AIAA 2003–3429 - Stanford Universityaero-comlab.stanford.edu/Papers/alonso.aiaa.03-3429.pdf · 2019. 6. 23. · AIAA 2003–3429 High-Fidelity Aero-Structural Design Using a Parametric

AIAA 2003–3429High-Fidelity Aero-Structural DesignUsing a Parametric CAD-BasedModelJuan J. AlonsoStanford UniversityStanford, CA 94305, U.S.A.

Joaquim R. R. A. MartinsUniversity of TorontoToronto, Ontario, Canada, M3H 5T6

James J. ReutherNASA Ames Research CenterMoffett Field, CA 94035, U.S.A.

Robert Haimes and Curran A. CrawfordMassachusetts Institute of TechnologyCambridge, MA 02139, U.S.A.

16th AIAA Computational Fluid Dynamics ConferenceJune 23–26, 2003/Orlando, FL

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AIAA 2003–3429

High-Fidelity Aero-Structural Design Using aParametric CAD-Based Model

Juan J. Alonso∗

Stanford UniversityStanford, CA 94305, U.S.A.

Joaquim R. R. A. Martins†

University of TorontoToronto, Ontario, Canada, M3H 5T6

James J. Reuther‡

NASA Ames Research CenterMoffett Field, CA 94035, U.S.A.

Robert Haimes§ and Curran A. Crawford¶

Massachusetts Institute of TechnologyCambridge, MA 02139, U.S.A.

This paper presents two major additions to our high-fidelity aero-structural designenvironment. Our framework uses high-fidelity descriptions for both the flow around theaircraft (Euler and Navier-Stokes) and for the structural displacements and stresses (afull finite-element model) and relies on a coupled-adjoint sensitivity analysis procedure toenable the simultaneous design of the shape of the aircraft and its underlying structureto satisfy the measure of performance of interest. The first of these additions is a directinterface to a parametric CAD model that we call AEROSURF and that is based onthe CAPRI Application Programming Interface (API). This CAD interface is meantto facilitate designs involving complex geometries where multiple surface intersectionschange as the design proceeds and are complicated to compute. In addition, the surfacegeometry information provided by this CAD-based parametric solid model is used as thecommon geometry description from which both the aerodynamic model and the structuralrepresentation are derived. The second portion of this work involves the use of the FiniteElement Analysis Program (FEAP) for the structural analyses and optimizations. FEAPis a full-purpose finite element solver for structural models which has been adapted towork within our aero-structural framework. In addition, it is meant to represent thestate-of-the-art in finite element modeling and it is used in this work to provide realisticaero-structural optimization costs for structural models of sizes typical in aircraft designapplications. The capabilities of these two major additions are presented and discussed.The parametric CAD-based geometry engine, AEROSURF, is used in aerodynamic shapeoptimization and its performance is compared with our standard, in-house, geometrymodel. The FEAP structural model is used in optimizations using our previous versionof AEROSURF (developed in-house) and is shown to provide realistic results with detailedstructural models.

Introduction

DURING the past decade the advancement of nu-merical methods for the analysis of complex engi-neering problems such as those found in fluid dynamicsand structural mechanics has reached a mature stage:

∗Assistant Professor, AIAA Member†Assistant Professor, AIAA Member‡Research Scientist, AIAA Associate Fellow§Senior Research Scientist, AIAA Member¶Graduate Student, AIAA MemberCopyright c© 2003 by the authors. Published by the American

Institute of Aeronautics and Astronautics, Inc. with permission.

many difficult numerically intensive problems are nowreadily solved with modern computer facilities. In fact,the aircraft design community is increasingly usingcomputational fluid dynamics (CFD) and computa-tional structural mechanics (CSM) tools to replacetraditional approaches based on simplified theories andwind tunnel testing. With the advancement of thesenumerical analysis methods well underway, the focusfor engineers is shifting toward integrating these anal-ysis tools into numerical design procedures.

These design procedures are usually based on com-

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American Institute of Aeronautics and Astronautics Paper 2003–3429

putational analysis methods that evaluate the relativemerit of a set of feasible designs. The merit of a designis normally based on the value of an objective functionthat is computed using analysis methods such as CFDand CSM programs, while the design parameters arecontrolled by an optimization algorithm.

Despite revolutionary accomplishments in single-discipline applications, progress towards the develop-ment of high-fidelity, multidisciplinary design opti-mization (MDO) methods has been slow. The level ofcoupling between disciplines is highly problem depen-dent and significantly affects the choice of algorithm.Multiple difficulties also arise from the heterogeneityamong design problems: an approach that is applica-ble to one discipline may not be compatible with theothers.

An important feature that characterizes the varioussolution strategies for MDO problems is the allow-able level of disciplinary autonomy in the analysis andoptimization components. The allowable level of disci-plinary autonomy is usually inversely proportional tothe bandwidth of the interdisciplinary coupling. Thus,for highly coupled problems it may be necessary to re-sort to fully integrated MDO, while for more weaklycoupled problems, modular strategies may hold an ad-vantage in terms of ease of implementation.

In the particular case of high-fidelity aero-structuraloptimization, the coupling between disciplines has avery high bandwidth. Furthermore, the values of theobjective functions and constraints depend on highlycoupled multidisciplinary analyses (MDA). As a result,we believe that a tightly coupled MDO environment ismore appropriate for aero-structural optimization.

The difficulty in formulating this kind of MDO prob-lem is that there are significant technical challengeswhen implementing tightly coupled analysis and de-sign procedures. Not only must MDA be performedat each design iteration but, in the case of gradient-based optimization, the coupled sensitivities must alsobe computed at each iteration.

Our previous work has presented and validateda tightly-coupled approach to high-fidelity aero-structural MDO that uses CFD and CSM. In addition,a coupled adjoint procedure was developed to produceinexpensive gradients of aero-structural cost functions.The main advantage of this approach resides in the factthat the cost of sensitivity calculations is completelyindependent of the total number of design variables inthe problem and their effect on the system (they mayinfluence either the aerodynamic shape of the aircraftor the shapes and thicknesses of the finite elements inthe structure).

During the validation process we have come to real-ize that there are two significant problems that mustbe tackled in order to make this design environmenttruly multi-purpose and industrial-strength. Firstly,high-fidelity design involving multiple disciplines re-

quires that a central, coherent, and accurate descrip-tion of the geometry of all participating disciplines becarefully computed and stored. This central geoemtryrepository changes as both the external shape of theconfiguration and the properties of the structure vary.Secondly, given that most of our prior experience wasbased on aerodynamic shape optimization work, wehad used simplified structural finite element model-ing techniques which we had developed in-house atStanford. In order to prove the versatility of this newdesign environment, more accurate and larger finiteelement models need to be used for the description ofthe structural behavior.

