Copyright ©1997, American Institute of Aeronautics and Astronautics, Inc.

AIAA Meeting Papers on Disc, January 1997A9715189, AF-AFOSR-91-0391, AIAA Paper 97-0101

Optimum aerodynamic design using the Navier-Stokes equations

A. JamesonPrinceton Univ., NJ

N. A. PierceOxford Univ., United Kingdom

L. MartinelliPrinceton Univ., NJ

AIAA, Aerospace Sciences Meeting & Exhibit, 35th, Reno, NV, Jan. 6-9, 1997

This paper describes the formulation of optimization techniques based on control theory for aerodynamic shape design inviscous compressible flow, modelled by the Navier-Stokes equations. It extends previous work on optimization for inviscidflow. The theory is applied to a system defined by the partial differential equations of the flow, with the boundary shapeacting as the control. The Frechet derivative of the cost function is determined via the solution of an adjoint partialdifferential equation, and the boundary shape is then modified in a direction of descent. This process is repeated until anoptimum solution is approached. Each design cycle requires the numerical solution of both the flow and the adjointequations, leading to a computational cost roughly equal to the cost of two flow solutions. The cost is kept low by usingmultigrid techniques, in conjunction with preconditioning to accelerate the convergence of the solutions. The power of themethod is illustrated by designs of wings and wingbody combinations for long range tranport aircraft. Satisfactory designsare usually obtained with 20-40 design cycles. (Author)

Page 1

Optimum Aerodynamic Designusing the Navier-Stokes Equations

A.JAMESON*, N.A. PIERCE t AND L. MARTINELLI §f' 3 Department of Mechanical and Aerospace Engineering

Princeton UniversityPrinceton, New Jersey 08544 USA

and

T Oxford University Computing LaboratoryNumerical Analysis Group

Oxford OX1 3QD UK

ABSTRACT

This paper describes the formulation of optimiza-tion techniques based on control theory for aerody-namic shape design in viscous compressible flow,modelled by the Navier-Stokes equations. It ex-tends previous work on optimization for inviscidflow. The theory is applied to a system defined bythe partial differential equations of the flow, with theboundary shape acting as the control. The Frechetderivative of the cost function is determined via thesolution of an adjoint partial differential equation,and the boundary shape is then modified in a di-rection of descent. This process is repeated until anoptimum solution is approached. Each design cyclerequires the numerical solution of both the flow andthe adjoint equations, leading to a computationalcost roughly equal to the cost of two flow solutions.The cost is kept low by using multigrid techniques,in conjunction with preconditioning to acceleratethe convergence of the solutions. The power of themethod is illustrated by designs of wings and wing-body combinations for long range tranport aircraft.Satisfactory designs are usually obtained with 20-40design cycles.

Copyright ©1997 by the Authors.Published by the AIAA Inc. with permission

* James S. McDonnell Distinguished University Professor ofAerospace Engineering, AIAA Fellow

t Doctoral Candidate, Student Member AIAA§ Assistant Professor, Member AIAA

1 INTRODUCTION

The ultimate success of an aircraft design dependson the resolution of complex multi-disciplinarytrade-offs between factors such as aerodynamic effi-ciency, structural weight, stability and control, andthe volume required to contain fuel and payload. Adesign is finalized only after numerous iterations,cycling between the disciplines. The developmentof accurate and efficient methods for aerodynamicshape optimization represents a worthwhile inter-mediate step towards the eventual goal of full multi-disciplinary optimal design.

Early investigations into aerodynamic optimizationrelied on direct evaluation of the influence of eachdesign variable. This dependence was estimatedby separately varying each design parameter andrecalculating the flow. The computational cost ofthis method is proportional to the number of designvariables and consequently becomes prohibitive asthe number of design parameters is increased.

An alternative approach to design relies on the factthat experienced designers generally have an intu-itive feel for the type of pressure distribution thatwill provide the desired aerodynamic performance.The resulting inverse problem amounts to deter-mination of the shape corresponding to a specifiedpressure distribution. This approach has the advan-tage that only one flow solution is required to obtainthe desired design. However, the problem must beformulated carefully to ensure that the target pres-

sure distribution corresponds to a physically realiz-able shape.

