IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS 1

View-Dependent Multiscale Fluid SimulationYue Gao, Chen-Feng Li, Bo Ren, and Shi-Min Hu

Abstract—Fluid flows are highly nonlinear and non-stationary, with turbulence occurring and developing at different length and timescales. In real-life observations, the multiscale flow generates different visual impacts depending on the distance to the viewer. Wepropose a new fluid simulation framework that adaptively allocates computational resources according to the viewer’s position. First, a3D empirical mode decomposition scheme is developed to obtain the velocity spectrum of the turbulent flow. Then, depending on thedistance to the viewer, the fluid domain is divided into a sequence of nested simulation partitions. Finally, the multiscale fluid motionsrevealed in the velocity spectrum are distributed non-uniformly to these view-dependent partitions, and the mixed velocity fields definedon different partitions are solved separately using different grid sizes and time steps. The fluid flow is solved at different spatial-temporalresolutions, such that higher-frequency motions closer to the viewer are solved at higher resolutions and vice versa. The new simulatorbetter utilizes the computing power, producing visually plausible results with realistic fine-scale details in a more efficient way. It isparticularly suitable for large scenes with the viewer inside the fluid domain. Also, as high-frequency fluid motions are distinguishedfrom low-frequency motions in the simulation, the numerical dissipation is effectively reduced.

Index Terms—fluid simulation, Hilbert-Huang transform, fluid velocity spectrum, flow field decomposition, view-dependent partition.

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1 INTRODUCTIONFluid simulations based on the Navier-Stokes equationshave achieved great success in computer graphics. Manycompelling methods with impressive animations havebeen reported in the past decade. However, fluid simu-lation remains a challenging task where improving thevisual effect of fine-scale fluid motions and reducingthe demand of computational resources are the mainconcerns. Unlike computational physics, the focus ofgraphics applications is on the visual effect of the finalrendered images and animations. This implies a highpotential value for exploiting the unique viewing in-formation to improve existing fluid simulators. In thiswork, we propose a novel approach which incorporatesthe viewing information into the fluid solver and adap-tively simulates the fluid at multiple scales, such that thecomputational resources are allocated to the key regionsand to the key scales that have important impacts onthe visual impression of turbulent flows. This approachis particularly suitable for large scenes with the viewerimmersed in the fluid domain. Such kind of scenes arerare to be seen in previous publications, but are oftenmost desired by movie directors and game designers.

Generally, the techniques considering the viewer arereferred as the levels of details, which has become astandard tool widely used in 3D geometry representationand texture rendering. The basic idea is that whenthe object is far from the viewer, a reduced geometryrepresentation or a reduced texture is applied. Thissimplification is supported by the fact that, for human

• Y. Gao and B. Ren are with TNList, Department of Computer Science,Tsinghua University, Beijing 100084, China.

• C.F. Li is with the College of Engineering, Swansea University, SwanseaSA2 8PP, UK.

• S.M. Hu is corresponding author and with TNList, Department ofComputer Science, Tsinghua University, Beijing 100084, China.

visual perception, the higher frequency signals play amore important role when the viewer is nearby, whilethe lower frequency signals are more important whenthe viewer is at distance [1].

Inspired by view-dependent rendering techniques, wefirst decompose the fluid velocity field into a series offrequency components using a modified empirical modedecomposition (EMD) method. Higher frequency com-ponents represent smaller scale fluid motions (typicallylocal turbulent flows), while lower frequency compo-nents represent motions at larger scales (typically largeeddies and global laminar flows). Also, the fluid domainis divided into a series of nested partitions centered withrespect to the viewing frustum. Different grid sizes andtime steps are assigned to different partitions dependingon their distances to the viewer. The control of levelsof details is then applied to each frequency compo-nent by distributing it non-uniformly to the simulationpartitions. Higher frequency components closer to theviewer are allocated to partitions with finer grids andsmaller time steps, while lower frequency componentsmore away from the viewer are allocated to partitionswith coarser grids and larger time steps. As a result,the effective velocity field defined on each simulationpartition is a mixture of frequency components, and thevisible evolution of the mixed velocity can be sufficientlycaptured by the space-time resolution associated withthe specific partition. To obtain the final solution, theeffective velocity fields are solved semi-independentlyon different partitions, which provides richer visual de-tails to the viewer in a more efficient way. Although thenested simulation partitions differ in size and resolution,they are all meshed with uniform rectangular grids,which makes the solver robust and efficient.

This novel view-dependent multiscale simulationframework distributes the computational resources ac-cording to the viewing settings of the target fluid motion,

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Fig. 1. Four snapshots of a view-dependent multiscale fluid simulation with moving camera positions. (6 partitions,grid sizes: 1/400 – 1/100, time steps: 1/120 s – 1/30 s)

and is adaptive in both space and time dimensions. Themain technical innovations include:

• Using a novel approach of space-filling curves, theEMD is efficiently extended to 3D and applied todecompose the velocity field into a small numberof frequency components, which represent the fluidmotion at different length scales.

• A spectrum based simulation pipeline is proposed,in which different frequency components evolveat different space-time resolutions. By doing so, itsignificantly reduces the numerical diffusion thatcauses damping of high frequency turbulence inprevious methods, and preserves more fine-scaleturbulence details in the result.

• The fluid domain is adaptively partitioned accord-ing to the camera position, and the fluid is simulatedat different space-time resolution depending on itsdistance to the viewer. This approach considers bothrendering and simulation together, and efficientlyutilizes the computational resources in the placesthat most affect the final rendered result.

