A NUMERICAL METHOD FOR NONLINEAR

AEROELASTIC ANALYSIS OF WINGS WITH

LARGE DEFORMATION

Paul G. A. Cizmas1, Joaquin I. Gargoloff2, Thomas W. Strganac3

Department of Aerospace Engineering, Texas A&M University

College Station, Texas 77843-3141

e-mail: 1cizmas@tamu.edu, 2gargoloff@tamu.edu and 3strag@tamu.edu

Philip S. Beran

Air Force Research Laboratory, Wright-Patterson AFB, Ohio 45433-7531

e-mail:philip.beran@wpafb.af.mil

Keywords: Limit cycle oscillations, Computational Fluid Dynamics, Continuation method.

Abstract. This paper presents a numerical method for nonlinear aeroelastic analysis. The algo-rithm was developed to handle wings with large deformations. The aeroelastic model consists of:(1) a nonlinear structural model that captures in-plane, out-of-plane, and torsional couplings, (2)an unsteady, viscous aerodynamic model that captures compressible flow effects for transonic flowswith shock/boundary layer interaction, and (3) a solution methodology that assures synchronousinteraction between the nonlinear structure and the fluid flow, including a consistent geometricinterface between the highly-deforming structure and the flow field. A formulation for a wing mod-eled as a cantilevered beam is used to analyze motion about all axes, including in-plane motion.The kinematic nonlinearities due to curvature and inertia are retained in their exact form. A par-allel computation algorithm based on message-passing interface is developed for the flow solver.This parallel computation algorithm and the grid generator of the flow solver were developed con-currently to improve the efficiency of parallel computation. A three-level multigrid algorithm wasimplemented in the flow solver, to further reduce the computational time. The aeroelastic solver isutilized to investigate variants of the Goland wing. In addition, the nonlinear aeroelastic responseof a wing constrained by trim conditions is examined using the method of numerical continuation.An aerodynamic stall nonlinearity is included using a quasi-steady approach. Trim state, stability,and associated bifurcation characteristics are rapidly determined as design parameters are varied.Limit cycle oscillations are investigated and are characterized in terms of amplitude of response.The effect of variation of design parameters such as stiffness ratio, aspect ratio and root angle ofattack is studied. Both subcritical and supercritical bifurcation branches are detected, and theirpresence is dependent upon design parameters. The importance of the in-plane degree of freedomon the stability characteristics of the wing in trim is also investigated.

1 INTRODUCTION

Studies of nonlinear fluid-structure interactions have been motivated by evidence of adverse aeroe-lastic responses attributed to system nonlinearities. Limit-cycle oscillations (LCOs) are one example

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of these nonlinear interactions. LCOs occur in nonlinear aeroelastic systems and remain a persistentproblem on fighter aircraft with store configurations [1, 2, 3, 4, 5].

LCOs are unacceptable because aircraft performance, aircraft certification, mission capability,and human factor issues such as pilot fatigue are adversely affected. The ability to fully analyzeand optimally design air vehicles with inherent nonlinear features is limited by the computationalcost. The small frequencies associated with wing flutter require a long integration time, sometimesof the order of tens of seconds, to properly capture the physics of the problem. As a consequenceof this limitation, the physics of nonlinear responses are not fully understood. Although there isagreement in the flutter engineering community that LCO arises from the nonlinear interaction ofthe structural and aerodynamic forces, there is significant disagreement as to which of these sourcesis the major contribution to the phenomenon.

This paper presents a fully nonlinear aeroelastic solver that is multigrid, parallel and unstruc-tured. This solver is utilized to predict LCO for the Goland wing. LCO prediction results, ob-tained using a continuation method, are also presented. The continuation method, however, useda simplified aerodynamics model, in which an aerodynamic stall nonlinearity was included using aquasi-steady approach. The continuation method results were compared against direct numericalsimulation results.

2 AEROELASTIC MODEL

This section presents the aerodynamics model, the structural model and the coupling between theaerodynamic and structural models. The aerodynamics model solved the Reynolds-averaged Navier-Stokes equations and a two-equation eddy-viscosity turbulence model. A nonlinear structural modelwas used to model the wing as a nonlinear beam.