The problems of optimal and inverse design canboth be systematically treated within the mathemat-ical theory for the control of systems governed bypartial differential equations [I] by regarding thedesign problem as a control problem in which thecontrol is the shape of the boundary. The inverseproblem then becomes a special case of the opti-mal design problem in which the shape changes aredriven by the discrepancy between the current andtarget pressure distributions.

The control theory approach to optimal aerody-namic design, in which shape changes are basedon gradient information obtained by solution of anadjoint problem, was first applied to transonic flowbyjameson [2,3]. He formulated the method for in-viscid compressible flows with shocks governed byboth the potential equation and the Euler equations[2, 4, 5]. With this approach, the cost of a designcycle is independent of the number of design vari-ables and the method has been employed for wingdesign in the context of complex aircraft configura-tions [6, 7], using a grid perturbation technique toaccommodate the geometry modifications.

Pironneau had earlier studied the use of control the-ory for optimum shape design of systems governedby elliptic equations [8]. Ta'asan, Kuruvila and Salashave proposed a one shot approach in which theconstraint represented by the flow equations needonly be satisfied by the final converged design so-lution [9]. Adjoint methods have also been used byBaysal and Eleshaky [10], and by Cabuk and Modi[11,12].

The objective of the present work is the extensionof adjoint methods for optimal aerodynamic de-sign to flows governed by the compressible Navier-Stokes equations. While inviscid formulations haveproven useful for the design of transonic wings atcruise conditions, the inclusion of boundary layerdisplacement effects with viscous design providesincreased realism and alleviates shocks that wouldotherwise form in the viscous solution over the finalinviscid design. Accurate resolution of viscous ef-fects such as separation and shock/boundary layerinteraction is also essential for optimal design en-compassing off-design conditions and high-lift con-figurations.

The computational costs of viscous design are atleast an order of magnitude greater than for designusing the Euler equations because a) the number of

mesh points must be increased by a factor of two ormore to resolve the boundary layer, b) there is theadditional cost of computing the viscous terms anda turbulence model, and c) Navier-Stokes calcula-tions generally converge much more slowly thanEuler solutions due to discrete stiffness and direc-tional decoupling arising from the highly stretchedboundary layer cells. The computational feasibil-ity of viscous design therefore hinges on the de-velopment of a rapidly convergent Navier-Stokesflow solver. Pierce and Giles have developed apreconditioned multigrid method that dramaticallyimproves convergence of viscous calculations byensuring that all error modes inside the stretchedboundary layer cells are either damped or expelled[13,14]. The same acceleration techniques are appli-cable to the adjoint calculation, so that a substantialreduction in the cost of each design cycle is achiev-able.

2 GENERAL FORMULATION OF THE AD-JOINT APPROACH TO OPTIMAL 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 [2, 3].

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 T', say. Then

and a change in T results in a change

61 = dw 6w + —— ' (1)

in the cost function. Here, the subscripts I and // areused to distinguish the contributions due to the vari-ation 5w in the flow solution from the change associ-ated directly with the modification 67 in the shape.This notation is introduced to assist in grouping thenumerous terms that arise during the derivation ofthe full Navier-Stokes adjoint operator, so that it re-mains feasible to recognize the basic structure of theapproach as it is sketched in the present section.

Using control theory, the governing equations of theflow field are introduced as a constraint in such away that the final expression for the gradient doesnot require multiple flow solutions. This corre-sponds to eliminating 6w from (1).

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

(2)

Then 6w is determined from the equation

..57 = -^— 6waw

T \9R]\- ^ \^-\([aw] J /

(3)

Next, introducing a Lagrange Multiplier if>, we have

(4)

(5)

(6)

(7)

(8)

(9)

Choosing i[> to satisfy the adjoint equation

dRdw

dl

the first term is eliminated, and we find that

SI =

where

aiT T dR-The advantage is that (9) is independent of 5w, withthe result that the gradient of / with respect to anarbitrary number of design variables can be deter-mined without the need for additional flow-fieldevaluations. In the case that (2) is a partial differ-ential equation, the adjoint equation (8) is also apartial differential equation and determination ofthe appropriate boundary conditions requires care-ful mathematical treatment.

