[American Institute of Aeronautics and Astronautics 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization - St. Louis,MO,U.S.A. (02 September 1998 - 04

Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

AIAA-98-4783A98-39760A CFD/CSD Interaction Methodology for

Aircraft WingsManoj K. Bhardwaj * and Rakesh K. Kapania t

Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0203

Eric Reichenbach *The Boeing Company, St. Louis, MO 63166

Guru P. Guruswamy §NASA Ames Research Center, Moffett Field, CA 940356

AbstractWith advanced subsonic transports and military air-

craft operating in the transonic regime, it is becomingimportant to determine the effects of the coupling be-tween aerodynamic loads and elastic forces. Sinceaeroelastic effects can significantly impact the designof these aircraft, there is a strong need in the aerospaceindustry to predict these interactions computationally.Such an analysis in the transonic regime requires highfidelity computational fluid dynamics (CFD) analysistools, due to the nonlinear behavior of the aerodynam-ics in the transonic regime and also high fidelity com-putational structural dynamics (CSD) analysis tools.Also, there is a need to be able to use a wide varietyof CFD and CSD methods to predict aeroelastic ef-fects. Since source codes are not always available, it isnecessary to couple the CFD and CSD codes withoutalteration of the source codes. In this study, an aeroe-lastic coupling procedure is developed to determinethe static aeroelastic response of aircraft wings usingany CFD and CSD code with little code integration.The aeroelastic coupling procedure is demonstratedon an F/A-18 Stabilator using NASTD (an in-houseMcDonnell Douglas CFD code) and NASTRAN. Inaddition, the Aeroelastic Research Wing (ARW-2) isused for demonstration of the aeroelastic coupling pro-cedure by using ENSAERO (NASA Ames ResearchCenter CFD code) and a finite element wing-box code.The results obtained from the present study are com-pared with those available from an experimental studyconducted at NASA Langley Research Center and a

* Former Graduate Student, Member AIAA, Now LimitedTerm Staff, Sandia National Laboratories, P.O. Box 5800 MS0439, Albuquerque, NM 87185-0439.

t Professor, Associate Fellow AIAA.* Senior Project Engineer, Member AIAA§ Senior Research Scientist, Associate Fellow AIAA, Applied

Computational Aerodynamics Branch, MS 258-1Copyright © 1998 by Bhardwaj et al. Published by the An

of Aeronautics and Astronautics, Inc. with permission.erican Institute

study conducted at NASA Ames Research Center us-ing ENSAERO and modal superposition. The resultscompare well with experimental data.

IntroductionTraditionally, aircraft designers have viewed aeroe-

lastic effects as undesirable. To avoid aeroelasticphenomena, the stiffness of the wing was increasedby adding weight to the structure. Recently, therehas been an increased interest in taking advantageof aeroelastic effects for roll control, load alleviation,and drag reduction while reducing the wing weight, asin the Active Flexible Wing1'2 (AFW) and the Ac-tive Aeroelastic Wing3 (AAW) programs. In addition,the accurate prediction of wind tunnel model staticaeroelastic deformations is becoming increasingly im-portant for transonic testing of transport aircraft.4

Whether viewed as undesirable or desirable, it is be-coming more important to predict static aeroelasticbehavior of transport and fighter aircraft especially inthe the transonic regime.

Advanced CFD tools are necessary to capture thenonlinear behavior of the aerodynamics in the tran-sonic regime (shocks, vortices, separation). In thisregime, the nonlinear nature of the aerodynamicsmakes load prediction difficult. The accuracy of theloads on a wing depends on the accuracy of the shockwaves prediction.5 Coupling of high fidelity CFD andCSD tools to solve aeroelastic problems has receivedinterest only in the past few years. Huge computa-tional power is required to make the use of such toolsfeasible. Continuous improvements in computer speed,memory, and architecture have, however, made solvingthese computationally intensive problems more cost ef-fective.

Both uncoupled and coupled methods for solvingthese nonlinear systems of equations6 exist. Aeroelas-tic problems of aerospace vehicles are often dominatedby flow nonlinearities and at times by large structural

581AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS


deformations. Therefore, coupled approaches are nec-essary to solve such problems accurately.7

Coupled approaches for solving aeroelastic prob-lems are usually categorized in two ways: loosely orstrongly coupled. The loosely coupled approaches canbe integrated or modular. Integrated, loosely coupledmethods alter the source code of either the CSD orCFD analysis tool by including the coupling schemesin either code. Though the codes are integrated, theCFD and CSD equations are not being altered and aresolved independently. Modular, loosely coupled meth-ods do not integrate the coupling schemes into eitherthe CFD or CSD code. This allows the use of a varietyof CFD/CSD codes.

