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Modeling Fluid Phenomena - PowerPoint PPT Presentation

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

Vinay Bondhugula

(25th & 27th April 2006)

• Solve the PDE describing fluid dynamics.

• Simulate the fluid as a collection of particles.

SIGGRAPH 1990

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 and Miller

• 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 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… and Miller

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

Limitation:

Inaccurate simulation for steep waves.

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.

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.

The Equations and Miller

• F = ma, gives the following:

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

• Volume conservation gives:

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.

Discretization and Miller

• Finite-difference technique is applied:

Integration and Miller

• Implicit techniques are used:

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.

The linear equation: and Miller

• Symmetric tridiagonal matrices can be solved very efficiently.

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:

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.

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!

Rendering and Miller

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

• Real-time simulation!!

• 30 fps on a 32x32 grid

Miscellaneous and Miller

• Walls are simulated by having a steep incline.

Results and Miller

Water flowing down a hill…

More Images and Miller

Wave speed depends on the depth of the water…

• Matthias Muller et. al.

SCA 2003

Motivation and Miller

Limitations of grid based simulation:

• No splashing or breaking of waves

• Cannot handle multiple fluids

• Cannot handle multiple phases

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).

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.

Contd… and Miller

• mjis the mass, rj is the density, Aj is the quantity to be interpolated and W is the smoothing kernel

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.

Contd… and Miller

• Momentum equation:

Three components:

• Pressure term

• Force due to gravity

• Viscosity term (m is the viscosity of the liquid)

Pressure Term and Miller

• 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 and Miller

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

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.

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”.

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:

Contd… and Miller

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

• l is a threshold parameter.

Results and Miller

• Interactive Simulation (5fps)

• Videos from Muller’s site:

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

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