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Simulation of SmokeSimulation of Smokebased on based on

Vortex Filament PrimitivesVortex Filament PrimitivesAlexis AngelidisAlexis Angelidis

Graphics & Vision Research LabGraphics & Vision Research Lab

Otago - New ZealandOtago - New Zealand

Fabrice NeyretFabrice NeyretGRAVIR / IMAG - INRIAGRAVIR / IMAG - INRIA

Grenoble - FranceGrenoble - France

Tangled-SpaghettisTangled-SpaghettisAlexis AngelidisAlexis Angelidis

Graphics & Vision Research LabGraphics & Vision Research Lab

Otago - New ZealandOtago - New Zealand

Fabrice NeyretFabrice NeyretGRAVIR / IMAG - INRIAGRAVIR / IMAG - INRIA

Grenoble - FranceGrenoble - France

BackgroundBackgroundBackgroundBackgroundFluid animation approaches: Fluid animation approaches: Lagrangian vs EulerianLagrangian vs Eulerian

Popular: Eulerian velocity gridPopular: Eulerian velocity grid

Fluid animation approaches: Fluid animation approaches: Lagrangian vs EulerianLagrangian vs Eulerian

BackgroundBackgroundBackgroundBackground

[McNamara et al.04][McNamara et al.04] [Fattal et al.04][Fattal et al.04][Fedkiw et al.01][Fedkiw et al.01] [Pighin et al.04][Pighin et al.04]

A.A. Velocity grid Velocity grid B.B. Update rules Update rules

ijkv

0

1

)( 2

v

gvvvv

p

t

0

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Fluid animation approaches: Fluid animation approaches: Lagrangian vs EulerianLagrangian vs Eulerian

Popular: Eulerian velocity gridPopular: Eulerian velocity grid

BackgroundBackgroundBackgroundBackground

[McNamara et al.04][McNamara et al.04] [Fattal et al.04][Fattal et al.04][Fedkiw et al.01][Fedkiw et al.01] [Pighin et al.04][Pighin et al.04]

One Alternative – Vortex MethodsOne Alternative – Vortex Methodsvelocityvelocity vs. vorticityvs. vorticity


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vorticity vorticity vorticity vorticity velocity vvelocity vvelocity vvelocity v

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BIOT-SAVARTBIOT-SAVART

One Alternative – Vortex MethodsOne Alternative – Vortex MethodsOne Alternative – Vortex MethodsOne Alternative – Vortex Methods

What’s induced by these curves?What’s induced by these curves?What’s induced by these curves?What’s induced by these curves?

dxxp

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Fluid described with curvesFluid described with curvesFluid described with curvesFluid described with curves

Fine simulationsFine simulations• FilamentsFilaments• FeaturesFeatures

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Geometric InterpretationGeometric InterpretationGeometric InterpretationGeometric Interpretation

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VortexVortex


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Rotation magnitudeRotation magnitude


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Lagrangian Vortex MethodsLagrangian Vortex MethodsLagrangian Vortex MethodsLagrangian Vortex Methods

CC00

CC22

CC33

• Entire fluid = curves of vortices !Entire fluid = curves of vortices !

DynamicsDynamics

• Curves induce movement Curves induce movement

• Curves are animated with this movementCurves are animated with this movement

CC11

ConsequenceConsequence• Cheap storageCheap storage• Dynamic-keyframed curveDynamic-keyframed curve

VideoVideo#2-5#2-5

VideoVideo#2-5#2-5

dxv

dxv



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• EfficiencyEfficiency• stable vortex + noisestable vortex + noise• closed-form integralclosed-form integral• O(O(NN22)), accelerated with LOD, accelerated with LOD• time integrationtime integration

• Define smoke particlesDefine smoke particles

ContributionsContributionsContributionsContributions

Sum of vortices along curvesSum of vortices along curvesSum of vortices along curvesSum of vortices along curvesA more convenient amplitudeA more convenient amplitude

Biot-SavartBiot-Savart CauchyCauchy

There are There are closed-formsclosed-forms for the Cauchy kernel for the Cauchy kernel integral along a integral along a circlecircle and a and a segmentsegment

3 4

1

d 3 4

1

d )/1(

122 rd )/1(

122 rd

Discrete segmentsDiscrete segments

[MS.98][MS.98]

Large time steps: high order schemeLarge time steps: high order schemeLarge time steps: high order schemeLarge time steps: high order scheme

• Traditional forward Euler , BSTraditional forward Euler , BStrajectory = sum of velocities of rotationtrajectory = sum of velocities of rotation

