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AIAA 2002–1483High-Fidelity Aero-Structural DesignOptimization of a SupersonicBusiness JetJoaquim R. R. A. Martins, Juan J. AlonsoStanford University, Stanford, CA 94305

James J. ReutherNASA Ames Research Center, Moffett Field, CA 95035

43rd AIAA/ASME/ASCE/AHS/ASC Structures,Structural Dynamics, and Materials Conference

April 22–25, 2002/Denver, CO

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics1801 Alexander Bell Drive, Suite 500, Reston, VA 20191–4344

AIAA 2002–1483

High-Fidelity Aero-Structural DesignOptimization of a Supersonic Business Jet

Joaquim R. R. A. Martins∗, Juan J. Alonso†

Stanford University, Stanford, CA 94305

James J. Reuther‡

NASA Ames Research Center, Moffett Field, CA 95035

This paper focuses on the demonstration of an integrated aero-structural method forthe design of aerospace vehicles. Both aerodynamics and structures are represented us-ing high-fidelity models such as the Euler equations for the aerodynamics and a detailedfinite element model for the primary structure. The aerodynamic outer mold line (OML)and a structure of fixed topology are parameterized using a large number of design vari-ables. The aero-structural sensitivities of aerodynamic and structural cost functions withrespect to both aerodynamic shape and structural variables are computed using an accu-rate and efficient coupled-adjoint procedure. K-S functions are used to reduce the numberof structural constraints in the problem. Sample optimization results of the aerodynamicshape and structure of a natural laminar flow supersonic business jet are presented to-gether with an assessment of the accuracy of the sensitivity information obtained usingthis procedure.

Introduction

A considerable amount of research has been con-ducted on Multi-Disciplinary Optimization (MDO)and its application to aircraft design. The surveypaper by Sobieski20 provides a comprehensive discus-sion of much of the work in this area. The effortsdescribed therein range from the development of tech-niques for inter-disciplinary coupling to applicationsin real-world design problems. In most cases, soundcoupling and optimization methods were shown tobe extremely important since some techniques, suchas sequential discipline optimization, were unable toconverge to the true optimum of a coupled system.Wakayama,21 for example, showed that in order to ob-tain realistic wing planform shapes with aircraft designoptimization, it is necessary to include multiple disci-plines in conjunction with a complete set of real-worldconstraints.

Aero-structural analysis has traditionally been car-ried out in a cut-and-try basis. Aircraft designers havea pre-conceived idea of the shape of an “optimal” loaddistribution and then tailor the jig shape of the struc-ture so that the deflected wing shape under a 1-g loadgives the desired distribution. While this approachis typically sufficient for traditional swept-back wingdesigns, the complexity of aero-structural interactionscan be such that, in more advanced designs where lit-

∗Doctoral Candidate, AIAA Member†Assistant Professor, AIAA Member‡Research Scientist, AIAA Associate FellowCopyright c© 2002 by the authors. Published by the American

Institute of Aeronautics and Astronautics, Inc. with permission.

tle experience has been accumulated or where multipledesign points are part of the mission, it can resultin sub-optimal designs. This is the case in the de-sign of both small and large supersonic transports,where simple beam theory models of the wing can-not be used to accurately describe the behavior of thewing structure. In some cases, these aircraft must evencruise for significant portions of their flight at differentMach numbers. In addition, a variety of studies showthat supersonic transports exhibit a range of undesir-able aeroelastic phenomena due to the low bendingand torsional stiffness that result from wings with lowthickness to chord ratio. These phenomena can only besuppressed when aero-structural interactions are takeninto account at the preliminary design stage.3

Unfortunately, the modeling of the participating dis-ciplines in most of the work that has appeared so farhas remained at a relatively low level. While use-ful at the conceptual design stage, lower-order mod-els cannot accurately represent a variety of nonlin-ear phenomena such as wave drag, which can playan important role in the search for the optimum de-sign. An exception to this low-fidelity modeling isthe recent work by Giunta6 and by Maute et al.14

where aero-structural sensitivities are calculated usinghigher-fidelity models.

The ultimate objective of our work is to developan MDO framework for high-fidelity analysis and op-timization of aircraft configurations. The frameworkis built upon prior work by the authors on aero-structural high-fidelity sensitivity analysis.11,17 Theobjective of this paper is to present the current ca-

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American Institute of Aeronautics and Astronautics Paper 2002–1483

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Spanwise coordinate, y/b

Lift

Aerodynamic optimum (elliptical distribution)

Aero−structural optimum (maximum range)

Student Version of MATLAB

Fig. 1 Elliptic vs. aero-structural optimum liftdistribution.

pability of this framework and to demonstrate it byperforming a simplified aero-structural design of a su-personic business jet configuration.

The following sections begin with the description ofthe aircraft optimization problem we propose to solve.We then introduce the adjoint method for the calcula-tion of sensitivities and derive the sensitivity equationsfor the aero-structural system. A detailed study ofthe accuracy of the aeroelastic sensitivity informationis also presented for validation purposes. Finally wepresent results of the application of our sensitivityanalysis method to two supersonic business jet designcases.

