Course 14Course 14

Fluid SimulationFluid SimulationMonday, Half Day, 8:30 am - 12:15 pmMonday, Half Day, 8:30 am - 12:15 pm

Course ScheduleCourse Schedule

• The Basics of Fluid Flow (Robert, 105 min)

• Break (15 min)

• The Cutting-Edge in Film (Eran, 45 min)

• Real Time Fluids in Games (Matthias, 45 min)

• Non-Newtonian Fluids (Robert, 15 min)

Talk OutlineTalk Outline

• Particle level set method

• Vorticity confinement & vortex particles

• Fire simulation

• Solid-fluid coupling

Particle Level Set MethodParticle Level Set Method

[Enright et al. ’02]

Level SetsLevel Sets

• We like level sets:

– Smooth surface for rendering

– Geometric information (normals, curvature)

– Handle topological changes

simple rigid body rotation vortex stretching

Problem: Bad Mass ConservationProblem: Bad Mass Conservation

• Even with high order solvers!

SolutionSolution

Lagrangian marker particlesEulerian level set

0

0

COMBINE

ParticlesParticles

• Passively advected with flow

• Help correct interface

– Especially areas of high curvature

• Hybrid surface model [Foster & Fedkiw ’01]

– Particles inside water

• Particle level set method [Enright et al. ’02]

– Particles on both sides

ParticlesParticles

Rigid Body

Original shape

Level set only One-sided particles Particle level set method

ParticlesParticles

• Seeded within band of interface

• Carry sign and radius

• Periodically reseed particles

– Add to under-resolved regions

– Delete unnecessary particles

Error IdentificationError Identification

• Escaped particles (on wrong side of interface)

pr

0

0

Error QuantificationError Quantification

• Particles behave like little level set spheres

pr

Error CorrectionError Correction

• Compute corrections to >0 and <0 regions

• Merge into final corrected level set

– Take minimum magnitude

0

0

E

0 0

0

0

Particle Level Set UpdateParticle Level Set Update

• Advect particles & level set

• Particle correction

• Reinitialize

• Particle correction

• Adjust radii

simple rigid body rotation vortex stretching

Reduced Mass LossReduced Mass Loss

Fast and Accurate PLSFast and Accurate PLS

• Original PLS:

– 3rd order RK; 5th order HJ-WENO

• [Enright et al. ‘04]: Good results with lower order methods

– Semi-Lagrangian (advection)

– Fast marching method (redistancing)

– 2nd order RK for particles

• Easier for adaptive grids

Vorticity Confinement & Vorticity Confinement & Vortex ParticlesVortex Particles

[Selle et al. ’05]

VorticityVorticity

• Vorticity:

– Local rigid rotation

• Want to simulate turbulent phenomena

– Problem: numerical dissipation (boring flows)

Fixing Boring FlowsFixing Boring Flows

Boring Flow Vorticity Confinement Vortex Particle Method

Vorticity ConfinementVorticity Confinement

2. Compute vorticity

4. Compute a force

1. Start with a velocity field

3. Compute vectors directed toward local maxima in vorticity magnitude

[Steinhoff & Underhill ’94; Fedkiw et al. ’01]

[Andrew Selle]

Confinement ParameterConfinement Parameter

=0.25 =0.50

Limitations of Vorticity ConfinementLimitations of Vorticity Confinement

• Uniformly amplifies vorticity ) grid artifacts

• Can only amplify vorticity already on the grid

• Unstable with larger

=2

Vortex Equations of FlowVortex Equations of Flow

Navier-Stokes Equations

Vortex Equations of Flow

Vortex Particle MethodVortex Particle Method

• Hybrid method:

– Vortex particles & grid-based fluid solver

– Evolve both and u

• Easier than trying to compute u from

[Selle et al. ’05]

Vortex Particle MethodVortex Particle Method

• Vortex particle update:

– Move particles (advection)

– Change vorticity (vortex stretching)

• Adding vorticity to flow:

– Apply analytic confinement force

– Ensures vorticity is conserved

Results - SmokeResults - Smoke

Hybrid Method(6000 particles)

Results - Smoke Results - Smoke

Results - WaterResults - Water

[320x128x320 effective octree, 600 particles]

Fire SimulationFire Simulation

[Nguyen et al. ’02]

FireFire

Gaseous fuel

Blue core

Blackbody radiation

Soot & smoke

Heating

Cooling

Emission due to chemical reaction

Assumed premixed with air

Yellowish/orange glow

Visible after cooling

ignition

Fire SimulationFire Simulation

• 2 phase flow

– Gaseous fuel

– Hot gaseous product

• Level set captures interface (blue core)

• Incompressible & inviscid

• Model gas expansion

[Nguyen et al. ’01,’02]

Flame ModelingFlame Modeling

hot gaseous product

gaseous fuel

thin flame

Blue CoreBlue Core

• Track using level set (don’t need PLS)

Varying flame reaction speed S (smaller on right)

Jump ConditionsJump Conditions

• Using:

• e.g. Conservation of mass:

Shorthand:

Mass flux entering flame front

Mass flux exitingflame front

Jump ConditionsJump Conditions

• Conservation of mass and momentum:

Jump ConditionsJump Conditions

• Can rewrite as

Varying Density RatioVarying Density Ratio

Larger f / h on right

Ghost ValuesGhost Values

Fuel

Ghost ValueHot Gas

Ghost Fluid Method: [Fedkiw et al. ’99]

Velocity JumpVelocity Jump

Solve for fuel phase Solve for products phase

Pressure JumpPressure Jump

• Incorporate into pressure solve (projection step)

• e.g. In 1D:

pi+2

pi pi-1

xi xi-1 xi+1 xi+2 xI

pi+1

ghost

When All is Said and Done…When All is Said and Done…

• Still symmetric, positive definite!