2.1 Aerodynamic model

The flow was modeled as unsteady, compressible and viscous using the mass, momentum andenergy conservation equations [6]. The turbulence effects were modeled by using the two-equationeddy-viscosity Shear Stress Transport model [7]. The time-dependent integral form of the two-equation eddy-viscosity model was expressed in a vectorial form similar to the form used for theNavier-Stokes equations.

2.2 Structural Model

The wing shown in Fig. 1 was modeled as a nonlinear beam that could bend about y- and z-axes,and twist about ξ-axis. The governing equations contained structural coupling terms and includedboth quadratic and cubic nonlinearities due to curvature and inertia [8].

The equations of motion for in-plane bending w, out-of-plane bending v, and torsion φ responsewere expressed as [9]

mw − Iηw′′ + Dηw

IV = G′

w + FAw

mv − meφ − Iζ v′′ + Dζv

IV = G′

v + FAv(1)

Iξφ − mev − Dξφ′′ = Gφ + M

where m is the mass of the wing per unit span, I is the moment of inertia per unit span, D is thesectional stiffness coefficient, e is the center of gravity offset from the elastic axis. G′

w, G′

v and Gφ

are the structural nonlinear components [9]. FAwand FAv

are the aerodynamic forces and M isthe aerodynamic moment.

2

Figure 1: Coordinate systems for the wing structural model.

The terms of the equations of motion were expanded in Taylor-series, and nonlinearities upto third order were retained. Longitudinal extension of the wing was not permitted. Thus, theequations were derived with an inextensionality constraint along the span.

2.3 Coupling of aerodynamic and structural models

The coupling of aerodynamic and structural models was done using a tightly-coupled solution whichallowed the two models to communicate during every time step. The aerodynamic nonlinear loadswere generated by integrating the pressure over the wing surface. The pressure values were obtainedfrom the unsteady flow solver. The aerodynamic loads were then passed to the structural solverand were used to calculate the wing deformation. The coordinates of the new position of the wingwere passed to the grid deformation algorithm which updated the grid of the flow solver. The flowsolution was then calculated and the aerodynamic loads generated for a new wing position.

3 NUMERICAL METHOD

This section describes the numerical implementation of the aeroelastic solver. The section presentsthe methodology for the grid generation and deformation algorithm, followed by the spatial andtemporal discretization, and the parallel and multigrid algorithms.

3.1 Grid generation and deformation algorithm

The computational domain was discretized to facilitate large deformations and parallel processing.A hybrid grid was used, which consisted of hexahedra and triangular prisms. The computationaldomain was divided into layers that were topologically identical, and that spanned from the wingroot to past the wing tip. The topologically identical nodes of adjacent layers were interconnectedto generate the volume elements that were either triangular prisms or hexahedra. Each layer hada structured O-grid around the wing, and an unstructured grid at the exterior of the O-grid. TheO-grid allowed good control over the mesh size near the airfoil and permitted clustering cells in thedirection normal to the wing to properly capture boundary layer effects. The unstructured gridwas flexible in filling the rest of the domain.

While the computational grid deformed as the wing bent and twisted under aerodynamic loads,the grid connectivity remained unchanged. To maintain a good quality grid, the mesh deforming

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algorithm was required to generate a moving grid that was both perpendicular to the wing surfaceand to the outer boundary of the computational domain [10].

Figure 2 shows the undeformed and deformed grids for the Goland wing [11, 12]. In this case,the deformation at the tip is approximately 60% of the span length, illustrating that the griddeformation algorithm is robust and can be applied to large deformations. A detail of the grid seenalong the x-axis (Fig. 3) shows the structured and unstructured grids.

X

Y

Z

Figure 2: Deformed and undeformed gridsof the Goland wing (tip deformation equalto 60% of wing semi-span).

X

Y

Z

Figure 3: Detail of a lateral view of the Golandwing grid.

3.2 Spatial and temporal discretization

The governing equations were solved using a finite volume method implemented for an unstructuredgrid [13]. The cell-averaged variables were stored at the nodes of the grid, that is, the vertices of thecells. The governing equations were discretized using mesh duals as control volumes. The mediandual-mesh was adopted herein because of its flexibility to handle unstructured mixed meshes.

