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RAeS 3

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Aircraft Structural Design Conference, 9-11

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October 2012, Delft, The Netherlands.

1

A Bi-Level High Fidelity Aero-Structural Integrated Design

Methodology - A Focus on the Structural Sizing Process

Christophe Blondeau

1

François-Xavier Irisarri

2

, François-Henri Leroy

2

Onera - The French Aerospace Lab, Châtillon, France

Itham Salah El Din

3

Onera - The French Aerospace Lab, Meudon, France

The optimization of wing-body configurations using

standard disciplinary optimization algorithms and

high fidelity numerical tools is today mature enough

to be used for real aircraft design process by aircraft

manufacturers. Among the challenges that arises

today is the capability to design optimal multi-

disciplinary feasible configurations within a single

global design process run. In the present paper a bi-

level and bi-disciplinary optimization, based on a

BLISS-like algorithm, of a transonic wing-body

configuration is presented. The aero-structure

optimization process shows that it is possible to

improve the performance of the design while

maintaining the feasibility for both aerodynamic and

structure disciplines at each step of the optimization

algorithm.

Allthough rather dedicated to the detailed description

of the high fidelity structural optimization process

(aeroelastic loads, structural sizing), this paper also

introduces the ARTEMIS (Advanced R&T Enablers

for Multidisciplinary Integrated Systems) project

which objective was to set up an original strategy to

efficiently couple a conceptual design level optimi-

zation tool and a high fidelity aero-structure bi-level

optimization capability.

I. Introduction

This paper presents recent developments and results

achieved at ONERA on multi-disciplinary wing

optimization of a wing-body configuration using a bi-

level algorithm.

______________________________________________

1

Research Engineer, Aeroelasticity and Structural Dynamics Dept.,

Christophe.Blondeau@onera.fr

2

Dr. Research Engineer, Composite Materials and Structures Dept.,

Francois-Xavier.Irisarri@onera.fr, Francois-Henri.Leroy@onera.fr

3

Dr. Research Engineer, Applied Aerodynamics Dept.,

Itham.Salah_el_Din@onera.fr

This work is part of the development of a general

framework named ARTEMIS (Advanced R&T Enablers

for Multidisciplinary Integrated Systems) in collabora-

tion between Airbus and ONERA. The ARTEMIS

project is meant to demonstrate the possibility to couple

two processes enabling the optimization of an aircraft,

considering design space at the system level (an

important number of disciplines and high level

requirements of the aircraft) along with the use of high

fidelity models and tools to achieve preliminary MDO of

the aircraft up to local sizing of components. In the

present study the two disciplines considered are the

aerodynamic of the flexible wing and the wing box

structural mechanics, the aim being to reach a feasible

optimum with respect to these two disciplines.

The problem definition, including the reference

configuration and the optimization problem formulation

will be presented. Then, a section will be dedicated to the

description of the optimization algorithm, focusing on the

composite wing structural optimization. The focus on the

aerodynamic sizing process will be the subject of a

separate communication [24]. Finally, the results of the

bi-level optimization process will be presented and

analyzed.

In the frame of the collaboration of Airbus with the R&T

community, ONERA proposed a roadmap on the

necessary steps to gradually implement the Airbus’

MDDC (Multi-Disciplinary Design Capability). This

roadmap focused on two main research axes:

• Bi-Disciplinary Process (BDP): developments concer-

ning aerodynamic / structure optimization with high

fidelity models and tools to be used at the

preliminary/detailed design level,

• Global Aircraft Process (GAP): developments concer-

ning the complete aircraft optimization (large number

of disciplines involved and lower fidelity tools) to be

used at the conceptual design level.

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This paper introduces the ARTEMIS project which

objective is to propose a methodology to couple BDP and

PAG processes, in order to achieve an optimal design

which satisfies the global aircraft performance for a

whole mission profile as well as the disciplinary

(aerodynamic, structure) constraints at the preliminary

design level (see Figure 1). This organization/architecture

is a complete breakthrough compared to the classical

sequential iterative design lines classically operated by

aircraft manufacturers.

In the present context, we will focus on the bi-

disciplinary aero-structural optimization process. A

BLISS-like approach has been chosen (Bi-Level

Integrated System Synthesis) [26]. An advantage of this

method is to preserve disciplinary design autonomy

which is of prime importance to the aircraft’s

manufacturer. This ensures that the design teams keep

responsibility of their own design methodology and

environment at the low price of changing their specific

objective function for the contribution to the global

system performance.

At the upper level, the wing planform shape is designed

(typically sweep angle, span, outer wing taper ratio,

relative thicknesses).

At the disciplinary level, the aerodynamic task is

responsible for airfoils and twist law optimal design (up

to 100 design variables, geometry and global

performance constraints) while the structural task

optimises the wingbox stiffness through composite skins

and spars physical properties (several hundreds of design

variables, up to 20000 design constraints).

Both disciplinary processes are based on up-to-date high

fidelity tools.

