Optimization Under Uncertainty with Applications to Aerospace Engineering

✍ Scribed by Massimiliano Vasile (editor)

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 568
• Category: Library

✦ Synopsis

In an expanding world with limited resources, optimization and uncertainty quantification have become a necessity when handling complex systems and processes. This book provides the foundational material necessary for those who wish to embark on advanced research at the limits of computability, collecting together lecture material from leading experts across the topics of optimization, uncertainty quantification and aerospace engineering.
The aerospace sector in particular has stringent performance requirements on highly complex systems, for which solutions are expected to be optimal and reliable at the same time. The text covers a wide range of techniques and methods, from polynomial chaos expansions for uncertainty quantification to Bayesian and Imprecise Probability theories, and from Markov chains to surrogate models based on Gaussian processes. The book will serve as a valuable tool for practitioners, researchers and PhD students.

✦ Table of Contents

Contents
1 Introduction to Spectral Methods for Uncertainty Quantification
1.1 Motivation
1.1.1 Typical UQ Questions
1.2 Illustrative Problem
1.2.1 The Deterministic Heat Diffusion Equation
1.2.2 The Stochastic Heat Diffusion Equation
1.3 The Sampling Process
1.4 Sampling Techniques
1.4.1 Karhunen–Loève Expansion
1.4.2 Mathematical Reformulation of the Dirichlet Problem
1.5 Monte Carlo Methods
1.5.1 Mean and Variance
1.5.2 PDF Reconstruction for Different Correlation Lengths
1.6 Spectral Methods
1.6.1 Polynomial Chaos Expansion
1.6.2 Non-Intrusive Spectral Projection Methods
1.6.2.1 Numerical Approaches for NISP
1.6.2.2 Linear Regression
1.6.3 Galerkin Methods
1.6.3.1 Weak Formulation and Deterministic Discretisation
1.6.3.2 Stochastic Discretisation
1.6.3.3 Computational Cost of Stochastic Galerkin Method
1.6.4 Application of Surrogate Models: A Sensitivity Analysis Using PC Expansions
1.6.4.1 MC Approach
1.6.4.2 Surrogate Approach
1.7 Concluding Remarks
References
2 Introduction to Imprecise Probabilities
2.1 Introduction
2.2 Some Models of Uncertainty
2.2.1 A Point Estimate
2.2.2 An Interval
2.2.3 A Probability Distribution
2.2.4 A Set of Probability Distributions
2.3 Probability Theory
2.3.1 Measure-Theoretic Probability
2.3.2 Probability via Expectation
2.4 Imprecise Probabilities
2.4.1 A Set of Measures
2.4.2 Capacities
2.4.3 Neighbourhood Models
2.4.4 Random Sets
2.4.5 Probability Boxes
2.5 Lower Previsions
2.5.1 Desirability
2.5.1.1 Axioms
2.5.2 Lower Previsions
2.5.3 Natural Extension
2.5.4 Duality
2.6 Constructing the Laws
2.6.1 Statistical Inference with Precise Probabilities
2.6.2 Robust Bayesian Inference
2.6.3 Frequentist Inference with Imprecise Probabilities
2.7 Concluding Remarks
References
3 Uncertainty Quantification in Lasso-Type RegularizationProblems
3.1 Introduction
3.1.1 Statistical Modeling
3.1.2 Statistical Inference
3.1.3 Linear Models
3.1.4 Strong Duality and the Karush–Kuhn–Tucker Conditions
3.2 Parameter Estimation
3.2.1 Ordinary Least Squares
3.2.2 Non-Negative Garrote
3.2.3 Regularization Under lq Penalty
3.3 The LASSO
3.3.1 Solving the LASSO Optimization Problem
3.3.2 Cross-Validation
