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Stampede Problem
Meghan Galiardi, Jordan Hasler, Joe Nance
September 25, 2012
There were two main parts to this project. Meghan Galiardi and Jordan Hasler worked with different methods on
solving the system of partial differnetial equations. They collaborated closely and the conttributions from sections
2-5 cannot be distinguished. Joe Nance programmed a model of the system and all the work in section 6 is a
result of his work. The REGS project was completed under the supervision of Professor Vadim Zharnitsky. The
authors acknowledge support from National Science Foundation grant DMS 08-38434 “EMSW21-MCTP: Research
Experience for Graduate Students”. We would also like to acknowlede Shenghui Yang who helped clean up code and
contributed to the simulation.
Abstract
Several models exist for describing the motion of a crowd. Two equations involved for describing the motion
are the law of conservation of momentum and the continuity equation. We studied the crowd dynamics for a
densely packed crowd that experiences a stampede effect. We first developed and analyzed a model for the 1D
motion and started exploring 2D motion. For a detailed introduction to the crowd safety literature see the article
by Still [6].
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Introduction
Assume the distribution of people in the system is tightly packed. The motion of the crowd becomes similar to
that of a fluid. The continuity equation and the conservation of momentum equation can be used to describe the
movement of the system. For the continuity equation, suppose that the velocity is given by v, and the density is
given by ρ. The rate of change in mass is the difference between the mass flow in and the mass flow out. For a steady
state process, there is no change in the rate of mass.
The continuity equation states that
∂ρ
+ ∇(ρv) = 0
∂t
The momentum equation states that
ρut + ρuux = FA
where FA is the forces acting on the object.
To study the dynamics of a crowd of people in motion, the force equation F = π4 E ∗ LD where E is elasticity,
D is the indentation (from contact force) between two objects, and L is the length of the objects was used. Using
the contact force equation, one is able to derive the momentum equation for the study of a crowd in motion. For a
more detailed introduction to the equations of motion, see Anderson [1].
Using the contact force equation, the equation
vt + vvx = −
px
ρ
for momentum was derived, where p is pressure. To describe the equation of motion, we assumed that pressure
p = f (ρ) is a function of density.
Meg and Jordan first studied one dimensional motion. Constitutive equations where p = cργ were analyzed for
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various integral values of γ. The goal of this project is to analyze the distribution of denisity in the system as time
progresses. We want to see if at anytime the density in the system becomes too large, which may cause injuries or
even death.
Assume that p = cρ2 , then the equations of motion for 1D are:
We can rewrite this as
2
ρ
v
ρt + (uρ)x = 0
(1)
ut + uux = −2cρx
(2)
+
t
v
2c
ρ
v
ρ
v
=0
(3)
x
Riemann Invariants
One way we tried to solve (3) is using Riemann Invariants. By letting ∇r · v1 = 0, and ∇s · v2 = 0, we transformed
the system into
r
λ2 0
r
+
=0
(4)
s t
0 λ1
s x
where the Riemann invariants are
p
r = v + 2 2cρ
p
s = v − 2 2cρ
and the eigenvalues are
p
2cρ
p
λ2 = v + 2cρ
λ1 = v −
For a detailed explaination of Riemann Invariants, see McOwen [5]. The system in (4) is now diagonalized, but still
hard to solve. We the used the Hodogrph Transformation to help.
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Hodograph Transformation
One method to help solve the system of partial differential equations is called the hodograph transformation, , [5],
[2], [4]. We have two equivalent systems of partial differential equations described earlier.
λ2 0
r
r
ρ
v ρ
ρ
+
=0
= 0 and
+
0 λ1
s t
s x
v t
2c v
v x
The hodograph transformation can be used on either system. A hodograph transformation interchanges the roles
of the independent and dependent variables. In the case of this problem, a hodograph transformation will take a
system of nonlinear partial differential equations and transform it into a linear system. A hodograph transformation
of the first system will give a system with independent variables ρ and v and dependent variables t and x. In order
for the hodograph transformation to work J = ρt vx − ρx vt 6= 0. Through multivariable calculus it can be seen that
vx = −Jtρ , vt = Jxρ , ρt = −Jxv and ρx = Jtv
These relations transform the nonlinear system into the linear system,
−xv − ρtρ + vtv = 0
xρ − vtρ + 2ctv = 0
Differentiate the first equation with respect to ρ and the second with respect to v, and add to get
tvv =
ρtρρ
tρ
+
c
2c
2
Equivalently a hodograph transformation could be applied to the system involing the riemann invariants r and s to
yeild
3(ts − tr )
trs =
2(s − r)
Either of these equivalent second order equations may be expressed by means of an integral operator involving
Riemann fuctions, see [3]. However the solution will be a solution for t. Once t is determined the solution for x is
obtained. Then the values of either ρ and v or r and s are obtained from the jacobian equations. This method does
not lead to a solution that can be analyzied easily. Recall we want to analyze the distribution of denisity in the
system as time progresses. This solution will not allow us to do this.
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General Solution
One should note that the equation
γ−1
ux = 0
2
2
aax = 0
ut + uux +
γ−1
is similar to ours. Using Hodograph transformations, Whitham [7] finds that this has the solution
at + uax +
F (u + a) + G(u − a)
a
where F, G are arbitrary functions. If we apply a similar technique to (3) we find that
√
√
F (v + 3cρ) + G(v − 3cρ)
t=
ρ
t=
Similary to the previous section, we need to then find solutions for v and ρ. Again this does not lead to a solution
that is easy to analyze. We need to do more numerical analysis of this solution to analyze the distribution of density
in the solution.
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Perturbation Method
Methods of perturbabtion take a known solution of a problem and perturb it slightly and then study the new solution.
A trivial solution to the problem is a constant solution ρ(x, t) = ρ0 and v(x, t) = v0 where ρ0 and v0 are constants.
We looked at the perturbation of this solution, where now
ρ(x, t) = ρ0 + ερ1 (x, t) and v(x, t) = v0 + εv1 (x, t)
where ρ(x, 0) = r0 and v(x, 0) = v0 . Let’s also assume at time 0, people are not moving, thus v0 = 0. We now want
to study ρ1 (x, t) and v1 (x, t). Substituting these perturbed equations into the pdes yields,
ερ1t + ερ0 v1x + ε2 ρ1 v1x + ε2 v1 ρ1x = 0
εv1t + ε2 v1 v1x + 2cεv1 ρ1x = 0
Now we study the first order terms in ε. This yeilds the equations
ρ1t + ρ0 v1x = 0
v1t + 2cp1x = 0
Differentiating the first equation with respect to x and the second with respect to t and solving for v1tt yeilds
ρ0
v1tt = v1xx
2c
Similarly
ρ0
ρ1tt = ρ1xx
2c
These are both forms of the wave equation. The next step in the reseach is to look into the wave equation to furher
study this perturbation solution.
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6
Simulation
To help visualize the motion of the System Joe prgrammed a simulation of the motion in Mathematica. Consider the
following system. In an interval of length L with boundary, there sit n balls of radius R with uniform mass m. Their
distances along the x-axis are denoted by xi . We assume elastic collisions between the balls and their neighbors and
with the boundary. We define the repulsive force of xi on xj to be proportional to the overlap of the balls given by
(
2R − |xj − xi |, if 2R > |xj − xi |
fij (2R − |xj − xi |) =
0,
if 2R ≤ |xj − xi |.
Note that this can be written as f (x) = xunitstep[x], where unitstep[x] is a step function which is equal to 0 for
negative values and equal to 1 for positive values. So we have a Hooke-esque law relating the force, F , to the overlap,
f : Fij = Efij , where E is a constant of elasticity. This system is governed by Newton’s second law of motion:
P
Fij = mẍi . We now have a system of second-order differential equations

