Simultaneous Excitation of Multiple-Input Multiple … Excitation of Multiple-Input Multiple-Output CFD-Based Unsteady Aerodynamic Systems ... Four diﬀerent types of

Simultaneous Excitation of Multiple-InputMultiple-Output CFD-Based Unsteady

Aerodynamic Systems

Walter A. Silva ∗

NASA Langley Research CenterHampton, Virginia 23681-0001

A significant improvement to the development of CFD-based unsteady aerodynamicreduced-order models (ROMs) is presented. This improvement involves the simultaneousexcitation of the structural modes of the CFD-based unsteady aerodynamic system thatenables the computation of the unsteady aerodynamic state-space model using a singleCFD execution, independent of the number of structural modes. Four different types ofinputs are presented that can be used for the simultaneous excitation of the structuralmodes. Results are presented for a flexible, supersonic semi-span configuration using theCFL3Dv6.4 code.

Introduction

EARLY mathematical models of unsteady aerody-namic response capitalized on the efficiency and

power of superposition of scaled and time-shifted fun-damental responses, also known as convolution. Clas-sical models of two-dimensional airfoils in incompress-ible flow1 include Wagner’s function2 (response to aunit step variation in angle of attack), Kussner’s func-tion3 (response to a sharp-edged gust), Theodorsen’sfunction4 (frequency response to sinusoidal pitchingmotion), and Sear’s function5 (frequency response toa sinusoidal gust). As geometric complexity increasedfrom airfoils to wings to complete configurations, theanalytical derivation of these types of response func-tions became impractical and the numerical computa-tion of linear unsteady aerodynamic responses, in thefrequency domain, became the method of choice.6

When geometry- and flow-dependent nonlinear aero-dynamic effects became significant, appropriate non-linear aerodynamic equations were solved using time-integration techniques. Coupling the nonlinear aero-dynamic equations with a linear structural model pro-vides a direct simulation of aeroelastic phenomena.This direct simulation approach for solving nonlinearaeroelastic problems has yielded a very powerful sim-ulation capability with two primary challenges. Thefirst challenge is the associated computational costof this simulation, which increases with the fidelityof the nonlinear aerodynamic equations to be solved.Computational cost may be reduced via the imple-mentation of parallel processing techniques, advanced

∗Senior Research Scientist, Aeroelasticity Branch, NASALangley Research Center, Hampton, Virginia; AIAA AssociateFellow

algorithms, and improved computer hardware process-ing speeds. The second, more serious, challenge is thatthe information generated by these simulations cannotbe used effectively within a preliminary design envi-ronment. Any attempt to incorporate the output ofthese aeroelastic simulations within a design environ-ment inevitably becomes design by trial-and-error. Asa result, the integration of traditional, computationalaeroelastic simulations into preliminary design activi-ties involving disciplines such as aeroelasticity, aeroser-voelasticity (ASE), and optimization is, at present, acostly and impractical venture.

The goal behind the development of reduced-ordermodels (ROMs) is aimed precisely at addressing thesetwo challenges. Development of a ROM entails thedevelopment of a simplified mathematical model thatcaptures the dominant dynamics of the original sys-tem. This alternative mathematical representation ofthe original system is, by design, in a mathematicalform suitable for use in a multidisciplinary, prelimi-nary design environment. As a result, interconnectionof the ROM with other disciplines is possible, therebyaddressing the second challenge. The simplicity ofthe ROM yields significant improvements in compu-tational efficiency as compared to the original system,thereby addressing the first challenge.

At present, the development of CFD-based ROMsis an area of active research at several government,industry, and academic institutions.7–11 Developmentof ROMs based on the Volterra theory is one of sev-eral ROM methods currently under development.12–16

Reduced-order models based on the Volterra theoryhave been applied successfully to Euler and Navier-Stokes models of nonlinear unsteady aerodynamic and

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American Institute of Aeronautics and Astronautics Paper 2007-XXX

https://ntrs.nasa.gov/search.jsp?R=20070018341 2018-06-04T09:27:04+00:00Z

aeroelastic systems. Volterra-based ROMs are basedon the creation of linearized and nonlinear unsteadyaerodynamic impulse responses that are then used in aconvolution scheme to provide the linearized and non-linear responses of the system to arbitrary inputs. Inthis setting, the linearized and nonlinear impulse re-sponses may be considered to be the ROMs of theparticular nonlinear system under investigation. Al-ternatively, upon transformation of the linearized andnonlinear impulse responses into state-space form, thestate-space models generated may also be consideredROMs.

Various inputs can be used in the time domain(CFD code) to generate generalized aerodynamicforces (GAFs) in the frequency domain in order toperform standard, frequency-domain aeroelastic anal-yses. But if time-domain aeroservoelastic (ASE) anal-yses are desired, the frequency-domain GAFs aretransformed back into the time domain using tradi-tional rational function approximation (RFA) tech-niques. These techniques include, for example, thewell-known Rogers approximation17 and the MinimumState technique.18 The RFA techniques transformfrequency-domain GAFs into state-space (time do-main) models amenable for use with modern controltheory and optimization. The process just describedtransforms time-domain information (CFD results)into frequency-domain information only to have thefrequency-domain information transformed back intothe time domain.