All multidisciplinary design relies on a parameteri-zation of the system that is being optimized: the shapeof the outer mold line (OML) is often described as afunction of a number of scalar parameters (design vari-ables) such as the aspect ratio, span, reference area,airfoil shape perturbations, etc., while the shape ofthe structure typically includes its appropriate param-eterization with the addition of elemental thicknessesand areas. As the models become more complex (com-plete aircraft configurations and detailed finite elementmodels), the task of reconstructing the underlying ge-ometry can become quite tedious and error prone:multiple intersections between aerodynamic surfaceshave to be computed, surface meshes must be regen-erated, and both CFD meshes and structural modelshave to be modified to accommodate the newly per-turbed shape dictated by the current values of thedesign variables.

Although in the past we have relied on efficientmethods that were developed in-house11,18 to re-intersect components and reconstruct the shape of theconfiguration, we have been limited to typical wing-fuselage, wing-pylon, and pylon-nacelle configurations.The true value of a high-fidelity multidisciplinary en-vironment will be brought to bear on unconventionalconfigurations with different arrangements of liftingand non-lifting surfaces and with propulsive systems(nacelles, in particular) that are tightly integratedwith the rest of the aircraft.

It seems obvious that this type of complex geometrymanagement can be handled routinely by all ComputerAided Design (CAD) packages used in industry. Sincethe latest generations of all the major CAD packagessupport parametric design, we have opted to interfaceour design environment with CAD-based parametricaircraft models of arbitrary complexity. This approachcan treat the aero-structural models we are interestedin and will allow us to focus on the development of themodules that are more specific to realistic design.

The result of a high-fidelity aero-structural opti-mization is only as good as the quality of the indi-vidual components. For this reason, we have replacedour in-house finite element structural model,8 whichwe used for all of our initial development with the

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full-featured, multi-purpose, Finite Element AnalysisProgram (FEAP) of Taylor.20 FEAP includes all thenecessary infrastructure to carry out linear and non-linear analyses on very complicated models. In addi-tion, it has interfaces to various parallel sparse matrixsolvers that can reduce the computational burden pre-sented by extremely large models. We have enhancedFEAP with a full Application Programming Interface(API) that allows us to integrate FEAP into our de-sign environment. We have also enhaced FEAP so thatit may be used to compute sensitivities of structuralfunctionals via an adjoint method.

The following sections begin with a description ofthe aero-structural design framework we have previ-ously developed and the kind of design problem thatwe intend to solve. We then present the philosophyand details of our new CAD-neutral interface togetherwith the details of the parametric aircraft model thatwe use in our optimizations. After a brief dscriptionof the additions and modifications that we have madeto the FEAP software, we present results of the aero-dynamic optimization of a small supersonic jet usingthe new additions to our aero-structural framework.We finally conclude with some remarks regarding theusability and effectiveness of this new tool and withour view of the potential future use of this framework.

Aero-Structural Design FrameworkThe main objective of this framework is to calculate

the sensitivity of a multidisciplinary cost function withrespect to a number of design variables. The functionof interest can be either the objective function (typ-ically the drag coefficient at fixed lift or the emptyweight of the structure) or any of the constraints spec-ified in the optimization problem (such as elementstresses or lumped versions of the element stresses likeKS functions). In general, such functions depend notonly on the design variables, but also on the physicalstate of the multidisciplinary problem. Thus we canwrite the function as

I = I(x, y), (1)

where x represents the vector of design variables andy is the state variable vector.

For a given set of design variables x, the solution ofthe governing equations of the multidisciplinary sys-tem yields a state y, thus establishing the dependenceof the state of the system on the design variables. Wedenote these governing equations by

R (x,y (x)) = 0. (2)

The first instance of x in the above equation indicatesthe fact that the residual of the governing equationsmay depend explicitly on x. In the case of a structuralsolver, for example, changing the size of an element hasa direct effect on the stiffness matrix. By solving the

governing equations we determine the state, y, whichdepends implicitly on the design variables through thesolution of the system.

Since the number of equations must equal the num-ber of state variables, R and y have the same size.For a structural solver, for example, the size of y isequal to the number of unconstrained degrees of free-dom, while for a computational fluid dynamics (CFD)solver, this is the number of mesh points multipliedby the number of state variables stored at each point.For a coupled system, R represents all the governingequations of the different disciplines, including theircoupling. This can be a rather large set of equationsthat need to be solved to obtain the equilibrium stateof the multidisciplinary system.

�

� � � ��

Fig. 1 Schematic representation of the govern-ing equations (R = 0), design variables (x), statevariables (y), and objective function (I), for an ar-bitrary system.

A graphical representation of the system of govern-ing equations is shown in Figure 1, with the designvariables x as the inputs and I as the output. Thetwo arrows leading to I illustrate the fact that theobjective function typically depends on the state vari-ables and may also be an explicit function of the designvariables.

When solving the optimization problem using agradient-based optimizer, we require the total varia-tion of the objective function with respect to the designvariables, dI/ dx. As a first step towards obtainingthis total variation, we use the chain rule to write thetotal variation of I as

δI =∂I

∂xδx +

∂I

∂yδy. (3)

If we were to use this equation directly, the vectorδy would have to be calculated by solving the govern-ing equations for each component of δx. If there aremany design variables and the solution of the govern-ing equations is costly (as is the case for large couplediterative analyses), using equation (3) directly can beimpractical.

We now observe that the variations δx and δy in thetotal variation of the objective function (3) are not in-dependent of each other since the perturbed systemmust always satisfy the governing equations (2). A re-lationship between these two sets of variations can be

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obtained by realizing that the variation of the residu-als (2) must be zero, i.e.

δR = ∂R∂x

δx +∂R∂y

δy = 0. (4)

Since this residual variation (4) is zero we can addit to the objective function variation (3) without mod-ifying the latter, i.e.

δI =∂I

∂xδx +

∂I

∂yδy + ΨT

(∂R∂x

δx +∂R∂y

δy

), (5)

where Ψ is a vector of arbitrary scalars that we callthe adjoint vector. This approach is identical to theone used in nonlinear constrained optimization, whereequality constraints are added to the objective func-tion, and the arbitrary scalars are known as Lagrangemultipliers. The problem then becomes an uncon-strained optimization problem, which is more easilysolved.

We can now group the terms in equation (5) thatcontribute to the same variation and write

δI =(

∂I

∂x+ ΨT

∂R∂x

)δx+

(∂I

∂y+ ΨT

∂R∂y

)δy. (6)

If we set the term multiplying δy to zero, we are leftwith the total variation of I as a function of the de-sign variables and the adjoint variables, removing thedependence of the total variation on the state vari-ables. Since the adjoint variables are arbitrary, we canaccomplish this by solving the adjoint equations

∂R∂y

Ψ = − ∂I∂y

. (7)

These equations depend only on the partial derivativesof both the objective function and the residuals of thegoverning equations with respect to the state variables.Since these partial derivatives do not depend on thedesign variables, the adjoint equations (7) only needto be solved once for each I and their solution is validfor all the design variables.

