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A Study on Particle Level Sets and Navier-StokesSolver For Water Simulations

Koray Balci, Isik Baris Fidaner

Abstract— In this paper we present a survey on simulation offluids in computer graphics domain, especially focusing on theones with the level-set approach. We overview the mathematicalfoundations that governs the behaviour of fluids under theinfluence of physics in an environment, present state-of-the-artin surface tracking methods that drives the data for animation.

We focus on a specific paper from Fedkiw et. al. [1] usingparticle level sets for surface tracking as basis for our own studies.We also show our early results and present our progress in thedomain.

Index Terms— Computational Fluid Dynamics, Level-Sets, An-imation, Simulation.

I. I NTRODUCTION

A NIMATION of fluids is a challenging problem involvingboth physical simulation of fluid behaviour in an environ-

ment using computational methods, and visual representationusing computer graphics techniques. Despite computationaland theoretical difficulties, with the advances in computerhardware and numerical methods, realistic simulation of fluidshas become a popular field, thanks to the support from appli-cation areas such as simulation, entertainment/movie industry,medical imagery and video games.

Realistic representation of fluids as physical entities andtheir interaction with other objects in a scene requires com-plex methods and computationally expensive techniques. Fluiddynamics is a field of engineering which studies the physicallaws governing the flow of fluids under various conditions suchas simultaneous flow of heat, phase change, mass transfer, etc.

Great effort has gone into understanding the governing lawsand the nature of fluids themselves, resulting in a complexyet theoretically strong field of research. Computational FluidDynamics (CFD), is used to generate flow simulations with thehelp of computers. CFD involves the solutions of governinglaws of fluid dynamics numerically. The complex set of partialdifferential equations are solved on a geometrical problemdivided into small volumes, commonly known as mesh (orgrid).

Once the fluids and their interaction with the surroundingis modeled using CFD methods, realistic animation under thelaws of physics is the next problem to be solved. In orderto tackle that, many approaches commonly known assurfacetracking methodologiesare proposed in the literature. Finally,photorealistic rendering using computer graphics techniqueshas to be applied in order to achieve visual appeal for theviewers.

In recent years, two pioneering works changed the course ofapplications of fluid simulations in computer graphics domain.In 1999, J. Stam came up with a stable, simple and fast

computational method for solving Navier-Stokes equations [2].Second, R. Fedkiw et.al. applied level sets to tracking watersurface problem and achieved impressive results [1]. Thesetwo studies also inspired our work presented here.

In this study, we first overview the CFD techniques com-monly used in computer graphics domain. Note that, incomputer graphics, some of the algorithms in CFD domainis avoided in general because of computational complexityinvolved. Next, state-of-the-art in surface tracking method-ologies are presented. In this study, we will not cover theavailable photorealistic rendering techniques that can be usedfor visualizing the results, the reader is invited to the surveypaper by A. Iglesias [3]. Then, we will summarize the workof Fedkiw et. al. on the subject and our own progress inimplementing water simulation. Finally, we conclude with ageneral overview of the material presented and disscuss aboutpossible future work.

II. M ATHEMATICAL FOUNDATIONS

Fluid dynamics can be classified in two main approaches.In one approach, the calculations are based on changes atconstant positions. This is called the Euler approach. But theLagrangian approach focuses on moving particles.

For example, consider a waterfall in a river. The waterflows slowly in the river and quickly in the waterfall, but themovement is stable. If we use Euler approach, we find out thatthe water flows at constant speed at every constant position,there is no acceleration. But if we use Lagrangian approach,we follow the particles, and the particles accelerate before andafter the waterfall.

The techniques used in computational fluid dynamics gen-erally correspond to one of these two approaches.

A. Eulerian approach

In this approach, the fluid flows through a 3D continuousspace. This space is computationally simulated by a 3D griddivided into small cells with constant positions. These smallcubes are an approximation to the differential cells. Level setevolution is used for evolving the liquid surface mesh.

There are two grids in the same space.• Level-set grid: This is a high resolution grid. An implicit

function is defined by scalar numbers on every grid cell.Surface of the fluid is defined by level set zero of thisimplicit function.

• Navier-Stokes grid: This is a lower resolution grid. Everycell has a pressure value. Between every adjacent cell (inother words, at every face) a velocity value is assigned.

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Summing up velocities through a certain cell gives thecell-centered velocity vector at that point.