The computational cost of a single design cycle isroughly 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

theory approach over traditional approach, whichrequires direct evaluation of the gradients by indi-vidually varying each design variable and recom-puting the flow field, becomes compelling.

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

In these definitions, p is the density, 111,112,113 arethe Cartesian velocity components, E is the totalenergy and

SFvl = (20)

The inviscid contributions arc easily evaluated as

[Fiw]r -dfi.

*dw'&Fvin =

The details of the viscous contributions are compli-cated by the additional level of derivatives in thestress and heat flux terms and will be derived inSection 6. Multiplying by a co-state vector ?/>, whichwill play an analogous role to the Lagrange mul-tiplier introduced in equation (7), and integratingover the domain produces

(21)

If tj) is differentiable this may be integrated by partsto give

(22)

(23)

Since the left hand expression equals zero, it maybe subtracted from the variation in the cost function(17) to give

61 =

+

f [6MJB

S (Ft - Fvi)}

. (24)

Now, since t/J is an arbitrary differentiable function,it may be chosen in such a way that 51 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 6w resultingfrom the perturbation of each design variable.

Comparing equations (18) and (20), the variation 6wmay be eliminated from (24) by equating all fieldterms with subscript "/" to produce a differentialadjoint system governing ifj

- Fvjw] r + Pw = 0 in V. (25)

The corresponding adjoint boundary condition isproduced by equating the subscript "/" boundaryterms in equation (24) to produce

(26)

The remaining terms from equation (24) then yielda simplified expression for the variation of the costfunction which defines the gradient

SI = I {6Mn - n.^r (F, - Fvi}}

ness. Taking the transpose of equation (25), theinviscid adjoint equation may be written as

C'f— = 0 in£>, (28)' d&

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

The transformed velocity components have theform

where the quantity

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 reducesto

1/j.jrij =p-pd (30)where rij are the components of the surface normal

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

so that

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

5F2 = Sp <

0

S2j5225230

55225523

0

(29)

Since 6F2 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 cancellationof the terms containing 5w in the boundary inte-gral would be impossible. One may, for example,include arbitrary measures of the forces amd mo-ments in the cost function, since these are functionsof the surface pressure.

In order to design a shape which will lead to a de-sired pressure distribution, a natural choice is to set

= 5/0-z JB PdfdS

where p,i is the desired surface pressure, and theintegral is evaluated over the actual surface area. Inthe computational domain this is transformed to

I=\JI (P-P

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 if} at theboundary.

In the presence of shock waves, neither p nor pj. arenecessarily continuous at the surface. The bound-ary condition is then in conflict with the assump-tion that i/i is differentiable. This difficulty can becircumvented by the use of a smoothed boundarycondition [15].

6 DERIVATION OF THE VISCOUS ADJOINTTERMS

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

Computing the variation 5w resulting from a shapemodification of the boundary, introducing a co-statevector t/} and integrating by parts following the stepsoutlined by equations (19) to (23) produces

- I ̂JT> "w

where the shape modification is restricted to thecoordinate surface £2 — 0 so that n\ — ris = 0,and ri2 = 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 [4, 15].

The viscous terms will be derived under the as-sumption that the viscosity and heat conduction

coefficients \i and k are essentially independent ofthe flow, and that their variations may be neglected.In the case of turbulent flow, if the flow variationsare found to result in significant changes in the tur-bulent viscosity, it may eventually be necessary toinclude its variation in the calculations.