When adjoint variables are found in this manner,we can use them to calculate the total sensitivity of Iusing the first term of equation (6), i.e.,

dIdx

=∂I

∂x+ ΨT

∂R∂x

. (8)

The cost involved in calculating sensitivities using theadjoint method is practically independent of the num-ber of design variables. After having solved the gov-erning equations, the adjoint equations (7) are solvedonly once for each I, and the vector products in thetotal derivative in equation (8) are relatively inexpen-sive.

It is important to realize the difference between thetotal and partial derivatives in this context. Partial

�

�

��

Fig. 2 Schematic representation of the aero-structural governing equations.

derivatives can be evaluated without regard to the gov-erning equations. This means that the state of thesystem is held constant when partial derivatives areevaluated, except, of course, when the denominatorhappens to be a state variable, in which case all butthat particular state variable can kept constant. Totalderivatives, on the other hand, take into account thesolution of the governing equations which change thestate y. Therefore, when using finite differences, thecost of computing partial derivatives is usually a verysmall fraction of the cost involved in estimating totalderivatives.

The partial derivative terms in the adjoint equationsare therefore relatively inexpensive to calculate. Thecost of solving the adjoint equations is similar to thatinvolved in the solution of the governing equations.

The adjoint method has been widely used in severalindividual disciplines and examples of its applicationinclude structural sensitivity analysis1 and aerody-namic shape optimization.9,15,16

Aero-Structural Sensitivity AnalysisWe now use the equations derived in the previous

section to write the adjoint sensitivity equations spe-cific to the aero-structural system. In this case we havecoupled aerodynamic and structural governing equa-tions, and two sets of state variables: the flow statevector and the vector of structural displacements. Fig-ure 2 shows a diagram representing the coupling in thissystem. In the following expressions, we split the vec-tors of residuals, states and adjoints into two vectorscorresponding to the aerodynamic and structural sys-tems, i.e.

R =[ A

S]

, y =[

wu

], Ψ =

[ψφ

]. (9)

Using this notation, the adjoint equations (7) for anaero-structural system can be written as

[∂A∂w

∂A∂u

∂S∂w

∂S∂u

]T [ψφ

]= −

[∂I∂w∂I∂u

]. (10)

In addition to the diagonal terms of the matrix that ap-pear when we solve the single-discipline adjoint equa-tions, we also have off-diagonal terms that express the

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sensitivity of the governing equations of one disciplinewith respect to the state variables of the other. Theresidual sensitivity matrix in this equation is identi-cal to that of the global sensitivity equations (GSE)introduced by Sobieski.19 Considerable detail is hid-den in the terms of this matrix and their calculationis not always straightforward. The meaning of all ofthese terms and the procedures that we have used inour work to compute them have been reported previ-ously.10–14 The reader is referred to these publicationsfor in-depth explanations.

Note that the diagonal entries in (10) representthe single discipline adjoint solutions for both aerody-namics and structures. Our previous work has dealtwith the development of aerodynamic adjoint formu-lations in detail15–17 and we have in place the abilityto perform aerodynamic shape optimization of com-plete configurations using our SYN107-MB solver. Forlinear structures, The adjoint operator is simply thetranspose of the stiffness matrix. Since this matrixis symmetric, the linear structures adjoint problem isessentially the same as a structural solution with adifferent right hand side that is often called a pseudo-load.

The off-diagonal terms in (10) introduce the effectsof the aero-structural coupling into the calculation offunctional sensitivities. These are the terms that areresponsible for the differences between truly-coupledaero-structural sensitivities and those obtained via se-quential single-discipline optimizations.

The right-hand side terms in the aero-structural ad-joint equation (10) depend on the function of interest,I. In our case, we are interested in two differentfunctions: the coefficient of drag, CD, and the KSfunction, a lumped version of the stress constraints.When I = CD we have,

• ∂CD/∂w: The direct sensitivity of the drag co-efficient to the flow variables can be obtainedanalytically by examining the numerical integra-tion of the surface pressures that produce CD.

• ∂CD/∂u: This term represents the change in thedrag coefficient due to the displacement of thewing while keeping the pressure distribution con-stant. The structural displacements affect thedrag directly, since they change the wing surfacegeometry over which the pressure distribution isintegrated.

When I = KS,

• ∂KS/∂w: This term is zero, since the stresses donot depend explicitly on the loads.

• ∂KS/∂u: The stresses depend directly on the dis-placements since σ = Su. This term is thereforeequal to [∂KS/∂σ] S.

Since the factorization of the full matrix in thecoupled-adjoint equations (10) would be extremelycostly, our approach uses an iterative solver, much likethe one used for the aero-structural solution, where theadjoint vectors are lagged and the two different sets ofequations are solved separately. For the calculationof the adjoint vector of one discipline, we use the ad-joint vector of the other discipline from the previousiteration, i.e., we solve

[∂A∂w

]Tψ = − ∂I

∂w−

[∂S∂w

]Tφ̃, (11)

[∂S∂u

]Tφ = − ∂I

∂u−

[∂A∂u

]Tψ̃, (12)

where ψ̃ and φ̃ are the lagged aerodynamic and struc-tural adjoint vectors. The final result given by thissystem, is the same as that given by the originalcoupled-adjoint equations (10). We call this procedurethe lagged-coupled adjoint (LCA) method for comput-ing sensitivities of coupled systems. Note that theseequations look like the single discipline adjoint equa-tions for the aerodynamic and the structural solvers,with the addition of forcing terms in the right-hand-side that contain the off-diagonal terms of the resid-ual sensitivity matrix. Note also that, even for morethan two disciplines, this iterative solution procedureis nothing but the well-known block-Jacobi method.This iterative procedure is guaranteed to converge toan aero-structural adjoint solution as long as the ma-trix representing the aero-structural operator in equa-tion (10) remains diagonally dominant. Obviously,this may become an issue in problems that are stronglycoupled. However, our experience has shown that evenin the case of extremely large structural deflections (ofthe order of 1/3 of the span) the lagging approach con-tinues to converge properly.

Once both adjoint vectors have converged, we cancompute the final sensitivities of the objective functionby using the following expression

dIdx

=∂I

∂x+ ψT

∂A∂x

+ φT∂S∂x

, (13)

which is the coupled version of the total sensitivityequation (8). For more details regarding the meaningand calculation of these partial derivative terms, theredear is again referred to our previous work.11,12

It must be noted that, as is the case of the partialderivatives in equations (10), all these terms in (8) canbe computed without incurring a large computationalcost, since none of them involve the solution of thegoverning equations.

