AIAA 2003–3438CFD for Aerodynamic Design andOptimization: Its Evolution overthe Last Three DecadesAntony JamesonDept of Aeronautics and AstronauticsStanford University, Stanford, CA

16th AIAA CFD ConferenceJune 23–26, 2003/Orlando, FL

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

CFD for Aerodynamic Design and Optimization: Its Evolution over

the Last Three Decades

Antony Jameson ∗

Dept of Aeronautics and Astronautics

Stanford University, Stanford, CA

1. Palm Springs

The AIAA First Computational Fluid Dynamics Con-ference, held in Palm Springs in July 1973, signifiedthe emergence of computational fluid dynamics (CFD)as an accepted tool for airplane design. The meetingwas a great success, despite the extreme heat. I havea lasting memory of the presentations of Jay Boris,who displayed the perfect advection of square waves byhis flux corrected transport (FCT) algorithm,1, 2 andof Joe Thompson, who showed meshes around rocksgenerated by the solution of elliptic equations. As aparticipant in the Palm Springs meeting who has re-mained active in the field, I welcome the opportunityto offer some remarks on the evolution of CFD duringthe last three decades.

My emphasis is on the development of computa-tional algorithms which can be used both for flowanalysis and aerodynamic design. I was interested inboth issues from the start of my own work in 1970.At that time we had no computational capability influid dynamics at all at Grumman Aerospace, where Iwas working, although Hess and Smith had announcedtheir panel method several years earlier. In order toget started I wrote two computer programs for idealtwo-dimensional potential flow, flo1 and syn1, bothbased on conformal mapping. The names were re-stricted to the three characters ‘flo’ and ‘syn’ becauseat that time fortran program names were restricted tosix characters, and since I already anticipated a seriesof codes, I wanted to allow for three numeric digits.

Flo1 calculates the flow past a given profile by The-ordorsen’s method. Syn1 solves the inverse problemof finding the profile corresponding to a specified tar-get pressure distribution by an extension of Lighthill’smethod. In developing syn1 I had the benefit of talk-ing to Malcolm James, who had written an inverseprogram at McDonnell Douglas which was used byLiebeck for the design of his well known high lift air-foils. My programs were written for the IBM 1130.This was an early precursor of the class of machineswhich came to be called minicomputers. It was aboutthe size of a refrigerator, and had only a few thousandwords of memory. Coding was restricted to a subsetof Fortran. Input was by punched cards, and output

∗Professor, Stanford UniversityCopyright c© 2003 by Antony Jameson. Published by the Amer-

ican Institute of Aeronautics and Astronautics, Inc. with permission.

by a line printer. There was no graphics capability.The calculations took 5-10 minutes. These codes havesurvived, and now run on a laptop computer in about1/50 second. Figure 1 illustrates a direct calculationby flo1 of the flow past a NACA 0012 airfoil. Figure 2illustrates an inverse calculation by syn1 in which theWhitcomb airfoil is recovered from its subsonic pres-sure distribution. The conformal mapping techniquesyield essentially exact results with quite a small num-ber of mesh points, of the order of 72.

NACA 0012 ALPHA 3.000

CL 0.3614 CD -0.0003 CM -0.0042

GRID 72

1.

20

0.80

0.

40

0.00

-0.

40 -

0.80

-1.

20 -

1.60

-2.

00

Cp

+

++

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

+

+

+

+

+

++

+

+

+

+++

++

++

++

++

++

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

+

Fig. 1 Direct calculation of flow past a NACA0012airfoil by flo 1

W 100-0 AIRFOIL ALPHA 0.000

CL 0.5924 CD -0.0009 CM -0.1486

GRID 80

1.

20

0.80

0.

40

0.00

-0.

40 -

0.80

-1.

20 -

1.60

-2.

00

Cp

-

+

++

+++++

+

+

+

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

++++++

+

++++++

+

+

+

+

+

+

+++++++

+

++

++

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

++

++

+

Fig. 2 Inverse calculation, recovering Whitcombairfoil

1 of 11

American Institute of Aeronautics and Astronautics Paper 2003–3438

2. The Importance of Transonic Flow

Flo1 and syn1 were never used at Grumman. Manyyears later I found them very useful for the develop-ment of hydrofoils designed to delay the onset of cav-itation. They were, however, a first step towards thedevelopment of methods to calculate transonic flow,which was the major challenge at that time. The com-pelling need both to predict transonic flow, and to gaina better understanding of its properties and character,continued to be a driving force for the development ofCFD through the period 1970-1990.

In the case of military aircraft capable of supersonicflight, the high drag associated with high g maneuversforces them to be performed in the transonic regime.In the case of commercial aircraft the importance oftransonic flow stems from the Breguet range equation.This provides a good first estimate of range as

R =V

sfc

L

Dlog

W0 + Wf

W0

(1)

Here V is the speed, L/D is the lift to drag ratio,SFC is the specific fuel consumption of the engines,W0 is the landing weight and Wf is the weight ofthe fuel burnt. The Breguet equation clearly exposesthe multi-disciplinary nature of the design problem.A light weight structure is needed to minimize W0.The specific fuel consumption is mainly the provinceof the engine manufacturers, and in fact the largest ad-vances during the last thirty years have been in engineefficiency. The aerodynamic designer should try tomaximize V L/D. This means that the cruising speedshould be increased until the onset of drag-rise due tothe formation of shock waves. Consequently the bestcruising speed is the transonic regime.

