Aerodynamic shape optimization techniques based on …aero-comlab.stanford.edu/Papers/AIAA-1998-2538-773.pdf · bination of computational fluid dynamics with nu- ... Anderson and

Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

Aerodynamic Shape Optimization TechniquesBased On Control Theory

Antony Jameson* and Juan J. Alonso^Department of Aeronautics & Astronautics

Stanford UniversityStanford, California 94305 USA

James J. Reuther*Research Institute for Advanced Computer Science

NASA Ames Research Center, MS 227-6Moffett Field, CA 94035, USA

Luigi Martinelli§Department of Mechanical & Aerospace Engineering

Princeton UniversityPrinceton, NJ 08544, USA

John C. Vassberg^Boeing Commercial Airplane Group

Long Beach, CA 90846, USA

AbstractThis paper reviews the formulation and applicationof optimization techniques based on control theoryfor aerodynamic shape design in both inviscid andviscous compressible flow. The theory is applied toa system defined by the partial differential equa-tions of the flow, with the boundary shape actingas the control. The Frechet derivative of the costfunction is determined via the solution of an ad-joint partial differential equation, and the bound-ary shape is then modified in a direction of descent.This process is repeated until an optimum solutionis approached. Each design cycle requires the nu-merical solution of both the flow and the adjointequations, leading to a computational cost roughlyequal to the cost of two flow solutions. Representa-tive results are presented for viscous optimizationof transonic wing-body combinations and inviscid

*AIAA Fellow, T. V. Jones Professor of EngineeringtAIAA Member, Assistant Professor*AIAA Member, Research Scientist§AIAA Member, Assistant Professor'ALAA Senior Member, Senior Principal Engineer

optimization of complex configurations.

1 IntroductionThe definition of the aerodynamic shapes of mod-ern aircraft relies heavily on computational simula-tion 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 cy-cles of computational analysis followed by testingof a preferred design may be used in the evolutionof the final configuration. Wind tunnel testing alsoplays a crucial role in the development of the de-tailed loads needed to complete the structural de-sign, and in gathering data throughout the flightenvelope for the design and verification of the sta-bility and control system. The use of computationalsimulation to scan many alternative designs hasproved extremely valuable in practice, but it stillsuffers the limitation that it does not guarantee the


identification of the best possible design. Generallyone has to accept the best so far by a given cutoffdate in the program schedule. To ensure the real-ization of the true best design, the ultimate goal ofcomputational simulation methods should not justbe the analysis of prescribed shapes, but the auto-matic determination of the true optimum shape forthe intended application.

This is the underlying motivation for the com-bination of computational fluid dynamics with nu-merical optimization methods. Some of the earlieststudies of such an approach were made by Hicksand Henne [1, 2]. The principal obstacle was thelarge computational cost of determining the sensi-tivity of the cost function to variations of the designparameters by repeated calculation of the flow. An-other way to approach the problem is to formulateaerodynamic shape design within the framework ofthe mathematical theory for the control of systemsgoverned by partial differential equations [3]. Inthis view the wing is regarded as a device to pro-duce lift by controlling the flow, and its design isregarded as a problem in the optimal control of theflow equations by changing the shape of the bound-ary. If the boundary shape is regarded as arbitrarywithin some requirements of smoothness, then thefull generality of shapes cannot be defined with afinite number of parameters, and one must use theconcept of the Frechet derivative of the cost withrespect to a function. Clearly such a derivative can-not be determined directly by separate variation ofeach design parameter, because there are now aninfinite number of these.

Using techniques of control theory, however, thegradient can be determined indirectly by solving anadjoint equation which has coefficients determinedby the solution of the flow equations. This directlycorresponds to the gradient technique for trajectoryoptimization pioneered by Bryson [4]. The cost ofsolving the adjoint equation is comparable to thecost of solving the flow equations, with the conse-quence that the gradient with respect to an arbi-trarily large number of parameters can be calcu-lated with roughly the same computational cost astwo flow solutions. Once the gradient has been cal-culated, a descent method can be used to determinea shape change which will make an improvement inthe design. The gradient can then be recalculated,and the whole process can be repeated until thedesign converges to an optimum solution, usuallywithin 50 to 100 cycles. The fast calculation ofthe gradients makes optimization computationallyfeasible even for designs in three-dimensional vis-cous flow. There is a possibility that the descentmethod could converge to a local minimum ratherthan the global optimum solution. In practice this

has not proved a difficulty, provided care is takenin the choice of a cost function which properly re-flects the design requirements. Conceptually, withthis approach the problem is viewed as infinitely di-mensional, with the control being the shape of thebounding surface. Eventually the equations mustbe discretized for a numerical implementation ofthe method. For this purpose the flow and adjointequations may either be separately discretized fromtheir representations as differential equations, or,alternatively, the flow equations may be discretizedfirst, and the discrete adjoint equations then de-rived directly from the discrete flow equations.

The effectiveness of optimization as a tool foraerodynamic design also depends crucially on theproper choice of cost functions and constraints.One popular approach is to define a target pressuredistribution, and then solve the inverse problem offinding the shape that will produce that pressuredistribution. Since such a shape does not necessar-ily exist, direct inverse methods may be ill-posed.This difficulty is removed by reformulating the in-verse problem as an optimization problem, in whichthe deviation between the target and the actualpressure distribution is to be minimized accordingto a suitable measure such as the surface integral

1= =•

where p and pr are the actual and target pres-sures. This still leaves the definition of an appropri-ate pressure architecture to the designer. One mayprefer to directly improve suitable performance pa-rameters, for example, to minimize the drag at agiven lift and Mach number. In this case it is im-portant to introduce appropriate constraints. Forexample, if the span is not fixed the vortex drag canbe made arbitrarily small by sufficiently increasingthe span. In practice, a useful approach is to fix theplanform, and optimize the wing sections subject toconstraints on minimum thickness.

Studies of the use of control theory for optimumshape design of systems governed by elliptic equa-tions were initiated by Pironneau [5]. The con-trol theory approach to optimal aerodynamic de-sign was first applied to transonic flow by Jame-son [6, 7, 8, 9, 10, 11]. He formulated the methodfor inviscid compressible flows with shock wavesgoverned by both the potential flow and the Eu-ler equations [7]. Numerical results showing themethod to be extremely effective for the design ofairfoils in transonic potential flow were presentedin [12], and for three-dimensional wing design us-ing the Euler equations in [13]. More recentlythe method has been employed for the shape de-sign of complex aircraft configurations [14, 15],


using a grid perturbation approach to accommo-date the geometry modifications. The methodhas been used to support the aerodynamic designstudies of several industrial projects, including theBeech Premier and the McDonnell Douglas MDXXand Blended Wing-Body projects. The applica-tion to the MDXX is described in [9]. The experi-ence gained in these industrial applications made itclear that the viscous effects cannot be ignored intransonic wing design, and the method has there-fore been extended to treat the Reynolds Aver-aged Navier-Stokes equations [11]. Adjoint meth-ods have also been the subject of studies by a num-ber of other authors, including Baysal and Ele-shaky [16], Huan and Modi [17], Desai and Ito [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 applicationsof design techniques based on control theory to vis-cous transonic wing design, and also to transonicand supersonic wing design for complex configura-tions.

2 General Formulation of theAdjoint Approach to Opti-mal Design

Before embarking on a detailed derivation of theadjoint formulation for optimal design using theNavier-Stokes equations, it is helpful to summa-rize the general abstract description of the adjointapproach which has been thoroughly documentedin references [7, 21].

The progress of the design procedure is measuredin terms of a cost function /, which could be, forexample the drag coefficient or the lift to drag ratio.For flow about an airfoil or wing, the aerodynamicproperties which define the cost function are func-tions of the flow-field variables (w) and the physicallocation of the boundary, which may be representedby the function f, say. Then

and a change in J- results in a change

(1)

in the cost function. Here, the subscripts / andII are used to distinguish the contributions dueto the variation Sw in the flow solution from thechange associated directly with the modification SJ-"

in the shape. This notation is introduced to assistin grouping the numerous terms that arise duringthe derivation of the full Navier-Stokes adjoint op-erator, so that it remains feasible to recognize thebasic structure of the approach as it is sketched inthe present section.

Using control theory, the governing equations ofthe flow field are introduced as a constraint in sucha way that the final expression for the gradientdoes not require multiple flow solutions. This cor-responds to eliminating Sw from (1).

Suppose that the governing equation R which ex-presses the dependence of w and T within the flow-field domain T> can be written as

R(w,f) = Q. (2)

Then Sw is determined from the equation

,„Next, introducing a Lagrange Multiplier V>, we have

dIT ,dIT T / r d f l l ,\9RSI = -z— Sw + — -=-6F - V I 5— \Sw+ —

aw dT \ l9t tfJ id?fdIT

H -fT~-|̂ dwdI

-V- ^ f <5^-(4)

Choosing $ to satisfy the adjoint equation

[dw\ dw

the first term is eliminated, and we find that

SI = QSF, (6)

where

'dR

The advantage is that (6) is independent of Sw,with the result that the gradient of / with respectto an arbitrary number of design variables can bedetermined without the need for additional flow-field evaluations. In the case that (2) is a partialdifferential equation, the adjoint equation (5) is alsoa partial differential equation and determination ofthe appropriate boundary conditions requires care-ful mathematical treatment.