Aircraft Optimization ProblemFor maximum lift-to-drag ratio, it is a well know

result from classical inviscid aerodynamics that a wingmust exhibit an elliptic lift distribution. For aircraftdesign, however, it is usually not the lift-to-drag ratiowe want to maximize but an objective function thatreflects the overall mission of the particular aircraft.Consider, for example, the Breguet range formula

R =V

c

CL

CDln

Wi

Wf, (1)

where V is the cruise velocity and c is the specific fuelconsumption of the powerplant. CL/CD is the ratio oflift to drag, and Wi/Wf is the ratio of initial and finalcruise weights of the aircraft.

The Breguet range equation expresses a trade-off be-tween the drag and the empty weight of the aircraftand constitutes a reasonable objective function to usein aircraft design. If we were to parameterize a de-sign with both aerodynamic and structural variablesand then maximized the range for a fixed initial cruiseweight, subject to stress constraints, we would obtaina lift distribution similar to the one shown in Figure 1.

This optimum lift distribution trades off the dragpenalty associated with unloading the tip of the wing,where the loading contributes most to the maximumstress at the root of the wing structure, in order toreduce the weight. The end result is an increase in

Fig. 2 Natural laminar flow supersonic businessjet configuration.

range when compared to the elliptically loaded wingthat results from an increased weight fraction Wi/Wf .The result shown in Figure 1 illustrates the need fortaking into account the coupling of aerodynamics andstructures when performing aircraft design.

The aircraft configuration that is the focus of thiswork is that of the supersonic business jet shown inFigure 2. This configuration is being developed bythe ASSET Research Corporation and it is designedto achieve a large percentage of laminar flow on thelow-sweep wing, resulting in decreased friction drag.9

The aircraft is to fly at Mach 1.5 and have a range of5,300 nautical miles.

Detailed mission analysis for this aircraft has deter-mined that one count of drag (∆CD = 0.0001) is worth310 pounds of empty weight. This means that to opti-mize the configuration we can minimize the objectivefunction

I = αCD + βW, (2)

where CD is the drag coefficient, W is the structuralweight in pounds and α/β = 3.1× 106.

We will parameterize the design using an arbitrarynumber of shape design variables that modify theOuter Mold Line (OML) of the aircraft (xA) and struc-tural design variables that dictate the thicknesses ofthe structural elements (xS). In this work, the topol-ogy of the structure remains unchanged, i.e. the num-ber of spars and ribs and their planform-view locationis fixed. However, the depth and thickness of the struc-tural members is still allowed to change with variationsof the OML.

Among the constraints to be imposed, the most ob-vious one is that during cruise the lift must equalthe weight of the aircraft. In our optimization prob-lem we constrain the CL by periodically adjusting theangle-of-attack within the aero-structural solution andassume that the aircraft will adjust its altitude to ob-tain the desired lift.

We also must constrain the stresses so that the yieldstress of the material is not exceeded at a number

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of load conditions. There are typically thousands offinite-elements describing the structure of the aircraft,and it can become computationally very costly to treatthese constraints separately. The reason for this highcost is that although there are efficient ways of com-puting sensitivities of a few functions with respect tomany design variables, and for computing sensitivi-ties of many functions with respect to a few designvariables, there is no known efficient method for com-puting sensitivities of many functions with respect tomany design variables.

For this reason, we lump the individual elementstresses using Kreisselmeier-Steinhauser (K-S) func-tions. In the limit, all element stress constraints canbe lumped into a single K-S function, thus minimizingthe cost of a large-scale aero-structural design cycle.Suppose that we have the following constraint for eachstructural finite element,

gi = 1− σiV M

σyield≥ 0, (3)

where σiV Mis the element von Mises stress and σyield

is the yield stress of the material. The correspondingK-S function is defined as

KS (gi(x)) = −1ρ

ln

[∑

i

e−ρgi(x)

]. (4)

This function represents a lower bound envelope of allthe constraint inequalities and ρ is a positive parame-ter that expresses how close this bound is to the actualminimum of the constraints. This constraint lump-ing method is conservative and may not achieve theexact same optimum that a problem treating the con-straints separately would. However, the use of K-Sfunctions has been demonstrated and it constitutes aviable alternative, being effective in optimization prob-lems with thousands of constraints.2

Having defined our objective function, design vari-ables and constraints, we can now summarize the air-craft design optimization problem as follows:

minimize I = αCD + βW

xA, xS ∈ Rn

subject to CL = CLT

KS ≥ 0xS ≥ xSmin .

The stress constraints in the form of K-S functionsmust be enforced by the optimizer for aerodynamicloads corresponding to a number of flight conditions.Finally, a minimum gage is specified for each structuralelement thickness.

Analytic Sensitivity AnalysisWhen solving an optimization problem the goal is

typically to minimize an objective function, I, by care-fully choosing the values of a set of design variables

and satisfying the constraints of the problem. In gen-eral, the objective function depends not only on thedesign variables, but also on the physical state of theproblem that is being modeled. Thus we can write

I = I(xj , yk), (5)

where xj represents the design variables and yk is thesymbol for the state variables.

For a given vector xj , the solution of the govern-ing equations of the system yields a vector yk, thusestablishing the dependence of the state of the systemon the design variables. We will denote the governingequations as

Rk′ (xj , yk (xj)) = 0. (6)

The first instance of xj in the above equation signalsthe fact that the residual of the governing equationsmay depend explicitly on xj . In the case of a structuralsolver, for example, changing the size of an element hasa direct effect on the stiffness matrix. By solving thegoverning equations we determine the state, yk, whichdepends implicitly on the design variables through thesolution of the system.