3. Transonic Potential Flow

Transonic flow had proved essentially intractable toanalytic methods. Garabedian and Korn had demon-strated the feasibility of designing airfoils for shock-free flow in the transonic regime by the method ofcomplex characteristics.3 Their method was formu-lated in the hodograph plane, and it required greatskill to obtain solutions corresponding to physicallyrealizable shapes. It was also known from Morawetz’stheorem4 that shock free transonic solutions are iso-lated points.

A major breakthrough was accomplished by Mur-man and Cole5 with their development of type-dependent differencing in 1970. They obtained stablesolutions by simply switching from central differenc-ing in the subsonic zone to upwind differencing inthe supersonic zone, and using a line-implicit relax-ation scheme. Their discovery provided major impetusfor the further development of CFD by demonstrat-ing that solutions for steady transonic flows could becomputed economically. Figure 3 taken from theirlandmark paper, illustrates the scaled pressure distri-

bution on the surface of a symmetric airfoil. Effortswere now underway to extend their ideas to more gen-eral transonic flows.

Fig. 3 Pressure distribution on the surface of asymmetric airfoil in transonic flow

In Palm Springs I presented the rotated differencescheme for the transonic potential flow equation forthe first time in two papers. The first11 was on thecalculation of the flow past a yawed wing, which wasthen being advocated by R.T. Jones as the most ef-ficient solution for supersonic transport aircraft. Thesecond12 was a joint paper with Jerry South on thecalculation of axisymmetric transonic flow.

The rotated difference scheme proved to be a veryrobust method, and it provided the basis for flo22, de-veloped with David Caughey during 1974-75 to predicttransonic flow past swept wings. At the time we wereusing the CDC 6600, which had been designed by Sey-mour Cray, and was the world’s fastest computer at itsintroduction, but had only 131000 words of memory.This forced the calculation to be performed one planeat a time, with multiple transfers from the disk. Flo22was immediately put into use at McDonnell Douglas.A simplified in-core version of flo22 is still in use atBoeing Long Beach today. Figure 4, supplied by JohnVassberg, shows the result of a recent calculation usingflo22 of transonic flow over the wing of a proposed air-craft to fly in the Martian atmosphere. The result wasobtained with 100 iterations on a 192x32x32 mesh in 7seconds, using a typical modern workstation. John in-forms me that when flo22 was first introduced at LongBeach the calculations cost $3000 each. Neverthelessthey found it worthwhile to use it extensively for theaerodynamic design of the C17.

By this time I had moved to the Courant Institute towork with Paul Garabedian and his group. We contin-ued to look for more efficient and accurate methods,and to try to gain a better understanding of issuessuch as numerical shock structure and prediction ofwave drag. This motivated the switch to equations inconservation form,15 and also the use of multigrid tech-

2 of 11

American Institute of Aeronautics and Astronautics Paper 2003–3438

SYMBOL

SOURCE FLO-22 + L/NM+S

ALPHA 6.700

CD .0319

CM -0.01225

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSBASELINE MARS00 FLYING WING CONFIGURATION

MACH = 0.650 , CL = 0.615

John C. Vassberg17:36 Fri13 Jun 03

COMPPLOTVer 2.00

Solution 1 Upper-Surface Isobars

( Contours at 0.05 Cp )

0.2 0.4 0.6 0.8 1.0

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 0.0% Span

0.2 0.4 0.6 0.8 1.0

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 14.6% Span

0.2 0.4 0.6 0.8 1.0

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 24.4% Span

0.2 0.4 0.6 0.8 1.0

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 39.0% Span

0.2 0.4 0.6 0.8 1.0

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0C

p

X / C 53.7% Span

0.2 0.4 0.6 0.8 1.0

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 63.4% Span

0.2 0.4 0.6 0.8 1.0

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 78.0% Span

0.2 0.4 0.6 0.8 1.0

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 92.7% Span

Fig. 4 Pressure distribution over the wing of aMars Lander using FLO22

niques, which were already being advocated by AchiBrandt.16 Many of the resulting improvements wereembodied in flo36, which solves the fully conservativepotential flow equations by a multigrid alternating di-rection method. Figure 5 shows a result for the NACA64A410 calculated in just three multigrid cycles.

NACA 64A410 MACH 0.720 ALPHA 0.000

CL 0.6609 CD 0.0028 CM -0.1469

GRID 96X16 NCYC 3 RES0.607E-05

1.

20

0.80

0.

40

0.00

-0.

40 -

0.80

-1.

20 -

1.60

-2.

00

Cp

++++

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

+

+++

+

+

+

++++++

++

++

++ + + +

++

++

+ + + + +

+

+

+

+ + + + ++

++

++++

+++++++

NACA 64A410 MACH 0.720 ALPHA 0.000

CL 0.6641 CD 0.0032 CM -0.1476

GRID 192X32 NCYC 3 RES0.938E-06

1.

