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Proceedings of KSAS’03KSAS 1st International Sessions in 2003 Fall Conference

GyeongJu Korea, November 14-15, 2003

KSAS2003

AERODYNAMIC SHAPE OPTIMIZATION :EXPLORING THE LIMITS OF DESIGN

Antony JamesonDepartment of Aeronautics and Astronautics

Stanford University [email protected]

Prepared with the assistance ofSangho Kim Sriram Shankaran Kasidit Leoviriyakit

Department of Aeronautics and Astronautics Department of Aeronautics and Astronautics Department of Aeronautics and AstronauticsStanford University Stanford University Stanford University

[email protected] [email protected] [email protected]

AbstractThis paper discusses the role that computational uid dy-

namics (CFD) plays in the design of aircraft. An overview of the design process is provided, covering some of the typical de-cisions that a design team addresses within a multi-disciplinaryenvironment. On a very regular basis trade-offs between disci-plines have to be made where a set of conicting requirementsexists. Within an aircraft development project, we focus on theaerodynamic design problem and review how this process hasbeen advanced, rst with the improving capabilities of traditionalcomputational uid dynamics analyses, and then with aerody-namic optimizations based on these increasingly accurate meth-ods.

1 BackgroundThe past 25 years have seen a revolution in the entire en-

gineering design process as computational simulation has cometo play an increasingly dominant role. Most notably, computeraided design (CAD) methods have essentially replaced the draw-ing board as the basic tool for the denition and control of theconguration. Computer visualization techniques enable the de-signer to verify that no interferences exist between different parts

in the layout.Similarly, structural analysis is now almost entirely carried

out by computational methods, typically nite element meth-ods. Commercially available software systems have been pro-gressively developed and augmented with new features, and cantreat the full range of requirements for aeronautical structures,including the analysis of stressed skin into the nonlinear range.

The concept of a numerical wind tunnel, which might even-tually allow computers “to supplant wind tunnels in the aerody-namic design and testing process”, was already a topic of discus-sion in the 1970-1980. In their celebrated paper of 1975, Chap-man, Mark and Pirtle [1] listed three main objective of computa-tional aerodynamics:

1. To provide ow simulations that are either impractical

or impossible to obtain in wind tunnels or other ground basedexperimental test facilities.2. To lower the time and cost required to obtain aerodynamic

ow simulations necessary for the design of new aerospace vehi-cles.

3. Eventually, to provide more accurate simulations of ightaerodynamics than wind tunnels can.

There have been major advances towards these goals. De-spite these, CFD is still not being exploited as effectively as one

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would like in the design process. This is partially due to the longset-up time and high costs, both human and computational, as-sociated with complex ow simulations. This paper examinesways to exploit computational simulation more effectively in theoverall design process, with the primary focus on aerodynamicdesign, while recognizing that this should be part of an integratedmulti-disciplinary process.

With the availability of high performance computing plat-forms and robust numerical methods to simulate uid ows, it ispossible to shift attention to automated design procedures whichcombine CFD with optimization techniques to determine opti-mum aerodynamic designs. The feasibility of this is by now wellestablished, [2–8] and it is actually possible to calculate optimumthree dimensional transonic wing shapes in a few hours, account-ing for viscous effects with the ow modeled by the Reynoldsaveraged Navier-Stokes (RANS) equations. By enforcing con-straints on the thickness and span-load distribution one can makesure that there is no penalty in structure weight or fuel volume.Larger scale shape changes such as planform variations can alsobe accommodated [9]. It then becomes necessary to include astructural weight model to enable a proper compromise betweenminimum drag and low structure weight to be determined.

Aerodynamic shape optimization has been successfully per-formed for a variety of complex congurations using multi-block structured meshes [10,11]. Meshes of this type can be relativelyeasily deformed to accommodate shape variations required in theredesign. However, it is both extremely time-consuming and ex-pensive in human costs to generate such meshes. Consequentlywe believe it is essential to develop shape optimization methodswhich use unstructured meshes for the ow simulation.

Typically, in gradient-based optimization techniques, a con-trol function to be optimized (the wing shape, for example) isparameterized with a set of design variables and a suitable costfunction to be minimized is dened. For aerodynamic problems,the cost function is typically lift, drag or a specied target pres-sure distribution. Then, a constraint, the governing equationscan be introduced in order to express the dependence betweenthe cost function and the control function. The sensitivity deriva-tives of the cost function with respect to the design variables arecalculated in order to get a direction of improvement. Finally, astep is taken in this direction and the procedure is repeated un-til convergence is achieved. Finding a fast and accurate way of calculating the necessary gradient information is essential to de-

veloping an effective design method since this can be the mosttime consuming portion of the design process. This is particu-larly true in problems which involve a very large number of de-sign variables as is the case in a typical three dimensional shapeoptimization.

The control theory approach [12–14] has dramatic computa-tional cost advantages over the nite-difference method of calcu-lating gradients. With this approach the necessary gradients areobtained through the solution of an adjoint system of equations

of the governing equations of interest. The adjoint method is ex-tremely efcient since the computational expense incurred in thecalculation of the complete gradient is effectively independent of the number of design variables.

In the following sections we rst examine the fundamen-tal design trade-offs between aerodynamic efciency and struc-ture weight. Then the design process itself is surveyed in Sec-tion 3. We discuss the formulation of shape optimization tech-niques based on control theory in Section 4–9. In Section 10 wepresent several case studies which highlight the potential bene-ts of aerodynamics shape optimization. Finally in Section 11we suggest some future directions.

2 Aerodynamic Design Trade-offsFocusing on the design of long range transport aircraft, a

good rst estimate of performance is provided by the Breguet

range equation:

Rn =V L D

1 E s f c

logW 0 + W f

W 0=

V L D

1 E s f c

logW 1W 2

. (1)

Here V is the speed, L/ D is the lift to drag ratio, E s f c is thespecic fuel consumption of the engines, W 0 is the landingweight(empty weight + payload+ fuel resourced), and W f is theweight of fuel burnt.

Equation (1) already displays the multi-disciplinary natureof design. A light weight structure is needed to reduce W 0. The

specic fuel consumption is mainly the province of the enginemanufacturers, and in fact the largest advances in the last 30years have been in the engine efciency. The aerodynamic de-signer should try to maximize V L/ D, but must consider the im-pact of shape modications on structure weight.

The drag coefcient can be split into an approximately xedcomponent C D0 , and the induced drag due to lift as

C D = C D0 +C 2 L

πε AR(2)

where AR is the aspect ratio, and ε is an efciency factor close tounity. C D0 includes contributions such as friction and form drag.It can be seen from this equation that L/ D is maximized by yingat a lift coefcient such that the two terms are equal, so that theinduced drag is half the total drag. Moreover, the actual drag

Dv =2 L2

περV 2b2

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due to lift varies inversely with the square of the span, b. Thusthere is a direct conict between reducing the drag by increasingthe span and reducing the structure weight by decreasing it.

It also follows from equation (1) that one should try to max-imize V L/ D. This means that the cruising speed V should be in-creased until it approaches the speed of sound C , at which pointthe formation of shock waves causes the onset of “drag-rise” .Typically the lift to drag ratio will drop from around 19 at a Machnumber = V / C in the neighborhood of 0.85, to the order of 4 atMach 1. Thus the optimum cruise speed will be in the transonicregime, when shock waves are beginning to from , but remainweak enough only to incur a small drag penalty.

The designer can reduce shock drag and delay the onset of drag-rise by increasing the sweep back of the wing or reducingits thickness. Increasing the sweepback increases the structureweight, and may incur problems with stability and control. De-creasing the thickness both reduces the fuel volume (since thewing is used as the main fuel tanks), and increases the structureweight, because for a given stress level in the skin and a givenskin thickness, the bending moment that can be supported is di-rectly proportional to the depth of the wing. In the absence of winglets, the optimum span load distribution is elliptic, givingan efciency factor ε = 1. When, however, the structure weightis taken into account, it is better to shift the load distribution in-board in order to reduce the root bending moment. It may also benecessary to limit the section lift coefcient in the outboard partof the wing, in order to delay the onset of buffet when the liftcoefcient is increased to make a turn at a high Mach number.

3 Design ProcessThe design process can generally be divided into threephases: conceptual design, preliminary design, and nal detaileddesign, as illustrated in Figure 1.

The conceptual design stage denes the mission in the lightof anticipated market requirements, and determines a generalpreliminary conguration capable of performing this mission, to-gether with rst estimates of size, weight and performance. Aconceptual design requires a staff of 15-30 people. Over a periodof 1-2 years, the initial business case is developed. The costsof this phase are the range of 6-12 million dollars, and there isminimal airline involvement

In the preliminary design stage the aerodynamic shape and

structural skeleton progress to the point where detailed perfor-mance estimates can be made and guaranteed to potential cus-tomers, who can then, in turn, formally sign binding contractsfor the purchase of a certain number of aircraft. At this stage thedevelopment costs are still fairly moderate. A staff of 100-300people is generally employed for up to 2 years, at a cost of 60-120 million dollars. Initial aerodynamic performance is exploredthrough wind tunnel tests.