2 PREVIOUS WORK

Jos Stam’s unconditionally stable solver [2] made thegrid-based fluid simulation popular in the graphics com-munity. Since then, many different techniques have beendeveloped to add details to the fluid. The basic approachis to reduce the numerical dissipation. Fedkiw et al. usedthe vorticity confinement technique to prevent the rapiddissipation of vortices. Artificial divergence sources areintroduced in [3] to simulate gas explosion. Vortex parti-cles are used in [4] to add the vorticity more accurately.Zhu et al. [5] introduced FLIP to overcome advectiondissipation. Other methods including BFECC [6], QUICK[7], MacCormack [8] suggested using higher-order spacediscretization schemes and higher-order time integrationschemes (e.g. Runge-Kutta methods). These methodsdiscretize the whole fluid domain using uniform grids,thus they are all limited by the Nyquist frequency.

For 3D fluid simulation, a small increase of the gridresolution by a factor of k will cause a dramatic increaseto the computational cost by a factor of k4 [9]. Therefore,various techniques have been investigated in order to

increase the simulation resolution while controlling thecomputational expense. Different methods [10], [11], [12],[13] have been proposed to generate divergence freefields from random noise, and then used these artificialvelocity fields to represent the turbulent flow. Othermethods [14], [15], [16], [17], [18] simulated the fluidon a low-resolution grid to obtain the macro-scale flow,which was then combined with the artificial divergence-free velocity fields to mimic the turbulent flow at themicro scale. Instead of adding noise, Yoon et al. [19]used the vortex particle method [4] to directly generatea high-resolution turbulent flow, and Rasmussen et al.[12] synthesized the 3D velocity field from 2D slices.These synthesis methods do not perform high-resolutioncomputation on the Navier-Stokes equations, and in-stead attempt to produce plausible results using artificialmeans. Thus, their results are nonphysical, but can becombined with any grid-based method.

Although grid-based fluid solvers are often preferredin the graphics community, other numerical schemes in-cluding finite volume [20], [21] and finite element meth-ods [22] have also been used in many specific graphicsapplications. Some researchers have also exploited theviewing information in fluid simulations. Until recently,there have been mainly two types of approaches: (a)octree [23], adaptive mesh refinement [24] and meshcoupling [25] methods, which use non-uniform meshesto distinguish different levels of details for the fluids; and(b) multi-grid methods [7], [9], [26], which use multiplelayers of meshes to represent fluid motions at differentlength scales. The idea of multi-grid simulations has alsobeen adopted in the framework of smoothed-particlehydrodynamics to accelerate fluid simulation [27]. Ina wider context, it is also noted that Horvath et al.[28] presented a view-dependent multiscale simulationframework for fire simulations, and for accelerating sea-wave simulations [29], the viewing information is usedto adaptively determined the surface mesh and to filterout invisible wave lengths.

3 ALGORITHM OVERVIEW

Fluid phenomena are interesting and visually attractivebecause of turbulent fluid motions. It is well known

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VelocityField +

+

=N-S

Solver

N-SSolver

N-SSolver

+

+

=

... ...

...

Ω1

Ω2

Ωm

*1u

*2u

*mu

1u

2u

mu

Fig. 2. The algorithm framework of view-dependent multiscale fluid simulation

in fluid dynamics that turbulence occurs and developsat different length and time scales, with the extent ofscale difference indicated by the Reynolds number. Inorder to capture fine-scale features of turbulent flows,it is normally necessary to use fine simulation gridsand small time steps, or to use higher-order space dis-cretization and time integration schemes. This creates ahuge computational burden to the fluid simulator, inparticular when simulating a large fluid domain. Onthe other hand, objects in a large scene are observed atdifferent resolutions by human eyes depending on theirdistances to the viewer. Fine local fluid motions havea significant visual impact when the viewer is nearby,and as the distance to the viewer increases, these fine-scale features become less and less visible, while globalfluid motions at larger scales becoming more and moredominant. Thus, for the purpose of achieving visuallyappealing results, there is a clear potential of benefitof utilizing the viewing information to improve theperformance of fluid simulators.

We propose a view-dependent multiscale simulationframework as shown in Fig. 2. First, the fluid velocityfield is decomposed into a series of frequency com-ponents u1, u2, · · ·, um, representing the fluid motionat different length scales ranging from small to large.Next, according to the position of the viewer, m nestedsimulation partitions Ωi are constructed, with Ω1 indicat-ing the vicinity of the viewer and Ωm representing thewhole fluid domain. These simulation partitions are allmeshed into uniform rectangular grids, and each parti-tion Ωi is set with a different grid size depending on thelength scale of the corresponding frequency componentui. Then, each frequency component ui is sequentiallyallocated to partitions Ωi, Ωi+1, · · ·, Ωm, such that foreach partition, it only carries the component quantitythat has not been supported by the previous ones. Thus,the effective velocity field u∗

i defined on partition Ωi

is a mixture of velocity components u1, · · ·, ui that

share a similar visual significance determined by theirintrinsic length scales and distances to the viewer. Tosolve this combined velocity field u∗

i with uniform visualsignificance, a separate fluid simulation is performed onpartition Ωi with individually assigned grid size andtime step. Finally, the total fluid motion in the wholefluid domain is constructed by adding up the resultsobtained on all simulation partitions. Depending on thefluid evolution, the velocity spectrum is repeatedly com-puted to ensure that the new fluid motion is efficientlyrepresented by the frequency components u1, · · ·, um.

The proposed view-dependent multiscale fluid simu-lation framework can be viewed as a multi-grid methodcombined with spectral decomposition. The idea of us-ing spectral analysis in CFD applications is not entirelynew, and a remarkable example is the large eddy sim-ulation [30] that introduces spatial-temporal filters toreduce the range of length scales of the solution, hencereducing the computational cost. The feasibility of thisnew simulation framework relies on two assumptions:(a) the fluid velocity field can be decomposed into asmall number of meaningful frequency components atdifferent length scales; and (b) the Navier-Stokes equa-tions can be linearized to allow separately solving eachfrequency component with varying grids and varyingtime steps. The consideration and solution of these twoissues are addressed in Sections 4 and 5 respectively.

