aerodynamic shape optimization part ii: sample...

Aerodynamic Shape OptimizationPart II: Sample Applications

John C. Vassberg ∗

Phantom WorksThe Boeing Company

Huntington Beach, CA 92647-2099, USA

Antony Jameson †

Department of Aeronautics and AstronauticsStanford University

Stanford, CA 94305-3030, USA

Von Karman Institue

Brussels, Belgium9 March, 2006

Nomenclature

A Hessian Matrix / Operator

AR Wing Aspect Ratio = b2

Sref

b Wing Span

B Shape Function Basis

CFD Computational Fluid Dynamics

CD Drag Coefficient = Dragq∞Sref

CL Lift Coefficient = Liftq∞Sref

Cref Wing Reference Chord

count Drag Coefficient Unit = 0.0001

F Surface Defining Function

G Gradient of Cost Function

H Estimate of Inverse Hessian Matrix

HP Horse Power

I Objective or Cost Function

KEAS Knots Equivalent Air Speed

MPH Mile Per Hour

N Number of Design Variables

R Flow-Equation Function

RANS Reynolds-Averaged Navier-Stokes

RCS Reaction Control System

Re Wing Reynolds number based on Cref

Reθ Attachment Line Reynolds number

Sref Wing Reference Area

UAV Unmanned Aerial Vehicle

x Independent Spatial Variable

q Dynamic Pressure = 1

2ρV 2

w Flow Variable

λ Wing Taper Ratio; Search Step Parameter

Λc/4 Wing Quarter-Chord Sweep

∞ Infinity

δ∗ First Variation of

O(∗) Order of

(∗)−1 Inverse Matrix of

∗Boeing Technical Fellow†T. V. Jones Prof. of Engineering

Copyright c© 2006 by Vassberg & Jameson.

Published by the VKI with permission.

1 Introduction

This is the second of two lectures prepared by theauthors for the von Karman Institute that deal withthe subject of aerodynamic shape optimization. Inour first lecture we introduced some theoretical back-ground on optimization techniques commonly usedin the industry, applied these approaches to a coupleof very simple model problems, compared the resultsof these schemes, and discussed their merits and def-ficiencies as they relate to the class of aerodynamicshape optimization problems the authors deal withon a regular basis. In this lecture, we illustrate howthe gradient of a complex system of nonlinear partialdiffererential equations can be obtained for about thesame computational cost as that of the cost function,and we provide a set of sample applications.

In an airplane design environment, there is no needfor an optimization based purely on the aerodynam-ics of the aircraft. The driving force behind (almost)every design change is related to how the modifica-tion improves the vehicle, not how it enhances anyone of the many disciplines that comprise the design.Although we focus this lecture on the aerodynamicsof an airplane, we also include the means by whichother disciplines are linked into and affect the aero-dynamic shape optimization subtask. Another char-acteristic of the problems we typically (but not al-ways) work on is that the baseline configuration isitself within 1-2% of what may be possible, given theset of constraints that we are asked to satisfy. Thisis certainly true for commercial transport jet aircraftwhose designs have been constantly evolving for thepast half century or more. The sample applicationsprovided herein do not fall into this category. Quiteoften the problem can be very constrained; this isthe case when the shape change is required to be aretrofitable modification that can be applied to air-

Vassberg & Jameson, VKI Lecture-II, Brussels, Belgium, 9 March, 2006 1 of 41

craft already in service. Occasionally, we can beginwith a clean slate, such as in the design of an all-newairplane. And the problems cover the full spectrumof studies in between these two extremes.

Let’s note a couple of items about this setting.First, in order to realize a true improvement to thebaseline configuration, a high-fidelity and very ac-curate computational fluid dynamics (CFD) methodmust be employed to provide the aerodynamic met-rics of lift, drag, pitching moment, spanload, etc.Even with this, measures should be taken to estimatethe possible error band of the final analyses; this dis-cussion is beyond the scope of these lectures. Fig-ures 1-2 illustrate the class of aircraft and the levelof detail the first author addresses every day. TheseNavier-Stokes CFD solutions are conducted on full-up cruise configurations, complete with wing, fuse-lage, engine groups, empennage, flap-support fair-ings, and winglets. The engine groups include a py-lon, nacelle, core-cowl, shelf, and bifurcation flows.Although not obvious in these images, various filletsare also included. Finally, the CFD calculations areperformed at prescribed lifting conditions by alteringangle-of-attack, and are trimmed to specified center-of-gravity locations by adjusting the horizontal tailincidence. This level of detail is needed to achieve anaccuracy on the absolute performance of the aircraftthat is within 1% of flight test data. However, this iswhat is required to improve the performance of theaircraft by 1-2% without a numerical optimizationyielding a false positive. The second item to con-sider is related to the definition of the design space.A common practice is to use a set of basis functionswhich either describe the absolute shape of the geom-etry, or define a perturbation relative to the baselineconfiguration. In order to realize an improvement tothe baseline shape, the design space should not beartificially constrained by the choice of the set of ba-sis functions. This can be accomplished with either asmall set of very-well-chosen basis functions, or witha large set of reasonably-chosen basis functions. Theformer approach places the burden on the user to es-tablish an adequate design space, the later approachplaces the burden on the optimization software toeconomically accommodate problems with large de-grees of freedom. Over the past decade, the authorshave focused on solving the problem of aerodynamicshape optimization utilizing a design space of verylarge dimension. Our principal motivation for ad-dressing the problem of large number of design vari-ables is two fold. The first is to provide a situationwhere the design space never needs to be artificiallyconstrained. The second is to allow us the flexibilityto automatically set up the design space within theoptimization software at the highest dimensionalitysupported by the discrete numerical simulation. In

doing so, the aerodynamic shape optimization soft-ware based on these concepts allow the user to runoptimizations immediately after set up of the anal-ysis inputs are complete. This speeds time-to-first-optimization and minimizes the human errors asso-ciated with defining a design space. Furthermore,aerodynamic shape optimizations based on either theEuler or Navier-Stokes equations can be run on rel-atively inexpensive computer equipment.

The next section provides an overview of aerody-namic optimization. We develop an efficient evalua-tion for the gradient; this is based on solving an ad-joint equation. A brief review of the search methodswe utilize are then included. Following this discus-sion, we present a few selected case studies. Thesesample applications are all on design activities thatwe have been involved with; they include a Mars air-craft, a Reno Racer, and an aero-structural optimi-zation of a generic B747 wing/body configuration.

2 Aerodynamic Design Trades

The objective of aerodynamic design is to producea structurally feasible shape with sufficient carryingcapacity, which achieves good aerodynamic perfor-mance.

For example, consider the generic task of deliveringa payload between distant city pairs. The BreguetRange equation, which aptly applies to long-rangemissions of jet aircraft, is:

Range =ML

D

a

SFCln

(

W0 +Wf

W0

)

. (1)

Here, M is the cruise Mach number, L & D are theaerodynamic forces of lift and drag, respectively, ais the acoustic speed, SFC is the specific fuel con-sumption of the engines, W0 is the aircraft landingweight, and Wf is the weight of fuel burned duringthe flight. The Breguet Range equation illustratesthe importance of drag prediction as a function oflift and Mach number in the context of aerodynamicdesign; it also provides a glimpse into the interplaybetween the various disciplines.

Referring to Eqn (1), one might assume that theaerodynamic efficiency of an aircraft is representedby ML

D, the propulsion efficiency is embedded in

SFC, and that the structural efficiency directly im-pacts W0. Interestingly, historical trends of in-service transport aircraft indicate that very little im-provement in the ML

Dmetric has been accomplished

in the past 40 years. Yet it would be somewhatnaive to state that no aerodynamic advances havebeen made during this period. In actuality, advancesin aerodynamics have better served aircraft designswhen traded for improvements in other disciplines.


For example, the ability to increase the thickness-to-chord ratio of a wing while maintaining ML

Dnot only

reduces the structural weight of the wing, it also pro-vides additional fuel volume. In terms of Eqn (1),an aerodynamic improvement of this nature wouldmanifest itself as a decrease in W0 and an increasein Wf with the net result being an increase in range.Reducing the aircraft’s empty weight has the addedbenefit of reducing the cost of the vehicle. Obvi-ously, this aerodynamic improvement would not beapparent in the trend charts of ML

D.

Assume that an airline would like to provide a ser-vice between two cities with an aircraft that, whenfully loaded with payload and fuel, is 1% short onrange. Since the aircraft is fuel-volume limited, theonly recourse is to reduce the payload weight. Inrelative terms, a typical ratio of weights might haveWf = 2

3W0 and Wpayload = 1

6W0. In this scenario,

Eqn (1) shows that the operator would have to re-duce the payload (read revenue) by 7.6% to recoverthe 1% shortfall on range. Since most airlines op-erate on very small margins, this service most likelywill no longer be a profit-generating venture. Thisexample illustrates that in the current business offlight, a 1% delta in aircraft performance is a sig-nificant change. While improving an aircraft’s per-formance by 1% may not be a trivial task given theusual constraints, losing 5% is easily done if attentionis not paid to details such as juncture flows, externaldoublers, gaps, etc.

Now consider a more typical case where the air-craft does not suffer from a shortfall on range. Inround numbers, the Direct Operating Costs (DOC)of a transport aircraft can be itemized as: 50% forthe cost of ownership, 20% for fuel burn, 20% forcrew salaries and maintanence, and 10% for miscel-laneous other items. From an airline’s perspective, ifthe DOC of its fleet of aircraft could be reduced by5% with a new design (while providing the same setof services to its customers), the airline would mostlikely retire its entire fleet and replace it with thenew aircraft [1].

So how can aerodynamics be leveraged to improvethe economics associated with a flight-based mission?By enabling the development of a simplified high-lift-system design that for a given L

Dand CL max reduces

manufacturing and maintanence costs as well as partcount. By increasing the cruise Mach number with-out reducing ML

Dreduces the time-dependent costs

such as crew and maintanence. Also, by increasingMLD

without penalizing the other disciplines reducesfuel burn. These are just a few examples that illus-trate how aerodynamic advances would impact onDOC. A common requirement for achieving thesegoals is the accurate prediction of drag.

To push aerodynamic technologies forward, it is

becoming more important that accurate drag predic-tion become a consistent product of the CFD com-munity. Once this prerequisite is accomplished, thefull benefits of automated aerodynamic shape opti-mization may begin to be realized.

With the various on-going design programs, theseare exciting times for the aircraft industry. A primeexample is the Blended-Wing-Body (BWB) whichhas established a renaissance in the design of a familyof all-new aircraft [2]. This revolutionary concept isenabling aerodynamic advances in all of the aboveareas, and then some. It presents challenges, yetoffers significant opportunities, and as a result, a 5%reduction in DOC is within grasp. Suffice it to saythat aerodynamics is not a sunset technology, butrather, it is as important today as it was a centuryago; only the stakes have changed.

3 Aerodynamic Optimization

Traditionally the process of selecting design varia-tions has been carried out by trial and error, rely-ing on the intuition and experience of the designer.The degree of success with this classical approach de-pends directly on the level of expertise of the aero-dynamic designer, which can take over a decade ofapprenticeship to develop. It is not at all likely thata similar process of repeated trials in an interactivedesign and analysis procedure can lead to a truly op-timum design. In order to take full advantage of thepossibility of examining a large design space, the nu-merical simulations need to be combined with auto-matic search and optimization procedures. This canlead to automatic design methods which will fullyrealize the potential improvements in aerodynamicefficiency.