20

0.80

0.

40

0.00

-0.

40 -

0.80

-1.

20 -

1.60

-2.

00

Cp

+++++++

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

+

+

+

+

+++

+

+

+

+

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

++++

+++++

+++++

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

+

+

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

Fig. 5 Pressure distribution over NACA 64A410in transonic flow after three multigrid cycles

David Caughey and I also developed a scheme tosolve transonic potential flow on arbitrary grids.17

The discretization formulas could be derived from theBateman variational principle that the integral of thepressure over the domain

I =

∫D

pdξ

is stationary.18 While we called it a finite volumescheme, it was essentially a finite element scheme usingtrilinear isoparametric elements, stabilized by the in-troduction of artificial viscosity to produce an upwindbias. The flow solvers (flo27-30) were subsequently in-corporated in Boeing’s A488 software, which was used

in the aerodynamic design of Boeing commercial air-craft throughout the eighties.19

In the same period Pierre Perrier was focusing theresearch efforts at Dassault on the development of fi-nite element methods using triangular and tetrahedralmeshes, because he believed that if CFD software wasto be really useful for aircraft design, it must be ableto treat complete configurations. Although finite ele-ment methods were more computationally expensive,and mesh generation continued to present difficulties,finite element methods offered a route towards theachievement of this goal. The Dassault/INRIA groupwas ultimately successful, and they performed tran-sonic potential flow calculations for complete aircraftsuch as the Falcon 50 in the early eighties.20 This wasa major achievement which had a significant impacton the thinking of the CFD community world wide. Itplaced Dassault clearly at the fore-front in the indus-trial application of CFD.

4. The Euler and Navier-Stokes Equations

By the eighties advances in computer hardware hadmade it feasible to solve the full Euler equations us-ing software which could be cost-effective in industrialuse. The idea of directly discretizing the conservationlaws to produce a finite volume scheme had been in-troduced by MacCormack.13 Most of the early flowsolvers tended to exhibit strong pre- or post-shockoscillations. Also, in a workshop held in Stockholmin 1979,14 it was apparent that none of the existingscheme converged to a steady state. These difficultieswere resolved during the following decade.

The Jameson-Schmidt-Turkel21 scheme, which usedRunge-Kutta time stepping and a blend of second- andfourth-differences (both to control oscillations and toprovide background dissipation), consistently demon-strated convergence to a steady state, with the con-sequence that it has remained one of the widely usedmethods to the present day.

A fairly complete understanding of shock captur-ing algorithms was achieved, stemming from the ideasof Godunov, Van Leer, Harten and Roe. The issueof oscillation control and positivity had already beenaddressed by Godunov in his pioneering work in the1950s (translated into English in 1959). He had intro-duced the concept of representing the flow as piecewiseconstant in each computational cell, and solving aRiemann problem at each interface, thus obtaining afirst-order accurate solution that avoids non-physicalfeatures such as expansion shocks. When this workwas eventually recognized in the West, it became veryinfluential. It was also widely recognized that nu-merical schemes might benefit from distinguishing thevarious wave speeds, and this motivated the develop-ment of characteristics-based schemes.

The earliest higher-order characteristics-basedmethods used flux vector splitting,6 but suffered from

3 of 11

American Institute of Aeronautics and Astronautics Paper 2003–3438

oscillations near discontinuities similar to those ofcentral difference schemes in the absence of numericaldissipation. The Monotone Upwind Scheme forConservation Laws (MUSCL) of Van Leer7 extendedthe monotonicity-preserving behavior of Godunov’sscheme to higher order through the use of limiters.The use of limiters dates back to the flux-correctedtransport (FCT) scheme of Boris and Book.2 A gen-eral framework for oscillation control in the solution ofnon-linear problems is provided by Harten’s conceptof Total Variation Diminishing (TVD) schemes.

Roe’s introduction of the concept of locally lineariz-ing the equations through a mean value Jacobian8

had a major impact. It provided valuable insightinto the nature of the wave motions and also enabledthe efficient implementation of Godunov-type schemesusing approximate Riemann solutions. Roe’s flux-difference splitting scheme has the additional benefitthat it yields a single-point numerical shock struc-ture for stationary normal shocks. Roe’s and otherapproximate Riemann solutions, such as that dueto Osher, have been incorporated in a variety ofschemes of Godunov type, including Essentially Non-Oscillatory (ENO) schemes of Harten, Engquist, Osherand Chakravarthy.9 It finally proved possible to givea rigorous justification of the JST scheme.21

Fast multigrid solution methods were also devel-oped, typically using generalized Runge Kutta24 26

or LU25 implicit methods with some type of pre-conditioning. It has recently proved possible to re-fine the LUSGS multigrid method to the point wheresteady state Euler solutions can be obtained in 3-5cycles.27 This allows two dimensional calculations ona 160 x 32 grid to be performed in 1/2 second on aPC with a 2GHz Pentium 4 processor, and three di-mensional calculations on a 192 x 32 x 32 grid in 23seconds. Figure 6 shows a result for the RAE 2822airfoil.