Additionally, the structure disciplinary design task

embeds a critical loads computation module which is

responsible for providing the correct set of aeroelastic

sizing loads. The wing box design is performed using a

bi-level scheme. At the upper level, only macroscopic

parameters and constraints are addressed while the lower

level is dedicated to the assessment of buckling criteria.

The buckling reserve factors are then approximated using

a response surface approach which is then fed back to the

macroscopic wing design level.

The link between optimal disciplinary designs and upper

level optimization problem is performed through a post-

optimal sensitivity analysis.

The process makes use of advanced numerical tools like

discrete adjoint technique for computing the flexible

aerodynamic gradients, analytic parameterisation for

CFD grid deformation and morphing techniques for FEM

mesh shape optimization.

Preliminary aero-structural optimization results for a

realistic civil aircraft transport with a composite wing

(typically A350 like) will be discussed and especially

how such an MDO approach helps in understanding the

key drivers of a candidate design.

Figure 1 – Organization of the ARTEMIS project.

II. Bi-level bi-disciplinary optimization problem

formulation

While the GAP follows a Multi-Discipline Feasible

(MDF) approach coupled with a global optimizer, the

BDP is centered around a bi-level, bi-disciplinary (aero-

structure) architecture with the aim of addressing the

issue of high fidelity modelling and numerical

computations.

The optimization problem is defined using the BLISS

MDO formulation. This approach uses two design levels

which are qualified respectively as high-level and

disciplinary-level. The high level design phase consists in

optimizing the aircraft using a common parameterization

for both disciplines. The disciplinary level optimization

is performed at fixed high level design variables

separating the aerodynamic and structures design spaces.

Disciplinary level calls high fidelity models, using

Computational Fluid Dynamics (CFD) and Compu-

tational Structural Mechanics (CSM) tools for the

analysis of the different models.

MDO methods were first used to simulate, predict and

optimise complex aerospace systems and products [28],

and has now matured into a critical aerospace technology

within Europe, see for instance [21]. Several MDO

methodologies have been proposed over the last two

decades, ranging from relatively straightforward

formulations (All-At-Once and Individual Discipline

Feasible [6], Multiple-Discipline Feasible [17]) to those

aiming at partitioning large systems into sub-system

optimization problems where a coordination algorithm

drives the sub-problem designs towards an optimal

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solution for the overall system. These decomposition

methods exploit the hierarchy in MDO problems by

reformulating them as a set of independent disciplinary

sub-problems and introducing a master problem to

coordinate the sub-problems towards the optimal system

design. The most promising decomposition methods

include Bi-Level Integrated System Synthesis (BLISS)

and its variants, Collaborative Optimization and

Analytical Target Cascading [27, 4, 16, 31]. A classifi-

cation based on the problem formulation structure has

been proposed in [32].

The Bi-Disciplinary process, as explained earlier, aims at

optimizing the shape of the wing (planform, airfoils and

twist law) and its internal structure. The process is based

on high fidelity tools used at the detailed design level.

For CFD, the elsA/AEL aeroelastic analysis capability

[13, 5] is used while for CSM, MSC/NASTRAN

software applies. An advantage of the BLISS

methodology is the possibility to carry out the

disciplinary optimizations in an independent manner. The

different teams keep the responsibility for their design at

the low price of changing their specific objective

function for the contribution to the global system

performance [1].

A general description of BLISS is given in the next

figure knowing that:

- X

a

corresponds to aerodynamic variables,

- X

s

corresponds to structural variables,

- Z corresponds to system or shared variables.

Post-Optimal

sensitivities or RSMs

System

Optimisation / Z

X

a

+∆

∆∆

∆X

a

Z+∆

∆∆

∆Z

MDA Aero-Structure

Adjoint Aero-Structure

Gradient / X

a

, X

s

, Z

CSM

Optim.

/ X

s

CFD

Optim.

/ X

a

X

s

+∆

∆∆

∆X

s

Post-Optimal

sensitivities or RSMs

System

Optimisation / Z

X

a

+∆

∆∆

∆X

a

Z+∆

∆∆

∆Z

MDA Aero-Structure

Adjoint Aero-Structure

Gradient / X

a

, X

s

, Z

CSM

Optim.

/ X

s

CFD

Optim.

/ X

a

X

s

+∆

∆∆

∆X

s

Figure 2 – BLISS-based aero-structural optimization

algorithm.

This MDO method for decomposition based optimization

of engineering systems involves system optimization

with relatively small number of design variables and a

number of sub-system optimizations that could each have

a large number of local variables. The algorithm

alternates between disciplinary and global optimization

steps.

The first step consists in an elimination of the

interdisciplinary state variables through a Multi-

Disciplinary Analysis (MDA). Subsequently, the coupled

total derivatives of relevant state responses are computed

with respect to local and shared variables.

At the local level disciplinary parameters X

a

and X

s

, are

optimized to improve the contribution of each discipline

to the global performance. This is done through a simple

separation technique :

0 0

(1 )

d d

d d

D W

T T

a s

a s

J C W

J J

J J J J

X X

X X

α α

= + −

≈ + ∆ + ∆ = +∆

(1)

Once that the optimal X

a

and X

s

are obtained, the system

optimization is performed with respect to the shared

variables Z, while preserving the disciplinary optimality.