In Eulerian grids, basic problem is volume loss. In levelset evolution, very detailed turbulences such as splashing mayresult with volume loss or volume gain. This is because gridresolution is limited and mass conservation is not guaranteed.In other words, since the Eulerian grid is too coarse, andmarked as filled or empty only, even if the cell can be partiallyfilled, we have the chance of losing or gaining extra volumeduring every step.

B. Lagrangian approach

In contrast with the Eulerian approach, Lagrangian approachcalculates the motion of movable particles, instead of a gridwith constant positions. As moving particles define the liquidvolume, updating the liquid state is updating particle positionsand velocities.

Particle-based implementation solves the problem of massconservation. Every particle represents a constant mass, andmass is conserved if no particles are inserted or removed. Butthis also brings some limitations. As particle-particle interac-tions are more complex than cell-cell interactions, there canbe a limited number of particles. This kind of implementationis more appropriate for liquids that have rougher surface andhigher viscosity such as mud.

A method based on fluid particles is Smoothed ParticleHydrodynamics (SPH). In SPH, liquid is composed of particleswith certain positions. Variables like pressure, density aredistributed in the space around the particles by a smoothingkernel, for example Gaussian distribution.

In addition to fully Lagrangian and fully Eulerian ap-proaches, there are methods that use a combination of thesetechniques.

C. Semi-Lagrangian approach

The mathematics used in computational fluid dynamics isdifferential calculus in finite difference forms. Navier-Stokesequations are the differential equations of fluid motion.

In the case of incompressible fluids such as water, we haveto define a surface between liquid and air. This is defined asthe zero level-set of an implicit function of the fluid.

To simulate Newtonian incompressible fluids, two key vari-ables are pressure and velocity. Pressure is defined as the scalarfield with values at the center points of cells. Velocity is avector field that have values at the faces that connect cells toeach other.

In the case of non-Newtonian and compressible fluids,viscosity and density must also be defined as functions oftime and space. But they are generally assumed to be uniformthroughout the fluid volume for fluids like water.

In Figure 1, ui, uj , uk define the velocities at the facesof cells. At the center of each cell is the pressure value. Sixvelocities from the faces are added to find the cell-centeredvelocity. In this grid structure, finite difference versions ofgradient, divergence and Laplacian operators are used.

In the grid, some cells are empty meaning that there isno fluid inside. These are ignored for performance. Surface

Fig. 1. Grid cell used for Navier-Stokes.

cells (that contain both air and fluid) are given a constantatmospheric pressure.

D. Navier-Stokes equations

The motion of fluids are governed by the laws definedby Navier-Stokes equations. These are used in simulationsfor obtaining the fluid velocity field at discrete time steps.Navier-Stokes equations are differential equations obtained bycombining the conservation laws in fluid motion.

∇ · u = 0 (1)

ut = −(u · ∇)u + v∇2u− 1ρ∇p + f (2)

First equation is the conservation of mass. It states thatdivergence of velocity field is zero at every point. This meansthe mass flowing into a cell is equal to the mass flowing out ofit. This is because it is assumed that the fluid is incompressible.

Second equation is the conservation of momentum.ut is thetime derivative of velocity field. In the right hand side, thereare four terms:

• Advection term−(u · ∇)u is the effect of nearby cells’velocity to this cell’s velocity.

• Diffusion term v∇2u defines how fast variations in ve-locity are damped. Higher viscosityv makes dampingquicker.

• Pressure term− 1ρ∇p pushes the fluid from high pressure

to low pressure.• Body forces in termf are the external forces on the fluid

such as gravity or wind.

E. Level sets

As fluid motion is defined as the evolution of an implicitfunction φ with time, the fluid surface is defined mathemati-cally as the zero level set of an implicit function. This functionis defined on a grid in the space fluid is flowing. This is amore detailed sub-grid that sits inside the Navier-Stokes grid.φ evolves in time according to the following equation:

φt + u · ∇φ = 0 (3)

In this equation,u is the liquid velocity field, calculated bysolving Navier-Stokes equations at discrete time steps.

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Surface mesh (zero isocontour) of a level set is createdby using the Marching Cubes algorithm [4]. In the March-ing Cubes algorithm, new vertices are created on the edgesconnecting air points to liquid points. Then, these vertices areconnected to form a triangulated mesh structure in each surfacecell throughout the grid.

