Aerodynamic Design & Optimization

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CFD Open Series Revision 1.65 1.1.1.1

Aerodynamic Design & Optimization A Review Ideen Sadrehaghighi, Ph.D.

Optimized

Baseline

Optimized

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ANNAPOLIS, MD

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Contents 1.1.1.1 ................................................................................................................................... 1 List of Figures: ..................................................................................................................................................................... 5

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Introduction .................................................................................................................................. 8

1.2 1.3 1.4

Complexity of Flow ................................................................................................................................................ 8 Computational Cost ............................................................................................................................................... 8 Aerodynamic Optimization ............................................................................................................................. 10

Airplane Design ......................................................................................................................... 11

2.1 Role of CFD in the Design Process ................................................................................................................ 11 2.1.1 Conceptual Design ............................................................................................................. 11 2.1.2 Preliminary Design ............................................................................................................. 11 2.1.3 Detailed or Final Design ..................................................................................................... 11 2.2 Conceptual Aerodynamic Design Process as Applied to Airplanes ............................................... 11 2.2.1 Purpose and Scope of Conceptual Airplane Design .......................................................... 11 2.2.2 Cost Estimation .................................................................................................................. 12 2.2.3 Preliminary Weight Estimation .......................................................................................... 12 2.2.4 Range Estimation ............................................................................................................... 12 2.2.5 Aerodynamic Considerations............................................................................................. 13 2.2.6 Wing Design and Selection of Wing Parameters ............................................................... 13 2.2.6.1 Airfoil Selection ....................................................................................................... 14 2.2.6.2 Presentation of Aerodynamic Characteristics of Airfoils ........................................ 14 2.2.6.3 Geometrical Characteristics of Airfoils .................................................................... 14 2.2.6.4 Airfoil Shape and Ordinates..................................................................................... 14 2.2.6.5 Airfoil Nomenclature ............................................................................................... 16 2.2.6.6 NACA Four-Digit Series Airfoils ................................................................................ 16 2.2.6.7 NACA Five-Digit Series Airfoils ................................................................................. 16 2.2.6.8 Six Series Airfoils...................................................................................................... 17 2.2.6.9 NASA Airfoils ............................................................................................................ 17 2.2.7 Estimation of Wing Loading & Thrust Loading .................................................................. 18 2.2.8 Structural Considerations .................................................................................................. 18 2.2.9 Environmental Impacts ...................................................................................................... 19 2.2.9.1 Airplane Noise ......................................................................................................... 19 2.2.9.2 Emissions ................................................................................................................. 19 2.2.10 Performance Estimation .................................................................................................... 19 2.2.10.1 General Remarks on Performance Estimation ........................................................ 20 2.2.10.2 Fuselage and Tail Sizing ........................................................................................... 21 2.2.10.3 Tail cone/Rear Fuselage: ......................................................................................... 21 2.2.11 Estimation of Wing and Thrust Loading Based on Conception Design ............................. 22 2.2.11.1 Remarks on for choosing Wing Loading and Thrust Loading or Power Loading ..... 23 2.2.11.2 Selection of Wing Loading based on Landing Distance ........................................... 23 2.2.11.3 Wing Loading from Landing Consideration based on Take-off Weight .................. 24 2.2.12 Stability and controllability ................................................................................................ 24 2.2.12.1 Static longitudinal stability and control................................................................... 24 2.3 Aerodynamic Design and Analysis Coupling for Wing ........................................................................ 24 2.3.1 The Straight Wing Configuration ....................................................................................... 25

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2.3.2 The Swept Wing Configuration .......................................................................................... 26 2.3.2 The Rear Fuselage mounted Engine Configuration ........................................................... 26 2.4 Control Theory Approach to Airplane Design ......................................................................................... 27 2.5 Thought on Hierarchal Design Approach .................................................................................................. 29 2.6 Classification of Design Optimization Methods ...................................................................................... 30

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Optimization Problem ............................................................................................................. 32

3.1 Role of Optimization .......................................................................................................................................... 32 3.2 Types of Optimization ....................................................................................................................................... 32 3.2.1 Continuous Optimization versus Discrete Optimization ................................................... 33 3.2.2 Unconstrained versus Constrained Optimization.............................................................. 33 3.2.3 None, One or Many Objectives ......................................................................................... 33 3.2.3.1 Single vs. Multi-Objective Optimization .................................................................. 33 3.2.3.2 Various Methods to Solve Multiple Objective Optimization................................... 35 3.2.3.3 Analysts, Decision Makers and Optimization Techniques....................................... 35 3.2.4 Deterministic vs. Stochastic Optimization ......................................................................... 35 3.3 Statement of Optimization Problem ............................................................................................................ 36 3.3.1 Design Variables and Geometric Representation using Parametrization ......................... 36 3.3.2 Geometric Representation using Parametrization ............................................................ 37 3.3.2.1 Discrete Approach ................................................................................................... 38 3.3.2.2 Analytical Approach................................................................................................. 39 3.3.2.3 Partial Differential Equation Approach ................................................................... 39 3.3.2.4 CST Method ............................................................................................................. 39 3.3.2.5 Case Study - Airfoil Optimization............................................................................. 41 3.3.2.6 Spline Based Parameterization with eye on Literature Survey ............................... 42 3.3.2.7 Constraint Handling ................................................................................................. 44 3.3.3 Application of Gradient-Based Methods to Aerodynamic Optimizations ......................... 46 3.3.4 Sensitivity Analysis............................................................................................................. 47 3.3.5 Aero-Elastic Optimization ................................................................................................. 48 3.3.6 Multi-Point Optimization ................................................................................................... 49 3.4 Application of Gradient-Free Methods to Aerodynamic Optimizations ....................................... 50 3.4.1 Genetic Algorithms (GA) .................................................................................................... 51 3.4.1.1 Case Study - Framework for the Shape Optimization of Aerodynamic Profiles Using Genetic Algorithms (GA) .................................................................................................... 52 3.4.2 Application of Hybrid Algorithms to Aerodynamic Optimizations .................................... 53 3.4.3 Application of Surrogate Modelling to Aerodynamic Optimization .................................. 57 3.5 Case Study - Aerodynamic Shape Optimization sing a Gradient Based applied to a Common Wing .................................................................................................................................................................. 60 3.5.1 Methodology ..................................................................................................................... 61 3.5.1.1 Geometric Parametrization ..................................................................................... 61 3.5.1.2 Mesh Perturbation .................................................................................................. 62 3.5.1.3 CFD Solver................................................................................................................ 62 3.5.1.4 Optimization Algorithm ........................................................................................... 63 3.5.2 Problem Formulation......................................................................................................... 63 3.5.2.1 Baseline Geometry .................................................................................................. 63 3.5.2.2 Mesh Convergence Study ........................................................................................ 64 3.5.2.3 Optimization Problem Formulation ........................................................................ 65 3.5.2.4 Surface Sensitivity on the Baseline Geometry ........................................................ 66

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3.5.3 Single-Point Aerodynamic Shape Optimization ................................................................. 66 3.5.4 Effect of the Number of Shape Design Variables .............................................................. 68 3.5.5 Multi-Level Optimization Acceleration Technique ............................................................ 69 3.5.6 Multi-Point Aerodynamic Shape Optimization.................................................................. 70 3.6 Effect of Variable Cant Angle Winglet in Aircraft Control .................................................................. 72 3.6.1 Results and Discussion....................................................................................................... 73 3.6.2 Concluding Remarks .......................................................................................................... 73

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Sensitivity Analysis for Aerodynamic Optimization ..................................................... 76

4.1 Background............................................................................................................................................................ 76 4.2 Aerodynamic Sensitivity .................................................................................................................................. 77 4.3 Flow Analysis and Sensitivity Equation ..................................................................................................... 78 4.4 Optimization.......................................................................................................................................................... 79 4.5 Surface Modeling Using NURBS .................................................................................................................... 80 4.6 Case Study -2D Study of Airfoil Grid Sensitivity using Direct Differentiation (DD) ................ 81 4.6.1 2D Case Study- Airfoil Grid, Flow Sensitivity, and Optimization....................................... 82 4.6.2 Discussions......................................................................................................................... 83 4.7 Extension to 3D using Automatic Differentiation (AD) ....................................................................... 84 4.8 The Adjoint Method Duality............................................................................................................................ 85 4.8.1 Optimization ...................................................................................................................... 86 4.8.2 Gradient Calculation as Related to Ajoint Variable (AV) Scheme...................................... 86 4.8.3 Classical Formulation of the Adjoint Approach to Optimal Design ................................... 87 4.8.4 Limitations of the Adjoint Approach ................................................................................. 89 4.8.4.1 Constraints .............................................................................................................. 89 4.8.4.2 Limitations of Gradient-Based Optimization ........................................................... 90 4.8.5 Case Study – Adjoint Aero Design Optimization for Multi-Stage Turbomachinery Blades 90

Turbo-Machinery Design and Optimization .................................................................... 92

5.1 A Road Map to Turbo-Machine Design and Optimization .................................................................. 92 5.1.1 Wu’s Pioneering (S1 and S2) Scheme ................................................................................ 93 5.1.2 Concept of Streamline Curvature Method ........................................................................ 94 5.2 Case Study 1 - Aerodynamic Design of Compressors ........................................................................... 95 5.2.1 Statement of Problem ....................................................................................................... 95 5.2.2 Different Compressors Objectives ..................................................................................... 96 5.2.3 Design Techniques for Compressor ................................................................................... 97 5.2.4 Preliminary Design Techniques (1D).................................................................................. 99 5.2.5 Through Flow Design Techniques (2D) .............................................................................. 99 5.2.6 Detailed Design Techniques (3D)..................................................................................... 101 5.2.6.1 Direct Methods ...................................................................................................... 101 5.2.6.2 Inverse Methods.................................................................................................... 102 5.2.7 Concluding Remarks ........................................................................................................ 103 5.3 Case Study 2 – Turbine Airfoil Optimization using Quasi 3D Analysis Codes ......................... 103 5.3.1 Parametric Representation of Airfoil Design Process...................................................... 105 5.3.2 Constraints and Problem Formulation ............................................................................ 105 5.3.3 Quasi-3D CFD Analysis and Results ................................................................................. 108 5.3.4 Concluding Remarks ........................................................................................................ 109

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Multi-Disciplinary Optimization (MDO) ........................................................................ 111

6.1 Background......................................................................................................................................................... 111 6.2 Computational Cost Associated with MDO ............................................................................................ 111 6.3 Organizational Complexity ........................................................................................................................... 112 6.4 Clarification of Some Terminology ........................................................................................................... 112 6.5 Categories of MDO Analysis ......................................................................................................................... 112 6.6 MDO Components ............................................................................................................................................ 113 6.6.1 MDO Components as Environed by [Sobieszczanski-Sobieski] ....................................... 113 6.6.1.1 Mathematical Modeling of a System .................................................................... 113 6.6.1.2 Trade Off between Accuracy and Cost in MDO..................................................... 114 6.6.1.3 Design-Oriented Analysis ...................................................................................... 114 6.6.1.4 Approximation Concepts Applicable to MDO ....................................................... 115 6.6.1.5 System Sensitivity Analysis .................................................................................... 116 6.6.1.6 Optimization Procedures with Approximations and Decompositions .................. 117 6.6.1.7 Human Factor ........................................................................................................ 118 6.6.2 MDO Formulation as Depicted by Wikipedia .................................................................. 119 6.6.2.1 Design Variables .................................................................................................... 119 6.6.2.2 Constraints ............................................................................................................ 119 6.6.2.3 Objective................................................................................................................ 119 6.6.2.4 Models ................................................................................................................... 120 6.6.2.5 Simple Optimization .............................................................................................. 120 6.6.2.6 Problem Solution ................................................................................................... 120 6.7 Approaches to MDO for Turbomachinery Engine Applications ................................................... 120 6.7.1 Overall Design Process..................................................................................................... 121 6.7.2 Single Discipline Optimization ......................................................................................... 122 6.7.3 Aerodynamic Design Optimization for Turbomachinery ................................................. 122 6.7.3.1 Axial Compressor Gas path Optimization.............................................................. 123 6.7.3.2 Turbine Gas path Optimization ............................................................................. 124 6.7.4 Concluding Remarks ........................................................................................................ 124

List of Tables: Table 1 Performance comparison of initial an optimize airfoils (Courtesy of 44)................................... 41 Table 2 Mesh convergence study for the baseline CRM wing ......................................................................... 65 Table 3 Aerodynamic shape optimization problem............................................................................................ 65 Table 4 Design improvement for an Airfoil ............................................................................................................ 84 Table 5 Aerodynamic Sensitivity Coefficient ......................................................................................................... 84 Table 6 Axial Flow Compressor Design.................................................................................................................... 96 Table 7 Airfoil Geometry Parameters .................................................................................................................... 107 Table 8 Airfoil Design Variables ............................................................................................................................... 108

List of Figures: Figure 1 Hierarchy of models for industrial flow simulations ......................................................................... 8 Figure 2 Cp Contours on High Lift Configuration with 22 M cells model..................................................... 9 Figure 3 Aerodynamic characteristics of an airfoil ............................................................................................. 15 Figure 4 Rear fuselage shape........................................................................................................................................ 22 Figure 5 Payload – Range diagram............................................................................................................................. 22 Figure 6 Pressure distribution for wing-pylon-nacelle configuration; (Initial left), (refined right) ....................................................................................................................................................................................................... 25

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Figure 7 Over wing mounted engines configuration .......................................................................................... 25 Figure 8 Under wing mounted engines configuration ....................................................................................... 26 Figure 9 Rear fuselage mounted engines configuration ................................................................................... 27 Figure 10 leading edge droop and vortilons .......................................................................................................... 27 Figure 11 Redesigned Boeing 747 wing at Mach 0.86, Cp distributions .................................................... 29 Figure 12 Tightly coupled two level design process........................................................................................... 30 Figure 13 Global Maximum of f (x, y) ........................................................................................................................ 32 Figure 14 Different search and optimization techniques ................................................................................. 34 Figure 15 Contours of Each number for the initial airfoil (Left) an the optimize airfoil (Right)44 .. 42 Figure 16 NURBS surfaces parametrizing surface blend on fuselage (Courtesy of Vecchia & Nicolosi)..................................................................................................................................................................................... 43 Figure 17 Free-form deformation (FFD) parametrizing wing with 720 control points ...................... 44 Figure 18 Concept of using parallel evaluation strategy of feasible and infeasible solutions to guide optimization direction in a GA ............................................................................................................................. 45 Figure 19 Schematic diagram of a gradient-based aerodynamic optimization process ...................... 47 Figure 20 High performance low drag solutions found for Single and Multiple design points ........ 50 Figure 21 Non-dominated solution information for a winglet design ................................................... 51 Figure 22 Optimization Scheme (GA) ....................................................................................................................... 52 Figure 23 Hybrid Organic-Optimization Algorithm ............................................................................................ 55 Figure 24 Hybrid GA-SA Organic-Optimization Algorithm .............................................................................. 56 Figure 25 Geometry and aerodynamic performance of optimized wind turbine blades, ................... 57 Figure 26 Example structure for surrogate based optimization with a standard genetic .................. 59 Figure 27 The shape design variables are the z-displacements of 720 FFD control points ............... 62 Figure 28 Wing mesh of varying sizes ...................................................................................................................... 64 Figure 29 Sensitivity study of the baseline wing w.r.t z-direction for CD and CMY .................................. 66 Figure 30 Optimized wing is shock-free with 8.5% lower Drag. ................................................................... 67 Figure 31 Mesh Convergence study........................................................................................................................... 68 Figure 32 Insensitivity of number of optimization iterations to number of design parameters ..... 69 Figure 33 Multipoint Optimization flight conditions.......................................................................................... 70 Figure 34 Multipoint Optimized.................................................................................................................................. 71 Figure 35 Comparison of Baseline, Single, and Multipoint Optimization .................................................. 72 Figure 36 An un-symmetric wing-tip arrangement for a sweptback wing to initiate .......................... 72 Figure 37 Lift-to-Drag Ratio, L/D (Wind Tunnel and CFD Comparison) ................................................... 74 Figure 38 Optimization Strategy Loop ..................................................................................................................... 79 Figure 39 Seven control point representation of a generic Airfoil ............................................................... 80 Figure 40 Free form deformation (FFD) volume with control points (Courtesy of Kenway et al.) 81 Figure 41 Sample grid and grid sensitivity............................................................................................................. 82 Figure 42 Optimization Cycle History....................................................................................................................... 83 Figure 43 Original and Optimized Airfoil ................................................................................................................ 83 Figure 44 3D Volume grid and sensitivty w.r.t. wing root chord .................................................................. 85 Figure 45 Aerodynamic shape optimization procedure ................................................................................... 89 Figure 46 Pressure Contours ........................................................................................................................................ 91 Figure 47 General Description of Computational Planes.................................................................................. 92 Figure 48 Turbomachine Design Process................................................................................................................ 93 Figure 49 Sketch of a Compressor stage (left) and cascade of geometries at mid- span (right) ...... 97 Figure 50 Compressor design flow chart ................................................................................................................ 98 Figure 51 Preliminary Estimation of Number of Stages in Compressor .................................................... 99 Figure 52 Optimization Procedures proposed in [Massardo et al.] .......................................................... 100 Figure 53 Mach number contours a) Base line b) Max. PR. Ratio c) Max. Efficiency ............... 101 Figure 54 Comparison of blade loading prescribed by inverse mode...................................................... 102

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Figure 55 Figure 56 Figure 57 Figure 58 Figure 59 Figure 60 Figure 61 Figure 62 Figure 63

The turbine design process ................................................................................................................... 104 Parametric representation of an Airfoil .......................................................................................... 105 A sample Mach number distribution ................................................................................................. 106 Flow path of the turbine ......................................................................................................................... 109 Schematics of an airfoil showing stream lines along the radial direction ......................... 109 3D model of an airfoil showing the passage between adjacent airfoils .............................. 110 Product, components and the supporting disciplines ................................................................ 121 Aerodynamic design process for Turbomachinery ..................................................................... 122 Comparison of "baseline" and "optimized" turbine mean line results................................ 124

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1 Introduction 1.2 Complexity of Flow The complexity of fluid flow is well illustrated in Van Dyke’s Album of Fluid Motion. Many critical phenomena of fluid flow, such as shock waves and turbulence, are essentially nonlinear and the disparity of scales can be extreme. The flows of interest for industrial applications are almost invariantly turbulent. The length scale of the smallest persisting eddies in a turbulent flow can be estimated as of order of 1/Re3/4 in comparison with the macroscopic length scale. In order to resolve such scales in all three spatial dimensions, a computational grid with the order of Re 9/4 cells would be required. Considering that Reynolds numbers of interest for airplanes are in the range of 10 to 100 million, while for submarines they are in the range of , the number of cells can easily overwhelm any foreseeable supercomputer. [Moin and Kim] reported that for an airplane with 50-meter-long fuselage and wings with a chord length of 5 meters, cruising at 250 m/s at an altitude of 10,000 meters, about 10 quadrillions (1016) grid points are required to simulate the turbulence near the surface with reasonable details. They estimate that even with a sustained performance of 1 Teraflops, it would take several thousand years to RANS (1990s) simulate each second of flight time. Spalart has estimated that if computer performance continues to increase at the present rate, the Direct Numerical Euler (1980s) Simulation (DNS) for an aircraft will be feasible in 2075. Non-linear Consequently mathematical models Potential (1970s) with varying degrees of simplification have to be introduced in order to make Linear Potential computational simulation of flow (1960s) feasible and produce viable and costeffective methods. Figure 1 indicates a hierarchy of models at different levels of simplification which have proved Figure 1 Hierarchy of models for industrial flow useful in practice. Inviscid calculations simulations with boundary layer corrections can provide quite accurate predictions of lift and drag when the flow remains attached. The current main CFD tool of the Boeing Commercial Airplane Company is TRANAIR, which uses the transonic potential flow equation to model the flow. Procedures for solving the full viscous equations are needed for the simulation of complex separated flows, which may occur at high angles of attack or with bluff bodies. In current industrial practice these are modeled by the Reynolds Average Navier Stokes (RANS) equations with various turbulence models1.

1.3 Computational Cost In external aerodynamics most of the flows to be simulated are steady, at least at the macroscopic scale. Computational costs vary drastically with the choice of mathematical model. Studies of the dependency of the result on mesh refinement, performed by this author and others, have Antony Jameson and Massimiliano Fatica, “Using Computational Fluid Dynamics for Aerodynamics”, Stanford University. 1

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demonstrated that inviscid transonic potential flow or Euler solutions for an airfoil can be accurately calculated on a mesh with 160 cells around the section, and 32 cells normal to the section. Using a new non-linear Symmetric Gauss-Siedel (SGS) algorithm, which has demonstrated “text book” multigrid convergence (in 5 cycles), two-dimensional calculations of this kind can be completed in 0.5 seconds on a laptop computer (with a 2Ghz processor). A three dimensional simulation of the transonic flow over a swept wing on a 192x32x32 mesh (196,608 cells) takes 18 seconds on the same laptop. Moreover it is possible to carry out an automatic redesign of an airfoil to minimize its shock drag in 6.25 seconds, and to redesign the wing of a Boeing 747 in 330 seconds2. Viscous simulations at high Reynolds numbers require vastly greater resources. On the order of 32 mesh intervals are needed to resolve a turbulent boundary layer, in addition to 32 intervals between the boundary layer and the far field, leading to a total of 64 intervals. In order to prevent degradations in accuracy and convergence due to excessively large aspect ratios (in excess of 1,000) in the surface mesh cells, the chord wise resolution must also be increased to 512 intervals. Translated to three dimensions, this implies the need for meshes with 5-10 million cells (for example, 512x64x256 = 8,388,608 cells) for an adequate simulation of the flow past an isolated wing. When simulations are performed on less fine meshes with, say, 500,000 to 1 million cells, it is very hard to avoid mesh dependency in the solutions as well as sensitivity to the turbulence model. Currently Boeing uses meshes with 15-60 million cells for viscous simulations of commercial aircraft with their high lift systems deployed3. Figure 2 show the Cp contours on High Lift Configuration (wing) with 22 M Cells (Courtesy of Boeing). Using a multigrid algorithm, 2000 or

Figure 2

Cp Contours on High Lift Configuration with 22 M cells model

Antony Jameson and Massimiliano Fatica, “Using Computational Fluid Dynamics for Aerodynamics”, Stanford University. 3 Boing using 22 M Cells on High lift Configuration. 2

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more cycles are required to reach a steady state, and it takes 1-3 days to turn around the calculations on a 200 processor Beowulf cluster.

1.4 Aerodynamic Optimization The use of computational simulation to scan many alternative designs has proved extremely valuable in practice, but it is also evident that the number of possible design variations is too large to permit their complete evaluation. Thus it is very unlikely that a truly optimum solution can be found without the assistance of automatic optimization procedures. To ensure the realization of the true best design, the ultimate goal of computational simulation methods should not just be the analysis of prescribed shapes but the automatic determination of the true optimum shape for the intended application. The need to find optimum aerodynamic designs was already well recognized by the pioneers of classical aerodynamic theory. A notable example is the determination that the optimum span-load distribution that minimizes the induced drag of a monoplane wing is elliptic [Glauert]4, [Prandtl and Tietjens]5. There are also a number of famous results for linearized supersonic flow. The body of revolution of minimum drag was determined by Sears6, while conditions for minimum drag of thin wings due to thickness and sweep were derived by Jones7. The problem of designing a two-dimensional profile to attain a desired pressure distribution was studied by [Lighthill]8, who solved it for the case of incompressible flow with a conformal mapping of the profile to a unit circle. As an vital “ingredients” in Gradient Base Optimization, the sensitivities may now be estimated by making a small variation in each design parameter in turn and recalculating the flow. The gradient can be determined directly or indirectly by number of available methods, including Direct Differentiation (DD), Adjoin Variable(AV), Symbolic Differentiation (SD), Automatic Differentiation (AD), and Finite Difference (FD). Once the gradient has been calculated, a descent method can be used to determine a shape change that will make an improvement in the design. The gradients can then be recalculated, and the whole process can be repeated until the design converges to an optimum solution, usually within 50–100 cycles. The fast calculation of the gradients makes optimization computationally feasible even for designs in three-dimensional viscous flow. However, there is a possibility that the descent method could converge to a local minimum rather than the global optimum solution.

Glauert, H. (1926), “The Elements of Aero foil and Airscrew Theory”, Cambridge University Press. Prandtl, L. and Tietjens, O.G. (1934), “Applied Hydro and Aerodynamics”, Dover Publications. 6 Sears, W.D. (1947), ”On projectiles of minimum drag”, Q. Appl. Math., 4, 361–366. 7 Jones, R.T. (1981), ”The minimum drag of thin wings in frictionless flow”. J. Aerosol Sci., 18, 75–81. 8 Lighthill, M.J. (1945), ”A new method of two dimensional aerodynamic design”. Rep. Memor. Aero. Res. Coun. Lond., 2112, 143–236. 4 5

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2 Airplane Design 2.1 Role of CFD in the Design Process

The actual use of CFD by Aerospace companies is a consequence of the trade-off between perceived benefits and costs. While the benefits are widely recognized, computational costs cannot be allowed to swamp the design process9. The need for rapid turnaround, including the setup time, is also crucial. In current industrial practice, the design process can generally be divided into three phases: 1. Conceptual Design, 2. Preliminary Design, and 3. Detailed Design. 2.1.1 Conceptual Design The conceptual design stage, typically carried out by a staff of 15 - 30 engineers, defines the mission in the light of anticipated market requirements, and determines a general preliminary configuration, together with first estimates of size, weight and performance. The costs of this phase, depending on application (i.e., airplane configuration), and costs in the range of $M 6-12. 2.1.2 Preliminary Design In the preliminary design stage the aerodynamic shape and structural skeleton progress to the point where detailed performance estimates can be made and guaranteed to potential customers, who can then, in turn, formally sign binding contracts for the purchase of a certain number of aircraft. A staff of 100-300 engineers is generally employed for up to 2 years, at a cost of $M60 - 120 (again the same application). Initial aerodynamic performance is explored by computational simulations and through wind tunnel tests. While the costs are still fairly moderate, decisions made at this stage essentially determine both the final performance and the development costs. 2.1.3 Detailed or Final Design In the final design stage the structure must be defined in complete detail, together with complete systems, including the flight deck, control systems (involving major software development for flyby-wire systems), avionics, electrical and hydraulic systems, landing gear, weapon systems for military aircraft, and cabin layout for commercial aircraft. Major costs are incurred at this stage, during which it is also necessary to prepare a detailed manufacturing plan. Thousands of engineers define every part of the aircraft. Total costs are $B 3-10. Therefore, the final design would normally be carried out only if sufficient orders have been received to indicate a reasonably high probability of recovering a significant fraction of the investment10.

2.2 Conceptual Aerodynamic Design Process as Applied to Airplanes

In the development of commercial aircraft, aerodynamic design plays a leading role during the conceptual and preliminary design stage, Ultimately, the definition of the external aerodynamic shape is typically finalized after a detailed analysis. 2.2.1 Purpose and Scope of Conceptual Airplane Design The process of design of a device or a vehicle, in general involves the use of knowledge in diverse fields to arrive at a product that will satisfy requirements regarding functional aspects, operational safety and cost. The design of an airplane, which is being dealt in this course, involves synthesizing Antony Jameson and, assisted by, Kui Ou, “Optimization Methods in Computational Fluid Dynamics”, Aeronautics and Astronautics Department, Stanford University, Stanford, CA, USA. 10 See Previous. 9

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knowledge in areas like aerodynamics, structures, propulsion, systems and manufacturing techniques. The aim is to arrive at the configuration of an airplane, which will satisfy abovementioned requirements. The design of an airplane is a complex engineering task. which generally involves the following among others:            

Obtaining the specifications of the airplane, selecting the type and determining; Aerodynamic Considerations; Wing design; Optimization of wing loading and thrust loading; Fuselage design, preliminary design of tail surface and preliminary layout; Center of Gravity calculation; the geometric parameters for different surfaces; Preliminary weight estimation; Estimates of areas for horizontal and vertical tails; Engine Selection; Detail Structural design; Determination of airplane performance, stability, and structural integrity from flight tests.

The aerodynamic lines of the Boeing 777 were frozen, for example, when initial orders were accepted, before the initiation of the detailed design of the structure. The starting point is an initial CAD definition resulting from the conceptual design. The inner loop of aerodynamic analysis is contained in an outer multi-disciplinary loop, which is in turn contained in a major design cycle involving wind tunnel testing. In recent Boeing practice, three major design cycles, each requiring about 4-6 months, have been used to finalize the wing design. Improvements in CFD, might allow the elimination of a major cycle, would significantly shorten the overall design process and reduce costs. Moreover, the improvements in the performance of the final design, which might be realized through the systematic use of CFD, could have a crucial impact. 2.2.2 Cost Estimation An improvement of 5 percent in lift to drag (L/D) ratio directly translates to a similar reduction in fuel consumption. With the annual fuel costs of a long-range airliner in the range of $5-10 million, a 5 percent saving would amount to a saving of the order of $10 million over a 25 year operational life, or $5 billion for a fleet of 500 aircraft. In fact an improvement in L/D enables a smaller aircraft to perform the same mission, so that the actual reduction in both initial and operating costs may be several times larger. Furthermore a small performance advantage can lead to a significant shift in the share of a market estimated to be more than $1 trillion over the next decades. 2.2.3 Preliminary Weight Estimation An accurate estimate of the weight of the airplane is required for the design of the airplane. This is arrived at in various stages. In the last chapter, the procedure to obtain the first estimate of the gross weight was indicated. This was based on the ratio of the payload to the gross weight of similar airplanes. This estimate of the gross weight is refined by estimating (a) the fuel fraction i.e. weight of fuel required for the proposed mission of the airplane, divided by gross weight and (b) empty weight fraction i.e. empty weight of airplane divided the gross weight. 2.2.4 Range Estimation A good first estimate of performance is provided by the Breguet range equation:

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Range 

W  Wf VL 1 log 0 D SFC W0

Eq. 2.1

Here V is the speed, L/D is the lift to drag ratio, SFC is the specific fuel consumption of the engines, W0 is the loading weight (empty weight + payload + fuel resourced), and Wf is the weight of fuel burnt. Eq. 2.1 displays the multidisciplinary nature of design. A light structure is needed to reduce W0. SFC is the province of the engine manufacturers. The aerodynamic designer should try to maximize VL/D. This means the cruising speed V should be increased until the onset of drag rise at a Mach Number M = V/C ∼ 0.85. But the designer must also consider the impact of shape modifications in structure weight11. An excellent discussion of these and other factors in design of airplane, is documented by [Tulapurkara]12 and readers encourage to consult the on-line material. 2.2.5 Aerodynamic Considerations A poorly designed external shape of the airplane could result in undesirable flow separation resulting in low CLmax, low lift to drag ratio and, large transonic and supersonic wave drag. Following remarks can be deducted:  Minimization of wetted area is an important consideration as it directly affects skin friction drag and in turn parasite drag. One way to achieve this is to have smallest fuselage diameter and low excellence ratio (between 3 and 4). However, proper space for payload, ease of maintenance and tail arm also needs to be considered.  To prevent flow separation, the deviation of fuselage shape from free stream direction should not exceed 10 – 12 degrees .  Proper fillets should be used at junctions between  wing and fuselage,  fuselage and tails and  wing and pylons.  Base area should be minimum.  Canard, if used, should be located such that its wake does not enter the engine inlet as it may cause engine stalling.  Area ruling The plan view of supersonic airplanes indicates that the area of cross section of fuselage is decreased in the region where wing is located. This is called area ruling. A brief note on this topic is presented below. It was observed that the transonic wave drag of an airplane is reduced when the distribution of the area of cross section of the airplane, in planes perpendicular to the flow direction, has a smooth variation. In this context, it may be added that the area of cross section of the fuselage generally varies smoothly. However, when the wing is encountered there is an abrupt change in the cross sectional area. This abrupt change is alleviated by reduction in the area of cross section of fuselage in the region where the wing is located. Such a fuselage shape is called ‘Coke-bottle shape’. 2.2.6 Wing Design and Selection of Wing Parameters In the context of wing design the following aspects need consideration:

Antony Jameson, “Airplane Design with Aerodynamic Shape Optimization”, Aeronautics & Astronautics Department, Stanford University, 2010. 12 E.G. Tulapurkara, ”Airplane design(Aerodynamic)”, Dept. of Aerospace Engineering., Indian Institute of Technology, Madras, India. 11

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 Wing area (S) : This is calculated from the wing loading and gross weight which have been already decided i.e. S = W/(W/S)  Location of the wing on fuselage : High-, low- or mid-wing  Airfoil : Thickness ratio, camber and shape  Sweep : Whether swept forward, swept backward, angle of sweep, cranked wing, variable sweep.  Aspect ratio : High or low, winglets  Taper ratio : Straight taper or variable taper.  Twist: Amount and distribution  Wing incidence or setting  High lift devices : Type of flaps and slats; values of CLmax, Sflap/S  Ailerons and spoilers : Values of Saileron/S; Sspoiler/S  Leading edge strakes if any;  Dihedral angle.  Other aspects : Variable camber, planform tailoring, area ruling, braced;  Wing, aerodynamic coupling (intentionally adding a coupling lifting surface like canard). 2.2.6.1 Airfoil Selection Large airplane companies like Boeing and Airbus may design their own airfoils. However, during the preliminary design stage, the usual practical is to choose the airfoil from the large number of airfoils whose geometric and aerodynamic characteristics are available in the aeronautical literature. To enable such a selection it is helpful to know the aerodynamic and geometrical characteristics of airfoils and their nomenclature. 2.2.6.2 Presentation of Aerodynamic Characteristics of Airfoils Figure 5.1 shows typical experimental characteristics of an airfoil. The features of the three plots in this figure can be briefly described as follows.   



