A Coupled Aero-Structural Optimization Method

For Complete Aircraft Configurations

James J. Reuther∗

MCAT Institute

Juan J. Alonso†

Stanford University

Joaquim R. R. A. Martins‡

Stanford University

Stephen C. Smith§

NASA Ames Research Center

1 Abstract

This paper presents a new framework for the cou-pled optimization of aero-structural systems. Theframework permits the use of high-fidelity modelingof both the aerodynamics and the structures andrepresents our first step in an effort towards the de-velopment of a high-fidelity multidisciplinary opti-mization capability. The approach is based on effi-cient analysis methodologies for the solution of theaerodynamics and structures subproblems, an ad-joint solver to obtain aerodynamic sensitivities, anda multiprocessor parallel implementation. We haveplaced a geometry database representing the outermold line (OML) of the configuration of interest atthe core of our framework. Using this geometry de-scription, the information exchange between aero-dynamics and structures is accomplished throughan independent coupling of each discipline with theOML database. The framework permits the laterinclusion of other disciplines, such as heat transferand radar signatures, with relative ease. Specific re-sults from the coupling of a finite volume flow solverfor the Euler and Reynolds Averaged Navier-Stokes

∗AIAA Member, Research Scientist, NASA Ames Re-search Center, MS 227-6, Moffett Field, CA 94035, U.S.A.

†AIAA Member, Assistant Professor, Department of Aero-nautics and Astronautics, Stanford University, Stanford, CA94305, U.S.A

‡AIAA Student Member, Graduate Student, Departmentof Aeronautics and Astronautics, Stanford University, Stan-ford, CA 94305, U.S.A

§AIAA Member, Research Scientist, NASA Ames Re-search Center, MS 227-6, Moffett Field, CA 94035, U.S.A.

Copyright c©1999 by the American Institute of Aeronauticsand Astronautics, Inc. No Copyright is asserted in the UnitedStates under Title 17, U.S. Code. The U. S. Government hasa royalty-free license to exercise all rights under the copyrightclaimed herein for Governmental purposes. All other rightsare reserved by the copyright owner.

equations with two different linear finite elementstructural models are explored. Care is taken in thetreatment of the coupling of the disciplines such thata consistent and conservative scheme is achieved.Direct comparisons with wind-tunnel data are pre-sented to demonstrate the importance of aeroelasticsolutions. In addition, simplified design examplesare presented to illustrate the possible advantages ofthe new aero-structural design methodology in eval-uating trade-offs between aerodynamic performanceand structural weight for complete aircraft configu-rations.

2 Introduction

Considerable research has already been conductedon the multidisciplinary optimization (MDO) offlight vehicles. The survey paper by Sobieski [1]provides a comprehensive discussion of much ofthe work completed to date. These efforts haveranged from the development of techniques for dis-cipline coupling to actual demonstrations on real-world design problems. In most cases, these re-search efforts have shown the importance of inter-disciplinary coupling, as well as the inability ofsequential disciplinary optimization to achieve thetrue global optimum of a coupled system. For ex-ample, Wakayama [2, 3] has shown that in orderto obtain realistic planform shapes in the designof aircraft configurations it is necessary to includeboth multiple disciplines and a complete set of real-world constraints. Meanwhile, in the design of novelconfigurations such as a joined-wing aircraft, Gall-man [4] demonstrated that only multidisciplinarymethods are capable of revealing the relevant de-sign trade-offs; single-discipline optimization oftenleads to incorrect design choices. Unfortunately, the

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fidelity in the modeling of the various componentdisciplines in these preliminary design tools has re-mained at a relatively low level. Therefore, whileuseful at the conceptual design stage, these toolscannot accurately represent a variety of nonlinearphenomena, such as wave drag, which can play akey role during the detailed design phase.

On the other hand, recent applications of aerody-namic shape optimization using high-fidelity CFDmethods have resulted in substantial improvementsin the aerodynamic performance of complex air-craft configurations [5, 6, 7, 8, 9]. Jameson, etal. [10, 11, 12, 13, 14] have developed a mathematicalframework for the control of systems governed by theEuler and Navier-Stokes equations that has resultedin significant reductions in the computational costof aerodynamic shape optimization (ASO). Despitethe broad possibilities that these new ASO methodshave brought about, they also have had their shareof problems. In the case of aerodynamic wing design,planform and thickness constraints have often beenartificially imposed so that structural weight, fuelvolume, and takeoff/landing requirements would notbe adversely affected by the changes in the wingshape. These constraints were typically guided bythe result of low-fidelity multidisciplinary modelsand individual decisions made by experts from se-lected disciplines. By neglecting the coupling be-tween various disciplines, design constraints have of-ten been too restrictive to permit significant perfor-mance improvements, or not restrictive enough, thusallowing ASO to produce infeasible designs. In ad-dition, improvements in aerodynamic performanceresulting from span load changes cannot be accu-rately quantified in view of their unknown impacton the structural weight.

Enabled by recent advances in single-disciplineoptimization, novel restructuring of the multidisci-plinary design process [15, 16], and affordable super-computing alternatives [17, 9], the opportunity nowexists to develop an MDO framework which allowsthe participation of various relevant disciplines withhigh-fidelity modeling. The goal here is not to usehigh-fidelity modeling to construct a response sur-face [18] or train a neural network [19] but to useit directly during design. This kind of MDO envi-ronment has yet to be developed, but promises toimprove upon existing design methodologies by in-creasing the level of confidence in the final resultsfrom preliminary design. A higher confidence levelat an earlier stage in the design process holds out thepossibility of dramatically reducing the developmentcosts of the detailed design phase. Furthermore, theoverall quality and performance of the resulting de-sign will be improved when compared with tradi-tional sequential design strategies.

The goal of the current research is to establisha new framework for high-fidelity MDO. The im-

portant contributions presented to support such aframework are:

• The use of high-fidelity modeling of two dis-ciplines (RANS aerodynamics and linear FEMstructures).

• An OML geometry database which serves asboth an interface to the optimization algorithmand an interface for communication betweendisciplines.

• Sophisticated coupling algorithms that linkeach discipline to the OML such that informa-tion transfer between the disciplines is consis-tent and conservative.

• A framework for the computation of coupledsensitivities.

An excellent demonstration problem which illus-trates the strong coupling that can occur betweendisciplines is the case of aeroelastic wing design. Theoptimized shape and structure are the result of com-promises among numerous requirements and con-straints. Changes in the span load may lead to im-provements in induced drag but they can also incura structural weight penalty. Similarly, an increasein the thickness-to-chord ratio of the wing sectionsmay substantially improve the structural efficiencyof the configuration, but it may also lead to an unde-sirable increase in compressibility drag. Moreover,design constraints are often set by off-design condi-tions, such as protection from high-speed pitch-up,leading to the need to simulate these conditions aswell.

The complete aero-structural design problem in-volves the simultaneous optimization of the aerody-namic shape of a configuration and the structurethat is built to support its loads. The cost func-tion to be optimized requires a combination of aero-dynamic performance and structural weight, in or-der to address two of the main components of theBreguet range equation. Design variables are set upto parameterize the external aerodynamic shape ofthe configuration and the shape and material prop-erties of the underlying structure (spar cap areas,skin thicknesses, etc.). The design problem mustalso impose various constraints on the details of thestructure, such as the yield stress criterion (the max-imum stress in any part of the structure may notexceed the yield stress of the material at a num-ber of critical load conditions with the appropriatesafety margin), minimum skin thickness constraints,and fuel volume requirements. On the aerodynam-ics side, equality and inequality constraints may beimposed on both the total lift and pitching moment.Details of the pressure distribution for a transonicwing design problem, such as the location of the up-per surface shock, the slope of the pressure recovery,

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and the amount of aft loading, may also be imposedas design constraints.

The desired high-fidelity MDO framework forflight vehicle design suggested by this work must ad-dress the following issues:

1. Level of accuracy of disciplinary models.

2. Coupling between disciplines.

3. Computation of sensitivities.

In order to obtain the necessary level of accuracy,we intend to use high-fidelity modeling for both theaerodynamic and structural subsystems. For thispurpose, an Euler and Reynolds Averaged Navier-Stokes (RANS) flow solver has been used to modelthe aerodynamics. The details of the multiblocksolver, FLO107-MB, can be found in Ref. [7] andits parallel implementation on a variety of comput-ing platforms has been described in Ref. [9, 17]. Twodifferent Finite Element Methods (FEM) have beenused for the description of the behavior of the struc-ture. The first is a linear FEM model that uses brickelements which are appropriate for solid wind tun-nel configurations. The second is a linear FEM thatuses truss and triangular plate elements to modelthe structural components of aircraft configurations.Given these choices of the physical models for thedisciplines involved, it will be possible to capture allof the key trade-offs present in the aero-structuraldesign problem.