3.3.2.1 Example: Gaia Dataset
3.4 Uncertainty Quantification
3.4.1 Refit-LASSO
3.4.1.1 Example: Gaia Dataset
3.4.2 Bootstrap Method
3.4.2.1 Bootstrap for LASSO
3.4.2.2 Example: Gaia Dataset
3.4.3 Bayesian LASSO
3.4.3.1 Example: Gaia Dataset
3.5 LASSO for Classification
3.5.1 Logistic Regression
3.5.1.1 Cross-Validation
3.5.2 Uncertainty Quantification
3.5.2.1 Refit-LASSO
3.5.2.2 Bootstrap
3.5.2.3 Bayesian Approach
3.6 Conclusion
References
4 Reliability Theory
4.1 Reliability and Risk
4.2 Mathematical Theory of Reliability
4.2.1 Structural Reliability
4.2.2 Survival Analysis
4.3 System Reliability
4.3.1 Structure Function
4.3.2 Graphical Models
4.3.2.1 Reliability Block Diagrams
4.3.2.2 Fault Trees
4.3.2.3 Bayesian Networks
4.3.3 Phased Missions
4.3.4 Signatures
4.3.4.1 System Signature
4.3.4.2 Survival Signature
4.4 Statistical Inference in Reliability
4.4.1 Censored Datasets
4.4.2 Accelerated Life Testing
4.4.3 Proportional Hazards Model
4.4.4 Quality Control
4.5 Designing Highly Reliable Systems
4.5.1 Redundancy Allocation
4.5.2 System Maintenance
4.6 Concluding Remarks
References
5 An Introduction to Imprecise Markov Chains
5.1 Introduction
5.2 (Precise) Stochastic Processes
5.2.1 Probability Trees
5.2.2 Bayesian Networks
5.2.3 Transition Graphs
5.3 Imprecise Discrete-Time Markov Chains
5.3.1 Imprecise Probability Trees
5.3.2 Credal Networks
5.3.3 Limits of Homogeneous IDTMCs
5.4 Imprecise Continuous-Time Markov Chains
5.4.1 Imprecise Continuous-Time Markov Chains
5.4.2 Limits of ICTMCs
5.5 Literature and Further Reading
References
6 Fundamentals of Filtering
6.1 The State Estimation Problem
6.1.1 Building Blocks
6.1.1.1 State Marginalization
6.1.1.2 Markov and Independence Assumptions
6.1.2 Filtering Problem Formulation
6.1.3 Bayesian Approach for Filtering
6.1.3.1 Conditional Probability Evolution Between Observations
6.1.4 Batch Processor vs. Sequential Filtering
6.1.5 Optimal Estimate
6.2 Probability Distribution Propagation
6.2.1 Linear Transformation
6.2.2 Nonlinear Transformation
6.2.2.1 Taylor Expansion
6.2.2.2 Unscented Transform
6.2.2.3 Monte Carlo Methods
6.3 Filtering Algorithms
6.3.1 Kalman Filter
6.3.2 Extended Kalman Filter
6.3.3 Unscented Kalman Filter
6.3.4 Gaussian Filter Framework
6.3.5 Particle Filter
6.4 Conclusions
References
7 Introduction to Optimisation
7.1 Introduction
7.1.1 Solving an Optimisation Problem
7.1.2 Local vs Global Optimisation
7.1.3 Single- vs Multi-Objective
7.2 Continuous Optimisation
7.2.1 Local Optimisation
7.2.1.1 Optimality Conditions
7.2.1.2 Algorithms
7.2.2 Global Optimisation
7.2.2.1 Deterministic Strategies
7.2.2.2 Stochastic Strategies
7.2.3 Multi-Objective Optimisation
7.2.4 Optimal Control
7.2.4.1 Indirect Methods
7.2.4.2 Direct Methods
7.2.4.3 Comparison of Direct and Indirect Methods
7.2.4.4 Practical Techniques for Optimal Control
Single Shooting
Multiple Shooting
Collocation
7.3 Combinatorial and Network Optimisation
7.3.1 Pure Integer Optimisation
7.3.1.1 Special Case: 0-1 Integer Programming
7.3.2 Mixed-Integer Programming
7.3.2.1 MIP vs MINLP
7.3.2.2 Methods
Exact Methods
Heuristic Methods
7.3.3 Network Optimisation
Standard Network Flow Formulation and Notation
7.4 Summary
References
8 An Introduction to Many-Objective Evolutionary Optimization
8.1 Introduction
8.1.1 From Single- to Many-Objective Optimization