Gupta et al19 and Cowan et al20,21 applied a setof flight testing inputs to an unsteady CFD code andused the information to create a linear autoregressivemoving average (ARMA) model that was transformedinto state-space form. Although this technique is ap-plied entirely within the time domain, the shape of theinputs applied to the CFD code requires tailoring inorder to excite a specific frequency range, resulting inan iterative process. In a similar vein, Rodrigues22 de-veloped a state-space model for an airfoil in transonicflow using a transonic small-disturbance algorithm.

In the present paper, a direct approach for effi-ciently generating linearized unsteady aerodynamicstate-space models is presented. Although the presentapplication of the method deals with linearized re-sponses based on linearized impulse responses (lin-earized Volterra kernels), the method can be formallyextended to address nonlinear aeroelastic phenomenavia the use of nonlinear impulse responses (nonlinearVolterra kernels).

Silva and Bartels8 introduced the development oflinearized, unsteady aerodynamic state-space modelsfor prediction of flutter and aeroelastic response usingthe parallelized, aeroelastic capability of the CFL3Dv6code. The results presented provided an important

validation of the various phases of the ROM devel-opment process. The Eigensystem Realization Algo-rithm (ERA),23 which transforms an impulse response(one form of ROM) into state-space form (anotherform of ROM), was applied for the development ofthe aerodynamic state-space models. The ERA ispart of the SOCIT (System/Observer/Controller Iden-tification Toolbox). Flutter results for the AGARD445.6 Aeroelastic Wing using the CFL3Dv6 code werepresented as well, including computational costs. Un-steady aerodynamic state-space models were gener-ated and coupled with a structural model within aMATLAB/SIMULINK24 environment for rapid calcu-lation of aeroelastic responses including the predictionof flutter. Aeroelastic responses computed directlyusing the CFL3Dv6 code showed excellent compari-son with the aeroelastic responses computed using theCFD-based ROM.

Previously, the aerodynamic impulse responses thatwere used to generate the aerodynamic state-spacemodels were computed using CFL3Dv6.0 via the exci-tation of one mode at a time. For a four-mode system,these computations are not very expensive. However,for more realistic cases where the number of modescan be an order of magnitude or more larger, theone-mode-at-a-time method becomes impractical. To-wards the solution of this problem, Raveh,11 Kim25

and Kim et al10 developed methods that enable the si-multaneous excitation of the structural modes, greatlyreducing the cost of identifying the aerodynamic im-pulse responses from the CFD code. Raveh’s methodconsists of using filtered white Gaussian noise as si-multaneous excitation to the structural modes. Ravehthen presents three methods for generating ROMs: afrequency-domain method, an ARMA method, and astate-space method. Kim’s method consists of usingstaggered step inputs, one per mode, and then ap-plying several techniques for recovering the individualresponses from this type of simultaneous excitationleading to the creation of state-space models.

This paper introduces four different types of ex-citations that can be used to simultaneously excitethe structural modes while enabling the recovery ofthe individual responses. This will enable the com-putation of aerodynamic impulse responses for anynumber of structural modes using a single CFD ex-ecution. Reduced-order models generated using theone-mode-at-a-time method are compared to ROMsusing the simultaneous excitation inputs. Recent ad-ditional enhancements to the development of unsteadyaerodynamic and aeroelastic ROMs, which includesthe use of ROMs for static aeroelastic responses atmatched-point atmospheric conditions, are presentedin a separate paper.26

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Description of CFD and SystemIdentification Methods

The following subsections describe the parallelized,aeroelastic version of the CFL3Dv6.4 code, the phasesof the original and improved ROM development pro-cesses, and a description of the four types of functionsused for simultaneous excitation of the CFD unsteadyaerodynamic system.

CFL3Dv6.4 Code

The computer code used in this study is theCFL3Dv6 code, which solves the three-dimensional,thin-layer, Reynolds averaged Navier-Stokes equationswith an upwind finite volume formulation.27–29 Thecode uses third-order upwind-biased spatial differenc-ing for the inviscid terms with flux limiting in thepresence of shocks. Either flux-difference splitting orflux-vector splitting is available. The flux-differencesplitting method of Roe30 is employed in the presentcomputations to obtain fluxes at cell faces. There aretwo types of time discretization available in the code.The first-order backward time differencing is used forsteady calculations while the second-order backwardtime differencing with subiterations is used for staticand dynamic aeroelastic calculations. Furthermore,grid sequencing for steady state and multigrid and lo-cal pseudo-time stepping for time marching solutionsare employed.

One of the important features of the CFL3D code isits capability of solving multiple zone grids with one-to-one connectivity. Spatial accuracy is maintainedat zone boundaries, although subiterative updating ofboundary information is required. Coarse-grained par-allelization using the Message Passing Interface (MPI)protocol can be utilized in multiblock computations bysolving one or more blocks per processor. When thereare more blocks than processors, optimal performanceis achieved by allocating an equal number of blocks toeach processor. As a result, the time required for aCFD-based aeroelastic computation can be dramati-cally reduced.

In this paper, multiblock MPI parallel aeroelasticcomputations, including flutter, for a flexible, semi-span supersonic configuration are performed using 28flowfield blocks. In order to achieve an optimal di-vision of grid points, it is necessary to place flowfield block boundaries near a moving solid surface (thewing). The multiblock boundary and interior move-ment scheme allows the user to place block boundariesnear surfaces as necessary for optimal parallelization.Boundaries interior to the fluid domain near a sur-face respond to the local surface motion. As the wingmoves, block boundaries move to maintain integrity ofblock interfaces and the airfoil surface.