In the nal design stage the structure must be dened in

complete detail, together with complete systems, including theight deck, control systems (involving major software develop-ment for y-by-wire systems), avionics, electrical and hydraulicsystems, landing gear, weapon systems for military aircraft, andcabin layout for commercial aircraft. Major costs are incurred atthis stage, during which it is also necessary to prepare a detailedmanufacturing plan, together with appropriate facilities and tool-ing. Thousands of people dene every part of the aircraft. WindTunnel validation of the nal design is carried out. Signicantdevelopment costs are incurred over a 3 year period, plus an addi-tional year of Flight Testing and Structural Qualication Testingfor Certication. Total costs are in the range of 3-12 billion dol-lars. Thus, the nal design would normally be carried out onlyif sufcient orders have been received to indicate a reasonablyhigh probability of recovering a signicant fraction of the invest-ment. For a commercial aircraft there are extensive discussionswith airlines.

ConceptualDesign

PreliminaryDesign

Final Design

Defines MissionPreliminary sizingWeight, performance

Figure 1. The Overall Design Process

In the development of commercial aircraft, aerodynamic de-sign plays a leading role during the preliminary design stage,during which the denition of the external aerodynamic shape

is typically nalized. The aerodynamic lines of the Boeing 777were frozen, for example, when initial orders were accepted be-fore the initiation of the detailed design of the structure. Figure 2illustrates the way in which the aerodynamic design process isembedded in the overall preliminary design. The starting pointis an initial CAD denition resulting from the conceptual design.The inner loop of aerodynamic analysis is contained in an outermulti-disciplinary loop, which is in turn contained in a major de-sign cycle involving wind tunnel testing. In recent Boeing prac-

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tice, three major design cycles, each requiring about 4-6 months,have been used to nalize the wing design. Improvements inCFD which would allow the elimination of a major cycle wouldsignicantly shorten the overall design process and therefore re-duce costs.

{PropulsionNoiseStabilityControlLoadsStructuresFabrication

ConceptualDesign

CADDefinition

MeshGeneration

CFDAnalysis

Visualization

PerformanceEvaluation

Multi−DisciplinaryEvaluation

Wind TunnelTesting

Central Database

DetailedFinal

Design

Release toManufacturing

ModelFabrication

O u t e r L o o p

M a j o r D e s i g n

C y c l e

I n n e r

L o o p

Figure 2. The Aerodynamic Design Process

The inner aerodynamic design loop is used to evaluate nu-merous variations in the wing denition. In each iteration it isnecessary to generate a mesh for the new conguration prior toperforming the CFD analysis. Computer graphics software isthen used to visualize the results, and the performance is evalu-ated. The rst studies may be conned to partial congurationssuch as wing-body or wing-body-nacelle combinations. At thisstage the focus is on the design of the clean wing. Key pointsof the ight envelope include the nominal cruise point, cruise athigh lift and low lift to allow for the weight variation between

the initial and nal cruise as the fuel is burned off, and a longrange cruise point at lower Mach number, where it is importantto ensure there is no signicant drag creep. Other dening pointsare the climb condition, which requires a good lift to drag ratioat low Mach number and high lift coefcient for the clean wing,and buffet conditions. The buffet requirement is typically takenas the high lift cruise point increased to a load of 1.3 g to allowfor maneuvering and gust loads.

The aerodynamic analysis interacts with the other disci-

plines in the next outer loop. These disciplines have their owninner loops, not shown in Figure 2. For an efcient design pro-cess the fully updated aero-design database must be accessibleto other disciplines without loss of information. For example,the thrust requirements for the power plant design will dependon the drag estimates for take-off, climb and cruise. In orderto meet airport noise constraints a rapid climb may be requiredwhile the thrust may also be limited. Initial estimates of the liftand moments allow preliminary sizing of the horizontal and ver-tical tail. This interacts with the design of the control system,where the use of a y-by-wire system may allow relaxed staticstability, hence tail surfaces of reduced size.

First estimates of the aerodynamic loads allow the design of an initial structural skeleton, which in turn provides a weight es-timate of the structure. One of the main trade-offs is betweenaerodynamic performance and wing structure weight. The re-quirement for fuel volume may also be an important considera-tion. Manufacturing constraints must also be considered in thenal denition of the aerodynamic shape. For example, the cur-vature in the spanwise direction should be limited. This avoidsthe need for shot peening which might otherwise be required toproduce curvature in both the spanwise and chordwise directions.

From the foregoing considerations, it is apparent that in or-der to carry out the inner loop of the aerodynamic design processthe main requirements for effective CFD software are:

1. Sufcient and known level of accuracy2. Acceptable computational and manpower costs3. Fast turn around time

4 Aerodynamic Shape OptimizationTraditionally the process of selecting design variations has

been carried out by trial and error, relying on the intuition andexperience of the designer. It is also evident that the numberof possible design variations is too large to permit their exhaus-tive evaluation, and thus it is very unlikely that a truly optimumsolution can be found without the assistance of automatic opti-mization procedures. In order to take full advantage of the pos-sibility of examining a large design space, the numerical sim-ulations need to be combined with automatic search and opti-mization procedures. This can lead to automatic design methodswhich will fully realize the potential improvements in aerody-

namic efciency.Ultimately there is a need for multi-disciplinary optimiza-

tion (MDO), but this can only be effective if it is based on suf-ciently high delity modeling of the separate disciplines. Asa step in this direction there could be signicant pay-offs fromthe application of optimization techniques within the disciplines,where the interactions with other disciplines are taken into ac-count through the introduction of constraints. For example thewing drag can be minimized at a given Mach number and lift

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coefcient with a xed planform, and constraints on minimumthickness to meet requirements for fuel volume and structureweight.

An approach which has become increasingly popular is tocarry out a search over a large number of variations via a geneticalgorithm. This may allow the discovery of (sometimes unex-pected) optimum design choices in very complex multi-objectiveproblems, but it becomes extremely expensive when each eval-uation of the cost function requires intensive computation, as isthe case in aerodynamic problems.

In order to nd optimum aerodynamic shapes with reason-able computational costs, it is useful to regard the wing as adevice which controls the ow in order to produce lift withminimum drag. As a result, one can draw on concepts whichhave been developed in the mathematical theory of control of systems governed by partial differential equations. In particu-lar, an acceptable aerodynamic design must have characteristicsthat smoothly vary with small changes in shape and ow condi-tions. Consequently, gradient-based procedures are appropriatefor aerodynamic shape optimization. Two main issues affect theefciency of gradient-based procedures; the rst is the actual cal-culation of the gradient, and the second is the construction of anefcient search procedure which utilizes the gradient.

4.1 Gradient CalculationFor the class of aerodynamic optimization problems under

consideration, the design space is essentially innitely dimen-sional. Suppose that the performance of a system design can bemeasured by a cost function I which depends on a function F ( x)that describes the shape, where under a variation of the design,δF ( x), the variation of the cost is δ I . Now suppose that δ I canbe expressed to rst order as

δ I =Z

G ( x)δF ( x)dx

where G ( x) is the gradient. Then by setting

δF ( x) = − λG ( x)

one obtains an improvement

δ I = − λZ

G 2( x)dx

unless G ( x) = 0. Thus thevanishing of the gradient is a necessarycondition for a local minimum.

Computing the gradient of a cost function for a complex sys-tem can be a numerically intensive task, especially if the number

of design parameters is large and the cost function is an expen-sive evaluation. The simplest approach to optimization is to de-ne the geometry through a set of design parameters, which may,for example, be the weights α i applied to a set of shape functionsB i( x) so that the shape is represented as

F ( x) = ∑α iB i( x).

Then a cost function I is selected which might be the drag co-efcient or the lift to drag ratio; I is regarded as a function of the parameters α i. The sensitivities ∂ I

∂α imay now be estimated

by making a small variation δα i in each design parameter in turnand recalculating the ow to obtain the change in I . Then

∂ I ∂α i

≈I (α i + δα i) − I (α i)

δα i.

The main disadvantage of this nite-difference approach isthat the number of ow calculations needed to estimate the gra-dient is proportional to the number of design variables [15].Similarly, if one resorts to direct code differentiation (ADI-FOR [16, 17]), or complex-variable perturbations [18], the costof determining the gradient is also directly proportional to thenumber of variables used to dene the design.

