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JlfJIJIA01-16888

AIAA2001-1107

Simultaneous AerodynamicAnalysis and Design Optimization(SAADO) for a 3-D Flexible Wing

Clyde R. GumbertNASA Langley Research CenterHampton, Va 23681-2199

Gene J. -W. HouOld Dominion UniversityNorfolk, VA 23529 - 0247

Perry A. NewmanNASA Langley Research CenterHampton, VA 23681-2199

39th Aerospace SciencesMeeting & Exhibit

January 8-11, 2001 / Reno, NVFor permission to copy or republish, contact the American Institute of Aeronautics and Astronautics1801 Alexander Bell Drive, Suite 500, Reston, VA 20191

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

AIAA2001-1107

Simultaneous Aerodynamic Analysis and Design Optimization (SAADO)for a 3-D Flexible Wing

Clyde R. Gumbert, Gene J.-W. Hou , Perry A. Newman

ABSTRACTThe formulation and implementation of anoptimization method called SimultaneousAerodynamic Analysis and Design Optimization(SAADO) are extended from single disciplineanalysis (aerodynamics only) to multidisciplinaryanalysis - in this case, static aero-structural analysis -and applied to a simple 3-D wing problem. Themethod aims to reduce the computational expenseincurred in performing shape optimization usingstate-of-the-art Computational Fluid Dynamics (CFD)How analysis, Finite Element Method (FEM)structural analysis and sensitivity analysis tools.Results for this small problem show that the methodreaches the same local optimum as conventionaloptimization. However, unlike its application to therigid wing (single discipline analysis), the method, asimplemented here, may not show significantreduction in the computational cost. Similarreductions were seen in the two-design-variable (DV)problem results but not in the 8-DV results givenhere.

NOMENCLATUREb wing semispanCD drag coefficientCl rolling moment coefficientC, lift coefficientCin pitching moment coefficientCp pressure coefficientc wing root chord

ct wing tip chordF design objective functiong design constraintsK stiffness matrixL aerodynamic loadsMoo free-stream Mach numberP compliance, the work done by the

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

CFD mesh pointAQ, change in flow solver field variables due to

better analysis convergenceAQ2 change in flow solver field variables due to

design changesR state equation residuals at each CFD mesh

point|R/R()I norm of the residual ratio, current/initialS semispan wing planform areau structural deflectionsAu, change in deflections due to better analysis

convergenceAu2 change in deflections due to design changesW wing weightX CFD mesh coordinatesx, r vector location of wing root leading edgex/c chordwise location normalized by local wing

section chordxt longitudinal location of wing tip trailing edgezr root section camber parametera free-stream angle-of-attack

Research Scientist, Multidisciplinary Optimization Branch,M/S 159, NASA Langley Research Center, Hampton, VA23681-2199, c.r.gumbert@larc.nasa.gov.

Professor, Department of Mechanical Engineering, OldDominion University, Norfolk, VA 23529-0247,ghou@lions.odu.edu, Member AIAA."Senior Research Scientist, M/S 159, NASA LangleyResearch Center, Hampton, VA 23681-2199,p.a.newman@larc.nasa.gov.

Copyright €> 2001 by the American Institute of Aeronauticsand Astronautics, Inc. No copyright is asserted in theUnited States under Title 17 U. S. Code. The U. S.Government has a royalty- free license to exercise all rightsunder the copyright claimed herein for GovernmentalPurposes. All other rights are reserved by the copyrightholder.

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(3 design variablesA operator which indicates a change in a

variable8, e' convergence tolerancesy line search parameteri twist angle at wing tip, positive for leading

edge upsubscripts:b baseline volume meshd deflected shapej jig (undetected) shapes wing surfacev volumesuperscripts:* designates updated value' gradient with respect to design variables

INTRODUCTIONSimultaneous Aerodynamic Analysis and DesignOptimization (SAADO) is a procedure that incorporatesdesign 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 rather than arigid wing, the linear FEM solution is iteratively coupledwith the nonlinear CFD solution. Overall computationalefficiency is achieved because the many expensive iterative(nonlinear) solutions for non-optimal design parameters arenot converged (i.e., obtained) at each optimization step.One can obtain the design in the equivalent of a few (ratherthan many) multiples of computational time for a single,fully converged coupled aero-structural analysis. SAADOand similar procedures for simultaneous analysis anddesign (SAND) developed by others are noted anddiscussed by Newman et al.' These SAND proceduresappear best suited for applications where disciplineanalyses involved in the design are nonlinear and solvediteratively. Generally, convergence of these disciplineanalyses (i.e., state equations) is viewed as an equalityconstraint in an optimization problem. From this latterpoint of view, the SAADO method proceeds throughinfeasible regions of the ((3, Q, u) design space. A furtheradvantage of SAADO is the efficient use of existingdiscipline analysis codes (without internal changes),augmented with sensitivity or gradient information, and yeteffectively 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 shapeoptimization2 discusses both NAND and SAND proceduresin the context of current steady aerodynamic optimizationresearch.

For single-discipline design problems, the distinctionbetween NAND and SAND procedures is fairly clearand readily seen. With respect to discipline feasibility(i.e., convergence of the generally nonlinear,iteratively solved state equations), these procedurescan be viewed as accomplishing design by using onlywell converged discipline solutions (NAND) or as asequence of discipline solutions converged frompoorly to well as the design progresses (SAND).However, the problem formulation and solutionalgorithms may differ considerably. About twentySAND references are quoted by Newman et al.1 andNewman et al.2; these references discuss a variety offormulations, algorithms, and results for single-discipline problems (mostly CFD applications) in thesense of SAND defined above. For multidisciplinarydesign optimization problems, the distinction betweenNAND and SAND is somewhat blurred because thereare feasibility considerations with respect to allindividual discipline state equations as well as withrespect to multidisciplinary system compatibility andconstraints. A number of the papers in Ref. 3 discussMDO formulations and algorithms that are calledSAND-like. However, not all of these latter MDOprocedures appear to agree with the sense of SANDdefined above and used here; one that does is Ref. 4.