-2

-1.5

-1

-0.5

0

0.5

1

1.50 0.2 0.4 0.6 0.8 1

Pre

ssur

e C

oeffi

cien

t, C

_p

Chordwise station, x/c

RAE 2822 Airfoil; Mach 0.75, 3.0 degrees

Pressure Coeff.Delta p_0 (x 10)

Fig. 6 Transonic flow past RAE 2822 airfoil atMach 0.75, 3.0 degrees incidence. 2 Solution withH-CUSP scheme after three multigrid cycles. Solidline (——): Fully converged solution.

In 1980 I had moved to Princeton. There, motivated

AIRPLANE DENSITY from 0.6250 to 1.1000

Fig. 7 Density contours for the A-320

AIRPLANE DENSITY from 0.6250 to 1.1000

Fig. 8 Density contours for the MD-11

by the successes at Dassault, we also mounted a majoreffort to develop a method to solve the Euler equationson unstructured meshes, and were finally able to cal-culate the flow past a complete Boeing 747, includingflow through the nacelles, at the end of 1985 with the”AIRPLANE” code.28 This software was heavily usedin the NASA supersonic transport program and con-tinues to be used at the present time. Current versionsuse a multigrid algorithm with fully parallel operationon multiple CPUs. This enables an airplane calcula-tion on a mesh with 2 million cells to be performed inabout 30 seconds. Figures 7, 8, 9 show flow simulationsof some commercial aircraft in transonic flight.

4 of 11

American Institute of Aeronautics and Astronautics Paper 2003–3438

AIRPLANE CP from -0.8000 to 0.5000

Fig. 9 Pressure contours for the Boeing 747-200

Solution methods for the Reynolds averaged Navier-Stokes (RANS) equations had been pioneered in theseventies by MacCormack and others, but at that timethey were extremely expensive. By the nineties com-puter technology had progressed to the point whereRANS simulations could be performed with manage-able costs, and they began to be fairly widely usedby the aircraft industry, using codes such as Buning’sOVERFLOW. There were also major efforts on bothsides of the Atlantic to improve the ability to pre-dict hypersonic flow, stemming from the Hermes andNASP projects. Figures 10 and 11 shows a Hermessimulation performed with the LUSGS scheme.25

Fig. 10 CFD calculation of Hermes Spacecraft,Mach 8 and 30 degrees angle of attack, black is free-stream, yellow-red the Mach number range from3-6, and green-white the range from Mach numberrange from 3 to 0

Fig. 11 CFD calculation of Hermes Spacecraft,Comparison of Mach number distribution for in-viscid (A) and viscous (B) flow

5. Aerodynamic Shape Design

The effective use of CFD for design ultimately requiresanother level of software which can guide the designerin the search for improved aerodynamic shapes onthe basis of the predicted aerodynamic performance.Hicks and Henne made a first attempt at using nu-merical optimization techniques in the late seventies.29

Pironneau had also investigated the problem of opti-mum shape design for elliptic equations by 1984.30

I had revisited the issue of shape design several timessince I originally wrote syn1 in 1970, and I actuallywrote a program for transonic inverse design which wasused by Grumman. In my first years at Princeton I su-pervised a thesis by John Fay31 on inverse design usingthe Euler equations. In 1988 I realized that one couldcombine CFD with control theory to calculate opti-mum shapes after attending a meeting on flow controlsponsored by ICASE. I was able to derive the adjointequations for transonic potential flow and the Eulerequations which allowed the extraction of the Frechetderivative (infinitely dimensional gradient) at the costof one flow and one adjoint solution.33

I was certain these ideas would work and publishedthem without attempting to demonstrate them numer-ically. In the following year I implemented the adjointmethod for design in transonic potential flow and thefirst result appeared in Science.32 This is reproducedin figure 12, which shows the redesign of the RAE 2822airfoil to minimize its drag coefficient, subject to theconstraints that the lift coefficient is held constant atapproximately 1.0, and the thickness is not reduced.As can be seen, an almost shock-free profile was ob-tained in 5 cycles. In order to guarantee a sequence ofsmooth profiles, I smoothed the gradient by an implicitprocedure at each step. This process, which is equiv-alent to redefining the gradient to correspond to aninner product in a Sobolev space35 is a key ingredientin the success of the method.

The adjoint method has been refined over the lastdecade,40 43 42 44 39 41 and extended to the Eulerand Navier Stokes equations with numerous collabora-tors including Luigi Martinelli, James Reuther, Juan

5 of 11

American Institute of Aeronautics and Astronautics Paper 2003–3438

Fig. 12 Redesign of the RAE 2822 airfoil by meansof control theory to reduce its shock-induced pres-sure drag. (A) Initial profile. Drag coefficient of0.0175. (B) Redesigned profile after five cycles.Drag coefficient of 0.0018.

