[American Institute of Aeronautics and Astronautics 15th AIAA Computational Fluid Dynamics...

(c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)1 Sponsoring Organization.

15th AlAA Computational FluidDynamics Conference

11-14 June 2001 Anaheim, CA A01-31042AlAA 2001-2527

Simultaneous Aerodynamic and StructuralDesign Optimization (SASDO) for a 3-D Wing

Clyde R. Gumbert*NASA Langley Research Center, Hampton, VA 23681

Gene J. -W. HoutOld Dominion University, Norfolk, VA 23529-0247

Perry A. Newman*NASA Langley Research Center, Hampton, VA 23681

The formulation and implementation of an optimization method called SimultaneousAerodynamic and Structural Design Optimization (SASDO) is shown as an extension ofthe Simultaneous Aerodynamic Analysis and Design Optimization (SAADO) method. Itis extended by the inclusion of structure element sizing parameters as design variables andFinite Element Method (FEM) analysis responses as constraints. The method aims toreduce the computational expense incurred in performing shape and sizing optimizationusing state-of-the-art Computational Fluid Dynamics (CFD) flow analysis, FEM struc-tural analysis and sensitivity analysis tools. SASDO is applied to a simple, isolated, 3-Dwing in inviscid flow. Results show that the method finds the same local optimum asa conventional optimization method with some reduction in the computational cost andwithout significant modifications to the analysis tools.

Nomenclature6 wing semispanCD drag coefficientCL lift coefficientCm pitching moment coefficientcr wing root chordct wing tip chordF design objective functiong design constraintsK stiffness matrixL aerodynamic loadsMOO free-stream Mach numberh unit normal vectorp local aerodynamic pressureP compliance, the work done by the aerody-

namic load to deflect the structuregoo free-stream dynamic pressureQ flow-field variables (state variables) at each

CFD mesh point

* Research Engineer, Multidisciplinary Optimization Branch,M/S 159, [email protected]

tProfessor, Department of Mechanical Engineering, AlAAmember, [email protected]

* Senior Research Scientist, Multidisciplinary OptimizationBranch, M/S 159, [email protected]

Copyright (c) 2001 by the American Institute of Aeronautics andAstronautics, Inc. No copyright is asserted in the United Statesunder Title 17, U.S. Code. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed hereinfor Governmental Purposes. All other rights are reserved by thecopyright owner.

AQ2

R

\R/RO\sSuAT/I

WX

a

7A

change in flow solver field variables due tobetter analysis convergencechange in flow solver field variables due todesign changesaerodynamic state equation residuals ateach CFD mesh pointnorm of the residual ratio, current/initialsurface areasemispan wing planform areastructural deflections (state variables)change in deflections due to better analysisconvergencechange in deflections due to design changeswing weightCFD volume mesh coordinateslocation of wing root leading edgechordwise location normalized by local wingsection chordlongitudinal location of wing tip trailingedgeroot section maximum camberfree-stream angle-of-attackdesign variablesstructural element size factorline search parameteroperator which indicates a change in a vari-able

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AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2001-2527


£, e' convergence tolerances for function and gra-dients

A adjoint variable for Q// adjoint variable for ur twist angle at wing tip, positive for leading

edge up

Subscriptsd deflected shapej jig (undeflected) shapes wing surface meshv wing volume mesh

Superscripts* designates updated value1 gradient with respect to design variables