Strongly or fully (single domain) coupled approachesrequire the solution of the CFD and CSD equationssimultaneously which necessitates the reformulationof the equations of each discipline.8 The numericalmatrices associated with the structures are orders ofmagnitude stiffer than those associated with fluids.Thus, it is numerically inefficient or even impossibleto solve both systems using a monolithic numericalscheme.7 Recently, there have been renewed attemptsto solve both fluids and structures in a single computa-tional domain.9'10 However, they have been limited tosimple two-dimensional problems and have not provento be better than the loosely coupled approach.

Guruswamy and Yang6 demonstrated a loosely cou-pled approach to aeroelasticity. The fluids and struc-tures were modeled independently and exchangedboundary information to obtain aeroelastic solutions.The fluids were modeled using finite-difference basedtransonic small disturbance (TSD) equations. Thestructures were modeled using finite element equa-tions. The two disciplines were coupled to solveaeroelastic problems of two-dimensional airfoils. Thisloosely coupled or domain decomposition approachwas shown to be efficient and accurate. This approachhas been extended to three-dimensional problems andis incorporated into advanced aeroelastic codes asXTRAN3S,11 ATRAN3S,12 and CAP-TSD.13 Gu-ruswamy14'15 also demonstrated the same techniqueby modeling fluids with Euler/Navier-Stokes equationson moving grids. Matching the CFD grid displace-ments with the CSD or finite element model responsemaintains the accuracy of this loosely coupled ap-proach.

Several papers4' 16~25 have presented techniques forcalculating aeroelastic solutions using loosely coupledhigh fidelity CFD and CSD methods. Often the cou-pling is integrated, allowing the two disciplines to ex-change information at the boundaries in an efficientmanner. However, this usually requires either the CFDor CSD code to be rewritten to add for the communi-cation between the two separate disciplines.

The CSD analysis, in some of this work,22'23 is per-formed using a modal analysis approach; this makesthe exchange of boundary information easier. Theloads need only to be calculated on the CFD gridpoints. As a direct result, not many algorithms havebeen presented for accurate transformation of pres-sures on the CFD grid to loads on the CSD nodes.Future work in analyzing complex wing-body struc-tures will require the use of detailed finite elementmodels and the use of direct finite element equations,not modal analyses. Therefore, an accurate load trans-formation scheme is needed.

Macmurdy et al.26 obtained a static aeroelastic solu-tion of an intermediate complexity wing (ICW) usingEuler flow equations (ENSAERO) coupled with finiteelement equations. The finite element wing-box wasmodeled using a Wright Patterson Air Force Basestructural analysis code, ANALYZE.27 Static aeroe-lastic solutions were obtained by loosely coupling EN-SAERO with ANALYZE in a modular manner. Thetwist and leading edge plunge are obtained from thestructural response which are then applied to the CFDgrid. The loads are calculated at the CFD grid pointsand are transferred to the CSD nodes using variousschemes. The schemes do not transfer the loads accu-rately since some of the information is extrapolated.

Tzong et al.28 presented a general method for cal-culating aero-structure interactions. An interfacemethod based on finite element technology was usedto exchange information between the CFD and CSDcodes. The CFD analysis was performed using OVER-FLOW29 and a Douglas panel code.30 The CSDresponse was calculated using a McDonnell DouglasCorporation finite element code. The interface methodmaps each CFD grid point to a host finite element.The displacements and loads are transferred betweenthe CFD grid point and the CSD nodes using the shapefunctions of the host finite element. A disadvantage ofthis approach is that the shape functions of the fi-nite elements in the model might not be available tothe user. In addition, the normal degrees of freedommight not be contained in the host finite element totransfer the boundary information accurately. Thisinterface method has been integrated into the finiteelement code at Douglas. This again restricts a user'sability to use a CFD or a CSD code of her/his choice.

Two ways of transferring the pressures on the CFDgrid to the CSD nodes are possible.28 In the firstmethod, pressures on the CFD grid are interpolatedonto the CSD model and are integrated to obtain theforces on the CSD nodes. Tzong et al.28 state thatthe inconsistency between the CFD and CSD modelsmakes this conversion improper. The pressures can beconverted to the CSD model, but the loads may notbe integrated accurately since information about the



true surface areas is often not available from the CSDmodel. In the second method, the forces at the CFDgrid points are calculated by using the CFD grid in-formation and then are transferred to the CSD nodes.This transfer calculates loads on the CSD nodes moreaccurately and is easier to implement. This is themethod chosen in this study.