In our work, the inter-disciplinary coupling is per-formed using a geometry database of the outer moldlines (OML). All exchanges of information betweendisciplines are accomplished by independent com-munication with this OML database. This has theadvantage of standardizing the communication pro-cess and facilitates the inclusion of other disciplines.For the specific case of aero-structural coupling, wehave chosen to follow the work of Brown [20] in orderto carry out the bidirectional transfer of loads anddisplacements between the structure and the CFDmesh via the OML database. Careful attention hasbeen paid to the consistency and conservativenessof the load transfer, to the point that we believethe current setup will be suitable even for unsteadyaeroelastic flutter analysis. A consistent transfer isone that preserves the resultant forces and moments.If, in addition, the total work and energy are con-served, the transfer method is said to be conserva-tive.

The strong interdependence between aerodynam-ics and structures makes the computation of sen-sitivities of cost functions and constraints a diffi-cult task. In our past works, we have obtainedthe sensitivities of aerodynamic cost functions usingthe solution of an adjoint equation. This techniqueproduced aerodynamic sensitivities at a fraction of

the cost of traditional methods such as the finite-difference approach. The advantage of using the ad-joint approach was due, in large part, to the factthat the number of design variables was much largerthan the number of functions for which sensitivitieswere needed.

In the case of combined aero-structural design,a similar approach can be pursued: a set ofaero-structural adjoint equations can be formulatedwhich considerably reduce the cost of coupled sen-sitivity analysis. However, the nature of the aero-structural design problem is such that the number ofdesign variables is not always larger than the numberof cost functions and constraints. In particular, thisproblem is often characterized by a large number ofstructural stress constraints (one per element in thecomplete finite element model). Thus, by using acoupled adjoint approach directly it will be neces-sary to calculate a separate adjoint system for eachof these structural constraints. The straightforwardalternative to the adjoint approach is to use finitedifferencing. For cases in which the number of de-sign parameters is relatively small, this alternativemay indeed prove more cost-effective. However, thedesired goal of admitting a large number of designvariables makes the computational cost of the finite-difference approach unaffordable. Similarly prob-lematic is the use of the “direct” approach often usedefficiently in structural optimization. A prefactoredCFD Jacobian matrix is simply too large to computewith reasonable resources. Given these constraints,the sensitivity analysis aspect of high-fidelity MDOwill require much further future research. Details ofthe simplified sensitivity analysis used here, as wellas a framework to obtain coupled sensitivities, arepresented in Section 5.3.

3 Structural Finite ElementModels

In order to allow for the possibility of utilizing anarbitrary finite element model for the description ofthe structure, a detailed Application ProgrammingInterface (API) has been developed. This API ex-plicitly outlines both the content and format of theinformation that must be provided by a Computa-tional Structural Mechanics (CSM) solver intendedfor aeroelastic design. The API definition has alsobeen kept general enough to allow for a variety ofelement types within the same model.

The integration of existing and future structuralsolvers with the design code is therefore accom-plished through the use of this API. A typical se-quence of calls to the structural model is as follows:the first function call in the API consists of an ini-tialization process that builds the structural modeland all ancillary arrays, matrices, and matrix de-

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compositions. Additional functions in the API pro-vide the design algorithm with the complete geom-etry description of the external surface of the struc-tural model and the interpolation functions for boththe coordinates and displacements at any point ofthe structural model surface. Simple function callsexist in the API to obtain the structural displace-ment vector and a list of element principal stresses.Finally, since the design module continuously up-dates the OML geometry, an additional API call isused to update the structural model geometry andits stiffness matrix such that they conform to theOML.

For the results presented in this paper, we chose todevelop our own CSM solvers so that any necessarychanges to the source code could be made readily.Retrospectively, it became clear that once a couplinginterface was defined, no source code for the CSMsolver needed to be examined. The only adaptationto existing CSM methods that will be required is thecreation of a conforming interface (see Section 4).Thus, in future works we intend to couple the sameMDO framework with commercially available CSMcodes such as ANSYS and MSC-NASTRAN. Thetwo CSM solvers developed here use different finiteelement types and meshing strategies. They werebuilt to reflect accurately the behavior of the types ofwing structures present both in wind tunnel modelsand in real aircraft. Both solvers require the solutionof the classical structural equilibrium equation,

Kq = f . (1)

Here, K is the global stiffness matrix of the struc-ture, q is the vector of nodal displacements, and f isthe vector of applied nodal forces. With the appro-priate boundary conditions, matrix K is symmetricand non-singular. For the problem sizes of interesthere, a Cholesky factorization is appropriate. Thisfactorization can be stored and used multiple timeswith changing load vectors during an aeroelastic cal-culation. The stresses in each element can then berelated to the displacements by the following equa-tion:

σ = Sq, (2)

where S represents the product of the constitutivelaw matrix, the nodal displacement-strain matrixand the local-to-global coordinate transformationmatrix.

3.1 Wind Tunnel Model CSM Solver

A simple CSM solver was developed to compute de-flections of wind tunnel model wings. Because windtunnel models are typically machined from a sin-gle billet, 8-node isoparametric hexahedral solid el-ements were chosen to represent this type of solidstructure. These “brick” elements have 24 degrees

of freedom, representing the 3 components of thedisplacement at each node. The stiffness matrix foreach element is found using an 8-point (2 pointsin each coordinate direction) Gauss quadrature ofthe strain energy distribution within the element.These elements are called “isoparametric” becausethe same interpolation functions are used to describethe displacement field and the metric Jacobians usedfor the global coordinate transformation.

The CSM solver was designed to exploit the con-venience of an ordered arrangement of elements; ele-ment connectivity is implied by the point ordering ofthe input CSM mesh. This approach greatly simpli-fies input, and allows the flexibility of modeling thechannels typically cut in the wing surface to installpressure orifices and route pressure tubing. For thispurpose, finite element nodes can be located alongthe channel edges, so that distinct brick elementsoccupy the volume of the pressure channels. Themodulus of elasticity is then set to zero for theseelements, thus simulating the missing material.

For this study, the geometries of two typical busi-ness jets were chosen since wind tunnel measure-ments and CFD computational meshes were alreadyavailable in both cases. For each of the wings, afinite element model was constructed using 8-nodebrick elements. To avoid zero-height elements at theleading and trailing edges, the wing surface defini-tion was truncated at 4% and 96% of the wing chord.The motion of all nodes at the side of the fuselageis fully constrained. The remaining enclosed volumewas modeled by an ordered mesh of 4 nodes throughthe wing thickness, 6 nodes in the chordwise direc-tion, and 44 nodes spanwise from the side of thefuselage to the wing tip. For both cases, this re-sults in 645 elements and 3, 168 degrees of freedom.A typical wing CSM mesh is shown in Figure 1 to-gether with the location of the points on the surfaceof the OML and short segments indicating the pointson the CSM surface from which the OML derives itsdisplacements.

3.2 Aircraft Structure CSM Solver

A different CSM solver was used to model the behav-ior of realistic aircraft structures. This solver modelsa wing with multiple spars, shear webs, and ribs lo-cated at various spanwise stations, and the skins ofthe upper and lower surfaces of the wing box. Thestructural solver is based on a finite element code,FESMEH, developed by Holden [21] at Stanford.

Two types of finite elements are used: truss andtriangular plane-stress plate elements. Both elementtypes have 3 translational degrees of freedom pernode, so the truss has a total of 6 degrees of free-dom and the plate has 9 degrees of freedom. Fig-ure 2 shows a graphical representation of these twoelement types. Neither of these elements can carry

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Figure 1: Brick-element mesh of wind tunnel modelwing.

1

y

z

x

3

2

1

2

Figure 2: Truss and Triangular Plane Stress Plateelements

a bending moment, since their nodes do not haverotational degrees of freedom. The wing bending,however, is still well-captured since the contribu-tions of the second moments of inertia for the platesand trusses due to their displacement from the wingneutral axis is dominant when compared to their in-dividual moments of inertia about their own neutralaxes. The only limitation when using these kinds ofelements is that each of the nodes must be simplysupported, implying that we can have only one setof plate elements between any two spars.

In the modeling of a typical wing structure, trian-gular plates are used to model the wing skins. Platesare also used for the shear webs of spars and ribs,while the upper and lower spar caps are modeled us-ing trusses. The wing model in our case consisted of6 spars and 10 ribs, adding up to a total of 132 nodesand 640 elements. Figure 3 shows the geometry ofthe finite element discretization used.

4 Aero-Structural CouplingTechniques

Within the framework described previously, the op-timization of aero-structural systems requires, atleast, the solution of the coupled aeroelastic analysisproblem. The interaction between these two disci-

Figure 3: Wing Structural Model

plines, aerodynamics and structures, is quite strongsince the former provides the necessary loads to thelatter in order to determine the displacement field ofthe structure. In return, the structure provides sur-face deflections that change the aerodynamic prop-erties of the initial configuration.

Two issues in this transfer of information betweendisciplines are of utmost importance to the successof an automatic design technique: first, the level offidelity in the coupling of both disciplines has to becarefully considered in order to guarantee that theaccuracy of the individual disciplines is not jeopar-dized, and second, the evolving disciplinary designsmust have exact geometric agreement by the end ofthe design process.