A more cost effective technique is to compute the gradientthrough the solution of an adjoint problem, such as that devel-oped in references [3, 19, 20]. The essential idea may be sum-marized as follows. For ow about an arbitrary body, the aero-dynamic properties that dene the cost function are functions of the oweld variables (w) and the physical shape of the body,which may be represented by the function F . Then

I = I (w, F )

and a change in F results in a change of the cost function

δ I =∂ I T

∂wδw +

∂ I T

∂F δF .

Using a technique drawn from control theory, the governingequations of the oweld are introduced as a constraint in sucha way that the nal expression for the gradient does not requirereevaluation of the oweld. In order to achieve this, δw must beeliminated from the above equation. Suppose that the governingequation R, which expresses the dependence of w and F withinthe oweld domain D , can be written as

R(w, F ) = 0. (3)

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Then δw is determined from the equation

δ R =∂ R∂w

δw +∂ R∂F

δF = 0.

Next, introducing a Lagrange multiplier ψ , we have

δ I =∂ I T

∂wδw +

∂ I T

∂F δF − ψ T ∂ R

∂wδw +

∂ R∂F

δF . (4)

With some rearrangement

δ I =∂ I T

∂w− ψ T ∂ R

∂wδw +

∂ I T

∂F − ψ T ∂ R

∂F δF .

Choosing ψ to satisfy the adjoint equation

∂ R∂w

T

ψ =∂ I T

∂w(5)

the term multiplying δw can be eliminated in the variation of thecost function, and we nd that

δ I = G δF ,

where

G =∂ I T

∂F − ψ T ∂ R

∂F .

The advantage is that the variation in cost function is independentof δw, with the result that the gradient of I with respect to anynumber of design variables can be determined without the needfor additional ow-eld evaluations.

In the case that (3) is a partial differential equation, the adjoint equation (5) is also a partial differential equation and ap-propriate boundary conditions must be determined. It turns out

that the appropriate boundary conditions depend on the choiceof the cost function, and may easily be derived for cost functionsthat involve surface-pressure integrations. Cost functions involv-ing eld integrals lead to the appearance of a source term in theadjoint equation.

The cost of solving the adjoint equation is comparable to thatof solving the ow equation. Hence, the cost of obtaining thegradient is comparable to the cost of two function evaluations,regardless of the dimension of the design space.

5 Design using the Euler EquationsThe application of control theory to aerodynamic design

problems is illustrated in this section for the case of three-dimensional wing design using the compressible Euler equationsas the mathematical model. The extension of the method to treatthe Navier-Stokes equations is presented in references [4,8, 21].It proves convenient to denote the Cartesian coordinates and ve-locity components by x1 , x2 , x3 and u1 , u2 , u3 , and to use theconvention that summation over i = 1 to 3 is implied by a re-peated index i. Then, the three-dimensional Euler equations maybe written as

∂w∂t

+∂ f i∂ xi

= 0 in D, (6)

where

w =

ρρu1ρu2ρu3ρ E

, f i =

ρuiρuiu1 + pδi1ρuiu2 + pδi2ρuiu3 + pδi3

ρui H

(7)

and δi j is the Kronecker delta function. Also,

p = ( γ − 1) ρ E −12

(uiui) , (8)

and

ρ H = ρ E + p (9)

where γ is the ratio of the specic heats.In order to simplify the derivation of the adjoint equations,

we map the solution to a xed computational domain with coor-dinates ξ1, ξ2 , ξ3 where

K i j =∂ xi

∂ξ j, J = det (K ) , K − 1

i j =∂ξ i

∂ x j,

and

S = JK − 1.

The elements of S are the cofactors of K , and in a nite volumediscretization they are just the face areas of the computationalcells projected in the x1 , x2 , and x3 directions. Using the permu-tation tensor εi jk we can express the elements of S as

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S i j =12

ε j pqεirs∂ x p

∂ξr

∂ xq

∂ξs. (10)

Then

∂∂ξ i

S i j =12

ε j pqεirs∂2 x p

∂ξr ∂ξ i

∂ xq

∂ξs+

∂ x p

∂ξr

∂2 xq

∂ξs∂ξ i

= 0. (11)

Also in the subsequent analysis of the effect of a shape vari-ation it is useful to note that

S 1 j = ε j pq∂ x p

∂ξ2

∂ xq

∂ξ3,


∂ξ3

∂ xq

∂ξ1,


∂ξ1

∂ xq

∂ξ2. (12)

Now, multiplying equation(6) by J and applying the chainrule,

J ∂w∂t

+ R(w) = 0 (13)

where

R(w) = S i j∂ f j∂ξi

=∂

∂ξi(S i j f j) , (14)

using (11). We can write the transformed uxes in terms of thescaled contravariant velocity components

U i = S i j u j

as

F i = S i j f j =ρU iρU iu1 + S i1 p

ρU iu2 + S i2 pρU iu3 + S i3 p

ρU i H

.

For convenience, the coordinates ξi describing the xedcomputational domain are chosen so that each boundary con-forms to a constant value of one of these coordinates. Variations

in the shape then result in corresponding variations in the map-ping derivatives dened by K i j . Suppose that the performance ismeasured by a cost function

I =

Z

B M (w, S ) dB ξ +

Z

D P (w, S ) dD ξ ,

containing both boundary and eld contributions where dB ξ anddD ξ are the surface and volume elements in the computationaldomain. In general, M and P will depend on both the ow vari-ables w and the metrics S dening the computational space. Thedesign problem is now treated as a control problem where theboundary shape represents the control function, which is cho-sen to minimize I subject to the constraints dened by the owequations (13). A shape change produces a variation in the owsolution δw and the metrics δS which in turn produce a variationin the cost function

δ I =Z

B δM (w, S ) dB ξ +

Z

D δP (w, S ) dD ξ . (15)

This can be split as

δ I = δ I I + δ I II , (16)

with

δM = [M w] I δw + δM II ,δP = [P w] I δw + δP II , (17)

where we continue to use the subscripts I and II to distinguishbetween the contributions associated with the variation of theow solution δw and those associated with the metric variationsδS . Thus [M w] I and [P w] I represent ∂M

∂w and ∂P ∂w with the met-

rics xed, while δM II and δP II represent the contribution of themetric variations δS to δM and δP .

In the steady state, the constraint equation (13) species thevariation of the state vector δw by

δ R = ∂∂ξ iδF i = 0. (18)

Here also, δ R and δF i can be split into contributions associatedwith δw and δS using the notation

δ R = δ R I + δ R II

δF i = [F iw] I δw + δF i II . (19)

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where

[F iw] I = S i j∂ f i∂w

.

Multiplying by a co-state vector ψ , which will play an anal-ogous role to the Lagrange multiplier introduced in equation (4),and integrating over the domain produces

Z

D ψ T ∂

∂ξ iδF id D ξ = 0. (20)

Assuming that ψ is differentiable, the terms with subscript I maybe integrated by parts to give

Z

B n

iψ T δF

i I d

B ξ−

Z

D

∂ψ T

∂ξ iδF

i I d

D ξ+

Z

D ψ T δ R

II d

D ξ= 0. (21)

This equation results directly from taking the variation of theweak form of the ow equations, where ψ is taken to be an arbi-trary differentiable test function. Since the left hand expressionequals zero, it may be subtracted from the variation in the costfunction (15) to give

δ I = δ I II −Z

D ψ T δ R II d D ξ +

Z

B δM I − niψ T δF i I d B ξ

+Z

D δP I +

∂ψ T

∂ξiδF i I d D ξ . (22)

Now, since ψ is an arbitrary differentiable function, it may bechosen in such a way that δ I no longer depends explicitly onthe variation of the state vector δw. The gradient of the costfunction can then be evaluated directly from the metric variationswithout having to recompute the variation δw resulting from theperturbation of each design variable.

Comparing equations (17) and (19), the variation δw may beeliminated from (22) by equating all eld terms with subscript“ I ” to produce a differential adjoint system governing ψ

∂ψ T

∂ξi [F iw] I + [P w] I = 0 in D . (23)

Taking the transpose of equation (23), in the case that there is noeld integral in the cost function, the inviscid adjoint equationmay be written as

C T i

∂ψ ∂ξi

= 0 in D , (24)

where the inviscid Jacobian matrices in the transformed space aregiven by

C i = S i j∂ f j∂w

.

The corresponding adjoint boundary condition is produced byequating the subscript “ I ” boundary terms in equation (22) toproduce

niψ T [F iw] I = [M w] I on B . (25)

The remaining terms from equation (22) then yield a simpliedexpression for the variation of the cost function which denes thegradient

δ I = δ I II −

Z

D ψ T δ R II d D ξ , (26)

which consists purely of the terms containing variations in themetrics, with the ow solution xed. Hence an explicit formulafor the gradient can be derived once the relationship betweenmesh perturbations and shape variations is dened.