An approach which has become increasingly pop-ular is to carry out a search over a large numberof variations via a genetic algorithm. This mayallow the discovery of (sometimes unexpected) opti-mum design choices in very complex multi-objectiveproblems, but it becomes extremely expensive wheneach evaluation of the cost function requires intensivecomputation, as is the case in aerodynamic problems.

In order to find optimum aerodynamic shapes withreasonable computational costs, it pays to embed theflow physics within the optimization process. In fact,one may regard an aerodynamic shape as a device tocontrol the flow in order to produce a specified liftwith minimum drag. As a result, one can draw onconcepts which have been developed in the math-ematical theory of control of systems governed bypartial differential equations. In particular, an ac-ceptable aerodynamic design must have characteris-tics that do not abruptly vary with small changes in


shape and flow conditions. Consequently, gradient-based procedures are appropriate for aerodynamicshape optimization. Two main issues affect the effi-ciency of gradient-based procedures; the first is theactual calculation of the gradient, and the secondis the construction of an efficient search procedurewhich utilizes the gradient.

3.1 Gradient Calculation

For the class of aerodynamic optimization problemsunder consideration, the design space is essentiallyinfinitely dimensional. Suppose that the perfor-mance of a system design can be measured by a costfunction I which depends on a function F(x) thatdescribes the shape, where under a variation of thedesign, δF(x), the variation of the cost is δI. Nowsuppose that δI can be expressed to first order as

δI =

∫

G(x)δF(x)dx

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

δF(x) = −λG(x)

one obtains an improvement

δI = −λ

∫

G2(x)dx

unless G(x) = 0. Thus the vanishing of the gradientis a necessary condition for a local minimum. Here,λ is a positive value that scales the step size of thesearch trajectory through the design space.

Computing the gradient of a cost function for acomplex system can be a numerically intensive task,especially if the number of design parameters is largeand if the cost function is an expensive evaluation.The simplest approach to optimization is to definethe geometry through a set of design parameters,which may, for example, be the weights αi appliedto a set of shape functions Bi(x) so that the shape isrepresented as

F(x) =∑

αiBi(x).

Then a cost function I is selected which might be thedrag coefficient or the lift-to-drag ratio; I is regardedas a function of the parameters αi. The sensitivities∂I∂αi

may now be estimated by making a small varia-tion δαi in each design parameter in turn and recal-culating the flow to obtain the change in I. Then

∂I

∂αi

≈I(αi + δαi) − I(αi)

δαi

.

The main disadvantage of this finite-difference ap-proach is that the number of flow calculations needed

to estimate the gradient is proportional to the num-ber of design variables [3]. Similarly, if one resortsto direct code differentiation (ADIFOR [4, 5]), orcomplex-variable perturbations [6], the cost of de-termining the gradient is also directly proportionalto the number of variables used to define the design.Even small problems of aerodynamic shape optimiza-tion based on these approaches can require computeresources that are measured in CPU-Years, whichcan only be completed in reasonable elapsed timethrough utilization of massively-parallel computerscosting millions of dollars.

A more cost effective technique is to compute thegradient through the solution of an adjoint problem,such as that developed in [7, 8, 9]. The essentialidea may be summarized as follows. For flow aboutan arbitrary body, the aerodynamic properties thatdefine the cost function are functions of the flowfieldvariables (w) and the physical shape of the body,which may be represented by the function F . Then

I = I(w,F)

and a change in F results in a change of the costfunction

δI =∂IT

∂wδw +

∂IT

∂FδF .

Using a technique drawn from control theory, thegoverning equations of the flowfield are introducedas a constraint in such a way that the final expres-sion for the gradient does not require reevaluation ofthe flowfield. In order to achieve this, δw must beeliminated from the above equation. Suppose thatthe governing equation R, which expresses the de-pendence of w and F within the flowfield domain D,can be written as

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

Then δw is determined from the equation

δR =

[

∂R

∂w

]

δw +

[

∂R

∂F

]

δF = 0.

Next, introducing a Lagrange multiplier ψ, we have

δI =∂IT

∂wδw+

∂IT

∂FδF−ψT

([

∂R

∂w

]

δw +

[

∂R

∂F

]

δF

)

.

With some rearrangement

δI =

(

∂IT

∂w− ψT

[

∂R

∂w

])

δw+

(

∂IT

∂F− ψT

[

∂R

∂F

])

δF .

Choosing ψ to satisfy the adjoint equation

[

∂R

∂w

]T

ψ =∂IT

∂w(3)


the term multiplying δw can be eliminated in thevariation of the cost function, and we find that

δI = GδF ,

where

G =∂IT

∂F− ψT

[

∂R

∂F

]

.

The advantage is that the variation in cost function isindependent of δw, with the result that the gradientof I with respect to any number of design variablescan be determined without the need for additionalflow-field evaluations.

In the case that equation (2) is a partial differ-ential equation, the adjoint equation (3) is also apartial differential equation and appropriate bound-ary conditions must be determined. As it turns out,the appropriate boundary conditions depend on thechoice of the cost function, and may easily be de-rived for cost functions that involve surface-pressureintegrations. Cost functions involving field integralslead to the appearance of a source term in the adjointequation.

The cost of solving the adjoint equation is com-parable to that of solving the flow equation. Hence,the cost of obtaining the gradient is comparable tothe cost of two function evaluations, regardless of thedimension of the design space. The downside of thisapproach is that it can take man-months to developan adjoint code for a given cost function. However,there is on-going research at Rice University to de-velop ADJIFOR [10] which automatically generatesa discrete adjoint code from existing analysis soft-ware. So far, however, this has not realized the samelevel of efficiency. In the present work, the adjointequations have been derived analytically and thenapproximated in discrete form.

3.2 Search Procedure

The remaining cost issue is related to finding a loca-tion in the design space where the gradient vanishes,and hence where there is a local optimum. Nor-mally, this search starts from a baseline design andthe design space is traversed by a search method.The final state of the search may be subject to con-straints imposed on the design space, yet there is norequirement that the trajectory adhere to these ex-cept at its end point. The efficiency of the searchdepends on the number of steps it takes to find alocal minimum as well as the cost of each step.

In order to accelerate the search, one may resort tousing the Newton method. Here, the search directionis based on the equation represented by the vanish-ing of the gradient, G(F) = 0, and is solved by thestandard Newton iteration for nonlinear equations.

Suppose the Hessian is denoted by

A =∂G

∂F

then the result of a step δF may be linearized as

G(F + δF) = G(F) +A δF

This is set to zero for a Newton step; therefore

δF = −A−1G.

The Newton method is generally very effective if theHessian can be evaluated accurately and cheaply.Unfortunately, this is not the case with aerodynamicshape optimization.

Quasi-Newton methods estimate A or A−1 fromthe changes of G recorded during successive steps.For a discrete problem with N design variables, itrequires N steps to obtain a complete estimate ofthe Hessian, and these methods have the propertythat they can find the minimum of a quadratic formin exactly N steps. Thus in general, the cost of aquasi-Newton search scales with the dimension of thedesign space. For the class of optimizations underconsideration, this is an undesirable property.

Rank-2 quasi-Newton schemes have an additionalcost associated with the line search of each step,where multiple function evaluations (5-10) are re-quired to locate the minimum along a fixed searchdirection. This additional cost is further amplified bya requirement that each of these function evaluationsbe converged to tighter tolerance than that typicallyneeded for engineering-level accuracy. As a conse-quence, it has been our experience that quasi-Newtonsearches are not particularily suited for the class ofoptimizations that are addressed here. However, al-ternative techniques have been developed that cannavigate the design space quite effectively; the foun-dation of our search method is described next.

Efficient aerodynamic shapes are predominatelysmooth. This suggests a natural alternative ap-proach to the search method. In order to make surethat each new shape in the optimization sequenceremains smooth, one may smooth the gradient andreplace G by its smoothed value G in the descent pro-cess. This is equivalent to reformulating the gradientin a Sobelov space [11], and acts as a preconditionerwhich allows the use of much larger steps. To ap-ply smoothing in the ξ1 direction, for example, thesmoothed gradient G may be calculated from a dis-crete approximation to

G −∂

∂ξ1ǫ∂

∂ξ1G = G (4)

where ǫ is the smoothing parameter. Then, if one setsδF = −λG, assuming the modification is applied on


the surface ξ2 = constant, the first order change inthe cost function is

δI = −

∫ ∫

GδF dξ1dξ3

= −λ

∫ ∫ (

G −∂

∂ξ1ǫ∂G

∂ξ1

)

G dξ1dξ3

= −λ

∫ ∫

(

G2 + ǫ

(

∂G

∂ξ1

)2)

dξ1dξ3

< 0.

Thus, an improvement is assured for arbitrarychoices of the smoothing parameter ǫ if λ is suf-ficiently small and positive, unless the process hasalready reached a stationary point at which G = 0,and therefore, according to Equation 4, G = 0. (Notethat the λ acting on G can be significantly larger thanthat acting on G, with respect to stability limits.)

It turns out that this approach is extremely toler-ant to the use of approximate values of the gradient,so that neither the flow solution nor the adjoint solu-tion need be fully converged before making a shapechange. This results in very large savings in the com-putational cost of the complete optimization process,which is discussed next.

3.3 Computational Costs

In order to address the issues of the search costs,a variety of techniques were investigated in Refer-ence [12] using a trajectory optimization problem(the brachistochrone) as a representative model. Thestudy verified that the search cost (i.e., number ofsteps) of a simple steepest descent method appliedto this problem scales as N2, where N is the numberof design variables, while the cost of quasi-Newtonmethods scaled linearly with N as expected. On theother hand, with an appropriate amount of smooth-ing, the smoothed descent method converged in afixed number of steps, independent of N . Consid-ering that the evaluation of the gradient by a finitedifference method requires N + 1 flow calculations,while the cost of its evaluation by the adjoint methodis roughly that of two flow calculations, one arrivesat the estimates of total computational cost given inTables 1-2.

Table 1: Cost of Search Algorithm.

Steepest Descent O(N2) stepsQuasi-Newton O(N) stepsSmoothed Gradient O(K) steps(Note: K is independent of N)

Table 2: Total Computational Cost of Design.

Finite Difference Gradients+ Steepest Descent O(N3)Finite Difference Gradients+ Quasi-Newton Search O(N2)Adjoint Gradients+ Quasi-Newton Search O(N)Adjoint Gradients+ Smoothed Gradient Search O(K)

(Note: K is independent of N)

Other methods have been investigated which fur-ther improve the dimension-independent conver-gence rate, including multigrid and postconditioningwith a Krylov subspace acceleration. Implementingthese in the current aerodynamic shape optimizationsoftware consistently converges the design to a lo-cal optimum within 30-60 steps, even for problemswith thousands of design variables. Moreover, be-cause they do not require either the flow or adjointsolutions to be fully converged, complete optimiza-tions are routinely completed with a computationalcost equivalent to 2-10 converged flow solutions. Asa consequence, our standard practice is to allow ev-ery discrete surface point within the CFD grid to beits own design function, aligned with the grid lineemanating from the surface. The amplitude of thisdesign variable corresponds to the signed distancefrom the original baseline surface node. This typi-cally leads to design space dimensions of N > 4000for three-dimensional aerodynamic shape optimiza-tions.