A trust region approach controls the validity of the

approximate models involved at the system level.

When considering high-fidelity CFD/CSM, all the steps

in an MDO process become very challenging. The

objective here is to set up a technological demonstrator

for the BLISS procedure which takes advantage of the

most recent research developments in the following

areas:

•

Bi-disciplinary aeroelasic simulation using the Onera

elsA-AEL module.

•

Computation of total flexible aero-structural gradients

with respect to shared and local disciplinary variables

through adjoint techniques. All the direct and cross

dependencies of weight and drag are to be adressed.

•

Parametric CFD grid generation using free-form

techniques and FEM mesh deformation based on

morphing techniques.

•

Identification and computation of critical transonic

load cases with advanced CFD/CSM aeroelastic

analyses.

Taking a closer look at equation (1),

J

∆

contains the

classical

D a

dC d

X

and

W s

dW d

X

terms and the

D s

dC d

X

and

W a

dW d

X

cross-terms. The former one

represents the sensitivity of the arodynamic flow with

respect to structural design parameters (i.e. flexibility),

the latter the dependency of the structural weight to the

local aerodynamic shape parameters (through critical

loads redistribution). These cross-terms have not been

explicitly computed in the present work, hence the

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aerodynamic and structural objective functions only

depend on direct sensitivities. However, computation of

D s

dC d

X

through the discrete adjoint technique is still

an active research area [10]. This term is of prime interest

when considering the design of highly flexible wings.

The demonstrator has been applied to optimize the

Airbus XRF1 wing-body based geometry which

dimensions are typical of a transonic airliner.

Disciplinary optimization problems

The composite bi-disciplinary objective function

considered is as follows:

2

2

(1 )

D W p

J C W r Z

α α δ

= + − +

∑

(2)

The coefficient α represents the trade-off between the

disciplinary objective functions and r

p

a penalty factor

used to impact the global aircraft design phase, which is

beyond the scope of the present paper but detailed in the

companion reference [18]. The high level parameters

selected are: the wing span, the leading edge sweep

angle, the outboard wing taper ratio and the relative

thickness in two control sections.

The aerodynamic performance vs. structural integrity

compromise comes from the trade-off between

conflicting objectives of lightness and flexibility.

Aerodynamic performance is driven by large aspect ratio

and thin wing box while structural strengh requires thick

wing box and small aspect ratio.

With the increasing part of composite components in

primary structure, flexibility plays a major role in aircraft

performance. A more flexible wing yields more wash-out

(induced airfoil nose-down due to back sweep angle),

favours more inboard lift distribution (see Figure 3),

Figure 3 – Typical aerodynamic vs. aeroelastic optimum

spanwise lift distribution.

and reduce wing bending moment hence stress level and

structural weight.

The table below summarizes the differences between the

considered aerodynamic and structural optimization

problems.

Aerodynamics Structures

- Maximize aerodynamic perfor-

mance (C

L

/C

D

): Maximize Lift,

minimise Drag

- Design parameters: Twist angle

and profile shape at several span

stations

- Loadcases: Cruise segment of

the mission

- Constraints: Minimum lift coef-

ficient, bound constraints on

design variables.

- Features: Highly nonlinear cost

function (multimodal)

- Minimise weight

- Design parameters: physical

properties of structural members

(thickness, section)

- Loadcases: Critical conditions

to prevent structural failure

(+2.5G pull up, static gust,

descent, …)

- Constraints: deformations in

composite plies, bound cons-

traints on design variables and

physical properties, buckling

reserve factor, …

- Features: Linear cost function

with respect to design variables,

huge set of nonlinear constraints.

Post-optimality and system level optimization

problem

The disciplinary optimizations lead to optimal design

variables

*

a

X

and

*

s

X

, for prescribed global variables

0

Z

.

To build up the system objective function, it is necessary

to obtain the disciplinary contributions to the overall

aircraft performance, and their total derivatives with

respect to the global design variables. The system level

optimization problem is then formulated as:

* * 0 * 0

( , ) ( , ) ( , )

T

Z

dJ

min J J

d

δ

∆

= +

X Z X Z X Z Z

Z

(4)

* 0

( , ) (1 )

T T

WD

dW dC

J

d d

α δ α δ

= + − +

X Z Z Z

Z Z

0

*

. .

( , ) 0

L U

L U

st

δ

δ δ δ

≤ + ≤

≤ ≤

≤

Z Z Z Z

Z Z Z

G X Z

(4.1)

(4.2)

(4.3)

The first set of constraints corresponds to the allowed

bounds for the global aircraft geometrical modifications.

The second set manages the trust region size which

controls the validity of the linear approximation of the

objective function. The set G comprises the nonlinear

constraints that can be formulated at the overall aircraft

design level (typically general geometrical layout

constraints involving Z variables only) and also

potentially the disciplinary constraints that are still

unfeasible at the end of the disciplinary optimizations.

