Optimal aircraft design is impossible without a parametric representation of the geometry of the airframe. We need a mathematical model equipped with a set of controls, or design variables, which generates different candidate airframe shapes in response to changes in the values of these variables.

Series Preface xi Preface xiii 1 Prologue 1 2 Geometry Parameterization: Philosophy and Practice 7 2.1 A Sense of Scale 7 2.1.1 Separating Shape and Scale 7 2.1.2 Nondimensional Coefficients 9 2.2 Parametric Geometries 11 2.2.1 Pre-Optimization Checks 13 2.3 What Makes a Good Parametric Geometry: Three Criteria 15 2.3.1 Conciseness 15 2.3.2 Robustness 16 2.3.3 Flexibility 16 2.4 A Parametric Fuselage: A Case Study in the Trade-Offs of Geometry Optimization 18 2.4.1 Parametric Cross-Sections 18 2.4.2 Fuselage Cross-Section Optimization: An Illustrative Example 22 2.4.3 A Parametric Three-Dimensional Fuselage 27 2.5 A General Observation on the Nature of Fixed-Wing Aircraft Geometry Modelling 29 2.6 Necessary Flexibility 30 2.7 The Place of a Parametric Geometry in the Design Process 31 2.7.1 Optimization: A Hierarchy of Objective Functions 31 2.7.2 Competing Objectives 32 2.7.3 Optimization Method Selection 35 2.7.4 Inverse Design 37 3 Curves 41 3.1 Conics and B´ezier Curves 41 3.1.1 Projective Geometry Construction of Conics 42 3.1.2 Parametric Bernstein Conic 43 3.1.3 Rational Conics and B´ezier Curves 49 3.1.4 Properties of B´ezier Curves 50 3.2 B´ezier Splines 51 3.3 Ferguson’s Spline 52 3.4 B-Splines 57 3.5 Knots 59 3.6 Nonuniform Rational Basis Splines 60 3.7 Implementation in Rhino 64 3.8 Curves for Optimization 65 4 Surfaces 67 4.1 Lofted, Translated and Coons Surfaces 67 4.2 B´ezier Surfaces 69 4.3 B-Spline and Nonuniform Rational Basis Spline Surfaces 74 4.4 Free-Form Deformation 76 4.5 Implementation in Rhino 82 4.5.1 Nonuniform Rational Basis Splines-Based Surfaces 82 4.5.2 Free-Form Deformation 82 4.6 Surfaces for Optimization 84 5 Aerofoil Engineering: Fundamentals 91 5.1 Definitions, Conventions, Taxonomy, Description 91 5.2 A ‘Non-Taxonomy’ of Aerofoils 92 5.2.1 Low-Speed Aerofoils 93 5.2.2 Subsonic Aerofoils 93 5.2.3 Transonic Aerofoils 93 5.2.4 Supersonic Aerofoils 94 5.2.5 Natural Laminar Flow Aerofoils 94 5.2.6 Multi-Element Aerofoils 95 5.2.7 Morphing and Flexible Aerofoils 98 5.3 Legacy versus Custom-Designed Aerofoils 98 5.4 Using Legacy Aerofoil Definitions 99 5.5 Handling Legacy Aerofoils: A Practical Primer 101 5.6 Aerofoil Families versus Parametric Aerofoils 102 6 Families of Legacy Aerofoils 103 6.1 The NACA Four-Digit Section 103 6.1.1 A One-Variable Thickness Distribution 104 6.1.2 A Two-Variable Camber Curve 105 6.1.3 Building the Aerofoil 105 6.1.4 Nomenclature 106 6.1.5 A Drawback and Two Fixes 107 6.1.6 The Distribution of Points: Sampling Density Variations and Cusps 107 6.1.7 A MATLAB® Implementation 109 6.1.8 An OpenNURBS/Rhino-Python Implementation 111 6.1.9 Applications 112 6.2 The NACA Five-Digit Section 113 6.2.1 A Three-Variable Camber Curve 113 6.2.2 Nomenclature and Implementation 116 6.3 The NACA SC Families 118 6.3.1 SC(2) 118 7 Aerofoil Parameterization 123 7.1 Complex Transforms 123 7.1.1 The Joukowski Aerofoil 124 7.2 Can a Pair of Ferguson Splines Represent an Aerofoil? 125 7.2.1 A Simple Parametric Aerofoil 125 7.3 Kulfan’s Class- and Shape-Function Transformation 127 7.3.1 A Generic Aerofoil 128 7.3.2 Transforming a Legacy Aerofoil 130 7.3.3 Approximation Accuracy 132 7.3.4 The Kulfan Transform as a Filter 135 7.3.5 Computational Implementation 137 7.3.6 Class- and Shape-Function Transformation in Optimization: Global versus Local Search 139 7.3.7 Capturing the Shared Features of a Family of Aerofoils 140 7.4 Other Formulations: Past, Present and Future 142 8 Planform Parameterization 145 8.1 The Aspect Ratio 145 8.1.1 Induced Drag 148 8.1.2 