aiaa 98–2538 · 2019. 5. 8. · aiaa 98–2538 aerodynamic shape optimization techniques based on...

AIAA 98–2538Aerodynamic Shape OptimizationTechniques Based On ControlTheoryAntony Jameson and Juan J. AlonsoStanford University, Stanford, CA 94305 USA

James J. ReutherResearch Institute for Advanced Computer Science,Moffet Field, CA 96035

L. MartinelliPrinceton University, Princeton, NJ 08544

J. C. VassbergBoeing Commercial Airplane Group, LongBeach, CA 90846

29th AIAA Fluid Dynamics ConferenceJune 15–18, 1998/Albuquerque, NM

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

AIAA 98–2538

Aerodynamic Shape Optimization TechniquesBased On Control Theory

Antony Jameson∗ and Juan J. Alonso†

Stanford University, Stanford, CA 94305 USA

James J. Reuther‡

Research Institute for Advanced Computer Science, Moffet Field, CA 96035

L. Martinelli§

Princeton University, Princeton, NJ 08544

J. C. Vassberg¶

Boeing Commercial Airplane Group, Long Beach, CA 90846

This paper reviews the formulation and application of optimization techniques basedon control theory for aerodynamic shape design in both inviscid and viscous compressibleflow. The theory is applied to a system defined by the partial differential equations of theflow, with the boundary shape acting as the control. The Frechet derivative of the costfunction is determined via the solution of an adjoint partial differential equation, and theboundary shape is then modified in a direction of descent. This process is repeated untilan optimum solution is approached. Each design cycle requires the numerical solutionof both the flow and the adjoint equations, leading to a computational cost roughlyequal to the cost of two flow solutions. Representative results are presented for viscousoptimization of transonic wing-body combinations and inviscid optimization of complexconfigurations.

Introduction

THE definition of the aerodynamic shapes of mod-ern aircraft relies heavily on computational sim-ulation to enable the rapid evaluation of many alter-native designs. Wind tunnel testing is then used toconfirm the performance of designs that have beenidentified by simulation as promising to meet theperformance goals. In the case of wing design andpropulsion system integration, several complete cyclesof computational analysis followed by testing of a pre-ferred design may be used in the evolution of the finalconfiguration. Wind tunnel testing also plays a crucialrole in the development of the detailed loads needed tocomplete the structural design, and in gathering datathroughout the flight envelope for the design and ver-ification of the stability and control system. The useof computational simulation to scan many alternativedesigns has proved extremely valuable in practice, butit still suffers the limitation that it does not guaranteethe identification of the best possible design. Gen-erally one has to accept the best so far by a given

∗AIAA Fellow, T. V. Jones Professor of Engineering†AIAA Member, Assistant Professor‡AIAA Member, Research Scientist§AIAA Member, Assistant Professor¶AIAA Senior Member, Senior Pricipal EngineerCopyright c© 1998 by the authors. Published by the American

Institute of Aeronautics and Astronautics, Inc. with permission.

cutoff date in the program schedule. To ensure the re-alization of the true best design, the ultimate goal ofcomputational simulation methods should not just bethe analysis of prescribed shapes, but the automaticdetermination of the true optimum shape for the in-tended application.

This is the underlying motivation for the combina-tion of computational fluid dynamics with numericaloptimization methods. Some of the earliest studies ofsuch an approach were made by Hicks and Henne.1, 2

The principal obstacle was the large computationalcost of determining the sensitivity of the cost func-tion to variations of the design parameters by repeatedcalculation of the flow. Another way to approachthe problem is to formulate aerodynamic shape de-sign within the framework of the mathematical theoryfor the control of systems governed by partial differen-tial equations.3 In this view the wing is regarded as adevice to produce lift by controlling the flow, and itsdesign is regarded as a problem in the optimal con-trol of the flow equations by changing the shape of theboundary. If the boundary shape is regarded as ar-bitrary within some requirements of smoothness, thenthe full generality of shapes cannot be defined witha finite number of parameters, and one must use theconcept of the Frechet derivative of the cost with re-spect to a function. Clearly such a derivative cannot

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American Institute of Aeronautics and Astronautics Paper 98–2538

be determined directly by separate variation of eachdesign parameter, because there are now an infinitenumber of these.Using techniques of control theory, however, the

gradient can be determined indirectly by solving anadjoint equation which has coefficients determined bythe solution of the flow equations. This directly cor-responds to the gradient technique for trajectory op-timization pioneered by Bryson.4 The cost of solvingthe adjoint equation is comparable to the cost of solv-ing the flow equations, with the consequence that thegradient with respect to an arbitrarily large number ofparameters can be calculated with roughly the samecomputational cost as two flow solutions. Once thegradient has been calculated, a descent method can beused to determine a shape change which will make animprovement in the design. The gradient can then berecalculated, and the whole process can be repeateduntil the design converges to an optimum solution,usually within 50 to 100 cycles. The fast calculation ofthe gradients makes optimization computationally fea-sible even for designs in three-dimensional viscous flow.There is a possibility that the descent method couldconverge to a local minimum rather than the globaloptimum solution. In practice this has not proved adifficulty, provided care is taken in the choice of a costfunction which properly reflects the design require-ments. Conceptually, with this approach the problemis viewed as infinitely dimensional, with the controlbeing the shape of the bounding surface. Eventuallythe equations must be discretized for a numerical im-plementation of the method. For this purpose theflow and adjoint equations may either be separatelydiscretized from their representations as differentialequations, or, alternatively, the flow equations maybe discretized first, and the discrete adjoint equationsthen derived directly from the discrete flow equations.The effectiveness of optimization as a tool for aero-

dynamic design also depends crucially on the properchoice of cost functions and constraints. One popu-lar approach is to define a target pressure distribu-tion, and then solve the inverse problem of findingthe shape that will produce that pressure distribution.Since such a shape does not necessarily exist, directinverse methods may be ill-posed. This difficulty isremoved by reformulating the inverse problem as anoptimization problem, in which the deviation betweenthe target and the actual pressure distribution is to beminimized according to a suitable measure such as thesurface integral

I =12

∫(p− pT )2dS

where p and pT are the actual and target pressures.This still leaves the definition of an appropriate pres-sure architecture to the designer. One may prefer todirectly improve suitable performance parameters, for

example, to minimize the drag at a given lift and Machnumber. In this case it is important to introduce ap-propriate constraints. For example, if the span is notfixed the vortex drag can be made arbitrarily small bysufficiently increasing the span. In practice, a usefulapproach is to fix the planform, and optimize the wingsections subject to constraints on minimum thickness.Studies of the use of control theory for optimum

shape design of systems governed by elliptic equa-tions were initiated by Pironneau.5 The control the-ory approach to optimal aerodynamic design was firstapplied to transonic flow by Jameson.6–11 He formu-lated the method for inviscid compressible flows withshock waves governed by both the potential flow andthe Euler equations.7 Numerical results showing themethod to be extremely effective for the design of air-foils in transonic potential flow were presented in,12

and for three-dimensional wing design using the Eu-ler equations in.13 More recently the method has beenemployed for the shape design of complex aircraft con-figurations,14, 15 using a grid perturbation approachto accommodate the geometry modifications. Themethod has been used to support the aerodynamicdesign studies of several industrial projects, includ-ing the Beech Premier and the McDonnell DouglasMDXX and Blended Wing-Body projects. The appli-cation to the MDXX is described in.9 The experiencegained in these industrial applications made it clearthat the viscous effects cannot be ignored in transonicwing design, and the method has therefore been ex-tended to treat the Reynolds Averaged Navier-Stokesequations.11 Adjoint methods have also been the sub-ject of studies by a number of other authors, includingBaysal and Eleshaky,16 Huan and Modi,17 Desai andIto,18 Anderson and Venkatakrishnan,19 and Peraireand Elliot.20

This paper reviews the formulation and develop-ment of the compressible viscous adjoint equations,and presents some examples of recent applications ofdesign techniques based on control theory to viscoustransonic wing design, and also to transonic and su-personic wing design for complex configurations.