• Biot-Savart tells more than velocityBiot-Savart tells more than velocity

• Our schemeOur schemetrajectory = sum of Rotationtrajectory = sum of Rotation

Levels of detailLevels of detailLevels of detailLevels of detail

• We define a bound to the error We define a bound to the error between a segment and split segmentsbetween a segment and split segments

• We precompute a binary tree for each filamentWe precompute a binary tree for each filament

Too detailedToo detailed Too coarseToo coarseAlrightAlright

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ijkLODijkLOD

• 3 types of noise vortices :3 types of noise vortices :– Tangent vortexTangent vortex

– Normal vortexNormal vortex

– Binormal vortexBinormal vortex

Divergence-free NoiseDivergence-free NoiseDivergence-free NoiseDivergence-free Noise

tt J' tt J'

nn 1 ' JT nn 1 ' JT

)()(' 1 ntb JJ T )()(' 1 ntb JJ T

Good distribution Good distribution of directionsof directions

Noise

FilamentsSmoke

SmokeSmokeSmokeSmoke• ParticlesParticles

– accumulate deformationaccumulate deformation– split when accumulated deformation too bigsplit when accumulated deformation too big

• Rendering Rendering – 2D ellipses2D ellipses– Self-shadowingSelf-shadowing

VideoVideoVideoVideo

• Filaments induce movement (everywhere)Filaments induce movement (everywhere)

• Filaments are animated with the movementFilaments are animated with the movement

• Smoke-particles are animated with LOD- Smoke-particles are animated with LOD- filaments and divergence-free noisefilaments and divergence-free noise

Smoke solver overviewSmoke solver overviewSmoke solver overviewSmoke solver overview

ImprovementsImprovementsImprovementsImprovements

• Smoke particle mergingSmoke particle merging

• Curve split/collapse or resamplingCurve split/collapse or resampling

• Currently, limited boundary conditionsCurrently, limited boundary conditions

ConclusionConclusionConclusionConclusion• Separated dynamics & renderingSeparated dynamics & rendering

• Efficient & hi-resolutionEfficient & hi-resolution

• Not bounded in spaceNot bounded in space

• Compact: easy to load and saveCompact: easy to load and save

• Dynamics or keyframesDynamics or keyframes

THANK YOUQuestions ?THANK YOUQuestions ?

A new integration schemeA new integration schemeA new integration schemeA new integration scheme

• Traditional forward EulerTraditional forward Eulerptpvtpp '

pv

• With our closed form, induced velocity is With our closed form, induced velocity is given by a 4x4 matrixgiven by a 4x4 matrix

• Our schemeOur scheme... ) exp(' ptpptp

a translation is a translationa translation is a translationa rotation is a rotationa rotation is a rotationa twista twist is a twistis a twist

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M

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Simple rotation algebraSimple rotation algebraSimple rotation algebraSimple rotation algebra

• Rotation of center c around axis Rotation of center c around axis of angle of anglegiven by the magnitude of given by the magnitude of

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MotivationMotivationMotivationMotivation

A fluid is A fluid is not an actornot an actor

Existing fluid-directing techniques areExisting fluid-directing techniques areslowslow OR OR tedioustedious

A technique for keyframing fluid animationA technique for keyframing fluid animation• Not boundedNot bounded in a cube in a cube

• Predictable fluid-editing Predictable fluid-editing primitivesprimitives

• FastFast//RobustRobust

AimAimAimAim



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vorticity vorticity vorticity vorticity

Biot-SavartBiot-Savart

velocity vvelocity vvelocity vvelocity v

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To get the motion: computeTo get the motion: compute velocity from vorticityvelocity from vorticity

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What does the Biot-Savart Law mean?What does the Biot-Savart Law mean?What does the Biot-Savart Law mean?What does the Biot-Savart Law mean?

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VortexVortex

Rotation magnitudeRotation magnitudevortexvortex

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The domain of the BS integralThe domain of the BS integralThe domain of the BS integralThe domain of the BS integral

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2.2. Integral over a curveIntegral over a curve of a slice :of a slice :

3. Integral on many curves3. Integral on many curves

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In 3D, vortices concentrate along In 3D, vortices concentrate along tubestubes(with a distribution profile around axis)(with a distribution profile around axis)

CC11

1.1. Integral over a slice of vortices :Integral over a slice of vortices :

change

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Documents

curves of vortices

induced velocity

lagrangian vs eulerianmcnamara

movement curves

actorexisting fluid

movementc0c2c3entire

closed form

velocity gridb