The computational cost of a single design cycleis roughly equivalent to the cost of two flow so-lutions since the the adjoint problem has similarcomplexity. When the number of design variablesbecomes large, the computational efficiency of thecontrol theory approach over traditional approach,which requires direct evaluation of the gradients by


individually varying each design variable and re-computing the flow field, becomes compelling.

Once equation (3) is established, an improvementcan be made with a shape change

&f = -\gwhere A is positive, and small enough that the firstvariation is an accurate estimate of SI. The varia-tion in the cost function then becomes

si = -\QTQ < o.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 vi-olating constraints, such as a minimum acceptablewing thickness, the gradient may be projected intoan allowable subspace within which the constraintsare satisfied. In this way, procedures can be devisedwhich must necessarily converge at least to a localminimum.

3 The Navier-Stokes Equa-tions

For the derivations that follow, it is convenient touse Cartesian coordinates (x\,x-2,xz) and to adoptthe convention of indicial notation where a re-peated index "j" implies summation over i = 1 to3. The three-dimensional Navier-Stokes equationsthen take the form

dw dfi _ dfvi~37" ' "3— ~ ~3— lnat axi axi (7)

where the state vector w, inviscid flux vector / andviscous flux vector /„ are described respectively by

w = < P«2PU3

(8)

(Willy + p6i2 (9)

/«,' = (10)

In these definitions, p is the density, u\,u^, 1*3 arethe Cartesian velocity components, E is the totalenergy and Sfj is the Kronecker delta function. Thepressure is determined by the equation of state

P= ~f~

and the stagnation enthalpy is given by

where 7 is the ratio of the specific heats. The vis-cous stresses may be written as

where \i and A are the first and second coefficientsof viscosity. The coefficient of thermal conductivityand the temperature are computed as

K — Pr' -Rp' (12)

where Pr is the Prandtl number, cp is the specificheat at constant pressure, and R is the gas con-stant.

For discussion of real applications using a dis-cretization on a body conforming structured mesh,it is also useful to consider a transformation to thecomputational coordinates (£1,^2 ,£3) defined by themetrics

The Navier-Stokes equations can then be writtenin computational space as

—E:— + —^—~ = ° in ^' (13)

where the inviscid and viscous flux contributionsare now defined with respect to the computationalcell faces by F{ = Sijfj and Fv{ = Sijfvj, andthe quantity 5,-j = J^i] represents the projectionof the £,• cell face along the Xj axis. In obtainingequation (13) we have made use of the propertythat

which represents the fact that the sum of the faceareas over a closed volume is zero, as can be read-ily verified by a direct examination of the metricterms.


4 Formulation of the Opti-mal Design Problem for theNavier-Stokes Equations

Aerodynamic optimization is based on the deter-mination of the effect of shape modifications onsome performance measure which depends on theflow. For convenience, the coordinates £,• describingthe fixed computational domain are chosen so thateach boundary conforms to a constant value of oneof these coordinates. Variations in the shape thenresult in corresponding variations in the mappingderivatives defined by K{j.

Suppose that the performance is measured by acost function

/= / M (w, S) dBs + f P(w,S)dP(,JB J-Dcontaining both boundary and field contributionswhere dB$ and dD^ 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 5 defining the computational space. Inthe case of a multi-point design the flow variablesmay be separately calculated for several differentconditions of interest.

The design problem is now treated as a controlproblem where the boundary shape represents thecontrol function, which is chosen to minimize / sub-ject to the constraints defined by the flow equations(13). A shape change produces a variation in theflow solution Sw and the metrics SS which in turnproduce a variation in the cost function

81= f 8M(w, S) dBt: + I SP(w, S) dVs, (15)JB J-D

with

SM =8P = (16)

where we continue to use the subscripts / and //to distinguish between the contributions associatedwith the variation of the flow solution 8w and thoseassociated with the metric variations SS. Thus[MW]I and [Pw]j represent ^£ and f£ with themetrics fixed, while SMu and SPn represent thecontribution of the metric variations SS to SM andSP.

In the steady state, the constraint equation (13)specifies the variation of the state vector Sw by

(17)

Here SF{ and SFvi can also be split into contribu-tions associated with Sw and SS using the notation

]j Sw + 6 Fia

SFvi = (18)

The inviscid contributions are easily evaluated as

= -[Fiw]f = Sij = 6 Sij fj .

The details of the viscous contributions are com-plicated by the additional level of derivatives inthe stress and heat flux terms and will be derivedin Section 6. Multiplying by a co-state vector ^,which will play an analogous role to the Lagrangemultiplier introduced in equation (4) , and integrat-ing over the domain produces

f rp 3

J-D d&(19)

If ip is differentiable this may be integrated by partsto give

In*JB

t - Fvi)

(20)

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

SI = f [SM - n^TS (Ft - Fvi)} dB<:JB

(21)

Now, since if> is an arbitrary differentiable function,it may be chosen in such a way that SI no longer de-pends explicitly on the variation of the state vectorSw. The gradient of the cost function can then beevaluated directly from the metric variations with-out having to recompute the variation Sw resultingfrom the perturbation of each design variable.

Comparing equations (16) and (18), the varia-tion Sw may be eliminated from (21) by equatingall field terms with subscript "/" to produce a dif-ferential adjoint system governing ij>

^r- [Fiw - FViw]r + Pw = 0 in V. (22)

The corresponding adjoint boundary condition isproduced by equating the subscript "/" boundaryterms in equation (21) to produce

[Fiw - Fviw]T = Mv on B. (23)


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

inviscid adjoint equation may be written as

SI i - SFvi] n= I {5Mn - n,-V>'J &

f f dibT )+ / \SPn + -^-[SFi-SFvi]a \ttDf. (24)JT> I a?' J

The details of the formula for the gradient dependon the way in which the boundary shape is parame-terized as a function of the design variables, and theway in which the mesh is deformed as the bound-ary is modified. Using the relationship between themesh deformation and the surface modification, thefield integral is reduced to a surface integral by inte-grating along the coordinate lines emanating fromthe surface. Thus the expression for SI is finallyreduced to the form of equation (6)

SI -L dBe

where f represents the design variables, and Q isthe gradient, which is a function defined over theboundary surface.

The boundary conditions satisfied by the flowequations restrict the form of the left hand side ofthe adjoint boundary condition (23). Consequently,the boundary contribution to the cost function Mcannot be specified arbitrarily. Instead, it must bechosen from the class of functions which allow can-cellation of all terms containing Sw in the bound-ary integral of equation (21). On the other hand,there is no such restriction on the specification ofthe field contribution to the cost function P, sincethese terms may always be absorbed into the ad-joint field equation (22) as source terms.

It is convenient to develop the inviscid and vis-cous contributions to the adjoint equations sepa-rately. Also, for simplicity, it will be assumed thatthe portion of the boundary that undergoes shapemodifications is restricted to the coordinate surface£2 = 0. Then equations (21) and (23) may be sim-plified by incorporating the conditions

n x = n3 = 0, n-2 = 1, dB$ = d£id£3,

so that only the variations SF2 and SFv2 need to beconsidered at the wall boundary.

5 Derivation of the InviscidAdjoint Terms

The inviscid contributions have been previously de-rived in [12, 22] but are included here for complete-ness. Taking the transpose of equation (22), the

(25)

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

IJ dw 'The transformed velocity components have theform

and the condition that there is no flow through thewall boundary at £2 = 0 is equivalent to

so thatSU2 = 0

when the boundary shape is modified. Conse-quently the variation of the inviscid flux at theboundary reduces to

SF-2 = 5p <

S-23

0

+ P ££226823

0

(26)

Since SF2 depends only on the pressure, it is nowclear that the performance measure on the bound-ary M (w, S) may only be a function of the pressureand metric terms. Otherwise, complete cancella-tion of the terms containing Sw in the boundaryintegral would be impossible. One may, for exam-ple, include arbitrary measures of the forces andmoments in the cost function, since these are func-tions of the surface pressure.

In order to design a shape which will lead to adesired pressure distribution, a natural choice is toset

I =- •dS

where pd is the desired surface pressure, and theintegral is evaluated over the actual surface area.In the computational domain this is transformedto


denotes the face area corresponding to a unit el-ement of face area in the computational domain.Now, to cancel the dependence of the boundary in-tegral on Sp, the adjoint boundary condition re-duces to

T p j n j = P ~ P d (27)where HJ are the components of the surface normal

This amounts to a transpiration boundary condi-tion on the co-state variables corresponding to themomentum components. Note that it imposes norestriction on the tangential component of ip at theboundary.

