reduced-order aeroelastic modeling using proper-orthogonal...

REDUCED-ORDER AEROELASTIC MODELING USINGPROPER-ORTHOGONAL DECOMPOSTIONS

Jeffrey P. Thomas�, and Kenneth C. Hall

�

, and Earl H. Dowell�

Duke UniversityDurham, NC 27708–0300

AbstractA new method for modeling small-disturbance unsteady

aerodynamics is presented. The method works in conjunctionwith a time-linearized (frequency-domain)computational flowsolver and can be used for any level of fluid dynamic approx-imation. For this study, we consider inviscid transonic Eu-ler flows for an isolated airfoil. The procedure begins withthe computation of the unsteady flow at a number of discretefrequencies over a selected interval for airfoil pitching andplunging motions. From this ensemble of previously com-puted flows, a newly developed frequency-domain form ofthe proper orthogonal decomposition (POD) technique is thenused to determine a set of shape vectors containing the mostdominant unsteady flow characteristics. These shape vectorsare next used in a Ritz expansion to form a reduced-ordermodel (ROM) of the unsteady flow. This reduced-order rep-resentation of the fluid dynamics is finally coupled with thetwo-degree-of-freedom typical airfoil structural model to forma reduced-order aeroelastic model. The resulting aeroelasticequations represent a generalized eigenvalue problem of suchgreatly reduced-order that aeroelastic stability may be rapidlydetermined. One of the primary advantages of this approachover more traditional rational polynomial curve-fit methods isthat the actual fluid dynamic operator is used in forming theunsteady aerodynamic model as opposed to tabulated aerody-namic data. We present a number of numerical examples todemonstrate the capability and accuracy of this new approach.

I. IntroductionIn recent years, the expanding power of modern com-

puters has enabled the simulation of fluid flows of ever in-creasing geometric and fluid dynamic complexity. Unsteadythree-dimensional flows involving full aircraft configurationsin transonic and viscous flow environments are becoming a re-ality. Although considerable effort has gone into developingtime-accurate methods for nonlinear motions of finite ampli-tude within such flows, the computation of small-disturbancefrequency-domain flows is also of great interest. Linearized

�Research Assistant Professor, Department of Mechanical Engineering

and Materials Science, Member AIAA.�Associate Professor, Department of Mechanical Engineering and Materi-

als Science, Associate Fellow AIAA.�Professor, Department of Mechanical Engineering and Materials Science,

and Dean Emeritus, School of Engineering, Fellow AIAA.Presented at the CEAS/AIAA/ICASE/NASA Langley International Fo-

rum on Aeroelasticity and Structural Dynamics 1999, Williamsburg, Virginia,June 1999.Copyright �

�1999 by Jeffrey P. Thomas, Kenneth C. Hall, and Earl H.

Dowell.

flow models, when coupled with linear structural models, al-low for aeroelastic stability analyses that may be rapidly per-formed over a wide range of structural parameters. For suchlinearized frequency-domain approaches, a model of the aero-dynamics as function of complex valued excitation frequencyis typically required. Although neutral stability conditions canbe determined using aerodynamic models valid for only purelyreal frequencies (e.g. an iterative V-g analysis), the capabilityto model aerodynamics for complex frequencies allows for amore direct approach to aeroelastic stability analysis. This isdue to the fact the aeroelastic equations represent an eigen-value problem whose eigenvalues are the system’s natural fre-quencies. The position of these eigenvalues within the com-plex plane dictates aeroelastic stability.

Although great improvements in computational algorithmsfor computing unsteady flows have been made in recent years,methods for modeling unsteady aerodynamics which are validfor complex valued frequencies are still typically based oncurve-fitting approaches developed nearly two decades ago.These techniques are typically rational polynomial curve-fitmethods using tabulated aerodynamic data computed for realfrequencies to then model unsteady aerodynamics for com-plex frequencies. These methods have generally fallen intothree different categories. These include Roger’s [1] commondenominator least-squares technique, various matrix Pade ap-proximant methods as proposed by Vepa [2], Edwards [3], andKarpel [4], and finally the “Minimum State” technique devel-oped by Karpel [4]. For each of these techniques, best approx-imations are determined in a least squares sense to tabulatedunsteady aerodynamic force data. Tiffany and Adams [5] inmore recent years have developed some nonlinear program-ming techniques which can further optimize these rationalpolynomial approximations.

This article presents a new method for unsteady aerody-namic modeling also valid for complex frequencies which webelieve is more straight forward and rigorous in its approach.Earlier work related to this new method has been presentedin [6]. This new approach has essentially the same cost as anyof the curve-fit methods in that the greatest expense of con-structing the model comes from the computational expense ofhaving to compute a number of unsteady flowfield solutions(we denote each of the solutions as a flowfield “snapshot”)over the range of frequencies and excitations of interest. How-ever, this new method is much more rigorous in that there isonly one way to implement it. For the rational polynomialcurve-fit procedures, in addition to there being a number ofdifferent expansion forms, each expansion has various optionsas to how it may be constrained. The three different forms

1

of the same second-order matrix Pade expansion of Vepa [2],Edwards [3], and Karpel [4] is a prime example of this char-acteristic.

Our method is applicable to any modern CFD method, andit can be considered a pre- and post-processor about the CFDsolver which then provides a method for quickly determiningthe unsteady aerodynamics for arbitrary complex valued fre-quencies. The method can then be coupled with a structuraldynamic model so that aeroelastic stability analyses may be ef-ficiently conducted. Our technique is essentially comprised oftwo aspects the first of which is the development of a reduced-order model (ROM) using previously computed flowfield so-lution “snapshots” as shape vectors for a Ritz-like expansionof the unknown solution. The second aspect is the use of theproper orthogonal decomposition (POD) as a method to de-termine from the ensemble of flow snapshots, a reduced num-ber of shape vectors containing the most dominant unsteadyflow characteristics which further enhances the efficiency ofthe overall procedure. We have thus named the procedure thePOD/ROM method for unsteady aerodynamic modeling.

In this paper, we begin by developing the aeroelastic gov-erning equations which for our method are combination of thefluid dynamic and structural dynamic systems of equations.We next proceed to describe the reduced-order modeling ap-proach along with an explanation of how the proper orthog-onal decomposition is used to determine optimal shape vec-tors. How the reduced-order model is used to reduce the or-der of the full aeroelastic system is then addressed. Althoughthe POD/ROM technique can be used for any level of fluiddynamic approximation, to initially demonstrate the methodfor a specific aeroelasticity problem, we investigate using theapproach for a benchmark two-dimensional transonic airfoilaeroelasticity test case. In the results section, we study a num-ber of POD/ROM models as compared to the full system solu-tion. In order to further evaluate POD/ROM performance, weinvestigate its accuracy in modeling some purely aerodynamicquantities and also compare the fluid dynamic eigenspectrumsfor the full and reduced-order systems.

