Jaroslaw Sobieszczanski-Sobieski NASA Langley Research Center, USAAlan Morris Cranfield University, UKMichel van Tooren University of South Carolina, USA

• Preface xiiiAcknowledgment xvStyles for Equations xvi1 Introduction 11.1 Background 11.2 Aim of the Book 31.3 The Engineer in the Loop 31.4 Chapter Contents 41.4.1 Chapter 2: Modern Design and Optimization 41.4.2 Chapter 3: Searching the Constrained Design Space 41.4.3 Chapter 4: Direct Search Methods for Locating the Optimum of a Design Problem with a Single-Objective Function 51.4.4 Chapter 5: Guided Random Search and Network Techniques 51.4.5 Chapter 6: Optimizing Multiple-Objective Function Problems 61.4.6 Chapter 7: Sensitivity Analysis 61.4.7 Chapter 8: Multidisciplinary Design and Optimization Methods 71.4.8 Chapter 9: KBE 71.4.9 Chapter 10: Uncertainty-Based Multidisciplinary Design and Optimization 81.4.10 Chapter 11: Ways and Means for Control and Reduction of the Optimization Computational Cost and Elapsed Time 81.4.11 Appendix A: Implementation of KBE in Your MDO Case 91.4.12 Appendix B: Guide to Implementing an MDO System 92 Modern Design and Optimization 102.1 Background to Chapter 102.2 Nature and Realities of Modern Design 112.3 Modern Design and Optimization 122.3.1 Overview of the Design Process 132.3.2 Abstracting Design into a Mathematical Model 152.3.3 Mono-optimization 172.4 Migrating Optimization to Modern Design: The Role of MDO 202.4.1 Example of an Engineering System Optimization Problem 212.4.2 General Conclusions from the Wing Example 242.5 MDO’s Relation to Software Tool Requirements 252.5.1 Knowledge-Based Engineering 26References 263 Constrained Design Space Search 273.1 Introduction 273.2 Defining the Optimization Problem 293.3 Characterization of the Optimizing Point 323.3.1 Curvature Constrained Problem 323.3.2 Vertex Constrained Problem 343.3.3 A Curvature and Vertex Constrained Problem 363.3.4 The Kuhn–Tucker Conditions 373.4 The Lagrangian and Duality 393.4.1 The Lagrangian 403.4.2 The Dual Problem 41Appendix 3.A 44References 464 Direct Search Methods for Locating the Optimum of a Design Problem with a Single-Objective Function 474.1 Introduction 474.2 The Fundamental Algorithm 484.3 Preliminary Considerations 494.3.1 Line Searches 504.3.2 Polynomial Searches 504.3.3 Discrete Point Line Search 514.3.4 Active Set Strategy and Constraint Satisfaction 534.4 Unconstrained Search Algorithms 544.4.1 Unconstrained First-Order Algorithm or Steepest Descent 554.4.2 Unconstrained Quadratic Search Method Employing Newton Steps 564.4.3 Variable Metric Search Methods 584.5 Sequential Unconstrained Minimization Techniques 594.5.1 Penalty Methods 604.5.2 Augmented Lagrangian Method 644.5.3 Simple Comparison and Comment on SUMT 644.5.4 Illustrative Examples 664.6 Constrained Algorithms 684.6.1 Constrained Steepest Descent Method 704.6.2 Linear Objective Function with Nonlinear Constraints 744.6.3 Sequential Quadratic Updating Using a Newton Step 784.7 Final Thoughts 79References 795 Guided Random Search and Network Techniques 805.1 Guided Random Search Techniques (GRST) 805.1.1 Genetic Algorithms (GA) 815.1.2 Design Point Data Structure 815.1.3 Fitness Function 825.1.4 Constraints 875.1.5 Hybrid Algorithms 875.1.6 Considerations When Using a GA 875.1.7 Alternative to Genetic-Inspired Creation of Children 885.1.8 Alternatives to GA 885.1.9 Closing Remarks for GA 895.2 Artificial Neural Networks (ANN) 895.2.1 