In the loosely coupled modular approach, boundaryinformation between the CFD and CSD codes is ex-changed through the codes' native files. Native filesare the files required by the code as input and the filesto which the output is written. The forces are obtainedfrom the output of pressures from the CFD code. Apressure mapping algorithm transfers the pressuresfrom the CFD grid to the CSD nodes. The CSD codecalculates the response of the structure. The result-ing output, the displacements, are interpolated to theCFD grid using a displacement mapping algorithm.The CFD code calculates the flow field about this newCFD grid. The procedure is repeated in an iterativemanner until a specified convergence criterion is met.Therefore, two mappings are necessary to obtain staticaeroelastic solutions in a loosely coupled and modularmanner. The mappings used are described later.

The mapping of the displacements from the CSDnodes to the CFD grid requires an interpolationscheme. Smith et al.31 presented a review of themethodologies used to do this mapping in interfacingCFD/CSD codes. A significant literature review andan industry/government survey narrowed the searchto six schemes: (i) the Infinite-plate spline; (ii) Finite-plate spline; (iii) Multiquadric-Biharmonic; (iv) Thin-Plate Spline; (v) Inverse Isoparametric Mapping; and(vi) Non-Uniform B-Splines. These methods wereanalyzed by a series of mathematical test cases andselected applications. The infinite-plate spline (IPS)method, commonly referred to as the Harder and Des-marais surface spline,32 was chosen to interpolate dis-placements from the CSD nodes to the CFD grid.The IPS method provides reasonable results withouthaving the requirement that the input grid be a rect-angular array. In addition, its ease of use and imple-mentation make it one of the better methods as canbe seen by its use in several codes. More details ofthe other methods can be found in an excellent reviewgiven in Ref. [31].

Several researchers have investigated either artifi-cial structural damping33 or under-relaxation tech-niques21'28 to converge the solution faster and/or tokeep it stable. In this paper, an initial rigid steadystate solution of the lifting surface is used to decreasethe time to calculate a static aeroelastic solution as op-posed to starting impulsively from free stream bound-ary conditions. In addition, the CFD solution need notbe fully converged after each grid deformation before

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exchanging information with the structural analysiscode. This has a similar effect as an underrelaxationscheme and has been used effectively as seen in Ref.[34].

Static aeroelastic solutions are obtained in this pa-per assuming a linear structural model. The loadsobtained from the pressures are applied to the orig-inal finite element model to obtain the displacements.The finite element model is not regenerated using thedisplacements in the previous iteration although thiscapability is not difficult to include in the aeroelasticcoupling procedure.

In this work, an aeroelastic coupling procedure ispresented by which static aeroelastic solutions of air-craft wings are obtained. The aeroelastic coupling pro-cedure requires only the grid point coordinates of theCFD and CSD grids to create the interface mappings.To demonstrate this procedure, a static aeroelastic so-lution of the F/A-18 Stabilator is calculated by usingEuler flow equations as available in NASTD (an in-house McDonnell Douglas Aerospace - East code) andfinite element equations as available in the structuralanalysis tool NASTRAN.35 The solution is obtainedin the highly nonlinear transonic range at Mach 0.95,and at one degree angle of attack. Next, two differ-ent CFD and CSD codes are used to obtain a staticaeroelastic solution for the Aeroelastic Research Wing(ARW-2). Navier-Stokes equations, as available inENSAERO,36 are coupled with a finite element wing-box code to obtain a static aeroelastic solution in thetransonic regime at Mach 0.85, at one and two degreesangle of attack. The flexible solutions are also com-pared with experimental results, and good agreementis obtained. The examples use direct finite elementequations, not modal analysis equations, to obtain thestructural response. The advantage of the proposedaeroelastic coupling procedure is thus shown by usingtwo different sets of CFD/CSD codes to perform staticaeroelastic analyses.

Due to space restrictions, all of the details and fig-ures are not shown as these can be found in Ref. [37].