II. Aeroelastic ModelingA. Structural Dynamic Model

Mid Chord

c

+h

+α

b

a b

Elastic Axis

α

b

Mean Position Mass Center

k

x b

αk

h

Figure 1: Geometry for “Typical” (Pitch/Plunge) Two-Degree-of-Freedom Airfoil Section Aeroelastic Model

The structural model follows the “typical” (Figure 1) air-foil section plunge (1a) and elastic axis pitching moment (1b)equations which can be expressed as�� "!#� � �� %$"&(' �(�#)* where+ � , = airfoil plunge (positive downward) and pitch (pos-

itive nose up) coordinates+ � ,$"&('

= airfoil sectional lift and sectional moment aboutthe elastic axis+�� ,

��,!#�

= airfoil sectional mass and first and secondmoments of inertia+ � � , �,� = airfoil bending and torsional mechanical resis-tances+ � � , � � = airfoil bending and torsional stiffnesses

The elastic axis is measured a distance -/. positive aft of themid-chord where . �%0�132 is semi-span of the airfoil chordlength

0. Dividing the plunge and pitching moment equations

by respectively � . and � .54 , one can express the structuralequations in a more conventional form as��*1 . �"6*� � �879 � ��*1 . �;: 4� �<1 . �=��>1/� � . �?23��

6*� ��<1 . ��@ 4� � � 79 � � ��: 4� @ 4� �%$"&('31/� � . 4 �A2B)* where+ 6 � �C� � 1D� � . static balance+ @ 4� �E! � 1/� � . 4 radius of gyration (squared)+ : 4� �C��F1 � plunging natural frequency (squared)+ : 4� �C��<1B!#� pitching natural frequency (squared)+ 79 �G�E�,�F1/� � bending damping+ 79 �H�E�,�<1/� � . 4 torsional damping

Considering harmonic motion such that�� 7�JI#K�L�MON � 7 I#K�L�MONP�� 7�HI#K�LFMQNH$ &(' � 7$ &�' I5K�LFMR�TS� where

7�, 7 ,

7�, and

7$�&('denote the frequency dependent

complex amplitudes of their time-dependent counterparts, thefrequency-domain form of the structural equations may be ex-pressed in matrix form asUCV �XW��>: 4 U 4 �ZY[: U]\ � UE^`_V �ba �dcF where V �fe 7�*1 .7hg N ai�fe � 7�j1/� � . 7$ &(' 1D� � .O4 g N �Ak� andU 4 �Rl �,6*�6 � @ 4�nm N U \��ol 79 �,pp 79 � m N U ^i�Rl : 4� pp�: 4� @ 4�/m,q �Tr� The aerodynamic loading

ais also a function of

7�and 7 and

its formulation is developed as follows.

2

B. Fluid Dynamic Model1. Governing Equations

Since we wish to study transonic airfoil aeroelasticityproblems where strong shocks might exist within the flowfield,we consider the two-dimensional Euler equations which maybe expressed as � ��

� � � �� 6 � � � � �� TS� where

6and � are the Cartesian coordinates,

�is time, and �

is the vector of conservation variables

�� d6GN � N � P� �� M� �� dcF

where�� ,�� ,�� , and

�� M are the static density,6

and � compo-nents of velocity, and total total energy, respectively. The fluxvectors

�and

�are

� � �� P� �� 4 �� M ��

� �� N�� H��

�� 4 �� M �� q �Ak�

The total energy�� M which is the sum of the internal (

�Iis the

specific internal energy) and kinetic energy is�� M �� I>� �� 2 W�� 4 �� 4 _ N �Tr� and considering a calorically perfect gas, the system of equa-tions can be completed by noting that the pressure is related tothe other flow variables via�� =��E� �` l �� M � �� 2 W�� 4 �� 4 _ m q �! [ In the present investigation, we are interested in small-disturbance, harmonically varying unsteady flows about somenonlinear mean operating condition. Thus, we assume the con-servative variables

�� may be expanded in a perturbation seriesof the form �� T6 N � N � P�#" �d6GN � �%$ � �T6 N � �I K�LFM �'&� where

"Z�T6 N � represents the nonlinear steady backgroundflow, and � �T6 N � is the complex amplitude of the unsteadyflow which arises from an external perturbation of order

$and frequency

:. Substituting Eq. (8) into the nonlinear Euler

equations (3) and expanding the result in a perturbation serieswith respect to

$, one finds to zeroth-order in

$, the governing

nonlinear steady flow equations are given by� � �("C � 6 � � � �'"] � � �#� q �')� This vector equation is just the steady Euler equations whichdescribe the steady background flow. The first-order equationis Y[: � � �� 6+*

� �� " �-, �� *

� �� " �., �� p�

where� �� " � � � � �� ////(01�2-3 and

� �� " � � � � �� ////!014253��[�

are the6

and � coordinate flux Jacobians.In the case of inviscid transonic aerodynamics, Eq. (9) is

solved first for a specified steady flow angle-of-attack ^ andfreestream Mach number

$76. Equation (10) can then be used

to describe the small-disturbance unsteady flow due to excita-tions created by airfoil harmonic motion of frequency

:about

this steady background flow.

2. Numerical ProcedureThe starting point of the POD/ROM procedure is any con-

ventional CFD method. For this investigation, the flow solversused to solve the steady (9) and unsteady (10) flow equationsare both second-order, explicit, cell-centered, finite-volumeGodunov [7] based methods utilizing Roe’s[8] approximateRiemann solver in conjunction with van Leer’s [9] techniquefor preserving monotonicity and better than first-order accu-racy.

As a first step to solving the steady and unsteady equations,the solution domain is divided into 8 quadrilateral cells. At thecenter of the

Yth cell of the computational grid, the estimate of

the solution � is stored, and is denoted by � K . The steady andunsteady solution for the entire computational domain may bethought of as 9 dimensional vectors of the form

: � �� " \" 4..."<;

� �� and = �� \� 4...� ;� �� `2�

where 9 � 8?> c since there are four dependent variables foreach computational cell in the case of the two-dimensional Eu-ler equations. Next, the steady and unsteady Euler equationsare marched (iterated) to a steady-state solution using the CFDalgorithms. The resulting discretization of the steady Eulerequations can be expressed as:A@4B \ ��:C@ �#DZ�!:C@D ��S� where

Dis the residual operator representing the discretiza-

tion of Eq. (9) and E is the iteration number. In a simi-lar fashion, the frequency-domain linearized unsteady Eulerequations (10) are marched to a steady-state to yield the com-plex valued unsteady flow = using= @�B \ � = @ �+F�� = @HGI:]N�:�NKJ q ��#cF Here,

Fis the residual operator representing the discretization

of Eq. (10), andJ

is a shorthand notation for the particulartype of external source of excitation which for this aeroelas-ticity study is the motion of the airfoil.

