Introduction to Optimum Design

Introduction to Optimum Design
اسم المؤلف
Jasbir S. Arora
التاريخ
23 أكتوبر 2019
المشاهدات
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Introduction to Optimum Design
Second Edition
Jasbir S. Arora
The University of Iowa
Table of Contents
Preface ix
Chapter 1 Introduction to Design 1
1.1 The Design Process 2
1.2 Engineering Design versus Engineering Analysis 4
1.3 Conventional versus Optimum Design Process 4
1.4 Optimum Design versus Optimal Control 6
1.5 Basic Terminology and Notation 7
1.5.1 Sets and Points 7
1.5.2 Notation for Constraints 9
1.5.3 Superscripts/Subscripts and Summation Notation 9
1.5.4 Norm/Length of a Vector 11
1.5.5 Functions 11
1.5.6 U.S.-British versus SI Units 12
Chapter 2 Optimum Design Problem Formulation 15
2.1 The Problem Formulation Process 16
2.1.1 Step 1: Project/Problem Statement 16
2.1.2 Step 2: Data and Information Collection 16
2.1.3 Step 3: Identification/Definition of Design
Variables 16
2.1.4 Step 4: Identification of a Criterion to
Be Optimized 17
2.1.5 Step 5: Identification of Constraints 17
2.2 Design of a Can 18
2.3 Insulated Spherical Tank Design 20
2.4 Saw Mill Operation 22
2.5 Design of a Two-Bar Bracket 24
2.6 Design of a Cabinet 30
2.6.1 Formulation 1 for Cabinet Design 30
2.6.2 Formulation 2 for Cabinet Design 31
2.6.3 Formulation 3 for Cabinet Design 31
xi2.7 Minimum Weight Tubular Column Design 32
2.7.1 Formulation 1 for Column Design 33
2.7.2 Formulation 2 for Column Design 34
2.8 Minimum Cost Cylindrical Tank Design 35
2.9 Design of Coil Springs 36
2.10 Minimum Weight Design of a Symmetric
Three-Bar Truss 38
2.11 A General Mathematical Model for Optimum Design 41
2.11.1 Standard Design Optimization Model 42
2.11.2 Maximization Problem Treatment 43
2.11.3 Treatment of “Greater Than Type” Constraints 43
2.11.4 Discrete and Integer Design Variables 44
2.11.5 Feasible Set 45
2.11.6 Active/Inactive/Violated Constraints 45
Exercises for Chapter 2 46
Chapter 3 Graphical Optimization 55
3.1 Graphical Solution Process 55
3.1.1 Profit Maximization Problem 55
3.1.2 Step-by-Step Graphical Solution Procedure 56
3.2 Use of Mathematica for Graphical Optimization 60
3.2.1 Plotting Functions 61
3.2.2 Identification and Hatching of Infeasible
Region for an Inequality 62
3.2.3 Identification of Feasible Region 62
3.2.4 Plotting of Objective Function Contours 63
3.2.5 Identification of Optimum Solution 63
3.3 Use of MATLAB for Graphical Optimization 64
3.3.1 Plotting of Function Contours 64
3.3.2 Editing of Graph 64
3.4 Design Problem with Multiple Solutions 66
3.5 Problem with Unbounded Solution 66
3.6 Infeasible Problem 67
3.7 Graphical Solution for Minimum Weight
Tubular Column 69
3.8 Graphical Solution for a Beam Design Problem 69
Exercises for Chapter 3 72
Chapter 4 Optimum Design Concepts 83
4.1 Definitions of Global and Local Minima 84
4.1.1 Minimum 84
4.1.2 Existence of Minimum 89
4.2 Review of Some Basic Calculus Concepts 89
4.2.1 Gradient Vector 90
4.2.2 Hessian Matrix 92
4.2.3 Taylor’s Expansion 93
4.2.4 Quadratic Forms and Definite Matrices 96
4.2.5 Concept of Necessary and
Sufficient Conditions 102
xii Contents4.3 Unconstrained Optimum Design Problems 103
4.3.1 Concepts Related to Optimality Conditions 103
4.3.2 Optimality Conditions for Functions of
Single Variable 104
4.3.3 Optimality Conditions for Functions of
Several Variables 109
4.3.4 Roots of Nonlinear Equations Using Excel 116
4.4 Constrained Optimum Design Problems 119
4.4.1 Role of Constraints 119
4.4.2 Necessary Conditions: Equality Constraints 121
4.4.3 Necessary Conditions: Inequality Constraints—
