August 1999

NASA/TM-1999-209367

Sensitivity Analysis for CoupledAero-structural Systems

Anthony A. GiuntaLangley Research Center, Hampton, Virginia

The NASA STI Program Office ... in Profile

Since its founding, NASA has been dedicated tothe advancement of aeronautics and spacescience. The NASA Scientific and TechnicalInformation (STI) Program Office plays a keypart in helping NASA maintain this importantrole.

The NASA STI Program Office is operated byLangley Research Center, the lead center forNASAÕs scientific and technical information. TheNASA STI Program Office provides access to theNASA STI Database, the largest collection ofaeronautical and space science STI in the world.The Program Office is also NASAÕs institutionalmechanism for disseminating the results of itsresearch and development activities. Theseresults are published by NASA in the NASA STIReport Series, which includes the followingreport types:

· TECHNICAL PUBLICATION. Reports of

completed research or a major significantphase of research that present the results ofNASA programs and include extensivedata or theoretical analysis. Includescompilations of significant scientific andtechnical data and information deemed tobe of continuing reference value. NASAcounterpart of peer-reviewed formalprofessional papers, but having lessstringent limitations on manuscript lengthand extent of graphic presentations.

· TECHNICAL MEMORANDUM. Scientific

and technical findings that are preliminaryor of specialized interest, e.g., quick releasereports, working papers, andbibliographies that contain minimalannotation. Does not contain extensiveanalysis.

· CONTRACTOR REPORT. Scientific and

technical findings by NASA-sponsoredcontractors and grantees.

· CONFERENCE PUBLICATION. Collected

papers from scientific and technicalconferences, symposia, seminars, or othermeetings sponsored or co-sponsored byNASA.

· SPECIAL PUBLICATION. Scientific,

technical, or historical information fromNASA programs, projects, and missions,often concerned with subjects havingsubstantial public interest.

· TECHNICAL TRANSLATION. English-

language translations of foreign scientificand technical material pertinent to NASAÕsmission.

Specialized services that complement the STIProgram OfficeÕs diverse offerings includecreating custom thesauri, building customizeddatabases, organizing and publishing researchresults ... even providing videos.

For more information about the NASA STIProgram Office, see the following:

· Access the NASA STI Program Home Pageat http://www.sti.nasa.gov

· E-mail your question via the Internet to

help@sti.nasa.gov · Fax your question to the NASA STI Help

Desk at (301) 621-0134 · Phone the NASA STI Help Desk at

(301) 621-0390 · Write to:

NASA STI Help Desk NASA Center for AeroSpace Information 7121 Standard Drive Hanover, MD 21076-1320

National Aeronautics andSpace Administration

Langley Research CenterHampton, Virginia 23681-2199

August 1999

NASA/TM-1999-209367

Sensitivity Analysis for CoupledAero-structural Systems

Anthony A. GiuntaLangley Research Center, Hampton, Virginia

Available from:

NASA Center for AeroSpace Information (CASI) National Technical Information Service (NTIS)7121 Standard Drive 5285 Port Royal RoadHanover, MD 21076-1320 Springfield, VA 22161-2171(301) 621-0390 (703) 605-6000

Sensitivity Analysis for Coupled Aero-structural Systems

Anthony A. Giunta∗

National Research Council/NASA Langley Research Center18d West Taylor Street, Mail Stop 139, Hampton, Virginia 23681–2199

Abstract

A novel method has been developed for calculatinggradients of aerodynamic force and moment coeffi-cients for an aeroelastic aircraft model. This methoduses the Global Sensitivity Equations (GSE) to ac-count for the aero-structural coupling, and a reduced-order modal analysis approach to condense the cou-pling bandwidth between the aerodynamic and struc-tural models. Parallel computing is applied to reducethe computational expense of the numerous high fi-delity aerodynamic analyses needed for the coupledaero-structural system. Good agreement is obtainedbetween aerodynamic force and moment gradientscomputed with the GSE/modal analysis approachand the same quantities computed using brute-force,computationally expensive, finite difference approxi-mations. A comparison between the computationalexpense of the GSE/modal analysis method and apure finite difference approach is presented. Theseresults show that the GSE/modal analysis approachis the more computationally efficient technique if sen-sitivity analysis is to be performed for two or moreaircraft design parameters.

Nomenclature

CD drag coefficientCL lift coefficientCMo pitching moment coefficient (about y-axis)CFD computational fluid dynamicsCSM computational structural mechanicsE Young’s modulusF vector of aerodynamic loadsGSE Global Sensitivity EquationsI area moment of inertiaI identity matrixK finite element stiffness matrixL beam lengthLE leading edgeLSM Local Sensitivity MatrixLSV Local Sensitivity Vector

M finite element mass matrixMDO multidisciplinary design optimizationOML outer mold lineq vector of mode shape scale factorsTSV Total Sensitivity Vectort/c thickness-to-chord ratioX vector of independent design parametersα angle-of-attackδ beam tip deflection∆ vector of structural deflectionsλ eigenvalue from modal analysisΛ diagonal matrix of eigenvaluesΛLE inboard leading edge sweep angleφ eigenvector from modal analysisΦ column matrix of eigenvectors

1 Introduction

In Paul Rubbert’s 1994 AIAA Wright Brother Lec-ture [1] he states a vision for the future of the aircraftdesign process as follows,

“My vision is to be able to carry out thedetailed aerodynamic design of any por-tion of an airplane within a handful ofdays at most, and to do it in concertwith the loads engineer, the structural de-signer, the systems person and the manu-facturing expert sitting side by side in thesame room, with computer systems thattalk well with one another.”

Implicit in this statement is that the computationalmodels used in aircraft design accurately capture theimportant physical phenomena of interest. That is,high fidelity analysis models are available for aerody-namic analysis, structural analysis, and for the otheraircraft design disciplines. Rubbert’s vision thatthe computer systems “talk well with one another”supposes that a capability exists to exchange thediscipline-specific high fidelity analysis data amongthe different engineering disciplines. The commu-nication links among the engineering disciplines arenecessary to perform the sensitivity analyses neededto quantify the impact of design perturbations (e.g.,changes in wing thickness) on the performance of the

∗Postdoctoral Fellow, National Research Council, currently Member of the Technical Staff (limited term), Sandia NationalLaboratories, PO Box 5800, Mail Stop 0847, Albuquerque, NM 87185, aagiunt@sandia.gov

1National Aeronautics and Space Administration

aircraft (e.g., aerodynamic performance, structuralweight, manufacturing costs). Such sensitivity stud-ies are the foundation for improving, or optimizing,an aircraft design.

Progress toward Rubbert’s vision is occurringgradually as high fidelity analysis tools, such asEuler/Navier-Stokes computational fluid dynamics(CFD) solvers and finite element computationalstructural mechanics (CSM) codes, are employed in-creasingly early in the aircraft design process. How-ever, there are several impediments to Rubbert’s vi-sion of efficient and accurate computing in aircraftdesign. One of these impediments is the compu-tational burden incurred when expensive CFD andCSM codes are employed in the analysis of coupledaero-structural systems such as modern transportand fighter aircraft. Analyzing these coupled systemstypically requires the repeated use of CFD and CSMcodes to obtain an accurate solution. Thus, the com-putational expense quickly mounts if many coupledaero-structural analyses must be performed for sen-sitivity analysis and/or optimization.

The focus of this work is the development of acomputational method that permits the rapid evalu-ation of sensitivity derivatives (gradients) for a cou-pled aero-structural system. This study employs theGlobal Sensitivity Equation (GSE) method devel-oped by Sobieski [2] which provides a mathemati-cal expression for the total sensitivity derivatives ofa general coupled system. For the aero-structuralsystem considered here, the GSE method requiresthe computation of interdisciplinary coupling terms(partial derivatives) between the aerodynamic andstructural models. To reduce the number of partialderivatives terms in the GSE, structural deflectionsare approximated using a superposition of structuralmode shapes (basis vectors). Additional computa-tional savings are realized by using coarse grainedparallel computing for the CFD evaluations needed tocompute some of the partial derivatives in the GSE.

