Water

1
Water

• Computational Fluid Dynamics
• Volumes
• Lagrangian vs. Eulerian modelling
• Navier-Stokes equations
• Solving Navier-Stokes
• Papers only

Foster and Fedkiw, 2001
2
Computational Fluid Dynamics

• CFD
• Describes the characteristics of fluids in a
  volume
• Animation vs. CFD
• CFD Initial conditions, let it run
• Animation Looks real and we have control
• CFD correctness
• Animation efficiency

3
Describing volumes

• We need to describe the volume and whats in it
• Volumes are typically described using regular
  voxels (3D cells) (128 x 128 x 128 is common)
• Boolean for each voxel

Foster and Metaxas, 2000
4
Describing whats in the volume

• How do we describe water?
• Ideas?

5
Two approaches

• Derives from Computational Fluid Dynamics
• Lagrangian models
• What happens at points in space
• Eulerian models
• Where does stuff go

6
Lagrangian Models

• Break space into volumes
• Describe what is in each volume
• Example Incompressibility
• What goes in must match what goes out

7
Eulerian Models

• We track motion of actual things over time
• Particles are usually used to represent water
• Example Incompressibility
• Particles must adhere to some packing rule

8
Navier-Stokes equation

• Describes the motion of incompressible fluids
• Water, oil, mud, etc.

9
Divergence

• u is the liquid velocity field
• Operator is called the divergence operator.
• Divergence of a vector field F, denoted div(F) or
  as above is a scalar field
• When equal to zero (zero vector), we have a
  divergenceless field

10
Gradient vs. Divergence

• I hate operator overloading

11
Divergenceless field

• Implies incompressiblity
• Mass is conserved
• What goes out a point equals what goes in
• Note This is inside the water, not what
  happens when air mixes in

12
More equation parts
Gravity and force
viscous drag
convection
pressure
pressure
velocity
viscosity
density
13
Problems with Navier-Stokes

• Acceleration is associated with moving elements,
  so Eulerian models make sense
• Pressure and boundary conditions are at
  locations, so Lagrangian models make sense
• Some methods use semi-Lagrangian models

14
Foster and Fedkiw method

• 1. Model environment as a voxel grid
• 2. Model liquid volume using particles and
  implicit surface
• 3. Update velocity field by solving
  Navier-Stokes using finite differences and a
  semi-Lagrangian method
• 4. Apply velocity constraints from moving
  objects
• 5. Enforce incompressibility
• 6. Update the liquid volume using new velocity
  field

15
The volume representation

• Each cell has
• Flag for filled or available for water
• Pressure variable at center (optional)
• Velocity vectors on 3 sides
• Shared with adjacent cell

16
Particles

• Water is represented by particles
• Introduced from source or initially placed
• Motion for particle is determined by tri-linear
  interpolation

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Isocontour

• Imagine each particle with a sphere around it

Isocontour
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Practical system issues

• The isocontour is all we need for creating the
  image
• High particle densities are needed to make good
  isocontours, but create lots of complexity
• Create initial isocontour at high density, then
  use lower density motion fields to deform the
  isocontour
• Other option dynamically create/destroy
  particles

19
Solving Navier-Stokes

• Step 1 Choose a time step
• Good rule of thumb Nothing can jump over any
  cells
• ???

20
Solving

• Step 2 Solve for the velocity
• This is so much fun
• Let w0(x) be a solution at location x at time t1
• Well move to a solution at time t2 using four
  steps

21
Getting to w1 Add force

• w1(x) w0(x) Dt f

22
Getting to w2Add convection

• At each time step, all particles are moved by the
  velocity of the fluid
• The particle at location x was at location
  p(x,t-Dt) before
• Let w2(x)w1(p(x,t-Dt))
• All that is required is a way to track particles
  and some linear interpolation.

23
Getting to w3Viscosity

• Can be shown that

24
Finite differences

• How do we get from or to code
  on arrays?
• Finite differences

25
Solving for viscosity
This is a standard problem formulation called the
Poisson Problem and can be solved using numerous
solving packages like FISHPAK.
26
From FISHPAK
C Subroutine POIS3D solves the linear system of
equations C C C1(X(I-1,J,K)-2.X(I,J,K)X(I1,J
,K)) C C2(X(I,J-1,K)-2.X(I,J,K)X(I,J1,K))
C A(K)X(I,J,K-1)B(K)X(I,J,K)C(K)X(I,J,K1)
F(I,J,K) C
27
Getting to w4Incompressiblity

• Our solution will have a divergence part and a
  divergence-free part. We need to solve for the
  divergence part and subtract it out.

28
Estimating the divergence of the velocity field
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Cont
Again, we have a Poisson Problem.
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What about pressure?

• Issues of incompressibility
• Stam claims that pressure drops out in his
  solution. This would be a consequence of the way
  he deals with incompressibility.

31
Boundary conditions

• What about where we run out of space?
• Set velocity to zero? (paper says this)
• Or what other option?