Neurons and Weights 915.2.2 Training via Gradient Calculation and Back-Propagation 935.2.3 Considerations on the Use of ANN 97References 976 Optimizing Multiobjective Function Problems 986.1 Introduction 986.2 Salient Features of Multiobjective Optimization 996.3 Selected Algorithms for Multiobjective Optimization 1026.4 Weighted Sum Procedure 1046.5 ε-Constraint and Lexicographic Methods 1086.6 Goal Programming 1116.7 Min–Max Solution 1116.8 Compromise Solution Equidistant to the Utopia Point 1136.9 Genetic Algorithms and Artificial Neural Networks Solution Methods 1136.9.1 GAs 1146.9.2 Ann 1146.10 Final Comment 115References 1157 Sensitivity Analysis 1167.1 Analytical Method 1167.1.1 Example 7.1 1187.1.2 Example 7.2 1217.2 Linear Governing Equations 1227.3 Eigenvectors and Eigenvalues Sensitivities 1247.3.1 Buckling as an Eigen-problem 1257.3.2 Derivatives of Eigenvalues and Eigenvectors 1257.3.3 Example 7.3 1277.4 Higher Order and Directional Derivatives 1297.5 Adjoint Equation Algorithm 1317.6 Derivatives of Real-Valued Functions Obtained via Complex Numbers 1337.7 System Sensitivity Analysis 1357.7.1 Example 7.4 1397.8 Example 1447.9 System Sensitivity Analysis in Adjoint Formulation 1457.10 Optimum Sensitivity Analysis 1467.10.1 Lagrange Multiplier λ as a Shadow Price 1497.11 Automatic Differentiation 1507.12 Presenting Sensitivity as Logarithmic Derivatives 153References 1548 Multidisciplinary Design Optimization Architectures 1558.1 Introduction 1558.2 Consolidated Statement of a Multidisciplinary Optimization Problem 1568.3 The MDO Terminology and Notation 1588.3.1 Operands 1598.3.2 Coupling Constraints 1598.3.3 Operators 1608.4 Decomposition of the Optimization Task into Subtasks 1618.5 Structuring the Underlying Information 1628.6 System Analysis (SA) 1678.7 Evolving Engineering Design Process 1708.8 Single-Level Design Optimizations (S-LDO) 1738.8.1 Assessment 1758.9 The Feasible Sequential Approach (FSA) 1768.9.1 Implementation Options 1778.10 Multidisciplinary Design Optimization (MDO) Methods 1788.10.1 Collaborative Optimization (CO) 1798.10.2 Bi-Level Integrated System Synthesis (BLISS) 1898.10.3 BLISS Augmented with SM 1928.11 Closure 1998.11.1 Decomposition 1998.11.2 Approximations and SM 2008.11.3 Anatomy of a System 2008.11.4 Interactions of the System and Its BBs 2018.11.5 Intrinsic Limitations of Optimization in General 2028.11.6 Optimization across a Choice of Different Design Concepts 2028.11.7 Off-the-Shelf Commercial Software Frameworks 203References 2059 Knowledge Based Engineering 2089.1 Introduction 2089.2 KBE to Support MDO 2099.3 What is KBE 2109.4 When Can KBE Be Used 2139.5 Role of KBE in the Development of Advanced MDO Systems 2149.6 Principles and Characteristics of KBE Systems and KBE Languages 2209.7 KBE Operators to Define Class and Object Hierarchies 2229.7.1 An Example of a Product Model Definition in Four KBE Languages 2269.8 The Rules of KBE 2309.8.1 Logic Rules (or Conditional Expressions) 2309.8.2 Math Rules 2319.8.3 Geometry Manipulation Rules 2329.8.4 Configuration Selection Rules (or Topology Rules) 2349.8.5 Communication Rules 2359.8.6 Beyond Classical KBS and CAD 2369.9 KBE Methods to Develop MMG Applications 2369.9.1 High-Level Primitives (HLPs) to Support Parametric Product Modeling 2379.9.2 Capability Modules (CMs) to Support Analysis Preparation 2389.10 Flexibility and Control: Dynamic Typing, Dynamic Class Instantiation, and