The combined GSE/modal analysis approach de-veloped in this study enables an aircraft design en-gineer to perform a sensitivity analysis of a coupledaero-structural system in a manner that is more ef-ficient than using traditional finite difference meth-ods. In addition, the GSE/modal analysis approachemploys the CFD and CSM solvers as black-boxes.Thus, virtually any CFD or CSM code may be usedwith this sensitivity analysis method.

The GSE/modal analysis approach is intended foruse in the preliminary phase of aircraft design whenboth detailed CFD and CSM (finite element) modelshave been created for an aircraft. The use of a lin-ear superposition of mode shapes to represent struc-

tural deflections of a finite element model limits theGSE/modal analysis approach to the exploration ofperturbations of an existing aircraft design. That is,one assumption of the GSE/modal analysis method isthat the natural frequencies and mode shapes remainessentially unchanged for small perturbations in theaircraft design parameters. For this reason, this ap-proach is not intended for use in the conceptual phaseof aircraft design when major configuration choiceshave yet to be finalized.

In this study, the Langley-developed CFD solverCFL3D [3] is used for aerodynamic analysis and thecommercial CSM solver GENESIS [4] is used forstructural analysis. The aircraft examined here isa generic supersonic transport configuration. Theparametric model for this aircraft contains 104 designvariables (64 planform and airfoil variables, 40 struc-tural variables). Gradients of aerodynamic force andmoment coefficients are computed for three of thesevariables using the GSE/modal analysis approach.Validation of the accuracy of the GSE/modal anal-ysis gradients is performed through comparisons togradients computed using a pure finite difference ap-proach.

The remainder of this paper is arranged as fol-lows. Section 2 contains some background infor-mation on related research using GSE methods andreduced-order modal analysis methods in computa-tional aeroelasticity. Section 3 covers the mathemat-ics of the GSE formulation and the modal analy-sis methods employed in this study. Section 4 con-tains a simple problem involving a cantilever beam todemonstrate the GSE/modal analysis method. Sec-tions 5 and 6 cover the modeling and aeroelastic anal-ysis of a supersonic transport aircraft, along with re-sults obtained from applying the GSE/modal analy-sis method to calculate sensitivities for this aircraftmodel. In Section 7 there is a comparison of the com-putational expense of the GSE/modal analysis ap-proach and the traditional finite difference approachfor calculating sensitivity derivatives. A summary ofthis work is contained in Section 8.

2 Background

This study builds on the past research efforts ofBarthelemy et al [5] and Dovi et al [6] who em-ployed the Global Sensitivity Equations in design op-timization of a supersonic transport aircraft. Re-lated research by Kapania et al [7] and Eldred etal [8] used a variation of the GSE method for thesensitivity analysis of a simple forward-swept wingmodel. These studies employed inexpensive low fi-

2National Aeronautics and Space Administration

delity analysis methods such as linear aerodynamics(panel codes) and equivalent plate structural models(ELAPS [9, 10]). These inexpensive methods per-mitted the use of finite difference approximations toevaluate the interdisciplinary coupling derivatives inthe GSE (described in Section 3).

The use of computationally expensive high fidelityanalysis software in this study greatly increases theamount of interdisciplinary coupling derivatives inthe GSE. That is, the number of coupling terms isrelated to the amount of force and deflection dataexchanged between the aerodynamic and structuralmodels. With the high fidelity CFD and CSM toolsemployed in this work, the number of coupling termsin the GSE is O(102−103). Thus, a pure finite differ-ence approach for approximating the partial deriva-tive coupling terms is not computationally affordable.

Fortunately, there has been considerable researchin the aeroelasticity community in developing meth-ods that simplify the coupling between aerodynamicand structural models. Numerous reduced-ordermodeling approaches are described in the survey pa-pers by Friedmann [11], Livne [12], and Karpel [13].Recent examples of the application of reduced-ordermodeling methods are found in the work of Ravehand Karpel [14] and Cohen and Kapania [15].

The reduced-order modeling approach followed inthis study employs a linear superposition of basis vec-tors to approximate the structural deflections of anaircraft finite element model. Here, the basis vectorsare supplied by a normal modes (eigenvalue) analysisof the finite element model. The number of basis vec-tors used in this study is O(101). One advantage ofthis reduced-order approach is that it supplies ana-lytic expressions for some of the interdisciplinary cou-pling terms in the GSE. Another advantage of this ap-proach is that it becomes computationally affordableto compute the remaining partial derivatives in theGSE using finite difference methods. The derivationand application of this GSE/modal analysis approachis described below.

3 Sensitivity Analysis

3.1 Global Sensitivity Equations

Consider a generic coupled aero-structural system de-picted in Figure 1. The coupling between the aero-dynamic and structural models is accomplished bycalculating external loads, F, on the aircraft struc-ture along with the resulting structural deflections,∆. The aerodynamic and structural data neededto compute F and ∆ are obtained from a varietyof computer programs along with suitable pre- and

post-processing software to transfer the load and de-flection data between the models.

The input into the aero-structural system is a vec-tor of independent parameters X. This vector con-tains aerodynamic design parameters, such as wingplanform and shape parameters, as well as struc-tural design parameters, such as internal rib and sparthicknesses. The output quantities from the coupledsystem include the aerodynamic load distribution onthe deflected structure, F, along with the force andmoment coefficients CL, CD, and CMo . In addition,the output includes the final structural deflections,∆, and the internal stresses in the structure.

For an aircraft design application, quantities ofinterest such as the aerodynamic force and momentcoefficients are needed to ensure that the aircraft willsatisfy the specified mission requirements. In addi-tion, the gradients of these quantities are needed todetermine the sensitivity of the aerodynamic perfor-mance to small perturbations in the design parame-ters. Analytic expressions for these gradients usuallyare not available and they are estimated using finitedifference approximations. This requires a separatesolution of the coupled system for each perturbationof X; a potentially prohibitive undertaking if eitherthe aerodynamic model or the structural model iscomputationally expensive to evaluate.

Following the notation employed by Sobieski[2], the loads and deflection transfer in the aero-structural system are represented in functional formas

F = F(X,∆), (1)

and∆ = ∆(X,F). (2)

Differentiating Equations 1 and 2 with respect to thevector of independent parameters, X, yields

dFdX

=∂F∂X

+∂F∂∆

d∆dX

, (3)

andd∆dX

=∂∆∂X

+∂∆∂F

dFdX

. (4)

Equations 3 and 4 are coupled and may be rearrangedinto a linear system of the form I − ∂F

∂∆

−∂∆∂F

I

dFdXd∆dX

=

∂F∂X∂∆∂X

. (5)

Equation 5 is known as the Global SensitivityEquation (GSE), which provides a convenient frame-work for grouping related terms in a system of cou-pled sensitivity equations. Olds [16] provides a use-ful lexicon for describing the components of the GSE.

3National Aeronautics and Space Administration

Following Olds’ approach, the matrix on the left sideof Equation 5 is termed the Local Sensitivity Matrix(LSM) and it contains the partial derivatives of theaerodynamic loads and structural deflections with re-spect to each other. Note that the vector of indepen-dent parameters, X, does not appear in the LSM. Thevector on the right side of Equation 5 is termed theLocal Sensitivity Vector (LSV). This vector containsthe partial derivatives of the loads and deflectionswith respect to the independent parameters. In theLSV there are no coupling derivatives. The vector onthe left side of Equation 5 is termed the Total Sen-sitivity Vector (TSV) and it contains the total sen-sitivity derivatives which are the unknown quantitiesin this system of equations.

