2023年 4月 21日2023年 4月 21日

A Semi-Lagrangian CIP Fluid Solver Without Dimensional

Splitting

A Semi-Lagrangian CIP Fluid Solver Without Dimensional

Splitting

2008.09.122008.09.12

Doyub KimDoyub Kim

Oh-Young SongOh-Young Song

Hyeong-Seok KoHyeong-Seok Ko

presented by ho-young presented by ho-young LeeLee

EUROGRAPHICS 2008

AbstractAbstractUSCIP : a new CIP methodUSCIP : a new CIP method

More stable, more accurate, less amount of More stable, more accurate, less amount of computation compared to existing CIP solvercomputation compared to existing CIP solver

Rich details of fluidsRich details of fluidsCIP is a high-order fluid advectionCIP is a high-order fluid advection

AbstractAbstractTwo shortcomings of CIPTwo shortcomings of CIP

Makes the method suitable only for simulations with a Makes the method suitable only for simulations with a tight CFL restrictiontight CFL restriction

CIP does not guarantee unconditional stabilityCIP does not guarantee unconditional stability

introducing other undesirable featureintroducing other undesirable feature

This proposed method (USCIP) brings This proposed method (USCIP) brings significant improvements in both accuracy significant improvements in both accuracy and speedand speed

IntroductionIntroductionAttempts for the accuracy of the advectionAttempts for the accuracy of the advection

Eulerian frameworkEulerian frameworkMonotonic cubic spline method Monotonic cubic spline method

CIP method (CIP, RCIP, MCIP)CIP method (CIP, RCIP, MCIP)

Back and force error compensation and correction(BFECC)Back and force error compensation and correction(BFECC)

Hybrid method (Eulerian and Largrangian framework)Hybrid method (Eulerian and Largrangian framework)Particle levelset methodParticle levelset method

Vortex particleVortex particle

Derivative particlesDerivative particles

IntroductionIntroduction

This paper develops a stable CIP method This paper develops a stable CIP method that does not employ dimensional splittingthat does not employ dimensional splitting

Original CIP

Rational CIP MCIP

Stability Unstable More stable than Origin CIP

More stable than Rational CIP

Computation time

lower than MCIP

lower than MCIP

high

Related WorkRelated Work““Visual simulation of smoke”, Fedkiw R., Visual simulation of smoke”, Fedkiw R., Stam J., Jensen H. W. Computer Graphics. Stam J., Jensen H. W. Computer Graphics. 20012001

Monotonic cubic interpolationMonotonic cubic interpolation

Related WorkRelated WorkCIP MethodsCIP Methods

““A universal solver for hyperbolic equations by cubic-A universal solver for hyperbolic equations by cubic-polynomial interpolation”, Yabe T., Aoki T. Computer polynomial interpolation”, Yabe T., Aoki T. Computer Physics. 1991.Physics. 1991.

Original CIPOriginal CIP

““Stable but non-dissipative water”, Song O.-Y., Shin H., Stable but non-dissipative water”, Song O.-Y., Shin H., Ko H.-S. ACM Trans Graph. 2005.Ko H.-S. ACM Trans Graph. 2005.

Monotonic CIPMonotonic CIP

““Derivative particles for simulating detailed movements Derivative particles for simulating detailed movements of fluids”, Song O.-Y., Kim D., Ko H.-S. IEEE Transactions of fluids”, Song O.-Y., Kim D., Ko H.-S. IEEE Transactions on Visualization and Computer Graphics. 2007.on Visualization and Computer Graphics. 2007.

Octree data structure with CIPOctree data structure with CIP

Related WorkRelated WorkEtc..Etc..

““Animation and rendering of complex water surfaces”, Animation and rendering of complex water surfaces”, Enright D., Lossaso F., Fedkiw R. ACM Trans. Graph. Enright D., Lossaso F., Fedkiw R. ACM Trans. Graph. 2002.2002.