The computational feasibility of SAADO for quasi 1-Dnozzle shape design based on the Euler equation CFDapproximation was demonstrated by Hou et al: andMani.6 Application of SAADO for turbulent transonicairfoil shape design based on a 2-D thin-layer Navier-Stokes CFD approximation was demonstrated andreported in a later paper by Hou et al.7 Both of theseapplication results are summarized and briefly discussedin Ref. 1. The application of SAADO for rigid 3-D wingdesign based on the Euler CFD approximation waspresented in Ref 8. These SAADO procedures utilizedquasi-analytical sensitivity derivatives obtained fromhand-differentiated code for the initial quasi 1-Dapplication,"6 and from automatically differentiatedcode for both the 2-D airfoil application7 and the 3-Drigid wing application.s Different optimizationtechniques have also been used in these SAADOprocedures.

The flexible wing studied here is formulated as a staticaeroelastic problem. Similar problems have been usedas examples in Refs. 9-14 to study various solutionstrategies for multidisciplinary analysis andoptimization. In particular, Arian" analyzed the Hessianmatrix for the system equations to derive mathematicalconditions under which the aeroelastic optimization

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problem can be solved in a "loosely" coupled manner.Multidisciplinary research of Walsh et al.12'13

emphasized the engineering aspects of integrating highfidelity disciplinary analysis software and distributedcomputing over a network of heterogeneous computers.The aeroelastic analysis results of Reuther et al.14 wereverified with experimental data.

Only a limited amount of literature related to aeroelasticproblems has elaborated on the coupled sensitivityanalysis. Kapania, Eldred and Barthelemy"; Arslan andCarlson16; and Giunta and Sobieszczanski-Sobieski17

derived global sensitivity equations (GSEs), with somematrix coefficients in these GSEs evaluated by finitedifferencing. Guinta^ later introduced modalcoordinates to approximate the elastic displacementvector in order to reduce the size of the GSE. Newman,Whitfield, and Anderson10 used the complex variableapproach to obtain aeroelastic sensitivity derivatives,whereas Reuther et al.14 employed the adjoint variableapproach to derive aeroelastic sensitivity equations. Amathematical study of the coupled nonlinear,incompressible aeroelastic analysis and sensitivityanalysis problems was performed by Ghattas and Li.20

Recent results on aeroelastic sensitivity analysis andoptimization can be found in Refs. 21-23. Particularly,Farhat22 and Hou and Satyanarayana23 explicitlyformulated deflection updates and load transfersbetween the separate flow and structures solvers as partof the coupled sensitivity equations. In the presentstudy, coupled sensitivity equations are constructed bydifferentiating the aeroelastic state equations andsolving them by a Generalized Gauss-Seidel (GGS)method." The present SAADO concept is very similarto that of Ghattas and others, Refs. 4, 20, 24, and 25, butdiffers in derivation and implementation details asdescribed later.

Our initial 3-D flexible wing results from SAADO aregiven in this paper. The problem is the same simplewing planform as used in Ref. 8 for rigid wing designstudies. Here, changes in design variables are sought toproduce improvement in the lift-to-drag ratio subject toboth aerodynamic and structural solution-dependentconstraints. These constraints are the differencebetween the lift and weight, the pitching momentcoefficient, and the compliance. The latter is a functionrepresenting work done by the aerodynamic load todeflect the structure. There are also geometricconstraints. Note that the structural interaction at boththe function (analysis) and derivative (sensitivityanalysis) levels must be included even when onlyaerodynamic design variables are involved. Theultimate goal of our work is to extend the SAADOprocedure to flexible wing design problems that also

involve structural design variables and additionalstructural responses in the problem formulation.

PROBLEM DESCRIPTIONTo evaluate efficacy of the SAADO procedure for aproblem involving multidisciplinary analysis, it isapplied herein to a simple, isolated, flexible wing.The wing 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 planform parameters is shownin Fig. 1 . The baseline wing section varied linearlyfrom an NACA 0012 at the root to an NACA 0008 atthe tip. The specific parameters selected as designvariables in sample optimization problems areidentified in the section entitled Results. Theobjective function to be minimized was the negativeof the lift-to-drag ratio, -L/D. Both coupled solution-dependent and geometric constraints were imposed.

The solution-dependent constraints were:

- lower limit on the difference between total liftand structural weight, (C, * S*qoo - W)(W=constant for rigid problem)

- upper limit on compliance, P (for flexible wingproblem)

- upper limit on rolling moment coefficient, C,, inlieu of bending moment limits (for rigid wingproblem)

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

The purely geometric constraints were:

- minimum leading edge radius, in lieu of amanufacturing requirement

- side constraints (bounds) on active designvariables

SAADQ PROCEDURE

FormulationThe flexible SAADO approach formulates the design-optimization problem as follows:

minF(Q,X d v((3,u) ,u , (3)P.Q.U

subject to

g i (Q ,X d v (p ,u ) , u ,p )<0 ; i = l,2,...,m

where flow field variables Q and structuraldeflections u are a solution of the coupled flowequation

(1)

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R(Q,Xd v(p,u),p) = 0 (3)

and finite element structural equation

K(Xj(P) )u=L(Q,X d v (p ,u) ) (4)

The deflected volume mesh, Xdv, is determined by thedeflected surface mesh, Xds, as Xdv=Xdv(Xds, Xhv). Thisdeflected surface mesh is a result of the jig shapeaugmented by elastic deflections, u, as Xds=X.(p)+u.The two disciplines are coupled through deflections,u, and loads, L.

