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.

Rapid Stable Fluid Dynamics for Computer Graphics – Kass and Miller

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

1) 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.

ResultsWater flowing down a hill…

More Images

Wave speed depends on the depth of the water…

Particle-Based Fluid Simulation for Interactive Applications

- Matthias Muller et. al.

SCA 2003

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…

• mj is the mass, j 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 (is the viscosity of the liquid)

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*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/

Fluid-Fluid Interaction Results

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