1 Modeling Highly- Deformable Liquid Chih-Wei Chiu Computer Graphics and Geometry Modeling Laboratory National Chiao Tung University June 25, 2002 Advisors:

1

Modeling Highly-Modeling Highly-Deformable LiquidDeformable Liquid

Chih-Wei ChiuChih-Wei ChiuComputer Graphics and Geometry Modeling LaboratoryComputer Graphics and Geometry Modeling Laboratory

National Chiao Tung UniversityNational Chiao Tung University

June 25, 2002June 25, 2002

Advisors: Cheng-Chung Lin and Jung-Hong Chuang

2

AgendaAgenda

IntroductionIntroduction Previous WorkPrevious Work MethodMethod ResultResult ConclusionConclusion

3

Modeling FluidsModeling Fluids

ParticlesParticles A particle represent a A particle represent a

fixed-mass fluid elementfixed-mass fluid element There may be There may be interaction interaction

forcesforces between particles between particles

VolumeVolume Partition the scene into many Partition the scene into many

cellscells Every cell carries a Every cell carries a density density

functionfunction representing the representing the proportionproportion of air and liquid of air and liquid

Surface can be Surface can be inferredinferred from from density functionsdensity functions

4

Computational AspectComputational Aspect ParticlesParticles

Number grows with Number grows with cubecube of resolution of resolution Computationally Computationally expensiveexpensive

VolumeVolume Only Only oneone equation for each cell to evolve equation for each cell to evolve

the density function (the density function (more on this latermore on this later)) Computationally Computationally economicaleconomical

5

Geometry AspectGeometry Aspect

ParticlesParticles No straightforward way to No straightforward way to

extract a extract a smoothsmooth surface surface Resolve details Resolve details

independentindependent of grid of grid resolutionresolution

VolumeVolume Iso-surfaceIso-surface can be extracted can be extracted

from the volume datafrom the volume data Potentially under-resolve Potentially under-resolve

details if the details if the grid is toogrid is too coarsecoarse

smooth

not smooth

6

Proposed MethodProposed Method

Since Since volume methodvolume method and and particlesparticles have have nearly “complemented” strength and nearly “complemented” strength and weakness…weakness…

CombinationCombination of particles and volume of particles and volume methodmethod Track initial fluid by Track initial fluid by volumevolume Resolve highly-deformed regions by Resolve highly-deformed regions by

particlesparticles

7

Previous WorkPrevious Work

Wave SimulationWave Simulation Surface is a parametric function can be animation Surface is a parametric function can be animation

over timeover time Fourier synthesis Fourier synthesis [Mastin ’87][Mastin ’87]

Wave-tracing Wave-tracing [Ts’o ‘87][Ts’o ‘87]

Particle SystemsParticle Systems Very heuristic, Very heuristic, CheapCheap computation computation Ship wakes Ship wakes [Goss ‘90][Goss ‘90]

Thin-film water splash Thin-film water splash [Ashraf ’99][Ashraf ’99]

Splash “subsystem”Splash “subsystem” – particles are generated when – particles are generated when the water surface is under impact the water surface is under impact [O’Brien ’95] [Mould ’97][O’Brien ’95] [Mould ’97]

8

MMolecular Dynamicsolecular Dynamics Simulate Simulate elasticelastic and and inelasticinelastic objects and viscous objects and viscous

fluidsfluids Computational expensive for Computational expensive for large volumelarge volume of fluid of fluid LackLack of coupling between velocity and pressure of coupling between velocity and pressure

SPH (Smoothed Particle Hydrodynamics)SPH (Smoothed Particle Hydrodynamics) Deformable object Deformable object [Desbrun’96][Desbrun’96]

Lava Lava [Stora ‘99][Stora ‘99]

AdvantagesAdvantages More More economicaleconomical than molecular dynamics than molecular dynamics CoupleCouple pressure and velocity pressure and velocity


That’s why we choose it!

9

3-D Navier-Stokes Equations3-D Navier-Stokes Equations Produce the most realistic motion but are very Produce the most realistic motion but are very

computationally computationally expensiveexpensive InitiatedInitiated by Foster ’96 by Foster ’96

Used Used MACMAC formulation dated by 1965 formulation dated by 1965 Semi-Lagrangian integrationSemi-Lagrangian integration allows allows largelarge time step time step

without harming the accuracy without harming the accuracy [Stan’99][Stan’99]

Implicit SurfaceImplicit Surface Represent fluid volume by Represent fluid volume by a density functiona density function

[Kunimatsu ’01][Kunimatsu ’01] Hybrid model Hybrid model [Foster ’01][Foster ’01]

Dynamic Dynamic implicit surfaceimplicit surface Allow Allow particlesparticles to locally correct surface value to locally correct surface value


10

OutlineOutline

Fluid dynamics solver

pn & vn

Partition the scene by a grid (only in the first

time-step)

grid

Evolve density functions

vn+1

Generate particles in the drastic motion region

Fn+1

Move particles by SPH

existing particles

new particles

Performed per time-step

Subdivision

Performed during rendering

Interpolation

high resolution grid

high resolution Fn+1

Convert particle

volumes to Fparticles positions

high resolution Fn+1 with particle volumes

Extract iso-surface

Rendered using a ray-tracer

polygon mesh

Our Methods !!!

