NASA Technical Reports Server (NTRS) 20010071780: Reduced-Order Modeling: Cooperative Research and Development at the NASA Langley Research Center

16 Pages·1.4 MB·English

by  NASA Technical Reports Server (NTRS)

Reduced-Order Modeling: Cooperative
Research and Development at the NASA
Langley Research Center
Walter A. Silva *
NASA Langley Research Center
Hampton, Virginia 23681-0001
Philip S. Beran t
Air Force Research Laboratory
Wright-Patterson Air Force Base, Ohio 4543:3-7531
Carlos E. S. Cesnik *and Randal E. Guendel
Massachusetts Institute of Technology
Cambridge, Massachusetts
Andrew Kurdila _and Richard J. Prazenica II
University of Florida
Gainesville, Florida 32611-6250
Liviu Librescu *_nd Piergiovanni Marzocca *t
Virginia Polytechnic Institute and State University
Blacksburg, Virginia 24061-0219
Daniella E. Raveh _
Georgia Institute of Technology
Atlanta, Georgia 30aa2-0150
Cooperative research and development activities at the NASA Langley Re-
search Center (LaRC) involving reducedoorder modeling (ROM) techniques are
presented. Emphasis is given to reduced-order methods and analyses based on
Volterra series representations, although some recent results using Proper Of
thogonal Decomposition (POD) are discussed as well. Results are reported for a
variety of computational and experimental nonlinear systems to provide clear
examples of the use of reduced-order models, particularly within the field of
computational aeroelasticity. The need for and the relative performance (speed,
accuracy and robustness) of reduced-order modeling strategies is documented.
The development of unsteady aerodynamic state-space models directly from CFD
analyses is presented in addition to analytical and experimental identifications of
Volterra kernels. Finally, future directions for this research activity are summa-
rized.
*Senior Fiesearch Scientist, Aeroelasticity Branch, _lprofessor, Department of Aerospace Engineering, Me-
NASA Langley' Besearch Center, Hampton, Virginia; chanics, and Engineering Science, University of Florida,
lSenior Research Aerospace Engineer, Structural Design Galnesville, Florida 32611-6250
and Development Branch, AFRL/VASD, Wright-Patterson IIGraduate Researcher, Department of Aerospace Engi-
AFB neering, Mechanics, and Engineering Science, University of
IAssistant Professor of Aeronautics and Astronau- Florida, GMnesville, Florida 32611-6250
tics,Departmeut of Aeronautics and Astronautics, Mas- **Professor of Aeronautical and Mechanical Engineer-
sachusetts Institute of Technology, Cambridge, Mas- ing,Department of Engineering Science and Mechanics, Vir-
sachusetts ginia Polytechnic Institute and State University, Blacks-
§Graduate Research Assistant, Department of Aeronau- burg, Virginia 24061-0219
tics and Astronautics, Massachusetts Institute of Technol- ttAerospace Engineer, PhD, Virginia Polytechnic Insti-
ogy, Cambridge, Massachusetts tute and State University, Blacksburg, Virginia 24061-0219
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Introduction second challenge is that simulations cannot be used
effectively for preliminary aeroservoelastic design.
cHipEliniensclussiounch oafs CaFerDo-eblaassteidcity, analayesreosservinoteolas-dis- Design by simulation inevitably becomes design
ticity, and optimization is currently not, routine by trial-and-error, an impractical approach. The
due to the high computational costs of the CFD development of ROMs is targeted precisely at. ad-
analyses. One solution to this problem is the dressing these two challenges. While a ROM pro-
development of reduced-order models (ROMs). vides increased computational efficiency over the
Reduced-order models capture tile essence of the original more complex system, it is the malhe-
dynamical system under investigation, resulting in matical model that is extracted from the original
a significantly less complex model. The reduced system that enables the intercomlect.ion with other
complexity of the ROM translates into improved disciplines. This simplified reduced-order mathe-
computational efficiency. The ROM can then be matical model transforms a highly complex "black
used for subsequent analyses at reduced computa- box" into a system with distinct physical and
tional costs. It should be stated that the ROMs mathematical properties suitable for design analy-
discussed within this context are a generalization ses. Although the first challenge can be improved
of the traditional ROMs that involve the reduc- via parallel processing and advanced algorithms.
tion of the dimensions of a given matrix. So, for the direct time-domain approach does not address
example, the lift response of a three-dimensional the second challenge.
aeroelaslic CFD system undergoing plunging mo- Attempts t.oaddress the problem of high compu-
tions can be characterized, as will be shown, using tational cost. include the development of transonic
a single impulse response flmetion that relates lift indieial responses. 3-5 Transonic indicial (st.ep) re-
due to plunge. In this ease, the single impulse re- sponses are responses due to a step excitation of
sponse function is the HeM. Therefore, a ROM a particular input, such as angle of attack, about
can also be a fimclional condensation of the origi- a transonic (or nonlinear) steady slate condition.
nal system in addition to the traditional interpre- More recently. Marzocca et al'_ have analytically
tations with respect to matrices. derived indicial functions for three-dimensional
Early mathematical models of unsteady aerody- wings in compressible flow. Neural networks have
namic response capitalized on the efficiency and also been used to develop nonlinear models of un-
power of superposition of scaled and time shifted steady aerodynamics r and nonlinear models of ma-
fundamental responses, or convolution. Classical neuvers (using an experimental database).5 Neural
models of two-dimensional airfoils in incompress- networks and Volterra series have some similari-
ible flow 1include SVagner's function (response to ties. since each involves the characterization of a
a unit step variation in angle of attack) Kuss- system via an input-output mapping. 7,8 In par-
net's ftmction (response to a sharp-edged gust), ticular, there is a direct relationship between the
Theodorsen's function (['requency response to sinu- weights of a neural network and the kernels of a
soidal pitching motion), and Scar's flmction (fie- Volterra series representation for a particular sys-
teln.9
quency response to a sinusoidal gust). As geomet-
ric complexity increased fl'om airfoils to wings to A major difference between Volterra series and
complete configurations, the analytical derivation neural networks, however, is in the training effort.
