Simulating Fluids in Computer Graphics

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Simulating Fluids in Computer Graphics

• Fluid Control

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Last Time

• More accurate curved boundaries
• How to handle boundaries not aligned with the
  grid
• Strong coupling of solids and fluids

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This Lecture

• Fluid Control
• Motivation
• Divergence control
• Explosions
• Driving simulations towards target shapes
• Characters and objects made of smoke

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Motivation

• Library in The Day After Tomorrow
• Sewer in Ratatouille (YouTube)
• Horse-river in Lord of the Rings
• Gandalfs smoke ring in Lord of the Rings
  (YouTube)
• Harry Potter 6 (Video)
• Explosions

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Divergence Control I

• More accurate buoyancy modeling
• Variable densities

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Divergence ControI II

• If the volume stays the same, but the density
  changes, what happens to the mass?
• We violate conservation of mass!
• We do want the fluid to change volume

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Divergence Control III

• Using Gauss theorem and Euler integration the
  change in volume is
• Combining the previous two expressions

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Divergence Control IV

• The divergence only changes the right-hand side
  of the equation system

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Divergence Control V

• Create a control field d(i,j,k) at grid cell
  centers, which defines a desired rate of
  fractional volume change
• Enforcing the corresponding divergence is
  achieved simply by adding d to the right-hand
  side

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Driving Simulations Towards Target Shapes
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Idea
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Modified Incompressible Euler Equations

• The fluid will flow to locations with lower
  potential.
• The control and design of the fluid simulation
  boils down to designing the potential field U.
• How should it look?

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Determining U I

• We will specify an initial shape pstart and a
  target shape ptarget

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Determining U II

• Solving Laplaces equation gives us a harmonic
  potential function
• Boundary conditions

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Determining U III
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Summing Up

• Preprocessing for NSSolver
• Preprocessing for GeometricPotential
• getControlInput pstart, ptarget and pcheck
• makePotentialField U
• determineGradientField U
• While(Simulating)
• add -s U while adding other body forces to u
• otherwise simulate as usual to update u and s
• U can be changed underway to allow for keyframe
  animation

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Results I
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Results II
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Limitations

• No direct control over how fast the smoke
  approximates the target. Only able to specify the
  potential field U.

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Another Method

• Different modification
• becomes

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Driving Force F

• F should drive s towards s
• Once the target is reached, F should enable the
  fluid to reach a rest state v vt 0
• What about the rest state?

Check!
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Example
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Smoke Gathering G I

• What happens if s contains higher values and
  gradients than s?
• Numerical dissipation makes it impossible
• Due to incompressibility, advection can never
  generate new values of s
• Assume that the total mass of the simulated smoke
  s equals the total mass of the target s
• The error is
• Smooth e until it is constant

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Smoke Gathering G II

• Formally

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Example
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Summing Up

• While(Simulating)
• Apply driving forces
• Attenuate momentum
• Advect velocity
• Pressure project
• Advect smoke
• Gather smoke

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Videos
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Limitations

• No direct control over how well a target is
  approximated at a given time. Only indirect using
  the parameters.
• The smoke gathering term and driving forces must
  be used carefully in order to preserve a
  natural-looking fluid motion.

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Next Time

• More fluid control