IntroductionSimultaneous Aerodynamic Analysis and Design

Optimization (SAADO) is a procedure that incorpo-rates design improvement within the iteratively solved(nonlinear) aerodynamic analysis so as to achieve fullyconverged flow solutions only near an optimal design.When SAADO is applied to a flexible wing ratherthan a rigid wing, the linear Finite Element Method(FEM) solution is iteratively coupled with the nonlin-ear Computational Fluid Dynamics (CFD) solution.When design variables that control structural elementsize are included, it is renamed Simultaneous Aerody-namic and Structural Design Optimization (SASDO).Overall computational efficiency is achieved becausethe many expensive iterative (nonlinear) solutions fornon-optimal design parameters are not converged (i.e.,obtained) at each optimization step. One can ob-tain the design in the equivalent of a few (ratherthan many) multiples of the computational time fora single, fully converged coupled aero-structural anal-ysis. SAADO and similar procedures for simultaneousanalysis and design (SAND) developed by others arenoted and discussed by Newman et al.1 These SANDprocedures appear best suited for applications wherethe discipline analyses involved in the design are non-linear and solved iteratively. Generally, convergenceof these discipline analyses (i.e., state equations) isviewed as an equality constraint in an optimizationproblem. From this latter point of view, the SASDOmethod proceeds through infeasible regions of the de-sign space which includes not only the design variables/?, but also the state variables Q and u. A further ad-vantage of SASDO is the efficient utilization of existingdiscipline analysis codes (without internal changes),augmented with sensitivity or gradient information,and yet effectively coupled more tightly than is done inconventional gradient-based optimization procedures,referred to as nested analysis and design (NAND)procedures.1 A recent overview of aerodynamic shape

optimization2 discusses both NAND and SAND pro-cedures in the context of current steady aerodynamicoptimization research.

For single-discipline design problems, the distinc-tion between NAND and SAND procedures is fairlyclear and readily seen. With respect to discipline fea-sibility (i.e., convergence of the generally nonlinear,iteratively solved state equations), these procedurescan be viewed as accomplishing design by using onlyvery well converged discipline solutions (NAND), oras converging a sequence of discipline solutions frompoorly to well as the design progresses (SAND). How-ever, the problem formulation and solution algorithmsmay differ considerably. About twenty SAND refer-ences are quoted by Newman et al.1 and Newman etal.;2 these references discuss a variety of formulations,algorithms, and results for single-discipline problems(mostly CFD applications) in the sense of SAND as de-fined above. For multidisciplinary design optimizationproblems, the distinction between NAND and SANDis somewhat blurred because there are feasibility con-siderations with respect to all the individual disciplinestate equations, as well as with respect to the multi-disciplinary system compatibility and constraints. Anumber of the papers in Ref. 3 discuss MDO for-mulations and algorithms that are called SAND-like;however, not all of these latter MDO procedures ap-pear to agree with the sense of SAND defined aboveand used herein; one that does is Ref. 4.

The computational feasibility of SAADO for quasi1-D nozzle shape design based on the Euler equationCFD approximation was demonstrated by Hou et al.5and Mani.6 Application of SAADO for turbulent tran-sonic airfoil shape design based on a 2-D thin-layerNavier- Stokes CFD approximation was demonstratedand reported in a later paper by Hou et al.7 Both ofthese application results are summarized and brieflydiscussed in Ref. 1. The application of SAADO forrigid 3-D wing design based on the Euler CFD ap-proximation was presented in Ref. 8. These SAADOprocedures utilized quasi-analytical sensitivity deriva-tives obtained from hand-differentiated code for theinitial quasi 1-D application5'6 and from automaticallydifferentiated code for both the 2-D airfoil applica-tion7 and the 3-D rigid wing application.8 Differentoptimization techniques have also been used in theseSAADO procedures.

The extension to multidisciplinary analysis withshape design variables only was presented in Ref. 9.Our initial results from SASDO are given in this pa-per. The analysis problem, the objective function, andthe constraints are the same as those used in Ref. 9.That is, changes in design variables are sought toproduce improvement in the lift-to-drag ratio of a sim-ple wing subject to both aerodynamic and structuralsolution-dependent constraints. These constraints arethe difference between the lift and weight, the pitch-

2 OF 11



ing moment coefficient, and the compliance, a functionrepresenting work done by the aerodynamic load todeflect the structure. There are also geometric con-straints.

The flexible wing studied here is formulated as astatic aeroelastic problem. Similar problems havebeen used as examples in Refs. 10-15 to study var-ious solution strategies for multidisciplinary analysisand optimization. In particular, Arian12 analyzed theHessian matrix of the system equations to derive themathematical conditions under which the aeroelasticoptimization problem can be solved in a "loosely" cou-pled manner. The multidisciplinary research of Walshet al.13'14 emphasized engineering aspects of integrat-ing high fidelity disciplinary analysis software and dis-tributed computing over a network of heterogeneouscomputers. The aeroelastic analysis results of Reutheret al.15 were verified with experimental data.