Because the CFD and computational structural me-

chanics (CSM) meshes usually do not match at theinterface, CFD/CSM coupling requires a surface splineinterpolation between the two domains. The interpo-lation of CSM mode shapes to CFD surface grid pointsis done as a preprocessing step. Modal deflections atall CFD surface grids are first generated. Modal dataat these points are then segmented based on the split-ting of the flow field blocks. Mode shape displacementslocated at CFD surface grid points of each segment areused in the integration of the generalized modal forcesand in the computation of the deflection of the de-formed surface. The final surface deformation at eachtime step is a linear superposition of all the modaldeflections.

System Identification Method

In structural dynamics, the realization of discrete-time state-space models that describe the modal dy-namics of a structure has been enabled by the de-velopment of algorithms such as the EigensystemRealization Algorithm (ERA)23 and the ObserverKalman Identification (OKID)31 Algorithm. Thesealgorithms perform state-space realizations by us-ing the Markov parameters (discrete-time impulseresponses) of the systems of interest. These algo-rithms have been combined into one package known asthe System/Observer/Controller Identification Tool-box (SOCIT)32 developed at NASA Langley ResearchCenter.

There are several algorithms within the SOCIT thatare used for the development of unsteady aerody-namic discrete-time state-space models. The PULSEalgorithm is used to extract individual input/outputimpulse responses from simultaneous input/output re-sponses. For a four-input/four-output system, simul-taneous excitation of all four inputs33 yields four out-put responses. The PULSE algorithm is used to ex-tract the individual sixteen (four times four) impulseresponses that associate the response in one of the out-puts due to one of the inputs. Details of the PULSEalgorithm are provided in the references. Once the in-dividual sixteen impulse responses are available, theyare then processed via the Eigensystem RealizationAlgorithm (ERA) in order to transform the sixteenindividual impulse responses into a four-input/four-output, discrete-time, state-space model. A brief sum-mary of the basis of this algorithm follows.

A finite dimensional, discrete-time, linear, time-invariant dynamical system has the state-variableequations

x(k + 1) = Ax(k) + Bu(k) (1)

y(k) = Cx(k) + Du(k) (2)

where x is an n-dimensional state vector, u an m-dimensional control input, and y a p-dimensional out-

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put or measurement vector with k being the discretetime index. The transition matrix, A, characterizesthe dynamics of the system. The goal of system real-ization is to generate constant matrices (A, B, C) suchthat the output responses of a given system due to aparticular set of inputs is reproduced by the discrete-time state-space system described above.

For the system of Eqs. (1) and (2), the time-domainvalues of the systems discrete-time impulse responseare also known as the Markov parameters and are de-fined as

Y (k) = CAk−1B (3)

with B an (n x m) matrix and C a (p x n) matrix.The ERA algorithm begins by defining the generalizedHankel matrix consisting of the discrete-time impulseresponses for all input/output combinations. The al-gorithm then uses the singular value decomposition(SVD) to compute the A, B, and C matrices.

In this fashion, the ERA is applied to unsteadyaerodynamic impulse responses to construct unsteadyaerodynamic state-space models.

Original ROM Development ProcessesA CFD-based aeroelastic system can be viewed as

the coupling of a nonlinear unsteady aerodynamic sys-tem (flow solver) with a structural system as depictedin Figure 1. The present study focuses on the devel-opment of a linearized unsteady aerodynamic ROM(in state-space form), using the general procedure de-picted in Figure 2, that is then coupled to a structuralmodel (also in state-space form) for aeroelastic analy-ses. For the discussions that follow, the term ROMwill refer to the unsteady aerodynamic state-spacemodel. When the unsteady aerodynamic state-spacemodel (ROM) is connected to a state-space model ofthe structure within the SIMULINK environment, thissystem is often also referred to as a ROM. However,to avoid confusion, the SIMULINK aeroelastic systemwill be referred to as the aeroelastic simulation ROM.

An outline of the original (one-mode-at-a-time)ROM development process8 is presented as back-ground for the new enhancements. The original ROMdevelopment process is as follows:

1. Implementation of impulse/step response tech-nique into aeroelastic CFD code;

2. Computation of impulse/step responses for eachmode, one mode at a time, of an aeroelastic systemusing the aeroelastic CFD code; these responses arecomputed about a static aeroelastic solution (or agiven dynamic pressure);

3. Impulse responses generated in Step 2 are trans-formed into an unsteady aerodynamic state-space sys-tem using the ERA (within SOCIT);

4. Evaluation/validation of the state-space modelsgenerated in Step 3 via comparison with CFD results

Fig. 1 Coupling of structure and aerodynamicswithin an aeroelastic CFD code.

Fig. 2 Identification of generalized aerodynamicforces (GAFs).

(i.e., ROM results vs. full CFD solution results);In the original ROM process, since each mode is be-

ing excited individually, the response in each outputdue to a particular input (e.g., the impulse responsein output 2 due to input 1) is generated almost di-rectly. If the function being used to excite the systemis an impulse (or a unit pulse for discrete-time sys-tems), then the output from the CFD solution willconsist of the impulse response of each output dueto that one input. If the function being used to ex-cite the system is not an impulse function (such as astep input or a random input), then the impulse re-sponse needs to be extracted from the input/output

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data. One method for extraction is deconvolution8

but there are other methods that can be used. For theoriginal ROM process, since each input to a systemis being excited individually, the exact nature of theinput function is defined, to a certain extent, basedon user preference (frequency range of interest, easeof implementation into a CFD solver, etc.). The stepinput has emerged as a convenient input for the orig-inal ROM process due to its ease of implementationinto a CFD solver and to the wide range of frequencyexcitation that it can generate.