Introducing the prototypical parameterized optimization

problem

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( , )

. . ( , ) 0,

n

m

min f p

st p

∈

≤ ∈

x

x

g x g

ℝ

ℝ

,

p

∈

ℝ

(5)

it is possible to derive a tractable expression of the total

derivative of the optimal value function

*

( ( ), )

f p p

x

.

This result is due to Fiacco [9], provided that

*

x

is a local

minimum of problem (5), sufficient optimality conditions

hold at

*

x

and gradients of the active constraints are

linearly independent (LICQ). Then, for all values of p

that do not change the set of active constraints we get :

i

i

i A

g

df f

dp p p

λ

∈

∂

∂

= +

∂ ∂

∑

(6)

Turning back to our system level problem, and recalling

that we consider a strong interaction aeroelastic problem

with

Y

denoting the interdisciplinary state variables, we

can rewrite equation (6) as :

* 0

( , )

T

Ti i

i

i A

g g

dJ J J d d

dZ Z dZ Z dZ

λ

∈

∂ ∂

∂ ∂

= + + +

∂ ∂ ∂ ∂

∑

Y Y

X Z Y Y

(7)

where p is replaced with Z and the active constraint set is

the union of the aerodynamic and structural active

constraints at the optimum

* 0 0

( , , )

a s

X X Z

for aerodyna-

mics and

0 * 0

( , , )

a s

X X Z

for structures. In this expression

the implicit dependency of

Y

and

*

X

with respect to Z is

taken into account. The first two terms on the right hand

side of equation (7) represent the total derivative of the

objective function for the coupled aeroelastic system.

Same remark applies for the gradient of the active

constraints. These terms can be obtained through a new

MDA at a perturbed configuration

0

δ

±

Z Z

followed by

a finite difference approximation or preferably directly

solving the aeroelastic discrete adjoint system. Details

about the adjoint formulation are beyond the scope of this

article but the interested reader should refer to [22, 20, 8].

At the disciplinary optimum

* 0

( , )

X Z

, the three sets of

feasible, active and violated constraints have to be

managed at the system level.

a) Feasible or inactive inequality constraints can be

used to derive the trust region size. The constraints are

linearized around the disciplinary optimum and a

perturbation vector

δ

Z

is computed such that

*

( ) 0

k k

g g Z

δ

+ ∂ ∂ =

Z

,

k I

∈

where I is the set of

inactive inequality constraints

*

{ | 0}

i

I i g

= <

. The

elements of

δ

Z

with the smallest magnitude then define

the confidence radius for constraint activation

max( | 0, )

L i

Z Z i I

δ δ δ

= ≤ ∈

Z

min( | 0, )

U i

Z Z i I

δ δ δ

= ≥ ∈

Z

(8)

Similarly, the same argument can be used to predict the

tendency of an active constraint to turn inactive by using

a linear predictor of the associated positive Lagrange

multiplier and solving for a perturbation that sets the

multiplier to zero:

*

( ) 0

k k

Z

λ λ δ

+ ∂ ∂ =

Z

. This expression

requires the sensitivity of the Lagrange multiplier with

respect to Z. Writing the stationarity condition

( , , ) 0

λ

∇ =

X

X Z

L

at

* 0

( , )

X Z

of the Lagrangian function

( , , ) ( , ) ( , )

i i

i A

f g

λ λ

∈

= +

∑

X Z X Z X Z

L

and differentiating

with respect to Z we get :

* *

Z

d d

dZ dZ

λ

∇ + = −∇

XX X

XN

L L

, with

j j

N g

= ∂ ∂

X

(9)

Solving this equation for

d dZ

λ

is practical if an

accurate Hessian matrix of the objective and constraints

is available, which is not a standard output of common

optimization algorithms.

b) The set of active constraints at the optimum

contributes to the computation of the Lagrange

multipliers [11, 12]. From the stationarity of the

Lagrangian function, the Lagrange multipliers satisfy the

over-determined set of linear equations:

,

n m

f

λ

×

= −∇ ∈

X

N N

ℝ

(10)

Assuming that a constraint qualification such as LICQ

holds, the efficient way to compute vector

λ

is based on

the QR factorization of

N

:

[ ]

1

1 2 1

T

f

λ

−

=⇒= ∇

X

R

N Q Q R Q

0

(11)

In order to build the matrix

N

of active constraints

sensitivities, let’s first recall the different types of

constraint that enter a practical structural optimization

problem such as the one considered in the ARTEMIS

project. The objective is to minimize the wing box

structural weight W

W

with respect to design parameters

X

:

min ( , ( )) with { , }

W I D

W r =

X

X X X X X

(12)

The set of design parameters is divided into independent

X

I

and dependant

X

D

design variables. This functional

dependence for a particular variable X

D

is usually defined

as a linear equation :

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01

i

ndvi D

D M i Ii M i

iI

X

X C C C X C C

X

=

∂

= + ⇒=

∂

∑

, (13)

where ndvi is the number of independent design

variables.