General Formulation of the AdjointApproach to Optimal Design

Before embarking on a detailed derivation of the ad-joint formulation for optimal design using the Navier–Stokes equations, it is helpful to summarize the generalabstract description of the adjoint approach which hasbeen thoroughly documented in references.7, 21

The progress of the design procedure is measured interms of a cost function I, which could be, for exam-ple the drag coefficient or the lift to drag ratio. Forflow about an airfoil or wing, the aerodynamic proper-ties which define the cost function are functions of theflow-field variables (w) and the physical location of theboundary, which may be represented by the function

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F , say. ThenI = I (w,F) ,

and a change in F results in a change

δI =[∂IT

∂w

]I

δw +[∂IT

∂F]II

δF , (1)

in the cost function. Here, the subscripts I and II areused to distinguish the contributions due to the varia-tion δw in the flow solution from the change associateddirectly with the modification δF in the shape. Thisnotation is introduced to assist in grouping the numer-ous terms that arise during the derivation of the fullNavier–Stokes adjoint operator, so that it remains fea-sible to recognize the basic structure of the approachas it is sketched in the present section.Using control theory, the governing equations of the

flow field are introduced as a constraint in such a waythat the final expression for the gradient does notrequire multiple flow solutions. This corresponds toeliminating δw from (1).Suppose that the governing equation R which ex-

presses the dependence of w and F within the flow-field domain D can be written as

R (w,F) = 0. (2)

Then δw is determined from the equation

δR =[∂R

∂w

]I

δw +[∂R

∂F]II

δF = 0. (3)

Next, introducing a Lagrange Multiplier ψ, we have

δI =∂IT

∂wδw +

∂IT

∂F δF − ψT([∂R

∂w

]δw +

[∂R

∂F]δF)

=

{∂IT

∂w− ψT

[∂R

∂w

]}I

δw +

{∂IT

∂F − ψT[∂R

∂F]}

II

δF . (4)

Choosing ψ to satisfy the adjoint equation

[∂R

∂w

]Tψ =

∂I

∂w(5)

the first term is eliminated, and we find that

δI = GδF , (6)

where

G = ∂IT

∂F − ψT

[∂R

∂F].

The advantage is that (6) is independent of δw, withthe result that the gradient of I with respect to an ar-bitrary number of design variables can be determinedwithout the need for additional flow-field evaluations.In the case that (2) is a partial differential equation,the adjoint equation (5) is also a partial differential

equation and determination of the appropriate bound-ary conditions requires careful mathematical treat-ment.The computational cost of a single design cycle is

roughly equivalent to the cost of two flow solutionssince the the adjoint problem has similar complex-ity. When the number of design variables becomeslarge, the computational efficiency of the control the-ory approach over traditional approach, which requiresdirect evaluation of the gradients by individually vary-ing each design variable and recomputing the flow field,becomes compelling.Once equation (3) is established, an improvement

can be made with a shape change

δF = −λG

where λ is positive, and small enough that the firstvariation is an accurate estimate of δI. The variationin the cost function then becomes

δI = −λGTG < 0.

After making such a modification, the flow field andcorresponding gradient can be recalculated and theprocess repeated to follow a path of steepest descentuntil a minimum is reached. In order to avoid violatingconstraints, such as a minimum acceptable wing thick-ness, the gradient may be projected into an allowablesubspace within which the constraints are satisfied. Inthis way, procedures can be devised which must nec-essarily converge at least to a local minimum.

The Navier-Stokes EquationsFor the derivations that follow, it is convenient to

use Cartesian coordinates (x1,x2,x3) and to adopt theconvention of indicial notation where a repeated in-dex “i” implies summation over i = 1 to 3. Thethree-dimensional Navier-Stokes equations then takethe form

∂w

∂t+∂fi∂xi

=∂fvi∂xi

in D, (7)

where the state vector w, inviscid flux vector f andviscous flux vector fv are described respectively by

w =

ρρu1ρu2ρu3ρE

, (8)

fi =

ρuiρuiu1 + pδi1ρuiu2 + pδi2ρuiu3 + pδi3

ρuiH

, (9)

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fvi =

0σijδj1σijδj2σijδj3

ujσij + k ∂T∂xi

. (10)

In these definitions, ρ is the density, u1, u2, u3 are theCartesian velocity components, E is the total energyand δij is the Kronecker delta function. The pressureis determined by the equation of state

p = (γ − 1) ρ{E − 1

2(uiui)

},

and the stagnation enthalpy is given by

H = E +p

ρ,

where γ is the ratio of the specific heats. The viscousstresses may be written as

σij = µ(∂ui∂xj

+∂uj∂xi

)+ λδij

∂uk∂xk

, (11)

where µ and λ are the first and second coefficients ofviscosity. The coefficient of thermal conductivity andthe temperature are computed as

k =cpµ

Pr, T =

p

Rρ, (12)

where Pr is the Prandtl number, cp is the specific heatat constant pressure, and R is the gas constant.For discussion of real applications using a discretiza-

tion on a body conforming structured mesh, it is alsouseful to consider a transformation to the computa-tional coordinates (ξ1,ξ2,ξ3) defined by the metrics

Kij =[∂xi∂ξj

], J = det (K) , K−1ij =

[∂ξi∂xj

].

The Navier-Stokes equations can then be written incomputational space as

∂ (Jw)∂t

+∂ (Fi − Fvi)

∂ξi= 0 in D, (13)

where the inviscid and viscous flux contributions arenow defined with respect to the computational cellfaces by Fi = Sijfj and Fvi = Sijfvj , and the quan-tity Sij = JK−1ij represents the projection of the ξicell face along the xj axis. In obtaining equation (13)we have made use of the property that

∂Sij∂ξi

= 0 (14)

which represents the fact that the sum of the face areasover a closed volume is zero, as can be readily verifiedby a direct examination of the metric terms.

Formulation of the Optimal DesignProblem for the Navier-Stokes

EquationsAerodynamic optimization is based on the determi-

nation of the effect of shape modifications on someperformance measure which depends on the flow. Forconvenience, the coordinates ξi describing the fixedcomputational domain are chosen so that each bound-ary conforms to a constant value of one of these coor-dinates. Variations in the shape then result in corre-sponding variations in the mapping derivatives definedby Kij .Suppose that the performance is measured by a cost

function

I =∫BM (w, S) dBξ +

∫DP (w, S) dDξ,

containing both boundary and field contributionswhere dBξ and dDξ are the surface and volume ele-ments in the computational domain. In general, Mand P will depend on both the flow variables w andthe metrics S defining the computational space. In thecase of a multi-point design the flow variables may beseparately calculated for several different conditions ofinterest.The design problem is now treated as a control prob-

lem where the boundary shape represents the controlfunction, which is chosen to minimize I subject to theconstraints defined by the flow equations (13). A shapechange produces a variation in the flow solution δwand the metrics δS which in turn produce a variationin the cost function

δI =∫BδM(w, S) dBξ +

∫DδP(w, S) dDξ, (15)

with

δM = [Mw]I δw + δMII ,δP = [Pw]I δw + δPII , (16)

where we continue to use the subscripts I and II todistinguish between the contributions associated withthe variation of the flow solution δw and those associ-ated with the metric variations δS. Thus [Mw]I and[Pw]I represent ∂M∂w and ∂P∂w with the metrics 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 the variation of the state vector δw by

∂

∂ξiδ (Fi − Fvi) = 0. (17)

Here δFi and δFvi can also be split into contributionsassociated with δw and δS using the notation

δFi = [Fiw]I δw + δFiIIδFvi = [Fviw]I δw + δFviII . (18)

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The inviscid contributions are easily evaluated as

[Fiw]I = Sij∂fj∂w

, δFiII = δSijfj.

The details of the viscous contributions are compli-cated by the additional level of derivatives in the stressand heat flux terms and will be derived in Section .Multiplying by a co-state vector ψ, which will play ananalogous role to the Lagrange multiplier introducedin equation (4), and integrating over the domain pro-duces ∫

DψT

∂

∂ξiδ (Fi − Fvi) = 0. (19)

If ψ is differentiable this may be integrated by partsto give ∫

Bniψ

T δ (Fi − Fvi) dBξ

−∫D

∂ψT

∂ξiδ (Fi − Fvi) dDξ = 0. (20)

Since the left hand expression equals zero, it may besubtracted from the variation in the cost function (15)to give

δI =∫B

[δM− niψT δ (Fi − Fvi)

]dBξ

+∫D

[δP + ∂ψ

T

∂ξiδ (Fi − Fvi)

]dDξ. (21)

Now, since ψ is an arbitrary differentiable function, itmay be chosen in such a way that δI no longer de-pends explicitly on the variation of the state vectorδw. The gradient of the cost function can then beevaluated directly from the metric variations withouthaving to recompute the variation δw resulting fromthe perturbation of each design variable.Comparing equations (16) and (18), the variation δw

may be eliminated from (21) by equating all field termswith subscript “I” to produce a differential adjointsystem governing ψ

∂ψT

∂ξi[Fiw − Fviw]I + Pw = 0 in D. (22)

The corresponding adjoint boundary condition is pro-duced by equating the subscript “I” boundary termsin equation (21) to produce

niψT [Fiw − Fviw]I = Mw on B. (23)

The remaining terms from equation (21) then yielda simplified expression for the variation of the costfunction which defines the gradient

δI =∫B

{δMII − niψT [δFi − δFvi] II

}dBξ

+∫D

{δPII + ∂ψ

T

∂ξi[δFi − δFvi] II

}dDξ. (24)

The details of the formula for the gradient dependon the way in which the boundary shape is param-eterized as a function of the design variables, and theway in which the mesh is deformed as the boundaryis modified. Using the relationship between the meshdeformation and the surface modification, the field in-tegral is reduced to a surface integral by integratingalong the coordinate lines emanating from the surface.Thus the expression for δI is finally reduced to theform of equation (6)

δI =∫BGδF dBξ

where F represents the design variables, and G is thegradient, which is a function defined over the boundarysurface.The boundary conditions satisfied by the flow equa-

tions restrict the form of the left hand side of theadjoint boundary condition (23). Consequently, theboundary contribution to the cost function M can-not be specified arbitrarily. Instead, it must be chosenfrom the class of functions which allow cancellation ofall terms containing δw in the boundary integral ofequation (21). On the other hand, there is no such re-striction on the specification of the field contributionto the cost function P , since these terms may always beabsorbed into the adjoint field equation (22) as sourceterms.It is convenient to develop the inviscid and vis-

cous contributions to the adjoint equations separately.Also, for simplicity, it will be assumed that the portionof the boundary that undergoes shape modifications isrestricted to the coordinate surface ξ2 = 0. Then equa-tions (21) and (23) may be simplified by incorporatingthe conditions

n1 = n3 = 0, n2 = 1, dBξ = dξ1dξ3,so that only the variations δF2 and δFv2 need to beconsidered at the wall boundary.