AlthoughF

is an operator for solving a linearized systemof equations,

Fis not strictly speaking a linear operator due

to the presence of inhomogeneous boundary conditions (i.e.F��(�* ML��). Nevertheless, with = being a converged result to

(14), the operatorF

may be expressed asF�� = G�:�NKJ i�ONi� = GP: N(:j �RQ>�!:]N�:�NKJ �TS �!: N(:j = �RQ �U:]N�:�NKJ C�� N ��`k� 3

or more simply, S �!: N(:j = �#Q>�!:]N�:�NKJ ��r� where

Nis the strictly linear portion of the

Foperator,

Qis a

vector representing the inhomogeneous boundary conditions,and

Sis the 9 > 9 fluid dynamic influence matrix.

S �!: N(:j is

in general a complex valued non-symmetric matrix. Further-more, for most numerical schemes,

S �!: N(:j can be written

as a series in:

. For a method which is second-order in:

,S �U:]N�:j may for example be written asS �U:]N(:j P�+S ^��!:� � Y[: S]\B�!:� �Z: 4 S 4 �!:� q �� [

For a problem employing a dense computational grid,S

may be so large that it is simply too expensive to determinemuch less store. Furthermore, solving an eigenvalue problemfor a system the size of a large

Smight simply be impos-

sible as standard eigenvalue determination methods typicallyrequire

� � 9�� operations. For these reasons, a reduced-ordermodeling technique can be of great benefit as will be shownsubsequently.

3. Aerodynamic LoadingOnce one has calculated an unsteady flow = , the corre-

sponding aerodynamic loada

may be determined. In fact, onecan define � as the

2 > 9 matrix which relates the airfoil lift7�and moment

7$"&('to the unsteady flow = . This matrix is a

discrete representation of the integral of the pressure loadingon the airfoil which in turn leads to the overall lift and momentdue to any particular flow = . Namelya,�8� �?= where � � l � 7N��J1D� � . �� 7� �/1D� � . 4 �� m q �� &� Each column element � of

7N �and

7� �can be determined by

setting = �C� except for the the � th row of = which is set tounity. The resulting lift and moment as computed by the flowsolver are then the values of the � th column of

7N �and

7� �respectively.

Next, since we are working with the two-degree-of-freedom airfoil structural model, we define � as the 9 > 2matrix which relates the boundary condition vector

Qto the

pitching7�

and plunging 7 motions. NamelyQ>� :j [� � � � :j V where � �d:j [� � �Q � �� :j Q��G� :j � �� q �� )�

Here,Q � �� d:j

andQ��d:j

denote the boundary condition vec-tors due to unit plunge and pitch motions respectively. Thereason we choose the negative sign will become apparentwhen we proceed to the overall aeroelasticity formulation.From Eqs. (18), (19), and (16), it can be shown thatai� � S�� \ � V �� V �?23p� where

� � � S � \ � is the aerodynamic influence matrix thatconsists of the frequency dependent transfer functions whichrelate the airfoil lift and moment to the pitch and plunge mo-tions. Namely,

��d:j P� �� . 4 l � . 7�� :j . 7� �� d:j 7$�� :j 7$ �� d:j m,q �?2/�

With the sectional lift and moment coefficients are customarilydefined as70�� 7�� 6�� 46 �?2 . 132 and 70�� 7$� 6�� 46 �A2 . 4 132 N �?2[2� we may rewrite the aerodynamic influence matrix as

� � :j P� � 6�� 46� l 70��! "�#%$ � :j 70�� & � :j 2 70 �' "�#%$ � :j 2 70 � & � :j m �?23S� Later, we assess the accuracy of the POD/ROM technique inmodeling some of the aerodynamic coefficient transfer func-tion elements of

�.

C. Coupled Fluid/Structural Aeroelastic ModelAfter constructing the coupling matrices � and � , one

may express the overall fluid dynamic equations asS = � � V �� N �?2BcF and the airfoil structural dynamic equations as

� = � UEV �+� q �?2[k� When combined together, these two systems of equations formthe full aeroelastic system of equations which written in asomewhat more expanded form are((((((((((((((((

(((((( S9 > 9�*))))))�(((((( �9 > 2

�*))))))�

l �2 > 9 m l U2 > 2 m

�*))))))))))))))))�

��

�= �7�<1 .7

� ��

��

�� pp

� ��q �?23r�

Considering a flow solver which in no higher than second-order in frequency

:, Eq. (26) can be expanded in

:as

* l S�^ � ^� ^ U ^ m � Y[: l SC\ � \� \ U \ m � : 4 l S 4 � 4� 4 U 4 m , e =V g � e �� g q �?2 [ We can reduce the system back to first-order in

:by employ-

ing state-space variables and coefficient matrices defined by

+ � �� =VY[: =Y[: V� �� N �?24&�

and

,]^ �((((((((l S ^ � ^� ^BU�^ m l S]\ � \� \#U \ ml �.- �0/�01M�02 m l43 - �./�01 3 2 m

�*))))))))� , \ �((((((((l � - � /� 1 � 2 m � l S 4 � 4� 4 U 4 ml53 - �0/�.1 3 2 m l �.- �./�.1 �02 m

�*))))))))� �?24)�

4

where 3 - and 3 2 represent square identity matrices which arethe same dimensions as

Sand

Urespectively, and

� -,� /

,�.1, and

� -denote a null matrices which are of the same di-

mension asS

, � , � , andU

respectively. The resulting systemcan be written as � ,]^ � Y3: , \ + �#� q �AS[p� This equation represents a generalized eigenvalue problem ofthe form

� ,]^ �� , \ �� where

� �bY[:is an eigenvalue

with corresponding eigenvector�. Since the time-dependent

aerodynamics and structural displacements are proportional toI K�LFM � I�� M, any eigenvalue

�with a positive real component

value implies aeroelastic instability.As a special note, in this current study, we have employed

a CFD method which happens to be first-order in:

. Namely,�'S ^ � Y[: S \ = �=�� ^ � Y[: � \ V �ASD� as opposed to the second-order example of (17). This means inour particular case that it is unnecessary to employ state-spacevariables for the fluid dynamics = . We can in fact express ouroverall aeroelastic system as��' S ^ � ^5� /� ^ U ^ U \� 1 � 2 3 2

�� JY[: � S]\ � � \ � /� � \ �.2 � U 4� 1 3 2 � 2�� =VY[: V

� ��

� �� AS�2�

which is an eigenvalue problem for an 9 �Xcdegree-of-

freedom system. This can still be quite large depending on 9 .Equation (28) has been presented for the sake of completenessas many numerical methods may be second-order in

:.