Alonso, John Vassberg, Sangho Kim, Siva Nadarajah,Kasidit Leoviriyakit and Sriram. Theoretical issuesconnected with the treatment of shock waves and prop-erties of the Hessian have been addressed by Gilesand Pierce,36 Matsuzawa and Hafez37 and Arian andTa’asan.38

Control theory now provides an effective tool forwing design. Figures 13,14 show the results of NavierStokes redesigns of the Boeing 747 wing at its presentcruising Mach number of .86, and also at a higher Machnumber of .90. These calculations are for the wing-fuselage combination, with shape changes restrictedto the wing. In each case the planform was held fixed,while section changes was subject to the constraint ofmaintaining the same thickness. The lift coefficientand also the span load distribution were constrainedto be fixed during the optimization, so that the rootbending moment would not be increased, and the sus-ceptibility to buffet would not be impaired due to anincrease in the lift coefficient of the outboard sections.

At Mach .86 the drag coefficient is reduced from126.9 counts (.01269) to 113.6 counts, a reduction ofabout 5 percent of the total drag of the aircraft. AtMach .90 it is reduced from 181.9 counts to 129.3counts. Thus the redesigned wing has about the samedrag at Mach .90 as the original wing at Mach .86,suggesting the potential for a significant increase inthe cruise Mach number, provided that other prob-lems such as engine integration could also be solved.Since both the wing thickness and span-load distri-bution are maintained there should be no penalty instructure weight or fuel volume. The required changesare quite subtle and there would be no hope of findingthem by wind tunnel testing.

Recently, we have extended this design methodol-ogy to unstructured grids, using the same discretiza-

tion scheme as the AIRPLANE code. The newsoftware SYNPLANE has been used to redesign theFalcon business jet in the cruise condition. Fig-ures 15, 16, 17, 18 show the density contours on thesurface of the aircraft and pressure distribution atthree span-wise locations on the existing wing. Theresults of a drag minimization that removes the shockson the wing are shown in figures 19, 20, 21, 22. Thedrag has been reduced from 235 counts to 215 countsin about 8 design cycles,while the lift is held fixed at0.4 and the thickness is maintained.

6. Reflections on the Future

Today CFD can be routinely used for the analysis ofcomplex flows, and CFD simulation of attached flowsare certainly accurate enough for performance predic-tions. The overall progress that has been achievedduring the last 30 years was unimaginable in 1970.A major factor has been the astonishing rate of im-provement of computers, so that modern laptops havea performance equivalent to the super-computers offifteen years go. But intellectual contributions suchas advances in algorithms have had a roughly equalimpact.

I consider the problems of both transonic wing anal-ysis and design to be essentially solved, although thereis clearly room for improvement. In the light of thevast volume of ongoing research world-wide, we cancertainly anticipate continuing advances in algorithms,particularly in the areas of higher order methods anderror estimation. Higher order reconstruction meth-ods become very complex and expensive on the generalunstructured meshes which are likely to be needed totreat very complex geometries. Consequently the dis-continuous Galerkin method is currently attracting alot of interest as a way to achieve high order accuracywith a compact discretization stencil. Methods basedon kinetic gas models such as the lattice Boltzmannmethod may also offer advantages for the treatment ofsome complex flows.

There are also numerous engineering applicationsthat have yet to be adequately solved. These in-clude three dimensional high lift systems, the flowthrough a helicopter rotor in forward flight, internalflows through jet engines (including compressor, com-bustor, turbine, and cooling flows), and the externalaerodynamics of automobiles. These flows are partic-ularly challenging because they are generally unsteady(at least in the smaller scales), and involve transition,turbulence and separation.

As computers continue to become more powerful,it is likely that there will be a shift to the wider useof Large Eddy Simulation (LES) and Direct NumericalSimulation (DNS) methods for turbulent flows. It maybe hard, however, for engineers to interpret the hugevolumes of data generated by these methods in a waythat will provide them with the insights needed to en-

6 of 11

American Institute of Aeronautics and Astronautics Paper 2003–3438

able better designs. It also remains an open questionwhether more rational turbulence modeling procedurescan be devised.

In choosing a direction of research I believe that itis generally useful to consider four main criteria. Theresearch should be generic, not limited to a single spe-cial case. It should be intellectually challenging. Itshould be feasible, and it should be useful. Viewedin this light I think it is evident that shape optimiza-tion procedures based on control theory can be appliedto a variety of important engineering problems (forexample, reduction of the resistance of a ship hull,or radar and sonar signatures). The general aerody-namic shape optimization problem is hard, presentinga true intellectual challenge, but by now it has beenclearly demonstrated that it is feasible. In fact wing re-designs using the Euler equations can be accomplishedin 5 minutes on a laptop computer. If it is effectivelyexploited in the design process, I believe that aerody-namic shape optimization can be really useful.

The accumulated experience of the last decade sug-gests that most existing aircraft which cruise at tran-sonic speeds are amenable to a drag reduction of theorder of 3-5 percent, or an increase in the drag-riseMach number of at least 0.02. The potential economicbenefits are substantial, considering the fuel costs ofthe entire airplane fleet. Moreover, if one were to takefull advantage of the improvement in the lift to dragratio during the design process, a smaller aircraft couldbe designed to perform the same task, with consequentfurther cost reductions. It seems inevitable that somemethod of this type will provide a basis for aerody-namic designs in the future.