Lift coefficient vs angle of attack. Drag coefficient (CD) vs Lift Coefficient (CL) Pitching moment coefficient about quarter-chord vs. Angle of attack . Stall pattern : Variation of the lift coefficient with angle of attack near the stall is an indication of the stall pattern

2.2.6.3 Geometrical Characteristics of Airfoils In this procedure, the camber line or the mean line is the basic line for definition of the airfoil shape (Figure 3A). The line joining the extremities of the camber line is the chord. The leading and trailing edges are defined as the forward and rearward extremities, respectively, of the mean line. Various camber line shapes have been suggested and they characterize various families of airfoils. The maximum camber as a fraction of the chord length (ycmax/c) and its location as a fraction of chord (xycmax/c) are the important parameters of the camber line. Various thickness distributions have been suggested and they characterize different families of airfoils Figure 3B. The maximum ordinate of the thickness distribution as fraction of chord (ytmax/c) and its location as fraction of chord (xytmax/c) are the important parameters of the thickness distribution. 2.2.6.4 Airfoil Shape and Ordinates The airfoil shape is obtained by combining the camber line and the thickness distribution in the following manner, according to Figure 3-C. First, draw the camber line shape and draw lines perpendicular to it at various locations along the chord. Then, lay off the thickness distribution along the lines drawn perpendicular to the mean line. Finally, the coordinates of the upper surface (xu, yu)

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and lower surface (xl, yl) of the airfoil are given by the four equations presented as :

x u  x  y t sinθ

y u  y c  y t cosθ

x l  x  y t sinθ

y l  y c  y t cosθ

Eq. 2.2

where yc and yt are the ordinates, at location x, of the camber line and the thickness distribution respectively; tan θ is the slope of the camber line at location x (see also Figure 3-C & D). The leading edge radius is also prescribed for the airfoil. The center of the leading edge radius is located along the tangent to the mean line at the leading edge. Depending on the thickness distribution, the trailing

A&B

C&D Figure 3

Aerodynamic characteristics of an airfoil

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edge angle may be zero or have a finite value. In some cases, thickness may be non-zero at the trailing edge. 2.2.6.5 Airfoil Nomenclature Early airfoils were designed by trial and error. Royal Aircraft Establishment (RAE), UK and Gottingen laboratory of the German establishment which is now called DLR(Deutsches Zentrum fϋr Luft-und Raumfahrt – German Centre for Aviation and Space Flight) were the pioneers in airfoil design. Taking advantage of the developments in airfoil theory and boundary layer theory, NACA (National Advisory Committee for Aeronautics) of USA systematically designed and tested a large number of airfoils in 1930’s. These are designated as NACA airfoils. In 1958 NACA was superseded by NASA (National Aeronautics and Space Administration). This organization has developed airfoils for special purposes. These are designated as NASA airfoils. A brief description of their nomenclature is presented below. 2.2.6.6 NACA Four-Digit Series Airfoils Earliest NACA airfoils were designated as four-digit series. The thickness distribution was based on successful RAE & Gottigen airfoils. It is given as :  yt 



t 0.2969 x  0.1260x  0.3516x 2  0.2843x 3  0.1015x 5 20



Eq. 2.3

where, t = maximum thickness as fraction of chord. The leading radius is : rt = 1.1019 t2. Figure 3-b shows the shape of NACA 0009 airfoil. It is a symmetrical airfoil by design. The maximum thickness of all four-digit airfoils occurs at 30% of chord. In the designation of these airfoils, the first two digits indicate that the camber is zero and the last two digits indicate the thickness ratio as percentage of chord. The camber line for the four-digit series airfoils consists of two parabolic arcs tangent at the point of maximum ordinate. The expressions for camber(yc) are :





m 2px - x 2 x  x ycmax 2 p m  (1 - 2p)  2px - x 2 2 (1 - p)

yc 





x  x ycmax

Eq. 2.4

Where m = maximum ordinate of camber line as fraction of chord and p = chord wise position of maximum camber as fraction of chord. The camber lines obtained by using different values of m & p are denoted by two digits, e.g. NACA 64 indicates a mean line of 6% camber with maximum camber occurring at 40% of the chord. A cambered airfoil of four-digit series is obtained by combining mean line and the thickness distribution as described in the previous subsection. For example, NACA 2412 airfoil is obtained by combining NACA 24 mean line and NACA 0012 thickness distribution. This airfoil has (a) maximum camber of 2% occurring at 40% chord and (b) maximum thickness ratio of 12%. 2.2.6.7 NACA Five-Digit Series Airfoils During certain tests it was observed that CLmax (Max. Lift Coeficient) of the airfoil could be increased by shifting forward the location of the maximum camber. This finding led to development of five-digit series airfoils. The new camber lines for the five-digit series airfoils are designated by three digits. The same thickness distribution was retained as that for NACA four-digit series airfoils. The camber line shape is given as :

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

1 y c  k1 x 3 - 3mx 2  m 2 (3  m)x 6 1  k1m3 1 - x  m  x  1 6



0xm Eq. 2.5

The value of ‘m’ decides the location of the maximum camber and that of k1 the design lift coefficient. A combination of m = 0.2025 and k1 = 15.957 gives li C = 0.3 and maximum camber at 15% of chord. This mean line is designated as NACA 230. The first digit ‘2’ indicates that CL = 0.3 and the subsequent two digits (30) indicate that the maximum camber occurs at 15% of chord. A typical five-digit cambered airfoil is NACA 23012. The digits signify : First digit(2) indicates that li CL = 0.3. Second & third digits (30) indicate that maximum camber occurs at 15% of chord. Last two digits (12) indicate that the maximum thickness ratio is 12%. 2.2.6.8 Six Series Airfoils As a background to the development of these airfoils the following points may be mentioned. In 1931 [T.Theodorsen] presented ’Theory of wing sections of arbitrary shape’ NACA TR 411, which enabled calculation flow past airfoils of general shape. Around the same time the studies of [Tollmien and Schlichting] on boundary layer transition, indicated that the transition process, which causes laminar boundary layer to become turbulent, depends predominantly on the pressure gradient in the flow around the airfoil. A turbulent boundary layer results in a higher skin friction drag coefficient as compared to when the boundary layer is laminar. Hence, maintaining a laminar boundary layer over a longer portion of the airfoil would result in a lower drag coefficient. Inverse methods, which could permit design of mean line shapes and thickness distributions, for prescribed pressure distributions were also available at that point of time. Taking advantage of these developments, new series of airfoils called low drag airfoils or laminar flow airfoils were designed. These airfoils are designated as 1-series, 2-series,…….,7-series. Among these the six series airfoils are commonly used airfoils. When the airfoil surface is smooth, these airfoils have a CDmin which is lower than that for four-and five-digit series airfoils of the same thickness ratio. Further, the minimum drag coefficient extends over a range of lift coefficient. This extent is called drag bucket. The thickness distributions for these airfoils are obtained by calculations which give a desired pressure distribution. Analytical expressions for these thickness distributions are not available. However, the camber lines are designated as : a = 0, 0.1, 0.2 …., 0.9 and 1.0. For example, the camber line shape with a = 0.4 gives a uniform pressure distribution from x/c = 0 to 0.4 and then linearly decreasing to zero at x/c = 1.0. If the camber line designation is not mentioned, ‘a’ equal to unity is implied. It is obtained by combining NACA 662 – 015 thickness distribution and a = 1.0 mean line. 2.2.6.9 NASA Airfoils NASA has developed airfoil shapes for special applications. For example GA(W) series airfoils were designed for general aviation aircraft. The ‘LS’ series of airfoils among these are for low speed airplanes. A typical airfoil of this category is designated as LS(1) - 0417. In this designation, the digit ‘1’ refers to first series, the digits ‘04’ indicate CLOPT of 0.4 and the digits ‘17’ indicate the thickness ratio of 17%. Figure 5.3e shows the shape of this airfoil. For the airfoils in this series, specifically designed for medium speed airplanes, the letters ‘LS’ are replaced by ‘MS’. NASA NLF series airfoils are ‘Natural Laminar Flow’ airfoils. NASA SC series airfoils are called ‘Supercritical airfoils’. These airfoils have a higher critical Mach number.

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2.2.7 Estimation of Wing Loading & Thrust Loading The wing loading (W/S) and the thrust loading (T/W) or power loading (W/P) are the two most important parameters affecting the airplane performance. It may be recalled that for airplanes with jet engines, the parameter characterizing engine output is the thrust loading (T/W) and for airplane with engine-propeller combination the parameter characterizing the engine output is the power loading (W/P). It is essential that good estimates of (W/S) & (T/W) or (W/P) are available before the initial layout is begun. The approaches for estimation of (W/S) and (T/W) or (W/P) can be divided into two categories.  In the approach given by [Lebedinski], the variations, of the following quantities are obtained when the wing loading is varied.  (T/W) or (W/P) required for prescribed values of flight speed, absolute ceiling, (R/C)max and output of a piston engine.  Weight of the fuel (Wf) required for a given range.  Distance required for landing. From these variations, the wing loading which is optimum for each of these items is obtained. However, the optimum values of W/S in various cases are likely to be different. The final wing loading is chosen as a compromise.  In the approach followed by [Raymer], (T/W) or (P/W) is chosen from statistical data correlations and then W/S is obtained from the requirements regarding V max, (Range)max, maximum based on rate of climb, absolute ceiling, maximum rate of turn, landing distance and take-off distance. Finally, W/S is chosen such that the design criteria are satisfied. 2.2.8 Structural Considerations Primary concern in the design process is to obtain an airplane with low structural weight. This is achieved by provision of efficient load path i.e. structural elements by which the opposing forces are connected. It may be recalled that the structural members are of the following types.     

Struts which take tension Columns which take compressive load Beams which transfer normal loads Shafts which transmit torsion Levers which transfer the load along with change of direction.

The most efficient way of transmitting the load is when the force is transmitted in an axial direction. In the case of airplane the lift acts vertically upwards and the weights of various components and the payload act vertically downwards. In this situation, the sizes and weights of structural members are minimized or the structure is efficient if opposing forces are aligned with each other. This has led to the flying wing or blended wing-body concept in which the structural weight is minimized as the lift is produced by the wing and the entire weight of the airplane is also in the wing. However, in a conventional airplane the payload and systems are in the fuselage. The wing produces the lift and as a structural member it behaves like a beam. Hence to reduce the structural weight, the fuel tanks, engines and landing gears are located on the wing, as they act as relieving load. Reduction in number of cutouts and access holes, consistent with maintenance requirements, also reduces structural weight.

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2.2.9 Environmental Impacts In recent years factors like aircraft noise, emissions and ecological effects have acquired due importance and have begun to influence airplane lay out. Following remarks can be made: 2.2.9.1 Airplane Noise Noise during the arrival and departure of the airplane affects the community around the airport. The noise is generated by:  The engines,  Parts of the airframe like control surfaces and high lift devices which significantly change the airflow direction.  Projections in airflow like landing gear and spoilers. Considerable research has been carried out to reduce the engine noise. High by-pass ratio engines with lobed nozzle have significantly lowered the noise level. Noise level inside the cabin has to be minimal. This is achieved by suitable noise insulation. Further, the clearance between cabin and the propeller should not be less than the half of the radius of the propeller. 2.2.9.2 Emissions Combustion of the fuel in an engine produces carbon dioxide, water vapor, various oxides of nitrogen (NOx), carbon monoxide, unburnt hydrocarbons and Sulphur dioxide (SO2). The components other than carbon dioxide and water vapor are called pollutants. The thrust setting changes during the flight and hence the emission levels have to be controlled during landing, takeoff and climb segment up to 3000 ft (1000 m). At high altitudes the NOx components may deplete ozone layer. Hence, supersonic airplanes may not be allowed to fly above 50000 ft (15 km) altitude. It may be noted that cruising altitude for Concorde was 18 km. Improvements in engine design have significantly reduced the level of pollutants. The amount of pollution caused by air transport is negligible as compared to that caused by road transport, energy generation and industry. However, the aircraft industry has always been responsive to the ecological concerns and newer technologies have emerged in the design of engine and airframe. 2.2.10 Performance Estimation The performance analysis includes the following:  The variation of stalling speed (VS) at various altitudes.  Variations with altitude of maximum speed (Vmax) and minimum speed from power output consideration (Vmin)Power. The minimum speed of the airplane at an altitude will be the higher of VS and (Vmin)Power. The maximum speed and minimum speed will decide the flight envelope.  Variations with altitude of the maximum rate of climb and maximum angle of climb ; the flight being treated as steady climb.  To arrive at the cruising speed and altitude, choose a range of altitudes around the cruising altitude mentioned in the specifications. At each of these altitudes obtain the range in constant velocity flights choosing different velocities. The information on appropriate values of specific fuel consumption (SFC) can be obtained from the engine charts. The values of range obtained at different speeds and altitudes be plotted as range vs velocity curves with altitude as parameter. Draw an envelope of these curves. The altitude and velocity at which the range is maximum can be considered as the cruising speed (Vcruise) and cruising altitude (hcruise). These curves also give information about the range of flight speeds and altitudes around Vcruise and hcruise at which near optimum performance is obtained.

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 The maximum rate of turn and the minimum radius of turn in steady level turn depend on the thrust available, and the permissible load factor. The value of CLmax used here is that without the flaps. For high speed airplanes the value of CLmax depends also on Mach number;  Take - off run and take - off distance: During take-off an airplane accelerates on the ground. For an airplane with nose wheel type of landing gear, around a speed of 85% of the take-off speed, the pilot pulls the stick back. Then, the airplane attains the angle of attack corresponding to take-off and the airplane leaves the ground. The point at which the main wheels leave the ground is called the unstick point and the distance from the start of take-off point to the unstick point is called the ground run. After the unstick, the airplane goes along a curved path as lift is more than the weight. This phase of take-off is called transition at the end of which the airplane climbs along a straight line. The take-off phase is said to be over when the airplane attains screen height which is generally 15 m above the ground. The horizontal distance from the start of the take off to the where the airplane attains screen height is called take off distance. The takeoff run and the take-off distance can be estimated by writing down equations of motion in different phases.  Landing Distance: The landing flight begins when the airplane is at the screen height at a velocity called the approach speed. During the approach phase the airplane descends along a flight path of about 3 degrees. Subsequently the flight path becomes horizontal in the phase called ‘flare’. In this phase the pilot also tries to touch the ground gently. The point where the main wheels touch the ground is called touch down point. Subsequent to touch down, the airplane rolls along the ground for about 3 seconds during which the nose wheel touches the ground. This phase is called free roll. After this phase the brakes are applied and the airplane comes to halt. In some airplanes, thrust in the reverse direction is produced by changing the direction of jet exhaust or by reversible pitch propeller. In some airplanes, the drag is increased by speed brakes, spoilers or parachutes. For airplanes which land on the deck of the ship, an arresting gear is employed to reduce the landing distance. The horizontal distance from the start of approach at screen height till the airplane comes to rest is called landing distance. 2.2.10.1 General Remarks on Performance Estimation I) operating envelope: The maximum speed and minimum speed can be calculated from the level flight analysis. However, the attainment of maximum speed may be limited by other considerations. The operating envelope for an airplane is the range of flight speeds permissible at different altitudes. Typical operating envelope for a military airplane is shown in [Tulapurkara]13 where reader are encouraged for detailed view of subject. II) Energy height technique for climb performance: The analysis of a steady climb shows that the velocity corresponding to maximum rate of climb increases with altitude. Consequently, climb with involves acceleration and the rate of climb will actually be lower than that given by the steady climb analysis. This is because a part of the engine output would be used to increase the kinetic energy. Secondly, the aim of the climb is to start from velocity near and at and attain a velocity near at h. To take these aspects into account, it is more convenient to work in terms of energy height (he) instead of height(h). The quantity he is defined as :

 V2  h e  h     2g 

 WV 2   Multiplyby W  Wh e  Wh    2g 

Eq. 2.6

E.G. Tulapurkara, ”Airplane design(Aerodynamic)”, Dept. of Aerospace Engineering., Indian Institute of Technology, Madras, India. 13

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The right hand side of the Eq. 2.6 is the sum of the potential energy and the kinetic energy of the airplane. It is denoted by E. The energy height (he) which is E / W, is also called specific energy. It can be shown that (dhe/ dt) = (TV – DV)/W and is referred to as specific excess power (Ps). Using energy height concept the optimum climb path for fastest climb or economical climb can be worked out. III) Range performance: For commercial airplanes the range performance is of paramount importance. Hence, range performance with different amounts of payload and fuel on board the airplane, needs to be worked out. In this context the following three limitations should to be considered. a) Maximum payload: The number of seats and the size of the cargo compartment are limited. Hence maximum payload capacity is limited. b) Maximum fuel: The size of the fuel tanks depends on the space in the wing and the fuselage to store the fuel. Hence, there is limit on the maximum amount of fuel that can be carried by the airplane. c) Maximum take-off weight: The airplane structure is designed for a certain load factor and maximum take-off weight. This value of weight cannot be exceeded. Keeping these limitations in mind a typical payload vs. range curve is shown in Figure 4 where a brief explanation as follows: Point A represents the maximum payload. As the fuel is added the range can increase as represented by line AB. At point B, the limit of the maximum take-off weight is reached. If it is desired that the range should increase further, then the payload has to be decreased so that the maximum take-off weight limit is not exceeded as represented by line BC. At point C, the maximum fuel capacity limit is also reached. If it is desired to increase the range further, the payload has to be decreased as fuel volume cannot be increased. Point D represents zero-payload condition. The range represented by point D is called ‘ferry range’. This type of analysis is used to obtain various combinations of payload and range under different flight operations. 2.2.10.2 Fuselage and Tail Sizing The primary purpose of the fuselage is to house the payload. As mentioned earlier, the payload is the part of useful load from which the revenue is derived or for which the airplane is designed. In transport airplanes the payload includes the passengers, their luggage and cargo. In military airplanes it is the ammunition and /or special equipment. In addition to the payload, the fuselage accommodates the following. In addition, the flight crew and the cabin crew in the transport airplane and the specialist crew members in airplanes used for reconnaissance, patrol and remote sensing. Also, fuel, engine and landing gear when they are housed inside the fuselage. Systems like airconditioning system, pressurization system, hydraulic system, electrical system, pneumatic system, electronic systems, emergency oxygen, floatation vests and auxiliary power unit. Jet airplanes cruise at altitudes of 10 to 14 km. The temperature and pressure are low at these altitudes. For the a pressure corresponding 8000 ft (2438 m) in ISA is maintained in these portions of the fuselage. The shell of the fuselage has to be designed to withstand the pressure difference between inside and outside the cabin. Secondly, to isolate the cockpit and cabin, from ambient conditions, the cabin is terminated with a pressure bulk head. The auxiliary power unit to engines and to supply power to accessories when the engines are off. 2.2.10.3 Tail cone/Rear Fuselage: At the end of subsection 6.2.1, some remarks have been made regarding the tail cone of a general aviation aircraft. Further, in the case of a passenger airplane the mid-fuselage has a cylindrical shape and is followed by the tail cone or rear fuselage of a tapering shape. In passenger airplanes the tail cone is of substantial length and the cabin layout extends into the rear fuselage. Galleys, toilets and storage compartments are also located here along with the auxiliary power unit (APU). The rear fuselage also supports the horizontal and vertical tail surfaces and the engine installation for rear

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mounted engines. The lower side of the rear fuselage should provide adequate clearance (about 0.15 m) for airplane during take-off and landing attitude (Figure 5). The length of the rear fuselage and upsweep angle are also affected by (a) the height of the main landing gear and (b) the length of the mid-fuselage after the main landing gear. For passenger airplanes (a) the ratio of length of the rear fuselage to the equivalent diameter of the mid-fuselage is between 2.5 to 3.5 and (b) the upsweep angle is between 15 to 20 degrees. For Boeing 777-300 this angle is 17 degrees.

Figure 5

Rear fuselage shape

2.2.11 Estimation of Wing and Thrust Loading Based on Conception Design The wing loading and the thrust loading or the power loading influence a number of performance items like take-off distance, maximum speed (Vmax) , maximum rate of climb (R/C)max, absolute ceiling (Hmax) and maximum rate of turn. Thus, they are the two most important parameters affecting the airplane performance. It may be recalled that for airplanes with jet engines, the parameter characterizing engine output is the thrust loading (T/W) and for airplane with engine-propeller combination the parameter characterizing the engine output is the power loading (W/P). It is essential that good estimates of (W/S) & (T/W) or (W/P) are available before the initial layout is begun. The approaches for estimation of (W/S) and (T/W) or (W/P) can be divided into two categories.  In the approach given by [Lebedinski], the variations, of the following quantities are obtained when the wing loading is varied. 





(T/W) or (W/P) required for prescribed values of Vp, Hmax (R/C)max and sto. Weight of the fuel (Wf) required for a given range (R). Distance required for landing.

Figure 4

Payload – Range diagram

23



From these variations, the wing loading which is optimum for each of these items is obtained. However, the optimum values of W/S in various cases are likely to be different. The final wing loading is chosen as a compromise.

 In the approach followed by [Raymer], (T/W) or (P/W) is chosen from statistical data correlations and then W/S is obtained from the requirements regarding Vmax, Rmax, (R/C)max, Hmax, max ψ , landing distance and take-off distance. Finally, W/S is chosen such that the design criteria are satisfied. These two approaches are described in the subsequent sections. 2.2.11.1 Remarks on for choosing Wing Loading and Thrust Loading or Power Loading It is felt that the approach presented by [Lebedinski] about 50 years ago, is still relevant. The main features are:     

Derive simplified relations between the chosen performance parameter and the wing loading. Obtain the wing loading which satisfies/optimizes the chosen parameter e.g. landing distance, thrust required for Vp, fuel required for range. Examine the influence of allowing small variations in wing loading from the optimum value and obtain a band of wing loadings. This would give an estimate of the compromise involved when (W/S) is non-optimum. After all important cases are examined, choose the final wing loading as the best compromise. With the chosen wing loading, obtain (T/W) or (W/P) which satisfy requirements of Vmax, (R/C)max, ceiling (Hmax), take-off field length ( to s ) and maximum turn rate (ψmax). If the requirements of engine output in these cases are widely different, then examine possible compromise in specification. After deciding the (T/W) or (W/P) obtain the engine output required. Choose the number of engine(s) and arrive at the rating per engine. Finally choose an engine from the engines available from different engine manufacturers.

During the process of optimizing the wing loading, a reasonable assumption is to ignore the changes in weight of the airplane (W0). However, when W0 is constant but W/S changes, the wing area and in turn, the drag polar would change. This is taken into account by an alternate representation of the drag polar. 2.2.11.2 Selection of Wing Loading based on Landing Distance Landing distance (Sland) is the horizontal distance the airplane covers from being at the screen height till it comes to a stop. The approach to landing begins at the screen height of 50’(15.2 m). The flight speed at this point is called ‘Approach speed’ and denoted by VA. The glide angle during approach is generally 3o. Then, the airplane performs a flare to make the flight path horizontal and touches the landing field at touch down speed (VTD). Subsequently, the airplane rolls for a duration of about 3 seconds and then the brakes are applied. The horizontal distance covered from the start of the approach till the airplane comes to a halt is the landing field length. (i) It may be added that in actual practice the airplane does not halt on the runway. After reaching a sufficiently low speed the pilot takes the airplane to the allotted parking place. (ii) Landing ground run is the distance the airplane covers from the point the wheels first touch the ground to the point the airplane comes to a stop. (iii) VA = 1.3(Vs)land, VTD = 1.15(Vs) land (4.1) (Vs)land is the stalling speed in landing configuration. Exact estimation of landing distance (sland) is difficult as some phases like flare depend on the piloting technique. based on consideration of landing distance.

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2.2.11.3 Wing Loading from Landing Consideration based on Take-off Weight The wing loading (W/S) of the airplane is always specified with reference to the take-off weight (WTO). Hence, the wing loading from landing consideration, based on take-off weight, is

W  (W/S) land  p land  T0   Wland 

Eq. 2.7

The weight of the airplane at the time of landing (Wland) is generally lower than WTO. The difference between the two weights is due to the consumption of fuel and dropping of any disposable weight. However, to calculate Wland only a part of the fuel weight is subtracted, from the takeoff weight. 2.2.12 Stability and controllability The ability of a vehicle to maintain its equilibrium is termed stability and the influence which the pilot or control system can exert on the equilibrium is termed its controllability. The basic requirement for static longitudinal stability of any airplane is a negative value of dCmcg /dCL. Dynamic stability requires that the vehicle be not only statically stable, but also that the motions following a disturbance from equilibrium be such as to restore the equilibrium. Even though the vehicle might be statically stable, it is possible that the oscillations following a disturbance might increase in magnitude with each oscillation, thereby making it impossible to restore the equilibrium (like weather). 2.2.12.1 Static longitudinal stability and control The horizontal tail must be large enough to insure that the static longitudinal stability criterion, dCmcg/dCL is negative for all anticipated center of gravity positions. An elevator should be provided so that the pilot is able to trim the airplane (maintain Cm = 0) at all anticipated values of CL. The horizontal tail should be large enough and the elevator powerful enough to enable the pilot to rotate the airplane during the take-off run, to the required angle of attack. This condition is termed as the nose wheel lift-off condition. For detailed view of this topics and more, please consult [Tulapurkara]14.

2.3 Aerodynamic Design and Analysis Coupling for Wing The Reynolds-Averaged Navier-Stokes equations can represent most of the flow phenomena of practical interest associated with complex aircraft configurations. In this respect, they could easily handle the analysis and design of transport wings in the transonic and in the low-speed, high-lift, separated-flow regimes. Their disadvantages lay usually on the long times involved in preparing suitable computational grids and solving the equations themselves. If viscous phenomena are not important for a particular case, the viscosity can be set to zero and an Euler analysis can be performed on a coarser grid in a considerably reduced time. A 2D/3D RANS/Euler code for detailed analyses of complex geometric configurations such as wing plus pylons and nacelles. Figure 6 present an example of this application where transonic winglet design is performed with the 3D Euler and N-S code. 2D analyses are also performed for airfoils in situations where large separated regions are present, such as an airfoil with a deployed spoiler15.

E.G. Tulapurkara, ”Airplane design(Aerodynamic)”, Dept. of Aerospace Engineering., Indian Institute of Technology, Madras, India. 15 O. C. de Resende, “The Evolution of the Aerodynamic Design Tools and Transport Aircraft Wings at Embraer”, J. of the Brazilian Soc. of Mech. Sci. & Engineering, October-December 2004. 14

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Figure 6

Pressure distribution for wing-pylon-nacelle configuration; (Initial left), (refined right)

According to [EMBRAER], a civil transport aviation company out of Brazil, three major different aerodynamic configurations were extensively studied during the development phase; straight wing with over wing mounted engines (Figure 7), swept wing with underwing mounted engines and swept wing with rear fuselage mounted engines. underwing engine configuration to be abandoned (Figure 8). and the third major aerodynamic configuration had the engines mounted on pylons on the rear fuselage (Figure 9). 2.3.1

The Straight Wing Configuration The initial configuration was directly derived from the turbofan engines mounted over the wings approximately at the same position of the original turboprops (see Figure 7). The center fuselage was stretched to carry 45 passengers, thus resulting in its designation. The straight tapered un-swept wing was derived from the Brasilia's. Figure 7 Over wing mounted engines configuration The wing rear part was kept, including the original rear and front spars, but the entire leading edge was extended to reduce the airfoil maximum relative thickness from 16% to 14% at the root and from 12% to 10% at the tip. An additional front spar was also introduced, the wing span was increased and winglets were installed. Initially, the design cruise Mach number was M=0.70, but during development it was raised to M=0.75 to provide a cruise performance differential in respect to that of competing new generation turboprops. The advantages of the configuration would be its low development and production costs (using modified Brasilia tooling and jigs), superior performance and comfort in respect to turboprops and reduced acquisition and operating costs in respect to other regional jets then in development. However, transonic wind tunnel tests indicated higher than expected drag at M= 0.75 and the modified wing was also found to be heavier than originally estimated. The aerodynamic analysis and design tools available at the time

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(full potential 2D airfoil code with coupled boundary layer, 3D inviscid wing full potential code and a 3D panel method) were not capable of calculating the unfavorable aerodynamic interference between the jet exhaust and the supersonic flow on the wing upper surface. Although there was prior qualitative knowledge of the phenomenon and the associated risks, it took a transonic wind tunnel test. 2.3.2 The Swept Wing Configuration The second configuration had an entirely new wing with approximately 26 degrees of leading edge sweep. The engines were mounted in pylons under the wings, requiring taller landing gears (see Figure 8). The fuselage was stretched to carry 48 passengers and the nose was extended to accommodate the longer landing gear leg. The design cruise Mach number was raised to around M = 0.80 to 0.82. The wing was designed using the available full potential transonic 2D and 3D codes. The resulting transonic airfoils had moderate rear loading, being of the type commonly called 'supercritical' due to the large region (typically from 10% to 70% chord) of supersonic flow on their upper surface at cruise conditions. This type of airfoil has been used in transonic transport aircraft since the late 1970's/early 1980's. Low transonic drag is obtained by keeping the flow on its upper surface at low supersonic Mach numbers, avoiding the presence of strong shock waves that could cause boundary layer Figure 8 Under wing mounted engines configuration separation. However, these low supersonic Mach numbers on the 'supercritical' region do not allow very large pressure differences to be generated between the upper and lower airfoil surfaces, resulting in reduced local lift. The required additional lift is achieved by increasing the camber at the rear part of the airfoil. The resulting wing profile shape is fairly flat on the upper surface from 10% to 60 or 70% of the chord, curving downward from that point until the trailing edge. In the lower surface, a concave region is present in the rear 30% to 40% of the chord. Figures 19 and 20 show the typical geometric and pressure distribution differences between a 'conventional' and a 'supercritical' airfoil. The pylon and underwing engine installation were evaluated using a 3D panel code which, in spite of being formally incapable of handling transonic problems, gave useful qualitative subsonic design indications. The configuration was successfully tested in the transonic Boeing Transonic Wind tunnel and met the performance expectations. Although the aerodynamic configuration was successful, the problems associated with the longer landing gear proved harder to solve. The cost of the fuselage nose modification would have been high, the longer landing gears would have required the installation of emergency escape slides, leading to the loss of space for two passenger seats and the close proximity of the engines to the ground would still have posed considerable risks of foreign object ingestion and damage. All these problems caused the underwing engine configuration to be abandoned. 2.3.2

The Rear Fuselage mounted Engine Configuration

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The third major aerodynamic configuration had the engines mounted on pylons on the rear fuselage (see Figure 9). There was a further increase in fuselage length to accommodate 50 passengers and the wing was initially the same as that of the underwing configuration. This configuration, with the changes described below, was the one finally chosen for production. Although good transonic wind tunnel results had been obtained for the cruise wing at the Boeing Transonic Wind tunnel, low speed wind tunnel tests at CTA indicated that the maximum lift coefficient values would not meet the short take-off and landing field lengths required for regional airline operations. At about the same time, market surveys indicated that the potential clients would not require cruise speeds in excess of Mach 0.75 to 0.78. This provided design margins to allow the leading edge to be modified with a fixed 'droop' and the wing root flap chord to be extended by 0.15 m. The droop was designed using the 2D and 3D full potential methods. Additionally, four vortilons were installed on the lower surface leading edge of the outboard wing panel. During the initial flight test campaign, some adverse yaw Figure 9 Rear fuselage mounted engines configuration (aileron roll command to the left would produce a slight yawing moment to the right and vice-versa) was noticed during climb. The ailerons already possessed differential gearing (the aileron whose trailing edge is going up always deflects more than the one whose trailing edge is going down) to counter the theoretically predicted adverse yaw, but the effect in flight was found to be larger than expected. Flow visualizations with wool tufts showed that the aileron going down had some regions of separated flow. This produced additional drag at that wingtip, which in turn produced the increased adverse yaw. The problem was solved by placing a row of vortex generators in front of the aileron to 'energize' the Figure 10 leading edge droop and vortilons boundary layer and delay its separation.

2.4 Control Theory Approach to Airplane Design

A wing is a device to control the flow where applying the theory of controlling partial differential equations in conjunction with CFD [Jameson]16. The simplest approach to optimization is to define the geometry through a set of design parameters, which may, for example, be the weights αi applied to a set of shape functions bi(x) so that the shape is represented as

Antony Jameson, “Optimum Aerodynamic Design Using CFD and Control Theory”, Department of Mechanical and Aerospace Engineering, Princeton University, AIAA 95-1729-CP. 16

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f(x)   αi bi (x)

Eq. 2.8

Then a cost function (I) is selected which might, for example, be the drag coefficient or the lift to drag ratio, and I is regarded as a function of the parameters αi. The sensitivities I may now be estimated by making a small variation Sai in each design parameter in turn and recalculating the flow to obtain the change in I. An alternative approach is to cast the design problem as a search for the shape that will generate the desired pressure distribution. This approach recognizes that the designer usually has an idea of the kind of pressure distribution that will lead to the desired performance. Thus, it is useful to consider the inverse problem of calculating the shape that will lead to a given pressure distribution. The method has the advantage that only one flow solution is required to obtain the desired design. Unfortunately, a physically realizable shape may not necessarily exist, unless the pressure distribution satisfies certain constraints. Thus the problem must be very carefully formulated. The shape changes in the section needed to improve the transonic wing (shock free) design are quite small. However, in order to obtain a true optimum design larger scale changes such as changes in the wing planform (sweepback, span, chord, and taper) should be considered. Because these directly affect the structure weight, a meaningful result can only be obtained by considering a cost function that takes account of both the aerodynamic characteristics and the weight. Consider a cost function (I) is defined as

I  α1CD  α 2

1 (p  p d ) 2dS  α 3C W  2B

Eq. 2.9

where pd is the target pressure and the integral is evaluated over the actual surface area (S). Maximizing the range of an aircraft provides a guide to the values for α1 and α3 as weight functions. In order to realize these advantages it is essential to move beyond flow simulation to a capability for aerodynamic shape optimization (a main focus of the first author research during the past decade) and ultimately multidisciplinary system optimization. Figure 11 illustrates the result of an automatic redesign of the wing of the Boeing 747, which indicates the potential for a 5 percent reduction in the total drag of the aircraft by a very small shape modification. It is also important to recognize that in current practice the setup times and costs of CFD simulations substantially exceed the solution times and costs. With presently available software the processes of geometry modeling and grid generation may take weeks or even months. In the preliminary design of the F22 Lockheed relied largely on wind-tunnel testing because they could build models faster than they could generate meshes. It is essential to remove this bottleneck if CFD is to be more effectively used. There have been major efforts in Europe to develop an integrated software environment for aerodynamic simulations, exemplified by the German “Mega Flow” program. Figure 11 also displays Redesigned Boeing 747 wing at Mach 0.86. Cp distributions In the final-design stage it is necessary to predict the loads throughout the flight envelope. As many as 20000 design points may be considered. In current practice wind-tunnel testing is used to acquire the loads data, both because the cumulative cost of acquisition via CFD still exceeds the costs of building and testing properly instrumented models, and because a lack of confidence in the reliability of CFD simulations of extreme flight conditions17.

Antony Jameson and, assisted by, Kui Ou, “Optimization Methods in Computational Fluid Dynamics”, Aeronautics and Astronautics Department, Stanford University, Stanford, CA, USA. 17

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Figure 11

2.5

Redesigned Boeing 747 wing at Mach 0.86, Cp distributions

Thought on Hierarchal Design Approach

Aero-engines and other large turbomachine components are very complex engineering systems. Viewed as a single entity there might be hundred thousands of components. This is obviously too large task to be handed by single designer and the computational costs are prohibit thought of global back box optimization concept. To overcome these problems, is to use a hierarchal representation in which the components is defined at different levels18. To avoid the huge computational cost of analyzing the entire engine, the design of all aero-engines is carried out at two levels, preliminary design and detailed component design. The preliminary design group considers the engine as an entire system, thinking about the customer’s requirements, sizing the major components, deciding which subsystems to retain from previous products, and aiming to maximize product over the lifetime of the entire project. At preliminary design process, many crucial design decisions have been made, such as engine thrust, mass ow and fan radius. The second level of the design hierarchy is the design of individual components within each subsystem, such as the HP turbine. The design intent for each component has been fairly tightly specified in preliminary design, and many constraints have been imposed. The task of the component design team is to full the design intent as well as possible (good aerodynamic performance, good structural integrity, low weight, etc.); subject to the constraints. To a large extent, this is a matter of shape optimization, the non-geometric design parameters having been set in preliminary design. It is worth mentioning that in some circles, there are also a conceptual design box before preliminary. As described above, and illustrated in Figure 12 (left), the current hierarchical design approach is sequential, preliminary design followed by M. B. Giles, “Some thoughts on exploiting CFD for turbomachinery design”, Oxford University Computing Laboratory, 1998. 18

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component design. Except in exceptional circumstances, the decisions made in preliminary design are not changed during component design. This is due to preliminary design being rely based on empiricism rom past experience, so major surprises are unlikely to arise during the component design process. There are two weaknesses to this sequential design process. The first is that its success depends on the new design not being too different from past designs, so that the empiricism in the modelling remains valid. This makes it very difficult to develop radically new designs. The second drawback is that the empiricism in the preliminary design system represents the collective experience of past projects, but no two projects are ever identical. Even if the customer requirements are identical, technological advances mean that the best engine or aircraft of today would be different from that designed twenty years ago. To some extent this technological progress can be accounted for in the empiricism, but inevitably preliminary design is based on only an approximate model of the system. In the future, there may be a shift to a more tightly-coupled two-level design system, as illustrated in Figure 12 (right). The overall system design will begin, as now, with a preliminary design based on past empiricism. This will provide the starting point for the detailed component design. The main reason a tightly coupled design system is not used today is time. The design time for an engine or aircraft project is strictly limited.