The boundary conditions satised by the ow equations re-strict the form of the left hand side of the adjoint boundary con-dition (25). Consequently, the boundary contribution to the costfunction M cannot be specied arbitrarily. Instead, it must bechosen from the class of functions which allow cancellation of all terms containing δw in the boundary integral of equation (22).

On the other hand, there is no such restriction on the specicationof the eld contribution to the cost function P , since these termsmay always be absorbed into the adjoint eld equation (23) assource terms.

For simplicity, it will be assumed that the portion of theboundary that undergoes shape modications is restricted to thecoordinate surface ξ2 = 0. Then equations (22) and (25) may besimplied by incorporating the conditions

n1 = n3 = 0, n2 = 1, d B ξ = d ξ1d ξ3 ,

so that only the variation δF 2 needs to be considered at the wall

boundary. The condition that there is no ow through the wallboundary at ξ2 = 0 is equivalent to

U 2 = 0,

so that

δU 2 = 0

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when the boundary shape is modied. Consequently the varia-tion of the inviscid ux at the boundary reduces to

δF 2 = δ p

0S 21

S 22

S 23

0

+ p

0δS 21

δS 22

δS 23

0

. (27)

Since δF 2 depends only on the pressure, it is now clear that theperformance measure on the boundary M (w, S ) may only be afunction of the pressure and metric terms. Otherwise, completecancellation of the terms containing δw in the boundary integral

would be impossible. One may, for example, include arbitrarymeasures of the forces and moments in the cost function, sincethese are functions of the surface pressure.

In order to design a shape which will lead to a desired pres-sure distribution, a natural choice is to set

I =12

Z

B ( p − pd )2 dS

where pd is the desired surface pressure, and the integral is eval-uated over the actual surface area. In the computational domainthis is transformed to

I =12

Z Z

B w( p − pd )2 |S 2 | d ξ1d ξ3 ,

where the quantity

|S 2 | = S 2 jS 2 j

denotes the face area corresponding to a unit element of face areain the computational domain. Now, to cancel the dependenceof the boundary integral on δ p, the adjoint boundary conditionreduces to

ψ jn j = p − pd (28)

where n j are the components of the surface normal

n j =S 2 j

|S 2 |.

This amounts to a transpiration boundary condition on the co-state variables corresponding to the momentum components.Note that it imposes no restriction on the tangential componentof ψ at the boundary.

We nd nally that

δ I = −Z

D

∂ψ T

∂ξiδS i j f jd D

−Z Z

BW (δS 21 ψ 2 + δS 22 ψ 3 + δS 23 ψ 4) pd ξ1d ξ3 . (29)

Here the expression for the cost variation depends on the meshvariations throughout the domain which appear in the eld inte-gral. However, the true gradient for a shape variation should notdepend on the way in which the mesh is deformed, but only onthe true ow solution. In the next section we show how the eld

integral can be eliminated to produce a reduced gradient formulawhich depends only on the boundary movement.

6 The Reduced Gradient FormulationConsider the case of a mesh variation with a xed boundary.

Then,

δ I = 0

but there is a variation in the transformed ux,

δF i = C iδw + δS i j f j.

Here the true solution is unchanged. Thus, the variation δw isdue to the mesh movement δ x at each mesh point. Therefore

δw = w · δ x =∂w∂ x j

δ x j (= δw )

and since

∂∂ξi

δF i = 0,

it follows that

∂∂ξ i

(δS i j f j) = −∂

∂ξi(C iδw ) . (30)

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It is veried in the following paragraph that this relation holds inthe general case with boundary movement. Now

Z

D ψ T δ Rd D =

Z

D ψ T ∂

∂ξiC i (δw − δw ) d D

=Z

B ψ T C i (δw − δw ) d B

−Z

D

∂ψ T

∂ξiC i (δw − δw ) d D . (31)

Here on the wall boundary

C 2δw = δF 2 − δS 2 j f j. (32)

Thus, by choosing ψ to satisfy the adjoint equation (24) and theadjoint boundary condition (25), we reduce the cost variation toa boundary integral which depends only on the surface displace-ment:

δ I =Z

B W

ψ T (δS 2 j f j + C 2δw ) d ξ1d ξ3

−Z Z

BW (δS 21 ψ 2 + δS 22 ψ 3 + δS 23 ψ 4) pd ξ1d ξ3 . (33)

For completeness the general derivation of equation (30) ispresented here. Using the formula (10), and the property (11)

∂∂ξi

(δS i j f j)

=12

∂∂ξi ε j pqεirs

∂δ x p

∂ξr

∂ xq

∂ξs +∂ x p

∂ξr

∂δ xq

∂ξs f j

=12

ε j pqεirs∂δ x p

∂ξr

∂ xq

∂ξs+

∂ x p

∂ξr

∂δ xq

∂ξs

∂ f j∂ξi

=12

ε j pqεirs∂

∂ξr δ x p

∂ xq

∂ξs

∂ f j∂ξi

+12

ε j pqεirs∂

∂ξsδ xq

∂ x p

∂ξr

∂ f j∂ξ i

=∂

∂ξr δ x pε pq j εrsi

∂ xq

∂ξs

∂ f j∂ξi

. (34)

Now express δ x p in terms of a shift in the original computational

coordinates

δ x p =∂ x p

∂ξk δξk .

Then we obtain

∂∂ξ i

(δS i j f j) =∂

∂ξr ε pq j εrsi

∂ x p

∂ξk

∂ xq

∂ξs

∂ f j∂ξ i

δξk . (35)

The term in ∂∂ξ1

is

ε123 ε pq j∂ x p

∂ξk

∂ xq

∂ξ2

∂ f j∂ξ3

−∂ xq

∂ξ3

∂ f j∂ξ2

δξk .

Here the term multiplying δξ1 is

ε jpq∂ x p

∂ξ1

∂ xq

∂ξ2

∂ f j∂ξ3

−∂ x p

∂ξ1

∂ xq

∂ξ3

∂ f j∂ξ2

.

According to the formulas(12) this may be recognized as

S 2 j∂ f 1∂ξ2

+ S 3 j∂ f 1∂ξ3

or, using the quasi-linear form(14) of the equation for steadyow, as

− S 1 j∂ f 1∂ξ1

.

The terms multiplying δξ2 and δξ3 are

ε j pq∂ x p

∂ξ2

∂ xq

∂ξ2

∂ f j∂ξ3

−∂ x p

∂ξ2

∂ xq

∂ξ3

∂ f j∂ξ2

= − S 1 j∂ f 1∂ξ2

and

ε jp q∂ x p

∂ξ3

∂ xq

∂ξ2

∂ f j∂ξ3

−∂ x p

∂ξ3

∂ xq

∂ξ3

∂ f j∂ξ2

= − S 1 j∂ f 1∂ξ3

.

Thus the term in ∂∂ξ1

is reduced to

−∂

∂ξ1S 1 j

∂ f 1∂ξk

δξk .

Finally, with similar reductions of the terms in ∂∂ξ2

and ∂∂ξ3

, weobtain

∂∂ξi

(δS i j f j) = −∂

∂ξ iS i j

∂ f j∂ξk

δξk = −∂

∂ξi(C iδw )

as was to be proved.In order to validate the concept, the new gradient equations

have been tested for various aerodynamic shape optimization

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problems and the accuracy of the gradients using the reduced for-mula are assessed by comparing with nite-difference gradients,complex-step gradients, and gradients calculated by the previousadjoint method which includes a volume integral [22].

The reduced gradient is crucial for unstructured meshes. If the gradient depends on the form of the mesh modication, thenthe eld integral in the gradient calculation has to be recomputedfor mesh modications corresponding to each design variable.This would be prohibitively expensive if the geometry is treatedas a free surface dened by the mesh points. Consequently inorder to reduce the computational cost with this approach, thenumber of design variables would have to be reduced by param-eterizing the geometry. However, this reduced set of design vari-ables could not recover all possible shape variations.

7 The Viscous Adjoint Equations

The derivation of the viscous adjoint equations is presentedin detail in [21,23]. Here we summarize the main results, underthe assumption that the viscosity and heat conduction coefcients µ and k are essentially independent of the ow, and that their vari-ations may be neglected. This simplication has been success-fully used for may aerodynamic problems of interest. However,if the ow variations could result in signicant changes in the tur-bulent viscosity, it may be necessary to account for its variationin the calculation.

