemij , Fmarkj =∑

i

Felemij .

This resulting force is directly taken into account bythe solids as one of the many gathered forces, such asgravity or internal cohesion. No further consideration isnecessary since this raw expression of the new force per-fectly fits the evolution scheme of solids.

However, if a lagrangian simulator can make direct useof this force, an eulerian simulator can not. Although notof a straightforward use, this force should not be wastedsince it carries all the information to update the euleriansimulator to make it react to the solids. The interactionforce is thus stored on the interface marker basis to bereintroduced in the eulerian simulator after suitable trans-formation.

3.2 Interface / Fluid relationshipOn the other side of the interface, half of the relation be-tween the interface and the fluid is already built. Indeed,since the interface is formed out of the fluid advectedmarkers, the interface is already under the influence offluid in a physically realistic way, and fluid velocity isable to affect solids through the interface.

The last link that must be devised to complete the in-teraction scheme is the link between the interface and thefluid, to keep the solid to liquid force flowing from inter-face to the fluid model. Recalling the last term of equa-tion 1, the Navier-Stokes equation related to momentumconservation, external forces, such as the interaction one,can be incorporated inside the simulation. However, un-like gravity, the interaction is spatially dependant, whichmeans that last term of the the equation should be updatedto F(x) instead ofF, as depicted in Figure 5.

To populate this force fieldFinter(x), a simple yet ef-fective method, summarized in Figure 6 is used. Forcesstored at interface markers are summed on a per compu-tational grid cell basis, so as to obtain the forceF̄interglobally applied to the fluid located inside each cellC:

F̄inter(C) =∑i∈C

Fmarki .

Figure 5: The interaction is defined as a spatially depen-dant force field for the fluid simulator.

The key point of this method is that partially filled cellsdo not require special handling with respect to completelysubmerged ones. Indeed, the relative number of interfacemarkers per cell is automatically able to handle this fillingdiscrepancy.

Once these per-cell forces have been computed, theforce field is ready to use. Due to the staggered natureof the grid, the actual forceFinter used during time evo-lution of the velocity carried by a given face inside thegrid is obtained by averaging the two forces stored in theneighboring cells. Using the Kronecker’s deltaδpq, forcoordinates denoted asn = 0, 1, 2, this can be expressedas:

Finter(x(i,j,k)+δ/2) = F̄inter(Ci,j,k)/2� δ+ F̄inter(C(i,j,k)+δ)/2� δ

where δ = [δ0n, δ1n, δ2n] and where� stands forcomponent-wise multiplication:(xi)� (yi) = (xi · yi).

Figure 6: Processing of raw per-marker interaction forcesto take account of interaction into eulerian simulator. —Left: Forces of markers are summed on a grid cell basis.Right: Adjacent cell forces are used to compute force re-quired in speed evolution of faces.

4 Limitations of the method

Despite the clear improvement of allowing a simple twoway interaction, the method still suffers from some draw-backs.

Indeed, the discretization of the force field is extremelycoarse and imprecise. This imprecision can sometimes

lead to ”spurious currents” that may appear at the inter-face between the solids and the fluid. However, it wasfound that these currents are most of the times unnotice-able and without tangible influence on the liquid surfaceposition. It should be noted though that this problem isintrinsic to the discretization of a force field in euleriansimulation methods and that the Computational Fluid Dy-namics community is still trying to address this problem.

Another limitation of the method is its relatively diffi-cult setup. Indeed, multiple interlocked parameters needto be devised, each having dramatic influence on the re-sults.

5 Results

This section presents results obtained from the previouslydescribed simulator. Pictures are rendered using the Ra-diance photo-realistic renderer1. The liquid surface is ex-tracted from the simulation using the technique describedin section 2.1. All simulations were run on an Athlon XP1800+ with 512MB RAM. Meaningful parameters of thedifferent examples are gathered in Table 1. The timestepused for all the simulations is1/240 second, but an exportof the surface is only performed every1/24 second.

