Introduction to Optimum Design phần 1 pdf

For undergraduundergradu-ate students, the key question is, “What should be taught on the subject of optimization?” I feel that the material thoroughly covered should be: optimum design

Introduction to Optimum Design

Introduction to Optimum Design

Second Edition

Jasbir S Arora

The University of Iowa

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Library of Congress Cataloging-in-Publication Data

Arora, Jasbir S.

Introduction to optimum design / Jasbir S Arora.—2nd ed.

p cm.

Includes bibliographical references and index.

ISBN 0-12-064155-0 (acid-free paper)

1 Engineering design—Mathematical models I Title.

TA174.A76 2004

620 ¢.0042¢015118—dc22 2004046995

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Printed in the United States of America

04 05 06 07 08 09 9 8 7 6 5 4 3 2 1

Jasbir S Arora

F Wendell Miller Distinguished Professor of EngineeringDepartment of Civil and Environmental EngineeringDepartment of Mechanical and Industrial Engineering

Center for Computer Aided Design

College of EngineeringThe University of IowaIowa City, Iowa 52242-1527

Ruhee Rita Balwant Kaur Wazir Singh

I have based the material of the Second Edition on the comments that I had received fromthe students over the years and on input from colleagues around the world The text has been

rewritten, reorganized, and expanded for the second edition Particular attention has been

paid to the pedagogical aspect of the material Each chapter starts with a list of learning

objectives that the students can keep in mind while studying the material of the chapter The

basic philosophy of the text remains the same as before: to describe an organized approach

to engineering design optimization in a rigorous and yet simplified manner, illustrate various concepts and procedures with simple examples, and demonstrate their applicability to engi- neering design problems Formulation of a design problem as an optimization problem is

emphasized and illustrated throughout the text Some computational algorithms are presented

in a step-by-step format to give the students a flavor of the calculations needed for solving

optimum design problems The new material covered in the second edition includes: use of

Excel and MATLAB as learning and teaching aids, discrete variable optimization, geneticalgorithms, multiobjective optimization, and global optimization

The text can be used in several ways to design different types of courses for

undergradu-ate and graduundergradu-ate studies For undergraduundergradu-ate students, the key question is, “What should be

taught on the subject of optimization?” I feel that the material thoroughly covered should be:

optimum design problem formulation, basic concepts that characterize an optimum design,basic concepts of numerical methods for optimization, and simple but illustrative examples

of optimum design In addition, some exposure to the use of optimization software would bequite beneficial With this background, the students would be able to formulate and use soft-ware properly to optimize problems once they go into industry The basic knowledge gainedwith this material can serve as a life-long learning tool on the subject of optimum design

Such a course for junior and senior students in most branches of engineering can include the

following material, augmented with 2- to 3-week-long team projects (project type exercises

and sections with advanced material are marked with an “*” in the text):

Appendix A Economic Analysis

Chapter 1 Introduction to Design

Chapter 2 Optimum Design Problem Formulation

Chapter 3 Graphical Optimization Method

Chapter 4 Optimum Design Concepts

Chapter 6 Linear Programming Methods for Optimum Design

Chapter 8 Numerical Methods for Unconstrained Optimum Design

Chapter 10 Numerical Methods for Constrained Optimum Design

ix

Another intermediate level course for seniors and first year graduate students can be

designed to augment the above material with Chapter 12 on MATLAB along with moreadvanced design projects and introduction to discrete variable optimization using the mate-rial contained in Chapters 15 and 16 The pace of material coverage can be a little faster than

the course designed for undergraduates only A two-course sequence for graduate students

may be designed using the material from Chapters 1 to 10 and 12 in the first course and thematerial from Chapters 11 and 13 to 18 for the second course

I have been fortunate to have received advice, encouragement, and help from numerouspeople around the globe to undertake and complete this project Without that, a project ofthis magnitude would not have been possible I would like sincerely to thank all of them fortheir input, in particular, Professor Tae Hee Lee of Hanyang University, and my graduate stu-

dents Tim Marler and Qian Wang for their special contributions to the text material

Pro-fessor Tae Hee Lee provided me with a first draft of the material for Chapter 12 on

Introduction to Optimization with MATLAB He developed all the examples and the

corre-sponding m-files Tim Marler provided me with first draft of the material for Chapter 17 on Multiobjective Optimum Design Concepts and Methods, and Qian Wang provided me with

material related to the use of Excel Without their contributions this material would not be inthe good shape it is now In addition, Tim Marler, Qian Wang, and Ashok Govil proofread severalchapters and provided me with suggestions for improving the presentation of the material.Along with all the individuals mentioned in the first edition, I would like to sincerely thank

the following colleagues and friends who provided me with specific suggestions on the material

for the second edition of the text: Rick Balling, Ashok Belegundu, Scott Burns, Alex Diaz, DanFrangopol, Ramana Grandhi, Don Grierson, Rafi Haftka, Gene Hou, Tae Hee Lee, T.C Lin, KuniMatsui, Duc Nguyen, Makoto Ohsaki, G.J Park, Subby Rajan, David Thompson, Mats Tinnsten,and Ren-Jye Yang In addition, the useful exchange of ideas on the subject of optimum designover the years with many colleagues are acknowledged: Santiago Hernández, Hans Eschenauer,

Ed Haug, Niels Olhoff, H Furuta, U Kirsch, J Sobieski, Panos Papalambros, Colby Swan, V.K.Goel, F.Y Cheng, S Pezeshk, D.H Choi, Dan Tortorelli, H Yamakawa, C.M Chan, LucienSchmit, V Kumar, Kwan Rim, Hasan Kamil, Mike Poldneff, Bob Benedict, John Taylor, MarekRysz, Farrokh Mistree, M.H Abolbashari, Achille Messac, J Herskovits, M Kamat, V Venkayya,