The primary issue,then, with the original ROM pro-cess was the identification of the unsteady aerody-namic impulse responses one mode at a time (Step 2).Clearly, for a large number of modes, this procedurebecomes impractical.

Improved ROM Development ProcessesAn outline of the improved simultaneous modal ex-

citation ROM development process with the recentenhancements is as follows:

1. Generate the number of functions (from a se-lected family) that corresponds to the number of struc-tural modeshapes;

2. Apply the generated input functions simulta-neously via one CFD execution; these responses arecomputed directly from the restart of a steady rigidCFL3D solution (not about a particular dynamic pres-sure);

3. Using the simultaneous input/output responses,identify the individual impulse responses using thePULSE algorithm (within SOCIT);

4. Transform the individual impulse responses gen-erated in Step 3 into an unsteady aerodynamic state-space system using the ERA(within SOCIT);

5. Evaluate/validate the state-space models gener-ated in Step 4 via comparison with CFD results (i.e.,ROM results vs. full CFD solution results);

An important difference between the original ROMprocess and the improved ROM process is stated insteps (2) of the outlines above. For the original ROMprocess, if a static aeroelastic condition existed, thena ROM was generated about a selected static aeroe-lastic condition. So a static aeroelastic condition ofinterest was defined (typically a dynamic pressure) andthat static aeroelastic condition was computed usingCFL3D as a restart from a converged steady, rigid so-lution. Once a converged static aeroelastic solutionwas obtained, the ROM process was applied aboutthat condition. This implies that the resultant ROMis, of course, limited in some sense to the neighbor-hood of that static aeroelastic condition. Moving ”toofar away” from that condition could result in loss ofaccuracy.

The reason for generating ROMs in this fashion

was because no method had been defined to enablethe computation of a static aeroelastic solution usinga ROM. Any ROMs generated in this fashion were,therefore, limited to the prediction of dynamic re-sponses about a static aeroelastic solution includingthe methods by Raveh11 and by Kim et al.10 The im-proved ROM method, however, includes a method forgenerating a ROM directly from a steady, rigid solu-tion. As a result, these improved ROMs can then beused to predict both static aeroelastic and dynamicsolutions for any dynamic pressure. In order to cap-ture a specific range of aeroelastic effects (previouslyobtained by selecting a particular dynamic pressure),the improved ROM method relies on the excitationamplitude to excite aeroelastic effects of interest. Thedetails of the method for using a ROM for computingboth static aeroelastic and dynamic solutions is pre-sented in another reference by the present author.26

For the present results, all responses were computedfrom the restart of a steady, rigid CFL3D solution,bypassing the need (and additional computational ex-pense) to execute a static aeroelastic solution usingCFL3D.

Simultaneous Excitation Input Functions

In the situation where the goal is the simulta-neous excitation of a multiple-input multiple-output(MIMO) system, system identification techniques34–36

dictate that the nature of the input functions usedto excite the system must be properly defined if ac-curate input/output models of the system are to begenerated. The most important point to keep in mindwhen defining these input functions is that these func-tions need to be different, in some sense, from eachother. This makes sense since, if the excitation inputsare identical and they are applied simultaneously, itbecomes practically impossible for any system iden-tification algorithm to relate the effects of one inputon a given output. This, in turn, makes it practicallyimpossible for that algorithm to extract the individ-ual impulse responses for each input/output pair. Ashas already been well established, the individual im-pulse responses for each input/output pair are nec-essary ingredients towards the development of state-space models. With respect to unsteady aerodynamicMIMO systems, these individual impulse responsescorrespond to time-domain generalized aerodynamicforces (GAFs), critical to understanding unsteadyaerodynamic behavior. The Fourier-transformed ver-sion of these GAFs are the frequency-domain GAFswhich provide an important link to more traditionalfrequency-domain-based aeroelastic analyses.

The question is how different should these inputfunctions be and how can we quantify a level of ”dif-ference” between each input function? Kim10 uses the

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familiar step function as the input function but witheach step input being applied at a different point intime (lagged) in order to maintain some difference be-tween the input signals. Kim refers to this family oflagged step functions as ”almost orthogonal”. As Kimpoints out, the greater the lag between successive stepinputs, the greater the difference between input sig-nals. This then presents a spectrum of possibilitieswhere, at one end, one has step input functions withno lags (identical functions, not orthogonal) and atthe other end the step functions are separated by anextremely large lag (in the limit, these are the indi-vidual step inputs from the original ROM process).Raveh11 uses filtered white Gaussian noise signals asthe input functions. In order to maintain the neces-sary level of difference amongst each input function,these functions need to be i.i.d. (independent, iden-tically distributed). The application of i.i.d. ran-dom functions for identification of MIMO systems is atraditional approach within the system identificationcommunity, especially in experimental settings. Sinceorthogonality (linear independence) is the most pre-cise mathematical method for guaranteeing the ”dif-ference” between signals, the present research focuseson the application of families of orthogonal functionsas candidate input functions. Using orthogonal func-tions directly provides a mathematical guarantee thatthe input functions are as different as mathematicallypossible. These orthogonal input functions then canbe considered optimal input functions for the identifi-cation of a MIMO system.