An edge-based data structure was utilized for the discretization of the governing equations dueto its flexibility and efficiency in handling different element types. The numerical scheme, however,was node-based and consequently the solution was obtained at the nodes of the mesh. The surfaceintegral of the inviscid and viscous fluxes was approximated as a sum over the faces of each controlvolume. The source terms were calculated using the control volume-averaged solution and thederivatives of flow variables at the cell centroid.

The Navier-Stokes equations were rewritten in a semi-discrete form as

∂ (Qi · Vi)

∂t=

∂Qi

∂tVi +

∂Vi

∂tQi = −

∮

Si

~F · ~n · dS + Ei · Vi (2)

where Qi denotes the volume-averaged flow variable Q over the volume Vi. The surface integralterm on the right-hand side of this equation was approximated as

∮

S

~F · ~n · dS =∑

j=k(i)

(

~Finv −~Fvis

)

· Sij (3)

4

where k(i) is the set of vertices adjacent to node i,(

~Finv −~Fvis

)

is the flux normal to the dual-mesh

cell face and Sij is the cell face surface area. Both the surface area and the flux were associatedwith the dual mesh face that was associated with the edge (i, j).

Equation (2) and the turbulence model equations were written for a grid node i as ∂Qi

∂tVi = Ri

where the residual Ri was

Ri = −

∮

Si

~F · ~n · dS + Ei · Vi −∂Vi

∂tQi (4)

The term ∂Vi/∂t is the time evolution of the cell volume and relates to the deformation of thegrid. This term was calculated using the Geometric Conservation Law [14] which relates the timederivative of the volume to the motion of the faces of the cell:

∂Vi

∂t=

∮

Si

~VS · ~n · dS =∑

j=k(i)

~VS · ~nij · Sij, (5)

where ~VS denotes the velocity vector corresponding to the center of the face Sij whose unitarynormal vector is ~nij. The integral was solved using the same edge-based numerical scheme used tosolve the inviscid fluxes. A time-marching algorithm was used to calculate the unsteady solution.

3.3 Parallel algorithm

The flow solver was parallelized to reduce the turnaround time. The message-passing interface(MPI) standard libraries were used for the interprocessor communication. As mentioned in thegrid generation section, the computational domain was divided into topologically identical layersthat spanned from the root to past the tip of the wing. Having topologically identical layerssimplified the parallel communication. In addition, the grid deformation algorithm was designedso that the connectivity of the grid did not change during wing deformation.

The runs up to 16 processors were done on a 128-processor (1.3 GHz Intel Itanium) SGI Altix3700 computer at the Texas A&M Supercomputing Center. 16- and 32-processor runs were done ona 2068-processor (2.4-GHz AMD Opteron) Cray XT3 at the Pittsburgh Supercomputing Center.The efficiency of the parallel computation was in excess of 90% for 16 processors or less. Moredetails on the parallel processing were presented in [6].

3.4 Multigrid technique

The parallel algorithm reduced the turnaround time by dividing the computing effort over severalprocessors. The number of operations slightly increased due to the processor communication, butthe overall effect of parallel computation was a reduction of the wall clock time. To further reducethe computational time, a multigrid technique was implemented in the flow solver algorithm.

The multigrid technique solves the governing equations on a series of successively coarser grids,accelerating the convergence of the solution on the finer grid while maintaining the overall accuracyof the flow solver. Therefore, multigrid methods achieve a reduction in the computational time byreducing the number of iterations necessary to obtain a converged solution. The numerical testingshowed that the multigrid solver reduced the residuals 1.9 times faster than the one-level gridsolver [6].

4 RESULTS

The aeroelastic solver was validated on several test cases, including the F-5 wing and the NATAwing [6]. This section presents results for different variants of the Goland wing.

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4.1 Direct Numerical Simulation

The stability boundary of limit-cycle oscillation for the Original Goland wing [15] was calculatedby computing the time response of the system for different velocities and standard atmosphericconditions. As the velocity increased, the system became unstable and the disturbances wereamplified. It was found that at a velocity of 332 ft/sec the system was stable, whereas at a velocityof 338 ft/sec the system became unstable. Figures 4 and 5 show the modal amplitudes for out-of-plane and torsional deformations at velocities ranging from 300 to 350 ft/sec.