4 3D VELOCITY FIELD DECOMPOSITION

The fluid velocity field can be viewed as a time-varyingsignal defined in a 3D domain. From the viewpoint ofphysics, it is clear that the 3D velocity signal consists ofintrinsic structures at different length scales. However,as turbulence is highly nonlinear and non-stationary,standard data analysis tools such as singular valuedecomposition [31], Fourier and wavelet analysis etc.typically produce many spurious frequency components

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causing energy spreading, which makes the resultingspectrum have little physical meaning. An exceptionis the empirical mode decomposition (EMD) [32], alsoknown as Hilber-Huang transform, which was originallydeveloped for processing nonlinear and non-stationarytime series. Over the past decade, the EMD method hasbeen extremely successful in engineering and success-fully applied in various complicated data sets, includingsea waves and earthquake signals etc.

For the sake of completeness, the standard EMD pro-cedure is briefly reviewed in Section 4.1, after which itis extended into 3D cases in Section 4.2 for processingthe fluid velocity field.

4.1 EMD BasicsThe standard EMD method is designed for the analysisof one-dimensional signals, in particular time series. Themain idea of EMD is to decompose the signal into asmall number of intrinsic mode functions (IMF), whichare based on and derived from the data. An IMF isany function with the same number of extrema andzero crossings, and with zero mean of the upper andlower envelops defined respectively by the local maximaand minima. From this definition, the IMF is a generaloscillatory function, with possibly varying amplitudeand frequency along the time axis. Thus, for representingsignals, an IMF is much more powerful than the simpleharmonic function, which has constant amplitude andfrequency. Given a one-dimensional signal f , its EMDrepresentation is

f =K∑

k=1

ck + rK , (1)

where ci, i = 1, 2, ..,K are the IMFs with the frequencyranging from high to low and rK is the residual. TheEMD algorithm sequentially extracts its IMFs via a “sift-ing” procedure as follows:

1) Initialization r0 = f , set index k = 12) Compute the k-th IMF, ck

a) Initialization h0 = rk−1, set index j = 1b) Find all local maxima and local minima of

hj−1

c) Build the upper envelope Emax,j−1 by con-necting all local maxima with a cubic spline,and build the lower envelope Emin,j−1 byconnecting all local minima with a cubic spline

d) Compute the mean of the upper and lower en-velopes, Emean,j−1 = 1

2 (Emin,j−1 + Emax,j−1)e) hj = hj−1 − Emean,j−1

f) If the IMF stopping criterion is satisfied, thenck = hj , else j = j + 1 and go to step 2(b)

3) rk = rk−1 − ck

4) If rk is monotonic, the decomposition stops, elsek = k + 1 and go to step 2

For the IMF stopping criterion in step 2(f), differentcriteria have been suggested in the literature based on

the definition of IMFs. In our applications, it is foundthat there is no visible difference in the final result ifwe simply fix the iteration number as 8 to 10. There isno rigorous convergence proof for the above algorithm,but practically it always converges very quickly [32]. Thephysical justification of the above EMD procedure is verysolid and has been verified and validated in numerousexperiments by various real data sets (see e.g. [33]).

4.2 3D EMD of Velocity Fields

The main challenge of extending the EMD into higher di-mensional signals arises in the construction of the upperand lower envelopes (step 2(c) in the EMD algorithm).Unlike the simple closed-form solution of the 1D cubicspline interpolation, higher dimensional surface inter-polation is complex and often involves time-consumingcomputation. For the 2D case, 2D radial basis functionsare introduced [34], [35] to transform the interpolationproblem into a global optimization problem. It requiresto solve a m×m linear system, where m is the total num-ber of extrema. The associated computation is affordablefor 2D image applications with hundreds of pixels alongeach axis, but is too slow for our 3D fluid simulationsthat require the EMD to be repeatedly performed in alarge 3D space as turbulence develops. A fast bidimen-tional EMD algorithm is proposed in [36], which is basedon the Delaunay triangulation and cubic interpolation ontriangles. In order to ensure the Delaunay triangulationto cover the whole domain, this method has to introducea bunch of artificial extrema, and as a result it is notsuitable for our 3D fluid simulations that require thehighest level of automation and robustness. Liu et al.[37] tested a tensor-product based 2D EMD approachthat applies separately 1D EMD on each row and columnof an image, after which averaging the envelopes fromdifferent directions. Although it is much faster to do so,our experiments led to a similar conclusion as [37]: theresult is generally worse in that each slice of data onlycontain a small portion of samples and the connectioninformation contained in the original data has beenseriously lost. As the EMD will be repeatedly performedin our fluid simulation framework, a more efficient andmore robust 3D algorithm is needed.

We propose to use space-filling curves to flatten 3Ddata into 1D. First, a space filling curve is constructedto fill the fluid domain, and moving along the curvean index is assigned to each grid cell and saved in atemplate. Then, the 3D velocity field is rearranged into a1D signal array according to the index template. Finally,the reshaped 1D signal is decomposed by using the 1DEMD algorithm, and the decomposition result is mappedback to the 3D space by using the same index template.In this simple 3D EMD approach, the EMD operation isessentially performed on the flattened 1D data set, andtherefore it converges in the same way as the standard1D EMD method [32]. As the index template of space-filling curve can be pre-computed, the CPU expense of

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(a) (b) (c) (d)

Fig. 3. Decomposition comparison of a 2D fluid velocity field. The top row shows Fourier decomposition, and thebottom row shows the EMD result. (a)-(d) are the 1st, 3rd, 4th, and 5th components from low frequency to highfrequency. The EMD and Fourier results are obtained with the same sampling resolution.

the 3D EMD is essentially the same as 1D EMD, whichis linearly proportional to the sampling density. Owingto the analytic cubic spline interpolation, our space-filling curve EMD technique is extremely fast. For 2Dcases, we have compared with the RBF method. It isfound that our approach is at least 20 times faster in alltest examples, and the new method also provides betteraccuracy because it avoids the numerical error causedby the least squares approximation required in the RBFapproach. In the context of fluid simulation, the CPUcost of an individual 3D EMD is about half of a singlepressure solver executed on the same sampling grid.