While Equation 5 provides a simple expression forthe interdisciplinary coupling of an aero-structuralsystem, the partial derivative terms in Equation 5may be particularly difficult to obtain. For example,consider the term ∂F/∂∆ in Equation 5. The vec-tor of aerodynamic forces applied to the structuralmodel, F, is computed using an expensive CFD code(assume for this example that suitable force calcula-tion and interpolation methods are available). If oneattempts to calculate ∂F/∂∆ using a finite differ-ence approximation, then a CFD code evaluation isrequired for each perturbation of the vector ∆. Evenwith parallel computing, this approach is not attrac-tive when the length of ∆ is large, e.g., O(102−103),as would be typical for a finite element model used inthe aircraft industry.

For this reason there is motivation to exploremethods that may reduce the computational expenseof computing the term ∂F/∂∆. One approach to thisproblem is to represent the nodal deflections using alinear superposition of a set of basis functions. Aversion of such a reduced basis approach is employedin this study, where the the basis functions are themode shapes of the structural finite element model.

An important aspect of this work is that thereduced basis method used here is employed whiletreating the CFD and CSM codes as black boxes.That is, access to the source code of the CFD andCSM analysis software is not needed. While this ap-proach may appear restrictive, it is a realistic modelof the aircraft design industry in which both com-mercial software and legacy software are used in theaircraft analysis and design process. An attractiveoutcome of this restriction is that successful meth-ods developed by treating the CFD and CSM codesas black boxes will be broadly applicable to aircraftdesign practices in industry.

3.2 Modal Analysis

In linear finite element structural analysis the vectorof structural displacements, ∆, is found by solvingthe equation

F = K∆, (6)

where F is the vector of applied loads and K is thestiffness matrix assembled from the individual shapefunctions of the finite elements. Another commoncomputation is a modal analysis of the finite elementmodel. This analysis provides the natural frequen-cies of the structure along with the associated modeshapes. The modal analysis is performed by solvingthe eigenvalue problem

Kφ = Mφλ, (7)

where M is the mass matrix of the finite elementmodel, and φ is an eigenvector (mode shape) corre-sponding to a particular eigenvalue, λ, that satisfiesthe linear system.

Equation 7 is solved to obtain the desired numberof eigenvalues and eigenvectors for the finite elementmodel. These n eigenvalues are expressed using adiagonal matrix of the form

Λ = [λ1, λ2, . . . , λn]I, (8)

where I is an n× n identity matrix. The n eigenvec-tors are grouped using the matrix Φ as

Φ = [φ1 φ2 . . . φn], (9)

where the n columns of Φ correspond to the n eigen-values in Λ. Note that in this study the eigenvec-tors are scaled to meet the K-orthogonality and M-orthonormality conditions described by Bathe [17],such that

ΦTKΦ = Λ, (10)

andΦTMΦ = I. (11)

This scaling is performed internally in the CSM solverGENESIS [4] used in this study.

The mode shapes in Φ are used to approximatethe structural deflections, ∆, where

∆ = Φq, (12)

and q is a vector of unknown scale factors. The vectorq is computed using a least squares approach where

q = (ΦTΦ)−1ΦT∆. (13)

Note that it is typical to use 10 – 30 mode shapesin this approximation. Thus, the length of q can bemuch smaller than the length of the vector of struc-tural deflections ∆.

4National Aeronautics and Space Administration

To take advantage of this reduced basis approachinvolving q, the Chain Rule is applied to the term∂F/∂∆ as follows

∂F∂∆

=∂F∂q

∂q∂∆

, (14)

where, from Equation 13

∂q∂∆

= (ΦTΦ)−1ΦT. (15)

Thus, Equation 14 becomes

∂F∂∆

=∂F∂q

(ΦTΦ)−1ΦT. (16)

The advantage of this approach is that q is O(101)whereas ∆ typically is O(102−103). It is much morereasonable to compute ∂F/∂q than ∂F/∂∆ if usingfinite difference approximations.

The other partial derivative term in the Local Sen-sitivity Matrix of Equation 5 is ∂∆/∂F. The exactvalue of this derivative is K−1, however, this matrixis not available to the user in many finite elementanalysis codes, particularly those that are commer-cially developed. For this reason, an approximationfor ∂∆/∂F is needed. Using the modal analysis ap-proach described above, it is possible to compose anexplicit expression relating ∆ and F. This is de-scribed below.

Substituting Equation 12 into Equation 6 yields

F = KΦq. (17)

Pre- and post-multiplying this expression by ΦT andΦ, respectively, gives

ΦTFΦ = ΦTKΦqΦ. (18)

The K-orthogonality condition in Equation 10 is usedto simplify Equation 18 to

ΦTFΦ = ΛqΦ, (19)

which reduces to

ΦTF = Λq. (20)

Rearranging this equation yields an expression for qwhere

q = Λ−1ΦTF. (21)

Finally, substituting Equation 21 into Equation 12gives

∆ = ΦΛ−1ΦTF. (22)

With this explicit relationship between ∆ and F, theanalytic expression for ∂∆/∂F is

∂∆∂F

= ΦΛ−1ΦT, (23)

and the Local Sensitivity Matrix in Equation 5 canbe rewritten as

LSM =

I −∂F∂q

(ΦTΦ)−1ΦT

−ΦΛ−1ΦT I

.(24)

The remaining terms in the Global SensitivityEquation are ∂F/∂q in the LSM, along with ∂F/∂Xand ∂∆/∂X in the LSV. In this study, these termsare evaluated using finite difference approximations.However, clearly it is advisable to use analytic formsof these partial derivatives if such information is avail-able. Current CFD solvers such as CFL3D.ADII [18]and SENSE [19], as well as CSM solvers includingMSC/NASTRAN [20] and GENESIS [4], can providesome of the needed partial derivative terms.

4 Beam Example

A simple example problem is shown below to demon-strate the GSE/modal analysis method. Consider acantilever beam of square cross section as shown inFigure 2. The vertical force on the tip of the beamacts in the positive z-direction with a magnitude thatdepends on the amount of deflection. The magnitudeof the load, F , in units of pounds, is

F = 900 lb − (2000 lb/ft) δ. (25)

Using basic mechanics of materials, the vertical de-flection at the tip of the beam is

δ =FL3

3EI, (26)

where the length of the beam, L, is 15 ft, Young’smodulus, E, is 2.3 × 109 lb/ft2 (for titanium alloyTi-6Al-4V), and the area moment of inertia, I, is0.00521 ft4 (h = w = 0.5 ft). The solution for F andδ to this set of coupled equations is F = 757.725 lband δ = 0.071137 ft.

4.1 Exact Gradients

The total derivatives of F and δ with respect to an in-dependent variable, e.g., beam length, may be foundanalytically as

dF

dL=−1.2× 107EIL2

(3EI + 2000L3)2≈ −23.9567 lb/ft, (27)

5National Aeronautics and Space Administration

and

dδ

dL=

8100EIL2

(3EI + 2000L3)2≈ 0.011978 lb/ft. (28)

The exact gradients also may be found using the GSEformulation. The GSE for this coupled system is I −∂F

∂δ

− ∂δ∂F

I

dF

dLdδ

dL

=

∂F

∂L∂δ

∂L

. (29)

Substitute the following partial derivatives into theGSE

∂F

∂δ= −2000, (30)

∂δ

∂F=

L3

3EI, (31)

∂F

∂L= 0, (32)

and,∂δ

∂L=FL2

EI. (33)

Solving for the unknown Total Sensitivity Vectorterms gives

dF

dL=−6000FL2

3EI + 2000L3≈ −23.9567 lb/ft, (34)

and,

dδ

dL=

3FL2

3EI + 2000L3≈ 0.011978 lb/ft, (35)

which, as expected, are identical to the total deriva-tive values computed explicitly.