Aeroelastic Coupling ProcedureA static aeroelastic solution of a wing is obtained

using the following aeroelastic coupling procedure:

1 Obtain an intermediate or rigid steady state CFDsolution for the wing

2 Calculate the pressures at the CFD grid points onthe aerodynamic surface

3 Map pressures at the CFD grid points to forceson the CSD nodes

4 Obtain the structural response of the wing



5 Map displacements at the CSD nodes to the dis-placements on the CFD grid points of the aerody-namic surface

6 Deform the entire CFD grid

7 Repeat steps 1-6 until preselected convergence cri-teria is met

The above steps, also shown in Fig. 1, are repeatedin an iterative manner until a converged solution isobtained. This fixed-point iteration scheme is usedfor its simplicity and for its application to obtainingloosely coupled CFD/CSD solutions. To use a methodwhich converges faster, like Newton's method,38 largeamounts of computational time would have to be spentin calculating sensitivities of pressure with respectto deformations. Direct finite element analysis, notmodal analysis, determines the structural response;thus the number of unknowns makes this process ineffi-cient. Therefore Newton's method is computationallytoo expensive to make this approach feasible.

In obtaining the static aeroelastic solution of a wing,either a fully converged rigid steady state solutionis obtained or an intermediate solution is obtainedbefore initiating the aeroelastic coupling procedure.In this paper, both methods were used. However,the aeroelastic solution converges faster if the aeroe-lastic coupling is started with the CFD rigid steadystate solution as opposed to starting impulsively fromfree stream boundary conditions. Alternatively, intro-ducing the structural coupling into the CFD solutionprocess from the start, before obtaining even an inter-mediately converged CFD solution on the rigid wing,can cause serious problems leading to the possibilityof a divergent solution.

The aerodynamic pressures are calculated using anyCFD code. The forces are calculated at each CFD gridpoint using the pressures and calculated areas. Theforces at the CFD grid points of the wing are thenmapped onto the CSD nodes. To do this, each CFDgrid point is mapped to a structural triangle. UsingFig. 2, step 1 shows the area used to obtain the forceat CFD grid point i, j as indicated by the dotted box.Here it is assumed that the CFD grid is denser thanthe CSD grid. The four closest structural nodes areobtained using the upper or lower surface structuralgrid depending on which surface the CFD grid pointis located. All possible triangles are formed using thefour CSD nodes. Triangles that do not contain theCFD point as an interior point are eliminated. Thearea coordinates of the CFD point i,j with respectto the structural triangle determine whether the pointis an interior point. If the area coordinates sum to1.0 ± 0.01, the CFD grid point is interior to the struc-tural triangle. From Fig. 2, there are four trianglesand triangles 1 and 2 do not contain the CFD grid

point and therefore are eliminated. Of the remainingtriangles, the distance, Vi, between the CFD grid pointi,j and each CSD node of triangle m is calculated as,

/(-rm — v }\j\xp xa)

For i = 1,3

—iiVa)

(1)

where (xa, ya,za) are the coordinates of the CFD gridpoint i, j and (x™, y™, z™) are the coordinates of CSDnode p of triangle m. The largest vertex distance foreach triangle m is obtained as

it,V2 (2)

where max is the maximum of the values v™, v™, v™.The triangle with the smallest value of wmax is the"smallest" structural triangle for CFD point i,j; thusthe forces at CFD grid point i, j are mapped to thistriangle.

Four CSD nodes were used to show this mapping al-gorithm, but this number, nci0, can be increased to anynumber depending on the density of the finite elementgrid. It is possible not to find a structural triangle fora CFD grid point if this number is too low. For ex-ample, if all four nodes in the previous example are tothe same side of the CFD grid point, then none of theformed triangles would contain the CFD grid point.In this paper, nci0 = 20 is used. This number wasvalidated by graphically viewing the mapping of theCFD grid points to the structural triangles for variouschoices of nc/0.

The structural response of the system is calculatedusing the forces obtained above on the CSD nodes.The following system of equations are solved,

= {/.} (3)

where {us} are the displacements at the CSD nodes,and {K} is the stiffness matrix of the CSD or finiteelement model. This can be solved by any structuralanalysis tool to obtain the displacements, {us}, on theCSD nodes.

