March 2014 

NASA/CR–2014-218185 

A Comparison of Metallic, Composite and 

Nanocomposite Optimal Transonic 

Transport Wings 

Graeme J. Kennedy, Gaetan K. W. Kenway, and Joaquim R. R. Martins 

University of Michigan, Ann Arbor, Michigan 
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March 2014 

NASA/CR–2014-218185 

A Comparison of Metallic, Composite and 

Nanocomposite Optimal Transonic 

Transport Wings 

Graeme J. Kennedy, Gaetan K. W. Kenway, and Joaquim R. R. Martins 

University of Michigan, Ann Arbor, Michigan 
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 

Acknowledgments 

Funding for this research was provided by NASA under grant number NNX11AI19A. 

The authors would like to thank Christine Jutte, Bret Stanford, and Karen Taminger for 

their help in defining this project and for their suggestions in the writing of this report. 
Contents

1 Introduction 1

2 Analysis tools 1

2.1 Aerodynamic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 Load transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.3 Structural analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.4 Approximate Newton–Krylov method . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.5 Adjoint-based gradient computation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 The QCRM wing 5

4 Structural design parametrizations 7

4.1 SP1: Basic metallic sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.2 SP2 and SP3: Metallic, composite and CNT-based composite sizing with smeared

stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 Aerostructural design formulation 11

5.1 Mission parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5.2 Structural constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.3 Aerostructural optimization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6 Results and Discussion 17

6.1 Aerostructural optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6.2 RANS-based aerodynamic shape optimization . . . . . . . . . . . . . . . . . . . . . . 21

7 Conclusions 23

Abstract

Current and future composite material technologies have the potential to greatly improve the performance of large transport aircraft. However, the coupling between aerodynamics and structures

makes it challenging to design optimal flexible wings, and the transonic flight regime requires

high fidelity computational models. We address these challenges by solving a series of high-fidelity

aerostructural optimization problems that explore the design space for the wing of a large transport

aircraft. We consider three different materials: aluminum, carbon-fiber reinforced composites and

an hypothetical composite based on carbon nanotubes. The design variables consist of both aerodynamic shape (including span), structural sizing, and ply angle fractions in the case of composites.

Pareto fronts with respect to structural weight and fuel burn are generated. The wing performance

in each case is optimized subject to stress and buckling constraints. We found that composite wings

consistently resulted in lower fuel burn and lower structural weight, and that the carbon nanotube

composite did not yield the increase in performance one would expect from a material with such

outstanding properties. This indicates that there might be diminishing returns when it comes to

the application of advanced materials to wing design, requiring further investigation.

iii
1 Introduction

The goal of NASA’s high aspect ratio wing work under the Fixed Wing project is to explore ways

to increase wing span and thus reduce induced drag without an excessive increase in structural

weight. The exploration of the wing design space requires the consideration of both aerodynamics

and structures due to wing flexibility and the inherent trade-offs between drag and structural weight

in the design. This is especially important in the high aspect ratio wings considered in this study,

since wings become more flexible as span is increased. In this report we explore the use of three

different materials: aluminum, carbon-fiber reinforced composites and an hypothetical composite

based on carbon nanotubes.

In the following sections, we describe a series of structural and aerostructural design problems

that we solved. The objective of this sequence of design problems is to evaluate the impact of

new structural material technologies on the trade-offs in the design of a conventional aircraft wing.

In order to perform this assessment, we use a series of structural and coupled aerostructural optimization problems that are described in detail below. Optimization of each candidate design is

necessary to make a fair comparison between potential technologies, so that the full benefit of the

potential technology is realized. Comparing sub-optimal designs may lead to incorrect conclusions

regarding the relative merits of different technologies. However, the design optimization problem

must also adequately capture the requirements and objective of a realistic aircraft design problem,

otherwise the predicted performance improvements may be a result of an incomplete analysis or an

inadequate design.

The remainder of this document is structured as follows: in Section 2 we describe the essential

aspects of the analysis and design tools used for this study. In Section 3 we present the geometry

we use in this study. In Section 4 we describe a series of structural design parametrizations used

in the following studies. In Section 5 we present the aerostructural design problem formulation. In

Section 6 we present some results and in the last section we present conclusions and future work

for the project.

2 Analysis tools

In the studies performed for this project, all the functions of interest in the design problem, such

as objectives or constraints, are evaluated at converged aerostructural solutions. In this report we

use the term aerostructural analysis to refer to the static aeroelastic analysis of an aircraft such

that the aerodynamic forces on the deformed flying shape of the aircraft are in equilibrium with the

inertial forces. In this study, aerostructural analysis is necessary due to the flexibility of the designs

obtained using advanced materials. As a result, it is necessary to fully capture the aerostructural

effects within the design space to fully model the performance of the aircraft. Since gradient-based

design optimization often requires several hundred gradient and function evaluations, an efficient

aerostructural solution and gradient evaluation technique is required. In this work, we use both

a medium fidelity method that uses a panel method, with compressibility, viscous and wave drag

corrections coupled to a high-fidelity finite-element solver, and a CFD solver that can solve either the

Euler or RANS equations coupled to the same finite-element solver. Due to the large dimensionality

of the design space and the computational cost of the analysis, we exclusively use adjoint-based

gradient evaluation techniques [1, 2]. The following section outlines the aerostructural analysis and

adjoint-based gradient evaluation techniques used within this work. Additional details about the

1
solution methods and the efficiency of the approach can be found in Kennedy and Martins [3] and

Kenway et al. [4].

2.1 Aerodynamic analysis

For the medium-fidelity aerostructural cases, the aerodynamic analysis is performed using TriPan,

an unstructured, three-dimensional parallel panel code for calculating the aerodynamic forces, moments and pressures for inviscid, incompressible, external lifting flows using the Prandtl–Glauert

equation [3]. TriPan uses constant first-order source and doublet singularity elements distributed

over the entire lifting surface and doublet elements distributed over the wake [5]. The source

strengths are determined based on the onset flow conditions while the boundary conditions for the

doublet strengths constitute a dense linear system of equations, represented here by

RA(w, u) = 0, (1)

where u and w are vectors of the structural and aerodynamic state variables, respectively. The

linear system represented by Equation (1) is solved in parallel using PETSc [6]. A dense matrix

format is used for the matrix-vector products, while a sparse approximate-Jacobian is used to form

a incomplete LU (ILU) preconditioner. The linear system is solved using the Krylov subspace

method GMRES.

For the high-fidelity aerodynamic cases, we use the CFD solver SUmb, which is a secondorder structured block-based finite-volume solver for the Euler, Navier–Stokes and RANS equations. SUmb has both a multi-grid solution method with an explicit Runge–Kutta time integration

method with residual smoothing, and an inexact Newton–Krylov solution method with a nonmonotone linesearch strategy. For the Newton–Krylov method, the linear systems arising at each

Newton step are solved using the Krylov subspace methods with the parallel algorithms available in

PETSc [6]. Typically, the multi-grid scheme is used during the start-up phase, while the Newton–

Krylov method is employed once the relative residual norm is reduced below a target tolerance.

The combination of these two methods provides a fast and robust solution strategy that can handle

significant design modifications during an optimization. Kenway et al. [4, 7] describe further details

of the most important aspects of the solver.

2.1.1 Drag build up

When using TriPan it is essential that the aerodynamic effects not captured by linear potential

methods be modeled as accurately as possible, so that the drag calculations provide reasonably

accurate estimates of the total aircraft drag. Here, we outline the drag build-up calculations

performed in this study. The drag build-up consists of the induced drag, computed directly from

TriPan using a Trefftz-plane integration, with additional empirical drag corrections including a

surface friction correction, profile drag corrections and drag-divergence.

The profile drag is computed based on a quadratic model of the sectional drag coefficient:

cdp = cd0 + cd2c

2

l

, (2)

where cdpis the profile drag, clis the sectional lift coefficient, and cd0 and cd2 are coefficients

defined below. The coefficient cd0is based on the skin friction estimate

cd0 = Fccf ,

2
where Fc is a profile drag form-factor, and cf is the turbulent skin-friction coefficient determined

using the van Driest II method [8]. The form-factor, Fc, is computed using the thickness to chord

ratio, t/c, as follows:

Fc = 1 + 2.7



t

c



+ 100 

t

c

4

.

Finally, based on the method presented by Wakayama and Kroo [9], the quadratic coefficient in

Equation (2), is computed based on the expression:

cd2 =

0.38

cos2 Λ

cd0

,

where Λ is the leading-edge sweep angle.