N Khot, Gary Vanderplaats, B.M Kwak, George Rozvany, N Kikuchi, Prabhat Hajela, Z.Gürdal, Nielen Stander, Omar Ghattas, Peter Eriksson, Olof Friberg, Jan Snyman, U Kirsch, P.Pedersen, K Truman, C Mota Soares, Igbal Rai, Rajbir Samra, Jagir Sooch, and many more

I appreciate my colleagues at The University of Iowa who used the first edition of thebook to teach an undergraduate course on optimum design: Karim Abdel-Malek, AsgharBhatti, Kyung Choi, Ray Han, Harry Kane, George Lance, and Emad Tanbour Their dis-cussions and suggestions have greatly helped in improving the presentation of the material

of first 11 chapters of the second edition

I would like to acknowledge all my former graduate students whose thesis work on varioustopics of optimization contributed to the broadening of my horizon on the subject The recentwork of Mike Huang, C.C Hsieh, Fatos Kocer, and Ossama Elwakeil has formed the basisfor the material of Chapters 15, 16, and 18

I would also like to thank Carla Kinney, Christine Kloiber, Joel Stein, Shoshanna man and Brandy Palacios of Elsevier Science, and Dan Fitzgerald of Graphic World Pub-lishing Services for their support and superb handling of the manuscript for the book

Gross-I am grateful to the Department of Civil and Environmental Engineering, College of neering, and The University of Iowa for providing me with time, resources, and support forthis very satisfying endeavor

Engi-Finally, I would like to thank all my family and friends for their love and support

Jasbir Singh Arora

Iowa City

x Preface

Table of Contents

1.2 Engineering Design versus Engineering Analysis 41.3 Conventional versus Optimum Design Process 41.4 Optimum Design versus Optimal Control 61.5 Basic Terminology and Notation 7

1.5.2 Notation for Constraints 91.5.3 Superscripts/Subscripts and Summation Notation 91.5.4 Norm/Length of a Vector 11

1.5.6 U.S.-British versus SI Units 12

2.1 The Problem Formulation Process 162.1.1 Step 1: Project/Problem Statement 162.1.2 Step 2: Data and Information Collection 162.1.3 Step 3: Identification/Definition of Design

2.3 Insulated Spherical Tank Design 20

2.5 Design of a Two-Bar Bracket 24

2.6.1 Formulation 1 for Cabinet Design 302.6.2 Formulation 2 for Cabinet Design 312.6.3 Formulation 3 for Cabinet Design 31

xi

2.7 Minimum Weight Tubular Column Design 322.7.1 Formulation 1 for Column Design 332.7.2 Formulation 2 for Column Design 342.8 Minimum Cost Cylindrical Tank Design 352.9 Design of Coil Springs 362.10 Minimum Weight Design of a Symmetric

2.11 A General Mathematical Model for Optimum Design 412.11.1 Standard Design Optimization Model 422.11.2 Maximization Problem Treatment 432.11.3 Treatment of “Greater Than Type” Constraints 432.11.4 Discrete and Integer Design Variables 44

2.11.6 Active/Inactive/Violated Constraints 45

3.1 Graphical Solution Process 553.1.1 Profit Maximization Problem 553.1.2 Step-by-Step Graphical Solution Procedure 563.2 Use of Mathematica for Graphical Optimization 603.2.1 Plotting Functions 613.2.2 Identification and Hatching of Infeasible

Region for an Inequality 623.2.3 Identification of Feasible Region 623.2.4 Plotting of Objective Function Contours 633.2.5 Identification of Optimum Solution 633.3 Use of MATLAB for Graphical Optimization 643.3.1 Plotting of Function Contours 643.3.2 Editing of Graph 643.4 Design Problem with Multiple Solutions 663.5 Problem with Unbounded Solution 66

3.7 Graphical Solution for Minimum Weight

3.8 Graphical Solution for a Beam Design Problem 69

4.1 Definitions of Global and Local Minima 84

4.1.2 Existence of Minimum 894.2 Review of Some Basic Calculus Concepts 89

4.2.3 Taylor’s Expansion 934.2.4 Quadratic Forms and Definite Matrices 964.2.5 Concept of Necessary and

Sufficient Conditions 102

xii Contents

4.3 Unconstrained Optimum Design Problems 1034.3.1 Concepts Related to Optimality Conditions 1034.3.2 Optimality Conditions for Functions of

Karush-Kuhn-Tucker (KKT) Conditions 1284.4.4 Solution of KKT Conditions Using Excel 1404.4.5 Solution of KKT Conditions Using MATLAB 1414.5 Postoptimality Analysis: Physical Meaning of

4.5.1 Effect of Changing Constraint Limits 1434.5.2 Effect of Cost Function Scaling on

Lagrange Multipliers 1464.5.3 Effect of Scaling a Constraint on Its

Lagrange Multiplier 1474.5.4 Generalization of Constraint Variation

Sensitivity Result 148

4.6.2 Convex Functions 1514.6.3 Convex Programming Problem 1534.6.4 Transformation of a Constraint 1564.6.5 Sufficient Conditions for Convex

Programming Problems 1574.7 Engineering Design Examples 1584.7.1 Design of a Wall Bracket 1584.7.2 Design of a Rectangular Beam 162Exercises for Chapter 4 166

