Stable Fluids

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Stable Fluids

• A paper by Jos Stam

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Contributions

• Real-Time unconditionally stable solver for
  Navier-Stokes fluid dynamics equations
• Implicit methods allow for large timesteps
• Excessive damping damps out swirling vortices
• Easy to implement
• Controllable (?)

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Some Math(s)

• Nabla Operator
• Laplacian Operator
• Gradient

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More Math(s)

• Vector Gradient
• Divergence
• Directional Derivative

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Navier-Stokes Fluid Dynamics

• Velocity field u, Pressure field p
• Viscosity v, density d (constants)
• External force f
• Navier-Stokes Equation
• Mass Conservation Condition

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Navier-Stokes Equation

• Derived from momentum conservation condition
• 4 Components
• Advection/Convection
• Diffusion (damping)
• Pressure
• External force (gravity, etc)

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Mass Conservation Condition

• Velocity field u has zero divergence
• Net mass change of any sub-region is 0
• Flow in flow out
• Incompressible fluid
• Comes from continuum assumption

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Enforcing Zero Divergence

• Pressure and Velocity fields related
• Say we have velocity field w with non-zero
  divergence
• Can decompose into
• Helmholtz-Hodge Decomposition
• u has zero divergence
• Define operator P that takes w to u
• Apply P to Navier-Stokes Equation
• (Used facts that and
  )

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Operator P

• Need to find
• Implicit definition
• Poisson equation for scalar field p
• Neumann boundary condition
• Sparse linear system when discretized

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Solving the System

• Need to calculate
• Start with initial state
• Calculate new velocity fields
• New state

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Step 1 Add Force

• Assume change in force is small during timestep
• Just do a basic forward-Euler step
• Note f is actually an acceleration?

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Step 2 - Advection
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Method of Characteristics

• p is called the characteristic
• Partial streamline of velocity field u
• Can show u does not vary along streamline
• Determine p by tracing backwards
• Unconditionally stable
• Maximum value of w2 is never greater than
  maximum value of w1

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Step 3 Diffusion

• Standard diffusion equation
• Use implicit method
• Sparse linear system

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Step 4 - Projection

• Enforces mass-conservation condition
• Poisson Problem
• Discretize q using central differences
• Sparse linear system
• Maybe banded diagonal
• Relaxation methods too inaccurate
• Method of characteristics more precise for
  divergence-free field

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Complexity Analysis

• Have to solve 2 sparse linear systems
• Theoretically O(N) with multigrid methods
• Advection solver is also O(N)
• However, have to take lots of steps in particle
  tracer, or vortices are damped out very quickly
• So solver is theoretically O(N)
• I think the constant is going to be pretty high

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Periodic Boundaries

• Allows transformation into Fourier domain
• In Fourier domain, nabla operator is equivalent
  to ik
• New Algorithm
• Compute force and advection
• Transform to Fourier domain
• Compute diffusion and projection steps
• Trivial because nabla is just a multiply
• Transform back to time domain

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Diffusing Substances

• Diffuse scalar quantity a (smoke, dust, texture
  coordinate)
• Advected by velocity field while diffusing
• ka is diffusion constant, da is dissipation rate,
  Sa is source term
• Similar to Navier-Stokes
• Can use same methods to solve equations, Except
  dissipation term

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The End