4.2 Approximate Gradients UsingGSE/Modal Analysis

Using the GSE/modal analysis approach described inSection 3, a four node, three element finite elementmodel was created for the cantilever beam using thecommercial CSM code GENESIS [4]. In this model,node points 1-4 are placed at 0.0, 5.0, 10.0 and 15.0ft, respectively, with the applied load at node #4.The first three natural frequencies and mode shapesare listed in Table 1.

The partial derivatives in the GSE were calculatedusing nodes 2-4, since these were free to deflect. Notethat the subscripts used below indicate node number.The partial derivatives of force with respect to deflec-tion are

∂Fi∂δj

= 0, for i = 2, . . . , 4 and j = 2, 3 , (36)

and∂F4

∂δ4= −2000. (37)

The remaining partial derivatives in the GSE are

∂δ

∂F= ΦΛ−1Φ, (38)

∂F2−4

∂L= {0, 0, 0}, (39)

and,

∂δ2−4

∂L= {0.00212893, 0.007618, 0.0143702}. (40)

Note that the term ∂δ/∂L was computed using a first-order, forward-step, finite difference approximationwith a step size of 1.0% (∆L = 0.15).

Solving the GSE for the terms in the Total Sensi-tivity Vector yields

dF4

dL≈ −24.2028 lb/ft, (41)

anddδ4dL≈ 0.012101 lb/ft, (42)

These estimates for the total derivatives dF/dL anddδ/dL agree to within 3.0 percent of the exact valuesfor the total derivatives.

5 Aircraft Analysis Example

The GSE/modal analysis method is applied to an air-craft analysis and design problem which involves com-putationally expensive CFD and CSM codes. Highfidelity static aeroelastic analysis is performed for asupersonic transport aircraft (see Figure 3) at Mach2.4, 1.0g, cruise conditions, at an angle-of-attack, α,of 3.5◦. The application of the GSE/modal analysismethods enables the efficient evaluation of gradientsof the aerodynamic lift, drag, and pitching momentcoefficients (CL, CD, and CMo) with respect to anyof the design parameters. These gradients take intoaccount the aero-structural interaction in the aeroe-lastic system.

5.1 Aircraft Parametric Model

A parametric wing/fuselage model of a supersonictransport was developed for this study and is detailedin Reference [21]. This model contains 64 parameterswhich define the outer mold line (OML) of the wing(i.e., planform and shape parameters). Six of the pa-rameters which define the wing planform are shownin Figure 4. Other parameters define the thickness,

6National Aeronautics and Space Administration

camber, and dihedral at various spanwise locationson the wing.

In addition to the 64 OML parameters, there are40 parameters that define the structural model, in-cluding the thickness of the wing skin panels and thesizes of various rib and spar structural elements.

5.2 Aircraft CFD Model

Surface and volume grids for CFD analysis are gen-erated based on the OML parameters using the codeCSCMDO (Coordinate and Sensitivity Calculator forMulti-disciplinary Design Optimization) [22]. Theoutput from CSCMDO is a structured grid with aC-O discretization scheme and dimensions of 121 ×41× 61, in the streamwise, circumferential, and sur-face normal directions, respectively. The volume gridis divided into two blocks with the interface splittingthe wing into upper and lower surfaces. There areapproximately 300,000 grid points in the model (seeFigure 5).

The CFD code CFL3D [3] was provided for thisstudy by the Aerodynamics and Acoustics MethodsBranch at NASA Langley Research Center. CFL3Dis a time-dependent, Reynolds-averaged, thin-layerNavier-Stokes flow solver for use with two- or three-dimensional structured grids. Both mesh sequencingand multigrid techniques are available in CFL3D forconvergence acceleration. In this study, CFL3D isused to resolve the inviscid, supersonic flow aroundthe aircraft configuration. Nominal cruise conditionsare Mach 2.4, 1.0g load factor, 3.5◦ angle-of-attack(with respect to the fuselage centerline), and an al-titude of 63,000 ft. An initial flow solution withCFL3D starting from a uniform flow field requiresapproximately 60 CPU minutes on an SGI work-station with a 250 MHz, IP27 R10000 processor.Subsequent analyses require approximately 30 CPUminutes through the use of the restart capability inCFL3D.

5.3 Aircraft CSM Model

A finite element model of the aircraft structure isgenerated using the 64 OML parameters and the 40structural parameters with the aid of software de-veloped by Balabanov [23] for related research in-volving supersonic transport configurations. Thewing/fuselage model of the aircraft is comprised of afixed number and topology of spar and rib elements,along with wing skin elements (Figure 6). The layoutof the structural elements is based on the OML pa-rameters, whereas the size (e.g., thickness and area)

of the finite elements is specified through the 40 struc-tural parameters.

The finite element model of the supersonic trans-port configuration has 226 nodes and 1130 elementswith a total of 1254 degrees-of-freedom. Note thatdue to structural symmetry only the starboard por-tion of the model is constructed. The FE model con-tains triangular membrane elements for the fuselageand wing skins, along with rod elements for the sparcap and rib caps. The spar and rib webs are modeledwith a combination of shear panels and rod elements.The material for all structural elements is titaniumalloy Ti-6Al-4V.

The CSM solver GENESIS [4] is used in this studyto perform linear structural analysis and modal anal-ysis of the aircraft model. The computational ex-pense for a single GENESIS analysis is approximatelyone CPU minute on an SGI workstation.

5.4 Loads Transfer and Structural De-formation

External aerodynamic loads for the GENESIS modelare obtained using the CFD-CSM loads transfer soft-ware code FASIT (Fluids and Structures InterfaceToolkit) developed by Smith et al [24, 25], and cur-rently maintained by the Air Force Research Labo-ratory at Wright-Patterson Air Force Base. FASITprovides a suite of interpolation schemes for trans-ferring the aerodynamic loads from the CFD modelto the CSM model. The thin plate spline method ofDuchon [26] was used in this study, as recommendedin the FASIT User’s Manual [25]. The loads trans-ferred from FASIT to the CSM surface grid are writ-ten in the NASTRAN Bulk Data format. Since thisNASTRAN format is compatible with the GENESISinput format, no translation was needed between FA-SIT and GENESIS. The computational expense ofrunning FASIT is negligible (less than 10 CPU sec-onds).

Displacement of the nodes in the CSM model andthe first 16 mode shapes are computed using GENE-SIS. A series of Mathematica [27] programs are usedto calculate the mode shape scale factors (see Equa-tion 13) needed to represent the node displacementsas a superposition of the 16 mode shapes. The ra-tionale for choosing 16 mode shapes is described inSection 5.6 below. Additional Mathematica programscalculate the changes in the 64 OML parameters dueto structural deformation. A new CFD grid is thengenerated from the updated list of 64 OML parame-ters.

7National Aeronautics and Space Administration

5.5 Aeroelastic Analysis

Aeroelastic analysis is performed by coupling theCFD, CSM, and interpolation codes as depicted inFigure 7. The box labeled “Geometry Manipulator”in Figure 7 contains the software used to generatethe CFD and CSM parametric models, including thevarious Mathematica programs described above. TheCFD, CSM, and interpolation software used in thestudy is loosely coupled using UNIX shell scripts andsome simple file manipulation codes. More detail onthe software coupling methods is provided in Refer-ence [21].

Due to the nonlinearity of the aero/structural in-teraction, the aeroelastic analysis involves an itera-tive procedure whereby the aerodynamic and struc-tural analyses are performed repeatedly until boththe aerodynamic loads and the structural deflectionsreach convergence. The convergence criterion usedin this loop is based on the z-direction displacementof the leading edge of the wing tip. If the differencein this z-displacement value between two successivepasses through the loop is less than 0.05 ft, then theaeroelastic analysis is considered to be converged.

