a semi-lagrangian cip fluid solver without dimensional splitting
DESCRIPTION
EUROGRAPHICS 2008. A Semi-Lagrangian CIP Fluid Solver Without Dimensional Splitting. 2008.09.12 Doyub Kim Oh-Young Song Hyeong-Seok Ko presented by ho-young Lee. Abstract. USCIP : a new CIP method More stable, more accurate, less amount of computation compared to existing CIP solver - PowerPoint PPT PresentationTRANSCRIPT
2023年 4月 20日2023年 4月 20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