Karush-Kuhn-Tucker (KKT) Conditions 128
4.4.4 Solution of KKT Conditions Using Excel 140
4.4.5 Solution of KKT Conditions Using MATLAB 141
4.5 Postoptimality Analysis: Physical Meaning of
Lagrange Multipliers 143
4.5.1 Effect of Changing Constraint Limits 143
4.5.2 Effect of Cost Function Scaling on
Lagrange Multipliers 146
4.5.3 Effect of Scaling a Constraint on Its
Lagrange Multiplier 147
4.5.4 Generalization of Constraint Variation
Sensitivity Result 148
4.6 Global Optimality 149
4.6.1 Convex Sets 149
4.6.2 Convex Functions 151
4.6.3 Convex Programming Problem 153
4.6.4 Transformation of a Constraint 156
4.6.5 Sufficient Conditions for Convex
Programming Problems 157
4.7 Engineering Design Examples 158
4.7.1 Design of a Wall Bracket 158
4.7.2 Design of a Rectangular Beam 162
Exercises for Chapter 4 166
Chapter 5 More on Optimum Design Concepts 175
5.1 Alternate Form of KKT Necessary Conditions 175
5.2 Irregular Points 178
5.3 Second-Order Conditions for
Constrained Optimization 179
5.4 Sufficiency Check for Rectangular Beam
Design Problem 184
Exercises for Chapter 5 185
Chapter 6 Linear Programming Methods for Optimum Design 191
6.1 Definition of a Standard Linear Programming
Problem 192
6.1.1 Linear Constraints 192
6.1.2 Unrestricted Variables 193
6.1.3 Standard LP Definition 193
Contents xiii6.2 Basic Concepts Related to Linear Programming
Problems 195
6.2.1 Basic Concepts 195
6.2.2 LP Terminology 198
6.2.3 Optimum Solution for LP Problems 201
6.3 Basic Ideas and Steps of the Simplex Method 201
6.3.1 The Simplex 202
6.3.2 Canonical Form/General Solution of Ax = b 202
6.3.3 Tableau 203
6.3.4 The Pivot Step 205
6.3.5 Basic Steps of the Simplex Method 206
6.3.6 Simplex Algorithm 211
6.4 Two-Phase Simplex Method—Artificial Variables 218
6.4.1 Artificial Variables 219
6.4.2 Artificial Cost Function 219
6.4.3 Definition of Phase I Problem 220
6.4.4 Phase I Algorithm 220
6.4.5 Phase II Algorithm 221
6.4.6 Degenerate Basic Feasible Solution 226
6.5 Postoptimality Analysis 228
6.5.1 Changes in Resource Limits 229
6.5.2 Ranging Right Side Parameters 235
6.5.3 Ranging Cost Coefficients 239
6.5.4 Changes in the Coefficient Matrix 241
6.6 Solution of LP Problems Using Excel Solver 243
Exercises for Chapter 6 246
Chapter 7 More on Linear Programming Methods for
Optimum Design 259
7.1 Derivation of the Simplex Method 259
7.1.1 Selection of a Basic Variable That Should
Become Nonbasic 259
7.1.2 Selection of a Nonbasic Variable That Should
Become Basic 260
7.2 Alternate Simplex Method 262
7.3 Duality in Linear Programming 263
7.3.1 Standard Primal LP 263
7.3.2 Dual LP Problem 264
7.3.3 Treatment of Equality Constraints 265
7.3.4 Alternate Treatment of Equality Constraints 266
7.3.5 Determination of Primal Solution from
Dual Solution 267
7.3.6 Use of Dual Tableau to Recover Primal Solution 271
7.3.7 Dual Variables as Lagrange Multipliers 273
Exercises for Chapter 7 275
Chapter 8 Numerical Methods for Unconstrained Optimum Design 277
8.1 General Concepts Related to Numerical Algorithms 278
8.1.1 A General Algorithm 279
8.1.2 Descent Direction and Descent Step 280
xiv Contents8.1.3 Convergence of Algorithms 282
8.1.4 Rate of Convergence 282
8.2 Basic Ideas and Algorithms for Step Size Determination 282
8.2.1 Definition of One-Dimensional
Minimization Subproblem 282
8.2.2 Analytical Method to Compute Step Size 283
8.2.3 Concepts Related to Numerical Methods to
Compute Step Size 285
8.2.4 Equal Interval Search 286
8.2.5 Alternate Equal Interval Search 288
8.2.6 Golden Section Search 289
8.3 Search Direction Determination: Steepest
Descent Method 293