The four families of functions to be applied towardsthe efficient identification of a CFD-based unsteadyaerodynamic state-space model are: lagged step, blockpulse, Haar, and Walsh. Although it has been statedby Kim that the lagged-step functions are not com-pletely orthogonal (almost orthogonal), these func-tions are used in the present study for comparisonpurposes. The other three functions each representan orthogonal family of functions. The block pulsefunctions, presented in Figure 3, are orthogonal andresemble a modified step input. An advantage of step-like functions is the broad frequency bandwidth that isexcited by the impulsive nature of these functions. Anadded benefit of the block pulse functions over tradi-tional step functions is that block pulse inputs provideexcitation to the system in both positive and nega-tive directions. From an unsteady aerodynamics pointof view, it is important that modes of realistic wingconfigurations (e.g., with a non-symmetric airfoil) beexcited in both directions for a more complete captureof relevant dynamics.

The Haar functions, presented in Figure 4, com-prise one of the simplest form of wavelet functions.These functions are orthogonal by the mathematical

Fig. 3 Block pulse functions.

definition of wavelet functions. Likewise, this familyof functions has a similarity to step inputs and there-fore embodies the impulsive (i.e., beneficial) nature ofstep inputs with regards to frequency bandwidth. Thefinal set of orthogonal functions is the Walsh function,presented in Figure 5.

Fig. 4 Haar (wavelet) functions.

It is important to mention that the Haar functionsare generated using formulas based on a power oftwo algorithm. These functions have yielded excellentresults when applied towards the identification of aCFD-based unsteady aerodynamic state-space modelusing a single CFL3D execution for a multi-mode con-figuration. However, due to the fact that these func-tions are generated based on a power of two algorithm,the number of modes drives the size and, subsequently,the record length of the input functions that are theninput to CFL3D. As a result, this can severely limit thenumber of modes that can be excited at one time. Forexample, if ten modes are to be excited simultaneously,then the record length of the ten Haar functions gener-

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Fig. 5 Walsh functions.

ated (one per mode) is defined by the record length ofthe Haar function that corresponds to mode ten. Thentaking two to the power of ten yields 1024 time steps.Twelve modes yields a record length for the Haar func-tions of 4096 time steps, thirteen modes yields 8192time steps, fourteen modes yields 16384 time steps,and so on. Clearly, for a very large number of modes,this family of functions is not practical. Therefore,only those results for the lagged step (not orthogonal),the block pulse (orthogonal), and the Walsh (orthog-onal) functions are presented.

Supersonic Semi-Span CFD ModelThe configuration used for the present analyses is

a supersonic semi-span wind-tunnel model known asthe Rigid Semi-span Model (RSM). This configurationhas been tested several times at the NASA Lang-ley’s Transonic Dynamics Tunnel (TDT). The actualwind-tunnel model was fabricated using graphite andis very rigid. However, as part of collaborative stud-ies between the NASA Langley Research Center andthe Boeing Company (Seattle), a ”softened” computa-tional model of the RSM was developed by Dr. MoeljoHong (Boeing).? The model was ”softened” by simplyreducing the four modal frequencies by a factor of four.

The results presented are for Euler (invis-cid)solutions at a Mach number of 0.7 and an angle ofattack of 3 degrees. This configuration does not have asymmetric airfoil and will, therefore, generate a staticaeroelastic response in addition to the dynamic aeroe-lastic response. The method for using an unsteadyaerodynamic state-space ROM for static aeroelastic re-sponses is not presented in this paper but is presentedin another paper by this author.26 Figure 6 presentsthe surface grid for the CFL3D RSM configuration.

ResultsAs previously stated, when using the lagged-step

functions as excitation inputs, it is important to con-sider the lag (number of time steps) between eachsuccessive step input. It is the time lag between each

Fig. 6 Computational grid of supersonic semi-spanconfiguration.

step input that determines the quality of the resultantmodel that is identified. A time lag of fifteen (15) timesteps was used initially in order to explore the effect ofthe time lag on the resultant ROM. Presented in Fig-ure 7 is the CFL3D response in the first generalizedcoordinate due to the excitation of the four lagged-step inputs. The effect of each of the lagged steps isevident in the response of the generalized coordinateas evidenced by the four pulses in the response.

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Fig. 7 CFL3D response in the first generalizedcoordinate due to excitation using four lagged-stepinputs with a lag of 15 time steps between eachinput.

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Presented in Figure 8 is the CFL3D response in thefirst generalized coordinate due to the excitation offour lagged-step inputs but this time with a relativetime lag of 50 time steps. The effect of each of thelagged steps is evident in the response of the general-ized coordinate as evidenced by the four pulses in theresponse.

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Fig. 8 CFL3D response in the first generalizedcoordinate due to excitation using four lagged-stepinputs with a lag of 50 time steps between eachinput.

Figure 9 presents the four block pulse functionsinput to the CFL3D RSM model for simultaneous ex-citation of the four modes.

Presented in Figure 10 is the CFL3D response inthe first generalized coordinate due to the excitation ofthe block pulse inputs. The effect of each of the blockpulse inputs is evident in the response of the general-ized coordinate as evidenced by the four pulses in theresponse. Note that the block pulse functions were de-fined over an interval of 2000 time steps. The laggedstep inputs were defined depending on the desired lagbetween each step input but all solutions (lagged step,block pulse, and Walsh) were executed for the samenumber of time steps.

