Introduction to optimum design

919 Pages·2011·18.14 MB·English

by  Jasbir S. Arora

INTRODUCTION
TO OPTIMUM
DESIGN
THIRD EDITION
J S. A
ASBIR RORA
TheUniversityofIowa
CollegeofEngineering
IowaCity,Iowa
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11 12 13 14 15 10 9 8 7 6 5 4 3 2 1
To
Ruhee
Rita
and
in memory of my parents
Balwant Kaur
Wazir Singh
Preface to Third Edition
The philosophy of this third edition of This edition can be broadly divided into
Introduction toOptimum Designistoprovide threeparts.PartI,Chapters1through5,pre-
readers with an organized approach to sents the basic concepts related to optimum
engineering design optimization that is design and optimality conditions. Part II,
both rigorous and simple, that illustrates Chapters 6 through 14, treats numerical
basic concepts and procedures with simple methods for continuous variable optimiza-
examples, and that demonstrates the appli- tionproblemsandtheirapplications.Finally,
cability of these concepts and procedures to Part III, Chapters 15 through 20, offers
engineering design problems. The key step advanced and modern topics on optimum
in the optimum design process is the for- design, including methods that do not
mulation of a design problem as an optimi- requirederivativesoftheproblemfunctions.
zation problem, which is emphasized and Introduction to Optimum Design, Third
illustrated with examples. In addition, in- Edition, can be used to construct several
sights into, and interpretations of, optimal- types of courses depending on the instruc-
ityconditionsarediscussedandillustrated. tor’s preference and learning objectives for
Two main objectives were set for the students. Three course types are suggested,
third edition: (1) to enhance the presenta- althoughseveral variationsare possible.
tion of the book’s content and (2) to include
advanced topics so that the book will be
Undergraduate/First-Year Graduate
suitable for higher-level courses on design
optimization. The first objective is achieved Course
by making the material more concise,orga-
Topicsfor anundergraduateand/orfirst-
nizing it with more second-, third-, and
yeargraduatecourseinclude
fourth-level headings, and using illustra-
tions in example problems that have more (cid:129) Formulation of optimization problems
details. The second objective is achieved by (Chapters1 and 2)
including several new topics suitable for (cid:129) Optimization concepts using the
bothalternatebasicandadvancedcourses. graphicalmethod (Chapter3)
New topics include duality in nonlinear (cid:129) Optimality conditionsfor unconstrained
programming, optimality conditions for the and constrained problems(Chapter4)
Simplex method, the rate of convergence of (cid:129) Use ofExcelandMATLABs illustrating
iterative algorithms, solution methods for optimum design of practicalproblems
quadratic programming problems, direct (Chapters6 and 7)
search methods, nature-inspired search (cid:129) Linearprogramming (Chapter8)
methods, response surface methods, design (cid:129) Numericalmethodsforunconstrained
of experiments, robust design optimization, andconstrainedproblems(Chapters10
and reliability-based design optimization. and12)
xiii
xiv PREFACETOTHIRDEDITION
The use of Excel and MATLAB is to be Second Graduate-Level Course
introduced mid-semester so that students
This course presents advanced topics on
have a chance to formulate and solve more
optimum design:
challenging project-type problems by seme-
ster’s end. Note that advanced project-type (cid:129) Duality theory in nonlinear
exercises and sections with advanced mate- programming,rate of convergenceof
rial are marked with an asterisk (*) next to iterative algorithms, derivation of
section headings, which means that they numerical methods, and direct search
may be omitted for thiscourse. methods (Chapters 1 through 14)
(cid:129) Methods for discrete variable problems
(Chapter15)
First Graduate-Level Course
(cid:129) Nature-inspired search methods
Topics for a first graduate-level course (Chapters 16and 19)
include (cid:129) Multi-objectiveoptimization
(cid:129) Theory andnumerical methods for (Chapter17)
(cid:129) Globaloptimization(Chapter 18)
unconstrained optimization (Chapters1
(cid:129) Response surfacemethods, robust
through 4 and10 and 11)
(cid:129) Theory andnumerical methods for design,and reliability-based design
optimization (Chapter 20)
constrainedoptimization (Chapters 4, 5,
12, and 13) During this course, students write com-
(cid:129) Linearandquadratic programming puter programs to implement some of the
(Chapters 8and9) numerical methods and to solve practical
problems.
The pace of material coverage should be
faster for this course type. Students can
code some of the algorithms into computer
programs andsolve practicalproblems.