To reduce the oscillations that occur in this under-damped system, a constant factor under-relaxationmethod is used to accelerate convergence. Thisunder-relaxation scheme follows the approach ofChipman et al [28] and Tzong et al [29]. A value of 0.7for the relaxation parameter is used for all aeroelas-tic calculations conducted in this study. The effect ofthis damping parameter is shown in the convergencehistory plot shown in Figure 8. This figure shows thedisplacement history of the wing tip versus the num-ber of passes through the aeroelastic loop in Figure 7.Typically, convergence is obtained in 6-10 iterations(2.7 CPU hours) when under-relaxation is used ascompared to 19 iterations (6.1 CPU hours) withoutthe relaxation method.

Results for an aeroelastic analysis at the nominalMach 2.4, 1.0g cruise conditions are shown in Fig-ures 9 and 10. In Figure 9 the final deformed wing isshown in comparison to the solid outline of the un-deformed wing. Note that the z-axis in this figurehas been scaled to exaggerate the wing deflection foreasier viewing. Figure 10 shows the aeroelastic de-formation of the aircraft wing at the wing tip, wingbreak (y = 33.7 ft), and wing root. At the 1.0gcruise conditions the z-direction (upward) wing tipdeflection is approximately 1.0 ft.

5.6 Mode Shape Selection Criteria

For this study 16 mode shapes were found to bemore than sufficient to represent the structural de-

formation of the aircraft finite element model. Thiswas demonstrated in a convergence study in whichthe number of modes used in the aeroelastic analy-sis procedure was varied from 1-16. Figure 11 showsthese results as compared to an aeroelastic analysisperformed using the exact structural deformations,i.e., without introducing the superposition of modeshapes. The correct final wing position was obtainedif at least four mode shapes were used in the linearsuperposition. All of the aeroelastic analysis casesshown in Figure 11 were started from the undeflectedaircraft shape with identical initial conditions.

6 Aircraft Sensitivity Analysis

Once a converged aeroelastic analysis was obtainedfor the aircraft model, the GSE method was used toestimate gradients of the coupled system with respectto perturbations in the independent design parame-ters. For this study, gradients of CL, CD, and CMo

were computed with respect to three different designparameters: (1) X = α, i.e., the angle-of-attack atcruise, (2) X = (t/c)break, the thickness-to-chord ra-tio at the wing leading edge break location (see Figure4), and (3) X = ΛLE, the inboard leading edge wingsweep angle.

Calculating gradients for this coupled aero-structural system involves computing the terms inthe Global Sensitivity Equation with the modifica-tion to the Local Sensitivity Matrix shown in Equa-tion 24. Solving the GSE yields the total derivativesdF/dX and d∆/dX, which estimate the change inthe aerodynamic loads and structural deflections dueto a perturbation in the parameter X . These gra-dient vectors are then used to compute the gradientsfor CL, CD, and CMo . This is possible since the aero-dynamic coefficients have a functional form identicalto that of the aerodynamic loads. That is,

CL = CL(X,∆), (43)

along with similar expressions for CD and CMo . Dif-ferentiating Equation 43 with respect to the param-eter X , yields

dCLdX

=∂CL∂X

+∂CL∂∆

d∆dX

. (44)

Following the approach shown in Equations 14-16yields

dCLdX

=∂CL∂X

+∂CL∂q

(ΦTΦ)−1ΦT d∆dX

. (45)

Thus, the total derivatives for CL, CD, and CMo

can be computed using the total derivative d∆/dX

8National Aeronautics and Space Administration

obtained from the GSE. These total derivatives forCL, CD, and CMo may then be used in Taylor Seriesapproximations of the form

CL(X + δX) ≈ CL(X) +dCLdX

δX. (46)

This Taylor Series approximation is an inexpensivetechnique for estimating the changes in CL, CD, andCMo due to small perturbations of X , in contrast toan expensive aeroelastic analysis for X + δX .

6.1 Interdisciplinary Coupling Terms

Computing the interdisciplinary coupling terms inEquation 24 involves the mode shape and eigenvaluedata from the finite element solver GENESIS. Someof this data is listed in Table 2 which includes the first16 natural frequencies and eigenvalues for the super-sonic transport model, along with the mode shapescale factors computed using Equation 13. Note thatthis eigenvalue analysis data is obtained from the un-deformed finite element model of the aircraft, i.e.,without application of any loads. The matrix of modeshapes, Φ, has 226 rows and 16 columns. That is,each row corresponds to one of the nodes in the fi-nite element model, and each column corresponds toa different mode shape.

The partial derivative term ∂F/∂q in the LocalSensitivity Matrix is calculated by perturbing eachmode shape coefficient by 20 percent, and then per-forming a CFD analysis for each of the new aircraftshapes. The 16 CFD analyses are performed usingcoarse grained parallel computing on an SGI Origin2000 computer. That is, the CFD analyses are ex-ecuted simultaneously, with each CFD analysis as-signed to a separate processor on the parallel com-puter. The SGI Origin 2000 computers at NASALangley Research Center and NASA Ames ResearchCenter were used in this study. The performance ofthis parallel computing approach is shown in Figure12, where “speedup” is defined as the ratio of thetotal serial computational time to the parallel execu-tion time. The discrepancy between ideal and actualspeedup is due primarily to file transfer operationsbetween the processors and disk storage. With 16 si-multaneous CFD analyses on the NASA Langley Ori-gin 2000, an aggregate parallel performance of 380MFLOPS was achieved. By using parallel comput-ing, 16 CFD analyses can be completed in about thesame wall-clock time as is needed for a single CFDanalysis.

6.2 Local Sensitivity Terms

The partial derivative terms in the Local SensitivityVector on the right side of Equation 5 must be com-puted for each design parameter X , before solvingthe GSE. Some CFD and CSM solvers may providethese partial derivatives based on analytic differentia-tion of the underlying CFD and CSM state equations.If analytic expressions for the partial derivatives areunavailable, finite difference approximations may beused to estimate the partial derivatives. The finitedifference approach is used in this study.

6.3 Global Sensitivity Calculations

6.3.1 Angle-of-Attack Sensitivity

For the angle-of-attack sensitivity calculations thepartial derivative term ∂F/∂X = ∂F/∂α was com-puted using finite differences. The aerodynamic loadswere obtained from the initial aeroelastic analysis re-sults at α = 3.5◦, and from one additional CFD anal-ysis at α = 4.0◦.

The nodal displacements from the CSM code arenot explicitly dependent on angle-of-attack. There-fore,

∂∆∂X

=∂∆∂α

= {0}. (47)

The GSE was then solved for the unknown to-tal derivatives dF/dα and d∆/dα. Gradients forCL, CD, and CMo were computed using Equation45, with the terms ∂CL/∂q, ∂CD/∂q, and ∂CMo/∂qcomputed using the results from the 16 CFD analy-ses for the mode shape perturbations. The gradientsof the aerodynamic coefficients are listed in Table 3.For comparison, the same gradients were computedusing finite differences, with an additional expensiveaeroelastic analysis at α = 4.0◦. Good agreement isobtained between the GSE-based gradients and thefinite difference gradients.

Taylor Series approximations for CL, CD, andCMo at α = 4.0◦ are listed in Table 4, along withthe exact values for CL, CD, and CMo computed inthe expensive aeroelastic analysis at α = 4.0◦. Goodagreement is obtained between the approximate andexact values of the aerodynamic coefficients.

6.3.2 t/c Ratio Sensitivity

For the thickness-to-chord ratio sensitivity calcula-tions the nominal value of the t/c ratio at the wingbreak was 2.36 percent, and the perturbed value ofthe t/c ratio was 2.60 percent. The partial deriva-tive terms ∂F/∂(t/c) and ∂∆/∂(t/c) were computed

9National Aeronautics and Space Administration

using finite difference approximations. The aerody-namic loads for t/c = 2.60 percent were obtained witha single additional CFD analysis. Similarly, the struc-tural deflections for t/c = 2.60 percent were obtainedfrom one additional CSM analysis.