In order to tackle the fidelity of the coupling, wehave chosen to ensure that the transfer of the dis-tributed pressure forces and moments from the CFDcalculation to the CSM nodal load vector is bothconsistent and conservative as defined in the ap-proach developed by Brown [20]. The property ofconsistency implies that the resultant forces and mo-ments imparted by the distributed pressure field, p,must be equal to the sum of the nodal forces andmoments in the CSM load vector, f . Conservationaddresses the important issue that the virtual workperformed by the load vector, f , undergoing a virtualdisplacement of the structural model (represented byδq) must be equal to the associated work performedby the distributed pressure field, p, undergoing theassociated displacement of the CFD mesh surface,δr. Thus, a procedure is devised that describes themotion of every surface point in the CFD mesh as afunction of the nodal displacements of the structuralmodel,

δr = [η]T · δq, (3)

where [η] is a matrix of linear weights on the dis-placement vector that is a combination of interpo-lations within the CSM mesh and extrapolations tothe OML as described by Brown [20]. The virtualwork in the CSM model can be represented as

δWCSM = fT · δq,

while the virtual work performed by the fluid acting

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on the surface of the CFD mesh is given by

δWCFD =∫

∂Ω

p nT · δr dS +∫

Ω

bT · δr dV.

Here, b represents a distributed body force per unitmass, if it exists, and ∂Ω is the CFD mesh surfacethat describes the interface between the fluid andthe structure. For a conservative scheme, δWCFD =δWCSM , and the consistent and conservative loadvector is given by:

FT =∫

∂Ω

p nT · [η]T dS +∫

Ω

bT · [η]T dV. (4)

For the two different structural models used in thiswork, the procedure used to obtain the relation inEq. 3 is implemented in a preprocessing step follow-ing Brown’s approach. The matrix [η] is thus pre-computed and stored for later use during the aeroe-lastic iteration procedure and plays a key role inboth the transfer of displacements and the compu-tation of the conservative and consistent load vector.

In order to enable communication between theaerodynamic and structural solvers, a standardizedOML surface representation of the configuration ofinterest is required. Solutions from each of the dis-ciplines (aerodynamics and structures) are interpo-lated onto this OML database so that they may beaccessed as needed by the other disciplines.

For this purpose, the OML geometry producedby AeroSurf has been used as the central database.AeroSurf is a geometry generation system that hasbeen specifically created for the analysis and de-sign of aircraft configurations including fuselage,wings, pylons, nacelles, and empennage [5, 6]. Aero-Surf preserves an aerodynamic geometry componentview of the complete configuration. These geome-try components are stored un-intersected in three-dimensional space. Typically, aerodynamic shapechanges are applied to these un-intersected compo-nents, and, once all modifications are completed, anew configuration is created by finding the inter-section(s) of the resulting surfaces. The intersectedgeometry is then decomposed into a series of well-defined parametric patches that constitute the OMLof the complete configuration. These patches (orthe points they are composed of) serve as the in-terface between aerodynamic and structural calcula-tions. It is our intention to expand the capability ofthis geometry-based interface to include additionaldisciplines in the future.

Each AeroSurf point is associated with a pointon the surface of the CSM model in a prepro-cessing step. During optimization, the displace-ments of each AeroSurf point are calculated by firstusing the CSM basis functions to interpolate theCSM nodal displacements at the projected Aero-Surf point. Then extrapolation functions are used

to carry the displacements from the CSM mesh tothe OML. When the CSM solver dictates a newposition for the structure, the locations in three-dimensional space of all the AeroSurf points areupdated by adding the deflections to the jig-shapepoints. This update process effectively constructsnew parametric patches to represent the surface ofthe perturbed configuration. In a similar fashion,during a preprocessing step, every point on the sur-face of the CFD mesh is associated with an Aero-Surf patch and a parametric location within thatpatch. The CFD points are assumed to be “tied”to these parametric locations, and, when the Aero-Surf database is altered, the location of the CFDsurface mesh points can be obtained by straightfor-ward evaluation of their parametric locations on thecorresponding AeroSurf patches. As can be seen,AeroSurf plays a central role in the transfer of dis-placements from CSM to CFD.

Furthermore, the AeroSurf database also plays asimilar role in the transfer of pressure informationfrom the CFD calculation to the structural load vec-tor. The transfer of surface pressure information tothe AeroSurf database is achieved by identifying the“donor” cells from the CFD mesh that contain thedesired information. The pressure integrations inEq. 4 are then performed with the same accuracyas can be achieved if the integration were to occurdirectly on the surface of the CFD mesh. The un-derlying assumption is that the mesh resolution ofthe AeroSurf database is comparable to, if not bet-ter than, that of the CFD surface mesh. This hasalways been the case in our design efforts. The cou-pling between aerodynamic and structural solvers inorder to obtain an aeroelastic solution is achieved inan explicit, sequential, iterative fashion by exchang-ing information at regular intervals during the con-vergence process. This coupling is greatly simplifiedby the fact that only static aeroelastic solutions areconsidered in this work, and the issue of time ac-curacy is inconsequential. For a typical complete,rigid configuration at fixed lift, an Euler solution re-quires in the neighborhood of 120 multigrid cyclesto reduce the average density residual by 5 ordersof magnitude. It has been found that, for fixed-liftaeroelastic calculations, the number of multigrid it-erations required increases by at most 10% if infor-mation is exchanged between the structural modeland the aerodynamics every 10 multigrid cycles. Ofcourse, in addition to the larger number of iterationsrequired, the cost of the structural solution has to beaccounted for. However, most of this cost is incurredin the decomposition of the stiffness matrix, and, asmentioned above, this can be accomplished in a pre-processing step. During the process of an updateto the structures, all that remains to be done is thecreation of a load vector and a back-solve operationwith the already factored stiffness matrix.

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The AeroSurf geometry database is currently a setof subroutines which are compiled together with themain optimization program. As the number of dis-ciplines increases, a desirable development would beto make the OML database a stand-alone programthat communicates directly with all the participat-ing disciplines. The AeroSurf OML can then takethe form of a daemon, and all communication canbe made via sockets.

Finally, although the current implementation ofAeroSurf relies on geometry creation and manipula-tion routines that we have developed, the ultimategoal is to use AeroSurf as a front-end to a ComputerAided Design (CAD) geometry kernel. This wouldgreatly facilitate the transfer of information back tothe working engineering model once the objectives ofthe design have been met. An interesting possibilityis to use the Computational Analysis PRogrammingInterface (CAPRI) developed by Haimes [22] whichenables individual discipline programs to interact di-rectly with a CAD solid model representation of thegeometry in question. However, even in this CAD-oriented scenario, the process of component-baseddesign with the necessary re-intersections will stillform the core of the methodology.

5 Sensitivity Analysis

The proposed high-fidelity MDO framework willalso need a strategy to perform design changes ina way that will minimize the need for expensivefunction evaluations. Detailed shape optimizationof aerodynamic surfaces for transonic wing designproblems requires a parameter space of O(100) orlarger [23, 24]. This requirement combined with theenormous cost of each function evaluation rendersthe use of zeroth-order methods, such as randomsearches and genetic algorithms, inefficient for thisproblem. The alternative of using a response surfacewhereby a polynomial fit of the design space is con-structed prior to optimization is also plagued withintractable computational costs since the number offunction evaluations required is proportional to thesquare of the number of design variables.

If we assume that the basic topology of the struc-ture (i.e., the number of spars, the number of ribs,the choice of materials, etc.) is not altered duringthe design, the design space should be smooth. Al-though many alternative global optimization strate-gies exist, for the aero-structural problem of in-terest, a gradient-based procedure holds the mostpromise. Gradient-based optimization algorithmscan be shown to converge only to a local optimum.If the cost function of the aero-structural problemis sufficiently multi-modal, these algorithms can failto achieve the global optimum. Nevertheless, whenused in conjunction with lower-fidelity MDO tools

that provide a reasonable starting point for the op-timization, they can yield significant and credibleimprovements in the design.

When compared with zeroth-order methods,gradient-based algorithms shift the computationalburden from evaluating the cost function to calculat-ing values of its gradient. The most direct way to es-timate gradients is the finite-difference approach inwhich a separate function evaluation is required foreach design variable in the problem. By using gradi-ent information, the total number of function eval-uations is greatly reduced. However, given the largecomputational cost involved in each function eval-uation, the finite-difference method has proven tobe unaffordable for the aerodynamic design of com-plete configurations. This limitation of the finite-difference method has provided the motivation todevelop new methods of obtaining sensitivity infor-mation for aerodynamic design problems. In partic-ular, the control theory adjoint technique has provenextremely valuable in making these kinds of calcu-lations possible.

5.1 Aerodynamic Sensitivities

The ground-breaking development of the adjointmethodology for both the Euler and Navier-Stokesequations was pioneered by Jameson [13, 14, 12, 10].Its extensions to treat complex configurations in-cluding the treatment of linear and non-linear con-straints and mesh deformations has been demon-strated by the first author [5, 6, 9].