Since the number of equations must equal the num-ber of state variables, the ranges of the indices k andk′ are the same, i.e., k, k′ = 1, . . . , nR. For a struc-tural solver, for example, nR is the number of degreesof freedom, while for a Computational Fluid Dynamics(CFD) solver, nR is the number of mesh points mul-tiplied by the number of state variables at each point(four in the two-dimensional case and five in three di-mensions.) For a coupled system, R′k represents allthe governing equations of the different disciplines, in-cluding their coupling.

R =0

x

Iy

Fig. 3 Schematic representation of the govern-ing equations (R = 0), design variables (x), statevariables (y), and objective function (I), for an ar-bitrary system.

A graphical representation of the system of govern-ing equations is shown in Figure 3, with the designvariables xj as the inputs and I as the output. Thetwo arrows leading to I illustrate the fact that theobjective function typically depends on the state vari-ables and may also be an explicit function of the designvariables.

When solving the optimization problem using agradient-based optimizer, the total variation of the ob-jective function with respect to the design variables,dIdxj

, must be calculated. As a first step towards ob-taining this total variation, we use the chain rule to

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write the total variation of I as

δI =∂I

∂xjδxj +

∂I

∂ykδyk, (7)

for k = 1, . . . , nR, j = 1, . . . , nx, where we use indicialnotation to denote the vector dot products. If we wereto use this equation directly, the vector of δyk’s wouldhave to be calculated for each δxj by solving the gov-erning equations nx times. If there are many designvariables and the solution of the governing equationsis costly (as is the case for large coupled iterative anal-yses), using equation (7) directly can be impractical.

We now observe that the variations δxj and δyk inthe total variation of the objective function (7) are notindependent of each other since the perturbed systemmust always satisfy equation (6). A relationship be-tween these two sets of variations can be obtained byrealizing that the variation of the residuals (6) mustbe zero, i.e.

δRk′ =∂Rk′

∂xjδxj +

∂Rk′

∂ykδyk = 0, (8)

for all k = 1, . . . , nR and j = 1, . . . , nx.Since this residual variation (8) is zero we can add

it to the objective function variation (7) without mod-ifying the latter, i.e.

δI =∂I

∂xjδxj +

∂I

∂ykδyk+

ψk′

(∂Rk′

∂xjδxj +

∂Rk′

∂ykδyk

), (9)

where ψk′ are arbitrary scalars we will call adjointvariables. This approach is identical to the one usedin non-linear constrained optimization, where equalityconstraints are added to the objective function, andthe arbitrary scalars are known as Lagrange multi-pliers. The problem then becomes an unconstrainedoptimization problem, which is more easily solved.

We can now group the terms in equation (9) thatcontribute to the same variation and write

δI =(

∂I

∂xj+ ψk′

∂Rk′

∂xj

)δxj+

(∂I

∂yk+ ψk′

∂Rk′

∂yk

)δyk. (10)

If we set the term multiplying δyk to zero, we are leftwith the total variation of I as a function of the designvariables and the adjoint variables, removing the de-pendence of the variation on the state variables. Sincethe adjoint variables are arbitrary, we can accomplishthis by solving the adjoint equations

∂Rk′

∂ykψk′ = − ∂I

∂yk. (11)

These equations depend only on the partial deriva-tives of both the objective function and the residualsof the governing equations with respect to the statevariables. Since these partial derivatives can be calcu-lated directly without solving the governing equations,the adjoint equations (11) only need to be solved oncefor each I and their solution is valid for all the designvariables.

When adjoint variables are found in this manner,we can use them to calculate the total sensitivity of Iwith the first term of equation (10), i.e.

dI

dxj=

∂I

∂xj+ ψk′

∂Rk′

∂xj. (12)

The cost involved in calculating sensitivities usingthe adjoint method is practically independent of thenumber of design variables. After having solved thegoverning equations, the adjoint equations are solvedonly once for each I. The terms in the adjoint equa-tions are inexpensive to calculate, and the cost ofsolving the adjoint equations is similar to that involvedin the solution of the governing equations.

The adjoint method has been widely used for sin-gle discipline sensitivity analysis and examples of itsapplication include structural sensitivity analysis1 andaerodynamic shape optimization.8,15,16

Aero-Structural Sensitivity Analysis

We will now use the equations derived in the previ-ous section to write down the adjoint sensitivity equa-tions specific to the aero-structural system. In thiscase we have coupled aerodynamic (RA) and struc-tural (RS) governing equations, and two sets of statevariables: the flow state vector, w, and the vector ofstructural displacements, u. In the following expres-sions, we no longer use index notation and we splitthe vectors of residuals, states and adjoints into twosmaller vectors corresponding to the aerodynamic andstructural systems, i.e.

Rk′ =[

RA

RS

], yk =

[wu

], ψk′ =

[ψA

ψS

]. (13)

Figure 4 shows a diagram representing the coupling inthis system.

R =0

x

I

wA R =0

uS

Fig. 4 Schematic representation of the aero-structural governing equations.

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Using this new notation, the adjoint equation (11)for an aero-structural system can be written as

[∂RA

∂w∂RA

∂u∂RS

∂w∂RS

∂u

]T [ψA

ψS

]= −

[∂I∂w∂I∂u

]. (14)

In addition to the diagonal terms of the matrix thatappear when we solve the single discipline adjointequations, we also have off-diagonal terms expressingthe sensitivity of one discipline to the state variablesof the other. The residual sensitivity matrix in thisequation is identical to that of the Global SensitivityEquations (GSE) introduced by Sobieski.19 Consider-able detail is hidden in the terms of this matrix andwe will describe each one of them for clarity.