III. SURFACE TRACKING METHODS

The fundamental problem of tracking a surface as it isadvected by some velocity field arises frequently in applica-tions such as surface reconstruction, image segmentation, andfluid simulation. Unfortunately, the naive approach of simplyadvecting the vertices of a polygonal mesh, or other explicitrepresentation of the surface, quickly encounters problemssuch as tangling and self-intersection. Instead, a family ofmethods, known as level-set methods, have been developedfor surface tracking. These methods represent the surfaceimplicitly as the zero set of a scalar field defined over theproblem domain. The methods are widely used, and the textsby Sethian [5] and Osher and Fedkiw [6], and Osher andSethian [7] provide an excellent introduction to the topic. Oneof the key issues that distinguishes various level-set and similarapproaches is the representation of the scalar field, which mustcapture whatever surface properties are important to a givenapplication.

Because surface tracking arises in a variety of contexts,the topic has received a significant amount of attention. Evenin the limited context of fluid animation, there has been agreat deal of excellent work on simulating fluids with freesurfaces [8], [1], [9], [10], [11], [12], [13]. The methods avail-able for tracking free surfaces of liquids can be roughly sortedinto four categories: level-set methods, particle-based methods,particle level-set methods, and semi-Lagrangian contouring. Inthis section we will briefly present these methods.

A. Level-Set Methods

Many of the most successful solutions to the surface track-ing problem are based on level-set methods, which wereoriginally introduced by Osher and Sethian [7]. A completereview of level-set methods is beyond the scope of thisarticle, and we recommend the surveys by Sethian [5] andOsher and Fedkiw [6]. Level-set methods represent a surfaceas the zero set of a scalar function which is updated overtime by solving a partial differential equation, known as thelevel-set equation. This equation relates change of the scalarfunction to an underlying velocity field. By using this implicitrepresentation, level-set methods avoid dealing with complextopological changes. However, the scalar function is definedand maintained in the embedding three-dimensional space,rather than just on the two-dimensional surface. In practice,scalar function values need only be accurately maintainedvery near the surface, resulting in a cost that is roughlylinear in the complexity of the surface. One difficulty withlevel-set methods is that they generally require very high-order conservation-law solvers, though fast semi-Lagrangianmethods have been shown to work in some cases [20], [15].The most significant drawback to using level-set methods

to track liquid surfaces is their tendency to lose volume inunderresolved, high-curvature regions. See Enright et al. [9]for an excellent discussion of the reasons for this volume loss.

Berentzen and Christensen [24] built a sculpting systemusing a level-set surface representation which could be ma-nipulated by a user with a variety of sculpting tools. Theyused adaptive grid structures to store the scalar field. However,they used a two-level structure rather than a full octree. Theyalso used semi-Lagrangian methods to update their level-setfunction. However, when evaluating the distance function afterthe semi-Lagrangian path tracing, they interpolated distancevalues stored on a regular grid. Sussman and Puckett [25] cou-pled volume-of-fluid and level-set methods to model dropletdynamics in ink-jet devices. Volume-of-fluid [27] techniquesrepresent the surface by storing, in each voxel, avolumefractionthe proportion of the voxel filled with liquid. Any cellwhose fraction is not one or zero contains surface. Unfor-tunately, this representation does not admit accurate curvatureestimates, which are essential to surface tension computations.However, accurate curvature estimates are easily computedfrom level-set representations. Thus, the authors combinedvolume-of-fluid and level-set representations to model surfacetension in ink droplets. Some volume-of-fluid methods buildan explicit surface representation from the volume fractionsstored in each voxel.

B. Particle-Based Methods

A number of researchers [26], [8], [30], [29], [28], [31],[32], [33], [34] have used particles to track surfaces. Inmany of these methods, the simulation elements are particles,which are already being tracked throughout the volume of thedeforming liquid or solid. The surface can then be implicitlydefined as the boundary between where the particles are andwhere they arent. The particles can be visualized directly, orcan be used to define an implicit representation using blobbiesor moving least-squares methods. Premoze et al. [34] wenta step further and used particle positions and velocities toguide a level-set solution. Mller et al. [32] and Pauly etal. [33] used special particles, called surfels, to represent thesurface. Surfels store a surface normal as well as positionand there are generally many more surfels than simulationparticles. The principal drawback of these methods is thatgenerating high quality time-coherent surfaces can be difficult:directly visualizing the particles is insufficient for high-qualityanimations, methods which convert the particles to someother representation on a per-frame basis often lack temporalcoherence, and methods which must run sequentially throughthe frames or run during the simulation are often quite costly.Additional difficulties arise when trying to ensure a goodsampling of the surface.

