MSc_Thesis_Mariens.

MSc_Thesis_Mariens.
Faculty of Aerospace Engineering
Wing Shape Multidisciplinary Design
Optimization
Jan Mariens
August 2, 2012
Wing Shape Multidisciplinary Design
Optimization
Master of Science Thesis
For obtaining the degree of Master of Science in Aerospace Engineering at
Delft University of Technology
Jan Mariens
August 2, 2012
Faculty of Aerospace Engineering
·
Delft University of Technology
Delft University of Technology
c Jan Mariens
Copyright All rights reserved.
Delft University Of Technology
Department Of
Flight Performance and Propulsion
The undersigned hereby certify that they have read and recommend to the Faculty of Aerospace Engineering for acceptance a thesis entitled “Wing Shape Multidisciplinary Design Optimization”
by Jan Mariens in partial fulfillment of the requirements for the degree of Master of Science.
Dated: August 2, 2012
Head of Department:
Dr. ir. Dries Visser
Supervisor:
Ali Elham, MSc.
Reader one:
Prof. dr. ir. Egbert Torenbeek
Reader two:
Dr. ir. Roelof Vos
Summary
Multidisciplinary design optimizations have shown great benefits for aerospace applications in the
past. Especially in the last decades with the advent of high speed computing. Still computational
time limits the desire for models with high level of fidelity cannot be always fulfilled. As a consequence, fidelity is often sacrificed in order to keep the computing time of the optimization within
limits. There is always a compromise required to select proper tools for an optimization problem.
In this final thesis work, the differences between existing weight modeling techniques are investigated. Secondly, the results of using different weight modeling techniques in multidisciplinary design
optimization of aircraft wings is compared. The aircraft maximum take-off weight was selected as
the objective function. The wing configuration of a generic turboprop and turbofan passenger
aircraft were considered for these optimizations. This should aid future studies of wing shapes in
early design stages to select a proper weight prediction technique for a given case. A quasi-threedimensional aerodynamic solver was developed to calculate the wing aerodynamic characteristics.
Various statistical prediction methods (low level of fidelity) and a quasi-analytical method (medium
level of fidelity) are used to estimate the structural wing weight. Furthermore, the optimal wing
shape was found using a local optimization algorithm and is compared to the results found using a
novel optimization algorithm to find the global optimum.
The quasi-three-dimensional aerodynamic solver was validated using experimental data and other
available aerodynamic tools. Compared to the results generated by other tools, the developed
solver has a wider range of validity. Most important of all, it is up to 10 times faster and the results
show good agreement with other data. Several test cases were used to prove the robustness and
effectiveness of the global optimization algorithm. A comparison of the different weight estimation
methods indicated that the lower level fidelity methods are insensitive for some wing parameters.
The results of the optimizations showed that the optimum wing shape is affected by the used weight
modeling technique. Use of different weight prediction methods strongly affects the computational
times and the convergence history. The global optimization algorithm was able to find the global
solution for the wing shape optimization. However, the search for the global optimum comes at a
cost: the computational time is significantly larger.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
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Jan Mariens
Summary
Wing Shape Multidisciplinary Design Optimization
Acknowledgements
This graduate thesis report is written as part of a Master Program in Aircraft Design at the Faculty
of Aerospace Engineering – Delft University of Technology. The report forms the end of my time as
an Aerospace Engineering student. Looking back at the last months, I realize that there a number
of people who made all this possible.
First of all, I would like to thank my supervisor Ali Elham, for his invaluable advise, expertise and
excellent support during the past year. Furthermore, I would also like to thank professor Egbert
Torenbeek for the interesting discussions we had that gave me a lot of new ideas and second
thoughts. I also would like to thank the members of my committee: Dries Visser, Ali Elham,
Egbert Torenbeek and Roelof Vos.
Many thanks to all of my friends and (ex) fellow students of room NB2.40, whom made it besides
a busy period also very pleasant one. A very special thanks goes to my parents, who have always
supported me in everything I did and encourage my dreams. Finally, I would like to address a special
person, my girlfriend Catherine for her continuing support, involvement and encouragements.
Jan Mariens
Deflt, July 2012
iv
Jan Mariens
Acknowledgements
Wing Shape Multidisciplinary Design Optimization
Contents
Summary
i
Acknowledgements
iii
List of Figures
ix
List of Tables
xiii
Nomenclature
xv
1 Introduction
1
2 Thesis background
2-1 Aircraft design process
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3
2-2 Role of MDO in aircraft design . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-2-1 MDO strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
2-2-2 Comparison of MDO strategies . . . . . . . . . . . . . . . . . . . . . . . .
2-2-3 Optimization algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-3 Wing design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
11
12
2-3-1
2-3-2
2-3-3
Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Aerodynamic flow models . . . . . . . . . . . . . . . . . . . . . . . . . . .
Numerical methods for solving the fluid flow models . . . . . . . . . . . . .
13
13
15
2-3-4
2-3-5
Aerodynamic methods comparison . . . . . . . . . . . . . . . . . . . . . .
Structural weight estimation methods . . . . . . . . . . . . . . . . . . . .
19
20
3 Local optima smoothing for global optimization
3-1 Optimization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-2 An algorithm for local optima smoothing . . . . . . . . . . . . . . . . . . . . . . .
23
23
24
3-2-1 Local optima smoothing principle . . . . . . . . . . . . . . . . . . . . . . .
3-2-2 The LOCSMOOTH framework . . . . . . . . . . . . . . . . . . . . . . . .
3-3 Test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
26
30
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vi
Contents
3-3-1
Ackley’s function (unconstrained)
. . . . . . . . . . . . . . . . . . . . . .
30
3-3-2
Rastrigin’s function (unconstrained) . . . . . . . . . . . . . . . . . . . . .
31
3-3-3
Rosenbrock’s function (unconstrained) . . . . . . . . . . . . . . . . . . . .
32
3-3-4
Schwefel’s function (unconstrained) . . . . . . . . . . . . . . . . . . . . .
32
3-3-5
Minimum induced drag of a wing . . . . . . . . . . . . . . . . . . . . . . .
34
4 Quasi-3D aerodynamic solver
39
4-1 Strip method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
4-2 Simple sweep theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
4-3 Wing taper implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4-4 Quasi-3D aerodynamic solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
4-5 Aerodynamic tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4-6 Selection panel density and number of strips . . . . . . . . . . . . . . . . . . . . .
47
4-6-1
Vortex lattice grid size . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
4-6-2
Number of strips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
4-7 Validation of the quasi-3D aerodynamic solver at low speeds . . . . . . . . . . . .
52
4-7-1
NACA 24-30-0 wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
4-8 Validation of the quasi-3D aerodynamic solver at high speeds . . . . . . . . . . . .
56
4-8-1
Drag coefficients comparison . . . . . . . . . . . . . . . . . . . . . . . . .
56
4-8-2
Pressure distribution comparison . . . . . . . . . . . . . . . . . . . . . . .
57
5 Wing weight estimation methods
61
5-1 Different methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-1-1 Torenbeek (1) method . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
61
5-1-2
Torenbeek (2) method . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
5-1-3
5-1-4
5-1-5
5-1-6
Shevell method . . . .
Howe method . . . . .
LTH method . . . . . .
Elham Modified Weight
.
.
.
.
63
64
65
66
5-2 Method comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
Estimation Technique
. . . . . . .
. . . . . . .
. . . . . . .
(EMWET) .
.
.
.
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5-2-1
Accuracy analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
5-2-2
Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
6 MDO of aircraft wings
73
6-1 Objective function, design vector and constraints . . . . . . . . . . . . . . . . . .
74
6-2 MDO strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
6-3 MDO results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-4 Additional constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-5 MDO results using additional constraints . . . . . . . . . . . . . . . . . . . . . . .
78
79
80
6-6 MDO results using the LOCSMOOTH algorithm . . . . . . . . . . . . . . . . . . .
82
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Contents
vii
7 Conclusions & recommendations
7-1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
85
86
Bibliography
89
A SQP algorithm
a
B Psuedo-code of LOCSMOOTH algorithm
c
C Taper implementation for simple sweep theory
e
D Validation quasi-3D aerodynamic solver at low speed
g
D-1 NACA24-0-0 wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
g
D-2 Tapered NACA24-15-0 wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i
D-3 NACA 24-30-85 airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
k
E Quasi-3D aerodynamic solver inputs and outputs
m
F Multidisciplinary design optimization modules
o
F-1 Weight module (We) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
o
F-2 Aerodynamic module (Ae) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p
F-3 Performance module (Pe) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
q
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viii
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Contents
Wing Shape Multidisciplinary Design Optimization
List of Figures
2-1 Schematic illustrating the difference between the design phases. (Adapted from [1])
5
2-2 Aircraft design processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2-3 Multidisciplinary feasible method . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2-4 Individual discipline feasible method . . . . . . . . . . . . . . . . . . . . . . . . .
7
2-5 Collaborative Optimization method . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2-6 Concurrent Subspace Optimization method . . . . . . . . . . . . . . . . . . . . .
10
2-7 Bi-Level Integrated System Synthesis method . . . . . . . . . . . . . . . . . . . .
11
2-8 Difference between Local minimum and Global optimum . . . . . . . . . . . . . .
12
2-9 Different types of drag, for both subsonic and supersonic flight regimes [14]. . . . .
14
2-10 Hierarchy of fluid flow models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2-11 Finite Difference mesh and examples of a 2D and 3D stencil . . . . . . . . . . . .
16
2-12 Finite Volume mesh and examples of a 2D and 3D volume blocks . . . . . . . . .
16
2-13 Finite Element mesh and examples of a 2D and 3D element
. . . . . . . . . . . .
17
2-14 Lifting-line model consisting of multiple horseshoe vortices . . . . . . . . . . . . .
18
2-15 Schematic of a lifting surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2-16 Multi-fidelity model in function of CFD method and geometry detail . . . . . . . .
19
3-1 Example of optimization function with a funnel structure. . . . . . . . . . . . . . .
24
3-2 Local and global optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3-3 Normal distribution curve (Gaussian), with standard deviation σ = 1.
. . . . . . .
26
3-4 Gaussian filtered step function L(x), with different standard deviations σ. . . . . .
26
3-5 Roadmap of the LOCSMOOTH algorithm. . . . . . . . . . . . . . . . . . . . . . .
29
3-6 3D plot of the Ackley’s function for n = 2 variables. . . . . . . . . . . . . . . . . .
30
3-7 A zoom near the origin of the Ackley’s function for n = 2 variables. . . . . . . . .
31
3-8 3D plot of the Rastrigin’s function for n = 2 variables. . . . . . . . . . . . . . . .
32
3-9 3D plot of the Rosenbrock’s function for n = 2 variables. . . . . . . . . . . . . . .
33
3-10 3D plot of the Schwefel’s function for n = 2 variables. . . . . . . . . . . . . . . .
33
3-11 Starting geometry of the wing with 5 bays, top view. . . . . . . . . . . . . . . . .
34
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
x
LIST OF FIGURES
3-12 Lift distribution on the Trefftz’s plane for the initial rectangular wing. . . . . . . .
35
3-13 Discontinuity in induced drag coefficient CDi calculated by AVL, fixed CL = 0.2. .
36
3-14 Lift distribution on the Trefftz’s plane of optimized wing using SQP. . . . . . . . .
37
3-15 Lift distribution on the Trefftz’s plane of optimized wing using LOCSMOOTH algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-16 Initial and optimized wing configurations. . . . . . . . . . . . . . . . . . . . . . .
37
38
4-1 Three-dimensional and local aerodynamic forces of a strip . . . . . . . . . . . . . .
40
4-2 Simple sweep theory of an infinite wing (untapered wing) . . . . . . . . . . . . . .
41
4-3 Friction and pressure drag forces on a swept tapered wing (adopted from [49]) . .
43
4-4 Definition of the forces and angles used to determine the inviscid downwash angle .
44
4-5 Overview of the sweep theory implementation in the strip method for one strip . .
46
4-6 Half wing with different grid sizes . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
4-7 Effect of number of spanwise (N b) and chordwise (N c) vortices on normalized
induced drag coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4-8 Effect of number of spanwise (N b) and chordwise (N c) vortices on computation
time. Each level indicates the computational time seconds. . . . . . . . . . . . . .
50
4-9 Effect of number of spanwise and chordwise elements on normalized induced drag
coefficient (separate plots). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4-10 Effect of number of strips (N w) on normalized profile drag coefficient. . . . . . . .
50
4-11 Effect of number of strips (N w) on computational time. . . . . . . . . . . . . . .
51
4-12 Graphical representation of the wing configurations used for low speed validation of
the quasi-three-dimensional aerodynamic solver . . . . . . . . . . . . . . . . . . .
53
4-13 CL − α curve of the NACA 24-30-0 wing . . . . . . . . . . . . . . . . . . . . . .
54
4-15 CD − CL curve of the NACA 24-30-0 wing . . . . . . . . . . . . . . . . . . . . .
55
4-14 CD − α curve of the NACA 24-30-0 wing . . . . . . . . . . . . . . . . . . . . . .
54
4-16 Drag differences of NACA 24-30-0 wing for quasi-3D solver and MATRICSV . . . .
55
4-17 CD versus Mach comparison with MATRICSV . . . . . . . . . . . . . . . . . . . .
56
4-18 CD difference between AVL-MSES/AVL-VGK and MATRICSV . . . . . . . . . . .
57
4-19 α difference between AVL-MSES/AVL-VGK and MATRICSV . . . . . . . . . . . .
57
4-20 Upper surface pressure coefficient distributions from MATRICSV (full potential 3D)
and VGK (full potential 2D) with MSES shockwave locations along span . . . . .
58
4-21 Minimum Cp distribution along span . . . . . . . . . . . . . . . . . . . . . . . . .
59
4-22 Pressure coefficient distribution comparison for three wing sections . . . . . . . . .
60
5-1 Statistical wing weight correlation (adopted from [31]). . . . . . . . . . . . . . . .
64
5-2 Howe method – actual and predicted wing masses for civil aircraft [69]. . . . . . .
65
5-3 Validation results of the LTH method [70]. . . . . . . . . . . . . . . . . . . . . . .
66
5-4 Actual total weight of the wing versus the analytically computer wing box weight
(ribs and non-optimum weight excluded) [34]. . . . . . . . . . . . . . . . . . . . .
67
5-5 Wing parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
5-6 Sensitivity analysis for turboprop aircraft. . . . . . . . . . . . . . . . . . . . . . .
71
5-7 Sensitivity analysis for turbojet aircraft. . . . . . . . . . . . . . . . . . . . . . . .
72
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
LIST OF FIGURES
xi
6-1 Turboprop and turbojet aircraft test cases. . . . . . . . . . . . . . . . . . . . . . .
73
6-2 Design structure matrix of MDO system with aerodynamics, performance and weight
modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-3 Normal flight mission definition with flight segments. . . . . . . . . . . . . . . . .
75
77
6-4 Planform geometry of the optimized wings. . . . . . . . . . . . . . . . . . . . . .
79
6-5 Sensitivity of the wing weight (calculated using EMWET) and the fuel weight with
respect to the wing span. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
6-6 Planform geometry of the optimized wings using new set of constraints. . . . . . .
81
6-7 Optimization convergency.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
6-8 Difference MDO results using SQP and LOCSMOOTH. . . . . . . . . . . . . . . .
83
C-1 Airfoil section perpendicular to sweep line of a tapered wing . . . . . . . . . . . .
e
C-2 Front part of the airfoil section perpendicular to sweep line . . . . . . . . . . . . .
f
C-3 Upper curve airfoil shape determination based on interpolation of coordinates . . .
f
D-1 CL − α curve of the NACA 24-0-0 wing . . . . . . . . . . . . . . . . . . . . . . .
g
D-2 CD − α curve of the NACA 24-0-0 wing . . . . . . . . . . . . . . . . . . . . . . .
h
D-3 CD − CL curve of the NACA 24-0-0 wing . . . . . . . . . . . . . . . . . . . . . .
h
D-5 CD − α curve of the NACA 24-15-0 wing . . . . . . . . . . . . . . . . . . . . . .
i
D-4 CL − α curve of the NACA 24-15-0 wing . . . . . . . . . . . . . . . . . . . . . .
i
D-6 CD − CL curve of the NACA 24-15-0 wing . . . . . . . . . . . . . . . . . . . . .
j
D-7 CL − α curve of the NACA 24-30-85 wing . . . . . . . . . . . . . . . . . . . . . .
k
D-9 CD − CL curve of the NACA 24-30-85 wing . . . . . . . . . . . . . . . . . . . . .
l
D-8 CD − α curve of the NACA 24-30-85 wing . . . . . . . . . . . . . . . . . . . . . .
Wing Shape Multidisciplinary Design Optimization
k
Jan Mariens
xii
Jan Mariens
LIST OF FIGURES
Wing Shape Multidisciplinary Design Optimization
List of Tables
2-1 CFD methods comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3-1 Numerical solutions of Ackley’s function with sphere radii ri = (ub − lb)/5, LOCSMOOTH algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
3-2 Numerical solutions of Rastrigin’s function, LOCSMOOTH algorithm. . . . . . . .
32
3-3 Numerical solutions of Rosenbrock’s function, LOCSMOOTH algorithm. . . . . . .
33
3-4 Numerical solutions of Schwefel’s function, number of sampled points K = 5, sphere
radii ri = (ub − lb)/2, LOCSMOOTH algorithm. . . . . . . . . . . . . . . . . . .
34
3-5 Starting aerodynamics of isolated wing. . . . . . . . . . . . . . . . . . . . . . . .
35
3-6 Wing optimization results for 5 bay geometry using SQP algorithm. . . . . . . . .
36
3-7 Optimized chord lengths of rectangular wing for minimum induced drag using the
SQP algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3-8 Wing optimization results for 5 bay geometry using LOCSMOOTH algorithm. . . .
36
3-9 Optimized chord lengths of rectangular wing for minimum induced drag. . . . . . .
37
4-1 NACA wing characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
4-2 Error analysis of different aerodynamic solver for the wing drag coefficient of the
tapered NACA 24-30-0 wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
4-3 Computational time per case . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
5-1 Estimated versus actual wing weight for several transport aircraft [65]. . . . . . . .
63
5-2 Coefficient C5 in equation (5-7). . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
5-3 Range of values for using the LTH method . . . . . . . . . . . . . . . . . . . . . .
66
5-4 Validation results of EMWET. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-5 Wing weight estimation method sensitivity to wing parameters. . . . . . . . . . . .
67
69
5-6 Wing weight estimation errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
6-1 Design vector for turboprop aircraft. . . . . . . . . . . . . . . . . . . . . . . . . .
74
6-2 Design vector for turbojet aircraft. . . . . . . . . . . . . . . . . . . . . . . . . . .
74
6-3 Flight conditions of test cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
xiv
LIST OF TABLES
6-4 Fuel fraction for each segment in simple flight mission, suggested values from
Roskam [72]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
6-5 Results of wing optimization for turboprop aircraft. . . . . . . . . . . . . . . . . .
78
6-6 Results of wing optimization for turbojet aircraft. . . . . . . . . . . . . . . . . . .
78
6-7 Results of wing optimization for turboprop aircraft using new set of constraints. . .
80
6-8 Results of wing optimization for turbojet aircraft using new set of constraints. . . .
81
6-9 MDO computational times using the new constraint set. . . . . . . . . . . . . . .
82
6-10 Comparison MDO results of turbojet aircraft with 1 constraint using SQP and LOCSMOOTH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-11 Comparison MDO results of turbojet aircraft with 3 constraints using SQP and
LOCSMOOTH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-12 MDO Computational times with EMWET using SQP and LOCSMOOTH. . . . . .
82
82
83
D-1 Error analysis of different aerodynamic solver for the wing drag coefficient of the
tapered NACA 24-0-0 airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
h
D-2 Error analysis of different aerodynamic solver for the wing drag coefficient of the
tapered NACA 24-15-0 airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
j
D-3 Error analysis of different aerodynamic solver for the wing drag coefficient of the
tapered NACA 24-30-85 wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
l
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Nomenclature
Latin symbols
(t/c)
A
b
bs
c̄
c
Cd
CD
CD c
CD i
CD f
CD p
CDprof
Cp
CL
d
D
g
h
K
l
L
M
Mff
nult
Nb
Nc
Nw
r
Airfoil thickness-to-chord ratio
Aspect ratio of the wing
Span width
Structural span
Mean aerodynamic chord
Chord length
Local two-dimensional drag coefficient
Drag coefficient
Compressibility (or wave) drag coefficient
Induced drag coefficient
Friction drag coefficient
Pressure drag coefficient
Profile drag coefficient
Local two-dimensional pressure coefficient
Lift coefficient
Local two-dimensional drag
Aerodynamic drag force
Gravitational acceleration
Flight altitude
Number of observations
Local two-dimensional lift coefficient
Aerodynamic lift force
Mach number
Mass fuel fraction
Ultimate load factor
Number of spanwise vortices
Number of chordwise vortices
Number of wing sections
Radius
Wing Shape Multidisciplinary Design Optimization
[-]
[-]
[m]
[m]
[m]
[m]
[-]
[-]
[-]
[-]
[-]
[-]
[-]
[-]
[-]
[N/m]
[N]
[m/s2 ]
[m]
[-]
[N/m]
[N]
[-]
[-]
[-]
[-]
[-]
[-]
[m]
Jan Mariens
xvi
Nomenclature
Re
S
Sref
V
V
W
x
Reynolds number
Wing planform area
Reference surface area
Velocity
Volume
Weight
Design vector
[-]
[m2 ]
[m2 ]
[m/s]
[m3 ]
[N] or [kg]
[-]
Greek symbols
α
αeff
ǫ
η
λ
µ
ρ
σ
ξc
Λ
Angle of attack
Effective angle of attack
Twist angle at local wing section
Dimensionless spanwise position from root till tip (0. . . 1)
Wing taper
Dynamic viscosity
Air density
Standard deviation
Constant chord percentage of sweep line
Sweep angle
[◦ ]
[◦ ]
[◦ ]
[-]
[-]
[kg/s/m]
[kg/m3 ]
[-]
[-]
[◦ ]
Subscripts
⊥
∞
av
ave
c/2
c/4
calc
des
eff
f
f
g
i
in
k
out
p
prof
r
Jan Mariens
Normal to sweep line (constant chord percentage line over wing)
Free-stream
Available
Average
Mid-chord
Quarter-chord
Calculated
Design
Effective
Fuel
Friction
Gaussian
Induced
Inner wing
Kink
Outer wing
Pressure
Profile
Root
Wing Shape Multidisciplinary Design Optimization
Nomenclature
ref
rest
t
to
w
zf
xvii
Reference
Rest
Tip
Take-off
Wing or wave (compressibility)
Zero-fuel
Abbreviations
AAO
BLISS
CAD
CFD
CO
CPU
CSSO
DNS
FDM
FEM
FVM
GSE
IDF
LLT
LOCSMOOTH
NS
MDA
MDF
MDO
RANS
SQP
VLM
All-at-once
Bi-Level Integrated System Synthesis Method
Computer Aided Design
Computations Fluid Dynamics
Collaborative Optimization
Central Processing Unit
Concurrent Subspace Optimization
Direct Numerical Solution
Finite Difference Method
Finite Element Method
Finite Volume method
Global sensitivity equation
Individual Design Feasible
Lifting-Line Theory
Local Optima Smoothing for Global Optimization
Navier-Stokes
Multidisciplinary Design Analysis
Multidisciplinary Design Feasible
Multidisciplinary Design Optimization
Reynolds-averaged Navier-Stokes
Sequential Quadratic Programming
Vortex Lattice Method
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
xviii
Jan Mariens
Nomenclature
Wing Shape Multidisciplinary Design Optimization
Chapter 1
Introduction
Aircraft wing design using Multidisciplinary Design Optimization (MDO) techniques is a complex
task which involves different disciplines, mainly aerodynamic and structure. Different levels of
analysis are used for wing design and optimization. Typically simple empirical methods are used
in the earliest stages of the concept design. The design task proceeds towards the final design by
increasing the complexity of the analysis methods. For instance, a variety of methods are available
for aerodynamic analysis of a wing; from a simple lifting line theory or a vortex lattice method
up to complex Euler and Reynolds-Average Navier-Stokes methods. Similarly for structural weight
estimation, various methods with different levels of fidelity are available. The difficulty lies in the
quest or development of analysis methods which are sufficiently simple to be used thousands of
times during the optimization. At the same time, these methods should be sophisticated enough
to capture changes in the local geometry. In this chapter, the thesis objective is stated, followed by
the approach used to answer the thesis objective.
As different analysis methods come with different levels of fidelity, this can influence the optimization
results using multidisciplinary design optimization techniques. Therefore, the main objective of this
thesis is to:
Investigate the effect of using different weight estimation methods on the outcome
in a wing design task using multidisciplinary design optimization techniques.
This main objective is accompanied by the following sub-goals:
• Develop a quasi-three-dimensional aerodynamic solver to calculate the wing aerodynamic
characteristics.
• Compare the different weight estimation methods, with low and medium levels of fidelity, by
analyzing their accuracy and sensitivity to wing parameters.
• Implement a global optimization algorithm that uses gradient-based techniques and local
optima smoothing. The results using this algorithm are also compared to a local solution.
The thesis context of this research is discussed in Chapter 2. This context presents a brief discussion
on aircraft design processes, the role of MDO in aircraft design and wing design. In Chapter 3,
an optimization algorithm to find the global solution is presented. Several test cases were used
to show the effectiveness and robustness of this algorithm. The development of the quasi-threedimensional aerodynamic solver is presented in Chapter 4. Different weight prediction methods are
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
2
Chapter 1. Introduction
discussed and compared to each other in Chapter 5. Consequently, Chapter 6 presents an aircraft
wing shape design task using MDO techniques. In this design task, different weight estimation
methods were used and the optimization results were compared. The MDO results generated using
a local optimization algorithm are presented and compared to the results found using the global
optimization algorithm. Finally, in Chapter 7 the conclusions and recommendations of this thesis
research are presented.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 2
Thesis background
Based on the thesis objectives, a preliminary research was done. This research serves as background
information for this thesis. First, the different phases of aircraft design processes are discussed in
Section 2-1. Multidisciplinary design optimization (MDO) has shown great benefits for aerospace
applications in the past decades. In Section 2-2 the role of multidisciplinary design optimization
(MDO) in aircraft design processes is discussed. The objective of this thesis concerns a wing design
task using MDO techniques. In Section 2-3, the different aerodynamic methods for calculating
the aerodynamic properties of a wing are discussed together with different wing weight prediction
techniques.