Not being constrained by the number of designvariables enables the optimization software to bewritten so that the end user is not burdened withthe task of defining a set of shape functions. Thisadded benefit is a significant improvement over thestandard approach. A common practice throughoutindustry is to require a user to specify shape func-tions of the bump-function class which must be tai-lored for each specific application. The main reasonfor using bump functions is to reduce the numberof design variables needed and yet obtain reasonableresults. Specifying an appropriate set of bump func-tions for a given problem is somewhat of a black artin itself, and getting an effective set frequently re-quires experimentation by the user. The process isalso prone to input error that may not be discovereduntil an optimization run fails to produce reasonableresults. This is a poor environment for both userand design, as the ”optimized” design will only be asgood as the user is at choosing his shape-function


set and accurately inputting it. Hence, the needfor an expert user is mandated by this approach.Optimization software should free the engineer frommundane tasks and allow him to focus on the moreglobal requirements of the system development. Thishas been achieved by the present aerodynamic shapeoptimization software.

The remaining sections of this lecture are devotedto sample applications associated with three casestudies: a Mars aircraft, a Reno Racer, and an aero-structural optimization of a generic B747 wing/bodyconfiguration.

4 MARES Development

Aircraft based remote sensing has proven to be an ex-cellent method for large scale geologic analysis andsurveying on earth. The same techniques can be ap-plied to planetary science for planets containing anatmosphere. Aircraft-based sensors can cover a muchgreater surface area than rovers, at a much greaterresolution than orbiting platforms, with flight-pathcontrol not possible using balloon systems. Ground-based aircraft operations usually require preparedlaunch sites, which are not normally available forextra-terrestrial exploration missions. Atmosphericaircraft deployment techniques, as developed by theNaval Research Labratory (NRL) during many yearsof unmanned aerial vehicle (UAV) research and de-velopment, can be used to eliminate the need for sur-face launch sites.

The Mars Airborne Remote Exploration Scout(MARES) is developed specifically for the con-straints of an extra-terrestrial exploration aircraftusing an atmospheric entry deployment. The over-all aircraft shape maximizes the airframe size thatcan be packaged in a traditional conical entry shelland requires a minimum number of deployable sur-faces, enhancing the system simplicity and reliabil-ity. The high angle-of-attack characteristics of thedelta wing coupled with a reaction control system(RCS) allow a non-parachute assisted deploymentwith a minimum loss of altitude and a minimumover speed, giving greater flexibility in the selectionof sites to be explored. To further minimize tech-nical risk, MARES is powered by a reliable light-weight rocket engine. Optimizing the airframe forrocket propulsion leads to a higher cruise speed anda more structurally robust aircraft than very low-speed propeller-driven gossamer-type aircraft. Thehigh durability enhances survival of the sensor andcommunications systems after landing, allowing sen-sor data to be stored during flight and transmittedafter landing, minimizing the impact of data commu-nication rates on sensor selection and resolution. The

higher flight speeds, coupled with the low density ofthe Martian atmosphere, also lead to the requirementfor wing design optimization in the unique environ-ment of transonic low Reynolds number flight.

The initial configuration layout and baseline geom-etry were created by the NASA/NRL/ITT devel-opment team. Further refinement of this geometrywas conducted by the authors and is documented inReference [13]. Aerodynamic design and shape opti-mization of the MARES wing-surface geometry wasconducted using a number of computational fluid dy-namics (CFD) methods, including FLO22 [14, 15],CFL3D [16], SYN88 and SYN107P; this lecture willconcentrate on the simulations associated with thesynthesis codes SYN88 and SYN107P, which are ca-pable of both analysis and design calculations.

The MARES aircraft configuration is shownstowed in the entry aeroshell in Figure 3 and in itsflight configuration in Figures 4-5. Deployment ofthe MARES begins with the folded aircraft being re-leased from the aeroshell, where it falls in a flat atti-tude enabled by the natural stability provided by thedelta planform and the location of the center of mass,which is augmented by the zero axial velocity controlcapability provided by the RCS. The main propul-sion system solid rocket motor is ignited to begintransition of the aircraft into forward flight withoutrequiring a nose-down body attitude. As the for-ward flight speed reaches approximately 30 m/s, thedrag forces acting on the vertical stabilizer surfacesare sufficient to begin the aerodynamic deploymentof the outer wing panels. As the forward speed ap-proaches the cruise flight speed (150 m/s), the RCSis no longer needed and flight control is transferredto conventional aerodynamic control surfaces. At thecompletion of the data collection flight, the MARESaircraft will be pitched up into a nose-high (deep-stall) attitude. The delta planform coupled with theRCS will maintain the aircraft in a stable attitudeand descent rate until reaching a predetermined al-titude above the ground, when an additional z-axisthruster will fire to minimize vertical velocity at themoment of impact. The MARES will contact theground ventral fins first, in a nose-high attitude. Thefins are designed to crush and provide additional en-ergy absorption during impact to provide a relativelysoft landing.

Because of the constraints associated with packag-ing the aircraft in the conical entry shell, the config-uration has evolved into a flying-wing design, withfolding wings. When deployed, the outboard wingplanform has a quarter-chord sweep of only 5.5 de-grees. Figure 6 provides the planform of MARES,and the following list specifies reference quantitiesand sweep angles.

In order to scope the problem and work out any


Table 3: MARES Planform Quantities.

Sref 36.38 ft2

b 13.38 ft

Cref 3.28 ft

Xref 3.28 ft

Y ref 1.51 ft

AR 4.9

λ 0.3

Λc/4 5.5◦

ΛLE 10.0◦

ΛLE.∆ 50.0◦

issues with the baseline geometry definition, SYN88was initially applied. SYN88 is an inviscid aerody-namic shape optimization code based on the Eulerequations and corresponding adjoint formulae. Theresults of this Euler-based optimization are providedin Figures 7-8. While the outcome of this exerciseis somewhat academic given the nature of the taskat hand, nonetheless, it is worth noting that SYN88is capable of solving this class of problem on a note-book computer in about 30 minutes of CPU time.Utilizing this code to sort out any possible issueswith the problem set-up can greatly accelerate thetime to final RANS-based design.

The final detailed aerodynamic design of theMARES wing is accomplished through application ofSYN107P by minimizing total drag at a fixed lift co-efficient, while maintaining the spanload and airfoilthickness distributions of the baseline configuration.The thickness constraint is imposed as a greater thanor equal to condition. The design flight condition forthe MARES mission is: M = 0.65, CL = 0.62, andRe = 170K, where Reynolds number is based onreference chord. The flow is assumed to be fully tur-bulent over the wing surface. The rarified Martianatmosphere design condition also has that the den-sity and viscosity are: ρ = 2.356 ∗ 10−5 slugs/ft3,and ν = 2.2517∗10−7 slugs/ft/sec. Our initial anal-ysis of the baseline configuration indicated that theoriginal selection of airfoils (with leading-edge droopsand aft-camber reflex modifications to trim the air-craft) yielded an undesirable, strong, unswept shocksystem.

SYN107P is an aerodynamic synthesis methodbased on the Reynolds-Averaged Navier-Stokes(RANS) equations and corresponding continuous-adjoint formulae. SYN107P runs in parallel ondistributed-memory computer systems by using theMessage-Passing Interface (MPI). The work per-formed herein utilizes a Linux cluster of 8 nodes,where each node is comprised of dual AMD Athlon1.7GHz processors and 2GB RAM. The nodes areinterconnected with a switched 100BaseT network.Running on 8 processors (half the cluster), a RANSanalysis requires less than 30 minutes of elapsedtime; optimizations complete in less than 5 hours.

These wall-clock times correspond to grid dimensionsof (257 x 64 x 49) and with a design space definedwith 5,313 basis functions.

Figure 9 provides the history of total drag duringthe Navier-Stokes optimization process. The base-line wing-only configuration has a cruise drag of 592counts, while the optimized design converges to acruise drag of 480 counts. Figure 10 gives the cor-responding history of the lift-to-drag ratio, whichstarts at about 10.4 and finishes at 12.8; an improve-ment of 23%. Figure 11 over-plots the chordwisepressure distributions of the baseline geometry (solidcurves) and optimized shape (dashed curves). It isevident from this comparison that the wing leading-edge pressure peaks are being reduced and movedforward, both of which act to soften the adversepressure gradient of the upper-surface recovery path.The effects of the upper-surface changes are also il-lustrated in the side-by-side comparison of isobars ofFigure 12. In addition, a lower-surface leading-edgepeak just outboard of the baseline wing midspan re-gion has been eliminated by the optimizer, whichreshaped the Cp distribution to be monotonically fa-vorable to about 35% chord.

Figure 13 gives a comparison of the drag loops forthe baseline and optimized wings. It is often useful toview pressure distributions in this manner to betterunderstand how changes in geometry and pressuresinterplay to effect an improvement in drag. For ex-ample, the thrust lobes (negative areas of the dragloops) of the optimized design are noticably largerthan those of the baseline configuration.

Although the drag minimization of the presentstudy is conducted as a single-point optimization,Figure 14 illustrates that the optimized design clearlyout-performs the baseline at all lifting conditions; thesolid symbol data is at the cruise design point. Thisresult is especially interesting as both planform andspanload are common to the initial and final designs,hence their associated induced drags must be compa-rable. This leaves shock drag, profile drag, and possi-bly skin-friction drag as the components contributingto the performance improvements. Upon closer in-spection of the results, the improvements are madein both the shock and profile drag components. Fig-ure 15 provides the lift curves for the baseline andredesigned configurations. This figure indicates thatthe boundary-layer health of the baseline has beengreatly improved with the optimized design, hencethe related improvement in profile drag.

Several baseline and optimized geometry airfoilsections are decomposed into thickness and camberdistributions and are compared in Figures 16-18. Al-though it is counter-intuitive to increase airfoil thick-ness when attempting to reduce the drag of a wing,that is precisely what SYN107P did at the wing root.


In addition, the camber levels across the span havebeen reduced, and in particular, the leading-edgedroop of the mid-to-outboard sections have been sig-nificantly reduced.

Regarding the utility of SYN107P as an aerody-namic shape optimization design tool, this exercisehighlights several key attributes that benchmark thestate-of-the-art. A few of these are as follows.

• Problem set-up time is essentially only thatwhich is required to prepare an input deck for ananalysis run. Since the set of basis functions isautomatically created within the code, the userexpends absolutely no effort on this task, and anoptimization run to minimize drag at a fixed liftis as simple as changing the number of designcycles in the input deck from a 0 (for analysis-only) to a non-zero number (say 50) for an opti-mization to be performed. If the optimizationis an inverse design, then the user will have tospecify the target pressure distribution.

• The automatically set-up design space is essen-tially infinitely dimensional. To be precise, thecontinuum design space is represented by thehighest possible discrete space that can be sup-ported by the surface-defining grid. The key at-tribute here is that the design space of the opti-mization is not artificially nor arbitrarily con-strained. This can occur when a user is requiredto specify a very low dimension design spacewith a much reduced set of basis functions.

• The cost of a fully converged optimization is lessthan or equivalent to that of about 10 convergedanalyses. To emphasize this point, consider thecomputer time required to generate the drag po-lars of Figure 14; this represents about twicethe computer time as that needed for the dragminimization run. Further, both analysis andoptimization capabilities run adequately fast onvery affordable computer equipment, allowingmultiple optimizations to be performed withina single day.

• Thickness constraints relative to the initialgeometry are automatically set-up by the codeand can be globally controlled by one input pa-rameter.