In the presence of shock waves, neither p norPd are necessarily continuous at the surface. Theboundary condition is then in conflict with the as-sumption that if} is differentiate. This difficultycan be circumvented by the use of a smoothedboundary condition [22].

6 Derivation of the ViscousAdjoint Terms

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

Computing the variation Sw resulting from a shapemodification of the boundary, introducing a co-state vector $ and integrating by parts followingthe steps outlined by equations (17) to (20) pro-duces

where the shape modification is restricted to thecoordinate surface £2 = 0 so that RI = n3 = 0,and r»a = 1- Furthermore, it is assumed that theboundary contributions at the far field may eitherbe neglected or else eliminated by a proper choiceof boundary conditions as previously shown for theinviscid case [12, 22].

The viscous terms will be derived under the as-sumption that the viscosity and heat conduction co-efficients n and k are essentially independent of theflow, and that their variations may be neglected.This simplification has been successfully used formay aerodynamic problems of interest. In the case

of some turbulent flows, there is the possibilitythat the flow variations could result in significantchanges in the turbulent viscosity, and it may thenbe necessary to account for its variation in the cal-culation.

Transformation to Primitive VariablesThe derivation of the viscous adjoint terms is sim-plified by transforming to the primitive variables

because the viscous stresses depend on the veloc-ity derivatives -^, while the heat flux can be ex-u j, j

pressed asd

where « = ^ = pJ!^.^. The relationship betweenthe conservative and primitive variations is definedby the expressions

Sw = MSw, Sw = M~lSw

which make use of the transformation matricesM = ff- and M"1 = ff. These matrices are pro-vided in transposed form for future convenience

MT =

' 10000

«1p000

"20p00

«300p0

2puipU-2PU3i-r-i -

' 10000

-^~f000

Mip

01.p00

p0070

(l- l)UiUi2

—(7 — 1)«1—(7 — 1)«2-(7 - I)w3

7-1 .The conservative and primitive adjoint operators Land L corresponding to the variations Sw and Sware then related by

/ SwTLi^ dT>( = I 6wTLil>Jv Jr>

withL = MTL,

so that after determining the primitive adjoint op-erator by direct evaluation of the viscous portionof (22), the conservative operator may be obtainedby the transformation L = M~l L. Since thecontinuity equation contains no viscous terms, itmakes no contribution to the viscous adjoint sys-tem. Therefore, the derivation proceeds by firstexamining the adjoint operators arising from themomentum equations.


Contributions from the MomentumEquationsIn order to make use of the summation convention,it is convenient to set if>j+i — <j)j for j = 1,2,3.Then the contribution from the momentum equa-tions is

LT>

(28)

The velocity derivatives in the viscous stresses canbe expressed as

duj __ duj d£i _ Sij duj

with corresponding variations

The variations in the stresses are then

As before, only those terms with subscript /, whichcontain variations of the flow variables, need beconsidered further in deriving the adjoint operator.The field contributions that contain Sui in equation(28) appear as

This may be integrated by parts to yield

d-8̂

where the boundary integral has been eliminatedby noting that <5u,- = 0 on the solid boundary. By

exchanging indices, the field integrals may be com-bined to produce

Sir,

which is further simplified by transforming the in-ner derivatives back to Cartesian coordinates

f Suk^sJJf^Jv ot,i { \ dxj(29)

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

f\

—Sui = Q if 1=1,3

on the boundary B so that they become

£• (30)

Together, (29) and (30) comprise the field andboundary contributions of the momentum equa-tions to the viscous adjoint operator in primitivevariables.

Contributions from the Energy Equa-tionIn order to derive the contribution of the energyequation to the viscous adjoint terms it is conve-nient to set

d (p\V>5 = 0, Qj = Ui<ri:i + K —— [ £ } ,OXj \pj

where the temperature has been written in terms ofpressure and density using (12). The contributionfrom the energy equation can then be written as

Lr do

JT> d&

+ dB

c. (31)

The field contributions that contain Su{,6p, andSp in equation (31) appear as

f 90Jv &£i >J

•/,%*>{P P P

(32)


The term involving Sffkj may be integrated by partsto produce

f Ac1-D Ukd& 'J

dd dd \c——— +Ui——dxj dxkJ

dB+\Sjkum-—dxm(33)

where the conditions u,- = Suf = 0 are used toeliminate the boundary integral on B. Notice thatthe other term in (32) that involves Su/, need notbe integrated by parts and is merely carried on as

-L' (34)

The terms in expression (32) that involve 6p andSp may also be integrated by parts to produce botha field and a boundary integral. The field integralbecomes

which may be simplified by transforming the innerderivative to Cartesian coordinates

/ (*_£«£) » f a "Jv \ P P P J d£i \ dxj

The boundary integral becomes

'8p _P P J

f (Sp pSpI \ — ~ ~"—JB \P P P

(36)

This can be simplified by transforming the innerderivative to Cartesian coordinates

J (37)

and identifying the normal derivative at the wall

— = 5 ——dn dxj

and the variation in temperature

'Sp

(38)

to produce the boundary contribution

JB kSTdn (39)

This term vanishes if T is constant on the wall butpersists if the wall is adiabatic.

There is also a boundary contribution left overfrom the first integration by parts (31) which hasthe form

(40)/.

where

since w,- = 0. Notice that for future convenience indiscussing the adjoint boundary conditions result-ing from the energy equation, both the Sw and SSterms corresponding to subscript classes / and 71are considered simultaneously. If the wall is adia-batic

so that using (38),

and both the Sw and SS boundary contributionsvanish.

On the other hand, if T is constant J£ =0 for 1= 1,3, so that

, dT

Thus, the boundary integral (40) becomes

Therefore, for constant T, the first term corre-sponding to variations in the flow field contributesto the adjoint boundary operator and the secondset of terms corresponding to metric variations con-tribute to the cost function gradient.

All together, the contributions from the energyequation to the viscous adjoint operator are thethree field terms (33), (34) and (35), and either oftwo boundary contributions ( 39) or ( 41), depend-ing on whether the wall is adiabatic or has constanttemperature.

The Viscous Adjoint Field OperatorCollecting together the contributions from the mo-mentum and energy equations, the viscous adjointoperator in primitive variables can be expressed as

de

1 d

for i = l , 2 , 3


The conservative viscous adjoint operator may nowbe obtained by the transformation

L = M~lTL.

7 Viscous Adjoint BoundaryConditions

It was recognized in Section 4 that the boundaryconditions satisfied by the flow equations restrictthe form of the performance measure that may bechosen for the cost function. There must be a di-rect correspondence between the flow variables forwhich variations appear in the variation of the costfunction, and those variables for which variationsappear in the boundary terms arising during thederivation of the adjoint field equations. Otherwiseit would be impossible to eliminate the dependenceof SI on Sw through proper specification of the ad-joint boundary condition. As in the derivation ofthe field equations, it proves convenient to considerthe contributions from the momentum equationsand the energy equation separately.

Boundary Conditions Arising fromthe Momentum EquationsThe boundary term that arises from the momen-tum equations including both the Sw and SS com-ponents (28) takes the form

/JB

Replacing the metric term with the correspondinglocal face area 8-2 and unit normal HJ defined by

rij = —L

fJB

Defining the components of the surface stress as

and the physical surface element

dS= \Sa\dBf,

the integral may then be split into two components

/ far* \SS,\ dBs + I (t>kSrkdS, (42)JB JB

where only the second term contains variations inthe flow variables and must consequently cancel theSw terms arising in the cost function. The first termwill appear in the expression for the gradient.

A general expression for the cost function thatallows cancellation with terms containing $Tk hasthe form

/= f M(r)dS, (43)JB

corresponding to a variation

SI = I j£JB ork

for which cancellation is achieved by the adjointboundary condition

Natural choices for M arise from force optimiza-tion and as measures of the deviation of the surfacestresses from desired target values.

For viscous force optimization, the cost functionshould measure friction drag. The friction force inthe x,- direction is

B

so that the force in a direction with cosines nt hasthe form

nf = / riiSijffijJB

Expressed in terms of the surface stress TJ , this cor-responds to

Cn} = / mndS,JB

so that basing the cost function (43) on this quan-tity gives

J\f = n,-r,-.

Cancellation with the flow variation terms in equa-tion (42) therefore mandates the adjoint boundarycondition

Note that this choice of boundary condition alsoeliminates the first term in equation (42) so that itneed not be included in the gradient calculation.

In the inverse design case, where the cost func-tion is intended to measure the deviation of thesurface stresses from some desired target values, asuitable definition is

fa ~

10


where TJ is the desired surface stress, including thecontribution of the pressure, and the coefficients a;^define a weighting matrix. For cancellation

Oik (n - Tdl) Srk.

This is satisfied by the boundary condition

4>k = aik (T) -Tdi). (44)

Assuming arbitrary variations in STk , this conditionis also necessary.

