Download Real-Time Fluid Dynamics

COMPUTATIONAL
FLUID DYNAMICS
IN REAL-TIME
An Introduction to
Simulation and Animation
of Liquids and Gases
Considered Papers
• Real-Time Fluid Dynamics for Games (2003)
• by Jos Stam
• previously worked on Maya; now with NVidia
• covers gases
• Realistic Animation of Liquids (1996)
• by Nick Foster and Dimitri Metaxas
• first attempt to solve Navier-Stokes in full 3-D
• Practical Animation of Liquids (2001)
• by Nick Foster and Ronald Fedkiw
• Particle-Based Fluid Simulation for Interactive Applications
(2003)
• by Matthias Mueller, David Charypar and Markus Gross
RealTime
Animated
Gas-Flow
Rendered using “density quads”
Grid-Based Water Animation
Hybrid Surface Model
Particle-Based Water Animation
Particles
Point Splatting
Iso-Surface
(Marching Cubes)
Difference between liquids and gases
• Gases have simpler free surfaces
• Density provides enough information for renderer
• For liquids we must also produce a smooth enough free surface
at boundaries
• Gases are compressible
• Density of liquids, generally, never changes
• Gases have often negligible viscosity
• Liquids observe more obvious friction at boundary interfaces
• But both are guided by the same equations
Physics of Fluids:
The Navier-Stokes Equations
• Velocity field:
• Density distribution:
u = velocity, ρ = density, κ = rate of diffusion, S = source influences
• Conservation of mass:
∇⋅u=0
• i.e., at any point in time, all velocities added together, cancel each
other
• Change of pressure:
• Caused by movement of the liquid
Three components
• Discretization
• Grid
• Particles
• Algorithm to solve the mathematical model
• Linear solver\
• More about this next time
• Rendering
• Billboarding
• Particles
• Point splatting
• Iso-surfaces
Discretization
• Old method: Propagate fluid state through a 2-D or 3-D grid
• Every cell has density, velocity and pressure information
• Liquid cells have 3 mutually exclusive states:
• Full
• Surface (boundary)
• Empty
• Object boundaries must coincide with the grid
• New method: Use Smoothed-particle hydrodynamics
• Particles that allow for smooth interpolation between their positions
• Define rules for boundaries
• At the system’s edges
• For interfaces with other objects or fluids
• Sources & Sinks
Grid-based algorithm for liquids (1996)
1. Define obstacles and initial configuration
2. Setup grid with initial configuration
3. Track surface in cells
4. Setup boundary conditions
5. Update velocity values
6. Update pressure values
7. Recalculate boundary information of surface cells
8. Update position of surface and objects
9. Go to step 3
Rendering (2001):
Hybrid Surface Model
• Move mass-less marker particles through the grid
• Use algorithm (e.g. Marching Cubes) to determine a
smooth iso-countour
• Use other particle information and isocountour to make
local changes
There is more to rendering
• Transparency
• Reflection
• Refraction
• Soft shadows
Fin
• That’s it – For now!