III. Model Reduction TechniqueA. Reduced-Order Fluid Dynamic Model

The basic concept of the reduced-order modeling (ROM)approach is that we assume a Ritz-like expansion where theunsteady flowfield = is expressed as a superpostion of a limitednumber of independent shape vectors. Namely,

=�� 2 \�� AS[S� where � � is a generalized coordinate sometimes referred to asan augmented aerodynamic state variable, and � � is the cor-responding Ritz vector. Equation (33) can also be written inmatrix form as

= ��N where� � � � �� \ � 4��

��and�j� ��

\� 4...�

� �� q �AS3cF Here,

�is an 9�>�� matrix whose

�th column is the shape

vector � � , and�

is the � dimensional vector of augmentedaerodynamic state variables � � . A reduced-order representa-tion of the fluid dynamics can thus be formulated by substitut-ing Eq. (34) into Eq. (16) and pre-multiplying by

��which is

the transpose of the complex conjugate (Hermitian transpose)of�

. Namely, � � S ��"!#� �$� � Q q �AS�k�

The central idea behind the expansion of Eq. (33) is that a suit-able set of Ritz vectors � � can be determined such that �&%�9while still preserving the accuracy of = . This is indeed thecase as will be shown subsequently, and actual unsteady flowsolutions (snapshots) computed for the excitations of interestat a number of discrete frequencies over a selected interval ofinterest in fact provide a starting point for just such a set ofshape vectors. Thus, Eq. (34) in effect allows a potentiallyvery large 9�> 9 fluid dynamic matrix equation

S = � Q tobe “compressed” to a much smaller � >'� matrix equation!��j��'� Q

.Please note that in practice for large systems, the matrixSis never actually computed. Instead, we compute the

�th

column ofS �

using the original linearized flow solver itself.That is, S � � �+F�� F��(�* q �AS[r� So for instance, in the case where the unsteady flow solver issecond-order in

:(i.e. Eq. (17)), the

�th column of

SC^(�,SC\��

, andS 4 � can be determined as follows. First, one de-

termines the columnsS �*)i +�

for say:C� p

,�, and

2, (i.e.S �d:��p� ,�

,S � : � �` +�

, andS �d:��2[ ,�

) from Eq. (36). Thefrequencies

: � p,�, and

2, are just chosen for demonstration

as any three distinct frequencies are acceptable. Next, withthe knowledge that the solver is second-order in

:, the

�th

columns ofS ^ �

,S \ �

, andS 4 � are respectivelyS�^(� �#SZ� : � p� ,�S \ � �=�HY*�TS S ^ �C� SZ� : �>2� +� � c SZ� : �i�` ,� 132S 4 � �#SZ� : �i�` ,� �ZY SC\��8� S ^�� AS [

So in the case of a CFD method which is second-order in:

,S >-� iterations of the flow solver are all that are necessary toassemble

S ^��,SC\��

, andS 4 � .

B. Proper Orthogonal Decomposition ShapeVectors

The next step in the reduced-order modeling procedure isto determine an appropriate set of shape vectors. As men-tioned in the previous section, actual solution vectors = makea good starting point. We thus calculate the small-disturbanceresponse of the fluid dynamic system at

$different combi-

nations of excitation and frequency. These solution snapshotsare denoted by = � for � � �[NQ2/N q�q#q NQ$ . Based on this en-semble of solution snapshots, the proper orthogonal decom-position is next used to determine a smaller number ( �/. $ )of shape vectors which contain the dominant unsteady aerody-namic characteristics of the flow.

The central idea behind the proper orthogonal decompo-sition is that from this ensemble of

$solution snapshots, a

smaller number of � shape vectors known as POD vectors(Refs. [10, 11]) can be determined which represent the dom-inate “directions” of the original

$solution vectors. The �

POD vectors in effect contain the most dominate physical at-tributes common to all unsteady flows within the snapshot en-semble. A newly published book by Holmes et. al. [12] pro-vides an overview of the POD method along with extensive de-tails of how the method has been used by researchers to studya wide variety of fluids problems.

Recently, Romanowski [13] has used the POD tech-nique to create a reduced-order aeroelastic model of a two-

5

dimensional isolated airfoil including compressible aerody-namics. Although Romanowski’s analysis takes place in thetime-domain, he has shown that very accurate unsteady flowmodels can be constructed that reduce the number of de-grees of freedom from the thousands associated with the orig-inal CFD flow solver to a few tens of degrees-of-freedom.Kim [14] has also recently developed a frequency domainform of the POD. Kim applied the technique to two rel-atively simple dynamic systems – a 12 degree-of-freedommass-spring damper system, and an incompressible three-dimensional vortex lattice model of a rectangular wing.

The POD vectors � � in effect are simply a linear rear-rangement of the original

$snapshots. Namely,

� � �� 2 \ = � � �� N � �8�[NQ2/N SDN q�q#q N � �AS�&� where � �� is the contribution of the � th snapshot to the

�th

shape vector. In matrix form, this can be written as� � �� AS�)� where

� � � � �= \ =n4 �� = ��

and� � �

�� \�� 4�...� �� q �Tc�p�

Here,�

is an 9 > $ matrix whose � th column is the � thsolution snapshot vector = � . The

$dimensional vector

� �contains the contribution weights � �� of the the � th snapshot= � to the

�th POD shape vector � � .

The key aspect of the proper orthogonal decomposition isthat the vectors

� � are selected so that they lie along the prin-cipal axes of the space spanned by

�. Furthermore, the vector� � is suitably scaled so that the POD vectors � � are of unit

length. Put together, these two conditions imply that it is nec-essary to find the extremum of the quantity� � � �� 4 �Tc � subject to the constraint that � � is of unit length. Thus, intro-ducing the Lagrange multiplier � � , the objective is to find thevector

� � that makes � stationary, where

� �� W�� _ q �Tc�2� Taking the variation of � and setting the result to zero yields� � �� q �Tc�S� Equation (43) defines an eigenvalue problem for the eigenvec-tors

� � and eigenvalues � � . The snapshots = � will thus tendto lie in a sub-space spanned by the POD shape vectors � �with the largest eigenvalues � � . Equation (43) can be quicklysolved as

$is typically on the order of 10 to 100.

C. Reduced-Order Aeroelastic ModelUsing the POD shape vectors for

�, we can now formu-

late the ROM of the aeroelastic system. This may be done bysubstituting e =V g �Rl � �� 4� 4 � 3 2 m e �V g �Tc[cF

into Eq. (27) where� 4 � and

� �� 4 are null sub-matrices di-mensioned

2 > � and 9�> 2 respectively, and pre-multiplyingby l �'� �

� 4� 4 �� 3 2 m q �Tc�k�

After some simplification, this results in the aeroelastic systemof equations having the reduce-order model form* l ! ^��i^� ^nUE^ m � Y[: l !Z\�� \� \ U \ m � : 4 l ! 4 � 4� 4 U 4 m , e �V g �"e �� g �Tc�r� where for example! ^ �&� � S ^ ��N�� ^ � � � � ^ N and

� ^ � � ^ � q �Tc [ We have thus reduced the potentially very large 9 � 2 degree-of-freedom system, to a much smaller � � 2

degree-of-freedom system expressed here in a somewhat expanded formas ((((((((((((

(( !� >�� ))� (( �� > 2

� ))�

l �2 > � m l U2 > 2 m

� ))))))))))))�

��

�� 7�J1 .7

� ��

�� pp

� ��Tc &�

which is computationally much less expensive to work with.Once again we can convert the second-order in frequency:

ROM system (46) to first-order by working with state spacevariables and coefficient matrices defined by

