Modeling Fluid Phenomena

1 / 40

# Modeling Fluid Phenomena - PowerPoint PPT Presentation

Modeling Fluid Phenomena. Vinay Bondhugula (25 th &amp; 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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Modeling Fluid Phenomena' - nova

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Modeling Fluid Phenomena

Vinay Bondhugula

(25th & 27th April 2006)

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

SIGGRAPH 1990

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
• 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
• 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…

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

Limitation:

Inaccurate simulation for steep waves.

Contd…

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

Assumption fails in some cases:

• Undercurrent
• Greater friction at the bottom.
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
• F = ma, gives the following:

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

• Volume conservation gives:
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
• Finite-difference technique is applied:
Integration
• Implicit techniques are used:
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:
• Symmetric tridiagonal matrices can be solved very efficiently.
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
• 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
• 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
• Rendered with caustics – the terrain was assumed to be flat.
• Real-time simulation!!
• 30 fps on a 32x32 grid
Miscellaneous
• Walls are simulated by having a steep incline.
Results

Water flowing down a hill…

More Images

Wave speed depends on the depth of the water…

Motivation

Limitations of grid based simulation:

• No splashing or breaking of waves
• Cannot handle multiple fluids
• Cannot handle multiple phases
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).
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.
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
• 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.

Contd…
• Momentum equation:

Three components:

• Pressure term
• Force due to gravity
• Viscosity term (m is the viscosity of the liquid)
Pressure Term
• It’s not symmetric! Can easily be observed when only two particles interact.
• 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
• Method used is similar to the one used for the pressure term.
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
• 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
• 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:
Contd…
• If |n(ri)| > l, then the point is a surface point.
• l is a threshold parameter.
Results
• Interactive Simulation (5fps)
• Videos from Muller’s site:

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

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