the role of cfd in preliminary aerospace...

March 23, 2004 17:49

Proceedings of FEDSM’034TH ASME JSME Joint Fluids Engineering Conference

Honolulu, Hawaii USA, July 6-11, 2003

FEDSM2003-45812

THE ROLE OF CFD IN PRELIMINARY AEROSPACE DESIGN

Antony JamesonDepartment of Aeronautics and Astronautics

Stanford [email protected]

AbstractThis paper discusses the role that computational fluid dy-

namics plays in the design of aircraft. An overview of the designprocess is provided, covering some of the typical decisions that adesign team addresses within a multi-disciplinary environment.On a very regular basis trade-offs between disciplines have to bemade where a set of conflicting requirements exists. Within anaircraft development project, we focus on the aerodynamic de-sign problem and review how this process has been advanced,first with the improving capabilities of traditional computationalfluid dynamics analyses, and then with aerodynamic optimiza-tions based on these increasingly accurate methods.

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 definition and control of theconfiguration. Computer visualization techniques enable the de-signer to verify that no interferences exist between different partsin the layout, and greatly facilitate decisions on the routing ofelectrical wiring and hydraulic piping.

Similarly, structural analysis is now almost entirely carriedout by computational methods, typically finite element methods.Commercially available software systems have been progres-sively developed and augmented with new features, and can treatthe full range of requirements for aeronautical structures, includ-

ing the analysis of stressed skin into the nonlinear range. Also,they are very carefully validated against a comprehensive suiteof test cases before each new release. Hence, engineers placecomplete confidence in their results. Accordingly, the structuraldesign is routinely committed on the basis of computational anal-ysis, while structural testing is limited to the role of verificationthat the design truly meets its specified requirements of ultimatestrength and fatigue life.

Computational simulation of fluid flow has not yet reachedthe same level of maturity. While commercial software for thesimulation of fluid flow is offered by numerous vendors, air-craft companies continue to make substantial investments in thein-house development of their own methods. At the same timethere are major ongoing efforts to develop the science of compu-tational fluid dynamics (CFD) in government research agenciesand Universities. This reflects the fact that fluid flow is generallymore complex and harder to predict than the behavior of struc-tures. The complexity and range of phenomena that characterizefluid flow is well illustrated in Van Dyke’s Album of Fluid Mo-tion [1].

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 [2] listed three main objective of computa-tional aerodynamics:

1. To provide flow simulations that are either impracticalor impossible to obtain in wind tunnels or other ground based

1 Copyright c© 2003 by A. Jameson

experimental test facilities.2. To lower the time and cost required to obtain aerodynamic

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

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

There have been major advances towards these goals. De-spite these, CFD is still not being exploited as effectively as onewould like in the design process. This is partially due to the longset-up times and high costs, both human and computational, as-sociated with complex flow 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.

The emphasis is on the second of the original goals for a nu-merical wind tunnel, but also on taking computational methodsa step further, to use them not only to predict the aerodynamicproperties, but also to find superior designs. The key idea is toembed both analysis and optimization methods in the design pro-cess, drawing from both CFD and control theory.

Design trade-offs and the design process itself are surveyedin the next sections. Section 4 discusses the state of the art forCFD in applied aerodynamics, in particular the accuracy of dragprediction. Sections 5-8 examine the way in which optimiza-tion techniques can be integrated with CFD. The paper concludeswith case studies which apply aerodynamic shape optimizationmethods within the design process.

2 Aerodynamic Design TradeoffsFocusing on the design of long range transport aircraft, a

good first estimate of performance is provided by the Breguetrange equation:

Range =VLD

1SFC

logW0 +Wf

W0. (1)

Here V is the speed, L/D is the lift to drag ratio, SFC is thespecific fuel consumption of the engines, W0 is the loadingweight(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 W0. Thespecific fuel consumption is mainly the province of the enginemanufacturers, and in fact the largest advances in the last 30years have been in engine efficiency. The aerodynamic designershould try to maximize VL/D, but must consider the impact ofshape modifications on structure weight.

The drag coefficient can be split into an approximately fixedcomponent CD0 , and the induced drag due to lift as

CD = CD0 +C2

LπεAR

(2)

where AR is the aspect ratio, and ε is an efficiency factor close tounity. CD0 includes contributions such as friction and form drag.It can be seen from this equation that L/D is maximized by flyingat a lift coefficient such that the two terms are equal, so that theinduced drag is half the total drag. Moreover, the actual drag dueto lift

Dv =2L2

περV 2b2

varies inversely with the square of the span b. Thus there is adirect conflict between reducing the drag by increasing the spanand 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 beincreased until it approaches the speed of sound C, at whichpoint the formation of shock waves causes the onset of ”drag-rise” . Typically the lift to drag ratio will drop from around 19at a Mach number V/C in the neighborhood of 0.85, to the orderof 4 at Mach 1. Thus the optimum cruise speed will be in thetransonic regime, when shock waves are beginning to form , butremain weak enough only to incur a small drag penalty.

The designer can reduce shock drag and delay the onset ofdrag-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 tank), 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 ofwinglets, the optimum span load distribution is elliptic, givingan efficiency 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 alsobe necessary to limit the section lift coefficient in the outboardpart of the wing in order to delay the onset of buffet, which mayoccur when the lift coefficient is increased to make a turn at ahigh Mach number.

3 Design ProcessThe design process can generally be divided into three

phases: conceptual design, preliminary design, and final detaileddesign, as illustrated in Figure 1.

2

The conceptual design stage defines the mission in the lightof anticipated market requirements, and determines a generalpreliminary configuration capable of performing this mission, to-gether with first 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 andstructural 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 final design stage the structure must be defined incomplete detail, together with complete systems, including theflight deck, control systems (involving major software develop-ment for fly-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. A staff of thousands of people define every part of the air-craft. Wind Tunnel validation of the final design is carried out.Significant development costs are incurred over a 3 year period,plus an additional year of Flight Testing and Structural Qualifica-tion Testing for Certification. Total costs are in the range of 3-12billion dollars. Thus, the final design would normally be car-ried out only if sufficient orders have been received to indicatea reasonably high probability of recovering a significant fractionof the investment. For a commercial aircraft there are extensivediscussions with airlines.

In the development of commercial aircraft, aerodynamic de-sign plays a leading role during the preliminary design stage,during which the definition of the external aerodynamic shapeis typically finalized. 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 definition 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-tice, three major design cycles, each requiring about 4-6 months,have been used to finalize the wing design. Improvements inCFD which would allow the elimination of a major cycle wouldsignificantly shorten the overall design process and therefore re-duce costs.