The displacements, {ua}, on the aerodynamic por-tion of the CFD grid are calculated using the structuralresponse, {us}. A surface spline32 is used to interpo-late the displacements from the CSD nodes to the CFDgrid points. Reasonable accuracy39 is obtained as longas extrapolation is avoided. The surface spline equa-tion is derived from the governing equations of a plateof infinite extent that deforms in bending only. Thesurface spline system of equations becomes

where [As] is dependent on the coordinates of thespline points, {c} is the vector of unknown coefficients



of the surface spline equation, and {uspi} are the dis-placements at the spline points. In the preprocessingstage, some of the structural nodes and CFD farfieldgrid points are chosen as the spline points. Matrix [As]is formed using the coordinates of the chosen splinepoints. The displacements for the CFD farfield splinepoints are fixed at zero while the remainder of thespline point displacements, {uspi}s, are extracted fromthe structural response, {us}, as

{uspl}* = [E}{us} (5)

Here [E], composed of zeroes and ones, is a nspi x nmax

matrix where nspi is the number of structural splinepoints and nmax is the number of CSD nodes. Matrix[As] is decomposed using an LU factorization. Thecoefficients of the surface spline, {c}, are solved byforward and backward substitutions.

The displacements at the CFD surface grid points,{ua}, are calculated by using the coordinates of theCFD grid points within the surface spline equation.The exterior CFD grid is deformed using the CFD sur-face grid displacements, {ua}, but the deformation ofthe exterior CFD grid depends on the aerodynamicanalysis code being used. Two separate codes for fluidanalysis are used in this paper. One of the codes,ENSAERO,36 has a built in scheme to move the gridonce the CFD surface grid is deformed. The other,NASTD,40 does not have such a scheme. So a simplegrid moving scheme was applied when NASTD wasused.

The aeroelastic coupling procedure is demonstratedby calculating a flexible solution of an F/A-18 Stabi-lator and the Aeroelastic Research Wing (ARW-2).

Test CasesNext, the details of the static aeroelastic analyses

of the F/A-18 Stabilator and the Aeroelastic ResearchWing (ARW-2) are presented and compared with ex-perimental and other available computational data.

F/A-18 Stabilator: CFD and CSD ModelingFor the F/A-18 Stabilator, Euler flow equations,

as available in NASTD, are used to demonstrate theaeroelastic coupling procedure. The analysis is per-formed at sea-level, one degree angle of attack, and atMach 0.95. The CFD grid of the F/A-18 Stabilator isapproximately 800,000 grid points. The CFD surfacegrid of the F/A-18 Stabilator only is shown in Fig. 3.

A general purpose finite element program, NAS-TRAN, is used in analyzing the structure. The stiff-ness matrix produced by NASTRAN is used to ob-tain the displacements for given aerodynamic loads.During the linear aeroelastic analysis procedure, NAS-TRAN is not directly involved, since the stiffness ma-trix does not change during the procedure. The finite

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element model of the F/A-18 Stabilator consists of2000 nodes and 12000 d.o.f.

F/A-18 Stabilator: Aeroelastic Coupling ProcedureThe first step in the aeroelastic coupling procedure

is obtaining the CFD solution for the lifting surface.For this case, the rigid steady state solution is obtainedbefore the aeroelastic analysis cycle begins. Once theCFD solution is obtained, the forces on the CSD gridare calculated using the preprocessed mapping. Themapping of the CFD points to the structural trian-gles as previously discussed is shown in Fig. 4. Herethe mapped structural triangle for each CFD point ispresented. The structural triangle does not refer toan actual structural element. So shape functions arenot necessary, and if linear displacements are assumedover each element, then energy is conserved during themapping. The actual structure of the wing does notextend to the wing root, but this was done to avoidcomputational problems with the CFD code NASTD.

S.ince NASTD does not have a built-in grid genera-tor the exterior CFD grid has to be deformed using thedeflections on the wing surface. There are two ways ofdoing this: (i) regenerate a completely new CFD exte-rior grid or (ii) deform the existing CFD grid. Often,the existing CFD grid is deformed. These methods re-distribute points along grid lines that are in the radialdirection normal to the surface by displacing them avalue equal to the surface value times a some spacingparameter. Guruswamy7 used a normalized arc lengthas the spacing parameter. Batina23 represented the ex-terior grid using a spring network, where the stiffnessof the spring is inversely proportional to the lengthof the side of the CFD cell. This prevents the CFDgrid from losing its quality. In this paper, only verti-cal displacements are taken into account. Therefore,a simple cosine spacing function is used to deform theexterior grid normal to the wing surface.