In essence, the adjoint approach is able to obtainthe gradient of a cost function with respect to an ar-bitrary number of design variables through the solu-tion of a co-state equation. Given an aerodynamiccost function, I, which depends on both the flowfield variables, w, and the physical location of theOML boundary, F ,

I = I (w,F) ,

a change in F results in a change

δI =∂IT

∂wδw +

∂IT

∂F δF (5)

in the cost function. The governing equation, R, andits first variation express the dependence of w andF within the flow field domain:

R (w,F) = 0, δR =[

∂R

∂w

]δw +

[∂R

∂F]

δF = 0.

(6)Next, introducing a Lagrange multiplier, ψ, we have

δI = ∂IT

∂wδw + ∂IT

∂F δF − ψT([

∂R∂w

]δw +

[∂R∂F

]δF

)

=

∂IT

∂w− ψT

[∂R∂w

]δw +

∂IT

∂F − ψT[

∂R∂F

]δF .(7)

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Choosing ψ to satisfy the adjoint or co-state equa-tion [

∂R

∂w

]T

ψ =∂I

∂w, (8)

the first term in Eq. (7) is eliminated, and we findthat the desired gradient is given by

GT =∂IT

∂F − ψT

[∂R

∂F]

. (9)

Since Eq. (9) is independent of δw, the gradient of Iwith respect to an arbitrary number of design vari-ables can be determined without the need for addi-tional flow field evaluations. The main cost incurredis in solving the adjoint equation. In general, thecomplexity of the adjoint problem is similar to thatof the flow solution. If the number of design vari-ables is large, it becomes compelling to take advan-tage of the cost differential between one adjoint so-lution and the large number of flow field evaluationsrequired to determine the gradient using finite dif-ferences. Once the gradient is obtained, any descentprocedure can be used to obtain design improve-ments. At the end of each optimization iteration,new flow and adjoint calculations are performed toobtain an updated gradient, and the process is re-peated until the cost function reaches a minimum.

It must be noted that in the case of aerodynamicdesign it is often the case that the problems are char-acterized by a large number of design variables anda small number of independent aerodynamic costfunctions and constraints. This ratio of design vari-ables to cost functions and constraints is often theopposite in structural optimization problems. If anaerodynamic problem were characterized by havinga larger number of aerodynamic constraints com-pared with the number of design variables, the finitedifference approach may be more suitable. The al-ternative direct approach, often used for structures,requires the solution of

[∂R

∂w

]δw = −

[∂R

∂F]

δF (10)

for δw, followed by a substitution into Eq. (5). Itis noted that δw must be calculated for each de-sign variable independently. For small problems, itis possible to factor and store ∂R

∂w and obtain allthe δw vectors by a series of back-substitutions [25].Unfortunately, for large three-dimensional Euler andNavier-Stokes problems, the cost of factoring ∂R

∂w isnot acceptable, leaving the advantage of the directapproach difficult to obtain. For many flow regimesof interest, the linearization of the CFD Jacobianmatrix introduced in Eq. (10) is an unacceptableapproximation. Most aerodynamic solvers make noattempt to compute the Jacobian matrix; it is sim-ply too large and prefactoring it does not yield theadvantage seen for linear systems. Thus, without

prefactoring, the cost of solving Eq. (10) for eachdesign variable is not too different from the cost offinite differencing.

The reader is referred to our earlier works for thedetailed derivation of the adjoint equations specificto either the Euler or Navier-Stokes equations as wellas the other elements necessary to create an overalldesign algorithm [5, 6, 9].

5.2 Structural Sensitivities

In the structural optimization subproblem, typicaldesign variables include the cross-sectional areas ofthe truss elements that are used to model the sparcaps, and the thicknesses of the plate elements thatmodel the shear webs and skins.

The functions for which we require sensitivity in-formation will typically be the total weight of thestructure and the maximum stress on a given ele-ment. These are used as part of the overall costfunction (aerodynamic performance plus structuralweight) and to impose constraints on the problem.The sensitivities of the total weight with respectto the element size are trivial, since the weight ofa given element is proportional to a given dimen-sion. The sensitivities of the element stresses canbe calculated in a straightforward fashion using fi-nite differences. However, this approach is not verycost-effective since it requires the assembly and fac-torization of the global stiffness matrix, along withthe solution of the structural equilibrium equationfor each of the design variables. Although the finite-difference method was used in the results presentedin this work, the method of choice is the directmethod which is more efficient for cases where thenumber of cost functions and constraints is largerthan the number of design variables [26]. For casesin which the number of design variables dominatesthe problem, a structural adjoint method analogousto the aerodynamic adjoint method can be used.

In the following, we are interested in obtainingthe sensitivity of a vector-valued function gi, (i =1, . . . , nelems) to the design parameters P. In otherwords, we are seeking the values for all the entriesin the matrix

[dgi

dP], where the cost function, say

structural weight, is but a single component of gi.The direct method is derived by taking the first

variation of Eq. (1):

K δq =∂f∂q

δq− ∂K∂P q δP +

∂f∂P δP. (11)

It must be noted that for static load conditions,where the load vector is assumed to be independentof the structural design variables and deflections, asis often the case for structural optimization as a sin-gle discipline,

∂f∂P = 0, and

∂f∂q

= 0. (12)

8

This reduces Eq. (11) to

K δq = −∂K∂P q δP. (13)

As is shown later, the assumption of a constant loadvector does not hold in the more general problem ofcoupled aeroelastic design.

To find δq, Eq. (13) can be solved using the previ-ously factorized stiffness matrix by the same methodused for the solution of Eq. (1). This procedureneeds to be repeated for each design variable.

To obtain the sensitivity of a vector of functionalsgi (where gi could represent the stress in an elementin addition to any cost functions), we write the to-tal variation with respect to the design variables asfollows:

δgi =∂gi

∂P δP +∂gi

∂qδq. (14)

Note that δq is valid for the evaluation of the sensi-tivity of any functional.

It is seen that the prefactored stiffness matrix ren-ders the solution with respect to a significant num-ber of design variables relatively inexpensive. In thework presented for this paper, where the cost of theaerodynamic state and co-state analyses are at least2 orders of magnitude more than that of the struc-tural analyses, the benefit of using the direct ap-proach has not as yet been pursued.

5.3 Coupled Sensitivities

The computation of sensitivities for the aero-structural problem has components of both ASOand structural optimization techniques. However,if the true sensitivities of the design problem areneeded, the coupling terms cannot be neglected. Forexample, the sensitivity of the stress in a given ele-ment of the CSM model to an aerodynamic twistvariable has a component that depends on thechange to the geometry of the structural model anda second component that depends on the changingload vector applied to the structure. Both of thesecontributions are significant and must be accountedfor. Although in the results presented in this papera simplified penalty function is used to obtain a firstcut at the aero-structural design problem, we feel itis important to place the mathematical frameworkfor coupled sensitivities on more solid footing. Itwill inevitably turn out that the choice of the useof an adjoint approach will depend upon the prob-lem at hand. Since we propose to establish a flexibledesign environment, the possibility of using a cou-pled adjoint must be considered. The remainder ofthis section has been developed in collaboration withLessoine [27].

Consider, for example, a cost function where bothaircraft weight and drag are included. Then, if q

and P denote respectively the structural displace-ment field and structural parameters of the struc-tural model, w denotes the flow solution, and F rep-resents the design parameters of the undeformed air-craft shape, the aeroelastic objective function whosesensitivity we are looking for becomes I (w,q,F ,P).The variations in I are subject to the constraint

Ras (w,q,F ,P) = 0, (15)

where Ras designates the set of aero-structural equa-tions and can be partitioned as

Ras =(

R (w,q,F ,P)S (w,q,F ,P)

). (16)

Here, R denotes the set of fluid equations and S theset of structural equations. The variation δI can beexpressed as

δI =∂IT

∂wδw +

∂IT

∂qδq +

∂IT

∂F δF +∂IT

∂P δP. (17)

In order to eliminate δw and δq from the aboveequation, the following constraint can be introduced:

δRas =[∂Ras

∂w

]δw +

[∂Ras

∂q

]δq

+[∂Ras

∂F]

δF +[∂Ras

∂P]

δP = 0,

which calls for the partitioned Lagrange Multiplier

ψas =(

ψa

ψs

), (18)

where ψa is the portion of the adjoint associatedwith the fluid, and ψs is the portion of the adjointassociated with the structure. It follows that thefirst expression of δI can be replaced by

δI = ∂IT

∂wδw + ∂IT

∂qδq + ∂IT

∂F δF + ∂IT

∂P δP−ψT

as

([∂Ras∂w

]δw +

[∂Ras

∂q

]δq +

[∂Ras∂F

]δF +

[∂Ras∂P

]δP

)

=

∂IT

∂w− ψT

as

[∂Ras∂w

]δw +

∂IT

∂q− ψT

as

[∂Ras

∂q

]δq

+

∂IT

∂F − ψTas

[∂Ras∂F

]δF +

∂IT

∂P − ψTas

[∂Ras∂P

]δP.