The four properties listed above have been de-veloped to the stage that they can be imple-mented into a general setting, such as that for anunstructured-mesh RANS aerodynamic shape opti-mization method. In addition to these, there area few other niceties related to SYN107P. These aremade possible because this code is tailored for very

specific aircraft geometries, namely wing-out-of-a-wall, or wing-body configurations. These featuresinclude the following.

• Automatic grid generation is available for bothEuler and RANS simulations.

• A variety of spanload constraints are available,which are either based on the initial distributionor a blending of elliptic and linear spanloads.

• Calculations can be performed at specified lift-ing conditions.

All of the properties listed above have enabled theauthors to perform the present detailed design studyfor the MARES wing under minimal funding levels.

5 Reno Racer

Once a year, Reno, Nevada plays host to an air showlike none other; this of course is the Reno air races.Spectators from all over the world converge on thisremote site to witness man and machine competewith one another in a series of races which culminatewith the unlimited class events. The only hard rulerequired of an unlimited class race plane is that it bea propeller design and powered by a piston engine.Most entries are modified warbirds of WW-II vintagesuch as North American P-51 Mustangs, or HawkerSea Furys, with power plants that produce well inexcess of 1, 000 HP. Miss Ashley II and Rare Bearof Figure 19 are representative of the unlimited classReno racer. These aircraft are now over 50 years oldand have very little service life left. To be competi-tive in the unlimited class, these aircraft see less thanone hour between engine rebuilds, and these over-hauls can cost upwards of $250K. Furthermore, thehistoric value of flying WW-II fighters has increasedso much that they are becoming unsuited for racinguse. If unlimited class racing is to continue throughthe next decade, new race plane designs are required.

A goal of a new unlimited class design would beto significantly push the performance envelope ofpropeller-aircraft technologies. The mission of thisspecial-purpose vehicle is to minimize the lap timearound the 8.3 mile unlimited race course at Reno,depicted as the largest oval in Figure 20. Previouswinners of this race have achieved average speedsaround the oval course in excess of 450 MPH. Thedesign requirements of the current development ef-fort called for an average race speed of about 550MPH. Because of this average velocity and the geom-etry of the race course, the aircraft pulls 4Gs about60% of the time, 1G about 20% of the time, andtransitions between these loads the remaining 20%of the time. Furthermore, because of the nature of


this racing environment, 7G maneuvers are typicallyencountered to avoid mid-air collisions. While thereare many other factors to consider during the designof this class an aircraft, the aforementioned loadingconditions set the stage for the aerodynamic shapeoptimization of this vehicle. More precisely, the mis-sion emphasizes that multi-point optimizations beperformed over a weighted range of lifting conditionsand on-set Mach numbers.

The design objectives of the Reno racer are givenwith respect to a standard day at the race locationwhich is at 5000’ MSL and ISA +20◦ C. The topspeed in straight and level flight is to exceed 600MPH TAS. The average lap speed around the 1999unlimited race course is to exceed 550 MPH TAS.The aircraft is to be capable of sustaining a 9G ma-neuver load, subject to a 5G gust load; yielding a 14Glimit load with a 1.5X safety factor. Roll rate shouldexceed 200◦ per sec at 350 KEAS. Stall speed shouldbe less than 90 KEAS. Landing distance should notexceed 1500’, dead stick. Note that some of these re-quirements are more stringent than the performanceof some state-of-the-art jet fighters.

The design requirements state that the aircraftmust be piston powered and propeller driven. Enginepower-to-weight ratio for reliability at continuousoutput should be about 2.5 HP/lb. for a turbo-charged piston engine with gear reduction and otheraccessories. The stability & control is to be providedby a manual, unboosted system with positive static& dynamic margins that exceed current unlimited-class race planes. There should be minimal changein stability between power on and off. For crew pro-visions, the design allows dual pilots in a tandemseating arrangement, with seats inclined 30◦ for Gtolerance, and include MIL-SPEC oxygen and G-suitconnections. Low altitude ejection for both pilots isalso required.

The development of this aircraft began from theground up, as an all-new design. Every major ele-ment of the airplane had to be engineered. This in-cluded the airplane’s general layout, a unique propul-sion system, the aerodynamic designs of the wing,fuselage and empennage, as well as the efficient in-tegration of these and other subsystems. Althougha unique propulsion system eventually became ourbaseline design, several systems were considered.These included a conventional tractor propeller witha front-mounted engine, and two mid-engine designs– one with a pusher prop aft of the tail and the othera body-prop design. While a tractor design is muchmore conventional, the design requirements favoreda mid-engine concept. Avoiding propeller strike forthe pusher design during rotation was a major issue.While there were many other factors that played intoour decision, the body-prop design became our base-

line configuration. With the propeller mounted aft ofthe wing, this concept also provided the possibility ofpromoting laminar flow on the forward fuselage andwing surfaces. A side view of the body-prop’s gen-eral layout is provided in Figure 21 and a computergraphics rendering of this configuration in flight isgiven in Figure 22. The highest risk item of thisdesign is definitely related to engineering the struc-ture to accommodate the load path between the tailand center wing box.

One can see from the general layout that the ver-tical tail (rudder) is rigged downward instead of ina normal upward position. This was done for tworeasons; the first to provide a skid at the rudder tipto prevent propeller strike, and the second to keep itin clean air during a high, positive G maneuver.

The complete aircraft design effort was conductedby a very small team; the authors were tasked withthe responsibility to design the outer-mold-line sur-faces of the wing and fuselage components.

During this multi-disciplinary design effort, thegeneral layout of the body-prop concept race planeevolved as the design team better understood howto maximize the performance of the integrated sys-tem. Normally, global changes such as those encoun-tered are very disruptive during the design of a high-performance, transonic wing. However, utilizationof the aerodynamic shape optimization software de-veloped by the authors allowed various aircraft sub-systems to be routinely modified without adverselyimpacting development costs or schedule; new wingdesigns occurred over night. Our ability to performnew optimizations over night, on affordable comp-uters, was a key factor which allowed this form ofsimulation-based aerodynamic design work to be em-braced by the rest of the design team. There weretwo other members of the team that are world-classwing designers, and they would have quickly relievedus of this duty if we were not providing quality de-signs in a timely manner. The complete evolution ofthe aircraft’s general layout was accomplished in avery compressed time frame: our aerodynamic shapeoptimizations played a pivotal role in this achieve-ment. More importantly, this evolution was requiredto meet all of the design goals imposed on the teamby our sponsor. For more detailed information onthis aircraft design, see References [17]-[18].

6 Reno Racer Wing Design

Any successful system design effort must accommo-date a changing set of requirements as the designersof the various subsystems learn more about how theirindividual efforts impact and are affected by the ac-tions of the other designers. This was certainly the


case with this wing design as we integrated it withthe fuselage, propulsion system, stability & control,manufacturing and overall packaging. Most of the as-sembled team has worked closely together for morethan a decade. Our style has been to allow the in-dividuals of the team to gravitate towards the workitems that they feel most comfortable with, however,each member loosely participates in all concurrentactivities in progress. On this project, participationwas usually in the form of daily discussions regardingthe overall design of the aircraft. By disseminatingeverybody’s findings on a very frequent basis, thegroup as a whole began to understand how best tomaximize the system’s performance. Hence, this wasvery much a multi-disciplinary (MD) effort with theteam members exploring the MD design space forincreasingly better aircraft designs. These informaldesign reviews also provided regular sanity checkssuch that a poor design direction was never ven-tured too far. A characteristic of this dynamic designenvironment was that the design constraints at thesubsystem levels were constantly and rapidly chang-ing. New constraints can be in unexpected direc-tions, and trying to program these dynamic changesin an MDO code, in a timely manner, can be quitedaunting. This is where the man-in-the-loop belongs.The interface between humans can easily adapt tothe ever-changing design requirements such that per-tinent information continues to be shared across theappropriate disciplines.

This group regularly works on tight schedules andunder small budgets. As a result, the most cost-effective tools are used at every stage of a designeffort. Initially, when the design is not very well un-derstood, design charts and rules-of-thumb dominatethe effort. As the design begins to evolve, and thesemethods no longer add value to the direction of thegroup, linear methods are drawn into the tool set.Then, as the ROI of linear methods begins to reduce,they are replaced by non-linear tools – starting withthe simple and finishing with the most sophisticated.Using the right tool at the right time helps managecosts and schedule, and allows the final designs tobe competitive at the highest level. This approach isin contrast to those efforts that start using the mostsophisticated numerical tools from the onset of thedesign activity.

6.1 Phase I: Conceptual Layout

The design of the wing geometry occurred in severalphases; the duration of each of the first five phaseslasted from 1 day to 1 week long. In most cases, therewas a lapse between phases, as time was required forthe team to digest the evolution of the aircraft designand formulate new ideas to investigate.

The basic requirements defined in phase I werebased on conceptual methods and design charts.These requirements included the general layout ofthe wing (planform & thickness distribution), thedesign cruise condition

M = 0.77, CLTotal = 0.32, Ren = 14.5M,

the off-design capabilities for buffet

CLBuffet = 0.64 at M = 0.72,

the drag divergence Mach number

Mdd = 0.80 at CL = 0.1,

the clean wing maximum lift coefficient

CLmaxCW ≥ 1.6 at M = 0.2,

and a pad on the divergence Mach number to allowroom for growth in out years.

The conceptual methods set the wing area,Sref = 75ft2, to provide a wing loading range of40 − 60 lbs./ft2. Despite the high dynamic pres-sures of the racing environment and the opportunityfor very high wing loading, the stall speed require-ment sized the wing area. The maneuver loads, sus-tained turn rate, and gust loads required that thewing have no buffet at CLTotal = 0.64,M = 0.72. Atrade study of wing thickness, sweep and taper ratiowas made using NACA SC(2) airfoils as a baseline.From this, a section thickness of 13.5% at the wingroot and 12% at the tip was chosen, combined witha quarter-chord sweep of 28◦ to meet the Mdd = 0.8requirement. An aspect ratio of AR = 8.3 and ataper ratio of λ = 0.45 were chosen to allow a wing-tip extension for a growth airplane. Conversely, aproduction break was included at 87% semi-span toallow a 4ft2 reduction in wing area if ever needed. Aplanform Yehudi (inboard chord extension) was in-corporated into the wing trailing edge to accommo-date the main landing gear. Inclusion of this Yehudialso helped reduce the wing downwash angle of theflow entering the propeller. The wing is a two-spardesign with spars at 15% and 65% chord, and is aug-mented with a secondary spar behind the main gearwells that parallel the Yehudi trailing edge. This sec-ondary spar provides structural support at the maingear pivots. A one-piece wing box construction willbe used to reduce weight and complexity.