of response fllnctions was no longer practical and Neural networks can require a substantial train-
Ihe numerical computation of linear unsteady aero- ing effort 5 while Volterra series require neither a
dynamic responses in the frequency domain be- training period nor curve fitting for model con-
came the method of choice. 2 And, when geometry- struetion. Also, \.:olterra kernels provide a direct
and/or flow-induced nonlinearities aerodynamic el- means for physical interpretation of a system's re-
Dots became significant, the nonlinear equations sponse characteristics in the time and fl'equency
were computed via tilne integration. The direct. domains. However, potential disadvantages of the
time-integration approach for solving aeroelast.ic Volterra theory method include input amplitude
problems via the coupling of the nonlinear aerody- limitations related to convergence issues and the
namic equations with the linear structural equa- need for higher order kernels.l°
tions has yielded a very powerful silnulation ca-
Another approach for reducing the computa-
pability with two primary challenges. The first
tional cost. of CFD analysis is to restrict atten-
challenge is the computational cost of this simu-
tion to linearized dynamics. The response of the
lation, which increases with the fidelity of the non-
linearized system about a nonlinear steady-state
linear aerodynamic equations to be solved. The condition can be obtained via several methods.
Some of these methods include the model or-
t/Research Engineer II. Aerospace Systems Design Labo-
der reduction of state-space models using various
ratory, School of Aerospace Engineering, Georgia Institute
techniques 11'12 One method for building a ]in-
of Technology, Atlanta, Georgia
earized, low-order, fl'equency-domain model from
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INTERNATIONAL FORUM ON AEROELASTICITY AND STRUCTURAL DYNAMICS PAPER IFASD 2001-008
CFD analysis is to apply the exponential (Gaus- aerodsmamic responses to compute non,linear sta-
sian) pulse input. 13 This method is used to excite bility derivatives. Jenkins 24 also investigates the
the aeroelastic system, one mode at a time, us- determination of nonlinear aerodynamic indicial
ing a broadband Gaussian pulse. The tinm-domain responses and nonlinear stability derivatives us-
responses are transformed into the frequency do- ing similar functional concepts. Stalford et at25
main to obtain the frequency-domain generalized develop Volterra models for simulating the behav-
aerodynamic force (GAF) influence coefficient ma- ior of a simplified nonlinear stall/post-stall aircraft
trix. These linearized GAFs can then be used model and the limit cycle oscillations of a simpli-
in standard linear aeroelastic analyses. 14 Raveh fied wing-rock model. In particular, they establish
et al ]5 applied this method while replacing the a straightforward analytical procedure for deriving
Gaussian pulse with step and impulse inputs. Re- the Volterra kernels from known nonlinear func-
cently, these time-domain, linearized impulse and tions.
step responses have been t.ransformed directly into
A particular response fl'om a CFD code may
state-space form for use in other disciplines such
provide information regarding the nonlinear aeroe-
as controls or optimization. I6 Guendal and Ces-
lastic behavior of a complex configuration due to
nik lr applied the Aerodynamic Impulse Response
a particular input, at a particular flight condition.
(AIR.) technique, based on the Volterra theory,
It does not, however, provide general information
to tile PMARC aerodynamic panel code. The
regarding the behavior of the configuration to a
PMARC/AIR code was applied to a simplified
variation of the input, or the flight condition, or
High Altitude Long Endurance (HALE) aircraft
both. As a result, repeated use of the CFD code is
for rapid linear and nonlinear aeroelast.ic analysis
necessary as input parameters and flight conditions
of tile vehicle. Yet another method, different from
are varied. A primary feature of the Volt.erra ap-
the Volterra-based ROM approach, is the Proper
proach is the ability to characterize a linearized or
Orthogonal Decomposition (POD) technique. The
nonlinear system using a small number of CFD-
POD is a method that is used extensively at. sev-
code analyses. Once characterized (via step or
eral research organizations for the development
impulse responses of various orders), the functions
of reduced-order models. A thorough review of
can be implemented in a computationally-efficient
POD research activities can be found in Beran and
convolution scheme for prediction of responses to
Silva. 18 A brief overview of the POD method and
arbitrary inputs without tile costly repeated use of
a representative result are included in this paper the CFD code of interest. Characterization of the
for completeness. In addition, a review of the is-
nonlinear response to an arbitrary input via the
sues involved in tile development of reduced-order
Volterra theory requires identification of the non-
models for fluid-structure interaction problems is
linear Volterra kernels for a specified configuration
provided by Dowell and Hall. 19 A topic of re-
and flight condition. Clearly, development and ap-
cent interest is tile potential development of hybrid
plication of Volterra-based ROM concepts depend
POD/Volterra methods. These hybrid techniques on the identification of the associated kernels for
would combine the spat ial resolution possible with
the problem of interest.
