Simulating Smoke and Water in CG

1
Simulating Smoke and Water in CG

• Simulating Free-Surface Water Based on the Navier
  Stokes Equations

2
Last Time

• The Particle Level Set Method for reducing
  numerical dissipation
• Methods for high resolution level sets.

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Todays Lecture

• The Water Surface
• The Water Solver
• Pressure boundary condition at surface-cells
• Velocity boundary conditions at surface-cells
• Surface tension
• Sources and drains
• The MAC grid

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How to Represent the Water Surface
We represent the water surface by a level set
(signed distance field).
?gt0
?lt0

• All grid cells where ?lt0 are water
• All grid cells where ?gt0 are air
• Air grid cells with fluid cell neighbors are
  called surface cells.

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Solver Overview

• Given an initial level set and velocity field
• Extrapolate/extend the velocity across the
  interface.
• Advect the level set and the particles.
• Apply sources and drains.
• Use the new level set to define the new region of
  fluid.
• Set boundary conditions for the free-surface,
  sources, drains and obstacles.
• Apply forces and advect and diffuse fluid.
• Compute pressure and divergence free velocity
  field.

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Boundary Condtions for the Water Surface

• We have discussed how to set pressure and
  velocities at the boundaries of solid objects.
  Here, fluid is not allowed to penetrate the
  boundary.
• How do we specify pressure and velocity in air
  and surface cells?
• It is important that the air excerts no stresses
  on the water surface, instead it should move
  freely.

Let us first recall how to solve for a divergence
free velocity field...
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Computing a Divergence Free Velocity Field III
So we wish to solve the following equation known
as a Poisson equation
where
And where w is known and p is unknown.
Lets first write out the equation
In this case all the quantities we use are
defined at time t.
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Computing a Divergence Free Velocity Field IV

• Next we discretize the equation using finite
  differences

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Computing a Divergence Free Velocity Field V
Now let
Multiplying the equation by ?x2 and re-arranging
terms, we get
Again there is a linear relations ship, so we can
solve a linear system Of equations of the form
Where A is a coefficient matrix, x is the
pressure we are solving for, and b Is the known
divergence of the velocity field.
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Pressure Boundary Condition

• In the case of a neighboring air cell we just set
  the air pressure to zero
• This ensures that fluid can accurately expand and
  contract (the pressure gradient is not always
  parallel to the surface).
• For more advanced surface effects, we can include
  surface tension...

pair 0
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Surface Tension

• Smooths out liquid surfaces.
• Breaks up fluid into small droplets.
• Caused by molecular cohesion.
• Modeled using jump condition in the pressure
  field ??

Dirichlet Boundary Condition set in air cells
bordering the water.
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Velocity Extrapolation/Extension

• In order to make the surface move freely we must
  ensure that all fluid cells are divergence free.
  But we need boundary conditions for the velocity
  on the surface.
• One option is to adjust the velocity in all
  surface cells such that all fluid cells become
  divergence free and use this as boundary
  condition. This is ambiguous.
• Instead we apply velocity extrapolation into the
  air which in the limit is divergence free.

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Velocity Extrapolation/Extension

• Recall the level set equation for moving a
  surface
• We can use a similar equation for moving a
  property in the normal of the level set surface
• Let the velocity field be defined as
  then velocity extension is

14
Sources and Drains I

• Modeled using CSG operations. Recall that the
  signed distance field is negative inside and
  positive outside

Union(x,y) min(x,y)
Intersect(x,y) max(x,y)
x-y subtract(x,y) Intersect(x,-y)
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Sources and Drains II

• A source is a collection of grid cells where
  there is always fluid
• We can model it as a CSG union.
• A drain is a collection of grid cells where there
  is never fluid
• We can model it as a CSG subtraction.

Velocities can be given to source and drain grid
cells by explicitly specifying non-zero
velocities on the source/drain boundary and e.g.
incorporating it into the poisson equation when
solving for the pressure.
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Solver Overview Revisited

• Given an initial level set and velocity field
• Extrapolate the velocity across the interface.
• Advect the level set and the particles.
• Apply sources and drains.
• Use the new level set to define the new region of
  fluid.
• Set boundary conditions for the free-surface,
  sources, drains and obstacles.
• Apply forces and advect and diffuse fluid.
• Compute pressure and divergence free velocity
  field.

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The MAC/Staggered Grid
Velocities are stored at cell faces.
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The MAC/Staggered Grid

• Advantages
• Conjugate gradient method converges in fewer
  iterations.
• The resulting velocity field conserves mass
  better.
• Central differences can be computed in the
  context of adjacent cells.

It is slightly more complicated to implement.
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Recent Work in MultiPhase Fluids

• Considers discontinuities in variables
• Pressure jump (surface tension and bubble shape)
• Density jump (rising bubbles)
• Viscosity jump (bubble shape)

Important for small scale simulations!
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Next Time

• Simulating Ocean Water