AIAA 2001-1526Three-Dimensional Transonic AeroelasticityUsing Proper Orthogonal DecompositionBased Reduced Order ModelsJeffrey P. Thomas, Earl H. Dowell, and Kenneth C. HallDuke University, Durham, NC 27708–0300

42nd AIAA/ASME/ASCE/AHS/ASCStructures, Structural Dynamics, and Materials

Conference and ExhibitApril 16–19, 2001/Seattle, WA

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics1801 Alexander Bell Drive, Suite 500, Reston, VA 20191–4344

Three-Dimensional Transonic Aeroelasticity

Using Proper Orthogonal Decomposition

Based Reduced Order Models

Jeffrey P. Thomas, ∗ Earl H. Dowell, † and Kenneth C. Hall ‡

Duke University, Durham, NC 27708–0300

The proper orthogonal decomposition (POD) based reduced order modeling (ROM)technique for modeling unsteady frequency domain aerodynamics is developed for a largescale computational model of an inviscid flow transonic wing configuration. Using themethodology, it is shown that a computational fluid dynamic (CFD) model with over athree quarters of a million degrees of freedom can be reduced to a system with just a fewdozen degrees of freedom, while still retaining the accuracy of the unsteady aerodynamicsof the full system representation. Furthermore, POD vectors generated from unsteadyflow solution snapshots based on one set of structural mode shapes can be used for differ-ent structural mode shapes so long as solution snapshots at the endpoints of the frequencyrange of interest are included in the overall snapshot ensemble. Thus, the snapshot com-putation aspect of the method, which is the most computationally expensive part of theprocedure, does not have to be fully repeated as different structural configurations areconsidered.

Nomenclature

AR = aspect ratio = wing span squared/wing areaA = matrix defining homogeneous part of

discretized fluid-dynamic operatorA = reduced-order form of A

B = matrix relating modal structural motioncoordinates ξ to CFD boundary conditions

B = reduced-order form of B

b, c = semi-chord and chord respectivelyC = matrix relating unsteady flow q to generalized

forces CQ acting on wing

C = reduced-order form of C

CQ = vector of nondimensional generalized forces

I = identity matrixj =

√−1

m = mass of wingM∞ = freestream Mach numberM = number of structural modesM = generalized mass matrixN = number of degrees of freedom of CFD modelNP = number of POD vectorsNS = total number of solution snapshots vectorsp = unsteady pressureq∞ = freestream dynamic pressureQ,q = vectors for steady and small-disturbance

∗Research Assistant Professor, Department of MechanicalEngineering and Materials Science, Member AIAA.

†J. A. Jones Professor, Department of Mechanical Engi-neering and Materials Science, and Dean Emeritus, School ofEngineering, Fellow AIAA.

‡Professor and Department Chairperson, Department of Me-chanical Engineering and Materials Science, Associate FellowAIAA.

Copyright c© 2001 by Jeffrey P. Thomas, Earl H. Dowell, andKenneth C. Hall. Published by the American Institute of Aeronau-tics and Astronautics, Inc. with permission.

flow solutionsQ = vector of generalized forcesr = magnitude of reduced Laplace variables ss = complex reduced frequency Laplace variable,

s = jωU∞ = freestream velocityv = volume of a truncated cone having streamwise

root chord as lower base diameter, streamwisetip chord as upper base diameter, and winghalf span as height

v,v = reduced order fluid dynamic variable andvector of reduced order fluid dynamic variables

V = reduced velocity, V = U∞/√

µωαbα0 = airfoil or wing root steady flow angle of attackθ = angle made by reduced Laplace variable s in

complex frequency plane, s = rejθ

λt = wing taper ratio λt = ct/cr

µ = mass ratio for wing configuration, µ = m/ρ∞vρ∞ = freestream densityφ,Φ = POD vector and matrix of POD vectorsψ = structural mode shapeω, ω = frequency and reduced frequency

ω = ωc/U∞ (airfoil) ω = ωb/U∞ (wing)ωα = wing first torsional mode natural frequencyΩ = matrix with structural frequency ratios

squared along main diagonal(i.e. (ω1/ωα)2, ..., (ωM/ωα)2)

ξ, ξ = structural coordinate and vector ofstructural coordinates

Subscripts/Superscriptsf = flutter (i.e. neutral stability) conditionH = Hermitian transposer, t = root and tip respectively

1 of 10

American Institute of Aeronautics and Astronautics Paper 2001-1526

IntroductionIn the following paper, we demonstrate how the

recently devised proper orthogonal decomposition(POD) based reduced order modeling (ROM) tech-nique1–3 can be used to model unsteady aerodynamicand aeroelastic characteristics of three-dimensional in-viscid transonic wing configurations. Although an in-compressible vortex lattice fluid model1 and transonicEuler CFD models2, 3 have been previously studied,these initial demonstrations of the POD/ROM methodhave been for two-dimensional airfoil configurationswith at most two structural degrees of freedom, e.gplunge and pitch.