These two types of design parameters are subjected to

absolute bounds and dynamic move limits (managed by a

trust region strategy) leading to the constraint set :

k k

L L U U

≤ ≤ ≤ ≤

X X X X X

, (14)

k denoting the current iteration.

Classical physical properties such as member thickness,

cross-section area or area moment of inertia, … are

derived from primal design variables through linear or

nonlinear relations. In addition to the bound constraints

that affect the primal design parameters, manufacturing

constraints can be associated directly to the physical

properties. Thus an additional set of constraints is defined

:

( , )

k k

L L I D U U

≤ ≤ ≤ ≤

p p p X X p p

(15)

In order to properly consider the contribution of these

constraints into the matrix

N

, the explicit dependence

with respect to

X

I

has to be exposed. For instance, for a

linear physical property definition we get :

0

( )

Dj

i i

i j

I I Dj I

X

dp p p

p C C X

dX X X X

∂

∂ ∂

= + ⇒= +

∂ ∂ ∂

∑ ∑

X

(16)

The structural responses r(

X

) denote the natural state

variables (or unknowns) associated to the problem

formulation and post-processing, e.g. displacements,

constraints, frequencies, temperatures … Once again

linear or nonlinear expressions of these natural outputs

can be formulated to enter the optimization problem

definition. These design responses may be used either as

constraints or as objective function. Fortunately their

derivatives are readily available as a standard post-

processing procedure of the structural optimization

solver.

Finally, gathering all contributions, the standard

structural optimization problem is formulated as :

min ( , ( )) with { , }

W I D

W r =

X

X X X X X

s.t. ( )

( )

k k

L L U U

k k

L L U U

k k

L L U U

≤ ≤ ≤ ≤

≤ ≤ ≤ ≤

≤ ≤ ≤ ≤

X X X X X

r r r X r r

p p p X p p

(17)

Thus completion of the active set information is not a

trivial task and the construction of the sensitivity matrix

N

may require a significant extra work.

In practice, the optimizer is not directly linked to the

finite element analysis solver. Rather information from

the sensitivity matrix is used to build explicit

approximations to the design responses in terms of the

independent design parameters. The resultant explicit

representation can then be used by the optimizer

whenever function or gradient evaluations are required

instead of the costly, implicit finite element structural

analysis.

Figure 4- Coupling analysis and optimization using

approximations.

The counterpart of this approach is that optimal

information at the optimizer level is computed with

respect to the approximation model. For instance, the

identification of the set of active constraints might differ

whether approximated or implicit constraint values are

considered, even if the convergence criteria are fully

satisfied at the optimum. The standard procedure is to

compute the constraint values at the optimum, identify

the active set based on the convergence criteria at the

finite element analysis level then assemble the matrix

N

from all sensitivity contributions with respect to

independent design variables. This way, it is likely that

columns of the resulting sensitivity matrix are linearly

dependent, mainly because a number of them are

numerically colinear. An additional rank revealing

technique based on a QR decomposition with pivoting

[14] has to be applied in order to restore a well posed

over-determined set of equations as in (10).

c) If any constraint is violated, it is linearized with

respect to

Z

and associated to the system level

optimization problem. This situation arises whenever a

particular set of global design variables leads to a non

feasible design at the end of the disciplinary optimization

process. For instance, it is known that an increase of the

wing box thickness induces heavier critical loads on the

structure that the current allowed stiffness budget for the

wing skins design might not be able to withstand. A way

to manage this is to bias the global layout in order to

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restore a feasible design by putting back the violated

constraints into the system level optimization problem.

As explained earlier, the shared design parameters

control the wing planform and thickness. The three

variables defining the planform (sweep angle, outer taper

ratio and span) are common to the GAP and BDP

process. Once the GAP submits a candidate design to the

BDP (the target vector

Z

GAP

), the objective function is

altered so as to avoid searching the design space in areas

that are not compatible with the results of the GAP. We

then formulate the following performance index for the

BDP :

2

* * 0

2

* 0

2

2

( , ) ( , ) (18)

( , ) (1 )

,with { , , }

TPAG

p

TT

WD

PAG

p

dJ

J X Z J X Z Z r

dZ dW dC

J X Z Z Z

dZ dZ

r Span Sweep Taper

α α

= + ∆ + −

= + − ∆ + ∆

+ − =

Z Z

Z Z Z

The penalty coefficient r

p

> 0 is monotonically increased

at each coupling step from GAP to BDP. Thus final

designs at GAP and BDP level gradually step to the same

configuration. Increase of r

p

means that each process gain

confidence in each other thanks to the data exchange and

enrichment process. Similarly, the objective function at

the GAP level is influenced by the same penalty term, but

this time

Z

BDP

being the target vector. Figure 5 below

illustrates qualitatively how the objective function of a

single parameter is modified for different values of r

p

.

A good initial choice for r

p

is is to penalize the system

level objective function such that the corresponding

optimal configuration at the outcome of the BDP first

iteration lies on the trust region boundary.