In summary, once an aero-structural analsyis hasbeen obtained, the aero-structural adjoint equationscan be solved with a cost that is close to that of themultidisciplinary analysis itself. With both the aero-dynamic and structural adjoint vectors in hand, the

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sensitivity of a given cost function with respect to anarbitrary number of design variables can be computedwithout significant additional cost. One additionalcoupled-adjoint calculation is required to compute thesensitivities of each additional cost function. Since thecost of the coupled adjoint procedure is proportionalto the number of functions for which we are seek-ing sensitivities (while the cost remains independentof the total number of design variables) the coupledadjoint procedure is more suitable for problems withlarge numbers of design variables and a handful of costfunctions. With appropriate use of constraint lumpingapproaches (K-S functions in our work, for example)this approach can offer very significant performanceadvantages with respect to the other available alter-natives.

CAD-Based Parametric ModelingAs mentioned in the introduction, the geometric

manipulations that are necessary to carry out mul-tidisciplinary design on complete configurations canbecome quite complicated. As the design evolves, itsshape changes, and the geometry must be regeneratedrepeatedly through a process of geometry creation, in-tersection, filleting, etc. Although in the past we havecreated our own geometry creation/intersection rou-tines, their applicability has been limited to a numberof types of configurations such as wing, wing-body, andwing-body empennage. Support for some classes ofpylons and nacelles was also added and the resultingframework was used with success during the High-Speed Research program for complete HSCT config-urations.

High-fidelity multidisciplinary techniques, however,can be used to design configurations that are non-standard with much more closely integrated wing/s,fuselages, propulsion system and with possibly multi-ple lifting surfaces. Although continued developmentof our in-house geometry framework was considered asa possibility, the advantages of a completely general,CAD-based, geometry engine far outweigh the devel-opment cost: arbitrarily complex parametric modelscan be handled by a design module (based on theCAD package of the designer’s choice) which has beenspecifically developed for that purpose. Moreover, allgeometric operations can be carried out more robustlyand, in addition, the existence of this central geometrymodel can be used as the medium for the exchange ofinformation between disciplines and as the provider ofinformation for mesh generation/perturbation for allthe participating disciplines.

It is a simple matter of economics (i.e., the enor-mous amount of work required) that the traditionalcoupling approach of constructing a direct interfacefor each software package to each available CAD en-gine becomes prohibitive, particularly for “smaller”disciplines where the interface development may con-

sume more resources than the discipline solver itself.CAPRI7 (Computational Analysis Programming In-terface) provides a solution to the CAD dependencyissue. Coupling to any supported CAD package isboth unified and simplified by using the CAPRI def-inition of geometry (with topology) and its API toaccess the geometry and topological data. This CAD-vendor neutral API is more than just an interface toCAD data; it is specifically designed for the construc-tion of complete analysis suites. CAPRI’s ’GeometryCentric’ approach allows access to the CAD part fromwithin all sub-modules (grid generators, solvers andpost-processors), facilitating such tasks as node en-richment by solvers and designation of mesh faces asboundaries (for the solver and the visualization sys-tem). CAPRI supports only manifold solids at its baselevel, eliminating problems associated with manuallyclosing surfaces outside of the underlying CAD kernel.Multidisciplinary coupling algorithms can use the ac-tual geometry as the medium to interpolate data fromdifferent grids. One clear advantage to this approachis that the geometry never needs to be translated andhence remains simpler and closed. The other majoradvantage is that writing and maintaining the gridgenerator (coupled to the CAD system) can be doneonce through CAPRI; all of the major CAD vendorsare then automatically supported.

Geometry Creation and Modification

At the beginning of the CAPRI project there wasalways the notion that design functionality wouldbe supported. At the time, it was thought thatCAPRI would support the direct construction of three-dimensional solid geometry in order to allow for themodification of said geometry. As the geometry read-ers were being implemented, it became obvious thatthis would not be possible. Each CAD system dealswith the low-level geometry construction in a very dif-ferent manner. There was certainly not a commonvendor-neutral perspective on direct construction. Infact, only those systems based on geometry kernels(and allowing the use of the kernel) could performconstruction. Therefore, only if one programmed inParasolid, Acis or OpenCASCADE could this kind ofconstruction be performed.

As it turns out, this limitation was fortunate; an-other type of construction was available that could bedriven by an API. Most modern CAD systems supportthe master-model concept of representing an object.A master model describes the sequence of topologicaloperations to build the geometry of a solid model. Ata basic level, it is an ordered list of extrude, revolve,merge, subtract and intersection operations. CAD sys-tems support more meaningful abstractions, such asblends, fillets, drilled holes and bosses. When the CADmodel is regenerated, the operation list is interpretedto sequentially build the geometry of the part. This

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gives the operator the ability to construct a familyof parts (or assemblies) by building a single instance.Many of the operations used in the construction canbe controlled by parameters that may be adjustable.By changing these values, a new member of the familycan be built by simply following the prescription out-lined in the master-model definition. We will refer tothis type of modifiable CAD part as a parametric CADpart. This kind of part is fundamental to the gener-ation of the full aircraft aerodynamic and structuralmeshes in the following sections.

The recipe may be simple, like a serial collection ofprimitive operations, but can also be complex, whereoperations are performed on previously or temporar-ily constructed geometry. The representation of thisconstruction in most CAD systems is in the form of atree, usually referred to as the feature tree. By sup-porting this method of construction, CAPRI providesboth simple and powerful access to the CAD system.This approach is clearly outside the static view tradi-tionally held of geometry. That is, this kind of accessand control is not possible from any type of file trans-fer.

Within CAPRI, this tree is presented to the pro-grammer in the form of branches. Each of these en-tities has an index to identify where in the tree thereference is made. All indices are relative (that is theycan occur anywhere in the tree; the assignment is usu-ally given during initial parsing of the CAD internalstructures). There is a special branch always giventhe index zero: the root of the tree. Therefore, theentire tree may be traversed starting at the root andmoving toward the end of each branch. The branchesterminate at leaves (branches that do not contain anychildren). To aid in traversing the tree toward the rootthe parent branch is always available. Unlike simplebinary trees, a branch in CAPRI’s feature tree maycontain zero or more children.

Currently, the structure of tree itself cannot beedited from within CAPRI (though this may changeat some future release). However, some branches maybe marked suppressible -these features may be turnedoff- in a sense removing that branch (and any childrenof the branch) from the regeneration. This is powerfulin that it allows for defeaturing the model, so that itmay be made appropriate for the type of analysis athand. For example: if fasteners are too small for a fluidflow calculation, they may be easily suppressed (if themaster-model was constructed with this in mind). Af-ter part regeneration the resultant geometry would besimplified and the details associated with the fastenerswould not be expressed.

Parameters are those components of the master-model that contain values (and should not be confusedwith the geometric parameterization). CAPRI exposesall of the adjustable (non-driven) parameters found inthe model. This is a separate list from the feature

tree, but references back to the associated branch fea-tures where the values are used or defined. Parametersmay be single- or multi-valued and can be booleans,integers, flooating-points or strings.