In extending the POD/ROM technique to three-dimensions, two primary issues have been of con-cern. First, the size of the computational fluid dy-namic (CFD) model will in general be at least anorder of magnitude greater than for two-dimensions.Whereas a typical CFD model for a realistic two-dimensional configuration might have on the order of10’s or even 100’s of thousands of degrees of freedom(DOF), a CFD model for a three-dimensional config-uration might easily have on the order of hundredsof thousands if not millions or more DOF’s. In two-dimensions, we have found that very accurate ROM’swith on the order of only a few dozen DOF’s can bedevised using the POD/ROM methodology. A firstissue to address has thus been whether or not in three-dimensions one can also generate accurate ROM’s,which also require at most a few dozen DOF’s.

The second concern is, for any variation of thestructural properties of a given wing under consid-eration, will a completely new ensemble of solutionvector “snapshots” have to be computed in order todevise an accurate POD/ROM. A basic aspect of thePOD/ROM method is that an ensemble of solutionvectors is first assembled by computing unsteady CFDsolutions at a number of discrete frequencies within afrequency range of interest for the unsteady structuralmotions that are also of interest. In two dimensions,this step is relatively straight forward since one onlyhas to consider a few possible motions, e.g. pitch andplunge.

In three-dimensions however, the wing vibratorymode shapes will be different for each different struc-tural configuration of a given wing. As such, therecan be a substantial number of unsteady motions (orat least a number of motions equivalent to the num-ber of DOF’s of the discrete structural model) onemust consider. So the second concern about extendingthe POD/ROM to three-dimensions has been whetheror not it is necessary to compute a completely differ-ent ensemble of solution snapshots for every possiblestructural configuration. For example, say one com-putes solution snapshots for a given wing configurationbased on the wing’s particular vibratory modes shapesin order to develop a POD/ROM to model the config-

uration’s aeroelastic characteristics. The question is,if the structural make-up of the wing changes, doesone have to compute a whole new ensemble of solutionsnapshots for the different set of vibratory structuralmode shapes.

Fortunately in addressing these two issues, and aswill be shown in the following, we have found thataccurate POD/ROM’s with just a few dozen degreesof freedom can in fact be created for a realistic tran-sonic three-dimensional configurations. This is trueeven though for the configuration to be shown subse-quently, the CFD model is easily an order of magni-tude larger than anything we have previously studiedin two-dimensions. Furthermore, we have discoveredthat a “fundamental” ensemble of solution snapshots,based on wing motions that need not be identical tothe structural modes under consideration, can be as-sembled as a first step. Accurate POD/ROM’s for agiven wing configuration can then be created by simplyadding to this “fundamental” ensemble, the snapshotscorresponding to actual wing structural modal mo-tions solely at the frequencies corresponding to the endpoints of the frequency range of interest. In general,these two snapshots prove to be sufficient to “lock in”the conditions corresponding to the particular struc-tural motion, and indeed this fundamental ensemble ofsolution snapshots is sufficient to reveal the unsteadydynamics of the fluid dynamic model. Consequently,this fundamental ensemble of snapshots can be usedagain and again even as the structural mode change,and the computational cost of having to compute anentirely new snapshot ensemble for every new struc-tural configuration is thus greatly reduced.

POD/ROM MethodologyIn the following, we will be considering inviscid tran-

sonic three-dimensional flows and, more specifically,linearized (about some nonlinear background steadyflow) unsteady frequency-domain CFD solutions to theEuler equations. The POD/ROM procedure can beapplied in principle to any conventional CFD method.The CFD method we have employed for the presentanalysis is a variant of Ni’s4 approach to the standardLax-Wendroff method. The frequency domain CFDmethod in effect represents a linear system formula-tion of the unsteady fluid dynamic model, i.e.

Aq = −Bξ (1)

where q is the N dimensional vector (N is the num-ber of mesh points times the number of dependentflow variables) of dependent unsteady flow variablesat each mesh point in the CFD domain, and ξ =ξ1, · · · , ξMT is the M dimensional vector (M is thenumber structural mode shapes) of modal coordinatesfor the structural model. A is the N × N fluid dy-namic influence matrix, and B is the N × M matrixthat relates the flow solver boundary conditions to

2 of 10

American Institute of Aeronautics and Astronautics Paper 2001-1526

each particular structural mode shape. A and B areboth functions of the background flow Q and unsteadyfrequency ω. The structural equations for the wingconfiguration can be written as

Dξ = −Cq (2)

where D is the M × M structural influence matrix,which in the case of the case of a swept delta wingconfiguration can be expressed as

Dξ = kwM

(

−ω2µ I +1

V 2Ω

)

ξ (3)

where kw is a constant dependent on the wing shapeand overall mass given by

kw =πAR(1+λt)(1+λt+λt

2)

6m. (4)

C is the M × N matrix that represents the discreteintegration operator used to obtain the generalizedforces associated with each structural mode shape andunsteady flow q, i.e.