5.1 Alternate Form of KKT Necessary Conditions 175

5.3 Second-Order Conditions for Constrained Optimization 1795.4 Sufficiency Check for Rectangular Beam

Exercises for Chapter 5 185

6.1 Definition of a Standard Linear Programming

6.1.1 Linear Constraints 1926.1.2 Unrestricted Variables 1936.1.3 Standard LP Definition 193

Contents xiii

6.2 Basic Concepts Related to Linear Programming

6.3.5 Basic Steps of the Simplex Method 2066.3.6 Simplex Algorithm 2116.4 Two-Phase Simplex Method—Artificial Variables 2186.4.1 Artificial Variables 2196.4.2 Artificial Cost Function 2196.4.3 Definition of Phase I Problem 2206.4.4 Phase I Algorithm 2206.4.5 Phase II Algorithm 2216.4.6 Degenerate Basic Feasible Solution 2266.5 Postoptimality Analysis 2286.5.1 Changes in Resource Limits 2296.5.2 Ranging Right Side Parameters 2356.5.3 Ranging Cost Coefficients 2396.5.4 Changes in the Coefficient Matrix 2416.6 Solution of LP Problems Using Excel Solver 243Exercises for Chapter 6 246

7.3.6 Use of Dual Tableau to Recover Primal Solution 2717.3.7 Dual Variables as Lagrange Multipliers 273Exercises for Chapter 7 275

8.1 General Concepts Related to Numerical Algorithms 2788.1.1 A General Algorithm 2798.1.2 Descent Direction and Descent Step 280

xiv Contents

8.1.3 Convergence of Algorithms 2828.1.4 Rate of Convergence 2828.2 Basic Ideas and Algorithms for Step Size Determination 2828.2.1 Definition of One-Dimensional

Minimization Subproblem 2828.2.2 Analytical Method to Compute Step Size 2838.2.3 Concepts Related to Numerical Methods to

Compute Step Size 2858.2.4 Equal Interval Search 2868.2.5 Alternate Equal Interval Search 2888.2.6 Golden Section Search 2898.3 Search Direction Determination: Steepest

8.4 Search Direction Determination: Conjugate

Exercises for Chapter 8 300

9.1 More on Step Size Determination 3059.1.1 Polynomial Interpolation 3069.1.2 Inaccurate Line Search 3099.2 More on Steepest Descent Method 3109.2.1 Properties of the Gradient Vector 3109.2.2 Orthogonality of Steepest Descent Directions 3149.3 Scaling of Design Variables 3159.4 Search Direction Determination: Newton’s Method 3189.4.1 Classical Newton’s Method 3189.4.2 Modified Newton’s Method 3199.4.3 Marquardt Modification 3239.5 Search Direction Determination:

9.5.1 Inverse Hessian Updating: DFP Method 3249.5.2 Direct Hessian Updating: BFGS Method 3279.6 Engineering Applications of Unconstrained Methods 3299.6.1 Minimization of Total Potential Energy 3299.6.2 Solution of Nonlinear Equations 3319.7 Solution of Constrained Problems Using Unconstrained

10.1 Basic Concepts and Ideas 34010.1.1 Basic Concepts Related to Algorithms for

Constrained Problems 34010.1.2 Constraint Status at a Design Point 34210.1.3 Constraint Normalization 343

Contents xv

10.1.4 Descent Function 34510.1.5 Convergence of an Algorithm 34510.2 Linearization of Constrained Problem 34610.3 Sequential Linear Programming Algorithm 35210.3.1 The Basic Idea—Move Limits 35210.3.2 An SLP Algorithm 35310.3.3 SLP Algorithm: Some Observations 35710.4 Quadratic Programming Subproblem 35810.4.1 Definition of QP Subproblem 35810.4.2 Solution of QP Subproblem 36110.5 Constrained Steepest Descent Method 36310.5.1 Descent Function 36410.5.2 Step Size Determination 366

10.5.4 CSD Algorithm: Some Observations 36810.6 Engineering Design Optimization Using Excel Solver 369Exercises for Chapter 10 373

11.1 Potential Constraint Strategy 37911.2 Quadratic Programming Problem 38311.2.1 Definition of QP Problem 38311.2.2 KKT Necessary Conditions for

11.2.3 Transformation of KKT Conditions 38411.2.4 Simplex Method for Solving QP Problem 38511.3 Approximate Step Size Determination 38811.3.1 The Basic Idea 38811.3.2 Descent Condition 38911.3.3 CSD Algorithm with Approximate Step Size 39311.4 Constrained Quasi-Newton Methods 40011.4.1 Derivation of Quadratic Programming

11.4.2 Quasi-Newton Hessian Approximation 40311.4.3 Modified Constrained Steepest

Descent Algorithm 40411.4.4 Observations on the Constrained

Quasi-Newton Methods 40611.4.5 Descent Functions 40611.5 Other Numerical Optimization Methods 40711.5.1 Method of Feasible Directions 40711.5.2 Gradient Projection Method 40911.5.3 Generalized Reduced Gradient Method 410Exercises for Chapter 11 411

12.1 Introduction to Optimization Toolbox 41312.1.1 Variables and Expressions 413

xvi Contents

12.1.2 Scalar, Array, and Matrix Operations 41412.1.3 Optimization Toolbox 41412.2 Unconstrained Optimum Design Problems 41512.3 Constrained Optimum Design Problems 41812.4 Optimum Design Examples with MATLAB 42012.4.1 Location of Maximum Shear Stress for Two

Spherical Bodies in Contact 42012.4.2 Column Design for Minimum Mass 42112.4.3 Flywheel Design for Minimum Mass 425Exercises for Chapter 12 429

13.1 Role of Interaction in Design Optimization 43413.1.1 What Is Interactive Design Optimization? 43413.1.2 Role of Computers in Interactive

Design Optimization 43413.1.3 Why Interactive Design Optimization? 43513.2 Interactive Design Optimization Algorithms 43613.2.1 Cost Reduction Algorithm 43613.2.2 Constraint Correction Algorithm 44013.2.3 Algorithm for Constraint Correction

13.2.4 Algorithm for Constraint Correction

at Specified Increase in Cost 44513.2.5 Constraint Correction with Minimum Increase