• ∂RA/∂w: This term represents the variation ofthe residuals of the CFD equations due to changesin the the flow variables. When a flow variableat a given cell center is perturbed, the sum ofthe fluxes on that cell is altered. Only that celland its neighbors are affected. Therefore, eventhough ∂RA/∂w is a large square matrix, it isalso extremely sparse and its non-zero terms canbe easily calculated. In our solvers this matrix isnot stored explicitly.

• ∂RA/∂u: This derivative represents the effectthat the structural displacements have on theresiduals of the CFD solution through the pertur-bation of the CFD mesh. When the wing deflects,the mesh must be warped, resulting in a changein the geometry of a subset of grid cells. Eventhough the flow variables are kept constant, thechange in the geometry has an influence on thesum of the fluxes, whose variation can be easilyobtained by re-calculating the residuals for thewarped cells.

• ∂RS/∂w: The linear structural equations can bewritten as RS = Ku−f = 0, where K is the stiff-ness matrix and f is the vector of applied forces.The only term that the flow variables affect di-rectly is the applied force, and therefore this termis equal to −∂f/∂w, which can be found by exam-ining the procedure that integrates the pressuresin the CFD mesh and transfers them to the struc-tural nodes to obtain the applied forces.

• ∂RS/∂u: Since the forces do not depend directlyon the displacements and neither does K (for alinear model), this term is simply the stiffness ma-trix, K.

The right-hand side terms depend on the function ofinterest, I. In our case, we have two different functionswe are interested in: the coefficient of drag, CD, andthe K-S function. When I = CD we have,

• ∂CD/∂w: The direct sensitivity of the drag coef-ficient to the flow variables can be obtained ana-lytically by examining the numerical integrationof the surface pressures that produce CD.

• ∂CD/∂u: This term represents the change in thedrag coefficient due to the displacement of thewing while keeping the pressure distribution con-stant. The structural displacements affect thedrag directly, since they change the wing surfacegeometry over which the pressure distribution isintegrated.

When I = KS,

• ∂KS/∂w: This term is zero, since the stresses donot depend explicitly on the loads.

• ∂KS/∂u: The stresses depend directly on the dis-placements since σ = Su. This term is thereforeequal to [∂KS/∂σ] S.

Since the factorization of the full matrix in the sys-tem of equations (14) would be extremely costly, ourapproach uses an iterative solver, much like the oneused for the aero-structural solution, where the ad-joint vectors are lagged and the two different sets ofequations are solved separately. For the calculationof the adjoint vector of one discipline, we use the ad-joint vector of the other discipline from the previousiteration, i.e., we solve

[∂RA

∂w

]T

ψA = − ∂I

∂w−

[∂RS

∂w

]T

ψS , (15)

[∂RS

∂u

]T

ψS = −∂I

∂u−

[∂RA

∂u

]T

ψA, (16)

where ψA and ψS are the lagged aerodynamic andstructural adjoint vectors. The final result given bythis system, is the same as that of the original coupled-adjoint equations (14). We will call this the Lagged-Coupled Adjoint (LCA) method for computing sensi-tivities of coupled systems. Note that these equationslook like the single discipline adjoint equations forthe aerodynamic and the structural solvers, with theaddition of forcing terms in the right-hand-side thatcontain the off-diagonal terms of the residual sensitiv-ity matrix. Note also that, even for more than twodisciplines, this iterative solution procedure is nothingbut the well-known Block-Jacobi method.

As noted previously, ∂RS/∂u = K for a linearstructural solver. Since the stiffness matrix is sym-metric (KT = K) the structural equations (16) areself-adjoint. Therefore, the structural solver can beused to solve for the structural adjoint vector, ψS , byusing the pseudo-load vector given by the right-hand-side of equation (16).

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Once both adjoint vectors have converged, we cancompute the final sensitivities of the objective functionby using

dI

dx=

∂I

∂x+ ψT

A

∂RA

∂x+ ψT

S

∂RS

∂x, (17)

which is the coupled version of the total sensitivityequation (12). We will now describe the last two par-tial derivatives in the above equation.

• ∂RA/∂x: The direct effect of aerodynamic shapeperturbations on the CFD residuals is similar tothat of the displacements on the same residu-als, ∂RA/∂u, that we mentioned previously. Thestructural thicknesses of the structural finite ele-ments do not affect the CFD residuals.

• ∂RS/∂x: The design variables have a direct ef-fect on both the stiffness matrix and the load.Although this partial derivative is taken for aconstant surface pressure field, a variation in theOML will affect the translation of these pressuresto structural loads. Hence this partial derivativeis equal to ∂K/∂x · u− ∂f/∂x.

For the ∂I/∂x term we have again two different cases:I = CD and I = KS. For each of these cases:

• ∂CD/∂x: This is the change in the drag coefficientdue to wing-shape perturbations, while keepingthe pressure distribution constant. This sensitiv-ity is analogous to the partial derivative ∂CD/∂uthat we described above and can be easily calcu-lated using finite-differences. For structural vari-ables that do not affect the OML, this term iszero.

• ∂KS/∂x: This term represents the variation of thelumped stresses for fixed loads and displacements.When the OML is perturbed, the stresses in agiven element can vary under these conditions ifits shape is distorted.