Cq = −CQ (5)

where CQ = Q/q∞c2r (Q = Q1, · · · ,QMT ) is

the vector of frequency dependent generalized aero-dynamic force coefficients acting on the wing. Thegeneralized aerodynamic force coefficients can also bewritten in matrix form as

CQ =

[∂CQi

∂ξj

]

︸ ︷︷ ︸

M×M

ξ = [Ei,j ] ξ (6)

where[

∂CQi

∂ξj

]

(¯ω) is the M×M matrix of aerodynamic

transfer functions (∂CQi/∂ξj being the coefficient of

the ith generalized force due to the jth structuralmodal coordinate) that we wish to approximate withthe reduced order modeling strategy.

Coupling Eqs. 1 and 2, e.g.

[A B

C D

] q

ξ

=

0

0

, (7)

yields the aeroelastic system of equations, which fornontrivial q and ξ, represents an eigenvalue problemwith ω being the eigenvalue. Any eigenvalues with apositive real part imply the aeroelastic system is un-stable.

The problem with constructing and solving thiseigenvalue problem is that A is simply too large forrealistic configurations. As mentioned in the intro-duction, N can easily be on the order of 104 or 105

for two-dimensional configurations, and on the orderof 105 to 106 or even more for three dimensional con-figurations. For such large cases, even attempting to

set up A is well beyond the memory limits of currentcomputers.

The basic premise of the POD/ROM technique isthat we assume the unknown flow-field solution vectorq can be expressed as a Ritz type expansion of theform

q ≈NP∑

n=1

vnφn NP N (8)

where vn is a generalized coordinate sometimes re-ferred to as an augmented aerodynamic state variable,and φn is the corresponding Ritz vector. NP is thenumber of Ritz vectors, which in the following will bePOD vectors, used in the expansion. Equation 8 canalso be written in matrix form as

q=Φv, where Φ=

| | |φ1φ2· · ·φNP

| | |

and v=

v1

v2

...vNP

.

(9)Here, Φ is an N×NP matrix whose nth column is theshape vector φn, and v is the NP dimensional vectorof augmented aerodynamic state variables vn.

A reduced-order representation of the fluid dynamicand aeroelastic systems can be formulated by sub-stituting Eq. 9 into Eq. 1 (and/or Eq. 7) and pre-multiplying by the Hermitian transpose of Φ. Forinstance

ΦHAΦv = ΦHBξ or Av = −Bξ (10)

and

ΦHI[A B

C D

] Φv

ξ

=

[A B

C D

] v

ξ

=

0

0

(11)

If the Ritz approximation is a good one (i.e. NP ismuch less than N with no essential loss in accuracy),then Eqs. 10 and 11 will be much smaller models of theoriginal fluid dynamic (Eq. 1) and aeroelastic (Eq. 7)systems that can be quickly assembled and readilysolved using conventional eigenvalue techniques.

As will be shown in the following, NP N , andEqs. 10 and 11 thus represent much smaller models forthe original fluid dynamic and/or aeroelastic systemsthat can readily be assembled and examined using con-ventional eigenvalue techniques.

The next question becomes what are good choicesfor the Ritz vectors φn that will in fact result ingood Ritz approximations. Previous detailed stud-ies2, 3 have demonstrated that shape vectors derivedvia the proper orthogonal decomposition technique5–7

are an excellent choice. For the sake of brevity, thedetails are omitted here. However a discussion of howthe shapes are derived can be found in References.2, 3

First, solution “snapshots” are directly computed fromthe original CFD model for several discrete frequencies

3 of 10

American Institute of Aeronautics and Astronautics Paper 2001-1526

and structural motions of interest. From this ensembleof solution vectors, the POD shapes are easily derivedby solving a small (the size of the number of snapshots)eigenvalue problem. Typically the first few dozen PODmodes describe the most dominant dynamic charac-teristics of the fluid dynamic system, and as such, thePOD shapes have proven to be an excellent set of Ritzvectors for fluid dynamic and/or aeroelastic models.

