Aerostructural Adjoint Method for Flexible Wing Optimization

I. Ghazlane� , G. Carriery and A. Dumontz

ONERA, The French Aerospace Lab, 92100 Meudon, France

J.-A. Desiderix

INRIA, Sophia-Antipolis Mediterannee, 06905 Sophia-Antipolis, France

This paper presents the current developments at ONERA on wing optimization via the aero-structuraladjoint method. The aero-structural adjoint is the extension of the aero-elastic adjoint already used in aero-dynamic function optimization.1 The aero-structural adjoint method allows the improvement of both aerody-namic and structural functions in the same design space. The internal structural element thicknesses (sparwebs, caps, skins), the structural characteristics (flexibility) and the planform parameters are all variable inthe aero-structural adjoint-based design process. A module for structural modelling, wing weight estimationand adjoint-compatible structural sensitivities computation is presented. The material stresses are aggregatedinto the Kreisselmeier-Steinhauser function to reduce the high number of design structural constraints. Thestructural module is integrated with the existing aero-elastic environment for adjoint-based optimizations inorder to perform drag and weight optimizations.

Nomenclature

�geom Shape parameters�struct Structural parametersJ(�) Cost functionX Aerodynamic meshXb Beam meshCD Aerodynamic drag coe�cientF Flexibility MatrixI Area moment of inertiaJ Torsion constantL Aerodynamic loadsD Structural displacementsW Aerodynamic conservative variables on X

Wps Structural weight of the primary structureWB Wing boxKS Kreisselmeier-Steinhauser function� aggregation parameterng Number of structural constraints�i Bending stress at section i�ti Torsion stress at section i�yield Maximum allowable bending stress�yield Maximum allowable torsion stressgi Dimensionless structural constraint at section i

I. Introduction

An optimal wing will be extremely rigid to resists aero-elastic e�ects and considerably light for economical rea-sons. Unfortunately these two properties evolve monotonously in opposite directions: the wings are flexible becauseof their high dimension (high aspect ratio), making them sti�er will come at the expense of structural weight. Thetask of the designer is then to find a balance in the design space, a wing light enough to meet the environmental andeconomical needs and rigid enough to meet the FAR25 safety standards. Wing structural and aerodynamic behaviourare not only linked through performance. Sobieski shows2 that a lift resultant located aft on an airfoil, results in anincrease in the torque on the wing box and also generates twisting movement of the structure. Therefore, to keep�Applied Aerodynamics Department, Ph.D candidateyApplied Aerodynamics Department, Research EngineerzApplied Aerodynamics Department, Research EngineerxResearch Scientist, Head of Opal Research Group

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AIAA 2012-1924

Copyright © 2012 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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the wing from excessive twist more structural material for strength and sti�ness is requisite. The aerodynamic loadsinfluence the structural wing deformations, and the wing planform influences both magnitude and distribution of aero-dynamic loads. The direct conclusion is: when optimizing aircraft performance in terms of drag, the structure cannotbe treated as an isolated system, it must interacts with Aerodynamics.For aerodynamic-only optimization the ONERA code elsA38 provides aerodynamic objective gradient evaluation withthe adjoint method3 and for structural-only optimizations the Aeroelasticity and Structures Dynamics Department atONERA have optimization capabilities that lays on mixed-fidelity for the aerodynamic loads modelling. But for aero-structural adjoint-based optimization procedures, the complexity increases and so the computation cost.From an aerodynamicist point of view, some questions must be asked before stepping into an elaborate process. Isa complete aero-structural optimization including the construction and sizing of a high-fidelity structural model basedon FEM (finite element model) realistic (scalable to industrial-relevant problems)? Can the computation of objectivefunctions sensitivities with respect to the structural model parameters be computed when a high-fidelity FEM com-mercial software is used for structure modelling? Can we access easily to the structural parameters? To answer thosequestions many activities are conducted at ONERA for the integration of the adjoint method in a multidisciplinarydesign process.5, 6

The section II focuses on the extension of the aero-elastic adjoint towards a aero-structural adjoint approach for flex-ible wings, in which aerodynamic shape and primary structure geometry are optimized simultaneously. Because bothweight and drag are unfavourable to flight performance, this work aims, through a unified formulation of the opti-mization problem, to minimize the weighted sum of drag and structural weight under aerodynamic and structuresconstraints for both monopoint and multipoint processes. This can be formalized as follows:

J(�) =Pn f

i=1 kiCDi + !Wps n f flight case8>>>><>>>>:� = (�geom + �struct) 2 Rn

Gi(X) 6 0�k(X) 6 0 k aero load case number

Where Gi states for the aerodynamic constraints imposed on each flight point condition (typically on lift and pitchingmoment), � are the internal structural stresses, ki and ! are weightening parameters. The structural part of this workrequired the development of a structural module: InAirSsi (Internal Aircraft structure Sensitivity based Sizing), pre-sented in Section III and applied to achieve a structural optimization of the Airbus wing-body XRF1 configuration.In the last section, the results of aerodynamic, aero-elastic and structural redesigns of the XRF1 configuration arepresented and discussed.

II. Adjoint Formulation for a coupled problem

The adjoint formulation calculates the gradients of any scalar cost function with respect to a large number ofdesign parameters at a very low computational cost. We first present the aero-elastic adjoint system, already availablein elsA.7 In this case the flexibility matrix F of the wing structure is kept constant.Let J be the aerodynamic function to minimize:

J = J(S v;Dp); Function of interest

where S v =

0BBBB@S v f

S vs

1CCCCA =

0BBBB@WD

1CCCCA; State variable vector and Dp =

0BBBBB@Dp f

Dps

1CCCCCA =

0BBBB@ X

Xb;Wb; I; J

1CCCCA; Dependencies of J

The aeroelastic system is described by:

R =

0BBBBB@ R f (S v f ;Dp f )

Rs(S vs; �;Dps)

1CCCCCA =

0BBBB@00

1CCCCA; Governing equations

R f and Rs are the discrete residual form of RANS equations and the structural equilibrium equation. Gradient basedoptimization requires the information of the sensitivity of the objective function with respect to the design variables:

dJd�

=

26666664 @J@Dp f

0

0 @J@Dp s

377777752666664@Dp f

@�@Dp s@�

3777775 +

2666664 @J@S v f

0

0 @J@S v s

3777775 266664@S v f

@�@S v s@�

377775 (1)

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To compute the total variation of J , @S v@�

must be computed for each �, the increase of computation cost with theincrease of design space dimension is obvious. Because the variation of the residuals is zero, the variation of the costfunction with respect to the design space can be expressed without being modified as:

dJd�

=

26666664 @J@Dp f

0

0 @J@Dp s

377777752666664@Dp f

@�@Dp s@�

3777775+2666664 @J@S v f

0

0 @J@S v s

3777775 266664@S v f

@�@S v s@�

377775+"� f 00 �s

# 266666664@R f

@Dp f

@R f

@Dp s@Rs@Dp f

@Rs@Dp s

3777777752666664@Dp f

@�@Dp s@�

3777775+"� f 00 �s

# 26666664 @R f

@S v f

@R f

@S v s@Rs@S v f

@Rs@S v s

37777775 266664@S v f

@�@S v s@�

377775(2)

Where �s and � f are two arbitrary vectors called the adjoint vectors.

�@J@Dp f

@Dp f

@�: The change in the cost function due to design space perturbation through the CFD mesh.

�@J@Dp s

@Dp s@�

: The change in the cost function due to design space perturbation through the structural dependencies.

�@R f

@Dp f

@Dp f

@�: The sensitivity of the residual form of flow equations with respect to wing shape changes through the

CFD mesh.

�@R f

@Dp s

@Ds f

@�: The sensitivity of the residual form of flow equations with respect to wing shape changes through

wing box geometry modification through the structural dependencies.

�@Rs@Dp f

@Dp f

@�: The sensitivity of the residual form of structural equations with respect to wing shape changes through

the CFD mesh. In other words, this term quantify the e�ect of design space perturbation on L and F through theCFD mesh. The linearisation of this term is now implemented in elsA and validated using finite di�erences.

�@Rs@Dp s

@Ds f

@�: The sensitivity of the residual form of structural equations with respect to wing shape changes through

wing box geometry modifications through the structural dependencies. It represents the e�ect of design spaceperturbation on aerodynamic loads through Xb, F , and the sensitivity of of design space perturbation on struc-tural rigidity through the structural characteristics I and J. The linearisation of this term is now implemented inelsA and validated using finite di�erences.