Object Quantification 2419.11 Declarative and Functional Coding Style 2419.12 KBE Specific Features: Runtime Caching and Dependency Tracking 2439.13 KBE Specific Features: Demand-Driven Evaluation 2469.14 KBE Specific Features: Geometry Kernel Integration 2479.14.1 How a KBE Language Interacts with a CAD Engine 2489.15 CAD or KBE? 2529.16 Evolution and Trends of KBE Technology 253Acknowledgments 256References 25610 Uncertainty-Based Multidisciplinary Design Optimization 25810.1 Introduction 25810.2 Uncertainty-Based Multidisciplinary Design Optimization (UMDO) Preliminaries 25910.2.1 Basic Concepts 25910.2.2 General UMDO Process 26310.3 Uncertainty Analysis 26410.3.1 Monte Carlo Methods (MCS) 26510.3.2 Taylor Series Approximation 26610.3.3 Reliability Analysis 26810.3.4 Decomposition-Based Uncertainty Analysis 27110.4 Optimization under Uncertainty 27210.4.1 Reliability Index Approach (RIA) and Performance Measure Approach (PMA) Methods 27310.4.2 Single Level Algorithms (SLA) 27510.4.3 Approximate Reliability Constraint Conversion Techniques 27810.4.4 Decomposition-Based Method 28010.5 Example 28210.6 Conclusion 285References 28511 Ways and Means for Control and Reduction of the Optimization Computational Cost and Elapsed Time 28711.1 Introduction 28711.2 Computational Effort 28811.3 Reducing the Function Nonlinearity by Introducing Intervening Variables 28911.4 Reducing the Number of the Design Variables 28911.4.1 Linking by Groups 29011.5 Reducing the Number of Constraints Directly Visible to the Optimizer 29211.5.1 Separation of Well-Satisfied Constraints from the Ones Violated or Nearly Violated 29211.5.2 Representing a Set of Constraints by a Single Constraint 29311.5.3 Replacing Constraints by Their Envelope in the Kreisselmeier–Steinhauser Formulation 29311.6 Surrogate Methods (SMs) 29811.7 Coordinated Use of High- and Low-Fidelity Mathematical Models in the Analysis 30111.7.1 Improving LF Analysis by Infrequent Use of HF Analysis 30111.7.2 Reducing the Number of Quantities Being Approximated 30311.7.3 Placement of the Trial Points X T in the Design Space X 30411.8 Design Space in n Dimensions May Be a Very Large Place 308References 309Appendix A Implementation of KBE in an MDO System 310Appendix B Guide to Implementing an MDO System 349Index 360

Jaroslaw Sobieszczanski-Sobieski NASA Langley Research Center, USAAlan Morris Cranfield University, UKMichel van Tooren University of South Carolina, USA

• Preface xiiiAcknowledgment xvStyles for Equations xvi1 Introduction 11.1 Background 11.2 Aim of the Book 31.3 The Engineer in the Loop 31.4 Chapter Contents 41.4.1 Chapter 2: Modern Design and Optimization 41.4.2 Chapter 3: Searching the Constrained Design Space 41.4.3 Chapter 4: Direct Search Methods for Locating the Optimum of a Design Problem with a Single-Objective Function 51.4.4 Chapter 5: Guided Random Search and Network Techniques 51.4.5 Chapter 6: Optimizing Multiple-Objective Function Problems 61.4.6 Chapter 7: Sensitivity Analysis 61.4.7 Chapter 8: Multidisciplinary Design and Optimization Methods 71.4.8 Chapter 9: KBE 71.4.9 Chapter 10: Uncertainty-Based Multidisciplinary Design and Optimization 81.4.10 Chapter 11: Ways and Means for Control and Reduction of the Optimization Computational Cost and Elapsed Time 81.4.11 Appendix A: Implementation of KBE in Your MDO Case 91.4.12 Appendix B: Guide to Implementing an MDO System 92 Modern Design and Optimization 102.1 