As in the case of the partial derivatives in equa-tions (14), all the terms can be computed withoutincurring a large computational cost since none ofthem involve the solution of the governing equations.

In order to solve the aircraft optimization problemwe proposed earlier on, we also need sensitivities of thestructural weight with respect to the design variables.Since the aero-structural coupling does not involve theweight, these sensitivities are easily computed.

ResultsIn this section we present the results of the ap-

plication of our sensitivity calculation method to theproblem of aero-structural design of a supersonic, nat-ural laminar flow, business jet. Before describing theresults of our design experience, we present the aero-structural analysis framework and the results of a sen-sitivity validation study.

Fig. 5 Aero-structural model and solution of thesupersonic business jet configuration, showing aslice of the grid and the internal structure of thewing.

Aero-Structural Analysis

The coupled-adjoint procedure was implemented asa module that was added to the aero-structural designframework previously developed by the authors.11,17

The framework consists of an aerodynamic analysisand design module (which includes a geometry engineand a mesh perturbation algorithm), a linear finite-element structural solver, an aero-structural couplingprocedure, and various pre-processing tools that areused to setup aero-structural design problems. Themulti-disciplinary nature of this solver is illustratedin Figure 5 where we can see the aircraft geometry,the flow mesh and solution, and the primary structureinside the wing.

The aerodynamic analysis and design module,SYN107-MB,16 is a multiblock parallel flow solver forboth the Euler and the Reynolds Averaged Navier-Stokes equations that has been shown to be accurateand efficient for the computation of the flow aroundfull aircraft configurations.18 An aerodynamic adjointsolver is also included in this package in order to per-form aerodynamic shape optimization in the absenceof aero-structural interaction.

The structural analysis package is FESMEH, a finiteelement solver developed by Holden.7 The package isa linear finite-element solver that incorporates two ele-ment types and computes the structural displacementsand stresses of wing structures. Although this solveris not as general as some commercially-available pack-ages, it is still representative of the challenges involvedin using large models with tens of thousands of de-grees of freedom. High-fidelity coupling between theaerodynamic and the structural analysis programs isachieved using a linearly consistent and conservativescheme.4,17

The structural model of the wing can be seen inFigure 5 and is constructed using a wing box with sixspars evenly distributed from 15% to 80% of the chord.

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Ribs are distributed along the span at every tenth ofthe semispan. A total of 640 finite elements were usedin the construction of this model. Appropriate thick-nesses of the spar caps, shear webs, and skins werechosen according to the expected loads for this design.

Aero-Structural Sensitivity Validation

In order to gain confidence in the effectiveness of thecoupled aero-structural adjoint sensitivities for use indesign optimization, we must ensure that the valuesof the gradients are accurate. For this purpose, wehave chosen to validate the four sets of sensitivitiesdiscussed below. The exact value of these sensitivitiesis calculated using the complex-step derivative approx-imation.

The complex-step is a relatively new method thatcomputes the sensitivities dI/dxj using the formula

dI

dxj≈ Im [I (xj + ih)]

h, i =

√−1 (18)

with second order accuracy. Details of this simple yetpowerful approximation can be found in earlier workpublished by the authors.10,12,13 As in the case offinite-differences, the cost of a full gradient calcula-tion scales linearly with the total number of designvariables and therefore we do not use this method inactual design. Unlike finite-differences, however, theaccuracy of the complex-step method is extremely in-sensitive to the step size, h, making it much morerobust.

In this sensitivity study two different functions wereconsidered: the drag coefficient, CD, and the K-S func-tion of the stresses. Sensitivities of these two quantitieswith respect to both OML and structural variablesare computed. The OML design variables are shapeperturbations in the form of Hicks-Henne bump func-tions,16 which not only control the aerodynamic shapeof the aircraft configuration, but also change the inter-nal structure. For example, if a perturbation increasesthe thickness of the wing at a specific location, theheight of the spars at that point will also increase.The second set of design variables — the structuralvariables — are the thicknesses of the triangular plateelements that model the skin and spars of the wing.

The values of the aero-structural sensitivities ofdrag coefficient with respect to shape perturbationsare shown in Figure 6. The ten shape perturbationswere chosen to be Hicks-Henne bumps distributedchordwise on the upper surface of two adjacent air-foils around the quarter span. The plot shows verygood agreement between the coupled-adjoint and thecomplex-step results, the average relative error be-tween the two being 3.5%.

Figure 7 also shows the sensitivity of the drag co-efficient, this time with respect to the thicknesses offive skin groups and five spar groups distributed alongthe span. This time the agreement is even better; the

average relative error is only 1.6%.The sensitivities of the K-S function with respect

to the two sets of design variables described aboveare shown in Figures 8 and 9. The results show thatthe coupled-adjoint sensitivities are extremely accu-rate, with average relative errors of 2.9% and 1.6%.In Figure 9 we observe that the sensitivity of the K-Sfunction with respect to the first structural thickness ismuch higher than the remaining sensitivities. This isbecause that particular structural design variable cor-responds to the thickness of the top and bottom skinsof the wing bay closest to the root, where the stress isthe highest for this design case.

Note that all these sensitivities are total sensitivitiesin the sense that they account for the coupling betweenaerodynamics and structures. For example, a pertur-bation of the shape of the wing results in changes inthe surface pressures and geometry that directly affectthe drag. Those changes in surface pressure create adifferent displacement field which itself has an effecton the flow field. This coupling is fully accounted forin the coupled-adjoint method.