In Equation:3 , the total variation of the cost function is reorganized, the first part represents the link to the depen-dencies of J and the second part is the dependence on the state variables:

dJd�

=

@J

@Dp+

266664� f 00 �s

377775266666664@R f

@Dp f

@R f

@Dp s@Rs@Dp f

@Rs@Dp s

377777775! 2666664 @Dp f

@�@Dp s@�

3777775 +

@J

@S v+

266664� f 00 �s

377775 26666664 @R f

@S v f

@R f

@S v s@Rs@S v f

@Rs@S v s

37777775 !2666664 @S v f

@�@S v s@�

3777775 (3)

Similarly to the aerodynamic adjoint, to eliminate the dependency of the total variation of the cost function on the statevariables, we solve for �s and � f :

26666664 @R f

@S v f

@R f

@S v s@Rs@S v f

@Rs@S v s

37777775T 266664� f 0

0 �s

377775 = �@J

@S v(4)

This system is called the aero-structural adjoint system and it is now implemented in the elsA code and solvedusing an iterative fixed-point like scheme.

In equation 3, the terms involving the sensitivities of the structural residuals or the sensitivities through the struc-tural dependencies requisite a structural solver that modifies the internal topology of the wing during the optimizationprocess and supply the adjoint solver with the necessary sensitivities. In this section we present the structural packagefor structural modelling and sensitivity computation implemented for solving aero-structural design problems.

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III. Structural Modeling for Aerostructural Optimization

In aero-structural optimization both structural parameters �struct and shape parameters �geom are modified to im-prove the total performance. The design variables �geom (span, camber, twist...) parametrize the aerodynamic envelopewhile �struct controls the thickness of the wing internal primary structure elements. The structural analysis is per-formed by the module InAirSsi (Internal Aircraft structure Sensitivity based Sizing) developed within the frameworkof aero-structural adjoint development. The CFD surfacic mesh is the main input of the structural module. The wingis modelled using a wing box (Fig 1) delimited by a front spar and a rear spar positioned at 25% and 75% of the chord,these stations can be modified by the user, or used as design variables. Each section has six degrees of freedom tovariate the element thicknesses of the primary structure along the span aiming to provide the locally required strengthand sti�ness with a minimum weight:

Figure 1. CFD mesh and the internal structural model of the pri-mary structure

* Upper wing skin thickness;* Lower wing skin thickness;* Upper wing cap thickness;* Lower wing cap thickness;* Front spar thickness;* Rear spar thickness.

The aerodynamic and the structure are coupled via theflexibility matrix (F) approach. The displacement D un-der aerodynamic loads L of the structural model nodes isD = F L. To take into account the variation of wing rigid-ity with the geometry changes, the coe�cients of bendingsti�ness EI and torsional sti�ness GJ are computed, withthe contribution of all primary structure elements, for eachgeometry during the aero-structural optimization, or thestructural optimization.

Structural weightThe structural weight Wps of the primary structure is ap-proximated by the module InAirSsi :

Wps = 2X�struct

�i

Z maxspan

minspanS i(y) dy (5)

Finite Di�erences InAirSsidW

d�S I15.2183e+04 5.2183e+04

dKSd�S I1

-1.1103e-07 -1.1104e-07dW

d�CI26.6702e+04 6.6702e+04

dKSd�CI2

-1.16e-02 -1.12e-02dW

d�CE71.0329e+04 1.0329e+04

dKSd�CE7

-1.3503e-09 -1.3497e-09dW

d�FS 71.0329e+04 1.0329e+04

dKSd�FS 7

-5.5511e-11 -5.5522e-11

Table 1. Gradients of Structural weight and KSfunction for a sample de design variables

�i and S i are the density and the surface of the component i. InAirssihas been fully di�erentiated by hand. The sensitivity of the structuralweight with respect to the design variables of the wing box are availableanalytically. Validation of these sensitivities with respect to a set of �struct

for the Airbus wing-body XRF1 configuration are presented in Table 1.

Kreisselmeier-Steinhauser function for constraint aggregation

The wing weight minimization process is constrained by the allow-able material stresses in the wing box elements. These stresses representthe numerical constraint for the optimization process that will providewing box elements that withstand the sizing loads. To limit assumption,each primary structure element is sized under both flexural and torsionalloads:The normal stress from the bending moment Mx at a section y is:

� (y) =MxzIx

(6)

For a wing box element of thickness t, the level of stress due to torsional loads is given by:

�t =My

2At(7)

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where A is the enclosed area of the cross section.The vertical force Fz (the integral of lift forces from the wing tip to current cross section) are used to size spar

webs of thickness tweb. The shear constraint is:

