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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 l.jpg

Modeling Fluid Phenomena

Vinay Bondhugula

(25th & 27th April 2006)


Two major techniques l.jpg
Two major techniques

  • Solve the PDE describing fluid dynamics.

  • Simulate the fluid as a collection of particles.



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Previous Work and Miller

  • Older techniques were not realistic enough:

    • Tracking of individual waves

    • No net transport of water

    • Can’t handle changes in boundary conditions


Introduction l.jpg
Introduction and Miller

  • 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 and Miller

  • 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 l.jpg
Contd… and Miller

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

Limitation:

Inaccurate simulation for steep waves.


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Contd… and Miller

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 and Miller

  • 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 and Miller

  • 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… and Miller

  • 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 and Miller

  • Finite-difference technique is applied:


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Integration and Miller

  • Implicit techniques are used:


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Another approximation and Miller

  • 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: and Miller

  • Symmetric tridiagonal matrices can be solved very efficiently.


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The linear equation and Miller

  • 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 and Miller

  • 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 and Miller

  • 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 and Miller

  • Rendered with caustics – the terrain was assumed to be flat.

  • Real-time simulation!!

    • 30 fps on a 32x32 grid


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Miscellaneous and Miller

  • Walls are simulated by having a steep incline.


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Results and Miller

Water flowing down a hill…


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More Images and Miller

Wave speed depends on the depth of the water…


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Particle-Based Fluid Simulation for Interactive Applications and Miller

  • Matthias Muller et. al.

    SCA 2003


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Motivation and Miller

Limitations of grid based simulation:

  • No splashing or breaking of waves

  • Cannot handle multiple fluids

  • Cannot handle multiple phases


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Introduction and Miller

  • 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 and Miller

  • 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… and Miller

  • mjis 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 and Miller

  • 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… and Miller

  • 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 and Miller

  • 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.


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Viscosity Term and Miller

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


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Miscellaneous and Miller

  • 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 and Miller

  • 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 and Miller

  • 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:


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Contd… and Miller

  • If |n(ri)| > l, then the point is a surface point.

  • l is a threshold parameter.


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Results and Miller

  • Interactive Simulation (5fps)

  • Videos from Muller’s site:

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



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References and Miller

  • 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