The cost of the sensitivity calculation using eitherthe finite-difference or complex-step methods is lin-early dependent on the number of design variables inthe problem, whereas the cost of the coupled-adjointprocedure is essentially independent of this number.In more realistic design situations where the numberof design variables chosen to parameterize the surfaceis much larger than 20, the computational efficiency ofthe adjoint sensitivity calculation would become evenmore obvious.

The cost of a finite-difference gradient evaluationfor the 20 design variables is about 15 times the costof a single aero-structural solution for computationsthat have converged 5.5 orders of magnitude. Noticethat one would expect this method to incur a compu-tational cost equivalent to 21 aero-structural solutions(the solution of the baseline configuration plus one flowsolution for each design variable perturbation.) Thecost is lower than this because the additional calcu-lations do not start from a uniform flow-field initialcondition, but from the previously converged solution.

The cost of the complex-step method is more thantwice of that of the finite-difference procedure sincethe function evaluations require complex arithmetic.We feel, however, that the complex-step calculationsare worth this cost penalty since there is no need tofind an acceptable step size a priori, as in the case ofthe finite-difference approximations.

Finally, the coupled-adjoint method requires theequivalent of 7.5 aero-structural solutions to computethe coupled gradient. As mentioned previously, thiscomputational cost is practically independent of thetotal number of design variables in the problem andwould therefore remain at a similar value, even in themore realistic case of 200 or more design variables. In

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1 2 3 4 5 6 7 8 9 10−8

−6

−4

−2

0

2

4

6

8

10x 10

−3

Shape variable, xj

d C

D /

d x A

Coupled adjoint Complex step Avg. rel. error = 3.5%


Fig. 6 Sensitivities of the drag coefficient with re-spect to shape perturbations.

11 12 13 14 15 16 17 18 19 20−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

d C

D /

d x S

Structural variable, xj



Fig. 7 Sensitivities of the drag coefficient with re-spect to structural thicknesses.

1 2 3 4 5 6 7 8 9 10−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Shape variable, xj

d C

D /

d x A



Fig. 8 Sensitivities of the K-S function with respectto shape perturbations.

11 12 13 14 15 16 17 18 19 20−10

0

10

20

30

40

50

60

70

80

d C

D /

d x S

Structural variable, xj



Fig. 9 Sensitivities of the K-S function with respectto structural thicknesses.

contrast, the finite-difference method would require acomputational effort of around 200

20 × 15 = 150 timesthe cost of an aero-structural solution to compute thesame gradient, which is impractical within industrialdesign environments, even with current state-of-the-art parallel computing systems.

Aerodynamic Design

In order to establish an upper limit for the po-tential improvements that can be realized with theuse of the aero-structural design framework, we firstcarried out an aerodynamic shape optimization thatassumes a rigid configuration. For the natural lam-inar flow supersonic business jet, the total invisciddrag at CL = 0.1 is CD = 0.0074. Of this totalinviscid drag, with an aspect ratio A = 3.0 and aspan efficiency factor of 0.95, the lift-induced dragis CDi = 0.0011. The remaining amount of drag

(∆CD = 0.0063) arises from the volume- and lift-dependent contributions of the wave drag. Using lineartheory, the lift-dependent portion of the drag turns outto be CDlift−dep

= 0.0047, and the volume-dependentpart is the remaining CDvol−dep

= 0.0016. Our aero-dynamic optimization cannot decrease the CDi sig-nificantly, since the spanload distribution is nearlyoptimal. If the volume of the wing-fuselage combina-tion remains constant, CDvol−dep

will not change. Theonly other option to reduce drag in an inviscid set-ting is to redistribute the total lift in such a way thatCDlift−dep

is minimized by arriving at a lift distribu-tion that is roughly elliptic in the streamwise direction.For these reasons, reductions in drag as small as onecount are quite significant in supersonic design. Ingeneral terms, a one count reduction corresponds toapproximately 1% of the total drag of the airplane.

In order to parameterize the shape of the aircraft, we

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SYMBOL

SOURCE Rigid Baseline

Rigid, constrained, aero only design

ALPHA 3.440

4.113

CD 0.00743

0.00700

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSNLF AERODYNAMIC DESIGN

MACH = 1.500 , CL = 0.100

MCDONNELL DOUGLAS

Solution 1 Upper-Surface Isobars

( Contours at 0.05 Cp )

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 8.6% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 24.3% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 40.8% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 58.9% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 76.5% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 91.5% Span

Fig. 10 Wing pressure distributions for rigid baseline and aerodynamic optimized cases.

have chosen sets of design variables that apply to boththe wing and the fuselage. The wing shape is modifiedby the design optimization procedure at six definingstations uniformly distributed from the side-of-bodyto the tip of the wing. The shape modifications ofthese defining stations are linearly lofted to a zerovalue at the previous and next defining stations. Oneach defining station, the twist, the leading and trail-ing edge camber distributions, and five Hicks-Hennebump functions on both the upper and lower surfacesare allowed to vary. The leading and trailing edge cam-ber modifications are not applied at the first definingstation. This yields a total of 76 design variables on thewing. Planform modifications are not allowed in thisparameterization. In order to prevent the optimizerfrom thinning the wing to an unreasonable degree, 5thickness constraints are added to each of the definingstations for a total of 30 linear constraints. The shapeof the fuselage is parameterized in such a way that itscamber is allowed to vary, but the total volume andcross-sectional shape remain constant. This is accom-plished with 9 bump functions evenly distributed inthe streamwise direction starting at the 10% fuselagestation. Fuselage nose and trailing edge camber func-tions are added to the fuselage camber distribution ina similar way to what was done with the wing sections.

