Introduction to Optimum Design (eBook)

Jasbir Singh Arora (Autor)

eBook Download: EPUB

2004 | 2. Auflage
728 Seiten
Elsevier Science (Verlag)
978-0-08-047025-2 (ISBN)

Infos

Introduction to Optimum Design is intended for use in a first course on engineering design and optimization. Virtually any problem for which specific parameters need to be determined to satisfy constraints can be formulated as a design optimization problem. The concepts and methods described in the text are quite general and applicable to all such formulations. Inasmuch, the range of application of the optimum design methodology is almost limitless, constrained only by the imagination and ingenuity of the user.

Throughout the text, simple design problems involving two to three design variables and three to four constraints are solved in detail to illustrate fundamental concepts and basic ideas. The necessary results from optimization theory are stated and their implications are studied through application to engineering design problems. Theory and concepts of optimum design are explained only through examples and simple engineering applications. Several of the numerical procedures and concepts described in the text are useful in many other engineering courses and applications.

* Allows engineers involved in the design process to adapt optimum design concepts in their work using the material in the text.

* Basic concepts of optimality conditions and numerical methods are described with simple examples, making the material high teachable and learnable.

* Classroom-tested for many years to attain optimum pedagogical effectiveness.

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Optimization is a mathematical tool developed in the early 1960's used to find the most efficient and feasible solutions to an engineering problem. It can be used to find ideal shapes and physical configurations, ideal structural designs, maximum energy efficiency, and many other desired goals of engineering. This book is intended for use in a first course on engineering design and optimization. Material for the text has evolved over a period of several years and is based on classroom presentations for an undergraduate core course on the principles of design. Virtually any problem for which certain parameters need to be determined to satisfy constraints can be formulated as a design optimization problem. The concepts and methods described in the text are quite general and applicable to all such formulations. Inasmuch, the range of application of the optimum design methodology is almost limitless, constrained only by the imagination and ingenuity of the user. The book describes the basic concepts and techniques with only a few simple applications. Once they are clearly understood, they can be applied to many other advanced applications that are discussed in the text. - Allows engineers involved in the design process to adapt optimum design concepts in their work using the material in the text- Basic concepts of optimality conditions and numerical methods are described with simple examples, making the material high teachable and learnable- Classroom-tested for many years to attain optimum pedagogical effectiveness