C. Particle Level-Set Methods

To address the volume loss of level-set methods, Enrightand his colleagues [9], [14], [15] built on the work ofFoster and Fedkiw [1] to develop particle level-set methods.These methods track the characteristics of the fluid flow with

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Lagrangian particles, which are then used to fix the level-set solution, essentially increasing the effective resolution ofthe method. Recently, these methods have been extended towork with octrees [15], [12], allowing for very high resolutionsurface tracking. These methods represent the current stateof the art on tracking liquid surfaces for animation, buthave some drawbacks. In particular, the published particlecorrection rules choose a single particle to provide the signed-distance value. Since there is no guarantee that the sameparticle will be chosen at subsequent timesteps, the method isextremely susceptible to high-frequency temporally incoherentperturbations of the surface. The artifacts are most noticeablewhen the surface thins out below the grid resolution andparticles happen to be near some of the sample points, but notothers. Also, the method has a large number of parameters andrules, such as the number of particles per cell and the reseedingstrategy, which need to be decided, often in an application-specific way. Finally, the method tends to produce very smoothsurfaces with very little detail, which is desirable in some,but not all, applications. Despite these drawbacks, the particlelevel-set methods have been very successful and represent asignificant step forward in the area of surface tracking forliquid simulations.

D. Semi-Lagrangian Contouring

Recently, Strain [19], [20], [21], [22], [23] has written aseries of articles building a theoretical framework culminatingin the formulation of surface tracking as a contouring problem.He demonstrated his semi-Lagrangian contouring method ona variety of two-dimensional examples. The method proposedby Bargteil et. al. [16] is based on the method presented byStrain [23], but with variations and extensions to deal withproblems that arise in three-dimensional computer animation.While this method bears a number of similarities to level-set methods and takes advantage of many techniques devel-oped for those methods, it does not directly attempt solvingthe level-set equation. By formulating surface tracking asa contouring problem, they avoid many of the issues thatcomplicate level-set methods. In particular, they do not havethe same volume loss issues which prompted the particle level-set methods: while the algorithm proposed does not explicitlyconserve volume, semi-Lagrangian path tracing tends to con-serve volume in the same way as the Lagrangian particles inthe particle level-set method.

IV. FOCUS

In 2001, Foster and Fedkiw presented a method for realisticanimation of fluids [1]. They proposed a hybrid method whereliquid volume is represented by a combination of level-set andparticles. By combining these techniques, they could preventthe drawbacks of both Eulerian and Lagrangian approacheswhile still having the benefits of them. They also used anintuitive way for simulating the interaction of liquid withmoving objects. In our study, this study will be the basis forour future research on the subject. Thus, we will focus on thiswork in detail.

Their method consists of the following steps:

1. Modelling the environment by a 3D grid of voxels. Thereare three types of voxels (or cells); that are full of liquid, thatare empty (full of air) and that contain both (surface cells). Ifexternal objects such as static glass container for the wateror any moving object exist in the same scene, some cellsare inside these objects and some are at the boundary of theobjects. The resulting grid is staggered, i.e. either occupied bywater completely, or filled with air or impermeable object.

2. Modelling the liquid by a combination of particles andan implicit surface (level-set). There is a signed distancefunction covering the space. Zero level-set of this functionis the liquid surface. There are also particles near the surfacein the liquid. These particles are automatically concentratedat surface regions with high curvature. Velocity and pressurefield that are updated by Navier-Stokes equations are definedon a seperate 3D grid. The level-set function is defined on ahigher resolution sub-grid that sits inside Navier-Stokes grid.

Following steps are repeated for every simulation frame:3. Updating the velocity field by solving the second Navier-

Stokes equation (see Eq 2). First, velocities through emptycells are initialised to zero, and pressures at empty cells andsurface cells are set to atmospheric pressure. Velocities atsurface cells are set directly by explicity enforcing incom-pressibility (first Navier-Stokes equation, see Eq 1). This is notenough, because every grid cell must satisfy incompressibility,but this is done in the 5th step. Finally, in this step, thevelocity-pressure equation is solved by using finite differencescombined with a semi-Lagrangian method, see Stam [2] fordetails. They use standard central differencing for viscousterms as described by Foster and Metaxas [8]. Time stepsare computed by using CFL condition. This condition statesthat a time step cannot be large enough to make a significantchange in the liquid. This is numerically cell width divided bymaximum speed. Navier-Stokes time steps are taken (aboutfive times) larger to increase performance, but level-set andparticle evolution is done in small time steps.