The complete configuration is therefore parameter-

ized with a total of 87 design variables and 30 linearconstraints. As mentioned in an earlier section, thecost of aero-structural gradient information using ourcoupled-adjoint method is effectively independent ofthe number of design variables: in more realistic fullconfiguration test cases that we are about to tackle,500 or more design variables are often necessary todescribe the shape variations of the configuration (in-cluding nacelles, diverters, and tail surfaces.)

The objective in this optimization is to minimize thevalue of CD of the configuration at constant CL = 0.1while maintaining the thickness of the wing. All op-timization work was carried out using the non-linearconstrained optimizer NPSOL.5 Euler calculations areperformed on a 36-block multiblock mesh that is con-structed from the decomposition of a 193×33×49 C-Hmesh. During the process of the optimization, all flowevaluations are converged to 5.5 orders of magnitudeof the average density residual and the CL constraintis achieved to within 10−6.

Figure 10 shows a comparison of the pressure distri-butions of the baseline (rigid) geometry and those ofthe optimized design after 10 design iterations. Thecoefficient of drag of the complete aircraft has de-creased by 5.8% from CD0 = 0.00743 to CD = 0.0070.This decrease in the drag of the configuration is par-tially achieved by streamwise redistribution of the lift

9 of 13


at all spanwise stations. The thickness distributionis maintained due to the imposition of the thicknessconstraints. Interestingly, the angle of attack of theoptimized configuration has increased by almost 0.7◦

which is an indication that the original wing mountingangle, along the axis of the fuselage, was not opti-mal. A substantial portion of the decrease in dragis obtained from the fuselage camber modifications.In fact, from the results of two separate design cal-culations where the fuselage and wing were modifiedwhile keeping the shape of the other fixed, it is clearthat nearly one-half of the drag decrease results fromfuselage camber modifications alone. The streamwiseredistribution of lift achieved with these modificationsis quite effective in spreading the lift along the lengthof the aircraft. The final shape of the fuselage (for adifferent test case) can be seen in Figure 14.

The exact same optimization was also carried outwhile allowing the configuration to undergo aeroelas-tic deformations. The sensitivities of CD with respectto the aerodynamic design variables are now computedusing the coupled structural adjoint procedure. Noticethat while no structural design variables are present inthis test case, the depths of the ribs and spars followthe shape of the OML of the aircraft. Figure 11 showsthe resulting pressure distributions after 9 design it-erations. Notice that an almost identical reduction inCD was obtained in this case. This fact is an indirectvalidation of the coupled-adjoint sensitivity analysisprocedure since the optimization is able to recover avery similar solution to the rigid case in the presence ofaeroelastic deformations. For an aircraft that operatesat a single cruise point, the typical approach would beto consider the outcome of the first design case the 1-gshape and to construct a jig shape that would deflectto this geometry under cruise loads.

Aero-Structural Design

The initial application of our design methodology tothe aero-structural design of a supersonic business jetis simply a proof-of-concept problem meant to validatethe sensitivities obtained with our method. Currentwork is addressing the use of multiple, realistic loadconditions, proper imposition of non-linear stress con-straints, and the addition of diverters, nacelles, andempennage.

As an intermediate step before we include non-linearconstraints based on load cases that result from a num-ber of different flight conditions, we decided to restrictourselves to the cruise condition and test the sensitiv-ities of the lumped stress function, KS, by using thefollowing objective function,

I = αCD + βW + γ max(0,−KS)2. (19)

In the results to be presented, the values α = 104,β = 3.226×10−3, and γ = 103 are used. Note that thescalars that multiply the structural weight, W , and the

2 4 6 8 10 12

80

100

120

140

160

180

200

220

0 2 4 6 8 10 12

70

70.5

71

71.5

72

72.5

73

73.5

74

74.5

75

2 4 6 8 10 123000

3200

3400

3600

3800

4000

4200

4400

4600

Objective function

Weight (lbs)

Drag counts

Iteration number

Fig. 12 Iteration history for the aero-structuraloptimization.

coefficient of drag, CD, reflect the correct trade-off be-tween drag and weight that was previously mentioned,i.e. that one count of drag is worth 310 pounds ofweight. Notice also that instead of imposing the stressconstraints as separate non-linear constraints (whichthe optimizer is able to handle), we have initially optedfor adding them as a large positive penalty functionto the objective of the optimization. Due to the na-ture of the KS function, it was necessary to zero outthe contribution of this penalty when the stress con-straints were not violated. This functional dependencemakes the curvature of the penalty function discontin-uous near the fully stressed condition and can thereforeconfuse the optimizer. This would not be the case hadthe constraints been properly imposed.

For this design case we use the same aerodynamicvariables we used for the previous design cases withthe addition of 20 structural design variables. The first10 are related to the skin thicknesses of the top andbottom surfaces of the wing. Each group consists ofthe plate elements located between two adjacent ribs.The remaining structural variables are the thicknessesof the plate elements that model the spars of the wing.Again, each group is composed of the plates betweentwo adjacent ribs. All structural design variables areconstrained to exceed a pre-specified minimum gagevalue.