2-1
Aircraft design process
The complete aircraft design process goes through three distinct phases that are carried out in
a sequence. These phases are, in chronological order: conceptual design, preliminary design and
detailed design. Discrimination between the three above-mentioned design phases is related to the
differences in activities, tools, amounts of people and expertise, time scales, etc. that take place in
each process.
Conceptual design
The design process starts with a set of specifications (mission requirements) for a new aircraft.
There is a rather concrete goal to which the designers are aiming. The first steps towards achieving
that goal constitute the conceptual design phase. Here, within a certain design freedom, the
overall shape, size, weight and performance of the new design are determined. The product of
the conceptual design phase is a layout (on paper or computer) of the aircraft configuration. This
concept might still be slightly changed during the second design phase. However, the conceptual
design phase determines such fundamental aspects as the the shape of the wings (swept back,
forward sweep or straight), the location of the wings relative to the fuselage, shape and location
of the horizontal and vertical tail, use of canard surface or not, etc. The major drivers during the
conceptual design process are aerodynamics, propulsion, weights and flight performance [1]. Typical
questions designers will have to answer in this phase can be:
• What is actually driving the design?
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
4
Chapter 2. Thesis background
• What are the most critical requirements?
• Can the design meet the specifications?
• Which of the possible aircraft configurations has the highest potential?
These questions are answered in the conceptual design phase by using tools primarily from aerodynamics, structures, propulsion and flight performance. No part of the design process is ever carried
out in a total vacuum unrelated to the other parts.
Preliminary design
The preliminary design phase starts at that point when the major changes in the design solutions
are over. The preliminary phase uses the baseline configuration that was elaborated and selected
during the conceptual phase. The purpose of this phase is to further develop and mature the
baseline design, until sufficient understanding (with confidence) of the design quality is achieved.
At that point, the design can be frozen and the detail design phase can start. In the preliminary
design phase, only minor changes are made to the conceptual design. Questions such as wether
to use a canard or an aft tail have been resolved. If major changes were demanded during this
phase, the conceptual design process would have been flawed to begin with. The purpose of this
phase is to further develop and mature the baseline design, until sufficient understanding (with
confidence) of the design quality is achieved. During this phase major computational fluid dynamic
(CFD), structural and stress calculations of the aircraft configuration will be made. Additionally,
substantial wind tunnel testing will be carried out. It is possible that the wind tunnel tests and/or
the CFD calculations will uncover some undesirable aerodynamic interference, or some unexpected
stability problems, which will promote changes to the configuration layout. The drawing process
called “lofting” is carried out which mathematically models the precise shape of the outside skin of
the aircraft, making certain that all sections of the aircraft fit together. Lofting is a term adopted
from ship design, where shipbuilders designed the shape of the hull in the loft (an area located above
the shipyard floor). At the end of the preliminary design phase, the configuration is frozen and
precisely defined. Moreover, the end of this phase brings a major decision – to commit the aircraft
to the manufacturer or not. The importance of this decision for modern aircraft manufacturers
cannot be understated, considering the tremendous costs involved in the design and manufacture
of a new aircraft.
Detailed design
The detail design phase is literally the “nuts and bolts” phase of aircraft design. The aerodynamic,
structural, propulsion, performance and flight control analyses have all been finished at the preliminary design phase. In this phase, the design of each individual spar, web, skin, panels, etc. can now
take place. The size, number and location of fasteners (rivets, joints, etc.) are determined. Manufacturing tools and jigs are designed. At the end of this phase, the aircraft is ready to be fabricated.
Figure 2-1 is intended to visualize, in a very simple manner, the distinction between the products
of the three design phases in aircraft design. The product of conceptual design is represented in
Figure 2-1a. Here, the basic configuration of the aircraft is determined within a certain (hopefully
small) “fuzzy” latitude. Figure 2-1b shows the product of the preliminary design phase (with precise
dimensions). Finally, the product of the detailed design is represented in Figure 2-1c. Here, the
precise fabrication details are determined, represented by the rivets sizes and locations.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 2. Thesis background
“Fuzzy” configuration
definition
5
Precise configuration
definition
(a) Product of conceptual design
(b) Product of preliminary design
Rivet size and
locations
(c) Product of detailed design
Figure 2-1: Schematic illustrating the difference between the design phases. (Adapted from [1])
2-2
Role of MDO in aircraft design
The traditional approach in aircraft design consists roughly out of 20% creative work and 80%
repetitive work [2]. The repetitive work load is too high compared to the creative work in engineering.
How can the designers productivity be improved while simultaneously reducing the whole design
process duration? Multidisciplinary design optimization (MDO) provides a good solution to improve
this. In this section, it is explained why this methodology provides a good solution and what different
strategies there are. This methodology is also compared to the traditional process in aircraft design,
and what improvements it could bring.
Multidisciplinary Design Optimization or MDO, is a methodology for the design of systems in which
strong interaction between disciplines motivates designers to simultaneously manipulate variables
in several disciplines [3]. The interdisciplinary coupling is inherent in MDO and shows tougher
computational and organizational challenges than single-discipline optimization. The goal of MDO,
according to Kroo [4], is to provide a more consistent, formalized method for complex system design
than is found in traditional approaches.
Aircraft design optimizations are complex processes because they contain a lot of design variables
and interdisciplinary operations. Here, the use and necessity of MDO in aircraft design is highlighted.
The traditional aircraft design process for conventional airplanes is shown in Figure 2-2a. In order
to cope with a more complex design procedure, the need of a different design process arises. A
new methodology is needed, that is able to facilitate the application of new technologies for aircraft
design. This methodology should also allow the user to acquire more design freedom and more
knowledge about the design during the conceptual design phase. Figure 2-2b shows graphically the
difference between the traditional design process and the preferred process. In order to establish
this methodology, the new design process will enhance MDO techniques.
2-2-1
MDO strategies
The MDO architectures those are used to solve a problem, can be divided into two classes: monolithic formulations and multilevel formulations [6]. The term architecture refers to how the multidisciplinary system is decomposed and the optimization formulation employed to meet design
requirements. Monolithic formulations, which among others include the multidisciplinary design
feasible (MDF), individual discipline feasible (IDF) and all-at-once (AAO), use a single-system level
optimizer for the whole problem. In all single level system architectures, the design task (also-called
the decision authority) is in the hands of the optimizer, that means that the optimizer controls
the design variables [2]. These approaches are the most straightforward to implement for small
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
6
Chapter 2. Thesis background
(a) Traditional aircraft design process
(b) Preferred aircraft design process
Figure 2-2: Aircraft design processes. (Source: [5])
problems. However, for real life design processes, it is often harder to implement these approaches,
which is partly due to the presence of centralized decision authority (a single system optimizer
that controls all the design variables). As the system scale increases, this problem becomes more
apparent [2].
Multilevel architectures such as collaborative optimization (CO), concurrent subspace optimization
(CSSO) and bi-level integrated systems synthesis (BLISS) make use of subspace optimization to
promote discipline autonomy [6]. This means that the system-level optimizer is then responsible
for managing the interactions between the discipline optimizations. Multilevel strategies are more
advanced compared to single system-level because they employ disciplinary optimizers in addition
to the system optimizer.
MDO can also be defined as a methodology together with a set of tools for assistance in the design
of complex coupled systems, that is, systems whose behavior is governed by many distinct but
interacting physical phenomena [4].
Initial applications of multidisciplinary design optimization involve the direct integration of multiple
disciplinary analyses and an optimizer [7] .
Multidisciplinary Feasible Design
The multidisciplinary feasible design (MDF) has the simplest formulation for solving MDO problems. The MDF formulation links a multidisciplinary design analysis (MDA) with an optimizer (see
Figure 2-3) to find the optimal global z and local variables x, for a given objective function and
constraints. The system optimizer guides the MDA and reaches a multidisciplinary feasible state
for an entire set of disciplines. In this method, the word feasible refers to consistency. A disadvantage of this approach is that the solution of the system could be expensive and it does not exploit
the potentially weak coupling between some of the disciplines that would enable the division into
different analyses modules that might run in parallel [8]. The MDF approach can be stated as:
min
z,x
subject to
f (z, x, y (x, y, z))
c (z, x, y (x, y, z)) ≤ 0
where f represents the objective function and c all the global and local system constraints.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 2. Thesis background
7
Optimizer
z, x
f, c
Multidisciplinary analysis
Discipline 1
b
b
Discipline 2
b
b
Discipline 3
Figure 2-3: Multidisciplinary feasible method
Individual discipline feasible
The main idea of the individual discipline feasible strategy is to use the optimizer to enforce interdisciplinary compatibility. Instead of iterating the multidisciplinary analysis to converge the coupling
variables y, these coupling variables are given by the optimizer as a guess, y ′ . The optimization
problem can then be stated as follows,
f z, y x, y ′ , z
c z, y x, y ′ , z ≤ 0
min′
z,x,y
subject to
y ′ − y(x, y ′ , z) = 0
The number of design variables has increased and equals the number of original design variables plus
the number of coupling variables. On the one side this increases the size of the optimization problem,
but on the other side it conveniently decouples all the analyses. This decoupling enables it to solve
the problem in parallel without intercommunication. Note that if the optimizer is gradient-based,
the gradient ∂f /∂y ′ and ∂c/∂y ′ must also be calculated. This method is particularly advantageous
for cases where there is a small number of coupling variables.
Optimizer
f, c, y ′ − y
z, x, y ′
Disciplinary analyses
Discipline 1
Discipline 2
Discipline 3
Figure 2-4: Individual discipline feasible method
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
8
Chapter 2. Thesis background
All-At-Once
The all-at-once architecture (also known as simultaneous analysis and design – SAND) goes a step
beyond IDF. It decomposes the multidisciplinary problem further by setting the governing equation
for each discipline as equality constraints in the optimization problem. The AAO strategy can be
written as a single optimization problem:
min
z,x,y
subject to
f (z, x, y (z, x))
c (z, x, y (z, x)) ≤ 0
Ri (z, xi , yi (z, x, yi )) = 0
i = 1, . . . , n
where n represents the number of disciplines and Ri the residuals of the governing equations for each
discipline. In general, this architecture is impractical for MDO that involves large sets of governing
equations due to the excessive number of design variables that it adds to the optimization problem
[6]. Another drawback to this architecture is that it evaluates the residuals of the analysis equations,
rather than solving some set of equations [9].
Collaborative Optimization
The collaborative optimization architecture is designed to provide disciplinary autonomy while
achieving interdisciplinary compatibility. The optimization problem is decomposed into a number of independent optimization subproblems, each corresponding to a discipline. The objective of
each subproblem is to agree on the values of the coupling variables with the other disciplines. Each
discipline is given control over its design variables and is responsible for satisfying its constraints.
The CO formulation at system level can be stated as:
min
z,y
subject to
f (z, y)
Ji∗ (z, z ∗ , y, y ∗ (x∗i , y, zi∗ )) = 0,
i = 1, . . . , n
where Ji∗ represents the measure of the interdisciplinary discrepancy for the i-th discipline after
solving the disciplinary subproblem:
min
z,yi ,zi
subject to
Ji (z, zi , y, yi (xi , y, zi )) =
n
X
i=1
c (xi , zi , yi (xi , y, zi )) ≤ 0
(z − zi )2 +
n
X
i=1
(y − yi )2
Figure 2-5 shows the CO method. From this figure it can be seen that there is no direct communication for each discipline with other disciplines to enforce the governing equations. Albeit that
CO does has several benefits, it also comes with some drawbacks. As the number of coupling
variables increases, the dimensionality of the system level problem increases as well as the number
of variables involved with the calculation of the system level compatibility constraints. For this
reason, CO tends to be most effective on problems having a low dimensionality of coupling.
Concurrent Subspace Optimization
The Concurrent Subspace Optimization (CSSO) is also a decomposition-based strategy that allows
for the disciplines to run decoupled from each other. Also for this approach, the multiple subspace
optimization problems are driven by a system-level optimizer that provides overall coordination.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 2. Thesis background
9
System Level Optimizer
z, y ′
b
b
J1∗
J2∗
Optimizer
z, y ′ , x
J3∗
Optimizer
z, y ′ , x
y1
Discipline 1
Discipline 1
y2
Optimizer
z, y ′ , x
Discipline 2
Discipline 1
y3
Discipline 3
Discipline 1
Figure 2-5: Collaborative Optimization method
Each subproblem in CSSO uses approximations to non-local disciplinary coupling variables to estimate the influence of these variables on the system-level objective and constraints. The subspace
optimization problem for the i-th discipline is given by
min
f (z, yi (xi , ỹj , zi ) , ỹj )
z,xi ,yi ,ỹj
subject to
c (xi , z, yi (xi , ỹj , zi )) ≤ 0
where ỹj = ỹj (z, xj ) represent the approximations to coupling variables (or states) of the other
disciplines. The system-level optimizer solves the following problem,
min
f (z, ỹ)
z,ỹ
subject to
c (xi , z, yi (xi , ỹj , zi )) ≤ 0
After each iteration of the system-level optimizer, a multidisciplinary analysis (MDA) is performed
to update the model which gives the approximate response of all coupling variables ỹ.
Bi-Level Integrated System Synthesis
The bi-level integrated system synthesis (BLISS) method is a decomposition of the global sensitivity
equations (GSE) method. It calculates the total derivative of the coupling values y with respect
to local sensitivities. Each discipline is optimized by varying their local variables x, while holding
the global variables z constant and simultaneously minimizing the disciplinary objective under local
constraints. The global variables are utilized by the system level optimization only. The total
derivatives (obtained from GSE) are used to predict the effects of each set of variables on the
objective function. The optimization of the i-th discipline takes the form:
min
subject to
d (f, xi )T ∆xi
gi (xi ) ≤ 0
where d (f, xi )T is the local derivative of the objective function with respect to the local variables
and disciplines. It includes the indirect effects of these variables on other disciplines. The term
d (f, xi )T ∆xi corresponds to the first order predicted objective function change caused by the
change in xi . The system level objective in the BLISS formulation is strongly related to the objective
functions of the disciplines and it is expressed in terms of a first order Taylor series expansion:
min
subject to
Φ = d (y1,i , xi )T ∆x1 + d (y1,i , x2 )T ∆x2 + d (y1,i , x3 )T ∆x3 + . . .
g (z, y (x, z) , x)
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
10
Chapter 2. Thesis background
System Analysis
Model Update
Approximation
b
System 1
Approximation
System 2
Approximation
System 3
Approximation
Optimizer
Optimizer
Optimizer
System 1
Analysis
System 2
Analysis
System 3
Analysis
b
Model Update
System Approx.
Sys. Optimizer
Figure 2-6: Concurrent Subspace Optimization method
2-2-2
Comparison of MDO strategies
As for any optimization problem, the choice of the MDO strategy (method) depends strongly on
the problem that is to be solved. It is almost impossible to say in advance which strategy should be
used, but nevertheless there are some guidelines which are helpful to decide which strategy would
be most likely better to use.
Multilevel strategies are often better suited for use in large complex design problems, because
they allow distribution decision authority. However, the multilevel strategies are more prone to
convergence issues. The following aspects have been identified that are important for successful
implementation in large-scale projects [10]. An ideal MDO strategy should have the following
characteristics (based on [2]):
• Disciplinary design autonomy: The strategy should allow the use of available expertise and
legacy design tools. Local decision authority should be respected.
• Flexibility: The strategy should be easily adaptable to a specific organization.
• Mathematical rigor: The strategy should yield reliable and consistent results, and the optimality of the results should be provable.
• Efficiency: The strategy should lead to an optimal solution in a minimum number of iterations
and it should minimize the design time (e.g. by the concurrency of tasks).
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 2. Thesis background
11
System Analysis
b
Subsystem 1
Analysis
Subsystem 2
Analysis
Subsystem 3
Analysis
b
Sys. Sensitivity
Analysis (GSE)
b
Optimizer
Optimizer
Optimizer
Discipline
Evaluation 1
Discipline
Evaluation 2
Discipline
Evaluation 3
b
Sys. Derivative
Calculation
Optimizer
Variables
Update
Figure 2-7: Bi-Level Integrated System Synthesis method
During the discussion of the MDO strategies, the optimizer itself was treated as a black box. However, the choice of the optimizer also influences the success of the MDO strategy. The optimization
algorithm decides how to move through the design space.
2-2-3
Optimization algorithms
Optimization methods can often yield good designs and sometimes even optimal designs. This
rather because they provide a systematic framework for considering some of the many decisions
designers are charged with taking. Therefore it is very helpful when looking at computational
approaches for designing to have a thorough understanding of the various classes of optimization
methods. Also for how they work and more importantly, where they are likely to fail.
When bringing together multiple disciplines in one single computational environment, including
the subject of design search and optimization, attention must be paid on the subject of formal
optimization methods and their underlying theory. The (numerical) optimization algorithms can be
divided into two main categories, the gradient-based and non gradient-based algorithms. In this
subsection, these algorithms are discussed. Furthermore, there also exist combinations of gradientbased and non gradient-based, so-called hybrid methods.
Gradient-based algorithms
The gradient-based optimization methods use gradient information of the objective functions to
drive the design into direction of improvement. Using the gradient information of the objective
functions, the direction of the steepest descent can be determined. Then a single move or an entire
line search is performed in the identified steepest direction obtained from the gradient information
of the objective function. Following this, the direction of the steepest descent is again determined
and the process repeated until optimization criteria is satisfied.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
12
Chapter 2. Thesis background
A major drawback of these methods is that they always (directly) steer towards a local minimum,
while the local minimum might not be the same as the global optimum (as illustrated in Figure 2-8).
Only convergence to a local minimum is guaranteed [11]. Therefore, it is of great importance to
chose the initial points in the vicinity of the global minimum. Although, for complex multidisciplinary
design optimization processes (such as aircraft design problems) it is not straightforward where this
global optimum is to be found within the design space. The method requires existence of continuous
first derivatives of the objective function and possibly higher derivatives [11]. The technique used
to obtain the gradient information is very important as it has a a large influence the required
number of function evaluations for the optimization [2]. Examples of these techniques are steepest
descent, conjugate gradients, Newton methods, Quasi-Newton methods, etc. A drawback of the
gradient-based method is the inability to explore the design space. This implies that many of the
gradient-based methods yield good performance in theory, but less good in practice [12]. This
method is significantly faster than non-gradient based algorithms and easier to implement.
f
Local minimum
Global minimum
x
Figure 2-8: Graphical representation of difference between Local minimum and Global optimum.
Non-gradient algorithms
Non-gradient algorithms overcome the problem related to the complexity of the computation of the
partial derivatives of the objective functions. Therefore they are more suitable for global optimization
problems. In contrary with the gradient-based methods, this method is able to explore the whole
design space and (depending on optimization criteria) to find the global optimum but requires
a large number of function evaluations [11]. Usually it is one or two orders of magnitude more
expensive (and slower) than gradient-based [13]. Another disadvantage of non-gradient algorithms
is that they usually show slow convergence. For aerospace applications, this method has been
proven to effective but time-consuming.
2-3
Wing design
In this research an optimization of aircraft wings is conducted. The different methods to predict
the wing aerodynamics and to calculate the structural wing weight, are discussed in this section.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 2. Thesis background
2-3-1
13
Aerodynamics
Aerodynamic drag
Drag is the aerodynamic force that opposes an aircraft’s motion through the air. The aim of
aerodynamicists (aerospace engineers) always has been to minimize the drag by direct and in-direct
methods. The drag force is mathematically defined as
D = 21 ρCD V 2 S
(2-1)
where V is the aircraft’s velocity, ρ the free-stream air density, Sref the reference surface area and
CD the dimensionless drag coefficient. Drag is what drives the aerodynamic design because it is
a measure of how much power is required from propulsion to overcome this drag during its cruise
stage (from Eq. (2-1), it can be deduced that the drag increases quadratically with the velocity).
Drag is thus directly related the amount of power required, the fuel consumed and resultantly the
overall weight of the aircraft.
The drag coefficient CD is a dimensionless term that quantifies the drag or resistance a particular
aircraft has. The lower this coefficient, the ”cleaner” the design. Next, an overview of the different
types of drag is given.
Types of drag
The drag forces can be divided into subsonic drag and supersonic drag. Figure 2-9 shows a schematic
overview of how the different types of drag are divided. An additional drag type that occurs in
supersonic and transonic flights is the so-called wave drag. Wave drag is the pressure drag that
arrises due to the effect of expansion and compressibility waves due to the body shape. In subsonic
flight, the drag may be divided into two major categories: profile drag and induced drag. The
profile drag can be subdivided into skin friction drag, parasite drag and pressure drag. Induced
drag is caused by lift producing surfaces such as the wing. Those parts of the aircraft that do not
produce lift produce non-lift dependent drag also known as parasite drag.
These different drag types form the total drag and can the be written in the following mathematical
coefficient form
CD = CDprof + CDi + CDw
(2-2)
where CDprof , CDi and CDw are respectively the profile drag, induced drag and wave drag coefficients. The profile drag is a direct function of aircraft wetted area and its aerodynamic contouring.
Meanwhile, induced drag account for the losses associated with lift generation.
2-3-2
Aerodynamic flow models
The most fundamental basis of computational fluid dynamics (CFD) problems are the Navier-Stokes
(NS) equations. The NS equations completely describe the aerodynamics of a fluid (except for the
chemical-reaction effects at high temperatures). Though the NS seem straightforward enough,
they cannot be analytically solved for any useful flow conditions [15]. While ”direct numerical
solution” (DNS) codes are the beginning to solve the full NS equations for simplified geometries
and conditions. For aircraft design however, there are currently no practical codes to solve the full
NS equations. This is mainly due to the difficulty in mathematically analyzing the aerodynamic
phenomena turbulence.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
14
Chapter 2. Thesis background
associated
with airfoils
Drag =
Supersonic
Profile Drag
due to viscous
pressure drag
induced change
of pressure
+
distribution, in
2D d'Alembert's skin friction
drag
paradox says
this is zero
+ Induced Drag
sometimes
called form drag,
associated with
form factors
due to lift
generated
vorticity shed
into wake
laminar or turb,
a big difference
+ Wave Drag
drag due to generation
of shock waves
wave drag wave drag
due to
due to
lift
volume
additional
profile drag
also known as
should
profile drag
independent
parasite drag,
be small
due to lift
of lift
associated
(the drag from 2D at
with entire
cruise
airfoils at lift)
aircraft polar,
and may include
component
interference drag
drag due to lift
zero lift drag
Figure 2-9: Different types of drag, for both subsonic and supersonic flight regimes [14].
Several simplifications can be made to the NS equations. The Reynolds-Averaged Navier-Stokes
(RANS) equations are the time-averaged equations of motion for fluid flows which models the
turbulence statistically. RANS codes are used for many projects to solve particular design problems
where no other methods provide correct results. Unfortunately, it still is really expensive to setup
and run the RANS codes.
By ignoring all viscous terms and assuming steady flow, the Euler equations can be derived from
the NS equations. Euler codes are cheaper in use for simulations (runs) than RANS codes, and
are used quite often. Moreover, the inviscid assumption performs well outside the boundary layer.
They also can handle vortex formations, and by adding a separate boundary-layer code, they can
also realistically estimate viscous and separation effects.
The Euler equations can be simplified further by ignoring rotational terms, yielding the potential
flow equations. This simplification prevents the analysis of vortex flow, which is important at higher
angle of attacks and of less importance at cruise conditions. The potential flow equations have the
advantage that they can handle transonic shock formation which makes it very useful for transonic
design compared to the linearized methods. Although these equations imply some simplifications,
they are widely used in aerodynamic codes that treat the entire flow field instead of just the surface
conditions [16].
The linearized potential flow equations are obtained by neglecting the higher-order terms in the
potential flow equations. Due to the linearization, these equations do not perform well at transonic
speeds since it neglects the non-linear terms.
Figure 2-10 shows the hierarchy of the above mentioned aerodynamic solvers. In order to solve the
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 2. Thesis background
15
hy
Flo
wP
4. RANS (1990s)
+ Rotation
Co
m
2. Potential flow (1970s)
sin
g
+ Non-linear Formulation
s
ost
ple
xit
y, M
ore
Ac
cu
3. Euler (1980s)
C
al
ion
tat
pu
om
rat
e
+ Viscosity
C
ng
asi
cre
De
sic
s
partial differential equations of the aerodynamic models, use can be made of numerical solution
techniques.
In c
rea
1. Linearized potential flow (1960s)
+ Inviscid, Irrotational, Linear Formulation
Figure 2-10: Hierarchy of fluid flow models. (Source: [17])
2-3-3
Numerical methods for solving the fluid flow models
The governing equations (partial differential equations) from the aerodynamic models can be solved
using numerical solution techniques. The techniques differ for nonlinear of linear simulations, as
will be explained in the next sections. Besides the linear and non-linear methods, semi-empirical
models (handbook models and DATCOM) can also be used [18]. Note that non-linear methods
usually require more detailed input geometries than linear methods.
Nonlinear ’field’ methods
The nonlinear CFD methods can be used for predicting complex flow fields, such as those associated
with transonic or separated flows. The nonlinear CFD methods are used in RANS, Euler and
(full) potential solvers. The numerical methods that are used most frequently for nonlinear CFD
simulations are the finite volume method (FVM), the finite element method (FEM) and the finite
difference method (FDM). Among these, there are several more available techniques, although
the three techniques mentioned have the broadest applicability (about 95%) [19]. These spatial
discretization techniques will be briefly discussed below. FEM, FDM and FVM are called ”field
methods”, because they all discretize the whole fluid domain (field).
Finite difference method Historically, this method is the oldest of the three. The finite difference
method is the method that is most used and it was the first one applied to the numerical solution
of differential equations. This method is directly applied to the differential form of the governing
equations. The first and second derivatives of the differential equations are approximated by a
difference formula that can be (easily) derived from a Taylor series expansion [20, 21]. An important
advantage of the finite difference methodology is its simplicity for numerical implementation [19].
