The document discusses computational water models used in computer graphics and computational fluid dynamics. It covers four main methods: procedural methods, Navier-Stokes based methods (Lagrangian and Eulerian), shallow water equations, and lattice Boltzmann method. While procedural methods are fastest, the other methods provide more realistic simulations but are more computationally expensive. Hybrid approaches allow additional water effects to be modeled.

1. Computational Water Models State-of-the-Art
Computational Water Models
State-of-the-Art
Goncalo Amador
¸
e-mail:gamador@it.ubi.pt
January, 2011

2. Computational Water Models State-of-the-Art
1 Introduction
2 Procedural Methods
3 Navier-Stoke-based Methods
Lagrangian Methods
Eulerian Methods
4 Lattice Boltzmann Method (LBM)
5 Conclusions
6 References

3. Computational Water Models State-of-the-Art
Introduction
Overview
Two major fields, that study fluid simulation:
Computational Fluid Dynamics (CFD)

4. Computational Water Models State-of-the-Art
Introduction
Overview (Cont.)
Computer Graphics (CG)
(real-time ≥ 30 fps) (off-line ≤ 30 fps)
Methods used in CG [1, 2, 3, 4]:
Procedural methods.
Navier-Stokes based methods.
Lattice Boltzmann Method (LBM).

5. Computational Water Models State-of-the-Art
Procedural Methods
Procedural Methods
Parametric function:
x distance to origin
C propagation speed
2π(x−Ct) A wave amplitude
f (x, t) = A cos L t time instant
L wave length
Only non-physically-based method in CG.
Faster than physically-based methods.
Large amounts of detail for several resolutions.
Integrated in most 3D modelling and animation tools.
Used in movie/animation and game/virtual-reality
industries.

6. Computational Water Models State-of-the-Art
Procedural Methods
Procedural Methods (Cont.)

7. Computational Water Models State-of-the-Art
Navier-Stoke-based Methods
Navier-Stokes equations for incompressible fluids
→
−
Mass conservation: u =0
Relation between external forces, pressure, temperature,
and density of a viscous moving fluid:
→
− →
−
∂u →
− → 1
− 2→
−
=− u · u − p+v u + f
∂t ρ
→
−
u : velocity field
v : fluids viscosity
ρ: density of the field
→
−
f : external forces
∂ ∂ ∂
= , , : gradient
∂x ∂y ∂z

8. Computational Water Models State-of-the-Art
Navier-Stoke-based Methods
Lagrangian Methods
Lagrangian Methods
Particles = sets of fluid molecules.
Track particles as if the observer was ridding them.
Particles have position, velocity, mass, phase, etc.
Allow splash, spray, foam, boiling, etc.

9. Computational Water Models State-of-the-Art
Navier-Stoke-based Methods
Eulerian Methods
Eulerian Methods
Space is discretized into a limited grid of cells.
The variations are analysed within each grid cell.
More memory consuming than procedural and
Lagrangian methods.

10. Computational Water Models State-of-the-Art
Navier-Stoke-based Methods
Eulerian Methods
Eulerian Methods (Cont.)
Two types of grid: coarse and the Marker-and-Cell
(MAC).

11. Computational Water Models State-of-the-Art
Navier-Stoke-based Methods
Eulerian Methods
Shallow Water Equations (SWE)
Simplified version (2D equations) of the 3D NS
equations, for e.g., ocean near shore, rivers, etc.
Used both in CG and CFD.
Less computationally demanding, than the 3D NS
equations.
an vertical acceleration
∂η
+ ( η) v = −η ·v h height above level 0
∂t η height above ground
∂v v horizontal velocity
+ ( v ) v = an h t time
∂t

12. Computational Water Models State-of-the-Art
Lattice Boltzmann Method (LBM)
LBM
CFD grid-based method.
Evolved from the Lattice Gas Automata (LGA), a mo-
del that describes gases in space.
A discrete-velocity Boltzmann equation
Approximates the solution of the NS equations with
good accuracy.

13. Computational Water Models State-of-the-Art
Lattice Boltzmann Method (LBM)
LBM (Cont.)

14. Computational Water Models State-of-the-Art
Conclusions
Conclusions
CFD simulates fluids to solve and analyse enginee-
ring problems involving fluid flows
CG simulates fluids to generate appealing, convin-
cing, plausible simulations of fluid effects, for the
game and movie industries.
Four models used to simulate fluids in CG.
LBM is the grid-based most memory consuming
method.
Procedural methods are the fastest and less memory
consuming.
Hybrids (Eulerian+Lagrangian) allow more effects.

15. Computational Water Models State-of-the-Art
Conclusions
Conclusions (Cont.)
All the four methods have been implemented using
CUDA and shading languages.
Aside from procedural methods, all methods have
been implemented using MPI.
LBM and NS-based Eulerian methods, have been im-
plemented using OpenMP.
With CUDA, CFD research for Supercomputer and
cluster fluid computing increased.
Some OpenMP/CUDA and MPI integration as already
been performed.
Apparently, there is no OpenMP and CUDA and MPI
integration, for fluid simulation.

16. Computational Water Models State-of-the-Art
References
References
J. Tessendorf, “Simulating Ocean Water,” in ACM
SIGGRAPH 2004 Course Notes.
R. Bridson, M. F. Muller, E. Guendelman, and R. Fedkiw,
¨
“Fluid simulation,” in ACM SIGGRAPH 2006 Course Notes.
R. Bridson and M. F. Muller, “Fluid simulation,” in ACM
¨
SIGGRAPH 2007 Course Notes.
M. F. Muller, J. Stam, D. James, and N. Thurey, “Real time
¨ ¨
physics,” in ACM SIGGRAPH 2008 Course Notes.