6.2 Phase II: Rough Detailed Design

The baseline wing of phase II was defined using air-foil sections derived from NACA 64 sections, scaled


to conform to the planform and thickness distribu-tion established in phase I. Some cursory 2D aero-dynamic optimizations were performed on these sec-tions to better tailor their characteristics for the ini-tial design conditions; the 2D conditions and geom-etry transformations used for this effort were basedon simple-sweep theory. SYN103 was run in Euler,drag-minimization mode for this 2D design effort.The remaining unspecified geometric quantity for thewing was it’s twist distribution. To set this, FLO22was used to provide the span load of the wing. Thiscode, which solves the three-dimensional transonicpotential flow equation, has been extensively usedsince its inception in 1976. (For reference, FLO22runs in about 5 seconds on an AMD Athlon 850MHz PC.) While FLO22 is a wing-only CFD code,pseudo-fuselage effects were included in the presentwork. The first pseudo-body influence is its accel-eration of the on-set Mach number at a critical sta-tion on the wing; typically this is around 50%-60%semi-span. Running the isolated fuselage geometryin a surface-panel method and interrogating the flow-field velocity at the critical wing station determinesthis acceleration. The second pseudo-body effect ishow the presence of the fuselage at an angle of at-tack warps the flowfield’s local angle of attack as afunction of span location. The third pseudo-body in-fluence is related to the carry-over lift of the wing’scirculation onto the fuselage. This ratio is defined asCLTotal/CLWing, is 1.22 for this configuration andwas determined by running a surface-panel methodon the wing/body combination. These pseudo-bodyeffects are included in FLO22’s wing-only solution byrunning the exposed wing in the code at the wing’sCL, at a higher Mach number and re-referencing theresults back to the original Mach, and adding a delta-twist distribution to the wing to simulate the flow-field warping. Using this procedure, a twist distri-bution was specified that yielded a near-elliptic spanloading. This initial design was done very rapidly,covering only a two-day period, and provided a pointto start the 3D design effort.

The initial FLO22 analyses indicated that thewing design requirements could be satisfied; theinitial wing had a Mach capability of 0.775 atCLTotal = 0.3. However, there was serious concernwith the body effects of the fuselage’s low finenessratio. The team was relatively sure that the baselinewing would have problems near the root region be-cause of the atypical contouring of the fuselage geom-etry.

6.3 Phase III: Aero Optimization

In phase III, the first step was to assess the issuesexisting with the baseline wing geometry, designated

Shark1, as it integrates with the fuselage. This anal-ysis was performed using SYN88, and is illustratedin Figure 23. SYN88 is a wing/body Euler methodwhich also incorporates an adjoint-based optimiza-tion procedure for aerodynamic shape design. In Fig-ure 23, pressure distributions at several stations onthe wing are provided. Adhering to standard aerody-namic practices, the pressure coefficient of the sub-plots are presented with the negative axis upward.The area trapped by the upper- and lower-surfacepressure-distribution curves is equivalent to the lo-cal sectional lift coefficient. Each subplot is linkedgraphically to its corresponding location on the wingdepicted in the center of the figure. Also included onthe wing-planform plot are the upper-surface isobarsof the first solution which is depicted by the solidcurves in the perimeter subplots. A shock is evi-dent with a concentration of contour lines in the iso-bar image and corresponds to a sharp discontinuityin the pressure-distribution subplots. The quanti-ties in the legend of this figure correspond to thewing forces. The drag listed is only the invisciddrag (induced+shock). Recall that the design liftwas CLTotal = 0.32 and that the carry-over lift ratiowas 1.22 for this configuration. Hence, the wing lift isCLWing = CLTotal/1.22 = 0.27. The other item tonote is that this analysis was performed at a Machnumber ofM = 0.78 rather than M = 0.77. The rea-son for this increase in freestream Mach number wasdue to the acceleration of the flowfield near the wingroot from the propulsion system. Methods based onactuator-disc and blade-element theories determinedthis acceleration to be ∆M ≃ 0.01. Since the wing-root region was of utmost concern at this stage in thedesign, the full level of propeller effects on the on-set Mach number was used. Referring to the wing-planform plot of Figure 23, notice the strong shockthat unsweeps as it nears the side-of-body. The mainpurpose of a swept-back wing is to reduce the normalMach number of the flow into a shock, however, if theshock unsweeps, this benefit is lost. As suspected,the contouring of the fuselage cross-sections had anadverse effect on the wing aerodynamics, unfortu-nately it was worse than expected. The inviscid drag(induced+shock) of the wing was CDWingINV = 180counts for the baseline configuration.

Phase III continued by running SYN88 in drag-minimization mode, constraining the wing modifi-cations to be thicker everywhere than the baselinegeometry to maintain structural depth; the fuselagegeometry was frozen. Initially, these were single-point optimizations at the 4G design condition, justto scope the potential benefit. (For reference, aSYN88 wing/body analysis takes about 15 minuteson a Sony VAIO notebook computer with a 500 MHzPentium II chip; a single-point optimization takes


about 75 minutes.) Eventually, all optimizationswere migrated to triple-point designs that considereda range of lifting conditions at the design Mach num-ber. This range corresponded to variation and persis-tence of G loads being pulled during a lap of the racecourse. The design Mach number corresponded toan average speed around the track. Within 30 designcycles, SYN88 dropped the wing’s inviscid drag from180 counts to 104 counts. The results of this optimi-zation are illustrated in Figure 24. Although fairlylarge improvements were realized, we felt we could dobetter if the fuselage contour near the wing trailingedge was allowed to be modified. Several concur-rent changes to the aircraft’s general layout were be-ing considered. The team was forming new ideas asthe complete system integration was beginning to bebetter understood. The changes that were directlyrelated to the wing design were the fuselage reshap-ing and a trailing-edge planform blending that wouldallow more room for stowing the landing-gear struc-ture. The planform modifications were made to thecurrent wing and three additional fuselages were de-fined that stretched it by 1, 2 and 4 feet aft of thewing-root mid-chord and consistent with the enginepackaging requirements. In fact, the trailing-edgemodification was also done in a manner to help allevi-ate the shock-unsweep problem, as well as accommo-date the landing gear. This planform change provedto be beneficial as another triple-point drag mini-mization was performed, which dropped the wing’sinviscid drag from 148 counts to 98 counts at thedesign point. For clarification, this wing redesignwas done with the original fuselage to establish a newbase for the parametric study stretching the fuselage.Repeating similar triple-point optimizations on the1-, 2- and 4-foot fuselage extensions provided suffi-cient data to show that the shock unsweep problemcould be completely eliminated with a 2-foot fuse-lage stretch. This optimization reduced the wing’sinviscid drag from 92 counts to 74 counts within 30design cycles; the resulting wing geometry was desig-nated Shark52. The pressure distributions and dragloops for Shark52 at M = 0.78 and CLWing = 0.27are shown in Figures 25-26.

It should be emphasized that within the course ofone week, the wing geometry had evolved from onethat produced 180 counts of inviscid drag to Shark52,which only had 74 counts at the design point. Duringthis week, the wing planform changed and the fuse-lage length stretched. This is an extremely large im-provement accomplished in a very compressed time!Furthermore, the database of CFD solutions O(100)had grown large enough that very informed modi-fications to the configuration could be made. Thisincluded the wing-planform change to better stowthe landing gear, as well as the fuselage reshaping to

eliminate the shock unsweep issue. Figure 26 showsa side-by-side comparison of the pressure isobars ofthe Shark52 wing and the initial wing that clearlyillustrates the reduction of the shock strength acrossthe entire wing. This improvement was a result ofall of these important changes to the configuration.

The final aspect of phase III was to rebalance theaircraft. This required a 6-inch fuselage stretch for-ward of the wing to compensate for the 2-foot stretchof the after-body. Once the fuselage geometry wasfrozen, the wing pressures were polished by runningSYN88 in inverse-design mode. A drag build-up ofthis design showed that, at 550 MPH, the maxi-mum L/D of the complete aircraft was estimated tobe 14.78. It occurs at CLTotal = 0.49, which corre-sponds to a 7G maneuver.

6.4 Phase IV: Laminar Flow

The team began to kick around the idea of a laminar-flow design. Quick calculations on the attachment-line Reynolds number, Reθ, indicated that theTollmien-Schlichting waves would decay rather thanamplify. The possibility of having runs of laminarflow was achievable. The dilemma, however, wascould laminar flow be achieved in the field? The pri-mary mission of this plane occurs just above groundlevel where bug strikes are sure to occur, thus con-taminating the wing’s leading edge. We decided toinvestigate whether or not the wing’s pressure distri-butions could be tailored to have favorable gradientsfor up to 40% chord without adversely affecting theaerodynamic performance of the fully-turbulent wingdesign. If it could, then the resulting design would beadopted, yet without taking credit for laminar-flowdrag reductions.

Phase IV concentrated on promoting laminar flowon the wing without degrading the performance ofthe wing if it was fully turbulent as compared withthe fully-turbulent design of Shark52. This objectivewas not limited to the design point, but rather wasexpanded to include a Mach number rangeM ≥ 0.74and a lift range of CLWing ≤ 0.27. The first taskwas to compute the viscous flow about the Shark52configuration at various flow conditions. This wasaccomplished using SYN107P, a wing/body Navier-Stokes method for analysis and design. The designReynolds number was Re = 14.5M , based on thereference chord. (For reference, SYN107P runs inparallel under MPI; on a 16 processor AMD Athlon650 MHz cluster, an analysis takes about 30 min-utes of wall-clock time, while an optimization of 30design cycles takes less than 3 hours.) Starting withthe computed pressure distributions of Shark52, aseries of inverse designs were performed, also withSYN107P. It was easier to redesign the wing at the


higher Mach number and accommodate the require-ments at M = 0.74, rather than the other wayaround. This study was completed with the winggeometry designated SharkNS7. Figure 28 illustratesthe pressure distributions for SharkNS7 at M = 0.78and a lift range of

0.18 ≤ CLWing ≤ 0.34.

At the design point CDWing = 128 counts whichis composed of CDWingForm = 77 counts andCDWingSkinFriction = 51 counts.

Note that favorable gradients exist on both up-per and lower surfaces for about the first 30%-40%chord, depending on span location and lifting con-dition. On the upper surface the shock will triggertransition provided any attachment line contamina-tion from the fuselage boundary layer is removed bya notch-bump, and Reθ < 200. Reθ varied fromapproximately 125 just outboard of the fuselage toaround 80 at the wing tip. The amount of laminarrun on the upper surface increases as Mach increasesdue to the shock moving aft on the airfoil as wellas the pressure gradients becoming more favorable.At race conditions the wing should have an appre-ciable extent of laminar flow, provided the surface ofthe wing is smooth and free of particulate contam-ination. The estimated benefit of the laminar flowruns is between 10 and 20 counts of drag reduction,depending on Mach number. This level of drag re-duction increases the aircraft’s performance by anadditional 5%, which is a significant improvement.

6.5 Phase V: Final Touches

The first task of phase V was to establish an ap-propriate leading-edge radius distribution, tailoredfor low-speed characteristics, without really changingthe wing pressure distributions at the cruise designconditions. This modification was accomplished withlocal, explicit geometry perturbations. An additionalmodification to the wing thickness distribution wasalso done. After these changes were incorporatedinto SharkNS7, the final wing was analyzed to ver-ify that these geometry changes did not adverselyeffect the pressure distributions. When overlaid onthe same plot, the curves nearly appeared as one.

Finally, clean wing CLmax, CLmaxCW , was com-puted to ensure the wing satisfied the required cleanwing stall speed. The design requirement was toprovide CLmaxCW > 1.6 at M = 0.2. This wasdetermined by finding the flow condition where thewing’s Cpmin distribution reached an empirically-determined critical value. The final wing provideda CLmaxCW = 1.64, just meeting the requirement.

The final wing geometric characteristics are shownin Figure 29 which illustrates the half-thickness and

camber distributions of the root and outboard airfoilsections. Note that the root airfoil is 13% thick @31.5% chord and has -1.2% camber. The outboardairfoil is 11.5% thick @ 38.6% chord and has +1%camber. While these thicknesses are about 0.5%thinner of that specified in the conceptual designstage, it was the team that reduced this thickness,not the optimization exercises.