Derivation of the Inviscid AdjointTerms

The inviscid contributions have been previously de-rived in12, 22 but are included here for completeness.Taking the transpose of equation (22), the inviscid ad-joint equation may be written as

CTi∂ψ

∂ξi= 0 in D, (25)

where the inviscid Jacobian matrices in the trans-formed space are given by

Ci = Sij∂fj∂w

.

The transformed velocity components have the form

Ui = Sijuj ,

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and the condition that there is no flow through thewall boundary at ξ2 = 0 is equivalent to

U2 = 0,

so thatδU2 = 0

when the boundary shape is modified. Consequentlythe variation of the inviscid flux at the boundary re-duces to

δF2 = δp

0

S21

S22

S23

0

+ p

0

δS21

δS22

δS23

0

. (26)

Since δF2 depends only on the pressure, it is nowclear that the performance measure on the boundaryM(w, S) may only be a function of the pressure andmetric terms. Otherwise, complete cancellation of theterms containing δw in the boundary integral would beimpossible. One may, for example, include arbitrarymeasures of the forces and moments in the cost func-tion, since these are functions of the surface pressure.In order to design a shape which will lead to a de-

sired pressure distribution, a natural choice is to set

I =12

∫B(p− pd)2 dS

where pd is the desired surface pressure, and the inte-gral is evaluated over the actual surface area. In thecomputational domain this is transformed to

I =12

∫ ∫Bw

(p− pd)2 |S2| dξ1dξ3,

where the quantity

|S2| =√S2jS2j

denotes the face area corresponding to a unit elementof face area in the computational domain. Now, tocancel the dependence of the boundary integral on δp,the adjoint boundary condition reduces to

ψjnj = p− pd (27)where nj are the components of the surface normal

nj =S2j|S2| .

This amounts to a transpiration boundary conditionon the co-state variables corresponding to the momen-tum components. Note that it imposes no restrictionon the tangential component of ψ at the boundary.In the presence of shock waves, neither p nor pd are

necessarily continuous at the surface. The boundarycondition is then in conflict with the assumption thatψ is differentiable. This difficulty can be circumventedby the use of a smoothed boundary condition .22

Derivation of the Viscous AdjointTerms

In computational coordinates, the viscous terms inthe Navier–Stokes equations have the form

∂Fvi∂ξi

=∂

∂ξi

(Sijfvj

).

Computing the variation δw resulting from a shapemodification of the boundary, introducing a co-statevector ψ and integrating by parts following the stepsoutlined by equations (17) to (20) produces∫

BψT

(δS2jfvj + S2jδfvj

)dBξ

−∫D

∂ψT

∂ξi

(δSijfvj + Sijδfvj

)dDξ,

where the shape modification is restricted to the coor-dinate surface ξ2 = 0 so that n1 = n3 = 0, and n2 = 1.Furthermore, it is assumed that the boundary contri-butions at the far field may either be neglected or elseeliminated by a proper choice of boundary conditionsas previously shown for the inviscid case.12, 22

The viscous terms will be derived under the assump-tion that the viscosity and heat conduction coefficientsµ and k are essentially independent of the flow, andthat their variations may be neglected. This simplifica-tion has been successfully used for may aerodynamicproblems of interest. In the case of some turbulentflows, there is the possibility that the flow variationscould result in significant changes in the turbulent vis-cosity, and it may then be necessary to account for itsvariation in the calculation.

Transformation to Primitive Variables

The derivation of the viscous adjoint terms is sim-plified by transforming to the primitive variables

w̃T = (ρ, u1, u2, u3, p)T ,

because the viscous stresses depend on the velocityderivatives ∂ui∂xj , while the heat flux can be expressedas

κ∂

∂xi

(p

ρ

).

where κ = kR =γµ

Pr(γ−1) . The relationship betweenthe conservative and primitive variations is defined bythe expressions

δw =Mδw̃, δw̃ =M−1δw

which make use of the transformation matricesM = ∂w∂w̃ andM

−1 = ∂w̃∂w . These matrices are providedin transposed form for future convenience

MT =

1 u1 u2 u3 uiui20 ρ 0 0 ρu10 0 ρ 0 ρu20 0 0 ρ ρu30 0 0 0 1γ−1

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M−1T=

1 −u1ρ −u2ρ −u3ρ (γ−1)uiui20 1ρ 0 0 −(γ − 1)u10 0 1ρ 0 −(γ − 1)u20 0 0 1ρ −(γ − 1)u30 0 0 0 γ − 1

.

The conservative and primitive adjoint operators Land L̃ corresponding to the variations δw and δw̃ arethen related by∫

DδwTLψ dDξ =

∫Dδw̃T L̃ψ dDξ,

withL̃ =MTL,

so that after determining the primitive adjoint op-erator by direct evaluation of the viscous portion of(22), the conservative operator may be obtained bythe transformation L = M−1T L̃. Since the continu-ity equation contains no viscous terms, it makes nocontribution to the viscous adjoint system. Therefore,the derivation proceeds by first examining the adjointoperators arising from the momentum equations.

Contributions from the Momentum Equations

In order to make use of the summation convention,it is convenient to set ψj+1 = φj for j = 1, 2, 3. Thenthe contribution from the momentum equations is∫

Bφk (δS2jσkj + S2jδσkj) dBξ

−∫D

∂φk∂ξi

(δSijσkj + Sijδσkj) dDξ. (28)

The velocity derivatives in the viscous stresses can beexpressed as

∂ui∂xj

=∂ui∂ξl

∂ξl∂xj

=SljJ

∂ui∂ξl

with corresponding variations

δ∂ui∂xj

=[SljJ

]I

∂

∂ξlδui +

[∂ui∂ξl

]II

δ

(SljJ

).

The variations in the stresses are then

δσkj ={µ[SljJ

∂∂ξlδuk + SlkJ

∂∂ξlδuj

]+ λ

[δjk

SlmJ

∂∂ξlδum

]}I

+{µ[δ(SljJ

)∂uk∂ξl

+ δ(SlkJ

) ∂uj∂ξl

]+ λ

[δjkδ

(SlmJ

)∂um∂ξl

]}II.

As before, only those terms with subscript I, whichcontain variations of the flow variables, need be con-sidered further in deriving the adjoint operator. The

field contributions that contain δui in equation (28)appear as

−∫D

∂φk∂ξi

Sij

{µ

(SljJ

∂

∂ξlδuk +

SlkJ

∂

∂ξlδuj

)

+λδjkSlmJ

∂

∂ξlδum

}dDξ.

This may be integrated by parts to yield∫Dδuk

∂

∂ξl

(SljSij

µ

J

∂φk∂ξi

)dDξ

+∫Dδuj

∂

∂ξl

(SlkSij

µ

J

∂φk∂ξi

)dDξ

+∫Dδum

∂

∂ξl

(SlmSij

λδjkJ

∂φk∂ξi

)dDξ,

where the boundary integral has been eliminated bynoting that δui = 0 on the solid boundary. By ex-changing indices, the field integrals may be combinedto produce∫

Dδuk

∂

∂ξlSlj

{µ

(SijJ

∂φk∂ξi

+SikJ

∂φj∂ξi

)

+ λδjkSimJ

∂φm∂ξi

}dDξ,

which is further simplified by transforming the innerderivatives back to Cartesian coordinates∫

Dδuk

∂

∂ξlSlj

{µ

(∂φk∂xj

+∂φj∂xk

)+ λδjk

∂φm∂xm

}dDξ.(29)

The boundary contributions that contain δui inequation (28) may be simplified using the fact that

∂

∂ξlδui = 0 if l = 1, 3

on the boundary B so that they become∫BφkS2j

{µ

(S2jJ

∂

∂ξ2δuk +

S2kJ

∂

∂ξ2δuj

)

+ λδjkS2mJ

∂

∂ξ2δum

}dBξ. (30)

Together, (29) and (30) comprise the field and bound-ary contributions of the momentum equations to theviscous adjoint operator in primitive variables.