Figure 12

2.6

Tightly coupled two level design process

Classification of Design Optimization Methods

According to different definitions of optimization objectives, the numerical design methods can be classified into two groups: Inverse design and Direct design. In inverse design, for example, the blade geometry is modified to minimize the difference of profiles of pressure or velocity between the designed and the specified. This method is widely used in 1980s -1990s since it provides a cheap way for numerical aerodynamic design, such as 2D and 3D blade optimization. Whereas, considerable experience of the designer is still needed to give a proper profile of pressure or velocity. In optimization design, the geometry of the blade is sought to maximize the overall performance parameters, such as efficiency, total pressures ratio, etc. In some sense, it is not as efficient as inverse design which has clear direction at the beginning, while it makes the design process less dependent on designers experience, which gives the potential possibility of better design than specified by designer. In practical optimization design, there often exist Multi-objectives optimization. The classical optimization method usually converts a multi-objective optimization problem into a single objective problem using penalty functions or weighting coefficients. However, for most cases, these objectives often are incompatible. It’s difficult to set the appropriate penalty functions or weighting functions which really depends on the experience and preferences of the designer. Moreover, only one optimal result is obtained after optimization. The designers have no alternative options to

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choose. In fact, in most cases, there is no “best” solution by nature, but an infinite number of feasible solutions which represent different levels of trade-off between the objectives. This set of solutions is called Pareto optimal set or Pareto Front. A couple of novel Multi-objective optimization algorithms based on Pareto optimal concept are proposed and applied into optimal design. They provide a set of non-inferior solutions rather than one “best” solution, which represents a more reasonable optimal nature. The practical application of multi-objective optimization algorithms in engineering design is still far away from success, especially in aerodynamic design. High computational cost and convergence are two critical problems needed to be investigated.19

Xiaodong Wang, “CFD Simulation of Complex Flows in Turbomachinery and Robust Optimization of Blade Design”, Submitted to the Department of Mechanical Engineering Doctor of Philosophy at the Vrije Universiteit Brussel July 2010. 19

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3 Optimization Problem 3.1 Role of Optimization In mathematics and computer science, an optimization problem is the problem of finding the best solution from all feasible solutions. In the simplest terms, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. Figure 13 shows a graph of a paraboloid given by z = f(x, y) = − (x² + y²) + 4. The global maximum at (x, y, z) = (0, 0, 4) is indicated by a blue dot. Optimization is the process of obtaining the most suitable solution to a given problem, while for a specific problem only a single solution may exist, and for other problems there may exist multiple potential solutions [Skinner and Zare-Behtash]20. Thus, optimization is the process of finding the `best' solution, where `best' implies that the solution is not the exact solution but is sufficiently superior. Optimization tools should be used for supporting decisions rather than for Figure 13 Global Maximum of f (x, y) making decisions, i.e. should not substitute decision-making process21. Most current design optimization approaches are heavily dependent on user training and experience requiring an array of specialized optimization tools and compact shape parametrization. This constitutes a major obstacle to robustness and reliability. Another persistent difficulty in aerodynamic optimization is the ability to define an analysis method that is capable of operating as many time as required (often thousands of times) and integrating it appropriately with an optimization strategy. The methods employed must execute with realistic run times, dependent on computational resource, but must also be sophisticated enough to capture enough information to analyses local geometry that feeds into a globally optimal system. Many optimization problems, especially those involved with large design spaces with coupled variables, inherently fall into the category of Multi-Disciplinary design Optimization (MDO), which in turn require a multi-objective (MO) compromise to be an effective design. The main motivation for applying MDO is that the performance of a real system is driven not only by the performance of individual disciplines but also by their coupled interactions. It is no longer acceptable to consider the aerodynamic analysis alone, its far reaching coupled effects to other disciplines must also be taken into account if a truly optimal design is to be reached [Sobieszczanski-Sobieski and Haftka]22.

3.2 Types of Optimization

An important step in the optimization process is classifying your optimization model, since algorithms for solving optimization problems are tailored to a particular type of problem. Here we S. N. Skinner and H. Zare-Behtash, ”State-of-the-Art in Aerodynamic Shape Optimisation Methods”, Article in Applied Soft Computing , September 2017, DOI: 10.1016/j.asoc.2017.09.030. 21 21 Dragan Savic,” Single-objective vs. Multi-objective Optimization for Integrated Decision Support”, Centre for Water Systems, Department of Engineering School of Engineering and Computer Science, University of Exeter, United Kingdom, 22 J. Sobieszczanski-Sobieski and R.T. Haftka. “Multidisciplinary Aerospace Design Optimization: Survey of Recent Developments”. In 34th Aerospace Sciences Meeting and Exhibit, AIAA 96-0711, volume 70, Reno, Navada, 1996. 20

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provide some guidance to help you classify your optimization model; for the various optimization problem types, we provide a linked page with some basic information, links to algorithms and software, and online and print resources. 3.2.1 Continuous Optimization versus Discrete Optimization Some models only make sense if the variables take on values from a discrete set, often a subset of integers, whereas other models contain variables that can take on any real value. Models with discrete variables are discrete optimization problems; models with continuous variables are continuous optimization problems. Continuous optimization problems tend to be easier to solve than discrete optimization problems; the smoothness of the functions means that the objective function and constraint function values at a point x can be used to deduce information about points in a neighborhood of x. However, improvements in algorithms coupled with advancements in computing technology have dramatically increased the size and complexity of discrete optimization problems that can be solved efficiently. Continuous optimization algorithms are important in discrete optimization because many discrete optimization algorithms generate a sequence of continuous sub problems. 3.2.2 Unconstrained versus Constrained Optimization Another important distinction is between problems in which there are no constraints on the variables and problems in which there are constraints on the variables. Unconstrained optimization problems arise directly in many practical applications; they also arise in the reformulation of constrained optimization problems in which the constraints are replaced by a penalty term in the objective function. Constrained optimization problems arise from applications in which there are explicit constraints on the variables. The constraints on the variables can vary widely from simple bounds to systems of equalities and inequalities that model complex relationships among the variables. Constrained optimization problems can be furthered classified according to the nature of the constraints (e.g., linear, nonlinear, convex) and the smoothness of the functions (e.g., differentiable or no differentiable). 3.2.3 None, One or Many Objectives Most optimization problems have a single objective function, however, there are interesting cases when optimization problems have no objective function or multiple objective functions. Feasibility problems are problems in which the goal is to find values for the variables that satisfy the constraints of a model with no particular objective to optimize. Complementarity problems are pervasive in engineering and economics. The goal is to find a solution that satisfies the complementarity conditions. Multi-objective optimization problems arise in many fields, such as engineering, economics, and logistics, when optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives. For example, developing a new component might involve minimizing weight while maximizing strength or choosing a portfolio might involve maximizing the expected return while minimizing the risk. In practice, problems with multiple objectives often are reformulated as single objective problems by either forming a weighted combination of the different objectives or by replacing some of the objectives by constraints. 3.2.3.1 Single vs. Multi-Objective Optimization Many real-world decision making problems need to achieve several objectives: minimize risks, maximize reliability, minimize deviations from desired levels, minimize cost, etc.23. The main goal of single-objective (SO) optimization is to find the “best” solution, which corresponds to the minimum Dragan Savic,” Single-objective vs. Multi-objective Optimization for Integrated Decision Support”, Centre for Water Systems, Department of Engineering School of Engineering and Computer Science, University of Exeter, United Kingdom, 23

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or maximum value of a single objective function that lumps all different objectives into one. This type of optimization is useful as a tool which should provide decision makers with insights into the nature of the problem, but usually cannot provide a set of alternative solutions that trade different objectives against each other. On the contrary, in a multi-objective optimization with conflicting objectives, there is no single optimal solution. The interaction among different objectives gives rise to a set of compromised solutions, largely known as the trade-off, non-dominated, non-inferior or Pareto-optimal solutions. The consideration of many objectives in the design or planning stages provides three major improvements to the procedure that directly supports the decision-making process [Cohon, 1978]:  



A wider range of alternatives is usually identified when a multi-objective methodology is employed. Consideration of multiple objectives promotes more appropriate roles for the participants in the planning and decision-making processes, i.e. “analyst” or “modeler” who generates alternative solutions, and “decision maker” who uses the solutions generated by the analyst to make informed decisions. Models of a problem will be more realistic if many objectives are considered.

Single-objective optimization identifies a single optimal alternative, however, it can be used within the multi-objective framework. This does not involve accumulating different objectives into a single objective function, but, for example, entails setting all except one of them as constraints in the optimization process. Those objectives expressed as constraints are assigned different levels of attainment of their respective objective functions (e.g. minimum reliability levels) and several runs are performed to obtain solutions corresponding to different satisfaction of constraints. However, most design and planning problems are characterized by a large and often infinite number of alternatives. Thus, multi-objective methodologies are more likely to identify a wider range of these alternatives since they do not need to specify for which level of one objective a single optimal solution is obtained for another. Figure 14 illustrates different optimization techniques. Be advised that in Direct search methods perform hill climbing in the function space by moving in a direction related to

Numerical Methods

Otimization Methods

Enumertive Methods

Direct

Local Gradient

Indirect

Set of Equations

Dynamic Programing Single Point Search

Guided Randum Search Techniques Multi-Point Search

Figure 14

Simulated Annealing (SA) Ant Colony Optimization

Genetic Algorithms (GA)

Evolutionary Algorithms

Generic Programming

Tabu Search

Evolutuionary Stategies

Different search and optimization techniques

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the local gradient. Where else in indirect methods, the solution is sought by solving a set of equations resulting from setting the gradient of the objective function to zero24. 3.2.3.2 Various Methods to Solve Multiple Objective Optimization A large number of approaches exist in the literature to solve multi-objective optimization problems. These are aggregating (combining), population-based non-Pareto, and Pareto-based techniques. In case of aggregating techniques, different objectives are generally combined into one using weighting or a goal-based method. One of the techniques in the population-based non-Pareto approach is the Vector Evaluated Genetic Algorithm (VEGA). Here, different subpopulations are used for the different objectives. Pareto-based approaches include multiple objective GA (MOGA), non-dominated sorting GA (NSGA), and positioned Pareto GA. Note that all these techniques are essentially non-exclusive in nature. Simulated annealing (SA) performs reasonably well in solving single-objective optimization problems. However, its application for solving multi-objective problems has been limited, mainly because it finds a single solution in a single run instead of a set of solutions. This appears to be a critical bottleneck in MOOPs. However, SA has been found to have some favorable characteristics for multimodal search. The advantage of SA stems from its good selection technique25. 3.2.3.3 Analysts, Decision Makers and Optimization Techniques A very simplified view of the decision-making process is that it involves two types of performers: analysts (modelers) and decision makers26. This is a crude simplification of the process since many stakeholders and actors may be involved, but simple enough to demonstrate shortcomings of assuming that in general one person can assume both (or many more) roles. Analysts are technically capable people who provide information about a problem to decision makers who decide which course of action to take. Modelling and optimization techniques are tools which analysts may use to develop useful information for the decision makers. However, single-objective models require that all design objectives must be measurable in terms of a single fitness function. This in turn requires some a priori ordering of different objectives (i.e., a weighting scheme) to allow easy integration of them into a single function of same units. Thus, single-objective approaches place the burden of decision making squarely on the shoulders of the analyst. For example, it is the analyst who must decide the cost equivalent of a specific risk of failure. Even if the decision makers are technically capable and willing to provide some a priori preference information, the decision making role is taken away from them. By providing a trade-off curve between different objectives and alternative solutions corresponding to the points on this curve, multi-objective approaches allow for the responsibility of assigning relative values of the objectives to remain where it belongs: with the decision maker. What it comes to is single-objective optimization can detect one optimal solution in a single run while Multi-Objective General Algorithm (MOGA) can detect a whole set of (Pareto) optimal solutions, i.e. it can detect the whole trade-off surface. As a consequence, Multiple SO runs are necessary to obtain the same level of information that can be obtained from a single MOGA run. When the SO model is used, the decision maker must express preferences before a model run, while in the MOGA approach one expresses preferences after a run. 3.2.4 Deterministic vs. Stochastic Optimization In deterministic optimization, it is assumed that the data for the given problem are known accurately. However, for many actual problems, the data cannot be known accurately for a variety of Dragan Savic,” Single-objective vs. Multi-objective Optimization for Integrated Decision Support”, Centre for Water Systems, Department of Engineering School of Engineering and Computer Science, University of Exeter, United Kingdom, 25 Bandyopadhyay, S., Saha, S, “Some Single- and Multi-objective Optimization Techniques”, Chapter 2”, ISBN: 973-3-642-35450-5, 2013. 26 See Previous. 24

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reasons. The first reason is due to simple measurement error. The second and more fundamental reason is that some data represent information about the future (e. g., product demand or price for a future time period) and simply cannot be known with certainty. In optimization under uncertainty, or stochastic optimization, the uncertainty is incorporated into the model. Robust optimization techniques can be used when the parameters are known only within certain bounds; the goal is to find a solution that is feasible for all data and optimal in some sense. Stochastic programming models take advantage of the fact that probability distributions governing the data are known or can be estimated; the goal is to find some policy that is feasible for all (or almost all) the possible data instances and optimizes the expected performance of the model.

3.3 Statement of Optimization Problem

By convention, the standard form defines a minimization problem. A maximization problem can be treated by negating the objective function. In CFD analysis, we mostly deal with Continues Optimization. Most optimization methods use an iterative procedure. The initial set x design variables, which in the context of aerodynamic optimization this is referred to as the baseline configuration, and is updated until a minimum of f(x) is identified or the optimization process runs out of allocated time/iterations. The standard form of optimization problem statement is

M inimize f(x) subject to ; g i (x)  0, i  1,...., m ; h i (x)  0, i  1,....., p where f(x) is the Objective function to be minimize over the variable x g i (x)  0 are called inequality constraints and, h i (x)  0 are called equality constraints In the initial set-up of the optimization problem consideration must be given to:      

the level of information fidelity required from the flow solver, depending on problem ; scope of parametrized design space; types of design variables, e.g. discrete and/or continuous; single or multi-objective optimization; constraints handling; properties of the design space, e.g. number of local optima, discontinuities.

It is important to note that no optimization procedure guarantees the global optima of the objective function f(x) will be found the process may only converge towards a locally optimal solution. Typically in this situation there are three possibilities:  restart the optimization process to investigate if the same solution is found;  approach the design problem with a different optimization methodology to compare solution quality at a high computational expense;  accept the optimum found knowing that while it is superior to the baseline configuration it may not be the optimal solution. 3.3.1 Design Variables and Geometric Representation using Parametrization Before we go any further, we counsel the readers that we are following closely the development in

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[Skinner and Zare-Behtash]27 for this particular subject. In general, it is necessary to implement geometric parametrizations in such a way that reduce the complexity and cost of the optimization process, but do not restrict communication of variables or the degree to which aerodynamic performance is optimized. Parametrization aims to balance the fundamental compromise between computational speed of the optimization run-time, favoring a tight parametrization. [Zhang et al.]28 show that the defined dimensionality of a problem for shape optimizations can restrict the optimal design. Using too few variables may prove certain potential improvements impossible. Conversely, if too many design variables are used, particularly if variables are strongly coupled, the search landscape can become intractably complex to navigate. Increasing the dimensionality of a given problem excessively leads to a paradox, first addressed by [Sobieszczanski-Sobieski]29-30 in which increasing the number of design variables leads to a decrease in the number of variables that can be manipulated as a direct result of increased coupling. It is often desirable to limit the allowable design variables to avoid geometries that cannot be evaluated with sufficient accuracy by the flow solver: due to meshing limitations for example. Furthermore, this can help to avoid geometries that are unacceptable in terms of some criteria, or simfly, restrict the optimization to geometries that are necessary for other criteria. Regardless of the user defined parametrization, the final design is most definitely suboptimal; often limited by parametrization. 3.3.2 Geometric Representation using Parametrization [Chernukin and Zingg]31 conducted one of the few studies on how the number of variables used, and the related modality, can affect aerodynamic designs and highlight that distinguishing between multimodality and poor optimizer convergence can prove problematic. By increasing the dimensionality of a design space it can be expected, but not guaranteed, to increase the modality of the search space. The planform shapes are distinct and so demonstrate that geometric variation is significant between local optima which share similar performance characteristics. The method of geometric parametrization used to communicate a set of variables plays an important role in identifying optimal aerodynamics. It determines what shapes and topologies can be represented, and how many design variables are necessary for sufficient representation of the geometry. Thus, parametrization dictates particular geometric requirements and has a strong influence on the design landscape. Therefore it cannot be precluded that different geometric parametrizations will increase or decrease the degree of modality, linearity, or discontinuity observed. Additionally, a complex geometry parametrization may impose distinct computational costs. Representations of a geometry can be broken down into a number of categories but in a more broad sense they can be considered to be constructive, reformative, or volume based. According to [J. A. Samareh]32, the shape parameterization must be compatible with and adaptable to various analysis tools ranging from low-fidelity tools, such as linear aerodynamics and equivalent laminated plate structures, to high-fidelity tools, such as nonlinear CFD and detailed CSM. For a multidisciplinary problem, the application must also use a consistent parameterization across all S. N. Skinner and H. Zare-Behtash, ”State-of-the-Art in Aerodynamic Shape Optimization Methods”, Article in Applied Soft Computing , September 2017, DOI: 10.1016/j.asoc.2017.09.030. 28 Y. Zhang, Z.H. Han, L. Shi, and W.P. Song, “Multi-Round Surrogate-based Optimization for Benchmark Aerodynamic Design Problems”, 54th AIAA Aerospace Sciences Meeting, AIAA 2016-1545. 29 J. Sobieszczanski-Sobieski. “A Linear Decomposition Method For Large Optimisation Problems - Blueprint For Development”. NASA TM-83248, 1982. 30 J. Sobieszczanski-Sobieski. “Optimisation by Decomposition: A Step from Hierarchic to Non-Hierarchic Systems”. NASA Technical Report: N89-25149, 1988. 31 O. Chernukhin and D.W. Zingg. ”Multimodality and Global Optimization in Aerodynamic Design”. AIAA Journal, 51(6):1342{1354, 2013. 32 Jamshid A. Samareh, “A Survey of Shape Parameterization Techniques”, NASA Langley Research Center, Hampton, Va. 27

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disciplines. An MDO application requires a common geometry data set that can be manipulated and shared among various disciplines. In addition, an accurate sensitivity derivative analysis is required for gradient-based optimization. The sensitivity derivatives are defined as the partial derivatives of the geometry model or grid-point coordinates with respect to a design variable. The sensitivity derivatives of a response, F, with respect to the design variable vector D, can be written as

F  F   R F   R S   R G      D  R F   R S   R G   D    I

II

III

Eq. 3.1

IV

The first term on the right-hand side of Eq. 3.1 represents the sensitivity derivatives of the response with respect to the field grid point coordinates. The second term on the right-hand side is vector of the field grid-point sensitivity derivatives with respect to the surface grid points. The sensitivity derivative vector must be provided by the field grid generator, but few grid generation tools have the capability to provide the analytical grid-point sensitivity derivatives? The third term on the righthand side of Eq. 3.1 denotes the surface grid sensitivity derivatives with respect to the shape design variables, which must be provided by the surface grid generation tools. The fourth term on the righthand side of Eq. 3.1 signifies the geometry sensitivity derivatives with respect to the design variable vectors; this must be provided by the geometry construction tools33. An important ingredient of shape optimization is the availability of a model parameterized with respect to the airplane shape parameters such as planform, twist, shear, camber, and thickness. The parameterization techniques, according to [J. A. Samareh], are divided into the following categories: basis vector, domain element, partial differential equation, discrete, polynomial and spline, CADbased, analytical, free form deformation (FFD), and modified FFD. Among those, we attend to Discrete , Analytical, PDE, CST, and Free From Deformation (FFD). 3.3.2.1 Discrete Approach The discrete approach is based on using the coordinates of the boundary points as design variables. This approach is easy to implement, and the geometry changes are limited only by the number of design variables. However, it is difficult to maintain a smooth geometry, and the optimization solution may be impractical to manufacture. To control smoothness, one could use multipoint constraints and dynamic adjustment of lower and upper bounds on the design variables. For a model with a large number of grid points, the number of design variables often becomes very large, which leads to high cost and a difficult optimization problem to solve. The natural design approach is a variation of the discrete approach that uses a set of fictitious loads as design variables. These fictitious loads are applied to the boundary points, and the resulting displacements, or natural shape functions, are added to the baseline grid to obtain a new shape. Consequently, the relationship between changes in design variables and grid-point locations is established through a finite element analysis. [Zhang and Belegundu]34 provided a systematic approach for generating the sensitivity derivatives and several criteria to determine their effectiveness. The typical drawback of the natural design variable method is the indirect relationship between design variables and grid-point locations. For an MDO application, grid requirements are different for each discipline. So, each discipline has a Jamshid A. Samareh, “A Survey of Shape Parameterization Techniques”, NASA Langley Research Center, Hampton, Va. 34 Zhang, S. and Belegundu, A. D., "A Systematic Approach for Generating Velocity Fields in Shape Optimization," Structural Optimization, Vol. 5, No. 1-2, 1993, pp. 84-94. 33

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different grid and a different parameterized model. Consequently, using the discrete parameterization approach for an MDO application will result in an inconsistent parameterization35. The most attractive feature of the discrete approach is the ability to use an existing grid for optimization. The model complexity has little or no bearing on the parameterization process. It is possible to have a strong local control on shape changes by restricting the changes to a small area. When the shape design variables are the grid-point coordinates, the grid sensitivity derivative analysis is trivial to calculate; the third and fourth terms in Eq. 3.1 can be combined to form an identity matrix. 3.3.2.2 Analytical Approach [Hicks and Henne]36 introduced a compact formulation for parameterization of airfoil sections. The formulation was based on adding shape functions (analytical functions) linearly to the baseline shape. The contribution of each parameter is determined by the value of the participating coefficients (design variables) associated with that function. All participating coefficients are initially set to zero, so the first computation gives the baseline geometry. The shape functions are smooth functions based on a set of previous airfoil designs. [Elliott and Peraire]37 and [Hager et al.]38 used a formulation similar to that of [Hicks and Henne], but a different set of shape functions. This method is very effective for wing parameterization, but it is difficult to generalize it for a complex geometry. 3.3.2.3 Partial Differential Equation Approach This method views the surface generation as a boundary-value problem and produces surfaces as the solutions to elliptic partial differential equations (PDE). [Bloor and Wilson]39 showed that it was possible to represent an aircraft geometry in terms of a small set of design variables. [Smith et al. ]40 extended the PDE approach to a class of airplane configurations. Included in this definition were surface grids, volume grids, and grid sensitivity derivatives for CFD. The general airplane configuration had wing, fuselage, vertical tail, horizontal tails, and canard components. Grid sensitivity was obtained by applying the automatic differentiation tool ADIFOR. Using the PDE approach to parameterize an existing complex model is time-consuming and costly. Also, because this method can only parameterize the surface geometry, it is not suitable for the MSO applications that must model the internal structural elements such as spars, ribs, and fuel tanks. As a result, this method is suitable for problems involving a single discipline with relatively simple external geometry changes. 3.3.2.4 CST Method In order to present a general parameterization technique for any type of geometries and to overcome the mentioned limits, [Kulfan and Bussoletti]41 developed the method of Class function / Shape

Jamshid A. Samareh, “A Survey of Shape Parameterization Techniques”, NASA Langley Research Center, Hampton, VA. 36 Hicks, R. M. and Henne, P. A., "Wing Design by Numerical Optimization," Journal of Aircraft, Vol. 15, No. 7, 1978, pp. 407-412. 37 Elliott, 3. and Peralre, J., "Practical Three-Dimensional Aerodynamic Design and Optimization Using Unstructured Meshes," AIAA Journal, Vol. 35, No. 9, 1997, pp. 1479-1486. 38 Hager, J. O., Eyi, S., and Lee, K. D., "A Multi-Point Optimization for Transonic Airfoil Design," Paper 92-4681-CP, AIAA, Sep. 1992. 39 Bloor, M. I. G. and Wilson, M. J., "Efficient Parameterization of Genetic Aircraft Geometry," Journal of Aircraft, Vol. 32, No. 6, 1995, pp. 1269-1275. 40 Smith, R. E., Bloor, M. I. G., Wilson, M. J., and Thomas, A. T., "Rapid Airplane Parametric Input Design (RAPID)," AIAA 12th Computational Fluid Dynamics Conference, AIAA, Jun. 1995, pp. 452-462, also AIAA-95-1687 41 Kulfan, B. M. and Bussoletti, J. E., “Fundamental Parametric Geometry Representations for Aircraft Component Shapes," 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 2006. 35

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function Transformation (CST). This method provides the mathematical description of the geometry through a combination of a shape function and class function. The class function provides for a wide variety of geometries. The shape function replaces the complex non-analytic function with a simple analytic function that has the ability to control the design parameters and uses only a few scalable parameters to define a large design space for aerodynamic analysis. The advantage of CST lies in the fact that it is not only efficient in terms of low number of design variables but it also allows the use of industrial related design parameters like radius of leading edge or maximum thickness and its location42. A) CST Airfoils Any smooth airfoil can be represented by the general 2D CST equations. The only things that differentiate one airfoil from another in the CST method are two arrays of coefficients that are built into the defining equations. These coefficients control the curvature of the upper and lower surfaces of the airfoil. This gives a set of design variables which allows for aerodynamic optimization. This method of parameterization captures the entire design space of smooth airfoils and is therefore useful for any application requiring a smooth airfoil. The upper and lower surface defining equations are as follows:

 ς U (ψ)  C N1 x z N2 (ψ).SU (ψ)  ψ.ΔςU  where ψ  and ς   c c ς L (ψ)  C N1 N2 (ψ).S L (ψ)  ψ.ΔςL  

Eq. 3.2

The last terms define the upper and lower trailing edge thicknesses. Equation uses the general class function to define the basic profile and the shape function to create the specific shape within that geometry class. The general class function is defined as: N1 N2 C N1 N2 (ψ)  ψ .(ψ)

Eq. 3.3

For a general NACA type symmetric airfoil with a round nose and pointed aft end, N1 is 0.5 and N2 is .0 in the class function. This classifies the final shape as being within the "airfoil" geometry class, which forms the basis of CST airfoil representation. This means that all other airfoils represented by the CST method are derived from the class function airfoil. Further details can be found in [Lane and Marshall]43, or [Ceze et al.]44. B) CST Wings The 2D process for airfoils is easily extended to wings as a simple extrusion of parameterized airfoils. This greatly increases the number of design variables for an optimization scheme. However, it is no less powerful. By controlling the distribution of airfoils, any smooth wing can be represented. Also, such characteristics as sweep, taper, geometric twist, and aerodynamic twist can be included. The

42 Arash Mousavi, Patrice Castonguay and Siva

K. Nadarajah, “Survey Of Shape Parameterization Techniques And its Effect on Three-Dimensional Aerodynamic Shape Optimization”, AIAA 2007-3837. 43 Kevin A. Lane and David D. Marshall, “A Surface Parameterization Method for Airfoil Optimization and High Lift 2D Geometries Utilizing the CST Methodology”, AIAA 2009-1461. 44 Marco Ceze, Marcelo Hayashiy and Ernani Volpe, “A Study of the CST Parameterization Characteristics”, AIAA 2009-3767.

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definition of a 3D surface follows a similar structure to that of a 2D surface. Again, for complete description of method, readers are encouraged to consult45. 3.3.2.5 Case Study - Airfoil Optimization As previously explained, the CST method gives equations for the upper and lower surfaces of an airfoil in terms of the curvature coefficients46. These coefficients can be used as design variables in an aerodynamic optimization scheme. Such a scheme is currently being developed to maximize the lift to drag ratio (L/D) of a supercritical airfoil for use on Initial Optimized a next generation commercial airliner. The optimization CL 0.329 0.318 scheme uses the MATLAB function fmincon as the CD 0.0273 0.0140 optimizer. This function was selected over fminunc so that constraints could be placed on the airfoil geometry. L/D 12.05 22.78 Computational fluid dynamics (CFD) was selected for AoA 0.96 1.5 the solution method because the optimization is Table 1 Performance comparison of performed in the transonic regime where many other initial an optimize airfoils (Courtesy of 44) solution methods are not valid. This complicates the process greatly. Since CFD is to be used by an optimizer, both the meshing and solution processes must be automated. Therefore, the meshing process must be robust enough to handle any airfoil selected by the optimizer. However, the meshing process will still be sensitive to the given airfoil geometry. If the airfoil selected by the optimizer is too unlike the airfoil used to develop the meshing automation, the meshing process is prone to errors. Therefore, constraints are used to force the optimizer to select airfoils that somewhat resemble the initial airfoil. Constraints implemented to ensure an airfoil successfully passes the meshing stage include limits on maximum thickness and minimum thickness. An additional constraint was placed so that the upper surface does not cross the lower surface. The optimizer is currently being tested for a cruise condition of Mach 0.8 at an altitude of 35,000 feet. To ensure that a constant CL is maintained, the objective function estimates the current airfoil's lift curve by fitting a line to CL values taken from CFD solutions at different angles of attack. This is used to obtain the angle of attack that should produce the desired Cl. This angle of attack is used for the final CFD solution of the objective function from which L/D is taken and read by the optimizer. The initial airfoil selected for the optimization scheme was the RAE 2822 transonic airfoil previously used in the class function coefficient optimization study. A CL of 0.322 was selected to correspond to the CL at cruise of the airfoil used by a next generation commercial airliner currently being studied at Cal Poly. Table 1 Displays a comparison between the performance of the initial and optimized airfoils. The CL values differ somewhat and displays some error in the selection of angle of attack. However, the CD of the optimized airfoil is dramatically lower than that of the initial airfoil. The CD value drops from 273 drag counts to 40, which is a reduction of 33 drag counts or about 49%. This also causes the L/D to increase from 2.05 to 22.78, which is an increase of about 89%. Figure 15 displays contours of Mach number over the initial and optimized airfoils. The initial airfoil is displayed on the left while the optimized airfoil is shown on the right. The maximum Mach number in the flow over the initial airfoil is much higher than that of the optimized airfoil. This is because the upper surface of the optimized airfoil is much flatter than that of the initial airfoil, which causes the flow to accelerate less over the upper surface of the optimized airfoil. This is the cause of the dramatic drag reduction. The lowest point of the lower surface of the optimized airfoil is forward from that of the initial airfoil. This allows for lower speed flow and therefore higher pressure.

45 46

See 40. See 40.

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Figure 15

Contours of Each number for the initial airfoil (Left) an the optimize airfoil (Right)44

3.3.2.6 Spline Based Parameterization with eye on Literature Survey Constructive models include functions which define basic body shapes, spline methods (such as Bezier splines, basis splines (B-splines), non-uniform rational basis spline (NURBS)) and partial differential equations. Jansen et al.24 used a medium-fidelity aero-structural panel code to perform optimization of a conceptual wing configurations. The basic wing topology was defined through a series of globally enforced geometric variables to manipulate a series of wing sections. Parametrizing the entire geometry in this way typically allows for global shape control with few basic variables. This method is well suited to low-fidelity aerodynamic models if a wide allowable design scope is necessary - no need for mesh deformations. Spline-based geometric parametrizations are used to represent two- or three- dimensional surfaces and are typically used in conjunction with higherfidelity flow solvers, such as Euler and Navier Stokes solvers, with the control points being the design variables. Bezier splines are most efficient to evaluate requiring few variables and have been used for efficient aero-foil definition by [Peigin and Epstein]47. Modification of any single control point defining a Bezier spline will modify the entire curve and thus is inherently effective for global shape definition, but has very limited local control. B-splines address this issue of local control allowing single control point modifications to modify small portions of the overall curve. This allows for more complex aero foil definitions, as demonstrated by [Koziel at al.]48, and can enable the use of hinged control surfaces to an otherwise rigid body. NURBS increase the local deformation control over surface definitions further in order to have more complex geometric shapes such as fairings or wingfuselage junctions. [Vecchia and Nicolosi]49 and [Hashimoto et al.]50 adopt NURBS to parametrize the entire aircraft configuration in order to reduce drag of the vehicle through steam-lining fillets and

S. Peigin and B. Epstein. “Multiconstrained Aerodynamic Design of Business Jet By CFD Driven Optimization Tool”. Aerospace Science and Technology, 12(2):125{134, mar 2008. 48 S. Koziel, Y. Tesfahunegn, A. Amrit, and L.T. Leifsson. “Rapid Multi-Objective Aerodynamic Design Using CoKriging and Space Mapping”. 57th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA 2016-0418, number January, pages 1-10, San Antonio, TX, USA, 2016. 49 P.D. Vecchia and F. Nicolosi. “Aerodynamic Guidelines in The Design and Optimization of New Regional Turboprop Aircraft”. Aerospace Science and Technology, 38:88{104, Oct 2014. 50 A.H. Hashimoto, S.J. Jeong, and S.O. Obayashi. “Aerodynamic Optimization of Near-Future High-Wing Aircraft”. Japan Society for Aeronautical and Space Sciences, 58(2):73{82, 2015. 47

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fairings. Figure 16 shows an example of NURBS control points re-defining the surface over the upper section of the fuselage/wing juncture [Vecchia and Nicolosi]51. Geometry definition through the use of partial differential equations (PDEs) are not as commonly used as well-established spline-based methods but are just as versatile for geometry surface definition. [Athanasopoulos et al.]52 show that for equivalently complex surface construction PDEs require fewer design variables, resulting in a more compact design space. Due to the small set of design parameters required by the PDE method the computational cost associated with the optimization of a given aerodynamic surfaces can be reduced. In a PDE-based method the parameters are boundary values to the PDE, hence the relationship between the value of the design parameter and the geometry can be unclear making method-official surface deformations tedious. This is likely why the aerodynamic definition of a body in an optimization scheme does not used PDE representation even though it may initially seem a more appropriate method. Comparatively, spline-based methods are conceptually simpler and will provide a more direct relationship between design parameters and the resulting geometry and thus allow better control over the range of geometries that can be generated. If optimization establishes performance metrics from CFD, the simplest methods for body surface definitions are reformative ones. In reformative methods the mesh points on the surface of the body are directly treated as design variables,53 and their position can be perturbed by the optimizer in order to generate new shapes. These approaches have Figure 16 NURBS surfaces parametrizing surface blend on the significant advantage that any fuselage (Courtesy of Vecchia & Nicolosi) geometry the mesh generation algorithm is capable of can be evaluated, however it is likely to require many hundreds of design variables; deformations are therefore usually limited to single-degree-of-freedom deformations. A common method used for aerodynamic optimization is the Free-Form Deformation (FFD) approach which is useful if the geometry manipulations are particularly complex; FFD is covered in depth by [Kenway and Martins]54. This approach embeds the solid geometry within a FFD hull volume (volumes are typically referends of Bezier splines, B-splines of NURBS), which are parametrized by a series of control points as shown in Figure 17.

P.D. Vecchia and F. Nicolosi. “Aerodynamic Guidelines in The Design and Optimization of New Regional Turboprop Aircraft”, Aerospace Science and Technology, Oct 2014. 52 M. Athanasopoulos, H. Ugail, and G.G. Castro. “Parametric Design of Aircraft Geometry Using Partial Diffferential Equations. Advances in Engineering Software”, 40(7):479-486, 2009. 53 A. Jameson, L. Martinelli, and N.A. Pierce. “Optimum Aerodynamic Design Using the Navier-Stokes Equations. Theoretical and Computational Fluid Dynamics”, 10(1-4):213{237, 1998. 54 G. Kenway, G. Kennedy, and J.R.R.A. Martins. “A CAD-Free Approach to High-Fidelity Aero-structural Optimization”. 13th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference, AIAA 2010-9231, pages 1-18, Fort Worth, TX, 2010. 51

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Figure 17

Free-form deformation (FFD) parametrizing wing with 720 control points

These control points deform the volume which translate to geometric changes of the solid geometry rather than redefining the whole geometry itself which can give a relatively more efficient set of design variables. A key assertion of the FFD approach, when applied within a CFD environment, is that a geometry has constant topology through-out the optimization process; this is typical of highfidelity optimizations where the initial geometry considered is sufficiently close to the optimal solution. Figure 17 shows the FFD hull volume enclosing a wing with 720 geometric control points used by [Lyu et al.]55 which control shape deformation in the vertical (z) axis. The initial random wing deformation and associated optimized wing cross-sections at select locations are also shown. A similar method is based on radial basis function (RBF) interpolation which defines data sets of design variables and their global relationships. [Fincham and Friswell]56 use radial basis functions to optimize morphing aero foils and report that they provide a means to deform both aerodynamic and structural meshes and interpolate performance metrics between two non-coincident meshes. Volumetric-based body representation have been used for optimization but rarely in the field of aerodynamics, a recent review of the applicability of volumetric parametrization for aerodynamic optimization is given by [Hall et al.]57. 3.3.2.7 Constraint Handling Constraint handling in aerodynamic, and indeed any industrial optimization problem, plays a consequential role in the quality and robustness of an optimized solution within the defined design space. Geometric parametrization itself poses a constrained optimization problem since, in addition to minimizing the objective f(x), the design variables must satisfy some geometric constraints.