�s =Fz

2htweb(8)

where h is the height of the wing box.The aero-structural optimization problem is a highly constrained problem, the yield stresses can not be exceeded

at each element of the structural model. Treating separately each constraint may not help the convergence of thealgorithm. The di�erent constraints can be unified into one constraint using the Kreisselmeier-Steinhauser8, 9 envelopefunction:

KS (gi(�)) = gmax(�) +1�

ln[ngXi

e(�(gi(�)�gmax(�)))] (9)

Where gmax = jjgi(�)jj1 and gi(�) = �i�yield� 1 or gi(�) = �i

�yield� 1. An alternate formulation, that allows to free from the

computation of high exponential values and thus o�er a better numerical behaviour is:

KS (gi(�)) =1�

ln[ngXi

e(�gi(�))] (10)

The KS function represents an envelope surface of all the constraints and represents the maximum of the setof constraints. The KS function o�ers the property of being continuous and di�erentiable function, it reduces thediscontinuities at the intersection of constraints and provide a smooth transition from one surface constraint to another.The parameter � controls the distance from the KS surface to the surface of maximum constraint value jjgi(�)jj1. Ahigh value of � draws closer the KS bound to the maximum value of constraints.To quantify the e�ect of the parameter � and prepare the ground for the aero-structural optimizations, several test casesare presented. All optimizations start from the same initial design point but with a di�erent multiplying factor �:

minimize:W(�struct) ; �struct 2 R

42

Subject to KS � 0

Figure 2. the span and sweep angle modified XRF1configuration

The chosen configuration is a perturbed XRF1 configuration (thespan and the sweep angle are modified), the structural weight is mini-mized for a KS function that corresponds to a manoeuvre condition forwhich CL=1.25. The design space is parametrized by 42 parameters (6thicknesses at 7 control sections). The thickness of a y-span section isinterpolated from the thicknesses of the surrounded control sections .Table 1 summarizes the gradients validation for fixed loads of both KSfunction and structural weight with respect to a selection of 4 structuralthicknesses . In this problem KS function is the aggregation of 176 mate-rial constraints from both bending and torsion types and the initial pointof the optimization has a maximum stress from a bending type. When theparameter � increases, the KS function approaches from the maximumconstraint. Table 2 summarizes the results for three cases and highlightsthe influence of the � parameters on the objective function that is notalways straightforward. G. A. Wrenn shows8 that for the minimizationproblem of a 1 dimensional case with two constraints, larger values of �results in a smaller optimum objective function. Our results shows thatthis conclusion must be considered carefully.

� Initial wing box Redesigned wing box50 KS=0.139 & W=4.38e+04 KS=-0.0007 & W=3.02e+04100 KS=0.121 & W=4.38e+04 KS=-0.0009 & W=3.08e+04200 KS=0.115 & W=4.38e+04 KS=-0.0004 & W=3.22e+04

Table 2. Redesign results with di�erent values of KS function parameter �

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The bending stress in the sets lower wing+caps and the upper wing+caps is initially almost homogeneously dis-tributed. The results (Fig 3&4, 5&6, 7&8) show the tendency of the optimization algorithm in reducing firstly thethicknesses of the elements sized for bending loads: the lower wing+caps and the upper wing+caps. As � increase,the thicknesses of these elements are reduced and so their structural weight. The torsion stresses in the front and rearspar webs are initially low in the outer wing and exceed the yield stress in the inner wing region. The straightforwarddrawn remark on sizing results is the di�erence between the sizing process in the inner and the outer wing regions.The thickness modification of rear and front spar webs increases as � increases. �yield � � in the inner wing region,thus e(�gi(�)) increases as � increases, at the opposite of the outer wing, where the torsion stress is initially low, lesssizing is achieved as � increases, ( �i

�yield� 1) � 0 thus e(�gi(�)) decreases for high values of �.

Initial WB Redesigned WB � = 50 Redesigned WB � = 100 Redesigned WB � = 200Upper wing skin (kg) 3,611.26 3,924.80 3,602.45 3,441.51Lower wing skin (kg) 4,352.96 4,001.71 3,691.33 3,518.56Upper wing caps (kg) 11,559.48 7,587.60 7,497.47 7,449.23Lower wing caps (kg) 10,574.96 7,652.70 7,511.13 7,457.42Front spar web (kg) 6,903.25 3,543.41 4,359.39 5,240.85Rear spar web (kg) 6,802.21 3,522.24 4,164.96 5,159.21

Table 3. Redesign results with di�erent values of KS function parameter �

The structural weight decomposition table 3 shows that for the elements where the initial stress material is almostuniformly distributed over the span (skins, caps). The structural weight of this set decreases as � increase, while the

front and rear spar structural weight decreases significantly for the smallest value of � that makes the term e( �i�yield

�1)

comparatively not negligible with respect to the one computed for higher values of the parameter. One solution will bethe identification of wing span region where the stresses are in the same order and sign, and introduce a KS functionfor each region.

