Numerical Analysis

2015 Fall

Course Materials

http://cvlab.khu.ac.kr/nagrad.html

Instructor [E&I Bldg. Room #223] Seungkyu Lee, Email: seungkyu(at)khu.ac.kr

Reference Books

Applied Numerical Methods W/MATLAB: for Engineers & Scientists

3rd Edition

Steven Chapra

Additional Materials Matlab Tutorial #1

Evaluation

Final Exam (Can be a Matlab project) 50%

Mid Exam 40%

Attendance 10%

Mathematical Model

Extraction

A Simple Mathematical Model • A mathematical model can be broadly defined as a

formulation or equation that expresses the essential features of a physical system or process in mathematical terms.

• Models can be represented by a functional relationship between dependent variables, independent variables, parameters, and forcing functions.

Model Function

• Dependent variable - a characteristic that usually reflects the

behavior or state of the system

• Independent variables - dimensions, such as time and space, along

which the system’s behavior is being determined

• Parameters - constants reflective of the system’s properties or

composition

• Forcing functions - external influences acting upon the system that

cannot be changed within the system

Dependentvariable

findependent

variables, parameters,

forcingfunctions

Model Function Example

mgFD

Newton’s 2nd raw

Air resistance 2vcF dU

maF

Model Function Example

UD FFF

Bungee Jumper

mavcmgF d 2

dt

dvmvcmg d 2

A differential eq.

Numerical Modeling • System models - can be solved either using analytical

methods or numerical methods.

• Example - the bungee jumper velocity equation from before is the analytical solution to the differential equation

where the change in velocity is determined by the gravitational forces acting on the jumper versus the drag force.

dv

dt g

cd

mv2

Analytical solution

• Assuming a bungee jumper is in mid-flight, an analytical model for the jumper’s velocity, accounting for drag, is

• Dependent variable - velocity v

• Independent variables - time t

• Parameters - mass m, drag coefficient cd

• Forcing function - gravitational acceleration g

v t gm

cdtanh

gcd

mt

Analytical solution • Using a computer (or a calculator), the model can be

used to generate a graphical representation of the system. For example, the graph below represents the velocity of a 68.1 kg jumper, assuming a drag coefficient of 0.25 kg/m

v t gm

cdtanh

gcd

mt

Model Results

v t gm

cdtanh

gcd

mt

Closed form solution

Analytical solution

dv

dt g

cd

mv2

of

Model Results

Numerical solution

dv

dt g

cd

mv2

of

In many cases..

analytical solution cannot be obtained ex) equation does not have a closed from solution

Numerical Methods

• To solve the problem using a numerical method, note that the time rate of change of velocity can be approximated as:

dv

dtv

tv ti1 v ti ti1 ti

Assume that we cannot solve the diff. eq.

analytically

Numerical Methods

2

1

1i

d

ii

ii tvm

cg

tt

tvtv

2vm

cg

dt

dv d

)( 1

2

1 iiid

ii tttvm

cgtvtv

Numerical solution

Next = present + slope time step

dv

dt g

cd

mv2

slope

Numerical solution • Using a computer (or a calculator), the model can be

used to generate a graphical representation of the system. For example, the graph below represents the velocity of a 68.1 kg jumper, assuming a drag coefficient of 0.25 kg/m

)( 1

2

1 iiid

ii tttvm

cgtvtv