Recall that Q, R, and Xdv are very large vectors. Thisformulation treats the state variables, Q and u, as partof the set of independent design variables, andconsiders the state equations as constraints. Becausesatisfaction of the equality constraints, Eqs. (3) and(4), is required only at the final optimum solution,coupled steady-state aero-structural field equationsare not converged at every design-optimizationiteration. Easing of this requirement is expected tosignificantly reduce excessively large computationalcosts incurred in the conventional approach.However, this advantage would most likely be offsetby the very large increase in the number of designvariables and equality constraint functions, unlesssome remedial procedure is adopted.

ApproximationsThe SAADO method begins with a linearized design-optimization problem which is solved for the mostfavorable change in design variables, A(3, as well asfor changes in state variables, AQ and Au; that is,

min F(Q,X,(3)Af3.AQ.Au

aq axd¥ axds^

(5)

axdv axds •subject to inequality constraints

(6)

and equality constraints

3Xdv 3XdsAu

(7),

axdv axds J ap

and

K(X i(p))u-L(Q,Xdv)+ K-•'

dL ^ dK—— AQ+ ——u-3Q V ax, axdv axds

Note that Eqs. (5) through (8) are linearizedapproximations of Eqs. (1) through (4), respectively.

In this formulation, neither the residual of the non-linear aerodynamic field equations, R(Q,X,p), northat of the structures equation, Ku-L, is required to bezero (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.This difficulty is overcome for the directdifferentiation method by using direct substitution toremove AQ, Au, and Eqs. (7) and (8) altogether fromthis linearized problem; that is, one expresses AQ andAu as functions of A(J.

AQ = AQj +AQ2A(3Au = A u , + A u 2 A p

(9)

where vectors AQ, and Au, and matrices AQ2 and Au2

are solutions of the following coupled sets ofequations, obtained from Eqs. 7 and 8,

axdv ax^-Au, = -Rds

ax (10)dv

l, ax ds

where, for the linear FEM, Ku - L = 0 at everyiteration, and

apaxdv axds

v ds

v,—————— X:UaX >

Note that the number of columns of matrices AQ2 andAu2 is equal to the number of design variables, p; thus

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the computational cost of Eq. (11) is directlyproportional to the number of design variables.

A new linearized problem with Ap as the only designvariables can be obtained by substituting Eq. (9) intoEqs. (5) and (6) for AQ and Au:

minF(Q,X,p)+vv

(12)

subject to

(13)

Note that the expressions inside large parentheses inEqs. (12) and (13) are approximated gradients of theobjective and constraint functions. Once established,this linearized problem can be solved using anymathematical programming technique for designchanges, A(3.

Line SearchA one-dimensional search on the step size parametery is then performed in order to find updated values ofP , X , Q , and u . This line search functions to adjustthe magnitude of A(3 so as to simultaneously ensurebetter results for both design and analysis (convergedsolutions). The step size parameter y plays the role ofa relaxation factor in the standard Newton's iteration.The search procedure employed solves a nonlinearoptimization problem of the form

minF(Q*,X : :,p ; : :)

subject to

g i (Q* ,X* ,p* ,u* )<0 ;

and

K(X*)u*=L(Q* ,X* ,p* )

(14)

(15)

(16)

(17)

where step size y is the only design variable. Again itis noted for emphasis that equality constraints, Eqs.(16) and (17), are not required to be zero (reachtarget) until the final optimum design; violations of

these equality constraints must simply beprogressively reduced until the SAADO procedureconverges.

The updated Q and u can be viewed as Q =Q+AQand u =u +Au where AQ and Au satisfy the firstorder approximations to Eqs. (16) and (17). That is,AQ and Au are the solutions of Eqs. (7) and (8)where, in Eq. (9), Ap is replaced by Ap = yA(3.Consequently, Q =Q+AQ,+yAQ2Ap and

u =u+Au,+yAu2Ap are readily available once y isfound. The A(), terms appearing in the above SAADOformulation are due to better convergence of thecoupled analysis, whereas A()2 terms are due tochanges in design variables. In fact, AQ, and Au,approach the flow field and deflection sensitivities, Q'and u', as the solution becomes better converged. Theappearance of AQ, and Au, in the formulation makesthe SAADO approach different from the conventionalNAND aerodynamic optimization method. The AQ,and Au, not only constitute changes in Q and u, butalso play important roles in defining the constraintviolation of Eq. (13). Since AQ, and Au,, as shown inEq. (10), represent a single Newton's iteration on thecoupled equations, it is possible to approximate themas the changes in Q and u as a result of severalNewton's iterations to improve quality of the solutionas was done in this study.

ImplementationThe following pseudocode shows algorithmically howthe method was implemented.

set initial analysis convergence tolerance, eset initial solution vectors, Q and uset initial design variables, pdo until converged

1. solve Eqs. (3) & (4) partially converged to e2. compute F and g3. solve Eq. (11) partially converged to e'4. compute Ap terms of Eqs (12) & (13)5. solve optimization problem Eq (12) & (13)

for AP6. solve Eqs (14) - (17) for line search

parameter, y7. update p, u, and Q8. tighten analysis convergence tolerance,

e=£*factor, factor < 1enddo

This pseudocode is similar to that used in the Birosand Ghattas24 SAND approach. Specifically, both

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approaches use an SQP method to solve the designequation (step 5) and an approximate factorizationmethod to solve the system equations (step 1). Step 3above uses an incremental iterative method withapproximate factorization to solve for derivatives indirect mode rather than as a solution of the adjointequation of Biros and Ghattas.24 In addition, the linesearch step (step 6) and the convergence tighteningstep (step 8) were not included in the Biros andGhattas method.