The compressibility drag is computed based on a crest-critical Mach number computed using

the Korn equation,

Mcrit =

κA

cos Λ −

t/c

cos2 Λ

−

cl

10 cos3 Λ

−



0.1

80 1/3

, (3)

where κA is a technology factor that we set to κA = 0.95, which is suitable for supercritical airfoil

sections commonly used in transport aircraft. The sectional contribution to the compressibility

drag is then computed using

cdc = 20(M − Mcrit)

4

(4)

for M > Mcrit.

2.2 Load transfer

The load and displacement transfer scheme follows the work of Brown [10]. The displacements from

the structures are extrapolated to the aerodynamic nodes using rigid links. These rigid links are

formed by locating the closest point on the structural surface to each of the aerodynamic nodes.

The structural surface is determined by interpolating between structural nodes using the finiteelement shape functions. The displacements uS and rotations θS on the structural surface, and the

rigid links r are used to determine the displacements of the aerodynamic nodes uA as follows:

uA = uS + θS × r. (5)

Note that this formula uses a small angle approximation. Equation (5) can be used, in conjunction

with the method of virtual work, to form the consistent force vector for the aerodynamic forces at

the structural nodes. More details of the approach are outlined in Kennedy and Martins [3].

2.3 Structural analysis

The structural analysis is performed using the Toolkit for the Analysis of Composite Structures

(TACS), a parallel, finite-element code designed specifically for the analysis of stiffened, thin-walled,

composite structures using either linear or geometrically nonlinear strain relationships [3]. In the

work to date, we have exclusively used linear structural analysis, however, geometrically nonlinear

analysis may be required to accurately predict the flying shape of very flexible aircraft. In addition,

we have focused on using a smeared stiffness approach in which the effect of stiffeners are included

in the stiffness of the skin as described in detail below, as a result, we have not used the full stiffened

3
shell analysis capabilities of TACS. When modeling stiffened shell structures, we typically use third

order finite-element discretizations that provide a good trade-off between solution accuracy and

gradient-evaluation costs.

The residuals of the structural governing equations are

RS(w, u) = Sc(u) − F(w, u), (6)

where u is a vector of displacements and rotations, Sc are the residuals due to conservative forces

and internal strain energy and F are the follower forces due to aerodynamic loads.

The Jacobian of the structural residuals involves two terms: the stiffness, or tangent stiffness

matrix K = ∂Sc/∂u and the derivative of the consistent force vector with respect to the structural

displacements. This results in the following expression for the Jacobian of the structural residuals:

∂RS

∂u

= K −

∂F

∂u

. (7)

While the matrices involved in structural problems are typically symmetric, the term ∂F/∂u is

non-symmetric due to the non-conservative nature of the aerodynamic forces. These non-symmetric

matrices require different solution algorithms than those typically employed in structural finiteelement codes. We use GMRES [11] to solve the non-symmetric, linear systems involving the

matrix in Equation (7).

2.4 Approximate Newton–Krylov method

The aerostructural residuals are the concatenation of the aerodynamic and structural residuals,

represented by:

R(q, x) = 

RA(w, u, x)

RS(w, u, x)



= 0, (8)

where RA and RS are the aerodynamic and structural residuals, w and u are the aerodynamic and

structural state variables, q is the full set of aerostructural state variables q

T = [wT

, u

T

], and x is

a vector of design variables.

Newton’s method applied to Equation (8) results in the following linear system of equations for

the update ∆q

(n)

,

∂R

∂q

∆q

(n) = −R(q(n)

), (9)

where the n

th update is applied as follows: q(n+1) = q(n) = ∆q(n)

. In our Newton–Krylov approach,

we solve an approximate linearization of the system (9) inexactly, to a loose tolerance, using a

Krylov subspace method. Additionally, we use an approximate linearization that is less expensive

to compute than the exact Jacobian, leading to better overall solution performance. We form a

precondition for the Krylov subspace method based on discipline-level preconditioners by dropping

the off-diagonal coupling terms. Note that these coupling terms are retained for the matrix-vector

products which are computed using a product-rule implementation that is discussed in further

detail in Kennedy and Martins [3].

4
2.5 Adjoint-based gradient computation

Efficient gradient-based optimization requires the accurate and efficient evaluation of gradients of

the objective function and constraints. In the aerostructural optimization problem described below,

there are typically far fewer objective and constraint functions than there are design variables.

While there are various sensitivity methods that can be used to compute accurate derivatives [12],

the coupled adjoint method [13, 14], is the most computationally efficient for coupled systems when

the gradients of a small number of functions of interest are required with respect to many design

variables. We have developed a coupled aerostructural adjoint that is based entirely on analytic

derivatives without the use of finite-difference computations. Further details of this approach are

presented in Kennedy and Martins [3]. The aerostructural adjoint equations can be written in the

following form:

∂R

∂q

T

ψ =

∂f

∂q

T

, (10)

where ψ is the adjoint vector and f(q, x) is either an aerodynamic or structural function of interest.

Once the adjoint vector has been determined using Equation (10), the total derivative is determined

using the additional computation:

∇xf =

∂f

∂x

− ψ

T ∂R

∂x

. (11)

We use a Krylov method to solve the linear coupled aerostructural adjoint equations (10) in

an analogous manner to the Krylov method applied to the linearized Newton system. In the

Krylov approach, the matrix-vector products are computed using the exact Jacobian-transpose of

the coupled aerostructural system. One iteration of a transpose block Jacobi iteration is used as

the preconditioner.

3 The QCRM wing

For this study we have developed what we call the Quasi-CRM (QCRM) geometry. The QCRM

is a wing and wing-tail geometry that has a planform roughly the same as the Common Research

Model (CRM) used in the DPW 4 and 5 workshops [15]. However, it should be emphasized that

the QCRM geometry is not derived from the CRM geometry directly.

The QCRM wing is shown in Figure 1. The QCRM wing is untwisted, has a root chord of 12 m,

a tip chord of 2.75 m, a semi-span of 30 m and a quarter-chord sweep of 35◦. The leading edge of

the wing is straight, while the trailing edge of the wing is quadratically interpolated between the

root trailing edge and the tip. The trailing edge is determined using a quadratic spline where a

control point is added at a span-wise location of 10.5 m, with a chord of 6.5 m. The curved trailing

edge corresponds roughly with a wing crank, but is entirely smooth. The wing is constructed using

a set of lofted super-critical NASA SC(2)-0414 and SC(2)-0610 airfoil sections. While the initial

aerodynamic performance of this wing is poor, we do not use its initial performance as a baseline.

Instead, we only compare optimized designs.

The QCRM geometry has several advantages over the CRM for parametric design studies.

The QCRM is based on the jig shape of the wing and does not implicitly assume any aeroelastic

deflection, as in the case of the CRM geometry. This greatly simplifies aerostructural design

studies since the wing deflections do not have to be removed from the wing geometry. In addition,

5
Figure 1: The Quasi-CRM (QCRM) wing geometry.

the QCRM geometry has a simple parametric description, and as a result, the wing planform can

easily be manipulated using a small number of geometric design variables. Since the QCRM wing is

untwisted, the aerodynamic performance of the wing will be poor, however, we always compare the

performance of optimized designs so the initial planform and twist distributions are unimportant.

The internal wing structure geometry is shown in Figure 2. The wing box structure consists

of two structural spars and 44 ribs as well as top and bottom wing skins. In addition, nonstructural elements are added to the model to improve the load and displacement transfer between

the aerodynamic and structural models. These non-structural elements consist of the leading edge

skin and a false trailing edge spar and skin that extend behind the true trailing edge spar. A

more detailed wing model could include the control surfaces and leading and trailing edge support

structures. However, fully modeling these additional features is not considered here due to the

additional complexity. Furthermore, it would be necessary to add additional load cases to the

design problem to properly size these portions of the structure, complicating the design problem

formulation. We have also experimented with the complete omission of the leading and trailing

edge surfaces, but this leads to difficulties, as the structural displacements must be extrapolated

to the aerodynamic surface. When the structural and aerodynamic surfaces are far apart, the

extrapolation of the deformed aerodynamic surface becomes inaccurate and maintaining a smooth

deformed surface can be difficult. As a result, we include these non-structural elements as a

compromise. One of the primary purposes of the structural model is to provide an accurate weight

estimate based on the primary structure within the wing. Therefore, we omit the non-structural

components from the weight estimate and set them with a minimum thickness determined by the

part type, which is described in more detail below. Furthermore, we reduce the stiffness of these

components and maintain a consistent Poisson’s ratio to avoid stress concentration effects.