Model ProblemThe configuration under consideration is the

AGARD model 445.6 wing.8, 9 This is a 45 degreequarter chord swept wing based on the NACA 64A004airfoil section that has an aspect ratio of 3.3 (for thefull span) and a taper ratio of 2/3. Figure 1 illustratesthe computational mesh employed for this configura-tion. The grid is an “O-O” topology that employs49 computational nodes about each airfoil section, 33nodes normal to the wing, and 33 nodes along the semi-span. The total number of fluid dynamic DOF for thisCFD mesh is thus 266,805. The outer boundary of thegrid extends five semi-spans from the mid-chord of thewing root section. The particular structural configu-ration of the wing under consideration is referred to asthe “2.5 ft. weakened model 3”.8, 9

Figure 2 shows the computed wing surface and sym-metry plane steady flow pressure contours for the back-ground flow freestream Mach numbers of M∞ = 0.960and M∞ = 1.141. A relatively weak shock can be seenat the trailing edge for M∞ = 0.960. This shock ap-pears to get stronger at M∞ = 1.141. The wing sectionis quite thin (4%), so a strong shock is not really ex-pected. Comparing contours, our flow-fields look verysimilar to those of Lee-Rausch and Batina,10 althoughthey employed a finer mesh.

Flutter ResultsFigure 3 shows the eigenvalue root-loci for various

steady background flow Mach numbers. The “gain”for the root-loci is the mass ratio or, equivalently, thereduced velocity for a fixed physical velocity. Solu-tion snapshots are computed for the first five givenwing structural mode shapes8 for reduced frequenciesfrom ω = 0.0 to ω = 0.5 in ∆ω = 0.1 increments.This particular structural configuration flutters for re-duced frequencies less than 0.5 thus making solutionsnapshots for ω > 0.5 unnecessary. Including the com-plex conjugate solutions for the corresponding negativevalued reduced frequencies in the overall ensemble re-sults in a total of 55 available POD shape vectors.In Fig. 3, the curves represent the eigenvalues corre-sponding to the primarily structural natural modes asmass ratio is varied. Our method also determines theaeroelastic modes originating from the fluid dynamicmodes of the POD/ROM. For the range of mass ratios(0 ≤ µ ≤ 300) considered in these parametric analy-ses, the fluid dynamic modes are very damped, and as

a) Wing Surface and Symmetry Plane Grids

b) Outer Boundary Grid

Fig. 1 AGARD 445.6 Wing Grid Topology.

such lie to the left and outside of the eigenspectrumrange shown. As can been, for each of the Mach num-bers, the first structural mode tends to be the criticalflutter mode. For the highest Mach number however,the third structural mode can go unstable if the massratio is large enough. Also from this figure, it can beseen that it is unnecessary to use all 55 of the avail-able POD shapes. If fact, with less than one half ofthe POD modes (25 for instance), well converged re-sults (in the sense of POD mode refinement) can beachieved.

Figure 4 shows the computed POD/ROM flutterspeed and flutter frequency ratios, along with experi-mental data,8 and data from two other computationalmethods,10, 11 as a function of Mach number. As canbe seen, using the POD/ROM methodology one canpredict the well known transonic flutter speed dip,and our results are all within the same tolerance to

4 of 10

American Institute of Aeronautics and Astronautics Paper 2001-1526

a) M∞ = 0.960

b) M∞ = 1.141

Fig. 2 AGARD 445.6 Wing Surface and Symme-try Plane Pressure Contours.

the experimental results as the other computationalmethods. Gupta11 does show better agreement withexperiment at the two supersonic Mach numbers, andGupta attributes this better agreement to better CFDgrid refinement.

As a check on mesh convergence, we examined threedifferent grid resolutions for M∞ = 1.141. The follow-ing are the mesh sizes and the corresponding computedflutter reduced velocities:

Mesh Size Vf

33× 25 × 25 0.642149× 33 × 33 0.659165× 49 × 49 0.6653

The 65 × 49 × 49 mesh in fact corresponds to a CFDmodel with 780,325 DOF that can remarkably be re-duced to a system with just 55 DOF. As can be seen,

−0.06 −0.04 −0.02 0.00 0.02

Real Frequency Ratio, Re(λ/ωα)

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5M=1.072

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Imag

Fre

quen

cy R

atio

, Im

(λ/ω

α) M=0.901

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5M=0.499

25 of 55 POD Vectors35 of 55 POD Vectors45 of 55 POD Vectors55 of 55 POD Vectors

−0.06 −0.04 −0.02 0.00 0.02

M=1.141

M=0.960

M=0.678

Fig. 3 Aeroelastic Root-Loci at Various MachNumbers for the AGARD 445.6 Wing “Weakened”Configuration, α0 = 0.0 (deg).

the results are changing only slightly for each meshrefinement. Note, our results do not match Yates’s8

experimental (Vf ≈ 0.41) or Gupta’s11 computational(Vf ≈ 0.40) results. However our result matches al-most exactly the computational result of Lee-Rauschand Batina.10 At other Mach numbers, there are moremodest differences among all the methods. Thus ourconclusion is that the differences among the differentcomputational methods are most likely due to the par-ticular CFD methods employed and the grid layout,rather than due to grid refinement per se.

