9. introduction to optimization in engineering...

Hae-Jin ChoiSchool of Mechanical Engineering,

Chung-Ang University

9. Introduction to

Optimization in Engineering Design

SCHOOL OF MECHANICAL ENG.

CHUNG-ANG UNIVERSITY-1-

Optimization is derived from the Latin word “optimus”, thebest.

Optimization characterizes the activities involved to find“the best”.

People have been “optimizing” forever, but the roots formodern day optimization can be traced to the SecondWorldWar.

DOE and Optimization

Optimization



Operations Research originated from the activities performed bymultidisciplinary teams formed in the British armed forces involvedin solving complex strategic and tactical problems inWorldWar II.

Waddington describes the main objectives of the OperationalResearch Section in the British armed forces as

“The prediction of the effects of new weapons and tactics.”

(Waddington, C.H. (1973). O.R. in World War 2 - Operational Research against the U-Boat,History of Science Series, C.W. Kilmist Ed., Elek Science, London.)


Operations Research (OR)



Design requires designer’s experience, intuition and ingenuity in

most field of engineering (aerospace, automotive, civil,

chemical, industrial, electrical, mechanical, etc.)

Design is iterative process

Iterative implies analyzing several trial systems in a sequence

before an acceptable design is obtained

Engineers strive to design the best system, which implies the

most cost effective, efficient, reliable, and durable system.


Engineering Design



5%

50%30%

15%

Cost

Design Materials Overhead Labor


Importance of Engineering Design

70%

20%

5% 5%

Performance and Productivity

Design Materials Overhead Labor



Analysis is

Finding behavior of an existing system

Calculating response of an existing system under the specified

input

Design is

Determining sizes and shapes of various parts of the system to

meet performance requirements


Engineering Design vs Analysis


CHUNG-ANG UNIVERSITY-6-DOE and Optimization

Engineering Design vs Analysis

Load

Maximum

StressHeight

Height

Analysis

Design

Height

H=100 mm

Maximum Stress

= ???

Maximum Stress

≤ 400 MPa

Height

H=???



Optimization in Engineering Design Identify

(1) Design variables

(2) Cost function to be minimized

(3) Constraints that must be satisfied

Collect data to describe

the system

Estimate initial design Analyze the system

Check the constraintsSatisfy convergence

criteria?

Optimum Design

Change the design

Yes

No



Optimization in Engineering Design Identify

(1) Height of the beam

(2) Mass (or Cost) to be minimized

(3) Stress constraint must be satisfied

Learn Beam Theory and

develop computer code

Estimate initial heightFind mass and analyze

Maximum stress

Check the stress

constraintsIs this minimum Mass?

Optimum Beam

Change the height

Yes

No



Role of Computers in Optimum Design

Analysis using computers allow us

more accurate calculation

Computer Aided Design (CAD) and

Computer Aided Engineering (CAE)

Iterative process of engineering

optimization requires computer

calculations

Large amount of data and

information to handle for us requires

extensive use of computer



Understand ‘OPTIMIZATION’ in Engineering Design

Internalize ‘Optimum Design Problem Formulation’ from

engineering problems

Understand basic theory of optimization

Practice software (e.g., MATLAB) for solving optimization

problem


Objectives in this Classes



SI unit

Sets and Points

Notation for Constraints

Superscripts/Subscripts and Summation Notation

Norm/Length of a Vector

Functions


Basic Terminology and Notation



Realistic systems generally involve several variables; thus,

dimensions: 1, 2, …., n

A vector (or point) in n-dimensional space (Rn) represents a

set of variable specifications.

We also use


Points and Sets

1

2

1 2. ...

.

T

n

n

x

x

x x x

x

x

1 2, ,..., nx x xx



A Set is collection of points satisfying certain conditions

Denoting the set by S , we can write, for example

Which can be read as ‘S equals the set of all points

with x3 =0

Right of the vertical bar indicates the characteristics a point

must possess to be in the set S


Points and Sets

1 2 3 3{ ( , , ) | 0}S x x x x x

1 2 3( , , )x x x



x is an element of ( belongs to ) S

y is not an element of (does not belong to ) S

For example

(3,3), (2,2), (3,2) belong to the set

(1,1), (8,8), (-1,2) does not belong to the set


Points and Sets

Sx

Sy

2 2

1 2 1 2{ ( , ) | ( 4) ( 4) 9}S x x x x x

(3,2) S

( 1,2) S



Constraints arise naturally in optimum design problems

For example

Material of the system must not fail

Demand must be met,

Resources must not exceed, etc.

A constraints may be expressed as

Less than or equal to type (≤)

Greater than or equal to type (≥)

For example


Notation for Constraints

2 2

1 2( 4) ( 4) 9x x



We need to describe a set of vectors, components of vectors, and

multiplications of matrices and vectors.

Superscripts are used to represent different vectors and matrices

Subscripts are used to represent components of vectors and

matrices.


Superscripts/Subscripts

( )

(k)

: th vector of a set

: th matrix

i i

k

x

A

(j) (1) (2) (k)

; 1 : numbers (components) of vector

; 1 : vectors , ,...,

where , : free variables

ix i to n

j to k k

i j

x

x x x x



Summation notation

Multiplication of an n-dimensional vector x by an m x n matrix A to

obtain an m-dimensional vector y is

y=Ax

Or

Or


Summation Notation

1 1 2 2

1

... will be written as n

n n i i

i

c x y x y x y c x y

1 1 2 2

1

...n

i ij j i i in n

j

y a x a x a x a x

( ) (1) (2) ( )

1 2

1

...n

j n

j n

j

x x x x

y a a a a



Double summation notation

Since Ax represent a vector, the triple product will be rewritten as a

dot product


Double Summation Notation

1 1 1 1

n n n n

i ij j ij i j

i j i j

c x a x a x x

T

x Ax

c Tx Ax (x Ax)



Dot product is defined as

or

where is the angle between the vectors x and y and represents

the length of the vector

The length (norm) of vector x

Dot product is a sum of product of corresponding elements of the vectors

x and y

x and y are Orthogonal (normal) if x ∙ y =0


Norm/Length of a Vector

1

( )n

i i

i

x y

T Tx y x y y x cos x y x y

x

2

1

n

i

i

x

Tx x x = x x



Function of a single variable, f(x)

Function of n independent variables x1, x2, …, xn

f(x) = f(x1, x2, …, xn)

Multiple function with vector variables

gi (x) = gi(x1, x2, …, xn)

If there is m functions gi (x) ; i=1 to m

In vector form g(x) = [g1 (x), g2 (x), …., gm (x)] T


Functions



Functions

Continuous and differentiable Continuous

Discontinuous Discontinuous

9. introduction to optimization in engineering...

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