13.2.6 Observations on Interactive Algorithms 44713.3 Desired Interactive Capabilities 44813.3.1 Interactive Data Preparation 44813.3.2 Interactive Capabilities 44813.3.3 Interactive Decision Making 44913.3.4 Interactive Graphics 45013.4 Interactive Design Optimization Software 45013.4.1 User Interface for IDESIGN 45113.4.2 Capabilities of IDESIGN 45313.5 Examples of Interactive Design Optimization 45413.5.1 Formulation of Spring Design Problem 45413.5.2 Optimum Solution for the Spring

13.5.3 Interactive Solution for Spring Design Problem 45513.5.4 Use of Interactive Graphics 457Exercises for Chapter 13 462

14.1 Formulation of Practical Design Optimization Problems 46614.1.1 General Guidelines 46614.1.2 Example of a Practical Design

Optimization Problem 467

Contents xvii

14.2 Gradient Evaluation for Implicit Functions 47314.3 Issues in Practical Design Optimization 47814.3.1 Selection of an Algorithm 47814.3.2 Attributes of a Good Optimization Algorithm 47814.4 Use of General-Purpose Software 47914.4.1 Software Selection 48014.4.2 Integration of an Application into General-

Optimal Control of System Motion 508Exercises for Chapter 14 508

16.1 Basic Concepts and Definitions 53216.2 Fundamentals of Genetic Algorithms 534

xviii Contents

16.3 Genetic Algorithm for Sequencing-Type Problems 538

Exercises for Chapter 16 540

17.1 Problem Definition 54317.2 Terminology and Basic Concepts 54617.2.1 Criterion Space and Design Space 54617.2.2 Solution Concepts 54817.2.3 Preferences and Utility Functions 55117.2.4 Vector Methods and Scalarization Methods 55117.2.5 Generation of Pareto Optimal Set 55117.2.6 Normalization of Objective Functions 55217.2.7 Optimization Engine 55217.3 Multiobjective Genetic Algorithms 55217.4 Weighted Sum Method 55517.5 Weighted Min-Max Method 55617.6 Weighted Global Criterion Method 55617.7 Lexicographic Method 55817.8 Bounded Objective Function Method 558

17.10 Selection of Methods 559Exercises for Chapter 17 560

18.1 Basic Concepts of Solution Methods 56518.1.1 Basic Concepts 56518.1.2 Overview of Methods 56718.2 Overview of Deterministic Methods 56718.2.1 Covering Methods 56818.2.2 Zooming Method 56818.2.3 Methods of Generalized Descent 56918.2.4 Tunneling Method 57118.3 Overview of Stochastic Methods 57218.3.1 Pure Random Search 57318.3.2 Multistart Method 57318.3.3 Clustering Methods 57318.3.4 Controlled Random Search 57518.3.5 Acceptance-Rejection Methods 57818.3.6 Stochastic Integration 57918.4 Two Local-Global Stochastic Methods 57918.4.1 A Conceptual Local-Global Algorithm 57918.4.2 Domain Elimination Method 58018.4.3 Stochastic Zooming Method 58218.4.4 Operations Analysis of the Methods 58318.5 Numerical Performance of Methods 58518.5.1 Summary of Features of Methods 58518.5.2 Performance of Some Methods Using

Unconstrained Problems 586

Contents xix

18.5.3 Performance of Stochastic Zooming and

Domain Elimination Methods 58618.5.4 Global Optimization of Structural

Exercises for Chapter 18 588

A.1 Time Value of Money 593A.1.1 Cash Flow Diagrams 594A.1.2 Basic Economic Formulas 594A.2 Economic Bases for Comparison 598A.2.1 Annual Base Comparisons 599A.2.2 Present Worth Comparisons 601Exercises for Appendix A 604

B.1 Definition of Matrices 611B.2 Type of Matrices and Their Operations 613

B.2.3 Addition of Matrices 613B.2.4 Multiplication of Matrices 613B.2.5 Transpose of a Matrix 615B.2.6 Elementary Row–Column Operations 616B.2.7 Equivalence of Matrices 616B.2.8 Scalar Product–Dot Product of Vectors 616B.2.9 Square Matrices 616B.2.10 Partitioning of Matrices 617B.3 Solution of n Linear Equations in n Unknowns 618B.3.1 Linear Systems 618

B.3.3 Gaussian Elimination Procedure 621B.3.4 Inverse of a Matrix: Gauss-Jordan Elimination 625B.4 Solution of m Linear Equations in n Unknowns 628B.4.1 Rank of a Matrix 628B.4.2 General Solution of m ¥ n Linear Equations 629B.5 Concepts Related to a Set of Vectors 635B.5.1 Linear Independence of a Set of Vectors 635

B.6 Eigenvalues and Eigenvectors 642B.7 Norm and Condition Number of a Matrix 643B.7.1 Norm of Vectors and Matrices 643B.7.2 Condition Number of a Matrix 644Exercises for Appendix B 645

Appendix C A Numerical Method for Solution of

C.1 Single Nonlinear Equation 647C.2 Multiple Nonlinear Equations 650Exercises for Appendix C 655

xx Contents

Appendix D Sample Computer Programs 657

D.1 Equal Interval Search 657D.2 Golden Section Search 660D.3 Steepest Descent Method 660D.4 Modified Newton’s Method 669

1 Introduction to Design

1

Upon completion of this chapter, you will be able to:

• Describe the overall process of designing systems

• Distinguish between engineering design and engineering analysis activity

• Distinguish between the conventional design process and optimum design process

• Distinguish between the optimum design and optimal control problems

• Understand the notations used for operations with vectors, matrices, and functionsEngineering consists of a number of well established activities, including analysis, design,fabrication, sales, research, and the development of systems The subject of this text—thedesign of systems—is a major field in the engineering profession The process of designingand fabricating systems has been developed over centuries The existence of many complexsystems, such as buildings, bridges, highways, automobiles, airplanes, space vehicles, andothers, is an excellent testimonial for this process However, the evolution of these systemshas been slow The entire process has been both time-consuming and costly, requiring substantial human and material resources Therefore, the procedure has been to design,

fabricate, and use the system regardless of whether it was the best one Improved systems

were designed only after a substantial investment had been recovered These new systemsperformed the same or even more tasks, cost less, and were more efficient