Figure 3. Comparison of material stresses in the initial config-uration (left) and the redesigned internal structure of the wing(right) with a parameter � = 50. At a y-span position, is repre-sented respectively the bending stress in upper wing caps andskin, bending stress in lower wing caps and skin and torsionstress in the front spar elements.

Figure 4. Comparison of material stresses in the initial config-uration (left) and the redesigned internal structure of the wing(right) with a parameter � = 50. At a y-span position, is rep-resented respectively the torsion stress in upper wing caps andskin, bending stress in lower wing caps and skin and torsionstress in the front spar elements.

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Figure 5. Comparison of material stresses in the initial config-uration (left) and the redesigned internal structure of the wing(right) with a parameter � = 100. At a y-span position, is rep-resented respectively the bending stress in upper wing caps andskin, bending stress in lower wing caps and skin and torsion stressin the front spar elements.

Figure 6. Comparison of material stresses in the initial configura-tion (left) and the redesigned internal structure of the wing (right)with a parameter � = 100. At a y-span position, is represented re-spectively the torsion stress in upper wing caps and skin, bendingstress in lower wing caps and skin and torsion stress in the frontspar elements.

Figure 7. Comparison of material stresses in the initial config-uration (left) and the redesigned internal structure of the wing(right) with a parameter � = 50. At a y-span position, is repre-sented respectively the torsion stress in rear spar,torsion stress inupper wing skin and torsion stress in lower wing skin.

Figure 8. Comparison of material stresses in the initial config-uration (left) and the redesigned internal structure of the wing(right) with a parameter � = 200. At a y-span position, is repre-sented respectively the torsion stress in rear spar,torsion stress inupper wing skin and torsion stress in lower wing skin.

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IV. Aeroelastic and structural redesign of the XRF1 configuration

This section presents the results on the steps leading to the optimization of the total aero-structural performances.Aero-elastic redesign of the XRF1 configuration is presented and compared briefly with the aerodynamic redesign. Theaero-elastic process assumes a structural model identical for all optimizer iterations, i.e the topology of the internalwing remains unchangeable. To free from this assumption and account for wing flexibility variations, the structuralpackage InAirSsi is solicited. The purely structural sizing of the internal topology of the wing, described in sectionIII, is performed under a single manoeuvre condition corresponding to a CL = 1:25.

The test case is the Airbus XRF1 wing body configuration 143-block mesh Fig 1.

Figure 9. Convergence history of aero-elastic and rigid pressure drag mini-mization

The flow characteristics are Mach = 0:83;Re =

10 � 106. All the optimizations are run usingan automated chain, InAirSsi for structural mod-elling, SeanDef an analytic tool for parametriza-tion and mesh generation, elsA code to performthe Euler computation and adjoint equation solu-tion. The drag extraction and breakdown is per-formed by FFD72 ONERA software that lays onthe farfield/nearfield balance.

The optimization problem has 11 degrees offreedom to redesign the aerodynamic shape of thewing. 5 sections distributed from the wing-body in-tersection to the tip of the wing control the twist an-gle and the camber of the control sections. The An-gle of attack is an extra ”design” parameter for CL

constraint satisfaction. For each control section, thetwist parameters evolve independently from �5� to5�, the camber varies from -5% of the local chordthickness to 5% and the angle of attack varies in[0; 3�]. The leading and trailing edge are not mod-ified. As previously said, the aero-elastic opti-mization process do not allow the internal structuremodification, thus the design variables were chosento have the less impact on the structural weight.