In order to control the surface pressure and nor-mal stress one can measure the difference

"j {*kj + faj (P - Pd)} ,

where pd is the desired pressure. The normal com-ponent is then

so that the measure becomes

~ Pd)}

This corresponds to setting

aik = nink

in equation (44) . Defining the viscous normal stressas

Tvn =

the measure can be expanded as

) (p - Pd) + (P - Pdf

For cancellation of the boundary terms

n? (p - pd) } nk (njS<rkj + nkSp)

leading to the boundary condition

<t>k = nk (rvn +p — pd) •

In the case of high Reynolds number, this is wellapproximated by the equations

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

requires

<j>knk6p = (p- Pd) nk6p = (p-

which is satisfied by equation (45), but which rep-resents an overspecification of the boundary con-dition since only the single condition (27) need bespecified to ensure cancellation.

Boundary Conditions Arising fromthe Energy EquationThe form of the boundary terms arising from theenergy equation depends on the choice of tempera-ture boundary condition at the wall. For the adia-batic case, the boundary contribution is (39)

f dOI kST—cJB dn

<t>k = nk(p- pd) , (45)

while for the constant temperature case the bound-ary term is (41). One possibility is to introduce acontribution into the cost function which dependson T or |̂ so that the appropriate cancellationwould occur. Since there is little physical intuitionto guide the choice of such a cost function for aero-dynamic design, a more natural solution is to set

in the constant temperature case or

£-•in the adiabatic case. Note that in the constanttemperature case, this choice of 0 on the boundarywould also eliminate the boundary metric variationterms in (40).

8 Implementation of Navier-Stokes Design

The design procedures can be summarized as fol-lows:

1. Solve the flow equations for p, u\, u^, «3, P-

2. Solve the adjoint equations for ^ subject toappropriate boundary conditions.

3. Evaluate Q .

11


4. Project Q into an allowable subspace that sat-isfies any geometric constraints.

5. Update the shape based on the direction ofsteepest descent.

6. Return to I until convergence is reached.

Practical implementation of the viscous designmethod relies heavily upon fast and accuratesolvers for both the state (w) and co-state (^) sys-tems. This work uses well-validated software for thesolution of the Euler and Navier-Stokes equationsdeveloped over the course of many years [23, 24, 25].

For inverse design the lift is fixed by the tar-get pressure. In drag minimization it is also ap-propriate to fix the lift coefficient, because the in-duced drag is a major fraction of the total drag, andthis could be reduced simply by reducing the lift.Therefore the angle of attack is adjusted during theflow solution to force a specified lift coefficient tobe attained, and the influence of variations of theangle of attack is included in the calculation of thegradient. The vortex drag also depends on the spanloading, which may be constrained by other consid-erations such as structural loading or buffet onset.Consequently, the option is provided to force thespan loading by adjusting the twist distribution aswell as the angle of attack during the flow solution.

DiscretizationBoth the flow and the adjoint equations are dis-cretized using a semi-discrete cell-centered finitevolume scheme. The convective fluxes across cellinterfaces are represented by simple arithmetic av-erages of the fluxes computed using values from thecells on either side of the face, augmented by arti-ficial diffusive terms to prevent numerical oscilla-tions in the vicinity of shock waves. Continuing touse the summation convention for repeated indices,the numerical convective flux across the interfacebetween cells A and B in a three dimensional meshhas the form

where

j (/Aj +

where SABJ is the component of the face area in thejtk Cartesian coordinate direction, (/A,-) and (fsj)denote the flux fj as defined by equation (12) and<IAB is the diffusive term. Variations of the com-puter program provide options for alternate con-structions of the diffusive flux.

The simplest option implements the Jameson-Schmidt-Turkel scheme [23, 26], using scalar dif-fusive terms of the form

- e(4)

and Aw+ and Atu~ are the same differences acrossthe adjacent cell interfaces behind cell A and be-yond cell B in the AB direction. By making thecoefficient e'2' depend on a switch proportional tothe undivided second difference of a flow quantitysuch as the pressure or entropy, the diffusive fluxbecomes a third order quantity, proportional to thecube of the mesh width in regions where the so-lution is smooth. Oscillations are suppressed neara shock wave because e^2' becomes of order unity,while e'4) is reduced to zero by the same switch. Fora scalar conservation law, it is shown in reference[26] that e(2) and e'4) can be constructed to makethe scheme satisfy the local extremum diminishing(LED) principle that local maxima cannot increasewhile local minima cannot decrease.

The second option applies the same constructionto local characteristic variables. There are derivedfrom the eigenvectors of the Jacobian matrix AABwhich exactly satisfies the relation

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 Jameson [28] which combinesdifferences fg — /A and WB — WA in a manner suchthat stationary shock waves can be captured witha single interior point in the discrete solution. Thisscheme minimizes the numerical diffusion as the ve-locity approaches zero in the boundary layer, andhas therefore been preferred for viscous calculationsin this work.

Similar artificial diffusive terms are introducedin the discretization of the adjoint equation, butwith the opposite sign because the wave directionsare reversed in the adjoint equation. Satisfactoryresults have been obtained using scalar diffusionin the adjoint equation while characteristic or H-CUSP constructions are used in the flow solution.

The discretization of the viscous terms of theNavier Stokes equations requires the evaluation ofthe velocity derivatives -£>- in order to calculate the

'viscous stress tensor <T,-J defined in equation (11).These are most conveniently evaluated at the cellvertices of the primary mesh by introducing a dualmesh which connects the cell centers of the primarymesh, as depicted in Figure (1). According to theGauss formula for a control volume V with bound-ary S

t dv'^ fI ^rdv ~ I UinicJv oxj Js

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Figure 1: Cell-centered scheme,vertices of the primary mesh

evaluated at

where HJ is the outward normal. Applied to thedual cells this yields the estimate

dxj volfaces

where 5 is the area of a face, and «,- is an esti-mate of the average of u,- over that face. In orderto determine the viscous flux balance of each pri-mary cell, the viscous flux across each of its facesis then calculated from the average of the viscousstress tensor at the four vertices connected by thatface. This leads to a compact scheme with a stencilconnecting each cell to its 26 nearest neighbors.

The semi-discrete schemes for both the flow andthe adjoint equations are both advanced to steadystate by a multi-stage time stepping scheme. Thisis a generalized Runge-Kutta scheme in which theconvective and diffusive terms are treated differ-ently to enlarge the stability region [26, 29]. Con-vergence to a steady state is accelerated by residualaveraging and a multi-grid procedure [30]. Thesealgorithms have been implemented both for singleand multiblock meshes and for operation on paral-lel computers with message passing .using the MPI(Message Passing Interface) protocol [8, 31, 32].

In this work, the adjoint and flow equations arediscretized separately. The alternative approachof deriving the discrete adjoint equations directlyfrom the discrete flow equations yields another pos-sible discretization of the adjoint partial differen-tial equation which is more complex. If the re-sulting equations were solved exactly, they couldprovide the exact gradient of the inexact cost func-

tion which results from the discretization of theflow equations. On the other hand, any consis-tent discretization of the adjoint partial differentialequation will yield the exact gradient as the meshis refined, and separate discretization has provedto work perfectly well in practice. It should alsobe noted that the discrete gradient includes bothmesh effects and numerical errors such as spuriousentropy production which may not reflect the truecost function of the continuous problem.

Mesh Generation and Geometry Con-trol

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 aC-H topology have been used for viscous optimiza-tion of wing-body combinations, while multiblockmeshes have been generated for complex configura-tions using GRIDGEN [33]. In either case geometrymodifications are accommodated by a grid pertur-bation scheme. For viscous wing-body design usingsingle block meshes, the wing surface mesh pointsthemselves are taken as the design variables. Asimple mesh perturbation scheme is then used, inwhich the mesh points lying on a mesh line project-ing out from the wing surface are all shifted in thesame sense as the surface mesh point, with a decayfactor proportional to the arc length along the meshline. The resulting perturbation in the face areasof the neighboring cells are then included in thegradient calculation. For complex configurationsthe geometry is controlled by superposition of an-alytic "bump" functions defined over the surfaceswhich are to be modified. The grid is then per-turbed to conform to modifications of the surfaceshape by the WARP3D and WARP-MB algorithmsdescribed 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 T represent the designvariable, and Q the gradient. Then the iteration

can be regarded as simulating the time dependentprocess

13


where A is the time step At. Let A be the Hessianmatrix with elements

Aijaft

£1 direction, for example, the smoothed gradient Qma be calculated from a discrete approximation to

Suppose that a locally minimum value of the costfunction /* = /(.F*) is attained when £ = T* .Then the gradient Q* = G(F*) must be zero, whilethe Hessian matrix A* = A(3:*} must be positivedefinite. Since Q* is zero, the cost function can beexpanded as a Taylor series in the neighborhood ofT* with the form

= r + (f - ?*) A (f - F] + . . .