7.1 Transformation to Primitive VariablesThe formulation of the viscous adjoint terms can be simpli-

ed by transforming to the primitive variables

wT = ( ρ, u1 , u2 , u3 , p),

because the viscous stresses depend on the velocity derivatives∂ui∂ x j

, while the heat ux can be expressed as

κ ∂

∂ xi

pρ

.

where κ = k R = γ µ

Pr (γ − 1) . The relationship between the conserva-tive and primitive variations is dened by the expressions

δw = M δw, δw = M − 1δw

which make use of the transformation matrices M = ∂w∂w and

M − 1 = ∂w∂w . These matrices are provided in transposed form for

future convenience

M T =

1 u1 u2 u3uiui

20 ρ 0 0 ρu10 0 ρ 0 ρu20 0 0 ρ ρu3

0 0 0 0 1γ − 1

M − 1T =

1 − u1ρ − u2

ρ − u3ρ

(γ − 1)uiui2

0 1ρ 0 0 − (γ − 1)u1

0 0 1ρ 0 − (γ − 1)u2

0 0 0 1ρ − (γ − 1)u3

0 0 0 0 γ − 1

.

The conservative and primitive adjoint operators L and ˜ L corre-sponding to the variations δw and δw are then related by

Z

D δwT Lψ d D ξ =

Z

D δwT ˜ Lψ d D ξ ,

with

˜ L = M T L,

so that after determining the primitive adjoint operator, the con-servative operator may be obtained by the transformation L = M − 1T ˜ L. Since the continuity equation contains no viscous terms,it makes no contribution to the viscous adjoint system. There-fore, the derivation proceeds by rst examining the adjoint oper-ators arising from the momentum equations and then the energyequation.

7.2 The Viscous Adjoint Field OperatorIn order to make use of the summation convention, it is con-

venient to set ψ j+ 1 = φ j for j = 1, 2, 3 and ψ 5 = θ. Collectingtogether the contributions from the momentum and energy equa-tions, the viscous adjoint operator in primitive variables can benally expressed as

( ˜ Lψ )1 = − pρ2

∂∂ξl

S l jκ ∂θ∂ x j

( ˜ Lψ )i+ 1 = ∂∂ξl

S l j µ ∂φi∂ x j

+ ∂φ j∂ xi

+ λδ i j∂φk ∂ xk

+ ∂∂ξl

S l j µ ui∂θ∂ x j

+ u j∂θ∂ xi

λδ i j uk ∂θ∂ xk

− σ i j S l j ∂θ∂ξl

for i = 1, 2, 3

( ˜ Lψ )5 = 1ρ

∂∂ξl

S l jκ ∂θ∂ x j

.

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The conservative viscous adjoint operator may now be obtainedby the transformation

L = M − 1T ˜ L.

7.3 Boundary Conditions for Force OptimizationDening the components of the total surface stress as

τk = n j δk j p + σk j

and the physical surface element

dS = |S 2 | d B ξ ,

the force in a direction with cosines qi has the form

C q =Z

B qiτidS .

Integrating the eld terms by parts, cancellation with the owvariation terms mandates the adjoint boundary condition

φk = qk .

7.4 Boundary Conditions for Inverse Design

In the case of high Reynolds number, the boundary conditionfor inverse design is well approximated by the equations

φk = nk ( p − pd ) , (36)

which should be compared with the single scalar equation de-rived for the inviscid boundary condition (28).

The inviscid boundary condition (28) is satised by equation(36), but this represents an over-specication of the boundarycondition since only the single condition (28) needs be specied,corresponding to the slip boundary condition for the inviscid owequations.

7.5 Boundary Conditions Arising from the EnergyEquation

The form of the boundary terms arising from the energyequation depends on the choice of temperature boundary con-dition at the wall. A natural solution is to set

θ = 0

in the constant temperature case or

∂θ∂n

= 0

in the adiabatic case.

8 Planform DesignThe shape changes in wing section needed to improve the

transonic wing design are quite small. However, in order to ob-tain a true optimum design larger scale changes such as changesin the wing planform (sweepback, span, chord, and taper) shouldbe considered. Because these directly affect the structure weight,a meaningful result can only be obtained by considering a costfunction that takes account of both the aerodynamic characteris-tics and the weight.

8.1 Cost Function for Planform DesignIn order to design a high performance transonic wing, which

will lead to a desired pressure distribution, and to still maintain arealistic shape, the natural choice is to set

I = α 1C D + α 212

Z

B ( p − pd )2dS + α 3C W (37)

with

C W =W wing

q∞S re f (38)

whereC D = drag coefcient,C W = dimensionless wing structural weight, p = current surface pressure, pd = desired pressure,q∞ = dynamic pressure,S re f = reference area,W wing = wing structure weight, andα 1 , α 2 , α 3 = weighting parameter for drag,

inverse design, and structuralweight respectively.

The constant α 2 is introduced to provide the designer some con-trol over the pressure distribution.

A practical way to estimate W wing is to use the so-calledStatistical Group Weights Method, which applies statisticalequations based on sophisticated regression analysis. For acargo/transport wing weight, one can use [24]

W weight = 0.0051 (W dg N z)0.557 S 0.649w A0.5

(t / c)− 0.4root (1 + λ)0.1cos (Λ)− 1.0S 0.1

csw (39)

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unless g = 0, the necessary condition for a minimum.Note that g is a function of y, y , y ,

g = g( y, y , y )

In the well known case of the Brachistrone problem, for example,which calls for the determination of the path of quickest descentbetween two laterally separated points when a particle falls undergravity,

F ( y, y ) = 1 + y 2

y

and

g = − 1 + y2

+ 2 yy2 y(1 + y 2) 3/ 2

It can be seen that each step

yn+ 1 = yn − λngn

reduces the smoothness of y by two classes. Thus the computedtrajectory becomes less and less smooth, leading to instability.

In order to prevent this we can introduce a weighted Sobolevinner product [25]

u, v =Z

(uv + εu v )dx

where ε is a parameter that controls the weight of the derivatives.We now dene a gradient g such that

δ I = g, δ y

Then we have

δ I =Z

(gδ y+ εg δ y )dx

=Z

(g −∂∂ x

ε∂g∂ x

)δ ydx

= ( g, δ y)

where

g −∂∂ x

ε∂g∂ x

= g

and g = 0 at the end points. Thus g can be obtained from g by asmoothing equation. Now the step

yn+ 1 = yn − λngn

gives an improvement

δ I = − λn gn , gn

but yn+ 1 has the same smoothness as yn , resulting in a stableprocess.

9.2 Sobolev Gradient for Shape OptimizationIn applying control theory to aerodynamic shape optimiza-

tion, the use of a Sobolev gradient is equally important for thepreservation of the smoothness class of the redesigned surface.Accordingly, using the weighted Sobolev inner product denedabove, we dene a modied gradient G such that

δ I = < G , δF > .

In the one dimensional case G is obtained by solving the smooth-ing equation

G −∂

∂ξ1ε

∂∂ξ1

G = G . (41)

In the multi-dimensional case the smoothing is applied in productform. Finally we set

δF = − λG (42)

with the result that

δ I = − λ < G , G > < 0,

unless G = 0, and correspondingly G = 0.When second-order central differencing is applied to (41),

the equation at a given node, i, can be expressed as

G i − ε G i+ 1 − 2G i + G i− 1 = G i, 1 ≤ i ≤ n,

where G i and G i are the point gradients at node i before and afterthe smoothing respectively, and n is the number of design vari-ables equal to the number of mesh points in this case. Then,

G = AG ,

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where A is the n × n tri-diagonal matrix such that

A− 1

=

1 + 2ε − ε 0 . 0ε . .

0 . . .. . . − ε0 ε 1 + 2ε

.

Using the steepest descent method in each design iteration, astep, δF , is taken such that

δF = − λ AG . (43)

As can be seen from the form of this expression, implicit smooth-ing may be regarded as a preconditioner which allows the use of

much larger steps for the search procedure and leads to a largereduction in the number of design iterations needed for conver-gence.

9.3 Outline of the Design ProcedureThe design procedure can nally be summarized as follows:

1. Solve the ow equations for ρ, u1, u2 , u3 , p.2. Solve the adjoint equations for ψ subject to appropriate

boundary conditions.3. Evaluate G and calculate the corresponding Sobolev gradi-

ent G .4. Project G into an allowable subspace that satises any geo-

metric constraints.5. Update the shape based on the direction of steepest descent.6. Return to 1 until convergence is reached.

Practical implementation of the design method relies heavilyupon fast and accurate solvers for both the state (w) and co-state(ψ ) systems. The result obtained in Section 10 have been ob-tained using well-validated software for the solution of the Eulerand Navier-Stokes equations developed over the course of manyyears [26–28]. For inverse design the lift is xed by the targetpressure. In drag minimization it is also appropriate to x thelift coefcient, because the induced drag is a major fraction of the total drag, and this could be reduced simply by reducing the

lift. Therefore the angle of attack is adjusted during each owsolution to force a specied lift coefcient to be attained, andthe inuence of variations of the angle of attack is included inthe calculation of the gradient. The vortex drag also depends onthe span loading, which may be constrained by other consider-ations such as structural loading or buffet onset. Consequently,the option is provided to force the span loading by adjusting thetwist distribution as well as the angle of attack during the owsolution.