Assuming the CFD grid for this case has the i in-dex varying circumferentially around the wing section,the j index varying in the normal direction, and thek index varying along the span. Once the surface de-flections are known at j =1, a cosine spacing functionis used to deform the exterior grid at each spanwise(k = constant face) location. The spacing function,dependent on the location along the normal direction, i.e. the j index, is defined as

oei cos- j -1) For j = (6)

where jmax is the maximum number of points extend-ing in the radial direction normal to the surface. Usingthe displacements at the CFD surface grid, i.e. j = 1,the exterior grid is deformed at each k = constant sur-face by multiplying the surface displacement by thespacing parameter, as. The new vertical coordinate



at some j section is,new _ rigid , j J=l

zi,k ~ zi,k + asui,k (7)

Only the vertical displacements are taken into account.Note that the zr^ coordinates are used and not the zcoordinates from the previous iteration. To avoid over-lapping of the CFD grid, a minimum spacing criteria,amin is chosen as,

__ f ( 1 2\ (Q\amin — Js * (O^s as) \°)

where a\ = 1 and fs is subjectively chosen to preventloss of grid quality. For this analysis, fs is chosen inthe range of 1-2. Parameter a^ depends on jmax. Thisassumes the grid is stretching smoothly away from thesurface. If the spacing between two consecutive pointsis smaller than amin, z^ — z\k < amin, then as isset to one for that entire j section. In this example,all the points within the j = 26 boundary are movedthe same amount as the aerodynamic surface at j =1. All the points exterior to j = 26, i.e. 26 < j <jmax, are moved using Eqn. 6. This enforces that theouter boundaries of the CFD grid do not move. Thisis done to take advantage of distributed computingcapabilities in the future where the grid can be brokeninto many zones. A Hewlett-Packard workstation wasused to perform the calculations.

Aeroelastic Research Wing (ARW-2): CFD andCSD Modeling

The Aeroelastic Research Wing (ARW-2), a super-critical airfoil with aspect ratio of 10.3 and a lead-ing edge sweep of 28.8°, is used to validate the forceand displacement mappings. The strong conservationlaw form of the thin-layer Reynolds-averaged Navier-Stokes equations are used to calculate the fluid flowabout the ARW-2 wing as available in ENSAERO. Thestructural response is calculated by the finite elementwing-box code. The aeroelastic solution is obtained atMach 0.85, a — 1 and 2 degrees, q = 200 psf, and iscompared with experimental results. In addition, theresults are also compared with another similar work,which uses modal analysis as opposed to the directfinite element analysis used in this study.

The CFD code uses a C-H type grid with a gridsize of 171 (circumferentially) x 51 (spanwise) x 45(normal) points. The wing CFD grid is shown in Fig.5. The wing has a grid size of 139 (circumferentially)x 39 (spanwise) points. The fluid flow equations aresolved for Mach 0.85, an angle of attack, a, of 1 and 2degrees, and a free stream dynamic pressure, q, of 200psf.

The finite element wing-box model of the ARW-2wing uses Airman's triangular elements in conjunctionwith axial bars to represent the wing's spars, ribs, andskins. Figure 6 shows the spars and ribs of the ARW-2

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wing. The wing is discretized into a 11 x 13 mesh, 312nodes, 1872 d.o.f. The ARW-2 wing consists of com-posite fiberglass skins, but the finite element wing-boxcode does not yet have composite capability. An equiv-alent isotropic wing is created by matching bendingand twisting properties with the ARW-2 wing madeof composite fiberglass skins. Details of the compositeskin ARW-2 wing finite element model can be obtainedin Ref. [41]. Details of the isotropic equivalent of thecomposite skin ARW-2 are available in Ref. [37].

Aeroelastic Research Wing (ARW-2): AeroelasticCoupling Procedure

The aeroelastic coupling procedure is more inte-grated using ENSAERO and the finite element wing-box code since the source codes were available. If onlythe vertical displacements are taken into account forthe F/A-18 Stabilator and ARW-2 wing, the CFD gridcan become distorted. Therefore, ENSAERO uses thevertical deflections to calculate a rigid body rotationand a deflection so as to avoid this problem when deal-ing with the ARW-2 wing. This was also done forthe F/A-18 Stabiltor using NASTD. This means thatchordwise rigidity is assumed for the wing. This is agood approximation for the ARW-2 wing. Byrdsong etal. measured experimental data for the flexible ARW-2, and state that the ARW-2 has sufficient chordwiserigidity. A Cray-90 was used to obtain the solution forthis case.