Transformation to Primitive Variables

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

wr = (p,u,v,w,p)T,

because the viscous stresses depend on the velocityderivatives ̂ , while the heat fluxes can be ex-pressed as

"''n —OXi /9,

The relationship between the conservative andprimitive variations are defined by the expressions

Sw = M5w, 6w = M~l6w

which make use of the transformation matricesM ~ if and M~l = if- Th686 matrices are pro-vided in transposed form for future convenience

MT =

" 10000

' 1 iap0 1p0 00 00 0

Mlp000

Hip

01p00

'20p00

U300p0

U3p

001p0

2 'pUl

pU2

PU31-y-1 -

(l-l)uiUi -\2

— (7 — !)MI\ / I— ("y — 1)«2

-(7 - 1)U3

7-1 .The conservative and primitive adjoint operators Land L corresponding to the variations 6w and 6ware then related by

T> T>with

L = MTL,so that after determining the primitive adjoint op-erator by direct evaluation of the viscous portion of(25), the conservative operator may be obtained bythe transformation L — M~l L. There is no contri-bution from the continuity equation so the deriva-tion proceeds by first examining the adjoint opera-tors arising from the momentum equations.

Contributions from the Momentum Equations

In order to make use of the summation convention,it is convenient to set i/Jj+i — j f°r J = 1)2,3. Thenthe contribution from the momentum equations is

/ k ('Jo

_[dfrJv dti

+ (31)

The velocity derivatives in the viscous stresses canbe expressed as

Jwith corresponding variations

The variation in the stresses are then

As before, only those terms with subscript 7, whichcontain variations of the flow variables, need be con-sidered further in deriving the adjoint operator. Thefield contributions that contain 6m in equation (31)appear as

, d „ Sik d „

S'"> d x i ,m-^-dum 1-d^.

This may be integrated by parts to yield

d

f a d/

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

af/Jv T —dxm

(32)

The boundary contributions that contain 6ui inequation (31) maybe simplified using the fact that

5 « i = 0 if 1 = 1,3oti

on the boundary B so that they become

f2j < /IB

(33)

Together, (32) and (33) comprise the field andboundary contributions of the momentum equa-tions to the viscous adjoint operator in primitivevariables.

Contributions from the Energy Equation

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

A; "» "-**_'t JJ dxj

and identifying the normal derivative at the wall

d ^s odn 3dxj'

and the variation in temperature

6T= k (— -5^"\7-1 V P P P / '

(40)

(41)

to produce the boundary contribution

L ~dn (42)This term vanishes if T is constant on the wall butpersists if the wall is adiabatic.

There is also a boundary contribution left over fromthe first integration by parts (34) which has the form

/ 95 (SyQj) dBs, (43)JB

where

since w; = 0. Notice that for future convenience indiscussing the adjoint boundary conditions result-ing from the energy equation, both the 5w and SSterms corresponding to subscript classes / and // areconsidered simultaneously. If the wall is adiabatic

so that using (41),

= 0

and both the Sw and SS boundary contributionsvanish.

On the other hand, if T is constant then it is moreconvenient to expand (43) into

(SS2jQj + Sy

where, since 1̂ = 0 for / - 1,3,

j dbj v JThus, the boundary integral (43) becomes

(44)

Therefore, for constant T, the first term correspond-ing to variations in the flow field contributes to theadjoint boundary operator and the second set ofterms corresponding to metric variations contributeto the cost function gradient.

All together, the contributions from the energyequation to the viscous adjoint operator are the threefield terms (36), (37) and (38), and either of twoboundary contributions ( 42) or (44), depending onwhether the wall is adiabatic or has constant tem-perature.

7 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

'

r (oe 09 \ ., de ] \•j k br~ + TT- + xs» ir~ f[ \dXj dxij dxk\\

(LB) =(7 - 1)

The conservative viscous adjoint operator may nowbe obtained by the transformation

L = M~lTL.

8 VISCOUS ADJOINT BOUNDARY CONDI-TIONS

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 51 on Sw through proper specification of the ad-joint boundary condition. As in the derivation of thefield equations, it proves convenient to consider thecontributions from the momentum equations andthe energy equation separately.

Boundary Conditions Arising from the Momen-tum Equations

The boundary term that arises from the momentumequations including both the Sw and 55 components(31) takes the form

/ faSJB

dBe.