Only a limited amount of literature related to aero-elastic problems has elaborated on the coupled sen-sitivity analysis. Kapania, Eldred and Barthelemy;16

Arslan and Carlson;17 and Giunta and Sobieszczanski-Sobieski18 derived global sensitivity equations (GSEs);some matrix coefficients in these GSEs were evaluatedby finite differencing. Guinta19 later introduced modalcoordinates to approximate the elastic displacementvector in order to reduce the size of the GSE. Newman,Whitfield, and Anderson20 used the complex variableapproach to obtain the aeroelastic sensitivity deriva-tives, whereas Reuther et al.15 employed the adjointvariable approach to derive the aeroelastic sensitivityequations. A mathematical study of the coupled non-linear, incompressible aeroelastic analysis and sensitiv-ity analysis problems has been given by Ghattas andLi.21 Recent results on aeroelastic sensitivity analysisand optimization can be found in Refs. 22-24. Partic-ularly, Maute et al.23 and Hou and Satyanarayana24

explicitly formulated the deflection update and theload transfer between the separate flow and structuressolvers as part of the coupled sensitivity equations.In the present study, the coupled sensitivity equa-tions are constructed by differentiating the aeroelasticstate equations and solving them by a GeneralizedGauss-Seidel (GGS) method.12 The present SASDOconcept is very similar to that of Ghattas and others,Refs. 4,21,25,26, but differs in the implementationdetails as described later.

Problem DescriptionTo evaluate the efficacy of the SASDO procedure for

a problem involving multidisciplinary analysis, it is ap-plied herein to a simple, isolated, flexible wing. Thewing shape consisted of a trapezoidal planform with arounded tip. It was parameterized by fifteen variables;five described the planform, and five each describedthe root and tip section shapes. A schematic of thewing and its associated shape parameters is shown in

Fig. 1 Description of semispan wing parameteri-zation.

Fig. 1. The baseline wing section varied linearly froman NACA 0012 at the root to an NACA 0008 at the tip.The wing structure consists of a skin, ribs, and spars.The ribs and spars consist of shear webs and trusses.Six spanwise zones of the structural model are definedas depicted in Fig. 1. The relative sizes of the skin andweb thicknesses and the truss cross section areas arefixed within each zone. Each zone is assigned a param-eter F which multiplies all the thicknesses and areasof the structural elements in that zone. The specificparameters selected as design variables in the sampleoptimization problems are identified in the section en-titled Results. The objective function to be minimizedwas the negative of the lift-to-drag ratio, —L/D. Bothcoupled solution-dependent and geometric constraintswere imposed.

The solution-dependent constraints were

• lower limit on the difference between the total liftand the structural weight, CL * S * goo — W

• upper limit on compliance, P — § pu • hds

• upper limit on pitching moment, Cm, in lieu of atrim constraint

The purely geometric constraints were

• minimum leading edge radius, in lieu of a manu-facturing requirement

• side constraints (bounds) on the active designvariables

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AMERICAN INSTITUTE OP AERONAUTICS AND ASTRONAUTICS PAPER 2001-2527


SASDO ProcedureFormulation

The flexible SASDO approach formulates the designoptimization problem as follows:

and equality constraints

minF(Q,

subject to

f = l , 2 , . . . , m (2 )

and the finite element structural equation

where the flow field Q and the structural deflection uare solutions of the coupled flow equation

dv(0>u),p) = Q (3)

(4)

The deflected volume mesh, Xdv, is determined bythe deflected surface mesh, Xds, as Xdv — Xdv(Xd$)-The deflected surface mesh is a result of the jig shapeaugmented by the elastic deflection, w, as Xds —Xj(/3) + u. The two disciplines are coupled throughthe deflection, u, and the load, L.

Recall that Q, .R, and Xdv are very large vectors.This formulation treats the state variables, Q and w,as part of the set of independent design variables, andconsiders the state equations to be constraints. Be-cause satisfaction of the equality constraints, Eqs. (3)and (4), is required only at the final optimum solution,the coupled steady-state aero-structural field equa-tions are not converged at every design-optimizationiteration. The easing of that restriction is expectedto significantly reduce the excessively large compu-tational cost incurred in the conventional approach.However, this advantage would likely be offset by thevery large increase in the number of design variablesand equality constraint functions, unless some reme-dial procedure is adopted.