This CAD perspective on parametric building ofparts and assemblies is fine for driving the part us-ing simple parameters but is problematic for detailedshape design of the kind necessary in high-fidelityaero-structural calculations. For example, simple pa-rameters may be used to define the planform of anaircraft, but are difficult to use to define the airfoilshape of the wing and tail components. The designerwould need to expose the curve/surface definition ata very fine and detailed level (i.e. knot points asthe parameters) to allow for the exact specificationof shapes. CAPRI avoids placing this burden on theCAD designer by exposing certain curves as multi-valued parameters. These curves are obtained from in-dependently sketched features in the model that laterare used in solid generation as the basis for rotation,extrusion, blending and/or lofting. The curves canbe modified, and, when regenerated, the new part ex-presses the changed shape(s). This functionality iscritical for shape design in general and specificallyaerodynamic shape design.

The authors have used this approach on a number ofdifferent environments and with several kinds of verydifferent geometries3,6 with success. Although the ob-ject of this paper is to use such an evironment forthe parametric design of aircraft, other examples ofthe use of CAPRI include the geometry representation(for meshing purposes) of highly complicated turbineblades with or without cooling passages6 and the mul-tidisciplinary design optimization of power-generatingwind turbines where the parametric CAD model is cen-tral to a number of different analyses (aerodynamics,structures, performance, cost, etc.) which were drivenby several kinds of optimizers.3

In the current work, the CAPRI interface also ap-pears to fulfill most of the requirements for geometrycreation and retrieval in a gradient-based design envi-ronment:

• Control: it allows both global and detailed controlover the shape of the geometry through the use ofuser-modifiable parameters and curves.

• Robustness: a large number of geometry modifi-cations can be created through the design processwithout failure.

• Accuracy: all of the geometries produced rep-resent the intended shape to a high degree ofaccuracy. This is particularly important whencomputing gradients, since very small geometryperturbations are required.

• Portability: the multidisciplinary design softwarethat is developed to interact with CAD can do so

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with all major vendors without the need to modifythe code.

For this purpose, we have developed a CAPRI-basedfront-end for the CAD package Pro/Engineer that al-lows for all the geometry manipulations that we havebeen accustomed to in our multiblock design programSYN107-MB.15,16

AEROSURF Parallel/Distributed GeometryServer

During the process of aero-structural optimization,the aircraft geometry is modified according to a pa-rameterization which is controlled by the values ofa number of independent design variables. In ouraerodynamic shape optimization work, for example,a number of wing and fuselage sections are modifiedto obtain an optimum-performing shape. In addition,scalar design variables such as the wing sweep angleor aspect ratio can be used to alter the global shapeof the aircraft. In the past, we have used a geometrykernel which was developed to support a number ofcomponent intersections typical in traditional aircraftshapes.18

The advantage of this approach is that the computa-tional cost of each geometry re-generation is extremelylow (a fraction of a second). The cost of gradientcomputation in adjoint methods is presumably inde-pendent of the number of design variables. However,the gradient formula typically includes a volume inte-gral term which requires that the volume mesh be per-turbed according to the changes in the surface geom-etry. For large numbers of design variables, this costcan become significant if the geometry re-generationprocedure is expensive. With our old geometry kernel,the impact of mesh regeneration was never more thana small portion of the overall cost of the optimization.

In this work, however, we are concerned with the useof an underlying CAD database to guide the processof the multidisciplinary design. At these early stagesof the integration process, the cost of CAD geome-try re-generation is still quite a bit higher than thatof our old method: for the parametric CAD modelof the aircraft that we will discuss in the followingsection, geometry regenerations that do not involvesectional changes (the airfoil and fuselage profiles arekept constant while the global shape of the configu-ration is altered) consume about 7 sec of CPU timeon a single SGI R14000 600 Mhz processor. Due toperceived internal Pro/Engineer limitations with re-generations that involve sectional changes, if airfoilprofiles or fuselage sections are changed, the actualcost of re-generation grows significantly to around 150sec. These figures are obviously a function of both thecomplexity of the parametric model (how many oper-ations are required to complete a re-generation) andthe processor being used. Although the performancequoted above is expected to improve drastically (par-

ticularly for sectional re-generations) with differentCAD packages, it is the current state of performancethat has driven the design of our new geometry kernel,AEROSURF.

AEROSURF is a CAPRI-based, parallel and dis-tributed application that acts as a broker betweenthe CAD package and the simulation software thatrequests the variation in the geometry (the designcode, for example). Figure 3 shows a schematic of thestructure of AEROSURF and its relation to an aero-dynamic shape optimization program. AEROSURFis built around the concept of a geometry server thatconstantly services multiple requests for parametric ge-ometry. AEROSURF starts a number of instances ofthe CAD kernel (Pro/Engineer in our case, but theprogramming, via CAPRI, is independent of the CADvendor) and maintains a series of queues for each ofthe CAD kernels that have been started.

The process is straightforward: AEROSURF isstarted on the server that is able to run the CADsoftware prior to the start of the design calculation.It itself starts a number N of CAD kernel instancesthat will do the actual re-generation work. Afterthe design software is started (typically on a differ-ent parallel computer), AEROSURF awaits until theneed for geometry re-generation arises. At periodicintervals during the design process, processors in thesimulation request geometry re-generations. Multiplerequests can and should be issed in parallel to min-imize the overall cost of the geometry re-generationand to take advantage of the parallel nature of oursystem. AEROSURF receives these re-generation re-quests, attaches a unique identifier to them, and eitherforwards them to a free CAD kernel, or queues themup if no CAD kernels are available. As the CAD ker-nels complete their re-generation work, they forwardtheir results (typically in the form of a surface grid)to AEROSURF, which, in turn, sends the results backto the requesting simulation processor. AEROSURFuses the Parallel Virtual Machine (PVM) interfaceto communicate with the simulation software. Thisenables the distributed running of the geometry andsimulation components. All communication betweenAEROSURF and the simulation software occurs acrossthe network (typically Ethernet). Since the size of eachgeometry description is (in our particular case) around315Kb, the cost of transmission (in both directions) ispractically negligible.