Z. Lyu, G. Kenway, and J.R.R.A. Martins. “Aerodynamic Shape Optimization Investigations of the Common Research Model Wing Benchmark”. AIAA Journal, 53(4):968{985, 2014. 56 J.H.S. Fincham and M.I. Friswell. “Aerodynamic Optimisation of a Camber Morphing Aerofoil”, Aerospace Science and Technology, 43:245{255, 2015. 57 J. Hall, D.J. Poole, T.C.S. Rendall, and C.B. Allen. Volumetric Shape Parameterisation for Combined Aerodynamic Geometry and Topology Optimisation. In 16th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, AIAA 2015-3354, number June, pages 1{29, Dallas, TX, 2015. 55

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Constraint management techniques found in literature which have been classified by [Koziel and Michalewicz]58 and [Sienz and Innocente]59 as:  strategies that preserve only feasible solutions with no constraint violations: infeasible solutions are deleted;  strategies that allow feasible and infeasible solutions to co-exist in a population, however penalty functions penalize the infeasible solutions (constraint based reasoning);  strategies that create feasible solutions only;  strategies that artificially modify solutions to boundary constraints if boundaries are exceeded; and  strategies that repair/modify infeasible solutions. Most commonly optimizations apply weighted penalties to the objective function if the constraint(s) are violated. The reason for this is that penalty functions are often deemed to ease the optimization process, and bring the advantage of transforming constrained problems into unconstrained one by directly enforcing the penalties directly to the objective function. With this method Pareto-optimal solutions with good diversity and reliable convergence for many algorithms can be obtained easily when the number of constraints are small; fewer than 20 constraints. It becomes more difficult to reach Pareto-optimal solutions efficiently as the number of constraints increase, and the number of analyses of objectives and constraints quickly becomes prohibitively expensive for many applications. This is because the selection pressure decreases due to the reduced region in which feasible solutions exist. [Kato et al.]60 suggest that in certain circumstances Pareto-optimal solutions may exist in-between regions of solution feasibility and infeasibility. This is illustrated in Figure 18, where it is seen that feasible and infeasible solutions could be evaluated in parallel to guide the optimization Figure 18 Concept of using parallel evaluation search direction towards feasible design strategy of feasible and infeasible solutions to guide spaces. This is intuitively true for single optimization direction in a GA discipline aerodynamic optimization problems where often small modifications to design variables can largely impact the performance rendering designs infeasible. Algorithm understanding of infeasible solutions can help in the betterment of feasible solutions though algorithm learning/training and constraint based reasoning.

58 S. Koziel

and Z. Michalewicz. “Evolutionary Algorithms, Homomorphous Mappings, and Constrained Parameter Optimization”. Evolutionary Computation, 7(1):19{44, 1999. 59 J. Siens and M.S. Innocente. “Particle Swarm Optimisation: Fundamental Study and its Application to Optimisation and to Jetty Scheduling Problems”. Trends in Engineering Computational Technology, pages 103126, 2008. 60 T. Kato, K. Shimoyama, and S. Obayashi. “Evolutionary Algorithm with Parallel Evaluation Strategy of Feasible and Infeasible Solutions Considering Total Constraint Violation”. IEEE, 1(978):986-993, 2015.

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[Robinson et al.]61, comparing the performance of alternative trust-region constraint handling methods, showed that reapplying knowledge of constraint information to a variable complexity wing design optimization problem reduced high-fidelity function calls by 58% and additionally compare the performance to alternative constraint managed techniques. Elsewhere, [Gemma and Mastroddi]62 demonstrated that for the multi-disciplinary, multi-objective aircraft optimizations the objective space of feasible and infeasible design candidates are likely to share no such definitive boundary. With the adoption of utter constraints, structural constraints, and mission constraints solutions defined as infeasible under certain conditions would otherwise be accepted, hence forming complex Pareto fronts. Interdisciplinary considerations such as this help to develop and balance conflicting constraints. For example, structural properties which may be considered feasible, but are perhaps heavier than necessary will inflict aero-elastic instabilities at lower frequencies. In the aerospace industry alone there are several devoted open-source aerodynamic optimization algorithms with built-in constraint handling capability. For example COBYLA41-43, DIRECT44-47, NOMAD48, and HAVOC49,50 each offer derivative-free optimization algorithms capable of handling constraints explicitly; each adaptable to the users aerodynamic solver whether commercial or in-house. Some studies have also adopted MATLAB's optimization tool-box for successful optimization constraint management. 3.3.3 Application of Gradient-Based Methods to Aerodynamic Optimizations Gradient-based optimization is a calculus-based point-by-point technique that relies on the gradient (derivative) information of the objective function with respect to a number of independent variables. The nature in which gradient-based methods (GBM) operate make them well suited to finding locally optimal solutions but may struggle to find the global optimal.68 With gradient-based algorithms an understanding of the design space is assumed, as an appropriately pre-conceived starting design point must be given. [Kenway and Martins]13 point out that with increasingly higher fidelity aerodynamic optimizations, a more refined initial design should be used so that the optimization does not diverge too far from the baseline. If large changes in topology are expected lower fidelity panel codes can facilitate useful optimization procedures. Typically, the higher the fidelity analysis used the more compact the design variables will need to be to allow effective optimization with a gradient based optimizer. Gradient-based optimization is, in its most basic form, a two-step iterative process which can be summarized mathematically as:

xnew  xold  hf

Eq. 3.4

where ⊽ f is the gradient of function f(x), and x is a vector of the design variables. The first step is to identify a search direction (gradient), ⊽ f, in which to move. The second step is to perform a onedimensional line search to determine a distance/step size h along ⊽f that achieves an adequate reduction of some cost function, i.e. define how far to move in the search direction until no more progress can be made. A schematic diagram illustrating the operation of a gradient-based optimization is shown in Figure 19. In-depth benchmarking of gradient based algorithms for

T.D. Robinson, K.E. Willcox, M.S. Eldred, and R. Haimes. “Multi-fidelity Optimization for Variable-Complexity Design”. 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, pages 1-18, Portsmouth, VA, 2006. AIAA 2006-7114. 62 S. Gemma and F. Mastroddi. “Multi-Disciplinary and Multi-Objective Optimization of an Unconventional Aircraft Concept”. 16th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, AIAA 2015-2327, pages 1-20, Dallas, TX, 2015. 61

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aerodynamic problems has been conducted by [Secanell and Suleman]63 and [Lyu et al.]64. Gradient based algorithms are extensively used in aerospace optimization as they exhibit low computational demands when handing many hundreds of design variables; this makes them well suited for optimizing shapes based on deformative geometric parametrizations.

Figure 19

Schematic diagram of a gradient-based aerodynamic optimization process

3.3.4 Sensitivity Analysis Significant difficulties arise if they are not applied within a restricted set of functions with welldefined slope values due to a dependency upon the existence of derivative information via some sensitivity analysis. Comparative studies on the numerical sensitivity analysis for aerodynamic optimization has been conducted The computational expense of evaluating gradients using finitedifference or the complex step method provide a simple and flexible means of estimating gradient information, but are considered excessive with respect to hundreds of variables. These approaches preserve discipline feasibility, but they are costly and can be unreliable. It is commented on that to increase the dimensionality of the problem an analytic sensitivity analysis would have to be adopted. Finite-differencing or complex-step methods employed for providing sensitivity analysis for lowfidelity codes can be considered appropriate due to low computational demand. Ning and Kroo66 optimize a series of wing topologies investigating fundamental wing design trade-offs for which sensitivity analysis of the objective and constraints were approximated by finite-differencing. Results provided by the sequential quadratic programming method show robust and quick convergence able to determine relative gradients between approximated area-dependent weight, effects of critical structural loading, and stall speed constraints. In the presence of several hundred design variables and constraints the analysis code will require a particularly long time to evaluate sensitivities. Automatic/algorithmic differentiation or analytic derivative calculations (direct or adjoint) can be used to avoid multi-discipline analysis evaluations. [Pironneau]65 pioneered the adjoint method in fluid dynamics, showing that the cost of computing sensitivity information was almost completely

M. Secanell and A. Suleman. “Numerical Evaluation of Optimization Algorithms for Low-Reynolds-Number Aerodynamic Shape Optimization”. AIAA Journal, 43(10):2262{2267, 2005. 64 Z. Lyu, Z. Xu, and J.R.R.A. Martins. “Benchmarking Optimization Algorithms for Wing Aerodynamic Design Optimization”. International Conference on Computational Fluid Dynamics, ICCFD8-2014-0203, pages 1-18, Chengdu, Sichuan, China, 2014. 65 O. Pironneau. “On Optimum Design In Fluid Mechanics”. Journal of Fluid Mechanics, 64(1): 97-110, 1974. 63

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independent of the number of design variables, and hence the overall cost of optimization is roughly linearly proportional to the number of design variables. The adjoint form of the sensitivity information is particularly efficient for aerodynamic optimization applications as the number of cost functions (outputs) is small, while the number of design variables (inputs) is relatively larger. The discrete adjoint method (as opposed to continuous adjoint method) is generally favored in aerospace-based optimization as it ensures that sensitivities are exact with respect to the discretized objective function. The implementation of the adjoint method for the governing equations of the ow analysis can often be difficult to derive and require direct manipulation; adjoint methods require much more involved detailed knowledge of the computational domain. One way to approach this difficulty is to use automatic/algorithmic differentiation, which is a method based on the systematic application of the differentiation chain rule to the source code to compute the partial derivatives required by the adjoint method. [Mader et al]66 developed a discrete adjoint method for Euler equations using automatic differentiation (AD), later followed by [Lyu et al.]67 who extended and developed this adjoint implementation to Reynoldsaveraged Navier-Stokes (RANS) equations and introduced simplifications to the automatic differentiation approach. Methods developed have shown robust an efficient application to highfidelity optimization. Others adopted similar methods for the high-fidelity aerodynamic optimization of non-planar wings addressing the non-linearity of wake shape and how it can impact the induced drag. Several non-planar geometries, inherently creating non-planar wake-wing interactions, are optimized using discrete adjoint sensitivities. This work illustrates the drawbacks in static-wake assumptions, demonstrating that higher-order effects must be included for accurate induced drag prediction and hence for meaningful optimizations. This work was followed by [Gagnon and Hicken]68 for the aerodynamic optimization of un-conventional aircraft configurations. These global geometric variables are generally not considered in high-fidelity simulation. The cost of allowing such geometric variation away from the baseline under high-fidelity optimization limited how many variables could be considered in any one optimization process. The authors observed limited optimization in some wing configurations because of this. 3.3.5 Aero-Elastic Optimization Aero elastic optimization requires the coupling of aerodynamic and structural models for most effective sensitivity analysis in optimization routines. Even small changes in aerodynamic shape can have a large influence on aerodynamic performance with various flow conditions resulting in multiple shapes. Wing flexibility impacts not only the static flying shape but also its dynamics, resulting in aero elastic phenomenon such as utter and aileron reversal. Based on this principle, to enable high-fidelity aero structural optimization while encompassing hundreds of design variables, [Martins et al.]69 proposed the use of a coupled adjoint method to compute sensitivities with respect to both the aerodynamic shape and the structural sizing. [Kenway et al.] subsequently made several developments and demon-started that the computation of coupled aero elastic gradient calculations were scalable to thousands of design variables and millions of degrees of freedom, and since applied it to the aero structural optimization of high aspect ratio wings with different structural properties.

C.A. Mader, J.R.R.A. Martins, J.J. Alonso, and E.V. Der Weide. “An Approach for the Rapid Development of Discrete Adjoint Solvers”. AIAA Journal, 46(4):863-873, 2008. 67 Z. Lyu, G. Kenway, C. Paige, and J.R.R.A. Martins. “Automatic Differentiation Adjoint of the Reynolds-Averaged Navier-Stokes Equations with a Turbulence Model”, 21st AIAA Computational Fluid Dynamics Conference, AIAA 2013-2581, pages 1-24, San Diego, California, 2013. 68 H. Gagnon and D.W. Zingg. “High-Fidelity Aerodynamic Shape Optimization of Unconventional Aircraft through Axial Deformation”. 52nd Aerospace Sciences Meeting, AIAA 2014-0908, pages 1-18, Maryland, 2014. 69 J.R.R.A. Martins, J.J. Alonso, and J.J. Reuther. “A Coupled-Adjoint Sensitivity Analysis Method for High-fidelity aero-structural design”. Optimization and Engineering, 6(1):33{62, 2005. 66

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More recently, [Burdette et al.]70 applied the coupled discrete adjoint method with the sparse nonlinear optimizer SNOPT for wing morphology optimization. This approach was capable of handling over a thousand design variables and constraints. The coupled adjoint method is also applicable to lower-fidelity model where they used a vortex lattice method and finite element analysis tool capable of accurately mimicking high-fidelity accuracy at a greatly reduced computational cost. The coupling of design constraints makes the optimizer additionally capable of considering more sophisticated criteria flight dynamics into the coupled adjoint sensitivity and explored the use of static and dynamic stability constraints. The result showed that coupling stability constraint sensitivities into the adjoint formulation had a significant impact on optimal wing shape. Elsewhere, structural dynamics were considered by [Zhang et al.]71 who used a coupled-adjoint formulation to include utter constraints. The utter constraints used the coupled aerodynamic/structural solver to suppress utter onset by identifying dominant modes and adjusting variables such as the wing stiffness. Others investigated using modular sensitivity analysis for aero structural sequential optimization of a sailplane. They showed that coupled aero structural optimization gave higher performance designs than those identified by sequential optimization of aerodynamics followed by structural optimization. Subsequently, [Grossman et al.]72 optimized the performance of a subsonic wing configuration showing that while modular sensitivity analysis for sequential optimization reduced the total number of function calls and sensitivity calculations, the wing performance gain was limited. When performing sequential optimization the optimizer does not have sufficient information necessary for aero elastic tailoring. This limitation of sequential optimization is further explained by [Chittick and Martins]73. A significant drawback of all gradient-based algorithms is the requirement for continuity and low-modality throughout the design space otherwise the algorithm may become sub-optimally trapped. The challenge is that an aerodynamic shape analysis throughout a geometrically varying search space will encounter both non-continuous topological and local flow changes, each providing local optima. Gradient-dependent algorithms' robustness significantly decreases in the presence of discontinuity and lack of convergence, usually related to turbulence modelling, making the objective function noisy.98 Kenway99 encountered such a problem with aerodynamic shape optimization with a separation-based constraint formulation to mitigate buffetonset behavior at a series of operating conditions. The discontinuity from the `separation sensor' function arose from monitoring the wing local surface for separated ow; this resulting in locally negative skin friction coefficients. To address this issue blending functions were to be implemented to smooth the discontinuity, smearing the separation sensor value around the separated ow region. 3.3.6 Multi-Point Optimization [Kenway and Martins]74, among several others, have used multi-point optimization strategies in order to consider several operating conditions simultaneously. For more realistic and robust design it is crucial to take into account more than one operating condition, especially off design conditions, which form additional multi-objective requirements into the optimization. The single-point D.A. Burdette, G. Kenway, Z. Lyu, and J.R.R.A. Martins. “Aero structural Design Optimization of an Adaptive Morphing Trailing Edge Wing”. 15th AIAA/ISSMO Multi-disciplinary Analysis and optimization Conference, AIAA paper 2014-3275, pages 1-13, 2014. 71 Z. Zhang, P.C. Chen, Z. Zhou, S. Yang, Z. Wang, and Q. Wang. “Adjoint Based Structure and Shape Optimization with Flutter Constraints”. 57th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA 2016-1176, pages 1-23, San Diego, California, 2016. 72 B. Grossman, R.T. Haftka, P.J. Kao, D.M. Polen, and M. Rais-Rohani. “Integrated Aerodynamic-Structural Design of a Transport Wing”. Journal of Aircraft, 27(12):1050-1056, 1990. 73 I.R. Chittick and J.R.R.A. Martins. “An Asymmetric Sub-optimization Approach to Aero-structural Optimization”. Optimization and Engineering, 10(1):133-152, 2009. 74 G. Kenway and J.R.R.A. Martins. “Aerodynamic Shape Optimization of the CRM Configuration Including BuffetOnset Conditions”, 54th AIAA Aerospace Sciences Meeting, AIAA 2016-1294, pages 1-23, San Diego, California. 70

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optimization achieved an 8.6 drag count reduction and the shock-wave over the upper surface of the wing is almost entirely eliminated. Drag divergence curves in this work show the nature of the singlepoint optimization presenting a significant dip in the drag at the design condition, but the performance is significantly deteriorated at off design conditions relative to the baseline condition. The multi-point optimization, accounting for 3 design conditions, found that drag at the nominal operating condition increased by 2.8 counts and produced double shocks on the upper surface of the wing visible in Figure 20. However, at the sacrifice of performance at the nominal operating condition, off design conditions for the multi-point optimization design was found to perform substantially better over the entire range of Mach numbers. Though biasing the optimization toward certain operating conditions the authors show that multi-point optimization with all conditions near the on-design condition is not sufficient for an overall robust design when considering operational envelopes.

Figure 20

High performance low drag solutions found for Single and Multiple design points

3.4 Application of Gradient-Free Methods to Aerodynamic Optimizations The principal source of difficulty in the application of gradient-based optimizers is the requirement for having a non-discontinuous and mathematically predictable design space. Non-gradient based methods can prove more complex to implement than GBM, but they do not require continuity or predictability over the design space, and usually increase the likelihood of finding a global optimum. Methods of optimization known as metaheuristics can offer robust methods of finding a solution, and increases the likelihood of converging onto a solution at the global optimum. These gradient-free methods are known to be capable of engaging with numerically noisy optimization problems that be difficult for GBM. This is because metaheuristics operate from a completely different paradigms usually based on the some naturally occurring phenomenon. Unlike gradient methods, derivatives of the cost functions are not necessary, allowing metaheuristics to easily cope with non-continuous or numerically noisy cost functions. Furthermore, no pre-defined baseline design or knowledge of the design space is required and gradient-free methods typically optimize several solutions in parallel.

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3.4.1 Genetic Algorithms (GA) GAs are a population-based optimization technique based on the Darwinian theory of survival of the fittest: a primary aspect of evolution. These algorithms are often praised for their ability to explore and exploit solutions simultaneously due to their inherent multi-start capability. They are easily capable of constructing insightful design trade-off relationships, referred to as Pareto fronts, between objectives. Figure 21 shows the resulting Pareto optimal solutions found by [Yamazaki et al.]75 in a winglet design problem illustrating the trade-off between pure drag (induced + wave+ profile) and

Figure 21 Non-dominated solution information for a winglet design problem using a Genetic Algorithm, Courtesy of [Yamazaki et al.]

root bending moment. GAs are well-suited to complex optimization tasks as they can easily use both discrete and continuous variables, and can easily handle non-linear, non-convex, and non-continuous objective functions. Chiba et al.105 suggest that GAs have four distinct advantages which encourage their use in aerodynamic/aero-structural optimizations:    

GAs have the ability to find multiple optimal solutions and design trade-offs; GAs process information in parallel, optimizing from multiple points within the design space; high-fidelity CFD codes can be adapted to GAs without any modifications; and GAs are insensitive to numerical noise that may be present in the computation.

The main drawbacks associated with these algorithms are high computational cost, poor constraints handling abilities, requirement for problem specific tuning and limitations in how many variables are feasible to handle. Studies have shown that GAs are very fast at identifying regions of optimality within a design space but demonstrate slow convergence as they moves nearer optimal solutions. W. Yamazaki, K. Matsushima, and K. Nakahashi. “Aerodynamic Design Optimization Using the DragDecomposition Method”. AIAA Journal, 46(5):1096-1106, 2008. 75

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Some studies have tried to build on the classical GA to enhance its applications to aerodynamic optimization. GA optimization is, in its most basic form, an iterative process which can be summarized as: Random generation of individuals to form the initial population. 1. Evaluation of the fitness/survivability of each individual in the population to the given environment. This would be done with the aerodynamic solver. 2. Selection of individuals to take part in genetic operations. 3. Apply genetic operations which mimic reproduction to define a new population. 4. Iterate over steps 2-4 are over multiple generations until some convergence criterion is met. Other noticeable techniques of Gradient-Free Methods to Aerodynamic Optimizations, includes:  

Particle Group Optimization Simulated Annealing (SA)

Details for these and other methods can be found in [Skinner and Zare-Behtash]76, as well as in Wikipedia. 3.4.1.1

Case Study - Framework for the Shape Optimization of Aerodynamic Profiles Using Genetic Algorithms (GA) To demonstrate the Genetic Algorithm GA, [López et al.]77 developed a framework for the shape optimization of aerodynamics profiles using computational fluid dynamics (CFD) and genetic algorithms. A genetic algorithm code and a commercial CFD code were integrated to develop a CFD shape optimization tool, as illustrated in Figure 22. The results obtained demonstrated the effectiveness of the developed tool. The shape optimization of airfoils was studied using different strategies to demonstrate the capacity of this tool with different GA parameter combinations. Details of procedures can be found in78. Optimization was performed using a simple GA that

Figure 22

Optimization Scheme (GA)

S. N. Skinner and H. Zare-Behtash, ”State-of-the-Art in Aerodynamic Shape Optimisation Methods”, Article in Applied Soft Computing , September 2017, DOI: 10.1016/j.asoc.2017.09.030. 77 D. López, C. Angulo, I. Fernández de Bustos, and V. García, “Framework for the Shape Optimization of Aerodynamic Profiles Using Genetic Algorithms”, Hindawi Publishing Corporation, Mathematical Problems in Engineering, Volume 2013, Article ID 275091, 11 pages http://dx.doi.org/10.1155/2013/275091. 78 See above. 76

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follows the sequence shown in Figure 22. First, a seed number 𝑠 is provided by the user, which is used to generate a univocal sequence of random numbers that form the genes of the P chromosomes in the initial generation population. This is denoted as P(𝑡 = 0), where P means population and 𝑡 = 0 is the number of the generation. The seed number could also be generated randomly, but the former option was preferred because it facilitated the performance of different experiments starting with the same initial population to investigate the separate effects of different combinations of GA parameters on exploratory and exploitative behaviors during search. The aim was to tune them to obtain the best results for a given objective function. After generating the initial population P(0), each individual was before selection, crossover, and mutation operators are applied to this population to obtain the next generation of individuals P(1). 3.4.2 Application of Hybrid Algorithms to Aerodynamic Optimizations Numerous hybrid algorithms which incorporate elements from different optimization algorithms exist and have shown successful application. Here we will only consider hybrid algorithm schemes applied to aerodynamic design problems; in an optimization context, hybridization can be defined as mixing two or more algorithms, with possibly, further complementary features. Deterministic approaches take advantage of the analytical properties of the search space to generate a sequence of candidate solutions with systematic improvements, usually resulting in the number of design iterations required to be small; a major short-coming is the dependency of a compatible design space and sufficient baseline geometry. Heuristic approaches do not take into account properties of the search space, but instead seek improvements to candidate solutions on the basis of experience and judgement with a probabilistic approach which can lead to large numbers of design iterations and long computing times. Therefore, hybrid approaches try to combine methods from these approaches in an attempt to mitigate the weaknesses each hold. Hybridization does not exclusively require deterministicheuristic combinations as heuristic-heuristic hybrid algorithms also exist. The various methods of hybridization between different types of algorithms can be classified into three main groups:  pre-hybridization, where, for example, the population of the (GA) is pre-optimized using the Gradient-Based Method (GBM);  organic-hybridization, in which the (GBM) is used as an operator within the (GA)s for improving each population member in each generation;  post-hybridization, in which the (GA)s final population is used to provide an initial design for the (GBM). It is should be noted that these classifications are not limited to the hybridization of GAs and GBMs, however, the most frequent hybrid algorithm found in aerospace application have been designed in attempt to combine the best characteristics of GAs and GBM. [Gage et al.]79 present the posthybridization of a classical GA with sequential quadratic programming for the topological design of wings and trusses. This work is notable for hybridization as it is one of the first hybrid methods (HM) employed in aerodynamic design optimization. They demonstrated that post-hybridization is effective for final refinement of the GA's candidate solutions. By switching to a GBM once the GAs population is sufficiently mature, computational demands can be reduced and superior solutions can be found relative to allowing the GA to continue. In more recent work, [Kim et al.]80 also use postoptimization, to improve the aerodynamic and acoustic performance of a axial-flow fan, by combining 79 P. Gage, I. Kroo, and P. Sobieski.

“Variable-Complexity Genetic Algorithm for Topological Design”. AIAA Journal, 33(11):2212-2217, 1995. 80 J. Kim, B. Ovgor, K. Cha, J. Kim, S. Lee, and K. Kim. “Optimization of the Aerodynamic and Aero acoustic Performance of an Axial-Flow Fan”. AIAA Journal, 52(9):2032-2044, 2014.

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the multi-objective real-encoded NSGA-II from which the Pareto-optimal solutions are further optimized using sequential quadratic programming. The specific difficulty in this method of hybridization is the transition from a multi-objective problem to a single-objective problem. There are typically two ways to transition:  combine all of the objectives into a composite objective using a weighted-sum approach for example;  sequential optimization, optimizing one objective at a time while treating all other objectives as equality constraints. [Kim et al.]81 adopted the latter method, pointing out that it did not preserve Pareto optimality and created a set of optimal solutions for each objective with many duplications forming. A preoptimization strategy is proposed in which takes advantage of the GA's stochastic search capability and the gradient-based optimizer SNOPT's ability to efficiently identify local optima and enforce constraints directly. They implement the developed HM into a series of different aerodynamic optimization problems including: aero foil optimization, transonic wing-section optimization, subsonic wing-section optimization, and blended-wing-body optimization. The HM's initial population is optimized by SNOPT for a limited number of design iterations on each candidate solution in order to balance biased and insufficient solution development. The improved candidate solutions are then stochastically perturbed by the GA's crossover operations to define a new population. Perturbing-mutation operations (perturbs random solution variables) had to be avoided as it was found to conflicted with the refinement capability of the GBM. Pure-mutation operations (random replacement of solution variables) were included to improve the stochastic search. Similar pre-optimization strategies have been employed by [Xing and Damodaran]82 which combine GA stochastic searching and GBMs analytical optimization procedures in the optimization of nozzle shapes. Compiled optimization results from [Chernukhin and Zingg]83 for a series of optimization problems was found to significantly outperform other algorithms for highly-multi-modal problems; namely the Griewank function which offers hundreds of locally optimal solutions. The classical GA has been extended for the purpose of more efficient aerodynamic shape design optimization by [Catalano et al.]84 to include two new operations based on gradient search in a hybrid organic-optimization algorithm. Figure 23 shows how the classical genetic algorithm has been modified to include such operators. The activation of each gradient operator is controlled probabilistically, inspired by classical crossover and mutation operation use. The first gradient-based operator, which acts on the whole population, has two further probabilistic controls determining behavior with each candidate solution. The first deter-mines the maximum number of iterations and the second determines the sensitivity analysis method the gradient optimizer uses. The second gradient operator is relatively simpler, applying only one optimization iteration to the current best solution according to the steepest descent rule with a random step size. Finally there is an exclusive mechanism which

See previous. X. Xing and M. Damodaran. “Design of Three-Dimensional Nozzle Shape using NURBS, CFD and Hybrid Optimization Strategies”. 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, AIAA4368, 2004. 83 O. Chernukhin and D.W. Zingg. “Multimodality and Global Optimization in Aerodynamic Design”. AIAA Journal, 51(6):1342-1354, 2013. 84 L.A. Catalano, D. Quagliarella, and P.L. Vitagliano. “Aerodynamic Shape Design Using Hybrid Evolutionary Computing and Multigrid-Aided Finite-Difference Evaluation of Flow Sensitivities”. International Journal for Computer Aided Engineering and Software, 32(2):178-210, 2014. 81 82

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preserve the test solutions from each generation and reintroduces the best known solution from all generations into the current population. For the single-objective optimization of an aero-foil with 24 variables, [Catalano et al.]85 find that the complex HM developed is comparable to the standard GBM in terms of the number of function evaluations only achieving a 0.4% better objective. Under different settings (population, number of iterations, sensitivity analysis, etc.), they find that the hybrid algorithm slightly outperforms the gradient method, reducing the objective function further by 2.2%, but requires many more function evaluations. The classical GA is outperformed by all other algorithms. Therefore, the hybridized GA-

Figure 23

Hybrid Organic-Optimization Algorithm

GBM showed accelerated performance in aerodynamic optimization relative to a standard GA and capable of matching, and in one case able to outperform, the GBM. The GBM (with a suitable sensitivity analysis) is shown overall more effective, the hybrid algorithm performed well however added unnecessary complexity to the optimization framework. PSO and GBM have also been combined for aerodynamic design. [Jansen et al.]86 used a PSO algorithm post-optimized with the gradient optimizer, SNOPT, in the aero-structural optimization on nonplanar wings using a panel method and potential flow theory. Little information is given on the hybridization however the reasoning behind the method employed was to compensate for instances when the PSO converged prematurely. This ensured the PSO solutions were at a local minimum and offered further solution refinement where possible.

L.A. Catalano, D. Quagliarella, and P.L. Vitagliano. “Aerodynamic Shape Design Using Hybrid Evolutionary Computing and Multigrid-Aided Finite-Difference Evaluation of Flow Sensitivities”. International Journal for Computer Aided Engineering and Software, 32(2):178-210, 2014. 86 P.W. Jansen, R.E. Perez, and J.R.R.A. Martins. “Aero-structural Optimization of Nonplanar Lifting Surfaces”. Journal of Aircraft, 47(5):1490-1503, 2010. 85

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Alternative hybrid methods based on hybridizing two heuristic approaches also exist in aerodynamic optimization literature. A modified binary genetic algorithm (GA) and SA have been used together for organic-hybridization using the optimization of wind turbine rotors. A schematic diagram of the hybrid algorithm procedure is shown in Figure 24 which shows that the hybrid procedure optimized the radial geometric distributions of the blade, using the SA algorithm, and the selection of aero-foil shape along the blade discretized span used the GA genetic operators. The GA did not optimize the combination of sectional aero-foil shape over the span, it acted to perturb the combination of

Figure 24

Hybrid GA-SA Organic-Optimization Algorithm

sectional aero-foil (specifically the camber distribution) along the blade for which there were 10017 possible combinations. To demonstrate the hybrid framework developed, a SA algorithm was used to optimise radial geometric of different blades with fixed camber distributions from which the results were compared to hybrid procedure. Figure 25 compares the different geometries and aerodynamic performance curves for different optimized NACA 4-digit wind turbine blades. The genetic operations in the hybrid algorithm gave the optimization process the ability to leverage the sectional aero-foil distribution to allow a nearly constant chord and thickness distribution. The obtained chord length attributed to aerodynamic benefits of operating at higher Reynolds numbers and the low thickness mitigated performance deterioration near stall. Furthermore, the results showed that the hybrid optimizer held distinct improvements in the blade design including: a reduction of the cut-in wind speed, increase aerodynamic efficiency, and an overall reduction of material used to manufacture the blades.

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Other heuristic hybrids that exist include hybridizations of operations between GAs and PSO, or SA and PSO. These studies typically attempt to handle highly multi-modal design spaces requiring large degrees of modification while addressing strongly stipulated aerodynamic constraints. For example, [Khurana et al.]87 show that for aero-foil shape optimization, the classical PSO algorithm was unable to optimize to a feasible optimal which met all constraints once the dimensionality of the problem exceeded 10 variables. Additionally, excessive computational resource was required for the evaluation of aerodynamically infeasible designs. Part of the problem was associated with the lack of search diversity attributing to sub-optimal solutions. To overcome this they introduced systematic mutation operators from GAs and observed significant convergence improvements in fewer iterations.

Figure 25

Geometry and aerodynamic performance of optimized wind turbine blades, optimized by the SA and hybrid GA-SA algorithms

3.4.3 Application of Surrogate Modelling to Aerodynamic Optimization Surrogate modelling can be viewed as a non-linear inverse problem with the aim of determining a continuous function that relates design variables to output responses from finite data. Surrogate assisted optimization aims to alleviate the computational burden of the aerodynamic optimization process by defining a simplified mathematical relationship allowing for fewer numerical simulations to be required. To interrogate the surrogate an optimization algorithm is needed to perform a global search of the design space relating to the response surface. It is seen from the literature that surrogates are almost exclusively coupled with gradient-free population based optimization M.S. Khurana, H. Winarto, and A.K. Sinha. “Airfoil Optimisation by Swarm Algorithm With Mutation and Arti_cial Neural Networks”. 47th AIAA Aerospace Sciences Meeting and Aerospace Exposition, (January), 2009. 87

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methods, such as genetic or particle swarm algorithms. Examples of surrogate assisted stochastic optimization would be as optimized a diffuser shape, or optimized the nacelle/pylon position for a wing, as well as, optimized ground vehicle aerodynamics. Key steps are outlined by [Hashimoto et al.]88 for a surrogate assisted aerodynamic optimization which is summarized in Figure 26. The construction of the surrogate generally consists of three steps:  design of experiment sample plan to generate initial sample points in the design space-point selection;  numerical simulations are performed to compute the output/performance of each sample point;  sample point data (input & output) are used by an approximation model to construct the surrogate. Replacing a particular problem analysis with a surrogate analysis does not affect the problem formulation, but it will strongly influence the solutions identified. Therefore, once the surrogate model is constructed it must be validated (sometimes considered a 4th step). This has the purpose of establishing the predictive capabilities of the surrogate model in design regions away from known sample data. There are both parametric and non-parametric alternatives in constructing a surrogate model. Parametric approaches (such as kriging or polynomial regression) are model dependent forming a functional relationship between the response variables and the design variable samples that are known. Non-parametric approaches (such as radial-basis functions or neural networks) use local models in different regions of the sample data to build-up an over fall frame work of the model. Furthermore, surrogates can also be classified into regression type (polynomial regression, radial basis functions) which tend to be better suited to noisy functions, and interpolation type (kriging) creating best-t response models. Usage of any of these models is not straight forward as the quantity and quality of information the user has to provide in the construction of the surrogate is not known a priori. Furthermore, the efficient exploitation of training data can be restricted by inherent problem complexity, constraints, design variable dimensionality, and accuracy Vs. computational cost. Hence selection of the most appropriate surrogate is considered problem dependent, as it will directly influence the optimization algorithms' decision making capability. There are no general rules leading to the choice of type of surrogate, generation of sample data for training and validation, and indeed the combination of surrogate model and optimization algorithm. Different surrogate models will be better suited to different data sets and care must be taken to not over-generalize the problem or false optimization my occur. Usage of any of these models is not straight forward as the quantity and quality of information the user has to provide in the construction of the surrogate is not known a priori. Furthermore, the efficient exploitation of training data can be restricted by inherent problem complexity, constraints, design variable dimensionality, and accuracy Vs. computational cost. Hence selection of the most appropriate surrogate is considered problem dependent, as it will directly influence the optimization algorithms' decision making capability. There are no general rules leading to the choice of type of surrogate, generation of sample data for training and validation, and indeed the combination of surrogate model and optimization algorithm. Different surrogate models will be better suited to different data sets and care must be taken to not over-generalize the problem or false optimization my occur. Parametric surrogates have been widely applied to aerodynamic optimizations due to their flexibility

A.H. Hashimoto, S.J. Jeong, and S.O. Obayashi. “Aerodynamic Optimization of Near-Future High-Wing Aircraft”. The Japan Society for Aeronautical and Space Sciences, 58(2):73-82, 2015. 88

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and ease of use. The resulting model will form a response surface that fits exactly to the sample data points. Therefore parametric models are very well suited to conditions where a design space is poorly/sparsely sampled, clustered, or noisy. These approaches do not rely on any specific model structure and are a successful statistical tool for modelling globally dispersed spatial observations. Perhaps the most straight-forward parametric surrogate is one formulated through polynomial regression. [Lian and Liou]89 use 2nd order polynomial response surface equations to construct the functional relationship between design variables and objectives enabling GAs to re-design centrifugal compressors and transonic compressor blades. As the surrogate greatly reduces the computational

Figure 26

Example structure for surrogate based optimization with a standard genetic algorithm, Courtesy of [Hashimoto et al.]