Background to Chapter 102.2 Nature and Realities of Modern Design 112.3 Modern Design and Optimization 122.3.1 Overview of the Design Process 132.3.2 Abstracting Design into a Mathematical Model 152.3.3 Mono-optimization 172.4 Migrating Optimization to Modern Design: The Role of MDO 202.4.1 Example of an Engineering System Optimization Problem 212.4.2 General Conclusions from the Wing Example 242.5 MDO’s Relation to Software Tool Requirements 252.5.1 Knowledge-Based Engineering 26References 263 Constrained Design Space Search 273.1 Introduction 273.2 Defining the Optimization Problem 293.3 Characterization of the Optimizing Point 323.3.1 Curvature Constrained Problem 323.3.2 Vertex Constrained Problem 343.3.3 A Curvature and Vertex Constrained Problem 363.3.4 The Kuhn–Tucker Conditions 373.4 The Lagrangian and Duality 393.4.1 The Lagrangian 403.4.2 The Dual Problem 41Appendix 3.A 44References 464 Direct Search Methods for Locating the Optimum of a Design Problem with a Single-Objective Function 474.1 Introduction 474.2 The Fundamental Algorithm 484.3 Preliminary Considerations 494.3.1 Line Searches 504.3.2 Polynomial Searches 504.3.3 Discrete Point Line Search 514.3.4 Active Set Strategy and Constraint Satisfaction 534.4 Unconstrained Search Algorithms 544.4.1 Unconstrained First-Order Algorithm or Steepest Descent 554.4.2 Unconstrained Quadratic Search Method Employing Newton Steps 564.4.3 Variable Metric Search Methods 584.5 Sequential Unconstrained Minimization Techniques 594.5.1 Penalty Methods 604.5.2 Augmented Lagrangian Method 644.5.3 Simple Comparison and Comment on SUMT 644.5.4 Illustrative Examples 664.6 Constrained Algorithms 684.6.1 Constrained Steepest Descent Method 704.6.2 Linear Objective Function with Nonlinear Constraints 744.6.3 Sequential Quadratic Updating Using a Newton Step 784.7 Final Thoughts 79References 795 Guided Random Search and Network Techniques 805.1 Guided Random Search Techniques (GRST) 805.1.1 Genetic Algorithms (GA) 815.1.2 Design Point Data Structure 815.1.3 Fitness Function 825.1.4 Constraints 875.1.5 Hybrid Algorithms 875.1.6 Considerations When Using a GA 875.1.7 Alternative to Genetic-Inspired Creation of Children 885.1.8 Alternatives to GA 885.1.9 Closing Remarks for GA 895.2 Artificial Neural Networks (ANN) 895.2.1 Neurons and Weights 915.2.2 Training via Gradient Calculation and Back-Propagation 935.2.3 Considerations on the Use of ANN 97References 976 Optimizing Multiobjective Function Problems 986.1 Introduction 986.2 Salient Features of Multiobjective Optimization 996.3 Selected Algorithms for Multiobjective Optimization 1026.4 Weighted Sum Procedure 1046.5 ε-Constraint and Lexicographic Methods 1086.6 Goal Programming 1116.7 Min–Max Solution 1116.8 Compromise Solution Equidistant to the Utopia Point 1136.9 Genetic Algorithms and Artificial Neural Networks Solution Methods 1136.9.1 GAs 1146.9.2 Ann 1146.10 Final Comment 115References 1157 Sensitivity Analysis 1167.1 Analytical Method 1167.1.1 Example 7.1 1187.1.2 Example 7.2 1217.2 Linear Governing Equations 1227.3 Eigenvectors and Eigenvalues Sensitivities 1247.3.1 Buckling as an Eigen-problem 1257.3.2 Derivatives of Eigenvalues and Eigenvectors 1257.3.3 Example 7.3 1277.4 Higher Order and Directional Derivatives 1297.5 Adjoint Equation Algorithm 1317.6 Derivatives of Real-Valued Functions Obtained via Complex Numbers 1337.7 System Sensitivity Analysis 1357.7.1 Example 7.4 1397.8 