8.4 Search Direction Determination: Conjugate
Gradient Method 296
Exercises for Chapter 8 300
Chapter 9 More on Numerical Methods for Unconstrained
Optimum Design 305
9.1 More on Step Size Determination 305
9.1.1 Polynomial Interpolation 306
9.1.2 Inaccurate Line Search 309
9.2 More on Steepest Descent Method 310
9.2.1 Properties of the Gradient Vector 310
9.2.2 Orthogonality of Steepest Descent Directions 314
9.3 Scaling of Design Variables 315
9.4 Search Direction Determination: Newton’s Method 318
9.4.1 Classical Newton’s Method 318
9.4.2 Modified Newton’s Method 319
9.4.3 Marquardt Modification 323
9.5 Search Direction Determination:
Quasi-Newton Methods 324
9.5.1 Inverse Hessian Updating: DFP Method 324
9.5.2 Direct Hessian Updating: BFGS Method 327
9.6 Engineering Applications of Unconstrained Methods 329
9.6.1 Minimization of Total Potential Energy 329
9.6.2 Solution of Nonlinear Equations 331
9.7 Solution of Constrained Problems Using Unconstrained
Optimization Methods 332
9.7.1 Sequential Unconstrained Minimization
Techniques 333
9.7.2 Multiplier (Augmented Lagrangian) Methods 334
Exercises for Chapter 9 335
Chapter 10 Numerical Methods for Constrained Optimum Design 339
10.1 Basic Concepts and Ideas 340
10.1.1 Basic Concepts Related to Algorithms for
Constrained Problems 340
10.1.2 Constraint Status at a Design Point 342
10.1.3 Constraint Normalization 343
Contents xv10.1.4 Descent Function 345
10.1.5 Convergence of an Algorithm 345
10.2 Linearization of Constrained Problem 346
10.3 Sequential Linear Programming Algorithm 352
10.3.1 The Basic Idea—Move Limits 352
10.3.2 An SLP Algorithm 353
10.3.3 SLP Algorithm: Some Observations 357
10.4 Quadratic Programming Subproblem 358
10.4.1 Definition of QP Subproblem 358
10.4.2 Solution of QP Subproblem 361
10.5 Constrained Steepest Descent Method 363
10.5.1 Descent Function 364
10.5.2 Step Size Determination 366
10.5.3 CSD Algorithm 368
10.5.4 CSD Algorithm: Some Observations 368
10.6 Engineering Design Optimization Using Excel Solver 369
Exercises for Chapter 10 373
Chapter 11 More on Numerical Methods for Constrained
Optimum Design 379
11.1 Potential Constraint Strategy 379
11.2 Quadratic Programming Problem 383
11.2.1 Definition of QP Problem 383
11.2.2 KKT Necessary Conditions for
the QP Problem 384
11.2.3 Transformation of KKT Conditions 384
11.2.4 Simplex Method for Solving QP Problem 385
11.3 Approximate Step Size Determination 388
11.3.1 The Basic Idea 388
11.3.2 Descent Condition 389
11.3.3 CSD Algorithm with Approximate Step Size 393
11.4 Constrained Quasi-Newton Methods 400
11.4.1 Derivation of Quadratic Programming
Subproblem 400
11.4.2 Quasi-Newton Hessian Approximation 403
11.4.3 Modified Constrained Steepest
Descent Algorithm 404
11.4.4 Observations on the Constrained
Quasi-Newton Methods 406
11.4.5 Descent Functions 406
11.5 Other Numerical Optimization Methods 407
11.5.1 Method of Feasible Directions 407
11.5.2 Gradient Projection Method 409
11.5.3 Generalized Reduced Gradient Method 410
Exercises for Chapter 11 411
Chapter 12 Introduction to Optimum Design with MATLAB 413
12.1 Introduction to Optimization Toolbox 413
12.1.1 Variables and Expressions 413
xvi Contents12.1.2 Scalar, Array, and Matrix Operations 414
12.1.3 Optimization Toolbox 414
12.2 Unconstrained Optimum Design Problems 415
12.3 Constrained Optimum Design Problems 418
12.4 Optimum Design Examples with MATLAB 420
12.4.1 Location of Maximum Shear Stress for Two
Spherical Bodies in Contact 420
12.4.2 Column Design for Minimum Mass 421
12.4.3 Flywheel Design for Minimum Mass 425
Exercises for Chapter 12 429
