Keyframe Control of Smoke Simulations SIGGRAPH 2003.
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Transcript of Keyframe Control of Smoke Simulations SIGGRAPH 2003.
Keyframe Control of SKeyframe Control of Smoke Simulationsmoke Simulations
SIGGRAPH 2003SIGGRAPH 2003
OverviewOverview
► IntroductionIntroduction►Basis equationBasis equation►Proposed methodProposed method►ResultsResults►Future workFuture work
IntroductionIntroduction
►Goal:Goal: Control of smoke simulationControl of smoke simulation
►DifficultiesDifficulties ComplexityComplexity Non-linearityNon-linearity
►Proposed method:Proposed method: Control the simulation by given density Control the simulation by given density
and velocityand velocity
Basis EquationsBasis Equations
►Navier-Stoke Equation:Navier-Stoke Equation:
Velocity diffusionVelocity advection
External forces
Smoke density advection
General procedureGeneral procedure
)(0 xw )(1 xw )(2 xw
)(3 xw)(4 xw
Add force Advect
Diffuse
Project
FrameworkFramework
► State consists of State consists of of densitiesof densities of velocity vectorof velocity vector
► Initial state:Initial state:
► State at time t:State at time t:
► Simulation: Simulation:
v
0q
ControlControl
► A set of keyframes that the smoke should achiA set of keyframes that the smoke should achieveeve
Specifies the density should match at timSpecifies the density should match at time te t
Specifies the constraint on Specifies the constraint on
► A set of parameterized forcesA set of parameterized forces Amount/directionAmount/direction
t
*tv tv
*t
Matching KeyframesMatching Keyframes► GoalGoal
Match the user-specified keyframeMatch the user-specified keyframe
Use as little force as possibleUse as little force as possible
Solve for the Solve for the equationequation
Computing DerivativesComputing Derivatives
►Calculating derivatives by simulating the Calculating derivatives by simulating the entire process in a space consisting ofentire process in a space consisting of A density and velocity fieldA density and velocity field Their derivativesTheir derivatives
► Initial state:Initial state:►State at time t:State at time t:
Computing DerivativesComputing Derivatives
► Standard solver process:Standard solver process:
Mass preservation stepMass preservation step Advects the smoke densityAdvects the smoke density Projects the resulting fieldProjects the resulting field Performs diffusionPerforms diffusion Advects the velocityAdvects the velocity External forcesExternal forces
M
DP
FvA
A
Calculating S
Computing DerivativesComputing Derivatives
►CalculatingCalculating Each operation induces a operationEach operation induces a operation Ex: Ex:
And similarly forAnd similarly for
Therefore, Therefore,
DerivativesDerivatives
►ProjectionProjection
►Diffusion Diffusion
DerivativesDerivatives
►AdvectionAdvection
DerivativesDerivatives
►Mass PreservationMass Preservation
►ForcesForces
Control ParametersControl Parameters
►Two types:Two types:
Wind forces Vortex forces
Wind forcesWind forces
►A single control vector scaled by a A single control vector scaled by a Gaussian falloff functionGaussian falloff function
►DerivativeDerivative
Vortex ForcesVortex Forces
►Using Gaussian falloff approachUsing Gaussian falloff approach
►DerivativesDerivatives
Objective FunctionObjective Function
►SmoothnessSmoothness
DerivativesDerivatives
Objective functionObjective function
►Keyframe-matchingKeyframe-matching Straightforward methodStraightforward method
Proposed methodProposed method
DerivativesDerivatives
ResultsResults
Future WorkFuture Work
►Drawbacks:Drawbacks: Computationally prohibitive with fine-grained Computationally prohibitive with fine-grained
controlcontrol Optimization might be caught in local minimuOptimization might be caught in local minimu
mm
►To paradigms other than keyframesTo paradigms other than keyframes