Different space-filling curves have been tested, includ-ing the Hilbert curve, the Z-order curve, the Koch curveand the Gosper curve. In 2D cases, the Hilbert curve andthe Z-order curve are found to have boundary artifactscaused by their regular quad fractal structures. By usingKoch or Gosper curves, the boundary artifacts can beeffectively removed. In 3D cases, all four curves givegood decomposition results without visible discontinu-ities. The reason is that both the 3D velocity filed and the3D space filling curves are sufficiently complex to avoidthe development of boundary artifacts. For the sake ofsimplicity, we use the Koch curve for 2D examples andthe Hilbert curve for 3D examples in this paper. It isnoted that the Z-order curve has recently been used inSPH simulations to compute SPH neighborhoods rapidly[38], which also demonstrates the benefit of using space-filling curves to accelerate 3D data processing.

The 3D Hilbert curve is defined on a cube, and whenusing the n-th approximation to the limiting curve, thelength of the curve is 2n. However, the fluid domain is

Fig. 4. Quadric KochCurve.

Fig. 5. Energy distribu-tion of the lowest 5 fre-quency components.

(a) (b) (c)

Fig. 6. Reconstruction comparison. (a) is the originalvelocity field, (b) is the sum of the first 5 EMD componentsand (c) is the sum of the first 5 Fourier components.

not necessarily a cube. Therefore, we build the Hilbertcurve with the smallest n such that 2n/3 is greaterthan the maximum velocity resolution in x-, y- and z-directions. When moving along the Hilbert curve, the cellindex is increased and saved if and only if the currentposition is located in the fluid domain. A similar method

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is applied to the Koch curve in 2D cases. In Fig. 4, thegray line is the whole Koch curve and the bold black lineis the space filling curve we used. This strategy preservesas much as possible the locality of the space filling curve.

To flatten 3D data into 1D for EMD operations isessentially an approximate re-sampling treatment. Sodoing inevitably causes some loss of local connectivityinformation presented in the original 3D data, and insome cases where the curve passes internal obstacles, itcan potentially bring in spurious noises. For use in fluidsimulation, we have tested the new space-fill curve EMDapproach in numerous examples, both in 2D and 3D. Fig.3 is an 2D example of our EMD result compared withFourier decomposition. The velocity field is generatedby using a 2D grid solver. Similar to image processing,the Fourier results are computed via 2D FFT, for whichthe velocity field must be discretized on a uniform 2Dgrid. For fair comparison, the same sampling grid isused in the EMD and the 2D FFT. As shown in Fig.3, the EMD frequency components concentrate in theareas where turbulence occurs, while the Fourier compo-nents have ring-shape vortices everywhere in the fluiddomain, which is non-physical. Fig. 5 shows the energydistribution of the frequency components obtained inthe EMD method and the Fourier decomposition. It isclear that fewer EMD components are needed in orderto recover the same amount of energy for the fluidmotion. A direct reconstruction comparison is given inFig. 6 (please zoom in to see the difference), whereFigs. 6(a-c) show respectively the original fluid velocityfield, the EMD and the Fourier reconstructions usingthe same number of components. It can be seen that byusing just five IMFs, the EMD method perfectly recov-ered the original velocity field with no visible defects,while a large amount fine-scale details are lost in theFourier reconstruction. These differences between EMDand Fourier results origin from their different functionalbasis. In Fourier transform, the base functions haveconstant frequencies and amplitudes. This representationis very economical for stationary signals, which havea periodic nature. However, as demonstrate in [32], itis very inefficient for non-stationary and nonlinear sig-nals, and will cause fake energy dispersion with numer-ous spurious high-frequency components. The empiricalmode decomposition uses adaptive base functions withvarying frequencies and amplitudes, and it is particu-larly suitable for non-stationary and nonlinear signals.This is why we chose the EMD for the decomposition ofturbulent flows.

For decomposing the fluid velocity field, an addedbenefit of the EMD method is on dealing with objectspresented inside the fluid domain, where the fluid ve-locity field can be discontinuous on the object boundary.Most standard data analysis tools use functional basiswith fixed amplitudes and frequencies, and consequentlythe signal discontinuity will cause many spurious fre-quency components due to energy spreading. However,the functional basis of the EMD method is adaptively

determined by the local features of the signal. TheIMFs have varying amplitudes and frequencies, so thatthe energy spreading caused by signal discontinuity isminimized. Indeed, this is one of the major advantagesof the EMD technique [32].

Using the EMD method, the velocity field u in thesimulation domain is represented as:

u =m∑

i=1

ui (2)

where ui, i = 1, 2, · · · , m, are frequency componentsrepresenting fluid motions at different length scales,ranging from small to large. In our implementation,m is a user specified constant controlling how manyIMFs to be extracted from the velocity field. Thus, ui,i = 1, · · · , m − 1, are IMF components, and um is a non-IMF component. As um consists of all lower frequencytail IMFs and the residual term, it carries the majorityof the kinetic energy of the fluid flow. Benefited fromthe adaptive and data dependent nature of IMFs, thenonlinear and non-stationary fluid velocity field can beeffectively represented with a small number of frequencycomponents. In our limited experiments, 4 to 6 frequencycomponents are sufficient to represent the velocity fields.Note that the EMD is performed separately for x-, y- andz- components, and then adding them together to obtainthe vector-valued decomposition (2).