Correspondingly,

As T approaches JF*, the leading terms becomedominant. Then, setting f = (F — J-*), the searchprocess approximates

-dt ~ -Also, since A* is positive definite it can be ex-panded as

A' = RMRT,where M is a diagonal matrix containing the eigen-values of A* , and

Setting

RR = R R = /.

v = RTF,the search process can be represented as

dv- = -Mv. •

The stability region for the simple forward Eulerstepping scheme is a unit circle centered at —1 onthe negative real axis. Thus for stability we mustchoose

^maxAt = /imaxA < 2,

while the asymptotic decay rate, given by the small-est eigenvalue, is proportional to

In order to make sure that each new shape in theoptimization sequence remains smooth, it provesessential to smooth the gradient and to replace Q byits smoothed value Q in the descent process. Thisalso acts as a preconditioner which allows the useof much larger steps. To apply smoothing in the

where e is the smoothing parameter. If one setsSJ- — —Xg, then, assuming the modification is ap-plied on the surface £2 = constant, the first orderchange in the cost function is

SI

< 0,

assuring an improvement if A is sufficiently smalland positive, unless the process has already reacheda stationary point at which Q = 0.

It turns out that this approach is tolerant tothe use of approximate values of the gradient, sothat neither the flow solution nor the adjoint solu-tion need be fully converged before making a shapechange. This results in very large savings in thecomputational cost. For inviscid optimization it isnecessary to use only 15 multigrid cycles for theflow solution and the adjoint solution in each de-sign iteration. For viscous optimization, about 100multigrid cycles are needed. This is partly becauseconvergence of the lift coefficient is much slower, soabout 20 iterations must be made before each ad-justment of the angle of attack to force the targetlift coefficient.

Our second main search procedure incorporatesa quasi-Newton method for general constrained op-timization. In this class of methods the step is de-fined as

s? = -xpg,where P is a preconditioner for the search. An idealchoice is P = A*~l, so that the corresponding timedependent process reduces to

_dt ~ '

for which all the eigenvalues are equal to unity, andT is reduced to zero in one time step by the choiceAt = 1 if the Hessian, A, is constant. The full New-ton method takes P = A~l, requiring the evalua-tion of the Hessian matrix, A, at each step. It corre-sponds to the use of the Newton-Raphson methodto solve the non-linear equation Q = 0. Quasi-Newton methods estimate A* from the change in

14


the gradient during the search process. This re-quires accurate estimates of the gradient at eachtime step. In order to obtain these, both the flowsolution and the adjoint equation must be fully con-verged. Most quasi-Newton methods also require aline search in each search direction, for which theflow equations and cost function must be accuratelyevaluated several times. They have proven quite ro-bust for aerodynamic optimization [34].

In the applications to complex configurationspresented below the optimization was carried outusing the existing, well validated software NPSOL.This software, which implements a quasi-Newtonmethod for optimization with both linear and non-linear constraints, has proved very reliable but isgenerally more expensive than the simple searchmethod with smoothing.

9 Industrial Experience andResults

The methods described in this paper have beenquite thoroughly tested in industrial applications inwhich they were used as a tool for aerodynamic de-sign. They have proved useful both in inverse modeto find shapes that would produce desired pres-sure distributions, and for direct minimization ofthe drag. They have been applied both to well un-derstood configurations that have gradually evolvedthrough incremental improvements guided by windtunnel tests and computational simulation, and tonew concepts for which there is a limited knowledgebase. In either case they have enabled engineers toproduce improved designs.

Substantial improvements are usually obtainedwith 20 — 200 design cycles, depending on the dif-ficulty of the case. One concern is the possibilityof getting trapped in a local minimum. In prac-tice this has not proved to be a source of difficulty.In inverse mode, it often proves possible to comevery close to realizing the target pressure distri-bution, thus effectively demonstrating convergence.In drag minimization, the result of the optimizationis usually a shock-free wing. If one considers dragminimization of airfoils in two-dimensional inviscidtransonic flow, it can be seen that every shock-freeairfoil produces zero drag, and thus optimizationbased solely on drag has a highly non-unique solu-tion. Different shock-free airfoils can be obtainedby starting from different initial profiles. One mayalso influence the character of the final design byblending a target pressure distribution with thedrag in the definition of the cost function.

Similar considerations applyto three-dimensional wing design in viscous tran-

sonic flow. Since the vortex drag can be reducedsimply by reducing the lift, the lift coefficient mustbe fixed for a meaningful drag minimization. Atypical wing of a transport aircraft is designed fora lift coefficient in the range of 0.4 to 0.6. Thetotal wing drag may be broken down into vortexdrag, drag due to viscous effects, and shock drag.The vortex drag coefficient is typically in the rangeof 0.0100 (100 counts), while the friction drag co-efficient 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. Witha fixed span, typically dictated by structural limitsor a constraint imposed by airport gates, the vor-tex drag is entirely a function of span loading, andis minimized by an elliptic loading unless wingletsare added. Transport aircraft usually have highlytapered wings with very large root chords to ac-commodate retraction of the undercarriage. An el-liptic loading may lead to excessively large sectionlift coefficients on the outboard wing, leading topremature shock stall or buffet when the load is in-creased. The structure weight is also reduced by amore inboard loading which reduces the root bend-ing moment. Thus the choice of span loading isinfluenced by other considerations. The skin fric-tion of transport aircraft is typically very close toflat plate skin friction in turbulent flow, and is veryinsensitive to section variations. An exception tothis is the case of smaller executive jet aircraft, forwhich the Reynolds number may be small enoughto allow a significant run of laminar flow if the suc-tion peak of the pressure distribution is moved backon the section. This leaves the shock drag as theprimary target for wing section optimization. Thisis reduced to zero if the wing is shock-free, leavingno room for further improvement. Thus the attain-ment of a shock-free flow is a demonstration of asuccessful drag minimization. In practice range ismaximized by maximizing M ||-, and this is likely tobe increased by increasing the lift coefficient to thepoint where a weak shock appears. One may alsouse optimization to find the maximum Mach num-ber at which the shock drag can be eliminated orsignificantly reduced for a wing with a given sweep-back angle and thickness. Alternatively one maytry to find the largest wing thickness or the mini-mum sweepback angle for which the shock drag canbe eliminated at a given Mach number. This canyield both savings in structure weight and increasedfuel volume . If there is no fixed limit for the wingspan, such as a gate constraint, increased thicknesscan be used to allow an increase in aspect ratio fora wing of equal weight, in turn leading to a reduc-tion in vortex drag. Since the vortex drag is usuallythe largest component of the total wing drag, this

15


is probably the most effective design strategy, andit may pay to increase the wing thickness to thepoint where the optimized section produces a weakshock wave rather than a shock-free flow [22].

The first major industrial application of an ad-joint based aerodynamic optimization method wasthe wing design of the Beech Premier [35] in 1995.The method was successfully used in inverse modeas a tool to obtain pressure distributions favorableto the maintenance of natural laminar flow over arange of cruise Mach numbers. Wing contours wereobtained which yielded the desired pressure distri-bution in the presence of closely coupled engine na-celles on the fuselage above the wing trailing edge.

During 1996 some preliminary studies indicatedthat the wings of both the McDonnell Douglas MD-11 and the Boeing 747-200 could be made shock-free in a representative cruise condition by usingvery small shape modifications, with consequentdrag savings which could amount to several percentof the total drag. This led to a decision to evaluateadjoint-based design methods in the design of theMcDonnell Douglas MDXX during the summer andfall of 1996. In initial studies wing redesigns werecarried out for inviscid transonic flow modelled bythe Euler equations. A redesign to minimize thedrag at a specified lift and Mach number requiredabout 40 design cycles, which could be completedovernight on a workstation.

Three main lessons were drawn from these ini-tial studies: (i) the fuselage effect is to large tobe ignored and must be included in the optimiza-tion, (ii) single-point designs could be too sensitiveto small variations in the flight condition, typicallyproducing a shock-free flow at the design point witha tendency to break up into a severe double shockpattern below the design point, and (iii) the shapechanges necessary to optimize a wing in transonicflow are smaller than the boundary layer displace-ment thickness, with the consequence that viscouseffects must be included in the final design.

In order to meet the first two of these consid-erations, the second phase of the study was con-centrated on the optimization of wing-body com-binations with multiple design points. These werestill performed with inviscid flow to reduce com-putational cost and allow for fast turnaround. Itwas found that comparatively insensitive designscould be obtained by minimizing the drag at a fixedMach number for three fairly closely spaced lift co-efficients such as 0.5, 0.525, and 0.55, or alterna-tively three nearby Mach numbers with a fixed liftcoefficient.

