Simulating Smoke and Water in CG

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Simulating Smoke and Water in CG

• Introduction to Water Simulation
• and
• The Level Set Method

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Last Time

• Hypertexturing
• Cellular Texturing
• Vortex Particle Method

3
This Lecture

• Introduction to Water Simulations
• The Level Set Method for Representing the fluid
  surface

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Intro to Water Simulations
We want to be able to simulate water like this
In fact you have already learned how water moves
The Navier Stokes Equations
In the case where we are not interested in the
fluid surface you can just use the methods you
have already learned.
There are three main open questions

• How do we represent the fluid surface? (this and
  next lecture)
• How do we make the fluid surface move? (this
  lecture and next Tuesday)
• How do we confine the volumetric velocity field
  to within the fluid (next Tuesday)

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How do we represent the surface?

• A water surface behaves in very complex ways
• It changes topology The surface merges and
  breaks up.
• It stretches and contracts.

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How do we represent the surface?
In an explict surface representation, the points
on the surface are specified explicitly. In this
example the representation is a surface mesh
consisting of connected triangles.
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How do we represent the surface?

• What happens when using an explicit surface
  representation?
• In an explicit representation, only the surface
  is
• maintained, not the inside and outside.

Loops are created and we have to remove them and
determine inside and outside. But what is inside
and what is outside? No general algorithm
proposed in 3D!
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How do we represent the surface?

• In an implicit surface representation, the
  surface is represented implicitly as the
  iso-contour (level set) of a higher dimensional
  scalar function (the surface has co-dimension
  one), and inside/outside information is
  unambiguous.

The signed distance field (a scalar field) is
one such surface representation...
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How do we represent the surface?
gt0
lt0
0
The unsigned distance field
The surface S is defined as
The signed distance field
where sign(x) is -1 inside and 1 outside.
Inside and outside is unambiguous A scalar
value cannot be both postive and negative!
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How do we represent the surface?

• Dynamic Implicit Surfaces (the signed distance
  field combined with a
• framework for advecting and propagating implicit
  surfaces) or Level Set Surfaces
• are a good candidate for representing fluid
  surfaces
• Inside/outside information is unambiguous
  (always a closed surface).
• No stretching and contraction artifacts as
  surface patches are not explicit.
• Topological changes are easily handled by the
  surface embedding (is just a
• matter of changing scalar values in the right
  way).
• The level set framework provides the vanishing
  viscosity solution with no loops
• which is the visual solution we desire.
• As we know, differential properties are easy to
  compute on a scalar grid.

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Intuition About Topology Change using Level Sets
Consider a 1D level set curve in 2D propagating
with time. We are solving the equation ?(x,y,t)
0 which in itself is a 2D iso-surface of a 3D
scalar function.
t
t2
t0
t0
t1
This means that the 1D curve is just the
intersection of a 2D Surface with a plane
orthogonal to the time, t, axis.
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Applications of Level Sets, 1

• Geometrical Modeling
• Morphing
• Medical Segmentation

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Applications of Level Sets, 2

• Water Simulations
• Fire Simulations
• Film production (Day after tomorrow, T3, Scooby
  Doo 2, Peter Pan ...)

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The Level Set Method

• Given a surface as a signed distance field and a
  velocity field (e.g. a fluid velocity field), how
  do we move the surface in the velocity field?

The surface, S, is given by
Written more compactly as
Consider a particle on the surface as a function
of time, x(t) is its path. We differentiate with
t
This is a linear Partial Differential Equation
(PDE)
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The Level Set Method

• Some problems are easier to model by motion in
  the normal direction Geometric Smoothing,
  Morphing, Geometric Modeling.
• How do we derive an equation for this?

Split V into its normal and tangential components
Insert it into the level set equation
Recall that
Which gives
This is a non-linear Partial Differential
Equation (PDE)
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The Level Set Method

• Let us develop some intuition about the level set
  equation
• How do we move a surface outwards in the
  direction of the normal (dilation)?