A schematic of the present SAADO procedure isshown in Fig. 2. The dashed box, labeled "PartiallyConverged System Analysis," depicts the coupledanalysis iteration loop, Steps 1 & 2 of thepseudocode; that labeled "Partially ConvergedSensitivity Analysis" depicts the coupled derivativeiteration loop, Step 3; and that labeled "PartiallyConverged Design" depicts the design steps, Steps 5 -8 of the pseudocode. Specific computational tools andmethods used to perform the tasks depicted by thesolid boxes in Fig. 2 are identified in the next section.

COMPUTATIONAL TOOLS AND MODELSMajor computations in this SAADO procedure areperformed using a collection of existing codes. Thesecodes are executed by a separate driver code andscripts that implement the SAADO procedure as justdiscussed. Each code runs independently, perhapssimultaneously, on different processors, and therequired I/O transfers between them, also directed bythe driver code, are accomplished by data files.

The aerodynamic flow analysis code used for thisstudy is a version of the CFL3D code.26 Only Euleranalyses are performed for this work, although thecode is capable of solving Navier-Stokes equationswith any of several turbulence models. The derivativeversion of this code, which was used for aerodynamicsensitivity analysis, was generated by anunconventional application27 of the automaticdifferentiation code ADIFOR282> to produce arelatively efficient, direct mode, gradient analysiscode, CFL3D.ADII.30 It should be pointed out that theADIFOR process produces a discretized derivativecode that is consistent with the discretized functionanalysis code. Addition of a stopping criterion basedon the norm of the residual of the field equations wasthe only modification made to the CFL3D.ADII codeto accommodate the SAADO procedure.

Surface geometry was generated based on parametersdescribed in a previous section by a code utilizing theRapid Aircraft Parameterization Input Design(RAPID) technique developed by Smith, et al / ' This

code was preprocessed with ADIFOR to generate acode capable of producing sensitivity derivatives, X',as well.

The CFD volume mesh needed by the flow analysiscode was generated using a version of the CSCMDO32

grid generation code. Associated grid sensitivityderivatives needed by the flow sensitivity analysiswere generated with an automatically differentiatedversion of CSCMDO." In addition to theparameterized surface mesh and accompanyinggradients, CSCMDO requires a baseline volume meshof similar shape and identical topology. The 45,000grid point baseline volume mesh of C-O topologyused in the present flexible wing examples wasobtained with the Gridgen™ code. The 41,000-pointbaseline volume mesh used in the rigid wingoptimization problem was generated using WTCO34.These meshes are admittedly particularly coarse bycurrent CFD analysis standards; the wing surfacemeshes are shown in Fig. 3.

The structural analysis code1" used to compute thedeflection of the elastic wing was a generic finiteelement code. The flexible structure for the wingshown in Fig. 3 was discretized by 583 nodes; therewere 2141 constant-strain triangle (CST) elementsand 1110 truss elements. Because the elasticdeformation was assumed to be small, linear elasticitywas deemed appropriate. The structural sensitivityequation was derived based upon the directdifferentiation method. Note that sensitivity of theaerodynamic forces appears as a term on the right-hand side (RHS) of the deflection sensitivityequation. The derivative of the stiffness matrix in thissensitivity equation was also generated36 by using theADIFOR2S2) technique. Since the coefficient matrix ofthe structural sensitivity equation was identical to thatof the structural equation, these sensitivity equationswere solved efficiently by backward substitution withdifferent RHSs for each sensitivity.

At the wing surface, i.e., the interface whereaerodynamic load and structural deflectioninformation is transferred, it was assumed that surfacenodes of the Finite Element Method (FEM) structuralmodel were a subset of CFD aerodynamic surfacemesh points (see Fig. 3) for the present SAADOapplication. This lack of generality allowed forsimplifications in data transfers and, although animportant issue, it was not deemed crucial for theseinitial flexible wing SAADO demonstrations. Futureapplications to more complex configurations shouldallow for transfer of conserved information between

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arbitrary meshes as required by individual disciplines.A recent review of such data transfer techniques andspecific recommendations are given in Ref. 37.

Conventional (NAND) and SAADO (SAND)procedures were implemented using the SequentialQuadratic Programming method of the DOT38

optimization software.. All computations wereexecuted on an SGI Origin 2000™ computer with25()Mhz R10000™ processors. The CFD sensitivitycalculations were partitioned and run on severalprocessors to reduce required memory and elapsedoptimization time. This partitioning, however, resultsin additional accumulated computational time due tothe nature of ADIFOR-generated sensitivity analysiscode.

RESULTSFigures 4 and 5 show the effect on convergence andcomputational cost of coupling the CFD and FEManalysis and sensitivity solvers, respectively. Themesh or mesh derivatives are updated with thedeflections or deflection derivatives, respectively, asindicated by the symbols. Even with a relativelyflexible wing, there is little effect on the convergencerate, i.e., residual reduction per CFD iteration.However, the computation (cpu) time does increase -rather dramatically for the coupled function analysisin Fig. 4 - due to repeated input and output of largemesh and restart files in the CFD flow solver andfrequent mesh regeneration. The cpu time spentperforming the FEM calculations and the interface ofcoupling data are too small to be visible in this figure.

The cpu time shown in Fig 5 is the cost for sensitivityanalysis for two design variables. For clarity only oneconvergence history was shown; the other was nearlyidentical. The cost of the function analysis relative tothe sensitivity analysis is greater than that anticipatedfrom the operation count. Since ADIFOR-generatedcode computes sensitivity analysis with the additionaloverhead of one function analysis one would expectthe ratio of sensitivity cost to function cost for twodesign variables to be 3. However, the compiler onthe SGI™ computer used in this study was able toperform more extensive code optimization to thefunction analysis portion than it could the sensitivityanalysis portion. As a result, the ratio is substantiallygreater.