Within the wing box, the first 3 ribs in the structure are parallel to the symmetry plane. This

section of the wing box lies within the fuselage and is designed to model the center wing-box.

6
Figure 2: The QRM structural wing box

The leading and trailing edge spars lie at 15% and 65% of the root chord, respectively, while the

non-structural trailing edge extends all the way to 90% of the chord.

We have used two structural finite-element models of the structure. The smaller model consists

of 39 134 second order MITC4 shell finite-elements with 36 417 nodes, and roughly 218 500 degrees

of freedom. The larger model is obtained by refining each second order element within the first

model to obtain a third order MITC9 shell element. This larger model consists of 39 134 third order

MITC9 shell finite-elements with 75 551 nodes and roughly 453 000 degrees of freedom.

4 Structural design parametrizations

In this section, we describe in detail the three structural parametrizations that we implemented.

These are presented in increasing degree of complexity.

4.1 SP1: Basic metallic sizing

In this structural parametrization we assign a structural thickness variable to all structural components within the structural model. The thickness variable is treated as the thickness of an

unstiffened component of the structure. This parametrization produces structural designs that

are heavier than a structure with stiffened construction since the unstiffened panels exhibit low

stiffness-to-weight ratios, particularly under bending loads.

In addition to structural failure constraints, which we discuss in detail below, we also impose

constraints on the change in thicknesses between adjacent structural components. For instance,

within the wing skin we limit both the span-wise and chord-wise change in thickness between

adjacent panels. These constraints are designed to address both manufacturing requirements and

un-modeled structural effects from out-of-plane shear. These adjacency constraints can be expressed

as follows:

− ∆ ≤ xi − xj ≤ ∆ (i, j) ∈ A (12)

which is equivalent to imposing |xi − xj | ≤ ∆. Here A is a set of paired indices as follows:

A = {(i1, j1),(i2, j2), . . . ,(in, jn)} ,

7
such that an adjacency constraint is imposed between design variables xik

and xjkfor k = 1, . . . , n.

In this case, we impose the adjacency constraint (12) to control how quickly the skin and spar

thickness may change along the wing.

4.2 SP2 and SP3: Metallic, composite and CNT-based composite sizing with

smeared stiffness

The issue with the structural parametrization SP1 is that each panel is treated as an unstiffened

shell. As a result, the structural components exhibit lower stiffness and strength-to-weight ratios

than an equivalent stiffened panel manufactured using the same technology. In order to include the

effect of stiffeners, we use a smeared stiffness approach, which avoids the more onerous modeling

requirements for discrete stiffeners, but has better predictive capability than the SP1 parametrization described above. We label these parametrizations SP2 and SP3 for the metallic and composite

smeared stiffness parametrizations, respectively. In addition, we model a Carbon-nanotube-based

composite with hypothetical properties using the SP3 parametrization as well. In the smeared

stiffener approach the effect of the stiffeners on the stiffness of the panels is included in the material properties of the panel directly rather than modeling each discrete stiffener individually. This

approach works reasonably well when the ratio of the panel side-length to stiffener pitch is high,

meaning that there is a high density of stiffeners [16].

In the parametrizations SP2 and SP3, we split the wing structure into approximately flat panels

that are analyzed and designed based on the stress state in global finite-element model under a

series of loading conditions. These panels consist of the structural components formed between the

ribs and spars. In order to obtain an accurate estimate of the overall wing-box weight it is necessary

to have a design tool that can correctly size panels over a wide range of loading conditions. The

panels range from relatively lightly loaded at the tip to heavily loaded at the wing root. Over this

range, it is most important to capture the behavior of the heavily loaded parts of the structure,

since these structural components will have the greatest impact on the structural wing weight.

In the present work, we have developed a design tool that predicts the stiffness of the panel

based on a smeared-stiffener approach. This tool can also be used to enforce strength, buckling,

and manufacturing constraints, as well as adjacency constraints that impose limits on the variation

of the design variables between adjacent panels. The sizing of the panel is based on the simplified

stiffness calculations and buckling criteria proposed by Stroud and Agranoff [16].

In order to avoid some of the more challenging aspects of composite design problems, we do

not design the exact stacking sequence of the structure. Instead, we obtain an approximate design

based on the fractions of plies at given angles. Here, we restrict the possible ply angles to 0o, ±45o

and 90o, where we enforce a balanced laminate by requiring that the ply fractions of the positive

and negative 45o plies are equal. The in-plane strength and stiffness can be determined exactly

using the ply fraction variables alone [17], however, the bending stiffness, bending strength, and

buckling characteristics of the laminate cannot be determined exactly without knowledge of the

laminate stacking sequence. Instead we use conservative estimates of the exact quantities, where

possible. Most importantly, these formula approach the exact formula for very thick panels.

4.2.1 Smeared panel stiffness calculations

The ply fractions of the skin-panel are denoted f

(p)

i

, while the ply fractions in the stiffener are

denoted f

(s)

i

. The stiffness of the panel and stiffener are based on a stiffness weighted by the ply

8
fractions:

Q(p) =

X

4

i=1

f

(p)

i Q ̄ (θi), Q(s) =

X

4

i=1

f

(s)

i Q ̄ (θi),

where θi = {−45o, 0

o

, 45o, 90o}, Q ̄ (θ) is the stiffness in the global axis, and Q(p) and Q(s) are

the weighted stiffness of the skin and stiffener, respectively. Note, that ply fractions are not

independent, and must satisfy the following linear constraints:

X

4

i=1

f

(p)

i = 1,

X

4

i=1

f

(s)

i = 1.

The stiffness of the overall panel is determined by accounting for the effect of the discrete

stiffeners by adding additional bending and shear stiffness to the skin stiffness. The panel stiffness

can be determined based on the panel thickness tp, the stiffener height hs, the stiffener width tw,

the stiffener pitch sp, and the stiffness of the skin and stiffener Q(p) and Q(s). The smeared panel

stiffness matrices are written as follows:

A = tpQ(p) + A(s), B = B

(s)

,

D =

t

3

p

12

Q(p) + D(s), As = A(p)

s + A(s)s

.

(13)

The non-zero components of the matrices A(s), B(s), D(s), and A

(s)

s are:

A

(s)

11 =

EsAs

sp

, B

(s)

11 = −

hs

2sp

EsAs,

D

(s)

11 =

Es(h

2

sAs + 4Is)

4sp

, A(s)

s 11 =

5GsAs

6sp

,

where, Es = Q

s

11 −

Qs

21Qs12

Qs

66

is the extension modulus of the stiffener, As is the area of the stiffener,

and Is is the second moment of area of the stiffener.

4.2.2 Panel-level failure analysis

Each panel in the structure must be constrained such that the response under each loading condition

lies within an allowable operational envelope. In this study, we impose this envelope by considering

both failure constraints and buckling constraints. For the metallic cases, the failure envelope

is modeled using a von Mises stress failure criterion, while for the composite and CNT-based

composite cases, the failure envelope is calculated based on a maximum strain failure criterion.

This maximum strain failure criterion can be written as follows:

max 

1

1t

,

1

1c

,

2

2t

,

2

2c

γ12

γ12s

, −

γ12

γ12s



(14)

where 1, 2 and γ12 are the normal, transverse and engineering shear strains in the local ply axis.

The constants 1t, 1c are the maximum allowable tensile and compressive strains along the fiber

direction, while 2t and 2c are the maximum tensile and compressive strains in the transverse

directions, and γ12sis the maximum in-plane shear strain allowable. This failure criteria is applied

at all ply angles for the outer-most fibers in the skin, and the lowest fiber in the stiffener. This

results in 12 separate failure criteria for each point in the panel where the failure constraint is

applied.

9
Overall buckling Skin buckling

N1,cr

π

2EIs

spL2

x

2π

2

s

2

p

p

D11D22 + D12 + 2D66

EIs = z

2

n

(tpspEp + tshsfbEs) + Es

 

ts

h

3

s

12

+ tshs



zn −

hs

2

2

!

zn =

Cn

An

Cn = Es

h

2

s

2

ts An = Epsptp + Eshsts(1 + fb)

N12,cr ξ =

√

D1D2

D3

ξ =

√

D11D22

D12 + 2D66

If ξ > 1

4

L2

x

(D3

1D2)

0.25 

8.125 +

5.045

ξ



4

s

2

p

(D11D3

22)

0.25 

8.125 +

5.045

ξ



If ξ ≤ 1

4

L2

x

p

D1D3(11.7 + 0.532ξ + 0.938ξ

2

)

4

s

2

p

p

D22(D12 + 2D66)(11.7 + 0.532ξ + 0.938ξ

2

)

Table 1: A summary of the critical load computations for the overall and skin buckling constraints.