The third phase of the project was focused on thedesign with viscous effects using as a starting pointwings which resulted from multipoint inviscid op-

timization. While the full viscous adjoint methodwas still under development, it was found that use-ful improvements could be realized, particularly ininverse mode, using the inviscid result to providethe target pressure, by coupling an inviscid adjointsolver to a viscous flow solver. Computer costs aremany times larger, both because finer meshes areneeded to resolve the boundary layer, and becausemore iterations are needed in the flow and adjointsolutions. In order to force the specified lift coeffi-cient the number of iterations in each flow solutionhad to be increased from 15 to 100. To achieveovernight turnaround a fully parallel implementa-tion of the software had to be developed. Finallyit was found that in order to produce sufficientlyaccurate results, the number of mesh points hadto be increased to about 1.8 million. In the finalphase of this project it was planned to carry out apropulsion integration study using the multiblockversions of the software. This study was not com-pleted due to the cancellation of the entire MDXXproject.

During the summer of 1997, adjoint methodswere again used to assist the McDonnell DouglasBlended Wing-Body project. By this time theviscous adjoint method was well developed, andit was found that it was needed to achieve trulysmooth shock-free solutions. With an inviscid ad-joint solver coupled to a viscous flow solver someimprovements could be made, but the shocks couldnot be entirely eliminated.

The next subsection shows a wing design usingthe full viscous adjoint method in its current form,implemented in the computer program SYN107.The remaining subsections present results of op-timizations for complete configurations in inviscidtransonic and supersonic flow using the multiblockparallel design program, SYN107-MB.

Transonic Viscous Wing-Body DesignA typical result of drag minimization in transonicviscous flow is presented below. This calculationis a redesign of a wing using the viscous adjointoptimization method with a Baldwin-Lomax tur-bulence model. The initial wing is similar to oneproduced during the MDXX design studies. Fig-ures 2-4 show the result of the wing-body redesignon a C-H mesh with 288 x 96 x 64 cells. The wing hassweep back of about 38 degrees at the 1/4 chord.A total of 44 iterations of the viscous optimiza-tion procedure resulted in a shock-free wing at acruise design point of Mach 0.86, with a lift coef-ficient of 0.61 for the wing-body combination at aReynolds number of 101 million based on the rootchord. Using 48 processors of an SGI Origin2000

16


parallel computer, each design iteration takes about22 minutes so that overnight turnaround for sucha calculation is possible. Figure 2 compares thepressure distribution of the final design with thatof the initial wing. The final wing is quite thick,with a thickness to chord ratio of about 14 percentat the root and 9 percent at the tip. The opti-mization was performed with a constraint that thesection modifications were not allowed to decreasethe thickness anywhere. The design offers excellentperformance at the nominal cruise point. A dragreduction of 2.2 counts was achieved from the ini-tial wing which had itself been derived by inviscidoptimization. Figures 3 and 4 show the results ofa Mach number sweep to determine the drag rise.The drag coefficients shown in the figures representthe total wing drag including shock, vortex, andskin friction contributions. It can be seen that adouble shock pattern forms below the design point,while there is actually a slight increase in the dragcoefficient at Mach 0.85. The tendency to producedouble shocks below the design point is typical ofsupercritical wings. This wing has a low drag co-efficient, however, over a wide range of conditions.Above the design point a single shock forms andstrengthens as the Mach number increases, a be-havior typical in transonic flow.

Transonic Multipoint ConstrainedAircraft DesignAs a first example of the automatic design capabil-ity for complex configurations, a typical businessjet configuration is chosen for a multipoint dragminimization run. The objective of the design isto alter the geometry of the wing in order to min-imize the configuration inviscid drag at three dif-ferent flight conditions simultaneously. Realisticgeometric spar thickness constraints are enforced.The geometry chosen for this analysis is a full con-figuration business jet composed of wing, fuselage,pylon, nacelle, and empennage. The inviscid multi-block mesh around this configuration follows a gen-eral C-O topology with special blocking to capturethe geometric details of the nacelles, pylons andempennage. A total of 240 point-to-point matchedblocks with 4,157,440 cells (including halos) areused to grid the complete configuration. This meshallows the use of 4 multigrid levels obtained throughrecursive coarsening of the initial fine mesh. Theupstream, downstream, upper and lower far fieldboundaries are located at an approximate distanceof 15 wing semispans, while the far field boundarybeyond the wing tip is located at a distance ap-proximately equal to 5 semispans. An engineering-accuracy solution (with a decrease of 4 orders of

magnitude in the average density residual) can beobtained in 100 multigrid cycles. This kind of so-lution can be routinely accomplished in under 20minutes of wall clock time using 32 processors ofan SGI Origin2000 computer.

The initial configuration was designed for Mach= 0.8 and CL = 0.3. The three operating pointschosen for this design are Mach = 0.81 with CL= 0.35, Mach = 0.82 with CL = 0.30, and Mach= 0.83 with CL = 0.25. For each of the designpoints, both Mach number and lift coefficient areheld fixed. In order to demonstrate the advan-tage of a multipoint design approach, the final so-lution at the middle design point will be comparedwith a single point design at the same conditions.As the geometry of the wing is modified, the de-sign algorithm computes new wing-fuselage inter-sections. The wing component is made up of sixairfoil defining sections. Eighteen Hicks-Henne de-sign variables are applied to five of these sectionsfor a total of 90 design variables. The sixth sectionat the symmetry plane is not modified. Spar thick-ness constraints were also enforced on each definingstation at the x/c = 0.2 and x/c = 0.8 locations.Maximum thickness was forced to be preserved atx/c = 0.4 for all six defining sections. To ensure anadequate included angle at the trailing edge, eachsection was also constrained to preserve thicknessat x/c — 0.95. Finally, to preserve leading edgebluntness, the upper surface of each section wasforced to maintain its height above the camber lineat x/c = 0.02. Combined, a total of 30 linear geo-metric constraints were imposed on the configura-tion.

Figures 5-7 show the initial and final airfoilgeometries and Cp distributions after 5 NPSOL de-sign iterations. It is evident that the new designhas significantly reduced the shock strengths onboth upper and lower wing surfaces at all designpoints. The transitions between design points arealso quite smooth. For comparison purposes, a sin-gle point drag minimization study (Mach = 0.81and CL = 0.25) is carried out starting from thesame initial configuration and using the same de-sign variables and geometric constraints.

Figures 8-10 show comparisons of the solutionsfrom the three-point design with those of the sin-gle point design. Interestingly, the upper surfaceshapes for both final designs are very similar. How-ever, in the case of the single point design, a stronglower surface shock appears at the Mach = 0.83,CL — 0.25 design point. The three-point design isable to suppress the formation of this lower surfaceshock and achieves a 9 count drag benefit over thesingle point design at this condition. However, ithas a 1 count penalty at the single point design

17


condition. The three-point design features a weaksingle shock for one of the three design points anda very weak double shock at another design point.Table 1 summarizes the drag results for the two de-signs. The CD values have been normalized by thedrag of the initial configuration at the second designpoint. Figure 11 shows the surface of the configu-ration colored by the local coefficient of pressure,Cp, before and after redesign for the middle designpoint. One can clearly observe that the strength ofthe shock wave on the upper surface of the config-uration has been considerably reduced.

Finally, Figure 12 shows the parallel scalabilityof the multiblock design method for the mesh inquestion using up to 32 processors of an SGI Ori-gin2000 parallel computer. Despite the fact thatthe multigrid technique is used in both the flow andadjoint solvers, the demonstrated parallel speedupsare outstanding.

Supersonic Constrained Aircraft De-signFor supersonic design, provided that turbulent flowis assumed over the entire configuration, the in-viscid Euler equations suffice for aerodynamic de-sign since the pressure drag is not greatly affectedby the inclusion of viscous effects. Moreover, flatplate skin friction estimates of viscous drag are of-ten very good approximations. In this study, thegeneric supersonic transport configuration used inreference [36] is revisited.

The baseline supersonic transport configurationwas sized to accommodate 300 passengers with agross take-off weight of 750,000 Ibs. The super-sonic cruise point is Mach 2.2 with a CL of 0.105.Figure 13 shows that the planform is a cranked-delta configuration with a break in the leading edgesweep. The inboard leading edge sweep is 68.5 de-grees while the outboard is 49.5 degrees. Since theMach angle at M = 2.2 is 63 degrees it is clear thatsome leading edge bluntness may be used inboardwithout a significant wave drag penalty. Blunt lead-ing edge airfoils were created with thickness rang-ing from 4% at the root to 2.5% at the leadingedge break point. These symmetric airfoils werechosen to accommodate thick spars at roughly the5% and 80% chord locations over the span up tothe leading edge break. Outboard of the leadingedge break where the wing sweep is ahead of theMach cone, a sharp leading edge was used to avoidunnecessary wave drag. The airfoils were chosento be symmetric, biconvex shapes modified to havea region of constant thickness over the mid-chord.The four-engine configuration features axisymmet-ric nacelles tucked close to the wing lower surface.

This layout favors reduced wave drag by minimiz-ing the exposed boundary layer diverter area. How-ever, in practice it may be problematic because ofthe channel flows occurring in the juncture region ofthe diverter, wing, and nacelle at the wing trailing

The computational mesh on which the design isrun has 180 blocks and 1,500,000 mesh cells (in-cluding halos), while the underlying geometry enti-ties define the wing with 16 sectional cuts and thebody with 200 sectional cuts. In this case, where wehope to optimize the shape of the wing, care mustbe taken to ensure that the nacelles remain properlyattached with diverter heights being maintained.