�+ � �� VY[: �Y[: V� �� N �Tc )�

and

��^ �((((((((l ! ^��i^� ^*UE^ m l ! \�� \� \ U]\ ml �� -�02 m l 3 � �� 3 2 m

�*))))))))� � \ �((((((((l �� 2 m � l ! 4 � 4� 4 U 4 ml 3 � �� 3 2 m l �� -�02 m

�*))))))))� �?k3p�

The resulting ROM aeroelastic system is� ��^ � Y[: � \ �+ �#� �?k/� which again is a generalized eigenvalue problem now of the

form � � ^J� �� \�� where��E��Y[:

is an eigenvalue with

corresponding eigenvector��. Again, if the real component of��

is positive, the aeroelastic system is unstable.As mentioned previously, our particular CFD method is

first-order in:

which means it is unnecessary employ state-space variables for the fluid dynamics. So our reduced-orderfluid dynamic system may be expressed as�*! ^ � Y[: ! \ ��8� � � � � ^ � Y[: � \ V N �?k[2�

6

and our POD/ROM aeroelastic system can thus be written as��' ! ^ � ^ � �� ^�UE^FU \�� .2 3 2�� JY[: � ! \ � � \ � �� \ � 2 � U 4�� 3 2 �.2

�� VY[: V� ��

� �� q �?k3S�

We present Eqs. (49-51) again for the sake of completeness asmany CFD methods may be second-order in

:.

IV. ResultsPOD/ROM Aeroelastic Stability Studies

To demonstrate the effectiveness of the POD/ROM proce-dure, we consider a “standard” benchmark airfoil aeroelastic-ity test case and examine a rather coarse mesh in order to keep9 small enough so that an eigenanalysis for the full aeroe-lastic system (32) can be performed. This full system eigen-analysis can then be considered an “exact” solution to whichvarious POD/ROM solutions may be compared against. It isemphasized that we use a coarse mesh so that we can computethe complete eigenspectrum of the original CFD model. ThePOD/ROM procedure can however readily treat much largersystems. For example, a system with on the order of

��p �degrees-of-freedom has been demonstrated in Reference [6].This is for a much finer mesh to the same test case to be de-scribed in the following. We discuss this large system config-uration at the end of this section.

The benchmark test case is referred to as “Isogai’s TestCase A” [15, 16, 17] which was chosen with structural param-eters to simulate “the vibrational characteristics of a typicalchordwise section of a swept wing.” The airfoil section con-sidered is the NACA 64 � 010 “Ames” configuration which is aslightly modified non-symmetric version of the actual NACA64 � 010. In the early 1980’s, an AGRAD advisory report [18]established this section as a benchmark test case to whichvarious numerical modeling methods could be studied. Ed-wards [19] and Ehlers & Weatherill [20] have also tested theirmethods for Isogai’s “Case A” arrangement whose structuralparameters are:- �8� 2DN 6<��=� q & N @ 4� � S q c &DN : �:�� =��N�� br�p �?kBcF where the mass ratio is defined as

� � � 1 � 6�� .54 .An eigenanalysis for a system with around 2000 degrees-

of-freedom is around the limit of our present computing capa-bilities. This limits our mesh size to approximately 8 �]k3p�pmesh cells ( 9 � 8 > c ). As such, we have chosen an “o-grid” mesh arrangement with 32 circumferential and 16 ra-dial cells. Figure (2) illustrates the layout, the outer radiusof which extends to five chord lengths. The total number ofdegrees-of-freedom is thus 2052 for the full aeroelastic sys-tem (32). Again, we consider this coarse mesh configurationso that we have the ability to compute full system solutions towhich we can then compare our POD/ROM solutions.

The background steady flow corresponds to a mean angle-of-attack of ^ � p��

and a Mach number$ 6 � p q &�k .

We have selected this Mach number since the aeroelasticstructural eigenvalues demonstrate interesting trajectories inthe complex plane as a function of non-dimensional velocity � � 6 1� ��:�� . as is shown in the following.

Figure (3) illustrates the aeroelastic structural eigenvaluetrajectories based on Eq. (53) for various POD/ROM shape

� �� !��"�#��$%�&��

Figure 2: Mesh Layout for NACA 64 � 010 “Ames” AirfoilCoarse Grid Test Case: ')(+*-,/. Computational Cells

vector arrangements against the full system (32) trajectoriesfor the Isogai test case. Here we examine how both the snap-shot sampling resolution and the fraction of the resulting num-ber of available POD shape vectors affect the overall accu-racy of the reduced-order model. The three columns of graphsin Figure (3) correspond to different snapshot sampling inter-vals in non-dimensional frequency 7:>�,:H0�1 � 6 . Each columncorresponds to unsteady flowfield snapshots for airfoil plung-ing and pitching motions over a frequency range from zeroto one at an incremental frequency of

J 7:C�Zp q � , J 7:E�Zp q 2 ,and

J 7:i� p q k respectively. This corresponds to a total of 21,11, and 5 snapshots respectively for each frequency increment.The 7: ��p q p solution for the plunging motion is trivial andis thus discarded for each column. For negative frequencies,the unsteady flow solutions are just the complex conjugates oftheir positive frequency counterparts. Thus an additional 20(J 7:i� p q � ), 10 (

J 7:i� p q 2 ), and 4 (J 7:,� p q k ) solution vectors

can be added to the overall ensembles which turn out to be re-spectively 41, 21 and 9 total unsteady flow solutions. Finally,the rows of Figure (3) represent POD/ROM results using all(row one), and fractions (rows two and three) of the total num-ber of available POD vectors.

We study the aeroelastic stability of Isogai’s configura-tion by making sweeps in non-dimensional velocity

�� 6 1 ��: � . over a range from zero to five. The open circlesrepresent the aeroelastic eigenvalues that are predominantlystructural in character for the full system calculated at 0.5 in-crements in airspeed

. These are the modes that correspond

to the purely structural eigenvalues when the airspeed is zero.It should be pointed out that each one of these calculationstook several hours to complete. The solid “filled” circles rep-resent the aeroelastic structural eigenvalues for the POD/ROMsystem also calculated at 0.5 increments in airspeed

. Each

of these calculations took on the order of a fraction of a sec-ond. In fact, what appear to be the solid line curves in each ofthe graphs, are actually the POD/ROM system results repre-sented as dots calculated at increments of 0.005 increments inairspeed

. These results have such close spacing, that when

placed together, they appear to trace out a line. The POD/ROMsystems of Figure (3) are all small enough that such dense tra-

7

jectory studies only take at most on the order of a few minutesto determine.

As can be seen, these two aeroelastic eigenvalues haverather interesting trajectories as a function of the airspeed

.