Another advantage is the possibility to obtain high-order approximations and thus achieving highorder accuracies of the spatial discretization [22]. On the other hand, the method requires a
structured grid, which restricts the range of application. And there is no conservation of momentum,
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
16
Chapter 2. Thesis background
energy and mass on coarse grids [19]. Figure 2-11 shows an example of a finite difference mesh
and a corresponding stencil (scheme) that can be used for discretization of the nonlinear fluid flow
equations.
i,j+1
i-1,j
b
b
i,j
2D stencil
b
b
b
i,j-1
i+1,j
b
b
b
b
3D stencil
i,j,k
b
b
b
Figure 2-11: Finite Difference mesh and examples of a 2D and 3D stencil
Finite Volume Method This method divides the problem domain into a set of finite volumes (or
cells, see Figure 2-12) and the resulting statement expresses the (exact) conservation of relevant
properties for each finite size cell (volume). The basic advantage of this method over FDM is that
it is applicable for any type of mesh (both structured and unstructured) where mass, momentum
and energy are conserved [19, 22]. The clear relationship between the numerical algorithm and the
underlying physical conservation principle forms one of the main attractions of the FVM. Therefore,
its concepts are much more simple to understand by engineers than the FEM [23]. This is why
this method is widely adopted for solving fluid flow problems and implemented in CFD tools. This
method is very flexible and so it can be rather easily implemented on structured grids as well as
on unstructured grids [22]. Just as the other methods, this method also has some disadvantages.
False diffusion often occurs in numerical predictions with FVM, especially when simple numerics are
engaged. Besides that, it is also difficult to develop schemes with higher than second-order schemes
accuracy for multidimensional problems.
2D volume
3D volume
Figure 2-12: Finite Volume mesh and examples of a 2D and 3D volume blocks
Finite Element Method FEM is mostly used for analyzing structural mechanics problems. However, it also found its applicability for fluid flow analyses. In general it is applied to the solution
of Euler and NS equations, and the physical space (domain) is subdivided into triangular (in 2D)
or into tetrahedral (in 3D) elements (see Figure 2-13). The classical FEM discretization is based
on these elements (refer to Figure 2-13) and a piecewise representation of the solution in terms of
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 2. Thesis background
17
basis functions. For improved numerical stability, more convincing finite element procedures can
be applied [21]. The strength of FEM is its ability to deal with arbitrary geometry using different
shapes and orders of element that are usually formed based on a set of unstructured mesh. And
just like the FVM, the discretization of the conservation laws allows the treatment of discontinuous
solutions such as shocks. FEM can give the highest accuracy on coarse mesh among the three
traditional numerical methods [19]. Although it can be shown that in certain cases the method is
mathematically equivalent to the FVM, the effort for the numerical implementation is much higher
[22]. Another advantage is that it is very effective for diffusion-dominated problems (viscous flow)
[19]. On the other hand, the coding of FEM is much more complex when it is used for solving
turbulent fluid flow problems [19, 21].
2D element
3D element
Figure 2-13: Finite Element mesh and examples of a 2D and 3D element
Linear methods
The simplest CFD solver methods are the linear solvers. Among the linear solvers, we have the
panel codes, the vortex lattice method (VLM) and the lifting-line method. These method make use
of singularity element methods and are not valid for compressible flows. Several corrections can be
applied such that these methods are valid within certain flight conditions.
Lifting-line method The simplest three-dimensional wing theory is that based on the concept of
Prandtl’s lifting line theory [24] and on linearized potential flow theory. In this theory, the wing
is replaced by a lifting line. The circulation about the wing associated with the lift is replaced
by a vortex filament. This vortex filament lies along the straight line. At each spanwise station
of this filament, the strength of the of the vortex is proportional to the local intensity of the lift.
The variation of the vortex strength along the straight line is therefore assumed to result from the
superposition of a number of horseshoe-shaped vortices, as shown in Figure 2-14. The portions
of the vortices lying along the the span are called the bound vortices, and the portions extending
downstream are called the trailing vortices (or wake). The effect of trailing vortices corresponding to
a positive lift is to induce a downward velocity component at and behind the wing. The downward
velocity is called downwash. From this rotation of the flow the effective angle of attack reduces, and
correspond in a rotation of the lift vector and produces on its turn an additional drag component
in stream-wise direction. The induced drag, downwash velocity, lift distribution, and vorticity
distribution can be calculated using this theory. This theory has several limitations: it does not take
compressibility, viscous flow and unsteady flow effects into account. Another important drawback
is that is does not account for the effect of wing sweep. The method is only valid for thin lifting
surfaces at small angles of attack and sideslip.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
18
Chapter 2. Thesis background
Bound vortices (lifting line)
z
Line of aerodynamic centers
y
Direction of airstream
Γ(y)
x
Trailing vortices (wake)
Figure 2-14: Lifting-line model consisting of multiple horseshoe vortices
Vortex Lattice Method Prandtl’s classical lifting line theory gives reasonable results for straight
wings at moderate to high aspect ratio. However, for low-aspect-ratio straight wings, swept wings
and delta wings, the lifting line theory is inappropriate [25]. For such planforms, more sophisticated
method models must be used. The vortex lattice method (VLM) is an extension of the lifting line
theory. The VLM places a series of lifting lines on the plane of the wing at different chordwise
stations. The wing is then represented by a lifting surface without thickness and discretized in
quadrilateral element, so-called panels. Figure 2-15 illustrates a swept wing with these panels. A
vortex ring is associated with each panel, placed on the quarter-chord line. The spanwise vorticity
is lumped on each panel into a discrete vortex along its quarter-chord line. This results in a system
of horseshoe vortices, one for every panel on the surface. Classical VLM formulations ignore the
thickness of the wing. With the addition of compressibility correction in the flow direction, the
VLM can also be used to a limited extent to compressible flow.
Figure 2-15: Schematic of a lifting surface
Panel method In the two above-mentioned methods, the singularities are located inside the body.
Unfortunately, an arbitrary body shape cannot be created using the singularities placed inside the
body. A more sophisticated method has to be used to determine the potential flow over arbitrary
shapes. Panel methods solve the linearized potential equations for inviscid, irrotational, incompressible flow. Since the equation is linear, superposition of the solutions can be used. The most
familiar singularities are the point source, doublet and vortex. With the addition of a compressibility
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 2. Thesis background
19
correction in the free-stream flow direction (e.g. Prandtl-Glauert, Karman-Tsien), panel methods
can also be applied to a limited extent to compressible flow around slim bodies. In general, with
compressibility corrections, panel methods are limited to free-stream Mach number of less than
0.7 [12]. For higher subsonic Mach numbers with small disturbance to the freestream flow, the
Prandtl-Glauert Equation can be used (see [26, 14] for more details).
Aerodynamic panel methods generally use quadrilateral panels to define the surface. Since three
points determine a plane, a quadrilateral may not necessarily define a consistent flat surface. In
practice however, the methods actually divide panels into triangular elements to determine an
estimate of the outward normal [14]. It is important that all the edges fit such that there is no gap
(or leakage) in the panel model representation of the surface.
Higher-order panel methods (advanced) use singularity distributions that are not constant on the
panel. Moreover, the may use panels which are non-planar. Higher-order methods were found
to be crucial in obtaining accurate solutions for the Prandtl-Glauert Equation at higher subsonic
and supersonic speeds. At these speeds, the Prandtl-Glaurt Equation becomes a wave equation
(hyperbolic) [14]. Such equation require more accurate numerical solutions than the subsonic case
in order to avoid pronounced errors. In theory, good results can be obtained using fewer panels with
higher-order methods [14]. More information about higher-order methods can be found in [27, 26].
2-3-4
Aerodynamic methods comparison
The most widely used CFD methods in industry and research are the RANS solver, Euler solver, the
panel method and VLM. These methods are compared to each other based on differences, such as
compressibility effects, shockwave prediction, etc. The comparison is shown in Table 2-1. A multifidelity model in function of the CFD method and the geometry detail of the model, is graphically
represented in Figure 2-16. The geometry detail increases when the level of CFD method increases.
Euler and RANS solvers require computer aided design (CAD) models for meshing.
CFD
High fidelity
RANS solver
Euler solver
Panel solver
VLM
Low fidelity
ng
fti
i
L
es
fac
r
su
m
Si
A
dC
e
ifi
pl
D
A
dC
e
fin
Re
D
Geometry detail representation
Figure 2-16: Multi-fidelity model in function of CFD method and geometry detail. (Adapted from:
[28])
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
20
2-3-5
Chapter 2. Thesis background
Structural weight estimation methods
Weight estimation methods used in aircraft design processes can be grouped based on their level
of fidelity. In general the lower the fidelity, the more the method is based on statistics. Higher
class methods rely less on statistical estimations and use more physics based calculations. These
calculations require more detailed input parameters. In a wing design task, three class of weight
estimation methods are used: statistical, quasi-analytical and analytical methods.
Statistical weight prediction techniques These methods use statistical coefficients, developed
based on existing aircraft, to estimate the wing weight as a function of several significant geometrical
parameters. A number of conceptually similar methods, tailored to different classes of aircraft, are
presented by Howe [29], Torenbeek [30], Shevell [31] and Raymer [15]. Implementing these methods
results in a fast and robust weight estimation with minimum amount of required data. The main
problem is the level of accuracy. Another problem is the validity margin, as these method are only
valid within the bounds of the original data. Note that such methods are limited when considering
new concepts that lie outside the original database.
Quasi-analytical weight estimations Another class of weight prediction techniques are the quasianalytical methods. These methods are based on elementary strength/stiffness analyses of aircraft
structures, augmented by empirical factors and statistical data. This approach enables weight
engineers to develop more accurate and design-sensitive results. Additionally they have a wider
validity margin compared to statistical methods. These methods are based on the ‘stationary
analysis’ of a simplified structure. In such methods the primary wing box weight is computed
by calculating the amount of material required to resist the applied loads. Contributions of the
secondary structures (i.e., flaps, slats, ailerons, etc.) are still estimated based on statistical methods.
Examples of wing weight estimations based on this approach are presented by Torenbeek [32], Macci
[33] and Elham [34].
Analytical methods These methods rely on finite element methods (FEM) to size various
components of the primary wing structure and compute their weight using the material density.
Examples of such methods are demonstrated by Bindolino [35] and Laban [36]. Generally, these
methods are more accurate than those mentioned above. However their implementation requires
a large amount of detailed geometrical information as well as structural and material data. This
is usually not available in early design stages. Furthermore, the time required to implement a
finite element model and perform the analysis, is significantly longer than computing the simple
empirical equations of a statistical method. This makes these methods unsuitable for both early
design phases and weight predictions within high level MDO processes.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 2. Thesis background
Wing Shape Multidisciplinary Design Optimization
Vortex-lattice method
Panel solver
Full potential solver
Euler solver
RANS solver
Linearized potential
flow equations
using compressibility
corrections
Yes
Full potential
equations
’Exact’
Euler equations
RANS equations
’Exact’
’Exact’
Lift coefficient
Linearized
potential
flow equations
using
compressibility
corrections
Yes
Yes
Yes
Yes
Shockwave prediction
No
No
Yes
Yes
Pressure distribution on surface
Maybe (inaccurate at
leading edge)
5 sec. - 1 min.
Yes
Inaccurate for strong
shocks
Yes
Yes
Yes
1 min. - 15 min.
5 min. - 1 hr.
1 - 15 hrs.
Multiple days
Governing equations
Compressibility
CPU calculation time per case
[28]
flow
Table 2-1: CFD methods comparison
21
Jan Mariens
22
Jan Mariens
Chapter 2. Thesis background
Wing Shape Multidisciplinary Design Optimization
Chapter 3
Local optima smoothing for global
optimization
In this chapter, a novel optimization algorithm to find the global solution is presented. This principle
of algorithm is extensively discussed and several benchmark test functions and a classic aerodynamic
problem were used to prove its effectiveness and accuracy.
First, a brief discussion is given on the formulation of optimization problems in Section 3-1. In the
same section, the need for the global solution is highlighted. In Section 3-2, a global optimization
algorithm is explained that uses local optima smoothing. Finally, in Section 3-3 several test cased
were used to prove the effectiveness, robustness and accuracy of the global optimization algorithm.
3-1
Optimization problem
The following mathematical formulation is used for a general constraint optimization problem:
min
f (x)
(3-1)
x
subject to
gi (x) ≤ 0
hj (x) = 0
i = 1, . . . , m
(3-2)
j = 1, . . . , n
(3-3)
where f (x) is the optimization objective function and x the design vector. Function gi (x) stands
for the inequality constraints and hj (x) for the equality constraints. There are different optimization
algorithms that can solve problem (3-1). In general gradient-based optimization algorithms only
seek local solutions. However, the search for global solutions is an important task for engineering
purposes, mainly for the following reasons (as addressed by Rizzo and Frediani [37]):
(a) The objective functions and constraints can include “black boxes”. This means that explicit
expressions for the objective function and constraints might not be available. The values
of these functions at point x can be (partly) provided by numerical code (so-called “black
boxes”). Therefore, the general properties of these functions might not be known in advance.
(See Figure 3-13 for an example of the discontinuity of a “black box”)
(b) Numerical issues associated to the evaluation of the objective and constraints introduce numerical noise. This is related to previous point, where the numerical evaluations of the function
values are often approximated. Approximations superimpose numerical errors that may affect
the function by high-frequency oscillations with local peaks (local minima).
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
24
Chapter 3. Local optima smoothing for global optimization
(c) When applying optimization algorithms in the study of new projects, a good exploration property of algorithms is mandatory. The effectiveness of the algorithm can be influenced by the
optimizing designs that have an almost unknown design domain. Furthermore, the initial point
for the optimization might lie far from the global solution.
Two algorithms are used in this research: one to find the local optimum and another one to find
the global solution. For the local optimization algorithm, the Sequential Quadratic Programming
(SQP, gradient-based) algorithm was selected. A brief overview of the SQP algorithm can be found
in Appendix A. The global optimization algorithm is explained in the following section.
3-2
An algorithm for local optima smoothing
Many algorithms to find the global optimum have been devised. Examples of global optimization
algorithms are: genetic algorithm, A*, dynamic programming, etc. The optimization algorithm
(framework) proposed by Addis et al. [38, 39] is capable of finding the global optima using a
local solver and it has shown good performances in terms of robustness and time consumption.
This algorithm is called “Local Optima Smoothing for Global Optimization” (LOCSMOOTH) and
is discussed here. The global algorithm is also used to compare the local optima results for a
multidisciplinary design optimization of aircraft wings in Chapter 6.
3-2-1
Local optima smoothing principle
The basic idea of the LOCSMOOTH algorithm is explained here. Suppose that the objective
function has the form represented by the solid blue line in Figure 3-1 and that it has an underlying
function like the dashed line. The objective function can then be viewed as the underlying function
plus some perturbation (noise) around it. A function with this property is known as a function
having a funnel structure.
10
9
8
7
f(x)
6
5
4
3
2
Objective function
Underlying funnel structure
1
0
0
2
4
x
6
8
10
Figure 3-1: Example of optimization function with a funnel structure.
When a local optimizer is used to find the optimum starting from a point in interval [a, b] in Figure 3-2, then it will return x1 as the solution. Only if the starting point belongs to the [c, d]
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 3. Local optima smoothing for global optimization
25
interval, the global solution will be found. This is a typical characteristic of gradient-based optimization algorithms. In Figure 3-1, the thickest line represents the local minima found by a local
optimization algorithm starting from different points. It can be deduced that the global optimum
is the minimum of the enveloping function containing all local minima (thickest, red line). The
idea of the LOCSMOOTH algorithm is to use this information from the local optimizations starting
from different points to construct the enveloping curve to find the global minimum. This (piecewise
constant) function is also known as step function.
10
Objective function
Underlying funnel structure
Step function, L(x)
9
8
7
f(x)
6
5
4
3
2
1
0
−1
0
a
x
b
1
2
c
4
x
*
x
d
6
8
10
Figure 3-2: Local and global optimization.
Gradient-based optimization algorithms (such as SQP) have problems with step functions. In order
to build a more reliable method for funnel-type global optimization problems, sampling the optimization function coupled with smoothing of the step function (denoted by L(x)) might provide a
good strategy [38]. Looking at Figure 3-2, it can immediately be noticed that a very good smoothing effect has already been achieved by simple observing L(x), the results of local optimizations,
instead of the objective function f (x). However, in order to fully exploit the funnel structure, a
smoothing method should be applied to L(x). This way, the piecewise constant structure of L(x)
will be replaced by a smooth function which contains information of descent directions. It is evident that in a piecewise constant function, no local information is available to provide guidelines
for gradient-based optimizations. Though, if smoothing were possible, the smoothed function could
help in finding appropriate descent directions and thus guide the search towards a global optimum.
The following smoothing function was derived by Addis et al. (see [38] for the complete derivation):
L̂B
g (x)
=
K
P
i=1
L(yi )g (||yi − x||)
K
P
i=1
(3-4)
g (||yi − x||)
where g(x) represents the Gaussian kernel, K the number of local minima in interval B. The ’kernel’
for smoothing, defines a shape of the function that is used to take the average of the neighboring
points. A Gaussian kernel is a kernel with the shape of a Gaussian (normal distribution) curve, see
Figure 3-3.
The basic process of smoothing is simple, as it proceeds through the data point by point. For each
data point, a new value is generated that is some function of the original value at that point and
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
26
Chapter 3. Local optima smoothing for global optimization
0.4
0.3
0.2
0.1
0
−6
−4
−2
0
2
4
6
Figure 3-3: Normal distribution curve (Gaussian), with standard deviation σ = 1.
the surrounding data points. The function that is used within the Gaussian kernel is the Gaussian
(or normal distribution) curve. In the standard statistical way, the width of the Gaussian shape is
defined by the standard deviation or σ (additional notes on σ are given in the following subsection).
The Gaussian kernel is defined as
2
−z
g(z) = e (2σ)2
(3-5)
An example of smoothing the step function L(x) is given by Figure 3-4. Now, the optimum of the
smoothed function can be found using a local optimization algorithm. At this minimum, a new local
search is performed on the real optimization function f (x) in order to find the global optimum.
10
L(x)
σ=0.25
σ=0.50
σ=0.75
σ=1.00
9
8
7
6
5
4
3
2
1
0
0
2
4
6
8
10
x
Figure 3-4: Gaussian filtered step function L(x), with different standard deviations σ.
3-2-2
The LOCSMOOTH framework
The main idea of the LOCSMOOTH algorithm is to take information from the local minima in order
to build a smoothed function giving a direction on the basin of attraction. When the starting point
changes, the minimum will generally change as well. By sampling multiple points and determine
their minima, a step function can be constructed.
The LOCSMOOTH algorithm is organized in two phases: an approximation phase and a displacement phase. In the approximation phase, the local minimum L (xh ) starting from point xh is
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 3. Local optima smoothing for global optimization
27
calculated. Then, K point are random uniformly sample inside a sphere B(xh , r) with radius r
center in xh , producing K observations. The step function L(x) can then be constructed based
on the minima of these observations. L(x) is then used to build the smoothed function L̂B
g (x).
Consequently, the smoothed function is minimized and its minimum gives directions on the basis
of attraction. The minimum of L̂B
g (x) is then taken as the next current point and the procedure is iterated (displacement phase). The whole procedure is interrupted at that point where no
improvements (MaxNoImp) are found.
In order to speed up this process, each time a record (the best local optimum observed so far) is
obtained, the procedure sets the current point to the record.
The definition for the standard deviation for the Gaussian operator is given by the following expression:
√
n
(3-6)
σ = r/ K
By this definition, the whole volume of the sphere is covered by the Gaussian weight. In order to
obtain an equal coverage for different radii values, the number of samples must be K = r n /σ n
[38]. This choice for standard deviation is less effective when the variables have a different range of
variation, which can lead to deterioration of the convergence speed of this algorithm, as addressed
by Rizzo [40]. This problem can be avoided by choosing different radii for the variables. Hence,
instead of sphere, the samples are chosen inside an ellipsoid. Equation (3-6) is then modified into
the following expression [40]:

1/n
K
Q
ri 

 i=1 
σ=
(3-7)

 K 
where ri represents the radii along different axes. For more details on this modified standard
deviation, one can refer to reference [40]. The number of observations (samples) to be made are
problem dependent. Parameter n influences the quality of the approximated function, but it is
rather time-consuming because a local optimization is performed for each sample. The variable ri
affects the exploration aptitude of the algorithm; smaller radii limit the search in a neighborhood
of the current solution, whereas larger radii could give too dispersed data with deterioration of the
approximated function [38, 40].
Below, the complete procedure of the LOCSMOOTH algorithm with the initialization, based on the
psuedo-code provided by Addis et al. [38], is discussed. Additionally, a visualization of the complete
procedure is shown in Figure 3-5.
Initialization
To start the optimization procedure, several initial user inputs are required: the initial starting point
x0 , the radius of the sphere r, the number of observations to perform in the current sphere K and
the maximum number of no improvement MaxNoImp. First, set NoImp = 0 and choose a starting
point x0 in which the sphere B(x0 , r) is centered with radius r. Choose a random uniform sample
point x in the sphere B(x0 , r) and calculate the minimum of the real-valued function f (x) (objective
function) resulting in x∗ = minf (x) = LS(x). Then the current is set to current = f (x∗ ) = L(x)
x
and the record point to current. Note that LS(x) denotes the local search of the objective function
starting from point x.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
28
Chapter 3. Local optima smoothing for global optimization
Procedure
The iterative procedure of the algorithm is given by the following steps:
1. When NoImp < MaxNoImp, then set i = 0 and continue with step 2.
2. If i < K and record ≤ current, then set i = i + 1 and observe a random uniform sample
point yi inside the sphere B(x∗ , r) centered in x∗ . Calculate the local minimum yi∗ = LS(x)
and set current = f (yi∗ ) = L(yi ).
3. If a new record has been found while sampling in B(x∗ , r) (current < record), then this is
set to record = current. Consequently, the center of the sphere is moved to yi∗ (x∗ = yi∗ )
and set NoImp = 0. At this point, the procedure starts again at step 1.
Else NoImp = NoImp + K and develop the smoothed function L̂B
g (x) using the stored
local minima yi∗ . Next, the smoothed function is optimized to find its minimum x =
arg min L̂B
g (x). Consequently, the local minimum y = LS(x) is to be found starting
x∈B(x∗ ,r)
from x and set current = f (y) = L(x). The procedure is continued with step 4.
4. If current < record, then a new record has been found and set to record = current. Furthermore, set x∗ = y and NoImp = 0. Else, set x∗ = x and go to step 1.
The algorithm terminates when NoImp = MaxNoImp. As soon as a new record is obtained, the
center of the sphere is moved. The LOCSMOOTH algorithm showed great performances in finding
global solutions as it is show in the following test cases.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 3. Local optima smoothing for global optimization
29
INPUT
x0 , r , K , MaxNoImp
NoImp = 0
x* = LS ( x) = min f ( x)
x
current = f ( x* ) = L( x)
record = current
WHILE
NoImp < MaxNoImp
i=0
WHILE
i<K
record ≤ current
i = i +1
yi = random uniform point in B ( x* , r )
yi* = LS ( yi )
current = f ( yi* ) = L( yi )
record = current
current < record
true
x* = yi*
NoImp = 0
false
NoImp = NoImp + K
x = arg min* LˆBg ( x)
( )
xÎB x ,r
y = LS ( x )
current = f ( y ) = L( x)
record = current
x* = y
current < record
true
NoImp = 0
false
x* = x
OUTPUT
x* , current
Figure 3-5: Roadmap of the LOCSMOOTH algorithm.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
30
Chapter 3. Local optima smoothing for global optimization
3-3
Test cases
In this section, some tests of the global search algorithm are performed by solving some benchmarking problems taken from literature [41, 42]. These problems include very noisy objective functions
without constraints. Another problem that was constituted, is a constrained classical aerodynamic
problem. All the test cases were solved by means of the global optimization algorithm LOCSMOOTH using a gradient-bad based algorithm for local searches. The user-defined parameters
for the LOCSMOOTH algorithm used throughout these test cases (unless stated otherwise) are:
• Sphere radius r = (ub − lb)/3, where ub and lb are the lower and upper bounds respectively
• Number of sampled points K within the sphere
• Maximum number of no improvements, MaxNoImp=3K
For the local search, the SQP algorithm of the Matlab Optimization Toolbox is used. All the test
case use the default settings of this algorithm.
3-3-1
Ackley’s function (unconstrained)
A classical benchmarking function to test optimization algorithms, is the Ackley’s function. A 3D
illustration of this figure is shown in 3-6. This function presents a deep basin (see Figure 3-7) of
attraction and is very noisy in minimum region. High frequency waves are superimposed over the
whole domain. Ackley’s function is defined by
s
−0.2
f (x) = −20e
1
2
n
P
i=1
x2i
−e
1
n
n
P
cos(2πxi )
i=1
+ 20 − e
(3-8)
where n represents the number of variables. The design space for this problem is bounded by
−32.768 ≤ xi ≤ 32.768 and the theoretical solution of the global optimum is f (x∗ ) = 0 where
x∗i = 0 for i = 1, . . . , n.
25
f (x)
20
15
10
5
0
50
50
0
0
x1
x2
−50
−50
Figure 3-6: 3D plot of the Ackley’s function for n = 2 variables.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 3. Local optima smoothing for global optimization
31
15
f (x)
10
5
0
5
x2
0
5
0
−5
x1
−5
Figure 3-7: A zoom near the origin of the Ackley’s function for n = 2 variables.
Initially, the radii of the spheres were defined by r = (ub − lb)/3 which gave results far from the
exact solution f (x∗ ) = 0. As mentioned before, larger radii could give too dispersed data with
deterioration of the approximated function. Therefore, the radii for the problem were redefined
to r = (ub − lb)/5. In Table 3-1 the minima calculated by the global optimization algorithm are
summarized.
Table 3-1: Numerical solutions of Ackley’s function with sphere radii ri = (ub − lb)/5, LOCSMOOTH algorithm.
n
f (x∗ )
NFval
2
5
10
15
20
1.0454 × 10−6
1.8150 × 10−6
2.8787 × 10−5
1.0664 × 10−4
1.6683 × 10−4
3062
8663
30496
48754
45268
It can be observed that in general the number of function evaluations NFval increase and the
solutions deteriorate as the space dimensions grow.
3-3-2
Rastrigin’s function (unconstrained)
This is typical example of a non-linear multimodal function (function with many local minima). A
typical funnel structure is present (see Figure 3-1) and consequently the LOCSMOOTH procedure
is expected to be effective.
f (x) = 10n +
n
X
i=1
x2i − 10 cos (2πxi )
(3-9)
The design space for this problem is bounded by −5.12 ≤ xi ≤ 5.12 and the theoretical solution
of the global optimum is f (x∗ ) = 0 where x∗i = 0 for i = 1, . . . , n. Table 3-2 provides the minima
found using LOCSMOOTH algorithm.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
32
Chapter 3. Local optima smoothing for global optimization
100
f (x)
80
60
40
20
0
5
5
0
x2
0
−5
x1
−5
Figure 3-8: 3D plot of the Rastrigin’s function for n = 2 variables.