Conceptual Design

Preliminary Design

Final Design

Defines MissionPreliminary sizingWeight, performance

Figure 1. The Overall Design Process

{PropulsionNoiseStabilityControlLoadsStructuresFabrication

Conceptual Design

CADDefinition

MeshGeneration

CFDAnalysis

Visualization

Performance Evaluation

Multi−Disciplinary Evaluation

Wind Tunnel Testing

Central Database

Detailed Final Design

Release toManufacturing

ModelFabrication

Out

er L

oop

Maj

or D

esig

n C

ycle

Inne

r Lo

op

Figure 2. The Aerodynamic Design Process

The inner aerodynamic design loop is used to evaluate nu-merous variations in the wing definition. In each iteration it isnecessary to generate a mesh for the new configuration prior toperforming the CFD analysis. Computer graphics software isthen used to visualize the results, and the performance is evalu-ated. The first studies may be confined to partial configurations

3

such as wing-body or wing-body-nacelle combinations. At thisstage the focus is on the design of the clean wing. Key pointsof the flight envelope include the nominal cruise point, cruise athigh lift and low lift to allow for the weight variation betweenthe initial and final cruise as the fuel is burned off, and a longrange cruise point at lower Mach number, where it is importantto ensure there is no significant drag creep. Other defining pointsare the climb condition, which requires a good lift to drag ratioat low Mach number and high lift coefficient 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 efficient 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 fly-by-wire system may allow relaxed staticstability, hence tail surfaces of reduced size.

First estimates of the aerodynamic loads allow the design ofan 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 thefinal definition 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. Sufficient and known level of accuracy2. Acceptable computational and manpower costs3. Fast turn around time

4 Performance PredictionThe discussion in section 2 illustrates how performance es-

timation at the cruise condition is crucial to the design of long-range transport aircraft. The error should be on the order of 1percent or less. The drag coefficient of a typical transport air-craft, such as the Boeing 747, is about 0.0275 (275 counts) fora typical lift coefficient of approximately 0.5. The drag coeffi-cient of proposed supersonic transport designs is in the range of

0.0120 to 0.0150 at much lower lift coefficients in the range of0.1 to 0.12. Thus one should aim to predict drag with an accuracyof 1 to 2 counts. Manufacturers have to guarantee performance,and errors can be very expensive due to the costs of redesign,penalty payments and lost orders.

In order to achieve this level of accuracy, it is ultimatelyessential to use the Reynolds-Averaged Navier-Stokes (RANS)equations. However, to accelerate their initial efforts, designerstypically use CFD methods based on less sophisticated flow mod-els, such as the full-potential or Euler equations coupled with aboundary-layer method. In order to allow the completion of themajor design cycle in 4-6 months, the cycle time for the multidis-ciplinary loop should not be greater than about 2 weeks. Consid-ering the need to examine the performance of design variationsat all the key points of the flight envelope, this implies the needto turn around aerodynamic analysis in a few hours. The compu-tational costs are also important because the cumulative costs oflarge numbers of calculations can become a limiting factor.

The state-of-the-art in CFD drag prediction was recently as-sessed by an international workshop on the subject. A summaryof the results of the workshop is presented in Reference [3]. Fig-ure 4 provides the 28 drag polars resulting from this drag pre-diction workshop (DPW). Also included in this figure are theavailable test data [4] for the DLR-F4 wing/body configuration,which was the subject of the study. With the exception of a fewout-layers, the computed polars fall within a band of about ±7%

the absolute level. The slopes, dC2L

dCD, are nearly identical. When

comparing the CFD results with the test data, we note that theCFD solutions were all run assuming fully-turbulent flow, whilethe test data were collected with laminar runs on the wing up totransition strips on both upper and lower surfaces. To quantifythe shift in drag associated with this difference, several indepen-dent calculations were performed yielding 12-13 counts higherdrag levels for the fully-turbulent flows. Accounting for this ad-justment, the center of the CFD drag polars band coincides withthe mean of the test polars. While this indicates that the industryas a whole is closing in on the ability to compute accurate abso-lute drag levels, in general, the errors are not to the level desiredby aircraft design teams. However, a few of the results submittedto the DPW fall within the uncertainty band of the experimentaldata. For example, the corresponding results of Reference [5]are shown in Figure 5. Achieving this level of accuracy is dom-inated by the quality of the underlying grid, but also dependson the turbulence model, the level of convergence, discretizationscheme, etc. It is imperative that each of these areas be studiedindependently of each other, otherwise ”accurate” results mightbe obtained as a consequence of cancellation of errors. Unfortu-nately, an optimization based on an analysis method containingsuch a cancellation of errors will most likely emphasize its weak-ness and probably yield a new design with a false performanceimprovement.

4

5 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 efficiency.

Ultimately there is a need for multi-disciplinary optimiza-tion (MDO), but this can only be effective if it is based on suf-ficiently high fidelity modeling of the separate disciplines. Asa step in this direction there could be significant 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 liftcoefficient with a fixed 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 find optimum aerodynamic shapes with reason-able computational costs, it pays to regard the wing as a devicewhich controls the flow in order to produce lift with minimumdrag. As a result, one can draw on concepts which have beendeveloped in the mathematical theory of control of systems gov-erned by partial differential equations. In particular, an accept-able aerodynamic design must have characteristics that smoothlyvary with small changes in shape and flow conditions. Conse-quently, gradient-based procedures are appropriate for aerody-namic shape optimization. Two main issues affect the efficiencyof gradient-based procedures; the first is the actual calculationof the gradient, and the second is the construction of an efficientsearch procedure which utilizes the gradient.

5.1 Gradient CalculationFor the class of aerodynamic optimization problems under

consideration, the design space is essentially infinitely 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 δI. Now suppose that δI canbe expressed to first order as

δI =

∫

G(x)δF (x)dx

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

δF (x) = −λG(x)

one obtains an improvement

δI = −λ∫

G2(x)dx

unless G(x) = 0. Thus the vanishing 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 numberof design parameters is large and the cost function is an expen-sive evaluation. The simplest approach to optimization is to de-fine the geometry through a set of design parameters, which may,for example, be the weights αi applied to a set of shape functionsBi(x) so that the shape is represented as

F (x) = ∑αiBi(x).

Then a cost function I is selected which might be the drag co-efficient or the lift to drag ratio; I is regarded as a function ofthe parameters αi. The sensitivities ∂I

∂αimay now be estimated

by making a small variation δαi in each design parameter in turnand recalculating the flow to obtain the change in I. Then

∂I∂αi

≈I(αi +δαi)− I(αi)

δαi.