The preceding discussion indicates that several systems can usually accomplish the sametask, and that some are better than others For example, the purpose of a bridge is to providecontinuity in traffic from one side to the other Several types of bridges can serve this purpose.However, to analyze and design all possibilities can be a time-consuming and costly affair.Usually one type has been selected based on some preliminary analyses and has beendesigned in detail

The design of complex systems requires data processing and a large number of

calcula-tions In the recent past, a revolution in computer technology and numerical computationshas taken place Today’s computers can perform complex calculations and process largeamounts of data rapidly The engineering design and optimization processes benefit greatlyfrom this revolution because they require a large number of calculations Better systems cannow be designed by analyzing and optimizing various options in a short time This is highly

desirable because better designed systems cost less, have more capability, and are easy tomaintain and operate.

The design of systems can be formulated as problems of optimization in which a measure

of performance is to be optimized while satisfying all constraints Many numerical methods

of optimization have been developed and used to design better systems This text describesthe basic concepts of optimization methods and their applications to the design of engineer-ing systems Design process is emphasized rather than optimization theory Various theoremsare stated as results without rigorous proofs However, their implications from an engineer-ing point of view are studied and discussed in detail Optimization theory, numerical methods,and modern computer hardware and software can be used as tools to design better engineer-ing systems The text emphasizes this theme throughout

Any problem in which certain parameters need to be determined to satisfy constraints can

be formulated as an optimization problem Once this has been done, the concepts and themethods described in this text can be used to solve the problem Therefore, the optimiza-tion techniques are quite general, having a wide range of applicability in diverse fields Therange of applications is limited only by the imagination or ingenuity of the designers It isimpossible to discuss every application of optimization concepts and techniques in this introductory text However, using simple applications, we shall discuss concepts, funda-mental principles, and basic techniques that can be used in numerous applications Thestudent should understand them without getting bogged down with the notation, terminol-ogy, and details of the particular area of application

The design of many engineering systems can be a fairly complex process Many assumptionsmust be made to develop models that can be subjected to analysis by the available methodsand the models must be verified by experiments Many possibilities and factors must be

considered during the problem formulation phase Economic considerations play an

impor-tant role in designing cost-effective systems Introductory methods of economic analysisdescribed in Appendix A are useful in this regard To complete the design of an engineeringsystem, designers from different fields of engineering must usually cooperate For example,the design of a high-rise building involves designers from architectural, structural, mechan-ical, electrical, and environmental engineering as well as construction management experts.Design of a passenger car requires cooperation among structural, mechanical, automotive,

electrical, human factors, chemical, and hydraulics design engineers Thus, in an

interdisci-plinary environment considerable interaction is needed among various design teams to

com-plete the project For most applications the entire design project must be broken down intoseveral subproblems which are then treated independently Each of the subproblems can beposed as a problem of optimum design

The design of a system begins by analyzing various options Subsystems and their ponents are identified, designed, and tested This process results in a set of drawings, calcu-lations, and reports by which the system can be fabricated We shall use a systems engineering

com-model to describe the design process Although a complete discussion of this subject is

beyond the scope of the text, some basic concepts will be discussed using a simple blockdiagram

Design is an iterative process The designer’s experience, intuition, and ingenuity are

required in the design of systems in most fields of engineering (aerospace, automotive, civil,

chemical, industrial, electrical, mechanical, hydraulic, and transportation) Iterative implies analyzing several trial designs one after another until an acceptable design is obtained The

concept of trial designs is important to understand In the design process, the designer

2 INTRODUCTION TO OPTIMUM DESIGN

How do I

begin to

design a

system?

estimates a trial design of the system based on experience, intuition, or some mathematicalanalysis The trial design is 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 thebest Depending on the specifications, “best” can have different connotations for differentsystems In general, it implies cost-effective, efficient, reliable and durable systems Theprocess can require considerable interaction among teams of specialists from different disci-plines The basic concepts are described in the text to aid the engineer in designing systems

at the minimum cost and in the shortest amount of time

The design process should be a well organized activity To discuss it, we consider a system

evolution model shown in Fig 1-1 The process begins with the identification of a need which

may be conceived by engineers or nonengineers

The first step in the evolutionary process is to define precisely specifications for the system.

Considerable interaction between the engineer and the sponsor of the project is usually

nec-essary to quantify the system specifications Once these are identified, the task of designing

the system can begin

The second step in the process is to develop a preliminary design of the system Various

concepts for the system are studied Since this must be done in a relatively short time,

simplified models are used Various subsystems are identified and their preliminary designs

estimated Decisions made at this stage generally affect the final appearance and performance

of the system At the end of the preliminary design phase, a few promising concepts that needfurther analysis are identified

The third step in the process is to carry out a detailed design for all subsystems using an

iterative process To evaluate various possibilities, this must be done for all previously tified promising concepts The design parameters for the subsystems must be identified Thesystem performance requirements must be identified and satisfied The subsystems must bedesigned to maximize system worth or to minimize a measure of the cost Systematic opti-mization methods described in this text can aid the designer in accelerating the detailed designprocess At the end of the process, a description of the system is available in the form ofreports and drawings

iden-The fourth and fifth blocks of Fig 1-1 may or may not be necessary for all systems iden-They

involve fabrication of a prototype system and testing These steps are necessary when thesystem has to be mass produced or when human lives are involved Although these blocksmay appear to be the final steps in the design process, they are not because the system maynot perform according to specifications during the testing phase Therefore, specificationsmay have to be modified or other concepts may have to be studied In fact, this re-

examination may be necessary at any step in the design process It is for this reason that

feed-back loops are placed at every stage of the system evolution process, as shown in Fig 1-1.