POD methods with the low dimensionality and
computational efficiency of Volterra methods. This The problem of Volterra kernel identification
is addressed by many investigators, including
is a very appealing concept that merits serious in-
vestigation. Rugh, -'6Clancy and Rugh, 'r Schetzen, -'s and more
recently by Boyd, Tang, and Chua. 29 There are
A valuable and important characteristic of tile several ways of identifying Volterra kernels in the
Volterra theory of nonlinear systems is that. the time and frequency domains that can be applied
theory is well defined in the time and frequency do- to continuous- or discrete-time systems. Tromp
mains for continuous- and discrete-time systems. and Jenkins 3° use indicial (step) responses from
In particular, this theory has found wide appli- a Navier-Stokes CFD code and a Laplace domain
cation in tile field of nonlinear discrete-time sys- scheme to identify the first-order kernel of a pitch-
tems 2° and nonlinear digital filters for telecommu- oscillating airfoil. Rodriguez al generates realiza-
nications and image processing, z] However, ap- tions of state-affine systems, which are related t.o
plication of nonlinear system theories, including discrete-thne Volterra kernels, for aeroelastic anal-
Volterra theory, t.o modeling nonlinear unsteady yses. Assuming high-frequency response, Silva 32
aerodynamic responses has not been extensive. introduces the concept of discrete-time, aerody-
One approach for modeling unsteady transonic namic impulse responses, or kernels, for a rect-
aerodynamic responses is Ueda and Dowell's 2e ap- angular wing under linear (subsonic) and non/in-
plication of describing functions, which is a har- ear (transonic) conditions. Silva aa improves upon
monic balance technique involving one harmonic. these results by extending the methodology to
Tobak and Pearson 23 apply the continuous-tinm arbitrary input frequencies, resulting ill the first
Volterra concept of functionals to Judicial (step) identification of discrete-time impulse responses of
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INTERNATIONAL FORUM ON AEROELASTICITY AND STRUCTURAL DYNAMICS PAPER IFASD 2001-008
an aerodynamic system. It. should be noted that
owing to separation of the downwash input terms,
Silva's first, approach had limited applicability for
the identification of nonlinear Volterra kernels 33 a
situation which has been resolved recent.ly. 1°
In his dissertation, Silva 1° discusses the funda-
mental differences between traditional, continuons-
time theories and modern discret.e-t.ime formula-
tions that allow the identification of discrete-time
kernels. The discrete-time methods are then ap-
plied to various nonlinear systems including a non-
linear Fliccat.i circuit, the viscous Burger's equa-
tion, an aeroelastic wing in transonic flow using a
transonic small-disturbance code. and a supercriti- Fig. 1 Schelnatle representation of the
cal airfoil undergoing large plunge motions at tran- Reduced-Order Modeling research program at
sonic conditions using a Navier-Stokes flow solver the NASA Langley Research Center.
with the Spatart-Alhnaras turbulence model. Al-
Center can be categorized into three disciplines:
though the majority of the research mentioned
computational fluid dynamics (('FD), system iden-
thus far involving aeroelastic Volterra kernel iden-
tification, and control law design. A schematic
tification has been of a computational/analytical
nature, the experimental identification of Volterra representation of the components of this research
program is presented as Figure 1.
kernels for a nonlinear aeroelastic syst.em has been
performed. Kurdila et a134 applied an efficient In addition to representing the vision for ROM
wavelet-based algorithm to the extraction of the research at. the NASA LalRC, Figure 1also serves
noulinear Volterra kernels of an aeroelast.ic system as an organizational outline for this paper. The
exhibiting limit cycle oscillations (LCO). There CFD portion consists of the development and ap-
is increased interest in the development of these plication of techniques implemented into various
CFD codes for the extraction of Volterra kernels.
experimental techniques for use in various experi-
mental settings. The identification of LCO during As will be discussed, the Volterra kernels represent
linear and nonlinear flmctional ROMs for these
flight flutter tests is a case in point.
CFD models. These functional ROMs can be used
One of the goals of the present paper is to pro-
vide enough information to motivate readers to within a digital convolution setting t.o provide in-
consult the refe,'ences. The methods and results creased computational efficiency over lhe original
CFD codes. These fimctional ROMs also can be
presented are not intended to be complete and all
inchlsive. Another goal of the present paperis to fa. transformed into the more traditional state-space
models, suitable for use in modern control the-
miliarize the reader with NASA Langley Research
Center's (LaRC's) vision for ROM research, unify- ory and optimization. The transformation of the
ing several of the methods aud results presented. Volterra kernels int.o state-space form is performed
It. should be stated that some of the research dis- within the System Identification category. In ad-
cussed in the paper is funded by NASA while the dition, the analytical derivation and the experi-
mental identification of aeroelast.ic kernels will be
other activities are purely of a cooperative nature.
discussed under this heading. The Control Law
Development of BOMs using POD methods is cur-
rently funded by the Air Force Research Labora- Design portion of Figure 1 is still in its early
stages and is left as a topic for future publications.
tory (AFBL) although a cooperative effort between
the NASA LaBC The paper begins with a brief Research involving POD could easily fit. within ei-
description of the methods discussed: \:olt.erra the- ther the CFD or Syst.em Identification topics but
ory of nonlinear systems, state-space models from is treated separately with various important and
aerodynamic impulse responses, and POD. This is valuable references for description of details.
followed by presentation of various COlnputational. Prior to discussion of the various results under
analytical, aim exl)e,'imental results, including re- /.he categories just described, a brief description of
cent results based on the application of the POD the various theoretical concepts follows.
method. The paper concludes with recommenda-
Volterra Theory
tions for tim, re research.