The main disadvantage of this finite-difference approach isthat the number of flow calculations needed to estimate the gra-dient is proportional to the number of design variables [6]. Simi-larly, if one resorts to direct code differentiation (ADIFOR [7,8]),or complex-variable perturbations [9], the cost of determining thegradient is also directly proportional to the number of variablesused to define 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 [10–12]. The essential idea may be summa-rized as follows. For flow about an arbitrary body, the aerody-namic properties that define the cost function are functions of

5

the flowfield 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

δI =∂IT

∂wδw+

∂IT

∂FδF .

Using a technique drawn from control theory, the governingequations of the flowfield are introduced as a constraint in sucha way that the final expression for the gradient does not requirereevaluation of the flowfield. 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 flowfield domain D, can be written as

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

Then δw is determined from the equation

δR =

[

∂R∂w

]

δw+

[

∂R∂F

]

δF = 0.

Next, introducing a Lagrange multiplier ψ, we have

δI =∂IT

∂wδw+

∂IT

∂FδF −ψT

([

∂R∂w

]

δw+

[

∂R∂F

]

δF)

. (4)

With some rearrangement

δI =

(

∂IT

∂w−ψT

[

∂R∂w

])

δw+

(

∂IT

∂F−ψT

[

∂R∂F

])

δF .

Choosing ψ to satisfy the adjoint equation

[

∂R∂w

]T

ψ =∂IT

∂w(5)

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

δI = GδF ,

where

G =∂IT

∂F−ψT

[

∂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 flow-field evaluations.

In the case that (3) is a partial differential equation, the ad-joint equation (5) is also a partial differential equation and ap-propriate boundary conditions must be determined. It turns outthat 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 field 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 flow equation. Hence, the cost of obtaining thegradient is comparable to the cost of two function evaluations,regardless of the dimension of the design space.

6 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 [13–15].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

+∂ fi

∂xi= 0 in D, (6)

where

w =

ρρu1ρu2ρu3ρE

, fi =

ρuiρuiu1 + pδi1ρuiu2 + pδi2ρuiu3 + pδi3

ρuiH

(7)

and δi j is the Kronecker delta function. Also,

p = (γ−1)ρ{

E −12

(

u2i)

}

, (8)

6

and

ρH = ρE + p (9)

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

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

Ki j =

[

∂xi

∂ξ j

]

, J = det(K) , K−1i j =

[

∂ξi

∂x j

]

,

and

S = JK−1.

The elements of S are the cofactors of K, and in a finite 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

Si j =12

ε jpqεirs∂xp

∂ξr

∂xq

∂ξs. (10)

Then

∂∂ξi

Si j =12

ε jpqεirs

(

∂2xp

∂ξr∂ξi

∂xq

∂ξs+

∂xp

∂ξr

∂2xq

∂ξs∂ξi

)

= 0. (11)

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

S1 j = ε jpq∂xp

∂ξ2

∂xq

∂ξ3,

S2 j = ε jpq∂xp

∂ξ3

∂xq

∂ξ1,

S3 j = ε jpq∂xp

∂ξ1

∂xq

∂ξ2. (12)

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

J∂w∂t

+R(w) = 0 (13)

where

R(w) = Si j∂ f j

∂ξi=

∂∂ξi

(Si j f j) , (14)

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

Ui = Si ju j

as

Fi = Si j f j =

ρUiρUiu1 +Si1 pρUiu2 +Si2 pρUiu3 +Si3 p

ρUiH

.

For convenience, the coordinates ξi describing the fixedcomputational domain are chosen so that each boundary con-forms to a constant value of one of these coordinates. Variationsin the shape then result in corresponding variations in the map-ping derivatives defined by Ki j. Suppose that the performance ismeasured by a cost function

I =∫

BM (w,S)dBξ +

∫

DP (w,S)dDξ,

containing both boundary and field contributions where dBξ anddDξ are the surface and volume elements in the computationaldomain. In general, M and P will depend on both the flow vari-ables w and the metrics S defining 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 defined by the flowequations (13). A shape change produces a variation in the flowsolution δw and the metrics δS which in turn produce a variationin the cost function

δI =

∫

BδM (w,S)dBξ +

∫

DδP (w,S)dDξ. (15)

This can be split as

δI = δII +δIII , (16)

with

δM = [Mw]I δw+δM II ,

δP = [Pw]I δw+δPII , (17)

7

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

∂w and ∂P∂w with the met-

rics fixed, while δMII and δPII represent the contribution of themetric variations δS to δM and δP .

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

δR =∂

∂ξiδFi = 0. (18)

Here also, δR and δFi can be split into contributions associatedwith δw and δS using the notation

δR = δRI +δRII

δFi = [Fiw]I δw+δFiII . (19)

where

[Fiw]I = Si j∂ fi

∂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

∫

DψT ∂

∂ξiδFidDξ = 0. (20)

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

∫

BniψT δFiI dBξ −

∫

D

∂ψT

∂ξiδFiI dDξ +

∫

DψT δRIIdDξ = 0. (21)

This equation results directly from taking the variation of theweak form of the flow 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

δI = δIII −

∫

DψT δRII dDξ −

∫

B

[

δMI −niψT δFiI]

dBξ

+∫

D

[

δPI +∂ψT

∂ξiδFiI

]

dDξ. (22)

Now, since ψ is an arbitrary differentiable function, it may bechosen in such a way that δI no longer depends explicitly on

the 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 field terms with subscript“I” to produce a differential adjoint system governing ψ

∂ψT

∂ξi[Fiw]I + [Pw]I = 0 in D. (23)

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

CTi

∂ψ∂ξi

= 0 in D, (24)

where the inviscid Jacobian matrices in the transformed space aregiven by

Ci = Si j∂ f j

∂w.

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

niψT [Fiw]I = [Mw]I on B . (25)

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

δI = δIII +

∫

DψT δRII dDξ, (26)

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

The details of the formula for the gradient depend on the wayin which the boundary shape is parameterized as a function of thedesign variables, and the way in which the mesh is deformed asthe boundary is modified. Using the relationship between themesh deformation and the surface modification, the field integralis reduced to a surface integral by integrating along the coordi-nate lines emanating from the surface. Thus the expression for

8

δI is finally reduced to the form

δI =

∫

BGδF dBξ

where F represents the design variables, and G is the gradient,which is a function defined over the boundary surface.

The boundary conditions satisfied by the flow 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 specified arbitrarily. Instead, it must bechosen from the class of functions which allow cancellation ofall terms containing δw in the boundary integral of equation (22).On the other hand, there is no such restriction on the specificationof the field contribution to the cost function P , since these termsmay always be absorbed into the adjoint field equation (23) assource terms.