Since the work in each geometry re-generation iscompletely independent of the others, the problem isembarrasingly parallel and AEROSURF achieves al-most perfectly linear scalability, which has been testedup to 32 simultaneous CAD kernels. In this way, theaverage time for a geometry re-generation can be aslow as 7 sec/32 ≈ 0.22 sec for scalar parameter manip-ulation, and 150 sec/32 ≈ 4.7 sec for a re-generationwith section changes. Note that we typically have

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Arbitrary Network Interconnect (Ethernet)

SYN107−MB−AE SYN107−MB−AESYN107−MB−AE SYN107−MB−AE SYN107−MB−AE



CAD Model CAD Model CAD Model

PVM Session

Task 1

ProE

Task 2 Task N

AEROSURF MASTER PROGRAM

MPI COMMUNICATION LAYER FOR SOLVER AND OPTIMIZER

ProE

ProE

v_1, v_2, ..., v_N v_1, v_2, ... , v_N v_1, v_2, ... , v_N

Parallel Computer 1

Parallel Computer 2

Fig. 3 Schematic representation of the CAPRI-based AEROSURF package.

run aerodynamic shape optimization in the past usingO(300-400) design variables that mostly affect the var-ious sections in the wing and fuselage components ofthe configuration. Given that the CAD re-generationcosts are quite a bit higher than those for our old geom-etry engine, we are at present limited as to the numberof design variables that we can handle. Work in thenear term will address these performance shortcomingsso that the CAD engine can fully replace the more lim-ited (albeit faster), older version.

A typical aerodynamic shape optimization calcu-lation requires a number of different geometry re-generations. After an initial re-generation to producethe baseline configuration (assuming that the initialvalues of the design variables are non-zero) the geom-etry needs to be perturbed in each and every one ofthe design variables (once per design cycle) when thegradient vector is completed. In addition, as we relyon the NPSOL4 Sequential Quadratic Programmingoptimizer, after a gradient is computed, the objectivefunction is minimized along this direction. Duringthese line searches, the geometry is perturbed (typi-cally three times to construct a quadratic fit) alongthe direction of the gradient. For a calculation withNdv design variables, a typical requirement if to haveapproximately Ndv + 5 CAD regenerations per designiteration. Typical calculations use around 50 design it-erations. The reader should be reminded that, becauseof the use of adjoint methods, we are able to afford theuse of very large numbers of design variables. Conse-quently, the number of required CAD re-generations

can indeed become very large.

Parametric Aircraft CAD Model

The basis of this work is an aicraft parametric modelconsisting of five components: fuselage, wing, verti-cal and horizontal tails (in a T-tail configuration) andnacelles. Although the nacelle definition is includedin the parametric CAD model, it is ignored in allsubsequent simulations. A total of 100 global shapeparameters can be changed to alter the configuration.In addition, a total of 36 sections (15 airfoils on thewing, 3 in both the horizontal and vertical tails, and15 fuselage sections) typically defined by 50-100 pointseach, can be modified to create exact geometry repre-sentations with the level of detail that is often requiredin aerodynamic shape optimization.

A top view of the parametric CAD model can beseen in Figure 4 where the top and bottom halvescorrespond to choices of section definitions and pa-rameters that create, using the same CAD paramet-ric part, both the baseline supersonic business jet forthis work, and an approximate definition of a Boeing717-200 jet. This figure is meant to show the versa-tility of the current parametric model which is ableto cover a family of wide-ranging geometries wherethe components are arranged with the same topol-ogy. In the future we intend to create a library ofparametric CAD models that span the range of our air-craft and spacecraft design interests (advanced super-sonic configurations with closely integrated nacelles,blended-wing-body aircraft, standard transonic trans-port configurations, reusable launch vehicles, and even

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Fig. 4 Aircraft parametric model for two sets of variables defining both a supersonic business jet andthe Boeing 717-200.

America’s Cup yachts).Each of the five components of the CAD model has

a number of design variables that can alter the shapeof that component (in addition to the section changesmentioned earlier). The three wing components haveidentical parameterizations: a single-crank planformmodel was adopted where the reference area, aspectratio, taper ratio, sweep angle, location of the leadingand trailing edge crank point, and leading and trailingedge extensions (inboard of the crank point) can becontrolled independently. In addition, the twist angleat three spanwise stations (root, crank, and tip loca-tions) can also be controlled. The fuselage shape canbe arbitrarily defined at 15 stations, whose shape andlocation can be changed, thus permitting both config-uration area ruling and fuselage camber modificationsthat can substantially help decrease both the volume-and lift-dependent portions of the wave drag of theaircraft. Finally, the nacelles are simply defined bytheir length and diameter, their toe-in and pitch ori-entation, their location, and the airfoil geometry thatis revolved to create the actual nacelle.

The design problems in the Results section use thisparametric CAD model and a subset of the availableparameters to carry out the aerodynamic shape opti-mization of a Mach 1.5 supersonic business jet.

FEAP - Finite Element AnalysisProgram

The Finite Element Analysis Program (FEAP),20

written by Prof. Robert L. Taylor at UC Berkeley,is a general purpose finite element package for theanalysis of complex structures. The program includesthe capability to construct arbitrarily complex finiteelement models using a library of one-, two-, and three-dimensional elements for linear and non-linear defor-mations. In addition, a number of material models

(isotropic, orthotropic, plasticity, etc.) are available tomodel the constitutive properties of the materials thatthe structure is built of. Once the model is assembled,a number of solution procedures are available for lin-ear, non-linear, and time-accurate problems. In addi-tion, for very large non-linear structural models, inter-faces are available for external parallel sparse solversthat can greatly improve the calculation turnaroundtimes. A number of advanced time-accurate integra-tion algorithms are also included with FEAP whichcan be of interest in the computation of aeroelasticresponses and constraints.

The problem solution step is constructed using acommand language concept in which the solution algo-rithm is completely written by the user. Accordingly,with this capability, each application may use a solu-tion strategy which meets its specific needs. There aresufficient commands included in the system for appli-cations in structural or fluid mechanics, heat transfer,and many other areas requiring solution of problemsmodeled by partial differential equations, includingthose for both steady-state and transient problems.Users also may add new routines for model descriptionand command language statements to meet specific ap-plications requirements. These additions may be usedto assist in the generation of meshes for specific classesof problems or to import meshes generated by othersystems.

Following our earlier approach with the in-housestructural model, we have developed an interface forFEAP that accesses the major data structures (nodes,elements, materials, displacements, stresses, etc.) andallows other programs (simulation software, optimiz-ers) to carry out the typical steps of a structural anal-ysis. In addition, we have expanded FEAP to includevarious modules that are necessary for structural op-timization. These modules include an adjoint solver

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and the calculation of the various terms that appear inthe coupled aero-structural adjoint equation describedearlier.

FEAP has surpassed our expectations and has be-haved consistently and robustly in a number of testcases that we have encountered. However, in orderto expedite our development, we have used finite dif-ferences in some of the FEAP-related terms of thecoupled aero-structural adjoint equation. This canlead to both inaccuracies and poor computational per-formance. For that purpose, we intend, in the nearfuture, to develop additional design modules for FEAPthat provide analytic sensitivities of some of the mostcommonly used elements in aircraft structures. In thisway we will by-pass the use of finite differences.

Structural Optimization

As a step toward the final goal of performing fully-coupled aero-structural optimization it was importantto perform structural optimization studies for a wingof fixed outer-mold line subject to constant loads.