In Figure 11, Figure 12, Figure 13, and Figure 14each of the four Walsh functions is presented sepa-rately for clarity. The CFL3D response in the first

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Fig. 9 Block pulse functions input to the CFL3DRSM model for modal excitation.

generalized coordinate due to the Walsh function in-puts is presented in Figure 15. Like the block pulsefunctions, the Walsh functions were generated over arecord length of 2000 time steps.

Processing the outputs for all four generalized coor-dinates for each input function through the PULSE al-gorithm in SOCIT, the time-domain generalized aero-dynamic forces (GAFs) can be identified. The firsttwenty-five time steps of the time-domain GAF inmode 1 due to an input in mode 1 (GAF 1,1) for thefour input functions (lagged step (15 steps), laggedstep (50 steps), block pulse, and Walsh) is presentedas Figure 16. Clearly, if the focus is on the initial por-tion of the GAF (high frequency), then all four inputfunctions yield identical GAFs. The results for theother GAFs (fifteen additional GAFs) is similar to theone shown in Figure 16.

However, an analysis of the time-domain GAF 1,1from the 20th to the 200th time step, presented in Fig-ure 17, reveals noticeable differences due to the fourinput functions. Note that the vertical scale has beengreatly expanded from that of Figure 16. This vari-ance can be considered an indication of the difficultythat the system identification algorithm is encounter-ing in attempting to identify the system. It can beseen that the greatest variance is for the lagged step

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Fig. 10 CFL3D response in the first generalizedcoordinate due to excitation using block pulse func-tions.

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Fig. 13 Walsh input function for the first, second,and third modes.

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−0.5

0

0.5

1

1.5x 10

−3

Time Steps

Gen

eral

ized

Coo

rdin

ate

Inpu

ts

Fig. 14 Walsh input function for the first, second,third, and fourth modes.

0 500 1000 1500 2000 2500 3000 3500400

450

500

550

600

650

700

Time Steps

Res

po

nse

in G

ener

aliz

ed C

oo

rdin

ate

1 D

ue

to W

alsh

Inp

uts

Fig. 15 CFL3D response in the first generalizedcoordinate due to excitation using Walsh functions.

0 5 10 15 20 25−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5x 10

5

Time Steps

Tim

e−D

omai

n G

AF

1,1

Lagged Step (15 time step lag)Lagged Step (50 time step lag)Block PulseWalsh

Fig. 16 Initial portion of the time-domain GAFin mode 1 due to an input in mode 1 (GAF 1,1).

input with 15 time step delays, compared to the oth-ers. It is interesting to note the similarity between thelagged step (50 time steps) and the block pulse inputfunctions.

20 40 60 80 100 120 140 160 180 200−20

−15

−10

−5

0

5

10

15

20

25

30

TIme Steps

Tim

e−D

omai

n G

AF

1,1

(T

ime

step

20−

>200

)

Lagged Step (15 steps)Lagged Step (50 steps)Block PulseWalsh

Fig. 17 Variation of GAF 1,1 for the four inputfunctions from the 20th to the 200th time step.

Additional comparison of these results can be ob-tained via the analysis of the frequency-domain version

10 of 15


of the GAFs. The frequency-domain GAFs from eachof four input functions are compared to the frequency-domain GAFs from the use of serial (one-at-a-time)step inputs. Comparison of the simultaneous input re-sponses with those from the one-mode-at-a-time doesnot imply that the one-mode-at-a-time results are thecorrect results. That is, it is quite possible that (andmost probable) that the responses obtained from thesimultaneous excitation will be different from the re-sponses obtained from the one-mode-at-a-time excita-tions due to potential nonlinear modal coupling effects.The reason for comparing these two sets of responses issimply due to the fact that the one-mode-at-a-time re-sponses represent the original ROM approach. Appli-cation of an FFT to each one of the time-domain GAFsdirectly yields the desired frequency-domain GAFs.8

Only a subset of the sixteen GAFs will be presented inthe figures below. The first comparison is for the realcomponent of GAF 1,1, due to the lagged step inputwith 15 time-step delays, presented in Figure 18. Theeffect of the variance exhibited in the time domain ismanifested as a difference at low reduced frequencies.

0 0.5 1 1.5 2−2500

−2000

−1500

−1000

−500

0

500

1000

Reduced Frequency

Rea

l Com

pone

nt o

f GA

F 1

,1

Serial (Step) InputsSimultaneous (Lagged Step) Inputs

Fig. 18 Real component of GAF 1,1 for the laggedstep (15 time steps) and serial step inputs.

The imaginary component of GAF 1,1 due to thelagged step with 15 time-step delays is presented inFigure 19. The comparison for the imaginary com-ponent is quite good, with slight variations at certainfrequency ranges.

The real and imaginary components of GAF 1,1 dueto the lagged step with 50 time-step delays are pre-sented in Figure 20 and Figure 21. Here again, some

0 0.5 1 1.5 2−9000

−8000

−7000

−6000

−5000

−4000

−3000

−2000

−1000

0

Reduced Frequency

Imag

inar

y C

ompo

nent

of G

AF

1,1


Fig. 19 Imaginary component of GAF 1,1 for thelagged step (15 time steps) and serial step inputs.

variations from the serial inputs are noticed as well,in particular with respect to the steady (zero reducedfrequency) value.