ApproximationsThe SASDO method begins with a linearized design

optimization problem solved for the most favorablechange in the design variables, A/3, as well as for thechanges in the state variables, AQ and Aw; that is,

(5)

subject to inequality constraints

(6)i = l , 2 , . . . , r a

(1) and

),p)u - L ( Q , X d v ( / 3 , u ) ) -

Note that Eqs. (5) through (8) are linearized approxi-mations of Eqs. (1) through (4), respectively.

In this formulation, neither the residual of the non-linear aerodynamic field equations, R(Q,X,/3), northat of the structures equation, Ku — L, is required tobe zero (reach target) until the final optimum design isachieved. The linearized problem of Eqs. (5) through(8) is difficult to solve directly because of the numberof design variables and equality constraint equations.

Direct differentiation methodOne way to overcome this difficulty is by the di-

rect differentiation method. In this method AQ, Aw,and Eqs. (7) and (8) are removed altogether fromthe linearized problem by direct substitution. Thisis achieved by expressing AQ and Aw as functions ofA/3.

Aw = Awi + Aw2A/3 ^

where vectors AQi and Awi are corrections in theaeroelastic solution due to the improvement of cou-pled aeroelastic analysis, while matrices A$2 and Aw2are corrections due to changes in the design variables.These vectors and matrices are solutions of the follow-ing coupled sets of equations, obtained from Eqs. (7)and (8):

dXdv dXds

, dL(10)

dxdv dxds

where, for the linear FEM, Ku — L = 0 at every iter-ation, and

dR

8K

Note that the number of columns of matrices AQ2and A^2 is equal to the number of design variables,/3. Thus the computational cost of Eq. (11) is directlyproportional to the number of design variables.

A new linearized problem with A/3 as the only designvariables can be obtained by substituting Eq. (9) into

4 OF 11



Eqs. (5) and (6) for A<3 and Aw.

111111

./0£1 ®Q

(12)

subject to

+ fa} A/? < 0; 1 = 1,2,...,*.(13)

The appearance of AQi and Awi in the formulationindicates the difference between the SASDO (SAND)method and the conventional (NAND) aerodynamicoptimization method. The AQi and A«i not onlyconstitute a change in Q and u, but also play an impor-tant role in defining the objective function of Eq. (12)and the constraint violation of Eq. (13). We can di-rectly solve Eqs. 10 for AQi and Ai/i, and, in fact,in previous SAADO applications that is how AQi wasdetermined. However, since AQi and Ai/i, as shownin Eq. (10), represent a single Newton's iteration onthe coupled flow and structures equations, it is possi-ble and less computationally expensive to approximatetheir influence on the solution Q and u by several New-ton's iterations of the coupled aeroelastic equations.That is, AQi and At«i are not determined explicitly,but rather the first three terms of Eqs. 12 and 13 areviewed as updated values of F and (/;. Note that theterms in parentheses in Eqs. (12) and (13) are ap-proximated gradients of the objective and constraintfunctions. Once established, this linearized problemcan be solved using any mathematical programmingtechnique for design changes, A/?. Results presentedin this paper are computed using this direct differen-tiation approach.

Adjoint method

An alternative way to remove AQ and Aw from thelinearized problems, Eqs. (5) and (6), is the adjointvariable method. The adjoint variables, A and ju, canbe introduced as the solutions of

(dR\T^ _ fdL_\T „ _ idF\T

( r) T f) JC \

(14)dxcdxc

OF"

so as to rewrite the expression of the objective functionin Eq. (5) in terms of A, //, and A/3 as

F(Q,Xdv(p,udF 9F\

(dKu , f dK U — dLQXdv

dxdvdXds

(15)Note that the terms in the brace represent the gra-dient of the objective function. The terms XTR andjj,T(Ku — L) indicate the effect on the design opti-mization formulation due to errors in the aeroelasticanalysis. Furthermore, in the linear sense, the adjointvariables and the solution errors can be related by thefollowing equations:

T

and

(16)

(17)

These equations have been mentioned, for example,by Pierce and Giles27 and Venditti and Darmofal28

for aerodynamic problems.In the typical optimization problem there are many

design variables. When one can also pose the op-timization problem such that there are only a fewoutput quantities for objectives and constraints or, inthe extreme, combine the constraints and objectivefunction into a single cost function, the adjoint ap-proach to sensitivity analysis has the advantage thatthe adjoint solutions are independent of the number ofdesign variables.29 However, when the disciplines areloosely coupled, this approach is impractical since thecoupled sensitivity analyses would require an adjointfor each disciplinary output being transferred, i.e., thediscretized loads and deflections. In a tightly or im-plicitly coupled multidisciplinary analysis the adjointapproach may prove practical since this system is anal-ogous to a single discipline.

Line SearchA one-dimensional search on the step size parame-

ter 7 is then performed in order to find the updatedvalues of A/?, AX, AQ, and Atz. Given the searchdirection A/? determined by either the direct differen-tiation method (Eqs. 12 and 13) or the adjoint methodequivalent, this line search functions to adjust its mag-nitude so as to simultaneously ensure better results forboth design and analysis (converged solutions). Thestep size parameter 7 plays the role of a relaxationfactor in the standard Newton's iteration. The searchprocedure employed solves a nonlinear optimizationproblem of the form

minF(Q*,X*, «*,/?*) (18)

5 OF 11



subject to

gi(Q*,X*,w*,/9*) < 0; « = l , 2 5 . . . , r a (19)

R(Q*,X*,/3*) = Q (20)and

where step size 7 is the only design variable. Againit is noted for emphasis that the equality constraints,Eqs. (20) and (21), are not required to be zero (reachtarget) until the final optimum design; violations ofthese equality constraints must simply be progressivelyreduced until the SASDO procedure converges.

The updated <2* and u* can be viewed as Q* = Q-\-A(J and u* — u + Atf, where AQ and Aw satisfy thefirst order approximations to Eqs. (20) and (21). Thatis, AQ and Aw are the solutions of Eqs. (7) and (8)where, in Eq. (9), A/? is replaced by A/?* = 7A/9.Consequently, Q* — Q 4- A<2i + "fAQzAfl and u* —u+Aui+7Au2 A/? are readily available once 7 is found.The A()i terms appearing in the above SASDO for-mulation are due to better convergence of the coupledanalysis, whereas the A02 terms are due to changes inthe design variables. In fact, AQ2 and At/2 approachthe flow field and deflection sensitivities, Q1 and u', asthe solution becomes better converged.

ImplementationThe following pseudocode shows algorithmically

how the method was implemented,set initial analysis convergence tolerance, eset initial solution vectors, Q and u,set initial design variables, /3do until converged

1. solve coupled aeroelastic analysis, Eqs. (3) fe (4),partially converged to e

2. compute F and g3. solve coupled aeroelastic sensitivity analysis,

Eq. (11), partially converged to e1

4. compute A/? terms of Eqs. (5) fe (6)5. solve optimization problem Eqs. (5) & (6) for A/?6. solve Eqs. (18) through (21) for line search pa-

rameter, 77. update /9, u, and Q8. tighten analysis convergence tolerance, e — e * Z,

enddo

This pseudocode is similar to that used in theBiros and Ghattas25 SAND approach. Specifically,both approaches use a Sequential Quadratic Program-ming (SQP) method to solve the design equations(step 5) and an approximate factorization method tosolve the system equations (step 1). Step 3 above usesan incremental iterative method with approximatefactorization to solve for derivatives in direct moderather than as a solution of the adjoint equation of

Specify:Design ProblemFinal Convergence

Initialize:GeometryFlow ConditionsFEM ModelConvergence Level

Partially ConvergedSystem Analysis

•*•Geometryand Mesh

AerodynamicAnalysis

t" Structural

Analysis

Partially ConvergedSensitivity Analysis

: ;j

i

TightenConvergence

Level -DesignUpdate

• Geometry1 and Mesh! Sensitivitie

llt:::::::::^•j Optimizer

Aerodynamic

ts ———— i ———

Sensitivities

^^ ̂ Improved™ "̂ Design

" Partially Con verged ~D~e~s!g~n~

Fig. 2 Diagram of SASDO procedure.