Y. Lian and M. Liou. “Multi-objective Optimization Using Coupled Response Surface Model and Evolutionary Algorithm”. AIAA Journal, 43(6):1316-1325, 2005. 89

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cost of function evaluations, the GA population and generation sizes were increased due to the freed computational resources. This facilitated increased exploration and identification of superior solutions. Other studies have used polynomial regression response surfaces for the optimization of helicopter rotor designs to increase efficiency at a reduced vibration over different operating conditions. [Collins et al.]90 report development of these methods and show that 4th order polynomials, requiring over 300 simulations to construct, are capable of accurately approximating performance metrics achieving a regression coefficient. This surrogate maintained robustness of the high fidelity numerical simulations at a greatly reduced computational time, however, scaling functions were required to map low fidelity results to the higher-fidelity domain. Non-parametric surrogates are becoming more popular in aerodynamic optimizations, but can be more complex to implement. The increasing popularity is based on their capability to approximate any continuous behavior with arbitrary precision of the host computing environment. For instance radial basis functions (RBF), although inferior to kriging models regarding interpolating accuracy235, are easier to characterize and modify, and its superior smoothness can make it more suitable for many design spaces. One of the advantages found was that once the initial computation cost of evaluating the design points was established, the computational cost through-out the optimization was constant (therefore predictable) and significantly reduced. Neural networks (multi-layered radial basis functions) form a more sophisticated behavioral modelling technique which may be unnecessary for most aerodynamic optimizations. The persistent difficulties associated with these methods is the need to choose the structure of the surrogate network and a suitable training sequence. In a situation where one may not know which surrogate model would perform best, the use of multiple or hybrid surrogates may be suitable and can offer several advantages to the optimization decision making capability. study the application of several open source surrogate modelling tool boxes demonstrating that hybridizing up to ten surrogates (one of which kriging) outperformed the stand alone kriging surrogate by substantially reducing the number cycles required for convergence. In certain instances however while the number of cycles for the optimization was reduced the number of functions calls required increased, i.e. computational demand increased as wall-clock time decreased, indicating parallel computation could be beneficial. Additionally results show that the rate of convergence did not scale with the number of surrogates in the hybrid model 5 and 10 surrogate hybrid models achieved comparable results. The particular aspect of this study attractive to aerodynamic optimization is that the use of multiple surrogates are shown to have a distinct advantage in highly dimensional design spaces with multiple design targets. Hybrid surrogate modelling under different techniques aid in design exploration and generate diversity as well as improve the surrogates prediction thus reducing the model uncertainty.

3.5 Case Study - Aerodynamic Shape Optimization sing a Gradient Based applied to a Common Wing

The design of transonic transport aircraft wings is particularly important because of the large number of such aircraft operating on a daily basis, and because small changes in the wing shape may have a large impact on fuel burn. This directly affects both the airlines' cash operating cost and the emission of green-house gases. Despite considerable research on aerodynamic shape optimization, there is no standard benchmark problem allowing researchers to compare results. This was also address by [Mavriplis]91 which he complained about the lack of certification by analysis. K.B Collins and D.N. Mavris. “Application of Low and High-Fidelity Simulation Tools to Helicopter Rotor Blade Optimization”. Journal of the American Helicopter Society, 042003: 1-11, 2013. 91 Dimitri Mavriplis, Department of Mechanical Engineering, University of Wyoming and the Vision CFD2030 Team, “Exascale Opportunities for Aerospace Engineering”, AIAA 2007-4048. 90

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Aerodynamic shape optimization can be dated back to the 16th century, when Sir Isaac Newton 92 used calculus of variations to minimize the fluid drag of a body of revolution with respect to the body's shape. Although there were many significant developments in optimization theory after that, it was only in the 1960s that both the theory and the computer hardware became advanced enough to make numerical optimization a viable tool for everyday applications. The application of gradient-based optimization to aerodynamic shape optimization was pioneered in the 1970s. The aerodynamic analysis at the time was a full-potential small perturbation inviscid model, and the gradients were computed using finite differences. Hicks et al93 first tackled airfoil design optimization problems. [Hicks and Henne]94 then used a three-dimensional solver to optimize a wing with respect to 11design variables representing both airfoil shape and the twist distribution. Because small local changes in wing shape have a large effect on performance, wing design optimization is especially effective for large numbers of shape variables. As the number of design variables increases, the cost of computing gradients with finite differences becomes prohibitive. The development of the adjoin method addressed this issue, enabling the computation of gradients at a cost independent of the number of design variables. For a review of methods for computing aerodynamic shape derivatives, including the adjoin method, see [Peter and Dwight]95. For a generalization of the adjoin method and its connection to other methods for computing derivatives, see [Martins and Hwang]96. 3.5.1 Methodology This section describes the numerical tools and methods that we used for the shape optimization studies. These tools are components of the framework for multidisciplinary design optimization (MDO) of aircraft configurations with high fidelity (MACH)97. MACH can perform the simultaneous optimization of aerodynamic shape and structural sizing variables considering aero-elastic directions98. However, here we use only the components of MACH that are relevant for aerodynamic shape optimization: the geometric parametrization, mesh perturbation, CFD solver, and optimization algorithm. 3.5.1.1 Geometric Parametrization Many different geometric parameterization techniques have been successfully used in the past for aerodynamic shape optimization. These include mesh coordinates (with smoothing), B-spline surfaces, Hicks–Henne bump functions, camber-line-thickness parameterization99, and free-form deformation (FFD)100. In this work is done using a Free Form Design (FFD) volume approach. The Newton, S. I., Philosophic Naturalis Principia Mathematica, Londini, jussi Societatus Regiaeac typis Josephi Streater; prostat apud plures bibliopolas, 1686. 93 Hicks, R. M., Murman, E. M., and Vanderplaats, G. N., “An Assessment of Airfoil Design by Numerical Optimization," Tech. Rep. NASA-TM-X-3092, NASA, 1974. 94 Hicks, R. M. and Henne, P. A., “Wing Design by Numerical Optimization," Journal of Aircraft, Vol. 15, 1978. 95 Peter, J. E. V. and Dwight, R. P., “Numerical Sensitivity Analysis for Aerodynamic Optimization: A Survey of Approaches," Computers and Fluids, Vol. 39, 2010, pp. 373-391. 96 Martins, J. R. R. A. and Hwang, J. T., “Review and Unification of Methods for Computing Derivatives of Multidisciplinary Computational Models," AIAA Journal, Vol. 51, No. 11, 2013, pp. 2582-2599. 97 Kenway, G. K. W., Kennedy, G. J., and Martins, J. R. R. A., “Scalable Parallel Approach for High-Fidelity SteadyState Aero-elastic Analysis and Adjoint Derivative Computations," AIAA Journal, Vol. 52, No. 5, 2014, pp. 935-951. 98 Kenway, G. K. W. and Martins, J. R. R. A., “Multi-point High-fidelity Aero-structural Optimization of a Transport Aircraft Configuration," Journal of Aircraft, Vol. 51, No. 1, 2014, pp. 144-160. 99 Carrier, G., Destarac, D., Dumont, A., Meheut, M., Din, I. S. E., Peter, J., Khelil, S. B., Brezillon, J., and Pestana, M., “Gradient-Based Aerodynamic Optimization with the elsA Software,” 52nd Aerospace Sciences Meeting, 2014. 100 Kenway, G. K., Kennedy, G. J., and Martins, J. R. R. A., “A CAD-free Approach to High-Fidelity Aero-structural Optimization,” Proceedings of the 13th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference, Fort Worth, TX, 2010. 92

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FFD approach can be visualized as embedding the spatial coordinates defining a geometry inside a flexible volume. The parametric locations (u; v; w) corresponding to the initial geometry are found using a Newton search algorithm. Once the initial geometry is embedded, perturbations made to the FFD volume propagate within the embedded geometry by evaluating the nodes at their parametric locations. Using B-spline volumes for the FFD implementation, and displacement of the control point locations as design variables. The sensitivity of the geometric location of the geometry with respect to the control points is computed efficiently using analytic derivatives of the B-spline shape functions101. The FFD volume parametrizes the geometry changes rather than the geometry itself, resulting in a more efficient and compact set of geometry design variables, thus making it easier to manipulate complex geometries. Any geometry may be embedded inside the volume by performing a Newton search to map the parameter space to the physical space. All the geometric changes are performed on the outer boundary of the FFD volume. Any modification of this outer boundary indirectly modifies the embedded objects. The key assumption of the FFD approach is that the geometry has constant topology throughout the optimization process, which is usually the case in wing design. In addition, since FFD volumes are B-spline volumes, the derivatives of any point inside the volume can be easily computed. Figure 27 shows the FFD volume and the geometric control points (red dots) used in the aerodynamic shape optimization. The shape design variables are the displacement of all FFD control points in the vertical (z) direction.

Figure 27

The shape design variables are the z-displacements of 720 FFD control points

3.5.1.2 Mesh Perturbation Since FFD volumes modify the geometry during the optimization, we must perturb the mesh for the CFD to solve for the revised geometry. The mesh perturbation scheme used here is a hybridization of algebraic and linear elasticity methods, developed by [Kenway et al.]102. The idea behind the hybrid scheme is to apply a linear-elasticity-based perturbation scheme to a coarse approximation of the mesh to account for large, low-frequency perturbations, and to use the algebraic warping approach to attenuate small, high-frequency perturbations. 3.5.1.3 CFD Solver We use a finite-volume, cell-centered multi-block solver for the compressible Euler, laminar Navier Stokes, and RANS equations (steady, unsteady, and time periodic). The solver provides options for a De Boor, C., A Practical Guide to Splines, Springer, New York, 2001. Kenway, G. K., Kennedy, G. J., and Martins, J. R. R. A., “A CAD-free Approach to High-Fidelity Aero-structural Optimization," Proceedings of the 13th AIAA/ISSMO Multi-disciplinary Analysis Optimization Conference”, 2010. 101 102

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variety of turbulence models with one, two, or four equations and options for adaptive wall functions. The Jameson-Schmidt-Turkel (JST) scheme augmented with artificial dissipation is used for the spatial discretization. The main ow is solved using an explicit multi-stage Runge-Kutta method, along with geometric multi-grid. A segregated Spalart-Allmaras turbulence equation is iterated with the diagonally dominant alternating direction implicit method. To efficiently compute the gradients required for the optimization, we have developed and implemented a discrete adjoin method for the Euler and RANS equations. The adjoin implementation supports both the full-turbulence and frozenturbulence modes, but in the present work we use the full-turbulence adjoin exclusively. We solve the adjoin equations with preconditioned [GMRES]103. The Euler-based Aerodynamic shape optimization and Aero-Structural optimization has been studies extensively earlier. However, preceding observation indicates serious issues with the resulting optimal Euler-based designs due to the missing viscous effects. While Euler-based optimization can provide design insights, it has found that the resulting optimal Euler shapes are significantly different from those obtained with RANS. Euler-optimized shapes tend to exhibit a sharp pressure recovery near the trailing edge, which is nonphysical because such flow near the trailing edge would actually separate. Thus, RANS-based shape optimization is necessary to achieve realistic designs. 3.5.1.4 Optimization Algorithm Because of the high computational cost of CFD solutions, we must choose an optimization algorithm that requires a reasonably low number of function evaluations. Gradient-free methods, such as genetic algorithms, have a higher probability of getting close to the global minimum for multi-nodal functions. However, slow convergence and the large number of function evaluations make gradient free aerodynamic shape optimization infeasible with the current computational resources, especially for large numbers of design variables. Since it usually require hundreds of design variables, the use of a gradient-based optimizer combined with adjoin gradient evaluations is recommended. The optimization algorithm to use in all the results presented herein is SNOPT (Sparse Non-linear OPTimizer)104 through the Python interface. SNOPT is a gradient-based optimizer that implements a sequential quadratic programming method; it is capable of solving large-scale nonlinear optimization problems with thousands of constraints and design variables. SNOPT uses a smooth augmented Lagrangian merit function, and the Hessian of the Lagrangian is approximated using a limitedmemory quasi-Newton method. 3.5.2 Problem Formulation The goal of this optimization case is to perform lift-constrained drag minimization of the NASA-CRM wing using the RANS equations. In this section, we provide a complete description of the problem. 3.5.2.1 Baseline Geometry The baseline geometry is a wing with a blunt trailing edge extracted from the CRM wing-body geometry. The NASA-CRM geometry was developed for applied CFD validation studies. The CRM is representative of a contemporary transonic commercial transport, with a size similar to that of a Boeing 777. The CRM has 3.5 degree more quarter chord wing sweep and 10.3% less wing area than the Boeing 777-200. The CRM geometry has been optimized in aerodynamic performance. However, several design features, such as an aggressive pressure recovery in the outboard wing, were introduced into the design to make it more interesting for research purposes and to protect intellectual property. This baseline geometry provides a reasonable starting point for the optimization, while leaving room for further performance improvements. In addition, the CRM was Saad, Y. and Schultz, M. H., “GMRES: A Generalized Minimal Residual Algorithm for Solving Non-symmetric Linear Systems," SIAM Journal on Scientific and Statistical Computing, Vol. 7, No. 3, 1986, pp. 856-869. 104 Gill, P. E., Murray, W., and Saunders, M. A., “SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization," SIAM Journal on Optimization, Vol. 12, No. 4, 2002, pp. 979-1006. 103

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designed together with the fuselage of the full CRM configuration, so its performance is degraded when only the wing is considered. All coordinates are scaled by the mean aerodynamic chord (275.8 in). The resulting reference chord is 1.0, and the half span is 4.758151. The moment reference point is at (x; y; z) = (1.2077; 0.0; 0.007669), while the reference area is 3.407014. 3.5.2.2 Mesh Convergence Study We generate the mesh for the CRM wing using an in-house hyperbolic mesh generator105. The mesh is marched out from the surface mesh using an O-grid topology to a far-field located at a distance of 25 times the span (about 185 mean chords). The nominal cruise ow condition is Mach 0.85 with a Reynolds number of 5 million based on the mean aerodynamic chord. The mesh we generated for the test case optimization contains 28.8 million cells. The mesh size and y+ max values under the nominal operating condition are listed in Table 2. We perform a mesh convergence study to determine the resolution accuracy of this mesh. It lists the drag and moment coefficients for the baseline meshes. We also compute the zero-grid spacing drag using Richardson's extrapolation, which estimates the drag value as the grid spacing approaches zero. The zero-grid spacing drag coefficient is 199.0 counts for the baseline CRM wing. We can see that the L0 mesh has sufficient accuracy: the difference in the drag coefficient for the L0 mesh and the zero-grid spacing drag is within one drag count. The surface and symmetry Figure 28 Wing mesh of varying sizes plane meshes for the L0, L1, and L2 grid levels are shown in Figure 28 where O-grids of varying sizes were generated using a hyperbolic mesh generator.

Zhoujie Lyu and J. R. R. A. Martins. “Aerodynamic Shape Optimization Investigations of the Common Research Model Wing Benchmark”. AIAA Journal, 2014. 105

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Table 2

Mesh convergence study for the baseline CRM wing

3.5.2.3 Optimization Problem Formulation The aerodynamic shape optimization seeks to minimize the drag coefficient by varying the shape design variables subject to a lift constraint (CL = 0.5) and a pitching moment constraint (CMy > -0.17). The shape design variables are the z-coordinate movements of 720 control points on the FFD volume and the angle-of-attack. The control points at the trailing edge are constrained to avoid any movement of the trailing edge. Therefore, the twist about the trailing edge can be implicitly altered by the optimizer using the remaining degrees of freedom. The leading edge control points at the wing root are also constrained to maintain a constant incidence for the root section. There are 750 thickness constraints imposed in a 25 chord wise and 30 span wise grid covering the full span and from 1% to 99% local chord. The thickness is set to be greater than 25% of the baseline thickness at each location. Finally, the internal volume is constrained to be greater than or equal to the baseline volume. The complete optimization problem is described in Table 3.

Table 3

Aerodynamic shape optimization problem

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3.5.2.4 Surface Sensitivity on the Baseline Geometry To examine the potential improvements of the baseline geometry, we performed a sensitivity analysis. The sensitivity of the drag and pitching moment with respect to the airfoil shape is shown in Figure 29 as a contour plot of the derivatives of CD and CMy with respect to shape variations in the z direction. Sensitivity study (CD and CMy w.r.t z-direction) of the baseline shows which shape changes yield the largest improvements. The regions with the highest gradient of CD are near the shock on the upper surface and near the wing leading edge. This indicates that leading-edge shaping and shock reduction through local shape changes should be the major drivers in C D reduction at the beginning of the optimization. As for CMy , the shape changes near the root and tip of the wing are the most effective in adjusting the pitching moment. Since these sensitivity plots are a linearization about the current design point, they provide no information about the constraints. Nonetheless, these sensitivity plots indicate what drives the design at this design point.

Figure 29

Sensitivity study of the baseline wing w.r.t z-direction for CD and CMY

3.5.3 Single-Point Aerodynamic Shape Optimization In this section, we present our aerodynamic design optimization results for the CRM wing benchmark problem (described in Figure 30) under the nominal flight condition (M = 0.85, Re = 5x106). We use the L0 grid (28.8M cells) for the optimization, thanks to a multilevel optimization acceleration technique that meaningfully reduces the overall computational cost of the optimization106. Our Zhoujie Lyu and J. R. R. A. Martins, “Aerodynamic Shape Optimization Investigations of the Common Research Model Wing Benchmark”, AIAA Journal, 2014. 106

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optimization procedure reduced the drag from 199.7 counts to 182.8 counts, i.e., an 8.5% reduction. The corresponding Richardson-extrapolated zero-grid spacing drag decreased from 199.0 counts to 181.9 counts. Given that the CRM configuration was designed by experienced aerodynamicists, this is a significant improvement (although they designed the wing in the presence of the fuselage, which we are ignoring in this problem). Figure 30 shows a detailed comparison of the baseline wing and the optimized wing. In this figure, the baseline wing results are shown in red and the optimized wing results are shown in blue. At the optimum, the lift coefficient target is met, and the pitching moment is reduced to the lowest allowed value. The lift distribution of the optimized wing is much closer to the elliptical distribution than that of the baseline, indicating an induced drag that is close to the theoretical minimum for a planar wake. This is achieved by fine-tuning the twist distribution and airfoil shapes. The baseline wing has a near-linear twist distribution. The optimized design has more twist at the root and tip, and less twist near mid-wing. The overall twist angle changed only slightly: from 8.06 degree to 7.43 degree. The optimized thickness distribution is significantly different from that of the baseline, since the thicknesses are allowed to decrease to 25% of the original thickness, and there is a strong incentive to reduce the airfoil thicknesses in order to reduce wave drag. The volume is constrained to be greater than or equal to the baseline volume, so the optimizer drastically decreases the thickness of the gained value drag trade off more promising. To ensure that the result of our single-point optimization has sufficient accuracy, we conducted a grid convergence study of the optimized design. The mesh convergence plot for both the baseline and optimized geometry meshes is shown in Figure 31. The zero-grid spacing drag, which was obtained using Richardson's extrapolation, is also plotted in the figure. We can see that the L0 mesh has sufficient accuracy: the

Figure 30

Optimized wing is shock-free with 8.5% lower Drag.

difference in the drag coefficient for the L0 mesh and the value obtained for the zero-grid spacing is within one drag count. The variation in drag coefficient between the baseline and optimized meshes is nearly constant for each grid level, which gives us confidence that the optimization using the coarse meshes represent the design space trends sufficiently well. Therefore, we perform the remaining

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optimization studies on the coarser mesh (L2), assuming that we capture the correct design trends. 3.5.4 Effect of the Number of Shape Design Variables The cost of computing gradients with an efficient adjoin implementation is nearly independent of the number of design variables. We took advantage of this efficiency by optimizing with respect to 720 shape design variables in the previous sections. However, we would like to determine the tradeoff

Figure 31

Mesh Convergence study

between the number of design variables and the optimal drag, and to examine the effect on the computational cost of the optimization. Thus, in this section we examine the effect of reducing the number of design variables. A series of new enlarged FFDs are created to ensure proper geometry embedding for small numbers of design variables. The shape design variables are distributed in a regular grid, where the finest grid has 15x48 = 720 design variables. The 15 chord wise stations correspond to 15 distinct airfoil shapes, while the shape of each airfoil is defined by 48 control points (half of these on the top, and the other half on the bottom). Figure 32 (top) shows the resulting optimized designs for different numbers of airfoil control points and a fixed number of defining airfoils. Reducing the airfoil control points from 48 to 24 has a negligible effect on the optimal shape and pressure distribution, and the optimum drag increases by only 0.1 counts. As we further reduce the number of airfoil points to 12 and 6, both the drag and pressure distribution show noticeable differences. Variation in the number of defining airfoils follows a similar trend to the variation in the number of airfoil control points, as shown in Figure 32 (middle). However, the drag penalty due to the number of airfoils is less severe than the penalty observed in the airfoil point reduction. Therefore, increasing the number of design variables in the chord wise direction is more advantageous than increasing the number of defining airfoils in the span wise direction. Also perform the optimization with a reduced number of shape design variables in both the chord wise and span wise directions simultaneously. From this study it can be concluded that an adequate optimized design can be achieved with a smaller number of design variables: with 8 x 24 = 192 shape variables, the difference in the optimal drag coefficient is only 0.6 counts. Any further reduction in the number of design variables has a much larger impact on the optimal drag. Figure 32 plots the convergence history for each optimization case. Note that number of optimization iterations does not decrease significantly as the number of defining airfoils is decreased.

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When we decrease the 20 number of airfoil control points, the number of optimization iterations required decreases drastically. However, the number of defining airfoils has little effect on the optimization effort. This is in part because the adjoin computational cost is independent of the number of design variables. In addition, the coupled effects between design variables are much stronger between variables within an airfoil than between variables in different airfoils. For an optimization process in which the computational cost scales with the number of design variables, such as when the gradients are computed vi a finite differences, or for gradient-free optimizers, a smaller number of design variables would significantly impact the optimized design. For example, for 3 x 6 = 18 variables, the drag of the optimized design would increase by 5.4 counts.

Figure 32

Insensitivity of number of optimization iterations to number of design parameters

3.5.5 Multi-Level Optimization Acceleration Technique An acceleration technique that reduced the overall computational cost of the optimization is presented. Aerodynamic shape optimization is a computational intensive endeavor, where the majority of the computational effort is spent in the flow solutions and gradient evaluations. Therefore, many CFD researchers have tried to develop more efficient flow and adjoin solvers. Commonly used methods, such as multigrid, pre-conditioning, and variations on Newton-type methods, can improve the convergence of the solver, thus reducing the overall optimization time. Our flow solver has been significantly improved over the years to provide efficient and reliable flow solutions. Another area of improvement is the efficiency of the gradient computation. As mentioned before, the adjoin method proficiently computes gradients with respect to large numbers of shape design variables. For our adjoin implementation, the cost of computing the gradient of a single function of interest with respect to hundreds or even thousands of shape design variables is lower

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than the cost of one flow solution. Here, we present a method that is inspired by the grid sequencing (multi-gridding) procedure in CFD. Since it is less costly to compute both the flow solution and the gradient on a coarser grid, we perform the optimization first on the coarsest grid until a certain level of optimality is achieved. Then, we move to the next grid level and start with the optimal design variables from the coarser grid. Since the drag and lift coefficients are generally different for each grid level, the approximate Hessian (used by the gradient-based optimizer) must be restarted. We repeat this process until the optimization on the finest grid has converged. Note that this procedure is different from traditional multigrid methods, where the coarse levels are revisited via multigrid cycles. We used this procedure to obtain the optimal wing presented in the previous section. We use three grid levels: L2 (451K cells), L1 (3.6M cells), and L0 (28.8M cells). We can see that most of the optimization iterations are performed on the coarse grid, and as a result, the number of the function and gradient evaluations on the successively finer grids is greatly reduced. Table 4 summarizes the computational time spent on each grid level. Thanks to the optimization with the coarser grids, only 18 Figure 33 Multipoint Optimization flight iterations are needed on the L0 grid to converge the conditions optimization. However, the L0 grid requires the largest computational effort, due to the high cost of the flow and adjoin solutions on this fine grid. Given that the cost per optimization iteration in the L0 grid is 770 process-hr (compared to 2.9 process-hr for the L2 grid) it is not feasible to perform an optimization using only the L0 grid. Assuming that the same number of iterations used for the L2 grid (638) would be needed for the L0 grid, the computational cost would be 23 times higher than that of the multilevel approach, which would correspond to 16 days using 1248 processors. 3.5.6 Multi-Point Aerodynamic Shape Optimization Transport aircraft operate at multiple cruise conditions because of variability in the flight missions and air traffic control restrictions. Single-point optimization under the nominal cruise condition could overstate the benefit of the optimization, since the optimization improves the on-design performance to the detriment of the off-design performance. In previous sections, the single-point optimized wing exhibited an unrealistically sharp leading edge in the outboard of the wing. This was caused by a combination of the low value for the thickness constraints (25% of the baseline) and the single-point formulation. A sharp leading edge is undesirable because it is prone to ow separation under off-design conditions. We address this issue by performing a multipoint optimization. The optimization is performed on the L2 grid. We choose five equally weighted flight conditions with different combinations of lift coefficient and the Mach number. The flight conditions are the nominal cruise, 10% of cruise CL, and 0.01 of cruise M, as shown in Figure 33. More sophisticated ways of choosing multipoint flight conditions and their associated weights can be used. The objective function is the average drag coefficient for the have flight conditions, and the moment constraint is enforced only for the nominal flight condition.

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A comparison of the single-point and multipoint optimized designs is shown in Figure 34. Unlike the shock-free design obtained with single-point optimization, the multipoint optimization settled on an optimal compromise between the flight conditions, resulting in a weak shock at all conditions. The leading edge is less sharp than that of the single-point optimized wing. Additional fight conditions, such as a low-speed flight condition, would be needed to further improve the leading edge. The overall pressure distribution of the multipoint design is similar to that of the single-point design. The twist and lift distributions are nearly identical. Most of the differences are in the chord wise Cp distributions in the outer wing section. The drag coefficient under the nominal condition is approximately two counts higher. However, the performance under the off design conditions is considerably improved.

Figure 34

Multipoint Optimized

To demonstrate the robustness of the multipoint design, we plot ML=D contours of the baseline, single-point, and multipoint designs with respect to CL and cruise Mach in Figure 35 where ML = D provides a metric for quantifying aircraft range based on the Breguet range equation with constant thrust specific fuel consumption. While the thrust-specific fuel consumption is actually not constant, assuming it to be constant is acceptable when comparing performance in a limited Mach number range. We add 100 drag counts to the computed drag to account for the drag due to the fuselage, tail, and nacelles, and we get more realistic ML = D values. The baseline maximum ML = D is at a lower Mach number and a higher CL than that of the nominal flight condition. The single-point optimization increases the maximum ML = D by 4% and moves this maximum toward the nominal cruise condition. If we examine the variation of ML=D along the CL = 0.5 line, we see that the maximum occurs at the nominal Mach of 0.85, which corresponds to a dip in a drag divergence plot. If we examine the variation of ML=D along the CL = 0.5 line, we see that the maximum occurs at the nominal Mach of 0.85, which corresponds to a dip in a drag divergence plot. For the multipoint optimization, the optimized flight conditions are distributed in the Mach-CL space, resulting in an attended ML=D

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variation near the maximum, which means that we have more uniform performance for a range of flight conditions. If we examine the variation of ML=D along the CL = 0.5 line, we see that the maximum occurs at the nominal Mach of 0.85, which corresponds to a dip in a drag divergence plot. For the multipoint optimization, the optimized flight conditions are distributed in the Mach-CL space, resulting in an attended ML=D variation near the maximum, which means that we have more uniform performance for a range of flight conditions. In aircraft design, the 99% value of the maximum ML = D contour is often used to examine the robustness of the design. The point with the highest Figure 35 Comparison of Baseline, Single, and Multipoint Mach number on that contour line Optimization corresponds to the Long Range Cruise (LRC) point, which is the point at which the aircraft can fly at a higher speed by incurring a 1% increase in fuel burn. In this case, we see that the 99% value of the maximum ML = D contour of the multipoint design is larger than that of the single-point optimum, indicating a more robust design.

3.6 Effect of Variable Cant Angle Winglet in Aircraft Control

Aircraft performance is highly affected by induced drag caused by wingtip vortices. Winglets, referred to as vertical or angled extensions at aircraft wingtips, are used to minimize vortices formation to improve fuel efficiency. Winglets application is one of the most noticeable fuel economic technologies on aircraft which defined as small fins or vertical extensions at the wingtips. They improve aircraft efficiency by reducing the induced drag caused by wingtip vortices, by improving the lift-to-drag ratio (L/D). Winglets function by increasing the effective aspect ratio of the wing without contributing significantly towards the structural loads. The effect of variable winglet investigated by [Beechook & Wang]107 where the analysis were to compare the aerodynamic characteristics and to investigate the Figure 36 An un-symmetric wing-tip performance of winglet at different cant for various arrangement for a sweptback wing to initiate angles of attack. Conventional winglets provide a coordinated turn. maximum drag cutback and improve L/D under A. Beechook1, J. Wang, “Aerodynamic Analysis of Variable Cant Angle Winglets for Improved Aircraft Performance”, Proceedings of the 19th International Conference on Automation & Computing, Brunel University, London, UK, 13-14 September 2013. 107

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cruise conditions only. During non-cruise conditions, these winglets are less likely to improve aircraft performance and subsequently, they do not provide optimal fuel efficiency during take-off, landing and climb. Non-cruise flight conditions add up to a significantly large fraction of a flight and therefore, winglet designs must be optimized to be able to function during both cruise and non-cruise flight conditions. In recent years, extensive research has been ongoing, aiming to improve the design of winglets in order to boost the aircraft performance during flight. Limited work has been carried out on winglet designs that can alter the cant angle. [Bourdin at al]108 has explore similar concept for ‘morphing’ the control of aircraft. The concept consists of a pair of winglets with adaptive cant angle, independently activated, mounted at the tips of a flying wing. The variable cant angle winglet appears to be a multi-axis effector with a favorable coupling in pitch and roll with regard to turning maneuvers. (See Figure 36). 3.6.1 Results and Discussion The comparison of lift-to-drag ratio values from wind tunnel test and CFD simulations is presented in Figure 37. The L/D values obtained from wind tunnel experiments were very low compared to those obtained from simulations. This is due to the high CD values from the wind tunnel results. From the simulations results, winglet with cant angle 45° has the highest L/D compared to all other configurations. From the wind tunnel results, the L/D for each winglet configuration varies for different angle of attack, e.g. the winglet at cant angle 45° has the highest L/D at 12° angle of attack. 3.6.2 Concluding Remarks The CFD simulations and wind tunnel test results showed that the different winglet configurations have different aerodynamic characteristics when the angle of attack is varied. At low angles of attack, ideally at cruise angle of attack, winglets at cant angle 45° and 60° showed improved the aerodynamic performance in terms of lift and drag coefficients. The winglets at cant 45° and 60° did not provide optimum performance at high angles of attack, for example, at higher angle of attack, the winglets at cant 45° produced more lift compared to other winglet configurations. Hence, varying the winglets’ cant angle at different flight phases can improve the aircraft efficiency and optimize performance. Therefore, it can be concluded that the investigated concept of variable cant angle winglets appears to be a promising alternative for traditional fixed winglets. However, this study involved only the flow study and there are many other important factors to consider while designing new devices for aircraft, e.g. structural weight and cost. Similar results obtained by [Bourdin at al.]109 for moments attainable by folding up or down the right winglet.

P. Bourdin_, A. Gatto_, and M.I. Friswell, “The Application of Variable Cant Angle Winglets for Morphing Aircraft Control”, AIAA 2006-3660. 109 See previous. 108

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Figure 37

Lift-to-Drag Ratio, L/D (Wind Tunnel and CFD Comparison)

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4 Sensitivity Analysis for Aerodynamic Optimization 4.1

Background

Sensitivity analysis is the study of how the uncertainty in the output of a mathematical model or system (numerical or otherwise) can be apportioned to different sources of uncertainty in its inputs. A related practice is uncertainty analysis, which has a greater focus on uncertainty quantification and propagation of uncertainty. Ideally, uncertainty and sensitivity analysis should be run in tandem. Sensitivity analysis can be useful for a range of purposes, including testing the robustness of the results of a model or system in the presence of uncertainty. Increased the understanding of the relationships between input and output variables in a system or model. Uncertainty reduction: identifying model inputs that cause significant uncertainty in the output and should therefore be the focus of attention if the robustness is to be increased (perhaps by further research). Searching for errors in the model (by encountering unexpected relationships between inputs and outputs). Model simplification – fixing model inputs that have no effect on the output, or identifying and removing redundant parts of the model structure. Enhancing communication from modelers to decision makers (e.g. by making recommendations more credible, understandable, compelling or persuasive). Finding regions in the space of input factors for which the model output is either maximum or minimum or meets some optimum criterion110. Quite often, some or all of the model inputs are subject to sources of uncertainty, including errors of measurement, absence of information and poor or partial understanding of the driving forces and mechanisms. This uncertainty, which previously discussed, imposes a limit on our confidence in the response or output of the model. Good modeling practice requires that the modeler provides an evaluation of the confidence in the model. This requires, first, a quantification of the uncertainty in any model results (uncertainty analysis); and second, an evaluation of how much each input is contributing to the output uncertainty. Sensitivity analysis addresses the second of these issues (although uncertainty analysis is usually a necessary precursor), performing the role of ordering by importance the strength and relevance of the inputs in determining the variation in the output. In models involving many input variables, sensitivity analysis is an essential ingredient of model building and quality assurance. National and international agencies involved in impact assessment studies have included sections devoted to sensitivity analysis in their guidelines. Examples are the European Commission (see e.g. the guidelines for impact assessment), the White House Office of Management and Budget, the Intergovernmental Panel on Climate Change and US Environmental Protection Agency's modeling guidelines. Modern engineering design makes extensive use of computer models to test designs before they are manufactured. Sensitivity analysis allows designers to assess the effects and sources of uncertainties, in the interest of building robust models. Sensitivity analyses have for example been performed in biomechanical models, tunneling risk models, amongst others. Sensitivity analysis is closely related with uncertainty analysis; while the latter studies the overall uncertainty in the conclusions of the study, sensitivity analysis tries to identify what source of uncertainty weighs more on the study's conclusions 111. However, despite considerable progress over twenty years, sensitivity analysis has only recently enjoyed widespread use in engineering practice. There are perhaps two principal causes of this: (i) (ii)

110 111

Questionable suitability of gradient-based optimization methods for many engineering problems, The lack of availability of cheap, accurate gradients.

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A close third might be the extra effort involved in setting up gradient-based optimizations, due to the need for smooth mesh deformation, preconditioning of design variables, and the lack of robustness to the failure of any step of the process. This last is not true of e.g. genetic algorithms, for which a flow code failure is simply a member of the population that is not evaluated. Issue (i) is an unavoidable consequence of the locality of gradient-based methods: in a design space with many local optima they will tend to find the nearest (with respect to the starting point), not the best, local optimum. Further there is evidence to suggest that configurations such as multi-element high-lift devices have just such highly oscillatory design spaces. This problem may be tackled with hybrid optimization algorithms that combine non-deterministic techniques for broad searches, with gradient-based methods for detailed optimization, a significant topic of current research112 113. However issue (ii) is perhaps more critical in general, and that with which this article is concerned. Inevitably the availability of sensitivity analysis tools for codes lags behind the codes themselves, a situation exacerbated by the considerable effort required to linearize complex solvers. A good example is the lack of reliable adjoin solvers for Navier-Stokes problems until recently. The situation deteriorates when multidisciplinary problems are considered, and the sensitivities of coupled multi-code systems are required. Nonetheless these problems are worth overcoming, as when they are available, gradientbased algorithms combined with adjoin gradients are the only methods which can offer rapid optimizations for extremely large numbers of design variables114, as needed for the aerodynamic shape design115.

4.2

Aerodynamic Sensitivity

Several methods concerning the derivation of sensitivity equations are currently available. Among the most frequently mentioned are     

Direct Differentiation (DD), Adjoin Variable (AV), Symbolic Differentiation (SD), Automatic Differentiation (AD), ( e.g. Odyssée or ADIFOR) Finite Difference (FD), (Brute Force)

Each technique has its own unique characteristics. For example, the Direct Differentiation, has the advantage of being exact, due to direct differentiation of governing equations with respect to design parameters, but limited in scope. There are two basic components in obtaining aerodynamic sensitivity. They are:  

Obtaining the sensitivity of the governing equations with respect to the state variables, Obtaining the sensitivity of the grid with respect to the design parameters.

The sensitivity of the state variables with respect to the design parameters are described by a set of linear-algebraic relation. These systems of equations can be solved directly by a LU decomposition of the coefficient matrix. This direct inversion procedure becomes extremely expensive as the problem G. Lombardi, G. Mengali, F. Beux, A hybrid genetic based optimization procedure for aircraft conceptual analysis, Optimization and Engineering 7 (2) (2006) 151–171. 113 V. Kelner, G. Grondin, O. Léonard, S. Moreau, Multi-objective optimization of a fan blade by coupling a genetic algorithm and a parametric solver, in: Proceedings of EUROGEN, Munich, 2005. 114 D. Mavriplis, Discrete adjoin-based approach for optimization problems on three-dimensional unstructured meshes, AIAA Journal 45 (4) (2007) 740–752. 115 Jacques E.V. Peter and Richard P. Dwight, “Numerical Sensitivity Analysis for Aerodynamic Optimization: A Survey of Approaches”, 2009. 112

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dimension increases. A hybrid approach of an efficient banded matrix solver with influence of offdiagonal elements iterated can be implemented to overcome this difficulty116.