Table 3-2: Numerical solutions of Rastrigin’s function, LOCSMOOTH algorithm.
3-3-3
n
f (x∗ )
NFval
2
5
10
15
20
2.1316 × 10−14
8.0291 × 10−13
1.0328 × 10−10
0
0
1745
5039
12628
21682
33511
Rosenbrock’s function (unconstrained)
Rosenbrocks’s valley is a classic optimization problem, also known as the “banana function”. A
3D plot of this function is given in Figure 3-9. The global optimum of this function lies inside a
long, narrow, parabolic-shaped flat valley. Finding the basin of attraction is a trivial task, since the
convergence of the global optimum is more difficult. The Rosenbrock’s function is defined as:
f (x) =
n h
X
i=1
100 xi+1 − x2i
2
+ (1 − xi )2
i
(3-10)
The design space for this problem is bounded by −2.48 ≤ xi ≤ 2.48 and the theoretical solution
of the global optimum is f (x∗ ) = 0 where x∗i = 1 for i = 1, . . . , n. The results using the
LOCSMOOTH algorithm are summarized in Table 3-3.
3-3-4
Schwefel’s function (unconstrained)
Schwefel’s function is deceptive in that the global minimum is geometrically distant from the next
best local minima. For this reason, search algorithm are potentially prone to convergence into the
wrong direction. The Schwefel’s function is illustrated by Figure 3-10 and given by the following
expression:
n h
p i
X
|xi | ,
−xi sin
f (x) =
(3-11)
i=1
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 3. Local optima smoothing for global optimization
33
f (x)
1500
1000
500
0
2
−2
−1
1
0
0
x1
x2
1
−1
−2
2
Figure 3-9: 3D plot of the Rosenbrock’s function for n = 2 variables.
Table 3-3: Numerical solutions of Rosenbrock’s function, LOCSMOOTH algorithm.
n
f (x∗ )
NFval
2
5
10
15
20
2.1384 × 10−12
3.5132 × 10−11
4.0188 × 10−11
4.8604 × 10−11
5.7358 × 10−11
5963
11794
51358
89519
63424
where n is the number of variables. The test area of this function is usually restricted to a hypercube,
bounded by −500 ≤ xi ≤ 500 for i = 1, ..., n. The global minimum is f (x∗ ) = −418.9829 · n
where x∗i = 420.9686 for i = 1, . . . , n.
1000
f (x)
500
0
−500
−1000
500
500
0
0
x2
x1
−500
−500
Figure 3-10: 3D plot of the Schwefel’s function for n = 2 variables.
The initial settings for the spheres’ radii was ri = (ub − lb)/3 which gives results far from the
theoretical global optimum. By upscaling this radii to ri = (ub − lb)/2 and downscaling the number
of samples to K = 5, the algorithm was able to find solutions close to the theoretical minimum.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
34
Chapter 3. Local optima smoothing for global optimization
Table 3-4 summarizes the numerical results of the optimization.
Table 3-4: Numerical solutions of Schwefel’s function, number of sampled points K = 5, sphere
radii ri = (ub − lb)/2, LOCSMOOTH algorithm.
3-3-5
n
f (x∗ )
f (x∗ ) − f (x∗ )theor.
2
5
10
15
20
−837.966
−2094.91
−4189.83
−6284.74
−8379.66
2.5455 × 10−5
6.3637 × 10−8
1.2728 × 10−4
1.9091 × 10−4
2.5455 × 10−4
NFval
922
2138
8555
4222
6670
Minimum induced drag of a wing
A classical problem of aerodynamics is to find the lift distribution over an isolated wing with the
minimum induced drag. It is well known that the optimum lift distribution is elliptical shaped. The
optimization problem considers an unswept wing. The wing span is divided into certain number of
trunks (or bays), see Figure 3-11.
Due to symmetry of the wing, only half the wing is considered for the optimization. The optimization
problem is to achieve a chord distribution to minimize the induced drag Di with a constraint on
the global lift L. The problem can be formulated as
min
Di (x)
x
subject to
L(x) = W
lb ≤ x ≤ ub
where W is the weight, lb the vector of the lower boundaries and ub the vector of the upper
boundaries. The variable x is the design vector containing the design variables. The starting
geometry has 5 bays and thus 6 equally spaced ribs. The design variables are the chord lengths at
each rib. The induced drag is defined as
Di = 21 CDi ρV 2 S
(3-12)
This configuration will be optimized at sea level conditions with a free-stream velocity V = 50
m/s and Mach number M = 0.2. Both parameters S and CDi are of importance for Di and will
be mentioned as well in the results. The weight W is fixed to 5000 N and the chords may vary
between 0.1 and 3 m. Each wing section has a NACA 0014 airfoil shape. The wing aerodynamics
x
12 m
y
bay 1
bay 2
bay 3
bay 4
bay 5
1m
V = 50 m/s
Figure 3-11: Starting geometry of the wing with 5 bays, top view.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 3. Local optima smoothing for global optimization
35
are calculated using AVL [43] (a vortex-lattice method, see Subsection 2-3-3), that is interfaced
with Matlab. This interface enables information exchange between AVL and the optimizer. The
starting aerodynamics of the initial wing configuration are shown in Table 3-5. AVL uses the Trefftz
plane (far-field analysis) to calculate the dimensionless induced drag (CDi ). The Trefftz plane result
for the starting geometry is given in Figure 3-12.
Table 3-5: Starting aerodynamics of isolated wing.
α
CL
CD i
S
Di
1.2196◦
0.2721
0.00209
12 m2
38.036 N
Figure 3-12: Lift distribution on the Trefftz’s plane for the initial rectangular wing.
The following optimization algorithms are used for this optimization problem:
• SQP algorithm, implemented in the optimization toolbox of Matlab
• LOCSMOOTH algorithm: algorithm based on local optima smoothing for global optimization
As was mentioned before, gradient-based optimization algorithms have problems calculating the
optimum of step functions. Low fidelity aerodynamic solvers, such as AVL (“black box”), can imply
noise or non-smooth functions for varying input parameters. An example of a piecewise constant
function is given in Figure 3-13. The function represents the induced drag coefficient as a function
of wing span. This graph was generated using the starting geometry and flight conditions, except
the wing span variable and the lift coefficient is fixed at CL = 0.2.
First, the wing is optimized using the SQP algorithm. The optimization results are summarized
in Table 3-6 and its lift distribution is shown in Figure 3-14. The optimized chord lengths of
each spanwise rib are given in Table 3-7, where the spanwise position η = y/(b/2) indicates
the (dimensionless) distance from root to tip section.. It can be noted that the optimized wing
configuration is close to and elliptical shaped wing.
Consequently, the same optimization problem is solved using the LOCSMOOTH algorithm. The
optimization results are summarized in Table 3-8. The design vector and the trefftz plane of the
optimized configuration are given by Table 3-9 and Figure 3-15.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
36
Chapter 3. Local optima smoothing for global optimization
−3
1.22
x 10
1.2
C D i [ -]
1.18
1.16
1.14
1.12
1.1
11
11.2
11.4
11.6
W i n g sp an [ m]
11.8
12
Figure 3-13: Discontinuity in induced drag coefficient CDi calculated by AVL, fixed CL = 0.2.
Table 3-6: Wing optimization results for 5 bay geometry using SQP algorithm.
CD i
CL
Di [N]
S [m2 ]
α [◦ ]
Initial
SQP
0.00207
0.2721
38.036
12
3.0562
0.00186
0.2566
36.245
12.726
2.8128
Table 3-7: Optimized chord lengths of rectangular wing for minimum induced drag using the SQP
algorithm.
Spanwise location
η = 0.0
η = 0.2
η = 0.4
η = 0.6
η = 0.8
η = 1.0
Initial
SQP
1
1.2574
1
1.4296
1
1.0955
1
1.0955
1
0.7596
1
0.5535
Table 3-8: Wing optimization results for 5 bay geometry using LOCSMOOTH algorithm.
CD i
CL
Di [N]
S [m2 ]
α [◦ ]
Initial
SQP
LOCSMOOTH
0.00207
0.2721
38.036
12
3.0562
0.00186
0.2566
36.245
12.726
2.8128
0.00183
0.2538
36.055
12.867
2.7845
Figure 3-16 illustrates the optimized wing configuration using both optimization algorithms. Worth
of notice is that the optimized wing planform using the LOCSMOOTH algorithm has an elliptical
shape providing an elliptical lift distribution.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 3. Local optima smoothing for global optimization
37
Figure 3-14: Lift distribution on the Trefftz’s plane of optimized wing using SQP.
Table 3-9: Optimized chord lengths of rectangular wing for minimum induced drag.
Spanwise location
η = 0.0
η = 0.2
η = 0.4
η = 0.6
η = 0.8
η = 1.0
Initial
SQP
LocSmooth
1
1.2574
1.3240
1
1.4296
1.2495
1
1.0955
1.2402
1
1.0955
1.0867
1
0.7596
0.8901
1
0.5535
0.4654
Figure 3-15: Lift distribution on the Trefftz’s plane of optimized wing using LOCSMOOTH algorithm.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
38
Chapter 3. Local optima smoothing for global optimization
Ch ord l en gth [ m]
1.5
1
0.5
0
0
Initial configuration
SQP optimized
LOCSMOOTH optimized
1
2
3
4
S p anwi se p osi ti on [ m]
5
6
Figure 3-16: Initial and optimized wing configurations.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 4
Quasi-3D aerodynamic solver
The total drag produced by a wing exists of induced drag and profile drag (see Figure 2-9). Profile
drag can be subdivided into friction drag and pressure drag. Friction drag is the component of the
resulting forces tangential to the wing surface acting in the direction of the flow. The pressure drag
is the component of the forces normal to the surface, in the flow direction.
Optimization of aircraft wings requires many repetitive aerodynamic evaluations of different planforms, which leads to time-consuming processes. Therefore, it is of great interest to find a fast
way for calculating the total wing drag. An interesting way for establishing such an approach,
is by combining (two-dimensional) viscous airfoil data with an inviscid three-dimensional wing lift
calculation. In reference [44], such a method has been developed and validated. Viscous airfoil data
can be found in experiments, or can be calculated using two-dimensional viscous airfoil calculation
tools. Three-dimensional lift distributions from wing configurations can be found using a vortex
lattice code.
First, an application of the strip method is presented in Section 4-1, because of its importance for
this research. A quasi-three-dimensional aerodynamic solver was developed, using the same principle
as the strip method in [44] extended with the simple sweep theory. The simple sweep theory is
discussed in Section 4-2. The developed solver is capable to analyze tapered swept wings. The
taper implementation into the simple sweep theory is discussed in Section 4-3. Consequently, the
procedure of the developed quasi-3D aerodynamic solver is elaborated in Section 4-4. As mentioned
before, the quasi-3D method can also used viscous airfoil calculation tools instead of performing
interpolations on experimental data. The different tools used by the developed solver are discussed
in Section 4-5. In Section 4-6, the selection of different input parameters regarding panel density
and number of strips are discussed. Finally, the developed tool was validated using low-speed and
high-speed test cases in Sections 4-7 and 4-8.
4-1
Strip method
The viscous drag of a wing can be found using a strip method. The strip method presented here
combines the strip theory with a vortex lattice method. The strip theory, also known as blade
element theory [45, 46], concerns dividing an aircraft wing geometry into discrete segments and
computing aerodynamic forces and moments on those segments based on their local velocities. A
more detailed explanation of the combination of strip theory and vortex lattice models can be found
in [47].
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
40
Chapter 4. Quasi-3D aerodynamic solver
In reference [44], an application of such a strip method is given. The principle of this method forms
the basis of the aerodynamic solver that was developed. This procedure combines two-dimensional
viscous airfoil data with a three-dimensional lift distribution over a rectangular wing, and was
validated using CFD results [44]. The principle of this method is of importance for the aerodynamic
solver that was developed and will therefore be presented here.
First, the three-dimensional lift distribution of the wing is calculated using a vortex lattice code.
Having the lift distribution along the span, the spanwise drag distribution can be determined by
calculating the drag at each strip. The process of calculating the drag at each strip (or wing
section), is explained using Figure 4-1.
leff
l
αi
Veff
αeff
α+ǫ
V∞
d
deff
αi
Figure 4-1: Three-dimensional lift (l), three-dimensional drag (d), local lift (leff ) and local drag
(deff ) of a strip.
The relationship between the three-dimensional lift force l (computed by the VLM), the threedimensional drag d, the local lift leff and the local drag deff is given in Figure 4-1. The local lift
and drag forces represent the viscous airfoil forces at the effective angle of attack αeff . Note that
the three-dimensional forces, shown in this figure, represent the forces at a particular section which
are deduced from the lift distribution obtained by the VLM. Based on this lift distribution, the
two-dimensional viscous drag force can be calculated. The following steps are required to calculate
this viscous drag at a given spanwise strip:
1. Set αi = 0 and deff = 0
2. leff = (l + deff sin αi ) / cos αi
3. Determine local velocity Veff = V∞ / cos αi and compute the local Reynolds number Reeff
4. Find αeff by two-dimensional interpolation of experimental data using leff , Veff and Reeff
5. Let αi = α + ǫ − αeff
6. Find deff by two-dimensional interpolation of experimental data using αeff , Veff and Reeff
7. Repeat steps 3-6 until αi converged
8. d = deff cos αi + leff sin αi
The total drag force of the wing is the sum over all strips computed spanwise three-dimensional
drag (d) distribution. This drag force is an approximation of the total drag of the wing including
induced drag, which is the drag due to the tip vortex on the wing. The strip method combines
experimentally measured profile drag and computationally predicted induced drag. This method
may give inaccurate results for flow regimes with dominant tip vortices. By adding the vortex drag
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 4. Quasi-3D aerodynamic solver
41
to the drag predicted by the strip method, one assumes that there is no influence of the tip vortex
suction on induced and profile drag (see [44] for more details).
4-2
Simple sweep theory
The simple sweep theory, described in reference [48], is presented here. This theory was derived
for wings with sweep and high aspect ratios, which is typical for most modern transport aircraft.
Such wings are often designed such that the flow (except in root and tip regions) is of quasi-twodimensional nature, in the sense that it varies sluggish across the span [48]. In these regions, the
semi-mid-span regions, the idealized concept of an “infinite yawed wing” is a useful starting point
that relates the swept wing flow to an equivalent two-dimensional flow over a transformed airfoil
section, taken normal to the sweep line of the actual wing (as indicated in Figure 4-2).
V∞
V⊥
y
c⊥
c Λ
sweep line
x
Figure 4-2: Simple sweep theory of an infinite wing (untapered wing)
The following relations were derived from geometric considerations:
c⊥ = c cos Λ,
so
and
z⊥ = z,
z z =
sec Λ
c ⊥ c t
t
=
sec Λ.
c ⊥
c
(4-1)
(4-2)
where z represents the ordinates perpendicular to the x − y plane. From the aerodynamics of the
flow cases, the following was derived:
M⊥ = M∞ cos Λ,
(4-3)
V⊥ = V∞ cos Λ,
(4-4)
2
Cp⊥ = Cp sec Λ.
(4-5)
Equation (4-5) can be derived from the pressure perturbations which are normalized by the dynamic
pressure. Based on the same equation, it can be deduced that the lift coefficients are related by
Cl⊥ = Cl sec2 Λ
(4-6)
while the pressure drag coefficients are related by
Cdp⊥ = Cdp sec3 Λ.
Wing Shape Multidisciplinary Design Optimization
(4-7)
Jan Mariens
42
Chapter 4. Quasi-3D aerodynamic solver
For the friction drag coefficient, it is assumed that they are about equal in free-stream direction
and in direction normal the sweep line (this assumption is based on reference [49], where the same
assumption has been made).
(4-8)
Cdf⊥ = Cdf
The profile drag, consisting of friction drag and pressure drag, is found by summing equations (4-7)
and (4-8). For the profile drag in free-stream flow direction, this yields:
(4-9a)
Cd = Cdf + Cdp
3
= Cdf⊥ + Cdp⊥ cos Λ
(4-9b)
Thus the air flow on a swept wing with sweep angle Λ, may be compared with the flow of an
equivalent airfoil at a lower free-stream Mach number (factor cos Λ), which is thicker (factor sec Λ)
and at a higher lift coefficient (factor sec Λ).
As the Reynolds number changes with varying chord length, the actual Reynolds number acting on
the airfoil section perpendicular to the sweep line needs to be calculated. The Reynolds number is
defined as follows:
ρV∞ c̄
(4-10)
Re∞ =
µ
where c̄ and µ represent respectively the mean aerodynamic chord (MAC) and the dynamic viscosity.
Only the velocity and the chord length vary in (4-10), whereas the density ρ and µ are altitude
dependent and are constant. Knowing that, the Reynolds number Re⊥ can be scaled by
Re⊥ = Re∞
V⊥ c⊥
V∞ c̄
(4-11)
It is important to note that the simple sweep theory is based on infinite wings and since it relates the
swept wing flow to an equivalent two-dimensional flow, it does not include real three-dimensional
flow characteristics such as boundary layer movements and others [48, 50].
4-3
Wing taper implementation
The simple sweep theory, needs some small modifications to account for taper (λ 6= 1). The two
major modifications to be made are: the selection of a proper sweep line and the interpolation of
the airfoil shape normal to this sweep line. The chord length of the airfoil shape normal to the
sweep line must also be derived.
The sweep line is taken at a constant chord percentage over the wing. Since the value of the constant
chord position defines the sweep line and thus the aerodynamic components and the shape of the
airfoil perpendicular to this line, it is important to make a proper selection of this value. According
to Obert [51], the velocity component perpendicular to the quarter-chord line with sweep angle Λ
can be used in practice. For transonic flows, it is more acceptable to chose this sweep line such
that it coincides with the shockwave line [49]. The reason for this is that the pressure drag acts
perpendicular to the isobars (or shockwave line), as is explained in [50]. Figure 4-3 shows an infinite
swept wing with the shock line taken as the sweep line for the sweep theory. Since the selection for
a proper chordwise position for this sweep line, an additional input variable was introduced, This
variable defines the chord percentage of the sweep line and is denoted by ξc . For this research, the
quarter-chord sweep line (ξc = 0.25) for subsonic conditions and mid-chord sweep line (ξc = 0.50)
for transonic conditions were assumed to be reasonable.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 4. Quasi-3D aerodynamic solver
43
Having a proper chord percentage for the sweep line of a tapered swept wing, the chord length c⊥
and airfoil shape perpendicular to this line can be determined. In Appendix C, it is explained how
c⊥ can be calculated. The airfoil shape is determined by interpolation between the surrounding
defined airfoil shapes. This interpolation is illustrated in Figure C-3 (see Appendix C).
V∞
V⊥
shockwave
line
Λ
Dp⊥
potential flow
streamline
Dp
Df
Figure 4-3: Friction and pressure drag forces on a swept tapered wing (adopted from [49])
4-4
Quasi-3D aerodynamic solver
The quasi-3D aerodynamic solver is a tool that was developed during this thesis. It is based on the
strip method, see Section 4-1, and combined with the simple sweep theory (Section 4-2). Additional
modifications for tapered wings are also implemented in this tool. Note that the implementation
of the sweep theory implies the limitation to model three-dimensional flow characteristics. In this
section, the procedure of the developed quasi-3D solver is presented. The overall procedure of the
developed aerodynamic solver, can be broken down into the three stages:
1. Run the vortex lattice code to calculate the lift distribution, the wing lift coefficient CL and
the wing induced drag coefficient CDi for the given wing.
2. Calculate the viscous drag distribution using the extended strip method.
3. Compute the total wing drag.
Instead of using experimental two-dimensional viscous airfoil data, two-dimensional viscous airfoil
analysis tools are used to ensure a wide range of applicability. A short overview of selected tools
for the developed quasi-3D solver is given in Section 4-5.
Next, the procedure of each stage is discussed. The second stage is explained in more detail, as it
combines the strip method with the sweep theory.
Stage 1
In the first stage, a vortex lattice method is used to calculate the three-dimensional inviscid lift
distribution of the given wing for a given angle of attack α or lift coefficient CL . The vortex lattice
method calculates the CL /α, CDi and lift distribution for the given wing configuration. Vortex
lattice methods calculate the induced drag at the Trefftz plane (“far-field” analysis) [27, 52]. The
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
44
Chapter 4. Quasi-3D aerodynamic solver
induced drag calculated in this stage is used for the determination of the total wing drag. Note
that the induced drag cannot be determined separately in “near-field” method, which are used in
the strip method to calculate the profile drag.
Stage 2
In this stage, the strip method is extended using the simple sweep theory and wing taper effect.
The implementation of the simple sweep theory into the strip method is described by the following
steps, for the viscous drag calculation at one spanwise strip:
1. Interpolation on the three-dimensional lift distribution of the wing (calculated by the VLM)
to find the lift coefficient Cl at the particular wing section (strip).
2. Apply sweep theory (considering taper effects) to find the geometric and aerodynamic characteristics of the wing section normal to the sweep line. Appendix C provides more information
on the determination of the chord length c⊥ and the corresponding airfoil shape. The aerodynamic characteristics of the section perpendicular to the sweep line (with angle Λ) are defined
as:
M⊥ = M∞ cos Λ,
(4-3)
V⊥ = V∞ cos Λ,
(4-4)
And the corresponding lift coefficient is found using (4-6) and the three-dimensional lift
distribution calculated by the VLM:
Cl⊥ = Cl sec2 Λ
(4-12)
3. Next, the induced angle of attack αi is found by using an iterative process:
(a) Set αi = 0 and Cdeff = 0
(b) The viscous lift coefficient can be derived using Figure 4-4:
leff =
l⊥ + deff sin αi
cos αi
(4-13)
leff
l⊥
αi
Veff
αeff
α+ǫ
V⊥
d⊥
deff
αi
Figure 4-4: Definition of the forces and angles used to determine the inviscid downwash angle
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 4. Quasi-3D aerodynamic solver
45
This method uses the lift and drag coefficients instead of the lift/drag forces. Therefore,
the following force-coefficient relations can be substituted in equation (4-13) together
with the effective velocity conversion:
2
SCleff
leff = 21 ρVeff
(4-14)
2
1
2 ρVeff SCdeff
2
1
2 ρV⊥ SCl⊥
(4-15)
deff =
l⊥ =
Veff =
which yields
Cleff =
V⊥
cos αi
(4-16)
(4-17)
Cl⊥ cos2 αi + Cdeff sin αi
cos αi
(4-18)
(c) The effective Reynolds number can be calculated using (4-11)
Reeff = Re∞
Veff c⊥
V∞ c̄
(4-19)
(d) Find the effective angle of attack αeff and the effective drag coefficient Cdeff of the
airfoil using a viscous airfoil analysis tool and the calculated values Reeff , M⊥ and Cleff .
The aerodynamic tools calculate besides the total drag coefficient, the friction drag
coefficient Cdfeff and pressure drag coefficient Cdpeff which will be used later on.
i
h
αeff , Cdeff , Cdpeff , Cdfeff = airfoil analysis tool (Cleff , Reeff , M⊥ )
(e) The induced angle of attack can now be recalculated using αeff .
αi = −αeff + (α + ǫ) cos Λ
(4-20)
where the cos Λ factor accounts for decrease of the inflow angle when the sweep angle
increases.
(f) Repeat steps (b)-(e) until αi has converged.
4. The viscous drag component normal the sweep line (in direction V⊥ ) can now be calculated.
First, the profile drag force can be derived from Figure 4-4, which yields
d⊥ = deff cos αi + leff sin αi
(4-21)
As this expression has been derived for the force, it can be rewritten for coefficients (analogue
to equation (4-18)):
cos αi
sin αi
Cd⊥ = Cdeff
+ Cleff
(4-22)
2
cos αi
cos2 α
{z
}
|
| {z }i
Profile drag
Induced drag
The drag coefficient Cdeff is the sum of both the friction and pressure drag coefficients. The
induced drag term is not taken into consideration here, as it is calculated by the VLM (see
Stage 1).
5. In order to find the profile drag coefficient in streamwise direction of the wing section, the
simple sweep theory can be used:
)
Cdf = Cdf⊥
(4-23)
Cdprof = Cdf + Cdp
Cdp = Cdp⊥ cos3 Λ
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
46
Chapter 4. Quasi-3D aerodynamic solver
Once all the profile drag coefficients on each wing section (N w sections) are calculated, they can
be integrated over the wing in order to find the three-dimensional profile drag coefficient.
CDprof
2
=
S
Zb/2
Cdprof c dy
(4-24)
0
where S and c represent the wing planform area and the chord length at a wing section.
Figure 4-5 gives an overview of the simple sweep theory implementation into the strip method.
Interpolation of the lift distribution
to find Cl at given spanwise section
Cl
Use simple sweep theory to find Cl , Mach
number and velocity normal to the sweep line:
Cl⊥ = Cl sec2 Λ
M⊥ = M∞ cos Λ
V⊥ = V∞ cos Λ
Find the induced angle of attack αi
and the effective drag coefficient Cdeff
(a) Set αi = 0 an Cdeff = 0
Cl⊥ cos2 αi + Cd⊥ sin αi
cos αi
Veff c⊥
V⊥
and Reeff = Re∞
(c) Veff =
cos αi
V∞ c̄
i
h
(d) αeff , Cdeff , Cdpeff , Cdfeff =
(b) Cleff =
airfoil analysis tool (Cleff , Reeff , M⊥ )
(e) αi = −αeff + (α + ǫ) cos Λ
(f) Repeat steps (b)-(e) until αi has converged.
Calculate drag tangential to V⊥
Cdeff
Cd⊥ =
cos αi
Profile drag coefficient in streamwise direction
Cdprof = Cdf⊥ + Cdp⊥ cos3 Λ
Figure 4-5: Overview of the sweep theory implementation in the strip method for one strip
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 4. Quasi-3D aerodynamic solver
47
Stage 3
The lift coefficient and induced drag coefficient of the given wing configuration are calculated by
the vortex lattice method at Stage 1. The profile drag is calculated as Stage 2 using a strip method
extended by the simple sweep theory. Finally, to conclude the procedure of the developed quasi-3D
aerodynamic solver, the total drag coefficient is determined by combining the induced drag (Stage
1) and the profile drag (Stage 2).
CD = CDi + CDprof
(4-25)
Finally, a complete overview of the inputs and outputs is given in Appendix E.
