Hae-Jin ChoiSchool of Mechanical Engineering,
Chung-Ang University
9. Introduction to
Optimization in Engineering Design
SCHOOL OF MECHANICAL ENG.
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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
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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.)
DOE and Optimization
Operations Research (OR)
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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.
DOE and Optimization
Engineering Design
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5%
50%30%
15%
Cost
Design Materials Overhead Labor
DOE and Optimization
Importance of Engineering Design
70%
20%
5% 5%
Performance and Productivity
Design Materials Overhead Labor
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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
DOE and Optimization
Engineering Design vs Analysis
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Engineering Design vs Analysis
Load
Maximum
StressHeight
Height
Analysis
Design
Height
H=100 mm
Maximum Stress
= ???
Maximum Stress
≤ 400 MPa
Height
H=???
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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
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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
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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
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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
DOE and Optimization
Objectives in this Classes
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SI unit
Sets and Points
Notation for Constraints
Superscripts/Subscripts and Summation Notation
Norm/Length of a Vector
Functions
DOE and Optimization
Basic Terminology and Notation
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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
DOE and Optimization
Points and Sets
1
2
1 2. ...
.
T
n
n
x
x
x x x
x
x
1 2, ,..., nx x xx
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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
DOE and Optimization
Points and Sets
1 2 3 3{ ( , , ) | 0}S x x x x x
1 2 3( , , )x x x
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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
DOE and Optimization
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
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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
DOE and Optimization
Notation for Constraints
2 2
1 2( 4) ( 4) 9x x
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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.
DOE and Optimization
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
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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
DOE and Optimization
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
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Double summation notation
Since Ax represent a vector, the triple product will be rewritten as a
dot product
DOE and Optimization
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)
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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
DOE and Optimization
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
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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
DOE and Optimization
Functions
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Functions
Continuous and differentiable Continuous
Discontinuous Discontinuous