4.3

Flow Analysis and Sensitivity Equation

The general equations can be written as

I Q  R(Q) J t

Q   ρ, ρu , ρv , ρw , E 

T

Eq. 4.1

Here, R is the residual and J is the Jacobean Transformation:

J

 (ξ, , ζ)  (x, y,z)

Eq. 4.2

Where R is the residual vector, Q is a four-element vector of conserved flow variables. Navier-Stokes equations are discretized in time using the Euler implicit method and linearized by employing the flux Jacobean. This results in a large system of linear equations in delta form at each time step as n  I  R     ΔQ  R n  Jt  Q    

Eq. 4.3

For a steady-state solution (i.e., t →∞) reduces to

R(Q (P), X (P), P)  0

Eq. 4.4

Where the explicit dependency of R on grid and vector of parameters P is evident. The parameters P control the grid X as well as the solution Q. The fundamental sensitivity equation containing ∂Q/∂P is obtained by direct differentiation of Eq. 4.4 as

 R   Q   R   X   R   Q   P    X   P    P   0        

Eq. 4.5

It is important to notice that Eq. 4.5 is a set of linear algebraic equation and matrices ∂R/∂Q and ∂R/∂X is well understood. The flow sensitivity, ∂Q/∂P can now be directly obtained as

 Q   R      P   Q 

1

  R   X   R              X   P   P  

Eq. 4.6

Further simplification could include the vector of grid sensitivity which is

Sadrehaghighi, I., Smith, R.E., Tiwari, S., N., “Grid Sensitivity And Aerodynamic Optimization of Generic Airfoils”, Journal of Aircraft, Volume 32, No. 6, Pages 1234-1239. 116

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 X   X   XB       P   XB   P 

Eq. 4.7

Where XB denotes the boundary nodes117.

4.4

Optimization

An objective of a multidisciplinary optimization of a vehicle design is to extremis a payoff function combining dependent parameters from several disciplines. Most optimization techniques require the sensitivity of the payoff function with respect to free parameters of the system. For a fixed grid and solution conditions, the only free parameters are the surface design parameters. Therefore, the sensitivity of the payoff function with respect to design parameters is needed. The optimization problem is based on the method of feasible directions and the generalized reduced gradient method. This method has the advantage of progressing rapidly to a near-optimum design with only gradient information of the objective and constrained functions required. The problem can be denned as finding the vector of design parameters XD, which will minimize the objective function f(XD) subjected to constraints

gi (XD )  0

j  1, m subject to

XLD  X D  X UD Eq. 4.8

Where superscripts denote the upper and lower bounds for each design parameter. The optimization process proceeds iteratively as

XnD  XnD1  γ S n

Figure 38

Optimization Strategy Loop

Eq. 4.9

Where n is the iteration number, Sn the vector of search direction, and γ a scalar move parameter. The first step is to determine a feasible search direction Sn, and then perform a one-dimensional search in this direction to reduce the objective function as much as possible, subjected to the

Sadrehaghighi, I., Smith, R.E., Tiwari, S., N., “Grid Sensitivity and Aerodynamic Optimization Of Generic Airfoils”, Journal of Aircraft, Volume 32, No. 6, Pages 1234-1239. 117

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constraints118. (See Figure 38).

4.5

Surface Modeling Using NURBS

Among many ideas proposed for generating any arbitrary surface, the approximate techniques of using spline functions are gaining a wide range of popularity. The most commonly used approximate representation is the Non-uniform Rational B-Spline (NURBS) function. They provide a powerful geometric tool for representing both analytic shapes (conics, quadrics, surfaces of revolution, etc.) and free-form surfaces119; or occasionally called Free From Deformations (FFD). The surface is influenced by a set of control points and weights to where unlike interpolating schemes the control points might not be at the surface itself. By changing the control points and corresponding weights, the designer can influence the surface with a great degree of flexibility without compromising the accuracy of the design. The relation for a NURBS Figure 39 Seven control point representation of a generic curve is Airfoil n

X (r)   R i,p (r) Di

i  0,........., n

i 0

R i,p (r) 

N i,p (r) ωi n

N i 0

i, p

(r) ωi

Eq. 4.10

where X(r) is the vector valued surface coordinate in the r-direction, Di are the control points (forming a control polygon), ωi are weights, Ni,p(r) are the p-th degree B-Spline basis function, and Ri,p(r) are known as the Rational basis functions satisfying n

R i 0

i, p

(r)  1 , R i,p (r)  0

Eq. 4.11

Figure 39 illustrates a six control point representation of a generic airfoil. The points at the leading and trailing edges are fixed. Two control points at the 0% chord are used to affect the bluntness of the section. Similar procedure can be applied to other airfoil geometries such as NACA four or five digit series.

Sadrehaghighi, I., Smith, R.E., Tiwari, S., N., “Grid Sensitivity and Aerodynamic Optimization Of Generic Airfoils”, Journal of Aircraft, Volume 32, No. 6, Pages 1234-1239. 119 Tiller, W., “Rational B-Splines for Curve and Surface Representation," Computer Graphics and Applications, Volume 3, N0. 10, September 1983. 118

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The procedure is easily applicable to 3D for example like the common wing & fuselage as designated in Figure 40 [G.K.W. Kenway et al.]120. The choice for number of control points and their locations are best determined using an inverse B-Spline interpolation of the initial data. The algorithm yields a system of linear equations with a positive and banded coefficient matrix. Therefore, it can be solved safely using techniques such as Gaussian elimination without pivoting. The procedure can be easily extended to cross-sectional configurations, when critical cross-sections are denoted by several circular conic sections, and the intermediate surfaces have been generated using linear interpolation. Increasing the weights would deform the circular segments to other conic segments (elliptic, parabolic, etc.) as desired for different flight regions. In this manner, the number of design parameters can be kept to a minimum, which is an important factor in reducing costs.

Figure 40

Free form deformation (FFD) volume with control points (Courtesy of Kenway et al.)

An efficient gradient-based algorithm for aerodynamic shape optimization is presented by [Hicken and Zingg]121 where to integrate geometry parameterization and mesh movement. The generalized B-spline volumes are used to parameterize both the surface and volume mesh. Volume mesh of Bspline control points mimics a coarse mesh where a linear elasticity mesh-movement algorithm is applied directly to this coarse mesh and the fine mesh is regenerated algebraically. Using this approach, mesh-movement time is reduced by two to three orders of magnitude relative to a nodebased movement.

4.6 Case Study -2D Study of Airfoil Grid Sensitivity using Direct Differentiation (DD)

The structured grid sensitivity of a generic airfoil with respect to design parameters using the NURBS parameterization is discussed in this section. The geometry, as shown in Figure 39, has six prespecified control points. The control points are numbered counter-clockwise, starting and ending with control points (0 and 5), assigned to the tail of the airfoil. A total of 18 design parameters (i.e., three design parameters per control point) available for optimization purpose. Depending on desired Gaetan K.W. Kenway, Joaquim R. R. A. Martins, and Graeme J. Kennedy, “Aero structural optimization of the Common Research Model configuration”, American Institute of Aeronautics and Astronautics. 121 Jason E. Hickenand David W. Zingg, “Aerodynamic Optimization Algorithm with Integrated Geometry Parameterization and Mesh Movement”, AIAA Journal Vol. 48, No. 2, February 2010. 120

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accuracy and degree of freedom for optimization, the number of design parameters could be reduced for each particular problem. For the present case, such reduction is achieved by considering fixed weights and chordlength. Out of the remaining four control points with two degrees of freedom for each, control points 1 and 5 have been chosen as a case study. The number of design parameters is now reduced to four with XD = {X1; Y1; X5; Y5}T, with initial values specified in Figure 39. Non-zero contribution to the surface grid sensitivity coefficients of these control points are the basis functions R1,3(r) and R5,3(r). The sensitivity gradients are restricted only to the region influenced by the elected control point. This locality feature of the NURBS parameterization makes it a desirable tool for complex design and optimization when only a local perturbation of the geometry is warranted. Similar results can be obtained for design control point 5 where the sensitivity gradients are restricted to the lower portion of domain. Figure 41 shows C-type dual blocks structured grid its sensitivity with respect to NURBS input polygon Y1. 4.6.1

2D Case Study- Airfoil Grid, Flow Sensitivity, and Optimization The second phase of the problem is Figure 41 Sample grid and grid sensitivity obtaining the flow sensitivity coefficients using the previously obtained grid sensitivity coefficients. In order to achieve this, a converged flow field solution about a fixed design point should be obtained. The two-dimensional, compressible, thin-layer Navier-Stokes equations are solved for a free stream Mach number of M = 0.7, Reynolds number Re = 106, and angle of attack = 0. The solution is implicitly advanced in time using local time step-ping as a means of promoting convergence toward the steady-state. The residual is reduced by ten orders of magnitude. All computations are performed on NASA Langley's Cray-2YMP mainframe with a computation cost of 0:1209x10-3 CPU seconds/iteration/grid point. Due to surface curvature, the flow accelerates along the upper surface to supersonic speeds, terminated by a weak shock wave behind which it becomes subsonic. The average relative error has been reduced by three orders of magnitude. The sensitivities of the aerodynamic forces, such as drag and lift coefficients with respect to design

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parameters, are obtained and results are presented in Table 5. An inspection, indicates the substantial influence of parameters Y1 and Y5 on the aerodynamic forces acting on the surface. The optimum design is achieved after 17 optimization cycles and a total of 8807 Cray-2 CPU seconds. These high computational costs make minimizing the number of design parameters in optimization cycle essential. Table 4 highlights the initial and final values of lift and drag coefficients with a 208% improvement in their ratio, as well as the initial and optimum design parameters with parameters Y1 and Y5 having the largest Figure 42 Optimization Cycle History change as expected. The history of design parameters deformation during the optimization cycles appears in Figure 42, where the oscillatory nature of design perturbations during the early cycles are clearly visible. Figure 43 compares the original and optimum geometry of the airfoil. 4.6.2 Discussions Several observations should be made at this point. First, although control points 1 and 5 demonstrated to have substantial influence on the design of the airfoil, they are not the only control points affecting the design. In fact, control points 2 and 4 near the nose might have greater affect due to sensitive nature of lift and drag forces on this region. The choice of control points 1 and 5 here was largely based on their camber like behavior. A complete design and optimization should include all the relevant control points (e.g., control points 1, 2, 4, and 5). For geometries with large number of control points, in order to contain the computational costs within a reasonable range, a criteria for selecting the most Figure 43 Original and Optimized Airfoil influential control points for optimization purposes should be established. This decision could be based on the already known sensitivity coefficients, where control points having the largest coefficients could be chosen as design parameters. Secondly, the optimum airfoil of is only valid for this particular example and design range. As a direct consequence of the nonlinear nature of governing equations and their sensitivity coefficients, the validity of this optimum design would be restricted to a very small range of the original design parameters. The best estimate for this range would be the finite-difference step size used to confirm the sensitivity coefficients (i.e., 10-3 or less). All the airfoils with the original control points within this range should conform to the optimum design of Table 4, while keeping the grid and flow conditions constant, and indicates the percentage in design improvement. Table 5 shows the Aerodynamic sensitivity coefficients. It is evident that grid sensitivity plays a significant role in the aerodynamic optimization process. The algebraic grid generation scheme presented here is intended to demonstrate the elements involved in obtaining the grid sensitivity from an algebraic grid generation system. Each

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grid generation formulation requires considerable analytical differentiation with respect to parameters which control the boundaries as well as the interior grid. It is implied that aerodynamic surfaces, such as the airfoil considered here, should be parameterized in terms of design parameters. Due to the high cost of aerodynamic optimization process, it is imperative to keep the number of design parameters as low as possible. Analytical parameterization, although facilitates this notion, has the disadvantage of being restricted to simple geometries. A geometric parameterization such as NURBS, with local sensitivity, has been Table 4 Design improvement for an Airfoil advocated for more complex geometries. Future investigations should include the implementation of present approach using larger grid dimensions, adequate to resolve full physics of viscous flow analysis. A grid optimization mechanism based on grid sensitivity coefficients with respect to grid parameters should be included in the overall optimization process. An optimized grid applied to present geometry, should increase the quality and convergence rate of flow analysis within optimization cycles. Other directions could be establishing a link between geometric design parameters (e.g., control points and weights) and basic physical design parameters (e.g., camber and thickness). This would provide a consistent model throughout the analysis which could easily be modified for optimization. Also, the effects including all the relevant control points on the design cycles should be investigated. Another contribution would be the extension of the current algorithm to three-dimensional space for complex applications. For threedimensional applications, even a geometric parameterization of a Table 5 Aerodynamic Sensitivity Coefficient complete aerodynamic surface can require a large number of parameters for its designation. A hybrid approach can be selected when certain sections or skeleton parts of a surface are specified with NURBS and interpolation formulas are used for intermediate surfaces.

4.7 Extension to 3D using Automatic Differentiation (AD) Traditionally, hand coding, Finite Difference (FD) and Symbolic Differentiation (SD) has been used for sensitivity derivatives in higher dimensions when Direct Differentiation (DD) is unusable. The issue with Finite Differencing (FD) is numerical unpredictability and human costs. In contrast, hand coding and symbolic approach cannot be applied to large codes due to extensive effort. The so called Automatic Differentiation (AD) is the only promising approach. From users perspective, AD tools should behave like black boxes which given the code describing the function to be differentiated and generate sensitivity argument. Functionally, language like C++ and FORTRAN 90 support a feature called operator overloading which allows the redefinition of behavior of the elementary arithmetic’s and hence can be employed to employed to attach under the rug, so to speak, sensitivity derivatives to original program variables. Figure 44 displays such an approach using an experimental High-

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Speed Civil Transport (HSCT) like configuration with a double-delta wing122.

Figure 44

3D Volume grid and sensitivty w.r.t. wing root chord

4.8 The Adjoint Method Duality

The only remaining option for calculating sensitivities are Adjoint Variable (AV) which we will considered it next. We shall begin by showing how the Adjoint method can be viewed as a special case of linear duality123. Then we will apply these ideas specifically to the problem of fluid control. At the heart of the adjoint method is a substitution of variables that allows us to compute the gradient of a function quickly. This substitution can be viewed in terms of linear duality [Giles and Pierce]124. Suppose that the matrix [A] and the vectors g and c are known, and that we would like to compute the vector product gTb such that [A]b = c in terms of the unknown vector b. A straightforward approach would be to first solve for b and then compute the vector product. An alternative would be to introduce a vector s and compute: sTc such that [A]T s = g. This is known as the dual of the problem. The equivalence can be shown through substitution: sTc = sT[A]b = ([A]T s)T b = gT b. Of course, this new linear system is not necessarily any easier to solve. However, consider a new case where the unknown vector b and the known vector c are actually matrices [B]and [C]. By the same logic, the vector-matrix product as

 g T B subject to AB  C g B  s C   T s T C subject to A s  g  where A  , C , g are known T

T

Eq. 4.12

Now these two linear systems look quite different! The former involves solving for the entire matrix [B], while the latter is the same single linear system as in our original example. Clearly, in this case,

Bischof, C. H., Jones, W. T., Mauser, A., Samareh, J. A., “Experience with Application of ADIC Automatic Differentiation tool to the CSCMDO 3-D Volume Grid generation Code”, AIAA 96-0716. 123 Michael B. Giles and Niles A. Pierce, ”An Introduction to the Adjoint Approach to Design”, Flow, Turbulence and Combustion 65: 393–415, 2000. 124 See Above. 122

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huge benefits can be reaped by solving the dual formulation. As it turns out, the problem we would like to solve involves calculating a vector-matrix product of this form. The adjoint method exploits this powerful aspect of duality to drastically improve the efficiency of this computation125. 4.8.1 Optimization The optimization problem is a continuous function minimization (or maximizing) φ(q0;u). There are many standard numerical methods for minimizing a continuous function. Since this is a derivativebased optimization, we must not only evaluate φ , but also compute its gradient, ∂φ/∂u. In computer numeric, this gradient computation has traditionally been the bottleneck for control because each parameter required its own derivative computation. Here, showing how the adjoint method, long used in the optimal control community, can be adapted to this and other derivative based problems. 4.8.2 Gradient Calculation as Related to Ajoint Variable (AV) Scheme Let us now dig more specifically into a particular type of optimization problem often seen of which physical simulation is one example. Suppose we have a fixed initial state q0, which we evolve into n subsequent states q1, , , , qn according to the update rule

qi 1  fi (qi , u)

Eq. 4.13

where each fi is an arbitrary differentiable function parameterized by a control vector u. We combined these into one long vector Q = [qT1, , , , , qTn]T and one long vector function: F(Q;u) = [f0(q0,u)T , , , , , fn-1(qn-1,u)T ]T . This allows us to write equation as

Q  F(Q, u)

Eq. 4.14

This equation is essentially a constraint on the optimization; for Q to represent a valid simulation generated by the sequence fi of functions, equation (6) must hold. Finally assume that we have a differentiable objective function φ(Q,u) and we would like to compute its derivative with respect to the control vector126:

d  dQ    du Q du u

Eq. 4.15

Computing this directly is extremely costly, as the matrix dQ/du consists of an entire state sequence for each control. As we shall see, the adjoint method provides a way of side-stepping this computation while still arriving at exact derivatives of φ. Differentiating the constraint equation (6) gives us a linear constraint on the derivative matrix dQ/du. Thus, the first term of equation (7) calls for calculating

125 Antoine McNamara, Adrien

Treuille, Zoran Popovi´c, and Jos Stam, “Fluid Control Using the Adjoint Method”, NSF grants CCR-0092970, ITR grant IIS-0113007. 126 Antoine McNamara, Adrien Treuille, Zoran Popovi´c, and Jos Stam, “Fluid Control Using the Adjoint Method”, NSF grants CCR-0092970, ITR grant IIS-0113007.

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 dQ  F  dQ F such that  I    Q du  Q  du u

Eq. 4.16

Notice that this is the exact situation described in equation (3), implying that the vector-matrix product might be much more efficiently calculated using the dual! Using the same substitution as equation (4), we introduce a vector R (equivalent to s above), and the product in equation (8) can instead be computed as

dQ  T  F  R such that  I  R  du Q  Q  T

Eq. 4.17

We call this new vector the adjoint vector. If we can calculate this adjoint R, the overall gradient can now be computed simply:

d dF   RT  du du u

Eq. 4.18

Now we must describe how to calculate R itself. First, rewrite the constraint in equation (9) as

  dF  R   R u  du  T

T

Eq. 4.19

The key to solving this lies in the sparse structure of dF/dQ, with its off-diagonal blocks representing each dfi/dqi. Much the same way that Q is an aggregate of a sequence of forward states q1, , , , qn, we may view R as an aggregate of a sequence of adjoint states r1 , , , , rn so that (10) implies that rn = (dφ/dqn)T and T

 df      ri   i  ri 1    dq i   q i 

T

Eq. 4.20

Note that each adjoint state depends on the subsequent state; therefore, whereas the regular simulation states are computed forward in time, these adjoint states must be computed in reverse 127. 4.8.3 Classical Formulation of the Adjoint Approach to Optimal Design Following the developments in [Jameson]128, a wing, for example, is a device to produce lift by controlling the flow, and its design can be regarded as a problem in the optimal control of the flow equations by variation of the shape of the boundary. If the boundary shape is regarded as arbitrary within some requirements of smoothness, then the full generality of shapes cannot be defined with a finite number of parameters, and one must use the concept of the Frechet derivative of the cost with Antoine McNamara, Adrien Treuille, Zoran Popovi´c, and Jos Stam, “Fluid Control Using the Adjoint Method”, NSF grants CCR-0092970, ITR grant IIS-0113007. 128 Antony Jameson, “Optimum Aerodynamic Design Using CFD and Control Theory”, Department of Mechanical and Aerospace Engineering, Princeton University, AIAA 95-1729-CP. 127

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respect to a function. Clearly, such a derivative cannot be determined directly by finite differences of the design parameters because there are now an infinite number of these. Using techniques of control theory, however, the gradient can be determined indirectly by solving an adjoint equation which has coefficients defined by the solution of the flow equations. The cost of solving the adjoint equation is comparable to that of solving the flow equations. Thus the gradient can be determined with roughly the computational costs of two flow solutions, independently of the number of design variables, which may be infinite if the boundary is regarded as a free surface. For flow about an airfoil or wing, the aerodynamic properties which define the cost function are functions of the flow-field variables (w) and the physical location of the boundary which may be represented by the function F. Then

I  I (w , F)

 I T   I T  or δI    δw    δF  w  F    

Eq. 4.21 changes in the cost function. Using control theory, the governing equations of the flow field are introduced as a constraint in such a way that the final expression for the gradient does not require re-evaluation of the flow field. In order to achieve this δw must be eliminated from Eq. 4.21. Suppose that the governing equation R which expresses the dependence of w and F within the flow field domain D can be written as

R  R ( w, F )

 R T   R T  or δR   δ w     δF  w  F    

Eq. 4.22

Next introducing a Lagrange Multiplier ψ 129, we have:

 I T   I T   R   T   R  δI   δw   δF   δw    δF - ψ     F     w   w   F   I T  I T  R   T  R    ψ   ψT  δw   δF  w F  w    F     

Eq. 4.23

0

Choosing ψ to satisfy the adjoint equation:

I  R   w  ψ  w T

Eq. 4.24

The first term is eliminated so we have

δI  GF

where

I T  R  G - ψT   F  F 

Eq. 4.25

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints. 129

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This equation is independent of ∂w, with the result that the gradient of I with respect to an arbitrary number of design variables can be determined without the need for additional flow-field evaluations. In the case that (1.2) is a partial differential equation, the adjoint equation Eq. 4.24 is also a partial differential equation and appropriate boundary conditions must be determined. After making a step in the negative gradient direction, the gradient can be recalculated and the process repeated to follow a path of steepest descent until a minimum is reached. In order to avoid violating constraints, such as a minimum acceptable wing thickness, the gradient may be projected into the allowable subspace within which the constraints are satisfied. In this way one can devise procedures which must necessarily converge at least to a local minimum, and which can be accelerated by the use of more sophisticated descent methods such as conjugate gradient or quasi-Newton algorithms130. There is the possibility of more than one local minimum, but in any case the method will lead to an improvement over the original design. Furthermore, unlike the traditional inverse algorithms, any measure of performance can be used as the cost function. Similar information can be found at [Oliviu Sugar-Gabor]131. All components just described are integrated into a computational framework capable of optimizing the aerodynamic shape, as shown in Figure 45. 4.8.4 Limitations of the Adjoint Approach 4.8.4.1 Constraints Engineering design applications often have a set of constraints which must be satisfied, in addition to the discrete flow equations132. Some of these may be geometric, such as airfoil design in which the length of the chord and the area of the airfoil are fixed. Others may depend on the flow variables, such as wing design in which one wishes to minimize the drag but keep the lift fixed. Geometric constraints

w

F

I dI/dF Fx 130 A. Jameson, L. Martinelli and

N.A. Pierce, “Optimum Aerodynamic Design Using the Navier–Stokes Equations”, Theoret. Comput. Fluid Dynamics (1998) 10: 213–237. 131 Oliviu SUGAR-GABOR, “Discrete Adjoint-Based Simultaneous Analysis and Design Approach for Conceptual Aerodynamic Optimization”, DOI: 10.13111/2066-8201.2017.9.3.11, August 2017. 132 Michael B. Giles and Niles A. Pierce, “An Introduction to the Adjoint Approach to Design”, Flow, Turbulence and Combustion 65: 393–415, 2000.

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are easily incorporated by modifying the search direction for the design variables to ensure that the geometric constraints are satisfied. It is the constraints which depend on the flow which pose a problem. If the constraint is taken to be ‘hard’ and so must be satisfied at all stages of the optimization procedure, then we need to know both the value of the constraint function, which we shall label J2(U(α), α), and its linear sensitivity to the design variables. The latter requires a second adjoint calculation; the addition of more flow-based hard constraints would require even more adjoint calculations. This type of constraint therefore undermines the computational cost benefits of the adjoint approach. If the number of hard constraints is almost as large as the number of design variables, then the benefit is entirely lost. To avoid this, the alternative is to use ‘soft’ constraints via the addition of penalty terms in the objective function, e.g. J(U)+λ(J2(U))2. The value of λ controls the extent to which the optimal solution violates the constraint J2(U, α) = 0. The larger the value of λ, the smaller the violation, but it also worsens the conditioning of the optimization problem and hence increases the number of steps to reach the optimum. 4.8.4.2 Limitations of Gradient-Based Optimization The adjoint approach is only helpful in the context of gradient-based optimization and such optimization has its own limitations. Firstly, it is only appropriate when the design variables are continuous. For design variables which can take only integer values (e.g. the number of engines on an aircraft) stochastic procedures such as simulated annealing and genetic algorithms are more suitable. Secondly, if the objective function contains multiple minima, then the gradient approach will generally converge to the nearest local minimum without searching for lower minima elsewhere in the design space. If the objective function is known to have multiple local minima, and possibly discontinuities, then again a stochastic search method may be more appropriate133. 4.8.5 Case Study – Adjoint Aero Design Optimization for Multi-Stage Turbomachinery Blades The adjoint method for blade design optimization will be described. The main objective is to develop the capability of carrying out aerodynamic blading shape design optimization in a multistage turbomachinery environment. To this end, an adjoint mixing-plane treatment has been proposed. In the first part, the numerical elements pertinent to the present approach will be described. Attention is paid to the exactly opposite propagation of the adjoint characteristics against the physical flow characteristics, providing a simple and consistent guidance in the adjoint method development and applications. The adjoint mixing-plane treatment is formulated to have the two fundamental features of its counterpart in the physical flow domain: conservation and non-reflectiveness across the interface. The adjoint solver is verified by comparing gradient results with a direct finite difference method and through a 2D inverse design. The adjoint mixing-plane treatment is verified by comparing gradient results against those by the finite difference method for a 2D compressor stage. The redesign of the 2D compressor stage further demonstrates the validity of the adjoint mixingplane treatment and the benefit of using it in a multi-blade row environment. Results for pressure contours are presented in Figure 46. In summary, method ingredients includes134:  A continuous Adjoint method has been developed based on a 3D time-marching finite volume RANS solver. This enables the performance gradient sensitivities to be calculated very efficiently, particularly for situations with a large number of design variables135. Michael B. Giles and Niles A. Pierce, “An Introduction to the Adjoint Approach to Design”, Flow, Turbulence and Combustion 65: 393–415, 2000. 134 Osney Thermo-Fluids Laboratory, “Adjoint Aerodynamic Design Optimization for Multi-stage Turbomachinery Blades”, University of Oxford. 135 D. X. Wang and L. He, “Adjoint Aerodynamic Design Optimization for Blades in Multi-Stage Turbomachines: Part I – Methodology and Verification”, ASME Journal of Turbomachinery, Vol.132, No.2, 021011, 2010. 133

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 A novel adjoint ‘mixing-plane’ model has been developed. This model makes it possible to carry out the adjoint design optimization in a multi-stage environment.  The optimization approach with the above capabilities has been implemented in a parallel mode, with which a simultaneous multi-point optimization can be conducted136.

Original

Figure 46

Optimized

Pressure Contours

D. X. Wang, L. He, Y.S Li and R.G. Wells, “Adjoint Aerodynamic Design Optimization for Blades in Multi-Stage Turbomachines: Part II – Verification and Application”, ASME Journal of Turbomachinery, Vol.132, No.2, 021012, 2010. 136

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5 Turbo-Machinery Design and Optimization 5.1 A Road Map to Turbo-Machine Design and Optimization The design of a turbomachine can be traced through three basic phases. First is the preliminary design phase in which the type of machine to be employed is determined. Additionally, the size, speed, and over-all geometry are determined. Since the entire design process is iterative, the preliminary design parameters and shapes are always subject to modification. Many aspects of this preliminary design process are empirical and/or arbitrary and are based on engineering experience, system or installation limitations, costs, and other factors of which the designer must be aware. However, to be able to proceed with a detailed design, a fairly complete conceptual form of the machine must be generated. The second phase is the detailed design of the machine duct and blade shapes. A design that is based on the equations of fluid motion requires the development of a mathematical description or model of the flow field and the machine geometry. This phase requires computerized methods to solve the equations of motion while accounting for as many of the physical properties and boundary conditions as possible. Figure 47 General Description of Computational Planes Since the direct solution of the equations of motion for viscous, turbulent flow is impossible, it becomes necessary to approximate some of the physics of the flow. However, the exact solution of the no viscous or ideal flow field with forces due to fluid accelerations, rotation, and non-uniform flows taken into account is possible. Figure 47 helps to visualize these various aspects of the flow geometry. The usual approach used is to solve for the inviscid flow field and superimpose the effects of real fluid flow which are difficult to treat analytically. The solution must include physical effects peculiar to each type of machine. Generally, the inflow conditions to the turbomachine will be no uniform in velocity and pressure. The chord wise and span wise loading or pressure on the blade rows will be no uniform as is the blockage due to blade thickness and boundary layer growth. If the flow field is unbounded, as for a nonconductor fan, additional boundary conditions are necessary to perform a flow analysis. The Effects of viscosity in the blade row passages must be modeled accurately to achieve the design performance requirements. The second phase terminates in an actual blade shape design that will produce the flow field specified by the analysis. Determination of this shape is difficult as there are a number of boundary conditions that must satisfy, and the performance of a blade row is highly subject to real flow phenomena which are not easily approached analytically. Again, a combination of ideal flow theory and empirical corrections are required to specify a blade row with desired performance.

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Phase 3

Phase 2

Phase 1

The third phase is essentially similar to the analysis and blade design except that the geometry of the system is fixed and it is desired to determine the effect of a blade row on the flow field. This should give the same solution as the original analysis at the design point. However, the results of performance testing and, in particular, measured velocity and pressure profiles in the vicinity of the blade row may be used to test the theoretical analysis and design. Where differences are found, corrections to the mathematical models or to the hardware may be required. In summary, the design and analysis of a turbomachine requires an accurate description of the internal flow field. A numerical simulation of the turbomachine, including the various components of the equations of motion in either exact or approximate (empirical) form, must be constructed. This simulation consists of two basic parts, a meridional plane or throughflow solution and a blade design method that includes an analytical blade-to-blade flow solution. Each of these solutions affects the other and must be developed iteratively to produce a consistent model of the flow. Once a model is generated, a check against the design requirements must be made to assure that no part of it fails the design requirements. Once complete, the model must be Figure 48 Turbomachine Design Process compatible with mechanical limitations and manufacturing capabilities. An additional iteration with the design constraints may be necessary to finally arrive at an acceptable engineering solution to the design of the turbomachine. Figure 48 illustrated these steps require in turbomachinery design and analysis137. 5.1.1 Wu’s Pioneering (S1 and S2) Scheme In the early 1950's, Wu138, as previously recognized, documented these problems and formulated a M. V. Casey, “Computational Methods for Preliminary Design and Geometry Definition in Turbomachinery”, Fluid Dynamics Laboratory, Sulzer Innotec AG, CH-8401, Winterthur, Switzerland. 138 Wu, C. H., "A General Theory of Three Dimensional Flow in Subsonic and Supersonic Turbomachines of Axial, Radial, and Mixed Flow Types," National Advisory Committee on Aeronautics, NACA TN 2604, 1952. 137

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set of equations which had the possibility of a solution. He broke the problem of 3D flow into a set of coupled 2D solutions. Figure 47 helps to explain Wu's analysis in which he broke the problem into two planes (S1 and S2) generally perpendicular to each other. One, the meridional plane, describes the flow on hub-to-tip stream surfaces. The other, the blade-to-blade solution, describes the flow on planes generally parallel to the hub surface of the machine and perpendicular to the blading. A complete solution by Wu's method would require a number of both parallel meridional and parallel blade-to-blade solutions. The solutions are coupled and must be solved iteratively to simultaneously satisfy the equations on all of the solution planes (Quasi-3D). At the time of formulation of Wu's analysis, computational methods and machines were not large or fast enough to give a comprehensive solution. As a result, many approximate methods evolved. [Wislicenus]139 summarized many of the design techniques in use at the time. Most of these techniques relied heavily on experimental data to be useful. Smith 140 rearranged the equations of motion in the meridional plane to give a time and spatially averaged picture of the flow in a blade row. At the same time, additional computerized techniques were developed to solve the through-flow problem. [Novak]141 formulated a solution (Streamline Curvature Method) that solved for the velocities and streamlines rather than the stream functions where solution was basically inviscid and non-turbulent. The problem of losses due to viscosity and turbulence was addressed by Bosman and Marsh142, but in general, experimental data are always required to adequately model the real fluid effects encountered in a turbo machine. 5.1.2 Concept of Streamline Curvature Method The Streamline Curvature Method (SCM) offers an advantage in that the equations and solution are in terms of physical variables of velocity and pressure rather than those of a stream function, which previously mentioned. Additionally, viscous and turbulence effects are much easier to incorporate into the (SCM) because their models are developed in terms of physical variables. Where the 3D nature of the flow field is required, determination effects of the blading on the meridional flow requires flow field solutions on the blade-to-blade surfaces. The design of blade sections also requires an accurate analysis technique. [Kansan’s]143 was one first to successfully compute the velocity and pressure distribution on the blade-to-blade plane. None of the above methods incorporates sufficient modeling of the turbulent boundary layer flow associated with turbomachine blade rows. [Raj and Lakshminarayana]144 conducted experiments which gave insight into the nature of the blade boundary layers and the structure of the wake shed from the trailing edge of a blade. These data will help in the formulation of more accurate models of this flow phenomenon. The availability of the various through-flow and blade-to-blade solutions leads to the possibility of synthesizing a threedimensional model of the turbomachine flow field. The interaction of the flow on the meridional plane and on the blade-to-blade planes becomes important in this case. The result of blade-to-blade analysis is that forces due to the geometry of the blading may be determined. [Novak and Hearsey]145 utilized the Streamline Curvature Method in a similar manner to generate a quasi-three-dimensional Wislicenus, C. F., Fluid Mechanics of Turbomachinery, New York, N. Y., Dover Publications Inc., 1965. Trans. ASME, J. Eng. Power, 88A, 1966. 141 Novak, R. A., "Streamline Curvature Computing Procedures for Fluid Flow Problems," Trans. ASME, J. Eng. Power, Vol. 89, 1967. 142 Bosman, C, and H. Marsh, "An Improved Method for Calculating the Flow in Turbomachines, including a Consistent Loss Model," J. Mech. Eng. Sci., 1974. 143 Katsanis, T., "Use of Arbitrary Quasi-Orthogonal for Calculating Flow Distribution on a Blade-to-Blade Surface in a Turbomachine." NASA TN D-2809, May 1965. 144 Raj, R., and B. Lakshminarayana, "Characteristics of the Wake Behind a Cascade of Airfoils," J. Fluid Mechanics, Vol. 61, Part 4, 1973. 145 Novak, R. A., and R. M. Hearsey, "A Nearly Three-Dimensional Intra Blade Computing System for Turbomachinery," Part I and TI, Trans. ASIT4, J. Fluids Eng., March 1977. 139

140 Smith, L. H., Jr., "The Radial Equilibrium Equation of Turbomachinery,"

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analysis. It should be stressed that the above techniques are for analysis of already designed blade rows and do not apply to the actual determination of a blade shape, i.e., the design problem. The aim here to address the design problem by using the Streamline Curvature Method to construct an averaged through flow picture that satisfies the general design requirements. Then two methods, the Mean Streamline Method and the Streamline Curvature Method, will be used to actually define blading that generates the flow field prescribed by the through-flow analysis. Combining the through-flow and the blade-to-blade analysis, a quasi-three-dimensional analytical representation of the flow field is generated.