Note that An and Cn are the modulus-weighted zeroth and first moments of area of the panel and

stiffener, and zn is the modulus-weighted centroid. The bending stiffness EIs can then be used to

determine the critical buckling load.

4.2.3 Panel-level buckling analysis

The buckling constraints are imposed by constraining several independent buckling modes including

buckling of the skin between stiffeners, buckling of the stiffeners, and overall panel buckling including stiffeners and skins. The overall critical buckling loads are determined based on the approach

of Stroud and Agranoff [16]. The skin and stiffener buckling loads are determined by assuming

that the panel ends are simply supported along the lines of attachment with adjacent structural

components. The critical loads are determined under the assumption that the panels are nearly

flat, and therefore the analysis ignores any curvature effects.

We assume that the interaction between the longitudinal and shear buckling modes collapses

into the following buckling envelope:

B(N1, N12) = N2

12

N2

12,cr

+

N1

N1,cr

≤ 1, (15)

where N1 and N12 are the longitudinal and shear loads respectively and N1,cr and N12,cr are the

critical longitudinal and shear buckling loads. Note that N1 and N12 are computed in a locally

aligned panel axis. Equation (15) is applied separately to the overall panel buckling, stiffener

buckling and inter-stiffener skin buckling. The critical loads, N1,cr and N12,cr are determined based

on the formula presented in Table 1.

10
b

wb

hs

tb ts

tw

Figure 3: The panel geometry and thickness design variables used in the structural design

parametrization.

5 Aerostructural design formulation

In this section, we describe the aerostructural design problem formulation and present the results

of a study comparing wings with different structural technologies. In the aerostructural design

problem, we have tried to develop a representative aircraft design problem formulation that captures

the most essential aspects of the aircraft design problem in order to capture the tradeoff between

aerodynamic performance and structural weight reduction. This is essential in order to make a fair

comparison between potential benefits of candidate structural technologies.

In this design formulation, we minimize a weighted combination of fuel burn and take-off gross

weight (TOGW) over a series of prescribed missions. The contribution to the objective from each

mission is weighted based on the proportion of flights flown for each distance. As a result, frequently

flown missions have a larger impact on the objective, while less frequently used missions have a

smaller impact. We compute the fuel consumption for each mission based on the Breguet range

equation as follows:

FB = LGWexp 

R TSFC

V (L/D)



− 1



, (16)

where FB is the fuel burn for the entire mission, LGW is the landing gross weight, R is the mission

range, TSFC is the thrust-specific fuel consumption, V is the cruise speed and L/D is the lift to

drag ratio. The issue with this equation, however, is that the L/D must be evaluated at some point

in the mission. However, due to aerostructural effects, the deformed, flying shape of the wing, and

thus L/D, will vary over the mission due primarily to changes in the aircraft weight, even at fixed

CL. A more accurate analysis would integrate the true aircraft L/D over the mission profile using

the Breguet range integral. We leave this refinement for future work.

The contribution to the objective function for each mission is a weighted combination of fuel

burn and takeoff gross-weight, written as follows:

fi(x) = βFBi + (1 − β)TOGWi

= βFBi + (1 − β)(LGWi + FBi)

= FBi + (1 − β)LGWi

= LGWi



exp 

Ri TSFCi

Vi (L/D)

i



− β



(17)

11
Parameter Value

Initial O.E.W. 138 100 kg

Fixed O.E.W. (no wing structure) 108 100 kg

Max payload 55 000 kg

span 60 m

root chord 13 m

Sref 430 m2

Table 2: Parameters used for the aircraft design problem.

where we have used the fact that TOGW = LGW+FB. Here, fi(x) is the un-weighted contribution

to the objective from the i-th mission. Finally, the design objective can be written as follows:

f(x) = X

N

i

wifi(x) = X

N

i

wiLGWi



exp 

Ri TSFCi

Vi (L/D)

i



− β



, (18)

where wi are the weights associated with each mission. Here we select weights such that:

X

N

i

wi = 1 (19)

however, any scalar multiple of these weights will yield the same design.

5.1 Mission parameters

In this section, we describe in detail the parameters that are used in the mission analysis. These

aircraft parameters are based roughly on a next-generation 777-sized aircraft. Table 2 shows the

parameters used in the aircraft design problem. Here we estimate the overall aircraft weight based

on a fixed operational empty weight (OEW) plus a variable OEW contribution that corresponds

to the wing structure. The fixed portion of the OEW is 108 100 kg based on the 777-200ER with a

30 000 kg wing weight estimated as 10% of MTOW.

The weight of the aircraft at any point in flight is the sum of the fixed portion of OEW,

the structural wing weight, a wing-area dependent weight factor, the payload weight and the fuel

weight. In addition, the fuel weight is broken up into mission fuel and reserve fuel. The reserve fuel

is calculated based on a 45 minute loiter at sub-optimal L/D and TSFC conditions, and a diversion

to the furthest alternate airport for which we use a constant 500 nautical mile mission. For ease of

presentation, we fix the reserve fuel at 14 100 kg throughout the design. For all missions considered

here, we evaluate the L/D performance of the aircraft at the point in the mission where half of the

mission fuel has been consumed.

In this study, we utilize the four missions outlined in Table 3. These are shown within a payloadrange diagram in Figure 4. We analyze the performance of the aircraft at two extreme points on

the payload-range diagram: the maximum range mission at the maximum payload weight and the

maximum range mission at maximum fuel capacity. The maximum payload is set at 55 000 kg for

this aircraft, based on the 777-200ER. Furthermore, we fix the payload of the maximum range

mission at maximum fuel capacity to 15 000 kg for a 9000 nautical mile mission.

12
Mission 1: Max payload at maximum range

payload 55 000 kg range 6000 nm

w1 0.1 L/D altitude 33 000 ft

M 0.85

Mission 2: Max range at max fuel capacity

payload 15 000 kg range 9000 nm

w2 0.1 L/D altitude 34 000 ft

M 0.85

Mission 3: New York to London

payload 30 000 kg range 3040 nm

w3 0.4 L/D altitude 38 000 ft

M 0.85

Mission 4: Los Angeles to London

payload 30 000 kg range 5400 nm

w4 0.4 L/D altitude 38 000 ft

M 0.85

Table 3: The design mission parameters.

In addition, we also analyze the performance of the aircraft for a 3040 and 5400 nautical mile

missions which correspond roughly with the distance from New York to London (JFK to LHR) and

Los Angeles to London (LAX to LHR). These two missions were chosen since they are frequently

flown long-range routes that this class of aircraft might be assigned. For the optimization problem,

these two routes are also assigned a payload of 30 000 kg. For the purposes of optimization, we

weight the JFK to LHR and LAX to LHR routes with a 0.4 weights, respectively, while for the

extreme maximum range, maximum weight missions we assign an objective weight of 0.1 for each

respectively. The rationale is that the performance of the aircraft for these practical missions is

more important than the performance in the extreme regions of the payload-range diagram.

5.2 Structural constraints

For the aerostructural problem, we use a series of failure and buckling constraints that are imposed

at two points within the design envelope. These are the 2.5g and -1g points illustrated in the V-n

diagram shown in Figure 5. For both the failure and buckling constraints we employ a KreisselmeierSteinhauser (KS) aggregation technique [18], where we group point-wise constraints from within

each element into a small number of global constraints. The KS function can be written as follows:

KS = cmax +

1

ρKS

ln "X

N

i=1

exp(ρKS(ci − cmax))#, (20)

where ci are all the point-wise failure or buckling constraints in the aggregation domain, cmax =

max{c1, c2, . . . , cN } is the maximum value of the failure or buckling constraint, and ρKS is a

13
1 2 3 4 5 6 7 8 9 10

Range (1000 nautical miles)

10

20

30

40

50

60

Payload (1000 kg)

1. Max payload

2. Max fuel capacity

3. JFK to LHR

4. LAX to LHR

Figure 4: Payload-range diagram for the aircraft

penalty parameter which we take as 80 in all calculations presented here. Since the KS function is

conservative, if the failure envelope of the KS function is not violated, then none of the individual

element-wise failure functions will be violated. Instead of aggregating all the failure and buckling

functions into a single constraint, we group the failure and buckling constraints separately over

common structural components. We separate the top and bottom wing skins, respectively and

group all spars and ribs into a single KS constraint. Since we do not utilize buckling constraints

for the spars and ribs, there are only two KS buckling constraints while there are 3 KS failure

constraints for each structural load condition.