The optimization results shown in this work are fordesign problems involving only two or eight out offifteen available wing design variables. These presentSAADO results are discussed in the context of otherSAND approaches at the end of this section. Flow

conditions for the wing optimization examples wereMoo = 0.8 and a- 1°.

Two-Design-Variable ProblemsTable 1 and Figs. 6 and 7 show results from severaloptimization problems involving two designvariables: the tip chord c( and the tip setback xr Twoof these problems are the conventional and SAADOoptimizations8 using rigid wing analysis. The otherproblems are optimizations using flexible winganalysis. The difference between the other two sets isdefinition of the constraints. One set uses the sameconstraints as the rigid wing optimization problem,denoted as "rigid" constraints in Table 1. That is,minimum total lift, maximum pitching moment andmaximum rolling moment. The other uses thepreviously defined "flexible" constraints; i.e., thoseconstraints that include structural responses.

Figure 6 shows wing planform and surface pressurecontours for the initial and optimized designs. TheSAADO and conventional "optimized" rigid wingsare essentially the same, with the DV differing onlyin the third significant figure as shown in Table 1.Resulting chordwise pressure distributions are thesame, so only results from the SAADO optimizationare shown. Similarly, the SAADO and conventional"optimized" flexible wings with either set ofconstraints show even smaller differences, so only theSAADO result is shown. The shock wave has beenweakened substantially in the optimized cases fromthat on the original wing, as would be expected. Thisis also evidenced in the chordwise pressurecoefficient distributions shown in Fig. 7.

Table 1 compares the values of design variables,objective functions and constraints for 2-DVproblems. Due to differences in the analyses causedby differences in meshes, comparisons betweenoptimization problems are made with objectivefunction values normalized by the value obtainedfrom analysis of the initial design. Overall, finaldesigns are very similar between the six problems.Since the problem is dominated by shock strength andthere are only two design variables available tochange, that is not surprising. The relativecomputational cost of SAADO optimizations andrespective conventional method optimizations isabout the same for the two cases with flexible winganalysis as it was for the rigid wing analysis fromRef. 8.

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Eight-Design-Variable ProblemsTable 2 and Fig. 8 show results from optimizationproblems using eight design variables as described inFig. 2. In this case, results for the optimizations usingrigid wing analysis with "rigid" constraints aresubstantially different from those using flexible winganalysis with "flexible" constraints. In particular,constraints on compliance and the difference betweenlift and weight do not allow the increase in span thatwas allowed in the rigid case. The results ofconventional and SAADO optimizations for the rigidwing analysis were so similar that only the SAADOresult is shown. The differences in conventional andSAADO results for the flexible wing are also small;but, the differences in planforms are noticeableenough to be shown in Fig. 8. In all of the optimizedresults, it is also seen that shock strength has beenreduced from that on the original wing.

Computation Cost ComparisonsIn view of the consistency of NAND and SANDoptimization results, measure of success or failure ofthe SAADO procedure is then its relativecomputational expense. Two-design-variable resultsin Table 1 show the relative cost of conventional andSAADO procedures based on accumulated CPU time.Geometry generator and mesh generator cost were notincluded for the rigid wing cases because theircontributions are minimal relative to cost of the flowsolver and flow sensitivity solver as shown in Figs. 4and 5. For the flexible wing cases, however, thosecontributions are significant for the coupled system,so they have been included along with the cost forCFD and FEM analyses and their respective gradientanalyses. Total cost has been normalized by the costof one full analysis to the target residual. TheSAADO method primarily reduces the cost of thecoupled function analysis. In this regard, the SAADOmethod does show improvement over its conventionalcounterpart for all methods applied to the two-designvariable case as shown in Table 1.

However, for the eight-design-variable flexible wingcase (Table 2), the SAADO optimization requiredmore function analysis computations than theconventional counterpart. In all other cases to date,SAADO and conventional optimization processesfollowed essentially the same path through designspace. But for this problem, the SAADO method"took a wrong turn" early in the process and spentmore time getting back to the "correct" answer. Thereare two factors affecting the path through designspace: function values and gradient values. That is,steps 1 and 3 affect step 5 in the algorithm described

earlier. Since, for SAADO, neither function valuesnor gradient values are expected to be well convergeduntil the end of the process, either could introduce theerror(s) that caused the "wrong turn". Previousexperience has shown that the gradient values tend tobe fairly reliable even at poor convergence levels;therefore, open questions remain concerning theseapproximations and how problem dependent theymight be.

The most computational time is spent computinggradients, even though none of the gradient residualratios were converged below three orders of magnitude.Early in the respective processes, gradients were not wellconverged. As the number of design variables isincreased, this proportion will grow nearly linearly. Theneed for faster gradient calculations is apparent. Hou etal.1 estimated a considerable speed-up attributed to usinghand-differentiated adjoint code for 2-D Euler equations.For a single discipline design, such as aerodynamicdesign, use of adjoint or co-state variables reducesgradient computational times significantly, as shown in anumber of the quoted references (See for example 1, 2,4, 14, 20, 24 and 25.). The SAADO formulation usingthe discrete adjoint method shown in the Appendix ofRef. 8 is easily extended to coupled aero/structuralanalysis. It is impractical, however, since the coupledsensitivity analyses would require adjoints for eachdisciplinary output being transferred, i.e., discretizedloads and deflections (See, for example, Ref. 18.). In atightly or implicitly coupled multidisciplinary analysis,adjoints may prove practical since this system would beanalogous to a single discipline.