4-5
Aerodynamic tools
Viscous airfoil data can be found in literature, e.g. see reference [53, 54], or from two-dimensional
viscous airfoil calculation tools. Selected airfoil calculation tools for the development of the quasi-3D
solver are
XFOIL is a program developed for the design and analysis of low Reynolds number airfoils (singleelement airfoils), see reference [55, 56]. It calculates the viscous aerodynamics using a higher
order panel method coupled with a viscous/inviscid interaction method. It uses the KarmanTsien compressibility correction, which extend the validity limit of free-stream Mach numbers
up to 0.7 [12].
VGK calculates the viscous aerodynamics of a single-element airfoil and is based on the full potential equations combined with integral boundary layer equations. The full potential flow
equations are accurate for shock-free flows, but are only approximate for flows with shocks of
appreciable strength, since the regions across the shock are generally rotational (and therefore
non-isentropic). For a good representation of such flows, solutions of the Euler equations are
required. However with a suitable choice of several parameters (refer to [57]), the solutions
yield surface pressure distributions that correspond closely to those of true physical shock
waves.
MSES is a coupled viscous/inviscid Euler method that features the design and analysis of single and
multi-element airfoils at low Reynolds numbers and transonic Mach numbers. Additionally,
MSES can also predict flows with transitional separation bubbles, shock waves, trailing edge
and shock-induced separations [58].
At subsonic flows, the aerodynamic solver uses XFOIL for viscous airfoil calculations. XFOIL shows
accurate results within this range and it is the fasted tool among the listed tools. For transonic
flows, the quasi-3D solver uses VGK or MSES. As MSES is an Euler based tool, it provides more
accurate results than VGK (full-potential) and comes at the cost of computation time.
The three dimensional inviscid lift calculation of the wing is performed by means of vortex lattice
method. AVL [43] was selected as VLM for this research.
4-6
Selection panel density and number of strips
The panel density (in vortex lattice code) and the number of strips (for strip method) have to
be selected properly. The panel density has an influence on the calculation of the induced drag,
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
48
Chapter 4. Quasi-3D aerodynamic solver
whereas the number of strips affects the profile drag prediction. In this section, a proper set for the
density and number of strips is derived based on convergency of these drag components.
4-6-1
Vortex lattice grid size
Vortex lattice codes model lifting surfaces (panels) by dividing them into a number of chordwise
and spanwise horseshoe vortices. Both these numbers can be changed independently, and in general
the results become more accurate by increasing the number of spanwise and chordwise vortices.
Increasing the number of horseshoe vortices comes at a cost: the amount of computational time
increases significantly, as well as the memory usage. Figure 4-6 shows a wing with different grid
sizes. At a certain certain point an increase in number of vortices will hardly effect the final results,
while the computational time will still increase. Therefore a number of runs was done, in which the
effect of the number of horseshoe vortices in spanwise and chordwise direction on computational
time and induced drag convergency was investigated. As a measure for accuracy, the induced drag
was normalized by the average of the four finest grids. Then the selection of the grid size was based
upon a balance between accuracy and computational time.
(a) Coarse grid
(b) Fine grid
Figure 4-6: Half wing with different grid sizes
Figure 4-7 shows the influence of the number of spanwise (N b) and chordwise vortices (N c) on the
normalized induced drag coefficient. From this figure it can be deduced that increasing N c yields
a more accurate induced drag coefficient. On the other hand, when N b increases, one can deduce
that by increasing N b, the resulting induced drag coefficient converges rather well. In Figure 4-8,
the time needed to compute each point is plotted against the grid size.
From the figures above, a grid size was selected to be N b = 24 and N c = 13 (indicated by a
small black box in Figure 4-7 and 4-8, with its corresponding values displayed). The selection
implies a deviation of 0.15% on the induced drag coefficients and requires about one fifth of the
computational times for the finest grid tested (N b = 30 and N c = 20).
In addition to Figure 4-7, the convergency plots for the number of spanwise and chordwise elements
are shown separately in Figure 4-9.
4-6-2
Number of strips
Using different number of strips strongly affects the computational times and final results. Consequently a number of runs was done, in which the effect of the number of strips on computational
time and profile drag convergency was investigated. The number of strips only affects the profile
drag in the developed aerodynamic solver. As a measure for accuracy, the profile drag coefficient
was normalized by the average of the three configurations having the highest number of strips.
Figure 4-10 shows the effect of the number of strips on the normalized profile drag coefficient. One
can deduce that when the number of strips (N w) increases, the resulting profile drag coefficient
converges rather well. In Figure 4-11, the computational times for each number of strips is plotted.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 4. Quasi-3D aerodynamic solver
49
1.006
1.004
N or maliz e d C D i [-]
1.015
1.002
1.01
1
1.005
X: 24
Y: 13
Z: 0.9985
1
0.998
0.996
0.995
0.994
0.99
0.992
0.985
30
0.99
5
25
10
20
Nb
15
15
10
Nc
0.988
0.986
20
Figure 4-7: Effect of number of spanwise (N b) and chordwise (N c) vortices on normalized induced
drag coefficient.
From these figures, a strip number was chosen to be N w = 8 (indicated by a black box in
Figure 4-10 and 4-11, with their values shown). This selection yields a deviation of the profile
drag coefficient smaller than 0.05% with a corresponding computational time of 2.25 seconds.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
50
Chapter 4. Quasi-3D aerodynamic solver
20
2
1
3
2.5
1.5
0.75
0.5
4
3.5
15
1
Nc
0.25
2
X= 24
Y= 13
Level= 0.89347
1.5
0.75
0.5
1
10
0.75
0.25
0.5
0.5
0.25
0.25
5
10
15
20
Nb
25
30
N or maliz e d C D i [-]
Figure 4-8: Effect of number of spanwise (N b) and chordwise (N c) vortices on computation time.
Each level indicates the computational time seconds.
1
X: 24
Y: 0.9985
0.95
0.9
Nor maliz e d C D i [-]
0
5
10
15
20
N b ( f or fix e d N c =13)
25
30
1
X: 13
Y: 0.9985
0.95
0.9
0
5
10
15
N c ( f or fix e d N b=24)
20
nor maliz e d C D
pr of
Figure 4-9: Effect of number of spanwise and chordwise elements on normalized induced drag
coefficient (separate plots).
1.02
1
X: 8
Y: 0.9993
0.98
0.96
0.94
0
5
10
Nw
15
20
Figure 4-10: Effect of number of strips (N w) on normalized profile drag coefficient.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
C omput at ion t ime [s ]
Chapter 4. Quasi-3D aerodynamic solver
51
6
4
2
0
0
X: 8
Y: 2.255
5
10
Nw
15
20
Figure 4-11: Effect of number of strips (N w) on computational time.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
52
4-7
Chapter 4. Quasi-3D aerodynamic solver
Validation of the quasi-3D aerodynamic solver at low speeds
Validation of the developed aerodynamic solver is done using experimental data from the NACA
Technical Report No. 572 [59]. This technical report contains experimental data of four wing
configurations having different sweep angles and twisted tip sections, as shown in Figure 4-12. The
results using the quasi-3D solver are also compared to two other aerodynamic solvers: VSAERO
[60, 61] and MATRICSV [62].
VSAERO is a commercial aerodynamic solver that calculated the linearized potential flow equation.
Compressible flow is analyzed by applying either a Karman-Tsien correction or a Prandtl-Glauert
linearization to the compressible flow. MATRICSV is a full potential transonic flow solver and
it can calculate both laminar and turbulent boundary layers. More information on VSAERO and
MATRICS can be found in references [61, 62].
Table 4-1 shows the characteristics that define each wing configuration. In the NACA24-AA-BB
wings denotation, the AA represents the quarter-chord sweep angle and BB the twist angle multiplied
by 10 at the tip section (positive downward). Both the sweep angle and the twist angle are defined
in degrees. All NACA wing configurations were tested at free-stream Reynolds number of 3,090,000
and freestream velocity of 69.9 ft/s.
Table 4-1: NACA wing characteristics
Parameter
Value
Unit
S
λ [-]
cr
6
150
0.5
6.6667
in2
ft
A
Only the validation results of the NACA 24-30-0 wing are discussed in this section. The results for
the other wing configurations are similar and can be found in Appendix D.
4-7-1
NACA 24-30-0 wing
The drag and lift coefficient curves as a function of the angle of attack for the NACA 24-30-0 wing,
are given by Figure 4-13 and 4-14. Figure 4-15 shows the drag coefficient curves as a function of
the lift coefficient.
Comparing the CL curves of the quasi-3D aerodynamic solver to experimental data and to results of
other solves, it can be concluded that the lift is slightly over-predicted. This over-prediction in lift
originates from the vortex lattice code. The drag coefficient on the other hand, is under-predicted
by the quasi-3D solver. MATRICSV, the three-dimensional CFD solver based on the full potential
equations, shows rather good results compared to the experimental data. Therefore, the induced
and profile drag coefficient curves of the quasi-3D solver are compared to those from MATRICSV
(as these data were available).
At the lower range angles of attack, the induced drag is about equal for both the quasi-3D solver
and MATRICSV. Figure 4-16 shows both the induced, profile and total drag coefficient curves of
the quasi-3D and the MATRICSV solver. At higher angles of attack however, the quasi-3D solver
shows higher values for induced drag coefficients. On the other hand, the profile (viscous) drag
is overall under-predicted compared to the MATRICSV data. Especially at higher α values, this
under-prediction becomes more significant.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 4. Quasi-3D aerodynamic solver
53
(a) NACA 24-0-0
(b) NACA 24-15-0
(c) NACA 24-30-0
(d) NACA 24-30-85
Figure 4-12: Graphical representation of the wing configurations used for low speed validation of
the quasi-three-dimensional aerodynamic solver
The numeric values values from the CD − CL curves are provided by Table 4-2. The error in
drag coefficient prediction (denoted by ∆CD ) represents the difference between the predicted and
the experimental values. The error in drag prediction by MATRICSV is smaller than those from
quasi-3D solver. However, the quasi-3D solver has a wider range of validity than MATRICSV.
Table 4-3 provides the computational times for a single case for each aerodynamic solver. Note
that all runs were performed on computers with similar hardware performance.1 Based on Table 4-2
and Figure 4-15, one can conclude that the developed solver shows slightly better results than the
3D panel method VSAERO for the NACA 24-30-0 wing. Moreover, the quasi-3D is significantly
faster than VSAERO and MATRICSV.
1
Using computers with better performance will reduce these computational times.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
54
Chapter 4. Quasi-3D aerodynamic solver
1.2
1
Experimental data
Quasi−3D
VSAERO
MATRICSV
C L [-]
0.8
0.6
0.4
0.2
0
−0.2
−4
−2
0
2
4
α [de g]
6
8
10
12
Figure 4-13: CL − α curve of the NACA 24-30-0 wing
0.07
0.06
Experimental data
Quasi−3D
VSAERO
MATRICSV
C D [-]
0.05
0.04
0.03
0.02
0.01
0
−4
−2
0
2
4
α [de g]
6
8
10
12
Figure 4-14: CD − α curve of the NACA 24-30-0 wing
Table 4-2: Error analysis of different aerodynamic solver for the wing drag coefficient of the tapered
NACA 24-30-0 wing
Experimental
Jan Mariens
Quasi-3D
∆CD
VSAERO
∆CD
MATRICSV
∆CD
CL
CD
CD
(×1000)
CD
(×1000)
CD
(×1000)
-0.129
0.013
0.156
0.298
0.439
0.580
0.718
0.855
0.990
0.0104
0.0094
0.0104
0.0136
0.0199
0.0279
0.0390
0.0530
0.0693
0.0072
0.0060
0.0068
0.0098
0.0154
0.0240
0.0351
0.0483
0.0649
-3.226
-3.420
-3.627
-3.837
-4.462
-3.899
-3.920
-4.667
-4.445
0.0060
0.0052
0.0065
0.0097
0.0153
0.0231
0.0330
0.0455
0.0617
-4.403
-4.147
-3.931
-3.906
-4.573
-4.705
-5.966
-7.490
-7.602
0.0079
0.0091
0.0128
0.0191
0.0284
-
-1.467
-1.315
-0.853
-0.769
0.532
-
Wing Shape Multidisciplinary Design Optimization
Chapter 4. Quasi-3D aerodynamic solver
55
0.07
0.06
Experimental data
Quasi−3D
VSAERO
MATRICSV
C D [-]
0.05
0.04
0.03
0.02
0.01
0
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
C L [-]
Figure 4-15: CD − CL curve of the NACA 24-30-0 wing
0.04
MATRICSV C
D
0.035
Quasi−3D C
D
MATRICSV C
0.03
Quasi−3D CD
D
i
i
C D [-]
0.025
MATRICSV CD
prof
Quasi−3D C
D
prof
0.02
0.015
0.01
0.005
0
−2
0
2
4
6
8
α [de g]
Figure 4-16: Drag differences of NACA 24-30-0 wing for quasi-3D solver and MATRICSV
Table 4-3: Computational time per case
Solver
Quasi-3D
VSAERO* [63]
MATRICSV
Computational time
20 sec. - 50 sec.
5 min. - 10 min.
4 min. - 5 min.
* from which about 4 min. were required for
input file generation and output processing
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
56
Chapter 4. Quasi-3D aerodynamic solver
4-8
Validation of the quasi-3D aerodynamic solver at high speeds
In addition to the low speed validation, the developed solver is also validated using a test case
at transonic conditions. The Fokker 100 wing is used for validation of the solver at high speeds.
Both the results of the solver using AVL-MSES and AVL-VGK are both illustrated and compared
to the results of MATRICSV. Note that the results of the developed solver are compared to a
three-dimensional CFD tool, as experimental data was not available.
4-8-1
Drag coefficients comparison
The Fokker 100 wing is analyzed at an altitude of 37,000 ft using different Mach numbers and lift
coefficient combinations. VSAERO is not valid in these flight regimes. As experimental data of
the flying wing was not available, a comparison was made using results from MATRICSV solver.
Figure 4-17 shows the resulting drag coefficients using both AVL-VGK and AVL-MSES combination.
Note that for all aerodynamic solvers, a fixed transition at 5% chord was used for the high-speed
cases to ensure convergency.
0.07
0.07
Quasi−3D
MATRICSV
Quasi−3D
MATRICSV
0.06
0.06
0.05
0.05
C =0.3
0.04
C =0.5
C D [−]
C D [−]
CL=0.3
CL=0.4
L
L
0.04
C =0.5
L
0.03
0.03
0.02
0.02
C =0.4
L
0.01
0.01
C =0.2
C =0.2
L
0
0.5
0.55
0.6
0.65
0.7
Mach numb e r [-]
(a) AVL-MSES
0.75
0.8
L
0.85
0
0.5
0.55
0.6
0.65
0.7
Mach numb e r [-]
0.75
0.8
0.85
(b) AVL-VGK
Figure 4-17: CD versus Mach comparison with MATRICSV
In general, the drag coefficient remains relatively constant all the way up to about M = 0.75. As the
Mach number increases slightly further, the drag usually rises slowly at first. By increasing the Mach
number even further, a point will be reached where the drag coefficient suddenly starts to increase.
This is the point where the wave drag (or compressibility drag) appears. For increasing CL , it can
be deduced that the under-prediction in comparison with the MATRICSV results becomes larger.
The drag coefficient of the quasi-3D solver using AVL-MSES, increases more than the MATRICSV
data at higher Mach numbers (M=0.75-0.78). For the quasi-3D solver using AVL-VGK, this higher
drag increase is only present for the lower CL values whereas for higher CL values, this drag increase
is negative. Figure 4-18 shows these differences between the solver and the MATRICSV data. The
difference in drag prediction between the quasi-3D solver and MATRICSV increase rapidly at Mach
numbers higher than 0.75.
Figure 4-19 shows the differences in the angle of attack between MATRICSV and the quasi-3D
solver (for both AVL-MSES and AVL-VGK). In general, the angles of attack of the quasi-3D solver
are under-predicted compared to MATRICSV. This means that for the same angle of attack, AVL
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 4. Quasi-3D aerodynamic solver
57
slightly over-predicts the lift. At the low-speed validation of the developed aerodynamic tool (see
Section 4-7), the lift was also over-predicted compared to both the experimental data and the
MATRICSV results.
−3
6
−3
x 10
6
2
0
D MatricsV
−6
∆C =C
−2
D
−C
D quasi−3D
∆ CD = CD MatricsV − CD quasi−3D [−]
4
[−]
4
−4
CL=0.2
CL=0.3
−8
x 10
C =0.4
2
0
−2
−4
−6
CL=0.3
−8
L
C =0.5
0.55
C =0.4
L
C =0.5
L
−10
0.5
CL=0.2
L
0.6
0.65
0.7
0.75
Mach number [−]
0.8
0.85
−10
0.5
0.9
0.55
(a) ∆CD from AVL-MSES
0.6
0.65
0.7
0.75
Mach number [−]
0.8
0.85
0.9
0.8
0.85
0.9
(b) ∆CD from AVL-VGK
Figure 4-18: CD difference between AVL-MSES/AVL-VGK and MATRICSV
1.8
1.6
1.8
CL=0.2
CL=0.3
1.6
[degr.]
−α
1.2
MatricsV
1
0.8
0.6
0.4
0.2
0.5
CL=0.3
CL=0.4
CL=0.5
quasi−3D
1.4
∆α=α
∆ α = αMatricsV − αquasi−3D [degr.]
CL=0.4
CL=0.2
1.4
CL=0.5
1.2
1
0.8
0.6
0.4
0.55
0.6
0.65
0.7
0.75
Mach number [−]
0.8
0.85
0.9
(a) ∆α from AVL-MSES
0.2
0.5
0.55
0.6
0.65
0.7
0.75
Mach number [−]
(b) ∆α from AVL-VGK
Figure 4-19: α difference between AVL-MSES/AVL-VGK and MATRICSV
4-8-2
Pressure distribution comparison
Three-dimensional aerodynamic effects, i.e., root and tip effect, are not modeled by the quasi-3D
solver (see Section 4-2). In this subsection the flow characteristics of the Fokker 100 wing flying
at M∞ = 0.8 at CL = 0.4 are shown for MATRICSV (full 3D CFD) and the quasi-3D solver. The
pressure contour plot of the upper wing surface from MATRICSV is provided by Figure 4-20a. This
figure shows the bending of the isobars near the root and tip section, which illustrates the so-called
root and tip effect. Note that the dark blue line indicates the position of the pressure drop (or
shock wave line). Figure 4-20b shows the pressure contour results from AVL-VGK, in which no
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
58
Chapter 4. Quasi-3D aerodynamic solver
root or tip effect is present. Both contour plots present the shockwave line (in red) calculated by
AVL-MSES. MATRICSV shows a more realistic representation of the actual flow over the wing.
The lack of these three-dimensional effect is due to the use of the simple sweep theory. As can be
deduced from Figure 4-20b, the shockwave lines of both VGK and MSES are quite similar. Note
the aft movement of the shockwave position near the kinked wing section in Figure 4-20b. This
aft movement can be explained by the effect of the different sweep angle (for the inner and outer
wing) on the simple sweep theory.
Fokker 100 at Mach 0.8 and CL=0.4
Fokker 100 at Mach 0.8 and CL=0.4
C upper surface MATRICSV
14
p
1
1
Shock wave AVL−MSES
12
12
0.5
10
8
0
6
−0.5
4
2
Spanwise position [m]
Spanwise position [m]
C upper surface VGK (2D)
14
p
Shock wave AVL−MSES
0.5
10
8
0
6
−0.5
4
2
−1
0
−2
0
2
4
6
Longitudinal position [m]
8
(a) Comparison of MSES shockline with
MATRICSV data
−1
0
−2
0
2
4
6
Longitudinal position [m]
8
(b) Comparison of MSES shockline with VGK
data
Figure 4-20: Upper surface pressure coefficient distributions from MATRICSV (full potential 3D)
and VGK (full potential 2D) with MSES shockwave locations along span
The minimum pressure along the half wing is shown in Figure 4-21. It can be deduced that the
minimum pressure near the tip station differs from the MATRICSV results. The minimum pressure
of MATRICSV towards the tip section decreases strongly in strength. This is caused by the tip
effect (three-dimensional aerodynamic effect) captured by MATRICSV.
Figure 4-22 shows the pressure distributions of each solver for 3 wing sections: one near the root,
a midspan section and one near the wing tip. As can be seen for the pressure distribution at the
near root section, the shock position (shock) for the MATRICSV solver is positioned more aft
than those from the quasi-3D solver. This is in agreement with Figure 4-20 and is caused by the
three-dimensional root effect.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 4. Quasi-3D aerodynamic solver
59
Min C along the span of Fokker 100 at M=0.8, C =0.4
p
L
−1.3
AVL−VGK
AVL−MSES
MATRICSV
−1.2
−1
p
min C [−]
−1.1
−0.9
−0.8
−0.7
−0.6
0
0.2
0.4
0.6
Spanwise position η [−]
0.8
1
Figure 4-21: Minimum Cp distribution along span
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
60
Chapter 4. Quasi-3D aerodynamic solver
Pressure distribution of wing section of Fokker 100 at η=0.1 (M=0.8, CL=0.4)
Pressure distribution of wing section of Fokker 100 at η=0.5 (M=0.8, CL=0.4)
−1.5
−1.5
VGK
MSES
MATRICSV
−1
−1
−0.5
C [−]
0
0
p
Cp [−]
−0.5
0.5
0.5
1
1
1.5
0
VGK
MSES
MATRICSV
0.2
0.4
0.6
0.8
1
1.5
0
0.2
0.4
x/c [−]
0.6
0.8
1
x/c [−]
(a) Cp distribution of wing section near root
(b) Cp distribution of wing section at
(η = 0.1)
midspan (η = 0.5)
Pressure distribution of wing section of Fokker 100 at η=0.9 (M=0.8, C =0.4)
L
−1.5
VGK
MSES
MATRICSV
−1
p
C [−]
−0.5
0
0.5
1
1.5
0
0.2
0.4
0.6
0.8
1
x/c [−]
(c) Cp distribution of wing section near tip
(η = 0.9)
Figure 4-22: Pressure coefficient distribution comparison for three wing sections
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 5
Wing weight estimation methods
In this chapter several existing statistical wing weight estimation methods are explained and compared with each other. This is done by analyzing their accuracy and sensitivity to different wing
design variables. These methods are compared with a quasi-analytical method (EMWET), based
on accuracy, design sensitivity and computational time.
5-1
5-1-1
Different methods
Torenbeek (1) method
Torenbeek has developed several weight estimation methods, of which two statistical methods were
used in present research. The first method is a simplified approximation, derived from a weight
estimation method by Torenbeek [30, 64]. This method can be used to predict the structural wing
weight of civil airplanes with cantilever wings and is presented by the following equation:
"
#
r
0.30
b
ref
0.75
0.55 bs / (cr · (t/c)r )
1+
Ww = WG · kw · bs
(5-1)
nult
bs
WG /S
where bs is the structural wing span. The factor of proportionality, kw , is as follows (for metric
units):
- For light aircraft, Wto ≤ 5670 kg: kw = 4.90 × 10−3 and WG = Wto in kg.
- Transport category aircraft, Wto > 5670 kg: kw = 6.67 × 10−3 and WG = Wzf in kg.
The weight given by Eq. (5-1) includes the weight of high-lift devices and ailerons. If spoilers and
speed brakes are incorporated an additional weight equal to 2% of the total wing weight should be
added. In the case of wing-mounted engines the wing weight must be reduced by 5% and 10% for
2 or 4 engines respectively. Additional reductions can be made for braced wings or in the case that
the main undercarriage is not mounted to the wing.
In this method, the primary wing structural weight is determined based on the requirement that in
a specified critical flight condition the bending moment due to the wing lift must be resisted by
the wing box structure. The weight of high-lift devices is estimated based on the critical loading
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
62
Chapter 5. Wing weight estimation methods
conditions at the flap design speed. The accuracy of this method has been assessed using the actual
wing weight of 46 aircraft (dating from about 1940-70s). A standard deviation of 9.64 percent was
achieved [30, 64]. Application of the method to several high-subsonic short-haul airplanes led to
the conclusion that the original method underestimates the wing weight. Main reasons for this
underestimation can be assumed to be caused by (a) the extra weight (up to 20%) required to
provide adequate stiffness against flutter and (b) the weight penalty due to long service life[64].
Since a simplified expression is used in this paper, a larger deviation is to be expected than the
indicated standard deviation.
5-1-2
Torenbeek (2) method
Another statistical method, also developed by Torenbeek, was used for this research. A practical wing weight formula presented by Torenbeek [65] was derived from a quasi-analytical method
described by the same author [32]:
Ww = 0.06
p
b3
1
1 + 2λ
w
nult Wto · Wzf
+ wtss S
2
σr
S (t/c)ave cos Λ 1 + λ
(5-2)
In this expression the term σwr represents the weight of the materials in the upper and lower wing
panels.√These panels are required to withstand the bending load due to lift, which is represented
by nult Wto · Wzf . The factor 0.06 is a non-dimensional constant of proportionality. This value
accounts for the average values of the bending relief due to e.g. wing and engine masses, structural
efficiency, spar locations, etc. The design weight Wdes and ultimate load factor nult refer to critical
combinations of manoeuvre/gust loads, weight distributions and operating speeds. The meaning
of the other values is as follows:
• The term σwr is the ratio of the specific wing material weight (w) to the mean stress level
at the wing root σr . Torenbeek [65] suggested that a statistical average value based on a
Al-alloy structure can be used:
h
−1/4 i
w
× 10−6
(5-3)
= 40 1 + 1.10 Wto /106
σr
• The average thickness-to-chord ratio t/c is found by suitable weighing of the wing thickness
in spanwise direction.
• The sweep angle Λ is theoretically related to the elastic axis. However, since its location is
usually unknown, it is for this method approximated by the mean quarter-chord line. Note
that textbooks usually indicate that the elastic axis lies at about 40% chord [66, 67].
• The smeared skin thickness tss is defined by
p
tss = 0.004 1 + Wto /106
(5-4)
As it is mentioned before, this method was derived from a more sophisticated method [32]. The
original method is characterized as follows:
• The wing structural weight is divided into functional components. The weight of each component is estimated by implementing either a statistical or a semi-analytical equation.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 5. Wing weight estimation methods
63
• The wingbox structure is sized in such a way to have minimum amount of material required
to resist the bending moment in the upper and lower equivalent panels and the minimum
amount of material to resist the shear force in the spar webs. These values are obtained
implementing an analytical structural analysis method.
• The effect of variation in the thickness-to-chord ratio and the compressive stresses (buckling
criteria) have been taken into account.
• A series of semi-empirical equations have been developed to estimate the ribs weight, the
non-optimum weight penalties, the weight of the fixed leading and trailing edge structures as
well as the weight of the high-lift devices and the control surfaces.