Aeroelastic redesignBoth aerodynamic and aero-elastic redesigns start from a feasible points. At the exception of AoA, twist and cambervariables are set to zero. The drag coe�cient of the initial design vector of the aero-elastic redesign is 167.44 DragCounts (DC) at AoA=1:9� and CL = 0:53. For the aerodynamic redesign the initial point at AoA=1� has CD = 158:12DC, CL = 0:51 with an angle of attack of 1�. The minimization of near-field drag coe�cient, constrained by the liftcoe�cient, is supervised by DOT algorithm and performed under the hypothesis of a constant sti�ness matrix. Eachstep of the gradient-based algorithm requires the computation of the aerodynamic flow and the evaluation of the adjointsystem for each function. 5 to 10 coupling iterations are performed to reach the aero-elastic equilibrium of the wing.Table 4 list, at CL = 0:5, CD for the initial and the redesigned configuration. The aero-elastic redesign converged to aconfiguration that has 9.02 DC comparing to the initial design, while the rigid configuration lead to a decrease of 5.64DC. For both minimizations, the decrease in the total drag is due to the wave drag component. The redesigned wingpresents a negative twist in the inner half wing and the last quarter. The optimal configuration has less camber thanthe initial. The load distribution over the optimum design [Fig.11] shows a redesigned configuration slightly loadedin the region between 50% to 72% of the span, corresponding to a positive twist. The loading decrease in the regionbetween the root to approximately 50% of the wing span is due to a negative twist.

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Table 4. Drag breakdown of both initial and optimum XRF1 designs at Cl=0.5

Baseline rigid Baseline Aerodynamic Aeroelastic Structure AerostructureCDp (Drag Counts) 147.72 146.27 142.08 137.25 * in progressKS function (CL = 1:25) * -7.22e-02 * * -9.23e-04 in progressWB weight (CL = 1:25) * 40351.82 * * 22761.61 in progress

Figure 10. Summary of Cdp aerodynamic optimization, performed with dot algorithm, of the XRF1 configuration. Parametrization with5 control twist section, 5 camber sections and the angle of attack

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Figure 11. Summary of Cdp aeroelastic optimization, performed with dot algorithm, of the XRF1 configuration. Parametrization with 5control twist section, 5 camber sections and the angle of attack

Structural redesign

The structural optimization of the XRF1 wing is presented in this section. The starting points is a feasible point,the KS function (� = 50), corresponding to CL = 1:25, is negative, which means that for all the sections �i � �yield.The initial thicknesses of the the internal structural element were chosen aleatory leading to a heavy structural weight.The redesign process, performed by CONMIN, results in a configuration two times lighter than the initial one. Fig13 shows the material bending and torsion stress distribution over the span for wing box elements. The increasein material bending stresses acting on the lower wing and the upper wing is generated by a decrease in caps&skinsthicknesses. In the wing tip region, as well as in front and rear spar webs, the redesign is modest. This is explainedby the low values of material stress in this regions and the di�culties of KS function in handling the aggregation ofdi�erent order functions. This highlight one more time the importance of the aggregation parameter �.

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Figure 12. Convergence history of the XRF1 structural weight minimization

Figure 13. Comparison of material stresses in the initialconfiguration (left) and the redesigned internal structureof the wing (right) with a parameter � = 50. At a y-spanposition, is represented respectively the bending stress inupper wing caps and skin, bending stress in lower wingcaps and skin and torsion stress in the front spar elements.

Figure 14. Comparison of material stresses in the initialconfiguration (left) and the redesigned internal structureof the wing (right) with a parameter � = 50. At a y-span position, is represented respectively the torsion stressin rear spar,torsion stress in upper wing skin and torsionstress in lower wing skin.

V. Conclusion and Near-Futur Work

This paper presents the fully-coupled adjoint, or the so called aero-structural ajoint method implemented within amultidisciplinary framework. This work is based on the aero-elastic adjoint method already available in the ONERACFD software elsA38 code. The fully-coupled adjoint method means that no assumption is made for the structuralmodelling. The internal structure of the wing is parametrized at the same level as planform design variables. This alsomeans that no assumption are made for wing aero-elastic response to aerodynamic loads. The structural characteristics(flexibility matrix) are redefined at each optimization step. The fully coupled adjoint, lays on a beam model. Althoughthis is not a high-fidelity model, the authors believe that such a model is a correct and trustful physical alternative thatwhen coupled with CFD presents all the quality of preliminary design tool: physically significant conclusions and lowcomputational cost. As shown in section II, the aero-structural method requisite the computation and the validation ofseveral coupling terms. For this purpose, as well as for structural modelling, InAirSsi a structural package is developedand integrated into the aero-structural optimization framework. Aerodynamic, aero-elastic and structural adjoint-basedoptimizations of the Airbus wing-body XRF1 configuration are presented and discussed.

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