We begin by reviewing key features of the
Volterra theory, as applied to time-invariant, non-
Theoretical Background linear, continuous- and discrete-time syst.ems. The
The research and development of reduced-order literature on Volterra theory is significant, includ-
models (ROMs) for applications in nonlinear ing several t.exts. 2<aS-at This section will con-
aeroservoelasticity at. the NASA Langley Research centrate on time-domain Volterra formulations.
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[NTERNATIONAL FORUM ON AEROELASTICITY AND STRUCTURAL DYNAMICS PAPER IFASD 2001-008
consistent with tile implied application to time- For linear systems, only the first-order kernel is
domain, computational aeroelasticity methods. non-trivial, and there are no limitations on input
Details regarding the foundations and applications amplitude.
of tile fl'equency-domain Volterra theory can be Silva 1° derives the first- and second-order ker-
found in several references. 26,29,a6-39 Marzocca nels, which are presented here in final form in
et al3s'3"_' present a thorough discussion of the terms of various response functions:
frequency-domain Volterra theory with respect to
1
nonlinear aeroelastic phenomena. This research is h:(t) = 2w:(tl) - (3)
discussed in a subsequent section.
We first consider time-invariant, nonlinear, 1
continuous-time, systems followed by the applica- h.,(tl, t2) = :_(wl(tl, t2) - wl(tt) - wl(t2)).
(4)
tion of Volterra theory to discrete-time systen_s
(e.g., systems arising in CFD). Of interest is the re- In (3), wl (tl) is the time response of the system to
sponse of the system about all initial state w(0) = a unit impulse applied at time 0 and w2(tl) is the
Wo due t.o an arbitrary input u(t) (we take u as a time response of the system to an impulse of twice
real, scalar input, such as pitch angle of an airfoil) unit magnitude at time O. If the system is linear,
then w2 = 2w0 and hi = w0. If the system is
for t >_ 0. As applied to these systems, Volterra
theory yields the response nonlinear, then this identification of the first-order
kernel captures an amplitude-dependent nonlinear
effect. The identification of tile second-order kernel
w(t) = h0 + h_(t - r)u(r)dr
t is more demanding, since it. is dependent oil two
paraineters. Assuming 12 > fl ill (4), Wl(f2) is
the response of the system to an impulse at time
+ JoJ' 0h'2(t - q,t - r2)u(rl)u(r..,)drldr2+ t2.
Time is discretized with a set of time steps of
N t t equivalent size. Discrete time increments are in-
dexed fl'om 0 (time 0) t.o n (time t), and tile
E _O ""_ hn('--Tl ,..,t--Tn)tt(T1)..U(Tn)dTl..d'Fn.
n=3 evaluation of w at. time 7_is denoted by w[n]. The
(i) convolution in discrete time is
The Volterra series in expression (1) contains three
N
classes of terms. The first is the steady-state term
w[,,]= h0+
satisfying the initial condition, h0 = W0. Next is
k=0
tile first response term, fo hl(t - r)u(rjdr, where
hi is known as the first-order kernel (or the lin- N N
+ ,h,,,- (6)
ear/linearized unit impulse response). This term
represents the convolution of the first-order ker- kl=0 k._=O
nel with the system input for times between 0 where N is tile total time record of interest.
and t. Lastly are the higher order terms involving It should be noted that an important conceptual
the second-order kernel, h2, through the nth-order breakthrough in the development and application
kernel, h,_. The existence of these terms is an in- of the discrete-time Volterra theory as a ROM
dication that the system is nonlinear, 4° technique is the distinction between a continuous-
The couvergence of the Volterra series is depen- time unit impulse response and a discrete-time
dent on input magnitude and the degree of system unit impulse response./°,4° The continuous-time
nonlinearity. Boyd 41 shows that the convergence unit impulse response is an abstract function typi-
of the Volterra series cannot be guaranteed when cally defined as a function with an amplitude that
the maximum value of the input exceeds a criti- reaches infinity while its width approaches zero but
cal value, which is system dependent. Of course, its integral is unity. This fuuction is difficult., if
the issue of convergence is important, since tile not impossible, to apply ill practical applications
Volterra series nmst be truncated for analysis of (i.e., discret.e-tinle problems). The discrete-tilne
practical systems. Silva 1°'4° and Raveh et. 31.15 unit impulse response (known as a unit sample
consider a weakly nonlinear formulation, where it response), on the other hand, is specifically de-
is assumed that the Volterra series can be accu- signed for discrete-time (i.e., numerical) applica-
rately truncated beyond the second-order term: tions. This function is defined as having a value
of unity at. one point in time and zero everywhere
else. This is clearly a simpler function to imple-
w(t) = h0 + ht(t - r)u(r)dr
_0 t ment in a numerical setting. The proof of this and
details regarding the very powerful unit sample re-
spouse can be found in any modern text. on digital
/0'f0'
+ ht,(f- rl,t -- ro,)u(q)u(r2)dqdr_. (2) signal processing. 4-_
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The identification of linearized and nonlinear
Volterra kernels is an essential step ill the deve]- v[,] = cx[,,] + D,,[,,] (s)
oplnent of ROMs based on Volterra theory, but it
where ,r is all m-dinlensional state vector, u an p-
is not the final step. Ultimately, these filnctiona]
dilnensional control input, and 9 a r-dinlensional
kernels can be transformed into linearized and
output or measurement vector with n being the
nonlinear (bilinear) state-space systems that can
discrete time index. The trallsition matrix, A,
be easily implenlented into other disciplines such
characterizes tile dynalnics of the system. The
as controls and opt.ilnization, m26 llecently, lin-
goal of systeln realization is to generate constant
earized state-space models of an unsteady aerody-
matrices (A, B. C. and D) such that tile output
namic systeln have been developed while research
responses of a given system due to a particular set
into tile developlnent of nonlinear state-space mod-
els continues. 1s of inputs is reproduced by the discrete-time slate-
space systenl described above.