Biros and Ghattas.25 In addition, the line search step(step 6) and the convergence tightening step (step 8)were not included in the Biros and Ghattas method.A schematic of the present SASDO procedure is shownin Fig. 2. The dashed box, labeled "Partially Con-verged System Analyis," depicts the coupled analysisiteration loop, Steps 1 and 2 of the pseudocode; thatlabeled "Partially Converged Sensitivity Analysis"depicts the coupled derivative iteration loop, Step 3;that labeled "Partially Converged Design" depicts thedesign steps, Steps 5-8 of the pseudocode. Specificcomputational tools and methods used to performthe tasks depicted by the solid boxes in Fig. 2 areidentified in the next section.

Computational Tools and ModelsMajor computations in this SASDO procedure are

performed using a collection of existing codes. Thesecodes are executed by a separate driver code andscripts that implement the SASDO procedure as justdiscussed. Each code runs independently, some simul-taneously, on different processors, and the requiredI/O transfers between them, also directed by thedriver, are accomplished via data files.

The aerodynamic flow analysis code used for thisstudy is a version of the CFL3D code.30 Only Eu-ler analyses are performed for this work, although thecode is capable of solving the Navier-Stokes equationswith any of several turbulence models. The gradientversion of this code, which was used for aerodynamicsensitivity analysis, was generated by an unconven-tional application31 of the automatic differentiationcode ADIFOR32'33 to produce a relatively efficient,direct mode, gradient analysis code, CFL3D.ADII.34

It should be pointed out that the ADIFOR processproduces a discretized derivative code consistent withthe discretized function analysis code. The addition ofa stopping criterion based on the norm of the residualof the field equations was the only modification of theCFL3D.ADII code made to accommodate the SASDOprocedure.

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CFD meshC-0 topology73x25x25 volume49x25 on the wing

FEM mesh3251 elements:

2240 CST1011 truss

583 points

Fig. 3 Computational meshes for wing analysis.

The surface geometry was generated based on theparameters described in a previous section by a codeutilizing the Rapid Aircraft Parameterization InputDesign (RAPID) technique developed by Smith, etal.35 This code was preprocessed with ADIFOR to gen-erate a code capable of producing sensitivity deriva-tives as well.

The CFD volume mesh needed by the flow analysiscode was generated using a version of the CSCMDO36

grid generation code. The associated grid sensitivityderivatives needed by the flow sensitivity analysis weregenerated with an automatically differentiated ver-sion of CSCMDO.37 In addition to the parameterizedsurface mesh and accompanying gradients, CSCMDOrequires a baseline volume mesh of similar shape andidentical topology. The 45,000 grid point baseline vol-ume mesh of C-0 topology used in the present flexiblewing examples was obtained with the Gridgen™ code.The wing surface portion of the mesh is shown inFig. 3. This mesh is admittedly particularly coarseby current CFD analysis standards.

The structural analysis code38 used to computethe deflection of the elastic wing was a generic fi-nite element code. The flexible structure for the wingshown in Fig. 3 was discretized by 583 nodes; therewere 2,141 constant-strain triangle (CST) elementsand 1,110 truss elements. Zone boundaries for the de-sign variables controlling element size are also shown inFig. 3. Because the elastic deformation was assumedto be small, linear elasticity was deemed to be ap-propriate. The structural sensitivity equations werederived based upon the direct differentiation method.We note that the sensitivity of the aerodynamic forcesappears as a term on the right-hand side (RHS) of thedeflection sensitivity equations. The derivative of thestiffness matrix in these sensitivity equations was alsogenerated39 by using the ADIFOR32'33 technique. Wenote that the coefficient matrix of the structural sensi-tivity equations was identical to that of the structuralequations. Consequently, these structural sensitivityequations were solved efficiently by backward substi-tution with different RHSs for each sensitivity.

