By Michael Su
04/16/2009
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Introduction
Fluid characteristics
Navier-Stokes equation
Eulerian vs. Lagrangian approach
Dive into the glory detail (A case study of the 2d fluid
simulation)
 Advection
 Diffusion
 Pressure solve
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Fluid object couple
 One-way and two-way coupling
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Real-time fluids
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Broad view of the fluid simulation in graphics
community and its potential applications
Basic knowledge about the grid-based fluid
simulation
Understanding the challenges of the existing
methods
Foundation for the following two fluidsrelated lectures (smoke & granular material).

Applications
 Games (Half Life, Crysis)
 Scientific visualization (Water sewage system, dam
construction)
 Movie special effects (Finding Nemo, Pirates of Caribbean)
 Medical simulation (Blood flow)

What can we achieve so far?


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
[Goktekin,
Bargteil,
[Fedkiw,
[Wang,
Mucha,
Stam,
Turk]
[Tessendorf]
[Zhu,
Bridson]
O'Brien]
A Method
for
Jensen]
Water
Drops
Visual
on
Simulating
Ocean
Sand
as a
Animating
Simulation
Surfaces,
of Smoke,
Water,
SIGGRAPH
Fluid,
SIGGRAPH
05
Viscoelastic
Fluids,
SIGGRAPH
01
05
01
SIGGRAPH 04
Smoke
Granular flow (Sand)
Newtonian fluid (Water, ocean)
Non-Newtonian fluid (Blood, honey, goop (viscoelaticity
flow))
 Microscopic phenomena
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Basic properties
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Pressure
Density
Viscosity (subject to shear stress)
Surface tension
Different types of fluids:
 Incompressible (divergence-free) fluids: Fluids doesn’t
change volume (very much).
 Compressible fluids: Fluids change their
volume significantly.
 Viscous fluids: Fluids tend to resist a
certain degrees of deformation
 Inviscid (Ideal) fluids: Fluids don’t have resistance
to the shear stress
 Turbulent flow: Flow that appears to have chaotic
and random changes
 Laminar (streamline) flow: Flow that has
smooth behavior
 Newtonian fluids: Fluids continue
to flow, regardless of the force
acting on it
 Non-Newtonian fluids: Fluids that have non-
constant viscosity
 Phase Transition: Fluids may change
physical behavior under different
environmental conditions.
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Modeling continuum fluids on discrete
systems – It’s all about approximations
Topological variations and different kinds of
behaviors with interacting subjects
Numerical stabilities, accuracy and
convergence issues
Performance
User control
By Michael Su
04/20/2009
Calculus Review (1)

Gradient (): A vector pointing
in the direction of the greatest
rate of increment
 u u u  u can be a scalar or a
u   , , 
 x y z  vector

 u u u 
u   , , 
 x y z 
Divergence ( ): Measure how the
vectors are converging or diverging
at a given location (volume
density of the outward flux)
u u u
 u 


u can only be a vector
x y z
Source,
Div(u) > 0
Sink,
Div(u) < 0
Calculus Review (2)

Laplacian (∆ or  2 ): Divergence of the gradient
2
2
2

u

u

u
 2u  2  2  2
x
y
z

u can be a scalar or a
vector
Finite Difference: Derivative approximation
u ui 1  ui

x xi 1  xi

Momentum equation
ut = k2u –(u)u – p + f
Change in Diffusion/Vi Advection Pressure Body
velocity
scosity
Forces

Incompressibility
Claude-Louis Navier George Gabriel
Stokes
(1785~1836)
(1819~1903)
u=0
u: the velocity field
k: kinematic viscosity


Borrowed from CFD (Computational Fluid
Dynamics)
Common techniques for solving Navier
Stoke’s equation:
 Eulerian approach (grid-based)
 Lagrangian approach (particle-based)
 Spectral method
 Lattice Boltzmann method
[Stam]
Stable
Fluids,
[Mueller,
Charypar,
SIGGRAPH
99
Gross] Particle-Based
Fluid Simulation for
Interactive
Applications, SCA03
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Discretize the domain using finite differences
Define scalar & vector fields on the grid
Use the operator splitting
technique to solve each
term separately
Evaluation:
Derivative approximation
Adaptive time step/solver
Memory usage & speed
Grid artifact/resolution limitation
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