ResultsF/A-18 Stabilator

The convergence of the aeroelastic solution for theF/A-18 Stabilator is monitored in several ways. The£2 norm of the residuals of the continuity, momen-tum, and energy equations are examined. The loadson the wing surface are also examined. Satisfying thesetwo criteria helps assure that the CFD solution is con-verged. In the CSD solution, the displacements atvarious locations are examined to assure convergence.One of the convergence checks for the structural anal-ysis is shown in Fig. 7, where the deflection of thewing tip of the F/A-18 Stabilator is plotted after eachcycle of the aeroelastic coupling procedure. The struc-tural solution converges very quickly. This is becausethe rigid steady state solution was obtained prior toinitiating the aeroelastic coupling procedure. In addi-tion, the aeroelastic effect is not significant; the largestdisplacement on the F/A-18 Stabilator is 1.55 inches.

The final converged flexible F/A-18 Stabilator isshown in Fig. 8 with the initial undeformed rigidF/A-18 Stabilator. The largest deflection occurs atthe trailing edge tip of the F/A-18 Stabilator, approx-imately 1.5 inches. From a previous analytical study(performed at McDonnell Douglas) using CAP-TSD,a transonic small disturbance CFD code, coupled with



modal analysis structures, the largest deflection of theF/A-18 Stabilator was calculated to be 1.56 inches.The deflection using NASTD coupled with NASTRANis also about 1.55 inches. The present results do com-pare well with existing data. Unfortunately, moredetails of the comparisons are not available.

Next, the Aeroelastic Research Wing (ARW-2), isused to determine the accuracy of the entire aeroelasticcoupling procedure, since experimental static aeroelas-tic data exist for it.

Rigid Steady State SolutionThe first step is to obtain the rigid steady state so-

lution for the two cases, i.e. the one and two degreeangle of attack cases. Intermediate rigid steady statesolutions were obtained by using Navier-Stokes flowequations as available in ENS AERO. Convergence ofthe rigid steady state solutions is checked by examiningthe L2 norm of the residuals of the fluids equations.The L2 norm is not sufficiently reduced, but this isdone since a completely converged solution is not nec-essary to start the aeroelastic coupling. This study andthe Farhangnia et a/.42 study start with the same rigidsteady state solution of the ARW-2 using ENSAERO.Farhangnia et al. use the first five mode shapes as op-posed to the direct finite element equations used in thework. Since final results are compared later, the start-ing rigid steady state solutions are also compared byexamining Fig. 9. The figure shows the Cp variationat the 70.7% semi-span location. Because both stud-ies used ENSAERO to obtain the rigid steady statesolution, the results match as expected.

After initiating the aeroelastic coupling procedure,the CFD solution convergence is checked by examin-ing the 1/2 norm of the residual of the fluids equations,while the CSD solution is checked by examing dis-placements at various locations on the wing structure.Flexible steady state solutions are obtained at a = 1and 2 deg. Cp variation at the 70.7% semi-span lo-cation, for the flexible ARW-2 wing, is shown in Fig.10 and plotted with experimental data from Ref. [42].The Cp variation compares well with the experimentaldata. The shock location for the experimental data is5% of chord aft of the computational data.

Due to the flexibility, the shock location has movedaft in both the one and two degree angle of attackcases. The Cp plot at the 70.7% semi-span location isshown in Fig. 11 verifies this for a = 2 deg case. Fora = 1 deg case, the shock movement is less.

Figures 12 and 13 show the deflections of the frontspar for one degree and two degree angle of attackcases, respectively. Experimental data from Byrdsonget a/.43 is also shown. The wing tip for the one degreecase deflects approximate six inches, while the wing tipfor the two degree case deflects approximately eightinches. Good agreement is obtained using direct finite

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element data coupled with Navier-Stokes flow equa-tions.

In addition, Fig. 14 also shows aeroelastic data fromFarhangnia et a/.42 where modal analysis was used forstructural analysis for the one degree case. Modalanalysis results are about 25% in error at the wingtip, where the first five mode shapes were used. Finiteelement equations results are 3% in error comparedto experimental data. Here it is shown the increasedaccuracy of using direct finite element displacementdata as opposed to modal analysis data. Again, theaccuracy of the aeroelastic coupling procedure and thefinite element wing-box code are demonstrated suc-cessfully.