Contributions from the Energy Equation

In order to derive the contribution of the energyequation to the viscous adjoint terms it is convenientto set

ψ5 = θ, Qj = uiσij + κ∂

∂xj

(p

ρ

),

where the temperature has been written in terms ofpressure and density using (12). The contribution

7 of 30


from the energy equation can then be written as∫Bθ (δS2jQj + S2jδQj) dBξ

−∫D

∂θ

∂ξi(δSijQj + SijδQj) dDξ. (31)

The field contributions that contain δui,δp, and δρin equation (31) appear as

−∫D

∂θ

∂ξiSijδQjdDξ =

−∫D

∂θ

∂ξiSij

{δukσkj + ukδσkj

+κSljJ

∂

∂ξl

(δp

ρ− pρ

δρ

ρ

)}dDξ. (32)

The term involving δσkj may be integrated by partsto produce∫

Dδuk

∂

∂ξlSlj

{µ

(uk

∂θ

∂xj+ uj

∂θ

∂xk

)

+λδjkum∂θ

∂xm

}dDξ, (33)

where the conditions ui = δui = 0 are used to elimi-nate the boundary integral on B. Notice that the otherterm in (32) that involves δuk need not be integratedby parts and is merely carried on as

−∫DδukσkjSij

∂θ

∂ξidDξ (34)

The terms in expression (32) that involve δp and δρmay also be integrated by parts to produce both a fieldand a boundary integral. The field integral becomes∫

D

(δp

ρ− pρ

δρ

ρ

)∂

∂ξl

(SljSij

κ

J

∂θ

∂ξi

)dDξ

which may be simplified by transforming the innerderivative to Cartesian coordinates∫

D

(δp

ρ− pρ

δρ

ρ

)∂

∂ξl

(Sljκ

∂θ

∂xj

)dDξ. (35)

The boundary integral becomes∫Bκ

(δp

ρ− pρ

δρ

ρ

)S2jSijJ

∂θ

∂ξidBξ. (36)

This can be simplified by transforming the innerderivative to Cartesian coordinates∫

Bκ

(δp

ρ− pρ

δρ

ρ

)S2jJ

∂θ

∂xjdBξ, (37)

and identifying the normal derivative at the wall

∂

∂n= S2j

∂

∂xj, (38)

and the variation in temperature

δT =1R

(δp

ρ− pρ

δρ

ρ

),

to produce the boundary contribution∫BkδT

∂θ

∂ndBξ. (39)

This term vanishes if T is constant on the wall butpersists if the wall is adiabatic.There is also a boundary contribution left over from

the first integration by parts (31) which has the form∫Bθδ (S2jQj) dBξ, (40)

where

Qj = k∂T

∂xj,

since ui = 0. Notice that for future convenience in dis-cussing the adjoint boundary conditions resulting fromthe energy equation, both the δw and δS terms corre-sponding to subscript classes I and II are consideredsimultaneously. If the wall is adiabatic

∂T

∂n= 0,

so that using (38),

δ (S2jQj) = 0,

and both the δw and δS boundary contributions van-ish.On the other hand, if T is constant ∂T∂ξl = 0 for l =

1, 3, so that

Qj = k∂T

∂xj= k

(Slj

J

∂T

∂ξl

)= k

(S2jJ

∂T

∂ξ2

).

Thus, the boundary integral (40) becomes

∫Bkθ

{S2j

2

J

∂

∂ξ2δT + δ

(S2j

2

J

)∂T

∂ξ2

}dBξ . (41)

Therefore, for constant T , the first term correspondingto variations in the flow field contributes to the adjointboundary operator and the second set of terms corre-sponding to metric variations contribute to the costfunction gradient.All together, the contributions from the energy

equation to the viscous adjoint operator are the threefield terms (33), (34) and (35), and either of twoboundary contributions ( 39) or ( 41), depending onwhether the wall is adiabatic or has constant temper-ature.

8 of 30


The Viscous Adjoint Field Operator

Collecting together the contributions from the mo-mentum and energy equations, the viscous adjointoperator in primitive variables can be expressed as

(L̃ψ)1 = − pρ2

∂

∂ξl

(Sljκ

∂θ

∂xj

)

(L̃ψ)i+1 =∂

∂ξl

{Slj

[µ

(∂φi∂xj

+∂φj∂xi

)+ λδij

∂φk∂xk

]}i = 1, 2, 3

+∂

∂ξl

{Slj

[µ

(ui∂θ

∂xj+ uj

∂θ

∂xi

)+ λδijuk

∂θ

∂xk

]}

− σijSlj ∂θ∂ξl

for i = 1, 2, 3

(L̃ψ)5 =1ρ

∂

∂ξl

(Sljκ

∂θ

∂xj

).

The conservative viscous adjoint operator may now beobtained by the transformation

L =M−1TL̃.

Viscous Adjoint Boundary ConditionsIt was recognized in Section that the boundary

conditions satisfied by the flow equations restrict theform of the performance measure that may be cho-sen for the cost function. There must be a directcorrespondence between the flow variables for whichvariations appear in the variation of the cost function,and those variables for which variations appear in theboundary terms arising during the derivation of theadjoint field equations. Otherwise it would be impos-sible to eliminate the dependence of δI on δw throughproper specification of the adjoint boundary condition.As in the derivation of the field equations, it provesconvenient to consider the contributions from the mo-mentum equations and the energy equation separately.

Boundary Conditions Arising from theMomentum Equations

The boundary term that arises from the momentumequations including both the δw and δS components(28) takes the form∫

Bφkδ (S2jσkj) dBξ.

Replacing the metric term with the corresponding lo-cal face area S2 and unit normal nj defined by

|S2| =√S2jS2j , nj =

S2j|S2|

then leads to ∫Bφkδ (|S2|njσkj) dBξ.

Defining the components of the surface stress as

τk = njσkj

and the physical surface element

dS = |S2| dBξ,

the integral may then be split into two components∫Bφkτk |δS2| dBξ +

∫BφkδτkdS, (42)

where only the second term contains variations in theflow variables and must consequently cancel the δwterms arising in the cost function. The first term willappear in the expression for the gradient.A general expression for the cost function that al-

lows cancellation with terms containing δτk has theform

I =∫BN (τ)dS, (43)

corresponding to a variation

δI =∫B

∂N∂τk

δτkdS,

for which cancellation is achieved by the adjointboundary condition

φk =∂N∂τk

.

Natural choices for N arise from force optimizationand as measures of the deviation of the surface stressesfrom desired target values.For viscous force optimization, the cost function

should measure friction drag. The friction force in thexi direction is

CDfi =∫BσijdSj =

∫BS2jσijdBξ

so that the force in a direction with cosines ni has theform

Cnf =∫BniS2jσijdBξ.

Expressed in terms of the surface stress τi, this corre-sponds to

Cnf =∫BniτidS,

so that basing the cost function (43) on this quantitygives

N = niτi.Cancellation with the flow variation terms in equation(42) therefore mandates the adjoint boundary condi-tion

φk = nk.

9 of 30


Note that this choice of boundary condition also elim-inates the first term in equation (42) so that it neednot be included in the gradient calculation.In the inverse design case, where the cost function

is intended to measure the deviation of the surfacestresses from some desired target values, a suitabledefinition is

N (τ) = 12alk (τl − τdl) (τk − τdk) ,

where τd is the desired surface stress, including thecontribution of the pressure, and the coefficients alkdefine a weighting matrix. For cancellation

φkδτk = alk (τl − τdl) δτk.

This is satisfied by the boundary condition

φk = alk (τl − τdl) . (44)Assuming arbitrary variations in δτk, this condition isalso necessary.In order to control the surface pressure and normal

stress one can measure the difference

nj {σkj + δkj (p− pd)} ,where pd is the desired pressure. The normal compo-nent is then

τn = nknjσkj + p− pd,so that the measure becomes

N (τ) = 12τ2n

=12nlnmnknj {σlm + δlm (p− pd)}

· {σkj + δkj (p− pd)} .

This corresponds to setting

alk = nlnk

in equation (44). Defining the viscous normal stress as

τvn = nknjσkj ,

the measure can be expanded as

N (τ) = 12nlnmnknjσlmσkj

+12(nknjσkj + nlnmσlm) (p− pd) + 12 (p− pd)

2

=12τ2vn + τvn (p− pd) +

12(p− pd)2 .

For cancellation of the boundary terms

φk (njδσkj + nkδp) ={nlnmσlm + n2l (p− pd)

}nk (njδσkj + nkδp)

leading to the boundary condition

φk = nk (τvn + p− pd) .

