Nonlinear Aeroelasticity Computations in Transonic Flows Using Tightly Coupling Algorithms

Proceedings of PVP2006-ICPVT-112006 ASME Pressure Vessels and Piping Division Conference

July 23-27, 2006, Vancouver, BC, CanadaPVP2006-ICPVT11-93244

NONLINEAR AEROELASTICITY COMPUTATIONS IN TRANSONIC FLOWS USINGTIGHTLY COUPLING ALGORITHMS

Zhengkun Feng and Azzeddine SoulaımaniDepartment of Mechanical Engineering, Ecole de Technologie Superieure

1100 rue Notre-Dame Ouest, Montreal(QC), H3C 1K3, CanadaEmail : [email protected], [email protected]

ABSTRACTA nonlinear computational aeroelasticity model based on

the Euler equations of compressible flows and the linear elasto-dynamic equations for structures is developed. The Euler equa-tions are solved on dynamic meshes using the ALE kinematic de-scription. Thus, the mesh constitutes another field governed bypseudo-elatodynamic equations. The three fields are discretizedusing proper finite element formulations which satisfy the geo-metric conservation law. A matcher module is incorporated forthe purpose of pairing the grids on the fluid-structure interfaceand for transferring the loads and displacements between thefluid and structure solvers. Two solutions strategies (Gauss Sei-del and Schur-Complement) for solving the nonlinear aeroelas-tic system are discussed. Using second order time discretiza-tion schemes allows us to use large time steps in the computa-tions. The numerical results on the AGARD 445.6 aeroelasticwing compare well with the experimental ones and show thatthe Schur-complement coupling algorithm is more robust thanthe Gauss-Seidel algorithm for relatively large oscillation am-plitudes.

NOMENCLATURECk1 Tuning parameterCm fourth order constitutive tensor of the fictitious materialCs fourth order constitutive tensor of the structural materialCv Constant volume specific heatDs Structural damping matrixe Total energy per unit massEm Elastic modulusF Solicitation vectorFadv

i Convective fluxg Internal volume forces

is Total nodes of a structural elementj f Total nodes of a fluid elementK f , K s Stiffness matrices of fluid and structureM f , Ms Mass matrices of fluid and structure respectivelyn Normal of the moving fluid boundaryNs

i , N fj Shape functions of structural elements and fluid ele-

ments respectivelyp Fluid pressurepf s Pressure at the fluid-structure interfaceP Matrix composed of mode vectorsu Fluid velocity vector or Nodal displacement vectoru f , um, us Fluid, Mesh, Structural displacement vectors re-

spectivelyU Conservative variables vectorw Grid velocity vectorW Weight function vectorx f , xm, xs Fluid, Mesh and Structural position vectors respec-

tivelyz Generalized coordinateγ Specific heat ratioΓ Boundaryεm Strain tensor of fictitious materialεs Green strain tensorη Damping coefficientλ Second viscosity coefficientµ First viscosity coefficientν Poisson ratioρ Densityσ Stress tensorσm Stress tensor of fictitious elastic materialσs Cauchy stress tensor of a structureω frequencyΩ Volume

1 Copyright c© 2006 by ASME

INTRODUCTIONTransonic airflows around a flexible structure are character-

ized by the existence of nonlinearities, such as shock waves andflow induced vibrations [1]. In the classical linear aeroelasticitytheory [2, 3], airflow is assumed inviscid, incompressible and ir-rotational. Corresponding numerical methods can be found in theearly literature, such as the well-known Doublet Lattice method,the Vortex-Lattice method, the panel method and the transonic-small-disturbance (TSD) method. However, the linear theorycannot accurately predict the transonic dip [4], therefore non-linear aeroelasticity theories are required. Discussions on the be-haviors of the aerodynamic nonlinearity can be found in Ref. [5].The full-potential equation, which describes the nonlinearity of atransonic flow, gives better solutions [6] when the shock is weak.But, for the flows with strong shocks, a high order model suchas the Euler equations that describe more completely the nonlin-earity in transonic regime are required to obtain accurate solu-tions [7, 8]. Indeed, the computational cost increases evidently.However, reduced models based on such full-order models areable to represent the essential features [9, 10]. This issue is be-yond the work of this paper.