Example 1447.9 System Sensitivity Analysis in Adjoint Formulation 1457.10 Optimum Sensitivity Analysis 1467.10.1 Lagrange Multiplier λ as a Shadow Price 1497.11 Automatic Differentiation 1507.12 Presenting Sensitivity as Logarithmic Derivatives 153References 1548 Multidisciplinary Design Optimization Architectures 1558.1 Introduction 1558.2 Consolidated Statement of a Multidisciplinary Optimization Problem 1568.3 The MDO Terminology and Notation 1588.3.1 Operands 1598.3.2 Coupling Constraints 1598.3.3 Operators 1608.4 Decomposition of the Optimization Task into Subtasks 1618.5 Structuring the Underlying Information 1628.6 System Analysis (SA) 1678.7 Evolving Engineering Design Process 1708.8 Single-Level Design Optimizations (S-LDO) 1738.8.1 Assessment 1758.9 The Feasible Sequential Approach (FSA) 1768.9.1 Implementation Options 1778.10 Multidisciplinary Design Optimization (MDO) Methods 1788.10.1 Collaborative Optimization (CO) 1798.10.2 Bi-Level Integrated System Synthesis (BLISS) 1898.10.3 BLISS Augmented with SM 1928.11 Closure 1998.11.1 Decomposition 1998.11.2 Approximations and SM 2008.11.3 Anatomy of a System 2008.11.4 Interactions of the System and Its BBs 2018.11.5 Intrinsic Limitations of Optimization in General 2028.11.6 Optimization across a Choice of Different Design Concepts 2028.11.7 Off-the-Shelf Commercial Software Frameworks 203References 2059 Knowledge Based Engineering 2089.1 Introduction 2089.2 KBE to Support MDO 2099.3 What is KBE 2109.4 When Can KBE Be Used 2139.5 Role of KBE in the Development of Advanced MDO Systems 2149.6 Principles and Characteristics of KBE Systems and KBE Languages 2209.7 KBE Operators to Define Class and Object Hierarchies 2229.7.1 An Example of a Product Model Definition in Four KBE Languages 2269.8 The Rules of KBE 2309.8.1 Logic Rules (or Conditional Expressions) 2309.8.2 Math Rules 2319.8.3 Geometry Manipulation Rules 2329.8.4 Configuration Selection Rules (or Topology Rules) 2349.8.5 Communication Rules 2359.8.6 Beyond Classical KBS and CAD 2369.9 KBE Methods to Develop MMG Applications 2369.9.1 High-Level Primitives (HLPs) to Support Parametric Product Modeling 2379.9.2 Capability Modules (CMs) to Support Analysis Preparation 2389.10 Flexibility and Control: Dynamic Typing, Dynamic Class Instantiation, and Object Quantification 2419.11 Declarative and Functional Coding Style 2419.12 KBE Specific Features: Runtime Caching and Dependency Tracking 2439.13 KBE Specific Features: Demand-Driven Evaluation 2469.14 KBE Specific Features: Geometry Kernel Integration 2479.14.1 How a KBE Language Interacts with a CAD Engine 2489.15 CAD or KBE? 2529.16 Evolution and Trends of KBE Technology 253Acknowledgments 256References 25610 Uncertainty-Based Multidisciplinary Design Optimization 25810.1 Introduction 25810.2 Uncertainty-Based Multidisciplinary Design Optimization (UMDO) Preliminaries 25910.2.1 Basic Concepts 25910.2.2 General UMDO Process 26310.3 Uncertainty Analysis 26410.3.1 Monte Carlo Methods (MCS) 26510.3.2 Taylor Series Approximation 26610.3.3 Reliability Analysis 26810.3.4 Decomposition-Based Uncertainty Analysis 27110.4 Optimization under Uncertainty 27210.4.1 Reliability Index Approach (RIA) and Performance Measure Approach (PMA) Methods 27310.4.2 Single Level Algorithms (SLA) 27510.4.3 Approximate Reliability Constraint Conversion Techniques 27810.4.4 Decomposition-Based Method 28010.5 Example 28210.6 Conclusion 285References 