Figure 5 again shows the computed POM/ROMflutter speed and flutter frequency ratios as a functionof Mach number. In this instance, however, the flutterresults are shown for various numbers of POD modesor Ritz vectors retained in the reduced order model.As can be seen, even with as few as one-half of theavailable shapes, well converged results are obtained.

5 of 10

American Institute of Aeronautics and Astronautics Paper 2001-1526

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

Mach Number, M∞

0.2

0.3

0.4

0.5

0.6

0.7

Flu

tter

Vel

ocity

, Vf=

U∞/µ

1/2 ω

αbYates − ExperimentalLee−Rausch and Batina − ComputationalGupta − ComputationalPOD/ROM 55 of 55 POD Shapes

a) Flutter Speed

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

Mach Number, M∞

0.3

0.4

0.5

0.6

0.7

0.8

Flu

tter

Fre

quen

cy R

atio

, ωf/ω

α Yates − ExperimentalLee−Rausch and Batina − ComputationalGupta − ComputationalPOD/ROM 55 of 55 POD Shapes

b) Flutter Frequency Ratio

Fig. 4 Mach Number Flutter Trend for theAGARD 445.6 Wing “Weakened” Configuration,α0 = 0.0 (deg).

Note however there is somewhat greater sensitivity tothe number of POD/ROM modes retained at the su-personic Mach numbers.

The Use of Alternate ModalExcitations for Snapshots

As mentioned before, one of the key concerns aboutusing the POD/ROM method for three-dimensionalconfigurations has been whether or not an entire setof solution snapshots must be computed for each possi-ble structural configuration of interest. That is, say wewish to consider a equivalently shaped wing that hasa different structural make-up, which in turn meansdifferent wing vibratory structural mode shapes. Thequestion has been whether or not this means that onehas to compute a whole new ensemble of solution snap-

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

Mach Number, M∞

0.2

0.3

0.4

0.5

0.6

0.7

Flu

tter

Vel

ocity

, Vf=

U∞/µ

1/2 ω

αb

POD/ROM 25 of 55 POD Shapes POD/ROM 35 of 55 POD ShapesPOD/ROM 45 of 55 POD ShapesPOD/ROM 55 of 55 POD Shapes

a) Flutter Speed

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

Mach Number, M∞

0.3

0.4

0.5

0.6

0.7

0.8

Flu

tter

Fre

quen

cy R

atio

, ωf/ω

α POD/ROM 25 of 55 POD Shapes POD/ROM 35 of 55 POD ShapesPOD/ROM 45 of 55 POD ShapesPOD/ROM 55 of 55 POD Shapes

b) Flutter Frequency Ratio

Fig. 5 POD/ROM Shape Vector RefinementsCharacteristic for Mach Number Flutter Trend forAGARD 445.6 Wing “Weakened” Configuration,α0 = 0.0 (deg).

shots based on these new structural motions in orderto do a flutter analysis for the new wing configuration.Fortunately, as we will demonstrate in the following,although a few snapshots based on the new modal mo-tion will need to be computed, the larger number ofsnapshots computed at all the numerous frequencieswill be unnecessary. The snapshots computed from aprevious wing structural configuration will still be ableto serve the purpose. Figure 6 demonstrates how thisworks.

Figure 6a shows the real and imaginary parts of thecoefficient of the generalized aerodynamic force corre-sponding to the first mode pressure acting through thefirst mode shape as a function of reduced frequency ata Mach number of 0.960. The coefficient of the gener-