5 VIEW DEPENDENT MULTISCALE SIMULA-TION

For incompressible ideal fluids, the Euler equations are:

ρ∂u∂t

+ ρ(u · ∇)u = −∇p + f , (3)

∇ · u = 0, (4)

where u is the velocity, p the pressure, ρ the fluiddensity, and f the effective body force including gravity,buoyancy and vorticity confinement etc.

Determined by human visual perception [1], real-lifeobservation of a large dynamic fluid scene has two mainfeatures:

• Fluid motions are observed at different resolutionsby human eyes, depending on the distance fromthe viewer to the location where the motion isdeveloping. The smaller the distance is, the higherresolution will be received; and vice versa.

• The fluid motion consists of intrinsic structures, i.e.frequency components, evolving at different lengthand time scales. These multiscale frequency compo-nents generate unequal visual impacts. When theviewer is nearby, the fast-developing small scalecomponents are more significant in our observation;and when the viewer is at distance, the slow-movinglarge scale components become more dominant.

In order to achieve the best visual effects with theminimum computational cost, the fluid solver needs to

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take into account both of the above aspects. This is doneby integrating spectral analysis and domain partitioninto a view driven simulation framework, whose detailsare explained in the following subsections.

5.1 Dynamics of Multiscale Flow

In the space dimension, the multiscale motion compo-nents of a turbulent flow are revealed in Eqn. (2) byusing the EMD method. Substituting Eqn. (2) into theN-S equations (3 - 4) and setting the fluid density tounit yields:

m∑i=1

∂ui

∂t+

m∑i=1

(u · ∇)ui = −∇p + f , (5)

m∑i=1

∇ · ui = 0. (6)

When an explicit solver is adopted, the total fluid veloc-ity u is computed using the results from the previoustime steps, thus u can be considered as semi-decoupledfrom ui in Eqn. (5).

Eqns. (5 - 6) show that multiscale fluid motions arecoupled together to satisfy momentum and mass con-servation. However, from the viewpoint of physics [39],fluid motions ui differ not only in their length scales, butthey also develop at different pace in the time dimension,with micro-scale motions developing fast and macro-scale motions developing relatively slow. Thus, if theobservation is fixed to a small window T in the timeaxis and if the inter-frequency exchange of momentumand mass can be ignored, this leads to

∂ui

∂t+ (u · ∇)ui = −∇pi i = 1, 2, · · · , m − 1, (7)

∂um

∂t+ (u · ∇)um = −∇pm + f , (8)

∇ · ui = 0 i = 1, 2, · · · , m, (9)

where pi, i = 1, 2, · · · , m are the unknown fluid pressurecorresponding to the motion components ui. Typicalbody forces, such as gravity and buoyancy, change muchslower comparing to the rapid development of micro-scale turbulent motions. This is particularly true for idealfluids [39], whose viscous force is zero and Reynoldsnumber is infinity. Therefore, in the momentum Eqn.(7), the influence from the slow changing body forcesto the fast developing micro-scale fluid motions is alsoignored, and the body force is only included in Eqn. (8)for the mixed low-frequency component um. By allowingall body forces to directly work on the um motion, thedominant energy carrier obtained in the EMD (2), theenergy transfer process occurring at the macro-scale levelis emphasized. However, if fast changing body forcesare involved, they should be likewise decomposed andapplied to the corresponding velocity component.

5.2 View-Dependent Simulation of Multiscale FlowEqns. (7 - 9) describe the dynamics of multiscale flow.Our aim is to solve these equations according to thecamera settings such that all visible fluid motions at bothmicro- and macro- scales are accurately captured withthe minimum computational cost.

First, the fluid domain is divided into m nested parti-tions Ωi, i = 1, 2, · · · m such that Ω1 ⊂ Ω2 ⊂ · · · ⊂ Ωm,where Ωm represents the whole fluid domain. Thesenested partitions are all centered with respected to theview frustum, so that partitions Ωi, i = 1, 2, · · · , mprovide a natural indication for the distance between theviewer and the fluid point, ranging from small to large. Itis noted that by building the partitions Ωi with respect tothe view frustum, the view direction and view angle arealso taken into account. As the viewer-to-fluid distanceincreases, the visibility of the fluid motion drops, whichsequentially reduces the accuracy requirement of thesimulation. Therefore, these simulation partitions arediscretized using different grid sizes and time steps, andwith the increase of index i, the space-time resolution ofΩi decreases. In particular, the grid size and time stepof each partition Ωi are set to allow an economical andyet sufficiently accurate description of the motion ui.

Next, depending on the viewer-to-fluid distance, eachmotion component ui is adaptively represented at dif-ferent space-time resolutions. This is achieved by dis-tributing the velocity quantities of ui to partitions Ωj ,j = i, i + 1, · · · , m such that the motion ui is discretizedon a composite grid Ωi∪{Ωi+1−Ωi}∪· · ·∪{Ωm−Ωm−1}.As shown in Fig. 2, after all frequency components ui

have been distributed to the simulation partitions, thevelocity field on each partition Ωi becomes a compositefield u∗

i as follows:

u∗i =

{ui for Ωi−1∑i

j=1 uj for Ωi − Ωi−1(10)

The effective velocity u∗i collects all visible fluid motions

measured at the space-time resolution of Ωi.Then, reorganizing Eqns. (7 - 9) according to Eqn. (10)

yields:

∂u∗i

∂t+ (u · ∇)u∗

i = −∇p∗i i = 1, 2, · · · , m − 1, (11)

∂u∗m

∂t+ (u · ∇)u∗

m = −∇p∗m + f , (12)

∇ · u∗i = 0 i = 1, 2, · · · , m, (13)

where p∗i , i = 1, 2, · · · , m are the unknown fluid pressurecorresponding to the composite velocity components u∗

i .Although similar in formulation, it should be notedthat Eqns. (11 - 13) and Eqns. (7 - 9) describe totallydifferent physical phenomena. Eqns. (11 - 13) are definedon partitions Ωi, i = 1, 2, · · · , m respectively, and foreach partition Ωi, they describe the evolution of all fluidmotions that are visible at the space-time resolutionassociated with Ωi. Eqns. (7 - 9) are defined in the wholefluid domain, and they describe the dynamics of the

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fluid motion at each individual length scale, regardlessof its visibility to the viewer.

