Computer Animation Algorithms and Techniques

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Computer AnimationAlgorithms and Techniques
Fluids
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Superficial models v. Deep models
(comes up throughout graphics, but particularly
relevant here)
OR
Directly model visible properties
Model underlying processes that produce the
visible properties
Water waves Wrinkles in skin and cloth Hair Clouds
Computational Fluid Dynamics Cloth weave Physical
properties of a strand of hair Computational
Fluid Dynamics
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Superficial Models for Water
Main problem with water Changes shape Changes
topology
Still waters Small amplitude waves The anatomy of
waves ocean waves running downhill
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Simple Wave Model - Sinusoidal
Distance-amplitude
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Simple Wave
Time-amplitude at a location
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Simple Wave
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Simple Wave
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Sum of Sinusoidals
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Sum of Sinusoidals
Height field
Normal vector perturbation
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Ocean Waves
s distance from source t time A maximum
amplitude C propogation speed L wavelength T
period of wave
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Movement of a particle
In idealized wave, no transport of water
Q average orbital speed S steepness of the
wave T time to complete orbit H twice the
amplitude
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Breaking waves
If Q exceeds C gt breaking wave If non-breaking
wave, steepness is limited Observed steepness
between 0.5 and 1.0
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Airy model of waves
Relates depth of water, propagation speed and
wavelength
g - gravity
As depth increases, C approaches
As depth decreases, C approaches
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Implication of depth on waves approaching beach
at an angle
Wave tends to straighten out closer sections
slow down
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Wave in shallow water
C an L are reduced
T remains the same
A (H) remain the same or increase
Q remains the same
Waves break
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See book for details of modeling ocean waves
from article by Peachey
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Model for Transport ofWater
h water surface b ground v water velocity
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Model for Transport ofWater
Relates Acceleration Difference in adjacent
velocities Acceleration due to gravity
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Model for Transport ofWater
d h(x) b(x)
Relates Temporal change in the height Spatial
change in amount of water
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Model for Transport ofWater
Small fluid velocity
Slowly varying depth
Use finite differences to model - see book
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Models for Clouds
Basic cloud types Physics of clouds Visual
characteristics, rendering issues Early
approaches Volumetric cloud modeling
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Basic Cloud Types
Example forces of formation Convection Convergence
lifting along frontal boundaries Lifting due to
mountains
Height water v. ice composition
cumulus stratus cirrus nimbus
heap layer curl of hair rain
cirr high alto mid-level
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Visual characteristics

• 3D
• Amorphous
• Turbulent
• Complex shading
• Semi-transparent
• Self-shadowing
• Reflective (albedo)

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Early Approach - Gardner
Early flight simulator research Static model for
the most part Sum of overlaping semi-transparent
hollow ellipsoids Taper transparency from edges
to center
See his paper from SIGGRAPH 1985
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Other approaches
Particle systems implicit functions Volumetric
representations
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Dave Ebert
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Models for Fire
Procedural 2D Particle system Other approaches
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Models for Fire - 2D
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Models for Fire - particle system
Derived from Reeves paper on particle systems
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Particle System Fire
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Combustion examples
Show Fedkiws work
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Computational Fluid Dynamics (CFD)
Fluid - a substance, as a liquid or gas, that is
capable of flowing and that changes its shape at
a steady rate when acted upon by a force.
Compressible changeable density
Steady state flow motion attributes are
constant at a point
Viscous resists flow Newtonian fluid has
linear stresss-strain rate
Vortices circular swirls
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General Approaches
Grid-based
Particle-based method
Hybrid method
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CFD equations
mass is conserved momentum is conserved energy is
conserved
Usually not modeled in computer animation
To solve discretize cells discrete
equations numerically solve
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CFD
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Conservation of mass (2D)
u
r
A
Small control volume Dx by Dy by 1 A Dx Dy
v
Time rate of mass change in volume rate of mass
entering
Mass inside CV
Time rate of mass change in volume
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Conservation of mass (2D)
u
r
A
Small control volume Dx by Dy by 1 A Dx Dy
v
Time rate of mass change in volume rate of mass
entering
Amount of mass entering from left
Difference between left and right
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Conservation of mass (2D)
Time rate of mass change in volume rate of mass
entering
Divergence operator
If incompressible
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Conservation of momentum
Momentum in CV changes as the result of
Mass flowing in and out
Collisions of adjacent fluid (pressure)
Random interchange of fluid at boundary
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Conservation of momentum in 2D
Rate of change of u-momentum within CV
u-Momentum entering
Difference in x
Difference in y
Pressure difference in x
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Conservation of momentum in 2D
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Conservation of momentum (3D)
x direction
y direction
z direction
Material derivative
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Navier-Stokes (for graphics)
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Viscosity, etc.
Hookes law in solid, stress is proportional to
strain
Fluid continuously deforms under an applied shear
stress
Newtonian fluid stress is linearly proportional
to time rate of strain
Water, air are Newtonian blood in non-Newtonian
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Stokes Relations
Extended Newtonian idea to multi-dimensional flows
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Stokes Hypothesis
Choose l so that normal stresses sum to zero
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Conservation of momentum with viscosity
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Incompressible, Steady 2-D flow
Kinematic viscosity
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2D Euler Equations no viscosity
If incompressible
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2D Equations review