Inhalt

Cover 1
Frontmatter 2
Half Title Page 2
Title Page 4
Copyright 5
Author Detail 6
Dedication Page 8
Preface 10
Contents 12
1. Introduction to Design 24
1.1 The Design Process 25
1.2 Engineering Design versus Engineering Analysis 27
1.3 Conventional versus Optimum Design Process 27
1.4 Optimum Design versus Optimal Control 29
1.5 Basic Terminology and Notation 30
2. Optimum Design Problem Formulation 38
2.1 The Problem Formulation Process 39
2.2 Design of a Can 41
2.3 Insulated Spherical Tank Design 43
2.4 Saw Mill Operation 45
2.5 Design of a Two-Bar Bracket 47
2.6 Design of a Cabinet 53
2.7 Minimum Weight Tubular Column Design 55
2.8 Minimum Cost Cylindrical Tank Design 58
2.9 Design of Coil Springs 59
2.10 Minimum Weight Design of a Symmetric Three-Bar Truss 61
2.11 A General Mathematical Model for Optimum Design 64
Exercises for Chapter 2 69
3. Graphical Optimization 78
3.1 Graphical Solution Process 78
3.2 Use of Mathematica for Graphical Optimization 83
3.3 Use of MATLAB for Graphical Optimization 87
3.4 Design Problem with Multiple Solutions 89
3.5 Problem with Unbounded Solution 89
3.6 Infeasible Problem 90
3.7 Graphical Solution for Minimum Weight Tubular Column 92
3.8 Graphical Solution for a Beam Design Problem 92
Exercises for Chapter 3 95
4. Optimum Design Concepts 106
4.1 Definitions of Global and Local Minima 107
4.2 Review of Some Basic Calculus Concepts 112
4.3 Unconstrained Optimum Design Problems 126
4.4 Constrained Optimum Design Problems 142
4.5 Postoptimality Analysis: Physical Meaning of Lagrange Multipliers 166
4.6 Global Optimality 172
4.7 Engineering Design Examples 181
Exercises for Chapter 4 189
5. More on Optimum Design Concepts 198
5.1 Alternate Form of KKT Necessary Conditions 198
5.2 Irregular Points 201
5.3 Second-Order Conditions for Constrained Optimization 202
5.4 Sufficiency Check for Rectangular Beam Design Problem 207
Exercises for Chapter 5 208
6. Linear Programming Methods for Optimum Design 214
6.1 Definition of a Standard Linear Programming Problem 215
6.2 Basic Concepts Related to Linear Programming Problems 218
6.3 Basic Ideas and Steps of the Simplex Method 224
6.4 Two-Phase Simplex Method—Artificial Variables 241
6.5 Postoptimality Analysis 251
6.6 Solution of LP Problems Using Excel Solver 266
Exercises for Chapter 6 269
7. More on Linear Programming Methods for Optimum Design 282
7.1 Derivation of the Simplex Method 282
7.2 Alternate Simplex Method 285
7.3 Duality in Linear Programming 286
Exercises for Chapter 7 298
8. Numerical Methods for Unconstrained Optimum Design 300
8.1 General Concepts Related to Numerical Algorithms 301
8.2 Basic Ideas and Algorithms for Step Size Determination 305
8.3 Search Direction Determination: Steepest Descent Method 316
8.4 Search Direction Determination: Conjugate Gradient Method 319
Exercises for Chapter 8 323
9. More on Numerical Methods for Unconstrained Optimum Design 328
9.1 More on Step Size Determination 328
9.2 More on Steepest Descent Method 333
9.3 Scaling of Design Variables 338
9.4 Search Direction Determination: Newton’s Method 341
9.5 Search Direction Determination: Quasi-Newton Methods 347
9.6 Engineering Applications of Unconstrained Methods 352
9.7 Solution of Constrained Problems Using Unconstrained Optimization Methods 355
Exercises for Chapter 9 358
10. Numerical Methods for Constrained Optimum Design 362
10.1 Basic Concepts and Ideas 363
10.2 Linearization of Constrained Problem 369
10.3 Sequential Linear Programming Algorithm 375
10.4 Quadratic Programming Subproblem 381
10.5 Constrained Steepest Descent Method 386
10.6 Engineering Design Optimization Using Excel Solver 392
Exercises for Chapter 10 396
11. More on Numerical Methods for Constrained Optimum Design 402
11.1 Potential Constraint Strategy 402
11.2 Quadratic Programming Problem 406
11.3 Approximate Step Size Determination 411
11.4 Constrained Quasi-Newton Methods 423
11.5 Other Numerical Optimization Methods 430
Exercises for Chapter 11 434
12. Introduction to Optimum Design with MATLAB 436
12.1 Introduction to Optimization Toolbox 436
12.2 Unconstrained Optimum Design Problems 438
12.3 Constrained Optimum Design Problems 441
12.4 Optimum Design Examples with MATLAB 443
Exercises for Chapter 12 452
13. Interactive Design Optimization 456
13.1 Role of Interaction in Design Optimization 457
13.2 Interactive Design Optimization Algorithms 459
13.3 Desired Interactive Capabilities 471
13.4 Interactive Design Optimization Software 473
13.5 Examples of Interactive Design Optimization 477
Exercises for Chapter 13 485
14. Design Optimization Applications with Implicit Functions 488
14.1 Formulation of Practical Design Optimization Problems 489
14.2 Gradient Evaluation for Implicit Functions 496
14.3 Issues in Practical Design Optimization 501
14.4 Use of General-Purpose Software 502
14.5 Optimum Design of a Two-Member Frame with Out-of-Plane Loads 504
14.6 Optimum Design of a Three-Bar Structure for Multiple Performance Requirements 506
14.7 Discrete Variable Optimum Design 514
14.8 Optimal Control of Systems by Nonlinear Programming 516
Exercises for Chapter 14 531
15. Discrete Variable Optimum Design Concepts and Methods 536
15.1 Basic Concepts and Definitions 537
15.2 Branch and Bound Methods (BBM) 539
15.3 Integer Programming 544
15.4 Sequential Linearization Methods 545
15.5 Simulated Annealing 545
15.6 Dynamic Rounding-off Method 547
15.7 Neighborhood Search Method 548
15.8 Methods for Linked Discrete Variables 548
15.9 Selection of a Method 549
Exercises for Chapter 15 550
16. Genetic Algorithms for Optimum Design 554
16.1 Basic Concepts and Definitions 555
16.2 Fundamentals of Genetic Algorithms 557
16.3 Genetic Algorithm for Sequencing-Type Problems 561
16.4 Applications 562
Exercises for Chapter 16 563
17. Multiobjective Optimum Design Concepts and Methods 566
17.1 Problem Definition 566
17.2 Terminology and Basic Concepts 569
17.3 Multiobjective Genetic Algorithms 575
17.4 Weighted Sum Method 578
17.5 Weighted Min-Max Method 579
17.6 Weighted Global Criterion Method 579
17.7 Lexicographic Method 581
17.8 Bounded Objective Function Method 581
17.9 Goal Programming 582
17.10 Selection of Methods 582
Exercises for Chapter 17 583
18. Global Optimization Concepts and Methods for Optimum Design 588
18.1 Basic Concepts of Solution Methods 588
18.2 Overview of Deterministic Methods 590
18.3 Overview of Stochastic Methods 595
18.4 Two Local-Global Stochastic Methods 602
18.5 Numerical Performance of Methods 608
Exercises for Chapter 18 611
Appendix A: Economic Analysis 616
A.1 Time Value of Money 616
A.2 Economic Bases for Comparison 621
Exercises for Appendix A 627
Appendix B: Vector and Matrix Algebra 634
B.1 Definition of Matrices 634
B.2 Type of Matrices and Their Operations 636
B.3 Solution of n Linear Equations in n Unknowns 641
B.4 Solution of m Linear Equations in n Unknowns 651
B.5 Concepts Related to a Set of Vectors 658
B.6 Eigenvalues and Eigenvectors 665
B.7 Norm and Condition Number of a Matrix 666
Exercises for Appendix B 668
Appendix C: A Numerical Method for Solution of Nonlinear Equations 670
C.1 Single Nonlinear Equation 670
C.2 Multiple Nonlinear Equations 673
Exercises for Appendix C 678
Appendix D: Sample Computer Programs 680
D.1 Equal Interval Search 680
D.2 Golden Section Search 683
D.3 Steepest Descent Method 683
D.4 Modified Newton’s Method 692
References 698
Bibliography 706
Answers to Selected Problems 710
Index 718