This concludes our discussion on the Reno Racer.Next, we will review an aero-structural optimizationof a generic 747 wing/body configuration.

7 Generic 747 Wing/Body

In this section, a simplified structural weight model isdirectly coupled with the high-fidelity aerodynamicshape optimization procedure. Further, the plan-form of the aircraft wing is allowed to be redesigned.The original work on this case study is documentedin detail in Reference [19].

In an airplane design, there are many criteria to besatisfied: multiple design points, fuel distributions,stability-and-control effects, aero-elastics, etc. Someof the general criteria that must be met in the designof any efficient transonic wing include:

1. Good drag characteristics (parasite, induced,compressibility) over a range of lift coefficients,i.e. CLdesign

± 0.1 at Mcruise.

2. No excess penalties for installation of nacelle-pylons, fairing, etc.

3. Sufficient buffet boundary margin at cruise liftcoefficients (1.3g margin).

4. No pitch-up tendencies near stall or buffet.

5. Maintain control surface effectiveness.

6. No unsatisfactory off-design performance.

7. Sufficient fuel volume for design range.

8. Structurally efficient (to minimize weight).

9. Sufficient space to house main landing gear.

10. Compatible with the high-lift system.

11. Consistent with airplane design for relaxedstatic stability.

12. Manufacturable at a reasonable cost.


7.1 Cost Function

In this optimization, the two relevant disciplines areaerodynamics and structural weight. Therefore weminimize a combination of aircraft drag and wingweight. This optimization not only makes the designmore realistic, but we can also relax some of the con-straints on thickness. Hence, we mainly target theminimization of

I = α1CD + α2CW . (5)

Here α1 and α2 are properly chosen weighting con-stants, and CW = weight

q∞Srefis a non-dimensional

weight coefficient. This choice of cost function em-phasizes the trade-off between aerodynamics andstructures. For design of a long range transport air-craft, Eqn (5) actually centers on the idea of improv-ing the range of the aircraft.

Consider the well known Breguet range equationwhich provides a good first estimate of the range ofthe airplane

R =V

C

L

Dln

(

We +Wf

We

)

(6)

where V is the speed, C is the specific fuel consump-tion of the engine, L/D is the lift to drag ratio,We is the landing weight, and Wf is the weight ofthe fuel burnt. During the last few decades, themeans to improve the efficiency of an airplane hascentered around reducing the fuel consumption C ofthe engine, increasing V L/D, and reducing the air-plane weight. The last two methods together implythat the constants α1 and α2 in Eqn (5) can be esti-mated from the range Eqn (6).

7.2 Structural Weight Model

Wing weight is directly given by the wing structure,which is sized by aerodynamic load, allowable de-flection, and failure criteria such as buckling. Dif-ferent methods to estimate wing weight have beenproposed, ranging from empirical expression to com-plex Finite Element Analysis. At the detailed designlevel, once the wing structure has been laid out,structure engineers use finite element analysis to sizethe interior structure to satisfy various criteria. Thenthe material weight of each component is added upto calculate the structural weight.

When the detailed structure layout is not known,which is usually the case for conceptual and pre-liminary design levels, it is very difficult to pre-dict the structural weight correctly. We select anemperically-influenced simple structure model thatcan be expressed analytically. Here, the wing struc-ture is modeled by a box beam whose primary struc-tural material is the upper and lower box skin. The

skin thickness (ts) varies along the span and resiststhe local bending moment caused by the wing lift.Then, the structural wing weight Wwing can be cal-culated on the basis of the skin material.

Consider the box structure of a swept wing whosequarter-chord sweep is Λ and its cross-section A-A isas shown in Figure 30. The skin thickness ts, struc-ture box chord cs, and overall thickness t vary alongthe span, such that the local stress is equal to themaximum allowable stress everywhere. The maxi-mum normal stress due to bending at section z∗ is:

σ =M(z∗)

t tscs.

The corresponding structural box-beam weight is:

Wwingbox= ρmatg

∫

structual span

2tscsdl

= 2ρmatg

σcos(Λ)

∫ b2

− b2

M(z∗)

t(z∗)dz∗

= 4ρmatg

σcos(Λ)

∫ b2

0

M(z∗)

t(z∗)dz∗,

and

CWb=

Wwingbox

q∞Sref

=β

cos(Λ)

∫ b2

0

M(z∗)

t(z∗)dz∗, (7)

where

β =4ρmatg

σq∞Sref

,

ρmat is the material density, and g is the accelerationdue to gravity.

The bending moment can be calculated by inte-grating pressure toward the wing tip. Ignoring theend effects due to the rotated axis of the box beam,

M(z∗) = −

∫ b2

z∗

p(x, z)(z − z∗)

cos(Λ)dA

= −

∫ b2

z∗

∮

wing

p(x, z)(z − z∗)

cos(Λ)dxdz.

Thus

CWb=

−β

cos(Λ)2

∫ b2

0

∫ b2

z∗

∮

wing

p(x, z)(z − z∗)

t(z∗)dxdzdz∗.

(8)However, CWb

must be expressed as∫

BdBξ in the

computational domain, or∫ ∫

dxdz in a physical do-main to match the adjoint boundary term.


To switch the order of integral of Eqn (8), intro-duce a Heaviside function

H(z − z∗) =

{

0, z < z∗

1, z > z∗

Then Eqn (8) can be rewritten as

CWb=

−β

cos(Λ)2

∫ b2

0

∮

wing

p(x, z)K(z)dxdz, (9)

where

K(z) =

∫ b2

0

H(z − z∗)(z − z∗)

t(z∗)dz∗

=

∫ z

0

z − z∗

t(z∗)dz∗

Finally, to account for the weight of other wingmaterial such as ribs, spars, webs, stiffeners, lead-ing and trailing edges, slats, flaps, main gear doors,primer and sealant, we multiply CWb

by a correctionfactor Kcorr,b.

Moreover, statistical correlation over the rangeof aircraft type indicates a relationship betweenWwing/S and Wwingbox

/S as a linear function, shownin Figure 31. Therefore we add another term to ac-count for area-dependent wing weight

CWs=

1

Sref

∮

B

|S22|dξ1dξ3, |S22| =√

S2iS2i,

(10)along with a correction factor Kcorr,s. Thus

CW = Kcorr,bCWb+Kcorr,sCWs

. (11)

7.3 Wing Weight Estimation

From our discussion in section 7.2, the expressionof wing weight contains terms that correspond tobending-load-carrying material and two correctionfactors; Kcorr,b and Kcorr,s. To estimate these cor-rection factors, we can rewrite the expression (11)as

Wwing

S= α1

Wb

S+ α2 (12)

where Wwing is the total weight of the wing, Wb isthe weight of the wing box, and S is a gross wingarea. When there is information from more than twoaircraft, we can use the “least-square” curve-fittingstrategy to calculate α1 and α2.

To compute α1 and α2, we solve a system of twoequations and two unknowns. These equations havethe form of Eqn (12) but with different values ofWwing

Sand Wb

S. This gives

α1 = 1.30 and α2 = 6.03 (lb/ft2).

However, the total wing weight estimation will beinexact because of the limitation of aircraft number.Finally, Kcorr,b and Kcorr,s can be computed by

Kcorr,b = α1 and Kcorr,s =α2

q∞

7.4 Planform Design Variables

From the trade study, the parameters that lead upto a basic design which satisfies the general designcriteria include:

• wing shape • area• span • sweep• aspect ratio • taper ratio• airfoil types • airfoil thickness

Because some of these parameters do not definethe wing geometry uniquely, we employ another setof design parameters that still represent these param-eters but can be extracted from surface mesh points.Here we model the wing of interest using six plan-form variables: root chord (c1), mid-span chord (c2),tip chord (c3), span (b), leading-edge sweep(Λ), andwing thickness ratio (t), as shown in Figure 32. Thischoice of design parameters will lead to an optimumwing shape that will not require an extensive struc-tural analysis and can be easily manufactured.

7.5 Maximizing Range

The choice of α1 and α2 in Eqn (5) greatly affectsthe optimum shape. We can interpret α1 and α2

as how much emphases we give to drag and wingweight. If α1

α2is high, we focus more on minimiz-

ing the drag than the weight and we tend to get anoptimum shape that has low CD but high CW .

An intuitive choice of α1 and α2 can be made byconsidering the problem of maximizing range of anaircraft. Consider the Breguet range Eqn (6)

R =V

C

L

Dln

(

We +Wf

We

)

where We is the gross weight of the airplane withoutfuel and Wf is weight of fuel burnt.

If we take

W1 = We +Wf = fixed

W2 = We

then the variation of the weight can be expressed as

δW2 = δWe.

With fixed VC

, W1, and L, the variation of R can bestated as

δR =V

C

(

δ

(

L

D

)

lnW1

W2

+L

Dδ

(

lnW1

W2

))


=V

C

(

−δD

D

L

DlnW1

W2

−L

D

δW2

W2

)

= −V

C

L

DlnW1

W2

(

δD

D+

1

lnW1

W2

δW2

W2

)

and

δR

R= −

(

δCD

CD

+1

lnW1

W2

δW2

W2

)

= −

δCD

CD

+1

lnCW1

CW2

δCW2

CW2

.

If we minimize the cost function defined as

I = CD + αCW ,

where α is the weighting multiplication, then choos-ing

α =CD

CW2ln

CW1

CW2

, (13)

corresponds to maximizing the range of the aircraft.

7.6 Optimization Results

In these calculations the flow is modeled by theReynolds Averaged Navier-Stokes equation, with aBaldwin Lomax turbulence model. This turbu-lence model was considered sufficient because thedesign point at cruise has predominately-attachedflow. The nominal cruise condition of the generic747 wing/fuselage combination has a Mach numberof 0.85 and a lift coefficient of CL = 0.45. On thecorresponding (256x64x48) grid, the wing is repre-sented by 4,224 surface mesh points and six plan-form variables. All coefficients are calculated witha fixed reference area based on the baseline config-uration. Thus an increase in skin friction due to anincrease in wetted area will appear as an increase inthe skin-friction drag coefficient.

As a reference point, we first modified only thewing sections to eliminate the shock drag, while keep-ing the planform fixed. Figure 33 shows the re-designed calculation. In 30 design cycles, the dragwas reduced from 137 counts to 127 counts (7.3%reduction), while the weight remained roughly con-stant.

Next, we implement both section and planformoptimization in a viscous redesign, using an invis-cid redesign as a starting point. Figure 34 showsthe effect of allowing changes in sweepback, span,root chord, mid-span chord, and tip chord. The pa-rameter α2/α1 is chosen to maximize the range of

the aircraft. In 30 design cycles, the drag was re-duced to 117 counts (14.5% reduction from the base-line), while the dimensionless structure weight wasdecreased from 546 counts to 516 counts (6.1% reduc-tion), which corresponds to a reduction of 4,800 lbs.The planform changes are shown in Figure 35. Thisviscous redesigned wing has less drag and structuralweight than the fixed-planform viscous redesignedwing.

When we compare this planform with the redesignby inviscid optimization, as seen in Figure 36, we cansee that the effect of viscosity is to shrink the area ofthe inviscidly redesigned planform. This trend is tobe expected because skin friction drag varies roughlylinearly with the area. By reducing the area, we canreduce the skin friction drag.