This small timestep is due to the solids introduced inthe simulation. Indeed, they must have small displace-ments during a timestep compared to the size of a cellof the fluid simulation grid. This is a requirement if onewants to avoid instabilities and to produce meaningfullresults.

Figure 7 illustrates buoyancy. On the beginning ofthe simulation, both the fluid and the ball are completelysteady, but buoyancy allows the ball to surface.

Figure 8 shows a ricochet of a ball thrown in a tank.Figure 9 involves a high speed jet striking a cube.

The cube is made of an elastic material that deformsbut quickly regains its shape such as rubber. It is mod-eled as thousand nodes arranged in a regular cubic lat-tice scheme. When the jet strikes the cube, water is ableto temporarily tear the cube, showing that the interactionforce can be precisely localized in space, even down to asub-object level. This can be seen in the close up shotsdepicted in Figure 10. Meanwhile, the cube is able todeflect the tip of the jet, before being put into move-ment. It should be noted that the noisy appearance thatfluid develops in complex motions areas is not inducedby the simulator itself but is rooted in the fact that thefluid surface evolves in a too convoluted mesh to be ren-dered smoothly.

Figure 11 depicts a U-shaped container, half filled withwater, where two different cubes are dropped on top ofthe water, in the two branches of the U container. The red

1http://radsite.lbl.gov/radiance/HOME.html

cube is five times as heavy as the yellow one and nearlythree times as rigid. The container has a square cross-section, almost identical to the cross-section of the twocubes. With such a situation, when a cube hits the fluid,the movement is entirely transmitted at the opposite endof the fluid without loss of momentum, like in a hydraulicbrake system. The interaction between the cubes and thefluid is clearly visible looking at the lighter yellow cube,on the left of the device. When it first hits the water, itpushes the fluid upward in the opposite part of the con-tainer, but when the second heavier cube hits the water abit later, it is itself propelled upward by the water. Thetwo way interaction is thus clearly demonstrated: solid tofluid forces are seen first and then fluid to solids forcesare visible.

6 Conclusions and future works

A new method that allows realistic interaction betweena free surface fluid and solids has been presented. Thismethod proved to be easy to implement, computationallyaffordable and efficient to fulfill its role.

This method is believed to provide a definite improve-ment in the visual realism of animations involving fluidby providing a simple way to add more life in thesescenes.

Since the method does not involve deep modificationsof classic eulerian fluid simulators, it should be possibleto integrate this extension into more sophisticated simula-tors that give better representation of the fluid’s surface tofurther increase realism of animation without much trou-ble.

Improvements could also be made by taking steps toinsure that simulation grid refinement does not affect dy-namic properties of the interaction, in order to allow anefficient transition between prototype and production an-imations. Moreover, more intuitive parameters would beof great help during the setup phase of the simulation.

Another possible enhancement lies in the solid repre-sentation. Indeed, not all objects are easy to representas a set of nodal masses fitted with spherical interactionfields. It would thus be interesting to extend the definitionof the interaction force to an interaction between fluid’smarkers and triangles, to deal with complex triangulatedmeshes in an obvious way.

References

[1] Jim X. Chen and Niels Da Vitoria Lobo. Towardinteractive-rate simulation of fluids with moving ob-stacles using Navier-Stokes equations.Graphicalmodels and image processing: GMIP, 57(2):107–116, March 1995.

http://radsite.lbl.gov/radiance/HOME.html

Figure Simulation Particles Solid Interaction Avg frame Avg surfacegrid size number or nodes particles simulation reconstruction

introduction rate subset time time7 40× 50× 40 4 676 065 1 1 / 8 45.4 sec 9.5 sec8 100× 40× 20 1 600 000 1 1 / 16 23.2 sec 4.1 sec9 40× 40× 20 800 000 / sec 1000 1 / 8 12 sec 9 sec11 20× 40× 5 250 000 2000 1 / 1 8 sec 1.6 sec

Table 1: Parameters of the different simulations.

[2] N. Chiba, S. Sanakanishi, K. Yokoyama,I. Ootawara, K. Muraoka, and N. Saito. Visualsimulation of water currents using a particle-basedbehavioural model. The Journal of Visualiza-tion and Computer Animation, 6(3):155–171,July–September 1995.