Further DiscussionRelative cost, based on CPU timing ratios, forSAADO (SAND) versus conventional (NAND)procedures applied to these present small 3-Daerodynamic shape design optimization problems areabout seven-tenths for all except the eight-design-variable SAADO case. This range is very similar tothat reported for 2-D nonlinear aerodynamic shapedesign optimization in Refs. 1 and 4, even thoughmany of the computational details differ. The resultsgiven in Ref. 1 were for a turbulent transonic flowwith shock waves computed using a Navier-Stokescode; a direct differentiation approach (usingADIFOR) was used for the sensitivity analysis. Theresults reported in Ref. 4 were for a compressibleflow without shock waves computed using anonlinear potential flow code; an adjoint approachwas used for the sensitivity analysis. Since these twooptimization problems were also not the same, then,no timing comparison between these adjoint and

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direct differentiation solution approaches would bemeaningful. As indicated earlier, an expected speed-up was estimated in Ref. 1 for using an adjointapproach instead of direct differentiation .

Ghattas and Bark2" recently reported 2-D and 3-Dresults for optimal control of steady incompressibleNavier-Stokes flow which demonstrate an order-of-magnitude reduction of CPU time for a SANDapproach versus a NAND approach. These resultswere obtained using reduced Hessian SQP methodsthat avoid converging the flow equations at eachoptimization iteration. The relationship of thesemethods with respect to other optimization techniquesis also discussed in Ref. 25.

Several other SAND-like methods for simultaneousanalysis and design are summarized and discussed byTa'asan. w These methods are called "One-Shot" and"Pseudo-Time" and have been applied toaerodynamic shape design problems at severalfidelities of CFD approximation, as noted in Ref. 39.These techniques have obtained an aerodynamicdesign in the equivalent of several analysis CPUtimes for some sample problems.

CONCLUDING REMARKSThis study has introduced an implementation of theSAADO technique for a simple, isolated, flexiblewing. Initial results indicate that SAADO

1. is feasible under dual simultaneity (i.e.simultaneity not only with respect to analysis anddesign optimization, but also simultaneity withrespect to flexible wing aero-structuralinteraction)

2. finds the same local minimum as a conventionaltechnique

3. can be computationally more efficient than aconventional gradient-based optimizationtechnique; however, the relative efficiency maybe dependent on the optimization problem

4. requires few modifications to the analysis andsensitivity analysis codes involved.

Perhaps improvements to this SAADO procedure orits implementation can be made with respect togradient-approximation and line-search techniques.

ACKNOWLEDGEMENTThe second author, G. J.-W. H., was supported in thiswork by NASA through several Tasks under contractNAS1-19858 and NASA P.O. No. L-9291 with theODU Research Foundation.

REFERENCES1. Newman, P. A., Hou, G. J.-W., and Taylor III, A.

C., "Observations Regarding Use of AdvancedCFD Analysis, Sensitivity Analysis, and DesignCodes in MDO," in Ref. 3, pp. 263-279; alsoICASE Report 96-16, NASA CR 198293,(available electronically at www.icase.edu).

2. Newman, III, J. C., Taylor, III, A. C., Barnwell,R. W., Newman, P. A., and Hou, G. J.-W.,"Overview of Sensitivity Analysis and ShapeOptimization for Complex AerodynamicConfigurations," Journal of Aircraft, Vol. 36,No. 1, 1999, pp. 87-96.

3. Alexandrov, N. M., and Hussaini, M. Y., Eds.,Multidisciplinary Design Optimization: State ofthe Art, SI AM Proceedings Series, SIAM,Philadelphia, 1997.

4. Ghattas, O. and Orozco, C. E., "A ParallelReduced Hessian SQP Method for ShapeOptimization," in Ref. 3, pp. 133-152.

5. Hou, G. J.-W., Taylor, III, A. C., Mani, S. V.,and Newman, P. A., "SimultaneousAerodynamic Analysis and DesignOptimization," Abstracts from 2nd U.S. NationalCongress on Computational Mechanics,Washington, DC, Aug., 1993, pp. 130.

6. Mani, S. V., "Simultaneous AerodynamicAnalysis and Design Optimization," M. S.Thesis, Old Dominion University, Norfolk, VA,Dec. 1993.

7. Hou, G. J.-W., Korivi, V. M., Taylor, III, A. C.,Maroju, V., and Newman, P. A., "SimultaneousAerodynamic Analysis and Design Optimization(SAADO) of a Turbulent Transonic AirfoilUsing a Navier-Stokes Code With AutomaticDifferentiation (ADIFOR)," ComputationalAerosciences Workshop 95, edited by W. J.Feiereisen, and A. K. Lacer, NASA CD CP-20010, Jan. 1996, pp. 82-85.

8. Gumbert, C. R., Hou, G.J.-W., and Newman, P.A., "Simultaneous Aerodynamic Analysis andDesign Optimization (SAADO) of a 3-D RigidWing", Proceedings, J4"' AlAA ComputationalFluid Dynamics Conference, Norfolk, June 1999,pp. 402-418;also AIAA Paper 99-3296.

9. Haftka, R. T., Sobieszczanski-Sobieski, J., andPadula, S. L., "On Options for InterdisciplinaryAnalysis and Design Optimization," StructuralOptimization, Vol. 4, 1992, pp. 65-74.

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10. Shubin, G. R., "Application of AlternativeMultidisciplinary Optimization Formulations to aModel Problem for Static Aeroelasticity."Journal of Computational Physics, Vol. 118, No.1, 1995, pp. 73-85.

11. Arian, E., "Analysis of the Hessian forAeroelastic Optimization," ICASE Report No.95-84, NASA CR 198245, Dec. 1995.

12. Walsh, J. L., Townsend, J. C, Salas, A. O.,Samareh, J. A., Mukhopadhyay, V., andBarthelemy, J.-F., "Multidisciplinary High-Fidelity Analysis and Optimization of AerospaceVehicles, Part 1: Formulation," AIAA Paper2000-0418, Jan. 2000.

13. Walsh, J. L., Weston, R. P., Samareh, J. A.,Mason, B. H., Green, L. L., and Biedron, R. T.,"Multidisciplinary High-Fidelity Analysis andOptimization of Aerospace Vehicles, Part 2:Preliminary Results," AIAA Paper 2000-0419,Jan. 2000.