• The mean stress level is computed as a function of the aircraft maximum take-off weight
using an empirical equation. This equation is valid for Al-alloy structures. A user-selected
design stress levels is also possible for alternative material applications.
The original method was validated using several transport aircraft. The results are shown in Table 5-1. Although the small number of aircraft considered does not permit to determine the mean
prediction error. Nevertheless the table indicates that the original method yields good results for the
size effect. The results from the simplified method will most-likely deviate more than the original
one.
Table 5-1: Estimated versus actual wing weight for several transport aircraft [65].
Wing weight
Aircraft
Boeing 747-100
Airbus A340
Fokker F-28 Mk4000
Cessna Citation II
5-1-3
Wto [kN]
computed [kN]
actual [kN]
error [%]
3158
2486
315.8
59.16
391.6
345.0
32.03
5.950
384.4
340.9
33.28
5.730
+1.9
+1.2
−3.8
+3.8
Shevell method
Shevell [31] developed a statistical method based on a wing weight index, which is related to the
weight of the fully-stressed wing box. This weight index is defined as follows (in imperial units):
√
nult b3 Wto · Wzf (1 + 2λ) −6
10
(5-5)
Iw =
(t/c)ave S cos2 Λ (1 + λ)
From Eq. (5-5) one can observe that the weight index is proportional to the compressive and tensile
loads due to the bending moments at the root of the wing in the skin and the spar caps. Figure 5-1
shows the variation of the ratio of the wing weight to the square foot of the wing planform area with
the wing weight index. Based on this figure the total wing weight is estimated as follows [31, 68]:
Ww
= 4.22 + 1.642Iw
S
(5-6)
The first term on the right hand side of this equation is a constant and represents the secondary
wing weight. The second term varies in proportion to the amount of material required to resist the
applied bending loads and represents the primary wing weight.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
64
Chapter 5. Wing weight estimation methods
2
Total wing weight / gross wing area [lb/ft ]
16
15
DC−10−10, status 39
DC−10−10, DTS 5300
DC−10−30, DTS 5400B
DC−10−30, ER version
DC−9−10, CAL
DC−9−20, SAS
DC−9−30, ACA
DC−8−61
DC−8−62
DC−8−63
990
C−141A
707−120
727−100
Trident 1C
14
13
12
11
10
9
8
2
3
4
5
6
Wing weight index [lb/ft]
7
8
Figure 5-1: Statistical wing weight correlation (adopted from [31]).
5-1-4
Howe method
A statistical weight estimation method was developed by Howe[29], in which the wing weight is
derived based on the number of passengers or the payload weight for a particular airplane. In this
fundamentally empirical method, the wing weight is calculated as a function of the main geometric
and operational parameters. The estimated wing weight is correlated to some actual wing weight
values. The equation for wing weight estimation is as follows:
"
0.5 #0.9
1 + 2λ Wto
VD
C1
0.5 1.5
S sec Λ
nult
(5-7)
Ww = g
C5
3 + 3λ gS
(t/c)r
A
Coefficient C1 in this expression is defined as follows:
a) Long range, s > 5000 km
C1 = 0.00072 − 0.0005(270 + 0.05s)P AX × 10−6
(5-8)
b) Short/medium range, Wto ≤ 46, 000 kg
C1 = 0.00167 − 0.016(370 + 0.03s)P AX × 10−6
c) Turboprop, Wto ≤ 46, 000 kg
C1 = 0.00149 − 5.8P AX × 10−6
(5-9)
(5-10)
d) Long range jet freight aircraft
C1 = 0.00072 − 0.0005(2.08 + 0.00038s)Wpay × 10−6
(5-11)
e) Turboprop freight aircraft
C1 = 0.00077 − 0.00053(2.08 + 0.00038s)Wpay × 10−6
Jan Mariens
(5-12)
Wing Shape Multidisciplinary Design Optimization
Chapter 5. Wing weight estimation methods
65
Table 5-2: Coefficient C5 in equation (5-7).
Type of aircraft
C5
Tailless delta
Long haul jet transports
Short/medium haul jet transports
Executive jet aircraft
All other types
1.10
1.16
1.20
1.30
1.24
The original weight estimation equation described above calculates the weight of all lifting surfaces.
Dividing this total weight by the lifting surface factor C5 (see Table 5-2) yields the individual wing
weight [29].
This method was tested on 118 aircraft for which sufficient data was available or could be reasonably
assumed [69]. These tested aircraft have take-off weight range from about 700 kg (Cessna 150) to
more than 377000 kg (Boeing 747-200). Figure 5-2 shows the results of the civil aircraft that were
used as the test cases, together with ±10% boundaries. For 76% of those aircraft, the error of the
wing weight prediction is less than 10%.
5
10
Predicted mass [kg]
General aviation types
Propjet transports
Subsonic jet transports and executive types
4
10
3
10
2
10 2
10
3
4
10
5
10
10
Actual mass [kg]
Figure 5-2: Howe method – actual and predicted wing masses for civil aircraft [69].
5-1-5
LTH method
The last statistical method which is used in this research is the empirical equation developed by
Dorbath on behalf of LTH (Luftfahrttechnisches Handbuch) organization [70]. The equation is
presented below:
"
#
Wto 1.1038
1.31
−4
1.5 1
401.146S
+
Ww = 2.20013g · 10
· (t/c)−0.5
(5-13)
rep
g
cos Λ
A
In this equation the representative thickness-to-chord ratio (t/c)rep is defined as 0.6(t/c)r +
0.3(t/c)k + 0.1(t/c)t . Table 5-3 shows the range of the values for which this method is valid.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
66
Chapter 5. Wing weight estimation methods
The validation results of this method are shown in Figure 5-3. This method has a standard deviation of 6.2%.
Table 5-3: Range of values for using the LTH method
Parameter
Wing mass
Reference wing area
Maximum take-off mass
Representative t/c ratio
Aspect ratio
Quarter-chord sweep angle
Range of values
Unit
4,100 – 50,300
75 – 550
40,000 – 400,000
0.10 – 0.15
6.9 – 9.6
15.0 – 37.5
[kg]
[m2 ]
[kg]
[-]
[-]
[◦ ]
9,#.E#$E!F&)(#,(1.G!HI?JK!=)&$#%&!F&)(#,(1.G!6IDJ!
!
Figure 5-3: Validation results of the LTH method [70].
5-1-6
Elham Modified Weight Estimation Technique (EMWET)
In addition to the statistical methods, a quasi-analytical weight estimation method is also used for
this research. The student version of the EMWET (Elham Modified Weight Estimation Technique)
tool [34] is used as a quasi-analytical weight estimation method. This tool makes use of an analytical
approach to size the wingbox primary structure. The structural and material parameters, the applied
loads and the geometrical data of the wing planform and relative airfoils are required to implement
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 5. Wing weight estimation methods
67
this method. A realistic aerodynamic loads distribution over the wing span is computed using
a commercial Vortex Lattice Method tool, AVL [43]. Several passenger aircraft from different
manufacturers (Fokker, Boeing and Airbus) have been used for validation. The validation study
done by Elham [34] has demonstrated an average accuracy in the order of 2% as shown in Table 5-4.
This level of accuracy is comparable to the higher class methods such as the FEM based weight
prediction methods, while the computational time is in the same order of magnitude as the empirical
methods.
Table 5-4: Validation results of EMWET.
Aircraft
Wto [kg]
Error [%]
Fokker 50
B737-200
B727-300
A300-600R
A330-300
B777-200
20,820
52,390
95,028
170,500
217,000
242,670
−0.72
+0.15
−2.71
+1.86
−2.18
+2.66
The EMWET tool calculates the primary and secondary weights separately. However, the student
version of EMWET uses a regression to find the total wing weight (Ww ) based on the analytically
computed wing box weight (Wcalc ), as shown in Figure 5-4. Applying a power regression results in
the following equation:
0.8162
Ww = 10.147Wcalc
(5-14)
2
with R = 0.9982
Note that by using the student version instead of the full version of EMWET the accuracy of the
results might slightly deviate from the average error of 2%, as will be shown later when comparing
the accuracy of the treated methods.
4
4
x 10
Actual wing weight [kg]
3.5
3
2.5
2
1.5
F50
B737
B727
A300
B777
A330
1
0.5
0
0
0.5
1
1.5
Calculated wing box weight [kg]
2
2.5
4
x 10
Figure 5-4: Actual total weight of the wing versus the analytically computer wing box weight (ribs
and non-optimum weight excluded) [34].
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
68
Chapter 5. Wing weight estimation methods
5-2
Method comparison
The above mentioned weight estimation methods are compared by analyzing their accuracy and sensitivity to several independent wing parameters. A distinction can be made among the independent
wing parameters for the sensitivity analyses:
• wing span b
• root chord length cr
• taper ratio λ (tip chord length divided by root chord length)
• thickness-to-chord ratio at root (t/c)r and tip section (t/c)t
• twist angle at tip section ǫt (downward negative)
• thickness-to-chord ratio at kink section (t/c)k , only for turbojet aircraft
• mid-chord sweep angle Λc/2 , only for turbojet aircraft
Note that the quarter-chord sweep angle of the turboprop aircraft is near 0◦ and consequently the
sensitivity to sweep for the turboprop aircraft is not considered.
flow
λ = ct/cr
tmax
cr
(t/c)max
cam b
c
erlin
e
ct
b/2
(a) Airfoil thickness-to-chord ratio
(b) Planform parameters for turboprop aircraft
flow
cr
Λin
Λc/4 = 0◦
λin = ck /cr
λout = ct /ck
ck
Λout
Λc/2
ct
b
(c) Planform parameters for turbojet aircraft
Figure 5-5: Wing parameters.
Figure 5-5 illustrates the wing parameters, of both the turboprop and turbojet aircraft, that are
used for the method comparison. Table 5-5 gives an overview of the dependency of each method to
those mentioned parameters. Each method depends on the maximum take-off weight, wing span,
sweep angle, taper ratio and the root thickness-to-chord ratio. Investigating the method presented
by Howe (Eq. (5-7)) closely, it can be concluded that the root chord length is canceled out by
the reference wing area and the aspect ratio (as they include cr ). Hence, the Howe method is the
only method that does not depend on cr . Another remarkable aspect, analyzing the parameter
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 5. Wing weight estimation methods
69
dependency of the method, is that only the Torenbeek (2) and EMWET methods require the
maximum zero-fuel weight as an input. Torenbeek (1) and Howe are the only methods independent
of the tip thickness-to-chord ratio.
Table 5-5: Wing weight estimation method sensitivity to wing parameters.
Wing weight as a function of ...
Method
Wto
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
LTH
5-2-1
✓
✓
✓
✓
✓
Wzf
b
cr
Λ
λ
(t/c)r
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
(t/c)k
(t/c)t
✓
✓
✓
✓
✓
✓
✓
✓
ǫt
✓
Accuracy analysis
The accuracy of all methods is analyzed using the actual wing weight of a turboprop and a turbojet
passenger aircraft. Fokker 50 is selected as the turboprop aircraft and Fokker 100 is selected as the
turbojet aircraft. The actual wing weight of those aircraft is available in [51]. Table 5-6 shows the
errors of both the turboprop and the turbojet aircraft. The error was calculated as follows:
ǫ=
Wcalc − Wref
Wcalc
(5-15)
Table 5-6: Wing weight estimation errors.
Method
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
LTH*
Fokker 50 [%]
Fokker 100 [%]
8.89
6.45
0.59
-3.30
12.20
-
-19.70
-19.69
-4.46
2.30
-17.53
-3.77
* this method is not valid for F50 class aircraft
From Table 5-6 one can observe that the quasi-analytical method under-predicts the wing weight
for the turboprop aircraft compared to the other methods. However for the turbojet aircraft this
is the opposite. The quasi-analytical method over-predicts the wing weight, while all the statistical
methods under-predict that.
In general, the weight prediction of Fokker 50 is for most methods within 10%. However, the
Shevell method predicts the wing mass with an error of 12.2% with respect to actual wing mass. A
possible cause of this slightly larger deviation can be explained by the fact that the relation between
the total wing weight and the weight index (Eq. (5-6)) was derived using only turbojet aircraft.
The mass estimated using EMWET, is slighty different than the one presented in Table 5-4. This
difference can be explained by the fact that the student version of EMWET method uses regression
to estimate the total weight based on the primary wing weight. In other words, in the student
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
70
Chapter 5. Wing weight estimation methods
version the secondary wing weight is estimated as a function of the primary wing weight. The Howe
method shows a good accuracy with an error of 0.59%, which can be clarified by the fact that this
method includes a term that depends on the airplane type.
As the quasi-analytical method showed accurate results for the turboprop aircraft, this is also the
case for the turbojet aircraft with an over-prediction of 2.30%. For the same reason as explained
before, the Howe method also estimates the weight accurately for the turbojet aircraft. On the
other hand, the Torenbeek (1), Torenbeek (2) and Shevell methods show an under-prediction of
about 18-20%. Application of these methods for short-haul airplanes could lead to underestimations
of the wing weight. As mentioned by Torenbeek [64], reasons for these underestimations can be
caused by the extra weight (up to 20%) required to provide adequate stiffness against flutter and
weight penalty due to long service life cost.
5-2-2
Sensitivity analysis
A study has been performed to investigate the sensitivity of the above mentioned weight estimation
methods to the design parameters. Consequently, gaining more insight about their behavior can
also serve as a foundation to explain their impact on optimizations. Both the turboprop and the
turbojet aircraft are used for sensitivity analysis.
The sensitivity study is performed for the independent wing parameters for which the wing weight
shows the greatest changes. The independent parameters are: wing span, root chord length, taper
ratio and sweep angle. Wing sweepback is (mainly) applied to increase the critical Mach and drag
divergence Mach numbers of the wing at transonic condition [31]. By increasing the critical Mach
number, the airplane can reach higher speeds. The considered turbojet aircraft flies at transonic
speeds. However, this is not the case for the turboprop aircraft. Therefore, there is no sensitivity
investigation of the sweep angle applied for the turboprop aircraft.
Figure 5-6 and 5-7 illustrate the sensitivity analysis of the mentioned parameters. From Figure 5-6a and 5-7a it can be concluded that a 10% decrease in the wing span results in a weight
reduction of about 10-15% for the turboprop aircraft and about 10-20% for the turbojet aircraft.
In Table 5-5 it is indicated that the Howe method is independent of cr , which can be clearly seen
in Figure 5-6b and 5-7c.
Observing the sensitivity graphs for both turboprop and turbojet aircraft, one can find a remarkable
different behavior for some design variables. For both test cases, the weight prediction methods
have a similar trend for sensitivity with respect to the wing span. The LTH method for the turbojet
case shows the largest change in the wing weight by varying the wing span, see Figure 5-7a. For
the turboprop case, the Howe method is insensitive to the root chord length, as can be derived
from Eq. (5-7). This explains the insensitivity to chord length of the Howe method illustrated in
Figure 5-6b. At the first sight the Torenbeek (1) methods looks insensitive to the root chord length,
although it is not the case as can be clearly seen for the turbojet aircraft in Figure 5-7c. Analysis
of the sensitivity to the wing taper ratio for the turboprop aircraft shows the same trend as the
wing span (see Figure 5-6c), in which the wing weight increases by increasing the taper ratio. In
Figure 5-7d, the wing weight sensitivity analysis with respect to the wing taper for turbojet aircraft
indicates a different trends compared to the turboprop case. To be more specific, the EMWET
and LTH method show opposite behavior than the other methods and to what was observed from
the turboprop aircraft. The sensitivity to the quarter-chord sweep shows a similar trend for all
methods, except the EMWET method is more prone to varying with the sweep angle. The reason
for the different behavior of the quasi-analytical method is that the design variables also affect the
aerodynamic loads required by the EMWET method. From the sensitivity to tip thickness-to-chord
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 5. Wing weight estimation methods
71
ratio graphs (Figure 5-6f and 5-7h), it can be deduced that the Torenbeek (1) and Howe method are
insensitive for (t/c)t . For the turboprop, Figure 5-7g shows that also the Torenbeek (1) and Howe
methods are insensitive for (t/c)k . The sensitivity to MTOW is also analyzed for both aircraft, as
shown by Figure 5-6d and 5-7e. As was indicated by Table 5-5, the Torenbeek (1) is not sensitive
to the MTOW as can be seen in the figures.
1.3
1.1
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
r e f.
1.1
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
1.08
W w /W w
W w /W w
r e f.
1.2
1
1.06
1.04
1.02
0.9
1
0.8
0.98
0.8
0.85
0.9
0.95
1
1.05
b/b r e f.
1.1
1.15
0.8
1.2
0.85
(a) Sensitivity to b
0.9
0.95
c r /c r
r e f.
1
1.05
1.1
(b) Sensitivity to cr
1.06
1.15
1.04
1.1
r e f.
1
W w /W w
W w /W w
r e f.
1.02
0.98
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
0.96
0.94
1.05
1
0.95
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
0.9
0.85
0.92
0.5
1
λ/λ r e f.
0.8
0.8
1.5
(c) Sensitivity to λ
0.95
1
W to/W to
1.05
1.1
1.15
1.2
r e f.
1.03
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
1.01
W w /W w
r e f.
1.01
1
1
0.99
0.99
0.98
0.98
0.85
0.9
0.95
1
1.05
( t/c) t / ( t/c) t
r e f.
1.1
(e) Sensitivity to (t/c)r
1.15
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
1.02
r e f.
1.02
W w /W w
0.9
(d) Sensitivity to MTOW
1.03
0.97
0.8
0.85
1.2
0.97
0.8
0.85
0.9
0.95
1
1.05
( t/c) t / ( t/c) t
r e f.
1.1
1.15
1.2
(f) Sensitivity to (t/c)t
Figure 5-6: Sensitivity analysis for turboprop aircraft.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
72
Chapter 5. Wing weight estimation methods
1.5
1.05
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
LTH
1.3
1.1
1.03
1.02
W w /W w
W w /W w
r e f.
1.2
1
1.01
1
0.9
0.99
0.8
0.98
0.7
0.97
0.6
0.8
0.85
0.9
0.95
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
LTH
1.04
r e f.
1.4
1
1.05
b/b r e f.
1.1
1.15
0.96
0.8
1.2
0.85
(a) Sensitivity to b
0.9
0.95
1
1.05
Λ c / 2 / Λ c/ 2 r e f.
1.1
1.15
1.2
(b) Sensitivity to Λ
1.2
1.03
r e f.
1.02
1.05
W w /W w
W w /W w
r e f.
1.1
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
LTH
1.04
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
LTH
1.15
1
1.01
1
0.99
0.95
0.98
0.9
0.8
0.97
0.85
0.9
0.95
1
c r /c r
1.05
1.1
1.15
1.2
0.9
0.95
r e f.
(c) Sensitivity to cr
1
λ/λ r e f.
1.1
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
LTH
1.08
1.06
r e f.
1.05
W w /W w
W w /W w
r e f.
1.1
1
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
0.9
0.85
0.8
0.8
0.85
0.9
0.95
1
W to/W to
1.05
1.1
1.15
1.04
1.02
1
0.98
0.96
0.94
0.92
0.8
1.2
0.9
0.95
1
1.05
( t/c) r / ( t/c) r
r e f.
1.1
1.15
1.2
(f) Sensitivity to (t/c)r
1.1
1.015
1.08
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
1.04
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
1.01
r e f.
1.06
1.02
W w /W w
r e f.
0.85
r e f.
(e) Sensitivity to MTOW
W w /W w
1.1
(d) Sensitivity to λ
1.15
0.95
1.05
1
1.005
1
0.98
0.995
0.96
0.94
0.92
0.8
0.99
0.85
0.9
0.95
1
1.05
( t/c) k / ( t/c) k
r e f.
1.1
(g) Sensitivity to (t/c)k
1.15
1.2
0.8
0.85
0.9
0.95
1
1.05
( t/c) t / ( t/c) t
r e f.
1.1
1.15
1.2
(h) Sensitivity to (t/c)t
Figure 5-7: Sensitivity analysis for turbojet aircraft.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 6
MDO of aircraft wings
A series of wing MDO assignments have been performed to evaluate the effect of using different
weight estimation on aircraft wing design and optimization. The turboprop and turbojet aircraft
mentioned in the previous chapter are used as test cases. In Figure 6-1 these test cases are
shown. The MDO system consists of three disciplines: aerodynamics, performance and weight. The
sequential quadratic programming (SQP) algorithm from Matlab software is used as the optimizer.
In addition to that, the LOCSMOOTH algorithm (see Chapter 3) was used in an MDO process to
investigate the difference between the global optima and the local optima (using the SQP algorithm).
(a) Fokker 50
(b) Fokker 100
Figure 6-1: Turboprop and turbojet aircraft test cases.
First, the objective function for the scope of this research is introduced together with the design
vectors (different for each test case) and constraints. Then the MDO strategy selected for this
assignment is discussed in Section 6-2. Having discussed the whole MDO procedure, the optimization results are shown and discussed in Section 6-3. In the same section a recommendation was
made to implement additional constraints. In Section 6-4, several additional constraints are examined and implemented. Consequently, new optimization were performed on the same test cases
with the implemented additional constraints. In Section 6-5 the MDO results are shown which are
generated using the additional constraints. Finally, the wing shape was also optimized using the
LOCSMOOTH algorithm and the results were compared in Section 6-6.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
74
6-1
Chapter 6. MDO of aircraft wings
Objective function, design vector and constraints
The optimization problem can be written in the following generic mathematical form:
min
x
subject to
f (x)
gi (x) ≤ 0 i = 1, . . . , m
hj (x) = 0 j = 1, . . . , n
The aircraft maximum take-off weight (MTOW) is considered as the objective function. The design
vectors for two different test cases test cases are as follows:
turboprop aircraft:
turbojet aircraft:
x = [ b, cr , λ, (t/c)r , (t/c)t , ǫt ]
(6-1)
x = [ b, cr , λin , λout , Λin , Λout , (t/c)r , (t/c)k , (t/c)t , ǫt ]
(6-2)
Important to note is that the turbojet aircraft has a kinked wing configuration, as illustrated in
Figure 5-5c, and therefore two taper ratios and two sweep angles are introduced to size the inner
and outer wing. The variable values are allowed to be changed according to engineering expertise
(see Tables 6-1 and 6-2) and must be restrained in order to generate a realistic design. The
quarter-chord sweep angle is kept constant (Λc/4 = 0◦ ) for the turboprop case.
The MTOW is considered to be the sum of three terms: the wing weight, the mission fuel weight
and the rest weight.
Wto = Ww + Wf + Wrest
(6-3)
The rest weight is initialized based on the reference aircraft actual weight and remains constant
throughout the optimization. The wing weight is estimated using different weight estimation methods. The fuel weight is calculated using the aircraft range performance equation (including the
aircraft aerodynamic efficiency) and a series of statistical data.
Table 6-1: Design vector for turboprop aircraft.
Design variables
Initial values
Lower bounds
Upper bounds
b
cr
λ
(t/c)r
(t/c)t
ǫt
[m]
[m]
[-]
[-]
[-]
[◦ ]
29.00
20
40
3.56
2.5
5
0.40
0.3
0.5
0.21
0.25
0.10
0.15
0.25
0.10
-2.0
-5
5
Table 6-2: Design vector for turbojet aircraft.
Design variables
Initial values
Lower bounds
Upper bounds
b
cr
λin
λout
Λin
Λout
(t/c)r
(t/c)k
(t/c)t
ǫt
[m]
[m]
[-]
[-]
[◦ ]
[◦ ]
[-]
[-]
[-]
[◦ ]
28.08
20
40
5.60
3.5
6.5
0.64
0.5
0.8
0.35
0.3
0.5
16.24
5
22
11.90
5
35
0.12
0.08
0.15
0.12
0.08
0.15
0.09
0.08
0.15
-2.00
-5
5
Optimizing the aircraft wing for cruise condition may result in a unrealistic value for wing loading
because the value of wing loading is affected by some other segments of flight such as take-off and
landing (the effect of wing loading on the aircraft stall speed). Hence, a constraint is defined to
keep the wing loading smaller or equal to that of the reference aircraft.
Wto
Wto
≤
(6-4)
S
S ref.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 6. MDO of aircraft wings
75
The flight conditions at which the turboprop and turbojet aircraft operate, are given in Table 6-3.
Based on the flight altitude, the other required parameters (such as air density and viscosity) can
be derived.
Table 6-3: Flight conditions of test cases.
Parameter
Turboprop aircraft
Turbojet aircraft
7620
0.55
10675
0.77
Cruise altitude [m]
Mach number at cruise
The following tolerances were defined for all optimizations performed using the fmincon function
in Matlab:
• Termination tolerance on the normalized objective function value of 1e-4 (default 1e-6).
• Violation tolerance on normalized constraints of 1e-3 (default 1e-6).
6-2
MDO strategy
The Individual Discipline Feasible (IDF) strategy [71] is selected for this MDO assignment. Using
this strategy the coupling between different disciplines is removed. In addition, the optimizer is
made responsible for the consistency of the coupling variables.
OPT
W f* ,Ww*
W f* ,Ww*
Geom.
W f* ,Ww*
Geom.
Loads (EMWET)
Ae
L/D
Pe
Wf
We
Ww
Figure 6-2: Design structure matrix of MDO system with aerodynamics, performance and weight
modules.
The design structure matrix of the MDO system is illustrated in Figure 6-2 with its three modules:
aerodynamics (Ae), performance (Pe) and weight (We). In this workflow, Wf∗ and Ww∗ represent
the coupling variables (also referred to as the surrogate values). Using IDF there is no feedback
between the disciplines and therefore there is no necessity for an iterative procedure to converge
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
76
Chapter 6. MDO of aircraft wings
the multidisciplinary system. A drawback of using this strategy is that additional design variables
are introduced. The optimization problem using IDF can be stated as follows:
min
w.r.t
subject to
Wto (X)
X = (x, Wf∗ , Ww∗ )
Wto
Wto
≤
S
S ini
Wf∗ = Wf
Ww∗ = Ww
As mentioned before, quasi-analytical weight prediction methods require the aerodynamic loads
in order to calculate the forces and moments applied to the structure. These aerodynamic loads
are calculated by the aerodynamics module. They are then passed to the weight module when
the EMWET is used for wing weight estimation. The flow diagram of each module is given in
Appendix F.