In this paper, the aeroelastic system is modeled by couplingnumerically three fields, which are respectively the structural, thefluid and the mesh fields, through an information transfer mod-ule. We assume that the flow is described by the Euler equationsand the structure is modeled by the linear elastodynamic equa-tions. The nonlinear coupling between the structure and the fluidis reinforced by imposing kinematic and dynamic compatibilityconditions at the fluid-structure interface (FSI). The outline ofthe following sections is as follows : First, the governing equa-tions of the complete system are presented; Second, their discreteforms are described; Third, the coupling algorithms will be pre-sented; Afterwards, the numerical results will be discussed, andfinally, we draw the conclusions.

GOVERNING EQUATIONS——

Structural dynamic modelThe displacement of a flexible structure satisfies the follow-

ing dynamic equilibrium equations :

ρsusi,tt +σs

i j , j(us) = gi in Ωs (1)

wheregi presents the body forces. Since the structural defor-mations are assumed small, the constitutive relation between theGreen strain tensorεs and the Cauchy stress tensorσs becomeslinear :

σs = Cs · εs

with

εsi j , j = (us

i, j +usj,i)/2

The kinematic and dynamic compatibility conditions at the fluid-structure interface must be respected.

Conservation equations of aerodynamicsThe Euler equations of unsteady viscous compressible flows

in the ALE (Arbitrary Lagrangian Eulerian) kinematic descrip-tion can be written in the following compact form [11] :

U,t +Fadvi,i −wiU,i = 0 (2)

with

U =< ρ, ρu, ρe>T

Fadvi =< ρui , ρuui , (ρe+ p)ui >T

wherewi denotes the grid velocity. Equations (2) satisfy the ge-ometric conservation law. That is, for any uniform flow, the flowfield remains unchanged for any grid motion. The pressure andthe temperature are two additional unknowns. For ideal gas, thefollowing two equations of state are added to close the system ofthese equations :

p = (γ−1)ρCvT

T = e− | u |22

Compatibility conditions at the fluid-structure inter-face

The coupling between the structure and fluid dynamicsis reinforced by imposing the boundary conditions on thefluid-structure interface. On this interface, each structural pointcorresponds to a fluid point. Thus, the compatibility conditionscan be described for inviscid flows by :

• kinematic conditions :

xs = x f ;

(∂us

∂t−u

)·n = 0

• dynamic compatibility conditions :

σs ·n =−pn.


Dynamic mesh modelIn order to satisfy the kinematic conditions, the fluid equa-

tions are solved on a dynamic mesh. On the FSI, the fluid meshnodes move with the structure skin whereas on the far field, thefluid nodes are fixed. In order to avoid large element distortions,the FSI nodes motion is smoothed throughout the fluid domain.Thus, the fluid mesh is considered as a third field described by itsown dynamic equations. This dynamic mesh is modeled using afictitious linear elastic material. Since this model is purely math-ematical, the density, the elastic modulus and the Poisson ratioof the material have no physical interpretation and can be freelychosen.

We have chosen zero density and zero body forces for thefictitious elastic material. So the mathematical equations whichdescribe the fluid domain motion can be written as :

∂σi j (um)∂x j

= 0 (3)

with the following boundary conditions :

um = 0 (on the far-field boundary) (4)

xm = x f (on the fluid-structure interface) (5)

(∂um

∂t−u

)·n = 0 (on the fluid-structure interface) (6)

The constitutive relation between the Green strain and theCauchy stress vectors reads :

σm= [Cm]εm (7)

DISCRETE FORM OF THE GOVERNING EQUATIONS——

Modal superposition for the structure motionThe discretization of the elastodynamic equations using the

finite element method leads to the following system :

Mu+Du+Ku = F (8)

The structural vibrations are usually dominated by fundamentalfrequencies. The displacement can be approached by the super-position of the first few modes. We can get the natural frequen-ciesω01, ω02, ω03, ...... ω0n and the following modal matrixP0

through a modal analysis :

P0 = [ p01 p02 p03 ...... p0n ]

If the first m modesp0i(i = 1,2, ......,m) are used, the time re-sponse can be approximated by :

us(t) =m

∑i=1

zi(t)p0i (9)

Substituting this displacement vector and its first and secondderivatives into equations (8) yields the following ordinary dif-ferential equations [12] :

zi(t)+2ηiω0i zi(t)+ω20izi(t) = s0i(pf s, t) (i = 1,2, ......,m)

(10)with the following initial conditions :

z0i = zi(0) = pT

0iMsu0 ; z0

i = zi(0) = pT0iM

su0

and the modal forces0i(pf s, t) (s0i(pf s, t) = pT0iF) which depends

on the pressure at the fluid-structure interface.Equations (10) are uncoupled and can easily be solved by

the Newmark algorithm [13].