Treat the fluid as discrete particles
Apply interaction forces (i.e.
pressure/viscosity) according to certain
pre-defined smoothing kernels
Evaluations:
Mass / Momentum conservation
More intuitive
Fast, no linear system solving
Connectivity information/Surface reconstruction
Case Study: A 2D Fluid Simulator
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We focus exclusively on incompressible,
viscous fluid
Assuming the gravity is the only external
force
No inflow or outflow
Constant viscosity, constant density
everywhere in the fluid
Advection
Body Force
Scalar/Vector
fields defined
on the grid
Diffusion
Pressure Solve
ut = k2u –(u)u – p + f
u=0
The Power of Operator Splitting
One complicated Multidimensional operator => A
series of simple, lower
dimensional operators
 Each operator can have its
own integration scheme
and different time step
sizes
 High modularity and easy
to debug

n
U +A+B+D+P
*
U
**
U
***
U
n+1
U
Sometimes called “Convection” or “Transport”
Define how a quantity moves with the underlying
velocity field
 This term ensures the conservation of momentum
 Advection equation:
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
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Approaches:
ut  (u  u)
 Forward Euler (unstable)
 Semi-Lagragian advection (stable for large time steps, but
suffers from the dissipation issue)
Forward Euler Advection
Semi-Lagragian Advection
Define how a quantity in a cell inter-changes with
its neighbors
 Diffusion = Blurring
 The viscous fluid can be achieved by applying
diffusion to the velocity field

Low Viscosity
High Viscosity
Figures from [Carlson, Mucha, Turk]
Melting and Flowing, SCA 02

Diffusion equation:

Approaches:
ut  k   u
2
1
 Explicit formulation
ut  k  t  (ui 1, j  ui 1, j  ui , j 1  ui , j 1  4  ui , j )
 Implicit formulation
1
-4
1
1
(for high viscosity)
u n i , j  u n1i , j  k  t  (u n1i 1, j  u n1i 1, j  u n1i , j 1  u n1i , j 1  4  u n1i , j )
Unknowns
0
0
0
0
0
5
0
0
0
0
0
0
0
2.5 0
2.5 -10 2.5
0
0
2.5
0
Before the diffusion
After the diffusion
(k = 0.5, time step
size =1)
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It’s sometimes called “Pressure Projection”
What does the pressure do?
 Keep the fluid at constant volume
(incompressible, conservation of mass).
 Make sure the velocity field stays divergence-free
 flux  0
all _ faces
Incompressible

Compressible

Equation to solve:
u

n 1
u 
n
1

p
s.t.
 u
n 1
0
• • • (1)
Unknowns
How to solve for pressure:
 Taking divergence of both sides of (1), we will
have 1  p    u (Poisson Equation)

 Build a system of equations and solve Ap = d using
an iterative method such as Conjugate Gradient
 Update the velocity field from the pressure
gradient
2
n

What about the pressure on boundary nodes?
 Free surface: The fluid can
Free surface
evolve freely (p = 0)
 Solid wall: The fluid can’t
penetrate the wall but can
flow freely in tangential
directions (Neumann BC)
uboundary  n  usolid  n
Solid wall

Possible reasons why your simulation doesn’t
look right:
 CFL condition violation





t  u  C  x
=> Smaller time steps / Implicit solver
Flux conservation => BCs may not be set correctly
Grid resolution/
 Memory => Adaptive grids
Numerical dissipation => Back and Forth Error Compensation
and Correction [4] / Vorticity confinement [5]
Handle the interface and complex topological changes =>
Level set method [6]
Volume loss => Particle level set [7]

One-way coupling:
 Solid-Fluid interaction: The fluid has no influence
on the solid
 Fluid-Solid interaction: The solid has no influence
on the fluid

Two-way coupling:
 Manipulate the boundary conditions
 Finite Element techniques: ALE & DLM
 Rigid Fluid: Treat the solid as fluids and enforce
the rigidity constraint [8]

Principles:
 Cheap to compute
 Low memory consumption
 Stability
 Plausibility
 Interactivity

Common techniques:
 Procedural water: Superimpose sine waves of a
variety of amplitudes and directions. [9]
Real-time Fluids (2)
 Heightfield approximations: If the surface is the
only interest, it can be represented using a 2d
heightfield and animated by 2d wave equations
with interaction forces.


k
k
1
2
 x i  x j
f (x i , x j )  

 x  x m x  x n  x i  x j
j
i
j 
 i
 Particle systems: This approach is

good at simulating a small amount of
water such as a puddle, a bubble,
or splashing fluids
H(x,
y)
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[1] R. Bridson and M. Müller-Fischer. Fluid Simulation. SIGGRAPH 07
Course Notes
[2] R. Bridson. Fluid Simulation for Computer Graphics. A K Peters,
2008
[3] J. Stam. Real-Time Fluid Dynamics for Games. GDC 2003
[4] B. Kim, Y. Liu, I. Llamas, and J. Rossignac. FlowFixer: Using BFECC
for Fluid Simulation. EGWNP 05
[5] R. Fedkiw, J. Stam, and H.W. Jenson. Visual Simulation of Smoke.
SIGGRAPH 01
[6] N. Foster, R. Fedkiw, Practical Animation of Liquids. SIGGRAPH 01
[7] D. Enright, S. Marschner, R. Fedkiw. Animation and Rendering of
Complex Water Surfaces. SIGGRAPH 02
[8] M. Carlson, P. J. Mucha, G. Turk. Rigid Fluid: Animating the
Interplay Between Rigid Bodies and Fluid. SIGGRAPH 04
References (2)

[9] D. Hinsinger, F. Neyret, M. Cani. Interactive Animation of Ocean
Waves. SCA 02