Acknowledgment

The results presented here were the outcome of col-laborations with many colleagues and friends both inuniversities and in industry. The author’s researchduring the last ten years on optimum aerodynamicshape design has also benefited greatly from the con-tinuing support of the Air Force Office of Scientificresearch under a series of grants. This paper has beenprepared with the assistance of Kasidit Leoviriyakitand Sriram.

References1Book D. L. and Boris J., Flux Corrected Transport: A Min-

imum Error Finite-Difference Technique Designed for VectorSolution of Fluid Equations, Proceedings of the AIAA Compu-tational Fluid Dynamics Conference, Palm Springs, July 1973,pp. 182-189.

2Book D. L. and Boris J., Flux Corrected Transport, 1SHASTA, A Fluid Transport Algorithm Works, Journal of Com-putational Physics, 11, 38-69.

3Bauer F., Garabedian P. and Korn D., A theory of Super-critical Wing Sections, with Computer Programs and Examples,Lecture Notes in Economics and Mathematical Systems 66,Springer Verlag, New York.

4Morawetz C. S., On the non-existence of Continuous Tran-

sonic Flows Past Profiles, Part 1, Communications in Pure andApplied Math, 9, pp. 45-68.

5Murman E. M., Cole J. D., Calculation of plane steadytransonic flows, AIAA 1974;12:626-33

6Steger J. and Warming R., Flux Vector Splitting of theInviscid Gas Dynamics Equations with Applications to FiniteDifference Methods Journal of Computational Physics, 40, pp.263-293.

7Van Leer B., Towards the Ultimate Conservative Differ-ence Scheme. II: Monotonicity and Conservation combined ina Second-order scheme, Journal of Computational Physics, 14,pp. 361-70.

8Roe P. L., Approximate Reimann Solvers, ParameterVectors, and Difference Schemes, Journal of ComputationalPhysics, 43, pp. 357-372.

9Chakravarthy S.,Harten A. and Osher S., Essentially Non-Oscillatory Shock Capturing Schemes of Uniformly Very HighAccuracy, AIAA Paper 86-0339, Reno, Nevada, 1986.

10Jameson A., Iterative solution of transonic flows over air-foils and wings, including flow at Mach 1., Commum Pure ApplMath 1974;27:238-309

11Jameson A., Numerical Calculations of the Three-Dimensional Flow over a Yawed Wing, Proceedings ofthe AIAA Computational Fluid Dynamics Conference, PalmSprings, July 1973, pp. 18-26.

12Jameson A. and South J.C., Relaxation Solutions for In-viscid Axisymmetric Transonic Flow over Blunt or PointedBodies, Proceedings of the AIAA Computational Fluid Dynam-ics Conference, Palm Springs, July 1973, pp. 8-17.

13MacCormack R. W and Paullay A. J, Computational Effi-ciency achieved by time splitting of finite difference operators,AIAA Paper 72-154, 1972.

14Rizzi A. and Viviand H. Eds, Numerical Methods for theComputation of Inviscid Transonic Flows with Shock Waves: AGAMM Workshop, Vieweg and Sohn, Braunschwig.

15Jameson A., Transonic Potential Flow Calculations UsingConservation Form, Proceedings of the Second AIAA Compu-tational Fluid Dynamics Conference, Hartford, June 1975, pp.148-161.

16Brandt. A, Multi-level adaptive solutions to boundary valueproblems, Math Comput 1977, 31:333-90

17A. Jameson and D. Caughey, A Finite Volume Methodfor Transonic Potential Flow Calculations, AIAA Paper 77-635,Proceedings of the Third AIAA Computational Fluid DynamicsConference, Alburquerque, June 1977, pp. 35-54.

18A. Jameson, Remarks on the Calculation of Transonic Po-tential Flow by a Finite Volume Method, Proceedings of IMAConference on Numerical Methods in Applied Fluid Dynamics,Reading, January 1978, edited by B. Hunt, Academic Press,1980, pp. 363-386.

19Rubbert P. E., The Boeing Airplanes that have benefitedfrom Antony Jameson’s CFD Technology, Frontiers of Compu-tational Fluid Dynamics, 1994, Eds. D.A. Caughey and M.M.Hafez.

20Bristeau M. O., Glowinski R., Periaux J., Perrier P., Piron-neau O., Poirier C., On the numerical solution of nonlinearproblems in fluid dynamics by least square and finite elementmethods(II), application to transonic flow simulations, ComputMethods Appl Mech Eng 1985;51:363-94.

21Jameson A, Schmidt W and Turkel E, Numerical Solutionof the Euler equations by finite volume methods using Runge-Kutta time stepping schemes, AIAA Paper 81-1259, June, 1981.

22Jameson A, Analysis and Design of Numerical Schemes forGas Dynamics-I, International Journal of Computational FluidDynamics, Vol. 4, 1995, pp. 171-218.

23Jameson A, Analysis and Design of Numerical Schemesfor Gas Dynamics-II, International Journal of ComputationalFluid Dynamics, Vol. 5, 1995, pp. 1-38.

7 of 11

American Institute of Aeronautics and Astronautics Paper 2003–3438

24Jameson A., Solution of the Euler Equations for Two Di-mensional Transonic Flow by a Multigrid Method MAE ReportNo. 1613, 1983.