5.3 Aerostructural optimization problem

The aerostructural optimization problem can now be summarized as follows: The objective is to

minimize a linear combination of the takeoff gross weight and fuel burn over a series of different

missions weighted by an imposed objective-weight distribution. Simultaneously, the structural

design must satisfy a series of failure and buckling constraints at the two off-design conditions

shown in Figure 5 such that the 3 KS failure constraints and 2 KS buckling constraints at each

maneuver load are within the admissible failure envelope. In addition, we impose steady-state lift

and trim constraints at each of the 6 operating conditions. This is an idealization for the maneuver

conditions since, in reality, a steady pull-up condition would require a finite turning radius.

The design variables consist of geometric variables, structural variables, aerodynamic design

variables and consistency variables. The aerodynamic design variables consist of the angles of

attack at each of the 6 operating conditions. The geometric variables consist of 4 twist variables

distributed along the wing span, with a fixed root twist, one span-scaling variable, 5 vertical scaling

variables, which modify the thickness to chord ratio, and a single chord scaling variable for the entire

wing. Together the span and chord scaling variables admit a series of planforms that are stretched

14
-1

0

1

2

2.5

3

Load factor

Equivalent air speed

VS VA VD

CL = CLmax

2.5g stall 2.5g dive

-1g dive

Figure 5: V-n diagram illustrating the points where the structural constraints are imposed.

in the chord-wise and span-wise directions, but share similar geometric features. In addition, we

employ 6 tail rotation angle variables in order to trim the aircraft at each flight condition.

For the SP2 and SP3 structural parametrizations, the structural design variables consist of a

top stiffener pitch and bottom stiffener pitch variable. Furthermore, this restricts the design such

that the stiffener pitch is constant along the entire chord-length of the top and bottom wing skins,

respectively. Each panel formed by the leading and trailing edge ribs is assigned the variables

shown in Figure 3. Each panel has 2 thickness variables for the skin and stiffener, respectively as

well as a panel height. The stiffener base width is fixed based on the stiffener height. For the SP2

parametrization, each material is isotropic and no further information is necessary. For the SP3

parametrization, each material is composite and there are an additional 3 variables that define the

ply-fraction for the skin. In this work, we fix the ply-fractions within the stiffeners to a 0o-ply

dominant laminate.

There are several consistency variables that are required in the design problem. First, we add a

fuel mass consistency variable for each of the 4 missions. These variables are used in the application

of the inertial relief from the fuel loads. They ensure that a fraction of the total fuel load is applied

to the wing. In addition, we add an equivalent fuel traction on either the top or bottom skins for

each maneuver load. These loads correspond to the surface force per-unit-area exerted by the fuel

on the wing. We add a consistency variable for each rib-bay for the entire wing, for each mission.

Finally, for the buckling constraints, it is often necessary to have an equivalent panel length, since

this information cannot be calculated for each element, we add a length variable for each panel in

the wing. We then add a geometric constraint that the physical panel length must match the panel

length variable for each component of the wing.

15
Parameter Value Units Parameter Value Units

Aluminum material data

E 70.0 GPa ν 0.3

σY S 380 MPa ρ 2780 kg/m3

Composite material data

E1 128 GPa E2 11 GPa

G12 4.5 GPa G13 4.5 GPa

G23 3.2 GPa ν12 0.25

Xt 1170 MPa Xc 1120 MPa

Yt 40 MPa Yc 170 MPa

S 48 MPa ρ 1522 kg/m3

Carbon nanotube-based composite material data

E1 1200 GPa E2 120 GPa

G12 45 GPa G13 45 GPa

G23 32 GPa ν12 0.25

Xt 6000 MPa Xc 5000 MPa

Yt 400 MPa Yc 1600 MPa

S 500 MPa ρ 1522 kg/m3

Table 4: Mechanical properties of the metallic, composite and CNT-based composite used in this

study.

The aerostructural optimization problem can be written as follows:

minimize X

4

i=1

wi (βFB + (1 − β)TOGW)

w.r.t. x

such that KS(i)

fail ≤ 1 i = 1, . . . , 6

KS(i)

buckling ≤ 1 i = 1, . . . , 4

L

(i) = n(i)W(i)

i = 1, . . . , 6

M(i) = 0 i = 1, . . . , 6

c(x) = 0

l ≤ Ax ≤ u

(21)

where KS(i)

fail and KS(i)buckling are the KS failure and KS buckling constraints, L

(i) = n(i)W(i) and

is the lift constraint for each analysis condition, and n(i)is the load factor, M(i) = 0 represents

the trim condition for each load condition, c(x) = 0 represents all the consistency constraints

and l ≤ Ax ≤ u represents the adjacency constraints. Note that the consistency constraints are

nonlinear but do not depend on the solution of the aerostructural problem directly, therefore their

derivatives are inexpensive to evaluate.

Table 4 shows the properties of the materials used in this study including the metallic, composite

16
130 140 150 160 170 180 190 200 210

61

63

65

67

69

71

73

75

77

79

81

Weighted fuel

burn (1000 kg)

OEW (1000 kg)

β = 0

β = 1

metallic

composite

CNT composite

Figure 6: The Pareto front of the weighted sum of the fuel burn vs. OEW of the aircraft.

and CNT-based composite. Note that the CNT-based composite material is based on a tensile

modulus of 1.2 TPa and a tensile strength of 6 GPa, while the remaining properties are scaled

to match the corresponding composite material data. We apply a 20% knock-down factor to all

strength properties to account for material degradation and strength after impact effects.

6 Results and Discussion

In this section, we present the results from a series of optimizations for the metallic, conventional

composite and CNT-based composite wings. In order to evaluate the effect of fuel burn and

structural weight reduction, we present a series of optimizations that examine the trade-offs between

structural weight reduction and fuel burn minimization.

6.1 Aerostructural optimization

Figure 6 shows the Pareto front comparing the optimization results with β = 0, 0.5, 0.625, 0.75,

0.875, and 1 for the metallic, composite and CNT-based composite wings. Here we plot the Pareto

fronts based on the OEW and the weighted sum of the fuel burn from each of the four missions

used in the objective function. We plot the OEW rather than TOGW since the TOGW varies

depending on the mission, while the OEW is independent of mission. In the context of this study,

the Pareto fronts based on OEW or TOGW and weighted fuel burn lie on top of one another since

the TOGW can be expressed as a sum of the OEW, the weighted fuel burn, and a constant offset

17
0

20

40

60

80

100

120

140

160

Fuel burn (1000 kg)

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

metallic

composite

CNT

β = 0 β = 0.5 β = 0.625 β = 0.75 β = 0.875 β = 1

Flight conditions

Figure 7: Fuel burn for each of the four missions for the metallic, composite and CNT-based

composite wing.

from the reserve mission fuel and payload. Note that the composite wing designed with β = 0.5 is

markedly different from the other composite designs resulting in a non-smooth Pareto front. This

discrepancy is due to a premature termination of this optimization which produces a feasible, but

not fully optimal design. We are currently investigating the cause of this premature termination.

From Figure 6 it is clear that the use of advanced materials for the wing leads to simultaneous

fuel burn and structural weight improvements. As a result, the Pareto fronts for each structural

technology shift towards the axis. For the fuel burn objective with β = 1, the best design for the

CNT-based composite wing has a 12% weighted fuel burn advantage over the metallic wing and a

7% weighted fuel burn advantage over the composite wing. However, as β decreases from 1, the

shape of the Pareto fronts change significantly. In particular there is a significantly smaller fuel

burn penalty for the minimum TOGW designs over the minimum fuel burn designs. This results

in Pareto fronts for the composite and CNT-based composite wings that are more compact along

the weighted fuel burn axis.

At the other end of the Pareto front, there is a significant increase in OEW between the

β = 0.875 point and the minimum fuel burn designs, while the fuel burn reduction is less than

1% for the metallic case and about 2% for both the composite and CNT-based composite designs,

respectively. In each case, the fuel burn reduction between the last two points comes at a significant

weight penalty of between 30 000 and 40 000 kg. This result illustrates the importance of correctly

modeling the potential drag reduction when considering fuel savings.

Note that the CNT results exhibit a relatively modest 7–11% improvement in the fuel burn

given the ten-fold improvement in the material properties of the CNT-based composite. This

modest improvement is due to a number of factors. First, only the structural weight of the wing

is reduced relative to the other designs. Structural material advances could also be used on other

parts of the structure, but assessing the impact of this weight savings is beyond the scope of the

current report.