n
P
E

(2R − |xj − xi |) unitstep[2R − |xj − xi|] sign[xj − xi ]
ẍ1 = (R − x1 ) unitstep[R − x1 ] + m


j=1


n


ẍi = E P (2R − |xj − xi |) unitstep[2R − |xj − xi |] sign[xj − xi ]
m
j=1



ẍn = (L − R − xn ) unitstep[(L − R − xn )]+


n

P

E


(2R − |xj − xi |) unitstep[2R − |xj − xi|] sign[xj − xi ].
m
j=1
In theory, solving this system will give us positions of each ball before the next collision. In practice, this is nearly
impossible to do analytically. We treat this as a difference equation and let mathematica do the heavy lifting. The
output is a nice simulation with a plot of the distribution of all pair-wise distances.
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7
Future Work
We will analyze this model in more detail and see how it corresponds to the physical world. Further, the solutions
are not easily tractable, and therefore more numerical analysis of solutions is needed. In particular, we will analyze
how large forces impact crowd dynamics on various 2D surfaces. Both numerical, analytical, and simulation of the
situation are needed. Many current equations for describing crowd dynamics modify traffic flow equations. Research
has currently focused on descriptive studies and more analytical and numerical research is needed.
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Bibliography
References
[1] J. Anderson, Fundamentals of Aerodynamics, 4th Edition, McGraw Hill, 2007.
[2] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Applied Mathematical Sciences, Vol 21,
Springer-Verlag, New York.
[3] P.R Garabedian, Partial Differential Equations, John Wiley & Son, New York, 1983.
[4] R. B. Guenther and J. W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations,
Prentice Hall, Englewood Cliffs, NJ, 1988.
[5] R. McOwen, Partial Differential Equations: Methods and Applications,Prentice-Hall, Inc. 2003.
[6] K. Still, Crowd Dynamics, 2012, available at http://www.safercrowds.com.
[7] G. B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, 1999.
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