28511 Ways and Means for Control and Reduction of the Optimization Computational Cost and Elapsed Time 28711.1 Introduction 28711.2 Computational Effort 28811.3 Reducing the Function Nonlinearity by Introducing Intervening Variables 28911.4 Reducing the Number of the Design Variables 28911.4.1 Linking by Groups 29011.5 Reducing the Number of Constraints Directly Visible to the Optimizer 29211.5.1 Separation of Well-Satisfied Constraints from the Ones Violated or Nearly Violated 29211.5.2 Representing a Set of Constraints by a Single Constraint 29311.5.3 Replacing Constraints by Their Envelope in the Kreisselmeier–Steinhauser Formulation 29311.6 Surrogate Methods (SMs) 29811.7 Coordinated Use of High- and Low-Fidelity Mathematical Models in the Analysis 30111.7.1 Improving LF Analysis by Infrequent Use of HF Analysis 30111.7.2 Reducing the Number of Quantities Being Approximated 30311.7.3 Placement of the Trial Points X T in the Design Space X 30411.8 Design Space in n Dimensions May Be a Very Large Place 308References 309Appendix A Implementation of KBE in an MDO System 310Appendix B Guide to Implementing an MDO System 349Index 360

Jaroslaw Sobieszczanski-Sobieski NASA Langley Research Center, USAAlan Morris Cranfield University, UKMichel van Tooren University of South Carolina, USA

• Preface xiiiAcknowledgment xvStyles for Equations xvi1 Introduction 11.1 Background 11.2 Aim of the Book 31.3 The Engineer in the Loop 31.4 Chapter Contents 41.4.1 Chapter 2: Modern Design and Optimization 41.4.2 Chapter 3: Searching the Constrained Design Space 41.4.3 Chapter 4: Direct Search Methods for Locating the Optimum of a Design Problem with a Single-Objective Function 51.4.4 Chapter 5: Guided Random Search and Network Techniques 51.4.5 Chapter 6: Optimizing Multiple-Objective Function Problems 61.4.6 Chapter 7: Sensitivity Analysis 61.4.7 Chapter 8: Multidisciplinary Design and Optimization Methods 71.4.8 Chapter 9: KBE 71.4.9 Chapter 10: Uncertainty-Based Multidisciplinary Design and Optimization 81.4.10 Chapter 11: Ways and Means for Control and Reduction of the Optimization Computational Cost and Elapsed Time 81.4.11 Appendix A: Implementation of KBE in Your MDO Case 91.4.12 Appendix B: Guide to Implementing an MDO System 92 Modern Design and Optimization 102.1 Background to Chapter 102.2 Nature and Realities of Modern Design 112.3 Modern Design and Optimization 122.3.1 Overview of the Design Process 132.3.2 Abstracting Design into a Mathematical Model 152.3.3 Mono-optimization 172.4 Migrating Optimization to Modern Design: The Role of MDO 202.4.1 Example of an Engineering System Optimization Problem 212.4.2 General Conclusions from the Wing Example 242.5 MDO’s Relation to Software Tool Requirements 252.5.1 Knowledge-Based Engineering 26References 263 Constrained Design Space Search 273.1 Introduction 273.2 Defining the Optimization Problem 293.3 Characterization of the Optimizing Point 323.3.1 Curvature Constrained Problem 323.3.2 Vertex Constrained Problem 343.3.3 A Curvature and Vertex Constrained Problem 363.3.4 The Kuhn–Tucker Conditions 373.4 The Lagrangian and Duality 393.4.1 The Lagrangian 403.4.2 The Dual Problem 41Appendix 3.A 44References 464 Direct Search Methods for Locating the Optimum of a Design Problem with a Single-Objective Function 474.1 Introduction 474.2 The Fundamental Algorithm 484.3 Preliminary Considerations 494.3.1 Line Searches 504.3.2 Polynomial Searches 