Figure 5 – Influence of the penalty factor.

III.

Wing box structural optimization process

The structural disciplinary optimization box is

detailed in Figure 6 and in particular how the high

fidelity aeroelastic computations are linked to the

structural optimization process. MSC/NASTRAN offers

some aeroelastic optimization capabilities but the

aerodynamic modeling is limited to linearized

formulations such as doublet lattice method which can

not take transonic nonlinear effects into account.

Figure 6 : Wing box structural sizing process overview.

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The objective is to set up a strategy for using

aerodynamic loads provided equally by elsA/AEL

computations or NASTRAN/Doublet Lattice Method

(DLM). In fact the CFD computations can not be

dynamically carried out during the optimization process

to cover all possible combinations of configurations and

flight conditions. They rather feed an initial Design of

Experiments database which is then combined with a

Proper Orthogonal Decomposition (POD) procedure to

derive the static load cases within the whole flight

envelope. This corresponds to the box entitled “Aero.

Tool for Static Loads” in Figure 6. For instance, the

aerodynamic field is parameterized with respect to Mach

number, angle of attack or sideslip angle.

The structural optimization process is converged for a

given set of external non-linear pressure fields. In this

optimization cycle the complete FEM is used and the

optimization parameters are the physical properties of

various structural members. An equivalent reduced order

model of the sized structural model is then automatically

derived in order to perform the adequate flexible static

aeroelastic computations at the current iteration. Once the

new flexible nonlinear loads are obtained, they are

transferred to the structural grid and a new optimization

process is performed. This organisation is very flexible as

it does not require any modification to the NASTRAN

core solver such as DMAP (Direct Matrix Abstraction

Programming) or SCA (Simulation Component

Architecture) programming.

The process depicted in Figure 6 has two levels of

convergence: one at the upper level on the critical load

cases and one at the optimiser level on physical

properties and design objective function. External sizing

criteria such as reserve factors for buckling, damage

tolerance or reparability of composite panels are included

in the form of several response surface models (RSM).

These RSM are built offline using a numerical DOE

approach coupled with dedicated expert tools.

Loads tool

CFD has improved the quality of aerodynamic design

in the industrial environment, but has not yet had as

much impact on the rest of the overall aeroplane

development process. Currently, most use of non-linear

CFD has been near the high-speed ‘cruise’ point as

illustrated in the V-n diagram in Figure 7. According to

[29], the high-speed lines development is less than 25%

of the total aerodynamics related aeroplane development

effort.

In transonic ﬂows where strong shock wave exists at the

wing, the aerodynamic modelling based on the RANS

equations should provide more realistic load distribution

than linear DLM methods. More specifically, noticeable

differences in the required force to trim the aircraft and

differences in the ratio between fuselage lift and wing lift

are known to occur [23].

Our goal is to have a good prediction for the trimmed

critical sizing loads distribution over the entire wing. To

achieve this, we do not consider CFD simulation of the

wing-fuselage-tail assembly for time consumption

reasons, but restrict the configuration to a clean wing-

fuselage. However, this simplification will be removed in

future demonstrations that will directly trim the complete

aircraft model. Thanks to the POD based reduction tool,

it is posssible to re-build the entire flow field, pressure

distribution or any aerodynamic coefficient in the whole

flight envelope for any combination of Mach and angle

of attack. Some authors have proposed a non-Galerkin

POD based reduction coupled with a surrogate model

approximation technique, see for instance [3]. More

intrusive strategies with respect to the CFD code also

investigated Galerkin POD projections to obtain a

reduced order model of the nonlinear aerodynamic

equations [32].

“+2.5g”

“-1.0g”

CFD mostly done

near cruise point

Load factor, n

Speed

V, keas

“+2.5g”

“-1.0g”

CFD mostly done

near cruise point

Load factor, n

Speed

V, keas

Figure 7 – Typical V-n maneuver diagram and domain of

classical CFD use.

As the interpolation procedure gives also access to

the zero lift pressure distribution, it is straightforward to

obtain a rather good estimate of the penalty on the

maximum lift coefficient due to trimming. Indeed, the lift

coefficient of the horizontal tail is directly linked to the

wing-fuselage moment coefficient. Thus, knowing the

different mass configurations to be investigated during

the loads process, we have access to the effective load

factor and the correct target lift coefficient to be used for

the CFD analysis. This use of CFD for critical aeroelastic

loads prediction near the stall conditions, see Figure 8, is

currently at the cutting edge of standard aerodynamic

simulations.

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Load distributions

Balanced load cases

alpha

C

L

Aerodynamic

conditions

close to stall

CFD/CSM aeroelasticity

LOADS

TOOL

C

LMax

Load distributions

Balanced load cases

alpha

C

L

Aerodynamic

conditions

close to stall

CFD/CSM aeroelasticity

LOADS

TOOL

C

LMax

Figure 8 – Components of the CFD loads capability.