Finally, the fluid simulation is performed by solvingthe Eqns. (11 - 13) on nested partitions Ωi, i = 1, 2, · · · , mrespectively. The initial values of u∗

i are computed withEqn. (10), in which the frequency components ui areobtained from the EMD (2). Starting from i = 1 andgoing through each simulation partition Ωi, the solu-tion u∗

i is obtained by using the standard advection-projection scheme [2]. Specifically, the advection stepsolves equation

∂u∗i

∂t+ (u · ∇)u∗

i = 0. (14)

Note that the background velocity field for advection isthe total velocity u instead of the velocity component u∗

i .Similarly, the projection step solves equations

∂u∗i

∂t= −∇p∗i , (15)

∇ · u∗i = 0. (16)

Note that for the last partition Ωm, the external bodyforce f is added into Eqn. (15). For the pressure solver,we use the standard preconditioned conjugate gradientmethod with the preconditioner obtained through theincomplete Cholesky decomposition. The final solutionof the fluid is:

u = u∗1 ⊕ u∗

2 ⊕ · · · ⊕ u∗m (17)

where ⊕ denotes the superposition of velocity fieldsu∗

i defined in different partitions. As Eqns. (11 - 13)hold only when the observation is fixed in a relativelysmall time window T , the EMD operation (2) needs tobe repeatedly performed after certain time steps to re-initialize the solution process (14 - 17). This EMD re-initialization step is necessary to ensure an adequate andtimely capture of the cross-scale motion transfer of thefluid.

Boundary conditions: For internal partitions Ωi, i =1, · · · , m − 1, the boundary conditions are set as u∗

i = 0on ∂Ωi. For the partition Ωm, the real boundary con-dition of the whole fluid domain is used on ∂Ωm.These simplifications practically over restrict the energyexchange between partitions. By doing so, we sacrificethe accuracy in order to minimize the coupling betweenpartitions and improve the efficiency of obtaining visu-ally plausible results. Our method also supports internalboundaries. For static obstacles in the fluid domain,each velocity component deals with the obstacle in thesame way as the traditional methods, e.g. using simpleobstacle discretization or some more precise models. Asthe obstacle is static, the final velocity field automaticallysatisfies the non-slip boundary condition. For dynamicobstacles, a practical approach is to add the dynamicboundary condition to the lowest frequency component,while adding static boundary conditions to all the othercomponents.

5.3 Computational Issues

Our multiscale fluid simulation is driven by the viewer.In standard rendering systems, such as the PBRT [40]used in this work, the fluid domain is defined in theobject space, then transformed into the view space bymodel and view matrices, and finally projected into theimage space according to camera parameters (projectionmatrix and viewport). We integrate the inverse of thispipeline into our simulator to control the levels of detailsin the simulation.

The fluid domain is divided into simulation partitionsaccording to the distance to the viewer and the viewdirection, and each partition is individually assignedwith a grid size and a time step. Thus, the partitionsmove when the viewer moves, which then requires thefluid velocity to be transferred between grids of differentsizes. For simplicity, we use linear interpolation for thevelocity transfer between coarse grids and fine grids.

In the current implementation, the grid sizes and timesteps are manually set by the user based on the size ofsimulation domain, the camera setting and the character-istics of IMFs. Separate velocity components communi-cate with each other through the advection term (14) andthe EMD re-initialization. It is possible to automaticallydetermine the spatial-temporal resolution. Specifically,the grid size can be associated to the dimension of thesimulation domain and the spatial frequencies of IMFs,which can be obtained via Hilbert transforms. Once thegrid size is fixed, the corresponding time step can thenbe determined in conjunction with the camera motion.This important adaptivity aspect will be pursued in ourfuture work as detailed in Section 7.

Given a target fluid and the camera settings, the view-dependent multiscale fluid simulation is performed asfollows:

1) Generate a Hilbert curve to cover the whole fluiddomain and build the 3D-to-1D index template

2) Compose an ordered work list consisting of fourtypes of jobs: partition, EMD, simulation and out-put

3) Follow the work list to do• For partition request: according to the current

camera settings, the whole fluid domain isdivided into simulation partitions Ωi with fixedgrid sizes and time steps

• For EMD request: compute the velocity spec-trum (2) with 3D EMD

• For simulation request: solve Eqns. (14 - 16) onthe specific simulation partition Ωi

• For output request: output the current velocityfield u

In step (2), time entries of the partition request aredetermined according to the camera motion; time entriesof the EMD request are set with a fixed time intervalspecified by the user; time entries of the simulationrequest are calculated according to the fixed time step ofeach simulation partition; and time entries of the output

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request are set according to the animation requirement.In the current implementation, the oldest time step isalways executed first in order to get the most up-to-dateinformation from the other fluid simulations. Also, in theadvection step, we simply use the latest total velocityfield as the background velocity. By doing so, we ignorethe numerical error caused by the simulations beingout of synchronization. The main computational cost ofour simulation framework is in the advection-projectionsolutions, which are performed separately on differentpartitions with different space-time resolutions. As highresolution solutions are only performed for the closestpartitions to the viewer, usually very small domains, oursimulation runs much faster comparing to the standardN-S solver using a uniform high resolution grid.