Since the terms in the Local Sensitivity Matrixdo not change with respect to perturbations in thedesign variables there was no need to recompute themode shape partial derivative term ∂F/∂q. Thus, thepreviously computed LSM was reused when solvingthe GSE for this t/c sensitivity analysis case.

The total derivatives dF/d(t/c) and d∆/d(t/c)were computed by solving the GSE, and gradientsfor the aerodynamic coefficients were estimated us-ing Equation 45. These values are listed in Table 5and show good agreement with gradients computedusing a finite difference approximation based on theexpensive aeroelastic analysis results.

Taylor Series approximations for CL, CD, andCMo for a wing break t/c ratio of 2.60 percent arelisted in Table 6. These approximations are in goodagreement with aerodynamic coefficients obtainedfrom the expensive aeroelastic analysis with the wingbreak t/c ratio of 2.60 percent.

6.3.3 Wing Sweep Angle Sensitivity

The sensitivity of the aerodynamic coefficients to per-turbations in the inboard wing sweep angle were com-puted using the GSE method. The nominal wingsweep for the aircraft was 74.0◦, and a perturbationof +2.0◦ was used in this study. A single CFD anal-ysis for a wing sweep of 76.0◦ was used to create afinite difference approximation for ∂F/∂ΛLE. Simi-larly, the results from a single CSM analysis were usedto estimate ∂∆/∂ΛLE. As before, the LSM does notchange with respect to the aircraft design parametersso it is reused in this sensitivity analysis.

After solving the GSE for the total sensitivityterms dF/dΛLE and d∆/dΛLE, gradients of the aero-dynamic coefficients were estimated using Equation45. These gradients are listed in Table 7. Taylor Se-ries approximations for CL, CD, and CMo for a sweepangle of 76.0◦ are listed in Table 8. Both the gradi-ents and the Taylor Series approximations obtainedfrom the GSE method are in good agreement withvalues obtained from expensive aeroelastic analysesfor an aircraft with an inboard wing sweep angle of76.0◦.

7 Computational Savings withthe GSE Approach

The advantage of computing gradients with theGSE/modal analysis approach rather than a pure fi-nite difference approach is illustrated in Figure 13.This plot shows the number of CFD evaluationsneeded to compute gradients of the coupled aero-structural system, versus the number of independentdesign parameters. Here, the number of CFD evalu-ation is used as a cost metric since a CFD evaluationis about 30 times more expensive than any other por-tion of the aeroelastic analysis process.

In the pure finite difference approach, computingthe gradients for each new parameter requires about10 CFD analyses. This cost stems from the need toperform an aeroelastic analysis for each perturbationof a design parameter. Thus, the cost of the pure fi-nite difference approach is linear with respect to thenumber of independent variables and is given by theexpression

COSTFD = 10nv, (48)

where nv is the number of independent variables.In the GSE/modal analysis approach there is an

initial cost of 16 CFD evaluations, i.e., one CFDevaluation for each mode shape coefficient pertur-bation in the partial derivative term ∂F/∂q. How-ever, after this initial cost only one new CFD evalu-ation is needed for each design parameter. Thus, theGSE/modal analysis approach quickly becomes moreattractive, from a computational standpoint, if gradi-ents are needed for more than two design parameters.The cost of the GSE/modal analysis approach is

COSTGSE/Modal = nv + 16. (49)

Note that the GSE approach without modal anal-ysis would require a separate CFD evaluation for eachof the elements of the partial derivative term ∂F/∂∆.With the models used in this aeroelastic analysis, thecost of a GSE-alone approach would be

COSTGSE−alone = nv + 226, (50)

where 226 is the number of nodes in the finite ele-ment model. For larger, more realistic finite elementmodels the value 226 would grow to be O(103−104).

In theory, the application of coarse grained par-allel computing renders the wall-clock time identicalfor both the GSE/modal analysis method and theGSE-alone method. However, the burden of file man-agement and serial file input/output make the GSE-alone approach unattractive, even when a parallelcomputer having hundreds or thousands of proces-sors is available.

10National Aeronautics and Space Administration

Although not attempted in this study, theGSE/modal analysis approach offers significant op-portunities for multi-level parallelization. For exam-ple, fine grained parallel computing could be used toperform a CFD analysis on an aircraft model havingseveral million grid points. With such a large grid thecomputational domain may be subdivided into hun-dreds of zones, each of which is assigned to a separateprocessor on a parallel computer. The coarse grainedGSE/modal analysis approach would then provide anadditional level of parallelism. As an illustration ofthis multi-level paradigm, consider a large CFD gridhaving 100 zones, each of which executes simultane-ously on a separate processor of a parallel computer.The GSE/modal analysis approach would replicatethis 100-zone grid 16 times, i.e., once for each per-turbation of a mode shape coefficient. Such a strat-egy would efficiently utilize 1600 processors. Further-more, this multi-level strategy readily accommodatesincreases in the number of zones (fine grained scal-ability) and in the number of mode shapes (coarsegrained scalability).

8 Conclusions

A method based on the Global Sensitivity Equationsand modal analysis has been developed to calculategradients of aerodynamic force and moment coeffi-cients for an aeroelastic aircraft model. The GlobalSensitivity Equations capture the the aero-structuralcoupling in the supersonic transport aircraft modelexamined in this study. Modal analysis is employedto reduce the coupling bandwidth between the aero-dynamic and structural models. Coarse grained par-allel computing is used with the high fidelity com-putational fluid dynamics solver, CFL3D, for the ef-ficient calculation of partial derivative terms in theGSE/modal analysis approach.

A sensitivity analysis was performed for the su-personic transport aircraft model at nominal Mach2.4 cruise conditions. The GSE/modal analysis ap-proach was used to estimate the gradients of CL, CDand CMo with respect to variations in three of theaircraft design parameters. Good agreement was ob-tained between the GSE-based gradients and finitedifference-based gradients of the aerodynamic coeffi-cients. In addition, approximations for CL, CD andCMo were computed with the GSE/modal analysisapproach for small perturbations in the three air-craft design parameters. These GSE-based approx-imations were in good agreement with exact valuesfor CL, CD and CMo computed using the high fidelityaerodynamic and structural models.

A comparison of the computational expense forthe GSE/modal analysis method and for the brute-force finite difference method demonstrated the ad-vantage of using the GSE/modal analysis approach ifgradients are needed for more than two aircraft designparameters. Thus, the initial cost of the GSE/modalanalysis approach is quickly recovered in a realisticaircraft sensitivity analysis which may involve tensor hundreds of independent parameters.

Acknowledgments

Much appreciation is extended to members of theComputational Aerosciences Team, the Multidisci-plinary Design Optimization Branch, the Aerody-namic and Acoustic Methods Branch, and the Com-putational Structures Branch at NASA Langley Re-search Center for their assistance with this project.In addition, the author is grateful for the assistance ofLt. Joel Luker, USAF, who provided the FASIT soft-ware package, and to the staff at VMA Engineering,Inc., for their assistance in using the GENESIS finiteelement analysis and optimization software package.

This study was performed while the author held aNational Research Council Research Associateship atNASA Langley. Funding for this work was providedby the Computational Aerosciences Team at NASALangley, under the direction of Dr. Jaroslaw Sobieski(j.sobieski@larc.nasa.gov).

References

[1] Rubbert, P. “CFD and the Changing World ofAircraft Design,” AIAA Wright Brothers Lec-ture, Anaheim, CA (1994).

[2] Sobieszczanski–Sobieski, J. “Sensitivity of Com-plex, Internally Coupled Systems,” AIAA J.,28(1), 153–160 (1990).

[3] Krist, S. L., Biedron, R. T., and Rumsey,C. L. “CFL3D User’s Manual, Version 5.0,”NASA/TM-1998-208444 (1998).