Now, if ψ is chosen as the solution of the aero-structural adjoint equation

(∂Ras

∂w

)T

(∂Ras

∂q

)T

(ψa

ψs

)=

( ∂I∂w∂I∂q

), (19)

the expression for δI simplifies to

δI = GFδF + GPδP, (20)

where

GF =∂IT

∂F − ψTas

[∂Ras

∂F]

, (21)

9

and

GP =∂IT

∂P − ψTas

[∂Ras

∂P]

. (22)

Hence, the sought-after objective, which is theelimination of δw and δq from the expression for δI,is attainable but requires the solution of the adjointcoupled aero-structural problem

(∂R∂w

)T (∂S∂w

)T

(∂R∂q

)T (∂S∂q

)T

(ψa

ψs

)=

( ∂I∂w∂I∂q

). (23)

Now, since the creation of a completely coupledaero-structural adjoint would compromise our ob-jective of developing a flexible MDO framework, wecan rewrite Eq. (23) as

(∂R

∂w

)T

ψa =∂I

∂w−

(∂S

∂w

)T

ψs

(∂S

∂q

)T

ψs =∂I

∂q−

(∂R

∂q

)T

ψa,

where ψs and ψa are lagged values which are up-dated via outer iterations. This implies that existingadjoint solvers for both the aerodynamics and struc-tures can be used subject to convergence of the iter-ation. The additional right-hand-side forcing termscan then be updated in the same way as has beendone here with the state equations. Thus, the OMLgeometry can serve to couple both the state and co-state equations.

Beyond employing a coupled adjoint, the alterna-tive of using a coupled direct approach also exists.The development follows the one above very closelyin terms of the coupling. The terms in Eq. (12)which were assumed to be zero become the couplingvariables. However, since prefactoring of the CFDJacobian matrix is problematic, the approach willnot be much cheaper than using finite differencing.An alternative to either the adjoint or the direct ap-proaches is the use of a decomposed optimizationstrategy such as multi-level optimization [28] or col-laborative optimization [15]. Exploring all of thesevarious possibilities will form the basis of our futurework.

For the purposes of the present paper where a cou-pled adjoint has yet to be implemented, the sensi-tivities are obtained without coupling. The aerody-namic adjoint is used to obtain aerodynamic sensi-tivities and finite differences are used to obtain thestructural sensitivities. This approximation inher-ently implies that gradient information for a com-bined aerodynamic plus structural objective func-tion will not be completely accurate. The earlierexample of exploring how wing twist affects struc-tural stress levels highlights our current limitation.Without the coupling, we will capture only the por-tions of the sensitivities that result from structural

changes. The loading will act as if it were frozen justas in Eq. (12). Future works will address this limi-tation by implementing the coupled adjoint as out-lined above. Finally, for a detailed treatment of theoverall design process, we refer to references [5, 6].

6 Results

The results of the application of our aero-structuraldesign methodology are presented in this section.These results are divided into two parts: results ofaeroelastic analysis of existing complete configura-tion wind tunnel models, and results of aeroelasticdesign for flight configurations. The two sets of re-sults use two different structural models. In addi-tion, some of the results presented used the Eulerequations, while others used the Reynolds AveragedNavier-Stokes equations to model the fluid flow. Theresults are intended to showcase the current capabil-ities of the design method.

6.1 Navier-Stokes Aeroelastic Anal-ysis of Complete ConfigurationWind Tunnel Models

In this section, results of the rigid and aeroelasticanalysis of two different wind tunnel models repre-senting typical complete configuration business jetsare presented and compared with the available ex-perimental data. The CFD meshes used for eachof the two models contain the wing, body, pylon,nacelle, and empennage components. The meshfor the first model (model A) uses 240 blocks witha total of 5.8 million cells while the second mesh(model B) contains 360 blocks and a total of 9 mil-lion cells. The large mesh sizes are required for ad-equate resolution of all the geometric features foreach of the configurations and the high Reynoldsnumber boundary layers on their wings. It shouldbe mentioned that viscous and structural effects areresolved only on the wing surface; all other surfacesin the model are assumed to be inviscid and rigid.All calculations were run using 48 processors of anSGI Origin2000 parallel computer. A total of 1.3hours (model A) and 2.0 hours (model B) of wallclock time were required for the rigid-geometry solu-tions, while 1.4 hours and 2.1 hours were required forthe aeroelastic calculations. The structural modelis the one described in Section 3.1 since the prop-erties of its elements more closely approximate thebehavior of the wind tunnel model structure. Ex-perimental wind tunnel data are available for thetwo models at flight conditions as follows: ModelA, M∞ = 0.80, Re = 2.5 million and cruise CL,and Model B, M∞ = 0.80, Re = 2.4 million andcruise CL. Aeroelastic updates are performed every10 multigrid iterations of the flow solver. A total of

10

400 iterations were used to ensure an aeroelasticallyconverged solution. All solutions were calculated ata fixed CL by incrementally adjusting the angle ofattack.

A view of model A colored by Cp appears in Fig-ure 4 showing the wing, body, pylon, nacelle, andempennage present in the calculation. Figure 5shows a comparison of the pressure distributionsfor the rigid wing, the aeroelastic wing, and thewind tunnel data for model A. The sectional cutis near mid-span where wind tunnel measurementswere available. The figure shows that for this casethe aeroelastic deformation of the wing is so smallthat virtually no difference between the two com-puted results exists. In fact, the maximum tip de-flection of the model was calculated to be only 0.3%of the wing span. Agreement with the sparse windtunnel data indicates that the CFD is capturing theright trends present in the tested configuration. Thefact that the differences between the computed rigidand elastic wings are so small leads to the conclusionthat the wind tunnel data from this test probablyneed not be corrected for aeroelastic deflections. Inretrospect, it can be noted that the model A con-figuration has low sweep so there is very little twistdue to bending. Thus, since the outboard wing tipis not twisting much, large differences in the pres-sure distribution do not appear. If these calculationshad been performed before test entry, the confidencelevel on the tunnel data could have been increased.Figure 6 shows the difference in the spanload of thetwo computed solutions.

Figure 7 shows a similar comparison of pressuredistributions for rigid, aeroelastic, and wind tunneldata from model B. It is immediately clear that thedeflections predicted by the aeroelastic calculationhave a much larger impact on the pressure distribu-tions than in the case of model A. The changes inthe pressure distributions show all the typical signsof aeroelastic relief in swept-back wings: a decreasein the twist of the outboard sections of the wing withthe consequent forward motion of the shock locationand alterations in the spanload distribution.

Although the aeroelastic solution does not agreefully with the experimental data for model B, it isclear that the aerolastic effects change the solutionin the correct direction to improve the agreement.Additional discrepancies are believed to be causedby inaccuracies in the Baldwin-Lomax turbulencemodel. It is also evident that this wind tunnel modelis flexible enough that significant aeroelastic effectsare present in the wind tunnel data. In view of thesmall increase in cost of the aeroelastic solutions, itis clear that this type of analysis is preferable for thecomparison between experimental and wind tunneldata in order to eliminate some of the uncertaintiescausing the differences.

6.2 Aerodynamic Shape Optimiza-tion of a Flight Wing-Alone Ge-ometry

The results presented in this section correspond to atypical aerodynamic shape optimization calculationon a rigid geometry. The structural model is com-pletely inactive. This calculation is representativeof many of our earlier works [5, 6] and is intendedto present a baseline for comparison with aeroelasticdesigns in subsequent sections.

The geometry to be optimized is the wing of a typ-ical business jet having the same planform as that ofthe airplane shown in Figure 4. The flow field is com-puted using the Euler equations. A multiblock meshfollowing a C-H topology is constructed around theconfiguration with a total of 32 blocks and 750, 000cells. A total of 133 design variables are used toparametrize the surface of the wing. Hicks-Henneperturbation functions combined with exponentialfunctions at the wing trailing edges were distributedacross the entire span of the wing to provide full ge-ometric flexibility. Thickness constraints typical ofour previous works are imposed in order to main-tain the structural soundness of the final outcome ofthe design process. These constraints include spardepth constraints at 10% and 80% chord, a lead-ing edge radius constraint ahead of the 2% loca-tion, a trailing edge included angle constraint behindthe 95% chord location, and an additional thicknessconstraint to maintain maximum thickness and fuelvolume at 40% chord. Note once more that thesethickness constraints are the results of low-fidelityanalyses and are derived from years of accumulatedexperience by aerodynamic and structural designers.The objective function is the wing coefficient of drag,CD, at a fixed cruise CL = 0.35 and a fixed Machnumber of M = 0.82. It must be said that theseflight conditions represent a significant increase inboth Mach number and lift coefficient over those forwhich the original baseline wing was designed. It istherefore expected that improved aerodynamic de-signs should be attainable with the use of optimiza-tion. All wing-alone design calculations presentedhereafter were carried out on an SGI Origin2000 par-allel computer using 16 processors.

The results of this single-point shape optimizationprocess can be seen in Figure 8 which shows the ini-tial and final pressure distributions for several spanstations along the wing. Similar results have beenpresented in [12]. Notable features are the decreasein induced drag due to the shifting of the spanloadtowards the tip (Figure 9) and the decrease of wavedrag that results from the weakening or disappear-ance of the shock waves on the upper and lower sur-faces. Note that at the location of the front spar(10% chord) where the thickness constraint is ac-tive, the lower surface pressure distribution at some

11

of the stations exhibits an oscillation and a loss oflift due to the requirement of maintaining thickness.The changes in airfoil shape are rather small, butthe overall effect on the CD of the configuration isdrastic: after 20 design iterations, the total value ofCD is reduced by 31%, or from 95.6 counts to 65.6counts.