6 of 10

American Institute of Aeronautics and Astronautics Paper 2001-1526

0.0 0.2 0.4 0.6 0.8 1.0

Reduced Frequency, ω− =ωb/U∞

−600

−400

−200

0

200

Gen

eral

ized

For

ce C

oeffi

cien

t, E

1,1

Real − ComputedImag − Computed

a) Snapshots

0.0 0.2 0.4 0.6 0.8 1.0

Reduced Frequency, ω− =ωb/U∞

−600

−400

−200

0

200

Gen

eral

ized

For

ce C

oeffi

cien

t, E

1,1,

Real − ComputedImag − ComputedReal − POD/ROMImag − POD/ROM

b) POD/ROM Based on ω = 0.0 and ω = 1.0 Modal Snapshots

0.0 0.2 0.4 0.6 0.8 1.0

Reduced Frequency, ω− =ωb/U∞

−600

−400

−200

0

200

Gen

eral

ized

For

ce C

oeffi

cien

t, E

1,1

Real − ComputedImag − ComputedReal − POD/ROMImag − POD/ROM

c) POD/ROM Based on “Basic” Motions Snapshots

0.0 0.2 0.4 0.6 0.8 1.0

Reduced Frequency, ω− =ωb/U∞

−600

−400

−200

0

200

Gen

eral

ized

For

ce C

oeffi

cien

t, E

1,1

Real − ComputedImag − ComputedReal − POD/ROMImag − POD/ROM

d) POD/ROM Based on ω = 0.0 and ω = 1.0 Modal Snapshotsand “Basic” Motions Snapshots

Fig. 6 Generalized Force Modeling with Unrelated Structural Mode Shape Snapshots.

alized aerodynamic force is given by

Ei,j(ω) =1

q∞c2r

∫∫

A

ψi pj(ω)nz dA. (12)

where the integral is evaluated over the surface of thewing, and nz is z component of the wing surface nor-mal vector (i.e. n = nxi + ny j + nzk and n is orientedto point towards the wing surface). In this definition,pj(ω) represents the frequency dependent unsteadypressure resulting from a wing deformation motion of

z

cr

= ψi (13)

where ψi is the ith structural mode shape.The results for E1,1 presented in Fig. 6a are based

on the actual solution snapshots and thus are what wedesire the POD/ROM to be able to model. In Fig. 6b,

the POD/ROM of E1,1 based on solution snapshotsfor each of the five structural mode shapes, but onlyat frequencies of ω = 0.0 and ω = 1.0 (for a total of10 snapshots), is compared to E1,1 obtained from theactual snapshots for all frequencies between ω = 0.0and ω = 1.0. As can be seen, the POD/ROM matchesat the end points of the frequency range as is expected,however this crude POD/ROM performs rather poorlyfor the intermediate frequencies. Of course, if we usesnapshots at all the frequencies between ω = 0.0 andω = 1.0, the POD/ROM would exactly reproduce thedata.

Next, in Fig. 6c, a new POD/ROM for E1,1 is shownbased on solution snapshots different from the actualmode shapes. These simple wing motion snapshotsare for a full wing plunge motion (up/down), full wingpitch about the quarter chord, a first bending type

7 of 10

American Institute of Aeronautics and Astronautics Paper 2001-1526

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

Mach Number, M∞

0.2

0.3

0.4

0.5

0.6

0.7

Flu

tter

Vel

ocity

, Vf=

U∞/µ

1/2 ω

αbPOD/ROM − Actual Wing Modal Motion Snapshots POD/ROM − Alternate Wing Motion Snapshots

a) Flutter Speed

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

Mach Number, M∞

0.3

0.4

0.5

0.6

0.7

0.8

Flu

tterF

lutte

r F

requ

ency

Rat

io, ω

f/ωα

POD/ROM − Actual Wing Modal Motion Snapshots POD/ROM − Alternate Wing Motion Snapshots

b) Flutter Frequency Ratio

Fig. 7 Mach Number Flutter Trends Using Alter-nate Snapshots: AGARD 445.6 Wing “Weakened”Configuration, α0 = 0.0 (deg).

of motion (wing is fixed at the root, and the z co-ordinate component of deflection varies linearly withspan), and a first twist type of motion (wing is fixedat the root, and the pitch varies linearly with span)for frequencies from ω = 0.0 to ω = 1.0 at ∆ω = 0.1increments for a total of 44 solution snapshots. As canbe seen, the POD/ROM in this case also perform verypoorly. Unbeknownst however, these solutions are infact helping to reveal the dynamics of the system. Infact, when one uses these snapshots in combinationwith the actual structural mode snapshots solely atthe end points of the frequency range of interest, onegets a POD/ROM that produces very accurate resultsfor E1,1 as is evident from Fig. 6d.

Fig. 7 shows a comparison of the POD/ROM Mach

number flutter trend for two cases, (1) the POD/ROMbased on solution snapshots corresponding to the ac-tual modal shapes of the wing, and (2) based on snap-shots using the simple wing motion mode shapes asdiscussed in the previous paragraph. As for the previ-ous Mach number flutter results (Figs. 4 and 5), thesnapshot reduced frequencies range from ω = 0.0 toω = 0.5 in ∆ω = 0.1 increments. Included in this sec-ond ensemble of solution snapshots, are the snapshotscorresponding to the actual mode shapes at the endpoints of the frequency range of interest.