Replacing the metric term with the correspondinglocal face area £2 and unit normal rij defined by

then leads to

S (\S2\njakj)

Defining the components of the surface stress as

and the physical surface element

dS= \S2\dBf,

the integral may then be split into two components

/ 04 T* \SS2\ dB,. + I k \S2 5rkdS, (45)JB JBwhere only the second term contains variations inthe flow variables and must consequently cancel the6w terms arising in the cost function. The first termwill appear in the expression for the gradient.

A general expression for the cost function that al-lows cancellation with terms containing STk has theform

/= I M(T}dS, (46)JB

corresponding to a variation

f 9AT/ "5—QTkdo,JB ork

for which cancellation is achieved by the adjointboundary condition

dtf*rk = ——— •

Natural choices for A/" 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 Xi direction is

CDfi = I ffijdSj = I S2jffij•IB JBBso that the force in a direction with cosines n-L hasthe form

= IJB

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

C,nf ~ I niTidS,JBso that basing the cost function (46) on this quantitygives

-A/" = HiTi.

Cancellation with the flow variation terms in equa-tion (45) therefore mandates the adjoint boundarycondition

4>k = nk.

Note that this choice of boundary condition alsoeliminates the first term in equation (45) so that itneed not be included in the gradient calculation.

In the inverse design case, where the cost functionis intended to measure the deviation of the surfacestresses from some desired target values, a suitabledefinition is

- rdk) ,

where Td is the desired surface stress, including thecontribution of the pressure, and the coefficients a//tdefine a weighting matrix. For cancellation

t

leading to the boundary condition

k = nk (rvn +p~ p,i).

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

Suppose that a locally minimum value of the costfunction/* - I(F*) is attained when T = f . Thenthe gradient Q* = G(F*) must be zero, while theHessian matrix A* — A(F*) must be positive defi-nite. Since Q* is zero, the cost function can be ex-panded as a Taylor series in the neighborhood of T*with the form

= r + (.F - F*) A (F - F*)

Correspondingly,

As T approaches T" ', the leading terms becomedominant. Then, setting T = (F — F*), the searchprocess approximates

dt -Also, since A" is positive definite it can be expandedas

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

RRT = RTR = /.

Settingv = RTF,

the search process can be represented as

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

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

In order to improve the rate of convergence, one canset

6F = -XPG,where P is a preconditioner for the search. An idealchoice is P = A"~l, so that the corresponding timedependent process reduces to

for which all the eigenvalues are equal to unity, andT is reduced to zero in one time step by the choiceAZ = 1. Quasi-Newton methods estimate ^4* fromthe change in the gradient during the search pro-cess. This requires accurate estimates of the gradi-ent at each time step. In order to obtain these, boththe flow solution and the adjoint equation must befully converged. Most quasi-Newton methods alsorequire a line search in each search direction, forwhich the flow equations and cost function mustbe accurately evaluated several times. They haveproven quite robust for aerodynamic optimization[6].An alternative approach which has also proved suc-cessful in our previous work [15], is to smooth thegradient and to replace Q by its smoothed value Qin the descent process. This both acts as a precondi-tioner, and ensures that each new shape in the opti-mization sequence remains smooth. It turns out thatthis approach is tolerant to the use of approximatevalues of the gradient, so that neither the flow solu-tion nor the adjoint solution need be fully convergedbefore making a shape change. This results in verylarge savings in the computational cost. For inviscidoptimization it is necessary to use only 15 multigridcycles for the flow solution and the adjoint solutionin each design iteration. For viscous optimization,about 100 multigrid cycles are needed. This is partlybecause convergence of the lift coefficient is muchslower, so about 20 iterations must be made beforeeach adjustment of the angle of attack to force thetarget lift coefficient. The new preconditioner forthe flow and adjoint calculations allows the numberof iterations to be substantially reduced in both theflow and the adjoint simulation.

The numerical tests so far have focused on theviscous design of wings for optimum cruise, forwhich the flow remains attatched, and the main vis-cous effect is due to the displacement thickness ofthe boundary layer. While some tests have beenmade with the viscous adjoint terms included, ithas been found that the optimization process con-verges when the viscous terms are omitted from theadjoint system. This may reflect the tolerance of thesearch process to inexact gradients.

