Animation%20of%20Fluids

1
Animation of Fluids
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Animating Fluid is Hard

• Too complex to animate by hand
• Surface is changing very quickly
• Lots of small details
• In short, a nightmare!
• Need automaticsimulations

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Ad-Hoc Methods

• Some simple algorithmsexist for special cases
• Mostly waves
• What about water glass?
• Too much work to comeup with empirical
  algorithmsfor each case

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Physically-Based Approach

• Look to Fluid Dynamics
• Long history. Back to Newton
• Equations that describe fluid motion
• Use numerical methods to approximate fluid
  equations, simulating fluid motion
• Like mass-spring systems

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Current State-of-the-art in CG

• Marker-And-Cell (MAC) Method
• fedkiw_fluid\glass00.avi
• fedkiw_fluid\splash-640.avi
• Smoothed Particle Hydrodynamics (SPH)
• muller_particles\sph.avi
• muller_particles\pool.avi

6
Applications

• Mostly Hollywood
• Shrek
• Antz
• Terminator 3
• Many others
• Games
• Engineering

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Fluid Dynamics

• (with as little math as possible)

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What do we mean by Fluid?

• liquids or gasses
• Mathematically
• A vector field u (represents the fluid velocity)
• A scalar field p (represents the fluid pressure)
• fluid density (d) and fluid viscosity (v)

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Vector Fields

• 2D Scalar function
• f(x,y) z
• z is a scalar value
• 2D Vector function
• u(x,y) v
• v is a vector value
• v (x, y)
• The set of valuesu(x,y) v is called avector
  field

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Fluid Velocity Vector Field

• Can model a fluid as a vector field u(x,y)
• u is the velocity of the fluid at (x,y)
• Velocity is different at each point in fluid!
• Need to compute change in vector field

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Conceptual Leap

• Particle Simulation
• Track particle positions x (x,y)
• Numerically Integrate change in position
• Fluid Simulation
• Track fluid velocities u (u,v) atall points x
  in some fluid volume D
• Numerically Integrate change in velocity

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Equations of Fluid Dynamics

• Navier-Stokes Equation
• Non-linear Partial Differential Equation
• Models fluid transport
• Derived from Newtons second law
• conservation of momentum all the forces go
  somewhere
• Mass-Conservation condition
• If we have a liter of water at the beginning of
  the solution, we have a liter at the end

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Change in Velocity

• Derivative of velocity with respect to time
• Change in velocity, or acceleration
• So this equation models acceleration of fluids

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Advection Term

• Advection term
• Force exerted on a particleof fluid by the other
  particlesof fluid surrounding it
• How the fluid pushes itself around

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Particle AdvectionVideo
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Diffusion Term

• Viscosity constant controls velocity
  diffusion
• Essentially, this term describes how fluid motion
  is damped
• Highly viscous fluids stick together
• Like maple syrup
• Low-viscosity fluids flow freely
• Gasses have low viscosity

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Weather Advection Diffusion

• Jet-Stream

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Pressure Term

• Pressure follows a diffusion process
• Fluid moves from high-pressureareas to
  low-pressure areas
• Moving velocity
• So fluid moves in direction oflargest change in
  pressure
• This direction is the gradient

Time
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Weather Pressure

• Fronts are the boundaries between regions of
  air with different pressure
• High Pressure Zones will diffuse into Low
  Pressure Zones

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Fluid Example

• Fast moving fluid is pulled towards
  slower-moving fluid

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Body Force

• Body force term represents external forces that
  act on the fluid
• Gravity
• Wind
• Etc

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Summary

• And 1 liter 1 liter constraint
• Need to simulate these equations

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Intermission

• Smoke
• fedkiw_octree\smoke_octree.avi
• Fire
• fedkiw_fire\flammable.avi

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Implementation Overview
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Fluid Representation

• Want to simulate motion of some fluid body
• fluid is represented by a vector field
• Two problems
• Need to compute change invector field (using
  Navier-Stokes equation)
• Need to track fluid position

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Solution Discretization

• Create regular grid

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Solution Discretization

• Create regular grid
• Discretize fluid into grid cells

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Solution Discretization

• Create regular grid
• Discretize fluid into grid cells
• Track single velocityvector in eachgrid cell

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Simulation Step

• Solve Navier-Stokes equation for each grid cell
  to compute change in velocity

Body Force is just like mass-spring systems
non-linear Advection term is difficult. Can
finite-difference, but is not robust
finite-difference Diffusion and Pressure terms
are linear systems of equations
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Free Surface Tracking with Marker Particles

• Want higher-resolution surface for rendering
• Add a bunch of particles
• Passively Advect thembased on fluid velocity

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So the rest is easy, right?

• No
• Still have to enforce mass-conservation
  constaint
• Standard equation does not take boundary
  conditions into account
• Boundary conditions are things like walls,
  fluid/air boundaries, rubber duckies, and so on
• Have to hack the equationsthis is hard
• Numerical Stability is elusive

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Intermission 2

• Melting
• carlson_melting\bunnyside.mpg
• Rigid Body Fluid
• carlson_rigidfluid\rigidfluid.avi

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Problems with Fluid Simulation
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Surface Resolution
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Weird Behavior.
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Water or Vegetable Oil?

• fedkiw_fluid\glass00.avi
• Oh, and its very, very slow
• 7 minutes per frame forwater glass

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Hard to Control

• Animators want to control fluid behavior
• Fluid simulation has a lot of free variables
• There has been limited success so far

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Numerical Stability
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Maya Fluid Effects
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What does fluid effects support?

• Naver-Stokes-based Fluids
• Smoke
• Clouds
• Explosions
• Fire
• Goo type-stuff
• Ad-Hoc / Mass-Spring fluids
• Oceans
• Ponds

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Fluid Effects Algorithms

• Unconditionally Stable Navier-Stokes simulation
• Means they never explode, even with large
  timesteps
• Jos Stam, Stable Fluids, SIGGRAPH 99
• Do not preserve volume very well
• Ok for smoke
• Problematic for water glass
• Gets worse w/ larger timestep

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Maya Smoke

• Smoke demo

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Maya Fire

• Looks like smoke

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More Fluid Effects Demos
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Maya Ocean

• 2D height field no crashing waves
• Can attach bouyant objects

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Maya Pond

• 2D height field
• No splashing
• Mass-spring system
• Bouys, Boats, Wakes
• Can run in real-time

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Fin

• (questions?)