At the wing surface, i.e., the interface where aero-dynamic load and structural deflection information istransferred, it was assumed that surface nodes of the

FEM structural model were a subset of the CFD aero-dynamic surface mesh points (see Fig. 3) for thepresent SASDO application. This lack of generalityallowed for simplifications in the data transfers and, al-though an important issue, it was not deemed crucialfor these initial SASDO demonstrations. Future ap-plications to more complex configurations should allowfor transfer of conserved information between arbitrarymeshes as required by the individual disciplines. Arecent review of such data transfer techniques and aspecific proposed one are given in Ref. 40.

Conventional (NAND) and SASDO (SAND) proce-dures were implemented using the SQP method of theDOT41 optimization software. All computations wereexecuted on an SGI Origin 2000™ workstation with250MHz R10000™ processors. The CFD sensitiv-ity calculations were partitioned and run on severalprocessors to reduce required memory and elapsed op-timization time. This partitioning, however, results inadditional accumulated computation time due to thenature of ADIFOR-generated sensitivity analysis code.

ResultsThe optimization results shown in this work are

for design problems involving only four or eight de-sign variables out of the 21 available wing parameters.The results shown by these authors in an earlier work9

used design variables that directly affected either theaerodynamic analysis alone or both the aerodynamicanalysis and the structural analysis. In this work, ad-ditional design variables are chosen that directly affectonly the structural analysis. The flow conditions forthe wing optimizations were MOO — 0.8 and a = 1°.

Fig. 4 Comparison of planform shapes and sur-face pressure contours for 4-design-variable cases,Moo =0.8, a = 1°.

Four-Design-Variable ProblemsTable 1 and Fig. 4 show results of several optimiza-

tion problems involving four design variables: the tipchord c$j the tip setback a?t, and the structural elementsize factor for the two most inboard zones, FI andF2. Two of the cases, designated Convc and SASDOcin Table 1, represent direct comparisons of SASDOand the conventional method for consistent accuracy

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Table 1 Summary of Four-Design-Variable Cases.

Ct

xtTir2

Fgig2

g3

e5^ state£]grad

baseline

1.1.1.1.

-7.149-.0302-.8882-.2647

9.6e-71

—

SASDOA1.077781.932591.066310.69510

-10.187-.000760

-0.880-.000639

9.9e-51.1

27.1

OptimizationsConvB SASDOc

1.072811.918801.080080.69187

-10.181-.00177

-0.876-.000159

8.9e-74.8

63.1

1.078271,930521.096370.68664

-10.186-.0000629

-0.880-.000149

6.6e-62.0

26.9

Convc1.076971.926141.099820.68691

-10.183-.000915

-0.879+.000108

4.0e-63.9

40.4

of function and gradient analyses. The other two casesshow effects of changes in the accuracy of the functionand gradient analyses. The resulting designs are es-sentially identical for all four cases. Figure 4 showsa comparison of the wing planform and the surfacepressure coefficient results for the baseline design andthe design designated SASDOc in Table 1. The shockwave has been weakened somewhat in the optimizedcases from that on the original wing, as one wouldexpect. As one can see qualitatively in Fig. 4 andnumerically from the values of the objective functionF, the constraints </;, and the final design variablesin Table 1, the final designs are very similar for thefour problems. The relative computational costs of

50 r

43(Q0O

•H.13id

conven-tional SASDO

Fig. 5 Comparison of computation cost of four-design-variable optimization problem using theconventional and SASDO methods.

the optimizations are shown in Table 1 and Fig. 5.The accumulated function and gradient analysis times(denoted as Estate and T^grad) are shown separately.They have been normalized by the cost of the base-line coupled function analysis. The components of theanalyses were lumped together in Table 1 but they

have been separated in Fig. 5. The total time forperforming this optimization problem was reduced by36 percent using the SASDO method. The analysisalone was reduced by 55 percent, but the gradient eval-uation was the dominant cost.

Fig. 6 Comparison of planform shapes and sur-face pressure contours for 8-design-variable cases,Mco =0.8, a= 1°.