The structural model of the wing — shown in Fig-ure 5 — is constructed using a wing box with six sparsevenly distributed from 15% to 80% of the local chordat the root and tip sections. Ribs are distributed alongthe span at every tenth of the semispan. A total of193 finite elements were used in the construction ofthis model. Appropriate thicknesses of the spar caps,shear webs, and skins were chosen to model the realstructure of the wing. The structural analysis is per-formed by FEAP.

The objective of this optimization case is to mini-mize the weight of the structure by varying the thick-nesses and cross-sectional areas of the finite elementswhile constraining the stresses in each of these ele-ments to be less than the yield stress of the material.

Because there is a significant number of elements(albeit not close to a realistic structure), it can be-come computationally very costly to treat the stressconstraints separately, especially in the case where thestructural optimization is coupled with aerodynamicshape optimization.

The sensitivities of KS functions (see below) withrespect to the finite-element sizes are efficiently com-puted by using an adjoint method.1,11 Since we areusing an adjoint method for computing sensitivities, itis convenient to lump the individual element stressesusing Kreisselmeier–Steinhauser (KS) functions. Sup-pose that we have the following constraint for eachstructural finite element,

gi = 1− σiσyield

≥ 0, (14)

where σi is the element von Mises stress and σyield isthe yield stress of the material. The corresponding KS

function is defined as

KS (gi(x)) = −1ρ

ln

[∑

i

e−ρgi(x)]

. (15)

This function represents a lower bound envelope ofall the constraint inequalities and ρ is a positive pa-rameter that expresses how close this bound is to theactual minimum of the constraints. This constraintlumping method is conservative and may not achievethe exact same optimum that a problem treating theconstraints separately would. However, the use of KSfunctions has been demonstrated and it constitutes aviable alternative, being effective in optimization prob-lems with thousands of constraints.2

The structure of the wing is parameterized with atotal 193 design variables representing the thicknessof the shells that model the spars, ribs and skins, andthe cross-sectional area of the frames that model thecaps for the spars. Although the structural model issmall, the design problem is rather large in comparisonto typical design space sizes. This compromise repre-sents the ideal spot for early development work sinceadditional model complexity and size would only in-crease the execution time, but would not increase thecomplexity of the design problem.

The structural optimization is performed bySNOPT, a nonlinear optimization package.5 The op-timization result shown in Figure 6 took 357 majoriterations to find the optimum solution. Note that thestructure is not as fully stressed as we would expectfor a fully optimized structure. This is due to the con-servative character of the KS function.

Results of Aerodynamic ShapeOptimization

The objective in this section is to both demonstrateand validate the outcome of our CAD-based aero-dynamic shape optimizations. For that purpose, wehave designed an efficient baseline configuration witha cruise weight of 100, 000 lbs, flying at a cruise alti-tude of 55, 000 ft at M∞ = 1.5. The cruise CL = 0.1is forced to remain constant throughout our optimiza-tions. The configuration wing planform is designedwith a cranked delta wing shape with the inboardleading edge swept behind the Mach cone, while theoutboard leading edge remains supersonic. The fuse-lage was sized to accommodate 10 passengers and arearuling was applied in an approximate manner. An Eu-ler analysis of this configuration results in an inviscidcruise drag coefficient of CD = 0.00858.

Aerodynamic Shape Optimization Using In-HouseGeometry Engine

Our first design test case modifies the detailed shapeof the wing and fuselage in order to minimize the invis-cid drag of the configuration at a constant CL = 0.1.Although the wing planform remains fixed, the twist

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Fig. 5 Baseline structure. Fig. 6 Optimized structure.

and shape of 7 defining stations evenly spread alongthe span can be altered. At each of these definingstations, in addition to the twist variable, 10 Hicks-Henne bump functions are added on the top and bot-tom surfaces. Additional leading and trailing edgecamber functions are also used. In order to preventimprovements in performance that simply result froma decrease in the wing volume, a total of 30 thicknessconstraints are added at six of the defining stations sothat the thickness may not decrease at the 2, 25, 50,75, and 98% chord locations.

The fuselage has circular cross-sections and its vol-ume is constrained to remain constant. A total of 11fuselage camber design variables are added to the opti-mization problem. Including the wing shape variables,a total of 136 design variables are considered in thismodel with 30 linear constraints for the wing thick-nesses.

Using the NPSOL optimizer, and after 9 design iter-ations, the drag of the configuration decreases by 9%to CD = 0.00781. This improvement in drag coeffi-cient has been achieved without decreases in either thewing or fuselage volume and it is about evenly dividedbetween improvements due to fuselage shape pertur-bations and wing shape perturbations. This fact canbe confirmed since we ran an identical optimizationwithout the fuselage design variables which achievedclose to 50% of the drag improvement reported here.

The optimizer changes the shape of the fuselagequite drastically: the originally axisymmetric body isgiven both fore and aft camber, presumably to spreadthe lift produced by the fuselage in the streamwisedirection so as to minimize the contribution of the lift-dependent wave drag. The wing geometry has alsochanged drastically: the originally untwisted wing nowhas nearly 0.5 deg of washout. In addition, the baselineconfiguration was created using a 4% thick RAE 2822airfoil in the inboard wing panel and a 3% thick bi-convex airfoil in the outboard panel. The wing shapedesign variables have drastically reduced the camberdistribution on the wing inboard sections (although

not eliminated it completely) and they have also mod-ified the shape of the outboard wing panels.

Side and top views of the resulting design with Machnumber contours superimposed (varying from M = 1.4to M = 1.7, blue to red) can be seen in Fig. 7.

Aerodynamic Shape Optimization UsingCAD-Based AEROSURF

The design problem setup in this case is identical tothe one before, except for two main differences. Firstly,all surface re-generations required during the designprocess are handled by our CAD-based AEROSURFgeometry engine. Secondly, in the interest of mini-mizing the CAD re-generation times at this stage ofvalidation process, we decided to maintain the origi-nal shapes of all of the sections in the geometry (bothfuselage and wing) and modify the twist distributionon the wing and the fuselage camber. Since the para-metric CAD model was constructed with control overthe twist angle of the wing root, crank, and tip sec-tions, only three twist design variables are used here.A piecewise linear variation is implied between thesethree defining stations. On the fuselage, 9 camber vari-ables such as the ones used before are used, for a totalof 12 design variables in this test case. Since the wingsections are not allowed to vary, it is not necessary toimpose thickness constraints.

After 6 design iterations and a total of 124 CAD re-generations, the coefficient of drag of the configurationis reduced to CD = 0.00809, an improvement of 5.7%compared with the earlier value of 9%. Since only thewing twist distribution is altered, we can observe thatthe wing de-cambering is responsible for the remaining3.3% improvement. This is a very significant amountin supersonic design and it highlights the need to usedetailed shape parameterizations to obtain the trueoptimum of such aircraft systems.

