Introduction Mathematical Modeling

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



A

mathematical model

system using mathematical concepts and language.



Mathematical modeling

Mathematical 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







/

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.

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 rM PP

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

       





i 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 f_ijrep

) (t f_ijatt

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



















) (

0

t v_i

) (t v_i

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

       





i 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