The results from the viscous planform optimiza-tions yield large drag reduction without structuralweight penalty in a meaningful way. They show thefollowing basic trends:

• Increase wing span to reduce vortex drag,

• Reduce sweep but increase section-thickness toreduce structural weight,

• Use section optimization to minimize shockdrag.

Although the suggested strategy tends to increasethe wing area, which increases the skin friction drag,the pressure drag drops at a faster rate, dominatingthe trade-off. Overall, the combined results yieldsimprovements in both drag and weight.

7.7 Pareto Front

In order to present the designer with a wider rangeof choices, the problem of optimizing both drag andweight can be treated as a multi-objective optimiza-tion problem. In this sense one may also view theproblem as a “game”, where one player tries to min-imize CD and the other tries to minimize CW . Inorder to compare the performance of various trial de-signs, designated by the symbol X in Figure 37, theymay be ranked for both drag and weight. A design isun-dominated if it is impossible either to reduce thedrag for the same weight or to reduce the weight forthe same drag. Any dominated point should be elim-inated, leaving a set of un-dominated points whichform the Pareto front. In Figure 37, for example,the point Q is dominated by the point P (same drag,less weight) and also the point R (same weight, lessdrag). So point Q will be eliminated. The Paretofront can be fit through the points P, R and otherdominating points, which may be generated by us-ing an array of different values of α1 and α2 in the


cost function to compute different optimum shapes.With the aid of the Pareto front the designer willhave freedom to pick the most useful design.

The problem of optimizing both drag and weightcan be treated as a multi-objective function optimi-zation. A different choice of α1 and α2 will resultin a different optimum shape. The optimum shapesshould not dominate each other, and therefore lie onthe Pareto front. The Pareto front can be very use-ful to the designer because it represents a set whichis optimal in the sense that no improvement can beachieved in one objective component that does notlead to degradation in at least one of the remainingcomponents.

Figure 38 shows the effect of the weighting param-eters (α1, α2) on the optimal design. As before, thedesign variables are sweepback, span, chords, sec-tion thickness, and mesh points on the wing sur-face. In Figure 38, each point corresponds to anoptimal shape for one specific choice of (α1, α2). Byvarying α1 and α2, we capture a Pareto front thatbounds all the solutions. All points on this front areacceptable solutions, and choosing the final designalong this front depends on the nature of the prob-lem and several other factors. The optimum shapethat corresponds to the maximum Breguet range isalso marked in the figure. In this test case, the Machnumber is the current normal cruising Mach numberof 0.85. We allowed section changes together withvariations of sweep angle, span length, chords, andsection thickness. Figure 33 shows the baseline wing.Figure 34 shows the redesigned wing. The parame-ter α2

α1was chosen according to formula (13) such

that the cost function corresponds to maximizingthe range of the aircraft. Here, in 30 design itera-tions the drag was reduced from 137 counts to 117counts and the structural weight was reduced from546 counts (88,202 lbs) to 516 counts (83,356 lbs).The large reduction in drag is the result of the in-crease in span from 212.4 ft to 231.7 ft, which reducesthe induced drag. The redesigned geometry also hasa lower sweep angle and a thicker wing section inthe inboard part of the wing, which both reduce thestructural weight. Moreover the section modifica-tion prevents the formation of shock. The baselineand redesigned planforms are shown in Figure 36,together with the planform which resulted from in-viscid optimization. Overall, the redesign with vari-ation planform gives improvements in both aerody-namic performance and structural weight, comparedto the previous optimization with a fixed planform.

8 Conclusions

In this lecture, sample applications of aerodynamicshape optimizations are provided through three casestudies in aircraft design. These include the designof a Mars aircraft, a Reno Racer, and a generic747-class wing/body configuration. The genericwing/body case included a simple structure-weightmodel to allow aero-structural optimizations to beperformed.

The utility of SYN107P has been demonstrated onthe Mars Airborne Remote Exploration Scout. Thisinvestigation has redesigned the baseline wing shapeto improve its aerodynamic performance. The re-design has yielded a 112-count reduction in total dragat its design point, and increased the lift-to-drag ra-tio 23% from 10.4 to 12.8. The cruise flight conditionof the MARES is M = 0.65, CL = 0.62, and Re =170K. Drag polars for the baseline and redesignedwing show that the performance improvement ex-tends over all lifting conditions and is well behavedat off-design points. All SYN107P calculations (anal-yses and optimizations) are performed on very af-fordable computer equipment with fast turn-aroundtimes, due in part to parallel processing. (In fact, aserial version of SYN107P will currently reproducethe present Navier-Stokes optimization on a 2GHznotebook computer within 27 hours.) Furthermore,essentially no additional set-up effort is required torun optimizations once an analysis input deck is de-fined. The effectiveness of this aerodynamic shapeoptimization has been made possible because thedevelopers have addressed the adjoint-based gradi-ent calculation and the design-space traversal pro-cesses from the perspective of what is required inthe infinite-dimensional continuum, then taken thesimple step towards discrete space.

Aerodynamic shape optimization methods weresuccessfully applied to the design of a new unlimitedclass Reno race plane. Very significant performancegains were achieved in very compressed time. Uti-lization of this software also allowed global changesto occur at the aircraft level without adversely af-fecting the efforts to aerodynamicly design its high-performance, transonic wing as new designs could beperformed over night. Normally, such major changeswould have had a very disruptive effect on the designof the wing. Yet, the evolution of the general lay-out was necessary for all of the design goals to beachieved.

Coupled aerodynamic-structural optimizationswere performed on a generic 747-class wing/bodyconfiguration. In this case study, a simple structure-weight model was developed and incorporated intothe aerodynamic shape optimization process. Thecost function in this investigation is a blending of


drag and structure-weight coefficients. The coef-ficients of blending were determined to maximizerange, as well as were allowed to vary to provide apareto front of optimum designs.

All optimizations utilized thousands of design vari-ables and were carried out on affordable compu-ters systems. These were key to the success of thissimulation-based design effort.

Exercises such as these have been very beneficialto the authors, as we get a better understandingand appreciation of the chaos and schedule pressuresthat exist in real-world design environments. We arealso pleased that the aerodynamic shape optimiza-tion software have had dramatic and positive effectson the outcome of the final designs.

9 Post Script

As can be seen from these studies, aerodynamicshape optimization can significantly streamline thedesign process. In addition to the case studies pre-sented herein, the present method has now beensuccessfully applied in a variety of projects, in-cluding the McDonnell-Douglas MDXX [20], theNASA HSCT studies, the Boeing Blended-Wing-Body project, and the Beech Premier [21].

There are many different examples on how air-craft design teams have utilized the rapidly providedinformation of aerodynamic shape optimization tomake improvements to their aircraft configurations.The diversity of these examples illustrate the artis-tic and creative nature of the thought processes bythe design teams. It is through these unpredictablepaths in design direction that dramatic improve-ments of the multi-disciplinary systems are accom-plished. Further, because unforeseen directions maybe required to accomplished the design goals, it ishighly unlikely that the designers will be replacedby a comprehensive, multi-disciplinary optimization(MDO) method. On the other hand, there are verywell established dependencies between certain disci-plines that can and should be coupled for MDO.

Although the search method is only guaranteed tofind a local minimum, it turns out in practice thatour aerodynamic optimizations are yielding resultsthat are in the neighborhood of the known lowerbounds for aerodynamic drag, as determined by opti-mum span loading, flat-plate skin-friction, and min-imum wave drag. Furthermore, in practice, the de-signer’s goal is not to determine the absolute bestdesign, but rather, is tasked to make the most im-provement to a design in a fixed amount of time spec-ified by program schedule.

Aerodynamic shape optimization will not replacethe judgement and insight of the aircraft designers.

Rather, it should properly be viewed as an enablingtool that allows the designers to focus their effortson the creative aspects of aircraft design, by reliev-ing them of the need to spend large amounts of timeexploring small variations. By intelligent choice ofthe cost function to measure the aerodynamic perfor-mance and perhaps also the deviation from a desiredpressure architecture, one can essentially eliminatethe need to carry out detailed section design. In-stead, the designers can concentrate their attentionon large scale parameters such as wing span, areaand sweep, knowing that the optimization processwill improve the performance for any given choice ofthese parameters.

References

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sentation, First M.I.T. Conference on Compu-tational Fluid and Solid Mechanics, Cambridge,MA, June 2001.

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[5] L. L. Green, P. A. Newman, and K. J. Haigler.Sensitivity derivatives for advanced CFD al-gorithm and viscous modeling parameters viaautomatic differentiation. AIAA paper 93-

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[6] W. K. Anderson, J. C. Newman, D. L. Whit-field, and E. J. Nielsen. Sensitivity analysisfor the Navier-Stokes equations on unstructuredmeshes using complex variables. AIAA pa-

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[7] A. Jameson, L. Martinelli, J. J. Alonso, J. C.Vassberg, and J. Reuther. Simulation basedaerodynamic design. IEEE Aerospace Confer-

ence, Big Sky, MO, March 2000.


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[9] A. Jameson. Optimum aerodynamic design us-ing CFD and control theory. AIAA paper 95-

1729, AIAA 12th Computational Fluid Dynam-ics Conference, San Diego, CA, June 1995.

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per 98-4807, 1998.

[11] A. Jameson, L. Martinelli, and J. C. Vassberg.Using computational fluid dynamics for aero-dynamics - A critical assessment. ICAS Pa-

per 2002-1.10.1, 23rd International Congressof Aeronautical Sciences, Toronto, Canada,September 2002.

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AIAA 2nd Computational Fluid Dynamics Con-

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[18] E. Ahlstrom, R. Gregg, J. Vassberg, andA. Jameson. G-Force: The design of an unlim-ited class Reno racer. AIAA paper 2000-4341,18th AIAA Applied Aerodynamics Conference,Denver, CO, August 2000.

[19] K. Leoviriyakit. Wing planform optimizationvia an adjoint method. Stanford University The-

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namics of the 21st Century, 78:135–178, July2000. A Symposium in honor of Professor Sato-fuka’s 60th anniversary.


Figure 1: CFD simulation of a commercial transport with aft-mounted engines.

Figure 2: CFD simulation of a commercial transport with wing-mounted engines.


Figure 3: MARES Packaging in the Aerodynamic-Shell Capsule.

Figure 4: MARES Configuration in Flight, Top-View Rendering.


Figure 5: MARES Configuration in Flight, Bottom-View Rendering.

Figure 6: MARES General Planform Layout.


SYMBOL

SOURCE Baseline Geometry

Optimized Geometry

ALPHA 4.316

4.167

CD 0.03567

0.02912

COMPARISON OF CHORDWISE PRESSURE DISTRIBUTIONSMARES AIRCRAFT MACH = 0.650 , CL = 0.620

John C. VassbergCOMPPLOTVer 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 1.6% 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 20.3% 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 35.9% 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 54.7% 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 73.4% 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 89.1% Span

Figure 7: Comparison of Baseline and Euler Optimized Wing Pressure Distributions.

John C. VassbergCOMPPLOTVer 2.00

COMPARISON OF UPPER SURFACE CONTOURSMARS00A LANDER (GSP ORIGINAL WING WITH EXTRA STATIONS)

MACH = 0.650 , CL = 0.620( Contours at 0.05 Cp )

Solution 1: Baseline GeometrALPHA = 4.32 , CD = 0.03567

Solution 2: Optimized GeometALPHA = 4.17 , CD = 0.02912

Figure 8: Comparison of Baseline and Euler Optimized Wing Pressure Contours.