The iterative process has to be continued until an acceptable system has evolved ing on the complexity of the system, the process may take a few days or several months

Depend-Introduction to Design 3

Final design

5.

System testing System

needs and

objective

Prototype system fabrication

4.

Detailed design

3.

Preliminary design

2.

System specification

1.

FIGURE 1-1 A system evolution model.

The previously described model is a simplified block diagram for system evolution Inactual practice, each block may have to be broken down into several sub-blocks to carry out

the studies properly and arrive at rational decisions The important point is that optimization

concepts and methods can help at every stage in the process The use of such methods along

with appropriate software can be extremely useful in studying various design possibilitiesrapidly These techniques can be useful during preliminary and detailed design phases as well

as for fabrication and testing Therefore, in this text, we discuss optimization methods andtheir use in the design process

At some stages in the design process, it may appear that the process can be completelyautomated, that the designer can be eliminated from the loop, and that optimization methodsand programs can be used as black boxes This may be true in some cases However, thedesign of a system is a creative process that can be quite complex It may be ill-defined and

a solution to the design problem may not exist Problem functions may not be defined incertain regions of the design space Thus, for most practical problems, designers play a keyrole in guiding the process to acceptable regions Designers must be an integral part of theprocess and use their intuition and judgment in obtaining the final design More details ofthe interactive design optimization process and the role of the designer are discussed inChapter 13

It is important to recognize differences between engineering analysis and design activities.

The analysis problem is concerned with determining the behavior of an existing system or atrial system being designed for a given task Determination of the behavior of the systemimplies calculation of its response under specified inputs Therefore, the sizes of various partsand their configurations are given for the analysis problem, i.e., the design of the system isknown On the other hand, the design process calculates the sizes and shapes of various parts

of the system to meet performance requirements The design of a system is a trial and errorprocedure We estimate a design and analyze it to see if it performs according to given spec-

ifications If it does, we have an acceptable (feasible) design, although we may still want to

change it to improve its performance If the trial design does not work, we need to change

it to come up with an acceptable system In both cases, we must be able to analyze designs

to make further decisions Thus, analysis capability must be available in the design process.This book is intended for use in all branches of engineering It is assumed throughout thatstudents understand analysis methods covered in undergraduate engineering statics and

physics courses However, we will not let the lack of analysis capability hinder the

under-standing of the systematic process of optimum design Equations for analysis of the system

will be given wherever needed

It is a challenge for engineers to design efficient and cost-effective systems without promising the integrity of the system The conventional design process depends on thedesigner’s intuition, experience, and skill This presence of a human element can sometimeslead to erroneous results in the synthesis of complex systems Figure 1-2(A) presents a self-explanatory flowchart for a conventional design process that involves the use of infor-mation gathered from one or more trial designs together with the designer’s experience andintuition

com-4 INTRODUCTION TO OPTIMUM DESIGN

Scarcity and the need for efficiency in today’s competitive world have forced engineers

to evince greater interest in economical and better designs The computer-aided design mization (CADO) process can help in this regard Figure 1-2(B) shows the optimum designprocess Both conventional and optimum design processes can be used at different stages

opti-of system evolution The main advantage in the conventional design process is that thedesigner’s experience and intuition can be used in making conceptual changes in the system

or to make additional specifications in the procedure For example, the designer can chooseeither a suspension bridge or an arched bridge, add or delete certain components of the struc-ture, and so on When it comes to detailed design, however, the conventional design processhas some disadvantages These include the treatment of complex constraints (such as limits

on vibration frequencies) as well as inputs (for example, when the structure is subjected to

a variety of loading conditions) In these cases, the designer would find it difficult to decidewhether to increase or decrease the size of a particular structural element to satisfy the constraints Furthermore, the conventional design process can lead to uneconomical designs

and can involve a lot of calendar time The optimum design process forces the designer to

identify explicitly a set of design variables, an objective function to be optimized, and the constraint functions for the system This rigorous formulation of the design problem helps

the designer gain a better understanding of the problem Proper mathematical formulation

of the design problem is a key to good solutions This topic is discussed in more detail in Chapter 2

Estimate initial design

Analyze the system

Check performance criteria

Is design satisfactory?

Change design based on experience/heuristics

Stop Yes

No (A)

FIGURE 1-2 Comparison of conventional and optimum design processes (A) Conventional

design process

The foregoing distinction between the two design approaches indicates that the tional design process is less formal An objective function that measures the performance ofthe system is not identified Trend information is not calculated to make design decisions forimprovement of the system Most decisions are made based on the designer’s experience and intuition In contrast, the optimization process is more formal, using trend information

conven-to make decisions However, the optimization process can substantially benefit from thedesigner’s experience and intuition in formulating the problem and identifying the criticalconstraints Thus, the best approach would be an optimum design process that is aided bythe designer’s interaction

Optimum design and optimal control of systems are two separate activities There are ous applications in which methods of optimum design are useful in designing systems Thereare many other applications where optimal control concepts are needed In addition, thereare some applications in which both optimum design and optimal control concepts must be

numer-used Sample applications include robotics and aerospace structures In this text, optimal

6 INTRODUCTION TO OPTIMUM 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 constraints

Does the design satisfy convergence criteria?