504.3.3 Discrete Point Line Search 514.3.4 Active Set Strategy and Constraint Satisfaction 534.4 Unconstrained Search Algorithms 544.4.1 Unconstrained First-Order Algorithm or Steepest Descent 554.4.2 Unconstrained Quadratic Search Method Employing Newton Steps 564.4.3 Variable Metric Search Methods 584.5 Sequential Unconstrained Minimization Techniques 594.5.1 Penalty Methods 604.5.2 Augmented Lagrangian Method 644.5.3 Simple Comparison and Comment on SUMT 644.5.4 Illustrative Examples 664.6 Constrained Algorithms 684.6.1 Constrained Steepest Descent Method 704.6.2 Linear Objective Function with Nonlinear Constraints 744.6.3 Sequential Quadratic Updating Using a Newton Step 784.7 Final Thoughts 79References 795 Guided Random Search and Network Techniques 805.1 Guided Random Search Techniques (GRST) 805.1.1 Genetic Algorithms (GA) 815.1.2 Design Point Data Structure 815.1.3 Fitness Function 825.1.4 Constraints 875.1.5 Hybrid Algorithms 875.1.6 Considerations When Using a GA 875.1.7 Alternative to Genetic-Inspired Creation of Children 885.1.8 Alternatives to GA 885.1.9 Closing Remarks for GA 895.2 Artificial Neural Networks (ANN) 895.2.1 Neurons and Weights 915.2.2 Training via Gradient Calculation and Back-Propagation 935.2.3 Considerations on the Use of ANN 97References 976 Optimizing Multiobjective Function Problems 986.1 Introduction 986.2 Salient Features of Multiobjective Optimization 996.3 Selected Algorithms for Multiobjective Optimization 1026.4 Weighted Sum Procedure 1046.5 ε-Constraint and Lexicographic Methods 1086.6 Goal Programming 1116.7 Min–Max Solution 1116.8 Compromise Solution Equidistant to the Utopia Point 1136.9 Genetic Algorithms and Artificial Neural Networks Solution Methods 1136.9.1 GAs 1146.9.2 Ann 1146.10 Final Comment 115References 1157 Sensitivity Analysis 1167.1 Analytical Method 1167.1.1 Example 7.1 1187.1.2 Example 7.2 1217.2 Linear Governing Equations 1227.3 Eigenvectors and Eigenvalues Sensitivities 1247.3.1 Buckling as an Eigen-problem 1257.3.2 Derivatives of Eigenvalues and Eigenvectors 1257.3.3 Example 7.3 1277.4 Higher Order and Directional Derivatives 1297.5 Adjoint Equation Algorithm 1317.6 Derivatives of Real-Valued Functions Obtained via Complex Numbers 1337.7 System Sensitivity Analysis 1357.7.1 Example 7.4 1397.8 Example 1447.9 System Sensitivity Analysis in Adjoint Formulation 1457.10 Optimum Sensitivity Analysis 1467.10.1 Lagrange Multiplier λ as a Shadow Price 1497.11 Automatic Differentiation 1507.12 Presenting Sensitivity as Logarithmic Derivatives 153References 1548 Multidisciplinary Design Optimization Architectures 1558.1 Introduction 1558.2 Consolidated Statement of a Multidisciplinary Optimization Problem 1568.3 The MDO Terminology and Notation 1588.3.1 Operands 1598.3.2 Coupling Constraints 1598.3.3 Operators 1608.4 Decomposition of the Optimization Task into Subtasks 1618.5 Structuring the Underlying Information 1628.6 System Analysis (SA) 1678.7 Evolving Engineering Design Process 1708.8 Single-Level Design Optimizations (S-LDO) 1738.8.1 Assessment 1758.9 The Feasible Sequential Approach (FSA) 1768.9.1 Implementation Options 1778.10 Multidisciplinary Design Optimization (MDO) Methods 1788.10.1 Collaborative Optimization (CO) 1798.10.2 Bi-Level Integrated System Synthesis (BLISS) 1898.10.3 BLISS Augmented with SM 1928.11 Closure 1998.11.1 Decomposition 1998.11.2 Approximations and SM 2008.11.3 Anatomy of a System 2008.11.4 