Euler equations discretized by the finite elementmethod

After the Euler equations are multiplied by the weight func-tionsW and integrated over the domainΩ, one obtains the fol-lowing Galerkin variational formulation :

∫Ω W · (U,t +Fadv

i,i −wiU,i)dΩ = 0 (11)

The local residual vector can be written as :

R(U) = U,t +Fadvi,i −wiU,i (12)

The fluid domain is discretized by linear tetrahedral ele-ments. The Galerkin finite element formulation often leads toserious numerical instability where the solutions is possibly cor-rupted by oscillations if the flow is dominated by convection.The SUPG formulation [14] introduces an integral term into thisformulation, so that the stability inside elements is reinforced :

∑e

∫

Ωe(A i −wi I)tW iτR(U)dΩ

whereτ is a matrix of time scale which depends on the elementsize [14] as :

τ = (∑ |A|)−1


A shock capturing Laplace-operator which depends also on thediscrete residualR(u) is added. This operator depends on a pa-rameter given by the following expression :

µc = Ck1hmin(‖τR(U)‖,‖u‖)/2 (13)

It adds more dissipation in the vicinity of shock where the resid-ual R(u) is higher than in smooth zones. After adding the shockcapturing operator and the influence term of the far-field bound-ary conditions, the stabilized variational formulation becomes :

∑e

∫

Ωe

[W · (U,t +Fadv

i,i −wiU,i)]

dΩ−∫

ΓeW ·An(U−U∞)dΓ

+∑e

∫

Ωeµc∇W ·∇UdΩ

+∑e

∫

Ωe(A i −wi I)tW,i · τ ·R(U)dΩ = 0

(14)where the second integral term represents the influence of the far-field boundary conditions. The third integral term represents theshock capturing operator whereas the forth term represents theSUPG contribution. The CFD model is completed by adding thefollowing boundary conditions on the moving fluid boundaries:

u ·n = w ·n with w =dxm

dt

Equations (14) satisfy the geometric conservation law as equa-tions (2). After replacingU andW by their interpolations andapplying integration over the elements, one obtains the follow-ing first order semi-discrete system :

M f U+K f (U)U = F f ;

The following implicit second order Gear-Scheme is chosen forthe time discretization :

dU(t)dt

' 3U(t +∆t)−4U(t)+U(t−∆t)2∆t

Discrete form of the mesh governing equationsSince strong gradients of aerodynamic variables appear in

the zone close to the structure, small elements are required to ob-tain computational accuracy. However, small elements are sus-ceptible to large distortion; therefore, their stiffness should belarge. The relation between the stress tensor and the strain ten-sor is assumed linear. If the structural displacement is small, thevirtual internal work for an element can be written as :

Weint =

∫

Ωe< εe∗ >T σmdΩ =

∫

Ωe< εe∗ >T [Cm]εedΩ (15)

Using the same shape functions for the virtual and the real dis-placements yields the following discrete formulation :

∑e

∫

Ωe([Bm]T [Cm][Bm]dΩe)um,e = ∑

eFm,e (16)

whereBm denotes the deformation matrix associated to the inter-polations of displacements,um,e denotes the nodal displacementvector,Fm,e represents the contribution of the condition of themoving boundaries. The solutions of this equation give the fluidmesh displacement vector. The stiffness is assumed to vary in-versely proportional to the element volume :

Cm,ei j ∼ 1

volumee

wherevolumee is the volume of each fluid element. The assem-bled form of equations (16) is :

Kmum = Fm (17)

whereFm represents the contribution of the displacements of themoving boundaries. In the case with strong structural displace-ment, the elements of matrixKm are functions of the grid dis-placementsum (for more details, see Ref. [15]). The approxi-mate solution of the displacement of the fluid nodes can be ob-tained using the iterative GMRES algorithm with ILUT precon-ditioner [16].

