Modeling and Simulation
Mathematics:
Pure Mathematics: is studied to develop the principles of
Mathematics for the sake of the principles of Mathematics.
Applied Mathematics:
concerns with mathematical methods that are typically used
in science, engineering, business, and industry.
is studied purely for the sake of application. Examples of
this lie in Economics, Computer Science, and Engineering.
focuses on the formulation and study of mathematical
Mathematical Modeling:
A
model
is a description of a system
A
mathematical model
is a description of asystem using mathematical concepts and language.
Mathematical modeling
is the process ofMathematical Modeling:
Problem
we want to understand some behaviors or phenomena
in the real world.
We may wish to make predictions about such behaviors
in the future
We want to examine its effects in various situations.
Why Mathematical Model
Mathematical Modeling:
The model allows us to reach mathematical
Mathematical Modeling
Let’s first understanding the process of Modeling:
Simplification
Verification Analysis
Interpretation
Mathematical Model
Mathematical Conclusion Prediction /
Mathematical Modeling
Simplification
Simplification: simplify the relationships between the variables of the phenomenon to construct a Model. One powerful technique for simplifying relationship is Proportionality:
(x and y are proportional (to each other) if one is multiple constant of the other)
where k is the constant of proportionality
Model Real world
phenomenon
y
Example: (population size)
Assumption:
( The rate of change of the population with respect to time t is
depends on the current size of P)
let assume a simple proportionality
where k is expressed as a percentage per unit of time:
The average rate of change: Difference Equation
P
t
P
/
kPt p
Example: (population size)
Solving the problem
The above problem after converting it to a differential equation (derivative) we have got analytical solution
where is the size of the population at time t = 0.
kt
e p t
p( ) 0
0 P ) ( ) ( lim
0 dt kp t
t dp t
p
t
Example: (population size)
Verifying the model
kt
e p t
p( ) 0
Mathematical Modeling
0.02(2026 2016)
(2026) 30000 36642
p e
0.02(2036 2016)
(2036) 30000 44754
p e
0.02(2066 2016)
(2066) 30000 81548
p e
0.02(2216 2016)
(2216) 30000 1637944
Example: (population size)
Refinement:
( The rate of change of the population with respect to time t is
depends on the current size of P). However, in most
populations, individuals compete with one another for food, living space, and other natural resources. Limited growth
P t
P
/ kP rM PP
t
p
Example: (population size)
Solving the problem
The above problem after converting it to a differential equation (derivative) we have got analytical solution
where is the size of the population at time t = 0.0
P ) ( )) ( ( ) ( lim
0 dt r M p t p t
t dp t
p
t
Introduction
Congestion: one of the common environmental problems
characteristics
Its increasing by the increasing of the growth rate of populations
It is found:
pedestrian transportation, shopping centers,
crowded buildings, areas of sport events,
introduction
characteristics of congestion:
It may cause :
Disasters
12, 2006. They reveal two subsequent, sudden transitions from
laminar to stop-and-go and “turbulent”
It can be solved by:
It can be solved by:
Overview
Mathematical Modeling
(Motion)
In physics, Motion is a change in the location of a body resulted from applied forces.
Motion is typically described in terms of velocity,
Mathematical Modeling
(Newton's second law)
Newton's second law states that the force applied to a body is proportional with the instantaneous rate
of change in velocity (acceleration):
where F is the force applied, m is the mass of the body, and a is the body's acceleration.
dt v d F ma dt v d m
F
Mathematical Modeling
(Newton's second law)
If the body is subject to multiple forces at the same time, then the acceleration is proportional to the
vector sum (that is, the net force):
Mathematical Modeling
(Newton's second law)
What about people? (As example pedestrians).
How to model the Pedestrian Motion?
Can we represent the pedestrian motion as a
Newtonian equation?
This is has been proposed in (1995) by Helbing in
The Social Force Model
The system of the pedestrian’s environment mainly consists of
pedestrians,
the physical environment,
repulsive and attractive sources (pedestrians or objects
such as walls or columns),
Social Force Model
Assumption:
The Social Force Model modeling the motion
wall wall i j ij preffered i i ii f f f f
dt v d
m ,
The collective forces exerted by the walls on
pedestrian i The collective forces
The Social Force Model
modeling the motivation
Assumption: Three motivations inside any individual while
he is in motion:
The Repulsion motivation:
the repulsive source motivates the individual i to avoid the source
The Attraction motivation:
the attractive source motivates individual i to change his direction toward the attractive source
The Driven motivation:
a motivation to adapt his actual velocity to the preferred one
) (t fijrep
) (t fijatt
The Social Force Model
modeling the motivation
The Driven motivation :
to adapt the actual velocity to the desired velocity :
The preferred force:
Helbing, molnar.(1995)
v
v
f
preferred
0
m) (
0
t vi
) (t vi
The Social Force Model
modeling the contact
Incorporating the physical forces into the model:
• f pushing works as a body force counteracting body compression
• f friction works as sliding friction force impeding
relative tangential motion.
J
i
pushing f
f riction
f
fric ij push ij att ij rep ij
ij
f
f
f
f
f
i
rep social
f ,
f riction
f
pushing f
pref f ered f
The Social Force Model
modeling the physical contact
fric wall i push wall i att wall i rep wall i wall
i
f
f
f
f
f
,
,
,
,
,rep social
f ,
f riction
f
pushing f
pref f ered f
The Social Force Model
The Social Force Model modeling the motion
wall wall i j ij preferred i i i ii f f f f
dt v d
m ,
The collective forces exerted by the walls on
pedestrian i The collective forces
Simulation
A computer simulation is an implementation of a model that allows us to test the model under
different conditions with the objective of learning about the model's behavior.
The compatibility of the model with the real
The Validation of the model
Does my model introduce the following:
Self-organization phenomena Reproduction of real life data Intelligence aspects
Check
Adjustment of parameters Assumptions
Self-organization phenomena
Oscillations at a bottleneck
Aspects of intelligence
-5 0 5 10 15 20 25 -5 0 5 10 15 20 25 30 35 40 45 a)
-5 0 5 10 15 20 25 -5 0 5 10 15 20 25 30 35 40 45
b) -5 0 5 10 15 20 25
Penetrating high dense crowd
Pilgrims are going outside the Sanctuary Mosque in Makkah after performing their
Muslim pilgrims perform circular movement around the Kaaba as a part of
6 8 10 12 14 16 -2 -1 0 1 2 3 4 5 6 7 (b)
6 8 10 12 14 16 -2 -1 0 1 2 3 4 5 6 7 (c)
6 8 10 12 14 16 -2 -1 0 1 2 3 4 5 6 7 (d) 6 8 10 12 14 16
Lane Joining behavior
Joining lane behavior by group of pedestrians in bidirectional
pedestrian flow.
0 5 10 15 20 25 30 35 40 45 50 0
2 4 6 8
100 5 10 15 20 25 30 35 40 45 50 0 2 4 6 8 10