10 RESULTS

Preconditioned Inverse Design

The first demonstration is an application of the pre-conditioning technique for inverse design with theEuler equations. The ONERA M6 (Figure Ib) wing

12

is recovered for a lifting case starting from a wingwith a NACA0012 section (Figure la) and usingthe ONERA M6 pressure distributions computed ata = 3.0 and M = 0.84 as the target (Fig. 2). Thus, asymmetric wing section is to be recovered from anasymmetric pressure distribution. The calculationswere performed on a 192x32x48 C-H mesh with294,912 cells. Each design cycle required 3 multi-grid cycles for the flow solver using characteristic-based matrix dissipation with a matrix precondi-tioner and 12 multigrid cycles for the adjoint solverusing scalar dissipation and a variable local timestep (scalar preconditioner). Compared to a test inwhich the 3 multigrid cycles using the matrix pre-conditioner were replaced by 15 multigrid cyclesusing a standard scalar preconditioner, and 15 cy-cles were used in adjoint solver, each design cyclerequired about 3/8 as much computer time, whilethe number of design cycles required to reach thesame level of error also fell from 100 to about 50.Use of the matrix preconditioner therefore reducedthe total CPU time on an IBM 590 workstation from97,683 sec (-27 hours) to 18,222 sec (-5 hours) forroughly equivalent accuracy.

Viscous Design

Due to the high computational cost of viscous de-sign, a two-stage design strategy is adopted. In thefirst stage, a design calculation is performed withthe Euler equations to minimize the drag at a givenlift coefficient by modifying the wing sections witha fixed planform. In the second stage, the pres-sure distribution of the Euler solution is used as thetarget pressure distribution for inverse design withthe Navier-Stokes equations. Comparatively smallmodifications are required in the second stage, sothat it can be accomplished with a small number ofdesign cycles.

In order to test this strategy it was used for the re-design of a wing representative of wide body trans-port aircraft. The results are shown in Figures 3and 4. The design point was taken as a lift coeffi-cient of .55 at a Mach number of .83. Figure 3 illus-trates the Euler redesign, which was performed ona mesh with 192x32x48 cells, displaying both thegeometry and the upper surface pressure distribu-tion, with negative Cp upwards. The initial wingshows a moderately strong shock wave across mostof the top surface, as can be seen in Figure 3a. Sixtydesign cycles were needed to produce the shock freewing shown in Figure 3b, with an indicated dragreduction of 15 counts from .0196 to .0181. Figure

4 shows the viscous redesign at a Reynolds num-ber of 12 million. This was performed on a meshwith 192x64x48 cells, with 32 intervals normal tothe wing concentrated inside the boundary layer re-gion. In Figure 4a it can be seen that the Euler designproduces a weak shock due to the displacement ef-fects of the boundary layer. Ten design cycles wereneeded to recover the shock free wing shown in Fig-ure 4b. It is interesting that the wing section modi-fications between the initial wing of Figure 3a andthe final wing of Figure 4b are remarkably small.

These results were sufficiently promising that itwas decided by McDonnell Douglas to evaluate themethod for industrial use, and it was used to sup-port design studies for the MDXX project. The re-sults of this experience are discussed in an accompa-nying paper [21]. It rapidly became apparent thatthe fuselage effects are too large to be ignored. Inviscous design it was also found that there were dis-crepancies between the results of the design calcula-tions, which were initially performed on a relativelycoarse grid with 192x64x48 cells, and the results ofsubsequent analysis calculations performed on finermeshes to verify the design.

In order to allow the use of finer meshes withovernight turnaround, the code was therefore mod-ified for parallel computation. Using the parallelimplementation, viscous design calculations havebeen performed on meshes with 1.8 million meshpoints. Starting from a preliminary inviscid design,20 design cycles are usually sufficient for a viscousre-design in inverse mode, with the smoothed in-viscid results providing the target pressure. Sucha calculation can be completed in about 7| hoursusing 48 processors of an IBM SP2.