In the case of high Reynolds number, this is well ap-proximated by the equations

φk = nk (p− pd) , (45)

which should be compared with the single scalar equa-tion derived for the inviscid boundary condition (27).In the case of an inviscid flow, choosing

N (τ) = 12(p− pd)2

requires

φknkδp = (p− pd)n2kδp = (p− pd) δp

which is satisfied by equation (45), but which repre-sents an overspecification of the boundary conditionsince only the single condition (27) need be specifiedto ensure cancellation.

Boundary Conditions Arising from the EnergyEquation

The form of the boundary terms arising from theenergy equation depends on the choice of temperatureboundary condition at the wall. For the adiabatic case,the boundary contribution is (39)∫

BkδT

∂θ

∂ndBξ,

while for the constant temperature case the boundaryterm is (41). One possibility is to introduce a con-tribution into the cost function which depends on Tor ∂T∂n so that the appropriate cancellation would oc-cur. Since there is little physical intuition to guide thechoice of such a cost function for aerodynamic design,a more natural solution is to set

θ = 0

in the constant temperature case or

∂θ

∂n= 0

in the adiabatic case. Note that in the constant tem-perature case, this choice of θ on the boundary wouldalso eliminate the boundary metric variation terms in(40).

Implementation of Navier-StokesDesign

The design procedures can be summarized as fol-lows:

1. Solve the flow equations for ρ, u1, u2, u3, p.

10 of 30


2. Solve the adjoint equations for ψ subject to ap-propriate boundary conditions.

3. Evaluate G .4. Project G into an allowable subspace that satisfies

any geometric constraints.

5. Update the shape based on the direction of steep-est descent.

6. Return to 1 until convergence is reached.

Practical implementation of the viscous design methodrelies heavily upon fast and accurate solvers for boththe state (w) and co-state (ψ) systems. This work useswell-validated software for the solution of the Eulerand Navier-Stokes equations developed over the courseof many years.23–25

For inverse design the lift is fixed by the target pres-sure. In drag minimization it is also appropriate tofix the lift coefficient, because the induced drag is amajor fraction of the total drag, and this could bereduced simply by reducing the lift. Therefore theangle of attack is adjusted during the flow solutionto force a specified lift coefficient to be attained, andthe influence of variations of the angle of attack is in-cluded in the calculation of the gradient. The vortexdrag also depends on the span loading, which may beconstrained by other considerations such as structuralloading or buffet onset. Consequently, the option isprovided to force the span loading by adjusting thetwist distribution as well as the angle of attack duringthe flow solution.

Discretization

Both the flow and the adjoint equations are dis-cretized using a semi-discrete cell-centered finite vol-ume scheme. The convective fluxes across cell inter-faces are represented by simple arithmetic averages ofthe fluxes computed using values from the cells on ei-ther side of the face, augmented by artificial diffusiveterms to prevent numerical oscillations in the vicinityof shock waves. Continuing to use the summation con-vention for repeated indices, the numerical convectiveflux across the interface between cells A and B in athree dimensional mesh has the form

hAB =12SABj

(fAj + fBj

)− dAB,where SABj is the component of the face area in thejth Cartesian coordinate direction,

(fAj

)and

(fBj

)denote the flux fj as defined by equation (12) anddAB is the diffusive term. Variations of the computerprogram provide options for alternate constructions ofthe diffusive flux.The simplest option implements the Jameson-

Schmidt-Turkel scheme,23, 26 using scalar diffusiveterms of the form

dAB = 1(2)∆w − 1(4)(∆w+ − 2∆w +∆w−) ,

where∆w = wB − wA

and ∆w+ and ∆w− are the same differences across theadjacent cell interfaces behind cell A and beyond cellB in the AB direction. By making the coefficient 1(2)

depend on a switch proportional to the undivided sec-ond difference of a flow quantity such as the pressureor entropy, the diffusive flux becomes a third orderquantity, proportional to the cube of the mesh widthin regions where the solution is smooth. Oscillationsare suppressed near a shock wave because 1(2) becomesof order unity, while 1(4) is reduced to zero by the sameswitch. For a scalar conservation law, it is shown inreference26 that 1(2) and 1(4) can be constructed tomake the scheme satisfy the local extremum diminish-ing (LED) principle that local maxima cannot increasewhile local minima cannot decrease.The second option applies the same construction to

local characteristic variables. There are derived fromthe eigenvectors of the Jacobian matrix AAB whichexactly satisfies the relation

AAB (wB − wA) = SABj(fBj − fAj

).

This corresponds to the definition of Roe .27 Theresulting scheme is LED in the characteristic vari-ables. The third option implements the H-CUSPscheme proposed by Jameson28 which combines dif-ferences fB − fA and wB −wA in a manner such thatstationary shock waves can be captured with a singleinterior point in the discrete solution. This schememinimizes the numerical diffusion as the velocity ap-proaches zero in the boundary layer, and has thereforebeen preferred for viscous calculations in this work.Similar artificial diffusive terms are introduced in

the discretization of the adjoint equation, but with theopposite sign because the wave directions are reversedin the adjoint equation. Satisfactory results have beenobtained using scalar diffusion in the adjoint equationwhile characteristic or H-CUSP constructions are usedin the flow solution.The discretization of the viscous terms of the Navier

Stokes equations requires the evaluation of the veloc-ity derivatives ∂ui∂xj in order to calculate the viscous

stress tensor σij defined in equation (11). These aremost conveniently evaluated at the cell vertices of theprimary mesh by introducing a dual mesh which con-nects the cell centers of the primary mesh, as depictedin Figure (1). According to the Gauss formula for acontrol volume V with boundary S∫

V

∂vi∂xj

dv =∫S

uinjdS

where nj is the outward normal. Applied to the dualcells this yields the estimate

∂vi∂xj

=1vol

∑faces

ūinjS

11 of 30


i jσ

dual cell

Fig. 1 Cell-centered scheme. σij evaluated at ver-tices of the primary mesh

where S is the area of a face, and ūi is an estimateof the average of ui over that face. In order to deter-mine the viscous flux balance of each primary cell, theviscous flux across each of its faces is then calculatedfrom the average of the viscous stress tensor at thefour vertices connected by that face. This leads to acompact scheme with a stencil connecting each cell toits 26 nearest neighbors.The semi-discrete schemes for both the flow and the

adjoint equations are both advanced to steady state bya multi-stage time stepping scheme. This is a general-ized Runge-Kutta scheme in which the convective anddiffusive terms are treated differently to enlarge thestability region.26, 29 Convergence to a steady state isaccelerated by residual averaging and a multi-grid pro-cedure.30 These algorithms have been implementedboth for single and multiblock meshes and for opera-tion on parallel computers with message passing usingthe MPI (Message Passing Interface) protocol.8, 31, 32

In this work, the adjoint and flow equations arediscretized separately. The alternative approach ofderiving the discrete adjoint equations directly fromthe discrete flow equations yields another possible dis-cretization of the adjoint partial differential equationwhich is more complex. If the resulting equations weresolved exactly, they could provide the exact gradientof the inexact cost function which results from the dis-cretization of the flow equations. On the other hand,any consistent discretization of the adjoint partial dif-ferential equation will yield the exact gradient as themesh is refined, and separate discretization has provedto work perfectly well in practice. It should also benoted that the discrete gradient includes both mesheffects and numerical errors such as spurious entropyproduction which may not reflect the true cost func-tion of the continuous problem.

Mesh Generation and Geometry Control

Meshes for both viscous optimization and for thetreatment of complex configurations are externallygenerated in order to allow for their inspection andcareful quality control. Single block meshes with a C-H topology have been used for viscous optimizationof wing-body combinations, while multiblock mesheshave been generated for complex configurations usingGRIDGEN.33 In either case geometry modificationsare accommodated by a grid perturbation scheme. Forviscous wing-body design using single block meshes,the wing surface mesh points themselves are takenas the design variables. A simple mesh perturbationscheme is then used, in which the mesh points lying ona mesh line projecting out from the wing surface areall shifted in the same sense as the surface mesh point,with a decay factor proportional to the arc lengthalong the mesh line. The resulting perturbation in theface areas of the neighboring cells are then included inthe gradient calculation. For complex configurationsthe geometry is controlled by superposition of analytic“bump” functions defined over the surfaces which areto be modified. The grid is then perturbed to conformto modifications of the surface shape by the WARP3Dand WARP-MB algorithms described in.31

Optimization

Two main search procedures have been used in ourapplications to date. The first is a simple descentmethod in which small steps are taken in the nega-tive gradient direction. Let F represent the designvariable, and G the gradient. Then the iteration

δF = −λG

can be regarded as simulating the time dependent pro-cess

dFdt

= −G

where λ is the time step ∆t. Let A be the Hessianmatrix with elements

Aij =∂Gi∂Fj =

∂2I

∂Fi∂Fj .