The objective of the design is to reduce the totaldrag 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 eachwing defining section. Similarly, instead of applyingthem to all 16 sections, they are applied to 8 of thesections and then lofted linearly to the neighboringsections. Spar thickness constraints are imposedfor all wing defining sections at x/c = 0.05 andx/c = 0.8. An additional maximum thickness con-straint is specified along the span at x/c = 0.5. Afinal thickness constraint is enforced at x/c = 0.95to ensure a reasonable trailing edge included angle.An iso-Cp representation of the initial and final de-signs is depicted in Figure 13 for both the upperand lower surfaces.

It is noted that the strong oblique shock evidentnear the leading edge of the upper surface on theinitial configuration is largely eliminated in the fi-nal design after 5 NPSOL design iterations. Also,it is seen that the upper surface pressure distribu-tion in the vicinity of the nacelles has formed anunexpected pattern. However, recalling that thick-ness constraints abound in this design, these uppersurface pressure patterns are assumed to be the re-sult of sculpting of the lower surface near the na-celles which affects the upper surface shape via thethickness constraints. For the lower surface, theleading edge has developed a suction region whilethe shocks and expansions around the nacelles havebeen somewhat reduced. Figure 14 shows the pres-sure coefficients and (scaled) airfoil sections for foursectional cuts along the wing. These cuts fur-ther demonstrate the removal of the oblique shockon the upper surface and the addition of a suc-tion region on the leading edge of the lower sur-face. The airfoil sections have been scaled by afactor of 2 so that shape changes may be seen moreeasily. Most notably, the section at 38.7% spanhas had the lower surface drastically modified suchthat a large region of the aft airfoil has a forward-

18


facing portion near where the pressure spike fromthe nacelle shock impinges on the surface. The fi-nal overall pressure drag was reduced by 8%, fromCD = 0.0088 to CD = 0.0081.

10 ConclusionsWe have developed a three-dimensional control the-ory based design method for the Navier Stokesequations and applied it successfully to the designof wings in transonic flow. The method representsan extension of our previous work on design withthe potential flow and Euler equations. The newmethod combines the versatility of numerical op-timization methods with the efficiency of inversedesign. The geometry is modified by a grid pertur-bation technique which is applicable to arbitraryconfigurations. Both the wing-body and multiblockversion of the design algorithms have been imple-mented in parallel using the MPI (Message PassingInterface) Standard, and they both yield excellentparallel speedups. The combination of computa-tional efficiency with geometric flexibility providesa powerful tool, with the final goal being to cre-ate practical aerodynamic shape design methods forcomplete aircraft configurations.

AcknowledgmentThis work has benefited from the generous supportof AFOSR under Grant No. AFOSR-91-0391, theNASA-IBM Cooperative Research Agreement, andthe DoD under the Grand Challenge Projects ofthe High Performance Computing ModernizationProgram.

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[2] R. M. Hicks and P. A. Henne. Wing design bynumerical optimization. Journal of Aircraft,15:407-412, 1978.

[3] J. L. Lions. Optimal Control of SystemsGoverned by Partial Differential Equations.Springer-Verlag, New York, 1971. Translatedby S.K. Mitter.

[4] A. E. Bryson and Y. C. Ho. Applied OptimalControl. Hemisphere, Washington, DC, 1975.

[5] 0. Pironneau. Optimal Shape Design for Ellip-tic Systems. Springer-Verlag, New York, 1984.

[6] A. Jameson. Optimum aerodynamic design us-ing CFD and control theory. AIAA paper 95-1729, AIAA 12th Computational Fluid Dy-namics Conference, San Diego, CA, June 1995.

[7] A. Jameson. Aerodynamic design via con-trol theory. Journal of Scientific Computing,3:233-260, 1988.

[8] A. Jameson and J.J. Alonso. Automatic aero-dynamic optimization on distributed mem-ory architectures. AIAA paper 96-0409,34th Aerospace Sciences Meeting and Exhibit,Reno, Nevada, January 1996.

[9] A. Jameson. Re-engineering the design processthrough computation. AIAA paper 97-0641,35th Aerospace Sciences Meeting and Exhibit,Reno, Nevada, January 1997.

[10] A. Jameson, N. Pierce, and L. Martinelli. Op-timum aerodynamic design using the Navier-Stokes equations. AIAA paper 97-0101,35th Aerospace Sciences Meeting and Exhibit,Reno, Nevada, January 1997.

[11] A. Jameson, L. Martinelli, and N. A.Pierce. Optimum aerodynamic design usingthe Navier-Stokes equations. Theoret. Corn-put. Fluid Dynamics, 10:213-237, 1998.

[12] A. Jameson. Automatic design of transonicairfoils to reduce the shock induced pressuredrag. In Proceedings of the 31st Israel AnnualConference on Aviation and Aeronautics, TelAviv, pages 5-17, February 1990.

[13] A. Jameson. Optimum aerodynamic design viaboundary control. In AGARD- VKI LectureSeries, Optimum Design Methods in Aerody-namics, von Karman Institute for Fluid Dy-namics, 1994.

[14] J. Reuther, A. Jameson, J. J. Alonso, M. J.Rimlinger, and D. Saunders. Constrained mul-tipoint aerodynamic shape optimization usingan adjoint formulation and parallel computers.AIAA paper 97-0103, 35th Aerospace SciencesMeeting and Exhibit, Reno, Nevada, January1997.

[15] J. Reuther, J. J. Alonso, J. C. Vassberg,A. Jameson, and L. Martinelli. An efficientmultiblock method for aerodynamic analysisand design on distributed memory systems.AIAA paper 97-1893, June 1997.

19


[16] O. Baysal and M. E. Eleshaky. Aerodynamicdesign optimization using sensitivity analy-sis and computational fluid dynamics. AIAAJournal, 30(3):718-725, 1992.

[17] J.C. Huan and V. Modi. Optimum design fordrag minimizing bodies in incompressible flow.Inverse Problems in Engineering, 1:1-25,1994.

[18] M. Desai and K. Ito. Optimal controls ofNavier-Stokes equations. SIAM J. Control andOptimization, 32(5): 1428-1446, 1994.

[19] W. K. Anderson and V. Venkatakrishnan.Aerodynamic design optimization on unstruc-tured grids with a continuous adjoint formu-lation. AIAA paper 97-0643, 35th AerospaceSciences Meeting and Exhibit, Reno, Nevada,January 1997.

[20] J. Elliott and J. Peraire. 3-D aerodynamic op-timization on unstructured meshes with vis-cous effects. AIAA paper 97-1849, June 1997.

[21] A. Jameson. Optimum aerodynamic design us-ing CFD and control theory. AIAA Paper 95-1729-CP, 1995.

[22] A. Jameson. Optimum aerodynamic design us-ing control theory. Computational Fluid Dy-namics Review, pages 495-528, 1995.

[23] A. Jameson, W. Schmidt, and E. Turkel. Nu-merical solutions of the Euler equations by fi-nite volume methods with Runge-Kutta timestepping schemes. AIAA paper 81-1259, Jan-uary 1981.

[24] L. Martinelli and A. Jameson. Validation ofa multigrid method for the Reynolds averagedequations. AIAA paper 88-0414, 1988.

[25] S. Tatsumi, L. Martinelli, and A. Jameson. Anew high resolution scheme for compressibleviscous flows with shocks. AIAA paper To Ap-pear, AIAA 33nd Aerospace Sciences Meeting,Reno, Nevada, January 1995.

[26] A. Jameson. Analysis and design of numeri-cal schemes for gas dynamics 1, artificial diffu-sion, upwind biasing, limiters and their effecton multigrid convergence. Int. J. of Comp.Fluid Dyn., 4:171-218, 1995.

[27] P.L. Roe. Approximate Riemann solvers, pa-rameter vectors, and difference schemes. Jour-nal of Computational Physics, 43:357-372,1981.

[28] A. Jameson. Analysis and design of numericalschemes for gas dynamics 2, artificial diffusionand discrete shock structure. Int. J. of Comp.Fluid Dyn., 5:1-38, 1995.

[29] L. Martinelli. Calculations of viscous flowswith a multigrid method. Princeton Univer-sity Thesis, May 1987.

[30] A. Jameson. Multigrid algorithms for com-pressible flow calculations. In W. Hackbuschand U. Trottenberg, editors, Lecture Notes inMathematics, Vol. 1228, pages 166-201. Pro-ceedings of the 2nd European Conference onMultigrid Methods, Cologne, 1985, Springer-Verlag, 1986.

[31] J. J. Reuther, A. Jameson, J. J. Alonso,M. Rimlinger, and D. Saunders. Constrainedmultipoint aerodynamic shape optimizationusing an adjoint formulation and parallel com-puters: Part i. Journal of Aircraft, 1998. Ac-cepted for publication.