[4] Vanderplaats, Miura and Associates, Inc., Col-orado Springs, CO. GENESIS User’s Manual,Version 4.0 (1997).

[5] Barthelemy, J.-F. M., Wrenn, G. A., Dovi, A. R.,Coen, P. G., and Hall, L. E. “Supersonic Trans-port Wing Minimum Weight Design Integrat-ing Aerodynamics and Structures,” J. Aircraft,31(2), 330–338 (1994).

11National Aeronautics and Space Administration

[6] Dovi, A. R., Wrenn, G. A., Barthelemy, J.-F. M., Coen, P. G., and Hall, L. E. “Multi-disciplinary Design Integration Methodology fora Supersonic Transport Aircraft,” J. Aircraft,32(2), 290–296 (1995).

[7] Kapania, R. K. and Eldred, L. B. “SensitivityAnalysis of Wing Aeroelastic Response,” J. Air-craft, 30(4), 496–504 (1993).

[8] Eldred, L. B. and Kapania, R. K. “SensitivityAnalysis of Aeroelastic Response of a Wing Us-ing Piecewise Pressure Representation,” J. Air-craft, 33(4), 803–807 (1996).

[9] Giles, G. L. “Equivalent Plate Analysis of Air-craft Wing Box Structures with General Plan-form Geometry,” J. Aircraft, 23(11), 859–864(1986).

[10] Giles, G. L. “Further Generalization of anEquivalent Plate Representation for AircraftStructural Analysis,” J. Aircraft, 26(1), 67–74(1989).

[11] Friedmann, P. P. “Renaissance of Aeroelastic-ity and Its Future,” J. Aircraft, 36(1), 105–121(1999).

[12] Livne, E. “Integrated Aeroservoelastic Opti-mization: Status and Directions,” J. Aircraft,36(1), 122–145 (1999).

[13] Karpel, M. “Reduced-Order Models for Inte-grated Aeroservoelastic Optimization,” J. Air-craft, 36(1), 146–155 (1999).

[14] Raveh, D. E. and Karpel, M. “StructuralOptimization of Flight Vehicles with CFD-Based Maneuver Loads,” AIAA Paper 98–4832, 7th AIAA/USAF/NASA/ISSMO Sympo-sium on Multidisciplinary Analysis and Opti-mization, St. Louis, MO (1998).

[15] Cohen, D. E. and Kapania, R. K. “Trim An-gle of Attack of Flexible Wings Using Non-linear Aerodynamics,” AIAA Paper 98–4833,7th AIAA/USAF/NASA/ISSMO Symposiumon Multidisciplinary Analysis and Optimization,St. Louis, MO (1998).

[16] Olds, J. “System Sensitivity Analysis Appliedto the Conceptual Design of a Dual-Fuel RocketSSTO,” Panama City Beach, FL, AIAA Paper94–4339 (Sept. 1994).

[17] Bathe, K.-J. Finite Element Procedures, pp.838–845, Prentice-Hall, Inc., Upper SaddleRiver, NJ (1996).

[18] Sherman, L., Taylor, A., Green, L., Newman,P., Hou, G., and Korivi, V. “First- and Second-Order Aerodynamic Sensitivity Derivatives viaAutomatic Differentiation with Incremental Iter-ative Methods,” J. Comp. Physics, 129, 307–331(1996).

[19] Godfrey, A. G. and Cliff, E. M. “Direct Cal-culation of Aerodynamic Force Derivatives: ASensitivity-Equation Approach,” AIAA Paper98–0393 (Jan. 1998).

[20] Anonymous. MSC/NASTRAN Version 70 Re-lease Guide, Los Angeles, CA (1997).

[21] Giunta, A. A. and Sobieszczanski-Sobieski, J.“Progress Toward Using Sensitivity Derivativesin a High-Fidelity Aeroelastic Analysis of aSupersonic Transport,” in Proceedings of the7th AIAA/USAF/NASA/ISSMO Symposium onMultidisciplinary Analysis and Design, pp. 441–453, St. Louis, MO, AIAA Paper 98–4763 (Sept.1998).

[22] Jones, W. T. and Samareh, J. A. “A Grid Gener-ation System for Multi-disciplinary Design Op-timization,” in Proceedings of the 12th AIAAComputational Fluid Dynamics Conference, pp.657–669, San Diego, CA, AIAA Paper 95-1689(June 1995).

[23] Balabanov, V. O. Development of Approxima-tions for HSCT Wing Bending Material WeightUsing Response Surface Methodology, Ph.D. the-sis, Virginia Polytechnic Institute and State Uni-versity, Blacksburg, VA (1997).

[24] Smith, M. J., Hodges, D., and Cesnik, C. AnEvaluation of Computational Algorithms to In-terface Between CFD and CSD Methodologies,Flight Dynamics Directorate, Wright Labora-tory, Wright-Patterson Air Force Base, OH, Re-port WL-TR-96-3055 (1995).

[25] Smith, M. J., Hodges, D., and Cesnik, C. Fluidsand Structures Interface Toolkit (FASIT), Ver-sion 1.0, Georgia Tech Research Institute, At-lanta, GA, Report GTRI A-9812-200 (1996).

[26] Duchon, J. Splines Minimizing Rotation-Invariant Semi-Norms in Sobolev Spaces, pp.85–100, Springer-Verlag, Berlin, eds. W.Schempp and K. Zeller (1977).

[27] Wolfram, S. The Mathematica Book, Cham-paign, IL, 3rd edition (1996).

12National Aeronautics and Space Administration

[28] Chipman, R., Waters, C., and MacKenzie,D. “Numerical Computation of AeroelasticallyCorrected Transonic Loads,” in Proceedings ofthe 20th AIAA/ASME/ASCE/AHS Structures,Structural Dynamics, and Materials Conference,pp. 178–184, St. Louis, MO, AIAA Paper 79–0766 (April 1979).

[29] Tzong, G., Chen, H. H., Chang, K. C., Wu,T., and Cebeci, T. “A General Method forCalculating Aero-Structure Interaction on Air-craft Configurations,” in Proceedings of the6th AIAA/USAF/NASA/ISSMO Symposium onMultidisciplinary Analysis and Optimization,pp. 14–24, Bellevue, WA, AIAA Paper 96–3982(Sept. 1996).

13National Aeronautics and Space Administration

Table 1: Natural frequencies, eigenvalues, and eigen-vectors for the four node beam finite element model.

Mode 1 Mode 2 Mode 3Frequency 5.86 Hz 36.77 Hz 103.69 HzEigenvalue 1354.0 53388.0 424454.6

EigenvectorComponents

Node 1 0.0 0.0 0.0Node 2 0.058156 -0.208162 0.262108Node 3 0.192160 -0.149674 -0.230501Node 4 0.351349 0.352866 0.351239

Table 2: Natural frequencies, eigenvalues, and scalefactors for the first 16 modes of the aircraft finiteelement model.

Mode Frequency Eigenvalue Scale(Hz) Factor

1 1.75 121.20 6.77409532 4.37 754.61 -1.90553023 7.05 1961.95 0.51207224 9.21 3350.76 0.05846075 10.94 4727.34 0.03033376 14.82 8675.36 -0.06757317 16.52 10768.64 -0.10064038 19.67 15277.35 0.05349379 24.42 23533.04 -0.046412510 25.33 25332.50 0.023342711 28.71 32539.54 -0.052898312 30.93 37765.35 -0.011346213 33.48 44241.15 0.026574514 34.47 46894.51 -0.003664615 37.59 55771.28 -0.006891516 41.37 67567.14 -0.0055217

Table 3: Gradients of CL, CD, and CMo due to per-turbations in cruise angle-of-attack (X = α). Thefinite difference values are computed from aeroelas-tic analyses at α = 3.5◦ (nominal condition) andα = 4.0◦. The gradients are in units of 1/deg.