Change the design using

an optimization method

Stop Yes

No (B)

FIGURE 1-2 continued (B) Optimum design process.

control problems and methods are not described in detail However, fundamental differencesbetween the two activities are briefly explained in the sequel It turns out that optimal controlproblems can be transformed into optimum design problems and treated by the methodsdescribed in the text Thus, methods of optimum design are very powerful and should beclearly understood A simple optimal control problem is described in Chapter 14 and is solved

by the methods of optimum design

The optimal control problem consists of finding feedback controllers for a system toproduce the desired output The system has active elements that sense fluctuations in theoutput System controls are automatically adjusted to correct the situation and optimize ameasure of performance Thus, control problems are usually dynamic in nature In optimumdesign, on the other hand, we design the system and its elements to optimize an objectivefunction The system then remains fixed for its entire life

As an example, consider the cruise control mechanism in passenger cars The idea behindthis feedback system is to control fuel injection to maintain a constant speed Thus thesystem’s output is known, i.e., the cruising speed of the car The job of the control mecha-nism is to sense fluctuations in the speed depending upon road conditions and to adjust fuelinjection accordingly

To understand and be comfortable with the methods of optimum design, familiarity withlinear algebra (vector and matrix operations) and basic calculus is essential Operations of

linear algebra are described in Appendix B Students who are not comfortable with that

mate-rial must review it thoroughly Calculus of functions of single and multiple variables mustalso be understood These concepts are reviewed wherever they are needed In this section,

the standard terminology and notations used throughout the text are defined It is important

to understand and memorize these, because without them it will be difficult to follow the rest

of the text

1.5.1 Sets and Points

Since realistic systems generally involve several variables, it is necessary to define and utilize

some convenient and compact notation Set and vector notations serve this purpose quite well and are utilized throughout the text The terms vector and point are used interchangeably and

lowercase letters in boldface are used to denote them Upper case letters in boldface sent matrices.

repre-A point means an ordered list of numbers Thus, (x1, x2) is a point consisting of the two numbers; (x1, x2, , x n ) is a point consisting of the n numbers Such a point is often called

an n-tuple Each of the numbers is called a component of the (point) vector Thus, x1is the

first component, x2 is the second, and so forth The n components x1, x2, , x ncan be lected into a column vector as

col-(1.1a)

where the superscript T denotes the transpose of a vector or a matrix, a notation that is used

throughout the text (refer to Appendix B for a detailed discussion of vector and matrixalgebra) We shall also use the notation

x=È

Î

ÍÍÍÍ

x x x n

n T

1 2

to represent a point or vector in n-dimensional space This is called an n-vector.

In three-dimensional space, the vector x= [x1x2x3]T

represents a point P as shown in Fig.

1-3 Similarly, when there are n components in a vector, as in Eqs (1.1a) and (1.1b), x is

interpreted as a point in the n-dimensional real space denoted as R n

prop-must possess to be in the set S).

Members of a set are sometimes called elements If a point x is an element of the set S,

then we write x Œ S The expression “x Œ S” is read, “x is an element of (belongs to) S.”

Conversely, the expression “yœ S” is read, “y is not an element of (does not belong to) S.”

If all the elements of a set S are also elements of another set T, then S is said to be a

“subset of T.” Symbolically, we write S Ã T which is read as, “S is a subset of T,” or “S is

contained in T.” Alternatively, we say T is a superset of S, written as T … S.

As an example of a set S, consider a domain of the xl –x2plane enclosed by a circle ofradius 3 with the center at the point (4, 4), as shown in Fig 1-4 Mathematically, all pointswithin and on the circle can be expressed as

(1.3)

S={xŒR2 (x - ) +(x - ) £ }

1 2 2 2

Thus, the center of the circle (4, 4) is in the set S because it satisfies the inequality in

Eq (1.3) We write this as (4, 4) ΠS The origin of coordinates (0, 0) does not belong to the

set since it does not satisfy the inequality in Eq (1.3) We write this as (0, 0) œ S It can be

verified that the following points belong to the set: (3, 3), (2, 2), (3, 2), (6, 6) In fact, set S

has an infinite number of points Many other points are not in the set It can be verified thatthe following points are not in the set: (1, 1), (8, 8), (-1, 2)

1.5.2 Notation for Constraints

Constraints arise naturally in optimum design problems For example, material of the systemmust not fail, the demand must be met, resources must not be exceeded, and so on We shalldiscuss the constraints in more detail in Chapter 2 Here we discuss the terminology and nota-tions for the constraints

We have already encountered a constraint in Fig 1-4 that shows a set S of points within and on the circle of radius 3 The set S is defined by the following constraint:

A constraint of this form will be called a less than or equal to type It shall be

abbrevi-ated as “£ type.” Similarly, there are greater than or equal to type constraints, abbreviated

as “≥ type.” Both types are called inequality constraints.

1.5.3 Superscripts/Subscripts and Summation Notation

Later we will discuss a set of vectors, components of vectors, and multiplication of matricesand vectors To write such quantities in a convenient form, consistent and compact notations

must be used We define these notations here Superscripts are used to represent different

vectors and matrices For example, x (i)

represents the ith vector of a set, and A (k)

represents

the kth matrix Subscripts are used to represent components of vectors and matrices For example, x j is the jth component of x and a ij is the i–j element of matrix A Double subscripts

are used to denote elements of a matrix

To indicate the range of a subscript or superscript we use the notation

(1.4)

x i; i= 1to n

2 2 2

x2

x1

FIGURE 1-4 Geometrical representation for the set S = {x | (x1 - 4) 2+ (x2 - 4) 2 £ 9}.

This represents the numbers x1, x2, , x n Note that “i = 1 to n” represents the range for the

index i and is read, “i goes from 1 to n.” Similarly, a set of k vectors, each having n

com-ponents, will be represented as

(1.5)

This represents the k vectors x(l)

, x(2)

, , x(k)

It is important to note that subscript i in Eq.

(1.4) and superscript j in Eq (1.5) are free indices, i.e., they can be replaced by any other variable For example, Eq (1.4) can also be written as x j ; j = 1 to n and Eq (1.5) can be

written as x(i)

; i = 1 to k Note that the superscript j in Eq (1.5) does not represent the power

of x It is an index that represents the jth vector of a set of vectors.