0 2 4 6 8 10 12Time [sec]

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Out

-of-

plan

e m

odal

coo

rdin

ate

Vel = 300 ft/secVel = 325 ft/secVel = 332 ft/secVel = 338 ft/sec

0 3 6 9 12 15 18 21 24Time [sec]

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Ou

t-o

f-p

lan

e m

od

al c

oo

rdin

ate

Vel = 338 ft/secVel = 350 ft/sec

Figure 4: Modal amplitudes of out-of-plane bending for Original Goland wing at velocities rangingfrom 300 to 350 ft/sec.

0 2 4 6 8 10 12Time [sec]

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

Tor

sion

al m

odal

coo

rdin

ate

Vel = 300 ft/secVel = 325 ft/secVel = 332 ft/secVel = 338 ft/sec

0 3 6 9 12 15 18 21 24Time [sec]

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

To

rsio

nal

mo

dal

co

ord

inat

e

Vel = 338 ft/secVel = 350 ft/sec

Figure 5: Torsional modal amplitudes for Original Goland wing at velocities ranging from 300 to350 ft/sec.

A comparison between results obtained using linear and nonlinear structural models showedthe importance of using a nonlinear model. Figure 6 shows the out-of-plane bending, torsionaland in-plane bending modal amplitudes of the Heavy Goland wing [11] at Mach = 0.7. Modalamplitudes were smaller when the nonlinear terms were taken into account. This suggests thatusing a linear structural model is a conservative approach, that is, the true (nonlinear) deformationis smaller than the deformation predicted by the linear model. However, a linear structural modelhas no coupling between the in-plane and out-of-plane modes. It is important to consider in-planeresponse as shown in Fig. 6c. The in-plane deformation oscillated about a non-zero equilibrium.When non-linear terms were taken into account, the deformation was approximately one order ofmagnitude higher than the deformation of the linear structural model.

6

0 3 6 9 12 15 18 21 24Time [sec]

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

Out

of

pla

ne

modal

am

pli

tude,

v

[−]

linear structurenonlinear structure

(a)

0 3 6 9 12 15 18 21 24Time [sec]

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

Out

of

pla

ne

modal

am

pli

tude,

v

[−]

linear structurenonlinear structure

(b)

0 3 6 9 12 15 18 21 24Time [sec]

-0.0125

-0.0100

-0.0075

-0.0050

-0.0025

0.0000

0.0025

In-p

lane

mod

al a

mpl

itude

, w [−

]

linear structurenonlinear structure

(c)

Figure 6: Modal amplitudes of linear and non-linear structural model: (a) out-of-plane bending,(b) torsional and (c) in-plane bending.

4.2 Continuation Method

A direct method for examining the bifurcation characteristics of the nonlinear aeroelastic equationshas been used herein to facilitate parametric studies and minimize time consuming numericalsimulations. The software AUTO2000 [16] was used for continuation and bifurcation analyses ofthe nonlinear ordinary differential equations. AUTO2000 automates the computation of solutionsof this parameter-dependent system of equations. Since the system possesses multiple solutions,it is valuable to compute the set of solutions and search for those solutions with specific desirableproperties as a system parameter is varied. This solution set forms a bifurcation diagram, i.e., asmooth curve or surface representing the solution for the varying system parameter. AUTO2000has the capability to compute periodic solutions such as the limit cycle oscillations discussed in thispaper. In addition, AUTO2000 determines the stability of the steady state solutions. AUTO2000permits a two-parameter continuation solution of Hopf and other bifurcation points [16].

The equations of motion must be presented in state space form [17, Eq. 51] and a known solutionmust be provided as an initial point. Continuation of solutions was conducted starting at this known

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solution, and all of the equilibrium points were computed automatically as a system parameter wasvaried. In this process, the critical value of parameters of interest at which bifurcation occurs wasalso computed. For computations presented herein, the free stream velocity was varied. The flutterpoint was detected as a bifurcation.

The extended Heavy Goland Wing (HGW) [17] with parameters shown in Table 1 was used asthe baseline configuration.