Figure 7 shows the variation of the fuel burn for the four missions analyzed in the design problem

18
0 0.25 0.5 0.75 1.0 50

55

60

65

70

75

80

85

90

95

Span (m)

β values

metallic

composite

CNT composite

Figure 8: The spans of the metallic and composite wing designs

for the metallic, composite and CNT-based composite wings for β = 0, 0.5, 0.625, 0.75, 0.875, and

1, respectively. For all missions and all values of β, the fuel burn decreases for increasingly advanced

structural technology so that all composite designs burn less fuel than the corresponding metallic

designs, and all CNT-based composite designs burn less fuel than the corresponding composite

designs. Note that the reduction in fuel burn, in an absolute sense, for increasing structural

technology is largest for the β = 0 designs and smaller for designs with β > 0. This effect can also

be observed in an aggregate sense in the Pareto fronts in Figure 6.

Figure 8 shows the optimal spans for the metallic, composite and CNT-based composite wings.

The trend is clear: the wing span increases as fuel burn objective becomes increasingly important.

It is reasonable to expect that the spans of the metallic, composite and CNT-based composite

wings would be successively larger for a given value of β. However, in the results shown here,

the metallic wing span is larger than the composite wing span for β = 1. This may be due to

the structural weight savings achieved by the composite designs which offset the fuel burn savings

achieved through a larger wing span, and thus higher L/D ratio.

Figure 8 shows that the wing spans of the final designs, particularly of the composite wings, do

not vary as smoothly as expected. This non-smooth behavior may be due in part to the termination

of the optimization before the design has reached the full optimality and may also be due to the

sensitive nature of the design to the wing span. Typically, the span only increases in the late stages

of the design optimization. Presently we are trying to determine the cause of the non-smooth

variation of the span and the larger metallic wing span for the fuel burn minimization case.

Figure 9 illustrates the L/D ratios achieved by the final designs at the mid-weight cruise con19
14

15

16

17

18

19

20

21

22

L/D at midmission fuel burn

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

β = 0 β = 0.5 β = 0.625 β = 0.75 β = 0.875 β = 1

metallic

composite

CNT

Flight conditions

Figure 9: L/D ratio evaluated at the half the mission fuel burn.

ditions for the metallic, composite and CNT-based composite designs. In this case, it would be

reasonable to assume that for a given mission, and a given value of β, the optimal L/D ratio should

increase with increasing level of technology, however the results presented here do not exhibit this

behavior. For the fuel burn minimization results with β = 1, the composite wings exhibit a lower

L/D ratio than the metallic or CNT-based composite designs. In fact, the L/D ratio correlates

more closely with the wing span.

Figure 10 shows the planforms of the metallic, composite and CNT-based composite designs

for β = 0, 0.5, 0.75, and 1, respectively. The results show the large span extensions exhibited by

the fuel burn optimized designs. In addition, there is relatively little change in the chord scaling

variable. As a result, the large span designs also exhibit a large increase in the wetted wing area

compared with the baseline QCRM wing.

Figure 11 shows the twist distributions at both the 2.5g maneuver condition and at the midweight point of the mission from Los Angeles to London for the metallic, composite and CNTbased composite designs for β = 0, 0.5, 0.75, and 1, respectively. In most cases, the composite and

metallic wings exhibit similar amounts of passive aeroelastic twist. However, the composite and

especially CNT-based composite wings are made with a much stiffer material and have much larger

wing spans.

Figure 12 shows the lift distributions normalized by dynamic pressure at both the 2.5g maneuver

condition and at the mid-weight point of the mission from Los Angeles to London for the metallic,

composite and CNT-based composite designs for β = 0, 0.5, 0.75, and 1, respectively. The designs

exhibit varying amounts of load alleviation where, in particular, the lift distribution for the fuel

burn designs exhibit a cross-over near the tip. This passive alleviation shifts the load in-board,

reducing the bending moment near the wing root.

Figures 13 and 14 show the metallic thickness distribution of the upper and lower wing skins

for the optimizations with beta = 0, 0.5, 0.75, and 1. For all cases, the thickness is largest at the

wing-body join, not the attachment point at the symmetry plane. As expected, the wing panel

thickness decreases towards the tip. The distribution of the upper and lower thicknesses are nearly

identical. For the design with β = 1 with the largest wing span, the thickness at the tip increases

20
slightly. This is due to a combination of inertial relief effects and reduced trim drag by aft-ward

motion of the center of gravity.

Figures 15 and 16 show the composite thickness distribution of the upper and lower wing skins

for the optimizations with beta = 0, 0.5, 0.75, and 1. The thickness of the panels is largest at the

side-of-body rather than the symmetry plane, except for the top skin for the β = 1 case. In the

composite cases, there is a non-uniform variation of the thickness on the upper wing skin while the

lower wing skin distribution appears to be smooth. We are currently investigating the cause of this

variation.

6.2 RANS-based aerodynamic shape optimization

In this section we examine the effects of performing a RANS-based aerodynamic shape optimization on the optimized design obtained from the coupled TACS and TriPan aerostructural design

optimization. The goal is to manipulate the airfoil cross sections slightly, improving the overall

drag characteristics without comprising the previously optimized structural design. To that end,

the planform remains fixed, and only small changes are allowed in the cross-sectional shape through

the use of shape design variables. All results in this section are purely aerodynamic and use the

undeformed jig shapes from the TACS-Tripan results.

6.2.1 Aerodynamic optimization problem formulation

Without the inclusion of the structural model, the aerodynamic optimization problem formulation is simplified considerably as compared to Equation (21). With a fixed structural model, the

Breguet range equation simplifies to the traditional aerodynamic optimization of lift-constrained

drag minimization. Our first aerodynamic problem formulation was as follows:

minimize CD

w.r.t. x

such that CL = C

∗

L

t > tinit i = 1, . . . , 425

(22)

where x is the vector of 192 shape design variables that perturb the surrounding free-form

deformation (FFD) box in the y direction. Figure 17 shows the FFD and the allowed motion of the

FFD points. C

∗

L

is the target lift coefficient that is set to 0.5 for all of the following optimizations.

In addition, there are 425 airfoil thickness constraints distributed over the entire wing surface and

ensure that the thickness of the wing cannot decrease anywhere. However, these constraints still

permit changes in the camber distribution at constant local thickness. Any decrease in the depth

of the spar box would immediately have a detrimental effect on the structural design and thus must

be avoided.

The first optimization we completed was the metallic design for β = 0.75. A comparison of the

initial and optimized lift distributions are given in Figure 18.

Twist variables were not included in the optimization under the assumption that with their

absence the optimization formulation would lack sufficient design freedom to significantly alter the

original lift distribution. Remarkably, this proved not to be case as shown in Figure 18. The

initial lift distribution is roughly similar to the corresponding β = 0.75 curve in Figure 12. We

do not expect these to match however, for two reasons: The current aerodynamic optimization

21
is performed on the jig shape whereas the aerostructural optimizations account for the deformed

shape. Secondly, the effects of the inboard shock are not captured in the panel method simulation

and this results in a more inboard-shifted load distribution than predicted by the TriPan simulation.

The aerodynamic optimization nearly recovers a perfect elliptical lift distribution. This large

change in lift distribution is even more unexpected when it is considered that both the initial and

optimized configurations are twisted by more than 14◦ along the span-wise direction. This result

highlights the capability of the RANS-based optimization framework: Even with a restrictive set

of design variables the optimization has produced a shock-free wing and elliptical lift distribution.

It is therefore clear that the new optimized aerodynamic design would require substantial changes

to the structural design to ensure that both the failure and buckling constraints can be met.

In an effort to prevent significant changes to the initial lift distribution, we modified the design

formulation to include a root bending moment constraint. This constraint is designed to act as a

very simple surrogate for the structure design by enforcing that the root bending moment of the

optimized design can be no larger than the initial bending moment. While there is no guarantee

that structural failure constraints will still be satisfied after the aerodynamic-only optimization, we

expect that the bending moment constraint surrogate will limit the amount of infeasibility in the

structural constraints. The modified aerodynamic optimization problem is given below:

minimize CD

w.r.t. x

such that CL = C

∗

L

t > tinit i = 1, . . . , 425

CMx < CMxinit

(23)

The initial value for the CMxconstraint is obtained by first performing a single flow solution at

C

∗

L

.

6.2.2 Metallic optimization results

Using the bending-moment constrained formulation, aerodynamic optimizations were completed

for each of the six metallic wing designs.

Figure 19 shows the surface mesh distribution, aerodynamic solution, and shock surface visualization (isosurface where (U/a) · (∇P/k∇Pk) = 1.0, where U is the velocity vector a is the speed

of sound and P is the pressure) for the metallic β = 0.75 design before optimization. The strong

shock on the inboard section of the wing is due to higher nose-up twist values near the wing root as

compared to the remainder of the wing. The grids and flow solutions for the other initial designs

show a similar shock pattern.