The frequency-dolnain version of the Volterra
For the system of (7) and (8), the tilne-domain
theory is silnply tile Fourier transforln of the se-
values of the systems unit salnple response are also
ries shown in (1). Therefore. tile Fourier transform
known as the Marker paralneters and at'(" defined
of the first-order kernel (for a linear system) is the
38
frequency response fllnction of the syst.em. Higher-
Y[,_]= (::A"- 1B (9)
order kernels are Fourier transforlned into higher-
order frequency response flllWtions, discussed ill with B an in by p matrix and C' a q by nl matrix.
most of the references already mentioned. Tile Syslem realization techniques provide the constant
primary benefit of these higher-order frequency inatl'ices A, B, C and D using Y(n).
response functions is that they provide informa- The gigensystem llealization Algorithm (ERA)
tion regarding the interaction of frequencies due algorit.hm 4s'4s begins by defining the generalized
to a nonlinear process. For example, bispectra Hankel matrix consisting of tile Markov parame-
(the frequency-domain version of the time-domain ters for all input/output combinations. The algo-
second-order kernel) have been used ill the study of rit.hna then uses the singular value decolnposition
grid-generated turbulence to identify the nonlinear (SVD) to COml)ute the A, ,B',and C matrices. Of-
exchange of energy fi'om one frequency to another. tell, the direct feedthrough matrix, D, is nonzero
Linear concepts, by definition, cannot provide this whenever the initial values of the Marker parame-
type of inforlnation. Ill addition, some very in- ters are nonzero.
t.eresting and fnndamenlal applications using the The ERA algorithnl has been used successfully
frequency-domain Volterra theory as'a9 and exper- for tile identification of several experilneutal struc-
ilnental applications of Volterra lnetllods 43'44 are tural dynamic systems. Although the algorithm
providing new "windows" on the world of nonlin- also has been used to extract damping and fie-
ear aeroelast icily. quency information fi'om CFD-generated aeroe-
lastic transients (no published references), this
Discrete-Tilne State-Space Models
research represents the first, time that the Ella
The basic fornmlation used in the development algorithm is applied to tile development of un-
of state-space models of the unsteady aerodynamic steady aerodynamic state-space models using aero-
system using the impulse responses (Volterra ker- dynamic pulse responses (Markov paralneters).
nels) obtained directly fi'om a CFD code is de- Additional details regarding the EI_A algorithm
scribed ill this subsection. Tile ability to generate and its nul]lerous applications are discussed ill the
state-space models of systems using ilnpulse re- references provided.
sponses was a primary ,notivation for tile develop-
Proper Orthogonal Decomposition (POD)
lnent of aerodynamic impulse response funct ions.l°
POD is a linear method for est.al)lishing all opti-
Although the method and results presented for this
Inal basis, or lnodal decomposition, of an enselnble
activity are linearized results, tile long-terln goal
of continuous or discrete functions. "File variables
is the developlnent of nonlinear state-space mod-
and paralneters defined ill this subsection are not
els directly from the nonlinear V'olterra kernels.
related t.o variables fi'oln any previous subsection.
The linearized results presented here are a starting
Beran is provides all excellent sunnnary of the
point for the nonlinear state-space lnodel develop-
POD lnethod and it.s origins. Detailed deriva-
ment activity. Details Call be found ill the reference
tions of tile POD and its properties are available
by Silva and Raveh. is
elsewhere 4r'4s and tier repeated herein. In this
A finite-dilnensional, discrete-tilne, linear, time-
discussion of POD, M basis vectors are used to rep-
invariant dynalnical systeln has the state-variable
resent deviations of w(/) froln a base solution, W0.
equal ions
These basis vectors are written as {e 1,e2..... eM},
and are referred to by many nalnes, inchlding POD
x[n + 1] = Ax[n] + Bu[n] (7) vectors, 49 empirical eigenfulwtions 4r or, simply,
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INTERNATIONAL FORUM ON AEROELASTICITY AND STRUCTURAL DYNAMICS PAPER IFASD 2001-008
modes. 4r'5° For the sake of brevity, we shall use
0.4 Identification amplitudes
the term "modes" to denote the POD basis vec-
tors. Each mode is normalized such that ciTe i ----1
(i = 1,..., M), and computed in a manner to be
described shortly. The modal decomposition of w 0.20_ _
using M modes, or wM, is given by hl -0.2
force/plunge -0.4
M
WM = W0 + E d'iei = W0 + Ow, (10)
-0.6
i=1
-0.8
0 0.005 0.010 0.015 0.020
where • is an N x:1.Imatrix containing the ordered Nondimensional time
set of modes, • = [el,e _-.... ,e M] and W is an
Fig. 2 Volterra kernels for CFD analysis
M-dimensional vector of modal amplitudes, _i, =
of RAE airfoil:First-order kernel and effect of
[ti,1, */,2,..., WM]. As a time-varying function, w is
identification amplitudes.
approximated by Wo + @W(t).
As stated by Hohnes et al., 47 "Linearit.y is the
0.040
source of the [POD] method's strengths as well as • 1st
its limitations ..." The method is linear owing to 0.030 • 2nd
• 3rd
the independence of the modes fi'om the modal am- 0.020
• 4th
plitudes, thereby allowing for the straightforward 0.010 4th • 5th
h2
construction of reduced-order equation sets from
force/plunge 0
the full equation sets following mode computation.