Parameter Value

semi-chord, b 0.914 m (3 ft)wing semi-span, L 6.096 m (20 ft)extended wing semi-span, L 30.48 m (100 ft)mass per unit span, m 538.57 kg/mout-of-plane bending stiffness, Dζ 9.772×106 Nm2

in-plane bending stiffness, Dη 1.4072×109 Nm2

torsional stiffness, Dξ 9.871×105 Nm2

pitch inertia per unit span, Iξ 111.955 kg melastic axis 0.6096 m from leading edgecentroidal axis 0.792 m from leading edge

Table 1: Baseline parameters of the (extended) Heavy GolandWing.

A trim model that required a balance of total lift and weight was introduced for longitudinalvehicle motion. This constraint was expressed as [17] α0 = (mL + M0)g/ρ(V 2

∞+ V 2

y )bCL0L where

α0 is the required angle of attack at the wing root, M0 is the total body mass, V∞ is the freestream velocity, Vy is the vertical velocity, g is the acceleration due to gravity, and CL0

is the liftcoefficient.

A sustained balance of pitching moment was assumed and was attained through instantaneoustail and elevator loads. The semi-span has been extended to 100 feet (30.48 m), providing anaspect ratio of approximately 33 based on the semi-span. This was done to more fully explore thenonlinear dynamics introduced by a highly flexible, high aspect ratio configuration.

In the form of a bifurcation diagram, Fig. 7 shows amplitudes of the out-of-plane modal coordi-nate equilibria and limit cycles as the free stream velocity was varied. βη was defined as the ratioof in-plane to out-of-plane bending stiffness, βη = Dη/Dζ . The location of the response in Fig. 7was approximately at the midspan of the wing. These results represent a continuation solution ofthe governing equations. A bifurcation point was indicated in the figure by the “+” symbol. Thethin solid line represents a branch of stable equilibria; the thin dashed line represents a branch ofunstable equilibria; and, the thick solid line represents a branch of stable LCOs. This notation isfollowed for all bifurcation diagrams presented herein.

Figure 7 shows bifurcation diagrams with supercritical branches which describe the response atvelocities higher than the bifurcation point. In linear analysis, the bifurcation point is the fluttervelocity. The branch describes limit cycle behavior. The nonlinear dynamics of this system aregoverned by the existence of a disturbance. With a disturbance, the system becomes entrained inthe LCO with amplitude as shown by the thick solid line.

Figure 7a shows the out-of-plane modal coordinate for the wing without the trim constraint, i.e.,the wing has a zero root angle of attack. The system lost stability at a velocity of approximately 60ft/sec. In this case the bifurcation point represents the traditional flutter velocity. LCOs emanatedfrom this bifurcation. For velocities greater than this velocity, the system underwent stable LCOswith amplitudes as shown by the thick solid line that represents the stable branch. The dashedline represents the unstable branch.

At flight conditions requiring a nonzero root angle of attack to satisfy the trim constraint, thestatic deformation and stability characteristics changed. For a root angle of attack α0=2.8o, the

8

(a) α0=0.0 (b) α0=2.8o

Figure 7: Bifurcation diagrams showing the out-of-plane bending response for the Heavy GolandWing with L = 100 and βη=144 (thin solid line represents stable equilibria, dashed line representsunstable equilibria, and thick solid line represents stable LCO amplitudes).

constraint associated with the trim condition leads to a statically deformed shape of the wing thatmust vary with the free stream velocity. This response is shown in Fig. 7b. Wing static deformationincreased with free stream velocity, as shown by the increase in the magnitude of the out-of-planecoordinate. The bifurcation occurred at a velocity higher than the case with α0=0.0.

The continuation solution, including the characteristics of the stable LCOs provided in Fig. 7,was verified by direct numerical simulation of the governing equations. Figure 8 shows the systemresponse for the two root angles of attack at V =70 ft/sec. This velocity was higher than the fluttervelocity (VF =60 ft/sec) for the α0=0.0 case. As shown in the bifurcation diagrams, the systemsettled into stable LCOs for α0=0.0. The amplitude of oscillation was consistent with Fig. 7a. Atα0=2.8o the wing undergoes a static deformation as shown in Fig. 8b. The value of the non-zerodeformation is again consistent with Fig. 7b.