To achieve accurate surface tracking in liquid animationTo achieve accurate surface tracking in liquid animation

““Texure liquids based on the marker level set”, Mihalef Texure liquids based on the marker level set”, Mihalef V., Metaxas D., Sussman M. In Eurographics. 2007.V., Metaxas D., Sussman M. In Eurographics. 2007.

The marker level set methodThe marker level set method

““Vortex particle method for smoke, water and Vortex particle method for smoke, water and explosions”, Selle A., Rasmussen N., Fedkiw R. ACM explosions”, Selle A., Rasmussen N., Fedkiw R. ACM Trans. Graph. 2005.Trans. Graph. 2005.

Simulating fluids with swirlsSimulating fluids with swirls

Original CIP MethodOriginal CIP MethodKey IdeaKey Idea

Advects not only the physical quantities but also their Advects not only the physical quantities but also their derivativesderivatives

The advection equation can be written asThe advection equation can be written as

Differentiating equation (1) with respect to the spatial Differentiating equation (1) with respect to the spatial variable x givesvariable x gives

Original CIP MethodOriginal CIP MethodThe value is approximated with the cubic-The value is approximated with the cubic-spline interpolationspline interpolation

Original CIP MethodOriginal CIP Method2D and 3D polynomials2D and 3D polynomials

In 2D caseIn 2D case

Original CIP MethodOriginal CIP Method2D Coefficients2D Coefficients

Original CIP MethodOriginal CIP MethodTakes x and y directional derivativesTakes x and y directional derivatives

Two upwind directionsTwo upwind directions

One starting pointOne starting point

Not use the derivative information at farthest cell cornerNot use the derivative information at farthest cell corner

The method is accurate only whenThe method is accurate only whenThe back-tracked point falls near the starting point of the The back-tracked point falls near the starting point of the semi-Lagrangian advectionsemi-Lagrangian advection

Original CIP MethodOriginal CIP MethodProblem for simulations with large CFL Problem for simulations with large CFL numbersnumbers

Stability is not guaranteedStability is not guaranteed

Monotonic CIP MethodMonotonic CIP MethodTo ensure stabilityTo ensure stability

Uses a modified version of the grid point derivativesUses a modified version of the grid point derivatives

Dimensional splittingDimensional splitting

Monotonic CIP MethodMonotonic CIP MethodA single semi-Lagrangian access in 2DA single semi-Lagrangian access in 2D

6 cubic-spline interpolations6 cubic-spline interpolations

Two along the x-axis for andTwo along the x-axis for and

Two along the x-axis for andTwo along the x-axis for and

One along the y-axis for andOne along the y-axis for and

One along the y-axis for andOne along the y-axis for and

In 3D, 27 cubic-spline interpolationsIn 3D, 27 cubic-spline interpolations

xyyx ,,,

xyx

y yx xy

Monotonic CIP MethodMonotonic CIP MethodTwo drawback of MCIP methodTwo drawback of MCIP method

First, High computation timeFirst, High computation timeThe computation time for MCIP is 60% higher than that of The computation time for MCIP is 60% higher than that of linear advectionlinear advection

Second, Numerical errorSecond, Numerical errorThe split-CIP-interpolation requires second and third The split-CIP-interpolation requires second and third derivativesderivatives

Must be calculated by central differencingMust be calculated by central differencing

This represents another source of numerical diffusionThis represents another source of numerical diffusion

Unsplit Semi-Lagrangian CIP MethodUnsplit Semi-Lagrangian CIP Method

To develop USCIPTo develop USCIPGo back to original 2D and 3D CIP polynomialsGo back to original 2D and 3D CIP polynomials

Make necessary modificationsMake necessary modifications

Utilize all the derivative information for each cellUtilize all the derivative information for each cell

12 known values in a cell12 known values in a cell

at the four cornersat the four corners

2 additional terms2 additional termsyx and ,

33 xyandyx

Unsplit Semi-Lagrangian CIP MethodUnsplit Semi-Lagrangian CIP Method

2 extra terms2 extra termsThe mismatch betweenThe mismatch between

The number of known values (12)The number of known values (12)

and the number of terms (10)and the number of terms (10)