Matcher module for information transferThe information exchange in the aeroelasticity solver is per-

formed on the fluid-structure interface. A constraint here is thatthe grids of the CFD and CSD solvers do not necessarily matchon this interface since the governing equations of the two modelsare usually discretized by elements with different type and size.Since the fluid and structural nodes do not coincide on the fluid-structure interface, a search procedure consisting of finding theassociated structural element of a fluid node is applied. A pro-cedure, known as surface tracking, performs the displacementtransfer from the CSD solver to the CFD solver. Another pro-cedure, known as loads projection, performs the transfer of theaerodynamic loads from the CFD solver to the CSD solver. For athin structure which is discretized by shell elements, there is onlyone element in the thickness direction and the nodes are locatedon the middle surface. Therefore, a gap exists between the struc-tural middle surface and the moving FSI. The distance betweeneach fluid node and its associated structural element is assumedunchanged during the structural motion. As Reissner/Mindlinhypothesis is applied on the shell structure, it is supposed thatthere is no deformation in the thickness direction (Fig. 1). Theinitial distance between a fluid node and its associated structuralelement is :


Figure 1. Gap between fluid nodes and structural elements

l0 = n · (x f −xp)

(x f −xp) is the vector between a fluid node and its projection onthe structure (with shell elements). Therefore, the updated fluidnode position vector is given by :

x f = x fp + l0n = x f

p(0)+us+ l0n

x f is also given by the following equations :

x f = x f (0)+u f

Since

x f (0) = x fp(0)+ l0n

one obtains

u f = us

As the displacement of the fluid node is equal to the dis-placement of its projected point, the displacement of the pro-jected point of a fluid node can be obtained using the followinginterpolation of the nodal displacement vectors of its associatedstructural element :

u f = us(xp,yp) =i=is

∑i=1

Nsi (xp,yp)us

i (18)

whereu f denotes the displacement of the projected point of fluidnode,Ns

m(xp,yp) denotes the value of the shape function of thestructural element at the projected point(xp,yp).

The loads projection is performed according to the conser-vation law of energy exchanged on the fluid-structure interface.The displacement of the projected point of the fluid node is givenby equation (18). The displacement of any point in the fluid ele-ment is interpolated by the shape functions of the fluid elements.The nodal forces vector of a structural element can be computed

by the interpolation of the fluid force of each node associatedwith this structural element [17].

f si =

j= j f

∑j=1

φ jNsi (χ j) (19)

with

φj =−∫Γ f

pnN fj ds

whereχ j denotes the projected point of thejth fluid node.

COUPLING ALGORITHMSThe nonlinear aeroelasticity model describing the coupling

(Fig. 2) of the fluid, mesh and structure fields can be presentedin the following form [15]:

R1(Y,Z(pf s))R2(Y,Z(pf s))

= 0 (20)

whereY is the set of fluid and mesh variables which consist ofvectorsU f andum, Z(pf s) is the set of structural variables,pf s

is the pressure at the fluid-structure interface,R1 andR2 are re-spectively the nonlinear discrete residuals of the fluid, mesh andstructural variables. This system of nonlinear equations can belinearized into the following system of equations :

[KT ]

δYδZ(pf s)

=−

R1(Y,Z(pf s))R2(Y,Z(pf s))

(21)

whereKT is the tangent matrix formally written as :

[KT ] =[

A CD B

]

Nonlinear Gauss-Seidel algorithmSince solving directly equations (21) is still difficult due

to the complexity of the coupling sub-matricesC andD whichare difficult to express analytically, approximate methods are ap-plied through iterative procedures. We consider here two iter-ative algorithms. First, we consider the Gauss-Seidel couplingalgorithm where the solution of the CFD solver at instanttn+1 isbased on the assumption that the solution of the CSD solver atthe same step is already known and vice versa, for the solution ofthe CSD solver. This procedure can be seen as solving iteratively


Interface

(Matcher)

Load

Projection

CFD SOLVER

Surface

Tracking

CSD SOLVER

Loads:

Pressure

Shear Stress

Surface Position

&

Velocity

Figure 2. Fluid-Structure coupling

equation (21) with the replacement of tangent matrixKT by thefollowing bloc diagonal matrix:

[M ] =[

A 00 B

]

By iterating until a numerical convergence, it is hoped that thecoupling is well satisfied. However, the convergence is condi-tioned by the time step used and the magnitude of the motion.