8.1.2 Optimality in Multi- and Many-Objective Optimization
8.2 Evolutionary Algorithm
8.2.1 Base Algorithm
8.2.2 Recombination
8.2.3 Mutation
8.2.4 Selection
8.3 Multi-Objective Optimization
8.3.1 Method Classifications Based on Preference-Imposing Timing
8.3.1.1 A Priori Method
8.3.1.2 A Posteriori Methods
8.3.1.3 Progressive Methods
8.3.2 Solution Quality Assessment
8.3.2.1 Hypervolume
8.3.2.2 Generational Distance
8.3.3 Algorithms Designed for Multi-Objective Optimization Problems
8.3.3.1 NSGA-II
8.3.3.2 SMS-EMOA
8.4 Many-Objective Optimization
8.4.1 Challenges in Many-Objective Optimization
8.4.1.1 Curse of Dimensionality
8.4.1.2 Expensive Evaluation
8.4.1.3 Visualization Challenge
8.4.2 Algorithms Designed for Many-Objective Optimization Problems
8.4.2.1 MOEA/D
8.4.2.2 NSGA-III
8.4.3 High-Dimension Visualization Techniques
8.4.3.1 Bubble Chart
8.4.3.2 Parallel Plot
8.4.3.3 Glyph Plot
8.5 Surrogate Model in Multi- and Many-Objective Optimization
8.5.1 ParEGO
8.5.2 Prescreening Method
8.5.3 Taxonomy of Surrogate Models for MOP
8.6 Test Problems for Many-Objective Optimization
8.6.1 Biobjective Test Problems
8.6.1.1 ZDT
8.6.1.2 Black-Box Optimization Benchmarking
8.6.2 Scalable Test Problems
8.6.2.1 DTLZ
8.6.2.2 WFG
8.7 Summary
References
9 Multilevel Optimisation
9.1 Introduction
9.2 Multilevel Optimisation Problem
9.3 Bilevel Optimisation Problem
9.3.1 Linear Bilevel Optimisation Example
9.4 Bilevel vs Biobjective Optimisation Problem
9.5 Special Cases of Bilevel Optimisation Problems
9.5.1 Bilevel Multiobjective Optimisation Problems
9.5.2 Bilevel Multileader and/or Multifollower Optimisation Problems
9.5.3 Bilevel Optimisation Problem Under Uncertainty
9.5.4 Minimax (Worst-Case Scenario) as Bilevel Optimisation Problem
9.6 Solution Algorithms
9.6.1 Classical Approaches
9.6.2 Metaheuristic Approaches
9.7 Applications
9.8 Summary
References
10 Sequential Parameter Optimization for Mixed-DiscreteProblems
10.1 Introduction
10.2 Problem Definition
10.3 Challenges in Real-World Optimization
10.3.1 Problem Features
10.3.2 High Dimensionality
10.3.2.1 Screening
10.3.2.2 Mapping
10.3.3 Uncertainty
10.4 Sequential Parameter Optimization
10.4.1 Initial Design
10.4.1.1 Strategies for Design of Experiment
10.4.1.2 Latin Hypercube Sampling
10.4.1.3 Factorial Designs
10.4.2 Modeling
10.4.2.1 Modeling in Mixed-Integer Space
10.4.2.2 The Naive Approach
10.4.2.3 Inherently Discrete Models
10.4.2.4 Similarity-Based Models
10.4.2.5 Handling Factor Variables in Kriging Model
10.4.3 Optimization Algorithms for the Metamodel
10.4.4 MIES
10.5 Case Study: Optimization of Composite Multi-Layered Plate
10.5.1 Overview
10.5.2 Optimization Problem
10.5.3 Methodology
10.5.4 Results
References
11 Parameter Control in Evolutionary Optimisation
11.1 Evolutionary Optimisation
11.1.1 Evolutionary Algorithms
11.1.2 Exploration and Exploitation
11.1.3 The Role of Control Parameters
11.2 Control Parameters
11.2.1 Typical Control Parameters
11.2.2 What Else can be Adapted
11.2.3 Influence on Algorithm Performance
11.2.4 Interaction of Control Parameters
11.3 Setting Approaches
11.3.1 Parameter Tuning
11.3.2 Parameter Control
11.4 Parameter Control Strategies
11.4.1 Deterministic Parameter Control
11.4.2 Adaptive Parameter Control
Model of Adaptive Parameter Control