As an illustration of the results that could be ob-tained, Figures 5 - 9 show a wing-body designwith sweep back of about 38 degrees at the 1/4chord. Starting from the result of an Euler design,the viscous optimization produced an essentiallyshock free wing at a cruise design point of Mach.86, with a lift coefficient of .6 for the wing bodycombination at a Reynolds number of 101 millionbased on the root chord. Figure 5 shows the designpoint, while the evolution of the design is shown inFigure 6, using Vassberg's COMPPLOT software.In this case the pressure contours are for the finaldesign. This wing is quite thick, with a thicknessto chord ratio of more than 14 percent at the rootand 9 percent at the tip. The design offers excellentperformance at the nominal cruise point. Figures 7and 8 show the results of a Mach number sweep

13

to determine the drag rise. It can be seen that adouble shock pattern forms below the design point,while there is actually a slight increase in the dragcoefficient of about 1 \ counts at Mach .85. Finally,Figure 9 shows a comparison of the pressure dis-tribution at the design point with those at alternatecruise points with lower and higher lift. The ten-dency to produce double shocks below the designpoint is typical of supercritical wings. This wing hasa low drag coefficient, however, over a wide rangeof conditions.

CONCLUSIONS

We 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 per-turbation technique which is applicable to arbitraryconfigurations. The combination of computationalefficiency with geometric flexibility provide a pow-erful tool, with the final goal being to create practicalaerodynamic shape design methods for completeaircraft configurations.

ACKNOWLEDGMENT

This work has benefited from the generous supportof AFOSR under Grant No. AFOSR-91-0391, theNASA-IBM Cooperative Research Agreement, andalso the Rhodes Trust.

REFERENCES

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

[4] A. Jameson. Automatic design of transonicairfoils to reduce the shock induced pressuredrag. In Proceedings of the 31st Israel Annual

Conference on Aviation and Aeronautics, Tel Aviv,pages 5-17, February 1990.

[5] J. Reuther and A. Jameson. Control based air-foil design using the Euler equations. /4MApaper 94-4272-CP, 1994.

[6] J. Reuther and A. Jameson. Aerodynamic shapeoptimization of wing and wing-body configu-rations using control theory. AIAA paper 95-0123, AIAA 33rd Aerospace Sciences Meeting,Reno, Nevada, January 1995.

[7] J. Reuther, A. Jameson, J. Farmer, L. Martinelli,and D. Saunders. Aerodynamic shape op-timization of complex aircraft configurationsvia an adjoint method. AIAA paper 96-0094,AIAA 34th Aerospace Sciences Meeting, Reno,Nevada, January 1996.

[8] O. Pironneau. Optimal Shape Design for EllipticSystems. Springer-Verlag, New York, 1984.

[9] S. Ta'asan, G. Kuruvila, and M. D. Salas. Aero-dynamic design and optimization in one shot.AIAA paper 92-005, 30th Aerospace SciencesMeeting and Exibit, Reno, Nevada, January1992.

[10] O. Baysal and M. E. Eleshaky. Aerodynamicdesign optimization using sensitivity anaysisand computational fluid dynamics. AIAA jour-nal, 30(3):718-725,1992.

[11] H. Cabuk, C.H. Shung, and V. Modi. Adjointoperator approach to shape design for internalincompressible flow. In G.S. Dulikravich, edi-tor, Proceedings of the 3rd International Conferenceon Inverse Design and Optimization in Engineer-ing Sciences, pages 391-404,1991.

[12] J.C. Huan and V. Modi. Optimum designfor drag minimizing bodies in incompressibleflow. Inverse Problems in Engineering, 1:1-25,1994.

[13] N.A. Pierce and M.B. Giles. Preconditioningcompressible flow calculations on stretchedmeshes. AIAA Paper 96-0889, 34th AerospaceSciences Meeting and Exhibit, Reno, NV, 1996.

[14] N.A. Pierce and M.B. Giles. Preconditionedmultigrid methods for compressible flow cal-culations on stretched meshes. Submitted to /.Comp. Phys., April, 1996.

[15] A. Jameson. Optimum aerodynamic design us-ing the Control Theory. Computational Fluid Dy-namics Review, pages 495-528,1995.

