Principles of Computer-Aided Design and Manufacturing Second Edition 2004 ISBN 0-13-064631-8 Author:...

Principles of Computer-Aided Design and Manufacturing

Second Edition 2004

ISBN 0-13-064631-8

Author: Prof. Farid. Amirouche

University of Illinois-Chicago

University of Illinois-Chicago

Chapter 6

Optimization Techniques

Principles of Computer-Aided Design and Manufacturing Second Edition 2004 – ISBN 0-13-064631-8

Author: Prof. Farid. Amirouche, University of Illinois-Chicago

CHAPTER 6 6.2 System Modeling

6.1 INTRODUCTION

6.2 SYSTEM MODELINGAnalysis and design

Figure 6.1 Simple beam

I

My

EI

PL

3

3

(6.2)

(6.3)




Example 6.1

F

2r

h

Consider a tree trunk, which can be modeled by a beam as shown in figure 6.2. On a windy day, one can determine the wind speed or force that would cause the tree to break.

3max

4

r

Fh

4

4rI

(6.4)




Example 6.2

A second example involves the modeling of an automobile wheel and tire.

rMKJ ..

Figure 6.3 (a) Automobile wheel and tire Figure 6.3 (b) Model representing the tire and shaft

(6.5)




l

GIK p

32

4dI p

29 /1080 mNG

radmNK /.455.2

4810006136.0 mI p

J

Kn

M = MO sin (ω t)

(6.6)

(6.7)

(6.8)



CHAPTER 6 6.3 Design Optimization

6.3 DESIGN OPTIMIZATIONFormulation of an optimum design

There are four steps to the formulation of an optimum design:1. Identifying the design parameters.2. Defining the design constraints.3. Defining the objective functions4. Evaluating alternatives.

Design Parameters Constraints

m)1,(i 0,,........., 21 nxxx

0),( 22121 Vxxxx

jnj Lxxx ,,........., 21

(6.9)

(6.10)

(6.11)



Evaluation Example 6.3

Consider a rectangular box used for storing important documents. Define the objective function if C denotes the cost per unit area of the metal used for fabrication of the box. Define the constraint equations and the limits on their design variables.




Solution:

A = 2LH + 2HW + 2WL

T = C (2LH + 2HW + 2WL)

V =LWH

= V – LWH = 0

1LL 1WW

1HH

As shown in the example, the steps that generally follow in formulation of a design problem are Identification of the design variables.Selection of a cost function and developing an expression for it in terms of the design variables.Identification of constraints and developing expressions for them in terms of design variables.




CHAPTER 6 6.4 Optimal Design Concept

6.4 OPTIMAL DESIGN CONCEPT

Questions: How is each design described?What is the criterion for best “design”?What are the available means?

Design optimization

Select a set of variables.Select an objective function.Determine a set of constraints.Solutions will be based on finding the values for the variables that would minimize or maximize the objective function and satisfy the constraints at the same time.






Example 6.4

Figure 6.5 A hollow cylinder with thickness t




tdd io 202 tdd io

tddtddh iio 2),,( 0 smax

J

dM ot )2/(maxmax

0max s

0)2/(max JdM ot

032

44

io ddJ

0)(

1644

0

sdd

dM

i

ot

(6.12)

(6.13)

(6.14)

(6.15)

(6.16)

(6.17)

(6.18)




CHAPTER 6 6.5 Unconstrained Optimization

6.5 UNCONSTRAINED OPTIMIZATIONSingle Variable minimization

0)( * xf

0)( * xf

0)( * xf

Figure 6.6 Function f showing local, global and strong minimum

),,....,,,,....,,( 2121 tqqqqqqf nn (6.20)

(6.19)



Example 6.5

Find whether the function 3xf has a minimum or maximum for ]1,1[x




Solution:

Figure 6.7 Tank




xL

xxL

xf

4

2201020*

4*2

22

xLxxLxf 20101010 22

xLxf 3020 2

120*4

2

Lx

2

480

xL

22 480

3020x

xxf

xxf

1440020 2

xL

(6.21)

(6.22)

(6.23)

(6.24)

(6.25)

(6.26)




Example 6.7

mx

x

x

xxxf

86.4

59.114

1440040

014400

40

3

3

2

(6.27)




(6.28)

(6.29)

(6.30)

(6.31)




(6.32)

(6.33)







(6.34)

(6.35)




6.6 CONSTRAINED OPTIMIZATION

j

m

jjoTT

1

Example 6.9

Solve for the optimal design problem of example 6.6 by the method of Lagrange multipliers.

(6.36)

(6.37)

CHAPTER 6 6.6 Constrained Optimization



(6.38)

(6.39)





(6.40)

(6.41)

(6.42)

(6.43)

(6.44)

(6.41)

(6.42)

(6.43) (6.44) (6.40)

(6.43) (6.44)




(6.45)

(6.46)

(6.47)

(6.48)

(6.49)




(6.50)

(6.51)

(6.52)

(6.53)

(6.54)

(6.55)












CHAPTER 6 6.7 Fibonacci Method

6.7 FIBONACCI METHOD

6.7.1 Fibonacci algorithm

Period 1

Period 2

Period 3

Period 4

Period 5

Figure 6.10 Rabbit’s multiplication at each mature period



Figure 6.11 Initial setup of the uncertainty interval







gx2

v0sinΘt




80







CHAPTER 6 6.8 Newton’s Method

6.8 NEWTON’S METHOD

k

kkk f

fxx

1 (6.72)



CHAPTER 6 6.9 Linear Programming

f

x

Linear approximationto f at x

k+1 k

-1

x = x - (f ) fk k

'

f fkk

'

xk

Figure 6.13 Newton’s method

6.9 LINEAR PROGRAMMING

i

ii xkT (6.74)



(6.75)







1 2 3 4 5 6

X1

X2

C

D

1

4

5

3

2

Figure 6.14 Graphical solution




CHAPTER 6 6.10 Geometric Programming

6.10 GEOMETRIC PROGRAMMING

33212

111

5

32 xxxxxxT

puuuT ......21

ps

p

p

ss

s

u

s

u

s

uG

...

21

2

2

1

1

ss p 'where the are chosen properly to minimize T.

(6.76)

(6.77)

(6.78)



CHAPTER 6



CHAPTER 6 6.11 Other Optimization Techniques

Search methodsExhaustive Search.Grid search.Random search.Simplex search.

Gradient methodsSteepest descentConjugate gradients

Second-derivative methodsNon-linear programming

Johnson’s methodPowell’s method for unconstrained situations.

6.11 OTHER OPTIMIZATION TECHNIQUES

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