-0.010
5th
The POD modes are constructed by first com-
-0.020 Ii
puting samples, or snapshots, of system behavior
-0.030 1
(solutions at different instants in time for dynamic
-0.040 I I I
problems, or equilibrium solutions at different pa-
0 0.005 0.010 0.015 0.020
rameter values for static problems) and storing Nondimensionaltime
these samples in a snapshot matrix, S. This sam-
pling process is often refered to as POD "training." Fig. 3 Volterra kernels for CFD analysis
For now, we assume that M snapshots are collected of RAE airfoil: First five COlnponents of the
second-order kernel for plunge amplitude of 0.1.
and colulnn-wise collocated into the N x M snap-
shot matrix: S = [w 1,w 2..... wM]. The basis Results
provided by the POD, known as the Karhunen-
The results discussed in this section address the
Loeve 51'_=' (or K-L) basis, has been shown to min-
top two disciplines (circles) of Figure 1. The first
imize the error of the approximation of functions
subsection discusses the approximation of CFD
using these basis functions 4r-49,sa. 54
results using ROMs and the second subsection dis-
In practice, fewer than M modes are retained
cusses results from system identification studies.
to simulate system behavior. These are selected
based on the size of the modal eigenvectors. Sim- Approximating CFD Results with ROMs
ply put, the K-L basis for a subspace of dimension This section describes some results obtained
M,. < M is obtained by retaining the modes asso- in the identification of CFD-based (time-domaiu)
ciated with the M,. largest eigenvalues computed. Volterra kernels. The first result presented is fl'om
The techniques described in Beran and Silva is Sih,a l° for a supercritical airfoil at a transonic
provide different means for obtaining reduced- Ma.ch number.
order equation sets governing w(¢) in the POD First.- and second-order kernels for the Navier-
subspace. There are several nlethods for accom- Stokes solution (with Spalart-Alhnaras turbulence
plishing this including the Galerkin projection, model) of an RAE airfoil undergoing plunging
"subspa.ce" projection (for linear and nonlinear motions at. a transonic Mach number using the
syst.ems), and collocation. These methods are dis- CFL3D code 5r are presented in Figures 2 and
cussed in detail by Beran and Silva is and will not 3. Shown in Figure 2 are two sets of first-
be repeated here. In addition, recent results by order kernels due to two different sets of excita-
Kim and Bussoletti 55 and Kim 5_ are indicative of tion amplitudes. Recall that the first-order kernel
the efficiency and suitability of these techniques. is computed using (3) and is tile result of two
However, due to the importance and value of POD pulse responses, one at a particular amplitude and
techniques, a sample result is provided in a sub- the second at double tile first amplitude. One
sequent section in order to familiarize the reader of the first-order kernels shown in Figure 2 was
with the appropriate references concerning POD computed using excitation phmge amplitudes of
research. w = 0.01 and w = 0.02, where w is a fraction
7oF 16
INTERNATIONAL FORUM ON AEROELASTICITY AND STRUCTURAL DYNAMICS PAPER IFASD 2001-008
of tile chord of the airfoil. The other first-order 1.5
keruel was computed using excitation plunge am-
1.0
plitudes ofw = 0.1 and w = 0.2. It. is clear that the
two kernels are not linearly related, demonstrating
0.5
[low tile first-order kernel can capture amplitude-
Plunge
dependent nonlinear effects. response
0
Shown in Figure 3 are five components of the
second-order kernel for this case. The second-order
-0,5
kernel is more complicated because it is a two- First-order _ne:¢_ _
dimensional function of time. The second-order
-1.0 I I I I I
kernel is presented as a family of ftmclions in Fig- 0 1 2 3 4 5
Nondimensional time
ure 3 for simplicity. The important point to
be made is that this kernel is readily generated
Fig. 4 Comparison of actual nonlhiear and
and its relatively sinaller values (compared with first-order Volterra responses for three differ-
the first-order kernel) and its rapid convergence ent plunge motions and a linear response for
indicate a. small (but not negligible) level of noE1- the largest motion.
linearity present. This information may be used
to determine if the first-order kernel is sufficient to course, once the kernel has been computed, it can
capture tile dominant nonlinear effects. This point be used to predict, tile response to any arbitrary
is demonstrated in Figure 4. input (stead),, any and all frequencies, random) of
Figure 4 is a comparison of plunge responses arbitrary length via digital convolution oil a work-
for three different phmge amplitudes for the same station. Using this method, there is no need for the
configuration. Specifically, three comparisons are repeated, and costly, execution of tile CFD code for
made between the flill ('FD solution due t.o a si- different inputs.
nusoidal l)hmging motion (labelled in the figure Raveh, Levy and Karpel have recently applied
as "actual nonlinear") and that obtained using the Volterra-based ROM approach to analysis
the first-order kernel from Figure 2 due t.o the of the AGARD 44,5.6 wing. 15 They simulate
larger exeitation amplitudes. As can be seen, the the flow field around the wing using the EZNSS
plunge response obtained nsing the Vo]terra first- guler/Navier-Stokes code. 58 This code provides a
order kernel compares almost identically with the choice between two implicit algorithms, tile Bean>
response obtained from the flill CFD solution foE' Warming algorithm59and the partially flux-vector
the two smaller amplitude responses. In fact., the splitting algorithm of Steger et. al. c'° Grid genera-
two curves for these responses, are almost not dis- tion and inter-grid connectivity are handled using
cernable in Figure 4. The comparison for the the Chimera approach. 61 Tile code was enhanced
largest, amplitude response is very good as well, with an elastic capability to compute trimmed ma-
with a slight but noticeable difference between the neuvers of elastic a.ircraft. 5. For the ('FD compu-
two results. The nonlinearity of the large- ampli- tations, the flow field around tile wing was evalu-
tude phmge responses is confirmed by linearly scal- ated oil a C-H type grid, with 193 points in the
ing the smallest amplitude (i.e., linear) response chordwise direction along the wing and its wake,
which, as shown in Figure 4, cannot capture the 65 grid points in the spanwise direction, and 41
amplitude-dependent nonlinear plunge dynamics grid points along the normal direction.