Sobolev Gradient

Gradient Calculation

Flow Solution

Adjoint Solution

Shape & Grid

Repeat the Design Cycleuntil Convergence

Modification

Figure 6. Design cycle

9.4 Computational Costs

In order to address the issue of the search costs, Jameson andVassberg investigated a variety of techniques in Reference [29]using a trajectory optimization problem (the brachistochrone) asa representative model. The study veried that the search cost(i.e., number of steps) of a simple steepest descent method ap-plied to this problem scales as N 2 , where N is the number of design variables, while the cost of quasi-Newton methods scaledlinearly with N as expected. On the other hand, with an appro-

priate amount of smoothing, the smoothed descent method con-verged in a xed number of steps, independent of N . Consideringthat the evaluation of the gradient by a nite difference methodrequires N + 1 ow calculations, while the cost of its evaluationby the adjoint method is roughly that of two ow calculations,one arrives at the estimates of total computational cost given inTables 1-2.

Table 1. Cost of Search Algorithm.

Steepest Descent O ( N 2) steps

Quasi-Newton O ( N ) steps

Smoothed Gradient O (K ) steps

(Note: K is independent of N )

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Table 2. Total Computational Cost of Design.

Finite Difference Gradients

+ Steepest Descent O ( N 3)

Finite Difference Gradients

+ Quasi-Newton Search O ( N 2)

Adjoint Gradients

+ Quasi-Newton Search O ( N )

Adjoint Gradients

+ Smoothed Gradient Search O (K )

(Note: K is independent of N )

10 Case StudiesSeveral design efforts which have utilized these methods in-clude: Raytheon’s and Gulfstream business jets, NASA’s High-Speed Civil Transport, regional jet designs, as well as severalBoeing projects such as the MDXX and the Blended-Wing-Body [30, 31]. Some representative examples of design calcu-lations are presented in this section to illustrate the present capa-bility.

10.1 Viscous Transonic Redesign of the Boeing 747wing

Over the last decade the adjoint method has been success-

fully used to rene a variety of designs for ight at both transonicand supersonic cruising speeds. In the case of transonic ight, itis often possible to produce a shock free ow which eliminatesthe shock drag by making very small changes, typically no largerthan the boundary layer displacement thickness. Consequentlyviscous effects need to be considered in order to realize the fullbenets of the optimization.

Here the optimization of the wing of the Boeing 747 is pre-sented to illustrate the kind of benets that can be obtained. Inthese calculations the ow was modeled by the Reynolds Av-eraged Navier-Stokes equations. A Baldwin-Lomax turbulencemodel was considered sufcient, since the optimization is for thecruise condition with attached ow. The computational mesh is

shown in Figure 7.The calculations were performed to minimize the drag co-

efcient at a xed lift coefcient, subject to the additional con-straints that the span loading should not be altered and the thick-ness should not be reduced. It might be possible to reduce theinduced drag by modifying the span loading to an elliptic dis-tribution, but this would increase the root bending moment, andconsequently require an increase in the skin thickness and struc-ture weight. A reduction in wing thickness would not only re-

B747 WING-BODY

GRID 256 X 64 X 48

K = 1

Figure 7. Computational Grid of the B747 Wing Fuselage

duce the fuel volume, but it would also require an increase inskin thickness to support the bending moment. Thus these con-straints assure that there will be no penalty in either structureweight or fuel volume.

Figure 8 displays the result of an optimization at a Machnumber of 0.86, which is roughly the maximum cruising Machnumber attainable by the existing design before the onset of sig-nicant drag rise. The lift coefcient of 0.42 is the contributionof the exposed wing. Allowing for the fuselage, the total lift co-efcient is about 0.47. It can be seen that the redesigned wingis essentially shock free, and the drag coefcient is reduced from0.01269 (127 counts) to 0.01136 (114 counts). The total drag co-efcient of the aircraft at this lift coefcient is around 270 counts,so this would represent a drag reduction of the order of 5 percent.

Figure 9 displays the result of an optimization at Mach 0.90.It might be expected that the Boeing 747 wing could be modiedto allow an increase in the cruising Mach number because it has ahigher sweep-back than later designs, and a rather thin wing sec-

tion with a thickness to chord ratio of 8 percent. In this case theshock waves are not eliminated, but their strength is signicantlyweakened, while the drag coefcient is reduced from 0.01819(182 counts) to 0.01293 (129 counts). Thus the redesigned winghas essentially the same drag at Mach 0.9 as the original wing atMach 0.86. Figures 10 and 11 verify that the span loading andthickness were not changed by the redesign, while Figures 12and 13 indicate the required section changes at 42 percent and 68percent span stations.

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10.2 Viscous Planform Redesign of the Boeing 747wing

We present results to show that the optimization can success-fully trade planform parameters. The case chosen is the Boeing

747 wing fuselage combination at Mach 0.90 and a lift coefcientC L = 0.42. We allowed section changes together with variationsof sweepback, span, root chord, mid-span chord, and tip chord.Figure 14 shows a baseline calculation with the planform xed.Here the drag was reduced from 181.9 counts to 127.9 counts(29.7% reduction) in 50 design iterations with relatively smallchanges in the section shape.

Figure 15 shows the effect of allowing changes in sweep-back, span, root chord, mid-span chord, and tip chord. The pa-rameter α 3 was chosen according to formula (40) such that thecost function corresponds to maximizing the range of the aircraft.In 50 design iterations the drag was reduced from 181.9 counts to

124.9 counts (31.3% reduction), while the dimensionless struc-ture weight was slightly increased from 0.02956 to .03047 (3.1%increase). This test case shows a good trade off among the plan-form variables to achieve an optimal performance for a realisticdesign. At Mach 0.9, which is an off design point, the drag isquite high. As a result, the optimizer increases the sweepback toweaken shock drag, increases the span to reduce vortex drag, andreduces the thickness to chord ratio (with the thickness xed) toalleviate shock drag. These changes cause a slight increase of wing weight. But if the wing structural weight is not includedin the cost function, the optimal shape will result in an excessivespan, chord-length, and sweep angle. As a result of the trade-off between drag reduction and increased wing weight, the over-

all drag reduction was more than in the previous gure, whilethe wing weight was slightly increased. These results verify thefeasibility of including the effects of planform variations in theoptimization.

Figure 16 shows the effect of varying the weighting param-eters α 1 and α 3 in the cost function (37). As before the designvariables are sweepback, span, chords at three different span lo-cations and mesh points on the wing surface. In Fig. 16 eachpoint corresponds to an optimal shape for one specic choiceof (α 1 , α 3). By varying α 1 and α 3 , we capture the Pareto frontwhich bounds all the non-dominated solutions. All points on thisfront are acceptable designs in the sense that no improvement can

be achieved in one objective that doesn’t lead to degradation inthe other objective. The optimum shape that corresponds to theoptimal Breguet range is also marked in the gure.

Figure 17 shows the change of planform when the α3α1

=1.This value of α3

α1is sufcient to cause the optimizer to reduce the

sweepback, reducing wing weight. But it allows the optimizer toincrease the span, reducing vortex drag. This yields an optimumshape which has low structure weight and moderate drag.

10.3 Shape optimization of complete aircraft congu-rations on unstructured meshes

We have recently extended the adjoint design method to un-structured meshes in order to facilitate the treatment of completeaircraft congurations [32]. Here we take advantage of the re-duced gradient formula (33) to reduced the computational costof the gradient calculation. The results for a transonic business jet are shown below. As shown in Figures 18, 19, 20, 21, theoutboard sections of the existing wing have a strong shock whileying at cruise conditions ( M ∞ = 0.80, α = 2o). The results ofa drag minimization that aims to remove the shocks on the wingare shown in Figures 22, 23, 24, 25. The drag has been reducedfrom 235 counts to 215 counts in about 8 design cycles. The liftwas constrained at 0.4 by perturbing the angle of attack. Further,the original thickness of the wing was maintained during the de-sign process ensuring that fuel volume and structural integritywill be maintained by the redesigned shape.

Thickness constraints on the wing were imposed on cuttingplanes along the span of the wing and by transferring the con-strained shape movement back to the nodes of the surface tri-angulation. The volume mesh was deformed to conform to theshape changes induced using the spring method. The entire de-sign process typically takes about 4 hours on a 1.7 Ghz Athlonprocessor with 1 Gb of memory. Parallel implementation of thedesign procedure has also been developed that further reducesthe computational cost of this design process.