Eight-Design-Variable Problems

Table 2 and Fig. 6 show results of three optimizationproblems involving eight design variables: the same setused in the four-design-variable cases with the inclu-sion of the span 6, the root section max camber zr,and the structural element size factor for two morezones, FS and IV Two of the cases, designated Conveand SASDOe in Table 2, represent direct comparisonsof SASDO and the conventional method for consis-tent accuracy of the function and gradient analyses.Figure 6 shows a comparison of wing planform andsurface pressure coefficient results for the baseline de-sign and the design designated SASDOe in Table 2.The relative computational costs of the optimizationsare shown in Table 2 and Fig. 7. The total time forperforming this optimization problem was reduced by

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Table 2 Summary of Eight-Design-Variable Cases.

Ct

xtbzr

Tir2r3r4

Fgig2

g3

e^ stateEgrad

baseline

1.1.1.1.1.1.1.1.

-7.149-.0302-.8882-.2647

9.6e-71

—

ConvA0.765742.600001.059921.115112.424442.095371.036350.65137

-21.255-.7536

-.00336-.000420

6.8e-713.5

146.7

OptimizationsSASDOB

0.759292.600001.075431.111732.552632.099241.018830.81046

-21.235-.7255

-.00671-.000962

1.8e-54.4

115.9

ConvB0.757942.600001.072691.111752.444982.093941.057780.76612

-21.244-.7316

-.000241-.000134

1.6e-511.3

148.6

150

0o<D

•rl43flj

100

50

conven- SASDOtional

Fig. 7 Comparison of computation cost of eight-design-variable optimization problem using theconventional and SASDO methods.

26 percent using the SASDO method. The analysisalone was reduced by 60 percent, but, as with the four-design-variable problem, the gradient evaluation wasthe dominant cost.

Further DiscussionThe relative costs, based on CPU timing ratios,

for the SASDO (SAND) procedures applied to thesepresent small 3-D aerodynamic/structural design opti-

mization problems are about seven-tenths of the costsof the corresponding conventional (NAND)procedures.This range is very similar to that reported for 2-Dnonlinear aerodynamic shape design optimization inRefs. 1 and 4, even though many of the computationaldetails differ. The results given in Ref. 1 were for aturbulent transonic flow with shock waves computedusing a Navier-Stokes code; a direct differentiation ap-proach (using ADIFOR) was used for the sensitivityanalysis. The results reported in Ref. 4 were for a com-pressible flow without shock waves computed using anonlinear potential flow code; an adjoint approach wasused for the sensitivity analysis. Since these two opti-mization problems were also not the same, no timingcomparison between these adjoint and direct differ-entiation solution approaches would be meaningful.As indicated earlier, an expected speed-up for usingan adjoint approach instead of the direct differentia-tion approach was estimated in Ref. 1. Ghattas andBark26 recently reported 2-D and 3-D results for op-timal control of steady incompressible Navier-Stokesflow that demonstrate an order-of-magnitude reduc-tion of CPU time for a SAND approach versus aNAND approach. These results were obtained usingreduced Hessian SQP methods that avoid convergingthe flow equations at each optimization iteration. Therelationship of these methods with respect to other op-timization techniques is also discussed in Ref. 26. The"Control Theory" approach of Jameson29 and severalother SAND-like methods for simultaneous analysisand design, which were summarized and discussed byTa'asan,42 have been applied to aerodynamic shapedesign problems at several fidelities of CFD approx-imation. These techniques have obtained an aerody-namic design in the equivalent of several analysis CPUtimes for some sample problems.

Concluding RemarksThis study has introduced an implementation of the

SASDO technique for a simple, isolated wing. Initialresults indicate that SASDO

1. is feasible under dual simultaneity (i.e. simultane-ity not only with respect to analysis and designoptimization, but also simultaneity with respectto flexible wing aero-structural interaction)

2. finds the same local minimum as a conventionaltechnique

3. is computationally more efficient than a conven-tional gradient-based optimization technique

4. requires few modifications to the analysis and sen-sitivity analysis codes involved

5. is effective at reducing the function analysis cost,but the gradient analysis time is the dominantcost

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AcknowledgmentThe second author, G. J.-W. Hou, was supported in

this work by NASA through several Tasks under con-tract NAS1-19858 and NASA P.O. No. L-9291 withthe Old Dominion University Research Foundation.

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