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The Level Set Method

• Let us develop some intuition about the level set
  equation
• How do we move a surface inwards in the direction
  of the normal (erosion)?

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The Level Set Method

• Let us develop some intuition about the level set
  equation
• How do we smooth a surface?

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The Level Set Method

• How do we do shape-metamorphosis between two
  surfaces?

The Original
More Stable
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Numerical Solution of the Level Set Equations

• The two level set equations are identical
  mathematically. But numerically we have to solve
  them differently.
• The linear equation
  is solved using upwind or semi-lagrangian
  methods.
• The non-linear equation
  is solved using the Godunov Scheme.

Today I will explain only the linear equation...
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Numerical Solution of the Level Set Equations

• Consider the following situation in 1D

V1
?
0.5
x
i
-0.5
We want to move the level set to the right, ie
V1. Assume we are at time n ?t, and we solve for
time (n1)?t. At position i at time (n1)?t, ?
-0.5.
We discretize this using a forward Euler
time-step
And re-arrange the terms
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Numerical Solution of the Level Set Equations
So we are looking at
How should we discretize the gradient at position
i? We have three standard ways
0.5
1
0
Computing these three in our case gives.
But which one is the right one? Recall that at
position i at time (n1)?t , ? -0.5. Inserting
into the equation
gives
-0.5
0.5
0
In general we should use the finite difference in
the direction opposite to The direction of the
velocity. This is called UPWIND DIFFERENCING.
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Numerical Solution of the Level Set Equations

• The numerical methods used for level set
  advection are typically not unconditionally
  stable (one exception is semi-Lagrangian
  advection)
• The CFL condition tells us the largest time-step
  we are allowed to take in order for the solution
  to remain stable. For the level set equation it
  looks like this
• You can also think of it as, the maximal distance
  travelled in one time-step by the surface must be
  less than the grid spacing

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Numerical Solution of the Level Set Equations

• It turns out that, in general, the signed
  distance function does not remain a signed
  distance function after advection.
• The signed distance function behaves nicely
  numerically (Lipschitz continous) and stability
  conditions are derived under the assumption of a
  signed distance field.
• Consequence We need a way to reset the level set
  function to a signed distance field
  Re-Initialization.

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Numerical Solution of the Level Set Equations

• Re-initialization itself can be formulated as a
  level set equation that is solved iteratively
  until convergence.
• Hint We have convergence when
• For outside (positive values)
• For inside (negative values)
• Values on the surface (zero) should not move!
• The combined equation
• Is solved numerically using the Gudonov Scheme.

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Numerical Solution of the Level Set Equations

• A loop for level set advection looks like this
• for (number of iterations)
• Advect(velocity field)
• Advect(level set, velocity field)
• Re-initialize(level set)
• end

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The Narrow Band Level Set Method

• The level set method as previously described
  solves the level set method in the entire domain
  of the embedding scalar function.
• BUT we are only interested in the zero-crossing
  of the scalar function.
• Narrow band level set methods track a narrow band
  of scalar values about the zero-crossing.

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The Narrow Band Level SetMethod
The narrow band of a circle. Outside of the
narrow band, values are clamped
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The Narrow Band Level Set Method

• The coordinates of the grid points inside the
  narrow band are kept in arrays.
• Advection is done in the gamma tube
• for (all (x,y) in gamma tube)
• Solve Level Set Equation
• End
• Re-initialization is done in the N tube (the
  gamma tube one additional layer of grid points)
• Computations at grid points in the gamma tube
  (but not in the beta tube) are damped to avoid
  oscillations/instabilities.

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The Narrow Band Level Set Method

• At each iteration the narrow band needs to be
  rebuilt
• Grid points are added to the narrow band in
  regions where the surface moved closer to the
  edge, and grid points are removed to the narrow
  band in regions where the surface moved further
  away from the edge (of the narrow band).
• This is determined by looking at the values of
  the grid points.

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Next Time

• Improving the accuracy of the level set method
  (the particle level set method)