11.4.3 Self-Adapting Parameter Control
11.4.4 Tuning vs. Control
11.5 Real-World Optimisation
11.5.1 Large-Scale Global Optimisation
11.5.2 Dynamic Optimisation
11.5.3 Optimisation Under Uncertainty
11.5.4 Multi-objective Optimisation
11.5.5 Multilevel Optimisation
11.6 Summary
References
12 Response Surface Methodology
12.1 Introduction
12.2 Response Surface Model Construction
12.2.1 Objective
12.2.2 Classification
12.2.3 Construction Stages
12.2.3.1 Data Preparation
12.2.3.2 Algorithm Choice
12.2.3.3 Model Training
12.2.3.4 Model Validation
12.3 Examples of Response Surface Models
12.3.1 Least Squares Method
12.3.2 Radial Basis Functions
12.3.3 Kriging
12.3.3.1 Variogram
12.4 Wing Structure Design Using Response Surface Models
12.4.1 Design Problem
12.4.2 Analytical Model
12.4.3 Comparison of Response Surface Models
12.4.4 RSM Construction on Noisy Data
12.4.5 Case-Study Conclusion and Take-Home Message
References
13 Risk Measures in the Context of Robust and Reliability Based Optimization
13.1 Introduction
13.2 Optimization Under Uncertainty
13.3 Risk Measures
13.4 Robust Optimization Problem Using Risk Functions
13.4.1 Estimation of Risk Functions Using ECDF
13.4.1.1 Value-at-Risk (Quantile) Estimation Using ECDF
13.4.1.2 Cumulative Value-at-Risk (Superquantile) Estimation Using ECDF
13.4.2 Estimation of Risk Functions Using WECDF
13.4.3 Bootstrap Error Analysis
13.5 Application Example
13.5.1 Results
References
14 Best Practices for Surrogate Based Uncertainty Quantification in Aerodynamics and Application to Robust Shape Optimization
14.1 Introduction
14.1.1 Deterministic Optimization
14.1.2 Motivation of Robust Design
14.2 Robust Design Approaches for Aerodynamic Shape Optimization
14.2.1 Multi-Point Optimization
14.2.2 Worst-Case Approach
14.2.3 Interval Analysis
14.2.4 Statistical Approach
14.2.4.1 Characterization of Input Uncertainty
14.2.4.2 Definition of Objective Function (I), Robust Design
14.2.4.3 Definition of Objective Function (II) Reliability Based
14.2.4.4 Evaluation of Statistics
14.3 Surrogate Models for Uncertainty Quantification
14.3.1 Surrogate Models Overview
14.3.1.1 Design of Experiments
14.3.1.2 Refinement Strategy for Uncertainty Quantification
14.3.2 Advantages of Surrogate Modelling for Uncertainty Quantification
14.3.3 Gradient-Enhanced Surrogates for Efficient UQ
14.3.3.1 Gradient-Enhanced Kriging
14.3.3.2 The Adjoint Method: Breaking the Curse of Dimensionality
14.3.4 Computing Statistics on Surrogate Models
14.4 Optimization of RAE2822 Airfoil Under Uncertainty
14.4.1 Problem Definition
14.4.1.1 Deterministic Optimization
14.4.1.2 Optimization Under Uncertainty
14.4.2 CFD Solver and Numerical Grid
14.4.3 CFD Process Chain
14.4.4 Parametrization of Deterministic Design Variables
14.4.5 Parametrization of Uncertainties
14.4.6 Optimizer
14.4.7 Robust Design Framework
14.4.8 Validation of the Framework
14.4.9 Deterministic Results
14.4.10 Robust Results
14.5 Conclusions
References
15 In-flight Icing: Modeling, Prediction, and Uncertainty
15.1 Introduction
15.2 In-Flight Ice Accretion
15.2.1 Icing Environment
15.2.1.1 Cloud Formations
15.2.1.2 Supercooled Large Droplets
15.2.1.3 Ice Crystals
15.2.1.4 Snow
15.2.2 Icing Relevant Parameters
15.2.2.1 Outside Air Temperature
15.2.2.2 Liquid Water Content
15.2.2.3 Airspeed
15.2.2.4 Altitude
15.2.2.5 Droplet Size