It is really not possible to distinguish one mode as being thepitching mode and the other being the plunging mode as thestructural equations are largely coupled due to the large staticunbalance

6 �. So we just label the modes as #1 and #2. It

can been be seen that the mode we have designated as #1 hasthe interesting behavior of going unstable at a

� p q 3k , butthen becomes stable again for

�� 2. Mode #2 becomes more

stable up to � 2 , but then goes unstable above

� S .In the case of the

J 7:��p q � snapshots (the first column ofgraphs), the POD/ROM technique is able to nicely match thefull system results with as few as 11 of the 41 available PODshape vectors. The same is true of the

J 7: � p q 2 snapshots (sec-ond column of graphs) with as few as 11 of the 21 availablePOD mode shapes. The results begin to deteriorate quite a bitwhen going to only 7 of the 21 POD shape vectors. For theJ 7: � p q k snapshot results (third column of graphs), even forthe 9 of 9 POD shape vectors case, the reduced-order model isstarting to deteriorate.

What can we conclude from this figure? We believe thesesnapshot interval refinement comparisons indicate that about11 POD/ROM mode shapes is the borderline for achieving adecent level of accuracy. This figure also shows how the POD,needless of the snapshot frequency sampling interval, is able tosort out these 11 best shapes. If there is not enough samplingresolution as in the 7:P�ip q k snapshots case, there is simply notenough modal information for an accurate expansion.

How might one then implement the POD/ROM procedurefor configurations using much larger meshes where an exact(full) solution for comparison is impossible to determine? Oursuggestion is the following. First, choose a large interval

J 7:in frequency over which to compute solution snapshots. Next,compute aeroelastic ROM solutions for all and one-half of theavailable POD mode shapes for this particular resolution ofsnapshots. If there is a large difference between the results,go back and collect solution snapshots for

J 7: 132 incrementsin the frequency. Once again compute ROM solutions for alland one-half the available POD mode shapes and check forthe level of difference in the results. Continue this procedureuntil the difference is to an acceptable level. At this point, thenumber of POD modes representing one-half of the total couldbe considered the number of degrees-of-freedom absolutelynecessary to model the system to the desired level of accuracy.

Finally, Figure (4) (which is similar to a figure presentedin Reference [6]) shows flutter speed results as a function ofMach number for Isogai’s test case except now using a muchfiner mesh. Namely a mesh arrangement with 128 circum-ferential and 32 radial cells for a total of 16,388 aeroelasticdegrees-of-freedom. The flutter speed is the airspeed

at

which one of the aeroelastic modes becomes neutrally stable.Shown are POD/ROM results using 11, 21, and 41 POD shapevectors from a

J 7: � p q � (41 total POD shape vectors) snap-shot sampling. All are in good agreement with one anotherespecially around

$ 6 � p q &�k where the course mesh studiesof this paper have taken place. Also shown for comparisonare flutter speeds predicted using transonic small disturbance(TSD) theories of Edwards [19] and Isogai [15, 16, 17], and

the time-linearized full potential theory of Ehlers & Weather-ill [20]. As can be seen, our method matches very nicely withthese other techniques. Interestingly enough, even for this finemesh, 11 POD shape vectors appear to be all that is necessaryto get good results especially for freestream Mach numbers$ 6 � p q &[p . Lower Mach numbers may need a few more PODshape vectors as can be seen.

0.70 0.75 0.80 0.85 0.90 0.95 1.00Freestream Mach Number, M

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Flu

tter

Spe

ed, V

F=

UF/(

µ1/2 ω

αb)

41 of 41 POD Modes21 of 41 POD Modes11 of 41 POD ModesIsogai (1980, 1981)Ehlers and Weatherill (1982)Edward et. al. (1982)

Figure 4: Computed Flutter Speed - Trend NACA 64 � 010“Ames” Airfoil Fine Grid Test Case: ��

, �� ( , �� ,�� , � �� '�� , �� , , � � . � , , (!� * ')(Computational Cells " ,/.�# '��$� Total Aeroelastic SystemDegrees-of-Freedom.

POD/ROM Purely Aerodynamic ConsiderationsAlthough the previous section demonstrated the capability

of the POD/ROM procedure to greatly reduce the number ofdegrees-of-freedom for an aeroelastic problem while still pre-serving accuracy, in this section we focus on the purely aero-dynamic aspects of the POD/ROM procedure in an attempt toget a better understanding of how the method is able to capturethe dominant physical aspects of the fluid dynamics while atthe same time greatly reducing the size of the system.

Considering an eigenanalysis of the fluid dynamics, the de-tails of which are presented in Appendix A, it can be shownthat the

�&%QN?YF th aerodynamic transfer function of the aerody-

namic influence matrix��('

(Eq. (23)) can be expressed as('*� Y[:

)

�*),+ K �(' [� �@42 \ - )&+ @ W/. ^ @ + K �0' . \ @ + K _� @ �1' �?k[k� 8

where � � � D N�� ^<� N � ^�N and��\ � N � \ q �?k3r� D

andN

represent 9 >�9 matrices whose columns and rows arecomprised respectively of the right and left eigenvectors of thegeneralized eigenvalue problem defined by the homogeneouscase of Eq. (31) suitably normalized so that

N SE^ D ��andN S \ D �"� 3 - where

�is the 9�> 9 matrix whose non-zero

entries are the eigenvalues� @ situated along the diagonal, and3 - is an 9�> 9 identity matrix.

Similarly, the POD/ROM version of the the aerodynamicaerodynamic influence matrix (denoted as

�� /' ) may be ex-

pressed as

�� )&+ K �/' H� �� 2 \ � )&+ � W��>^ � + K � '��\ � + K _�� ' �?k [ where

� � �� N � ^*�� ^[N and�Z\D�� \ �?k4&�

and in a similar fashion as the full system development,

and�

represent the now � >'� matrices who columns androws are comprised of the right and left eigenvectors of thegeneralized eigenvalue problem defined by the homogeneouscase of Eq. (52) suitably normalized so that

� SC^� � ��and� SC\�f�� 3 � where

��is the ��> � matrix whose non-zero

entries are the eigenvalues�� situated along the diagonal, and3 � is an ��> � identity matrix.

As can be seen from Eqs. (55) and (57), the eigenvalues� @ and�� represent the poles of the aerodynamic influence

matrix�

. A question of how well the POD/ROM procedureperforms might then be how well do the least damped eigen-values correlate with their full system counterparts. Sinceone typically wishes to model the aerodynamics in the vicin-ity of the imaginary axes for aeroelastic stability studies, theeigenmodes associated with the least damped eigenvalues willin general make the most dominant contributions to the un-steady aerodynamics. At the same time, we can also inves-tigate how well elements of the aerodynamic influence matrix(e.g.

0 � � �('` ) correlate for complex values of

'between the full

and ROM systems. This should tell how important it is that theeigenspectrums match.

In this section, we investigate how well the POM/ROMmethod does in modeling the unsteady aerodynamics for thesame NACA 64 � 010 “Ames” airfoil as studied in the previoussection. In addition to studying the ability of the POD/ROMmethod to model elements of aerodynamic transfer functionmatrix as a function of complex frequency

' � Y[:, we also

compare the eigenspectrums of the POD/ROM and full aero-dynamic systems. An angle-of-attack �Cp � and

$�6 �bp q &�kare again considered.