Comparing with octree and adaptive mesh refinement(AMR) methods, the proposed method differs mainly intwo aspects: 1) We distinguish the fluid flow not only byits distance to the viewer (resolved by setting multiplesimulation partitions), but also by its intrinsic motions atdifferent length scales (resolved by EMD). Both spatialand temporal resolutions are adaptive in our method,while the octree and AMR approaches are often adap-tive only in the space dimension. 2) Octree and AMRmethods use non-uniform grids, and we use multiplepartitions meshed into uniform grids. The use of uniformgrids and simple data structures significantly simplifiesthe implementation and computational complexity. In awider sense, the new method can be viewed as a multi-grid approach combined with spectral analysis. Unlikeother multi-grid methods using prefixed simulation res-olutions independent to the evolution of fluid flows, thespace-time resolutions for different simulation partitionsare determined according to the spectral decompositionresult of the fluid velocity field. Therefore, the newmethod is more adaptive, and can support moving cam-era positions and developing fluid flows in a uniformframework.

6 RESULT

Several examples are presented in this section to demon-strate the performance of the new fluid simulationframework. All numerical simulations are performed ona PC platform with an Intel Core2 2.4 GHz CPU and 8GB memory.

Assumption Verification 1 Our new method assumesthat the low frequency components of a velocity field canbe simulated on coarse grids to save computational cost.This is verified in Fig. 7, where a plume is developedin a velocity field that only contains low frequencycomponents. Figs. 7(a-c) are the results obtained usinga fine grid (128 × 256 × 128), and Figs. 7(d-f) are solvedon a coarse grid (64 × 128 × 64). The two groups ofresults are very similar up to frame 8 and they becomemore different as the simulation continues. This obser-vation confirms that the low frequency components ofa velocity field do generate high frequency motions as

(a) (b) (c)

(d) (e) (f)

Fig. 7. Comparison of a low-frequency velocity fieldsolved on fine and coarse grids. The same low-frequencyflow is solved respectively on a fine grid and a coarsegrid, where (a), (b) and (c) are the fine-grid results fromthe 1st, 8th and 16th frames, and (d), (e) and (f) are thecorresponding coarse-grid results.

(a) (b) (c)

Fig. 8. Comparison of the standard solver and the newmethod with flow field decomposition. (a) the standardmethod using a coarse grid; (b) the standard methodusing a fine grid; and (c) our method with high-frequencycomponents simulated on fine grids and low-frequencycomponents simulated on coarse grids.

time goes, but these newly generated high frequencycomponents are neglectable in the beginning period,during which the fluid motion can be well captured byusing a coarse grid. Therefore, we choose different gridresolutions to economically simulate different frequencycomponents, and periodically recompute the velocityspectrum to adjust the frequency-component allocationand ensure that every frequency component is alwayssimulated using the right grid resolution.

Assumption Verification 2 The new method alsoassumes that the evolution of turbulent flow can beeffectively simulated as the superposition of a series ofintrinsic flow modes developing separately at differentspatial frequencies. This assumption is examined in Fig.8, where Figs. 8(a-b) show a plume simulated using thestandard method on a coarse grid (64 × 128 × 64) anda fine grid (128 × 256 × 128) respectively. As expected,the result (b) obtained from the fine grid simulationcontains much more fine-scale features. Using the new

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(a) (b)

Fig. 9. Efficiency improvement. (a) standard methodusing a uniform grid size of 1/64; and (b) our new methodusing 5 partitions, grid size 1/128 for partitions 1-2, gridsize 1/64 for partitions 3-4 and grid size 1/32 for partition5.

(a) (b)

Fig. 10. Visual quality improvement. (a) standard method,(b) our new method using 4 partitions without view-control.

method, the same plume is simulated with 5 frequencycomponents as shown in Fig. 8(c). The first four higherfrequency components are simulated on the fine grid(128 × 256 × 128), while the last lower frequency com-ponents is simulated on the coarse grid (64 × 128 × 64).All five frequency components are simulated with thesame time step, and EMD is perform every 10 frames toupdate the velocity spectrum. Comparing the results (a-b) and (c), it can be seen that the new method with flowfield decomposition can produce similar level of detailsas the standard high-resolution simulation, and insteadof simulating the total velocity field at a high resolution,it only requires those high-frequency components to besimulated on fine grids.

Efficiency Improvement To examine the efficiency im-provement, a scene with two plumes is simulated usingthe standard method and the new method with view-control. Fig. 9(a) shows the standard result obtained witha uniform grid 64 × 128 × 256. The two plumes arecaptured at the same resolution, and the plume close tothe viewer is lack of fine-scale details, which makes thescene look unnatural. Fig. 9(b) shows the result obtainedusing the new method. It can be seen that different levelsof details are obtained in the scene, with more fine-scalefeatures captured for the plume closer to the viewer.Five simulation partitions with different dimensions anddifferent grid sizes are used in the simulation. Partitions

1-2 have a grid size of 1/128, partitions 3-4 have a gridsize of 1/64 and the last partition has a grid size of 1/32.Different time steps are used on different partitions, andfrom high to low, they are set respectively as 1/120,1/60, 1/40, 1/30 and 1/30. The EMD is performed every12 frames to update the velocity spectrum. As smallgrid sizes are only used on small partitions while largerpartitions using larger grid sizes, the total memory usedin the new method is similar to the standard solver. Foreach frame, the average CPU time cost for the standardsolver is 168.3 seconds, and 47.5 seconds for the newmethod. The efficiency improvement is achieved via theview-dependent partition, which combining with theflow field decomposition allows the simulation to be runat different spatial-temporal resolutions.