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[16] L. Martinelli and A. Jameson. Validation of amultigrid method for the Reynolds averagedequations. AlAA paper 88-0414,1988.

[17J 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. ofComp. FluidDyn., 4:171-218,1995.

[18] A. Jameson. Analysis and design of numericalschemes for gas dynamics 2, artificial diffusionand discrete shock structure. Int. }. of Comp.Fluid Dyn., To Appear.

[19] S. Tatsumi, L. Martinelli, and A. Jameson. Anew high resolution scheme for compressibleviscous flows with shocks. /i/AA paper 95-0466,AIAA 33nd Aerospace Sciences Meeting, Reno,Nevada, January 1995.

[20] 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.

[21] A. Jameson. Re-Engineering the design processthrough camputation. AIAA paper 97-0641, Jan-uary 1997.

15

Initial Wing. Cp on Upper Surface.

Figure la: M = .84, C, = .3000, Cd = .0205, Q = 2.935°.

Redisigned wing. Cp on Upper Surface.

Figure Ib: M = .84, C, = .2967, Cd = .0141, a = 2.935°

Figure 1: Redesign of the Onera M6 Wing. 1000 design cycles in inverse mode.

16

.•\

6 H

•'•-...

2a: span station z = 0.297

'

c2b: span station z = 0.484

* J

2c: span station 2 = 0.672

S J

2d: span station z = 0.859

Figure 2: Target and Computed Pressure Distributions of Redesigned Onera M6 Wing.M = 0.84, CL = 0.2967, CD = 0.0141, Q = 2.935°.

17

Initial Wing. Cp on Upper Surface.

Figure 3a: M = .83, Ct = .5498, Cd = .0196, a = 2.410°.

Redisigned wing. Cp on Upper Surface.

Figure 3b: M = .83, Cf = .5500, Cd = .0181, a = 1.959°.

Figure 3: Redesign of the wing of a wide transport aircraft. Stage 1 Inviscid design : 60 design cycles indrag reduction mode with forced lift.

18

Initial Wing. Cp on Upper Surface.

Figure 4a: M = 0.83, Ci = .5506, Cd = .0199, a = 2.317°

Redisigned wing. Cp on Upper Surface.iFigure 4b: M = 0.83, Ci = .5508, Cd = .0194, a = 2.355°

Figure 4: Redesign of the wing of a wide transport aircraft. Stage 2: Viscous re-design. 10 design cycles ininverse mode.

19

-1.0-0.8

-0.5

-O.J

U ().()0.2

0.5

0.8

1.0

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSMPX5X WING-BODY

REN = 101.00 , MACH = 0.860

SYMBOL SOURCE ALPHA CL CD——————— SYN»)7PI)hSIGN4« 2.0V4 (Ktlll O O l l ' f i

0.8 •

0.5 •

-0.2 -

Figure 5: Pressure distribution of the MPX5X at its design point.

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSMPX5X WING-BODY

REN = 101.00 , MACH = 0.860 , CL = 0.610 ,

SYMBOL SOURCE

SYNIDTPIJKSIGNO

ALPHA

2.251

CD

0.01 131

-0.8 •-0.5 •-0.2 '

J 0.00.2-0.5 '0.8

1.0

Figure 6: Optimization Sequence in the design of the MPX5X.

20

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSMPX5X WING-BODY

REN= lOl.(X) , CL =0.610

MHX5X [IKSIGN'10MI'XSX DKSIGN 40MPX5XUHSIGN4I)MPX5XDKSICN4I)MHX5XDKSIGN4I)

Figure 7: Off design performance of the MPX5X below the design point.

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSMPX5X WING-BODY

REN = 101.00 . CL = 0.610

Figure 8: Off design performance of the MPX5X above the design point.

21

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSMPX5X WING-BODYREN= 101.00 , MACH=0.«60

SYMBOL

—— •- —

SOURCE

MPX5X DHS1GN 40

ALPHA

2.3BO

CL

O.fifil

CD

0.01314 /

Figure 9: Comparison of the MPX5X at its design point and at lower and higher lift.

22