To overcome this mismatchTo overcome this mismatchLeat-squares solutionLeat-squares solution

Over-constrained problemOver-constrained problem

Insert extra termsInsert extra terms

Unsplit Semi-Lagrangian CIP MethodUnsplit Semi-Lagrangian CIP Method

Three principles for the two added termsThree principles for the two added termsNot create any asymmetryNot create any asymmetry

If is added, then must be addedIf is added, then must be added

Contain both x and yContain both x and yRotation and shearingRotation and shearing

The lowest order terms should be chosenThe lowest order terms should be chosenTo prevent any unnecessary wigglesTo prevent any unnecessary wiggles

The terms that pass all three criteria are The terms that pass all three criteria are and and

nm yx mn yx

yx3 3xy

Unsplit Semi-Lagrangian CIP MethodUnsplit Semi-Lagrangian CIP Method

To guarantee that the interpolated value To guarantee that the interpolated value will always be bounded by the grid point will always be bounded by the grid point valuesvalues

A provision to keep the USCIP stableA provision to keep the USCIP stableWhen the interpolated result is larger/smaller than the When the interpolated result is larger/smaller than the maximum/minimum of the cell node values,maximum/minimum of the cell node values,

Replace the result with the maximum/minimum valueReplace the result with the maximum/minimum value

Guarantees unconditional stability without over-stabilizingGuarantees unconditional stability without over-stabilizing

USCIP works on compact stencilsUSCIP works on compact stencilsNo need to calculate high-order derivativesNo need to calculate high-order derivatives

Reduce the computation timeReduce the computation time

Unsplit Semi-Lagrangian CIP MethodUnsplit Semi-Lagrangian CIP Method

USCIP requires fewer operations than MCIPUSCIP requires fewer operations than MCIPUnsplit polynomial is more complicatedUnsplit polynomial is more complicated

But split-CIP involves multiple interpolationsBut split-CIP involves multiple interpolations

MCIP : 693 operations for a 3D interpolationMCIP : 693 operations for a 3D interpolation

USCIP : 296 operations for a 3D interpolationUSCIP : 296 operations for a 3D interpolation

Only 43% of the total operation count needed for MCIPOnly 43% of the total operation count needed for MCIP

Experimental ResultsExperimental ResultsRigid Body Rotation of Zalesak’s DiskRigid Body Rotation of Zalesak’s Disk

Experimental ResultsExperimental ResultsRising Smoke Passing Through ObstaclesRising Smoke Passing Through Obstacles

Generate realistic swirling of smoke Generate realistic swirling of smoke Under complicated internal boundary conditionsUnder complicated internal boundary conditions

Without the assistance of vortex reinforcement mothodsWithout the assistance of vortex reinforcement mothods

Experimental ResultsExperimental ResultsDropping a Bunny-shaped Water onto Still Dropping a Bunny-shaped Water onto Still WaterWater

Generated complicated small-scale featuresGenerated complicated small-scale featuresDropletsDroplets

Thin water sheetsThin water sheets

Small wavesSmall waves

Experimental ResultsExperimental ResultsVorticity Preservation TestVorticity Preservation Test

FLIP vs USCIPFLIP vs USCIP

Noisy curl fieldNoisy curl field

ConclusionConclusionPresented a new semi-Lagrangian CIP Presented a new semi-Lagrangian CIP methodmethod

Stable, fast, accurate resultStable, fast, accurate result

Two additional fourth-order termsTwo additional fourth-order termsReflect all the derivative informationReflect all the derivative information

Stored at the grid pointsStored at the grid points

The proposed technique ran more thanThe proposed technique ran more thanTwice as fast as BFECC or MCIPTwice as fast as BFECC or MCIP

Clearly less diffusiveClearly less diffusive