11 ConclusionThe accumulated experience of the last decade suggests

that most existing aircraft which cruise at transonic speeds areamenable to a drag reduction of the order of 3 to 5 percent, or anincrease in the drag rise Mach number of at least .02. These im-provements can be achieved by very small shape modications,which are too subtle to allow their determination by trial anderror methods. The potential economic benets are substantial,considering the fuel costs of the entire airline eet. Moreover,if one were to take full advantage of the increase in the lift todrag ratio during the design process, a smaller aircraft could bedesigned to perform the same task, resulting in further cost re-ductions. Consequently we are condent that some methods of

this type will provide a basis for aerodynamic designs of the fu-ture.

AcknowledgmentThis work has beneted greatly from the support of the Air

Force Ofce of Science Research under grant No. AF F49620-98-1-2002.

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[7] S. Nadarajah and A. Jameson. Optimal control of un-steady ows using a time accurate method. AIAA paper 2002-5436 , 9 th AIAA/ISSMO Symposiumon Multidis-ciplinary Analysis and Optimization Conference, Atlanta,GA, September 4-6 2002.

[8] A. Jameson, L. Martinelli, and N. A. Pierce. Optimumaero-dynamic design using the Navier-Stokes equations. Theo-ret. Comput. Fluid Dynamics , 10:213–237, 1998.

[9] K. Leoviriyakit and A. Jameson. Aerodynamic shape op-timization of wings including planform variations. AIAA paper 2003-0210 , 41st Aerospace Sciences Meeting & Ex-hibit, Reno, Nevada, January 2003.

[10] J. Reuther, A. Jameson, J. J. Alonso, M. J. Rimlinger, andD. Saunders. Constrained multipoint aerodynamic shapeoptimization using an adjoint formulation and parallel com-puters. AIAA paper 97-0103 , 35th Aerospace SciencesMeeting and Exhibit, Reno, Nevada, January 1997.

[11] S. Cliff, J. Reuther, D. Sanders, and R. Hicks. Single and

multipoint aerodynamic shape optimization of high speedcivil transport. Journal of Aircraft , 38:997–1005, 2001.

[12] J. L. Lions. Optimal Control of Systems Governed by Par-tial Differential Equations . Springer-Verlag, New York,1971. Translated by S.K. Mitter.

[13] O. Pironneau. Optimal Shape Design for Elliptic Systems .Springer-Verlag, New York, 1984.

[14] A. Jameson. Optimum Aerodynamic Design Using ControlTheory, Computational Fluid Dynamics Review 1995 . Wi-

ley, 1995.[15] R. M. Hicks and P. A. Henne. Wing design by numerical

optimization. Journal of Aircraft , 15:407–412, 1978.[16] C. Bischof, A. Carle, G. Corliss, A. Griewank, and P. Hov-

land. Generating derivative codes from Fortran programs. Internal report MCS-P263-0991 , Computer Science Divi-sion, Argonne National Laboratory and Center of Researchon Parallel Computation, Rice University, 1991.

[17] L. L. Green, P. A. Newman, and K. J. Haigler. Sensitivityderivatives for advanced CFD algorithm and viscous mod-eling parameters via automatic differentiation. AIAA pa per 93-3321 , 11th AIAA Computational Fluid DynamicsConference, Orlando, Florida, 1993.

[18] W. K. Anderson, J. C. Newman, D. L. Whiteld, andE. J. Nielsen. Sensitivity analysis for the Navier-Stokesequations on unstructured meshes using complex variables. AIAA paper 99-3294 , Norfolk, VA, June 1999.

[19] A. Jameson. Optimum aerodynamic design using CFD andcontrol theory. AIAA paper 95-1729 , AIAA 12th Compu-tational Fluid Dynamics Conference, San Diego, CA, June1995.

[20] A. Jameson, L. Martinelli, J. J. Alonso, J. C. Vassberg, andJ. Reuther. Perspectives on simulation based aerodynamicdesign. In M. Hafez amd K. Morinishi and J. Periauz, ed-itors, Computational Fluid Dynamics for the 21 st Centurypages 135–178. Notes on Numerical Fluid Dynamics Vol.78, Springer-Verlag, 2001.

[21] A. Jameson. Aerodynamic shape optimization using the adjoint method. 2002-2003 lecture series at the von karmaninstitute, Von Karman Institute For Fluid Dynamics, Brus-

sels, Belgium, Febuary 3-7 2003.[22] A. Jameson and S. Kim. Reduction of the adjoint gradi-ent formula in the continuous limit. AIAA paper 2003-0040 , 41st Aerospace Sciences Meeting & Exhibit, Reno,Nevada, January 2003.

[23] S. Kim. Design Optimization of High–Lift CongurationsUsing a Viscous Adjoint-Based Method . Ph.d. dissertation,Department of Aeronautics and Astronautics, Stanford Uni-versity, Stanford, CA, August 2001.

[24] Daniel P. Raymer. Aircraft Design: A Conceptual Ap- proach, Third Edition . AIAA Education Series, Wright-Patterson Air Force Base, Ohio, 1990.

[25] A. Jameson. Optimum transonic wing design using control

theory. Technical report, IUTAM Symposium Transson-icum IV, Goettingen, Germany, September 2–6 2002.

[26] A. Jameson, W. Schmidt, and E. Turkel. Numerical solu-tions of the Euler equations by nite volume methods withRunge-Kutta time stepping schemes. AIAA paper 81-1259January 1981.

[27] L. Martinelli and A. Jameson. Validation of a multigridmethod for the Reynolds averaged equations. AIAA pa per 88-0414 , 1988.

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[28] S. Tatsumi, L. Martinelli, and A. Jameson. A new high reso-lution scheme for compressible viscous ows with shocks. AIAA paper To Appear, AIAA 33nd Aerospace SciencesMeeting, Reno, Nevada, January 1995.

[29] A. Jameson and J. C. Vassberg. Studies of alternativenumerical optimization methods applied to the brachis-tochrone problem. In Proceedings of OptiCON’99 , New-port Beach, CA, October 1999.

[30] R. H. Liebeck. Design of the Blended-Wing-Body subsonictransport. Wright Brothers Lecture, AIAA paper2002-0002 ,Reno, NV, January 2002.

[31] D. L. Roman, J. B. Allen, and R. H. Liebeck. Aerody-namic design challenges of the Blended-Wing-Body sub-sonic transport. AIAA paper 2000-4335 , Denver, CO, Au-gust 2000.

[32] A. Jameson, Sriram, and L. Martinelli. An unstructuredadjoint method for transonic ows. AIAA paper , 16 th AIAACFD Conference, Orlando, FL, June 2003.

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SYMBOL SOURCESYN107 DESIGN 50SYN107 DESIGN 0

ALPHA2.2582.059

CD0.011360.01269

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSB747 WING-BODY

REN = 100.00 , MACH = 0.860 , CL = 0.419

Solution 1Upper-Surface Isobars

( Contours at 0.05 Cp )

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

C p

X / C10.8% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

C p

X / C27.4% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

C p

X / C41.3% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

C p

X / C59.1% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

C p

X / C74.1% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

C p

X / C89.3% Span

Figure 8. Redesigned Boeing 747 wing at Mach 0.86, Cp distributions

SYMBOL SOURCESYN107 DESIGN 50SYN107 DESIGN 0

ALPHA1.7661.536

CD0.012930.01819

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSB747 WING-BODY

REN = 100.00 , MACH = 0.900 , CL = 0.421

Solution 1Upper-Surface Isobars

( Contours at 0.05 Cp )

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

C p

X / C10.8% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

C p

X / C

27.4% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

C p

X / C41.3% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

C p

X / C59.1% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

C p

X / C

74.1% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

C p

X / C89.3% Span

Figure 9. Redesigned Boeing 747 wing at Mach 0.90, Cp distributions21

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:

..........