This algorithm is summarized in the following steps :1. Loop over time steps.2. Loop over coupling iterations.3. Update the fluid field using the new boundary conditions.4. Project fluid forces on the structure.5. Update the structure displacements.6. Update the mesh configuration using the new

interface positions.7. Check for convergence criterion.8. Enddo time loop

Nonlinear Schur-complement algorithmEquations (20) can be restated in the form

ℜ(Z(pf s), t)) = 0 (22)

The idea now is to attack the above problem as a nonlinear equa-tion of Z. It can be seen thatℜ = R2−DA−1R1. ApplyingNewton algorithm to equations (22), leads to solving a system inthe form

ℜ′(Z)δZ =−ℜ(Z(pf s)) (23)

with ℜ′ = B−DA−1C the Jacobian ofℜ. Thus, if the prob-lem (22) converges then necessarily the original coupled fluid-

structure problem converges. That is, whenδZ = 0, the fluid-structure interface does not move which implies that the fluidforce does not change. However, the Jacobianℜ′ is not practi-cally computable. Since GMRES algorithm requires only the ac-tion of the product of this Jacobian times a vectorV, the matrixℜ′ is actually not needed. A nonlinear version of GMRES [16]can be used where the matrix-vector product is approximated bya simple forward finite difference :

ℜ′(Z) ·V ≈ ℜ(Z + εV)−ℜ(Z)ε

(24)

whereε is a small coefficient. The small-size linear system (23)(its size is actually the number of structural dry modesm) con-verges exactly inm iterations, even without any preconditioning.The main time consuming part of this algorithm consists in theconstruction of the Krylov subspace. A significant saving canbe obtained by freezing the Krylov subspace during few time it-erations. However, in order to generate the Krylov directions,the pressure at the interface must be computed by the followingcomputational steps :

1. Compute the coordinates of the structure correspondingto the perturbed modal coordinatesZ + εV.

2. Update the coordinates of the fluid nodes at the interface.3. Update the mesh coordinates.4. Update the fluid field.5. Project the fluid pressure on the interface.

NUMERICAL RESULTSNumerical simulations have been performed on the aeroe-

lastic wing AGARD 445.6. This wing with the symmetrical air-foil NACA 65A004 and a quarter-chord sweepback of 45 de-grees is immersed with zero angle of attack in a transonic air-flow. The wing is initialized with zero displacement and veloc-ity. It has a mass of0.1276slug (1.86227kg), a Young’s mod-ulus of 4.7072× 105 psi (3.2455× 109 N/m2), a Poisson ratioof 0.31 and a density of0.8088slug/ f t3 (416.86 kg/m3). Thestructural domain is discretized by 1176 quadrilateral shell ele-ments. The modal parameters are extracted with the commercialsoftware ANSYS. The natural frequencies of the first five modeswhich are respectively 9.6, 39.4, 49.6, 96.1, 126.3 Hz and themodal vectors presented in Fig. 3 are in good agreement withthe experimental results [4]. A coarse fluid mesh having 177042linear tetrahedral elements and 37965 nodes is used. Since thestrongest variation of the fluid variables occurs around the wing,the elements near the wing are much smaller than those in therest of the fluid domain. There are 12921 fluid nodes and 25684triangular elements on the wet surface of the wing on which slipboundary conditions are applied. Since the structural motion hasno influence on the far-field boundaries, the flow is imposed as


the oncoming flow. The initial solution for the CFD solver isobtained by considering the wing as a rigid structure. The corre-sponding fluid configuration is considered as the initial state ofthe unsteady flow. From this state, a Dirac force is applied onthe point located at the intersection between the wing tip and theleading edge. The Mach number of the oncoming flow is cho-sen as 0.96 in order to simulate the critical point of the transonicdip. The result of the flutter boundary prediction with our parallelcode can be found in Ref. [18].

Numerical simulations with nonlinear Gauss-Seidelcoupling algorithm

For the purpose of capturing the flutter dip with accuracy,first, a relatively small nondimensional time step of 0.1 whichcorresponds to a real time step of2.01× 10−4 secondsis usedin the numerical simulations. At the aerodynamic pressure of60 lb/ f t2 of the oncoming flow, the amplitudes of the lift and thegeneralized displacements of the first two modes, which domi-nate the responses of the structural displacements, are constantand the critical flutter is captured. When the aerodynamic pres-sure of the oncoming flow increases to61.3 lb/ f t2, the responsesof the wing get increasing amplitudes and the wing is beyond theflutter point (Fig. 4). For the purpose of reducing the computingtime, the simulation with the aerodynamic pressure of60 lb/ f t2

and an increased nondimensional time step of 0.3 is performed,the flutter point is also captured. Figure 5 represents the time his-tories of the generalized displacement of the first two modes incomparison with another case with a higher aerodynamic pres-sure of61.3 lb/ f t2.