As shown in Figure 10, a comparison of aeroelasticanalyses of the baseline and resulting designs revealsthat the maximum stress levels for the rear spar haveincreased substantially in the inboard wing region,especially near the crank point. Figure 9 shows thatthe reason for the increase in stress in the rear sparis that the span loading has been shifted outboardsubstantially for this rigid-wing design in an effortto reduce the induced drag. Since the optimizationalgorithm can not see a structural penalty in thisoutboard shift of the spanload, it simply maintainsthe required thickness constraints and redistributesthe load as it sees fit.

6.3 Aerodynamic Shape Optimiza-tion of a Flight Wing-Alone Ge-ometry Including Aeroelastic De-formations

The design example presented in this section is iden-tical to the one in the previous section with the onlydifference being that the structure is no longer as-sumed to be rigid. During the flow calculation pro-cess, the structural model is allowed to deform un-der aerodynamic loads. However, the cost functionto be minimized still remains the coefficient of dragof the wing, CD, and the same artificial thicknessconstraints are imposed on the problem. Since thisdesign case constitutes only a small perturbation ofthe previous problem, it is expected that its out-come will differ from the last case by only a smallamount. Indeed, this is the case: the resulting pres-sure distributions, as seen in Figure 11, are nearlyidentical across the span, with very similar changesin the aerodynamic shape when compared with Fig-ure 8. However, an expected trend is seen whenthis design is compared with the former design ana-lyzed aeroelastically. Because the rigid-wing designsettled on a span loading that minimized induceddrag while maintaining the geometric constraints inthe rigid mode, it becomes less than optimum whenit is analyzed in the presence of the aeroelastic re-lief. Meanwhile, the wing designed in the presence ofaeroelastic effects will compensate for this washoutand make the appropriate shift in the spanloading tooptimize performance despite the increase in twist.This is precisely the effect that can be seen in Fig-ure 12. Apart from this subtle difference, both de-signs are quite close to each other. Figure 13 depictsa comparison between the initial and final span-

wise stress distributions on the rear spar showingthe same trend of increased inboard stress near thecrank point as was seen in the rigid-wing design case.

6.4 Aero-Structural Shape Optimiza-tion of a Flight Wing-Alone Ge-ometry

The idea in this final wing-alone design case is to in-corporate some basic elements of the aero-structuralinteraction present in the existing design method-ology. Despite the fact that development of thecomplete coupled sensitivity analysis is not yet im-plemented, several results of interest can be shownwhich establish the soundness of the procedure. Inthis particular case, we utilize the geometry of theprevious two sections, the same CFD mesh andstructural model, and the same set of aerodynamicshape variables. The artificial thickness constraintsare removed, leaving only the leading edge radiusand included trailing edge angle constraints. The de-sign is now set up with both the coefficient of dragand the L2 norm of the stress in the structure asa combined cost function. This combined penaltyfunction method can be thought of as a first cut ap-proach to minimizing total drag in the presence ofstructural constraints. The ASO adjoint system isused to calculate the gradient of the aerodynamiccost function (CD) and finite differencing is used tocalculate the gradient contribution from the struc-tural changes. Despite the fact that these sensitiv-ities are not fully accurate because of the lack ofcoupling, they provide our first approximation forsolving the AESO (AeroElastic Shape Optimization)problem. The weights between the two componentsof the objective function were arbitrarily chosen suchthat the stress penalty was equal to about 40% ofthe drag penalty. This choice resulted in an opti-mized design where the L2 norm of the stress in thestructure remained largely unchanged.

Figure 14 depicts the pressure distributions be-fore and after the design process. Once more, theresulting pressure distributions and changes to thesections look similar to those from the previous twodesign cases. However, there are some noteworthydifferences. The oscillation in the lower surface pres-sure distribution seen in the earlier two solutionsnear the 10% span chord location is not present.Since we are no longer imposing artificial thicknessconstraints, the resulting design was able to thinthis region with some benefit to the aerodynam-ics and without a significant increase in the struc-tural stress distributions. The more clearly observ-able difference between this solution and the pre-vious two is the dramatic thickening of the airfoilsection near the crank point. This is the locationwhere the highest stress level is recorded in the rearspar. Figure 15 shows that the design has again

12

dramatically changed the loading distributions bymoving part of the load outboard. This has a corre-sponding tendency to increase the load at the crit-ical crank point rear spar location. The design al-gorithm has chosen to increase the airfoil thicknessat this station to compensate for the shift in loadoutboard. It is worth remembering that changes tothe wing thickness can have an effect on wave drag.Indeed a re-examination of Figure 14 reveals thatthe shock strength on the lower surface has beenincreased from the original design. However, sincethe final design in this case is less than one counthigher in drag than that achieved in the previoustwo cases, this weak lower surface shock must notbe incurring a significant drag penalty. Figure 16illustrates the benefit of adding the stress penaltyfunction to the design problem. The spanwise stresson the rear spar at the planform break has beenreduced slightly in the optimized configuration. As-suming that no other constraints were placed on theproblem, it would then be possible to shift the loadon the wing outboard, while thickening the inboardsections so as to keep the wing weight approximatelyconstant. With a more accurate description of thecost functions and constraints in the problem, thesekinds of trade studies will allow the designer to makebetter-informed choices about the development ofthe configuration.

6.5 Aero-Structural Shape Optimiza-tion of a Wing in the Presence ofa Complete Configuration

The final test case demonstrates the capability ofthe new design algorithm to treat complete aircraftconfigurations. The configuration modeled in theCFD analysis includes a wing, body, nacelle pylonand empennage. In this case the Euler equationswere used to model the flow during the design pro-cess. The mesh for the configuration contains 240blocks and 4.2 million cells. The structural modelused in the previous three design cases is used againto obtain the structural deformations of the wing.The rest of the configuration is assumed to be rigid.The design conditions were chosen to be M = 0.82and CL = 0.30. These numbers, just as in the wing-alone design cases, represent an increase in the Machnumber and lift coefficient over the design point forthe initial configuration. Hence, it is expected thatsignificant improvements to the drag should be re-alizable. The design calculations were carried outusing 48 processors of an Origin2000.

The design approach follows that used for the lastwing-alone design case: a combination of the dragcoefficient and the L2 norm of the stresses in thestructure is used as the objective function. The sen-sitivities are calculated separately using the adjointfor the aerodynamics and finite differences for the

structures and neglecting the coupling between thetwo.

Figure 17 displays the airfoil sections and pressuredistributions developed by the optimization proce-dure. It is evident that the optimized design hassignificantly reduced shock strengths, especially onthe upper surface when compared with the baselineconfiguration. However, unlike the third wing-alonetest case where we saw dramatic thickening of thelower surface of the configuration in the vicinity ofthe crank point in the planform, here only smallchanges are apparent. It is refreshing that no thick-ness constraints were necessary to ensure a realisticdesign and that the penalty function approach wassufficient to prevent the airfoil sections from becom-ing unreasonably thin. The differences between theresult of this design and that of the wing-alone casesmay seem at first puzzling. All of the geometry mod-ifications were very slight for the case of the com-plete configuration. An examination of Figure 18reveals a possible explanation: the loading distri-bution for the complete configuration was closer tothe ideal distribution, and therefore there was notmuch to be gained from shifting the load to the out-board stations. Since the loading is not changingsignificantly, the required thickness changes near thecrank point necessary to maintain reasonable stresslevels (as can be seen in Figure 19) need not be dras-tic. The conclusion is that for this configuration themajority of the improvements in the aerodynamicperformance (an 18.1% reduction in drag) are com-ing from a reduction in wave drag, and thereforethe trade-off between aerodynamics and structuresis not so significant.

7 Conclusions

The work presented in this paper represents ourfirst step towards the establishment of a high-fidelity multidisciplinary environment for the designof aerospace vehicles. The environment is in its in-fancy and will continue to evolve during the comingyear(s). At its core, it consists of the following keyelements:

• High-fidelity modeling of the participating dis-ciplines (RANS flow models for the aerodynam-ics and linear finite element model for the struc-ture).

• An OML geometry database which serves asthe interface between disciplines. This databasecontains information regarding the currentshape of the configuration and the physical so-lutions from the participating disciplines.

• A force- and work-equivalent coupling algo-rithm designed to preserve a high level of accu-

13

racy in the transfer of loads and displacementsbetween aerodynamics and structures.

• A framework for the computation of coupledsensitivities of the aero-structural design prob-lem.

This design environment has been used to performRANS aeroelastic analysis of complete configurationflight and wind-tunnel models with an additionalcost which is less than 10% of the cost of a tradi-tional rigid-geometry CFD solution. These solutionscan be used to determine a priori whether significantaeroelastic corrections will or will not be needed forthe resulting wind tunnel data.