Figure 8 shows several views of the resulting design.It is clear that the optimizer has chosen to shape thefuselage in a very similar way to the previous case,thus achieving the improvements that derive from liftre-distribution. Detailed examination of the values of

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the fuselage camber design variables reveals that thisis indeed true: the variations in fuselage camber arevery close to each other. Since the wing shape is notallowed to change, the optimizer changes the twistdistribution much more drastically than in the pre-vious case to achieve changes in lift that would haveotherwise resulted from the combination of twist andde-cambering. The total washout for the wing is nowalmost -1.2 deg.

The results are very close to our expectations andserve as validation of the CAD-based AEROSURF ge-ometry engine. In the near future we expect to extendthis validation to use the same number of design vari-ables as in the first optimization and will validateboth the gradients and the results obtained. Sincethe surface shape parameterization of the in-house andCAD-based engines are slightly different, exact agree-ment is not expected. However, the outcome of thedesign is likely to be quite close.

Conclusions and Future Work

In this paper we have reviewed the basis of ourcoupled-adjoint aero-structural design framework andhave provided details of the formulation of the op-timization problem. The coupled-adjoint design en-vironment allows for the calculation of coupled aero-structural sensitivities of aerodynamic and structuralcost functions with computational cost that is inde-pendent of the number of design variables. In order tofurther the applicability of this design environment,we have pursued the improvement of our geometrymanagement using a CAD-based geometry server ap-plication. This geometry server, AEROSURF, is madepossible in a CAD-vendor-neutral way through the useof the CAPRI API. In addition, we have replaced ourstructural analysis and design capability by the FEAPsolver of Taylor (UC Berkeley). FEAP has been shownto produce accurate and realistic results in both anal-ysis and design environments. Finally, aerodynamicshape optimizations have been carried out using theold and new geometry kernels to validate the use ofthe more sophisticated geometry re-generation tech-niques.

At the moment we are pursuing the full aero-structural optimization of the complete supersonicbusiness jet configuration which can be seen in Fig 9below. This work is now in its preliminary stages andwill be presented in the near future.

Acknowledgements

The authors wish to acknowledge the generosity ofProf. Robert L. Taylor (Civil Engineering, UC Berke-ley) for allowing the use and modification of his FEAPstructural solver. In addition, we would like to thankDr. David Saunders (NASA Ames Research Center)for his help in the preparation of both configurations

used in this work. The first author wishes to acknow-eldge the support of the Air Force Office of ScientificResearch under grant AF F49620-01-1-0291.

References1Howard M. Adelman and Raphael T. Haftka. Sensitiv-

ity analysis of discrete structural systems. AIAA Journal,24(5):823–832, May 1986.

2Mehmet A. Akgn, Raphael T. Haftka, K. Chauncey Wu,and Joanne L. Walsh. Sensitivity of lumped constraints usingthe adjoint method. AIAA Paper 99-1314, April 1999.

3Curran A. Crawford. An integrated CAD methodologyapplied to wind turbine optimization. Technical report, Cam-bridge, MA 02139, June 2001.

4P. E. Gill, W. Murray, M. A. Saunders, and M. H. Wright.User’s guide for NPSOL (version 4.0). a FORTRAN packagenonlinear programming. Technical Report SOL86-2, StanfordUniversity, Department of Operations Research, 1986.

5Philip E. Gill, Walter Murray, Michael A. Saunders, andM. A. Wright. User’s Guide for SNOPT 5.3: A Fortran Pack-age for Large-scale Nonlinear Programming. Systems Optimiza-tion Laboratory, Stanford University, California, 94305-4023,December 1998. Technical Report SOL 98-1.

6R. Haimes and C. Crawford. Unified geometry access foranalysis and design. Technical report, June 2003.

7R. Haimes and G. Follen. Computational analysis program-ming interface. Technical report, June 1998.

8M. E. Holden. Aeroelastic Optimization Using the Collo-cation Method. PhD thesis, Stanford University, Stanford, CA94305, 1999.

9A. Jameson. Aerodynamic design via control theory. Jour-nal of Scientific Computing, 3:233–260, 1988.

10J. R. R. A. Martins, J. J. Alonso, and J. Reuther. Aero-structural wing design optimization using high-fidelity sensitiv-ity analysis. In Proceedings — CEAS Conference on Multi-disciplinary Aircraft Design Optimization, Cologne, Germany,pages 211–226, June 2001.

11J. R. R. A. Martins, J. J. Alonso, and J. Reuther. High-fidelity aero-structural design optimization of a supersonic busi-ness jet. AIAA Paper 2002-1483, April 2002.

12J. R. R. A. Martins, J. J. Alonso, and J. Reuther. High-fidelity aero-structural design optimization of a supersonic busi-ness jet. AIAA Paper 2002-1483, April 2002.

13J. R. R. A. Martins, J. J. Alonso, and J. Reuther. Acoupled-adjoint sensitivity analysis method for high-fidelityaero-structural design. Submitted for Publication, Journal ofOptimization in Engineering, 2003.

14J. R. R. A. Martins, J. J. Alonso, and J. Reuther. High-fidelity aero-structural design optimization of a supersonic busi-ness jet. Accepted for Publication, Journal of Aircraft, 2003.

15J. Reuther, J. J. Alonso, A. Jameson, M. Rimlinger, andD. Saunders. Constrained multipoint aerodynamic shape opti-mization using an adjoint formulation and parallel computers:Part I. Journal of Aircraft, 36(1):51–60, 1999.

16J. Reuther, J. J. Alonso, A. Jameson, M. Rimlinger, andD. Saunders. Constrained multipoint aerodynamic shape opti-mization using an adjoint formulation and parallel computers:Part II. Journal of Aircraft, 36(1):61–74, 1999.

17J. Reuther, J. J. Alonso, J. C. Vassberg, A. Jameson, andL. Martinelli. An efficient multiblock method for aerodynamicanalysis and design on distributed memory systems. AIAA Pa-per 97-1893, June 1997.

18J. Reuther, M.J. Rimlinger, and D. Saunders. Geome-try driven mesh deformation. Technical Report Vol. 1, Part2, September 1999.

19J. Sobieszczanski-Sobieski. Sensitivity of complex, inter-nally coupled systems. AIAA Journal, 28(1):153–160, January1990.

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20O. C. Zienkiewicz and R. L. Taylor. The Finite ElementMethod, 5th Edition, Vol. I. Butterworth-Heinemann, 2000.

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Fig. 7 Optimized aerodynamic configuration. CL = 0.1, CD = 0.0078, M∞ = 1.5. 136 design variables usingin-house geometry engine.

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Fig. 8 Optimized aerodynamic configuration. CL = 0.1, CD = 0.0080, M∞ = 1.5. 12 design variables usingCAD-based geometry engine.

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Fig. 9 Top and perspective view of aero-structural optimization setup for supersonic business jet.

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