The aerodynamic mesh contains approximately 700 000 hexahedreal cells and optimizations

are performed using 64 processors. Optimizations required between 70 and 105 iterations and

approximately 7 hours of wall time to reduce the optimality to 5 × 10−5.

The evolution of the drag coefficient for each of the six optimizations is given in Figure 20. In

all cases, significant drag reduction is achieved, primarily due to the elimination of the shock on

the inboard section of the wing. The higher relative and absolute drag reductions on the lower β

values is due to the higher initial shock strength on these designs. A comparison of the initial and

optimized drag values (given in drag counts) is given in Table 5.

22
Table 5: Initial and optimized drag values in drag counts (ct).

β Initial Value (ct) Optimized Value (ct) Delta (ct) % Change

0.000 243.69 202.92 -40.77 -16.7

0.500 223.58 184.89 -38.69 -17.3

0.625 212.10 177.65 -34.45 -16.2

0.750 196.02 169.63 -26.39 -13.5

0.875 192.00 164.46 -27.54 -14.3

1.000 169.70 152.16 -17.54 -10.3

Lift distributions for each of six optimizations are given in Figure 21. The effect of the bending

moment constraint is now clear. In all cases the highly inboard loaded lift distributions have been

smoothed while retaining the same root bending moment. The optimized lift distributions indicate

that a nearly linear lift distribution is optimal for aerodynamic optimizations largely governed by

the fixed bending moment constraint. Note that even though each of the six designs have different

spans, the shape of the initial and optimized distributions are very similar.

It is also instructive to compare the drag values from the two β = 0.75 optimizations; one with

a bending moment (CMx) constraint and one without. The constrained optimization resulted in

169.63 drag counts while the unconstrained optimization (with a nearly elliptical lift distribution)

yielded 146.90 counts or approximately 14%. Since both solutions have no significant regions of

separated flow and are almost entirely shock free at the optimized design point, we can conclude

that this drag differential is due primarily to the different lift distributions. This drag differential

would represent approximately 10% of the total drag on a complete 777-like airframe.

Finally we examine a three dimensional view of the Cp contours on the optimized designs shown

in Figure 22. In all cases, the initial inboard shock has been completely eliminated and the smooth,

parallel Cp isobars indicate a smooth pressure recovery right up to the trailing edge.

A two dimensional slice of the wing at the parametric half-semi-span location shown in Figure 23

shows very precisely the power of the aerodynamic shape optimization to eliminate shocks and

reduce the resulting wave drag. It is particularly interesting to see just how little shape modification

is necessary to achieve these effects. Slight thickness increases and redistribution of camber is

enough to have a significant effect on the overall drag. Recall the twist distributions are fixed and

only thickness increases are allowed. Even with this very restrictive search space, all optimizations

yield significant aerodynamic improvements within the supplied constraints.

7 Conclusions

While some results raise new questions that require further investigation, the following conclusions

can be drawn from the results obtained so far:

1. The QCRM wing developed for this study seems suitable to evaluate the material trade-offs

for transonic transport wings.

2. Conventional composite wings exhibit a fuel burn that is 4–8% lower than metallic wings.

3. CNT composite wings, in spite of an elastic modulus and strength that is about ten times

higher than conventional composites, exhibits a relatively modest improvement in fuel burn

23
(7–11%). This is in part because the structural weight of components other than the wing

were assumed to be constant, and in part because many of the CNT structural thicknesses

converged to the lower bound.

4. The computation of fuel burn versus empty weight Pareto fronts provided valuable information in terms of the magnitude of the trade-offs involved for the various materials. The

span of the Pareto fronts in both axes decreased with better materials, as there the trade-off

between fuel-burn and weight became less dramatic.

5. CNT-based composite optimal wings have a span that is 13–30% larger than their metallic

and conventional composite counterparts.

6. At the optimum, the fuel burn and TOGW are very insensitive to the span. This needs to

be investigated further.

7. Using RANS-based aerodynamic shape optimization, it is possible to further reduce the drag

by up to 14%, with no structural weight penalty.

References

[1] Mader, C. A., Martins, J. R. R. A., Alonso, J. J., and van der Weide, E., “ADjoint: An

Approach for the Rapid Development of Discrete Adjoint Solvers,” AIAA Journal, Vol. 46,

No. 4, April 2008, pp. 863–873. doi:10.2514/1.29123.

[2] Lyu, Z., Kenway, G. K., Paige, C., and Martins, J. R. R. A., “Automatic Differentiation Adjoint

of the Reynolds-Averaged Navier–Stokes Equations with a Turbulence Model,” 21st AIAA

Computational Fluid Dynamics Conference, San Diego, CA, Jul 2013. doi:10.2514/6.2013-

2581.

[3] Kennedy, G. J. and Martins, J. R. R. A., “Parallel solution methods for aerostructural analysis

and design optimization,” Proceedings of the 13th AIAA/ISSMO Multidisciplinary Analysis

Optimization Conference, Fort Worth, TX, September 2010, AIAA 2010-9308.

[4] Kenway, G. K. W., Kennedy, G. J., and Martins, J. R. R. A., “Scalable Parallel Approach for

High-Fidelity Steady-State Aeroelastic Analysis and Derivative Computations,” AIAA Journal, 2014. doi:10.2514/1.J052255, (In press).

[5] Katz, J. and Plotkin, A., Low–Speed Aerodynamics, McGraw–Hill Inc., 1991.

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Computing, edited by E. Arge, A. M. Bruaset, and H. P. Langtangen, Birkh ̈auser Press, 1997,

pp. 163–202.

[7] Kenway, G. K. W. and Martins, J. R. R. A., “Multi-Point High-Fidelity Aerostructural Optimization of a Transport Aircraft Configuration,” Journal of Aircraft, Vol. 51, No. 1, 2014,

pp. 144–160. doi:10.2514/1.C032150.

[8] Hopkins, E. J., “Charts for predicting turbulent skin friction from the van Driest method (II),”

Tech. Rep. TN D-6945, NASA, October 1972.

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[9] Wakayama, S. and Kroo, I., “Subsonic wing planform design using multidisciplinary optimization,” Journal of Aircraft, Vol. 32, No. 4, 1995, pp. 746–753. doi:10.2514/3.46786.

[10] Brown, S., “Displacement extrapolation for CFD+CSM aeroelastic analysis,” 38th Structures, Structural Dynamics, and Materials Conference, April 1997. doi:10.2514/6.1997-1090,

AIAA97-1090.

[11] Saad, Y. and Schultz, M. H., “GMRES: A generalized minimal residual algorithm for solving

nonsymmetric linear systems,” SIAM Journal on Scientific and Statistical Computing, Vol. 7,

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[12] Martins, J. R. R. A. and Hwang, J. T., “Review and Unification of Methods for Computing

Derivatives of Multidisciplinary Computational Models,” AIAA Journal, Vol. 51, No. 11, 2013,

pp. 2582–2599. doi:10.2514/1.J052184.

[13] Martins, J. R. R. A., Alonso, J. J., and Reuther, J. J., “A Coupled-Adjoint Sensitivity Analysis

Method for High-Fidelity Aero-Structural Design,” Optimization and Engineering, Vol. 6,

No. 1, March 2005, pp. 33–62. doi:10.1023/B:OPTE.0000048536.47956.62.

[14] Martins, J. R. R. A., Alonso, J. J., and Reuther, J. J., “High-Fidelity Aerostructural Design

Optimization of a Supersonic Business Jet,” Journal of Aircraft, Vol. 41, No. 3, 2004, pp. 523–

530. doi:10.2514/1.11478.

[15] Vassberg, J. C., “A Unified Baseline Grid about the Common Research Model Wing-Body for

the Fifth AIAA CFD Drag Prediction Workshop,” 29th Applied Aerodynamics Conference,

Honolulu, Hawaii, June 2011. doi:10.2514/6.2011-3508, AIAA 2011-3508.

[16] Stroud, W. J. and Agranoff, N., “Minimum mass design of filamentary composite panels under

combined loads: Design procedure based on simplified buckling equations,” Tech. rep., NASA

Langley Research Center, Hampton, VA 23665, October 1976.

[17] Jones, R. M., Mechanics of Composite Materials, Technomic Publishing Co., 1996.

[18] Poon, N. M. K. and Martins, J. R. R. A., “An Adaptive Approach to Constraint Aggregation

Using Adjoint Sensitivity Analysis,” Structural and Multidisciplinary Optimization, Vol. 34,

No. 1, 2007, pp. 61–73. doi:10.1007/s00158-006-0061-7.