4. Adjusting liquid velocity near moving objects. The liquidvelocities at every object boundary cell is changed. The liquidvelocity must match the object velocity in the direction normalto object surface.

5. It is necessary to ensure that mass is conserved at everyliquid cell. Incompressibility should be enforced by applyingthe first Navier-Stokes equation. A method proposed by Fosterand Metaxas [8] is Successive Over Relaxation. But Fedkiwand Foster propose a more efficient algorithm. They derive anequation by using Laplacian operator to relate local pressurechange to velocity divergence in a cell. They discretize thisequation and reduce the problem to a symmetric and positivedefinite linear system. Then they solve it using PreconditionedConjugate Gradient method.

6. In this step, the liquid volume and particles are visited.Particles are updated by velocities obtained from the velocitiesof grid cells they belong. Trilinear interpolation is used forthis purpose. They are moved inertialessly by constant speed.Signed distance functions are defined for each particle asa sphere with negative distance values inside and positiveoutside. General implicit signed distance function is created byusing Fast Marching Method [5] with initial condition defined

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by particle spheres. Level-set evolution is done by a convectionequation that defines change ofΦ with respect to time by thevelocity field. Semi-Lagrangian methods can be used to solvethis equation, but they prefer to use a higher order upwindingdifferencing procedure [17]. After setting values of the implicitfunction, the polygon structure of the liquid surface can becreated by using Marching Cubes [18] algorithm.

V. PROGRESS

In this section, the implementation progress and resultsare presented. Our work can be explained by two differentsolvers, one for calculating the velocity vector field governedby Navier-Stokes equation, and the other for tracking thesurface using level sets under the influence of the vector fieldfound.

A. 3D Navier-Stokes Solver

As explained previously, the first step in our work is tosolve Navier-Stokes equations in 3D grid. For this task, weadopted Jos Stam’s original Navier-Stokes solver source codewhich is publicly available. Unfortunately, Stam’s code worksin a 2D grid and should be modified for 3D. In addition, hiscode involves a density scalar field representation to simulategaseous behaviour in the vector field which is unwanted in oursimulations. After studying the code, we managed to modifyit to fit our needs.

The resultant vector field in a 3D grid is shown in Fig. 2.The image shows the vector field under the initial influenceof a force created externally. In Fig. 3, you can see the 2DNavier-Stokes grid of the original Stam’s code.

Fig. 2. Our Navier-Stokes Solver in 3D.

Fig. 3. Jos Stam’s Navier-Stokes Solver in 2D.

B. Particle Level Set Solver

For particle level set simulations, we used the open sourcelibrary by Emud Mokhberi from UCLA. The library is capableof maintaing a level set under the influence of an externalvector field. We managed to integrate the Navier-Stokes solveras the vector field input. However, the library was not robustenough to achieve satisfactory results for different configura-tions and hard to fine tune as a general purpose solver. Oncewe modified the library to use the water surface (originally, itwas using a solid Zalesak’s disc as the level set boundaryinput), and use our time varrying Navier-Stokes solver asvector field input (it was working for constant vorticity field), itstarted to create issues. After studying the code and fine tuningthe parameters we got the following result. The image in Fig. 4shows the surface advected by Navier-Stokes equations underan initial external force.

One of the main reasons of the unsatisfactory results is thatobtained from Mokhberi’s library is that we could not manageto incorporate the effect of gravity in our Navier-Stokes solver.The original code by Stam did not take care of this since itwas written to work in 2D case.

We also implemented our own level set method, and ourearly result is shown in Fig. 5.

C. Visualization

For visualizing the results, we used VTK library. Especiallyfor viewing the vector field created using Navier-Stokes equa-tions and marching cube implementation for creating a meshfrom the implicit surface, we relied heavily on VTK.

VI. CONCLUSION

In this study, we presented an overview of the state of theart approaches in simulation of fluids by laying out the mathe-matical foundations used and recent surface tracking methodsproposed for animation under the influence of physics. Particle

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Fig. 4. Mokhberi’s modified particle level set.

Fig. 5. Our pseudo-particle level set.

level-sets is a recent approach yet to mature and although it isa complex method, it brings new insights to the field of fluidanimation. Therefore, among other surface tracking methods,Fedkiw and his colleagues work on the subject draws ourattention the most.

In the final sections, we focused more on Fedkiw et. al.’sinitial paper in their series of work on particle level sets. Westudied their foundations and did some progress.

In the future, we plan to study and explore further andimprove on the presented line of work.

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