At the flutter point with aerodynamic pressure of60.0 lb/ f t2

and Mach number of 0.96, the damping coefficients are verysmall positives or negatives (see simulation 1 and 3 in Table1), the coalesced frequency of13.5 Hz (84.8 rad/s) of the firsttwo modes is very close to the experimental one of13.9 Hz(87.3 rad/s) [4]. The spectral distributions of the first two modesobtained with the Fast Discrete Fourier Transform confirm wellthe frequency coalescence (Fig. 6). The frequencies of the liftand the first two modes have practically the same value at theflutter point. This flutter aerodynamic pressure also agrees withthe computational ones in Ref. [19, 20]. When the aerodynamicpressure was increased to61.3 lb/ f t2, the amplitudes of the gen-eralized displacements of the first two modes start to increase andso does the difference of the frequencies of the oscillation. Ta-ble 1 summarizes the damping coefficients and the frequencies ofthe oscillation of the lift and the generalized displacements of thefirst two modes of the numerical simulations [13]. In simulation1 and simulation 3 with aerodynamic pressure of60.0 lb/ f t2, thedamping coefficients are nearly zero and the frequencies are veryclose. The system reaches flutter point (Fig. 7). In simulation 2and simulation 4 with aerodynamic pressure of61.3 lb/ f t2, thedamping coefficients become negative and the difference of the

frequencies of the first two modes starts to increase. The re-sponses of the system indicate the flutter (Fig. 8). Simulation 3and simulation 4 show that the nondimensional time step can beincreased to 0.3, thus a computing speed is three times faster thansimulation 1 and simulation 2.

Yate's Resultscomputed Results

15

25

000

00

5

mode 1

0

0

0

25

25

-25

-15

5075

100-15

mode 4

-50

0

0

25

-25

0

mode 2

0

0

-25

25

25

mode 3

0

0

0

25

25

-25

-25

-500 0

-25

-25

mode 5

extracted with ANSYS

Figure 3. Modal vectors of wing AGARD 445.6 extracted with ANSYS

Numerical simulations with nonlinear Schur-complement coupling algorithm

The Gauss-Seidel coupling algorithm gives satisfying resultswhen the perturbation is small. However, this algorithm mayhave numerical instability and even may crash as the perturba-tion becomes strong. Schur-complement coupling algorithm canimprove the robustness of the code. However, as the structuraldisplacements increase, the linear mesh model is no longer sta-ble. Figure 9 shows the comparisons of the generalized displace-ments of the first two modes obtained by Schur-complement al-gorithm with linear and nonlinear mesh models. From this figurewe can see that the code gives the same results at the beginningof the simulations, but the simulation with linear mesh modelstops at one instant due to the difficulty of convergence. Figure10 shows the comparison of the numerical results obtained fromthe two nonlinear coupling algorithms at Mach number of 0.96and aerodynamic pressure of60 lb/ f t2. The strong force pertur-bation yields strong structural displacements which are 20 timeslarger than that of the test in the previous section. It is observed


-10

-8

-6

-4

-2

0

2

4

6

8

10

12

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Ge

ne

ralize

d

dis

pla

ce

me

nt

(10

-4)

Time (s)

Mode 1

61.3 lb/sqft

60.0 lb/sqft

-3

-2

-1

0

1

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Time (s)

Mode 2

Ge

ne

ralize

d

dis

pla

ce

me

nt

(10

-4)

61.3 lb/sqft

60.0 lb/sqft

Figure 4. Time history of the generalized displacements of the first two

modes under a load perturbation with nondimensional time step of 0.1 at

aerodynamic pressures 60 lb/ f t2 and 61.3 lb/ f t2

that the aeroelastic responses with Schur-complement couplingalgorithm are more stable than those with the Gauss-Seidel cou-pling algorithm.