ConclusionsAn aeroelastic coupling procedure was presented

whereby static aeroelastic analysis can be performedby allowing the coupling of a wide variety of computa-tional fluid dynamics (CFD) codes and computationalstructural dynamics (CSD) codes. The procedure wasdemonstrated by performing static aeroelastic analysison an F/A-18 Stabilator by using finite element capa-bility in NASTRAN coupled with Euler flow equationsas available in NASTD (an in-house McDonnell Dou-glas CFD code). In addition, the Aeroelastic ResearchWing (ARW-2) was used to validate the aeroelasticcoupling procedure by using a finite element wing-boxcode coupled with Navier-Stokes equations as availablein ENSAERO (NASA Ames Research Center CFDcode). Experimental data were used to compare thecomputational aeroelastic solution of the ARW-2 andgood agreement was obtained. The increased accu-racy of the use of direct finite element displacementdata as opposed to modal analysis was also shown.The advantage of this aeroelastic coupling procedureis that it only requires the grid points of the CSDand CFD grids. Using only the grid point locations,necessary mappings are created to be able to obtainstatic aeroelastic solutions. This procedure is mod-ular. Currently, only the vertical displacements areconsidered. Therefore, the interpolation scheme canbe changed to account for the in-plane displacements.The aeroelastic coupling procedure is not as efficientas a completely integrated scheme. This procedure isalso limited in that large amounts of deformation willcause the problems with CFD grid deformation. Thiswill occur at points near divergence speeds. However,for swept back wings, divergence is not a problem.

AcknowledgementsPortions of this work were performed under a NASA

Ames-Virginia Tech Consortium #NCC2-5097 and aMcDonnell Douglas Contract #PO Z60807. The au-thors would like to thank Vie Spain, Chansup Byun,



Rudy Yurkovich, and Mehrdad Farhangnia for theirhelp and guidance. Sandia is a multiprogram labo-ratory operated by Sandia Corporation, a LockheedMartin Company, for the United States Departmentof Energy under Contract DE-AC04-94AL85000.

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AIAA-98-4783

Input CFD grid points

CFD Analysis CodePressures

Map pressures from CFD grid pointsto forces on CSD nodes

Forces

CSD Analysis CodeDisplacements

Map displacements from CSD nodesto CFD grid points

New CFD grid

Converged? no

Fig. 1 Flow Chart of Aeroelastic Coupling Proce-dure

41Sanford, M. C., Seidel, D. A., Eckstrom, C. V., and Spain,C. V., "Geometrical and Structural Properties of an AeroelasticResearch Wing (ARW-2)," NASA TM 4110, April 1989.

42Farhangnia, M., Guruswamy, G. P., and Biringen, S.,"Transonic-Buffet Associated Aeroelasticity of a SupercriticalWing," AIAA Paper 96-0286, January 1996.

43Byrdsong, T. A., Adams, R. R., and Sanford, M. C.,"Close-range Photogrammetric Measurement of Static Deflec-tions of an Aeroelastic Supercritical Wing," NASA TM 4194,1990.

44Bisplinghoff, R. L., Principles of Aeroelasticity, DoverPublications, Inc.

Stepl

M iH

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Fig. 2 Mapping of a CFD Grid Point to a CSDTriangle

Fig. 3 CFD Grid for the F/A-18 Stabilator



740 750 760Streamwise Coordinate (in)

Fig. 4 Mapping of CFD Points to Structural Tri-angles for the F/A-18 Stabilator Fig. 7 Convergence of the Wing Tip of the F/A-18

Stabilator

Fig. 5 CFD Grid of the ARW-2 WingDeflections scaled bv a factor of 10

Fig. 8 Final Converged and Initial UndeflectedF/A-18 Stabilator

Fig. 6 Finite Element Model of the Spars and Ribsof the ARW-2 Wing

Comp. (Direct F.E.A.)O Camp. (Modal Analysis)

Fig. 9 Comparison of Cp Variation for RigidSteady State Solution at the 70.7% Semi-span Lo-cation for a = 1 deg



AIAA-98-4783

-1.5

-1.25

-1

-0.75

-0.5

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0

0.25

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1

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Fig. 10 Comparison of Cp Variation of Experi-mental Data Versus Computational Results at the70.7% Semi-span Location for a = 1 deg

. . . . . . wind-off zero positionO Experimental alpha .2 deg

———— Comp. (N-S) alpha- 2 dog

50 75Spanwise coordinate (In)

Fig. 13 Comparison of the Experimental andComputational Front Spar Deflections of the ARW-2 Wing at a = 2 deg

Fig. 11 Cp Variation for a = 2 deg at the 70.7%Semi-span Location

6362.5

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50 75Spanwise coordinate (In)

50 75Spanwise coordinate (in)

Fig. 14 Comparison of the Rear Spar DeflectionsUsing Modal Analysis Versus Finite Element Anal-ysis of the ARW-2 Wing at a = 1 deg

Fig. 12 Comparison of the Experimental andComputational Front Spar Deflections of the ARW-2 Wing at a — 1 deg


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