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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

- 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.
- 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

- 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.

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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