Interactions of the System and Its BBs 2018.11.5 Intrinsic Limitations of Optimization in General 2028.11.6 Optimization across a Choice of Different Design Concepts 2028.11.7 Off-the-Shelf Commercial Software Frameworks 203References 2059 Knowledge Based Engineering 2089.1 Introduction 2089.2 KBE to Support MDO 2099.3 What is KBE 2109.4 When Can KBE Be Used 2139.5 Role of KBE in the Development of Advanced MDO Systems 2149.6 Principles and Characteristics of KBE Systems and KBE Languages 2209.7 KBE Operators to Define Class and Object Hierarchies 2229.7.1 An Example of a Product Model Definition in Four KBE Languages 2269.8 The Rules of KBE 2309.8.1 Logic Rules (or Conditional Expressions) 2309.8.2 Math Rules 2319.8.3 Geometry Manipulation Rules 2329.8.4 Configuration Selection Rules (or Topology Rules) 2349.8.5 Communication Rules 2359.8.6 Beyond Classical KBS and CAD 2369.9 KBE Methods to Develop MMG Applications 2369.9.1 High-Level Primitives (HLPs) to Support Parametric Product Modeling 2379.9.2 Capability Modules (CMs) to Support Analysis Preparation 2389.10 Flexibility and Control: Dynamic Typing, Dynamic Class Instantiation, and Object Quantification 2419.11 Declarative and Functional Coding Style 2419.12 KBE Specific Features: Runtime Caching and Dependency Tracking 2439.13 KBE Specific Features: Demand-Driven Evaluation 2469.14 KBE Specific Features: Geometry Kernel Integration 2479.14.1 How a KBE Language Interacts with a CAD Engine 2489.15 CAD or KBE? 2529.16 Evolution and Trends of KBE Technology 253Acknowledgments 256References 25610 Uncertainty-Based Multidisciplinary Design Optimization 25810.1 Introduction 25810.2 Uncertainty-Based Multidisciplinary Design Optimization (UMDO) Preliminaries 25910.2.1 Basic Concepts 25910.2.2 General UMDO Process 26310.3 Uncertainty Analysis 26410.3.1 Monte Carlo Methods (MCS) 26510.3.2 Taylor Series Approximation 26610.3.3 Reliability Analysis 26810.3.4 Decomposition-Based Uncertainty Analysis 27110.4 Optimization under Uncertainty 27210.4.1 Reliability Index Approach (RIA) and Performance Measure Approach (PMA) Methods 27310.4.2 Single Level Algorithms (SLA) 27510.4.3 Approximate Reliability Constraint Conversion Techniques 27810.4.4 Decomposition-Based Method 28010.5 Example 28210.6 Conclusion 285References 28511 Ways and Means for Control and Reduction of the Optimization Computational Cost and Elapsed Time 28711.1 Introduction 28711.2 Computational Effort 28811.3 Reducing the Function Nonlinearity by Introducing Intervening Variables 28911.4 Reducing the Number of the Design Variables 28911.4.1 Linking by Groups 29011.5 Reducing the Number of Constraints Directly Visible to the Optimizer 29211.5.1 Separation of Well-Satisfied Constraints from the Ones Violated or Nearly Violated 29211.5.2 Representing a Set of Constraints by a Single Constraint 29311.5.3 Replacing Constraints by Their Envelope in the Kreisselmeier–Steinhauser Formulation 29311.6 Surrogate Methods (SMs) 29811.7 Coordinated Use of High- and Low-Fidelity Mathematical Models in the Analysis 30111.7.1 Improving LF Analysis by Infrequent Use of HF Analysis 30111.7.2 Reducing the Number of Quantities Being Approximated 30311.7.3 Placement of the Trial Points X T in the Design Space X 30411.8 Design Space in n Dimensions May Be a Very Large Place 308References 309Appendix A Implementation of KBE in an MDO System 310Appendix B Guide to Implementing an MDO System 349Index 360