Notice that given the construction of the cost func-tion, I, the incentive on the part of the optimizer is indecreasing the drag and weight of the structure in ap-propriate combinations. The large penalty imposed onthe violation of the stress constraints effectively pre-vents further weight reductions by decreasing the sizeof the structural variables. However, optimum designsmay not be those with minimum drag (if the weightcan be reduced significantly at the expense of a smallincrease in CD) or minimum weight (if a slightly higherweight can allow for a lower overall CD).

Figure 12 shows the evolution of this aero-structural

10 of 13


SYMBOL

SOURCE Baseline Aero-elastic

Aero-elastic, Constrained, aero only design

ALPHA 3.330

4.095

CD 0.00741

0.00701

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSNLF AEROELASTIC DESIGN - OML ONLY

MACH = 1.500 , CL = 0.100

MCDONNELL DOUGLAS



0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 8.6% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 24.3% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 40.8% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 58.9% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 76.5% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 91.5% Span

Fig. 11 Wing pressure distributions for aero-structural baseline and optimized cases, OML design vari-ables only.

design case for successive design iterations. In the fig-ure, we can see the values of the coefficient of drag (incounts), the wing structural weight (in lbs), and thevalue of the composite cost function. Notice that at it-eration 2, the stress constraint is violated, resulting ina large increase in the penalty function and the overallobjective function. As the iterations progress, the CD

decreases from 74.1 counts to 71.7 counts, while theweight of the structure also decreases from 4511 lbs to3150 lbs. In the last iteration the drag goes up but thisis offset by a substantial decrease in the weight of thestructure in such a way that the overall objective func-tion also decreases. Notice that the pitching momentcoefficient is practically unchanged, and, therefore, notrim drag penalty needs to be taken into account.

Since the loads for this simplified case are specifiedat the cruise condition, the optimizer attempts to de-crease the structural thicknesses to the point wherethe structure is nearly fully stressed. A close look atthe maximum stress of the structure reveals that thisis the case: the element with the largest von Misesstress achieves 95% of the yield stress of the material.

Figure 13 shows the resulting pressure distributionscompared to the baseline aeroelastic analysis. Noticethat the loading has changed substantially, particu-larly in the trailing edge region of the outboard portion

of the wing. We believe this is a consequence of thefact that our structural box only extends to the 80%chord location and the loading in the trailing edge sec-tion is creating a rapid change in camber in that area.

Finally, Figure 14 shows a comparison of the geome-tries and the corresponding surface Mach number dis-tributions of the baseline aero-structural analysis andthe aero-structural optimized configuration. In thisview the fuselage camber is obvious, but the changesin the wing shape and twist, as well as the aeroelasticdeflections are difficult to notice. In fact, the tip deflec-tions for this design case are on the order of 0.5% of thetotal span. The wing has thinned down considerablyand is prevented from further decreases in thickness bythe associated increase in structural weight that wouldresult due to the stress constraints.

ConclusionsA methodology for coupled sensitivity analysis of

high-fidelity aero-structural systems was presented.The sensitivities computed by the lagged-coupled ad-joint method were compared to sensitivities given bythe complex-step derivative approximation and shownto be extremely accurate, having an average relativeerror of 2%.

In realistic aero-structural design problems with

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SYMBOL

SOURCE Aero-elastic Baseline

Aero-elastic, Aero-structural design

ALPHA 3.330

3.689

CD 0.00741

0.00685

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSNLF FULL AERO-STRUCTURAL DESIGN

MACH = 1.500 , CL = 0.100

MCDONNELL DOUGLAS



0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 8.6% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 24.3% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 40.8% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 58.9% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 76.5% Span

0.2 0.4 0.6 0.8 1.0

-0.25

-0.15

-0.05

0.05

0.15

0.25

Cp

X / C 91.5% Span

Fig. 13 Wing pressure distributions for aero-structural baseline and optimized designs, OML and struc-tural design variables with K-S penalty function.

hundreds of design variables, there is a considerable re-duction in computational cost when using the coupled-adjoint method as opposed to either finite-differencesor the complex-step. This is due to the fact that thecost associated with the adjoint method is practicallyindependent of the number of design variables.

Sensitivities computed using the presented method-ology were successfully used to optimize the design of asupersonic business jet that was parameterized with alarge number of aerodynamic and structural variables.

Future work in the development of our aero-structural design framework is expected to add furthercapability such that more realistic aircraft design prob-lems can be solved. To achieve this we plan to addmultiple flight conditions each one involving the cal-culation of a separate KS function coupled adjoint toenforce the various structural constraints. We also in-tend to add even more shape and wing planform designvariables to the design cases as well as increase the ge-ometric complexity of the configuration to include thepresence of diverters and nacelles, as well as the em-pennage.

AcknowledgmentsThe first author acknowledges the support of the

Fundacao para a Ciencia e a Tecnologia from the

Portuguese government and the Stanford UniversityCharles Lee Powell Fellowship. The second au-thor has benefited greatly from the support of theAFOSR under Grant No. AF-F49620-01-1-0291 andthe Raytheon Aircraft Preliminary Design Group. Fi-nally we would like to thank the ASSET ResearchCorporation for providing the geometry and specifica-tions for the natural laminar flow supersonic businessjet.

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