The numerical investigation of the extended HGW indicated a remarkable sensitivity of thebifurcation to in-plane flexibility. To illustrate this, the response characteristics of the system wasinvestigated for wing designs with different ratios of out-of-plane to in-plane stiffness, βη. Highervalues of βη are representative of conventional low aspect ratio wings whereas lower values of βη

are expected in slender high aspect ratio wings typical for high-altitude long endurance vehicles(as well as fan blades). The set of bifurcations associated with different stiffness ratios, where thebifurcation in Fig. 7 represents one point, was extended to V -βη parameter space by a two parametercontinuation method. The set of bifurcations is shown in Fig. 9. As the stiffness ratio was decreased,the velocity for the bifurcation decreased. In addition, this boundary shifted upwards for increasingvalues of α0. Thus, larger trim angles of attack lead to increased stability. This effect was partiallyoffset by an increase in aerodynamic drag. There was a sharp increase in the bifurcation velocityfor values of βη < 20.

The influence of aspect ratio is illustrated in Fig. 10. For a given stiffness ratio, the systemwent from supercritical to subcritical when the aspect ratio was increased. This indicates that thesystem tends to become subcritical with decreasing stiffness ratio and increasing aspect ratio.

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(a) α0=0.0 (b) α0=2.8o

Figure 8: Simulation results at V =70 ft/s for two root angles of attack.

Figure 9: Set of Hopf bifurcations in V -βη parameter space for various α0.

5 CONCLUSIONS

This paper presented a multigrid parallel algorithm for a fully nonlinear aeroelastic analysis. Theaeroelastic model consisted of: (1) an unsteady, viscous aerodynamic model that captured com-pressible flow effects for transonic flows with shock/boundary layer interaction, (2) a nonlinearstructural model that captured in-plane, out-of-plane, and torsional couplings, and (3) a solutionmethodology that assured synchronous interaction between the nonlinear structure and fluid flow,including a consistent geometric interface between the highly-deforming structure and flow field.

The aeroelastic solver allowed to fully integrate a nonlinear structural model of a wing and anunsteady, compressible, viscous flow model. A parallel computational algorithm based on message-passing interface was developed for the flow solver. The grid generator and the parallel computationalgorithm were developed concurrently to improve the efficiency of parallel computation. A three-level multigrid algorithm was implemented in the flow solver, to further reduce the computationaltime. For the case presented herein the computational time was reduced by approximately a factorof two.

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(a) AR=33 (b) AR=40

Figure 10: Comparison of bifurcation diagrams showing out-of-plane response, βη=4 and α0=2.8o.

The numerical results for the Goland wing demonstrated the importance of using nonlinearstructural models. When the nonlinear terms were omitted, the in-plane wing deformation was ap-proximately one order of magnitude smaller than the in-plane deformation that included nonlinearterms.

The nonlinear aeroelastic response of a wing constrained by trim conditions was investigatedusing the method of numerical continuation. Kinematic nonlinearities due to curvature and inertiawere retained in their exact form. This nonlinear formulation allowed the analysis of dynamicsabout trim as the model did not require a priori assumption about the static trim shape of thewing. A continuation algorithm applied to this formulation facilitated the computation of nonzerotrims as a system parameter was varied. This represents a direct method for eigenanalysis of theaeroelastic system. The method automated the computation of trim states and system stability.The parameter space related to design was investigated for stability characteristics. Stiffness ratioand aspect ratio were found to be important parameters affecting the design. The in-plane degree offreedom participates in the dynamics instability phenomenon at low stiffness ratios. Eigenanalysisrevealed that the instability of the low stiffness ratio configuration was related to the in-plane mode.Thus, modeling of the in-plane degree of freedom is required for low stiffness ratio configurations.Such configurations are more susceptible to subcritical LCOs as a combination of trim angle ofattack, stiffness ratio and aspect ratio creates conditions for subcritical bifurcations.

6 ACKNOWLEDGMENTS

This work was sponsored by AFOSR under Grant No. FA9550-04-1-0174. The authors are thankfulto the Pittsburgh Supercomputing Center and the Texas A&M University Supercomputing Facilityfor making the computing resources available.

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