Simulating Fluids in Computer Graphics

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Simulating Fluids in Computer Graphics

• Vorticity and Particle Methods

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Last Lecture

• Smoke simulation
• Smoke concentration (previously called smoke
  densities but this is less confusing)
• Temperatures
• Sources
• Buoyancy
• Variable densities
• Simulator overview QA

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This Lecture

• Ways to combat numerical diffusion
• Vorticity
• Vorticity confinement
• Vortex particles
• Smoke particles
• Particle In Cell (PIC) method
• FLuid Implicit Particle (FLIP) method
• There are other ways BFECC, QUICK, CIP..

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Vorticity
The curl operator (vorticity) is defined as
What does this mean?
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Vorticity
The velocity field of a rigid motion
(translationrotation)
Differentiating with respect to position gives
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Vorticity
The velocity (vector) gradient (a matrix/tensor)
of rigid motion
Looking at the skew-symmetric part of general
velocity gradient
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Vorticity
By comparing we arrived at
Does this look familiar?
It is half the curl. We call the curl the
vorticity of the velocity field.
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Vorticity of Rotation Field
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Vorticity Dissipates

• Vorticity creates all the swirly motion, but
  dissipates
• Two main sources of dissipation
• Numerical diffusion
• Time splitting

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Vorticity Confinement
With no vorticity confinement
With vorticity confinement
Image by Selle et al. 2005
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Vorticity Confinement
(Slide by Jos Stam, SIGGRAPH 2003)
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Vorticity Confinement
(Slide by Jos Stam, SIGGRAPH 2003)
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Vorticity Confinement
(Slide by Jos Stam, SIGGRAPH 2003)
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Whats Wrong with Vorticity Confinement?

• Artifacts, difficulties rising
• Hard to control ?
• ? is the same everywhere

Image by Selle et al. 2005
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Vortex Particle Method
Image by Selle et al. 2005
We want rolling smoke effects that do not have
the artifacts of Vorticity confinement alone.
Vortex Particle Method was used in Star Wars
III.
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Vortex Particles Video
Selle et al.
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Vortex Particle Method

• The basic idea
• Use grid based solver
• Advect particles that carry dynamic vorticity
  (magnitude and direction)
• Transfer vorticity from particles to grid locally
  using vorticity confinement

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Vortex Particle Method

• Recall the momentum conserving part of the Navier
  Stokes equations
• This can be put into vorticity (curl) form by
  taking the curl on both sides

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Vortex Particle Method
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Vortex Particle Method
You should interpret
As a vector-matrix product
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Vortex Particle Method
We simplify the general vortex form of the
momentum equation
We ignore forces on vorticity
We solve vorticity stretching as an ODE
Vorticity advected in the velocity field. Just
particles carrying vorticity around!
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Vortex Particle Method
Finally the vorticity of the particles is
transferred to the grid using a localized
vorticify confinement
Where
is a Gaussian kernel
Since the vorticity is largest at the center of
the particle, xp, we have that
And finally
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Particle Advection
You already know how to do it!
Third order Runge Kutta
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Particle Advection
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Smoke Particles

• The smoke (density) can be represented by
  particles that each carry a smoke concentration
  sp to avoid numerical dissipation in smoke
  advection.
• It is interpolated to the grid as (for use by
  buoyancy, density, rendering)

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Smoke Particles

• Kernel function

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Smoke Particles

• Kernel weights

We choose the kernel weights such that these two
estimates are equal
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Smoke Particles

• Kernel weights

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Smoke Particles

• Particles are advected using Runge Kutta
• Particles are seeded in smoke sources, jittered
  in space and time.
• Small amount of smoke diffusion can be emulated
  with small pertubations

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PIC/FLIP

• To avoid numerical dissipation in velocity
  advection velocity can be carried by particles as
  well.
• Needs to be coupled to global pressure projection
  solve on the grid.

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PIC/FLIP
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Exercises