Visual Quality Improvement To examine the visualquality improvement, an example with smoke passingthrough multiple obstacles is designed. Fig. 10(a) is theresult of the standard solver on a 64×64×256 grid (gridsize 1/64). The relatively low resolution makes the resultlook flat. Fig. 10(b) is the result of our method usingfour partitions without view control. The grid sizes fromhigh frequency to low frequency are 1/128, 1/128, 1/64and 1/32. In Fig. 10(b), the non-potential flow structuresobserved at the upwind of the cylinders are caused bythe interaction of the closely placed cylinders and thenon-slip boundary effect. These high frequency struc-tures are not resolved by the standard solver (Fig. 10(a))due to numerical dissipation on the coarser mesh (gridsize 1/64), while they are emphasized in the proposedsolver (Fig. 10(b)) because the high frequency motionis solved separately on a finer mesh (grid size 1/128).The average CPU time costs of the standard methodand our new method are 56.7 seconds and 69.2 secondsrespectively. With a similar computational cost, the newmethod doubled the simulation resolution producingmuch more fine-scale details consistent with real-lifeobservations. The speedup gained in the new methodis because (a) the last partition is solved on the coarsegrid; and (b) the first four partitions do not have anybody force and as a result, their simulations convergevery fast, typically within 5 iterations. However, at eachtime step, the standard N-S solver typically takes 50 - 80iterations to converge to the error threshold 10−5.

Simulation with Moving Camera Positions The lastexample demonstrates the new method in a simulationwith a moving viewpoint. Fig. 1 shows the simulationresult, which is computed on six moving partitions. Themaximum grid size is 1/100, and the minimum is 1/400.The maximum time step is 1/30, which is used on thelargest partition, and the minimum time step is 1/120,which is used on the smallest partition. The space-timeresolution of each partition is fixed, but its positionand dimension change automatically as the viewpointmoves. It can be seen that the new method is robustand efficient, and it provides natural-looking results withmultiple levels of details that are consistent with real-lifeobservations.

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7 CONCLUSION AND LIMITATION

We propose a view-dependent multiscale fluid simu-lation framework that exploits both the viewing infor-mation in human visual perception and the multiscalevelocity spectrum of a turbulent flow. In the new simu-lation framework, the fluid is solved at different space-time resolutions according to its visual impact. Specifi-cally, high-resolution simulations are performed for thefluid regions closer to the viewer and for the frequencycomponents more visible to human eyes, and vice versa.The new simulator better utilizes the computing powersuch that (a) for the same simulation task, it is faster thanthe traditional grid-based N-S solver; and (b) with thesame computational resources (CPU time and memorystorage), it can simulate a larger fluid scene or producericher fine-scale details. Also, as the multiscale fluid mo-tions are distinguished in our simulation, the numericaldissipation is effectively reduced. By modulating thesimulation in the frequency space of the fluid motion,the new simulator can potentially provide the animatora simple way to modulate and enhance the visual effectsof fluid flows, which will be pursued in our future work.

The current implementation does not allow movinginternal obstacles. However, the extension to cope withmoving internal objects is relatively straightforward, andcare must be taken when the object moves across theboundary of simulation partitions. The main limitationof the proposed new framework is in three folds:

• For a fluid scene observed at multiple viewpoints,the simulation partitions become irregularly shaped,depending on the relative positions of differentviewpoints. The complicated geometry of simula-tion partitions make the grid-based solver morecomplex in implementation, and a finite-volumesolver might then become a better option.

• The proposed view-dependent simulation frame-work requires the camera to move continuouslywithout jump. Discontinuous camera positions willcause the algorithm lose its advantage of capturingfine-scale details. This is because fine-scale motionsare only simulated in the neighborhood of the previ-ous camera focus instead of the whole fluid domain.Similarly, the performance can potentially drop withrapidly moving cameras because a larger buffer areawill be needed for high resolution partitions.

• Our current implementation is purely sequential.As the simulations running on different partitionsare relatively independent, the algorithm can bereadily parallelized. Furthermore, as each partitionis meshed into uniform grids, its simulation can alsobe accelerated with GPU implementation.

These three important aspects will be pursued in ourfuture work.

ACKNOWLEDGMENT

We would like to thank the anonymous reviewers fortheir constructive comments. This work was supported

by the National Basic Research Project of China (ProjectNumber 2011CB302205) and the Natural Science Foun-dation of China (Project Number 61120106007). The au-thors would also like to thank the Royal Society and Nat-ural Science Foundation of China for support through anInternational Joint Project (No. JP100987/61111130210).

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Yue Gao is a Ph-D graduate at Department ofComputer Science and Technology of TsinghuaUniversity in 2011. His research interest is inphysical simulation and rendering.

Chen-Feng Li is currently a Lecturer at theCollege of Engineering of Swansea University,UK. He received his BEng and MSc degreesfrom Tsinghua University, China in 1999 and2002 respectively and received his Ph.D. degreefrom Swansea University in 2006. His researchinterests include computer graphics and numer-ical analysis.

Bo Ren is a Ph-D candidate at Department ofComputer Science and Technology of TsinghuaUniversity. His research interest is in physicalsimulation and rendering.

Shi-Min Hu received the PhD degree from Zhe-jiang University in 1996. He is currently a pro-fessor in the Department of Computer Scienceand Technology at Tsinghua University, Beijing.His research interests include digital geometryprocessing, video processing, rendering, com-puter animation, and computer-aided geometricdesign. He is associate Editor-in-Chief of TheVisual Computer (Springer), and on the editorialboards of Computer-Aided Design and Com-puter & Graphics (Elsevier). He is a member of

the IEEE and ACM.

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