Aerodynamics module
The strip theory [44] is used to develop an aerodynamic solver. The developed quasi-3D aerodynamic solver, see Chapter 4, is used to accurately calculate the total wing drag. Once the total lift
coefficient and the total wing drag coefficient are known, the wing lift-to-drag ratio can be calculated. In order to calculate the total lift to drag ratio of the aircraft, the drag of the other parts
of the aircraft (e.g. fuselage) should be added to the drag of the wing. The drag of the aircraft
without wing is considered as the “rest drag”. The rest drag of the reference aircraft is determined
and it is normalized using the wing area and dynamic pressure to obtain the drag coefficient of the
aircraft minus the wing. At each objective function evaluation, this coefficient is added to the wing
drag coefficient calculated by the aerodynamic solver.
Performance module
The required fuel for the flight mission is calculated using the method presented by Roskam [72]. In
this method the required fuel for the cruise is calculated using the Bréguet range equation [1, 73],
while some statistical factors are used to estimate the fuel weight of the other segments of the
flight mission, see Table 6-4. Each fuel weight fraction Mffi indicates the ratio of the total aircraft
weight at the end of the flight segment divided by the total aircraft weight at the beginning of the
segment, see Figure 6-3.
The total fuel weight fraction indicates the consumed fuel as a ratio of the total aircraft weight at
the end of the flight mission divided by the fuel weight at the beginning.
Mff = Mff1 · Mff2 · . . . · Mffn
(6-5)
Hence, the fuel weight can be determined including a 5% of the total fuel weight as reserve fuel
using the following equation:
Wf = 1.05 (1 − Mff ) Wto
(6-6)
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 6. MDO of aircraft wings
77
5
4
6
3
1
2
7
Figure 6-3: Normal flight mission definition with flight segments.
Table 6-4: Fuel fraction for each segment in simple flight mission, suggested values from Roskam
[72].
Fuel weight fraction, Mffi
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Start & warm-up
Taxi
Take-off
Climb
Cruise
Descent
Landing, taxi & shutdown
Turboprop aircraft
Turbojet aircraft
0.990
0.995
0.995
0.985
Calculated
0.985
0.995
0.990
0.990
0.995
0.980
Calculated
0.990
0.992
Weight module
In the weight module, the weight of the wing is predicted using one of the proposed wing weight
estimation methods in Chapter 5.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
78
6-3
Chapter 6. MDO of aircraft wings
MDO results
Tables 6-5 and 6-6 provide the results for the optimized wings for the turboprop and turbojet
aircraft. Each table also contains the percentage of the reduction in MTOW compared to the
reference design. The achieved reduction in MTOW is up to 3.45% for the turboprop aircraft and
up to 8.08% for the turbojet aircraft. The planforms of the optimized wings are shown in Figure 6-4.
Table 6-5: Results of wing optimization for turboprop aircraft.
Method
Reduction
Wto [%]
b
cr
λ
(t/c)r
(t/c)t
ǫt
[m]
[m]
[-]
[-]
[-]
[◦ ]
3.45
2.83
2.76
2.51
1.93
29.00
20.00
20.00
20.97
20.00
20.18
3.46
5.00
4.91
5.00
4.56
5.00
0.40
0.37
0.40
0.31
0.50
0.39
0.21
0.25
0.25
0.25
0.15
0.25
0.15
0.19
0.25
0.15
0.16
0.22
-2.00
-1.70
-0.70
-0.64
-2.45
-1.54
Reference wing
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
Table 6-6: Results of wing optimization for turbojet aircraft.
Method
Reference wing
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
LTH
Reduction
Wto [%]
b
cr
λin
λout
Λin
Λout
(t/c)r
(t/c)k
(t/c)t
ǫt
[m]
[m]
[-]
[-]
[◦ ]
[◦ ]
[-]
[-]
[-]
[◦ ]
4.44
4.41
4.54
4.84
4.03
8.08
28.08
21.62
21.11
24.69
20.01
20.71
20.00
5.60
6.50
6.50
6.50
6.49
6.50
6.50
0.64
0.67
0.70
0.59
0.80
0.74
0.80
0.35
0.38
0.36
0.30
0.32
0.31
0.42
16.24
21.13
21.10
20.09
19.75
17.64
19.78
11.90
13.58
13.78
14.30
11.58
15.92
12.33
0.12
0.14
0.14
0.15
0.13
0.12
0.15
0.12
0.08
0.08
0.08
0.08
0.09
0.08
0.09
0.10
0.10
0.08
0.08
0.12
0.08
-2.00
-2.15
-2.15
-2.51
-2.01
-2.45
-1.82
A noteworthy aspect of the optimization results, is that the wing span has been decreased towards
its lower bound value. Sensitivity analyses of the weight estimation methods demonstrated that the
wing span has a large impact on the wing weight. The objective function is affected by both the wing
weight and the fuel weight, where the fuel weight is a function of the aerodynamic efficiency and
the aircraft total weight. Increasing the wing span increases the wing aspect ratio (for a constant
chord length), which results in a lower induced drag but higher friction drag (as the wetted area
increases). Figure 6-5 shows the variation of the wing weight and the fuel weight of the turbojet
aircraft with the wing span. It is remarkable that the aerodynamic module shows little changes in
aerodynamic efficiency for varying wing span. The figure indicates that the wing weight shows a
greater impact on the objective function than the aerodynamic efficiency. This explains the fact that
the optimizer drives the wing span towards its lower bound. Other design variables (i.e. wing taper,
root chord length, kink thickness-to-chord ratio) are also driven towards lower/upper bounds by the
optimizer. In contrast to the statistical weight methods, the quasi-analytical method (EMWET)
does not drive the root thickness-to-chord ratio towards its upper bound. Note that the EMWET
method, is the only tool used that is sensitive to the wing twist. Hence, the value of the tip twist is
purely driven by the aerodynamic aspects in case of using one of the statistical weight estimation
methods.
As the optimization results show unrealistic configurations, additional constraints are required to
model a more realistic situation.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 6. MDO of aircraft wings
79
14
Reference wing
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
14
12
12
10
Spanwise position [m]
Spanwise position [m]
10
8
6
8
6
4
4
2
2
0
12
14
16
18
20
Reference wing
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Kroo
LTH
22
0
Longitudinal position [m]
2
4
6
8
Longitudinal position [m]
(a) Turboprop aircraft
(b) Turbojet aircraft
Figure 6-4: Planform geometry of the optimized wings.
14000
Weight [kg]
12000
Wing weight
Fuel weight
Wing + Fuel weight
10000
8000
6000
4000
2000
0.8
0.85
0.9
0.95
1
b/b r ef.
1.05
1.1
1.15
1.2
Figure 6-5: Sensitivity of the wing weight (calculated using EMWET) and the fuel weight with
respect to the wing span.
6-4
Additional constraints
In order to obtain more realistic wing geometry, two additional constraints are applied. A fuel
volume constraint is used to ensure that the required fuel for the flight mission can be stored in the
wing fuel tanks (available fuel volume). The fuel volume is calculated using the mission fuel weight
and the Jet-A fuel density (according to Air BP [74]).
(Vf )req. ≤ (Vf )av.
Another constraint is applied to keep aspect ratio
Wing Shape Multidisciplinary Design Optimization
(6-7)
A larger than typical values for turboprop and
Jan Mariens
80
Chapter 6. MDO of aircraft wings
turbojet aircraft categories [75, 76]. This constraint is implemented in order to take the performance
requirement for the second climb segment into account.
A≥
10
for turboprop aircraft
(6-8)
8
for turbojet aircraft
(6-9)
With these additional constraints, the optimization problem can be rewritten:
min
w.r.t
subject to
Wto (X)
X = (x, Wf∗ , Ww∗ )
Wto
Wto
≤
S
S ref.
(
10 for turboprop
≥
8
for turbojet
A
(Vf )req. ≤ (Vf )av.
Wf∗ = Wf
Ww∗ = Ww
6-5
MDO results using additional constraints
Tables 6-7 and 6-8 provide the MDO results for the turboprop and turbojet aircraft. Both cases
show smaller reductions in MTOW using three constraints instead of one.
Table 6-7: Results of wing optimization for turboprop aircraft using new set of constraints.
Method
Reference wing
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
Reduction
Wto [%]
[m]
b
[m]
cr
[-]
λ
(t/c)r
(t/c)t
[-]
[◦ ]
1.97
2.14
1.76
1.38
1.61
29.00
26.35
26.34
26.36
26.33
26.51
3.46
4.05
4.05
4.01
4.04
3.93
0.40
0.30
0.30
0.31
0.30
0.35
0.21
0.25
0.25
0.25
0.22
0.25
0.15
0.10
0.25
0.17
0.19
0.25
-2.00
-2.18
-1.36
-2.41
-2.07
1.11
[-]
ǫt
The optimizations convergence history is shown in Figure 6-7. Computational times of the MDO
processes are shown in Table 6-9. All optimizations were performed using Matlab R2010b on a
computer having an Intel Core2Duo E4400 (2.00Ghz), with 2Gb RAM and Windows 7 (64-bit
version). For the turboprop aircraft, the optimizer requires most iterations for the Torenbeek (1)
and Howe methods. Whereas, for the turbojet case most iterations are required for EMWET and
Howe method.
Each function evaluation using a statistical method takes about 39 seconds, whereas the EMWET
method takes 51 seconds. The additional time required for the quasi-analytical method is due to
the aerodynamic load calculation (see Figure 6-2).
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 6. MDO of aircraft wings
81
Table 6-8: Results of wing optimization for turbojet aircraft using new set of constraints.
Method
Reduction
Wto [%]
b
[m]
cr
[m]
λin
[-]
λout
[-]
Λ◦in
[ ]
Λ◦out
[ ]
(t/c)r
[-]
(t/c)k
[-]
(t/c)t
[-]
ǫt
[◦ ]
4.40
4.17
4.60
2.53
3.91
4.43
28.08
25.65
25.68
25.80
26.17
25.73
25.93
5.60
6.50
6.50
6.50
5.99
6.50
6.02
0.64
0.53
0.53
0.55
0.63
0.54
0.61
0.35
0.38
0.38
0.31
0.32
0.34
0.35
16.24
21.82
22.00
22.00
16.62
20.87
20.82
11.90
13.21
15.39
11.40
11.52
12.97
11.39
0.12
0.14
0.15
0.15
0.13
0.15
0.15
0.12
0.08
0.09
0.08
0.09
0.08
0.08
0.09
0.11
0.10
0.10
0.11
0.12
0.10
-2.00
-2.13
-2.05
-2.07
-2.62
-2.55
-1.89
Reference wing
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
LTH
12
12
10
10
Spanwise position [m]
Spanwise position [m]
14
Reference wing
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
14
8
6
8
6
4
4
2
2
0
12
14
16
18
20
Reference wing
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
LTH
22
0
Longitudinal position [m]
2
4
6
8
Longitudinal position [m]
(a) Turboprop aircraft
(b) Turbojet aircraft
Figure 6-6: Planform geometry of the optimized wings using new set of constraints.
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
1
0.995
0.99
0.985
0.98
0.975
0
10
20
30
40
Iteration
50
60
(a) Turboprop aircraft
70
normalized Objective function value
normalized Objective function value
1.005
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
LTH
1
0.99
0.98
0.97
0.96
0.95
0
10
20
Iteration
30
40
(b) Turbojet aircraft
Figure 6-7: Optimization convergency.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
82
Chapter 6. MDO of aircraft wings
Table 6-9: MDO computational times using the new constraint set.
Method
Torenbeek (1)
Torenbeek (2)
Howe
EMWET
Shevell
LTH
6-6
Turboprop
Turbojet
[hrs]
[hrs]
7.61
3.88
10.32
5.21
8.28
/
6.82
7.18
10.43
14.48
4.85
4.06
MDO results using the LOCSMOOTH algorithm
In this section, the optimized turbojet aircraft wing configurations using the SQP and LOCSMOOTH
algorithms are compared. Table 6-10 provides the MDO results using the wing loading constraint
for both optimization algorithms. The MDO results using the additional constraints are shown in
Table 6-11. The optimization results in this section were all generated using the EMWET wing
weight estimation method. From these tables, it can be deduced that the LOCSMOOTH algorithm
found higher reductions than the SQP algorithm. For the 1 constraint case (the wing loading
constraint) the SQP algorithm yields a reduction in MTOW of 4.84% and the LOCSMOOTH
algorithm a reduction of 5.08%. The difference 0.24% between these two reduction results in a
difference of 103 kg. Whereas for the 3 constraints case (wing loading constraint plus additional
constrains), this difference is 0.42% resulting in a difference of 173 kg.
Table 6-10: Comparison MDO results of turbojet aircraft with 1 constraint using SQP and LOCSMOOTH.
Method
Reduction
Wto [%]
b
cr
λin
λout
Λin
Λout
(t/c)r
(t/c)k
(t/c)t
ǫt
[m]
[m]
[-]
[-]
[◦ ]
[◦ ]
[-]
[-]
[-]
[◦ ]
Reference wing
EMWET:
-
28.08
5.60
0.64
0.35
16.24
11.90
0.12
0.12
0.09
-2.00
– SQP
– LOCSMOOTH
4.84
5.08
20.01
20.05
6.49
6.49
0.80
0.80
0.32
0.41
19.75
19.79
11.58
7.02
0.13
0.13
0.08
0.08
0.08
0.13
-2.01
-2.61
Table 6-11: Comparison MDO results of turbojet aircraft with 3 constraints using SQP and LOCSMOOTH.
Method
Reduction
Wto [%]
b
cr
λin
λout
Λin
Λout
(t/c)r
(t/c)k
(t/c)t
ǫt
[m]
[m]
[-]
[-]
[◦ ]
[◦ ]
[-]
[-]
[-]
[◦ ]
Reference wing
EMWET:
-
28.08
5.60
0.64
0.35
16.24
11.90
0.12
0.12
0.09
-2.00
– SQP
– LOCSMOOTH
2.53
2.95
26.17
26.11
5.99
5.85
0.63
0.66
0.32
0.30
16.62
14.79
11.52
6.68
0.13
0.14
0.09
0.09
0.11
0.10
-2.62
-4.59
Figure 6-8 shows the resulting wing planforms for both algorithms using both constraint cases.
It can be deduced that the LOCSMOOTH algorithm was able to find a better reduction (higher
reduction in MTOW) compared to the SQP algorithm. However, these further reductions come at
a cost: the computational time increases significantly, as indicated by Table 6-12.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 6. MDO of aircraft wings
Reference wing
EMWET−SQP
EMWET−LOCSMOOTH
12
12
10
10
8
6
8
6
4
4
2
2
0
−2
0
2
4
Reference wing
EMWET−SQP
EMWET−LOCSMOOTH
14
Spanwise position [m]
Spanwise position [m]
14
83
0
−2
6
0
2
4
Longitudinal position [m]
Longitudinal position [m]
(a) Using 1 constraint
(b) Using 3 constraints
6
Figure 6-8: Difference MDO results using SQP and LOCSMOOTH.
Table 6-12: MDO Computational times with EMWET using SQP and LOCSMOOTH.
EMWET
SQP
LOCSMOOTH
Wing Shape Multidisciplinary Design Optimization
1 constraint
3 constraints
[hrs]
[hrs]
10.02
179.80
14.48
330.92
Jan Mariens
84
Jan Mariens
Chapter 6. MDO of aircraft wings
Wing Shape Multidisciplinary Design Optimization
Chapter 7
Conclusions & recommendations
Based on the research and developments presented in the preceding chapters, different conclusions
are drawn. Additionally, several recommendations can be given for further developments of these
tools and further research on these topics.
7-1
Conclusions
The conclusions presented are based on the objectives stated in Chapter 1. The main objective of
this thesis research was to
Investigate the effect of using different weight estimation methods on the outcome
in a wing design task using multidisciplinary design optimization techniques.
which is accompanied by the following sub-goals:
• Develop a quasi-three-dimensional aerodynamic solver to calculate the wing aerodynamic
characteristics.
• Compare the different weight estimation methods, with low and medium levels of fidelity, by
analyzing their accuracy and sensitivity to wing parameters.
• Implement a global optimization algorithm that uses gradient-based techniques and local
optima smoothing. The results using this algorithm are also compared to a local solution.
First, the conclusions drawn from the sub-goals are addressed. When all the sub-goals are discussed,
the main objective can be elaborated as well.
Quasi-3D aerodynamic solver A strip method was combined with a vortex lattice method and
the simple sweep theory to develop an aerodynamic solver. The developed solver combines
combines the results of a three-dimensional vortex lattice method with the results of a viscous
two-dimensional airfoil analyzer to accurately calculate the total wing drag. The solver was
validated against experimental data and data from other CFD tools. From these validations,
it was shown that the quasi-three-dimensional shows good agreements with the experimental
data and other aerodynamic solvers. Except, at higher speeds (especially at free-stream Mach
numbers higher than 0.75) the developed solver is not able to simulate 3D aerodynamic effects
(such as root and tip effect). This difference of these effects was shown in Subsection 4-8-2
and is due to the use of the simple sweep theory.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
86
Chapter 7. Conclusions & recommendations
Weight estimation methods Both a quasi-analytical and statistical wing weight predictions methods were compared to each other by analyzing their accuracy and sensitivity to different wing
parameters. Most methods show results close to the reference weight for the turboprop
aircraft class. For the turbojet aircraft class, on the other hand, most statistical methods
under-predict the wing weight. This under-prediction of about 18-20% can be explained by
the fact that short-haul airplanes require extra weight (up to 20%) to provide appropriate
stiffness against flutter and a weight penalty due to long service lift cost (according to Torenbeek [30]). Furthermore, it was shown that the statistical methods are insensitive to some
design variables.
Local smoothing for global optimization The “local optima smoothing for global optimization”
algorithm was capable of finding the theoretical optimum for several benchmark test functions.
A classical aerodynamic problem was also tested, at which the global optimizer was able to
find the global optimum. It can be concluded that this hybrid algorithm, an appropriate
mixture of local approximation and global exploration, is efficient and robust in solving global
optimization problems. Another important is that the input settings (sphere radii and number
of sampled points) for this algorithm are problem dependent. For example, very noisy functions
mostly require smaller sphere radii.
Different wing weight estimation methods were implemented in a wing design task using MDO
techniques. Two test cases, a turboprop and a turbofan passenger aircraft, were used. Comparison
of the implemented weight prediction methods, based on their accuracy and sensitivity to wing
shape parameters, gave better insight into each method. MDO results for both test cases were
shown using one constraint (on wing loading), yielding optimized configurations where some design
variables were driven towards their lower/upper bounds. These results suggest that implementing
additional constraints to the optimization problem will yield a more realistic optimization. Doing
so, two more constraints concerning the aspect ratio and fuel volume were implemented. The
MDO results using three constraints, showed that the problem from the one constraint case was
partially solved. Realistic optimization problems contain significantly more constraints. However,
the number of constraints used was sufficient for the scope of this research.
When a sufficient geometry is available in early design stages, it is of great interest to use methods
with higher design-sensitivity. This allows engineers to gain more knowledge about the design and
the design-sensitivity. The statistical methods that were used for the wing weight calculations,
were insensitive to some design variables (tip twist, tip thickness-to-chord ratio and root chord).
EMWET, the quasi-analytical method, is sensitive to all used variables. However, using EMWET
(quasi-analytical weight predictions) comes at the cost of increasing the computational time for
each function evaluation by 30%.
7-2
Recommendations
Based on the work performed, several recommendations are given for further research and further
improvements that can be made.
Exploit parallel computing On a single computer, a multidisciplinary design optimization of an
aircraft wing takes several of hours. Especially, when using the global optimization algorithm
(LOCSMOOTH), this process can take up to two weeks. It is expected that the runtime
for these optimizations can be significantly reduced by using parallel computing capabilities.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Chapter 7. Conclusions & recommendations
87
To run the optimization process in parallel, several modification in the program are required.
Parallel computing can be applied in the quasi-3D aerodynamic solver, where the viscous
airfoil drag calculations can be done parallel.
Quasi-3D aerodynamic solver The quasi-3D aerodynamic solver could be further developed, such
that it can calculate more aerodynamic characteristics. Useful additions might be: calculations of maximum lift coefficient, stall characteristics, etc. Furthermore, the viscous airfoil
calculation that were used are capable of calculating the aerodynamics of wing sections with
high lift devices. Therefore, this tool can be further extended to analyze wings with high lift
devices.
Multi-stage MDO Multi-stage frameworks can be used to obtain high-fidelity design results. An
application of a multi-stage framework for this research might be to first optimize the wing
planform (first stage) and consequently to optimize the airfoil shapes (second stage). While
optimizing one stage, the design variables of the other stage are “frozen”.
More realistic optimizations For the scope of this research, a small amount of constraints were
used. For more realistic optimization, many more constraints should be implemented. For
example, additional constraints on performance, stability, control, etc.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
88
Jan Mariens
Chapter 7. Conclusions & recommendations
Wing Shape Multidisciplinary Design Optimization
Bibliography
[1] Anderson, J. D., Aircraft Performance and Design, WCB/McGraw-Hill, 1999.
[2] van Tooren, M., Steenhuizen, D., van Gerwen, D., La Rocca, G., and Schroijen, M., Advanced Design
Methodologies (TU Delft – Course notes, Chapter 2), Delft University of Technology, 2009.
[3] Sobieszczanski-Sobieski, J. and Haftka, R. T., “Multidisciplinary aerospace design optimization: survey
of recent developments,” Structural and Multidisciplinary Optimization, Vol. 14, 1997, pp. 1–23.
[4] Alexandrov, N. M. and Hussaini, M. Y., editors, Multidisciplinary Design Optimization: State of the
Art, SIAM, 1997.
[5] American Institute of Aeronautics and Astronautics, White Paper on Industrial Experience with MDO,
AIAA Technical Committee on Multidisiplinary Design Optimization (MDO), 1999.
[6] Martins, J. R. R. A. and Marriage, C. J., “An Object-Oriented Framework for Multidisciplinary Design
Optimization,” 3rd AIAA Multidisciplinary Design Optimization Specialist Conference, Vol. 36, No. 4,
2009, pp. 1–25.
[7] Kroo, I., “Distributed Multidisciplinary Design and Collaborative Optimization,” Lecture series on Optimization Methods and Tools for Multicriteria/Multidisciplinary Design, The von Karman Institute,
November 2004.
[8] Alonso, J. J. and Fike, J., Introduction to Multidisciplinary Design Optimization (Course material,
Chapter 7: MDO Architectures), Stanford University, http://adl.stanford.edu/aa222/Home.html,
2010.
[9] Cramer, E. J., Dennis, J. E. J., Frank, P. D., Lewis, R. M., and Shubin, G. R., “Problem Formulation
for Multidisciplinary Optimization,” SIAM Journal on Optimization, Vol. 4, No. 4, 1994, pp. 754–776.
[10] Sobieszczanski-Sobieski, J. and Haftka, R. T., “Multidisciplinary aerospace design optimization: survey
of recent developments,” Structural and Multidisciplinary Optimization, Vol. 14, 1997, pp. 1–23.
[11] Nadarajah, S., Review of numerical optimization methods, Brachistochrone problem, McGill University,
October 2004.
[12] Keane, A. J. and Nair, P. B., Computational Approaches for Aerospace Design: The Pursuit of Excellence, John Wiley & Sons Ltd., 2005.
[13] Whitney, E. J., Gonzalez, L. F., and Periaux, J., Multidisciplinary Methods for Analysis Optimization
and Control of Complex Systems, Vol. 6 of Mathematics in Industry , Springer Berlin Heidelberg, 2005.
[14] Mason, W., Applied Computational Aerodynamics, Volume 1: Foundations and Classical Pre-CFD
Methods, Department of Aerospace and Ocean Engineering - Virginia Polytechnic Institute and State
University, http://www.dept.aoe.vt.edu/~mason/Mason_f/CAtxtTop.html, 1992–1995.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
90
BIBLIOGRAPHY
[15] Raymer, D. P., Aircraft Design: a Conceptual Approach, AIAA (American Institute of Aeronautics and
Astronautics) Education Series, Reston, VA (USA), 4th ed., 2006.
[16] Mason, W., “On the Use of the Potential Flow Model for Aerodynamic Design at Transonic Speeds,”
AIAA paper 95-0741 , 33rd Aerospace Sciences Meeting & Exhibit, January 1995, pp. 1–6.
[17] Jameson, A., Encyclopedia of Computational Mechanics (Chapter 11:Aerodynamics), John Wiley &
Sons Ltd., 2004.
[18] Kroo, I., Aerodynamic Modeling for Simulation and Control (Course material, Chapter 3: Approaches to
Aerodynamic Modeling), Stanford University, http://adg.stanford.edu/aa208/amsc.html, 2002.
[19] Bakker, A., Computational Fluid Dynamics (Course material, Lecture 5: Finite volume solvers), Dartmout College, http://www.bakker.org/dartmouth06/engs150/05-solv.pdf, 2006.
[20] Kuzmin, D., Introduction to Computational Fluid Dynamics (Course material, Lecture 3: Discretization
techniques), Institute of Applied Mathematics – University of Dortmund, http://www.mathematik.
uni-dortmund.de/~kuzmin/cfdintro/lecture3.pdf, 2010.
[21] Liu, G. and Xu, G. X., “A gradient smoothing method (GSM) for fluid dynamics problems,” International
Journal for Numerical Methods in Fluids, Vol. 58, No. 10, March 2008, pp. 1101–1133.
[22] Blazek, J., Computational Fluid Dynamics: Principles and Applications, Elsevier Science Ltd., Oxford
(UK), 2001.
[23] Versteeg, H. and Malalasekera, W., An introduction to computational fluid dynamics: The finite volume
method, Longman Scientific & Technical, Essex (England), 1st ed., 1995.
[24] Prandtl, L., “Applications of modern hydrodynamics to aeronautics,” Tech. rep., NACA Rep. 116
(National Advisory Committee for Aeronautics), 1923.
[25] Anderson, J. D., Fundamentals of Aerodynamics, McGraw Hill, 4th ed., 2007.
[26] Moran, J., An Introduction to Theoretical and Computational Aerodynamics, Aeronautical Engineering
Series, Dover Publications, 1984.
[27] Katz, J. and Plotkin, A., Low-Speed Aerodynamics, Cambridge Aerospace Series, Cambridge University
Press, 2001.
[28] Tomac, M., Adaptive-fidelity CFD for Predicting Flying Qualities in Preliminary Aircraft Design, Master’s
thesis, KTH School of Engineering Sciences, Stockholm (Sweden), January 2011.
[29] Howe, D., Aircraft Conceptual Design Synthesis, Professional Engineering Publishing, London and Bury
St Edmunds (UK), 2000.