Chapter 13 Interactive Design Optimization 433
13.1 Role of Interaction in Design Optimization 434
13.1.1 What Is Interactive Design Optimization? 434
13.1.2 Role of Computers in Interactive
Design Optimization 434
13.1.3 Why Interactive Design Optimization? 435
13.2 Interactive Design Optimization Algorithms 436
13.2.1 Cost Reduction Algorithm 436
13.2.2 Constraint Correction Algorithm 440
13.2.3 Algorithm for Constraint Correction
at Constant Cost 442
13.2.4 Algorithm for Constraint Correction
at Specified Increase in Cost 445
13.2.5 Constraint Correction with Minimum Increase
in Cost 446
13.2.6 Observations on Interactive Algorithms 447
13.3 Desired Interactive Capabilities 448
13.3.1 Interactive Data Preparation 448
13.3.2 Interactive Capabilities 448
13.3.3 Interactive Decision Making 449
13.3.4 Interactive Graphics 450
13.4 Interactive Design Optimization Software 450
13.4.1 User Interface for IDESIGN 451
13.4.2 Capabilities of IDESIGN 453
13.5 Examples of Interactive Design Optimization 454
13.5.1 Formulation of Spring Design Problem 454
13.5.2 Optimum Solution for the Spring
Design Problem 455
13.5.3 Interactive Solution for Spring Design Problem 455
13.5.4 Use of Interactive Graphics 457
Exercises for Chapter 13 462
Chapter 14 Design Optimization Applications with
Implicit Functions 465
14.1 Formulation of Practical Design
Optimization Problems 466
14.1.1 General Guidelines 466
14.1.2 Example of a Practical Design
Optimization Problem 467
Contents xvii14.2 Gradient Evaluation for Implicit Functions 473
14.3 Issues in Practical Design Optimization 478
14.3.1 Selection of an Algorithm 478
14.3.2 Attributes of a Good Optimization Algorithm 478
14.4 Use of General-Purpose Software 479
14.4.1 Software Selection 480
14.4.2 Integration of an Application into GeneralPurpose Software 480
14.5 Optimum Design of Two-Member Frame with
Out-of-Plane Loads 481
14.6 Optimum Design of a Three-Bar Structure for Multiple
Performance Requirements 483
14.6.1 Symmetric Three-Bar Structure 483
14.6.2 Asymmetric Three-Bar Structure 484
14.6.3 Comparison of Solutions 490
14.7 Discrete Variable Optimum Design 491
14.7.1 Continuous Variable Optimization 492
14.7.2 Discrete Variable Optimization 492
14.8 Optimal Control of Systems by Nonlinear Programming 493
14.8.1 A Prototype Optimal Control Problem 493
14.8.2 Minimization of Error in State Variable 497
14.8.3 Minimum Control Effort Problem 503
14.8.4 Minimum Time Control Problem 505
14.8.5 Comparison of Three Formulations for
Optimal Control of System Motion 508
Exercises for Chapter 14 508
Chapter 15 Discrete Variable Optimum Design Concepts
and Methods 513
15.1 Basic Concepts and Definitions 514
15.1.1 Definition of Mixed Variable Optimum
Design Problem: MV-OPT 514
15.1.2 Classification of Mixed Variable Optimum
Design Problems 514
15.1.3 Overview of Solution Concepts 515
15.2 Branch and Bound Methods (BBM) 516
15.2.1 Basic BBM 517
15.2.2 BBM with Local Minimization 519
15.2.3 BBM for General MV-OPT 520
15.3 Integer Programming 521
15.4 Sequential Linearization Methods 522
15.5 Simulated Annealing 522
15.6 Dynamic Rounding-off Method 524
15.7 Neighborhood Search Method 525
15.8 Methods for Linked Discrete Variables 525
15.9 Selection of a Method 526
Exercises for Chapter 15 527
Chapter 16 Genetic Algorithms for Optimum Design 531
16.1 Basic Concepts and Definitions 532
16.2 Fundamentals of Genetic Algorithms 534
xviii Contents16.3 Genetic Algorithm for Sequencing-Type Problems 538
16.4 Applications 539
Exercises for Chapter 16 540
Chapter 17 Multiobjective Optimum Design Concepts and Methods 543