seen at the larger amplitude. A process of mode-by-mode excitation, dis-
The turnaround time ("wallclock") for the full cnssed previously, was performed for this wing
CFD solution was oil the order of a day whereas using four elastic modes at a Math number of
tile Volterra first-order solution was computed oil 0.96. The mode-by-mode excitation technique pro-
a. workstation in 30 seconds using digital convolu- vides tile unsl.eady time-domain generalized aero-
tion. The initial cost of computing the first-order dynamic forces (GAFs) in all four modes due to
kernel was trivial as well due to tile rapid conver- an excitation of one of the modes. In this fash-
gence of the pulse responses. In fact, since each ion, a matrix of four-by-four functions (sixteen
pulse (unit and double amplitudes) goes to zero in total) is developed. Two sets of excitation inputs
less than 100 time steps, the responses were gener- were used: the discrete-time pulse input and the
ated nsing a debugging mode option available on discrete-time step input. The cost of computing
the supercomputer system used. Using ihis option, these functions is minimal due t.o the rapid con-
computations requiring less than 300 time steps vergence of these flinctions and it, consists of only
are executed immediately, int.euded for debugging four code executions. Once these functions were
purposes. As a result, each pulse was computed defined, several full CFD solutions were generated
within five minutes, resulting in a first-order ker- that were due to various sinusoidal inputs at. vari-
nel that was computed in about t,en minutes. Of ous frequencies. Shown in Figure 5 is just one of
8OF 16
INTERNATiONAl, FORUM ON AEROELASTICITY AND STRUCTURAL DYNAMICS PAPER IFASD 2001-008
50 _ p _ I tool! , f '
sO_-..-i ._:.--/_-: _:-
Mod.,0l ,i / i\ Mode2'0_; _ 'i_ 0.9 r I_Air, dt=0.01 see ]
respoo. / : ?- 0.8 _ [........ Air, dI=0.015 see |
-slo-_ :A_u_,'t o.r /f '__, [I------4-APirM, dAt=RO.0C2sec II
resp°ns]50_ _ I I ' I- Firstorder| 0.6
-1001 I I ' ) • 1
'lz"% I I 0.5
Mod°30[ , \ : :.4 CL 0.4
0.3
,espo.seI _ \ /L
-1 -i .... 0.2
responset.4 _
0.1
o 50 loo 150 2O0 0 50 100 150 200
NonDimensiona;Time NonOimensionNTime 0 ,, _/ '_ ._
-0.1
Fig. 5 Colnparison of actual and first-order
-0,2-
Volterra responses to sinusoidal excitation at 5
Hz (Mach 0.96). .0.30 0.I1 0I.2 0.I3 0.I4 0.I5 0.I6 0I.7
Time,sec
these results for a 5 Hz input fi'equency, conlparing
the result obtained from the full CFD solution to Fig. 6 Pitch responses of the SHAR Wing to
that obtained via convolution of the step or pulse a prescribed pitch command.
responses with a 5 Hz sinusoid. As can be seen,
the comparison is exact t.o plotting accuracy for oped in the late 1980s at NASA Ames Research
Centerfi 2 This code is a low-order panel method,
most of the responses. The full CFD solution, con-
meaning that the source and doublet singulari-
sisting of 8000 iterations required approximately
ties are considered to be constant on each panel.
24 hours on an SGI Origin 2000 computer with 4
PMARC is one of the lnost advanced panel meth-
CPUs. By' comparison, the Volterra-based ROM
ods available, featuring an advanced time-stepping
response shown required about a minute. Even
relaxed wake, internal flow capabilities, jet plume
including the upfront, cost. of computing the pulse
(or step responses), the computational cost. savings modeling, and rudimentary unsteady capabilities.
The inaplementation of the Volterra-based ROM
are significant. More importantly, the same pulse
technique into the PMARC code is referred to
(or step) functions can now be used to predict the
as the Aerodynamic hnpulse Response (AIR)
response of the aeroelastic system to any arbitrary
technique. A particularly useful feature of
iUl)Ut Of any length.