[19] Lyu, Z., Kenway, G. K. W., and Martins, J. R. R. A., “RANS-based Aerodynamic Shape

Optimization Investigations of the Common Research Model Wing,” Proceedings of the AIAA

Science and Technology Forum and Exposition (SciTech), National Harbor, MD, January 2014.

doi:10.2514/6.2014-0567, AIAA-2014-0567.

25
40 35 30 25 20 15 10 5 0

40 35 30 25 20 15 10 5 0

0

5

10

15

20

25

30

35

40

0

5

10

15

20

25

30

35

40

(a) Metallic

40 35 30 25 20 15 10 5 0

40 35 30 25 20 15 10 5 0

0

5

10

15

20

25

30

35

40

0

5

10

15

20

25

30

35

40

(b) Composite

50 45 40 35 30 25 20 15 10 5 0

50 45 40 35 30 25 20 15 10 5 0

0

5

10

15

20

25

30

35

40

45

0

5

10

15

20

25

30

35

40

45

(c) CNT

Figure 10: The planforms of the metallic, composite and CNT-based composite wings.

26
span [m]

twist [degrees]

0 5 10 15 20 25 30 35 40 -18

-16

-14

-12

-10

-8

-6

-4

-2

0

β = 0

β = 0.5

β = 0.75

β = 1

2.5g maneuver

LAX to LHR cruise

(a) Metallic

span [m]

twist [degrees]

0 5 10 15 20 25 30 35 40 -18

-16

-14

-12

-10

-8

-6

-4

-2

0

β = 0

β = 0.5

β = 0.75

β = 1

2.5g maneuver

LAX to LHR cruise

(b) Composite

span [m]

twist [degrees]

0 5 10 15 20 25 30 35 40 45 -18

-16

-14

-12

-10

-8

-6

-4

-2

0

β = 0

β = 0.5

β = 0.75

β = 1

2.5g maneuver

LAX to LHR cruise

(c) CNT

Figure 11: Twist distributions for the metallic, composite and CNT-based composite wings at both

the 2.5g maneuver condition and the cruise condition from LAX to LHR.

27
span [m]

L’/q

0 5 10 15 20 25 30 35 40 0

2

4

6

8

10 β = 0

β = 0.5

β = 0.75

β = 1

2.5g maneuver

LAX to LHR cruise

(a) Metallic

span [m]

L’/q

0 5 10 15 20 25 30 35 40 0

2

4

6

8

10 β = 0

β = 0.5

β = 0.75

β = 1 2.5g maneuver

LAX to LHR cruise

(b) Composite

span [m]

L’/q

0 5 10 15 20 25 30 35 40 45 0

2

4

6

8

β = 0

β = 0.5

β = 0.75

β = 1 2.5g maneuver

LAX to LHR cruise

(c) CNT

Figure 12: Lift distributions for the metallic, composite and CNT-based composite wings for both

the 2.5g maneuver condition and the cruise condition from LAX to LHR.

28
upper panel thickness

0.016

0.014

0.013

0.011

0.009

0.007

0.005

0.004

(a) β = 0

upper panel thickness

0.020

0.018

0.015

0.013

0.011

0.008

0.006

0.004

(b) β = 0.5

upper panel thickness

0.024

0.021

0.018

0.015

0.012

0.009

0.006

0.004

(c) β = 0.75

upper panel thickness

0.023

0.020

0.017

0.015

0.012

0.009

0.006

0.004

(d) β = 1

Figure 13: The lower panel thickness distributions for the metallic wings in meters.

lower panel thickness

0.015

0.014

0.012

0.010

0.009

0.007

0.005

0.004

(a) β = 0

lower panel thickness

0.020

0.017

0.015

0.013

0.010

0.008

0.006

0.004

(b) β = 0.5

lower panel thickness

0.024

0.021

0.018

0.015

0.012

0.009

0.006

0.004

(c) β = 0.75

lower panel thickness

0.024

0.021

0.018

0.015

0.012

0.009

0.006

0.004

(d) β = 1

Figure 14: The lower panel thickness distributions for the metallic wings in meters.

29
upper stiffener thickness

0.025

0.022

0.019

0.016

0.013

0.010

0.007

0.004

(a) β = 0

upper stiffener thickness

0.020

0.018

0.015

0.013

0.011

0.008

0.006

0.004

(b) β = 0.5

upper stiffener thickness

0.024

0.021

0.018

0.015

0.012

0.009

0.006

0.004

(c) β = 0.75

upper stiffener thickness

0.038

0.033

0.028

0.023

0.018

0.013

0.008

0.004

(d) β = 1

Figure 15: The lower panel thickness distributions for the composite wings in meters.

lower panel thickness

0.047

0.041

0.034

0.028

0.022

0.016

0.010

0.004

(a) β = 0

lower panel thickness

0.045

0.039

0.033

0.027

0.021

0.015

0.009

0.004

(b) β = 0.5

lower panel thickness

0.041

0.036

0.030

0.025

0.020

0.014

0.009

0.004

(c) β = 0.75

lower panel thickness

0.049

0.042

0.036

0.029

0.023

0.016

0.010

0.004

(d) β = 1

Figure 16: The lower panel thickness distributions for the composite wings in meters.

30
Figure 17: Free form deformation (FFD) volume with 192 control points and the initial CFD

surface.

2z/b

Normalized LIft

0.0 0.2 0.4 0.6 0.8 1.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Initial Design

Optimized Design

Elliptical

Figure 18: Lift-constrained drag minimization of metallic design (β = 0.75) with 192 shape design

variables.

31
Figure 19: Initial design for metallic β = 0.75.

Iterations

C

D

0 20 40 60 80 100 0.0150

0.0170

0.0190

0.0210

0.0230

0.0250

=0.0

=0.5

=0.625

=0.75

=0.875

=1.0

Figure 20: Change in drag coefficient during optimization for each design.

32
2z/b

Normalized Lift

0.0 0.2 0.4 0.6 0.8 1.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Initial

Optimized

Ellptical

=0.0

2z/b

Normalized Lift

0.0 0.2 0.4 0.6 0.8 1.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

=0.5

2z/b

Normalized Lift

0.0 0.2 0.4 0.6 0.8 1.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

=0.625

2z/b

Normalized Lift

0.0 0.2 0.4 0.6 0.8 1.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

=0.750

2z/b

Normalized Lift

0.0 0.2 0.4 0.6 0.8 1.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

=0.875

2z/b

Normalized Lift

0.0 0.2 0.4 0.6 0.8 1.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

=1.0

Figure 21: Comparison of initial and optimized lift distributions for all designs.

33
Figure 22: Cp contours for initial (left) and optimized (right) designs.

34
x/c

y/c

Cp

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0 -1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

=0.50

x/c

y/c

Cp

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0 -1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

=0.625

x/c

y/c

Cp

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0 -1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

=0.750

x/c

y/c

Cp

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0 -1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

=0.875

x/c

y/c

Cp

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0 -1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

=1.0

x/c

y/c

Cp

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0 -1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

=0.0

Figure 23: Comparison of cross section and Cp distribution at 2z/b = 0.5. Red is initial design,

black is optimized design.

35
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A Comparison of Metallic, Composite and Nanocomposite Optimal 

Transonic Transport Wings 

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Kennedy, Graeme J.; Kenway, Gaetan K. W.; Martins, Joaquim R. R.

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14. ABSTRACT

Current and future composite material technologies have the potential to greatly improve the performance of large transport aircraft. However, the coupling 

between aerodynamics and structures makes it challenging to design optimal flexible wings, and the transonic flight regime requires high fidelity 

computational models. We address these challenges by solving a series of high-fidelity aerostructural optimization problems that explore the design space for 

the wing of a large transport aircraft. We consider three different materials: aluminum, carbon-fiber reinforced composites and an hypothetical composite based 

on carbon nanotubes. The design variables consist of both aerodynamic shape (including span), structural sizing, and ply angle fractions in the case of 

composites. Pareto fronts with respect to structural weight and fuel burn are generated. The wing performance in each case is optimized subject to stress and 

buckling constraints. We found that composite wings consistently resulted in lower fuel burn and lower structural weight, and that the carbon nanotube 

composite did not yield the increase in performance one would expect from a material with such outstanding properties. This indicates that there might be 

diminishing returns when it comes to the application of advanced materials to wing design, requiring further investigation. 

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Aerodynamics; Composite materials; Metal matrix composites; Transonic flight; Wings

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