Acknowledgments
I would like to give special thanks to my the subject of optimum design. I appreciate
colleague, Professor Karim Abdel-Malek, my colleagues at The University of Iowa
Director of the Center for Computer-Aided who used the previous editions of the book
Design at The University of Iowa, for his to teach an undergraduate course on opti-
enthuastic support for this project and for mum design: Professors Karim Abdel-
getting me involved with the very exciting Malek, Asghar Bhatti, Kyung Choi, Vijay
research taking place in the area of digital Goel, Ray Han, Harry Kane, George Lance,
human modeling under the Virtual Soldier andEmadTanbour.Theirinputandsugges-
ResearchProgram. tionsgreatlyhelpedmeimprovethepresen-
Iwouldalsoliketoacknowledgethecon- tation of material in the first 12 chapters of
tributions of the following colleagues: this edition. I would also like to acknowl-
Professor Tae Hee Lee provided me with a edge all of my former graduate students
first draft of the material for Chapter 7; whosethesisworkonvarioustopicsofopti-
Dr. Tim Marler provided me with a first mization contributed to the broadening of
draft of the material for Chapter 17; Profes- myhorizonsonthesubject.
sor G. J. Park provided me witha first draft I would like to thank Bob Canfield,
of the material for Chapter 20; and Drs. Hamid Torab, Jingang Yi, and others for
Marcelo A. da Silva and Qian Wang pro- reviewing various parts the third edition.
vided me with a first draft of some of the Their suggestions helped me greatly in its
material for Chapter 6. Their contributions fine-tuning.I would also like tothankSteve
were invaluable in the polishing of these Merken and Marilyn Rash at Elsevier for
chapters. In addition, Dr. Tim Marler, Dr. their superb handling of the manuscript
Yujiang Xiang, Dr. Rajan Bhatt, Dr. Hyun and production of the book. I also thank
Joon Chung, and John Nicholson provided MelanieLaverman for help with theediting
me with valuable input for improving the ofsomeofthebook’schapters.
presentation of material in some chapters. I I am grateful to the Department of Civil
would also like to acknowledge the help of and Environmental Engineering, Center for
Jun Choi, Hyun-Jung Kwon, and John Computer-Aided Design, College of Engi-
Nicholsonwithpartsofthebook’ssolutions neering, and The University of Iowa for
manual. providing me with time, resources, and
Iamgratefultonumerouscolleaguesand support for this very satisfying endeavor.
friends around the globe for their fruitful Finally, I would like to thank my family
associations with me andfor discussions on and friends fortheir loveand support.
xv
Key Symbols and Abbreviations
(cid:1)
(a b) Dot productof vectorsa and b; ACO Ant colonyoptimization
aTb BBM Branch-and-bound method
c(x) Gradientof cost function,rf(x) CDF Cumulative distribution
f(x) Cost function to beminimized function
g(x) jthinequality constraint CSD Constrained steepestdescent
j
h(x) ithequality constraint DE Differential evolution;Domain
i
m Numberof inequality elimination
constraints GA Geneticalgorithm
n Numberof design variables ILP Integer linear programming
p Numberof equality constraints KKT Karush-Kuhn-Tucker
x Designvariable vectorof LP Linearprogramming
dimension n MV-OPT Mixed variableoptimization
x ithcomponent of design vari- problem
i
able vector x NLP Nonlinear programming
x(k) kthdesignvariablevector PSO Particleswarm optimization
QP Quadratic programming
Note: A superscript (i) indicates optimum
RBDO Reliability-based design
value for a variable, (ii) indicates advanced
optimization
materialsection,and(iii)indicatesaproject-
SA Simulated annealing
typeexercise.
SLP Sequential linear programming
SQP Sequential quadratic
programming
TS Traveling salesman
(salesperson)
xvi
C H A P T E R
1
Introduction to Design Optimization
Upon completion of this chapter, you will be able to
(cid:129) Describetheoverallprocessofdesigning (cid:129) Distinguishbetweenoptimumdesignand
systems optimalcontrolproblems
(cid:129) Distinguishbetweenengineeringdesignand (cid:129) Understandthenotationsusedfor
engineeringanalysisactivities operationswithvectors,matrices,and
(cid:129) Distinguishbetweentheconventional functionsandtheirderivatives
designprocessandtheoptimumdesign
process
Engineering consists of a number of well-established activities, including analysis,
design, fabrication, sales, research, and development of systems. The subject of this text—
thedesignofsystems—isamajorfieldintheengineeringprofession.Theprocessofdesign-
ing and fabricating systems has been developed over centuries. The existence of many
complexsystems,suchasbuildings,bridges,highways,automobiles,airplanes,spacevehi-
cles, and others, is an excellent testimonial to its long history. However, the evolution of
such systems has been slow and the entire process is both time-consuming and costly,
requiring substantial human and material resources. Therefore, the procedure has been to
design, fabricate, and use a system regardless of whether it is the best one. Improved sys-
temshavebeendesignedonlyafterasubstantialinvestmenthasbeenrecovered.
The preceding discussion indicates that several systems can usually accomplish the
same task, and that some systems are better than others. For example, the purpose of a
bridge is to provide continuity in traffic from one side of the river to the other side.
Several types of bridges can serve this purpose. However, to analyze and design all possi-
bilities can be time-consuming and costly. Usually one type is selected based on some pre-
liminaryanalysesand isdesigned in detail.
The design of a system can be formulated as problems of optimization in which a perfor-
mance measure is optimized while all other requirements are satisfied. Many numerical
methods of optimization have been developedand used to designbettersystems. Thistext
IntroductiontoOptimumDesign 1 ©2012ElsevierInc.Allrightsreserved.