Suppose that a locally minimum value of the cost func-tion I∗ = I(F∗) is attained when F = F∗. Then thegradient G∗ = G(F∗) must be zero, while the Hessianmatrix A∗ = A(F∗) must be positive definite. SinceG∗ is zero, the cost function can be expanded as aTaylor series in the neighborhood of F∗ with the form

I(F) = I∗ + 12(F − F∗)A (F − F∗) + . . .

Correspondingly,

G(F) = A (F − F∗) + . . .12 of 30


As F approaches F∗, the leading terms become domi-nant. Then, setting F̂ = (F −F∗), the search processapproximates

dF̂dt

= −A∗F̂ .Also, since A∗ is positive definite it can be expandedas

A∗ = RMRT ,

whereM is a diagonal matrix containing the eigenval-ues of A∗, and

RRT = RTR = I.

Settingv = RT F̂ ,

the search process can be represented as

dv

dt= −Mv.

The stability region for the simple forward Euler step-ping scheme is a unit circle centered at −1 on thenegative real axis. Thus for stability we must choose

µmax∆t = µmaxλ < 2,

while the asymptotic decay rate, given by the smallesteigenvalue, is proportional to

e−µmint.

In order to make sure that each new shape in theoptimization sequence remains smooth, it proves es-sential to smooth the gradient and to replace G by itssmoothed value Ḡ in the descent process. This alsoacts as a preconditioner which allows the use of muchlarger steps. To apply smoothing in the ξ1 direction,for example, the smoothed gradient Ḡ ma be calculatedfrom a discrete approximation to

Ḡ − ∂∂ξ1

1∂

∂ξ1Ḡ = G

where 1 is the smoothing parameter. If one sets δF =−λḠ, then, assuming the modification is applied onthe surface ξ2 = constant, the first order change in thecost function is

δI = −∫ ∫

GδF dξ1dξ3

= −λ∫ ∫ (

Ḡ − ∂∂ξ1

1∂Ḡ∂ξ1

)Ḡ dξ1dξ3

= −λ∫ ∫ (

Ḡ2 + 1(∂Ḡ∂ξ1

)2)dξ1dξ3

< 0,

assuring an improvement if λ is sufficiently small andpositive, unless the process has already reached a sta-tionary point at which G = 0.

It turns out that this approach is tolerant to theuse of approximate values of the gradient, so that nei-ther the flow solution nor the adjoint solution need befully converged before making a shape change. Thisresults in very large savings in the computational cost.For inviscid optimization it is necessary to use only 15multigrid cycles for the flow solution and the adjointsolution in each design iteration. For viscous opti-mization, about 100 multigrid cycles are needed. Thisis partly because convergence of the lift coefficient ismuch slower, so about 20 iterations must be made be-fore each adjustment of the angle of attack to force thetarget lift coefficient.Our second main search procedure incorporates a

quasi-Newton method for general constrained opti-mization. In this class of methods the step is definedas

δF = −λPG,where P is a preconditioner for the search. An idealchoice is P = A∗−1, so that the corresponding timedependent process reduces to

dF̂dt

= −F̂ ,

for which all the eigenvalues are equal to unity, andF̂ is reduced to zero in one time step by the choice∆t = 1 if the Hessian, A, is constant. The full Newtonmethod takes P = A−1, requiring the evaluation ofthe Hessian matrix, A, at each step. It corresponds tothe use of the Newton-Raphson method to solve thenon-linear equation G = 0. Quasi-Newton methodsestimate A∗ from the change in the gradient duringthe search process. This requires accurate estimatesof the gradient at each time step. In order to obtainthese, both the flow solution and the adjoint equationmust be fully converged. Most quasi-Newton methodsalso require a line search in each search direction, forwhich the flow equations and cost function must beaccurately evaluated several times. They have provenquite robust for aerodynamic optimization.34

In the applications to complex configurations pre-sented below the optimization was carried out usingthe existing, well validated software NPSOL. Thissoftware, which implements a quasi-Newton methodfor optimization with both linear and non-linear con-straints, has proved very reliable but is generally moreexpensive than the simple search method with smooth-ing.

Industrial Experience and ResultsThe methods described in this paper have been quite

thoroughly tested in industrial applications in whichthey were used as a tool for aerodynamic design. Theyhave proved useful both in inverse mode to find shapesthat would produce desired pressure distributions, andfor direct minimization of the drag. They have been

13 of 30


applied both to well understood configurations thathave gradually evolved through incremental improve-ments guided by wind tunnel tests and computationalsimulation, and to new concepts for which there is alimited knowledge base. In either case they have en-abled engineers to produce improved designs.Substantial improvements are usually obtained with

20 − 200 design cycles, depending on the difficulty ofthe case. One concern is the possibility of gettingtrapped in a local minimum. In practice this has notproved to be a source of difficulty. In inverse mode,it often proves possible to come very close to realiz-ing the target pressure distribution, thus effectivelydemonstrating convergence. In drag minimization, theresult of the optimization is usually a shock-free wing.If one considers drag minimization of airfoils in two-dimensional inviscid transonic flow, it can be seen thatevery shock-free airfoil produces zero drag, and thusoptimization based solely on drag has a highly non-unique solution. Different shock-free airfoils can beobtained by starting from different initial profiles. Onemay also influence the character of the final design byblending a target pressure distribution with the dragin the definition of the cost function.Similar considerations apply to three-dimensional

wing design in viscous transonic flow. Since the vor-tex drag can be reduced simply by reducing the lift,the lift coefficient must be fixed for a meaningful dragminimization. A typical wing of a transport aircraftis designed for a lift coefficient in the range of 0.4 to0.6. The total wing drag may be broken down intovortex drag, drag due to viscous effects, and shockdrag. The vortex drag coefficient is typically in therange of 0.0100 (100 counts), while the friction dragcoefficient is in the range of 45 counts, and the shockdrag at a Mach number just before the onset of se-vere drag rise is of the order of 15 counts. With afixed span, typically dictated by structural limits or aconstraint imposed by airport gates, the vortex drag isentirely a function of span loading, and is minimized byan elliptic loading unless winglets are added. Trans-port aircraft usually have highly tapered wings withvery large root chords to accommodate retraction ofthe undercarriage. An elliptic loading may lead to ex-cessively large section lift coefficients on the outboardwing, leading to premature shock stall or buffet whenthe load is increased. The structure weight is also re-duced by a more inboard loading which reduces theroot bending moment. Thus the choice of span load-ing is influenced by other considerations. The skinfriction of transport aircraft is typically very close toflat plate skin friction in turbulent flow, and is veryinsensitive to section variations. An exception to thisis the case of smaller executive jet aircraft, for whichthe Reynolds number may be small enough to allow asignificant run of laminar flow if the suction peak ofthe pressure distribution is moved back on the section.

This leaves the shock drag as the primary target forwing section optimization. This is reduced to zero ifthe wing is shock-free, leaving no room for further im-provement. Thus the attainment of a shock-free flow isa demonstration of a successful drag minimization. Inpractice range is maximized by maximizing M LD , andthis is likely to be increased by increasing the lift coef-ficient to the point where a weak shock appears. Onemay also use optimization to find the maximum Machnumber at which the shock drag can be eliminated orsignificantly reduced for a wing with a given sweepbackangle and thickness. Alternatively one may try to findthe largest wing thickness or the minimum sweepbackangle for which the shock drag can be eliminated ata given Mach number. This can yield both savings instructure weight and increased fuel volume . If thereis no fixed limit for the wing span, such as a gate con-straint, increased thickness can be used to allow anincrease in aspect ratio for a wing of equal weight, inturn leading to a reduction in vortex drag. Since thevortex drag is usually the largest component of the to-tal wing drag, this is probably the most effective designstrategy, and it may pay to increase the wing thicknessto the point where the optimized section produces aweak shock wave rather than a shock-free flow.22

The first major industrial application of an adjointbased aerodynamic optimization method was the wingdesign of the Beech Premier35 in 1995. The methodwas successfully used in inverse mode as a tool toobtain pressure distributions favorable to the main-tenance of natural laminar flow over a range of cruiseMach numbers. Wing contours were obtained whichyielded the desired pressure distribution in the pres-ence of closely coupled engine nacelles on the fuselageabove the wing trailing edge.During 1996 some preliminary studies indicated that

the wings of both the McDonnell Douglas MD-11 andthe Boeing 747-200 could be made shock-free in a rep-resentative cruise condition by using very small shapemodifications, with consequent drag savings whichcould amount to several percent of the total drag.This led to a decision to evaluate adjoint-based de-sign methods in the design of the McDonnell DouglasMDXX during the summer and fall of 1996. In ini-tial studies wing redesigns were carried out for inviscidtransonic flow modelled by the Euler equations. A re-design to minimize the drag at a specified lift andMachnumber required about 40 design cycles, which couldbe completed overnight on a workstation.Three main lessons were drawn from these initial

studies: (i) the fuselage effect is to large to be ignoredand must be included in the optimization, (ii) single-point designs could be too sensitive to small variationsin the flight condition, typically producing a shock-freeflow at the design point with a tendency to break upinto a severe double shock pattern below the designpoint, and (iii) the shape changes necessary to opti-