15.2.3 Icing Types
15.2.3.1 Rime Ice
15.2.3.2 Glaze Ice
15.2.3.3 Mixed Ice Conditions
15.2.4 Aircraft Icing Interactions
15.2.4.1 Surface Interaction
15.2.4.2 Ice Shedding
15.2.4.3 Drop Impact
15.2.4.4 Crystal Bouncing
15.2.5 Existing Research Methods for Ice Accretion
15.2.5.1 Flight Test and Wind Tunnel Experiments
15.2.5.2 Uncertainties in Icing Tunnel Experiments
15.2.5.3 State-of-the-Art Computational Ice Accretion Methods
15.2.6 Ice Accretion and Performances
15.2.6.1 Fixed-Wing Icing Environment
15.2.6.2 Rotating-Wing Icing Environment
15.3 Modeling Ice Accretion
15.3.1 Flow Field Determination
15.3.1.1 Flow Solver
15.3.2 Governing Equations for Multiphase Flows
15.3.2.1 Mass Balance
15.3.2.2 Energy Balance
15.3.3 Droplet Solver
15.3.3.1 Eulerian and Lagrangian Specifications for Particle Tracking
15.3.3.2 Collection Efficiency
15.3.4 Unsteady Ice Accretion
15.3.4.1 Development of Ice Accretion Models
15.3.4.2 The Stefan Problem
15.3.4.3 Messinger Model
15.3.4.4 Myers Model
15.3.4.5 An Improved Myers Model
15.3.4.6 An Unsteady Ice Accretion Model
15.3.5 Mesh Morphing
15.3.6 Numerical Results
15.4 Ice Protection Systems and Certification
15.4.1 Mature Protection Technologies
15.4.1.1 Pneumatic-Thermal Protection
15.4.1.2 De-icing Boot Protection
15.4.1.3 Thermo-Electric protection
15.4.1.4 Chemical Protection
15.4.2 Alternative Protection Technologies
15.4.3 Regulations and Certification
15.4.3.1 FAA Code of Flight Regulations
15.4.3.2 EASA Certification Specifications
15.5 Concluding Remarks
References
16 Uncertainty Treatment Applications: High-Enthalpy Flow Ground Testing
16.1 Atmospheric Entry: A Complex Problem
16.1.1 Aerothermodynamics Testing
16.2 Ground Testing in High-Enthalpy Facilities
16.2.1 Inductively-Coupled Plasma Facilities
16.2.1.1 Material Characterization
16.2.1.2 Free Stream Characterization for Validation
16.2.2 Hypersonic Wind Tunnels
16.2.2.1 Non-equilibrium Effects
16.2.2.2 Shock Layer Radiation
16.2.3 Aleatory Uncertainties
16.3 Physico-Chemical Models and Computational Tools
16.3.1 Governing Equations for Atmospheric Flows
16.3.1.1 Resistive Magneto-Hydrodynamics (MHD) Model
16.3.2 Closure Models
16.3.2.1 Thermodynamic Properties
16.3.2.2 Transport Phenomena
16.3.2.3 Chemistry and Internal Energy
16.3.3 Radiative Heating: A Coupled Phenomenon
16.3.4 Gas-Surface Interactions
16.3.5 Epistemic Uncertainties
16.4 Putting It All Together: Extrapolation to Flight
16.4.1 Local Heat Transfer Simulation Methodology
16.4.2 Flight Extrapolation Uncertainties
16.5 Conclusions and Remarks
16.5.1 Current Margin Policies: Where Are We?
References
17 Introduction to Evidence-Based Robust Optimisation
17.1 Introduction
17.1.1 A Classification of Uncertainty
17.1.2 From Design by Analysis to Robust Design Optimisation
17.2 Evidence Theory
17.2.1 Frame of Discernment, Power Set and Evidence
17.2.2 Belief and Plausibility
17.3 Robust Optimisation with Evidence Theory
17.3.1 Optimising the Worst Case Scenario
17.4 Belief Curve Reconstruction
17.4.1 Belief Estimation by Sampling
17.4.2 Dimensionality Reduction
17.4.3 Outer Belief Estimation via Evolutionary Binary Tree
17.4.4 Outer Belief Estimation via Decomposition
17.4.4.1 Evidence Network Models
17.4.4.2 Decomposition Method
17.4.4.3 Complexity Analysis
17.4.5 Example
17.5 Conclusions
References