Structural Efficiency 150 8.1.3 Airport Compatibility 150 8.1.4 Handling 151 8.2 The Taper Ratio 152 8.3 Sweep 153 8.3.1 Terminology 153 8.3.2 Sweep in Transonic Flight 155 8.3.3 Sweep in Supersonic Flight 157 8.3.4 Forward Sweep 158 8.3.5 Variable Sweep 159 8.3.6 Swept-Wing ‘Growth’ 161 8.4 Wing Area 162 8.4.1 Constraints on the Wing Area 162 8.5 Planform Definition 167 8.5.1 From Sketch to Geometry 167 8.5.2 Introducing Scaling Factors: A Design Heuristic and a Simple Example 168 8.5.3 More Complex Planforms and an Additional Scaling Factor 169 8.5.4 Spanwise Chord Variation 171 9 Three-Dimensional Wing Synthesis 175 9.1 Fundamental Variables 175 9.1.1 Twist 175 9.1.2 Dihedral 176 9.2 Coordinate Systems 177 9.2.1 Cartesian Systems 177 9.2.2 A Wing-Bound, Curvilinear Dimension 181 9.3 The Synthesis of a Nondimensional Wing 181 9.3.1 Example: A Blended Box Wing 183 9.3.2 Example: Parameterization of a Blended Winglet 187 9.4 Wing Geometry Scaling. A Case Study: Design of a Commuter Airliner Wing 189 9.5 Indirect Wing Geometry Scaling 196 10 Design Sensitivities 199 10.1 Analytical and Finite-Difference Sensitivities 199 10.2 Algorithmic Differentiation 201 10.2.1 Forward Propagation of Tangents 201 10.2.2 Reverse Mode 203 10.3 Example: Differentiating an Aerofoil from Control Points to Lift Coefficient 204 10.4 Example Inverse Design 212 11 Basic Aerofoil Analysis: AWorked Example 217 11.1 Creating the .dat and .in files using Python 218 11.2 Running XFOIL from Python 219 12 Human-Powered Aircraft Wing Design: A Case Study in Aerodynamic Shape Optimization 223 12.1 Constraints 225 12.2 Planform Design 225 12.3 Aerofoil Section Design 226 12.4 Optimization 226 12.4.1 NACA Four-Digit Wing 227 12.4.2 Ferguson Spline Wing 229 12.5 Improving the Design 230 13 Epilogue: Challenging Topological Prejudice 237 References 239 Index 243

Optimal aircraft design is impossible without a parametric representation of the geometry of the airframe. We need a mathematical model equipped with a set of controls, or design variables, which generates different candidate airframe shapes in response to changes in the values of these variables. This model's objectives are to be flexible and concise, and capable of yielding a wide range of shapes with a minimum number of design variables. Moreover, the process of converting these variables into aircraft geometries must be robust. Alas, flexibility, conciseness and robustness can seldom be achieved simultaneously! Aircraft Aerodynamic Design: Geometry and Optimization addresses this problem by navigating the subtle trade-offs between the competing objectives of geometry parameterization. It begins with the fundamentals of geometry-centred aircraft design, followed by a review of the building blocks of computational geometries, the curve and surface formulations at the heart of aircraft geometry. The authors then cover a range of legacy formulations in the build-up towards a discussion of the most flexible shape models used in aerodynamic design (with a focus on lift generating surfaces). The book takes a practical approach and includes MATLAB, Python and Rhinoceros® code, as well as ‘real-life’ example case studies. Key features: Covers effective geometry parameterization within the context of design optimizationDemonstrates how geometry parameterization is an important element of modern aircraft designIncludes code and case studies which enable the reader to apply each theoretical concept either as an aid to understanding or as a building block of their own geometry modelAccompanied by a website hosting codes Aircraft Aerodynamic Design: Geometry and Optimization is a practical guide for researchers and practitioners in the aerospace industry and a reference for graduate and undergraduate students in aircraft design and multidisciplinary design optimization.