14 of 30


mize a wing in transonic flow are smaller than theboundary layer displacement thickness, with the con-sequence that viscous effects must be included in thefinal design.In order to meet the first two of these considerations,

the second phase of the study was concentrated on theoptimization of wing-body combinations with multipledesign points. These were still performed with invis-cid flow to reduce computational cost and allow forfast turnaround. It was found that comparatively in-sensitive designs could be obtained by minimizing thedrag at a fixed Mach number for three fairly closelyspaced lift coefficients such as 0.5, 0.525, and 0.55, oralternatively three nearby Mach numbers with a fixedlift coefficient.The third phase of the project was focused on the de-

sign with viscous effects using as a starting point wingswhich resulted from multipoint inviscid optimization.While the full viscous adjoint method was still underdevelopment, it was found that useful improvementscould be realized, particularly in inverse mode, usingthe inviscid result to provide the target pressure, bycoupling an inviscid adjoint solver to a viscous flowsolver. Computer costs are many times larger, bothbecause finer meshes are needed to resolve the bound-ary layer, and because more iterations are needed inthe flow and adjoint solutions. In order to force thespecified lift coefficient the number of iterations in eachflow solution had to be increased from 15 to 100. Toachieve overnight turnaround a fully parallel imple-mentation of the software had to be developed. Finallyit was found that in order to produce sufficiently ac-curate results, the number of mesh points had to beincreased to about 1.8 million. In the final phase ofthis project it was planned to carry out a propulsionintegration study using the multiblock versions of thesoftware. This study was not completed due to thecancellation of the entire MDXX project.During the summer of 1997, adjoint methods were

again used to assist the McDonnell Douglas BlendedWing-Body project. By this time the viscous adjointmethod was well developed, and it was found that itwas needed to achieve truly smooth shock-free solu-tions. With an inviscid adjoint solver coupled to aviscous flow solver some improvements could be made,but the shocks could not be entirely eliminated.The next subsection shows a wing design using the

full viscous adjoint method in its current form, im-plemented in the computer program SYN107. Theremaining subsections present results of optimizationsfor complete configurations in inviscid transonic andsupersonic flow using the multiblock parallel designprogram, SYN107-MB.

Transonic Viscous Wing-Body Design

A typical result of drag minimization in transonicviscous flow is presented below. This calculation is

a redesign of a wing using the viscous adjoint opti-mization method with a Baldwin-Lomax turbulencemodel. The initial wing is similar to one produced dur-ing the MDXX design studies. Figures 2-4 show theresult of the wing-body redesign on a C-H mesh with288× 96× 64 cells. The wing has sweep back of about38 degrees at the 1/4 chord. A total of 44 iterations ofthe viscous optimization procedure resulted in a shock-free wing at a cruise design point of Mach 0.86, witha lift coefficient of 0.61 for the wing-body combinationat a Reynolds number of 101 million based on the rootchord. Using 48 processors of an SGI Origin2000 par-allel computer, each design iteration takes about 22minutes so that overnight turnaround for such a cal-culation is possible. Figure 2 compares the pressuredistribution of the final design with that of the initialwing. The final wing is quite thick, with a thickness tochord ratio of about 14 percent at the root and 9 per-cent at the tip. The optimization was performed witha constraint that the section modifications were notallowed to decrease the thickness anywhere. The de-sign offers excellent performance at the nominal cruisepoint. A drag reduction of 2.2 counts was achievedfrom the initial wing which had itself been derived byinviscid optimization. Figures 3 and 4 show the re-sults of a Mach number sweep to determine the dragrise. The drag coefficients shown in the figures repre-sent the total wing drag including shock, vortex, andskin friction contributions. It can be seen that a dou-ble shock pattern forms below the design point, whilethere is actually a slight increase in the drag coefficientat Mach 0.85. The tendency to produce double shocksbelow the design point is typical of supercritical wings.This wing has a low drag coefficient, however, over awide range of conditions. Above the design point a sin-gle shock forms and strengthens as the Mach numberincreases, a behavior typical in transonic flow.

Transonic Multipoint ConstrainedAircraft Design

As a first example of the automatic design capabilityfor complex configurations, a typical business jet con-figuration is chosen for a multipoint drag minimizationrun. The objective of the design is to alter the geome-try of the wing in order to minimize the configurationinviscid drag at three different flight conditions si-multaneously. Realistic geometric spar thickness con-straints are enforced. The geometry chosen for thisanalysis is a full configuration business jet composedof wing, fuselage, pylon, nacelle, and empennage. Theinviscid multiblock mesh around this configuration fol-lows a general C-O topology with special blocking tocapture the geometric details of the nacelles, pylonsand empennage. A total of 240 point-to-point matchedblocks with 4, 157, 440 cells (including halos) are usedto grid the complete configuration. This mesh allowsthe use of 4 multigrid levels obtained through recur-

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sive coarsening of the initial fine mesh. The upstream,downstream, upper and lower far field boundaries arelocated at an approximate distance of 15 wing semis-pans, while the far field boundary beyond the wingtip is located at a distance approximately equal to 5semispans. An engineering-accuracy solution (with adecrease of 4 orders of magnitude in the average den-sity residual) can be obtained in 100 multigrid cycles.This kind of solution can be routinely accomplished inunder 20 minutes of wall clock time using 32 processorsof an SGI Origin2000 computer.The initial configuration was designed for Mach =

0.8 and CL = 0.3. The three operating points cho-sen for this design are Mach = 0.81 with CL = 0.35,Mach = 0.82 with CL = 0.30, and Mach = 0.83 withCL = 0.25. For each of the design points, both Machnumber and lift coefficient are held fixed. In order todemonstrate the advantage of a multipoint design ap-proach, the final solution at the middle design pointwill be compared with a single point design at the sameconditions. As the geometry of the wing is modified,the design algorithm computes new wing-fuselage in-tersections. The wing component is made up of sixairfoil defining sections. Eighteen Hicks-Henne designvariables are applied to five of these sections for a to-tal of 90 design variables. The sixth section at thesymmetry plane is not modified. Spar thickness con-straints were also enforced on each defining station atthe x/c = 0.2 and x/c = 0.8 locations. Maximumthickness was forced to be preserved at x/c = 0.4 for allsix defining sections. To ensure an adequate includedangle at the trailing edge, each section was also con-strained to preserve thickness at x/c = 0.95. Finally,to preserve leading edge bluntness, the upper surfaceof each section was forced to maintain its height abovethe camber line at x/c = 0.02. Combined, a total of30 linear geometric constraints were imposed on theconfiguration.Figures 5 – 7 show the initial and final airfoil ge-

ometries and Cp distributions after 5 NPSOL designiterations. It is evident that the new design has signif-icantly reduced the shock strengths on both upper andlower wing surfaces at all design points. The transi-tions between design points are also quite smooth. Forcomparison purposes, a single point drag minimizationstudy (Mach = 0.81 and CL = 0.25) is carried outstarting from the same initial configuration and usingthe same design variables and geometric constraints.Figures 8 – 10 show comparisons of the solutions

from the three-point design with those of the singlepoint design. Interestingly, the upper surface shapesfor both final designs are very similar. However, in thecase of the single point design, a strong lower surfaceshock appears at the Mach = 0.83, CL = 0.25 designpoint. The three-point design is able to suppress theformation of this lower surface shock and achieves a 9count drag benefit over the single point design at this

condition. However, it has a 1 count penalty at thesingle point design condition. The three-point designfeatures a weak single shock for one of the three designpoints and a very weak double shock at another designpoint. Table 1 summarizes the drag results for the twodesigns. The CD values have been normalized by thedrag of the initial configuration at the second designpoint. Figure 11 shows the surface of the configura-tion colored by the local coefficient of pressure, Cp,before and after redesign for the middle design point.One can clearly observe that the strength of the shockwave on the upper surface of the configuration hasbeen considerably reduced.Finally, Figure 12 shows the parallel scalability of

the multiblock design method for the mesh in ques-tion using up to 32 processors of an SGI Origin2000parallel computer. Despite the fact that the multigridtechnique is used in both the flow and adjoint solvers,the demonstrated parallel speedups are outstanding.