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

n1 = n3 = 0, n2 = 1, dBξ = dξ1dξ3,

so that only the variation δF2 needs to be considered at the wallboundary. The condition that there is no flow through the wallboundary at ξ2 = 0 is equivalent to

U2 = 0,

so that

δU2 = 0

when the boundary shape is modified. Consequently the varia-tion of the inviscid flux at the boundary reduces to

δF2 = δp

0

S21

S22

S23

0

+ p

0

δS21

δS22

δS23

0

. (27)

Since δF2 depends only on the pressure, it is now clear that theperformance measure on the boundary M (w,S) may only be a

function of the pressure and metric terms. Otherwise, completecancellation of the terms containing δw in the boundary integralwould 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

∫

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

∫ ∫

Bw(p− pd)

2 |S2|dξ1dξ3,

where the quantity

|S2| =√

S2 jS2 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 δp, the adjoint boundary conditionreduces to

ψ jn j = p− pd (28)

where n j are the components of the surface normal

n j =S2 j

|S2|.

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 find finally that

δI = −

∫

D

∂ψT

∂ξiδSi j f jdD

−∫ ∫

BW

(δS21ψ2 +δS22ψ3 +δS23ψ4) pdξ1dξ3. (29)

Here the expression for the cost variation depends on the meshvariations throughout the domain which appear in the field inte-gral. However, the true gradient for a shape variation should not

9

depend on the way in which the mesh is deformed, but only onthe true flow solution. In the next section we show how the fieldintegral can be eliminated to produce a reduced gradient formulawhich depends only on the boundary movement.

7 The Reduced Gradient FormulationConsider the case of a mesh variation with a fixed boundary.

Then,

δI = 0

but there is a variation in the transformed flux,

δFi = Ciδw+δSi j f j.

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

δw = ∇w ·δx =∂w∂x j

δx j (= δw∗)

and since

∂∂ξi

δFi = 0,

it follows that

∂∂ξi

(δSi j f j) = −∂

∂ξi(Ciδw∗) . (30)

It is verified below that this relation holds in the general case withboundary movement. Now

∫

DφT δRdD =

∫

DφT ∂

∂ξiCi (δw−δw∗)dD

=

∫

BφTCi (δw−δw∗)dB

−

∫

D

∂φT

∂ξiCi (δw−δw∗)dD. (31)

Here on the wall boundary

C2δw = δF2 −δS2 j f j . (32)

Thus, by choosing φ 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:

δI =∫

BWψT (δS2 j f j +C2δw∗)dξ1dξ3

−

∫ ∫

BW

(δS21ψ2 +δS22ψ3 +δS23ψ4) pdξ1dξ3. (33)

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

∂∂ξi

(δSi j f j)

=12

∂∂ξi

{

ε jpqεirs

(

∂δxp

∂ξr

∂xq

∂ξs+

∂xp

∂ξr

∂δxq

∂ξs

)

f j

}

=12

ε jpqεirs

(

∂δxp

∂ξr

∂xq

∂ξs+

∂xp

∂ξr

∂δxq

∂ξs

)

∂ f j

∂ξi

=12

ε jpqεirs

{

∂∂ξr

(

δxp∂xq

∂ξs

∂ f j

∂ξi

)}

+12

ε jpqεirs

{

∂∂ξs

(

δxq∂xp

∂ξr

∂ f j

∂ξi

)}

=∂

∂ξr

(

δxpεpq jεrsi∂xq

∂ξs

∂ f j

∂ξi

)

. (34)

Now express δxp in terms of a shift in the original computationalcoordinates

δxp =∂xp

∂ξkδξk.

Then we obtain

∂∂ξi

(δSi j f j) =∂

∂ξr

(

εpq jεrsi∂xp

∂ξk

∂xq

∂ξs

∂ f j

∂ξiδξk

)

. (35)

The term in ∂∂ξ1

is

ε123εpq j∂xp

∂ξk

(

∂xq

∂ξ2

∂ f j

∂ξ3−

∂xq

∂ξ3

∂ f j

∂ξ2

)

δξk.

Here the term multiplying δξ1 is

ε jpq

(

∂xp

∂ξ1

∂xq

∂ξ2

∂ f j

∂ξ3−

∂xp

∂ξ1

∂xq

∂ξ3

∂ f j

∂ξ2

)

.

10

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

S2 j∂ f1

∂ξ2+S3 j

∂ f1

∂ξ3

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

−S1 j∂ f1

∂ξ1.

The terms multiplying δξ2 and δξ3 are

ε jpq

(

∂xp

∂ξ2

∂xq

∂ξ2

∂ f j

∂ξ3−

∂xp

∂ξ2

∂xq

∂ξ3

∂ f j

∂ξ2

)

= −S1 j∂ f1

∂ξ2

and

ε jpq

(

∂xp

∂ξ3

∂xq

∂ξ2

∂ f j

∂ξ3−

∂xp

∂ξ3

∂xq

∂ξ3

∂ f j

∂ξ2

)

= −S1 j∂ f1

∂ξ3.

Thus the term in ∂∂ξ1

is reduced to

−∂

∂ξ1

(

S1 j∂ f1

∂ξkδξk

)

.

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

and ∂∂ξ3

, weobtain

∂∂ξi

(δSi j f j) = −∂

∂ξi

(

Si j∂ f j

∂ξkδξk

)

= −∂

∂ξi(Ciδw∗)

as was to be proved.

8 Optimization Procedure8.1 The Need for a Sobolev Inner Product in the Defi-

nition of the GradientAnother key issue for successful implementation of the con-

tinuous adjoint method is the choice of an appropriate inner prod-uct for the definition of the gradient. It turns out that there is anenormous benefit from the use of a modified Sobolev gradient,which enables the generation of a sequence of smooth shapes.This can be illustrated by considering the simplest case of a prob-lem in the calculus of variations.

Suppose that we wish to find the path y(x) which minimizes

I =

b∫

a

F(y,y′)dx

with fixed end points y(a) and y(b). Under a variation δy(x),

δI =

b∫

a

(

∂F∂y

δy+∂F∂y′ δy

′)

dx

=

b∫

a

(

∂F∂y

−ddx

∂F∂y′

)

δydx

Thus defining the gradient as

g =∂F∂y

−ddx

∂F∂y′

and the inner product as

(u,v) =

b∫

a

uvdx

we find that

δI = (g,δy).