As a validation, a comparison of this approach PMARC/AIR is the ability to predict, the response
to an arbitrary input. For the purpose of testing
to another result available in the literature is pre-
this feature, the following Fourier sine series with
sented as Figure 7. Shown in Figure 7 is a com-
three components was used:
parison of linear and nonlinear GAFs for the first
two modes of the AGARD 445.6 Aeroelastic Wing
at a Mach number of 0.96. Nonlinear GAFs refers
O[t]= a°sin(2rrt) + 3°sin(4rrt) + 3osin(67rt) (11)
to the GAFs computed using the Volterra pulse-
response technique about a nonlinear steady-state
The quantity theta It] is a signal with three fi'e-
value by exciting one mode at a time and obtaining quencies: 1Hz, 2 Hz, and 3Hz. The pitch response
the resultant responses in the other modes. The
of the SHAR wing is shown in Figure 6. The re-
CFD results are compared with those using the
sponse was calculated with the AIR method using
ZAERO code for a purely linear case. Frequency-
impulse responses generated with three time steps,
domain values were obtained by performing a con-
At = O.01 seconds, 0.015 seconds, and 0.02 sec-
volution of several frequencies of interest with the
onds, and with the full solution from the PMARC
computed CFD-based pulse responses. As can be
code at At = 0.015 seconds. All three time step
seen, the comparison is reasonable and shows the
choices produce a fairly accurate prediction of the
small (but not negligible) nonlinear aerodynamic
response. These results indicate that the method
effects identified using the Volterra pulse-response
is a useful one for predicting the response to arbi-
technique.
trary inputs, especially those that may not be as
Additional computational applications of the
easy to implement within the PMARC code. This
Volterra-based ROM technique include the appli-
is a significant improvement over the capabilities
cation to a ttigh Altitude Long Endurance (HALE)
of the original PMARC code.
wing using a panel method for the aerodynamics.it
The simplified wing is referred to as the Simple System Identification
High Aspect Ratio (SHAR) Wing. The particu- In this section, some of the results classified un-
lar panel method selected for this research was the der the system identification portion of Figure 1
PMARC aerodynamic panel code, a code deve]- are discussed. These results include the t.ranfor-
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INTERNATIONAL FORUM ON AEROELASTICITY AND STRUCTURAL DYNAMICS PAPER [FASD 2001-008
mation of Volterra kernels into state-space form, A11 AI2
and analytical and experimental identification of
Volterra kernels for various systems. The state-
space resuts follow the method described in an
earlier subsection. The analytical identification fm,lg
of Volterra kernels consists of two parts: 1) the
derivation of compressible aerodynamic indicial I--- ZCAFEDRO(ROliMr_)ear"l
02 04 06 08 02 O4 O6 08
flmctions for subsonic, supersonic, and hypersonic k W
Mach number regimes and 2) the derivation of A21 A22
4
freqnency-domain Volterra kernels for aeroelastic
2
systems with structural nonlinearities. It should Ima9
0
be pointed out that the relationship between in-
-2
dicial functions and Volterra kernels is the same ._--- Rea_
-'_---L__
as the relationship between a step and impulse re-
sponses: one is the derivative of the other. The -e
02 04 06 08 02 04 016 O;B
experimental identification of Volterra kernels con- k k
sists of the experimental extraction of second-order Fig. 7 Comparison of linear and nonlinear
Volterra kernels for an aeroelastic systeln under- GAFs for the first two wing modes.
going limit cycle oscillations (LCO). Results from
these categories are now discussed. :::[
In an earlier subsection, the generation of time-
domain GAFs using a CFD code was presented.
Validation of these fimctions was performed in the
time and frequency domains. Time-domain vali-
dation consisted of a comparison between the full 10a .......
10I 10° 10_ 102
CFD solution and the solution obtained via con-
volution for forced harmonic responses at. several 200
frequencies. Frequency-domain validation was per-
,oo_
formed by Fourier transforming the time-domain
GAFs to the frequency domain and comparing
these with results available in the literature for -100
that configuration. Because traditional flutter
200_
analyses are performed in the frequency domain 10' 100 10_ 102
Frequency. Hz
using frequency domain GAFs, the standard ap-
proach for using linearized GAFs froln a CFD Fig. 8 Comparison of frequency responses for
code is to transform the time- domain GAFs into AGARD 445.6 Aeroelastic Wing. CFD (Solid
Line), State-Space (Dotted Line).
the frequency domain and use the standard flutter
analysis routines. Then. if a time-domain model of
versus the frequency response for the state-space
the GAFs is needed for ASE analyses (i.e., state-
system for the AGARD 44.5.6 Aeroelastic Wing.
space form), then rational function approximations
Presented in the figure are the responses (output)
(RFAs) are employed to transform the frequency-
for all four modes due t,oan input in rite first mode.
domain GAFs back into the time domain. But
The frequency responses of the pulses COml)uted
rather than transforming the time-domain GAFs
directly front the CFD code (solid lines) compare
into the frequency domain only to transform them
very well with the frequency responses obtained
back into tile lime domain, discrete-time, state-
from 32nd-order state-space system generated to
space systems can be created using the Volterra
model this system. The responses due to inputs
kernels (time-domain GAFs) directly. 16
in the second, third, and fourth modes are just as
l!sing the ERA method previously discussed, a good as those shown in Figure 8, but are not pre-
32nd-order state-space mode] of the AGARD 44.5.6 sented here for brevity. The important point to be
Aeroelastic Wing was generated. This state-space made is that time-domain kernels, extracted from
model consists of four inputs attd four outputs, one a CFD code, can now be transformed directly into
for each of the four structural modes. Then, in state-space form, amenable for use in modern con-
order to validate the accuracy of the state-space trol theory and optimization.
model, the fl'equeney content of the state-space In terms of analytically-derived kernels (indi-
model is compared to tile fi'equency content of cial flmctions), Marzocca et. al'; have developed
the Volterra kernels extracted fi'om the CFD code. a unified approach, based Ul)On the use of aero-
Presented in Figure 8 is a comparison of the fre- dynamic indicial fimctious, that enables the de-
quency response for the CFD solution (kernels) termination of the aerodynamic lift and moment
10OF16
INTERNATIONAL FORUM ON AEROELASTICITY AND STRUCTURAL DYNAIvlICS PAPER IFASD 2001-008