0 0.5 1 1.5 2−2000

−1500

−1000

−500

0

500

1000

Reduced Frequency

Rea

l Com

pone

nt o

f GA

F 1

,1


Fig. 20 Real component of GAF 1,1 for the laggedstep (50 time steps) and serial step inputs.

The real and imaginary components of GAF 1,1 dueto the block pulse function inputs are presented as Fig-ure 22 and Figure 23. For both the real and imaginary

11 of 15


0 0.5 1 1.5 2−9000

−8000

−7000

−6000

−5000

−4000

−3000

−2000

−1000

0

Reduced Frequency

Imag

inar

y C

ompo

nent

of G

AF

1,1


Fig. 21 Imaginary component of GAF 1,1 for thelagged step (50 time steps) and serial step inputs.

components, noticeable improvement in the compari-son is achieved with the use of the block pulse function.

0 0.5 1 1.5 2−2500

−2000

−1500

−1000

−500

0

500

1000

Reduced Frequency

Rea

l Com

pone

nt o

f GA

F 1

,1

Serial (Step) InputsSimultaneous (Lagged Block Pulse) Inputs

Fig. 22 Real component of GAF 1,1 for the blockpulse and serial step inputs.

The real and imaginary components of GAF 1,1 dueto the Walsh function inputs are presented as Figure 24and Figure 25. For both the real and imaginary com-ponents, once again, noticeable improvement in the

0 0.5 1 1.5 2−9000

−8000

−7000

−6000

−5000

−4000

−3000

−2000

−1000

0

Reduced Frequency

Imag

inar

y C

ompo

nent

of G

AF

1,1

Serial (Step) InputsSimultaneous (Lagged Block Pulse) Inputs

Fig. 23 Imaginary component of GAF 1,1 for theblock pulse and serial step inputs.

comparison is achieved with the use of the Walsh func-tion. In particular, for both the block pulse and theWalsh functions, the steady value (value at zero re-duced frequency) is well captured.

0 0.5 1 1.5 2−2500

−2000

−1500

−1000

−500

0

500

1000

Reduced Frequency

Rea

l Com

pone

nt o

f GA

F 1

,1

Serial (Step) InputsSimultaneous (Walsh) Inputs

Fig. 24 Real component of GAF 1,1 for the Walshand serial step inputs.

A comparison of the frequency-domain GAF 2,2 ispresented for the lagged step (50 time-step delays) and

12 of 15


0 0.5 1 1.5 2−9000

−8000

−7000

−6000

−5000

−4000

−3000

−2000

−1000

0

Reduced Frequency

Imag

inar

y C

ompo

nent

of G

AF

1,1


Fig. 25 Imaginary component of GAF 1,1 for theWalsh and serial step inputs.

the Walsh function inputs. Figure 26 and Figure 27contain the comparison of the real and imaginary com-ponents of GAF 2,2 for the lagged step (50 time-stepdelays) and the serial (step) inputs. Some differencesare noticed, in particular in the low reduced frequencyrange for both the real and imaginary components.

0 0.5 1 1.5 2500

1000

1500

2000

2500

3000

Reduced Frequency

Rea

l Co

mp

on

ent

of

GA

F 2

,2


Fig. 26 Real component of GAF 2,2 for the laggedstep (50 time-step delays) and serial step inputs.

0 0.5 1 1.5 2−2500

−2000

−1500

−1000

−500

0

Reduced Frequency

Imag

inar

y C

ompo

nent

of G

AF

2,2


Fig. 27 Imaginary component of GAF 2,2 for thelagged step (50 time-step delays) and serial stepinputs.

Figure 28 and Figure 29 are the comparison of thereal and imaginary components of GAF 2,2 for theWalsh and the serial (step) inputs. An improvementin the comparison is once again noted.

The time-domain GAFs are then processed throughthe ERA (part of SOCIT) in order to generate a state-space model of the unsteady aerodynamic system. Theresultant state-space models (for the lagged step with50 time-step delays and the Walsh functions) were 12thorder in dimension. To demonstrate the effectivenessof the new input functions, a representative result ispresented as Figure 30, a comparison of responses inthe four generalized coordinates for the CFL3D solu-tion and the ROM solution. The CFL3D responses arefrom a CFL3D solution that includes both static anddynamic aeroelastic responses simultaneously. Typ-ically, static and dynamic aeroelastic responses arecomputed separately in order to reduce computationalcost. A static aeroelatic solution is computed as therestart solution from a steady, rigid solution using anartifically high structural damping value in order toaccelerate convergence to a static aeroelastic solution.Then, a dynamic aeroelastic solution is computed asthe restart solution from the converged static aeroe-lastic solution. In the paper by the present author,26

a method is introduced that enables the predictionof static and dynamic aeroelastic solutions from thesame ROM. For the present paper, the sample resultspresented in Figure 30 are for a full solution; that is,a solution that includes both static and dynamic re-

13 of 15


0 0.5 1 1.5 2500

1000

1500

2000

2500

3000

Reduced Frequency

Rea

l Co

mp

on

ent

of

GA

F 2

,2


Fig. 28 Real component of GAF 2,2 for the Walshand serial step inputs.