[32] J. J. Reuther, A. Jameson, J. J. Alonso,M. Rimlinger, and D. Saunders. Constrainedmultipoint aerodynamic shape optimizationusing an adjoint formulation and parallel com-puters: Part ii. Journal of Aircraft, 1998. Ac-cepted for publication.

[33] J.P. Steinbrenner, J.R. Chawner, and C.L.Fouts. The GRIDGEN 3D multiple block gridgeneration system. Technical report, FlightDynamics Laboratory, Wright Research andDevelopment Center, Wright-Patterson AirForce Base, Ohio, July 1990.

[34] J. Reuther and A. Jameson. Aerodynamicshape optimization of wing and wing-bodyconfigurations using control theory. AIAA pa-per 95-0123, AIAA 33rd Aerospace SciencesMeeting, Reno, Nevada, January 1995.

[35] J. Gallman, J. Reuther, N. Pfeiffer, W. For-rest, and D. Bernstorf. Business jet wing de-sign using aerodynamic shape optimization.AIAA paper 96-0554, 34th Aerospace SciencesMeeting and Exhibit, Reno, Nevada, January1996.

[36] J. Reuther, J.J. Alonso, M.J. Rimlinger,and A. Jameson. Aerodynamic shape op-timization of supersonic aircraft configura-tions via an adjoint formulation on paral-lel computers. AIAA paper 96-4045, 6thAIAA/NASA/ISSMO Symposium on Multi-disciplinary Analysis and Optimization, Belle-vue, WA, September 1996.

20


Design Conditions Initial Single Point Design Three Point DesignMach CL Relative CD Relative CD Relative CD0.810.820.83

0.350.300.25

1.002571.000001.08731

0.850030.773500.81407

0.854130.779150.76836

Table 1: Drag Reduction for Single and Multipoint Designs.

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSMPX5X WING-BODY

REN = 101.00 , MACH=0.860 , CL = 0.610

SYMBOL SOURCE ALPHA CDSYN107PDESION44 2.267 0.01496

Figure 2: Pressure distribution of the MPX5X before and after optimization.

21


-OJ •

O 0.0

0.5-

COMPARISON OF CHORD WISE PRESSURE DISTRIBUTIONSMPX5X WING-BODYREN= 101.00 . CL =0.610 ,

0.2 0.4 0.6 0* 80X/C ^^

42.7% Spin

SYMBOL SOURCE MACH

............. SYN1Q7P DESIGN 44V_. _._... SYN107P DESIGN 44V. ._.._.. SYN107P DESIGN 44Ju _ _ . . SYN107P DESIGN 44

0.8550.8500.8400.830

ALPHA

2.3552-4192-5182.607

CD

0.01523omsp~0.01 «40.01/53

/

-1.0 •

-OS •

& 0.0

0.5 '

1.0-

' 0.2 0.4 0.6 0.8\KOX/C

9.3% Spin

-0.5 '

& 0.0

OS

0.2 0.4 0.60*_/0X / C

92.0% Spin

Figure 3: Off design performance of the MPX5X below the design point.

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

SYMBOL SOURCE MACH ALPHA CDSYN107P DESIGN 44 0.860 2J67 0.014%SYN107P DESIGN 44 0.865 2201 0.01530

_._._._. SYN107PDESIGN44 0.8700.2 0.4 0.6

X / C42.7% S

0.2 0.4 0.6X/C

92.0% Spin

—•-•• SYN107P DESIGN 44 0.880 2.008 0.01807

SolutioUpper-Surface Isobars

(ContounMO-OSCp)

0.2 0.4 0.6 O.K\0X / C

26.1% Sp«i

0.2 0.4 0.6T>,i_/0X / C

76.1% Spin

0.2 0.4 0.6 0.8\KOX / C

9.3% Spin

0.2 0.4X / C

58.4% Spin

Figure 4: Off design performance of the MPX5X above the design point.

22


_ _ Original Configuration

030 0.30 0,« OJO 0.60 0.70 0.10

5a: span station z = 0.190

Original Configuration

5b: span station z = 0.475

0.00 0.10 020 OJO 0.40 OM OM 0.70 0.10 0.90 !.«

*c

5c: span station z = 0.665

Figure 5: Business Jet Configuration. Multipoint Drag Minimization at Fixed Lift.Design Point 1, M = 0.81, CL = 0.3590 Hicks-Henne variables. Spar Constraints Active.- - -, Initial Pressures————, Pressures After 5 Design Cycles.

0.00 0.10 0.20 OJO 0.40 OJO 0.60 0.70 0.10 0.90 l.OC

X/C

5d: span station z = 0.856

23



OJO 0.40 OJO 0.60 0.70 O.SO


0.00 0.10 0.10 OJO 0.40 0.50 0.60 O.TO OJO 0.90

6c: span station z — 0.665

0.10 0.20 OJO 0.40 O.SO 0.60 0.70 0.10 0.90 1.00



24


OJO 0.60 0.70 O.KJ 0.90

_ . Original Configuration X

7a: span station z — 0.190

0.10 0.20 0.30 0.40 OJO 060 0.70 0.10 0.90 1.00


0.10 0.20 OX OM OJO 0.60 0.70 O.M 0.90 1.00


0.20 OM O.«0 0.50 0.60 0.70 0.10 0.90 1.00



25


0.00 0.10 0.20 0.30 0.40


Single Point Deiign

0.00 0.10 0.20 OJO 0.40 OJO 0.60 0.70 O.W 0.90


0.10 0.20 0.30 0.40 OJO 0.60 0.70 OJO O.W


0.10 OJO O30 O.W OJO DM 0.70 O.tO O.W


Figure 8: Business Jet Configuration. Single Point vs. Multipoint Drag Minimization at Fixed Lift.Design Point 1, M = 0.81, CL = 0.3590 Hicks-Henne variables. Spar Constraints Active.- - -, Single Point Design Pressures.————, Multipoint Design Pressures.

26


Single Point Dciign

0.00 0.10 0.10 0.50 O.«0 0.50 0.60 0.70 O.M 0.90


_ . Single Point Dciign

0.10 0.30 O.«0 OJO 0.60 0.70 0.10 0,90 1.00


0.00 0.10 OJO O.JO 0.40 OJO 0.60 0.70 0.10 0.90

9c: span station z — 0.665

0.00 0.10 0.10 O.JO 0.40 OJO 0.60 0.70 0.10


Figure 9: Business Jet Configuration. Single Point vs. Multipoint Drag Minimization at Fixed Lift.Design Point 2, M = 0.82, CL = 0.3090 Hicks-Henne variables. Spar Constraints Active.- - -, Single Point Design Pressure.————, Multipoint Design Pressures.

27


0.00 0.10 0.20 OJO 0.60 0.70

10a: span station z — 0.190

0.00 0.10 0.20 OM 0.40 OJO 0,60 0.70 0.10 0.90 1.01


0.20 030 0.40 0.50 0.60 0.70 an 0.90

lOc: span station z = 0.665

0,00 0.10 0.20 0.30 0.40 030 040 0.70 O.U 0.90 1.00


Figure 10: Business Jet Configuration. Single Point vs. Multipoint Drag Minimization at Fixed Lift.Design Point 3, M = 0.83, CL = 0.2590 Hicks-Henne variables. Spar Constraints Active.- - -, Single Point Design Pressures.————, Multipoint Design Pressure.

28

Copyright © 1998, American Institute of Aeronautics and Astronautics, Inc.

Optimized Design

Figure 11: Geometry Surface Colored by local Cp Before and After Redesign.

29


35

30

25

20

=Il50-

10

Parallel Speedup of SYN87-MB

10 15 20Number of Processors

25 30 35

Figure 12: Scalability Study for Multiblock Design Method.

30

Copyright © 1998, American Institute of Aeronautics and Astronautics, Inc.

Baseline Optimized Raselina Optimized

Upper Surface Lower Surface

Figure 13: Supersonic Transport Configuration. lso-Cp Contours on Upper and Lower Surfaces. Baselineand Optimized Designs. M = 2.2, CL = 0.105.

31


i i

Original Configuration

Y stretch factor 2.0

0.00 0.10 0.20 O.JO O.«0 0.50 0.60 0.70 0.10 0.90 1.00


Original Coofigurttoo

Y stretch factor 2.0


Ori|iiul Confif ration

Y ttKtcb factor: 2.0

0.10 0.20 OJO 0.70 O.K 0.90 1,00


Original Coofiguntioa

Y itretch factor 2.0

0.20 030 0.40 0.50 0.60 0.70 O.tO 0.90


Figure 14: Supersonic Transport Configuration. Drag Minimization at Fixed Lift.M = 2.20, CL = 0.105144 Hicks-Henne variables. Spar Constraints Active.- - -, Initial Pressures————, Pressures After 5 Design Cycles.

32

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