Supersonic Constrained Aircraft Design

For supersonic design, provided that turbulent flowis assumed over the entire configuration, the inviscidEuler equations suffice for aerodynamic design sincethe pressure drag is not greatly affected by the in-clusion of viscous effects. Moreover, flat plate skinfriction estimates of viscous drag are often very goodapproximations. In this study, the generic supersonictransport configuration used in reference36 is revisited.The baseline supersonic transport configuration was

sized to accommodate 300 passengers with a grosstake-off weight of 750, 000 lbs. The supersonic cruisepoint is Mach 2.2 with a CL of 0.105. Figure 13 showsthat the planform is a cranked-delta configuration witha break in the leading edge sweep. The inboard lead-ing edge sweep is 68.5 degrees while the outboard is49.5 degrees. Since the Mach angle at M = 2.2 is 63degrees it is clear that some leading edge bluntnessmay be used inboard without a significant wave dragpenalty. Blunt leading edge airfoils were created withthickness ranging from 4% at the root to 2.5% at theleading edge break point. These symmetric airfoilswere chosen to accommodate thick spars at roughlythe 5% and 80% chord locations over the span up tothe leading edge break. Outboard of the leading edgebreak where the wing sweep is ahead of the Mach cone,a sharp leading edge was used to avoid unnecessarywave drag. The airfoils were chosen to be symmetric,biconvex shapes modified to have a region of constantthickness over the mid-chord. The four-engine con-figuration features axisymmetric nacelles tucked closeto the wing lower surface. This layout favors reducedwave drag by minimizing the exposed boundary layerdiverter area. However, in practice it may be prob-lematic because of the channel flows occurring in thejuncture region of the diverter, wing, and nacelle atthe wing trailing edge.

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The computational mesh on which the design is runhas 180 blocks and 1, 500, 000 mesh cells (includinghalos), while the underlying geometry entities definethe wing with 16 sectional cuts and the body with200 sectional cuts. In this case, where we hope tooptimize the shape of the wing, care must be takento ensure that the nacelles remain properly attachedwith diverter heights being maintained.The objective of the design is to reduce the to-

tal drag of the configuration at a single design point(Mach = 2.2, CL = 0.105) by modifying the wingshape. Just as in the transonic case, 18 design vari-ables of the Hicks-Henne type are chosen for each wingdefining section. Similarly, instead of applying them toall 16 sections, they are applied to 8 of the sections andthen lofted linearly to the neighboring sections. Sparthickness constraints are imposed for all wing definingsections at x/c = 0.05 and x/c = 0.8. An additionalmaximum thickness constraint is specified along thespan at x/c = 0.5. A final thickness constraint is en-forced at x/c = 0.95 to ensure a reasonable trailingedge included angle. An iso-Cp representation of theinitial and final designs is depicted in Figure 13 forboth the upper and lower surfaces.It is noted that the strong oblique shock evident near

the leading edge of the upper surface on the initial con-figuration is largely eliminated in the final design after5 NPSOL design iterations. Also, it is seen that theupper surface pressure distribution in the vicinity ofthe nacelles has formed an unexpected pattern. How-ever, recalling that thickness constraints abound inthis design, these upper surface pressure patterns areassumed to be the result of sculpting of the lower sur-face near the nacelles which affects the upper surfaceshape via the thickness constraints. For the lower sur-face, the leading edge has developed a suction regionwhile the shocks and expansions around the nacelleshave been somewhat reduced. Figure 14 shows thepressure coefficients and (scaled) airfoil sections forfour sectional cuts along the wing. These cuts fur-ther demonstrate the removal of the oblique shock onthe upper surface and the addition of a suction regionon the leading edge of the lower surface. The airfoilsections have been scaled by a factor of 2 so that shapechanges may be seen more easily. Most notably, thesection at 38.7% span has had the lower surface drasti-cally modified such that a large region of the aft airfoilhas a forward-facing portion near where the pressurespike from the nacelle shock impinges on the surface.The final overall pressure drag was reduced by 8%,from CD = 0.0088 to CD = 0.0081.

ConclusionsWe have developed a three-dimensional control the-

ory based design method for the Navier Stokes equa-tions and applied it successfully to the design of wingsin transonic flow. The method represents an extension

of our previous work on design with the potential flowand Euler equations. The new method combines theversatility of numerical optimization methods with theefficiency of inverse design. The geometry is modifiedby a grid perturbation technique which is applicableto arbitrary configurations. Both the wing-body andmultiblock version of the design algorithms have beenimplemented in parallel using the MPI (Message Pass-ing Interface) Standard, and they both yield excellentparallel speedups. The combination of computationalefficiency with geometric flexibility provides a powerfultool, with the final goal being to create practical aero-dynamic shape design methods for complete aircraftconfigurations.

AcknowledgmentThis work has benefited from the generous support

of AFOSR under Grant No. AFOSR-91-0391, theNASA-IBM Cooperative Research Agreement, and theDoD under the Grand Challenge Projects of the HighPerformance Computing Modernization Program.

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14J. Reuther, A. Jameson, J. J. Alonso, M. J. Rimlinger,and D. Saunders. Constrained multipoint aerodynamic shapeoptimization using an adjoint formulation and parallel comput-ers. AIAA paper 97-0103, 35th Aerospace Sciences Meeting andExhibit, Reno, Nevada, January 1997.

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30A. Jameson. Multigrid algorithms for compressible flowcalculations. In W. Hackbusch and U. Trottenberg, editors,Lecture Notes in Mathematics, Vol. 1228, pages 166–201. Pro-ceedings of the 2nd European Conference on Multigrid Methods,Cologne, 1985, Springer-Verlag, 1986.

31J. J. Reuther, A. Jameson, J. J. Alonso, M. Rimlinger, andD. Saunders. Constrained multipoint aerodynamic shape opti-mization using an adjoint formulation and parallel computers:Part i. Journal of Aircraft, 1998. Accepted for publication.

32J. J. Reuther, A. Jameson, J. J. Alonso, M. Rimlinger, andD. Saunders. Constrained multipoint aerodynamic shape opti-mization using an adjoint formulation and parallel computers:Part ii. Journal of Aircraft, 1998. Accepted for publication.

33J.P. Steinbrenner, J.R. Chawner, and C.L. Fouts. TheGRIDGEN 3D multiple block grid generation system. Tech-nical report, Flight Dynamics Laboratory, Wright Research and

Development Center, Wright-Patterson Air Force Base, Ohio,July 1990.

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36J. Reuther, J.J. Alonso, M.J. Rimlinger, and A. Jameson.Aerodynamic shape optimization of supersonic aircraft configu-rations via an adjoint formulation on parallel computers. AIAApaper 96-4045, 6th AIAA/NASA/ISSMO Symposium on Multi-disciplinary Analysis and Optimization, Bellevue, WA, Septem-ber 1996.

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Design Conditions Initial Single Point Design Three Point DesignMach CL Relative CD Relative CD Relative CD0.81 0.35 1.00257 0.85003 0.854130.82 0.30 1.00000 0.77350 0.779150.83 0.25 1.08731 0.81407 0.76836

Table 1 Drag Reduction for Single and Multipoint Designs.

SYMBOL

SOURCE SYN107P DESIGN44

SYN107P DESIGN 0

ALPHA 2.267

2.223

CD 0.01496

0.01518

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSMPX5X WING-BODY

REN = 101.00 , MACH = 0.860 , CL = 0.610

Solution 1 Upper-Surface Isobars

( Contours at 0.05 Cp )

0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 9.3% Span

0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 26.1% Span

0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 42.7% Span

0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 58.4% Span

0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 76.1% Span

0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 92.0% Span

Fig. 2 Pressure distribution of the MPX5X before and after optimization.

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SYMBOL

SOURCE SYN107P DESIGN 44

SYN107P DESIGN 44

SYN107P DESIGN 44

SYN107P DESIGN 44

SYN107P DESIGN 44

MACH 0.860

0.855

0.850

0.840

0.830

ALPHA 2.267

2.355

2.419

2.518

2.607

CD 0.01496

0.01523

0.01525

0.01494

0.01463

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSMPX5X WING-BODYREN = 101.00 , CL = 0.610



0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 9.3% Span

0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 26.1% Span

0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 42.7% Span

0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 58.4% Span

0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 76.1% Span

0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 92.0% Span

Fig. 3 Off design performance of the MPX5X below the design point.

SYMBOL

SOURCE SYN107P DESIGN 44

SYN107P DESIGN 44

SYN107P DESIGN 44

SYN107P DESIGN 44

MACH 0.860

0.865

0.870

0.880

ALPHA 2.267

2.201

2.130

2.008

CD 0.01496

0.01530

0.01596

0.01807

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSMPX5X WING-BODYREN = 101.00 , CL = 0.610



0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 9.3% Span

0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 26.1% Span

0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 42.7% Span

0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 58.4% Span

0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 76.1% Span

0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 92.0% Span

Fig. 4 Off design performance of the MPX5X above the design point.

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0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

X/C

--- Cp* ---

Original Configuration

a): span station z = 0.190

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

X/C

--- Cp* ---


b): span station z = 0.475

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

X/C

--- Cp* ---


c): span station z = 0.665

0.00

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