If we now set

δy = −λg, λ > 0

we obtain a improvement

δI = −λ(g,g)≤ 0

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 descent

11

between two laterally separated points when a particle falls undergravity,

F(y,y′) =

√

1+ y′2

y

and

g = −1+ y

′2 +2yy′′

2(

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 [16]

〈u,v〉 =

∫

(uv+ εu′v′)dx

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

δI = 〈g,δy〉

Then we have

δI =

∫

(gδy+ εg′δy

′)dx

=∫

(g−∂∂x

ε∂g∂x

)δydx

= (g,δy)

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

δI = −λn〈gn,gn〉

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

8.2 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 definedabove, we define a modified gradient G such that

δI =< G ,δF > .

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

G −∂

∂ξ1ε

∂∂ξ1

G = G . (36)

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

δF = −λG (37)

with the result that

δI =−λ < G , G > < 0,

unless G = 0, and correspondingly G = 0.When second-order central differencing is applied to (36),

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

Gi − ε(

Gi+1 −2Gi + Gi−1)

= Gi, 1 ≤ i ≤ n,

where Gi and Gi 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 ,

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

A−1 =

1+2ε −ε 0 . 0ε . .0 . . .. . . −ε0 ε 1+2ε

.

12

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

δF = −λAG . (38)

As can be seen from the form of this expression, implicit smooth-ing may be regarded as a preconditioner which allows the use ofmuch larger steps for the search procedure and leads to a largereduction in the number of design iterations needed for conver-gence.

8.3 Outline of the Design ProcedureThe design procedure can finally be summarized as follows:

1. Solve the flow 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 satisfies any geo-

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

Sobolev Gradient

Gradient Calculation

Flow Solution

Adjoint Solution

Shape & Grid

Repeat the Design Cycleuntil Convergence

Modification

Figure 3. Design cycle

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 9 have been ob-tained using well-validated software for the solution of the Eulerand Navier-Stokes equations developed over the course of manyyears [17–19]. For inverse design the lift is fixed by the targetpressure. In drag minimization it is also appropriate to fix thelift coefficient, because the induced drag is a major fraction ofthe total drag, and this could be reduced simply by reducing the

lift. Therefore the angle of attack is adjusted during each flowsolution to force a specified lift coefficient to be attained, andthe influence 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 flowsolution.

8.4 Computational CostsIn order to address the issues of the search costs, the au-

thors investigated a variety of techniques in Reference [20] usinga trajectory optimization problem (the brachistochrone) as a rep-resentative model. The study verified that the search cost (i.e.,number of steps) of a simple steepest descent method appliedto this problem scales as N2, where N is the number of designvariables, while the cost of quasi-Newton methods scaled lin-early with N as expected. On the other hand, with an appropriateamount of smoothing, the smoothed descent method convergedin a fixed number of steps, independent of N. Considering thatthe evaluation of the gradient by a finite difference method re-quires N + 1 flow calculations, while the cost of its evaluationby the adjoint method is roughly that of two flow calculations,one arrives at the estimates of total computational cost given inTables 1-2.

Table 1. Cost of Search Algorithm.

Steepest Descent O(N2) steps

Quasi-Newton O(N) steps

Smoothed Gradient O(K) steps

(Note: K is independent of N)

Table 2. Total Computational Cost of Design.

Finite Difference Gradients

+ Steepest Descent O(N3)

Finite Difference Gradients

+ Quasi-Newton Search O(N2)

Adjoint Gradients

+ Quasi-Newton Search O(N)

Adjoint Gradients

+ Smoothed Gradient Search O(K)

(Note: K is independent of N)

13

9 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 [21, 22]. Some representation examples of design calcu-lations are presented in this section to illustrate the present capa-bility.

9.1 Redesign of the Boeing 747 wingOver the last decade the adjoint method has been success-

fully used to refine a variety of designs for flight at both transonicand supersonic cruising speeds. In the case of transonic flight, itis often possible to produce a shock free flow 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 fullbenefits of the optimization.

Here the optimization of the wing of the Boeing 747-200 ispresented to illustrate the kind of benefits that can be obtained.In these calculations the flow was modeled by the Reynolds Av-eraged Navier-Stokes equations. A Baldwin Lomax turbulencemodel was considered sufficient, since the optimization is forthe cruise condition with attached flow. The calculation wereperformed to minimize the drag coefficient at a fixed lift coeffi-cient, subject to the additional constraints that the span loadingshould not be altered and the thickness should not be reduced. Itmight be possible to reduce the induced drag by modifying thespan loading to an elliptic distribution, but this would increasethe root bending moment, and consequently require an increasein the skin thickness and structure weight. A reduction in wingthickness would not only reduce the fuel volume, but it wouldalso require an increase in skin thickness to support the bend-ing moment. Thus these constraints assure that there will be nopenalty in either structure weight or fuel volume.

Figure 6 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-nificant drag rise. The lift coefficient of 0.42 is the contributionof the exposed wing. Allowing for the fuselage to total lift co-efficient is about 0.47. It can be seen that the redesigned wingis essentially shock free, and the drag coefficient is reduced from0.01269 (127 counts) to 0.01136 (114 counts). The total drag co-efficient of the aircraft at this lift coefficient is around 270 counts,so this would represent a drag reduction of the order of 5 percent.

Figure 7 displays the result of an optimization at Mach 0.90.In this case the shock waves are not eliminated, but their strengthis significantly weakened, while the drag coefficient is reducedfrom 0.01819 (182 counts) to 0.01293 (129 counts). Thus theredesigned wing has essentially the same drag at Mach 0.9 as theoriginal wing at Mach 0.86. The Boeing 747 wing could appar-

ently be modified to allow such an increase in the cruising Machnumber because it has a higher sweep-back than later designs,and a rather thin wing section with a thickness to chord ratio of8 percent. Figures 8 and 9 verify that the span loading and thick-ness were not changed by the redesign, while figures 10 and 11indicate the required section changes at 42 percent and 68 per-cent span stations.

9.2 Planform and Aero-structural OptimizationThe shape changes in the section needed to improve the tran-

sonic wing design are quite small. However, in order to obtain atrue optimum design larger scale changes such as changes in thewing planform (sweepback, span, chord, and taper) should beconsidered. 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.

In references [23–25] the cost function is defined as

I = α1CD +α212

∫

B(p− pd)

2dS+α3CW

where CW is a dimensionless measure of the wing weight, whichcan be estimated either from statistical formulas, or from a sim-ple analysis of a representative structure, allowing for failuremodes such as panel buckling. The coefficient α2 is introduced toprovide the designer some control over the pressure distribution,while the relative importance of drag and weight are representedby the coefficients α1 and α3. By varying these it is possible tocalculate the Pareto front of designs which have the least weightfor a given drag coefficient, or the least drag coefficient for agiven weight. The relative importance of these can be estimatedfrom the Breguet range equation (1).