25Rieger. H and Jameson A, Solution of Steady Three-Dimensional Compressible Euler and Navier-Stokes Equationsby an Implicit LU Scheme, AIAA Paper 88-0619, AIAA 26thAerospace Sciences Meeting, Reno, January, 1988

26Jameson A., Mavriplis D J and Martinelli L, Multi-grid Solution of the Navier-Stokes Equations on TriangularMeshes ICASE Report 89-11, AIAA Paper 89-0283, AIAA 27thAerospace Sciences Meeting, Reno, January, 1989.

27A. Jameson and D. A. Caughey, How Many Steps areRequired to Solve the Euler Equations of Steady CompressibleFlow: In Search of a Fast Solution Algorithm, AIAA 2001-2673,15th AIAA Computational Fluid Dynamics Conference, June11-14, 2001, Anaheim, CA.

28A. Jameson ,T. J. Baker, and N. P. Weatherill, Calculationof Inviscid Transonic Flow Over a Complete Aircraft, AIAAPaper 86-0103, AIAA 24th Aerospace Sciences Meeting, Reno,January 1986.

29R. M. Hicks and P. A. Henne, W ing design by numericaloptimization, Journal of Aircraft, Vol 15, pp. 407–412, 1978.

30Pironneau O., Optimal shape design for elliptic system,New York: Springer, 1984

31John Fay, Princeton University Thesis, 1985.32Jameson A., Computational Aerodynamics for Aircraft De-

sign, Science, Vol. 245, pp. 361-371.33Jameson A., Aerodynamic design via control theory, J Sci

Comput 1988;3:233-6034A. Jameson and J. C. Vassberg, Computational Fluid Dy-

namics for Aerodynamic Design: Its Current and Future Im-pact, AIAA 2001-0538, 39th AIAA Aerospace Sciences Meeting& Exhibit, January 8-11, 2001, Reno, NV.

35A. Jameson, Sriram and Luigi Martinelli, A continuous ad-joitn method for unstructured grids, AIAA 2003-3955, Orlando,Fl, 2003.

36M. B. Giles and N. A. Pierce, Analytic solutions for the qusione-dimensional Euler equations, Journal of Fluid Mechanics,426:327-345, 2001.

37T. Matsuzawa and M. Hafez, Optimum shape design usingadjoint equations for compressible flows with shock waves CFDJournal Vol. 7, No. 3, 1998, pp. 343-36.

38E. Arian and S. Ta’asan, Analysis of Hessian for Aerody-namic Optimization: Inviscid Flow, ICASE Report 96-28, 1996.

39Jameson A., Optimum Aerodynamic Design via Bound-ary Control, RIAC Technical Report 94.17, Princeton UniversityReport MAE 1996, Proceedings of AGARD FDP/Von KarmanInstitute Special Course on ”Optimum Design Methods in Aero-dynamics”, Brussels, April 1994, pp. 3.1-3.33.

40Jameson A., Optimum Aerodynamic Design Using ControlTheory, Computational Fluid Dynamics Review, 1995, pp. 495-528.

41Jameson A.,L. Martinelli,N. Pierce, Optimum Aerody-namic Design using the Navier Stokes Equation Theoretical andComputational Fluid Dynamics, Vol. 10, 1998, pp213-237.

42A. Jameson, J. Alonso, J. Reuther, L. Martinelli,J. Vass-berg, Aerodynamic Shape Optimization Techniques Based onControl Theory, AIAA paper 98-2538, 29th AIAA Fluid Dy-namics Conference, Alburquerque, June 1998.

43A. Jameson and Luigi Martinelli, Aerodynamic Shape Op-timization Techniques Based on Control Theory, CIME (Inter-national Mathematical Summer Center), Martina Fran-ca, Italy,June 1999.

44Siva K. Nadarajah and Antony Jameson, Optimal Con-trol of Unsteady Flows using a Time Accurate Method, AIAA-2002-5436, 9th AIAA/ISSMO Symposium on MultidisciplinaryAnalysis and Optimization Conference, September 4-6, 2002,Atlanta, GA.

8 of 11

American Institute of Aeronautics and Astronautics Paper 2003–3438

SYMBOL

SOURCE SYN107 DESIGN 50SYN107 DESIGN 0

ALPHA 2.258 2.059

CD 0.01136 0.01269

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSB747 WING-BODY

REN = 100.00 , MACH = 0.860 , CL = 0.419

Antony Jameson14:40 Tue

28 May 02COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13COMPPLOT

JCV 1.13MCDONNELL DOUGLAS

Solution 1 Upper-Surface Isobars

( Contours at 0.05 Cp )

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 10.8% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 27.4% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 41.3% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 59.1% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 74.1% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 89.3% Span

Fig. 13 Comparison of Chordwise pressure distributions before and after redesign, Re=100 million,Mach=0.86, CL=0.42