Wing box design model

Wing skins and spars are divided spanwise and

chordwise into regions which are then linked to finite

element physical properties. One design variable is

associated to each region. The demonstration presented

in this paper only considers total thickness of composite

laminates as design variables (ply proportion, stacking

sequence and material orientation were fixed).

Additionally, cross section area and first moment of

inertia of stringers were linked analytically to skin

thickness design parameters in order to preserve a

realistic material ratio between skins and stiffeners. This

allows to keep a realistic buckling behaviour with respect

to loads redistribution during the sizing process. Figure 9

shows the design variables definition.

Figure 9 – Wing box design variables definition.

A total number of 228 sizing variables was defined.

Introducing dependency between variables (symmetry

and analytical relations), a set of 66 independent

parameters was finally considered. Bound constraints on

each design variable and physical parameters were

defined. Allowables on equivalent homogeneous normal

and shear strains for skin laminates and on stringer axial

strain have been introduced at the finite element level. A

final constraint set related to the critical buckling reserve

factor computed at the panel level was added for wing

skin elements.

Component-level structural optimization

This is a brief description of the process that links

together the macroscopic wing box structural

optimization with the local stiffened composite panel

design. At the upper level, the typical characteristics of

an aeroelastic structural sizing problem for a production

type FEM are up to 400 design variables, 10 load cases,

more than 50000 design constraints (including bound

limits), about 5000 retained and up to 300 active. Thus

adding the panel design complexity to this problem is

obviously intractable. The bi-level approach adopted here

is similar to that presented in [19]. Figure 10 below

presents an overview of the process.

First, a detailed model is built for each candidate region

that is to be designed using advanced criteria. Then a

database of inputs at the panel boundary nodes is built

from a given design optimization history of the wing box

(typically 5 to 10 design cycles). This database is the

basis of the Principal Components Analysis coupled with

a numerical Design Of Experiments (DOE) that is used to

train the buckling reserve factor surogate model.

Figure 10 – The Component-level optimization process.

More details about this work are presented in [15]. The

five step procedure is depicted in Figure 11. A principal

component analysis in terms of displacements at the

boundary nodes has been selected. For this

demonstration, only the total thickness of the composite

laminate has been sized. Besides, the dimension

reduction leads to two principal components. Hence a

surrogate model with three input parameters has been

built.

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Figure 11 – Surrogate model construction for the Buckling Reserve Factor.

It is well known that the buclking reserve factor exhibits

a piecewise behaviour (depending on which buckling

mode is activated) that the surrogate model must be able

to tackle.

For the composite stiffened panel optimization task, two

operating modes can be considered:

• The first one is an analysis mode in which local

buckling reserve factors are fed back as additional

nonlinear constraints to the macroscopic wing box

optimization problem through a surrogate model. This

is the current operating mode presented in this paper.

• The second operating mode considers a true

optimization of the stacking sequence. In this case, a

bi-level scheme such as QSD has to be set up [25].

This scheme will be adopted for future

demonstrations more focused on aroelastic tailoring.

IV. Numerical experiments

GAP and BDP coupling scenario in the global process

The data flowchart between PAG and PBD is detailed

in Figure 12 below. Before running a design optimization

at the overall aircraft design (OAD) level, a number of

PBD experiments are performed around the initial

configuration to build response surfaces allowing the

PAG beforehand update of its constituent mass and drag

estimation modules.

Figure 12 – Flowchart between GAP and BDP.

Preliminary results

The BDP optimization process was run for five

iterations for each of which correspond separate

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disciplinary optimization processes with upgraded global

variables set. Hereafter we present selected results to

illustrate the BDP.

Let’s assume an optimal configuration (i.e. a set of global

variables

Z,

and a trade-off coefficient

α

) is available

from a previous run of the GAP. The first step of the

BDP procedure depicted in Figure 2 is to update CFD

and FEM meshes to match the new geometry. The

aeroelastic equilibrium is then computed using the

elsA/AEL code at cruise flight conditions, typically

Mach=0.83, altitude=35000 feet for this type of aircraft.

The CFD mesh corresponding to the flexible cruise flight

shape is then available (see Figure 13) along with the

total derivatives computed with the discrete adjoint

method. Then all information is available to build the

disciplinary objective functions and start the concurrent

optimization processes.

At the outcome of disciplinary optimizations the classical

integrated results such as drag decomposition and

structural mass breakdowns can be analyzed.

Figure 13 – 1G level flight and jig shape CFD mesh.

As an example a dedicated post-processing has been

developed to automatically derive the wing box masses

for several parts such as wing skins, stringers, ribs and

spars.

For this particular configuration, the final outer wing box

is thinner which leads to a decrease in “geometrical”

flexibility that favours an inboard aerodynamic loading.

Figure 14 –Mass breakdowns for initial and final design

at first run of the BDP (5 shared variables) .

However, the strain level in wing skins has increased

resulting in more material in panels and stringers while

spars and ribs become lighter, due to lower transverse

shear deformation.