0.046

0.048

0.050

0.052

0.054

0.056

0.058

0.060

0 5 10 15 20 25 30 35 40 45 500

Mach = 0.65 , CL = 0.62 , REN = 170K

MARES Wing Design

SYN107P Drag Minimization

Design Cycle

Tot

al D

rag

Figure 9: History of Drag Minimization during Navier-Stokes Optimization.

10.0

10.5

11.0

11.5

12.0

12.5

13.0

0 5 10 15 20 25 30 35 40 45 500

Mach = 0.65 , CL = 0.62 , REN = 170K

MARES Wing Design


Design Cycle

Lif

t / D

rag

Rat

io

Figure 10: History of Lift-to-Drag Ratio during Navier-Stokes Optimization.


SYMBOL


Optimized Geometry

ALPHA 5.270

5.752

CL 0.6151

0.6155

CD 0.05924

0.04800

CM -0.00418

-0.01545

MARES Wing Design - Pressure DistributionsSYN107P Drag Minimization

REN = 170K , MACH = 0.650

John C. Vassberg12:07 Sat

20 Dec 03COMPPLOTVer 2.00



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 1.6% 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 20.3% 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 35.9% 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 54.7% 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 73.4% 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 89.1% Span

Figure 11: Comparison of Baseline and Navier-Stokes Optimized Wing Pressure Distributions.



MARES Wing Design - Upper Surface IsobarsSYN107P Drag Minimization

REN = 170K , MACH = 0.650( Contours at 0.05 Cp )

Solution 1: Baseline GeometryALPHA = 5.27 , CL = 0.6151

CD = 0.05924

Solution 2: Optimized GeometryALPHA = 5.75 , CL = 0.6155

CD = 0.04800

Figure 12: Comparison of Baseline and Navier-Stokes Optimized Wing Upper-Surface Isobars.


SYMBOL


Optimized Geometry

ALPHA 5.270

5.752

CL 0.6151

0.6155

CD 0.05924

0.04800

CM -0.00418

-0.01545

MARES Wing Design - Drag LoopsSYN107P Drag Minimization

REN = 170K , MACH = 0.650





-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

Y/C

1.6% Span

-0.20

-0.15

-0.10

-0.05

0.00

0.05

Y/C

20.3% Span

-0.15

-0.10

-0.05

-0.00

0.05

0.10

Y/C

35.9% Span

-0.05

0.00

0.05

0.10

0.15

Y/C

54.7% Span

-0.10

-0.05

0.00

0.05

0.10

Y/C

73.4% Span

-0.10

-0.05

0.00

0.05

0.10

Y/C

89.1% Span

Figure 13: Comparison of Baseline and Navier-Stokes Optimized Wing Drag Loops.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

M = 0.65 , REN = 170K

MARES Wing Design - Drag Polars


Drag Coefficient

Lif

t C

oeff

icie

nt

Baseline Geometry

Optimized Geometry

Figure 14: Comparison of Baseline and Navier-Stokes Optimized Wing Drag Polars.


0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 1 2 3 4 5 6 7 80

M = 0.65 , REN = 170K

MARES Wing Design - Lift Curves


Angle of Attack (degrees)

Lif

t C

oeff

icie

nt

Baseline Geometry

Optimized Geometry

Figure 15: Comparison of Baseline and Navier-Stokes Optimized Wing Lift Curves.

MARES Wing DesignAirfoil Geometry -- Camber & Thickness Distributions

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0

0.0

Percent Chord

Air

foil

-3.0

-2.0

-1.0

0.0

1.0

2.0

Cam

ber

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

Hal

f-T

hick

ness

SYMBOL

AIRFOIL Baseline Wing

Optimized Wing

ETA 1.6

1.6

R-LE 0.500

0.465

Tavg 2.61

2.85

Tmax 4.14

4.37

@ X 35.74

38.62

Figure 16: Comparison of Baseline and Navier-Stokes Optimized Wing Root Airfoil Sections.



0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0

0.0

10.0

Percent Chord

Air

foil

0.0

1.0

2.0

3.0

4.0

5.0

Cam

ber

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

Hal

f-T

hick

ness

SYMBOL


Optimized Wing

ETA 48.4

48.4

R-LE 1.135

1.007

Tavg 3.82

3.97

Tmax 6.03

6.00

@ X 35.74

35.74

Figure 17: Comparison of Baseline and Navier-Stokes Optimized Wing Mid-Span Airfoil Sections.


0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0

0.0

10.0

Percent Chord

Air

foil

0.0

1.0

2.0

3.0

4.0

5.0

Cam

ber

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

Hal

f-T

hick

ness

SYMBOL


Optimized Wing

ETA 76.6

76.6

R-LE 1.202

1.151

Tavg 3.84

3.90

Tmax 6.04

6.00

@ X 35.74

35.74

Figure 18: Comparison of Baseline and Navier-Stokes Optimized Wing Outboard Airfoil Sections.


Figure 19: Miss Ashley II and Rare Bear en Route.

Figure 20: Reno Race Course Layout.


Figure 21: Side View of Body-Prop Design.

Figure 22: Rendering of Body-Prop Design in Flight.


SYMBOL

SOURCE SYN88 - Shark1

ALPHA 1.000

CL .2678

CD .01796

MACH = .780

Shark1 Upper-Surface Isobars

( Contours at .05 Cp )

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 18.0% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 32.5% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 47.8% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 61.6% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 75.7% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 90.0% Span

Figure 23: Pressure Distributions of Shark1 Baseline Wing.

SYMBOL


SYN88 - Shark1

ALPHA .706

1.000

CL .2721

.2678

CD .01043

.01796

MACH = .780



0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 18.0% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 32.5% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 47.8% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 61.6% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 75.7% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 90.0% Span

Figure 24: Comparison of Shark5 Wing on Baseline Fuselage with Baseline Configuration.


SYMBOL


SYN88 - Shark1

ALPHA .718

1.000

CL .2714

.2678

CD .00738

.01796

MACH = .780



0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 18.0% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 32.4% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 47.5% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 61.2% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 75.4% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 89.9% Span

Figure 25: Comparison of Shark52 Wing on Stretched Fuselage and Baseline Configuration.

SYMBOL


SYN88 - Shark1

ALPHA .718

1.000

CL .2714

.2678

CD .00738

.01796

MACH = .780



-0.10

-0.05

0.00

0.05

0.10

Y/C

18.0% Span

-0.10

-0.05

0.00

0.05

0.10

Y/C

32.4% Span

-0.10

-0.05

0.00

0.05

0.10

Y/C

47.5% Span

-0.10

-0.05

0.00

0.05

0.10

Y/C

61.2% Span

-0.10

-0.05

0.00

0.05

0.10

Y/C

75.4% Span

-0.10

-0.05

0.00

0.05

0.10

Y/C

89.9% Span

Figure 26: Comparison of Shark52 and Shark1 Wing Drag Loops.


MACH = .780( Contours at .05 Cp )

SYN88 - Shark52ALPHA = .72 , CL = .2714

CD = .00738

SYN88 - Shark1ALPHA = 1.00 , CL = .2678

CD = .01796

Figure 27: Comparison of Shark52 and Shark1 Wing Pressure Contours.

SYMBOL

SOURCE SYN107P - SharkNS7

SYN88 - Shark52

REN 14.50

.00

ALPHA .633

.718

CL .2605

.2714

CD .01283

.00738

MACH = .780

SharkNS7 Upper-Surface Isobars


0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 18.0% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 32.4% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 47.5% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 61.2% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 75.4% Span

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

Cp

X / C 89.8% Span

Figure 28: Result of Navier-Stokes Inverse Design.


Final Shark WingAirfoil Geometry -- Camber & Thickness Distributions

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

0.0

10.0

Percent Chord

Air

foil

-2.0

-1.0

0.0

1.0

Cam

ber

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

Hal

f-T

hick

ness

SYMBOL

AIRFOIL Outboard

Root

ETA 68.0

18.1

R-LE 1.055

1.488

Tavg 3.96

4.34

Tmax 5.74

6.48

@ X 38.62

31.51

Figure 29: Final Wing Airfoil Geometry - Thickness & Camber Plots.


cs

z*

b/2A

A

l

z

t

cs

ts

(a) swept wing planform (b) section A-A

Figure 30: Structural Model for a Swept Wing.

WwingS

S

boxWwing

indicates different airplanes

(0,0)

Figure 31: Statistical Correlation of Total Wing Weight and Box Weight.


t

b/2C2

C3

C1

Figure 32: Wing Planform Design Variables.


B747 WING-BODY Mach: 0.850 Alpha: 2.533 CL: 0.449 CD: 0.01270 CM:-0.1408 CW: 0.0546 Design: 30 Residual: 0.5305E+00 Grid: 257X 65X 49 LE Sweep:42.11 Span(ft): 212.43 c1(ft): 48.17 c2: 29.11 c3: 10.79 I: 0.02089

Cl: 0.373 Cd: 0.05530 Cm:-0.1449 T(in):66.1586 Root Section: 13.6% Semi-Span

Cp = -2.0

Cl: 0.647 Cd: 0.00557 Cm:-0.2398 T(in):23.8498 Mid Section: 50.8% Semi-Span

Cp = -2.0

Cl: 0.431 Cd:-0.02153 Cm:-0.1873 T(in):12.1865 Tip Section: 92.5% Semi-Span

Cp = -2.0

Baseline (Dashed) / Redesign (Solid).

Figure 33: Wing-Section Optimization of Generic 747 at Fixed Baseline-Planform.


B747 WING-BODY Mach: 0.850 Alpha: 2.287 CL: 0.448 CD: 0.01167 CM:-0.0768 CW: 0.0516 Design: 30 Residual: 0.3655E+00 Grid: 257X 65X 49 LE Sweep: 36.61 Span(ft): 231.72 c1(ft): 47.17 c2: 28.30 c3: 10.86 I: 0.01941

Cl: 0.347 Cd: 0.06011 Cm:-0.1224 T(in):74.0556 Root Section: 12.7% Semi-Span

Cp = -2.0

Cl: 0.582 Cd: 0.00213 Cm:-0.2154 T(in):25.3014 Mid Section: 50.5% Semi-Span

Cp = -2.0

Cl: 0.390 Cd:-0.01648 Cm:-0.1736 T(in):12.0445 Tip Section: 92.5% Semi-Span

Cp = -2.0

Figure 34: Complete Optimization of Generic 747 to Maximize Breguet Range.


(a) Isometric (b) Front View

(c) Side View (d) Top ViewBaseline (Green) / Redesigned (Blue).

Figure 35: Geometry Changes of Complete Navier-Stokes Optimization.

Figure 36: Comparison of Euler-Redesigned (Red) and NS-Redesigned (Blue) Planforms.


α 3 Pareto frontvs.

R Q

Drag

Weight

P1α

Figure 37: Cooperative Game Strategy with Drag and Weight as Players.

110 115 120 125 130 135 140460

480

500

520

540

560

580

CD

(counts)

CW

(co

un

ts)

Effects of the weighting constants on the optimal shapes

0.05

0.1

0.15

0.2

0.25

0.3

BaselineFixed Planform

Maximized Range

Figure 38: Pareto Front; Ratios of α2

α1Indicated.


aerodynamic shape optimization part ii: sample...

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