Multidisciplinary Design Optimization Supported by Knowledge Based Engineering

Jaroslaw Sobieszczanski-Sobieski NASA Langley Research Center, USA Alan Morris Cranfield University, UK Michel van Tooren University of South Carolina, USA

Preface xiii

Acknowledgment xv

Styles for Equations xvi

1 Introduction 1

1.1 Background 1

1.2 Aim of the Book 3

1.3 The Engineer in the Loop 3

1.4 Chapter Contents 4

1.4.1 Chapter 2: Modern Design and Optimization 4

1.4.2 Chapter 3: Searching the Constrained Design Space 4

1.4.3 Chapter 4: Direct Search Methods for Locating the Optimum of a Design Problem with a Single-Objective Function 5

1.4.4 Chapter 5: Guided Random Search and Network Techniques 5

1.4.5 Chapter 6: Optimizing Multiple-Objective Function Problems 6

1.4.6 Chapter 7: Sensitivity Analysis 6

1.4.7 Chapter 8: Multidisciplinary Design and Optimization Methods 7

1.4.8 Chapter 9: KBE 7

1.4.9 Chapter 10: Uncertainty-Based Multidisciplinary Design and Optimization 8

1.4.10 Chapter 11: Ways and Means for Control and Reduction of the Optimization Computational Cost and Elapsed Time 8

1.4.11 Appendix A: Implementation of KBE in Your MDO Case 9

1.4.12 Appendix B: Guide to Implementing an MDO System 9

2 Modern Design and Optimization 10

2.1 Background to Chapter 10

2.2 Nature and Realities of Modern Design 11

2.3 Modern Design and Optimization 12

2.3.1 Overview of the Design Process 13

2.3.2 Abstracting Design into a Mathematical Model 15

2.3.3 Mono-optimization 17

2.4 Migrating Optimization to Modern Design: The Role of MDO 20

2.4.1 Example of an Engineering System Optimization Problem 21

2.4.2 General Conclusions from the Wing Example 24

2.5 MDO’s Relation to Software Tool Requirements 25

2.5.1 Knowledge-Based Engineering 26

References 26

3 Constrained Design Space Search 27

3.1 Introduction 27

3.2 Defining the Optimization Problem 29

3.3 Characterization of the Optimizing Point 32

3.3.1 Curvature Constrained Problem 32

3.3.2 Vertex Constrained Problem 34

3.3.3 A Curvature and Vertex Constrained Problem 36

3.3.4 The Kuhn–Tucker Conditions 37

3.4 The Lagrangian and Duality 39

3.4.1 The Lagrangian 40

3.4.2 The Dual Problem 41

Appendix 3.A 44

References 46

4 Direct Search Methods for Locating the Optimum of a Design Problem with a Single-Objective Function 47

4.1 Introduction 47

4.2 The Fundamental Algorithm 48

4.3 Preliminary Considerations 49

4.3.1 Line Searches 50

4.3.2 Polynomial Searches 50

4.3.3 Discrete Point Line Search 51

4.3.4 Active Set Strategy and Constraint Satisfaction 53

4.4 Unconstrained Search Algorithms 54

4.4.1 Unconstrained First-Order Algorithm or Steepest Descent 55

4.4.2 Unconstrained Quadratic Search Method Employing Newton Steps 56

4.4.3 Variable Metric Search Methods 58

4.5 Sequential Unconstrained Minimization Techniques 59

4.5.1 Penalty Methods 60

4.5.2 Augmented Lagrangian Method 64

4.5.3 Simple Comparison and Comment on SUMT 64

4.5.4 Illustrative Examples 66

4.6 Constrained Algorithms 68

4.6.1 Constrained Steepest Descent Method 70

4.6.2 Linear Objective Function with Nonlinear Constraints 74

4.6.3 Sequential Quadratic Updating Using a Newton Step 78

4.7 Final Thoughts 79

References 79

5 Guided Random Search and Network Techniques 80

5.1 Guided Random Search Techniques (GRST) 80

5.1.1 Genetic Algorithms (GA) 81

5.1.2 Design Point Data Structure 81

5.1.3 Fitness Function 82

5.1.4 Constraints 87

5.1.5 Hybrid Algorithms 87

5.1.6 Considerations When Using a GA 87

5.1.7 Alternative to Genetic-Inspired Creation of Children 88

5.1.8 Alternatives to GA 88

5.1.9 Closing Remarks for GA 89

5.2 Artificial Neural Networks (ANN) 89

5.2.1 Neurons and Weights 91

5.2.2 Training via Gradient Calculation and Back-Propagation 93

5.2.3 Considerations on the Use of ANN 97

References 97

6 Optimizing Multiobjective Function Problems 98

6.1 Introduction 98

6.2 Salient Features of Multiobjective Optimization 99

6.3 Selected Algorithms for Multiobjective Optimization 102

6.4 Weighted Sum Procedure 104

6.5 ε-Constraint and Lexicographic Methods 108

6.6 Goal Programming 111

6.7 Min–Max Solution 111

6.8 Compromise Solution Equidistant to the Utopia Point 113

6.9 Genetic Algorithms and Artificial Neural Networks Solution Methods 113

6.9.1 GAs 114

6.9.2 ANN 114

6.10 Final Comment 115

References 115

7 Sensitivity Analysis 116

7.1 Analytical Method 116

7.1.1 Example 7.1 118

7.1.2 Example 7.2 121

7.2 Linear Governing Equations 122

7.3 Eigenvectors and Eigenvalues Sensitivities 124

7.3.1 Buckling as an Eigen-problem 125

7.3.2 Derivatives of Eigenvalues and Eigenvectors 125

7.3.3 Example 7.3 127

7.4 Higher Order and Directional Derivatives 129

7.5 Adjoint Equation Algorithm 131

7.6 Derivatives of Real-Valued Functions Obtained via Complex Numbers 133