Figures (5) and (6) illustrate the fluid dynamic eigenspec-trums and the coefficient of lift due to pitching motion transferfunction for both the full system and POD/ROM system con-figurations. The layout of the graphs in Figures (5) and (6)follows the same arrangement as in Figure (3). Again, sincethere are two issues, snapshot resolution and the number ofPOD modes employed for a given snapshot resolution, a ma-trix of graphs is a good way to assess the accuracy and capa-bility of the POD/ROM method.

Figure (5) shows the full system fluid dynamic eigenval-ues� @ (Eqs. (55) and (59)) along with the POD/ROM fluid

dynamic eigenvalues�� @ (Eqs. (57) and (68)). The full sys-

tem and POD/ROM eigenvalues are represented by open andclosed circles respectively. Interestingly, many of the full sys-tem eigenvalues have corresponding eigenmodes which pro-duce negligible lift on the airfoil and thus will not contributesignificantly to the aerodynamic transfer functions. Those thatdo however are marked in Figure (5) with a plus symbol. Sincethe NACA 64 � 010 “Ames” is a slightly non-symmetric airfoilsection, the unsteady pressure distributions on the upper andlower surfaces are not quite images of one another as theyare in the the case of a truly symmetric airfoil at a zero de-gree steady flow angle-of-attack. In such a symmetric airfoilcase, the pressure distributions over the upper and lower sur-faces are either the same, or they are an exact image of oneanother. For example, in the case of an eigenmode which pro-duces lift, a negative pressure distribution on the upper surfacewould have a corresponding yet positive pressure distributionon the lower surface. For a non-lifting eigenmode, the pres-sure distribution is the same on both the upper and lower sur-face. Since the NACA 64 � 010 “Ames” is only very slightlynon-symmetric, it is still easy to make the distinction betweenlifting and non-lifting modes. We have also confirmed thiswith a symmetric 64 � 010 configuration test case. As can beseen from Figure (5), the POD/ROM eigenspectrums typicallycorrelate with the lifting modes of the full system.

Figure (6) shows the full system (Eq. (55)) and POD/ROM(Eq. (57)) coefficient of lift due to pitch aerodynamic influencefunction as a function of

'along specific paths in the complex

plane. Namely, along the lines 7'B�i@BI K�� ( 7'B�>Y 7: ) for the angles� � )[p��,r[p �

, and�`2[p �

as@

ranges from zero to one. Note,these curves are for non-dimensional

'. Namely, 7'F�0'`0�1 � 6 .

The open circles represent full system results calculated fromEq. (55), and the filled circles are POD/ROM results fromEq. (57). As can be seen, the 41 of 41 POD vectors casematches nicely for all three angles of

�. All the POD/ROM

cases which though employ at least 11 POD shapes are rela-tively accurate for the

� � )[p/�,r[p��

angles in the complex'

plane. This trend compares nicely with the aeroelastic resultsof Figure (3) where the neutral stability states (

� � )[p �) of the

POD/ROM results correspond nicely with the full system withas few as 11 POD mode shapes.

Interestingly enough, when comparing Figures (5) and (6),it appears decent aerodynamic results can be obtained with justa few of the POD/ROM fluid dynamic eigenvalues correlat-ing to their full system counterparts. Furthermore, in all thePOD/ROM configurations, there is a POD/ROM eigenmode(or a complex pair as in the 21 of 21 POD vectors case), whichnearly corresponds to the least damped lifting eigenmode lyingon the real axis. Although not quite sufficient for an accurateROM, this does lead us to believe however that much of theunsteady flow physics may lie in a single mode shape.

V. ConclusionsA new method for unsteady small-disturbance aerody-

namic modeling is presented. The method works in conjunc-tion with a frequency-domain computational flow solver andcan be used for any level of fluid dynamic approximation. Aproper orthogonal decomposition is used to determine, from

9

an ensemble of previously computed flows, an optimal set ofshape vectors for a Ritz expansion of the fluid dynamic solu-tion. This allows for a reduction of the original matrix equa-tion representation of the unsteady fluid dynamics to such alarge degree that it can be coupled with the structural systemof equations and a standard eigenanalysis may be rapidly per-formed to determine aeroelastic stability. We have demon-strated the method for a benchmark two-dimensional tran-sonic aeroelasticity problem, and we have found that accuratereduced-order models with as few as 11-degrees-freedom-fora system with originally over 2000-degrees may be easily con-structed. The proper orthogonal decomposition has provento be a useful tool in sorting out the dominant shape vectorswhich best represent the unsteady flow.

VI. AcknowledgmentsThis work is part of NASA’s Advanced Subsonic Tech-

nology (AST) program managed by Peter Batterton and JohnRhode. Additional funding was provided by Air Force Officeof Scientific Research Grant No. F49620-97-1-0063, with Ma-jor Brian Sanders serving as program officer.

Appendix A - Fluid Dynamic EigenanalysisFull Fluid Dynamic System Model

Since our particular CFD model is first-order in:

, in thehomogeneous case (

Q;��), Eq. (31) defines a generalized

eigenvalue problem given by�!S ^J�� S]\5 � �� ?k4)� where an eigenvalue

�E��Y[:represents a natural frequency of

the unsteady flow, and�

is the corresponding eigenvector. WeletD

andN

denote the 9�> 9 matrices whose columns androws are comprised of the right and left eigenvectors of (59)

D�� \ � 4 �� N N�� ((( ��?\�� 4 �...�� h�

�*)))� �Ar[p� suitably normalized so thatNMS ^ DX� �

andN S]\ D �� 3 - �ArD�

where�

the 9 > 9 matrix whose non-zero entries are theeigenvalues

� @ of (59) which are situated along the diagonal,and 3 - is an 9+> 9 identity matrix. One may then express theunsteady flow = in terms of a modal expansion of the form

= � �@�2 \ � @ � @ � D �PN where� � ��

� \� 4...� � �� q �Ar�2�

Substituting (62) into (31) and multiplying by the matrix ofleft eigenvectors

Nyields (where

'*� Y[:)N �(S ^/�0' S]\5 D �,�Z� � �1' 3 - � � � N � � ^F�0' � \5 V q �Ar[S�

The unsteady flow = may then be written as= �#D � �� D � � �1' 3 �� \ N � � ^ �0' � \ V N �Ar3cF

and from Eq. (20), the aerodynamic influence matrix��('`

is� �/' H� � D � � � ' 3 � \ N � � ^��0' � \5 q �Ar�k� If we next let� � � D N � ^i�#N � ^�N and

��\ �#N � \BN �Ar[r� then the

�,%QN?YF th entry of

��('` can be written as

� )&+ K �('` �� @�2 \ - ),+ @ W . ^ @ + K �0' . \ @ + K _� @ �1' q �Ar [ This illustrates how the eigenvalues

� @ represent the polesof the unsteady aerodynamic influence functions and furtherleads to the question of how well the reduced-order modelpoles correspond to the complete (full) system poles. Thus weproceed to the development of the reduce-order aerodynamicinfluence matrix.