Fn

11

Modeling SplashModeling Splash

ProblemProblem Grid may be too Grid may be too coarsecoarse to resolve the detail to resolve the detail

features in a surface cellfeatures in a surface cell Put particles in Put particles in allall surface cell is too surface cell is too

computational expensivecomputational expensive SolutionSolution

Introduce particles in the cell with Introduce particles in the cell with drastic drastic motionmotion

12

Drastic Motion CriterionDrastic Motion Criterion Seed particles in the Seed particles in the surface cellsurface cell with with drasticdrastic motion motion

xtu

Limit the maximum number of particles in a cellLimit the maximum number of particles in a cell Existing particles provides more accurate approximationExisting particles provides more accurate approximation

Fluid particle should not travel more than one cell

size in a time step; 0<γ<1

Fluid traveling

distance in a time step Lower Lower γ tends to produce more particles tends to produce more particles

13

Volume FractionsVolume Fractions

14

Particle SeedingParticle Seeding

Particles are Particles are randomlyrandomly positioned inside positioned inside the interface (fluid side)the interface (fluid side)

Particles velocities are Particles velocities are linearly linearly interpolatedinterpolated in the cell in the cell

u

v

Reconstructed interface

15

Interface ReconstructionInterface Reconstruction Approximate the interface by a plane Approximate the interface by a plane

ax+by+cz+d=0ax+by+cz+d=0(a,b,c)

Vf

NormalNormal is easy to compute, e.g. by is easy to compute, e.g. by spatial gradientsspatial gradients

Find Find dd so that the so that the “cut volume”“cut volume” given by given by the interface is the interface is VVf f [Scardovelii ‘00][Scardovelii ‘00]

Cut volume

16

Partition the scene by a grid (only in the first

time-step)

grid

Evolve density functions

Fn+1

Move particles by SPH

Subdivision

Performed during rendering

Interpolation

high resolution grid

high resolution Fn+1

Convert particle

volumes to Fparticles positions

high resolution Fn+1 with particle volumes

Extract iso-surface

Rendered using a ray-tracer

polygon mesh

Next Slide…Next Slide…

My own method !!!

17

Subdivision Volume Subdivision Volume FractionsFractions

Coarse grid Fine grid

Smooth surface

Subdivision &

interpolation

iso-surface extraction

Convert particles

volumes to F

Non-smooth surface

Non-smooth surface

Without subdivision & interpolation

Without converting particles to F



Small particles are directly rendered as

blobs

18

Why Blobs Doesn’t Work?Why Blobs Doesn’t Work?

Non-smooth surface

19

3x3x3 subdivision

Convert particles to

volume fractions

20

ResultResult Computational grid - 28x24x28Computational grid - 28x24x28 Simulation time per frame – Simulation time per frame – 8 ~ 60 sec8 ~ 60 sec Cycles per frame - Cycles per frame - 55 Average number of particles - Average number of particles - 500500 Rendering time per frame – Rendering time per frame – 1.5 ~ 4 min1.5 ~ 4 min Subdivision –Subdivision – 3x3x3 3x3x3

AMD Athlon 1000 MHZAMD Athlon 1000 MHZ 512 MB DRAM512 MB DRAM

21

Control the Number of Control the Number of ParticlesParticles

γ=0.2 γ=0.1

22

Comparison 1Comparison 1O'Brien,J.F.; Hodgins,J.K., "Dynamic Simulation of Splashing Fluids", in Proceedings of Computer

Animation '95, pp.198-205, 1995.

Mould,D.; Yang,Y.H., "Modeling Water for Computer Graphics", Computers & Graphics, vol.21, no.6, pp.801-814, 1997.

Generate particles when the vertical velocities of a column exceed a predefined threshold

Extend O’Brien et al.

Allow droplets to split.

23

Comparison 2Comparison 2

Enright,D.; Marschner,S.; Fedkiw,R., "Animation and Rendering of Complex Water Surfaces", in Procceeding os SIGGRAPH 2002, 2002 (to

appear).

Foster,N.; Fedkiw,R., "Practical Animation of Liquids", Computer Graphics, pp.23-30, 2001.

Hybrid model:

•Dynamic level set implicit surface.

•Marker particles locally correct surface.

Extend Foster ’01:

Put particles on both sides of the surface.

24

Comparison 3Comparison 3•Heuristic Approach

•Marker Particles

Identify drastic motion region and generate particles.

Place particles near the surface to capture drastic motion.

Autonomous particles moves independently of the fluid.

Passive particles velocities are interpolated from the grid node velocities.

25

ConclusionConclusion

A hybrid approach integrating A hybrid approach integrating particles particles (Lagrangian method)(Lagrangian method) and and volume volume fractions (Eulerian method)fractions (Eulerian method)

A scheme to identify A scheme to identify drastic motion drastic motion area and area and seed particlesseed particles

Smooth surface by Smooth surface by convertingconverting particles particles to volume fractionsto volume fractions

Documents

1 Modeling Highly- Deformable Liquid Chih-Wei Chiu Computer Graphics and Geometry Modeling Laboratory National Chiao Tung University June 25, 2002 Advisors: