Modeling Fluid Phenomena

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Modeling Fluid Phenomena

• Vinay Bondhugula
• (25th 27th April 2006)

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Two major techniques

• Solve the PDE describing fluid dynamics.
• Simulate the fluid as a collection of particles.

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• Rapid Stable Fluid Dynamics for Computer Graphics
  Kass and Miller
• SIGGRAPH 1990

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Previous Work

• Older techniques were not realistic enough
• Tracking of individual waves
• No net transport of water
• Cant handle changes in boundary conditions

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Introduction

• Approximates wave equation for shallow water.
• Solves the wave equation using implicit
  integration.
• The result is good enough for animation purposes.

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Shallow Water Equations Assumptions

• Represent water by a height field.
• Motivation
• In an accurate simulation, computational cost
  grows as the cube of resolution.
• Limitation
• No splashing of water.
• Waves cannot break.

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Contd

• 2) Ignore the vertical component of the velocity
  of water.
• Limitation
• Inaccurate simulation for steep waves.

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Contd

• 3) Horizontal component of the velocity in a
  column is constant.
• Assumption fails in some cases
• Undercurrent
• Greater friction at the bottom.

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Notation

• h(x) is the height of the water surface
• b(x) is the height of the ground surface
• d(x) h(x) b(x) is the depth of the water
• u(x) is the horizontal velocity of a vertical
  water column.
• di(n) is the depth at the ith point after the nth
  iteration.

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The Equations

• F ma, gives the following
• The second term is the horizontal force acting
  on a water column.
• Volume conservation gives

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Contd

• Differentiating equation 1 w.r.t x and equation 2
  w.r.t t we get
• From the simplified wave equation, the wave
  velocity is sqrt(gd).
• Explains why tsunami waves are high
• The wave slows down as it approaches the coast,
  which causes water to pile up.

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Discretization

• Finite-difference technique is applied

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Integration

• Implicit techniques are used

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Another approximation

• Still a non-linear equation!
• d is dependent on h
• Assume d to be constant during integration
• Wave velocities only change between iterations.

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The linear equation

• Symmetric tridiagonal matrices can be solved very
  efficiently.

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The linear equation

• The linear equation can be considered an
  extrapolation of the previous motion of the
  fluid.
• Damping can be introduced if the equation is
  written as

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A Subtle Issue

• In an iteration, nothing prevents h from becoming
  less than b at a particular point, leading to
  negative volume at that point.
• To compensate for this the iteration creates
  volume elsewhere (note that our equations
  conserve volume).
• Solution After each iteration, compute the new
  volume and compare it with the old volume.

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The Equation in 3D

• Split the equation into two terms - one
  independent of x and the other independent of y -
  and solve it in two sub-iterations.
• We still obtain a linear system!

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Rendering

• Rendered with caustics the terrain was assumed
  to be flat.
• Real-time simulation!!
• 30 fps on a 32x32 grid

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Miscellaneous

• Walls are simulated by having a steep incline.

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Results
Water flowing down a hill
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More Images
Wave speed depends on the depth of the water
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• Particle-Based Fluid Simulation for Interactive
  Applications
• Matthias Muller et. al.
• SCA 2003

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Motivation

• Limitations of grid based simulation
• No splashing or breaking of waves
• Cannot handle multiple fluids
• Cannot handle multiple phases

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Introduction

• Use Smoothed Particle Hydrodynamics (SPH) to
  simulate fluids with free surfaces.
• Pressure and viscosity are derived from the
  Navier-Stokes equation.
• Interactive simulation (about 5 fps).

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SPH

• Originally developed for astrophysical problems
  (1977).
• Interpolation method for particles.
• Properties that are defined at discrete particles
  can be evaluated anywhere in space.
• Uses smoothing kernels to distribute quantities.

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Contd

• mj is the mass, rj is the density, Aj is the
  quantity to be interpolated and W is the
  smoothing kernel

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Modeling Fluids with Particles

• Given a control volume, no mass is created in it.
  Hence, all mass that comes out has to be
  accounted by change in density.
• But, mass conservation is anyway guaranteed in a
  particle system.

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Contd

• Momentum equation
• Three components
• Pressure term
• Force due to gravity
• Viscosity term (m is the viscosity of the liquid)

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Pressure Term

• Its not symmetric! Can easily be observed when
  only two particles interact.
• Instead use this
• Note that the pressure at each particle is
  computed first. Use the ideal gas state equation
• p kr, where k is a constant which depends on
  the temperature.

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Viscosity Term

• Method used is similar to the one used for the
  pressure term.

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Miscellaneous

• Other external forces are directly applied to the
  particles.
• Collisions In case of collision the normal
  component of the velocity is flipped.

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Smoothing Kernel

• Has an impact on the stability and speed of the
  simulation.
• eg. Avoid square-roots for distance computation.
• Sample smoothing kernel
• all points inside a radius of h are considered
  for smoothing.

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Surface Tracking and Visualization

• Define a quantity that is 1 at particle locations
  and 0 elsewhere (its called the color field).
• Smooth it out
• Compute the gradient of this field

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Contd

• If n(ri) gt l, then the point is a surface
  point.
• l is a threshold parameter.

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Results

• Interactive Simulation (5fps)
• Videos from Mullers site
• http//graphics.ethz.ch/mattmuel/

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Fluid-Fluid Interaction Results
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References

• Rapid, Stable Fluid Dynamics for Computer
  Graphics Michael Kass and David Miller
  SIGGRAPH 1990
• Particle-Based Fluid Simulation for Interactive
  Applications Muller et. al., SCA 2003
• Particle-Based Fluid-Fluid Interaction - M.
  Muller, B. Solenthaler, R. Keiser, M. Gross SCA
  2005