14. Reuther, J. J., Alonso, J. J., Martins, J. R., andSmith, S. C., "A Coupled Aero-StructuralOptimization Method for Complete AircraftConfigurations," AIAA Paper 99-0187, Jan.1999.

15. Kapania, R. K., Eldred, L. B., and Barthelemy,J.-F. M., "Sensitivity Analysis of a WingAeroelastic Response," Journal of Aircraft, Vol.30, No. 4, 1993, pp. 496-504; also AIAA Paper91-1103, Apr. 1991.

16. Arslan, A. E., and Carlson, L. A., "IntegratedDetermination of Sensitivity Derivatives for anAeroelastic Transonic Wing," 5th

AIAA/USAF/NASA/fSSMO Symposium onMultidisciplinary Analysis and Optimization,Panama City Beach, FL, Sept. 1994, pp. 1286-1300; also AIAA Paper 94-4400 CP.

17. Giunta, A. A., and Sobieszczanski-Sobieski, J.,"Progress Toward Using Sensitivity Derivativesin a High-Fidelity Aeroelastic Analysis of aSupersonic Transport," 7"'AIAA/USAF/NASA/ISSMO Symposium onMultidisciplinary Analysis and Optimization, St.Louis, MO, Sept. 1998, pp. 441-453; also AIAAPaper 98-4763 CP.

18. Giunta, A. A., "Sensitivity Analysis for CoupledAero-Structural Systems," NASA/TM-1999-209367, Aug. 1999.

19. Newman, III, J. C., Whitfield, D. L., andAnderson, K. W., "A Step-Size Independent

Approach for Multidisciplinary SensitivityAnalysis and Design Optimization," J7" AppliedAerodynamics Conference, Norfolk, VA, June-July 1999, pp. 12-22; also AIAA Paper 99-3101.

20. Ghattas, O., and Li, X., "Domain DecompositionMethods for Sensitivity Analysis of a NonlinearAeroelastic Problem," International Journal ofComputational Fluid Dynamics, Vol. 11, 1998,pp. 113-130.

21. Moller, H. and Lund, E., " Shape SensitivityAnalysis of Strongly Coupled Fluid-StructureInteraction Problems, " AIAA Paper AIAA-2000-4823.

22. Maute, K., Nikbay, M. and Farhat, C.,"Analytical Based Sensitivity Analysis andOptimization of Nonlinear Aeroelastic Systems"AIAA Paper AIAA-2000-4825.

23. Hou, G. and Satyanarayana, A., "AnalyticalSensitivity Analysis of a Static AeroelasticWing," AIAA Paper AIAA-2000-4824.

24. Biros, G. and Ghattas, O., "Parallel Newton-Krylov Algorithms For PDE-ConstrainedOptimization" Proceedings of ScientificComputing 99, Portland, OR, 1999.

25. Ghattas, O., and Bark, J.-H., "Optimal Control ofTwo- and Three-Dimensional Navier-StokesFlows," Journal of Computational Physics, Vol.136, No. 2, 1997, pp. 231-244.

26. Rumsey, C., Biedron, R., and Thomas, J.,"CFL3D: Its History and Some RecentApplications," NASA TM-112861, May 1997.

27. Sherman, L., Taylor, III, A., Green, L., Newman,P., Hou, G., and Korivi, M., "First- and Second-Order Aerodynamic Sensitivity Derivatives viaAutomatic Differentiation with IncrementalIterative Methods," Journal of ComputationalPhysics, Vol.129, No. 2, 1996, pp. 307-336.

28. Bischof, C. H., Carle, A., Corliss, G. F.,Griewank, A., and Hovland, P., "ADIFOR:Generating Derivative Codes from FortranPrograms," Scientific Programming, Vol. 1, No.1, 1992, pp. 1-29.

29. Bischof, C., and Griewank, A., "ADIFOR: AFortran System for Portable AutomaticDifferentiation," Proceedings, FourthAIAA/USAF/NASA/OAI Symposium onMultidisciplinary Analysis and Optimization,Cleveland, Sept. 1992, pp. 433-441; also AIAAPaper 92-4744 CP.

10American Institute of Aeronautics and Astronautics

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

30. Taylor, III, A. C, Oloso, A., and Newman, III,J.C., "CFL3D.ADII (Version 2.0): An Efficient,Accurate, General-Purpose Code for FlowShape-Sensitivity Analysis," AIAA Paper 97-2204, June 1997.

31. Smith, R. E., Bloor, M. I. G., Wilson, M. J., andThomas, A. T., "Rapid Airplane Parametric InputDesign (RAPID), " Proceedings, 12th AIAAComputational Fluid Dynamics Conference, SanDiego, June 1995, pp. 452-462; also AIAA Paper95-1687.

32. Jones, W. T., and Samareh-Abolhassani, J., "AGrid Generation System for MultidisciplinaryDesign Optimization," Proceedings, 12th AIAAComputational Fluid Dynamics Conference, SanDiego, June 1995, pp. 474-482; also AIAA Paper95-1689.

33. Bischof, C., Jones, W. T., Samareh-Abolhassani,J., and Mauer, A., "Experiences with theApplication of the ADIC AutomaticDifferentiation Tool to the CSCMDO 3-DVolume Grid Generation Code," AIAA Paper96-0716, Jan. 1996.

34. Vatsa, V. N., and Wedan, B. W., "Effect ofSidewall Boundary Layer on a Wing in a WindTunnel," Journal of Aircraft, Vol. 26, No. 2,1989, pp. 157-161; also AIAA Paper 88-1020,Jan. 1988.

35.

36.

37.