Textauszug

1 Introduction to Design

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 functions

Engineering consists of a number of well established activities, including analysis, design, fabrication, sales, research, and the development of systems. The subject of this text—the design of systems—is a major field in the engineering profession. The process of designing and fabricating systems has been developed over centuries. The existence of many complex systems, such as buildings, bridges, highways, automobiles, airplanes, space vehicles, and others, is an excellent testimonial for this process. However, the evolution of these systems has 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 systems performed the same or even more tasks, cost less, and were more efficient.

The preceding discussion indicates that several systems can usually accomplish the same task, and that some are better than others. For example, the purpose of a bridge is to provide continuity 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 been designed in detail.

The design of complex systems requires data processing and a large number of calculations. In the recent past, a revolution in computer technology and numerical computations has taken place. Today’s computers can perform complex calculations and process large amounts of data rapidly. The engineering design and optimization processes benefit greatly from this revolution because they require a large number of calculations. Better systems can now 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 to maintain 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 describes the basic concepts of optimization methods and their applications to the design of engineering systems. Design process is emphasized rather than optimization theory. Various theorems are stated as results without rigorous proofs. However, their implications from an engineering 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 engineering 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 the methods described in this text can be used to solve the problem. Therefore, the optimization techniques are quite general, having a wide range of applicability in diverse fields. The range of applications is limited only by the imagination or ingenuity of the designers. It is impossible to discuss every application of optimization concepts and techniques in this introductory text. However, using simple applications, we shall discuss concepts, fundamental principles, and basic techniques that can be used in numerous applications. The student should understand them without getting bogged down with the notation, terminology, and details of the particular area of application.

1.1 The Design Process

How do I begin to design a system?

The design of many engineering systems can be a fairly complex process. Many assumptions must be made to develop models that can be subjected to analysis by the available methods and the models must be verified by experiments. Many possibilities and factors must be considered during the problem formulation phase. Economic considerations play an important role in designing cost-effective systems. Introductory methods of economic analysis described in Appendix A are useful in this regard. To complete the design of an engineering system, designers from different fields of engineering must usually cooperate. For example, the design of a high-rise building involves designers from architectural, structural, mechanical, 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 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 independently. Each of the subproblems can be posed as a problem of optimum design.

The design of a system begins by analyzing various options. Subsystems and their components are identified, designed, and tested. This process results in a set of drawings, calculations, and reports by which the system can be fabricated. We shall use a systems engineering 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 block diagram.

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 estimates a trial design of the system based on experience, intuition, or some mathematical analysis. 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 the best. Depending on the specifications, “best” can have different connotations for different systems. In general, it implies cost-effective, efficient, reliable and durable systems. The process can require considerable interaction among teams of specialists from different disciplines. 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.

FIGURE 1-1 A system evolution model.

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 necessary 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 need further 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 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 can aid the designer in accelerating the detailed design process. At the end of the process, a description of the system is available in the form of reports and drawings.

The fourth and fifth blocks of Fig. 1-1 may or may not be necessary for all systems. They involve fabrication of a prototype system and testing. These steps are necessary when the system has to be mass produced or when human lives are involved. Although these blocks may appear to be the...

Produkt-Details

• Erscheint lt. Verlag: 2.6.2004
• Sprache: englisch
• Themenwelt: Sachbuch/Ratgeber
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Technik ► Bauwesen
• Themenwelt: Technik ► Maschinenbau
• ISBN-10: 0-08-047025-4 / 0080470254
• ISBN-13: 978-0-08-047025-2 / 9780080470252