Temperature & Smoke DensityTemperature & Smoke Density

Temperature

• Used for color map

Smoke Density

• Soot & smoke

CampfireCampfire

Flammable SolidsFlammable Solids

• Voxelize solid

• Track solid’s temperature (heat conduction)

• After ignition

– Change solid voxels to negative (fuel)

– Set injection velocity on faces of solid voxels

Flammable BallFlammable Ball

Multiple Interacting LiquidsMultiple Interacting Liquids[2nd talk – “Fluids” papers session – Wed. 8:30-10:15]

Solid-Fluid CouplingSolid-Fluid Coupling

[Guendelman et al. ’05]

Lagrangian vs. Eulerian MeshesLagrangian vs. Eulerian Meshes

Lagrangian (moving) mesh

• Good for solids

• Bad for fluids

– Significant deformation and topology change

Eulerian (static) mesh

• Good for fluids

• Bad for solids

– Harder to track moving material quantities

Strong vs. Weak CouplingStrong vs. Weak Coupling

Strong coupling

(simultaneous solution)

• Monolothic system

• More stable

Weak coupling

(staggered solution)

• Use existing simulators

• Less stable

SOLIDS SIM

FLUIDS SIMSOLIDS + FLUIDS SIM

e.g. [Chentanez et al. ’06]

Volumetric vs. Thin SolidsVolumetric vs. Thin Solids

Volumetric Thin shell

256x256x192 effective octree; 30k triangles

Coupling to Thin SolidsCoupling to Thin Solids[Guendelman et al. ’05]

Preventing Leaks: VisibilityPreventing Leaks: Visibility

Check visibility of interpolation nodesUse replacement ghost value when interpolating

One-Sided InterpolationOne-Sided Interpolation

One-Sided AdvectionOne-Sided Advection

• Clip semi-Lagrangian rays

u

x-ut

Additional ConsiderationsAdditional Considerations

• Crossed-over nodes

• Preventing particles from crossing solid

• Redistancing & velocity extrapolation

Simulation StepSimulation Step

un,n

Advance particle level set ! n+1

Advect un and add gravity ! u*

Project u* ! un+1

un+1,n+1

One-sided advection( and particles)

One-sided advection (u)

Solid Affecting FluidSolid Affecting Fluid

• Rasterize solid onto grid faces

• Enforce solid velocity with Neumann boundary conditions

Project u* ! un+1

One-Way Coupling StepOne-Way Coupling Step

un,n

Advance particle level set ! n+1

Advect un and add gravity ! u*

Project u* ! un+1

un+1,n+1

Advance solid

Enforce solid velocities at solid-fluid interface

160x192x160 effective octree

One Way Coupling ExampleOne Way Coupling Example

Fluid Coupling ForceFluid Coupling Force

• Want to use fluid pressure

• Incompressible pressure can be noisy

– Incompressibility = hard constraint

– Enforcing solid velocity = hard constraint

– Better for compressible fluids [Yngve et al. ‘00; Fedkiw ‘02]

Smoother Coupling PressureSmoother Coupling Pressure

• Don’t enforce solid velocity

– Treat solid as fluid

• Solve variable density fluid for pc

Two Pressure Solves!Two Pressure Solves!

• Incompressible pressure (projection):

– Enforce incompressibility & solid velocity

– Essential for reducing mass loss

• Coupling pressure:

– Does not modify fluid velocity

– Essential for smoother coupling force on solid

Computing Force on SolidComputing Force on Solid

• Fluid pressure pushes on both sides

p1

p2

Computing Force on SolidComputing Force on Solid

• Net force is proportional to pressure jump [pc]

p1 - p2

Computing Force on SolidComputing Force on Solid

Rasterize solid

Compute coupling pressure

Pressure jumps on faces

Average to nodes

Extrapolate

Interpolate at centroid

Compute force

Two-Way Coupling StepTwo-Way Coupling Step

un,n

Advance particle level set ! n+1

Advect un and add gravity ! u*

Advance solid

Project u* ! un+1

un+1,n+1

Compute coupling pressure and apply force to solid

148x148x111 uniform; 2.5k triangles

Rigid ShellRigid Shell

200x200x200 effective octree; 30k triangles

Water-Cloth CouplingWater-Cloth Coupling

210x140x140 uniform; 30k triangles

Coupling to SmokeCoupling to Smoke

Coupling to Volumetric SolidsCoupling to Volumetric Solids[Losasso et al. ’06]

ApproachApproach

• Rasterize solid onto fluid grid

– Compute object

• Fill with ghost fluid values

• Two-way coupling as for thin shells

Volumetric SolidsVolumetric Solids

Extrapolate fluid values[Houston et al. ’03;

Rasmussen et al. ’04]

Need grid values in solid

Volumetric SolidsVolumetric Solids

• Velocity boundary conditions

– Model object friction (slip to no-slip)

boundary conditions

– Extrapolate fluid into object

• Separation conditions

– Avoid pulling water out of solid

• Also collide fluid particles against objects

Two-Way CouplingTwo-Way Coupling

Solid Affecting Fluid

• Neumann boundary conditions

Fluid Affecting Solid

• Compute coupling pressure

• Apply force

Ice Cubes ExampleIce Cubes Example

100x100x100 uniform grid

SummarySummary

• Particle level set method

– Particles help conserve mass

• Vorticity confinement & vortex particles

– Help preserve “turbulence” in flow

• Fire simulation

– 2 phases with jump conditions

• Solid-fluid coupling

– Couple using solid velocity & fluid pressure