38

Nguyen, D. T., "Finite Element Software forMultidisciplinary Design Optimization," FinalReport, Old Dominion University ResearchFoundation, Contract NAS1 -19858, Task 69,NASA Langley Research Center, Hampton, VA,Nov. 1995.Hou, G., Arunkumar, S., and Tiwari, N. S.,"First- and Second-Order Sensitivity Analysis ofFinite Element Equations via AutomaticDifferentiation," 7il!AIAA/USAF/NASA/1SSMOSymposium on Multidisciplinary Analysis andOptimization, St. Louis, MO, Sept. 1998, pp.454_464; also AIAA Paper 98-4764.Samarah, J. A., and Bhatia, K. G., "A UnifiedApproach to Modeling MultidisciplinaryInteractions," accepted for presentation at the 8th

AIAA/NASA/USAF/ISSMO Symposium onMultidisciplinary Analysis and Optimization,Long Beach, CA, Sept. 6-8, 2000.Anon., DOT Users Manual: Version 4.20,Vanderplaats Research & Development, Inc.,Colorado Springs, May 1995.

39. Ta'asan, S., "Trends in Aerodynamic Design andOptimization: A Mathematical Viewpoint,"Proceedings, 12th AIAA Computational FluidDynamics Conference, San Diego, June 1995, pp.961-970; also, AIAA Paper 95-1

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Table 1. Comparison of two-design-variable results.

FoF/FO9i92

9sdv 1 (ct)dv 2 (xt)Cost

Analysis @ initialshape

Rigid(Ref. 8)

-8.431

-0.0822-0.9276

-0.532111

Flexible

-7.151

-0.0302-0.8882-0.2647

111

Optimization Results

Rigid (Ref. 8)

Conv

1.466-0.0003-0.9928-0.5671

1.072.0320.6

SAADO

1.4650.0003

-0.9931-0.5612

1.041.9715.1

Flexible w/ 'rigid1

constraints

Conv

1.260.0002

-0.9836-0.5752

1.111.8831.2

SAADO

1.260

-0.9843-0.5755

1.1 11.8919.1

Flexible w/ 'flexible1

constraints

Conv

1.3970

-0.9191-0.0968

1.131.9436.2

SAADO

1.396-0.0023-0.9246-0.0938

1.141.9522.2

Table 2. Comparison of 8-design-variable optimization results.

Fo

F/FO9i92

9sdv 1 (Ct)dv 2 (xt)dv 3 (b)dv 4 (t)dv 5 (tr)dv 6 ft)dv 7 (zr)dv 8 (xzr)Cost

Analysis @ initialshape

Rigid-8.43

1-0.0822-0.928-0.532

111111111

Flexible-7.15

1-0.0302-0.888-0.265

111111111

Optimization ResultsRigid

Conv

3.48-2.48

0.00090.00020.844

31.5

0.6250.5

0.9411.13

0.936152

SAADO

3.48-2.49

-0.00040.00050.836

31.5

0.620.5

0.5651.14

0.944128

FlexibleConv

2.83-0.464

-0.0003-0.0013

0.6052.29

0.8511.430.5

0.291.11

0.651120

SAADO

2.78-0.225

-0.0003-0.0007

0.5771.88

0.8781.450.5

0.3891.090.6117

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2DV: tip chord, tip setback8DV: 2 DV + semispan, twist, root thickness,

tip thickness, root camber, locationof max root camber

Figure 1. Description of semispan wing parameterization.

Specify:Design ProblemFinal Convergence

Initialize:GeometryFlow ConditionsFEM ModelConvergence Level

Flexible

Partially ConvergedSystem Analysis

ProcedurePartially ConvergedSensitivity Analysis

Geometryand Mesh

^*

StructuralAnalysis

i ""la

^ r \

-1

f

Geometryand Mesh

Sensitivities

AerodynamicSensitivities

StructuralSensitivities

TightenConvergence

i /~\\ i /-\ i-< — Design

Update -<— Optimizer ImprovedDesign

Partially Converged Design

Figure 2. Diagram of flexible wing SAADO procedure.

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97x17x25CFD mesh(rigid wing)

73x25x25CFD mesh

3251 elementFEM mesh

Figure 3. Computational meshes for rigid wing analysis and coupled flexible wing analysis.

Ft, Au,

10°

10-

max

10"

10,-6

10i-8

CFD residual, Rchange in deflection, Aurigid wing CFD residual, R

CFDFEMmeshinterface

1500

CO~o§1000CDCO

CDE

Q.O

100 200 300CFD iteration cycles

400 500 Rigid Flexible

Figure 4. Effect of aerodynamic/structural coupling on function analysis convergence, MM = 0.8, a- l°.

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10°

1(T1

io-2

10-4

io-5

10~6

10,-7

dfl/dx.——+A(du/dxt)- - dfl/dx, (Rigid)

0 100 200 300CFD iteration cycles

CFDFEMmeshinterface

6000

w•

o840000)

2000

0

I I

| |

400 Rigid Flexible

Figure 5. Effect of aerodynamic/structural coupling on sensitivity analysis convergence.

rigid analysis

exible analysis

optimized shapewith flexibleanalysisoptimized shapewith rigidanalysis

original shape

Figure 6. Comparison of planform shapes and surface pressure contours for two-design-variable cases,M.o-0.8, a- 1°.

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— - — rigid— — - flexible

original

1.0

Figure 1. Comparison of chord wise pressure coefficient distributions at section A-A for two-design-variable cases,

pressure contours and shapes -optimized flexible

original flexible

^0ji§j^^/^ ^optimized rigid

SAADO optimized shapewith flexible analysisconventional optimized shapewith flexible analysis

SAADO and conventionaloptimized shapes with rigidanalysis

Figure 8. Comparison of planform shapes and surface pressure contours for eight-design-variable cases,Moo = 0.8, a = 1°.

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