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Modeling Fluid Phenomena

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Rapid Stable Fluid Dynamics for Computer Graphics Kass and Miller. SIGGRAPH 1990 ... From the simplified wave equation, the wave velocity is sqrt(gd) ... – PowerPoint PPT presentation

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Title: Modeling Fluid Phenomena


1
Modeling Fluid Phenomena
  • Vinay Bondhugula
  • (25th 27th April 2006)

2
Two major techniques
  • Solve the PDE describing fluid dynamics.
  • Simulate the fluid as a collection of particles.

3
  • Rapid Stable Fluid Dynamics for Computer Graphics
    Kass and Miller
  • SIGGRAPH 1990

4
Previous Work
  • Older techniques were not realistic enough
  • Tracking of individual waves
  • No net transport of water
  • Cant handle changes in boundary conditions

5
Introduction
  • Approximates wave equation for shallow water.
  • Solves the wave equation using implicit
    integration.
  • The result is good enough for animation purposes.

6
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.

7
Contd
  • 2) Ignore the vertical component of the velocity
    of water.
  • Limitation
  • Inaccurate simulation for steep waves.

8
Contd
  • 3) Horizontal component of the velocity in a
    column is constant.
  • Assumption fails in some cases
  • Undercurrent
  • Greater friction at the bottom.

9
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.

10
The Equations
  • F ma, gives the following
  • The second term is the horizontal force acting
    on a water column.
  • Volume conservation gives

11
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.

12
Discretization
  • Finite-difference technique is applied

13
Integration
  • Implicit techniques are used

14
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.

15
The linear equation
  • Symmetric tridiagonal matrices can be solved very
    efficiently.

16
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

17
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.

18
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!

19
Rendering
  • Rendered with caustics the terrain was assumed
    to be flat.
  • Real-time simulation!!
  • 30 fps on a 32x32 grid

20
Miscellaneous
  • Walls are simulated by having a steep incline.

21
Results
Water flowing down a hill
22
More Images
Wave speed depends on the depth of the water
23
  • Particle-Based Fluid Simulation for Interactive
    Applications
  • Matthias Muller et. al.
  • SCA 2003

24
Motivation
  • Limitations of grid based simulation
  • No splashing or breaking of waves
  • Cannot handle multiple fluids
  • Cannot handle multiple phases

25
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).

26
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.

27
Contd
  • mj is the mass, rj is the density, Aj is the
    quantity to be interpolated and W is the
    smoothing kernel

28
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.

29
Contd
  • Momentum equation
  • Three components
  • Pressure term
  • Force due to gravity
  • Viscosity term (m is the viscosity of the liquid)

30
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.

31
Viscosity Term
  • Method used is similar to the one used for the
    pressure term.

32
Miscellaneous
  • Other external forces are directly applied to the
    particles.
  • Collisions In case of collision the normal
    component of the velocity is flipped.

33
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.

34
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

35
Contd
  • If n(ri) gt l, then the point is a surface
    point.
  • l is a threshold parameter.

36
Results
  • Interactive Simulation (5fps)
  • Videos from Mullers site
  • http//graphics.ethz.ch/mattmuel/

37
Fluid-Fluid Interaction Results
38
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39
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40
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
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