Figure 28 shows the Pareto front obtained from a study ofthe Boeing 747 wing [25], in which both the wing planformand section were varied simultaneously, with the planform de-fined by five parameters; sweepback, span, and the chord at threespan stations. In this case the planform variations were limited tochanges in sweepback, and the section changes were constrainedto maintain the same thickness to chord ratio. When α3

α1is rela-

tively large, the optimum wing has a reduced sweepback, leadingto a weight reduction, as illustrated in figure 29. The accompa-nying section shape changes keep the shock drag low.

Figure 28 also shows the point on the front that is estimatedto produce the maximum range according to the Breguet equa-tion.

9.3 Wing inverse design using an unstructured meshA major obstacle to the treatment of arbitrarily complex con-

figurations is the difficulty and cost of mesh generation. This can

14

be mitigated by the use of unstructured meshes. Thus it appearsthat the extension of the adjoint method to unstructured meshesmay provide the most promising route to the optimum shape de-sign of key elements of complex configurations, such as wing-pylon-nacelle combinations. Some results are presented below.These have been obtained with new software to implement theadjoint method for unstructured meshes which is currently underdevelopment [26]. Figures 12 and 13 shows the result of an in-verse design calculation, where the initial geometry was a wingmade up of NACA 0012 sections and the target pressure distri-bution was the pressure distribution over the Onera M6 wing.Figures 14, 15, 16, 17, 18, 19, show the target and computedpressure distribution at six span-wise sections. It can be seenfrom these plots the target pressure distribution is well recoveredin 50 design cycles, verifying that the design process is capableof recovering pressure distributions that are significantly differ-ent from the initial distribution. This is a particularly challeng-ing test, because it calls for the recovery of a smooth symmetricprofile from an asymmetric pressure distribution containing a tri-angular pattern of shock waves.

9.4 Shape optimization for a Transonic Business JetThe unstructured design method has also been applied to

complete aircraft configurations. The results for a business jetare shown below. As shown in figures 20, 21, 22, 23, the out-board sections of the wing have a strong shock while flying atcruise conditions (M∞ = 0.80, α = 2o). The results of a dragminimization that aims to remove the shocks on the wing areshown in figures 24, 25, 26, 27. The drag has been reduced from235 counts to 215 counts in about 8 design cycles. The lift wasconstrained at 0.4 by perturbing the angle of attack. Further, theoriginal thickness of the wing was maintained during the designprocess ensuring that fuel volume and structural integrity willbe maintained by the redesigned shape. Thickness constraintson the wing were imposed on cutting planes along the span ofthe wing and by transferring the constrained shape movementback to the nodes of the surface triangulation. The volume meshwas deformed to conform to the shape changes induced using thespring method. The entire design process typically takes about4 hours on a 1.7 Ghz Athlon processor with 1 Gb of memory.Parallel implementation of the design procedure has also beendeveloped that further reduces the computational cost of this de-sign process.

10 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 modifications,

which are too subtle to allow their determination by trial anderror methods. The potential economic benefits are substantial,considering the fuel costs of the entire airline fleet. 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, with consequent further costreductions. It seems inevitable that some method of this type willprovide a basis for aerodynamic designs of the future.

AcknowledgmentThis work has benefited greatly from the support of the Air

Force Office of Science Research under grant No. AF F49620-98-1-2002. I have drawn extensively from the lecture notes pre-pared by Luigi Martinelli and myself for a CIME summer coursein 1999 [14] and from a paper prepared for the 23rd Interna-tional Congress of Aeronautical Sciences, Toronto, September2002 [27]. I am also indebted to my research assistant KasiditLeoviriyakit for his assistance in preparing the Latex files forthis text.

REFERENCES[1] M. Van Dyke. An Album of Fluid Motion. The Parabolic

Press, Stanford, 1982.[2] D.R. Chapman, H. Mark, and M.W. Pirtle. Computers vs.

wind tunnels in aerodynamic flow simulations. Astronau-tics and Aeronautics, 13(4):22–30, 35, 1975.

[3] Levy D.W., Zickuhr T., Vassberg J., and al. Summary ofdata from the first aiaa cfd drag prediction workshop. AIAApaper 02-0841, AIAA 40rd Aerospace Sciences Meeting,Reno, Nevada, January 2002.

[4] G. Redeker. DLR-F4 wing-body configuration. In A Selec-tion of Experimental Test Cases for the Validation of CFDCodes, number AR-303, pages B4.1–B4.21. AGARD, Au-gust 1994.

[5] J. Vassberg, P.G. Buning, and C.L. Rumsey. Drag predic-tion for the dlr-f4 wing/body using overflow and cfl3d on anoverset mesh. AIAA paper 02-0840, AIAA 40rd AerospaceSciences Meeting, Reno, Nevada, January 2002.

[6] R. M. Hicks and P. A. Henne. Wing design by numericaloptimization. Journal of Aircraft, 15:407–412, 1978.

[7] 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.

[8] 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.

15

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

[10] A. Jameson, L. Martinelli, J. J. Alonso, J. C. Vassberg, andJ. Reuther. Simulation based aerodynamic design. IEEEAerospace Conference, Big Sky, MO, March 2000.

[11] A. Jameson. Optimum aerodynamic design using controltheory. Computational Fluid Dynamics Review, pages 495–528, 1995.

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

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

[14] A. Jameson and L. Martinelli. Aerodynamic shape op-timization techniques based on control theory. Lecturenotes in mathematics #1739, proceeding of computationalmathematics driven by industrial problems, CIME (Inter-national Mathematical Summer (Center), Martina Franca,Italy, June 21-27 1999.

[15] A. Jameson. Aerodynamic shape optimization using the ad-joint method. 2002-2003 lecture series at the von karmaninstitute, Von Karman Institute For Fluid Dynamics, Brus-sels, Belgium, Febuary 3-7 2003.

[16] A. Jameson, L. Martinelli, and J. Vassberg. Reduction ofthe adjoint gradient formula in the continuous limit. AIAApaper, 41st AIAA Aerospace Sciences Meeting, Reno, NV,January 2003.

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

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

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

[20] 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.

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

[22] 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.[23] K. Leoviriyakit and A. Jameson. Aerodynamic shape op-

timization of wings including planform variations. AIAApaper 2003-0210, 41st Aerospace Sciences Meeting & Ex-hibit, Reno, Nevada, January 2003.

[24] K. Leoviriyakit, S. Kim, and A. Jameson. Viscous aero-dynamic shape optimization of wings including planformvariables. AIAA paper 2003-3498, 21st Applied Aerody-namics Conference, Orlando, Florida, June 23-26 2003.

