Engineering Design via Surrogate Modelling

Dr. Alexander I. J. Forrester is Lecturer in Engineering Design at the University of Southampton. His main area of research focuses on improving the efficiency with which expensive analysis (particularly computational fluid dynamics) is used in design. His techniques have been applied to wing aerodynamics, satellite structures, sports equipment design and Formula One. Dr Andras Sobester is a Lecturer and EPSRC/ Royal Academy of Engineering research Fellow in the School of Engineering Sciences at the University of Southampton. His research interests include aircraft design, aerodynamic shape parameterization and optimization, as well as engineering design technology in general.Professor Andy J. Keane currently holds the Chair of Computational Engineering at the University of Southampton. He leads the University's Computational Engineering at the Design Research Group and directs the rolls-Royce University Technology centre for Computational Engineering. His interests lie primarily in the aerospace sciences, with a  focus on the design of aerospace systems using computational methods. He has published over two hundred papers and three books in this area, many of which deal with surrogate modelling concepts.

• Preface ixAbout the Authors xiForeword xiiiPrologue xvPart I Fundamentals 11 Sampling Plans 31.1 The ‘Curse of Dimensionality’ and How to Avoid It 41.2 Physical versus Computational Experiments 41.3 Designing Preliminary Experiments (Screening) 61.3.1 Estimating the Distribution of Elementary Effects 61.4 Designing a Sampling Plan 131.4.1 Stratification 131.4.2 Latin Squares and Random Latin Hypercubes 151.4.3 Space-filling Latin Hypercubes 171.4.4 Space-filling Subsets 281.5 A Note on Harmonic Responses 291.6 Some Pointers for Further Reading 30References 312 Constructing a Surrogate 332.1 The Modelling Process 332.1.1 Stage One: Preparing the Data and Choosing a Modelling Approach 332.1.2 Stage Two: Parameter Estimation and Training 352.1.3 Stage Three: Model Testing 362.2 Polynomial Models 402.2.1 Example One: Aerofoil Drag 422.2.2 Example Two: A Multimodal Testcase 442.2.3 What About the k-variable Case? 452.3 Radial Basis Function Models 452.3.1 Fitting Noise-Free Data 452.3.2 Radial Basis Function Models of Noisy Data 492.4 Kriging 492.4.1 Building the Kriging Model 512.4.2 Kriging Prediction 592.5 Support Vector Regression 632.5.1 The Support Vector Predictor 642.5.2 The Kernel Trick 672.5.3 Finding the Support Vectors 682.5.4 Finding μ 702.5.5 Choosing C and ε 712.5.6 Computing ε: v-SVR 732.6 The Big(ger) Picture 75References 763 Exploring and Exploiting a Surrogate 773.1 Searching the Surrogate 783.2 Infill Criteria 793.2.1 Prediction Based Exploitation 793.2.2 Error Based Exploration 843.2.3 Balanced Exploitation and Exploration 853.2.4 Conditional Likelihood Approaches 913.2.5 Other Methods 1013.3 Managing a Surrogate Based Optimization Process 1023.3.1 Which Surrogate for What Use? 1023.3.2 How Many Sample Plan and Infill Points? 1023.3.3 Convergence Criteria 1033.4 Search of the Vibration Isolator Geometry Feasibility Using Kriging Goal Seeking 104References 106Part II Advanced Concepts 1094 Visualization 1114.1 Matrices of Contour Plots 1124.2 Nested Dimensions 114Reference 1165 Constraints 1175.1 Satisfaction of Constraints by Construction 1175.2 Penalty Functions 1185.3 Example Constrained Problem 1215.3.1 Using a Kriging Model of the Constraint Function 1215.3.2 Using a Kriging Model of the Objective Function 1235.4 Expected Improvement Based Approaches 1255.4.1 Expected Improvement with Simple Penalty Function 1265.4.2 Constrained Expected Improvement 1265.5 Missing Data 1315.5.1 Imputing Data for Infeasible Designs 1335.6 Design of a Helical Compression Spring Using Constrained Expected Improvement 1365.7 Summary 139References 1396 Infill Criteria with Noisy Data 1416.1 Regressing Kriging 1436.2 Searching the Regression Model 1446.2.1 Re-Interpolation 1466.2.2 Re-Interpolation with Conditional Likelihood Approaches 1496.3 A Note on Matrix Ill-Conditioning 1526.4 Summary 152References 1537 Exploiting Gradient Information 1557.1 Obtaining Gradients 1557.1.1 Finite Differencing 1557.1.2 Complex Step Approximation 1567.1.3 Adjoint Methods and Algorithmic Differentiation 1567.2 Gradient-enhanced Modelling 1577.3 Hessian-enhanced Modelling 1627.4 Summary 165References 1658 Multi-fidelity Analysis 1678.1 Co-Kriging 1678.2 One-variable Demonstration 1738.3 Choosing Xc and Xe 1768.4 Summary 177References 1779 Multiple Design Objectives 1799.1 Pareto Optimization 1799.2 Multi-objective Expected Improvement 1829.3 Design of the Nowacki Cantilever Beam Using Multi-objective, Constrained Expected Improvement 1869.4 Design of a Helical Compression Spring Using Multi-objective, Constrained Expected Improvement 1919.5 Summary 192References 192Appendix: Example Problems 195A.1 One-Variable Test Function 195A.2 Branin Test Function 196A.3 Aerofoil Design 197A.4 The Nowacki Beam 198A.5 Multi-objective, Constrained Optimal Design of a Helical Compression Spring 200A.6 Novel Passive Vibration Isolator Feasibility 202References 203Index 205