CONCLUSIONSThe nonlinear computational aeroelasticity model is devel-

oped using tight coupling algorithms. This coupling techniquereuses the developed linear CSD solver based on the modal anal-ysis and the nonlinear CFD solver described by Euler equations.However, a matcher module and a mesh solver are required tomatch the CSD and CFD grids on the fluid-structure interfaceand to adapt the moving fluid boundaries. The use of the secondorder time discretization scheme and a nonlinear moving meshmodel allows us to use large time steps for the staggered cou-pling algorithms. This model is applied to the standard aeroe-

-10

-5

0

5

10

15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Ge

ne

ralize

d

dis

pla

ce

me

nt

(10

-4)

Time (s)

Mode 161.3 lb/sqft

60.0 lb/sqft

-3

-2

-1

0

1

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Ge

ne

ralize

d

dis

pla

ce

me

nt

(10

-4)

Time (s)

Mode 261.3 lb/sqft

60.0 lb/sqft


modes under a load perturbation with nondimensional time step of 0.3 at

aerodynamic pressures 60 lb/ f t2 and 61.3 lb/ f t2

lastic wing AGARD 445.6 to predict the wing flutter, especiallythe transonic dip. With a modest perturbation in transonic flowsat Mach number 0.96, the results of the flutter simulation by theGauss-Seidel coupling algorithm agree well with those in the ref-erences. The Schur-complement coupling algorithm improvesthe robustness of the code when the perturbation is strong.

REFERENCES[1] Dowell, E. H. (1995). A modern course in aeroelasticity

(3rd rev. and enlarged ed.). Dordrecht, Pays-Bas : KluwerAcademic.

[2] Bisplinghoff, R. L., Ashley, H. (1962). Principles of Aeroe-lesticity. John Wiley and Sons, Inc.

[3] Fung, Y. C. (1969). An Introduction to the Theory of Aeroe-lasticity. Dover publications.


0

100

200

300

400

500

600

0 5 10 15 20

0

20

40

60

80

100

120

Frequency (Hz)

Mo

de

1 (

x1

00

), M

od

e 2

Lif

t

Mode 1

Mode 2

Lift

Figure 6. Spectral distributions of the lift and the first two modes

No. of test

lift mode 1 mode 2 lift mode 1 mode 2

damping 0.00063 0.00063 0.00065 -0.0054 -0.00477 -0.00375

Freq (Hz) 13.51 13.53 13.53 13.75 13.6 13.63

state

No. of simulation

lift mode 1 mode 2 lift mode 1 mode 2

damping -0.0002 -0.00017 -0.00025 -0.00537 -0.00475 -0.00667

Freq (Hz) 13.52 13.52 13.52 13.58 13.58 13.6

state

Nondimensional time step

simulation 1

calculated flutter point

simulation 3

P_ref = 61.3 lb/sqftP_ref = 60 lb/sqft

calculated flutter point flutter with a small increasing amplitude

simulation 4

simulation 2

flutter with a small increasing amplitude

0.3

0.1

Table 1. Frequencies and damping coefficients of the osci-llations of wing AGARD 445.6

[4] Yates, E. C., Land, N. S., & Foughner, J. T. (1963). Measureand calculated subsonic and transonic flutter characteristicsof a 45 sweptback wing planform in air and in Freon-12in the Langley transonic dynamic tunnel. NASA Technicalnote, D-1616, March 1963.

[5] Dowell, E., Edwards, J., & Strganac, T. (2003). Nonlinearaeroelasticity.Journal of Aircraft, 40(5), 857-874.

[6] Shankar, V., & Ide, H. (1988). Aeroelastic computations offlexible configurations.Computers & Structures, 30(1-2),15-28.

[7] Guruswamy, G. P. (1988). Interaction of fluids and struc-tures for aircraft applications.Computers & Structures,30(1-2), 1-13.

[8] Snyder, R. D., Scott, J. N., Khot, N. S., Beran,P. S., & Zweber, J. V. (2003). Predictions of store-induced limit-cycle oscillations using Euler and Navier-Stokes fluid dynamics. Paper presented at the 44thAIAA/ASME/ASCE/AHS/ASC Structures, Structural Dy-namics, and Materials Conference, Apr 7-10 2003, Nor-folk, VA, United States.