5.2 Case Study 1 - Aerodynamic Design of Compressors Due to low cost and speed of CFD comparing to traditional testing, and the fact that it ability to simulated almost any testing, it can be useful tool in design and optimization. While experiment yields a discrete and limited data, CFD provide the entire region on interest or complete picture enabling complete design. In that respect, aerodynamic design techniques of gas turbine compressors have been dramatically changed in the last few years. While the traditional 1D and 2D design procedures are consolidated for preliminary calculations, emergence techniques have been developed and are being used almost routinely within industries and academia. The compressor design still remains a very complex and multidisciplinary task, where aero-thermodynamic issues traditionally considered prevalent, now become part of a more general design approach. Nowadays, interesting and alternative options are in fact available for compressor 3D design, such a new blade shapes for improved efficiency, end-wall contouring and casting treatment for enhanced stall margin, as well as many others. For this reason while experimental activity remains decisive for ultimate assessment of design choices, numerical optimization techniques, along CFD are assuming more and more importance for the detailed design. 5.2.1 Statement of Problem Qualitatively speaking, compressor aerodynamic design is the procedure by which the compressor geometry is calculated which fulfils the design cycle requirements in the best possible way146. Using a more certain statement, we can formulate the design problem by identifying objectives, boundary conditions, constraints, and decision variables as follows:  Objectives: Maximize Adiabatic Efficiency (η), Maximize stall margin (SM), both at nominal condition.  Boundary conditions: inlet conditions (pressure P, temperature T), flight Mach number, M, (in the case of an aero-engine).  Decision (Design) variables: number of stages, compressor and stage geometry parameters.  Functional constraints: mass flow rate, m, (based on engine Power or Thrust requirements), Pressure Ratio, PR, (from Cycle analysis), correct compressor component matching (i.e. intake-compressor, compressor-combustor, and above all compressor-turbine) as determined by a Matching Index (MI).  Side Constraints: each decision variables must be chosen within feasible lower and upper bounds (sides).  Multi-disciplinary constraints: structural and vibrational, weight, costs, manufacturability, accessibility, reliability. 146

Benini, E., “Advances in Aerodynamic Design of Gas Turbines Compressors “, University of Padova Italy.

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The aerodynamic design of an axial-flow compressor is inherently a Multi-Objective Constrained Optimization problem, which can be summarized on the as:

X  X (x1 , x 2 ,......, x n )

x i,min  x i  x i,max for i  1, 2,......n

Eq. 5.1

Where n is the number of decision variable. Table 6 summarizes these decision making criteria. It is worth underlying that this might not be the general formulation, as some constraints could be turned into objectives, mainly depending on compressor’s final destination and/or manufacturer’s strategies.

Axial Flow Compressor

For a given: P ,T , M

Maximize:

Minimize:

F(X) = η (X) , SM(X)

Stationary gas Turbine (Electric Power)

P ,T

F(X) = η (X) , Load – Response (X)

Aero-engine (military use)

P, T, M

F(X) = η (X) , SM(X)

Table 6

weight (X)

Subject to:

Comments:

m(X) , PR(X), MI(X) , Cost function (X), g(X) m(X) ,PR(X), MI(X), Cost function(X) , g(X) m(X) , PR(X) , q(X) , MI(X) Cost function(X), g(X)

g(X) = weight, structural, technological, other g(X) = weight, structural, technological, other g(X) = structural, technological, other q(X) = static trust at sea level

Axial Flow Compressor Design

5.2.2 Different Compressors Objectives In a stationary gas turbine used for electric power generation a great importance is given both to the compressor peak efficiency, and to the function called “load response” which quantifies the rapidity of the compressor in adjusting the airflow delivered by means of IGVs and/or VSVs. In this case, of course, an intervention aimed at regulating the delivered power of the gas turbine has an effect also on the mass flow conveyed by the compressor. An aero-engine for military use, a significant merit is attributed to reaching the best trade-off between performance and weight, objectives which are intuitively conflicting each other, while cost function is inevitably different to the one assigned to the civil application.

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Finally, a constraint based on the static thrust to be delivered by the overall engine at sea level is set which inevitably influences compressor design. From the problem formulations given above, remarkable importance is attributed to maximize or minimize some compressor performance indexes or figures of merit. Therefore, before examining how to deal with such problems, it is worth analyzing how performance can be significantly affected by the choice in the design variables. For instance, maximizing adiabatic efficiency requires a deep understanding of the physics governing stage losses, which have to be minimized either in design and off-design conditions. This, in turn, will have an important impact on the choice of stage geometrical and functional variables. On the other hand, maximizing stall margin involves acquiring a proper insight of stall physics and minimizing stall losses. Again, such problem can be tackled if proper stage geometry is foreseen. Lastly, minimizing compressor weight (at least from the aerodynamic point of view) implicates reducing the number of compressor stages and increasing individual stage loading, a fact which ultimately affects the choice of the blade shape, particularly cascade parameters. Based on the arguments above, in the following a brief summary of basic and advanced compressor aerodynamics is given147.

Figure 49

Sketch of a Compressor stage (left) and cascade of geometries at mid- span (right)

5.2.3 Design Techniques for Compressor148 Independently from the particular case under study, modern compressor design philosophy can be summarized as in Figure 50. A preliminary design is usually carried out at first, aiming at defining some basic features such as number of stages, inlet and outlet radii and length. Stage loading and reaction is established as well on the basis of preliminary criteria driven by basic theory and experiments. Such procedure is based on one-dimensional (1D) methods, where each stage characteristics are condensed into a single “design block”, to which basic thermo- and fluid dynamics equations are applied. Therefore, no effort is spent to account for flow variations other than those which characterize the main axial flow direction within each stage at a time. Then, stages are stacked together to determine the overall compressor design, regardless any mutual stage interaction. Within this process, which will be described later on, technological and process constraints, as well as 147 148

Benini, E., “Advances in Aerodynamic Design of Gas Turbines Compressors “, University of Padova Italy. Same source – see above.

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restrictions on weight and cost, play an important role that must be properly accounted for. In this framework, some early choices could be revisited and subject to aerodynamic criteria checking, so that an iterative process occurs until a satisfactory preliminary design is obtained. A second preliminary step, distinct from the 1D procedure, is the two-dimensional design (2D), which include both cascade and through flow models, from which a characterization of both design and off-design multi-stage compressor performance can be carried out after some iterations, if necessary. In this case, both direct and inverse design methodologies have been successfully applied. Numerical optimization strategies may be of great help in this case as the models involved are relatively simple to run on a computer. Often an optimization involves coupling a prediction tool, e.g. a blade to blade solver and/or a through flow code, and an optimization algorithm which assists the designer to explore the search space with the aim of obtaining the desired objectives. Finally, a fully three-dimensional (3D) design is carried out including all the details necessary to build the aerodynamic parts of the compressor. In this phase, some design intervention is needed to account for the real three-dimensional, viscous flows in the stages, especially tip clearance, secondary flows and casing treatment for stall delay. This is usually carried out using CFD models, where the running blade is modeled in its actual deformed shape, analyzed and, if necessary optimized. While traditional 3D analyses are aimed at evaluating and improving compressor performance of a single stage, the recent availability of powerful computers makes the analyses of multistage compressors an affordable task for most industries. Most advanced CFD computations include evaluation of complex unsteady effects due to successive full-span rotor-stator interaction149.

Figure 50

149

Compressor design flow chart

Benini, E., “Advances in Aerodynamic Design of Gas Turbines Compressors “, University of Padova Italy.

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5.2.4 Preliminary Design Techniques (1D) A simple mean or pitched line 1D method usually forms the basis of a preliminary design. For a given design condition, the compressor total pressure ratio is known from which the total number of compressor stages can be estimated (see Figure 51). To this respect, the designer can use statistical indications based on typical values of admissible peripheral speeds and stage loading. This is very useful for estimating the preliminary stage pressure ratio once the range for the other functional parameters has been settled. Result of stage stacking consists in the flow path definition, from which the distribution of stage parameters along the mean radii can be obtained. Because the stacking procedure is intrinsically iterative, a loop is required to satisfy all the design objectives and constraints. As a first check, the axial Mach distribution along the stages must be calculated and a value not exceeding 0.5 is tolerated for both subsonic and Figure 51 Preliminary Estimation of Number of Stages in transonic stages. By imposing such a Compressor constraint, the values of stage area passage can be derived from the continuity equation150. Next, the values of the hub-to-tip ratios must be defined. To this end, it is worth recalling that such value comes from a trade-off between aerodynamic, technological and economic constraints. For inlet stages, values between 0.45 and 0.66 can be assigned, while outlet stages often are given a higher value, say from 0.8 to 0.92, in order not to increase the exit Mach number (a condition that is detrimental for pneumatic combustor losses). Despite its relative simplicity, mean line 1D methods based on stage-stacking techniques still play an important role in the design of compressor stages. Recent works includes a numerical methodology used for optimizing a stator stagger setting in a multistage axial-flow compressor environment (seven-stage aircraft compressor), based on a stage-by-stage model to 'stack' the stages together with a dynamic surge prediction model. The absolute inlet and exit angles of the rotor are taken as design variables. Analytical relations between the isentropic efficiency and the flow coefficient, the work coefficient, the flow angles and the degree of reaction of the compressor stage were obtained. Numerical examples were provided to illustrate the effects of various parameters on the optimal performance of the compressor stage151. 5.2.5 Through Flow Design Techniques (2D) Through flow design allows configuring the meridional contours of the compressor, as well as all other stage properties in a more accurate way compared to 1D methods. They make use of cascade

150 151

Same as previous. Same as previous.

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correlations for total pressure loss/flow deviation and are based on through flow codes, which are two-dimensional inviscid methods that solve for axisymmetric flow (radial equilibrium equations) in the axial-radial meridional plane (Figure 52)152. A distributed blade force is imposed to produce the desired flow turning, while blockage factor that accounts for the reduced area due to blade thickness and distributed frictional force representing the entropy increase due to viscous stresses and heat conduction can be incorporated. Three methods are basically used for this purpose:   

Streamline Curvature Methods SCM (Novak, 1967), Matrix Through Flow Methods (MTFM) (Marsh, 1968) and Streamline Through Flow Methods (STFM) (Von Backström & Rows, 1993).

Figure 52

Optimization Procedures proposed in [Massardo et al.]

The SCM has the advantage of simulating individual streamlines, making it easier to be implemented because properties are conserved along each streamline but is typically lower compared to the other methods. On the other hand, MTFM uses a fixed geometrical grid, so that streamline conservation properties cannot be applied. However, despite stream function values must be interpolated throughout the grid, the MTFM is numerically more stable than SCM. Finally STFM is a hybrid approach which combines advantages of accuracy of SCM with stability of MTFM. These methods have recently been made more realistic by taking account of end-wall effects and span wise mixing by four aerodynamic mechanisms: turbulent diffusion, turbulent convection by secondary flows, span wise migration of airfoil boundary layer fluid and span wise convection of fluid in blade wakes. As a result of the application of through flow codes, the compressor map in both design and off design operation can be obtained exhibiting high accuracy153. Remarkable developments in the design techniques have been obtained using such codes. Among others, [Massardo] described a technique for the design optimization of an axial-flow compressor stage. The procedure allowed for optimization of the complete radial distribution of the geometry, being the objective function obtained using a through flow calculation (see Figure 52). Some examples were given of the 152 153

Benini, E., “Advances in Aerodynamic Design of Gas Turbines Compressors “, University of Padova Italy. Same Source – See Previous.

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possibility to use the procedure both for redesign and the complete design of axial-flow compressor stages. 5.2.6 Detailed Design Techniques (3D) 5.2.6.1 Direct Methods Advanced optimization techniques can be of great help in the design of 3D compressor blades when direct methods are used154. These are usually very expensive procedures in terms of computational cost such that they can be profitably used in the final stages of the design, when a good starting solution, obtained using a combination of 1D and/or 2D methods, is already available. Moreover, large computational resources are necessary to obtain results within reasonable industrial times. Examples of 3D designs of both subsonic and transonic compressor blading’s are today numerous in the open literature. For numerical optimization, searching direction was found by the steepest decent and conjugate direction methods, and it was used to determine optimum moving distance along the searching direction. The object of present optimization was to maximize efficiency. An optimum stacking line was also found to design a custom-tailored 3D blade for maximum efficiency with the other parameters fixed. The method combined a parametric geometry definition method, a powerful blade-to-blade flow solver and an optimization technique (breeder genetic algorithm) with an appropriate fitness function. Particular effort has been devoted to the design of the fitness function for this application which includes non-dimensional terms related to the required performance at design and off-design operating points. It has been found that essential aspects of the design (such as the required flow turning, or mechanical constraints) should not be part of the fitness function, but need to be treated as so-called "killer" criteria in the genetic algorithm. Finally, it has been found worthwhile to examine the effect of the weighting factors of the fitness function to identify how these affect the performance of the sections155.

Figure 53

Mach number contours a) Base line b) Max. PR. Ratio

c) Max. Efficiency

A multi-objective design optimization method for 3D compressor rotor blades was developed by [Benini, 2004], where the optimization problem was to maximize the isentropic efficiency of the rotor and to maximize its pressure ratio at the design point, using a constraint on the mass flow rate. Direct 154 155

Same Source – See Previous. Benini, E., “Advances in Aerodynamic Design of Gas Turbines Compressors “, University of Padova Italy.

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objective function calculation was performed iteratively using the 3D Navier-Stokes equations and a multi-objective evolutionary algorithm featuring a special genetic diversity preserving method was used for handling the optimization problem. In this work, blade geometry was parameterized using three profiles along the span (hub, mid span and tip profiles), each of which was described by camber and thickness distributions, both defined using Bezier polynomials. The blade surface was then obtained by interpolating profile coordinates in the span direction using spline curves. By specifying a proper value of the tangential coordinate of the first mid span and the tip profiles control point with respect to the hub profile, the effect of blade lean was achieved. Results of tip profiles control point with respect to the hub profile and the effect of blade lean was achieved. Performance enhancement severe shock losses (Figure 53). Computational time was enormous, involving about 2000 CPU hours on a 4-processor machine. 5.2.6.2 Inverse Methods In the last two decades, 3D inverse design methods have emerged and been applied successfully for a wide range of designs, involving both radial/mixed flow turbo-machinery blades and wings. Quite a new approach to the 3D design of axial compressor blading has been recently proposed by [Tiow 2002]. In this work, an inverse method was presented which is based on the flow governed by the Euler equations of motion and improved with viscous effects modeled using a body force model. However, contrary to the methods cited above, the methodology is capable of providing designs directly for a specific work rotor blading using the mass-averaged swirl velocity distribution. Moreover, the methodology proposed by [Tiow], joins the capabilities of an inverse design with the search potential of an optimization tool, in this case the simulated annealing algorithm. The entire computation required minimal human intervention except during initial set-up where constraints based Figure 54 Comparison of blade loading prescribed by on existing knowledge may be imposed inverse mode to restrict the search for the optimal performance to a specified domain of interest. Two generic transonic designs have been presented, one of which referred to compressor rotor, where loss reductions in the region of 20 per cent have been achieved by imposing a proper target surface Mach number which resulted in a modified blade shape. Figure 54 comparison of blade loading distributions of an original supersonic blade, a new design (prescribed by inverse mode), and a reference blade (R2-56 blade) for a given pressure ratio (left); comparison of passage Mach number distributions at 95% span. Results showed that an optimum combination of pressure-loading tailoring with surface aspiration can lead to a minimization of the amount of sucked flow required for a net performance improvement at design and off-design operations. By prescribing a desired loading distribution over the blade the placement of the passage shock in the new design was about the same as the original blade. However, the passage shock was weakened in the tip region where the relative Mach number is high.

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5.2.7 Concluding Remarks It can be concluded that Gas turbine compressors, either stationary or aeronautical, have reached a relatively mature level of development and performance. Nevertheless, the availability of advanced materials for blade construction makes it possible to rich levels of aerodynamic loading never experienced in the past while preserving high levels of on-off design efficiency. This holds especially for highly-subsonic and transonic blades, where tangential velocities are now becoming higher that 600 m/s, thus leading to stage pressure ratios of 2 and more. In fact, transonic blading’s make it possible to reduce the number of stages for a prescribed total compression ratio, thus leading to huge savings in compressor costs, weight and complexity. To properly design such machines, multi objective and multi criteria problems are to be dealt with which claim for rigorous and robust procedures, more often assisted by solid mathematical tools that help the designer to complete his/her skills and experience156. In view of the above, continuous effort is currently being spent in building advanced design techniques able to tackle the problem efficiently, cost-effectively and accurately. Plenty of design optimization techniques has been and are being developed including standard trial and-error 1D procedures up to the most sophisticated methods, such as direct or indirect methods driven by advanced optimization algorithms and CFD. Advanced techniques can be used in all stages of the design. In the field of 1D, or mean line methods, correlation-based prediction tools for loss and deviation estimation can be calibrated and profitably used for the preliminary design of multistage compressors. 2D methods supported by either through-flow or blade-to-blade codes in both a direct and an indirect approach, can be used afterwards, thus leading to a more accurate definition of the flow path of both meridional and cascade geometry. To enhance the potentialities of such methods, optimization algorithms can be quite easily used to drive the search toward optimal compressor configurations with a reasonable computational effort. Detailed 3D aerodynamic design remains peculiar of single stage analyses, although several works have described computations of multistage configurations, either in steady and unsteady operations. However, the latter is an approach suitable for verification and analysis purposes, thus with a limited design applicability. The 3D design optimization techniques can realistically be used if local refinement of a relatively good starting point is searched for. On the other hand, if more general results are expected, simplified design methods are mandatory, such as those based on supervised learning procedures, where surrogate models of the objective functions are constructed. Other very promising techniques include adjoin methods, where the number of design iteration can be potentially reduced by an order of magnitude if local derivatives of physical quantities with respect to the decision variables are carefully computed157.

5.3 Case Study 2 – Turbine Airfoil Optimization using Quasi 3D Analysis Codes

Turbine airfoil design has long been a domain of expert designers who use their knowledge and experience along with analysis codes to make design decisions. The turbine aerodynamic design is a three-step process that is pitch line analysis, through-flow analysis, and blade-to-blade analysis, as depicted in Figure 55. In the pitch line analysis, flow equations are solved at the blade pitch, and a free vortex assumption is used to get flow parameters at the hub and the tip. Using this analysis the flow path of the turbine is optimized, and number of stages, work distribution across stages, stage reaction, and number of airfoils in each blade row are determined. In the through-flow analysis, the calculation is carried out on a series of meridional planes where the flow is assumed to be axisymmetric and the boundary conditions of each stage are determined. The axisymmetric throughflow method allows for variation in flow parameters in the radial direction without using the free vortex assumption and accounts for interactions between multiple stages. In the blade-to-blade Benini, E., “Advances in Aerodynamic Design of Gas Turbines Compressors “, University of Padova Italy. Benini, E.,” Three-Dimensional Multi-Objective Design Optimization of a Transonic Compressor Rotor”. Journal of Propulsion and Power, Vol. 20, No. 3 (May/June), pp.559-565, ISSN: 0748-4658-2004. 156 157

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analysis, airfoil profiles are designed on quasi-3D surfaces using a computational fluid dynamics code. The design of airfoil profiles involves slicing the blade on quasi-3D surfaces, designing each section separately, and stacking the sections together to obtain a sooth radial geometry. The objective of airfoil design is to define the airfoil shape so as to ensure structural integrity and minimize losses. The primary sources of losses in an airfoil are profile loss, shock loss, secondary flow loss, tip

Figure 55

The turbine design process

clearance loss, and end-wall loss. Profile loss is associated with boundary layer growth over the blade profile causing viscous and turbulent dissipation. This also includes loss due to boundary layer separation because of conditions such as extreme angles of incidence and high inlet Mach number. Shock losses arise due to viscous dissipation within the shock wave which results in increase in static pressure and subsequent thickening of the boundary layer, which may lead to flow separation downstream of the shock. End-wall loss is associated with boundary layer growth on the inner and outer walls on the annulus. Secondary flow losses arise from flows, which are present when a wall boundary layer is turned through an angle by an adjacent curved surface. Tip clearance loss is caused by leakage flows in the tip clearance region of the rotor blade, where the leaked flow fails to contribute to the work output and also interacts with the end-wall boundary layer. The objective of the design is to create the most efficient airfoil by minimizing these losses. This often requires trading-off one loss versus another such that the overall loss is minimized. To compute all these losses a 3D viscous analysis is required; however, due to the computational load of such a code, a quasi-3D analysis code is often used in the design process. Thus the impact of the blade geometry on 3D losses cannot be determined and only 2D losses can be minimized, that is, profile and shock losses. A viscous quasi-3D analysis though less computationally intense is still too expensive for use in design optimization, and an inviscid quasi-3D code is used instead. Consequently, viscous losses are not computed from the analysis code and airfoil performance is gauged by the characteristics of the Mach number distribution on the blade surface. The most practical formulation for low-speed turbine airfoil designs still remains the direct optimization formulation based on 2D

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inviscid blade-to-blade solvers. This work automates the direct design process as described in the next section158. 5.3.1 Parametric Representation of Airfoil Design Process The parametric representations of the airfoils used in this work are based on the standard design tools and practices. There are separate models for the high-pressure and low pressure turbine blades. The high-pressure turbine blades are subject to very high temperatures and need to be cooled. The parametric representations of the airfoils used in this work are based on the standard design tools and practices. There are separate models for the high-pressure and low pressure turbine blades. The high-pressure turbine blades are subject to very high temperatures and need to be cooled. As a result, these airfoils are made thick to accommodate cooling passages inside the blades. For such (A) High Pressure Airfoil thick airfoils, suction and pressure surfaces need to be manipulated independently of each other. So the airfoil is represented as a combination of two separate curves, one for the pressure side and the other for the suction side (see Figure 56 (A)) Bezier curves are well suited for these airfoils. Low-pressure airfoils, on the other hand, have lower thermal stresses, are much longer, and have a lower speed of rotation compared to high-pressure airfoils. These airfoils are usually very thin and the two-surface model does not work very well as it is very difficult to (B) Low Pressure Airfoil vary the pressure and suction surfaces independently and still maintain a smooth thickness distribution. For such Figure 56 Parametric representation of an Airfoil airfoils, a mean line and thickness representation is used in which a thickness distribution is super imposed on the mean line of the airfoil as shown in Figure 56 (B). In this representation, the mean line and the thickness distribution can be varied independently, and good control of the thickness distribution is obtained. Here we discusses optimization of low-pressure turbine blades with the following parameters in Table 7. 5.3.2 Constraints and Problem Formulation Constraints are imposed on the airfoil geometry to ensure that the airfoil is manufacturable and structurally feasible as well as for ensuring high aerodynamic efficiency. The structural and manufacturing constraints are based on the airfoil geometry and the aerodynamic constraints are 158

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derived from the Mach number distribution on the airfoil. There aerodynamic constraints are defined, that is, peak-exit-ratio, peak-location, and inlet-valley-ratio. These are listed below and can be interpreted from the Mach number distribution shown in Figure 57. Peak-exit-ratio is defined as the ratio of the peak Mach number on the suction surface to the Mach number at the trailing edge of Constraints are imposed on the airfoil geometry to ensure that the airfoil is manufacturable and structurally feasible as well as for ensuring high aerodynamic efficiency. The structural and manufacturing constraints are based on the airfoil geometry and the aerodynamic constraints are derived from the Mach number distribution on the airfoil. There aerodynamic constraints are defined, that is, peak-exit-ratio, peak-location, and inlet-valley-ratio. These are listed below and can be interpreted from the Mach number distribution shown in Figure 57. Peak-exitratio is defined as the ratio of the peak Mach number on the suction surface to the Mach number at the trailing edge of the blade. This is a measure of flow acceleration on the unguided portion of the airfoil (between the throat and the trailing edge). A very high turning on the unguided portion of the airfoil can lead to separation of flow or the formation of a shock on the trailing edge. By putting a constraint on the maximum peakexit-ratio, chances of separation are minimized. Peak-location is the normalized location of the peak Mach number on the suction side. It is desirable to have an increasing Mach number as far along on the suction side as possible to prevent Figure 57 A sample Mach number distribution a thickening of the boundary layer. Imposing a constraint on which allows the peak to occur after 65% of the blade width guards against upstream diffusion and helps in achieving a smooth accelerating Mach number on the suction side. Inlet-valley-ratio is the ratio of the Mach number at the inlet of the airfoil on the pressure side, to the minimum Mach number on the pressure side. This constraint controls the diffusion near the inlet on the pressure side and restricts the thickening of the boundary layer, reducing chances of flow separation. Constraints are also imposed on:   

The curvature change on the unguided portion of the airfoil (unguided turning) The difference between the blades mean line angle and the flow angle at the trailing edge (over turning), and The difference between the inlet angle and the metal angle at the inlet (Δ1).

These additional checks further ensure that the designed airfoil stays within design practice guidelines159. To ensure mechanical and structural feasibility, constraints are imposed on the blade geometry. The primary geometry parameters are cross section area, maximum thickness of airfoil, wedge angle, and nose radius. In cooled airfoils, the constraints on the geometry stem from the 159

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necessity to construct cooling channels in the airfoil; these constraints are dictated by manufacturing requirements. In low-pressure airfoils, these constraints are primarily driven by s reassess and manufacturing limitations. Most of these constraints have soft limits on them; that is, it is best to have the responses within a given range, beyond a threshold of the range a penalty that increases nonlinearly with increased violation of the constraint is added to the objective function. The objective function includes the performance metrics and constraints where the violations are included via penalty functions. These factors can vary for different problems based on the requirements of the specific problem. The design variables and typical range of variations are listed 7

Table in and the constraints imposed on the problem are listed in Table 8. During the design of an airfoil, multiple sections are designed concurrently, and the objective function is a sum of the Geometry Variables

Definition

Stagger

Angle of line joining leading & trailing edge of the airfoil to axial

Tmaxx

Maximum thickness of the airfoil

C1

point of maximum thickness

C2

trailing edge included angle

C3

curvature of mean line

Ratu

curvature near leading edge on upper surface

Ratl

curvature near leading edge on lower surface

Pcttle

incidence angle

Ti

leading edge bluntness

E

ellipse ratio of the approximate ellipse fitted in the nose Table 7

Airfoil Geometry Parameters

objective functions of all the cross sections being designed. Constraints for all the sections are also included in the problem formulation. Polynomial fits are used to represent the radial variation of the design variables; thus the objective function becomes a combination of the coefficients of the fits across multiple sections rather than individual parameters for each section. A second-order polynomial fit is used in the formulation; so corresponding to each metric we have three coefficients. The solution to the problem can be attempted using a variety of optimization techniques including numerical optimization, genetic algorithms, simulated annealing, and heuristic search. In the current investigation, the BFGS variable metric method implemented in an optimization code ADS was used. A one-dimensional search technique was used in which the search was bounded followed by use of polynomial interpolation.

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5.3.3 Quasi-3D CFD Analysis and Results A quasi 3D CFD solver is used in the current investigation to analyze the flow on the airfoil, which is an isentropic that uses the streamline curvature method that computes the Mach Number/Pressure distribution on the airfoil surface 160. In the absence of a viscous code, designers usually estimate the quality of the airfoil by visually examining the Mach number distribution obtained from an in-viscid quasi 3D CFD solution. Since optimization techniques are driven by a numerical value of the objective function, and the visual perspective of the designer is the only proven metric available, it must be captured in a suitable numerical algorithm to provide a measure of quality of an airfoil. The current work employs curve fitting coupled with design heuristics to compute quality metrics from the Mach number distribution and the airfoil geometry. These metrics are weighted for different designs based on individual designer preferences. Primary evaluation metrics that have been defined are diffusion, deviation, incidence deviation, and leading edge crossover. A physical interpretation of these metrics is presented below. Diffusion is defined as the deceleration of the flow along the blade surface. It is measured as the cumulative aggregate of Design Lower Upper Initial Final all flow diffusions at each point along the Variables Bound Bound Value Value airfoil surface. As the flow diffuses, the boundary layer thickens, and the C1 0.2 0.5 0.35 0.35 momentum loss in the boundary layer C2 0.25 0.75 0.5 0.632 increases. In this case, the increased drag C3 0.25 0.75 0.5 0.569 causes a significant loss of momentum; Tmaxx 0.05 0.15 0.139 0.139 flow separation may result, causing much Stagger 8 40 31.643 39.464 larger losses. Thus, the objective of the Pcttle -0.25 0.25 0 0 design is to minimize the diffusion effect. Since the impact of diffusion on the Ratl 0 4 1.25 2.703 pressure and suction sides is different, Ratu 0 4 2.59442 2.727 separate terms are defined for the suction Ti 0 1 0.5 1 and pressure sides. In the test case E 1 5 3 2.044 presented here, a low-pressure turbine nozzle is optimized. The flow-path of the Table 8 Airfoil Design Variables low-pressure turbine used in the investigation is shown in Figure 58. The radial distances in the figure are measured with reference to the centerline of the engine and the axial distances are measured with reference to a point upstream of the first stage of the turbine. The horizontal lines in the figure represent the streamlines of the flow. Thirteen streamlines are shown, the top and bottom of which coincide with the casing and the hub respectively. The vertical lines represent the edges of the blade rows and the location of the frame. The turbine has six stages, each stage composed of two blade rows. The first blade row consists of nozzles and the second blade row consists of buckets. The stages are numbered from 1 to 6 in the Figure 58. In the current investigation, stage 5 nozzle was designed using sections from five streamlines equally spaced along the blade span (hub to tip). Figure 59 Shows the approximate locations of the streamlines for an airfoil in which the first and the last streamlines are shown at the hub and tip. In reality however streamlines at 5% and 95% span were used instead of streamlines directly on the hub and tip because Mach number distributions very close to the end walls are distorted by the end wall effects and not representative of the flow away from the walls. The starting solution for the test case was obtained by estimating the airfoil shape based on shapes of similar airfoils designed in the

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past. All the Mach number and airfoil geometry plots use the same reference radial and axial locations as shown in Figure 60. To ensure slope and curvature smoothness of the geometry, second- order polynomials were used to represent the radial distribution of geometry parameters.

Figure 58

Flow path of the turbine

Thus there are three design variables for each geometry parameter, that is, C0, C1, and C2. These are the coefficients of the 2nd polynomial representing the geometry parameter. The efficient of the fit match well with the starting design since the design is based on a previously designed airfoil. Subsequently the smoothness is maintained since the parameters are not changed directly but rather the coefficients of the polynomials are varied. The geometry parameters which describe the lowpressure turbine airfoil geometry are Stagger, Tmaxx, C1, C2, C3, Ratu, Ratl, Pcttle, ti, and E. These geometry parameters are varied within limits typically prescribed in design practice and on the basis of prior experience and manufacturing limitations. The limits for these parameters are described along with the results for each specific test case. 5.3.4 Concluding Remarks Here we presented a mathematical formulation for design of turbine airfoils using 2D geometry models and 2D inviscid analysis codes. The reduced computational complexity of the new formulation compared to 3D viscous analysis makes the airfoil design problem amenable to the use of formal optimization methods. The paper presents results from design of a lowpressure turbine nozzle. There are three primary contributions of this work: 

A numerical metric for emulating designer judgment in

Figure 59

Schematics of an airfoil showing stream lines along the radial direction

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 

evaluation of airfoil Mach number Distribution. An optimization formulation for design of airfoil sections, A methodology which allows design of 3D airfoils by simultaneous design of multiple 2D sections.

Designer heuristics are computed using curve fits and error norms. A set of penalty functions has been defined which allows for flexible constraint boundaries and influence constrained variables even within constraint limits. In the new approach multiple two-dimensional sections of the airfoil are designed with constraints on radial smoothness using polynomial fits on the parametric geometry variables in the radial direction (Figure 60)161. Airfoil design is a labor intensive, repetitive, and cumbersome task for the designers and is a bottleneck for both the design cycle and rapid generation of inputs for complex multistage analyses. Automating the design process significantly cuts down the design cycle time and facilitates the task of running multistage analysis by rapidly generating airfoil geometries. While designing an airfoil, it is hard to establish the existence of a unique optimum. Multiple evaluation criteria which are weighted together to define the objective function and the relative importance of these are determined based on designer experience. Furthermore, the analysis codes are not exact, and Figure 60 3D model of an even with precisely defined quality metrics, a significant margin of airfoil showing the passage error remains. In manual design the evaluation criteria are between adjacent airfoils implicitly considered by the designer, with weighting factors based on past experience and individual biases. Subjectivity is introduced into the design process since the evaluation criteria for the design are partially based on heuristics abstracted from designer experiences. Thus in order to completely understand the results of airfoil optimization, an evaluation of the qualitative changes to the design is essential after the optimization is completed. Over time as the metrics to evaluate airfoil design become more acceptable, a standard metric will emerge, till such time designers will need to tinker with the weights to suit their own preferences 162.

161 162

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6 Multi-Disciplinary Optimization (MDO) 6.1 Background The coupling schemes bring us to the essential subject of Multi-Disciplinary Optimization (MDO). The interdisciplinary coupling inherent in MDO tends to present additional challenges beyond those encountered in a single-discipline optimization163. It increases computational burden, and it also increases complexity and creates organizational challenges for implementing the necessary coupling in software systems. The increasing complexity of engineering systems has sparked increasing interest in multi-disciplinary optimization (MDO). The two main challenges of MDO are computational expense and organizational complexity. Accordingly the survey is focused on various ways different researchers use to deal with these challenges. The survey is organized by a breakdown of MDO into its conceptual components. Accordingly, the survey includes sections on Mathematical Modeling, Design-oriented Analysis, Approximation Concepts, Optimization Procedures, System Sensitivity, and Human Interface. With the increasing acceptance and utilization of MDO in industry, a number of software frameworks have been created to facilitate integration of application software, manage data, and provide a user interface with various MDO-related problem-solving functionalities. A list of frameworks that specialize in integration and/or optimization of engineering processes includes:     

iSIGHT (developed by Engineous Software), Model Center (developed by Phoenix Integration), Epogy (developed by Synaps), Infospheres Infrastructure (developed at the California Institute of Technology), DAKOTA (developed at Sandia National Laboratories).

And many others. An extensive evaluation of select frameworks has been performed at NASA Langley Research Center. The optimization problem is often divided or decomposed into separate suboptimizations managed by an overall optimizer that strives to minimize the global objectives. Examples of these techniques are Concurrent Optimization164, Collaborative Optimization165, and BiLevel System Synthesis166. Simpler optimization techniques, such as All-In-One optimization (in which all design variables are varied simultaneously) and sequential disciplinary optimization (in which each discipline is optimized sequentially) can lead to sub-optimal design and lack of robustness.

6.2 Computational Cost Associated with MDO

The increased computational burden may simply reflect the increased size of the MDO problem, with the number of analysis variables and of design variables adding up with each additional discipline. A case of tens of thousands of analysis variables and several thousands of design variables, reported in Berkes for just the structural part of an airframe design, illustrates the dimensionality of the MDO task one has to prepare for. Since solution times for most analysis and optimization algorithms Jaroslaw Sobieszczanski-Sobieskieski, Raphael T. Haftka, “Multidisciplinary Aerospace Design Optimization: Survey of Recent Developments”, AIAA 96-0711, 34th Aerospace Sciences Meeting and Exhibit, Reno, NV, 1995. 164 Sobieszczanski-Sobieski, J., "Optimization by Decomposition: A Step from Hierarchic to Non- hierarchic Systems", Proceedings, 2nd NASA/USAF Symposium on Recent Advances in Multidisciplinary Analysis and Optimization, Hampton, Virginia, 1988. 165 Braun, R.D., "Collaborative Optimization: An Architecture for Large-Scale Distributed Design", Ph.D. thesis, Stanford University, May 1996. 166 Sobieszczanski-Sobieski, J., Agte, J. and Sandusly, Jr., R., "Bi-Level Integrated System Synthesis (BLISS)", NASA/TM-1998-208715, NASA Langley Research Center, Hampton, Virginia, August 1998. 163

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increase at a super linear rate, the computational cost of MDO is usually much higher than the sum of the costs of the single-discipline optimizations for the disciplines represented in the MDO. Additionally, even if each discipline employs linear analysis methods, the combined system may require costly nonlinear analysis. For example, linear aerodynamics may be used to predict pressure distribution a wing, and linear structure analysis may be then used to predict is placement so waver, but the dependent pressure displacements may not be linear. Finally, for each disciplinary optimization we may be able to use as single-objective function, but for the MDO problem we may need to have multiple objective with an attendant increasing cost of optimizations167.

6.3 Organizational Complexity

In any type of MDO applications, the efficient solution of the problem depends greatly on the proper selection of a practical approach to MDO formulation. Six fundamental approaches are identified and compared by [Balling & Sobieszczanski-Sobieski]168: single-level vs. multilevel optimization, system-level simultaneous analysis and design vs. analysis nested in optimization, and discipline-level simultaneous analysis and design vs. analysis nested in optimization. From the results presented therein, two conclusions are apparent:  

No single approach is fastest for all implementation cases, and No single approach can be identified as being always the slowest.

Therefore, the choice of approach should be made only after careful consideration of all the factors pertaining to the problem at hand.

