Transcript Modeling Fluid Phenomena -Vinay Bondhugula (25 & 27

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 subiterations.
• 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…
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, 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.
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