SYMBOL

SOURCE SYN107 DESIGN 50SYN107 DESIGN 0

ALPHA 1.766 1.536

CD 0.01293 0.01819

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSB747 WING-BODY

REN = 100.00 , MACH = 0.900 , CL = 0.421

Antony Jameson18:59 Sun 2 Jun 02

COMPPLOTJCV 1.13

COMPPLOTJCV 1.13

COMPPLOTJCV 1.13

COMPPLOTJCV 1.13

COMPPLOTJCV 1.13

COMPPLOTJCV 1.13

COMPPLOTJCV 1.13

COMPPLOTJCV 1.13

COMPPLOTJCV 1.13

COMPPLOTJCV 1.13

MCDONNELL DOUGLAS

Solution 1 Upper-Surface Isobars

( Contours at 0.05 Cp )

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 10.8% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 27.4% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 41.3% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 59.1% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 74.1% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Cp

X / C 89.3% Span

Fig. 14 Comparison of Chordwise pressure distributions before and after redesign, Re=100 million,Mach=0.90, CL=0.42

9 of 11

American Institute of Aeronautics and Astronautics Paper 2003–3438

AIRPLANE DENSITY from 0.6250 to 1.1000

Fig. 15 Density contours for a business jet at M =

0.8, α = 2

FALCON MACH 0.825 ALPHA 2.000 Z 6.50

CL 0.5767 CD 0.0266 CM -3.4644

NNODE 353887 NDES 0 RES0.204E-03

0.1E

+01

0.8E

+00

0.4E

+00

-.1E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++++++++++

++++

++++

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

++

+++

+

+

+

++

+

+

+

+

++++++

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

+++++

+

+

+

++

++++ +++

++++++

+

Fig. 16 Pressure distribution at 66 % wing span

FALCON MACH 0.825 ALPHA 2.000 Z 7.00

CL 0.5694 CD 0.0235 CM -3.7159

NNODE 353887 NDES 0 RES0.204E-03

0.1E

+01

0.8E

+00

0.4E

+00

-.1E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++++++

++++++

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

++++

+

+

+

+++

+

+

+

+

+++++

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

++++

+

+

+

+

++

+ ++++ ++

++

Fig. 17 Pressure distribution at 77 % wing span

FALCON MACH 0.825 ALPHA 2.000 Z 7.50

CL 0.5513 CD 0.0194 CM -3.9202

NNODE 353887 NDES 0 RES0.204E-03

0.1E

+01

0.8E

+00

0.4E

+00

-.1E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

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

++++

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

++

++

+

+

+

++

++

+

++

++++++

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

+++ +

++

+

++

+ +++ ++ +++

Fig. 18 Pressure distribution at 88 % wing span

10 of 11

American Institute of Aeronautics and Astronautics Paper 2003–3438

AIRPLANE DENSITY from 0.6250 to 1.1000

Fig. 19 Density contours for a business jet at M =

0.8, α = 2.3, after redesign

FALCON MACH 0.800 ALPHA 2.298 Z 6.00

CL 0.5346 CD 0.0108 CM -0.1936

NNODE 353887 NDES 7 RES0.658E-03

0.1E

+01

0.8E

+00

0.4E

+00

0.0E

+00

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++++++

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

+++

++

++

++

+

+

+

++

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

+++

+++

++

+++

++

+++ ++++

++oooooo

ooooooooooooooooooooooooooooooooooooooooooo

oo

ooo

oo

o

oo

o

oo

oo

ooooo

ooooo ooo oo ooo oo oo o o ooo

oooo

oo

ooooo

ooo

ooooo

o

Fig. 20 Pressure distribution at 66 % wing span,after redesign, Dashed line: original geometry,solid line: redesigned geometry

FALCON MACH 0.800 ALPHA 2.298 Z 7.00

CL 0.5417 CD 0.0071 CM -0.2090

NNODE 353887 NDES 7 RES0.658E-03

0.1E

+01

0.8E

+00

0.4E

+00

0.0E

+00

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++

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

+

+

+++++

+

+

+

+

+

+

+

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

+++++

+ ++++

++

+++

+

+++ + ++++ooooooo

ooooooooooooooooooooooooooooooooooo

o

oo

oo

ooo

o

o

o

o

o

ooooo

oooo ooo oo oo o o o ooooo o

ooo

ooo

ooo

oooo

ooo

oo

Fig. 21 Pressure distribution at 77 % wing span,after redesign, Dashed line: original geometry,solid line: redesigned geometry

FALCON MACH 0.800 ALPHA 2.298 Z 8.00

CL 0.4909 CD 0.0028 CM -0.1951

NNODE 353887 NDES 7 RES0.658E-03

0.1E

+01

0.8E

+00

0.4E

+00

0.0E

+00

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++++++

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

+++

+

+

+

+

++++

+

+

+

++

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

+++ ++

+

+++

++

++

++++ +

++ooooooooooooooooooooooooooooooooooooo

oo

o

o

o

ooo

o

o

o

oo

oo

oo oo oo oo ooo oo oo o ooo ooo

ooo

ooo

oo

oooo

oo

Fig. 22 Pressure distribution at 88 % wing span,after redesign, Dashed line: original geometry,solid line: redesigned geometry

11 of 11

American Institute of Aeronautics and Astronautics Paper 2003–3438