7.7 System Sensitivity Analysis 135

7.7.1 Example 7.4 139

7.8 Example 144

7.9 System Sensitivity Analysis in Adjoint Formulation 145

7.10 Optimum Sensitivity Analysis 146

7.10.1 Lagrange Multiplier λ as a Shadow Price 149

7.11 Automatic Differentiation 150

7.12 Presenting Sensitivity as Logarithmic Derivatives 153

References 154

8 Multidisciplinary Design Optimization Architectures 155

8.1 Introduction 155

8.2 Consolidated Statement of a Multidisciplinary Optimization Problem 156

8.3 The MDO Terminology and Notation 158

8.3.1 Operands 159

8.3.2 Coupling Constraints 159

8.3.3 Operators 160

8.4 Decomposition of the Optimization Task into Subtasks 161

8.5 Structuring the Underlying Information 162

8.6 System Analysis (SA) 167

8.7 Evolving Engineering Design Process 170

8.8 Single-Level Design Optimizations (S-LDO) 173

8.8.1 Assessment 175

8.9 The Feasible Sequential Approach (FSA) 176

8.9.1 Implementation Options 177

8.10 Multidisciplinary Design Optimization (MDO) Methods 178

8.10.1 Collaborative Optimization (CO) 179

8.10.2 Bi-Level Integrated System Synthesis (BLISS) 189

8.10.3 BLISS Augmented with SM 192

8.11 Closure 199

8.11.1 Decomposition 199

8.11.2 Approximations and SM 200

8.11.3 Anatomy of a System 200

8.11.4 Interactions of the System and Its BBs 201

8.11.5 Intrinsic Limitations of Optimization in General 202

8.11.6 Optimization across a Choice of Different Design Concepts 202

8.11.7 Off-the-Shelf Commercial Software Frameworks 203

References 205

9 Knowledge Based Engineering 208

9.1 Introduction 208

9.2 KBE to Support MDO 209

9.3 What is KBE 210

9.4 When Can KBE Be Used 213

9.5 Role of KBE in the Development of Advanced MDO Systems 214

9.6 Principles and Characteristics of KBE Systems and KBE Languages 220

9.7 KBE Operators to Define Class and Object Hierarchies 222

9.7.1 An Example of a Product Model Definition in Four KBE Languages 226

9.8 The Rules of KBE 230

9.8.1 Logic Rules (or Conditional Expressions) 230

9.8.2 Math Rules 231

9.8.3 Geometry Manipulation Rules 232

9.8.4 Configuration Selection Rules (or Topology Rules) 234

9.8.5 Communication Rules 235

9.8.6 Beyond Classical KBS and CAD 236

9.9 KBE Methods to Develop MMG Applications 236

9.9.1 High-Level Primitives (HLPs) to Support Parametric Product Modeling 237

9.9.2 Capability Modules (CMs) to Support Analysis Preparation 238

9.10 Flexibility and Control: Dynamic Typing, Dynamic Class Instantiation, and Object Quantification 241

9.11 Declarative and Functional Coding Style 241

9.12 KBE Specific Features: Runtime Caching and Dependency Tracking 243

9.13 KBE Specific Features: Demand-Driven Evaluation 246

9.14 KBE Specific Features: Geometry Kernel Integration 247

9.14.1 How a KBE Language Interacts with a CAD Engine 248

9.15 CAD or KBE? 252

9.16 Evolution and Trends of KBE Technology 253

Acknowledgments 256

References 256

10 Uncertainty-Based Multidisciplinary Design Optimization 258

10.1 Introduction 258

10.2 Uncertainty-Based Multidisciplinary Design Optimization (UMDO) Preliminaries 259

10.2.1 Basic Concepts 259

10.2.2 General UMDO Process 263

10.3 Uncertainty Analysis 264

10.3.1 Monte Carlo Methods (MCS) 265

10.3.2 Taylor Series Approximation 266

10.3.3 Reliability Analysis 268

10.3.4 Decomposition-Based Uncertainty Analysis 271

10.4 Optimization under Uncertainty 272

10.4.1 Reliability Index Approach (RIA) and Performance Measure Approach (PMA) Methods 273

10.4.2 Single Level Algorithms (SLA) 275

10.4.3 Approximate Reliability Constraint Conversion Techniques 278

10.4.4 Decomposition-Based Method 280

10.5 Example 282

10.6 Conclusion 285

References 285

11 Ways and Means for Control and Reduction of the Optimization Computational Cost and Elapsed Time 287

11.1 Introduction 287

11.2 Computational Effort 288

11.3 Reducing the Function Nonlinearity by Introducing Intervening Variables 289

11.4 Reducing the Number of the Design Variables 289

11.4.1 Linking by Groups 290

11.5 Reducing the Number of Constraints Directly Visible to the Optimizer 292

11.5.1 Separation of Well-Satisfied Constraints from the Ones Violated or Nearly Violated 292

11.5.2 Representing a Set of Constraints by a Single Constraint 293

11.5.3 Replacing Constraints by Their Envelope in the Kreisselmeier–Steinhauser Formulation 293

11.6 Surrogate Methods (SMs) 298

11.7 Coordinated Use of High- and Low-Fidelity Mathematical Models in the Analysis 301

11.7.1 Improving LF Analysis by Infrequent Use of HF Analysis 301

11.7.2 Reducing the Number of Quantities Being Approximated 303

11.7.3 Placement of the Trial Points xT in the Design Space x 304

11.8 Design Space in n Dimensions May Be a Very Large Place 308

References 309

Appendix A Implementation of KBE in an MDO System 310

Appendix B Guide to Implementing an MDO System 349

Index 360