17.1 Problem Definition 543
17.2 Terminology and Basic Concepts 546
17.2.1 Criterion Space and Design Space 546
17.2.2 Solution Concepts 548
17.2.3 Preferences and Utility Functions 551
17.2.4 Vector Methods and Scalarization Methods 551
17.2.5 Generation of Pareto Optimal Set 551
17.2.6 Normalization of Objective Functions 552
17.2.7 Optimization Engine 552
17.3 Multiobjective Genetic Algorithms 552
17.4 Weighted Sum Method 555
17.5 Weighted Min-Max Method 556
17.6 Weighted Global Criterion Method 556
17.7 Lexicographic Method 558
17.8 Bounded Objective Function Method 558
17.9 Goal Programming 559
17.10 Selection of Methods 559
Exercises for Chapter 17 560
Chapter 18 Global Optimization Concepts and Methods for
Optimum Design 565
18.1 Basic Concepts of Solution Methods 565
18.1.1 Basic Concepts 565
18.1.2 Overview of Methods 567
18.2 Overview of Deterministic Methods 567
18.2.1 Covering Methods 568
18.2.2 Zooming Method 568
18.2.3 Methods of Generalized Descent 569
18.2.4 Tunneling Method 571
18.3 Overview of Stochastic Methods 572
18.3.1 Pure Random Search 573
18.3.2 Multistart Method 573
18.3.3 Clustering Methods 573
18.3.4 Controlled Random Search 575
18.3.5 Acceptance-Rejection Methods 578
18.3.6 Stochastic Integration 579
18.4 Two Local-Global Stochastic Methods 579
18.4.1 A Conceptual Local-Global Algorithm 579
18.4.2 Domain Elimination Method 580
18.4.3 Stochastic Zooming Method 582
18.4.4 Operations Analysis of the Methods 583
18.5 Numerical Performance of Methods 585
18.5.1 Summary of Features of Methods 585
18.5.2 Performance of Some Methods Using
Unconstrained Problems 586
Contents xix18.5.3 Performance of Stochastic Zooming and
Domain Elimination Methods 586
18.5.4 Global Optimization of Structural
Design Problems 587
Exercises for Chapter 18 588
Appendix A Economic Analysis 593
A.1 Time Value of Money 593
A.1.1 Cash Flow Diagrams 594
A.1.2 Basic Economic Formulas 594
A.2 Economic Bases for Comparison 598
A.2.1 Annual Base Comparisons 599
A.2.2 Present Worth Comparisons 601
Exercises for Appendix A 604
Appendix B Vector and Matrix Algebra 611
B.1 Definition of Matrices 611
B.2 Type of Matrices and Their Operations 613
B.2.1 Null Matrix 613
B.2.2 Vector 613
B.2.3 Addition of Matrices 613
B.2.4 Multiplication of Matrices 613
B.2.5 Transpose of a Matrix 615
B.2.6 Elementary Row–Column Operations 616
B.2.7 Equivalence of Matrices 616
B.2.8 Scalar Product–Dot Product of Vectors 616
B.2.9 Square Matrices 616
B.2.10 Partitioning of Matrices 617
B.3 Solution of n Linear Equations in n Unknowns 618
B.3.1 Linear Systems 618
B.3.2 Determinants 619
B.3.3 Gaussian Elimination Procedure 621
B.3.4 Inverse of a Matrix: Gauss-Jordan Elimination 625
B.4 Solution of m Linear Equations in n Unknowns 628
B.4.1 Rank of a Matrix 628
B.4.2 General Solution of m ¥ n Linear Equations 629
B.5 Concepts Related to a Set of Vectors 635
B.5.1 Linear Independence of a Set of Vectors 635
B.5.2 Vector Spaces 639
B.6 Eigenvalues and Eigenvectors 642
B.7 Norm and Condition Number of a Matrix 643
B.7.1 Norm of Vectors and Matrices 643
B.7.2 Condition Number of a Matrix 644
Exercises for Appendix B 645
Appendix C A Numerical Method for Solution of
Nonlinear Equations 647
C.1 Single Nonlinear Equation 647
C.2 Multiple Nonlinear Equations 650
Exercises for Appendix C 655
xx ContentsAppendix D Sample Computer Programs 657
D.1 Equal Interval Search 657
D.2 Golden Section Search 660
D.3 Steepest Descent Method 660
D.4 Modified Newton’s Method 669
References 675
Bibliography 683
Answers to Selected Problems 687
Index
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