6.4 Clarification of Some Terminology

Although the terms Multi-Physics and Multi-Disciplinary are used interchangeably, but there are distinctive different. While Multi-Physics refers to the cases when one solver is used in different physics, multi-disciplinary is referred to the cases when two or more solver is used in different physics, and the information in shared coupling data from separate analysis packages. In essence, difference is the way data is obtained for optimization process.

6.5 Categories of MDO Analysis

One may detect three categories of approaches to MDO problems depending on the way the organization challenges has been addressed. Two of these categories represents approaches that concentrate on problem formulation that evade the organization challenge while the third deals with attempts to address this challenge directly. 1. The first category includes problems with two or three interacting disciplines where a single analyst can acquire all the required expertise (multi-physics). This may lead to MDO where design variables in several disciplines have to be obtained simultaneously to ensure efficient design. Most of the papers in this category represent a single group of researchers or practitioners working with a single computer program, so that organizational challenges were minimized. Because of this, it is easier for researchers working on problems in this category to deal with some of the issues of complexity of MDO problems, such as the need for multi-objective optimization.

Jaroslaw Sobieszczanski-Sobieskieski, Raphael T. Haftka, “Multidisciplinary Aerospace Design Optimization: Survey of Recent Developments”, AIAA 96-0711. 168 Balling, R. J., Sobieszczanski-Sobieskieski, J. “An Algorithm for Solving the System-Level Problem in Multilevel Optimization: Structural Optimization”, Springer Verlag, 1995, pp.168-177. 167

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2. The second category includes works where the MDO of an entire system is carried out at the conceptual level by employing simple analysis tools. For aircraft design, the ACSYNT and FLOPS programs represent this level of MDO application. Because of the simplicity of the analysis tools, it is possible to integrate the various disciplinary analyses in a single, usually modular, computer program and avoid large computational burdens. As the design process moves on, the level of analysis complexity employed at the conceptual design level increases uniformly throughout or selectively. Therefore, some of these codes are beginning to face some of the organizational challenges encountered when MDO is practiced at a more advanced stage of design process. 3. The third category of MDO research includes works that focus on the organizational and computational challenges and develop techniques that help address these challenges. These include decomposition methods and global sensitivity techniques that permit overall system optimization to proceed with minimum changes to disciplinary codes. These also include the development of tools that facilitate efficient organization of modules or that help with organization of data transfer. Finally, approximation techniques are extensively used to address the computational burden challenge, but they often also help with the organizational challenge. This, accordingly includes sections on Mathematical Modeling, Design-oriented Analysis, Approximation Concepts, Optimization Procedures, System Sensitivity, Decomposition, and Human Interface.

6.6 MDO Components Several conceptual components combined to form MDO. We attempt to cover the most important ones, namely the one by Sobieszczanski-Sobieski169 (SS) of NASA Langley Research Center, and of course the one envisioned by Wikipedia. They are listed and characterized in following section. 6.6.1 MDO Components as Environed by [Sobieszczanski-Sobieski]170 6.6.1.1 Mathematical Modeling of a System For obvious pragmatic reasons, software implementation of mathematical models of engineering systems usually takes the form of assemblages of codes of modules, where each module representing a physical phenomenon, a physical part, or some other aspect of the system. Data transfers among the modules correspond to the internal couplings of the system. These data transfers may require data processing that may become a costly overhead. For example, if the system is a flexible wing, the aerodynamic pressure reduced to concentrated forces at the aerodynamic model grid points on the wing surface has to be converted to the corresponding concentrated loads acting on the structure finite-element model nodal points. Conversely, the finite-element nodal structural displacements have to be entered into aerodynamic model grid as shape corrections. The volume of data transferred in such couplings affects efficiency directly in terms of I/O cost. Additionally, many solution procedures require the derivatives of this data with respect to design variables, so that a large volume of data also increases computational cost. To decrease these costs, the volume of data may be reduced by various condensation (reduced basis) techniques. For instance, in the wing example one may represent the pressure distribution and the displacement fields by a small number of base functions defined over the wing planform and transfer only the coefficients of these functions instead of the large volumes of the discrete load and displacement data. In some applications, one may identify a cluster of modules in a system model that exchange very large Jaroslaw Sobieszczanski-Sobieskieski, Raphael T. Haftka, “Multidisciplinary Aerospace Design Optimization: Survey of Recent Developments”, AIAA 96-0711. 170 Jaroslaw Sobieszczanski-Sobieski, “Multidisciplinary an Emerging New Discipline Design Optimization Engineering”, NASA Technical Memorandum 107761. 169

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volumes of data that are not amenable to condensation. In such cases, the computational cost may be substantially reduced by unifying the two modules, or merging them at the equation level. A heattransfer-structural-analysis code is an example of such merger. Here, the analyses of the temperature field throughout a structure and of the associated stress-strain field share a common finite-element model. This line of development was extended to include fluid mechanics. Because of the increased importance of computational cost, MDO emphasizes the tradeoff of accuracy and cost associated with alternative models with different levels of complexity for the same phenomena. In single-discipline optimization it is common to have an “analysis model” which is more accurate and more costly than an “optimization model”. 6.6.1.2 Trade Off between Accuracy and Cost in MDO In MDO, this tradeoff between accuracy and cost is exercised in various ways. First, optimization models can use the same theory, but with a lower level of detail. For example, the finite-element models used for combined aero elastic analysis of the high-speed civil transport are much more detailed than the models typically used for combined aerodynamic structural optimization. Second, models used for MDO are often less complex and less accurate than models used for a single disciplinary optimization. For example, structural models used for airframe optimization of the HSCT are substantially more refined than those used for MDO. Aircraft MDO programs, such as FLOPS and ACSYNT use simple aerodynamic analysis models and weight equations to estimate structural weight. Similarly, an equivalent plate model instead of a finite-element models for structures-control optimization of flexible wings. Third, occasionally, models of different complexity are used simultaneously in the same discipline. One of them may be a complex model for calculating the discipline response, and a simpler model for characterizing interaction with other disciplines. For example, in many aircraft companies, the structural loads are calculated by a simpler aerodynamic model than the one used for calculating aerodynamic drag. Finally, models of various levels of complexity may be used for the same response calculation in an approximation procedure or fast reanalysis described in the next two sections. Recent Aerospace industry emphasis on economics will, undoubtedly, spawn generation of a new category of mathematical models to simulate man-made phenomena of manufacturing and aerospace vehicle operation with requisite support and maintenance. These models will share at least some of their input variables with those used in the product design to account for the vehicle physics. This will enable one to build a system mathematical model encompassing all the principal phases of the product life cycle: desired formulation of product design, manufacturing, and operation. Based on such an extended model of a system, it will be possible to optimize the entire life cycle for a variety of economic objectives, e.g., minimum cost or a maximum return on investment. There are numerous references that bring the life cycle issues into the MDO domain and discussion on the role of MDO in the Integrated Product and Process Development (IPPD), also known as Concurrent Engineering (CE). Mathematical modeling of an aerospace vehicle critically depends on an efficient and flexible description of geometry. 6.6.1.3 Design-Oriented Analysis The engineering design process moves forward by asking and answering "what if" questions. To get answers to these questions expeditiously, designers need analysis tools that have a number of special attributes. These attributes are: selection of the various levels of analysis ranging from inexpensive and approximate to accurate and more costly, "smart" re-analysis which repeats only parts of the original analysis affected by the design changes, computation of sensitivity derivatives of output with respect to input, and a data management and visualization infrastructure necessary to handle large volumes of data typically generated in a design process. The term "Design oriented Analysis" refers to analysis procedures possessing the above attributes. Sensitivity analysis discussed previously, and the issue of the selection of analysis level was discussed in the previous section, and will be returned

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to in the next section on approximations. An example of a design-oriented analysis code is the program LS-CLASS developed for the structures-control-aerodynamic optimization of flexible wings with active controls. The program permits the calculation of aero-servo-elastic response at different levels of accuracy ranging from a full model to a reduced one based on vibration modes. Additionally, various approximations are available depending on the response quantity to be calculated. A typical implementation of the idea of smart re-analysis. The code (called PASS) is a collection of modules coupled by the output-to-input dependencies. These dependencies are determined and stored on a data base together with the archival input/output data from recent executions of the code. When a user changes an input variable and asks for new values of the output variables, the code logic uses the data dependency information to determine which modules and archival data are affected by the change and executes only the modules that are affected, using the archival data as much as possible. One may add that such smart reanalysis is now an industry standard in the spreadsheets whose use is popular on personal computers. It contributes materially to the fast response of these spreadsheets171. 6.6.1.4 Approximation Concepts Applicable to MDO Direct coupling of a Design Space Search (DSS) code to a multidisciplinary analysis may be impractical for several reasons. First, for any moderate to large number of design variables, the number of evaluations of objective function and constraints required is high. Often we cannot afford to execute such a large number of exact MDO analyses in order to provide the evaluation of the objective function and constraints. Second, often the different disciplinary analyses are executed on different machines, possibly at different sites, and communication with a central DSS program may become unwieldy. Third, some disciplines may produce noisy or jagged response as a function of the design variables172. If we do not use a smooth approximation to the response in this discipline we will have to degrade the DSS to less efficient non-gradient methods. For all of the above reasons, most optimizations of complex engineering systems couple a DSS to easy-to-calculate approximations of the objective function and/or constraints. The optimum of the approximate problem is found and then the approximation is updated by the full analysis executed at that optimum and the process repeated. This process of sequential approximate optimization is popular also in single-discipline optimization, but its use is more critical in MDO as the principal cost control measure. Most often the approximations used in engineering system optimization are local approximations based on the derivatives. Linear and quadratic approximations are frequently used, and occasionally intermediate variables or intermediate response quantities173 are used to improve the accuracy of the approximation. A procedure for updating the sensitivity derivatives in a sequence of approximations using the past data was formulated for a general case in Scotti174. Global approximations have also been extensively used in MDO. Simpler analysis procedures can be viewed as global approximations when they are used temporarily during the optimization process, with more accurate procedures employed periodically during the process. For example, Unger et al. 175 developed a procedure where Jaroslaw Sobieszczanski-Sobieskieski, Raphael T. Haftka, “Multidisciplinary Aerospace Design Optimization: Survey of Recent Developments”, AIAA 96-0711. 172 Same as previous. 173 Kodiyalam, S., and Vanderplaats, G. N., “Shape Optimization of 3D Continuum Structures via Force Approximation Technique”, AIAA J., Vol. 27, No. 9, 1989, pp. 1256–1263. 174 Scotti, S. J., “Structural Design Utilizing Updated Approximate Sensitivity Derivatives”, AIAA Paper No. 931531, April 19–21, 1993. 175 Unger, E.; Hutchison, M.; Huang, X.; Mason, W.; Haftka, R.; and Grossman, B. “Variable-Complexity Aerodynamic-Structural Design of a High-Speed Civil Transport”, Proceedings of the 4th AIAA/NASA/USAF/OAI Symposium on Multidisciplinary Analysis and Optimization, Cleveland, Ohio, September 21–23, 1992. AIAA Paper No. 92-4695. 171

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both the simpler and more sophisticated models are used simultaneously during the optimization procedure. The sophisticated model provides a scale factor for correcting the simpler model where the scale factor is updated periodically during the design process. Another global approximation approach that is particularly suitable for MDO is the response-surface technique. This technique replaces the objective and/or constraints functions with simple functions, often polynomials, which are fitted to data at a set of carefully selected design points. Neural networks are sometimes used to function in the same role. The values of the objective function and constraints at the selected set of points are used to “train” the network. Like the polynomial fit, the neural network provides an estimate of objective function and constraints for the optimizer that is very inexpensive after the initial investment in the net training has been made. 6.6.1.5 System Sensitivity Analysis In principle, sensitivity analysis of a system might be conducted using the same techniques that became well-established in the disciplinary sensitivity analyses. However, in most practical cases the sheer dimensionality of the system analysis makes a simple extension of the disciplinary sensitivity analysis techniques impractical in applications to sensitivity analysis of systems. Also, the utility of the system sensitivity data is broader than that in a single analysis. An algorithm that capitalizes on disciplinary sensitivity analysis techniques to organize the solution of the system sensitivity problem and its extension to higher order derivatives was introduced in Sobieszczanski-Sobieski176-177. There are two variants of the algorithm: one is based on the derivatives of the residuals of the governing equations in each discipline represented by a module in a system mathematical model, the other uses derivatives of output with respect to input from each module. So far operational experience has accumulated only for the second variant. That variant begins with computations of the derivatives of output with respect to input for each module in the system mathematical model, using any sensitivity analysis technique appropriate to the module (discipline). The module level sensitivity analyses are independent of each other, hence, they may be executed concurrently so that the system sensitivity task gets decomposed into smaller tasks. The resulting derivatives are entered as coefficients into a set of simultaneous, linear, algebraic equation, called the Global Sensitivity Equations (GSE), whose solution vector comprises the system total derivatives of behavior with respect to a design variable. Solvability of GSE and singularity conditions have been examined in [Sobieszczanski-Sobieski]178. It was reported that in some applications, errors of the system derivatives from the GSE solution may exceed significantly the errors in the derivatives of output with respect to input computed for the modules. The system sensitivity derivatives, also referred to as design derivatives, are useful to guide judgmental design decisions, or they may be input into an optimizer. Application of these derivatives extended to the second order in an application to an aerodynamic-control integrated optimization was reported in Ide et al. 179. A completely different approach to sensitivity analysis has been introduced. It is based on a neural net trained to simulate a particular analysis (the analysis may be disciplinary or of a multidisciplinary system). In [Sobieszczanski-Sobieski et al.]180 and

Sobieszczanski-Sobieski, J., “Sensitivity of Complex, Internally Coupled Systems”, AIAA Journal, Vol. 28, No. 1, 1990, pp. 153–160. 177 Sobieszczanski-Sobieski, J.; Barthelemy, J.-F.M.; and Riley, K. M., “Sensitivity of Optimum Solutions to Problems Parameters”, AIAA Journal, Vol. 20, No. 9, September 1982, pp. 1291–1299. 178 See previous. 179 Ide, H.; Abdi, F. F.; and Shankar, V. J., “CFD Sensitivity Study for Aerodynamic/Control Optimization Problems”, AIAA Paper 88-2336, April 1988. 180 Sobieszczanski-Sobieski, J. Barthelemy, J.-F.M., and Riley, K. M. “Sensitivity of Optimum Solutions to Problems Parameters”, AIAA Journal, Vol. 20, No. 9, September 1982, pp. 1291–1299. 176

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[Barthelemy and Sobieszczanski-Sobieski]181 the concept of the sensitivity analysis was extended to the analysis of an optimum, which comprises the constrained minimum of the objective function and the optimal values of the design variables, for sensitivity to the optimization constant parameters. The derivatives resulting from such analysis are useful in various decomposition schemes (next section), and in assessment of the optimization results as shown in [Braun et al.]182 6.6.1.6 Optimization Procedures with Approximations and Decompositions Optimization procedures assemble the numerical operations corresponding to the MDO elements [Sobieszczanski-Sobieski]183, into executable sequences. Typically, they include analyses, sensitivity analyses, approximations, design space search algorithms, decompositions, etc. Among these elements the approximations and decompositions most often determine the procedure organization, therefore, this section focuses on these two elements as distinguishing features of the optimization procedures. The implementation of MDO procedures is often limited by computational cost and by the difficulty to integrate software packages coming from different organizations. The computational burden challenge is typically addressed by employing approximations whereby the optimizer is applied to a sequence of approximate problems. The use of approximations often allows us to deal better with organizational boundaries. The approximation used for each discipline can be generated by specialists in this discipline, who can tailor the approximation to special features of that discipline and to the particulars of the application. When response surface techniques are used, the creation of the various disciplinary approximations can be performed ahead of time, minimizing the interaction of the optimization procedure with the various disciplinary software. Decomposition schemes and the associated optimization procedures have evolved into a key element of MDO. One important motivation for development of optimization procedures with decomposition is the obvious need to partition the large task of the engineering system synthesis into smaller tasks. The aggregate of the computational effort of these smaller tasks is not necessarily smaller than that of the original undivided task. However, the decomposition advantages are in these smaller tasks tending to be aligned with existing engineering specialties, in their forming a broad work front in which opportunities for concurrent operations (calendar time compression) are intrinsic, and in making MDO very compatible with the trend of computer technology toward multiprocessing hardware and software. Three basic optimization procedures have crystallized for applications in aerospace systems. The simplest procedure is piece-wise approximate with the GSE used to obtain the derivatives needed to construct the system behavior approximations in the neighborhood of the design point. In this procedure only the sensitivity analysis part of the entire optimization task is subject to decomposition (i.e., Grid Sensitivity, Aerodynamic Sensitivity, etc.), and the optimization is a singlelevel, encompassing all the design variables and constraints of the entire system. Hence, there is no need for a coordination problem to be solved. This GSE-based procedure has been used in a number of applications. The cost of the procedure critically depends on the number of the coupling variables for which the partial derivatives are computed. Disciplinary specialists involved in a design process generally prefer to control optimization in their domains of expertise as opposed to acting only as analysts. This preference has motivated development of procedures that extend the task partitioning to optimization itself. Barthelemy, J.-F; and Sobieszczanski - Sobieski, J., “Optimum Sensitivity Derivatives of Objective Functions in Nonlinear Programming”, AIAA Journal, Vol. 21, No. 6, June 1983, pp. 913–915. 182 Braun, R. D., Kroo, I. M., and Gage, P. Y.,”Post-Optimality Analysis in Aircraft Design”, Proceedings of the AIAA Aircraft, Design, Systems, and Operations Meeting, Monterey, California”, AIAA Paper No. 93-3932, 1993. 183 Sobieszczanski-Sobieski, J.,”Multidisciplinary Design Optimization: An Emerging, New Engineering Discipline”. In Advances in Structural Optimization, Herskovits, J., (ed.), pp. 483–496, Kluwer Academic, 1995. 181

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A procedure called the Concurrent Subspace Optimization (CSO) introduced in [SobieszczanskiSobieski]184 allocates the design variables to subspaces corresponding to engineering disciplines or subsystems. Each subspace performs a separate optimization, operating on its own unique subset of design variables. In this optimization, the objective function is the subspace contribution to the system objective, subject to the local subspace constraints and to constraints from all other subspaces. The local constraints are evaluated by a locally available analysis, the other constraints are approximated using the total derivatives from GSE. Responsibility for satisfying any particular constraint is distributed over the subspaces using "responsibility" coefficients which are constant parameters in each subspace optimization. Post optimal sensitivity analysis generates derivatives of each subspace optimum to the subspace optimization parameters. Following a round of subspace optimizations, these derivatives guide a system-level optimization problem in adjusting the "responsibility" coefficients. This preserves the couplings between the subspaces. The system analysis and the system– and subsystem level optimizations alternate until convergence. Another procedure proposed is known as the Collaborative Optimization (CO). Its application examples for space vehicles are for aircraft configuration. This procedure decomposes the problem even further by eliminating the need for a separate system and system sensitivity analyses. It achieves this by blending the design variables and those state variables that couple the subspaces (subsystems or disciplines) in one vector of the system-level design variables. These variables are set by the system level optimization and posed to the subspace optimizations as targets to be matched. Each subspace optimization operates on its own design variables, some of which correspond to the targets treated as the subspace optimization parameters, and uses a specialized analysis to satisfy its own constraints. The objective function to be minimized is a cumulative measure of the discrepancies between the design variables and their targets. The ensuing systemlevel optimization satisfies all the constraints and adjusts the targets so as to minimize the system objective and to enforce the matching. This optimization is guided by the above optimum sensitivity derivatives. Each of the above procedures applies also to hierarchic systems. A hierarchic system is defined as one in which a subsystem exchanges data directly with the system only but not with any other subsystem. Such data exchange occurs in analysis of structures by sub structuring. One iteration of the procedure comprises the system analysis from the assembled system level down to the individual system components level and optimization that proceeds in the opposite direction. The analysis data passed from above become constant parameters in the lower level optimization. The optimization results that are being passed from the bottom up include sensitivity of the optimum to these parameters. The coordination problem solution depends on these sensitivity data. The current practice relies on the engineer's insight to recognize whether the system is hierarchic, nonhierarchic, or hybrid and to choose an appropriate decomposition scheme185. 6.6.1.7 Human Factor MDO, definitely, is not a push-button design. Hence, the human interface is crucially important to enable engineers to control the design process and to inject their judgment and creativity into it. Therefore, various levels of that interface capability is prominent in the software systems that incorporate MDO technology and are operated by industrial companies. Because these software systems are nearly exclusively proprietary no published information is available for reference and to discern whether there are any unifying principles to the interface technology as currently implemented. However, from personal knowledge of some of these systems we may point to features 184 Sobieszczanski-Sobieski, Jaroslaw, “Optimization by Decomposition: A Step from Hierarchic to Non-Hierarchic

Systems”, Hampton, VA, September 28–30, 1988. NASA TM-101494. NASA CP-3031, Part 1, 1989. 185 Jaroslaw Sobieszczanski-Sobieskieski, Raphael T. Haftka, “Multidisciplinary Aerospace Design Optimization: Survey of Recent Developments”, AIAA 96-0711, 34th Aerospace Sciences Meeting and Exhibit, Reno, NV, 1995.

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common to many of them. These are flexibility in selecting dependent and independent variables in generation of graphic displays, use of color, contour and surface plotting, and orthographic projections to capture large volumes of information at a glance, and the animation. The latter is used not only to show dynamic behaviors like vibration but also to illustrate the changes in design introduced by optimization process over a sequence of iterations. One common denominator is the desire to support the engineer's train of thought continuity because it is well known that such continuity fosters insight that stimulates creativity. The other common denominator is the support the systems give to the communication among the members of the design team. In the opposite direction, users control the process by a menu of choices and, at a higher level, by meta-programming in languages that manipulate modules and their execution on concurrently operating computers connected in a network. One should mention at this point, again, the nonprocedural programming introduced in [Kroo and Takai]186. This type of programming may be regarded as a fundamental concept on which to base development of the means for human control of software systems that support design. This is so because it liberates the user from the constraints of a prepared menu of preconceived choices, and it efficiently sets the computational sequence needed to generate data asked for by the user with a minimum of computational effort. A code representative of the state of the art was developed by General Electric, Engineous, for support of design of aircraft jet engines187. 6.6.2 MDO Formulation as Depicted by Wikipedia Problem formulation is normally the most difficult part of the process. It is the selection of design variables, constraints, objectives, and models of the disciplines. A further consideration is the strength and extent of the interdisciplinary coupling in the problem. 6.6.2.1 Design Variables A design variable is a specification that is controllable from the point of view of the designer. For instance, the thickness of a structural member can be considered a design variable. Another might be the choice of material. Design variables can be continuous (such as a wing span), discrete (such as the number of ribs in a wing), or Boolean (such as whether to build a monoplane or a biplane). Design problems with continuous variables are normally solved more easily. Design variables are often bounded, that is, they often have maximum and minimum values. Depending on the solution method, these bounds can be treated as constraints or separately. 6.6.2.2 Constraints A constraint is a condition that must be satisfied in order for the design to be feasible. An example of a constraint in aircraft design is that the lift generated by a wing must be equal to the weight of the aircraft. In addition to physical laws, constraints can reflect resource limitations, user requirements, or bounds on the validity of the analysis models. Constraints can be used explicitly by the solution algorithm or can be incorporated into the objective using Lagrange multipliers. 6.6.2.3 Objective An objective is a numerical value that is to be maximized or minimized. For example, a designer may wish to maximize profit or minimize weight. Many solution methods work only with single objectives. When using these methods, the designer normally weights the various objectives and sums them to form a single objective. Other methods allow multi-objective optimization, such as the calculation of a Pareto front. Kroo, I.; and Takai, M., “A Quasi-Procedural Knowledge Based System for Aircraft Synthesis”, AIAA-88-6502, AIAA Aircraft Design Conference, August 1988. 187 Jaroslaw Sobieszczanski-Sobieskieski, Raphael T. Haftka, “Multidisciplinary Aerospace Design optimization: Survey of Recent Developments”, AIAA 96-0711, 34th Aerospace Sciences Meeting and Exhibit, Reno, NV, 1995. 186

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6.6.2.4 Models The designer must also choose models to relate the constraints and the objectives to the design variables. These models are dependent on the discipline involved. They may be empirical models, such as a regression analysis of aircraft prices, theoretical models, such as from computational fluid dynamics, or reduced-order models of either of these. In choosing the models the designer must trade off fidelity with analysis time. The multidisciplinary nature of most design problems complicates model choice and implementation. Often several iterations are necessary between the disciplines in order to find the values of the objectives and constraints. As an example, the aerodynamic loads on a wing affect the structural deformation of the wing. The structural deformation in turn changes the shape of the wing and the aerodynamic loads. Therefore, in analyzing a wing, the aerodynamic and structural analyses must be run a number of times in turn until the loads and deformation converge. 6.6.2.5 Simple Optimization Once the design variables, constraints, objectives, and the relationships between them have been chosen, the problem can be expressed in the following form:

Find x that minimizes F(x) : subject to g(x)  0, h(x)  0, and x LB  x  x UB Where F is an objective, x is a vector of design variables, g is a vector of inequality constraints, h is a vector of equality constraints, and xLB and xUB are vectors of lower and upper bounds on the design variables. Maximization problems can be converted to minimization problems by multiplying the objective by -1. Constraints can be reversed in a similar manner. Equality constraints can be replaced by two inequality constraints. 6.6.2.6 Problem Solution The problem is normally solved using appropriate techniques from the field of optimization. These include gradient-based algorithms, population-based algorithms, or others. Very simple problems can sometimes be expressed linearly; in that case the techniques of linear programming are applicable.

6.7 Approaches to MDO for Turbomachinery Engine Applications A significant amount of MDO research has been conducted in the field of turbomachinery design. A number of reports have been published presenting the development of optimization environments, optimization methods, and procedures for turbine engine design188. Particular aspects of multidisciplinary optimization for different turbomachinery design stages are investigated by [Dornberger et al.]189. We describing the ongoing work related to the development and implementation of a MDO environment with a focus on its application to the conceptual design of the gas turbine engine. The effective introduction of MDO at the conceptual and preliminary design stage depends on adopting the appropriate strategy, as discussed before. Other requirements include adequate information infrastructure and robust design-oriented analysis tools. The use of high fidelity analyses has always been part of the detailed levels of design. The benefits of effective

Y. Panchenko, H. Moustapha, S. Mah, K. Patel, M.J. Dowhan, D. Hall, “Preliminary Multi-Disciplinary Optimization in Turbomachinery Design”, ADA415759. 189 Dornberger, R., Buch, D. and Stoll, P., "Multidisciplinary in Turbomachinery Design", Presented at the European Congress on Computational Methods in Applied Sciences and Engineering, September 11-14, 2000, Barcelona. 188

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inclusion of high fidelity data into the design optimization process at the conceptual stage have been investigated in190. 6.7.1 Overall Design Process Every design must be grounded in sound physical principles that are grouped into categories named disciplines191. Figure 61 illustrates the hierarchical breakdown of an engine into different engineering disciplines that govern the design of major engine components that, in turn, combine to make the final product. The process of engine design starts at the aircraft level. An engine is a system that seamlessly integrates into the larger system of an aircraft. Engine design is a top down procedure in which two processes, design and manufacturing, start and proceed from opposite ends of the system configuration. The design process starts at the overall system level and gradually moves down to the component level. The manufacturing process proceeds in the opposite direction. Traditionally, the design of the gas turbine engine follows three major phases: Conceptual Design, Preliminary Design, and Detailed Design that involves designing for manufacturing and assembly. Here, the application of MDO methodology to the conceptual stage of the design cycle will be referred to as Preliminary Multi-Disciplinary Design Optimization (PMDO). The aim is to explore the conceptual phase of the design process which involves the exploration of different concepts that satisfy engine

Figure 61

Product, components and the supporting disciplines

Lytle, J.K., "The Numerical Propulsion System Simulation: A Multidisciplinary Design System for Aerospace Vehicles", ISABE paper No. ISABE 99-7111, 1999. 191 Ryan, R., Blair, J., Townsend, J. and Verderaime, V., "Working on the Boundaries: Philosophies and Practices of the Design Process", NASA Technical Paper 3642, July 1996. 190

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design specifications and requirements. The interaction that takes place among the disciplines is a series of feedback loops and trades between conflicting requirements imposed on the system. 6.7.2 Single Discipline Optimization Optimization with a single tool has been investigated for two cases: axial compressor gas paths and turbine gas paths. In each of these cases, the tool has been linked with an optimizer and successful optimization runs have been accomplished. The purpose of the single discipline investigations was to:  Become familiar with the characteristics of various optimization methods  Determine the best optimization methods for each tool  Ensure that the selected tools are robust enough for use in optimization  Explore the effect of alternate sets of optimization variables on convergence and robustness of the solution The optimizer used is iSIGHT©, developed by Engineous Software192. The iSIGHT software is a generic shell environment that supports multidisciplinary optimization. The shell represents and manages multiple elements of a particular design problem in conjunction with the integration of one or more simulation programs. In essence, iSIGHT automates the execution of the different codes (in-house or commercial), data exchange and iterative adjustment of the design parameters based on the problem formulation and a specified optimization plan. 6.7.3

Aerodynamic Design Optimization for Turbomachinery The gas turbine design is a sequential and highly iterative process that is represented by a net of tightly coupled engineering disciplines as depicted in Figure 62 where a close-up view of the process that takes place within the discipline of aerodynamics was also shown. The aerodynamic characteristics of multi-stage axial compressors and turbines are predicted using 1D mean line programs. Flow prediction in a mean line program is based on the calculation of velocity triangles at the mid-span of the gas path with empirical models to account for losses. Further information on mean line programs and loss models is available in 193-194. Typical input to a mean line program includes geometric parameters and engine operating

Figure 62

Aerodynamic design process for Turbomachinery

iSIGHT V5.5 User's Guide, Engineous Software, Inc. Raw, J. A. and Weir, G. C., "The Prediction of Off - Design Characteristics of Axial and Axial / centrifugal Compressors ", SAE Technical Paper Series 800628, April, 1980. 194 Kacker, S.C. and Okapuu, U., “A Mean Line Prediction Method for Axial Flow Turbine Efficiency”, Journal of Engineering for Power; Vol. 104, Jan. 1982. 192 193

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conditions. The output from a mean line program includes a prediction of Mach numbers, pressure ratio and efficiency. Simple "layout" programs are used to predict the aerodynamic characteristics and geometric cross-sections of fans and centrifugal compressors. These programs are based on simple physics, design rules, and audits of previous engines. Losses in ducts such as the engine inlet, bypass duct, and inter-compressor ducts are modeled using either (i) simple correlations with geometric parameters and basic engine operating conditions as input, or (ii) the numerical solution of one-dimensional flow equations with calibrated source terms for blockages such as struts. In the traditional design process, these empirical correlations, "rules of thumb", and calibrated models have been applied manually195. 6.7.3.1 Axial Compressor Gas path Optimization A three-stage axial compressor optimization case was run at design point using mean line program with the following optimization variables:    

Shape of the hub and shroud Location and corner points of each rotor and stator Number of airfoils per blade row Airfoil angles

Constraints were imposed on the following variables:       

Diffusion factor Swirl angle at stator trailing edges Exit Mach number Ratio of hub to tip radius Blade angles Pressure ratio Choked flow

The objective of the optimization was to maximize efficiency. The optimization was run for approximately 1000 iterations which took about 1 hour on an HPC-class workstation using a Genetic Algorithm followed by a Direct Heuristic Search. The number of iterations required to achieve an optimum seems excessive and several opportunities are being explored to reduce the iteration count:  

alternate optimization strategies, and alternate sets of optimization variables based on "physical" quantities.

The iSIGHT optimizer has a suite of explorative and gradient-based optimization methods that can be applied in any sequence. Different combinations of optimization methods will be investigated in an attempt to improve the efficiency of the optimization process. The design variables used by the optimizer are expected to have a significant influence on the robustness and speed of optimization. In the current axial compressor mean line application, the optimizer alters the gas path shape by varying the coefficients of splines representing the hub and shroud curves. The dependence of the compressor pressure ratio and efficiency on the spline coefficients is not direct. An improved set of "physical" optimization variables has been suggested in which the optimizer varies axial Y. Panchenko, H. Moustapha, S. Mah, K. Patel, M.J. Dowhan, D. Hall, “Preliminary Multi-Disciplinary Optimization in Turbomachinery Design”, ADA415759. 195

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distributions of mean radius and area. The advantage of this formulation is that area and radius are "physical" variables that have a direct link to the pressure ratio and efficiency predicted by the mean line program. This direct link should result in a "cleaner" design space, a reduced number of iterations to converge to an optimal solution, and improved robustness of the optimization procedure. 6.7.3.2 Turbine Gas path Optimization A three-stage turbine optimization case was run with a mean line program in which the optimization variables included the number of airfoils per blade row, the location and cross-sectional shape of each blade and vane, and the shape of the hub and shroud. The only constraint on the output parameters was to keep the Zweifel Coefficient, which is a measure of airfoil loading, constant. The objective of the optimization was to maximize efficiency and minimize the Degree of Reaction which represents the proportion of the static temperature drop occurring in the rotor and, also, reduction in total relative temperature which results in a lower metal temperature for the airfoil. The optimization plan involved three optimization techniques available in the iSIGHT software: Genetic Algorithm followed by Hooke-Jeeves Direct Search Method followed by Exterior Penalty technique. The results of the optimization run were compared with "baseline" results, as shown in Figure 63. The baseline results were obtained by a turbine design expert in fraction of a day of. In contrast, the optimizer took twenty minutes to set up and two hours and Figure 63 Comparison of "baseline" and "optimized" turbine mean line results twenty minutes to run. The baseline solutions are shown as dotted lines in the Figure 63 and the optimizer solutions as solid lines. The gas path shape and number of airfoils per blade row obtained by the optimizer were close to the baseline results. The efficiencies were almost identical with slightly higher efficiencies obtained by the optimizer. Of most significance is order of magnitude reduction in human time required to obtain the solution196. 6.7.4 Concluding Remarks Multidisciplinary optimization (MDO) involves the simultaneous optimization of multiple coupled disciplines and includes the frequently conflicting requirements of each discipline. MDO is an active field of research and several methods have been proposed to handle the complexities inherent in systems with a large number of disciplines and design variables197. MDO can be described as an environment for the design of complex, coupled engineering systems, such as a gas turbine engine, the behavior of which is determined by interacting subsystems. It attempts to make the life cycle of a product and the design process less expensive and more reliable198. The optimization problem is

Y. Panchenko, H. Moustapha, S. Mah, K. Patel, M.J. Dowhan, D. Hall, “Preliminary Multi-Disciplinary Optimization in Turbomachinery Design”, ADA415759. 197 Sobieszczanski-Sobieski, J. and Haftka, R. T., "Multidisciplinary Aerospace Design Optimization: Survey of Recent Developments, Structural Optimization", AIAA Paper 96-0711, Jan. 1996. 198 See 99. 196

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often divided into separate sub-optimizations managed by an overall optimizer that strives to minimize the global objective. Examples of these techniques are Concurrent Sub-Space Optimization199, Collaborative Optimization200, and Bi-Level System Synthesis201. Simpler optimization techniques, such as All-In-One optimization (in which all design variables are varied simultaneously) and sequential disciplinary optimization (in which each discipline is optimized sequentially) can lead to sub-optimal design and lack of robustness. MDO eases the process of design and improves system performance by ensuring that the latest advances in each of the contributing disciplines are used to the fullest, taking advantage of the interactions between the subsystems. Although the potential of MDO for improving the design process and reducing the manufacturing cost of complex systems is widely recognized by the engineering community, the extent of its practical application is not as great as it should be due to the shortage of easily applied MDO tools.

Sobieszczanski-Sobieski, J., "Optimization by Decomposition: A Step from Hierarchic to Non-hierarchic Systems", Proceedings, 2nd NASA/USAF Symposium on Recent Advances in Multidisciplinary Analysis and Optimization, Hampton, Virginia, 1988. 200 Braun, R.D., "Collaborative Optimization: An Architecture for Large-Scale Distributed Design", Ph.D. thesis, Stanford University, May 1996. 201 Sobieszczanski-Sobieski, J., Agte, J. and Sandusly, Jr., R., "Bi-Level Integrated System Synthesis (BLISS)", NASA/TM-1998-208715, NASA Langley Research Center, Hampton, Virginia, August 1998. 199