[30] Torenbeek, E., Synthesis of Subsonic Airplane Design, Delft University Press, Delft (Netherlands), 1976.
[31] Shevell, R. S., Fundamentals of Flight, Prentice-Hall, Michigan, US, 1983.
[32] Torenbeek, E., “Development and Application of a Comprehensive, Design-sensitive Weight Prediction Method for Wing Structures of Transport Category Aircraft,” Report LR-693, Delft University of
Technology, Delft (Netherlands), September 1992.
[33] Macci, S. H., Semi-Analytical Method for Predicting Wing Structural Mass, 54th Annual Conference,
Society of Allied Weight Engineers, Inc., SAWE Paper No. 2282, May 1995.
[34] Elham, A., van Tooren, M. J. L., and La Rocca, G., An Advanced Quasi-Analytical Weight Estimation
Method for Airplane Lifting Surfaces, 71st International Conference on Mass Properties, Society of
Allied Weight Engineers, Inc., SAWE Paper No. 3571, May 2012.
[35] Bindolino, G., Ghiringhelli, G., Ricci, S., and Terraneo, M., “Multilevel Structural Optimization for
Preliminary Wing-Box Weight Estimation,” Journal of Aircraft, Vol. 47, No. 2, March-April 2010.
[36] Laban, M., Arendsen, P., Rouwhorst, W., and Vankan, W., “Multidisciplinary Design Optimization for
a Blended Wing Body Transport Aircraft with Distributed Propulsion,” AIAA paper 2001-5446 , 9th
AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, September 2002, pp. 1–11.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
BIBLIOGRAPHY
91
[37] Buttazzo, G., Frediani, A., Rizzo, E., and Frediani, A., “Application of Optimisation Algorithms to Aircraft Aerodynamics,” Variational Analysis and Aerospace Engineering , Vol. 33 of Springer Optimization
and Its Applications, Springer New York, 2009, pp. 419–446.
[38] Addis, B., Locatelli, M., and Schoen, F., “Local optima smoothing for global optimization,” Optimization Methods and Software, Vol. 20, No. 4-5, August 2005, pp. 417–437.
[39] Addis, B. and Leyffer, S., “A Trust-Region Algorithm for Global Optimization,” Computational Optimization and Applications, Vol. 35, No. 3, June 2006, pp. 287–304.
[40] Rizzo, E., Optimization Methods Applied to the preliminary design of innovative non conventional
aircraft configurations, Ph.D. thesis, Pisa University, June 2009.
[41] Yang, X. S., Engineering Optimization: An Introduction with Metaheuristic Applications, Wiley, 2010.
[42] Sarker, R., Mohammadian, M., and Yao, X., Evolutionary Optimization, International Series in Operations Research & Management Science, Kluwer Academic Publishers, 2002.
[43] Drela, M. and Youngren, H., AVL (Athena Vortex Lattice), Massachusetts Institute of Technology,
http://web.mit.edu/drela/Public/web/avl/, Last accessed: May 2011.
[44] Cosyn, P. and Vierendeels, J., “Numerical Investigation of Low-Aspect-Ratio Wings at Low Reynolds
Numbers,” Journal of Aircraft, Vol. 43, No. 3, 2006, pp. 713–722.
[45] Bramwell, A., Helicopter Dynamics, John Wiley & Sons, New York (US), 1976.
[46] Gessow, A. and Myers, G., Aerodynamics of the Helicopter , the Macmillan Company, New York (US),
1952.
[47] C.J.Sequira, D.J.Willis, and J.Peraire, “Comparing aerodynamic models for numerical simulation of
dynamics and control of aircraft,” AIAA paper 2006-1254 , 44th AIAA Aerospace Sciences Meething,
2006, pp. 1–20.
[48] Holt, D. R., “Introduction to transonic aerodynamics of aerofoils and wings,” Tech. rep., ESDU (Engineering Sciences Data Unit) 90008, April 1990.
[49] Drela, M., “N+3 Aircraft Concept Designs and Trade Studies – Volume 2 (Appendices): Design Methodologies for Aerodynamics, Structures, Weight, and Thermodynamic Cycles (Appendix A: TASOPT –
Transport Aircraft System OPTimization),” Final Report NASA/CR-2010-216794/VOL2, NASA Glenn
Research Center, December 2010.
[50] Desktop Aeronautics, Inc., Oblique Flying Wings: An Introduction and White Paper , http://www.
desktop.aero/library/ofwwhitepaper.pdf, June 2005.
[51] Obert, E., Slingerland, R., Leusink, D., van den Berg, T., Koning, J., and van Tooren, J., Aerodynamic
Design of Transport Aircraft, IOS Press, Amsterdam (The Netherlands), 2009.
[52] Kroo, I., “DRAG DUE TO LIFT: Concepts for Prediction and Reduction,” Annual Review of Fluid
Mechanics, Vol. 33, No. 1, 2001, pp. 587–617.
[53] Selig, M. S., Lyon, C. A., and Giguere, P., Summary of Low-Speed Airfoil Data, Vol. 2, SoarTech
Publications, Virgnia Beach, VA (US), 1995.
[54] Abbott, I. and Von Doenhoff, A., Theory of Wing Sections: Including a Summary of Airfoil Data, Dover
Books on Physics and Chemistry, Dover Publications, 1959.
[55] Drela, M. and Youngren, H., XFOIL: Subsonic Airfoil Development System, Massachusetts Institute of
Technology, http://web.mit.edu/drela/Public/web/xfoil/, Last accessed: May 2011.
[56] Drela, M., “XFOIL: An Analysis and Design System for Low Reynolds Number Airfoils,” Conference on
Low Reynolds Number Airfoil Aerodynamics, University of Notre Dame, 1989, pp. 1–12.
[57] The Royal Aeronautical Society, “VGK method for two-dimensional aerofoil sections (Part 1: principles
and results),” Tech. rep., ESDU (Engineering Sciences Data Unit) 96028, October 1996.
[58] Drela, M., MSES: Multi-Element Airfoil Design/Analysis Software, Massachusetts Institute of Technology, http://raphael.mit.edu/drela/msessum.ps, Last accessed: May 2011.
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
92
BIBLIOGRAPHY
[59] Anderson, R. F., “Determination of the characteristics of tapered wings,” Tech. Rep. 572, National
Advisory Commitee for Aeronautics, 1940.
[60] Maskew, B., PROGRAM VSAERO: A computer program for calculating the non-linear aerodynamic
characteristics of arbitrary configurations: User’s manual, NASA contractor report: CR-166476, NASA,
December 1982.
[61] Nathman, J. K., VSAERO: A Computer Program for Calculating the Nonlinear Aerodynamic Characteristics of Arbitrary Configurations – Users’ Manual version 7.2 , ANALYTICAL METHODS, INC.,
Washington (US), September 2007.
[62] van der Wees, A., van Muijden, J., and van der Vooren, J., A Fast and Robust Viscous-Inviscid Interaction Solver for Transonic Flow about Wing/Body Configurations on the Basis of Full Potential Theory ,
Technical Publication 93214 U, NLR – National Aerospace Laboratory of the Netherlands, 1993.
[63] Koning, J. H., Development of a KBE application to support aerodynamic design and analysis: Towards
a Next-Generation Multi-Model Generator , Master’s thesis, Delft University of Technology, Delft (The
Netherlands), May 2010.
[64] Torenbeek, E., “Prediction of wing group weight for preliminary design,” Aircraft Engineering and
Aerospace Technology , Vol. 43, No. 7, July 1971, pp. 16–21.
[65] Torenbeek, E., “Optimum Wing Area, Aspect Ratio and Cruise Altitude for Long Range Transport
Aircraft,” Report LR-775, Delft University of Technology, Delft (Netherlands), October 1994.
[66] Wright, J. and Cooper, J., Introduction to Aircraft Aeroelasticity and Loads, Vol. 18 of Aerospace Series
of AIAA education series, John Wiley, 2008.
[67] Bisplinghoff, R., Ashley, H., and Halfman, R., Aeroelasticity , Dover books on physics, Courier Dover
Publications, 1996.
[68] Kroo, I. and Shevell, R., Aircraft design: Synthesis and analysis, Digital Textbook Version 1.2, Desktop
Aeronautics, Inc., Stanford, CA (US), http://adg.stanford.edu/aa241/AircraftDesign.html,
September 2006.
[69] Howe, D., “The prediction of aircraft wing mass,” Proc. Instn Mech. Engrs, Part G, Journal of Aerospace
Engineering , 1996.
[70] Dorbath, F., “Large Civil Jet Transport (MTOM > 40t) Statistical Mass Estimation,” DLR-LY/Airbus,
LTH MA 401 12-01, 2011.
[71] Martins, J. R. and Marriage, C., “An Object-Oriented Framework for Multidisciplinary Design Optimization,” AIAA Paper 2007-1906 , 3rd AIAA Multidisciplinary Design Optimization Specialist Conference,
April 2007, pp. 1–15.
[72] Roskam, J., Airplane Design: Component Weight Estimation (Part V), Airplane Design, Design, Analysis
and Research Corporation (DARcorporation), Lawrence, KS (USA), 1999.
[73] Ruijrok, G., Elements of Airplane Performance, Delft University Press, 2009.
[74] Air BP Ltd., Handbook of products, http://www.bp.com/liveassets/bp_internet/aviation/
air_bp/STAGING/local_assets/downloads_pdfs/a/air_bp_products_handbook_04004_1.pdf,
Last accessed: June 2012.
[75] Jenkinson, L., Rhodes, D., and Simpkin, P., Civil jet aircraft design, AIAA education series, American
Institute of Aeronautics and Astronautics, 1999.
[76] Jenkinson, L., Rhodes, D., and Simpkin, P., “Civil Jet Aircraft Design – Aircraft Data Set,” http://
www.elsevierdirect.com/companions/9780340741528/appendices/data-a/default.htm, Last
accessed: June 2012.
[77] Bazaraa, S., Sherali, H., and Shetty, C., Nonlinear Programming: Theory And Algorithms, WileyInterscience, 3rd ed., 2006.
[78] Fletcher, R., Practical Methods of Optimization: Unconstrained Optimization, Wiley, 2nd ed., 1987.
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Appendix A
SQP algorithm
Sequential (or successive) Quadratics Programming (SQP) is a well-known and popular method
for non-linear constrained optimization. In addition to that, it is one of the robust methods. The
algorithm exists of two steps in which a quasi-Newton method and a line search are used iteratively
to find the best possible solution. The quasi-Newton method is used to locally, around the current
location, approximate the real problem with a quadratic function. The line connecting the solution
to this quadratic problem and the current location, is then used to determine the new-current
location, and is referred to as the line search.
Quadratic programming subproblem
The fundamental idea of sequential quadratic programming is to approximate the computationally
extensive full Hessian matrix using a quasi-Newton updating method. Subsequently, this generates a
subproblem of quadratic programming (a so-called QP subproblem) at each iteration. The solution
to this subproblem can be used to determine the search direction and next trial solution. Using
the Taylor expansion, the above problem can be approximated at each iteration, as the following
problem:
min
x
subject to
1 T 2
s ∇ L (xk ) s + ∇f (xk )T s + f (xk )
2
∇gi (xk )T s + gi (xk ) = 0,
i = 1, . . . , m
∇hj (xk )T s + hj (xk ) ≤ 0,
j = 1, . . . , n
(A-1)
(A-2)
(A-3)
where the Lagrangian function (also called merit function), is defined by
L (x) = f (x) +
i=1
X
m
T
λi gi (x) +
j=1
X
λj hj (x)
(A-4)
n
= f (x) + λ g(x) + µT h(x)
(A-5)
where λ = (λ1 , . . . , λm )T is the vector of Lagrange multipliers, and µ = (µ1 , . . . , µn )T is the
vector of KKT multipliers. The terms g and h represent g = (g1 (x), . . . , gm (x))T and h =
(h1 (x), . . . , hn (x))T .
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
b
Appendix A. SQP algorithm
To approximate the Hessian ∇2 L (xk ) by a positive definite symmetric matrix H k , the standard
Broydon-Fletcher-Goldfarbo-Shanno (BFGS) approximation of the Hessian can be used (see reference [41] for more details), which leads to
H k+1 = H k +
v k v Tk
H k uk uTk H Tk
+
v Tk uk
uTk H k uk
(A-6)
where
uk = xk+1 − xk
(A-7)
vk = ∇L (xk+1 ) − ∇L (xks )
(A-8)
and
Line search
The QP subproblem is solved to obtain the search direction
xk+1 = xk + αsk
(A-9)
using a line search method by minimizing a penalty function,


n
m
X
X
max {0, hj (x)}
|gi (x)| +
Φ(x) = f (x) + ρ 
i=1
(A-10)
j=1
where ρ is the penalty parameter. Any SQP method requires a good choice of H k as the approximate
Hessian of the Lagrangian L. Obviously, if H k is exactly calculated as ∇2 L, the SQP essentially
becomes Newton’s method for solving the optimality condition. A popular way to approximate
the Lagrangian Hessian is to use a quasi-Newton scheme as we used the BFGS formula mentioned
earlier.
The following psuedo-code summarizes the procedure of the sequential quadratic programming.
repeat k = 1, 2, . . .
Solve a QP subproblem: QPk to get the search direction sk
Given sk , find α so as to determine xk+1
Update the approximate Hessian H k+1 using BFGS scheme
k =k+1
until stop criterion
The built-in optimization routine of Matlab, fmincon uses the SQP algorithm to find a solution
for a constrained minimization problem. It can handle both equality and inequality constraints, and
bounds imposed on the design variables. The fmincon also uses BGFS to update the approximation
to the Hessian of the Lagrangian (H k ).
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Appendix B
Psuedo-code of LOCSMOOTH algorithm
The psuedo-code used to program the LOCSMOOTH algorithm is given on the next page. This
code is taken from reference [38, 39].
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
d
Appendix B. Psuedo-code of LOCSMOOTH algorithm
Procedure ALSO(r, MaxNoImprove, K)
// ALSO: an Algorithm based on Local Smoothing for Optimization
// see also: http://alanine.dsi.unifi.it/also
// MaxNoImprove: stopping criterion
// K: number of observations to perform in the current sphere
// r: radius used in local perturbation of the current point
NoImprove = 0;
x = random uniform point in S;
x ∗ = LS(x);
current= f (x ∗ ) = L(x);
record = current;
while (NoImprove < MaxNoImprove)
i = 0;
while (i < K and record ≤ current)
i = i + 1;
yi = random uniform point in B(x ∗ , r);
yi∗ = LS(yi );
current = L(yi );
end while
if (current < record)
// a new record has been found while sampling in B(x ∗ , r)
record = current
x ∗ = yi∗
NoImprove = 0;
else
NoImprove = NoImprove + K;
// Major iteration: optimization of the
//approximate smoothing based upon the observations placed in y1 , . . . , yK
x = arg minx∈ B(x ∗ ,r) L̂Bg (x);
// The current point is moved
y = LS(x)
current = L(x);
if (current < record)
// a new record has been found
record = current;
NoImprove = 0;
x∗ = y
else
x∗ = x
end if
end if
end while
end Procedure
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Appendix C
Taper implementation for simple sweep theory
The calculation of the chord length perpendicular to the quarter-chord sweep line for swept wings
without taper, can be done using equation (4-1). For tapered swept wings however, this calculation
is slight more complex. Based on Figure C-1, it is explained how the chord length perpendicular to
the sweep line, at given constant chord percentage (denoted by ξc ), can be found.
V∞
leading edge
V⊥
ΛLE
c⊥
Λ
c
ΛTE
sweepline
trailing edge
Figure C-1: Airfoil section perpendicular to sweep line of a tapered wing
The chord length of the airfoil section perpendicular to the sweep line is denoted by c⊥ . The
calculation of c⊥ consists of out two parts, the front part (between leading edge and sweep line)
and aft part (between sweep line and trailing edge).
c⊥ = cfront + caft
(C-1)
The front part can be derived using Figure C-2 and the law of sines, which yields the following
expressions for both the front part and the aft part (which can be calculated using the same
procedure as for the front part):
cfront =
caft =
sin (90◦ − ΛLE )
c · ξc
sin (90◦ + ΛLE − Λ)
sin (90◦ − ΛTE )
c · (1 − ξc )
sin (90◦ + ΛTE − Λ)
Wing Shape Multidisciplinary Design Optimization
(C-2a)
(C-2b)
Jan Mariens
f
Appendix C. Taper implementation for simple sweep theory
leading edge
ΛLE
90◦
− ΛLE
c · ξc
c⊥
90◦ + ΛLE − Λ
cfront
Λ
c
trailing edge
Figure C-2: Front part of the airfoil section perpendicular to sweep line
The airfoil shape at a given section normal to the sweepline, is interpolated between the neighboring
user-defined airfoils. Figure C-3 shows how this interpolation is done.
z
b
b
y
b
V∞
b
x
b
V⊥
b
b
b
b
b
b
b
b
b
b
b
b
b
Λ
b
b
b
b
b
sweep line
b
interpolated
airfoil
b
b
b
Figure C-3: Upper curve airfoil shape determination based on interpolation of coordinates
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Appendix D
Validation quasi-3D aerodynamic solver at
low speed
D-1
NACA24-0-0 wing
1.2
1
Experimental data
Quasi−3D
VSAERO
MATRICSV
C L [-]
0.8
0.6
0.4
0.2
0
−0.2
−4
−2
0
2
4
α [de g]
6
8
10
12
Figure D-1: CL − α curve of the NACA 24-0-0 wing
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
h
Appendix D. Validation quasi-3D aerodynamic solver at low speed
0.07
0.06
Experimental data
Quasi−3D
VSAERO
MATRICSV
C D [-]
0.05
0.04
0.03
0.02
0.01
0
−4
−2
0
2
4
α [de g]
6
8
10
12
Figure D-2: CD − α curve of the NACA 24-0-0 wing
0.07
0.06
Experimental data
Quasi−3D
VSAERO
MATRICSV
C D [-]
0.05
0.04
0.03
0.02
0.01
0
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
C L [-]
Figure D-3: CD − CL curve of the NACA 24-0-0 wing
Table D-1: Error analysis of different aerodynamic solver for the wing drag coefficient of the
tapered NACA 24-0-0 airfoil
Experimental
Jan Mariens
Quasi-3D
∆CD
VSAERO
∆CD
Matrix-V
∆CD
CL
CD
CD
(×1000)
CD
(×1000)
CD
(×1000)
-0.140
0.012
0.164
0.316
0.467
0.617
0.765
0.911
1.054
0.0100
0.0088
0.0100
0.0144
0.0216
0.0311
0.0440
0.0599
0.0766
0.0073
0.0059
0.0069
0.0106
0.0168
0.0260
0.0383
0.0529
0.0697
-2.647
-2.899
-3.100
-3.877
-4.794
-5.078
-5.741
-7.012
-6.895
0.0070
0.0059
0.0073
0.0113
0.0173
0.0258
0.0366
0.0505
0.0686
-2.922
-2.905
-2.722
-3.168
-4.315
-5.326
-7.362
-9.365
-7.940
0.0096
0.0082
0.0094
0.0135
0.0206
0.0308
0.0435
0.0598
-
-0.332
-0.629
-0.627
-0.909
-0.934
-0.331
-0.546
-0.080
-
Wing Shape Multidisciplinary Design Optimization
Appendix D. Validation quasi-3D aerodynamic solver at low speed
D-2
i
Tapered NACA24-15-0 wing
1.2
1
Experimental data
Quasi−3D
VSAERO
MATRICSV
C L [-]
0.8
0.6
0.4
0.2
0
−0.2
−4
−2
0
2
4
α [de g]
6
8
10
12
Figure D-4: CL − α curve of the NACA 24-15-0 wing
0.08
0.07
Experimental data
Quasi−3D
VSAERO
MATRICSV
0.06
C D [-]
0.05
0.04
0.03
0.02
0.01
0
−4
−2
0
2
4
α [de g]
6
8
10
12
Figure D-5: CD − α curve of the NACA 24-15-0 wing
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
j
Appendix D. Validation quasi-3D aerodynamic solver at low speed
0.08
0.07
Experimental data
Quasi−3D
VSAERO
MATRICSV
0.06
C D [-]
0.05
0.04
0.03
0.02
0.01
0
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
C L [-]
Figure D-6: CD − CL curve of the NACA 24-15-0 wing
Table D-2: Error analysis of different aerodynamic solver for the wing drag coefficient of the
tapered NACA 24-15-0 airfoil
Experimental
Quasi-3D
∆CD
VSAERO
∆CD
Matrix-V
∆CD
CL
CD
CD
(×1000)
CD
(×1000)
CD
(×1000)
-0.1375
0.0142
0.1659
0.3172
0.4678
0.6170
0.7646
0.9101
1.0532
0.00952
0.00844
0.00962
0.01376
0.02043
0.03066
0.04295
0.05793
0.07431
0.00723
0.00587
0.00691
0.01043
0.01659
0.02586
0.03800
0.05241
0.06911
-2.28849
-2.56521
-2.70861
-3.32916
-3.84575
-4.80759
-4.95374
-5.51947
-5.20098
0.00653
0.00544
0.00672
0.01060
0.01679
0.02546
0.03644
0.05049
0.06882
-2.99415
-2.99975
-2.89563
-3.15609
-3.64417
-5.20263
-6.51448
-7.44015
-5.48862
0.00807
0.00936
0.01344
0.02050
0.03058
-
-0.36780
-0.26057
-0.31479
0.06698
-0.08568
-
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Appendix D. Validation quasi-3D aerodynamic solver at low speed
D-3
k
NACA 24-30-85 airfoil
1
0.8
Experimental data
Quasi−3D
VSAERO
MATRICSV
C L [-]
0.6
0.4
0.2
0
−0.2
−0.4
−4
−2
0
2
4
α [de g]
6
8
10
12
Figure D-7: CL − α curve of the NACA 24-30-85 wing
0.05
0.045
0.04
Experimental data
Quasi−3D
VSAERO
MATRICSV
C D [-]
0.035
0.03
0.025
0.02
0.015
0.01
0.005
−4
−2
0
2
4
α [de g]
6
8
10
12
Figure D-8: CD − α curve of the NACA 24-30-85 wing
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
l
Appendix D. Validation quasi-3D aerodynamic solver at low speed
0.05
0.045
0.04
Experimental data
Quasi−3D
VSAERO
MATRICSV
C D [-]
0.035
0.03
0.025
0.02
0.015
0.01
0.005
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
C L [-]
Figure D-9: CD − CL curve of the NACA 24-30-85 wing
Table D-3: Error analysis of different aerodynamic solver for the wing drag coefficient of the
tapered NACA 24-30-85 wing
Experimental
Jan Mariens
Quasi-3D
∆CD
VSAERO
∆CD
MATRICSV
∆CD
CL
CD
CD
(×1000)
CD
(×1000)
CD
(×1000)
-0.312
-0.170
-0.028
0.115
0.257
0.398
0.538
0.677
0.814
0.0174
0.0127
0.0102
0.0102
0.0126
0.0185
0.0260
0.0356
0.0462
0.0143
0.0098
0.0075
0.0075
0.0096
0.0138
0.0213
0.0314
0.0435
-3.192
-2.944
-2.644
-2.685
-3.032
-4.643
-4.659
-4.244
-2.666
0.0130
0.0088
0.0068
0.0069
0.0091
0.0135
0.0203
0.0292
-
-4.456
-3.944
-3.394
-3.318
-3.517
-4.986
-5.663
-6.432
-
0.0099
0.0125
0.0181
-
-0.280
-0.135
-0.417
-
Wing Shape Multidisciplinary Design Optimization
Appendix E
Quasi-3D aerodynamic solver inputs and
outputs
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
n
Appendix E. Quasi-3D aerodynamic solver inputs and outputs
INPUT
Wing geometry
é x1
ê
ê
êë xn
y1
z1
c1
yn
zn
cn
e1 ù
L.E. position of root section
ú
ú
e n úû L.E. position of tip section
+ wing incidence angle
Airfoil definitions using CST coefficients
é [ Coeff. upper curve
ê
ê
ëê [ Coeff. upper curve
Coeff. lower curve ] ù
ú
ú
Coeff. lower curve ] ûú
+ normalized spanwise positions (η) of the airfoil
Quasi-3D aerodynamic solver inputs
Nc, Nb, Nw, xc
+ Mach number selection criteria to select airfoil analysis tool
Flight conditions
V , M , Re, r and CL or a
OR
M , h and CL or a
Quasi-3D aerodynamic solver
OUTPUT
Aerodynamics
CL , a , CD , CDi , CDprof
+ Lift, drag and moment distributions over wing
+ Aerodynamic forces and moments
+ Chord distribution
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Appendix F
Multidisciplinary design optimization modules
F-1
Weight module (We)
INPUT
Geom,W f* ,Ww*
Weights
Wto = W2restb/2+ W f* + Ww*
c = ò c( y ) 2 dy
S W0 rest + Ww*
Wzfw =
Weight prediction using
a weight estimation
method
OUTPUT
Ww
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
p
F-2
Appendix F. Multidisciplinary design optimization modules
Aerodynamic module (Ae)
INPUT
Geom,W f* ,Ww*
Mean Aerodynamic Chord
c=
2
S
b /2
ò c( y )
2
dy
0
Reynolds number
Re =
rV c
m
Weights
Wto = Wrest + W f* + Ww*
Wzfw = Wrest + Ww*
Rest drag coefficient
CDrest =
Lift coefficient
CL =
2 Wto × Wzfw
(
S
CDrest
S ref .
)
ref .
rV 2 S
2D
3D
Run AVL
CL , CDi
Cl - h
Strip method with
sweep theory
implementation
CDprof
Wing total drag coefficient
CDwing = CDi + CDprof
Aircraft total drag coefficient
CD = CDwing + CDrest
OUTPUT
L D of wing
Jan Mariens
Wing Shape Multidisciplinary Design Optimization
Appendix F. Multidisciplinary design optimization modules
F-3
q
Performance module (Pe)
INPUT
L D ,W f* ,Ww*
Weights
b /2
Wto = W
2 +W * +W *
c = restò c( y )f2 dy w
WzfwS= 0Wrest + Ww*
Cruise fuel fraction
(Bréguet range eqn.)
M ffcr
ì h RCL pD
ïe p ( )
=í
g R sfc
ï V ( L D)
e
î
for turboprop aircraft
for turbojet aircraft
Total mass fraction
n
M ff = Õ M ffi
i =1
Fuel weight (+5% reserve)
W f = (1 - M ff )Wto ×1.05
OUTPUT
Wf
Wing Shape Multidisciplinary Design Optimization
Jan Mariens
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