[25] K. Leoviriyakit and A. Jameson. Aero-structural wing plan-form optimization. Reno, Nevada, January 2004. Proceed-ings of the 42st Aerospace Sciences Meeting & Exhibit.

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

[27] A. Jameson. Using computational fluid dynamics for aero-dynamics - a critical assessment. Technical report, 23rd

International Congress of Aeronautical Sciences, Toronto,Canada, September 8-13 2002.

16

Figure 4. Composite Drag Results form Reference [3]

0.018 0.020 0.022 0.024 0.026 0.028 0.030 0.032 0.034 0.036 0.0380.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

CD_SHIFT = 0.00137 - 0.00023*CL**2

JC Vassberg18 MAY, 0109 JAN, 02 Rev

M = 0.75 , REN = 3M

DLR-F4 WING/BODY CONFIGURATIONBaseline Over-Set Grid

DRAG COEFFICIENT

LIF

T C

OE

FFIC

IEN

T S

QU

AR

ED

OVERFLOW_MPI_1.8m

OVERFLOW_SHIFTED

CFL3D_6.0

CFL3D_SHIFTED

TEST DATA

Figure 5. OVERFLOW Drag Polars with and without Transition Corrections.

17

SYMBOL

SOURCE SYN107 DESIGN 50SYN107 DESIGN 0

ALPHA 2.258 2.059

CD 0.01136 0.01269

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSB747 WING-BODY

REN = 100.00 , MACH = 0.860 , CL = 0.419

Antony Jameson14:40 Tue

28 May 02COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13MCDONNELL DOUGLAS

Solution 1 Upper-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

1.0

Cp

X / C 10.8% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

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

Cp

X / C 41.3% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 59.1% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

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

Cp

X / C 89.3% Span

Figure 6. Redesigned Boeing 747 wing at Mach 0.86, Cp distributions

SYMBOL


ALPHA 1.766 1.536

CD 0.01293 0.01819

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSB747 WING-BODY

REN = 100.00 , MACH = 0.900 , CL = 0.421

Antony Jameson18:59 Sun

2 Jun 02COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT


Solution 1 Upper-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

1.0

Cp

X / C 10.8% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

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

Cp

X / C 41.3% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 59.1% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

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

Cp

X / C 89.3% Span

Figure 7. Redesigned Boeing 747 wing at Mach 0.90, Cp distributions18

Antony Jameson18:59 Sun 2 Jun 02

COMPPLOTJCV 1.13

COMPPLOTJCV 1.13

COMPPLOTJCV 1.13

COMPPLOTJCV 1.13

COMPPLOTJCV 1.13

COMPPLOTJCV 1.13

COMPPLOTJCV 1.13

COMPPLOTJCV 1.13

COMPPLOTJCV 1.13

COMPPLOTJCV 1.13

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

SPA

NLO

AD

& S

EC

T C

L

PERCENT SEMISPAN

C*CL/CREF

CL

SYMBOL


ALPHA 1.766 1.536

CD 0.01293 0.01819

COMPARISON OF SPANLOAD DISTRIBUTIONSB747 WING-BODY

REN = 100.00 , MACH = 0.900 , CL = 0.421

MCDONNELL DOUGLAS

Figure 8. Span loading, Redesigned Boeing 747 wing at Mach 0.90

FOILDAGNSpanwise Thickness Distributions

Antony Jameson19:02 Sun

2 Jun 02FOILDAGN


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

Hal

f-Th

ickn

ess

SYMBOL

AIRFOIL Wing 1.030Wing 2.030

Maximum Thickness

Average Thickness

Figure 9. Spanwise thickness distribution, Redesigned Boeing 747wing at Mach 0.90

FOILDAGNAirfoil Geometry -- Camber & Thickness Distributions


FOILDAGNJCV 0.94

MCDONNELL DOUGLAS

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

0.0

10.0

Percent Chord

Air

foil

-1.0

0.0

1.0

2.0

Cam

ber

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

Hal

f-Th

ickn

ess

SYMBOL


ETA 42.0 42.0

R-LE 0.382 0.376

Tavg 2.78 2.78

Tmax 4.09 4.08

@ X 44.5044.50

Figure 10. Section geometry at η = 0.42, redesigned Boeing747 wing at Mach 0.90

FOILDAGNAirfoil Geometry -- Camber & Thickness Distributions


FOILDAGNJCV 0.94

MCDONNELL DOUGLAS

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

0.0

10.0

Percent Chord

Air

foil

-1.0

0.0

1.0

2.0

Cam

ber

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

Hal

f-Th

ickn

ess

SYMBOL


ETA 69.1 69.1

R-LE 0.379 0.390

Tavg 2.79 2.78

Tmax 4.06 4.06

@ X 43.0241.55

Figure 11. Section geometry at η = 0.68, redesigned Boeing 747wing at Mach 0.90

19

NACA 0012 WING TO ONERA M6 TARGET Mach: 0.840 Alpha: 3.060 CL: 0.325 CD: 0.02319 CM: 0.0000 Design: 0 Residual: 0.2763E-02 Grid: 193X 33X 33

Cl: 0.308 Cd: 0.04594 Cm:-0.1176 Root Section: 9.8% Semi-Span

Cp = -2.0

Cl: 0.348 Cd: 0.01749 Cm:-0.0971 Mid Section: 48.8% Semi-Span

Cp = -2.0

Cl: 0.262 Cd:-0.00437 Cm:-0.0473 Tip Section: 87.8% Semi-Span

Cp = -2.0

Figure 12. Initial(dashed lines) and final (solid lines) over a NACA 0012wing

NACA 0012 WING TO ONERA M6 TARGET Mach: 0.840 Alpha: 3.060 CL: 0.314 CD: 0.01592 CM: 0.0000 Design: 50 Residual: 0.1738E+00 Grid: 193X 33X 33

Cl: 0.294 Cd: 0.03309 Cm:-0.1026 Root Section: 9.8% Semi-Span

Cp = -2.0

Cl: 0.333 Cd: 0.01115 Cm:-0.0806 Mid Section: 48.8% Semi-Span

Cp = -2.0

Cl: 0.291 Cd:-0.00239 Cm:-0.0489 Tip Section: 87.8% Semi-Span

Cp = -2.0

Figure 13. Initial(dashed lines) and final (solid lines) pressure distribu-tion and modified section geometries