POD/ROM Fluid Dynamic System ModelAs for the full system, for trivial boundary conditions (

Q ��), Eq. (52) defines a generalized eigenvalue problem now of

the form � ! ^�� (� !Z\� � �� #� �Ar�&� where the tilde (

�) signifies the quantity is associated with the

POD/ROM approximation. We let

and�

represent the � >� matrices who columns and rows are comprised of the rightand left eigenvectors of (68)

� � � �� \ �� 4 �� N � � ((( � � � \ �� 4 �...� � �

��*)))� �Ar�)�

suitably normalized so that� S�^� � ��

and� S]\� �=� 3 � �U Bp�

where��

is now the � > � matrix who’s non-zero entries arethe eigenvalues of (68) which are situated along the diagonal,and 3 � is a ��> � identity matrix.

As with the development of the unsteady flow expan-sion using the actual eigen-information, one can express thePOD/ROM unsteady flow variables

�in terms of a modal ex-

pansion of the form

� � �� 2 \ �� HNwhere

��Z� �� \�� 4...��

� �� q �U F� Substituting (71) into (52) and multiplying by the matrix ofleft eigenvectors

�gives (

'*� Y[:)

� �*! ^F�0' !Z\5 ��,� � �� 1' 3 � � ��n� �� ^/�0' � \O V �U 32� which means the POD/ROM expansion of the unsteady flowcan be expressed as= ��j� � ��,�� 1' 3 � � � \ �� ^ �0' � \ V q �U BS�

10

Thus, the POD/ROM approximation of the aerodynamic influ-ence matrix

�� /' is

��('` P� � � � �� 1' 3 � � � \ �� ^F�0' � \O <N �U cF and if we let

� � � N � ^n� �� ^�N and�Z\J� �� \BN �U 3k�

the POD/ROM�&% N YF

th entry of��('`

is then given by

��*)&+ K �(' H� �� 2 \ � ),+ � W � ^ � + K � '�� \ � + K _�� ' q �U Br� References�� ! #"$��%& ('�)�%*�,+$�-��"�.%*�/'�)$�,+$0213�(�4��5/6

'��7��98:�#",';��#�=<>�/0;�!#"? @A��B>��<C6�84DE6�F#F#GH $��I�I$0J'K��LMM,�� F��N��-�O , $�9�� :�JPC"*'�)��KQ�0;�>�#1�D2 #+SR�9��$��TH��UA ('��-04'��V��-�H��/6

0;�-"W'4Q�"�0J'�� #+HX��/��,+,X$"� #U��5��?� +$0�13�(�Y��Z$��';�[ \��XA]HUA #��%*�#'��#"�0��#1�^_��"$0- @A��`��C�aD2 #�b�/�CM#c�dO��M, fe� ("?�?��L�M(cH�

� g\�Ah�+Hij \�[+$0- Ve$�>^k�� l�J��5� \'��!�#"�0l�#1m�= (��! #5/�on?�[ ("�0J1p�#��U%*�/'�)$�,+$0>'��&��;13��:%*�#'��"q #"O+l],'[ (Z��'�Xr84 #��5-I$�s \'��"�0- @��`��C�aD2 #�b�/�CM(L\d,tMM#F, u��H��E�-L�M(LH�

� v(��9 (��b�/�w b%l�� 4��<C�-0;��"*1p�#�C��5/'��7��9x��I$';'��>],I��$�$��-0;0;��"* #"�+BCI�0J'y��!��-7,�! ('��#"zQC0;��"�{]W'[ ('��/6�]H�� #5-��/��W�-�! (0J'��!5*%*�,+$�/��6��"�H @}|,~#�,�[�b��~J�K��3�J�/�;�J�[�� bNS�#��=��L, b�C�H�bg, ��L#G�F, f�$�?�OFFW��dFF#M,�

� ��An4��u #"WXW ?]f�f�.�� = ("O+}�C+$ #U�0- b^k�?%l�ue(��! ��"$�!��"$�� (�.D2��#6#�[ (U�U��"�yhET,'��-"$0;�!�#"�0j'��V�� ('��#"O #�?x�I$"�5/'��"*��$�$��TH�!UA \6'��"V%*�/'�)��,+H0=13�(�2QC"$0J'�� +,X9��,+HXH"O (U��!5�x��(��5-�-0- @K�>�C]$�njD26JF#MM(cH feI$��X{�-LL#GH�

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11

−0.35 −0.25 −0.15 −0.05 0.050

1

2

3

4

5

611 of 41 POD Vectors

0

1

2

3

4

5

6

Im(jω

/ωα)


0

1

2

3

4

5

6

∆ω=0.1 Snapshots (21 Total)41 of 41 POD Vectors

Full System POD/ROM

−0.35 −0.25 −0.15 −0.05 0.05

Re(jω/ωα)

7 of 21 POD Vectors



−0.35 −0.25 −0.15 −0.05 0.05

3 of 9 POD Vectors

5 of 9 POD Vectors


V=0

V=1

V=2

V=3V=4V=5

V=0V=1

V=2 V=3

Mode 1

Mode 2

Figure 3: Comparison of Various POD/ROM Arrangements Against a Full Aeroelastic System Eigenanalysis for the NACA64 � 010 “Ames” Airfoil Section: �� , �� , � � ( , � � ,�� , � �� ' �� , � � � � � � , , � � . � , '�(�* , .Computational Cells " ( � �)( Total Aeroelastic System Degrees-of-Freedom.

12

−1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2−0.2

0.0

0.2

0.4

0.6

0.8

1.011 of 41 POD Vectors

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

Im(jω

− )


−0.2

0.0

0.2

0.4

0.6

0.8

1.0


Full SystemPOD/ROMSnapshotLifting Eigenmode

−1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2

Re(jω − )

7 of 21 POD Vectors



−1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2

3 of 9 POD Vectors

5 of 9 POD Vectors


Figure 5: Fluid Dynamic Eigenspectrums for the NACA 64 � 010 “Ames” Airfoil Section: �� $�, �� , '�( * , .

Computational Cells " ( � �� Total Fluid Dynamic System Degrees-of-Freedom.

13

1 2 3 4 5 6 7 8 9 10−5

−4

−3

−2

−1

0

1

2

3


−5

−4

−3

−2

−1

0

1

2

3

4

Im(c

l α(s− ))


−5

−4

−3

−2

−1

0

1

2

3

4


Full SystemPOD/ROM

1 2 3 4 5 6 7 8 9 10

Re(clα(s

− ))

7 of 21 POD Vectors



1 2 3 4 5 6 7 8 9 10

3 of 9 POD Vectors

5 of 9 POD Vectors


r=0.2

r=0.0

r=0.4

r=0.6

r=0.8

r=1.0θ=90o

θ=60oθ=120o

Figure 6: Aerodynamic Lift Due to Airfoil Pitching Coefficient Transfer Function for the NACA 64 � 010 “Ames” Airfoil Section:�� , � � � �$� , � � � � � � � , ')( * ,/. Computational Cells " ( � �� Total Fluid Dynamic System Degrees-of-Freedom.

14

reduced-order aeroelastic modeling using proper-orthogonal...

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