We shall also use the summation notation quite frequently For example,

(1.6)will be written as

(1.7)

Also, multiplication of an n-dimensional vector x by an m ¥ n matrix A to obtain an

m-dimensional vector y, is written as

The sum on the right side of Eq (1.10) is said to be a linear combination of columns of the

matrix A with x j , j = 1 to n as multipliers of the linear combination Or, y is given as a linear

combination of columns of A (refer to Appendix B for further discussion on the linear

com-bination of vectors)

Occasionally, we will have to use the double summation notation For example,

assum-ing m = n and substituting yifrom Eq (1.9) into Eq (1.7), we obtain the double sum as

(1.11)

Note that the indices i and j in Eq (1.11) can be interchanged This is possible because c is

a scalar quantity, so its value is not affected by whether we first sum on i or j Equation

(1.11) can also be written in the matrix form as will be shown later

c x i a x ij j a x x

j n

i

n

ij i j j n

y=Ax

c x y i i i

n

=

=

Â1

1.5.4 Norm/Length of a Vector

If we let x and y be two n-dimensional vectors, then their dot product is defined as

(1.12)

Thus, the dot product is a sum of the product of corresponding elements of the vectors x and

y Two vectors are said to be orthogonal (normal) if their dot product is zero, i.e., x and y

are orthogonal if x · y= 0 If the vectors are not orthogonal, the angle between them can be

calculated from the definition of the dot product:

(1.13)where q is the angle between vectors x and y, and ||x|| represents the length of the vector x.

This is also called the norm of the vector (for a more general definition of the norm, refer to

Appendix B) The length of a vector x is defined as the square root of the sum of squares of

the components, i.e.,

We will deal with many functions of vector variables To distinguish between functions,

sub-scripts will be used Thus, the ith function is written as

(1.18)

If there are m functions g i (x); i = 1 to m, these will be represented in the vector form

(1.19)

Throughout the text it is assumed that all functions are continuous and at least twice

con-tinuously differentiable A function f (x) of n variables is called continuous at a point x* if

for any e > 0, there is a d > 0 such that

n

i

n

i i

n

ij j j

n

2 1

n

x y

1

Introduction to Design 11

whenever ||x - x*|| < d Thus, for all points x in a small neighborhood of the point x*, a

change in the function value from x* to x is small when the function is continuous A

con-tinuous function need not be differentiable Twice-concon-tinuous differentiability of a function

implies that it is not only differentiable two times but also that its second derivative is tinuous Figures 1.5(A) and 1.5(B) show continuous functions The function shown in Fig.1.5(A) is differentiable everywhere, whereas the function of Fig 1.5(B) is not differentiable

con-at points x1, x2, and x3 Figure 1-5(C) provides an example in which f is not a function because

it has infinite values at x1 Figure 1-5(D) provides an example of a discontinuous function.

As examples, f (x) = x3

and f (x) = sin x are continuous functions everywhere, and they are

also continuously differentiable However, the function f (x) = |x| is continuous everywhere

but not differentiable at x = 0

1.5.6 U.S.-British versus SI Units

The design problem formulation and the methods of optimization do not depend on the units

of measure used Thus, it does not matter which units are used in defining the problem.However, the final form of some of the analytical expressions for the problem does depend

on the units used In the text, we shall use both U.S.-British and SI units in examples andexercises Readers unfamiliar with either system of units should not feel at a disadvantagewhen reading and understanding the material It is simple to switch from one system of units

to the other To facilitate the conversion from U.S.-British to SI units or vice versa, Table 1-1 gives conversion factors for the most commonly used quantities For a complete list ofthe conversion factors, the ASTM (1980) publication can be consulted

x1

x1

x1

x2 x3 x

FIGURE 1-5 Continuous and discontinuous functions (A) Continuous function (B)

Continu-ous function (C) Not a function (D) DiscontinuContinu-ous function.

Introduction to Design 13

TABLE 1-1 Conversion Factors Between U.S.-British and SI Units

Bending moment or torque

pound force inch (lbf·in) Newton meter (N·m) 0.1129848 pound force foot (lbf·ft) Newton meter (N·m) 1.355818

mile (mi), International, nautical meter (m) 1852*

Mass

slug (lbf·s 2

(lbf/ft 2

(N/m 2

or Pa) 47.88026 pound/inch 2

14 INTRODUCTION TO OPTIMUM DESIGN

2 Optimum Design Problem Formulation

15

Upon completion of this chapter, you will be able to:

• Translate a descriptive statement of the design problem into a mathematical

statement for optimization using a five-step process

• Identify and define the problem’s design variables

• Identify a function for the problem that needs to be optimized

• Identify and define the problem’s constraints

It is generally accepted that the proper definition and formulation of a problem takes

roughly 50 percent of the total effort needed to solve it Therefore, it is critical to follow welldefined procedures for formulating design optimization problems It is generally assumed in

this text that various preliminary analyses have been completed and a detailed design of a

concept or a subproblem needs to be carried out Students should bear in mind that a

con-siderable number of analyses usually have to be performed before reaching this stage ofdesign optimization In this chapter, we describe the process of transforming the design of aselected system/subsystem into an optimum design problem using several simple and mod-erately complex applications More advanced applications are discussed in later chapters

The importance of properly formulating a design optimization problem must be stressed

because the optimum solution will only be as good as the formulation For example, if weforget to include a critical constraint in the formulation, the optimum solution will most likelyviolate it because optimization methods tend to exploit deficiencies in design models Also,

if we have too many constraints or if they are inconsistent, there may not be a solution to thedesign problem However, once the problem is properly formulated, good software is usually

available to solve it For most design optimization problems, we shall use the following

five-step formulation procedure:

Step 1: Project/problem statement

Step 2: Data and information collection

Step 3: Identification/definition of design variables

Step 4: Identification of a criterion to be optimized

Step 5: Identification of constraints