As can be seen in Fig. 7, one can obtain accuratePOD/ROM flutter results using solution snapshots dif-ferent from the actual wing motions (except at theend points of the frequency range of interest) thatcompare very well to the flutter results based on aPOD/ROM model using snapshots corresponding tothe actual motions. This is especially true at the lowerMach numbers. There is some difference at the highestMach number, again suggesting the supersonic case ismore sensitive for this wing.

Finally we present similar results for a simple two-dimensional configuration. Consider an unsteadyNACA 64A010A airfoil configuration that not onlyunder goes typical plunge and pitch motions (seeFigs. 8a and 8b), but also has motions where airfoilmean camber line distorts based on simple trigono-metric functions. i.e. zc(x) = σ1 cos(2πx/c) (Fig. 8c),zc(x) = σ2 sin(2πx/c) (Fig. 8d), zc(x) = σ3 cos(4πx/c)(Fig. 8e), etc. The initial question was, as one con-siders each subsequent motion, does one have to in-clude a number of snapshots based on the new motionthat is equal to the number of snapshots for each ofthe previous motions in order to produce an accuratePOD/ROM.

Figure 9 illustrates how after a sufficient number ofsnapshots have been included in the snapshot ensem-ble, only the end point frequencies are required foreach additional motion. In this instance, the NACA64A010A airfoil is modeled in a M∞ = 0.5, α0 = 0.0(deg) background flow, and shown on the abscissa areincrements in the number of overall motions consid-ered. Shown on the ordinate is the order in whichthe snapshots for each particular motion are added tothe overall ensemble. The reduced frequency range ofinterest is 0.0 ≤ ω ≤ 1.0, and thus the first two snap-shots considered for each motion correspond to theend points of this frequency range. Further snapshotsare added to the ensemble for a given motion to bestmodel the intermediate frequencies.

Shown in Fig. 9 is the accuracy achieved in modelingthe airfoil unsteady lift and moment along the pathss = rejθ (where θ = 90, 60, 120 and 0 ≤ r ≤ 1) inthe complex reduced frequency s plane. The curves il-lustrate the number of snapshots necessary to achievea given level of accuracy for the nth and all previousmotions. The accuracy is based on a comparison to a

8 of 10

American Institute of Aeronautics and Astronautics Paper 2001-1526

a) Plunge h b) Pitch α c) σ1 d) σ2

e) σ3 f) σ4 g) σ5 h) σ6

i) σ7 j) σ8 k) σ9 l) σ10

m) σ11 n) σ12 o) σ13 p) σ14

Fig. 8 Airfoil Sample Motions: NACA 64A010AAirfoil Section - 129 x 65 Mesh.

POD/ROM that is derived from a snapshot ensemblecomprised of all the possible motions at all the pos-sible frequencies. So for example, to achieve a 10−3

L2 norm accuracy when just considering plunge mo-tion, one needs a total of ten plunge snapshots for thefrequency values indicated on the ordinate of the plot.Next considering pitch motion, one would then need toadd only three pitch motion snapshots correspondingto the frequencies ω = 0.0, ω = 1.0, and ω = 0.5 tothe overall ensemble to get 10−3 L2 norm accuracy fornow both the pitch and plunge motions. Consideringnext the first airfoil bending motion σ1, one would thenneed to add a total of seven σ1 snapshots to achieve10−3 L2 norm accuracy for now plunge, pitch, and σ1

motion. Three σ2 snapshots would then be neededwhen also taking in account σ2 motions, two σ3 snap-shots when considering σ3 motions, and so on.

As can be seen, Fig. 9 illustrates the interesting re-sult that after a sufficient number of snapshots havebeen included in the overall ensemble, only the two endpoint frequency snapshots for each subsequent possiblemotion need be added to the ensemble to maintain agiven level of accuracy. Interestingly enough, this ap-pears to be an asymptotic limit. That is, the two endpoint frequency range snapshots always appear to benecessary when considering a large number of motions.

0 1 2 3 4 5 6 7 8 9 1011121314151617

Excitations

0

1

2

3

4

5

6

7

8

9

10

11

Num

ber

and

Fre

quen

cy o

f Add

ition

al S

naps

hots

10−2 L2 Error Norm Accuracy10−3 L2 Error Norm Accuracy

−σ−

1cos

(2πx

/c)

−α−

(P

itch)

−σ−

2sin

(2πx

/c)

−σ−

3cos

(4πx

/c)

−σ−

4sin

(4πx

/c)

−σ−

5cos

(6πx

/c)

−σ−

6sin

(6πx

/c)

ω− = 0.0

ω− = 1.0

ω− = 0.5

ω− = 0.625

ω− = 0.75

ω− = 0.25

ω− = 0.375

ω− = 0.875

ω− = 0.5625

−h−

(P

lung

e)

−σ−

7cos

(8πx

/c)

−σ−

8sin

(8πx

/c)

−σ−

9cos

(10π

x/c)

−σ−

10si

n(10

πx/c

)

−σ−

11co

s(12

πx/c

)

−σ−

12si

n(12

πx/c

)

−σ−

13co

s(14

πx/c

)

−σ−

14si

n(14

πx/c

)

ω− = 0.125

Fig. 9 Additional Snapshot Requirements toAchieve Two Different Levels of Accuracy for Liftand Moment Transfer Functions Along the Pathss = rejθ, where θ = 90, 60, 120 and 0 ≤ r ≤ 1, in theComplex Reduced Frequency s Plane for nth andAll Previous Motions: M∞ = 0.5, α0 = 0.0 (deg).

