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Modeling Fluid Phenomena. Vinay Bondhugula (25 th & 27 th April 2006). Two major techniques. Solve the PDE describing fluid dynamics. Simulate the fluid as a collection of particles. Rapid Stable Fluid Dynamics for Computer Graphics – Kass and Miller SIGGRAPH 1990. Previous Work.

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modeling fluid phenomena

Modeling Fluid Phenomena

Vinay Bondhugula

(25th & 27th April 2006)

two major techniques
Two major techniques
  • Solve the PDE describing fluid dynamics.
  • Simulate the fluid as a collection of particles.
previous work
Previous Work
  • Older techniques were not realistic enough:
    • Tracking of individual waves
    • No net transport of water
    • Can’t handle changes in boundary conditions
introduction
Introduction
  • Approximates wave equation for shallow water.
  • Solves the wave equation using implicit integration.
  • The result is good enough for animation purposes.
shallow water equations assumptions
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.
contd
Contd…

2) Ignore the vertical component of the velocity of water.

Limitation:

Inaccurate simulation for steep waves.

contd8
Contd…

3) Horizontal component of the velocity in a column is constant.

Assumption fails in some cases:

  • Undercurrent
  • Greater friction at the bottom.
notation
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.
the equations
The Equations
  • F = ma, gives the following:

The second term is the horizontal force acting on a water column.

  • Volume conservation gives:
contd11
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.
discretization
Discretization
  • Finite-difference technique is applied:
integration
Integration
  • Implicit techniques are used:
another approximation
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.
the linear equation
The linear equation:
  • Symmetric tridiagonal matrices can be solved very efficiently.
the linear equation16
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:
a subtle issue
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.
the equation in 3d
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!
rendering
Rendering
  • Rendered with caustics – the terrain was assumed to be flat.
  • Real-time simulation!!
    • 30 fps on a 32x32 grid
miscellaneous
Miscellaneous
  • Walls are simulated by having a steep incline.
results
Results

Water flowing down a hill…

more images
More Images

Wave speed depends on the depth of the water…

motivation
Motivation

Limitations of grid based simulation:

  • No splashing or breaking of waves
  • Cannot handle multiple fluids
  • Cannot handle multiple phases
introduction25
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).
slide26
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.
contd27
Contd…
  • mjis the mass, rj is the density, Aj is the quantity to be interpolated and W is the smoothing kernel
modeling fluids with particles
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.

contd29
Contd…
  • Momentum equation:

Three components:

    • Pressure term
    • Force due to gravity
    • Viscosity term (m is the viscosity of the liquid)
pressure term
Pressure Term
  • It’s 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 = k*r, where k is a constant which depends on the temperature.

viscosity term
Viscosity Term
  • Method used is similar to the one used for the pressure term.
miscellaneous32
Miscellaneous
  • Other external forces are directly applied to the particles.
  • Collisions: In case of collision the normal component of the velocity is flipped.
smoothing kernel
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”.

surface tracking and visualization
Surface Tracking and Visualization
  • Define a quantity that is 1 at particle locations and 0 elsewhere (it’s called the color field).
  • Smooth it out:
  • Compute the gradient of this field:
contd35
Contd…
  • If |n(ri)| > l, then the point is a surface point.
  • l is a threshold parameter.
results36
Results
  • Interactive Simulation (5fps)
  • Videos from Muller’s site:

http://graphics.ethz.ch/~mattmuel/

references
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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