[3] Eric Climent and Martin R. Maxey. Numericalsimulation of random suspensions at finite reynoldsnumbers.International Journal of multiphase flows,2001.

[4] D. S. Ebert, W. E. Carlson, and R. E. Parent. Solidspaces and inverse particle systems for controllingthe animation of gases and fluids.The Visual Com-puter, 10(4):179–190, March 1994.

[5] Douglas P. Enright, Stephen R. Marschner, andRonald P. Fedkiw. Animation and rendering of com-plex water surfaces. InSIGGRAPH 2002 Confer-ence Proceedings, pages 736–744, 2002.

[6] N. Foster and D. Metaxas. Controlling fluid anima-tion. Computer Graphics International 1997, June1997.

[7] Nick Foster and Ronald Fedkiw. Practical anima-tion of liquids. In SIGGRAPH 2001 ConferenceProceedings, pages 23–30, 2001.

[8] Nick Foster and Dimitri Metaxas. Realistic anima-tion of liquids. Graphical Models and Image Pro-cessing, 58(5):471–483, September 1996.

[9] Patrick Fournier, Arash Habibi, and Pierre Poulin.Simulating the flow of liquid droplets. InGraphicsInterface, pages 133–142, June 1998.

[10] A. Habibi, A. Luciani, and A. Vapillon. Aphysically-based model for the simulation of reac-tive turbulent objects. InWinter School of ComputerGraphics 1996, 1996.

[11] Michael Kass and Gavin Miller. Rapid, stable fluiddynamics for computer graphics.Computer Graph-ics, 24(4):49–57, August 1990.

[12] William E. Lorensen and Harvey E. Cline. March-ing cubes: A high resolution3D surface construc-tion algorithm. Computer Graphics, 21(4):163–169, July 1987.

[13] Gavin Miller and Andrew Pearce. Globular dynam-ics: A connected particle system for animating vis-cous fluids. Computers and Graphics, 13(3):305–309, 1989.

[14] Gavin S. P. Miller. The motion dynamics of snakesand worms. InSIGGRAPH 88 Conference Proceed-ings, pages 169–178, 1988.

[15] J. F. O’Brien and J. K. Hodgins. Dynamic simu-lation of splashing fluids. InComputer Animation’95, pages 198–205, 1995.

[16] Jos Stam. Stable fluids. InSIGGRAPH 99 Confer-ence Proceedings, pages 121–128, 1999.

[17] Jos Stam and Eugene Fiume. Turbulent wind fieldsfor gaseous phenomena. InSIGGRAPH 93 Confer-ence Proceedings, pages 369–376, 1993.

[18] Jos Stam and Eugene Fiume. Depicting fireand other gaseous phenomena using diffusion pro-cesses. InSIGGRAPH 95 Conference Proceedings,pages 129–136, 1995.

[19] Henrik Weimer and Joe Warren. Subdivisionschemes for fluid flow. InSIGGRAPH 99 Confer-ence Proceedings, pages 111–120, 1999.

[20] Jakub Wejchert and David Haumann. Animationaerodynamics. InSIGGRAPH 91 Conference Pro-ceedings, pages 19–22, 1991.

[21] Gary D. Yngve, James F. OBrien, and Jessica K.Hodgins. Animating explosions. InSIGGRAPH 00Conference Proceedings, pages 29–36, 2000.

Figure 7: Buyoancy — The ball is initially steady but automatically surface.

Figure 8: Ricochet of a ball inside a tank.

Figure 9: High speed collision of a water jet and a cube.

Figure 10: Close-up of the jet collision with the cube. — In upper row, regular scene is used. In lower row, only thecube is represented.

Figure 11: U-shaped container and two cubes.

IntroductionUnderlying simulationsLiquid simulationSolid simulation

Interfacing the two modelsInterface / Solids relationshipInterface / Fluid relationship

Limitations of the methodResultsConclusions and future works