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 110.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

S P A N L O A D &

S E C T C L

PERCENT SEMISPAN

C*CL/CREF

CL

SYMBOL SOURCESYN107 DESIGN50

SYN107 DESIGN 0

ALPHA1.766

1.536

CD0.01293

0.01819

COMPARISON OF SPANLOAD DISTRIBUTIONSB747 WING-BODY

REN = 100.00 , MACH = 0.900 , CL = 0.421

Figure 10. Span loading, Redesigned Boeing 747 wing at Mach 0.90

FOILDAGNSpanwise Thickness Distributions

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.00.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

Percent Semi-Span

H a l f - T h i c k n e s s

SYMBOL AIRFOILWing 1.030Wing 2.030

Maximum Thickness

Average Thickness

Figure 11. Spanwise thickness distribution, Redesigned Boeing 747wing at Mach 0.90

FOILDAGNAirfoil Geometry -- Camber & Thickness Distributions

Antony Jameson19:02 Sun

2 Jun 02

0. 0 10. 0 20. 0 3 0. 0 40 .0 50. 0 60. 0 7 0. 0 8 0. 0 90 .0 100 .0-10.0

0.0

10.0

Percent Chord

A i r f o i l

-1.0

0.0

1.0

2.0

C a m

b e r

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0



ETA42.042.0

R-LE0.3820.376

Tavg2.782.78

Tmax4.094.08

@ X44.5044.50

Figure 12. Section geometry at η = 0.42, redesigned Boeing747 wing at Mach 0.90

FOILDAGNAirfoil Geometry -- Camber & Thickness Distributions

Antony Jameson19:02 Sun

2 Jun 02

0. 0 1 0. 0 20 .0 30 .0 40. 0 50. 0 6 0. 0 70 .0 80. 0 90. 0 1 00. 0-10.0

0.0

10.0

Percent Chord

A i r f o i l

-1.0

0.0

1.0

2.0

C a m

b e r

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0



ETA69.169.1

R-LE0.3790.390

Tavg2.792.78

Tmax4.064.06

@ X43.0241.55

Figure 13. Section geometry at η = 0.68, redesigned Boeing 747wing at Mach 0.90

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Mach: 0.900 Alpha: 1.765CL: 0.419 CD: 0.01279 CM:-0.1384Design: 50 Residual: 0.3633E+00Grid: 257X 65X 49Sweep: 42.1138 Span(ft): 212.43C1(ft): 48.13 C2: 29.13 C3: 10.78CW: 0.02956 I: 0.01279

Cl: 0.346 Cd: 0.06819 Cm:-0.1354Root Section: 13.6% Semi-Span

Cp = -2.0

Cl: 0.614 Cd: 0.00146 Cm:-0.2404Mid Section: 50.8% Semi-Span

Cp = -2.0

Cl: 0.376 Cd:-0.02298 Cm:-0.1803Tip Section: 92.5% Semi-Span

Figure 14. Redesign of Boeing 747, xed planform

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Mach: 0.900 Alpha: 1.760CL: 0.419 CD: 0.01249 CM:-0.1612Design: 50 Residual: 0.4527E+00Grid: 257X 65X 49Sweep: 43.0188 Span(ft): 213.48C1(ft): 48.48 C2: 29.99 C3: 11.15CW: 0.03047 I: 0.02163

Cl: 0.341 Cd: 0.06667 Cm:-0.1332Root Section: 13.6% Semi-Span

Cp = -2.0

Cl: 0.592 Cd: 0.00072 Cm:-0.2293Mid Section: 50.8% Semi-Span

Cp = -2.0

Cl: 0.362 Cd:-0.02080 Cm:-0.1743Tip Section: 92.5% Semi-Span

Figure 15. Redesign of Boeing 747, variable planform and maximizing range

24

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120 130 140 150 160 170 180 19078,000

80,000

82,000

84,000

86,000

88,000

90,000

92,000

C D (count)

W w

i n g

( l b f )

Effects of the weighting constants on the optimal shapes

Maximize Breguet range

Baseline

Optimize only wing−sections

Figure 16. Pareto front of section and planform modications

Figure 17. Superposition of the baseline geometry (green/light) and the optimized planform geometry (blue/dark), using α 1=1 and α 3=1

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AIRPLANE

DENSITY from 0.6250 to 1.1000

Figure 18. Density contours for a business jet at M = 0.8, α = 2. Ex-isting wing.

FALCON

MACH 0.800 ALPHA 2.000 Z 6.00

CL 0.5351 CD 0.0 157 CM -0.2097

.

0 . 1

E + 0 1

0 . 8

E + 0 0

0 . 4

E + 0 0

0 . 0

E + 0 0

- . 4 E + 0 0

- . 8 E + 0 0

- . 1 E + 0 1

- . 2 E + 0 1

- . 2 E + 0 1

C p

+ + + + + + +

+ +

+ + + +

+ + + +

+ + + + + + + + + + + + + +

+ + + + + + + + + + +

+ + + + + + +

+ +

+ +

+

+

+

++

++

+

++

++ +++++

++++++++ ++ +++

++ ++ ++

+

++

+

++

+++

+++++

+++

+++ ++

+

Figure 19. Pressure distribution at 66 % wing span

FALCON

MACH 0.800 ALPHA 2.000 Z 7.00

CL 0.5283 CD 0.0135 CM -0.2117

.

0 . 1

E + 0 1

0 . 8

E + 0 0

0 . 4

E + 0 0

0 . 0

E + 0 0

- . 4 E + 0 0

- . 8 E + 0 0

- . 1 E + 0 1

- . 2 E + 0 1

- . 2 E + 0 1

C p

+ + + + + + +

+ +

+ + + + +

+

+ + + +

+ + + + + + + + + + + + + + + + + + + + +

+

+

+ + +

+

+

+

+

+

+

+

+

+

+

+++

++++ + ++

+++ ++ +++

+ ++++ +

+

+

+++

++

+ +++ ++++

+++

++


FALCON

MACH 0.800 ALPHA 2.000 Z 8. 00

CL 0.4713 CD 0. 0092 CM -0.1915

.

0 . 1

E + 0 1

0

. 8 E + 0 0

0 . 4

E + 0 0

0 . 0

E + 0 0

- . 4 E + 0 0

- . 8 E + 0 0

- . 1 E + 0 1

- . 2 E + 0 1

- . 2 E + 0 1

C p

+ + + + + +

+ +

+ +

+ +

+ + + + +

+ + + + + + + + + + + + + + + + + +

+ +

+ + +

+

+

+

+++

+

+

+

++

++++ ++ ++ ++ +++

++

+ ++

+++

+

+

+

+++++ + + +

++++

++


26

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AIRPLANE

DENSITY from 0.6250 to 1.1000

Figure 22. Density contours for a business jet at M = 0.8, α = 2.3.After redesign.

FALCON

MACH 0.800 ALPHA 2.298 Z 6.00

CL 0.5346 CD 0.0 108 CM -0.1936

.

0 . 1

E + 0 1

0 . 8

E + 0 0

0 . 4

E + 0 0

0 . 0

E + 0 0

- . 4 E + 0 0

- . 8 E + 0 0

- . 1 E + 0 1

- . 2 E + 0 1

- . 2 E + 0 1

C p

+ + +

+ + +

+ +

+ + + + +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+ + + +

+ +

+ + +

+ +

++

++

+

+

+

++

+++++ +++++ +++ ++ +++ ++ ++ +

++

+++

+++

++

+++

++

+++ +++ +

+

+oooooo

ooooooooooooooooooooooooooooooooooooooooo

oo

oo

ooo

oo

o

oo

o

oo

oo

oo

ooo

oo o oo ooo oo ooo oo oo o oo

o oo

ooo

oo

ooooo

ooo

oooo

oo

Figure 23. Pressure distribution at 66 % wing span, after redesign,Dashed line: original geometry, solid line: redesigned geometry

FALCON

MACH 0.800 ALPHA 2.298 Z 7. 00

CL 0.5417 CD 0. 0071 CM -0.2090

.

0 . 1

E + 0 1

0 . 8

E + 0 0

0 . 4

E + 0 0

0 . 0

E + 0 0

- . 4 E + 0 0

- . 8 E + 0 0

- . 1 E + 0 1

- . 2 E + 0 1

- . 2 E + 0 1

C p

+ + +

+

+

+ +

+ +

+ + + + + +

+ + + + + + + + + + + + + + + + + + + +

+ + + + + + +

+

+

+

+

+ + +++

+

+

+

+

+

+

+

++++ + ++ +++ ++ ++ + + ++++

+++

++++

++

+++

+

+++ + ++

++ooooooooo

ooooooooooooooooooooooooooooooooo

o

oo

oo

ooo

o

o

o

o

o

o

o

oo

oo o oo ooo oo oo o o o

ooooo

oooo

o

oo

ooo

o

oooo

ooo

o


FALCON

MACH 0.800 ALPHA 2.298 Z 8. 00

CL 0.4909 CD 0. 0028 CM -0.1951

.

0 . 1

E + 0 1

0 . 8

E + 0 0

0 . 4

E + 0 0

0 . 0

E + 0 0

- . 4 E + 0 0

- . 8 E + 0 0

- . 1 E + 0 1

- . 2 E + 0 1

- . 2 E + 0 1

C p

+ + +

+ + +

+ +

+ + + +

+ + + + +

+ + + + + + + + + + + + +

+ +

+ + +

+ + +

+

+

+

+

+ +++

+

+

+

++

++++ ++ ++ ++ +++ ++ + + +

+++

++

+

+++

++

++

++++ +

++ooo

oooo

ooooooooooooooooooooooooooooo

oo

o

o

o

o

ooo

o

o

o

oo

oo

ooo o oo oo ooo oo o o o

oo oo

oo

ooo

oo

oo

o

ooo

oo

o


27

jameson 2003

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