[9] Anttonen, J. S. R., King, P. I., & Beran, P. S. (2003). POD-Based reduced-order models with deforming grids.Mathe-

-10

-5

0

5

10

15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Ge

ne

raliz

ed

d

isp

lace

me

nt

(10

-4)

Time (s)


modes of the wing AGARD 445.6 under a load perturbation with nondi-

mensional time step of 0.1 (blue and red curves for mode 1 and mode2

respectively) and 0.3 (green and black curves for mode 1 and mode2 re-

spectively), aerodynamic pressure of q = 60 lb/ f t2

-10

-5

0

5

10

15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Time (s)

Ge

ne

raliz

ed

d

isp

lace

me

nt

(10

-4)


modes of the wing AGARD 445.6 under a load perturbation with nondi-

mensional time step of 0.1 (blue and red curves for mode 1 and mode2

respectively) and 0.3 (green and black curves for mode 1 and mode2 re-

spectively), aerodynamic pressure of q = 61.3 lb/ f t2

matical and Computer Modelling, 38(1-2), 41-62.[10] Lucia, D. J., Beran, P. S., & King, P. I. (2003). Reduced-

order modeling of an elastic panel in transonic flow.Journalof Aircraft, 40(2), 338-347.

[11] Elkadri E, N. E., Soulaimani, A., & Deschenes, C. (2000).A finite element formulation of compressible flows usingvarious sets of independent variables.Computer Methodsin Applied Mechanics and Engineering, 181(1-3), 161-189.

[12] Gmur, T. (1997). Dynamique des structures analyse modalenumrique. Lausanne, Suisse : Presses polytechniques etuniversitaires romandes.


-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

Genera

lized dis

pla

cem

ent (1

0-2

)G

enera

lized dis

pla

cem

ent (1

0-2

)

Mode 1

Time (s)

with linear mesh model

with nonlinear mesh model

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

Time (s)

Genera

lized dis

pla

cem

ent (1

0-2

)

Mode 2with linear mesh model

with nonlinear mesh model

Figure 9. Comparisons of the generalized displacements obtained by

Schur-complement algorithm with linear and nonlinear mesh models of

the first two modes under a load perturbation at aerodynamic pressures

60 lb/ f t2

[13] Feng, Z. (2005). A nonlinear computational aeroelasticitymodel for aircraft wings. Ph.D. Thesis, l’Ecole de technolo-gie superieure, University of Quebec, Montreal.

[14] Soulaimani, A., & Fortin, M. (1994). Finite element solu-tion of compressible viscous flows using conservative vari-ables.Computer Methods in Applied Mechanics and Engi-neering, 118(3-4), 319-350.

[15] Soulamani, A., Feng, Z., & Ben Haj Ali, A. (2005). Solu-tion techniques for multi-physics problems with applicationto computational nonlinear aeroelasticity.Nonlinear Anal-ysis, 63(5-7), 1585-1595.

[16] Saad, Y. (1996). Iterative methods for sparse linear systems.Boston, Mass. : PWS Publishing.

[17] Farhat, C., Lesoinne, M., & Le Tallec, P. (1998). Loadand motion transfer algorithms for fluid/structure interac-tion problems with non-matching discrete interfaces : Mo-

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

Genera

lized dis

pla

cem

ent (1

0-2

)

Mode 1

Time (s)

Schur-complement

Gauss-Seidel

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

Time (s)

Genera

lized dis

pla

cem

ent (1

0-2

)

Mode 2Schur-complement

Gauss-Seidel

Figure 10. Comparisons of the generalized displacements obtained by

Gauss-Seidel and Schur-complement algorithms of the first two modes

under a load perturbation at aerodynamic pressures 60 lb/ f t2

mentum and energy conservation, optimal discretizationand application to aeroelasticity.Computer Methods in Ap-plied Mechanics and Engineering, 157(1-2), 95-114.

[18] Soulamani, A., BenElHajAli, A., & Feng, Z. (2004). AParallel-Distributed Approach for Multi-Physic Problemswith Application to Computational Nonlinear Aeroelastic-ity. the Canadian Aeronautics and Space Journal, 50(4), p.221-235.

[19] Gupta, K. K. (1996). Development of a finite elementaeroelastic analysis capability.Journal of Aircraft, 33(5),995-1002.

[20] Lesoinne, M., Sarkis, M., Hetmaniuk, U., & Farhat, C.(2001). A linearized method for the frequency analysis ofthree-dimensional fluid/structure interaction problems in allflow regimes.Computer Methods in Applied Mechanicsand Engineering, 190(24-25), 3121-3146.


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