In addition, simplified design cases have been pre-sented that include the effect of aeroelastic defor-mations in the design process. These cases haveshown that our design methodology is able to pre-dict the correct trades between aerodynamic per-formance and structural properties present in thesetypes of wing design problems.

Finally, a structural stress penalty function wasadded to the coefficient of drag of the completeconfiguration to allow elimination of artificial thick-ness constraints that are typically imposed in aero-dynamic shape optimization methods. This rudi-mentary coupling of aerodynamics and structures inthe design not only eliminates the necessity to im-pose artificial constraints, but also produces designswhere trade-offs between aerodynamic and struc-tural performance are considered.

Further work will focus on the continued develop-ment of the proposed MDO framework. Topics re-quiring significant research include sensitivity anal-ysis, optimization strategy, Navier-Stokes based de-sign, use of commercially available CSM codes, mul-tipoint design, and CAD integration.

8 Acknowledgments

This research has been made possible by the sup-port of the MCAT Institute, the Integrated SystemsTechnologies Branch of the NASA Ames ResearchCenter under Cooperative Agreement No. NCC2-5226, and the David and Lucille Packard Founda-tion. Raytheon Aircraft is acknowledged for provid-ing relevant aircraft configurations and wind tunneldata as well as guiding the overall research effort.The authors would like to acknowledge the assis-tance of David Saunders in the review of the finalmanuscript and the help of Mark Rimlinger in thepreparation of CFD meshes and figures.

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[15] I. M. Kroo. Decomposition and collaborativeoptimization for large scale aerospace design.In N. Alexandrov and M. Y. Hussaini, editors,Multidisciplinary Design Optimization: Stateof the Art. SIAM, 1996.

[16] R. Braun, P. Gage, I. Kroo, and I. Sobieski.Implementation and performance issues in col-laborative optimization. AIAA paper 96-4017,6th AIAA/USAF/NASA/ISSMO Symposiumon Multidisciplinary Analysis and Optimiza-tion, Bellevue, WA, September 1996.

[17] A. Jameson and J.J. Alonso. Automaticaerodynamic optimization on distributed mem-ory architectures. AIAA paper 96-0409, 34thAerospace Sciences Meeting and Exhibit, Reno,Nevada, January 1996.

[18] A. A. Giunta, V. Balabanov, D. Haim,B. Grossman W. H. Mason, and L. T. Watson.Wing design for a high-speed civil transport us-ing a design of experiments methodology. AIAApaper 96-4001, 6th AIAA/NASA/ISSMO Sym-posium on Multidisciplinary Analysis and Op-timization, Bellevue, WA, September 1996.

[19] R. S. Sellar and S. M. Batill. Concurrentsubspace optimization using gradient-enhancedneural network approximations. AIAA pa-per 96-4019, 6th AIAA/NASA/ISSMO Sympo-sium on Multidisciplinary Analysis and Opti-mization, Bellevue, WA, September 1996.

[20] S. A. Brown. Displacement extrapolation forCFD+CSM aeroelastic analysis. AIAA pa-per 97-1090, 35th Aerospace Sciences Meetingand Exhibit, Reno, NV, January 1997.

[21] M. Holden. Optimization of dynamic systemsusing collocation methods. Ph. D. Disserta-tion, Stanford University, Stanford, CA, Febru-ary 1999.

[22] R. Haimes. CAPRI: Computational AnalysisPRogramming Interface. Massachusetts Insti-tute of Technology, March 1998.

[23] R. M. Hicks and P. A. Henne. Wing designby numerical optimization. Journal of Aircraft,15:407–412, 1978.

[24] R. Kennelly. Improved method for transonicairfoil design-by-optimization. AIAA paper 83-1864, AIAA Applied Aerodynamics Confer-ence, Danvers, Massachusetts, July 1983.

[25] G. W. Burgreen and O. Baysal. Three-dimensional aerodynamic shape optimizationof wings using sensitivity analysis. AIAA pa-per 94-0094, 32nd Aerospace Sciences Meetingand Exhibit, Reno, Nevada, January 1994.

[26] H. M. Adelman and Raphael T. Haftka. Sen-sitivity analysis of discrete structural systems.AIAA Journal, 24:823–832, 1986.

[27] J. Gallman, J. J. Alonso, J. Reuther, andM. Lessoine. Multi-disciplinary optimiza-tion using computational fluid dynamics (cfd).Topic Area Number 29: BAA 98-04-PRK, Re-search Proposal Submitted to the Air Force Re-search Laboratory, Wright-Patterson AirforceBase, Dayton, Ohio, 1998.

[28] M. Baker and J. Giesing. A practical approachto mdo and its application to an hsct aircraft.AIAA paper 95-3885, 1st AIAA Aircraft Engi-neering, Technology, and Operations Congress,Los Angeles, CA, September 1995.

15

Figure 4: Typical business jet configuration. FLO107-MB: Navier-Stokes, Baldwin-Lomax, M = 0.80, Re =2.5 million, 5.8. million mesh cells. Cp contours.

16

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

X/C

--- Cp* ---

Mach Alpha Re CL CD Z Cl Cd Cm Load0.801 1.190 2.52E+06 -0.28055 0.00486 152.060 0.40565 0.00686 -0.15171 0.347780.801 1.145 2.52E+06 -0.28308 0.00473 152.060

+

+

+

++

+

+

+

+

++

+

++

+ + +

+

++

+

+

+

Figure 5: Cp distribution at near wing tip station. Navier-Stokes calculations, M = 0.80, Re = 2.5 million———, Aeroelastic solution– – –, Solid geometry solution+ + +, Wind tunnel data

17

0.0 50.0 100.0 150.0 200.0 250.0 300.0

Span position

Loa

d

Aeroelastic AnalysisRigid Analysis

Figure 6: Spanwise Load Distribution.Comparison of the Rigid Analysis and Aeroelastic Analysis.Complete Configuration Navier-Stokes Solution.

18

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

X/C

--- Cp* ---

Mach Alpha Re CL CD Z Cl Cd Cm Load0.800 1.970 2.39E+06 -0.86257 -0.02482 259.280 0.45554 -0.00444 -0.16252 0.560600.800 1.970 2.39E+06 -0.90117 -0.02612 259.280

+

++

+

++

++++

+++

+

+

+

+

++

+

+

+ ++

+ +

+

+

+

++

++

Figure 7: Cp distribution at near wing tip station. Navier-Stokes calculations, M = 0.80, Re = 2.4 million———, Aeroelastic solution– – –, Solid geometry solution+ + +, Wind tunnel data

19

--- Cp* ---

Optimized Aeroelastic Euler Calcultion W25Mach Alpha Re CL CD Z Cl Cd Cm Load0.820 -0.133 1.45E+00 -0.37054 -0.00714 60.000 0.33272 0.00600 -0.15024 0.440070.820 -0.052 1.45E+00 -0.37057 -0.01020 60.000 0.35617 0.01047 -0.16076 0.47107

8a: span station z = 0.194

--- Cp* ---

Optimized Aeroelastic Euler Calcultion W25Mach Alpha Re CL CD Z Cl Cd Cm Load0.820 -0.133 1.45E+00 -0.37054 -0.00714 100.000 0.39833 0.00253 -0.19253 0.412360.820 -0.052 1.45E+00 -0.37057 -0.01020 100.000 0.38340 0.00350 -0.17055 0.39691

8b: span station z = 0.387

--- Cp* ---

Optimized Aeroelastic Euler Calcultion W25Mach Alpha Re CL CD Z Cl Cd Cm Load0.820 -0.133 1.45E+00 -0.37054 -0.00714 140.000 0.42006 -0.00053 -0.20178 0.377290.820 -0.052 1.45E+00 -0.37057 -0.01020 140.000 0.37990 0.00029 -0.16403 0.34122

8c: span station z = 0.581

--- Cp* ---

Optimized Aeroelastic Euler Calcultion W25Mach Alpha Re CL CD Z Cl Cd Cm Load0.820 -0.133 1.45E+00 -0.37054 -0.00714 180.000 0.40967 -0.00324 -0.19520 0.311810.820 -0.052 1.45E+00 -0.37057 -0.01020 180.000 0.35576 -0.00297 -0.15234 0.27077

8d: span station z = 0.775

Figure 8: Typical Business Jet Configuration. Drag Minimization at Fixed CL.Rigid Design, M = 0.82, CL = 0.35, 133 Hicks-Henne variables. Spar Constraints Active.Rigid Analysis at Fixed CL.– – –, Initial Pressures———, Pressures After 20 Design Cycles.

20

0.0 50.0 100.0 150.0 200.0 250.0

Span position

Loa

d

Rigid Design, Aeroelastic AnalysisBaseline Design, Aeroelastic Analysis

Figure 9: Spanwise Load Distribution.Comparison of the Rigid Design and the Baseline Design.Wing Alone Configuration.Rigid Design, Drag Minimization at Fixed CL.Aeroelastic Analysis at Fixed CL.

21

0.0 0.2 0.4 0.6 0.8 1.00.0

1.0

2.0

3.0

4.0

5.0

4*1

0

Span position

Str

ess

Rigid Wing DesignBaseline Wing Design

Figure 10: Spanwise Stress Distribution for the Rear Spar.Comparison of the Rigid Design and the Baseline Design.Wing Alone Configuration.Rigid Design, Drag Minimization at Fixed CL.Aeroelastic Analysis at Fixed CL.