Once an optimal set of aerodynamic and structural design

variables has been obtained, the identification of the

active constraints set is performed in order to compute

the associated Lagrange multipliers. For instance typical

results of a structural post-optimality analysis are

produced below:

For a number of 66 independent design variables, two of

them activate their bounds. 54 design responses among a

total of 3502, corresponding to equivalent strains in

composites, are active at the optimum. The optimal

designed properties, mainly manufacturing constraints

such as minimal thickness, are limited at their bounds for

280 of them. The union of active constraints related to

design responses, design variables and design properties

form the basis for the construction of the constraint

gradient matrix

N

.

A number of these constraints are linearly dependent (in

their formulation or numerically) and a preliminary

cleaning step is required in order to robustly compute the

Lagrange multipliers. Using Fiacco’s theorem, the total

derivative of the optimal structural weight with respect to

shared variables can be derived. Focusing on the

contribution of the relative thickness at wing crank and

tip we get (weight in tons) :

*

*

+

ss

i

i

i A

XX

g

dW W

dZ Z Z

λ

∈

∂

∂

=∂ ∂

∑

6.764 1.967 8.731

4.035 1.822 5.857

− −

= +

− −

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Several remarks can be formulated. Firstly, from the first

term on the right hand side, a wing box height increase

results in a structural weight increase (geometrical

effect). Secondly, the constraints satisfaction improves

when the global thickness increases. Thus there is a

competition between optimal weight reduction and active

constraints satisfaction relative to wing thickness.

Similarly it is numerically observed that a thickness

increase induces a drag rise as could be expected, along

with a lift rise at the same angle of attack. The Lagrange

multiplier related to the minimal lift coefficient

constraint,

0.50

L

C

≥

, represents the price to pay in

terms of drag degradation for a lift improvement.

The normalized structural weight and drag optimization

histories are given below. Each vertical segment

represents the step due to global variables increment

produced by a BDP system level optimization.

Figure 15 –Structural weight and total drag history.

Each oblique segment represents the variation due to a

disciplinary optimization.

The evolution of the composite objective function

(1 )

D W

J C W

α α

= + −

is given in Figure 16. The vertical

dotted line points out the best configuration according to

disciplinary experts knowledge to be fed back to the PAG

level in order to initiate a new cycle.

Figure 16 – Aircraft performance index history.

The upper curve corresponds to the system performance

computed using the aircraft total drag history while the

lower curve only considers the pressure drag con-

tribution.

It is important to consider that the aircraft configurations

produced so far are all feasible with respect to

aerodynamic and structural design criteria evaluated with

high fidelity tools.

It is thus the designer’s decision to pick up a particular

candidate configuration to be returned to the PAG for an

evaluation at the overall aircraft level. It might not be an

adequate choice to wait until the BDP process overall

convergence since the associated computational burden

would not necessarily worth the gain in performance

index.

The spanwise lift distributions for the initial and

optimized wing are given below, as well as the design

variables history.

Figure 17 – Lift distribution and shared variables

optimization history (% deviation from initial

configuration) for a BDP run.

V.

Conclusion

To answer the challenge of developing simulation-

based optimization processes aiming to deliver faster the

best aircraft performance, the ARTEMIS project has

proposed a multi-level multi-fidelity MDO capability and

demonstrated it on a realistic civil aircraft transport test

case. The optimization process has been achieved

successfully to optimize a wing-body configuration in

terms of wing drag and wing-box structural weight. The

flexible wing performance improvement obtained has

been analyzed thoroughly and shows very encouraging

results. The organization adopted here followed several

guidelines such as exploiting the state of the art

numerical optimization techniques available for CFD and

CSM along with ensuring as much as possible

disciplinary autonomy of the designer’s team and at least

fits a design process into the aircraft manufacturer’s

required timescale. To this end, a multi-site collaborative

environment based on Python language exploiting it’s

network programming interface through the valuable

PYRO package [7] has been set-up.

It must be advised that this demonstration is a first

attempt to a fully integrated multi-level multi-

disciplinary design capability that will support in the near

future many improvements in the disciplinary design

processes as well as considering several variants in the

BDP and GAP architectures.

Generally speaking, the benefits of MDO for classical

configurations (large aspect ratio wings) where a huge

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expert knowledge and efficient practices exist, lay in the

help for harmonisation of models and tools, coherent data

layout, and better understanding of interactions. Existing

aircraft manufacturer’s OAD and disciplinary design

processes do not need to be significantly altered.

For new configurations (for instance blended wing

body, joined-wing, strut-braced wing) MDO approach

helps in supporting all the interactions at the uppermost

design level. Demonstration through the ARTEMIS

project. MDO has to be considered as a great opportunity

for improving traditional design lines.

In any case, MDO, as presented in this paper, leads to

multilevel consistent and feasible designs, enables faster

and systematic parametric studies, helps in better

understanding of the disciplinary trade-offs and offers

many potential time savings for the design process.

VI.

Acknowledgments

The authors would like to thank Airbus partners for

their support as well as the French Civil Aviation

Authority (DGAC) for financing the ARTEMIS project.

VII.

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