2
1. INTRODUCTIONTODESIGNOPTIMIZATION
describes thebasic concepts of optimizationandnumericalmethods for thedesign ofengi-
neering systems. Design process, rather than optimization theory, is emphasized. Various
theorems are stated as results without rigorous proofs; however, their implications from
an engineering point of view are discussed.
Any problem in which certain parameters need to be determined to satisfy constraints
can be formulated as one optimization problem. Once this has been done, the concepts
andmethodsdescribedinthistextcanbeusedtosolveit.Forthisreason,theoptimization
techniques are quite general, having a wide range of applicability in diverse fields. It is
impossible to discuss every application of optimization concepts and techniques in this
introductory text. However, using simple applications, we discuss concepts, fundamental
principles, and basic techniques that are used in numerous applications. The student
should understand them without becoming bogged down with the notation, terminology,
and detailsof the particular area of application.
1.1 THE DESIGN PROCESS
How DoIBegin toDesigna System?
The design of many engineering systems can be a complex process. Assumptions
must be made to develop realistic models that can be subjected to mathematical analysis
by the available methods, and the models must be verified by experiments. Many possi-
bilities and factors must be considered during problem formulation. Economic considera-
tions play an important role in designing cost-effective systems. To complete the design
of an engineering system, designers from different fields of engineering usually must
cooperate. For example, the design of a high-rise building involves designers from archi-
tectural, structural, mechanical, electrical, and environmental engineering as well as con-
struction management experts. Design of a passenger car requires cooperation among
structural, mechanical, automotive, electrical, chemical, hydraulics design, and human
factors engineers. Thus, in an interdisciplinary environment considerable interaction is
needed among various design teams to complete the project. For most applications the
entire design project must be broken down into several subproblems, which are then
treated somewhat independently. Each of the subproblems can be posed as a problem of
optimum design.
The design of a system begins with the analysis of various options. Subsystems and
theircomponentsareidentified,designed, andtested.Thisprocessresultsinasetofdraw-
ings, calculations, and reports by which the system can be fabricated. We use a systems
engineering model to describe the design process. Although a complete discussion of this
subject is beyond the scope of this text, some basic concepts are discussed using a simple
block diagram.
Design is an iterative process. Iterative implies analyzing several trial designs one after
another until anacceptable design isobtained. It isimportant to understand the concept of
trialdesign. Inthedesignprocess,thedesigner estimatesatrialdesignofthesystembased
on experience, intuition, or some simple mathematical analyses. The trial design is then
analyzed to determine if it is acceptable. If it is, the design process is terminated. In the
optimization process, the trial design is analyzed to determine if it is the best. Depending
I. THEBASICCONCEPTS
3
1.1 THEDESIGNPROCESS
1 2 3 4 5
System Preliminary Detailed Prototype System Final
System specification design design system testing design
needs and fabrication
objectives
FIGURE1.1 Systemevolutionmodel.
on the specifications, “best” can have different connotations for different systems. In gen-
eral, it implies that a system is cost-effective, efficient, reliable, and durable. The basic con-
cepts are described in this text to aid the engineer in designing systems at the minimum
cost and inthe shortest amountof time.
The design process should be well organized. To discuss it, we consider a system evolu-
tion model, shown in Figure 1.1, where the process begins with the identification of a need
that may be conceived by engineers or non-engineers. The five steps of the model in the
figure are described in the following paragraphs.
The first step in the evolutionary process is to precisely define the specifications for the
system. Considerable interaction between the engineer and the sponsor of the project is
usually necessary to quantify thesystem specifications.
The second step in the process is to develop a preliminary design of the system. Various
system concepts are studied. Since this must be done in a relatively short time, simplified
models are used at this stage. Various subsystems are identified and their preliminary
designs estimated. Decisions made at this stage generally influence the system’s final
appearanceandperformance.At theendofthepreliminarydesign phase,afewpromising
concepts that needfurther analysisare identified.
The third step in the process is a detailed design for all subsystems using the iterative pro-
cess described earlier. To evaluate various possibilities, this must be done for all previ-
ously identified promising concepts. The design parameters for the subsystems must be
identified. The system performance requirements must be identified and satisfied. The
subsystems must be designed to maximize system worth or to minimize a measure of the
cost. Systematic optimization methods described in this text aid the designer in accelerat-
ing the detailed design process. At the end of the process, a description of the system is
availablein the form of reportsand drawings.
The fourth and fifth steps shown in Figure 1.1 may or may not be necessary for all sys-
tems. They involve fabrication of a prototype system and testing, and are necessary when
the system must be mass-produced or when human lives are involved. These steps may
appear to be the final ones in the design process, but they are not because the system may
not perform according to specifications during the testing phase. Therefore, the specifica-
tions may have to be modified or other concepts may have to be studied. In fact, this re-
examination may be necessary at any point during the design process. It is for this reason
that feedback loops are placed at every stage of the system evolution process, as shown in
I. THEBASICCONCEPTS