22

--- Cp* ---

Optimized Aeroelastic Euler Calcultion W25Mach Alpha Re CL CD Z Cl Cd Cm Load0.820 0.221 1.45E+00 -0.37060 -0.00731 60.000 0.34439 0.00657 -0.15187 0.455460.820 0.226 1.45E+00 -0.37052 -0.01046 60.000 0.36685 0.01171 -0.16241 0.48514

11a: span station z = 0.194

--- Cp* ---

Optimized Aeroelastic Euler Calcultion W25Mach Alpha Re CL CD Z Cl Cd Cm Load0.820 0.221 1.45E+00 -0.37060 -0.00731 100.000 0.39915 0.00189 -0.19197 0.413220.820 0.226 1.45E+00 -0.37052 -0.01046 100.000 0.38688 0.00336 -0.17102 0.40050

11b: span station z = 0.387

--- Cp* ---

Optimized Aeroelastic Euler Calcultion W25Mach Alpha Re CL CD Z Cl Cd Cm Load0.820 0.221 1.45E+00 -0.37060 -0.00731 140.000 0.41157 -0.00172 -0.20424 0.369600.820 0.226 1.45E+00 -0.37052 -0.01046 140.000 0.37243 -0.00084 -0.16317 0.33445

11c: span station z = 0.581

--- Cp* ---

Optimized Aeroelastic Euler Calcultion W25Mach Alpha Re CL CD Z Cl Cd Cm Load0.820 0.221 1.45E+00 -0.37060 -0.00731 180.000 0.38930 -0.00481 -0.19960 0.296170.820 0.226 1.45E+00 -0.37052 -0.01046 180.000 0.33828 -0.00444 -0.15126 0.25735

11d: span station z = 0.775

Figure 11: Typical Business Jet Configuration. Drag Minimization at Fixed CL.Design in the Presence of Aeroelastics. M = 0.82, CL = 0.35133 Hicks-Henne variables. Spar Constraints Active.Aeroelastic Analysis at Fixed CL.– – –, Initial Pressures———, Pressures After 13 Design Cycles.

23

0.0 50.0 100.0 150.0 200.0 250.0

Span position

Loa

d

Aeroelastic Design, Aeroelastic AnalysisRigid Design, Aeroelastic Analysis

Figure 12: Spanwise Load Distribution.Comparison of the Aeroelastic Design and the Rigid Design.Wing Alone Configuration.Design in the Presence of Aeroelastics, Drag Minimization at Fixed CL.Aeroelastic Analysis at Fixed CL.

24

0.0 0.2 0.4 0.6 0.8 1.00.0

1.0

2.0

3.0

4.0

5.0

4*1

0

Span position

Str

ess

Aeroelastic Wing DesignRigid Wing Design

Figure 13: Spanwise Stress Distribution for the Rear Spar.Comparison of the Aeroelastic Design and the Rigid Design.Wing Alone Configuration.Design in the Presence of Aeroelastics, Drag Minimization at Fixed CL.Aeroelastic Analysis at Fixed CL.

25

--- Cp* ---

Optimized Aeroelastic Euler Calcultion W25Mach Alpha Re CL CD Z Cl Cd Cm Load0.820 0.244 1.45E+00 -0.37057 -0.00732 60.000 0.33599 0.00551 -0.14302 0.444380.820 0.226 1.45E+00 -0.37052 -0.01046 60.000 0.36685 0.01171 -0.16241 0.48514

14a: span station z = 0.194

--- Cp* ---

Optimized Aeroelastic Euler Calcultion W25Mach Alpha Re CL CD Z Cl Cd Cm Load0.820 0.244 1.45E+00 -0.37057 -0.00732 100.000 0.39600 0.00013 -0.18930 0.409960.820 0.226 1.45E+00 -0.37052 -0.01046 100.000 0.38688 0.00336 -0.17102 0.40050

14b: span station z = 0.387

--- Cp* ---

Optimized Aeroelastic Euler Calcultion W25Mach Alpha Re CL CD Z Cl Cd Cm Load0.820 0.244 1.45E+00 -0.37057 -0.00732 140.000 0.41870 -0.00319 -0.20832 0.376020.820 0.226 1.45E+00 -0.37052 -0.01046 140.000 0.37243 -0.00084 -0.16317 0.33445

14c: span station z = 0.581

--- Cp* ---

Optimized Aeroelastic Euler Calcultion W25Mach Alpha Re CL CD Z Cl Cd Cm Load0.820 0.244 1.45E+00 -0.37057 -0.00732 180.000 0.40554 -0.00476 -0.20656 0.308530.820 0.226 1.45E+00 -0.37052 -0.01046 180.000 0.33828 -0.00444 -0.15126 0.25735

14d: span station z = 0.775

Figure 14: Typical Business Jet Wing Configuration. Drag + Stress Minimization at Fixed CL.Aeroelastic Design with Stress Penalty Function. M = 0.82, CL = 0.35133 Hicks-Henne variables. Spar Constraints Inactive.Aeroelastic Analysis at Fixed CL.– – –, Initial Pressures———, Pressures After 13 Design Cycles.

26

0.0 50.0 100.0 150.0 200.0 250.0

Span position

Loa

d

Aeroelastic Design, Aeroelastic AnalysisBaseline Design, Aeroelastic Analysis

Figure 15: Spanwise Load Distribution.Comparison of the Aeroelastic Design and the Rigid Design.Wing Alone Configuration.Aeroelstic Design, Drag + Stress Minimization at Fixed CL.Aeroelastic Analysis at Fixed CL.

27

0.0 0.2 0.4 0.6 0.8 1.00.0

1.0

2.0

3.0

4.0

5.0

4*1

0

Span position

Str

ess

Aeroelastic Wing DesignBaseline Wing Design

Figure 16: Spanwise Stress Distribution for the Rear Spar.Comparison of the Aeroelastic Design and the Rigid Design.Wing Alone Configuration.Aeroelstic Design, Drag + Stress Minimization at Fixed CL.Aeroelastic Analysis at Fixed CL.

28

--- Cp* ---

Optimized Aeroelastic Euler Calcultion W25Mach Alpha Re CL CD Z Cl Cd Cm Load0.820 0.698 1.45E+00 -0.30164 0.00156 50.000 0.28500 0.02391 -0.12118 0.435140.820 0.759 1.45E+00 -0.29924 -0.00106 50.000 0.28807 0.02630 -0.12218 0.43981

17a: span station z = 0.194

--- Cp* ---

Optimized Aeroelastic Euler Calcultion W25Mach Alpha Re CL CD Z Cl Cd Cm Load0.820 0.698 1.45E+00 -0.30164 0.00156 100.000 0.36070 0.00655 -0.16848 0.395930.820 0.759 1.45E+00 -0.29924 -0.00106 100.000 0.36091 0.01195 -0.16536 0.39616

17b: span station z = 0.387

--- Cp* ---

Optimized Aeroelastic Euler Calcultion W25Mach Alpha Re CL CD Z Cl Cd Cm Load0.820 0.698 1.45E+00 -0.30164 0.00156 150.000 0.38957 -0.00147 -0.18712 0.356800.820 0.759 1.45E+00 -0.29924 -0.00106 150.000 0.38570 0.00193 -0.17985 0.35328

17c: span station z = 0.581

--- Cp* ---

Optimized Aeroelastic Euler Calcultion W25Mach Alpha Re CL CD Z Cl Cd Cm Load0.820 0.698 1.45E+00 -0.30164 0.00156 200.000 0.36974 -0.00568 -0.18229 0.271390.820 0.759 1.45E+00 -0.29924 -0.00106 200.000 0.36494 -0.00392 -0.17357 0.26787

17d: span station z = 0.775

Figure 17: Typical Business Jet Full Configuration. Drag + Stress Minimization at Fixed CL.Aeroelastic Design with Stress Penalty Function. M = 0.82, CL = 0.30133 Hicks-Henne variables. Spar Constraints Inactive.Aeroelastic Analysis at Fixed CL.– – –, Initial Pressures———, Pressures After 7 Design Cycles.

29

0.0 50.0 100.0 150.0 200.0 250.0 300.0

Span position

Loa

d

Aeroelastic DesignBaseline Design

Figure 18: Spanwise Load Distribution.Comparison of the Aeroelastic Design and the Baseline.Complete Configuration.Aeroelastic Design, Drag + Stress Minimization at Fixed CL.Aeroelastic Analysis at Fixed CL.

30

0.0 0.2 0.4 0.6 0.8 1.00.0

1.0

2.0

3.0

4.0

5.0

4*1

0

Span position

Str

ess

Aeroelastic DesignBaseline Design

Figure 19: Spanwise Stress Distribution for the Rear Spar.Comparison of the Aeroelastic Design and the Baseline.Complete Configuration.Aeroelastic Design, Drag + Stress Minimization at Fixed CL.Aeroelastic Analysis at Fixed CL.

31