Conclusions

The POD/ROM method has been demonstrated forthe flutter analysis of a three-dimensional flow tran-sonic wing configuration. We have shown that thenumber of ROM DOF’s necessary to create accuratemodels is on the order of a few dozen as is the casein two-dimensions. We have also shown that it is un-necessary to compute a completely new ensemble ofsolution snapshots based on the vibratory mode shapesfor each new structural configuration that might beunder consideration. One can simply compute a setof snapshots based on some basic wing motions at anumber of frequencies. Then snapshots only at theend points of the frequency range of interest needto be computed for the specific mode shapes of theconfiguration of interest. These end point snapshots“lock in” the unsteady fluid dynamic characteristicsfor the particular mode shapes, and the simple motionsnapshots then act to resolve the dominant dynam-ics of the flow throughout the full frequency rangeof interest. Both of these observations suggest thatthe POD/ROM methodology will be very useful fordesign studies of transonic aeroelastic configurationswhen several structural configurations may be underconsideration, and where a range of structural and flowparameters must be examined.

9 of 10

American Institute of Aeronautics and Astronautics Paper 2001-1526

AcknowledgmentsThe authors would like to acknowledge with appre-

ciation the support of AFOSR and program director,Dr. Daniel Segalman, for this work.

References1Kim, T., “Frequency-Domain Karhunen-Loeve

Method and Its Application to Linear Dynamic Systems”,AIAA Journal, Vol. 36, No. 11, 1998, pp. 2117–2123.

2Hall, K. C., Thomas, J. P., and Dowell, E. H., “ProperOrthogonal Decomposition Technique for Transonic Un-steady Aerodynamic Flows”, AIAA Journal, Vol. 38,No. 10, 2000, pp. 1853–1862.

3Thomas, J. T., Hall, K. C. and Dow-ell, E. H, “Reduced-Order Aeroelastic Model-ing Using Proper-Orthogonal Decompositions”,CEAS/AIAA/ICASE/NASA Langley InternationalForum on Aeroelasticity and Structural Dynamics 1999,Williamsburg, Virginia, June 1999. (Under Review for theJournal of Aircraft)

4Ni, R. H., “A Multiple-Grid Scheme for Solving theEuler Equations”, AIAA Journal, Vol. 28, No. 12, 1982,pp. 2050–2058.

5Loeve, M., Probability Theory, D. Van Nostrand Com-pany, Inc., New York, 1955.

6Lumley, J. L., “The Structures of Inhomogeneous Tur-bulent Flow”, in Atmospheric Turbulence and Radio Wave

Propagation, edited by A. M. Yaglom and V. I. Tatarski,Nauka, Moscow, 1967, pp. 166–178.

7Holmes, P., Lumley, J. L., and Berkooz, G., Turbu-

lence, Coherent Structures, Dynamical Systems and Sym-

metry, Cambridge University Press, Cambridge, 1996.8Yates, E. C., Jr., “AGARD Standard Aeroelastic Con-

figurations for Dynamic Response I - Wing 445.6”, NASATM 100492, August 1987; also Proceedings of the 61st

Meeting of the Structures and Materials Panel, Germany,AGARD-R-765, 1985, pp. 1–73.

9Yates, E. C., Jr., Land, N. S., and Foughner, J. T. , Jr.,“Measured and Calculated Subsonic and Transonic FlutterCharacteristics of a 45 Deg Sweptback Wing Planform inAir and in Freon-12 in the Langley Transonic DynamicsTunnel”, NASA TN D-1616, March 1963.

10Lee-Rausch, E. M. and Batina, J. T., “Wing FlutterBoundary Prediction Using Unsteady Euler AerodynamicMethod”, Journal of Aircraft, Vol. 32, No. 2, 1995, pp. 416–422.

11Gupta, K. K., “Development of a Finite ElementAeroelastic Analysis Capability”, Journal of Aircraft,Vol. 33, No. 5, 1996, pp. 995–1002.

10 of 10

American Institute of Aeronautics and Astronautics Paper 2001-1526