Transcript swa2006 6267

Dynamics of a Continuous
Model for Flocking
Ed Ott
in collaboration with
Tom Antonsen
Parvez Guzdar
Nicholas Mecholsky
Dynamical Behavior in Observed Bird
Flocks and Fish Schools
Our Objectives- Introduce a model
and use it to investigate:
1. Flock equilibria
2. Relaxation to equilibrium
3. Stability of the flock
4. Response to an external stimulus, e.g. flight
around a small obstacle: poster of Nick
Mecholsky
Characteristics of Common Microscopic
Models of Flocks
1. Nearby repulsion (to avoid collisions)
2. Large scale attraction (to form a flock)
3. Local relaxation of velocity orientations to a
common direction
4. Nearly constant speed, v0
Continuum Model
• Many models evolve the individual positions and
velocities of a large number of discrete boids.
• Another approach (the one used here) considers
the limit in which the number of boids is large and
a continuum description is applicable.
• Let
 ( x, t ) 
v ( x, t ) 
The number density of boids
The macroscopic (locally
averaged) boid velocity field
Governing Equations:
Conservation of Boids:

v  0
t
1
Velocity Equation:
2
v
1
 v   v   p(  )  U
t

2
1
v
 W [ v]  (1 
)v
2

v0
3
4
(1).
Short range repulsion: 
p (  )  *T

1
1

p
*

*
This is a pressure type interaction that models the
short range repulsive force between boids. The
denominator prevents  from exceeding * so that
the boids do not get too close together.
(2).
Long-range attraction: U
u0  0
U  u0   ( x', t ) u ( x, x') d x'
u ( x, x ' )  
exp( k  | x  x '|)
4 | x  x '|
Here k-1 represents a ‘screening length’ past which the
interaction between boids at x' and x becomes
ineffective.
2
2
In this case, u( x, x' )satisfies:  u  k  u   ( x  x' )
and U satisfies:
2U  k 2U  u0  ( x)
These equations for u and U apply in 1D, 2D,
and 3D.
(3). Velocity orientation relaxation term:
W [ v]  w0  w( x  x' )  ( x' , t ) [ v( x' , t )  v( x, t )] d x'
Our choice for w( x, x' ) satisfies:
 w  k w   ( x  x' )
2
2
v
(4).
Speed regulation term:
1

(1 
v2
v0
2
)v
This term brings all boids to a common speed v0.
If |v| > v0 (|v| < v0 ), then this term decreases
(increases) |v|. If   0 , the speed |v| is clamped
to v0.
Equilibrium
We consider a one dimensional flock in which
the flock density, in the frame moving with the
flock, only depends on x. Additionally, v is
independent of x and is constant in time (v0):
v  v0 x 0
Equilibrium Equations:
1 d
d
 p ( x)  U ( x)  0
 dx
dx
2
d U ( x)
2

k
U
(
x
)

u

(
x
)

0
2
dx
Equilibrium Solutions
The equilibrium equations combine to give an energy
like form
1  d 

  (  )  0
2  dx 
2
where  depends on a dimensionless parameter a
defined below) and both  and x are made
dimensionless by their respective physical
parameters * and k
k T
a
*u0
An Example
a  0.2,   0.045
and the density at x = 0 is determined to be
0  0.5604
Solving the potential equation, we get
The profile is symmetric about k  x  0
Waves and Stability
Equilibrium:
Perturbations:
 0   0 ( x)
v0  v0 x 0
~ f ( x) exp( st  ik y y  ik z z)
Basic Equation:




2
2 1
s  0    Kˆ Hˆ Kˆ  sWˆ 
x 0 x 0   
 0



where:

   0 v
d
ˆ
K  k y y0  kz z 0  i x0
dx
and the notation
1
2

dp
(

)
1
d
2
2
0 u 
Hˆ 
0 2  k  k
 0 d 0
 dx
Wˆ  ...
 d2

 2  k 2 
 dx

1
signifies the operator

k x  x '
d

e
 2  k 2  f ( x)    dx' f ( x' )
2k
 dx


2





1
Long Wavelength Expansion
2π
,
k
2
 w
k
w  width of slab
Ordering Scheme:
k y ~ k z ~   1,
 ~2
   0    1   2 2
Hˆ  Hˆ 0   2U ,
Kˆ  Kˆ 0   Kˆ 1
Analysis:
Expand equations in  :
O( 0 ) equation     0  O( )
Inner product of equation for  with  0
annihilates higher order terms to give:
A s2  C s  k 4B  0
 1  d 0
A
 dx

 0 

2
 dx,

  d 0
C
 dx
 

2

d
p
 1
 dx,
B 1
u 0     0 dx 

 Wˆ  d 0  dx.
  dx 
Comment:
The eigenfunction from the analysis represents
a small rigid x-displacement whose amplitude
varies as exp(ikyy + ikzz).
  0 ( x  )  0 ( x) ~
d 0 ( x)
dx
Cylindrical Flock
We have also done a similar analysis for a
cylindrical flock with a long wavelength
perturbation along the cylinder axis.
Numerical Analysis of Waves and Stability
Linearized equations are a coupled system for ,
vx, and vy. Discretize these functions of position,
and arrange as one large vector.
AV   B V
Use a standard algorithm to determine eigenvalues and
eigenvectors. The solutions give all three branches of
eigenvalues and their respective eigenfunctions.
Preliminary Conclusions From Numerical
Stability Code:
All eigenmodes are stable (damped).
For small k (wavelength >> layer width), the
damping rate is much larger than the real
frequency.
For higher k (wavelength ~ layer width), the real
frequency becomes bigger than the damping rate.
Flock Obstacle Avoidance
We consider the middle of a very large flock
moving at a constant velocity in the positive x
direction. The density of the boids is uniform in all
directions.
The obstacle is
represented by a
repulsive Gaussian hill
Solution using Linearized System
Add the repulsive potential, linearize the original
equations   0
t 
Fourier-Bessel transform in r  x 2  y 2 and  .
(r , ) 