NACA 0012 WING TO ONERA M6 TARGET MACH 0.840 ALPHA 3.060 Z 0.00

CL 0.2814 CD 0.0482 CM -0.1113

GRID 192X32 NDES 50 RES0.162E-02

0.1E

+01

0.8E

+00

0.4E

+00

0.3E

-07

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++

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

++++

+

+

+

++

++++

+

+

+

+

+

+

+

++

+

+

++++

+++++

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

+ ++

+

++ + +

++

++

+oo

oo

ooooooooooooooooooooooooooooooooooooooooooooooooooo

oooo

o

o

o

oooo

o

o

o

o

o

o

oooo

o

oooo

ooooo

ooooooooooooooooooooooooo o o o o o

o oo

o

oo o o o

oo

oo

Figure 14. Attained(+,x) and target(o) pressure distributions at 0 % ofthe wing span


CL 0.2814 CD 0.0482 CM -0.1113


0.1E

+01

0.8E

+00

0.4E

+00

0.3E

-07

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++

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

++++

+

+

+

++

++++

+

+

+

+

+

+

+

++

+

+

++++

+++++

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

+ ++

+

++ + +

++

++

+oo

oo


oooo

o

o

o

oooo

o

o

o

o

o

o

oooo

o

oooo

ooooo

ooooooooooooooooooooooooo o o o o o

o oo

o

oo o o o

oo

oo


20


CL 0.3269 CD 0.0145 CM -0.0865


0.1E

+01

0.8E

+00

0.4E

+00

0.3E

-07

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++

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

++++

+

+

+

+

++++

+

+

+

+

+

+

+

+

+++++++++

++

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

+++

+

+

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

++o

oo


oooo

o

o

o

o

oooo

o

o

o

o

o

o

o

o

ooooooo

ooooooooooooooooooooooooooooo

oo

o

o

o

o o o o o o o o o oo

oo

o



CL 0.3356 CD 0.0081 CM -0.0735


0.1E

+01

0.8E

+00

0.4E

+00

0.3E

-07

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++

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

+

+

+

+

+

+++

+

+

+

+

+

+

+

+

+

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

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

+++

+

+

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

++

+oo

oooooooooooooooooooooooooooooooooooooooooooooooooooo

oooo

o

o

o

o

oooo

o

o

o

o

o

o

o

o

ooooooooooooooo

oooooooooooooooooo

ooo

o

o

o o o o o o o o o o o o o oo

oo



CL 0.3176 CD 0.0011 CM -0.0547


0.1E

+01

0.8E

+00

0.4E

+00

0.3E

-07

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++

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

+

+

+

+

+

+++

+

+

+

+

+

+

+

+

+

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

+++++

++

+

+

+

+++

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

++

+oo

oooooooooooooooooooooooooooooooooooooooooooooooooooo

oooo

o

o

o

o

oooo

o

o

o

o

o

o

o

o

ooooooooooooooooooooooo

oooooo

o

o

o

o

ooo

o o o o o o o o o o o o o o o oo

oo



CL 0.4846 CD 0.0178 CM -0.1518


0.1E

+01

0.8E

+00

0.4E

+00

0.3E

-07

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

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

+

+

+

++++

++

+

+

+

+

+

+

+

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

+

+

+

+

+++

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

+ +

+ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

o

o

o

oooo

o

o

o

o

o

o

o

o

oooooooooooooooooooooo

o

o

o

o

ooo

oooooooo o o o o o o o o o o o o o o o

o o

o

Figure 19. Final computed and target pressure distributions at 100 % ofthe wing span

21

AIRPLANE DENSITY from 0.6250 to 1.1000

Figure 20. Density contours for a business jet at M = 0.8, α = 2

FALCON MACH 0.800 ALPHA 2.000 Z 6.00

CL 0.5351 CD 0.0157 CM -0.2097

NNODE 353887 NDES 0 RES0.287E-05

0.1E

+01

0.8E

+00

0.4E

+00

0.0E

+00

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

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

++++

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

+++++

++

++

++

+

++

+++++++

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

++

+++

+++

+++ +++

++++++

++

Figure 21. Pressure distribution at 66 % wing span


CL 0.5283 CD 0.0135 CM -0.2117


0.1E

+01

0.8E

+00

0.4E

+00

0.0E

+00

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++++++

++++++

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

++++

+

+

+

+++

+

+

+

+

+++++

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

+++ ++

+

+++

++

+ +++ ++++

+++

++



CL 0.4713 CD 0.0092 CM -0.1915


0.1E

+01

0.8E

+00

0.4E

+00

0.0E

+00

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++++

+++++

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

++

+++

+

+

+

+++

+

+

+

++

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

++++ +

+++

+

+

+

+++++ + ++

++++

++


22

AIRPLANE DENSITY from 0.6250 to 1.1000

Figure 24. Density contours for a business jet at M = 0.8, α = 2.3,after redesign


CL 0.5346 CD 0.0108 CM -0.1936


0.1E

+01

0.8E

+00

0.4E

+00

0.0E

+00

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++++++

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

+++

++

++

++

+

+

+

++

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

+++

+++

++

+++

++

+++ ++++

++oooooo

ooooooooooooooooooooooooooooooooooooooooooo

oo

ooo

oo

o

oo

o

oo

oo

ooooo

ooooo ooo oo ooo oo oo o o ooo

oooo

oo

ooooo

ooo

ooooo

o

Figure 25. Pressure distribution at 66 % wing span, after redesign,Dashed line: original geometry, solid line: redesigned geometry


CL 0.5417 CD 0.0071 CM -0.2090


0.1E

+01

0.8E

+00

0.4E

+00

0.0E

+00

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++

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

+

+

+++++

+

+

+

+

+

+

+

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

+++++

+ ++++

++

+++

+

+++ + ++++ooooooo

ooooooooooooooooooooooooooooooooooo

o

oo

oo

ooo

o

o

o

o

o

ooooo

oooo ooo oo oo o o o ooooo o

ooo

ooo

ooo

oooo

ooo

oo



CL 0.4909 CD 0.0028 CM -0.1951


0.1E

+01

0.8E

+00

0.4E

+00

0.0E

+00

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++++++

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

+++

+

+

+

+

++++

+

+

+

++

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

+++ ++

+

+++

++

++

++++ +

++ooooooooooooooooooooooooooooooooooooo

oo

o

o

o

ooo

o

o

o

oo

oo

oo oo oo oo ooo oo oo o ooo ooo

ooo

ooo

oo

oooo

oo


23

86 88 90 92 94 96 98 100 102 10482000

84000

86000

88000

90000

92000

94000

96000

CD(count)

Ww

ing(lb

f)

Effects of the weighting constants on the optimal shapesBaseline geometryOptimized geometriesOptimized Braguet rangePareto front

Figure 28. Pareto front of section and sweep modifications

Figure 29. Superposition of the baseline geometry (green) and the optimized planform geometry (blue), using α1=1 and α3=0.5.

24

the role of cfd in preliminary aerospace...

Documents