GSE Finite DifferencedCL/dX 0.02432 0.02404dCD/dX 0.002486 0.002408dCMo/dX -0.02763 -0.02707

Table 4: Aerodynamic coefficients at an angle-of-attack of 4.0◦. The Taylor Series approximations usegradients from the GSE.

Taylor Series New Aeroelasticwith GSE Analysis

CL 0.090385 0.089027CD 0.006231 0.006129CMo -0.101000 -0.099099

Table 5: Gradients of CL, CD, and CMo due to per-turbations in t/c ratio at the wing break. The finitedifference values are computed from aeroelastic anal-yses at t/c = 2.36% (nominal value) and t/c = 2.60%.The gradients are nondimensional.

GSE Finite DifferencedCL/dX 6.317× 10−4 9.958× 10−4

dCD/dX 3.835× 10−4 4.033× 10−4

dCMo/dX −0.929× 10−3 −1.446× 10−3

Table 6: Aerodynamic coefficients for a new t/c ratioof 2.6 percent. The Taylor Series approximations usegradients from the GSE.

Taylor Series New Aeroelasticwith GSE Analysis

CL 0.077207 0.077295CD 0.005017 0.005022CMo -0.085789 -0.085913

14National Aeronautics and Space Administration

Table 7: Gradients of CL, CD, and CMo due to per-turbations in the leading edge sweep angle (X =ΛLE). The finite difference values are computed fromaeroelastic analyses at ΛLE = 74.0◦ (nominal condi-tion) and ΛLE = 76.0◦. The gradients are in units of1/deg.

GSE Finite DifferencedCL/dX -0.003385 -0.004111dCD/dX -0.000224 -0.000266dCMo/dX 0.003359 0.004355

Table 8: Aerodynamic coefficients for a leading edgewing sweep angle of 76.0◦. The Taylor Series approx-imations use gradients from the GSE.

Taylor Series New Aeroelasticwith GSE Analysis

CL 0.070285 0.068835CD 0.004478 0.004393CMo -0.078848 -0.076857

InputVariables

X

OutputF, ∆

CL, CD, CMstress

StructuralModel

AerodynamicModel

Aerodynamic LoadsF

∆Structural Deflections

Figure 1: Depiction of a coupled aero-structural sys-tem.

L

F(z)

z

x

z

yhw

h=w

Figure 2: Sample problem of a cantilever beam sub-ject to a tip load, where the load magnitude dependson the tip displacement.

Figure 3: Supersonic transport aircraft.

15National Aeronautics and Space Administration

InboardSweep

Outboard Sweep

Y-Wing Tip

Y-Break

ChordBreak

ChordTip

6 ft

X

YFu

sela

ge C

ente

rlin

e

Figure 4: Planform variables for the aircraft wing.

X

Y

Z

Figure 5: A view of one block in the aerodynamicmodel of the supersonic transport. This grid showsthe starboard wing, the x−z plane of symmetry, andthe exit plane.

Figure 6: The structural model of the aircraft show-ing the wing skin elements (port) and the rib/sparelements (starboard).

16National Aeronautics and Space Administration

Start

Converged?

ManipulatorGeometry

CSCMDO

CFL3D

FASIT

No

Yes

GENESIS

Stop

Figure 7: The arrangement of software used to per-form static aeroelastic analysis.

0 5 10 15Iteration Number

-5

-4

-3

-2

Z-V

alue

ofW

ing

Tip

Lea

ding

Edg

e(f

eet)

No RelaxationRelaxation Parameter = 0.7

Figure 8: Convergence history of the aeroelastic anal-ysis with and without relaxation.

X

Y

Z

ScaleFactors X:Y:Z = 1:1:2

Key:solid outline - undeformed wingmesh - final deformed wing

Figure 9: Orthographic view of the deformed wing(mesh) and the undeformed wing (solid outline).Note the X:Y:Z scaling of 1:1:2 used to show the wingdeformation.

50 100 150 200X (feet)-10

-5

0

5

10

Z(f

eet)

Wing Root

150 200X (feet)-8

-6

-4

-2

0

Z(f

eet)

Wing Leading Edge Break

202 210X (feet)-5

-4

-3

Z(f

eet)

Wing Tip

Figure 10: Airfoil sections at the (top-bottom) wingtip, wing break, and wing root for the initial un-deformed (dashed) and final deformed (solid) wingshapes.

17National Aeronautics and Space Administration

Iteration Number

Z-V

alue

ofW

ing

Tip

Lea

ding

Edg

e(f

eet)

0 1 2 3 4 5 6 7 8 9 10 11-6.0

-5.5

-5.0

-4.5

-4.0

-3.5

-3.0

-2.5

TrueStructural Deflections16 ModeShapes12 ModeShapes8 ModeShapes4 ModeShapes2 ModeShapes1 ModeShape

Figure 11: Convergence study showing the effect ofvarying the number of mode shapes used to approxi-mate the structural deformation of the aircraft finiteelement model.

Processors

Spe

edup

0 2 4 6 8 10 12 14 160

2

4

6

8

10

12

14

16

ActualIdeal

Figure 12: Speedup obtained using coarse grainedparallel execution of CFL3D on the NASA LangleySGI Origin 2000.

Number of Design Parameters

Num

ber

ofC

FD

Eva

luat

ions

1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120GSE / Modal AnalysisFiniteDifference

Figure 13: Cost of computing total sensitivityderivatives using finite differences (solid) and theGSE/modal analysis approach (dashed).

18National Aeronautics and Space Administration

REPORT DOCUMENTATION PAGE Form ApprovedOMB No. 0704–0188

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources,gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of thiscollection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson DavisHighway, Suite 1204, Arlington, VA 22202–4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704–0188), Washington, DC 20503.

NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)Prescribed by ANSI Std. Z39-18298-102

1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE

August 19993. REPORT TYPE AND DATES COVERED

Technical Memorandum

4. TITLE AND SUBTITLE

Sensitivity Analysis for Coupled Aero-structural Systems5. FUNDING NUMBERS

WU 509-10-11-04

6. AUTHOR(S)

Anthony A. Giunta

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research CenterHampton, VA 23681–2199

8. PERFORMING ORGANIZATIONREPORT NUMBER

L–17890

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space AdministrationWashington, DC 20546–0001

10. SPONSORING/MONITORINGAGENCY REPORT NUMBER

NASA/TM–1999–209367

11. SUPPLEMENTARY NOTES

12a. DISTRIBUTION/AVAILABILIT Y STATEMENT

Unclassified-UnlimitedSubject Category 05 Distribution: StandardAvailability: NASA CASI (301) 621–0390

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

A novel method has been developed for calculating gradients of aerodynamic force and moment coefficients for anaeroelastic aircraft model. This method uses the Global Sensitivity Equations (GSE) to account for theaero-structural coupling, and a reduced-order modal analysis approach to condense the coupling bandwidthbetween the aerodynamic and structural models. Parallel computing is applied to reduce the computationalexpense of the numerous high fidelity aerodynamic analyses needed for the coupled aero-structural system. Goodagreement is obtained between aerodynamic force and moment gradients computed with the GSE/modal analysisapproach and the same quantities computed using brute-force, computationally expensive, finite differenceapproximations. A comparison between the computational expense of the GSE/modal analysis method and a purefinite difference approach is presented. These results show that the GSE/modal analysis approach is the morecomputationally efficient technique if sensitivity analysis is to be performed for two or more aircraft designparameters.

14. SUBJECT TERMS

aeroelasticity, sensitivity analysis, modal analysis, aircraft design15. NUMBER OF PAGES

23

16. PRICE CODE

A03

17. SECURITY CLASSIFICATIONOF REPORT

Unclassified

18. SECURITY CLASSIFICATIONOF THIS PAGE

Unclassified

19. SECURITY CLASSIFICATIONOF ABSTRACT

Unclassified

20. LIMITATION OF ABSTRACT