 0
n  
where
An(k)  ...
An (k ) J n (k r ) exp( in ) dk ,
Black = Lower Density,
White = Higher Density
First Three Lowest Eigenmodes
k = 0.4, a = 0.3, 0/* = 0.8,  = 1,  = 2,  = 100
r=-1.072, i=-0.025
r=1.072, i=-0.025
0.15
Re(  )
Re(  )
0.075
Im (  )
0

Re(  )

0.075
0.15
0
0.15
10
Im (  )
0
x
0.15
10
Re(  )
0.15
0
x
10
0.075
0.15
0.15
0
10
10
0.01
0.01
0.015
Re vx 0.005
Re vx 0.005
Re vx 0.005
Re vx 0.005
Re vx 0.0075
0
0
0
0
0
Vx
Vx
0.005
0.01
 
Im  vx
0
0.01
10
10
x
0.01
10
0
0.015
 
Im  vx
Re vx 0.0075
10
0.015
0
10
0.015
10
0
10
10
0
10
x
x
x
x
0
0.0075
0.0075
10
10
Vx
0.3
0.3
0.3
0.3
Re vy 0.15
Re vy 0.15
Re vy 0.15
Re vy 0.15
0
0
0
0
 
Im  vy 
0
Vy
0.15
0.3
0.01
0
0.3
Re vy 0.15
Vy
10
0.005
10
x
Vx
0.005
x
0.3
 
Im  vy 
0
 
Im  vx
Vx
Vx
0.005
10
 
Im  vx
10
x
0.01
 
Im  vx
0
x
x
0

0.075
10
0.075
Im (  )
0
0.01
 
Im  vx
r=0.978, r=-0.122
Re(  )

0.075
10
0.075
Im (  )
0

0.075
10
0.075
0

0.075
10
Re(  )
i  0.122
0.15
0.15
0.15
0.075
Im (  )
0
r=-0.978, i=-0.122
r=1.025, i=-0.075
0.15
0.15
0.075
Im (  )
r=-1.025, i=-0.075
r  0.967
 
Im  vy 
Vy
0.15
 
Im  vy 
Vy
 
Im  vy 
Vy
0.15
0.15
0.15
0.3
0.3
0.3
 
Im  vy 
Re vy 0.15
Vy
0
0.15
0.3
10
0
x
10
0.3
10
0
x
10
10
0
x
10
10
0
x
10
10
0
x
10
10
0
x
10
ZERO REAL FREQUENCY EIGENMODES
k=0.01, a=0.3, =0.8, =1, =2, =100
r=0, r=-0.158i  0.158
r  0
0.3
0.3
0.3
0.15
0.15
Re(  )
0.1
0.3
Re(  )
0
Im (  )
0.3
10
0
10
0.15
10
0
0.05
0.12
0.025
0.06
 
 
0
0.12
0
10
 
Im  vx
0
10
0
10
0.12
x
10
0
10
x
0.2
0.1
0.2
 
Im  vy 
0.05
0.1
Re vy
Re vy
0
 
Im  vy 
0.1
Re vy
0
0.1
0.05
0
0.1
0.2
0.1
0.06
0.06
x
 
 
10
Re vx
0.06
10
0
0.12
0
Im vx
0.025
0.05
10
x
Re vx
Re vx
Im vy
0.3
10
x
x
Im vx
0
Im (  )
0.15
0.1
 
 
r=0, r=-0.091
i  0 .0 9 1
r  0
0.5
Re(  )
Im (  )
r=0, r=-0.123
i  0.123
r  0
10
0
10
10
0
10
x
0.2
10
0
x
10
x
NORMALIZED EQUATIONS
  k0 Vy  i
Vx  i
d
(0 Vx )
dx
a d  dP0 
a dP0
d
   i
  i []  i[0Vx ]  iVx

0 dx  d0  02 dx
dx
Vy  k
a d  dP0 
d
   k []  [0 Vx ]

0 dx  d0 
dx
[] 
1

 dx(x)exp(
k  1 
[0 V] 
2


k 
2
2
 dx exp(
k 2  1 x  x )
k 2   2 x  x  )0 (x )V(x ) 

V(x)

2
Dimensionless parameters
a
2Tk 2
*u 0
0 (0)=
0 (0)
1/ 2 
  *1/ 2 20
*
u0 k

4
(*u 0 )1/ 2
k=
ky
k
=
k
k

 dx exp( x  x ) (x)
0

k=0.01, a=0.3,  =0.8, =1, =2, =100
1
98
0 .5
9 8 .5
0
99
Im( z ) 0 .5
Im( z ) 9 9 .5
1
100
1 .5
1 0 0 .5
2
0 .03
0 .02
0 .01
0
Re( z )
0 .01
0 .02
0 .03
101
4
2
0
Re( z )
2
4