0 0.5 1 1.5 2−2500

−2000

−1500

−1000

−500

0

Time Steps

Imag

inar

y C

ompo

nent

of G

AF

2,2


Fig. 29 Imaginary component of GAF 2,2 for theWalsh and serial step inputs.

sponses for a given dynamic pressure. In this case, theresults presented are for a dynamic pressure of 0.1 psi.Notice that the solutions start at a generalized coordi-nate value of zero (static, rigid solution) and convergeto a non-zero mean value (static aeroelastic solution)with the dynamic solution superimposed on the staticsolution. As can be seen, the comparison is very goodand the ROM captures the combined static and dy-namic aeroelastic responses predicted by the CFL3Dsolution.

0 500 1000 1500 20000

0.05

0.1

Gen

Coo

rd 1

0 200 400 600 800 1000 1200 1400 1600−0.04

−0.02

0

Gen

Coo

rd 2

0 500 1000 1500 2000−5

0

5x 10

−3

Gen

Coo

rd 3

0 500 1000 1500 2000−1

0

1x 10

−3

Time Steps

Gen

Coo

rd 4

ROM (Walsh)CFL3D

Fig. 30 Full solution (static plus dynamic aeroe-lastic responses) generalized coordinate responsesat a dynamic pressure of 0.1 psi.

Concluding RemarksAn improved method for generating unsteady aero-

dynamic state-space reduced-order models (ROMs)was introduced that enables the simultaneous excita-tion of an unsteady aerodynamic CFD model, con-sisting of any number of modes, with a single CFDexecution. This is achieved by the introduction of or-thogonal functions as the input excitations to the sys-tem. Comparisons of generalized aerodynamic forces(GAFs) were presented for the lagged step input, blockpulse input, and the Walsh function input. Resultscompared well with the results from the one-mode-at-a-time method that, for four modes, required fourseparate CFD executions. Additional research is un-derway to continue the evaluation of these functions

14 of 15


on more complex configurations with a larger numberof modes in order to define the optimal input (andinput amplitudes) for the simultaneous excitation ofunsteady aerodynamic systems. In addition, researchwill be performed in the system identification arena inorder to minimize the error associated with the iden-tified state-space models.

References1Bisplinghoff, R. L. and Ashley, H., Principles of Aeroelas-

ticity, Dover Publications, Inc., New York, 1975.2Wagner, H., “Uber die Entstehung des dynamischen

Auftriebes von Tragflugeln,” Mathematical Mechanics, 1925.3Kussner, H. G., “Schwingungen von Flugzeugflugeln,”

Luftfahrtforschung, Vol. 4, 1929.4Theodorsen, T., “General Theory of Aerodynamic Instabil-

ity and the Mechanism of Flutter,” Tech. rep., NACA, Report496, 1935.

5Sears, W. R., “Some Aspects of Non-Stationary AirfoilTheory and Its Practical Applications,” Journal of the Aero-nautical Sciences, Vol. 8, No. 3, pp. 104–108.

6Giesing, J. P., Kalman, T. P., and Rodden, W. P.,“Subsonic Unsteady Aerodynamics for General Configurations,Part I. Direct Application of the Nonplanar Doublet LatticeMethod,” Report AFFDL-TR-71-5, Vol. I, November 1971.

7Silva, W. A., “Identification of Nonlinear Aeroelastic Sys-tems Based on the Volterra Theory: Progress and Opportuni-ties,” Journal of Nonlinear Dynamics, Vol. 39, Jan. 2005.

8Silva, W. A. and Bartels, R. E., “Development of Reduced-Order Models for Aeroelastic Analysis and Flutter PredictionUsing the CFL3Dv6.0 Code,” Journal of Fluids and Structures,No. 19, 2004, pp. 729–745.

9Beran, P. S. and Silva, W. A., “Reduced-Order Modeling:New Approaches for Computational Physics,” Presented at the39th AIAA Aerospace Sciences Meeting, 8-11 January 2001,Reno,NV , January 2001.

10Kim, T., Hong, M., Bhatia, K. G., and SenGupta,G., “Aeroelastic Model Reduction for Affordable Computa-tional Fluid Dynamics-Based Flutter Analysis,” AIAA Journal ,Vol. 43, 2005, pp. 2487–2495.

11Raveh, D. E., “Identification of Computational-Fluid-Dynamic Based Unsteady Aerodynamic Models for AeroelasticAnalysis,” Journal of Aircraft , Vol. 41, June 2004, pp. 620–632.

12Silva, W. A., Beran, P. S., Cesnik, C. E. S., Guendel,R. E., Kurdila, A., Prazenica, R. J., Librescu, L., Marzocca,P., and Raveh, D., “Reduced-Order Modeling: CooperativeResearch and Development at the NASA Langley ResearchCenter,” CEAS/AIAA/ICASE/NASA International Forum onAeroelasticity and Structural Dynamics, June 2001.

13Silva, W. A., “Reduced-Order Models Based onLinear and Nonlinear Aerodynamic Impulse Responses,”CEAS/AIAA/ICASE/NASA International Forum on Aeroe-lasticity and Structural Dynamics, June 1999.

14Silva, W. A., Discrete-Time Linear and Nonlinear Aero-dynamic Impulse Responses for Efficient CFD Analyses, Ph.D.thesis, College of William & Mary, December 1997.

15Silva, W. A., “Application of Nonlinear Systems Theory toTransonic Unsteady Aerodynamic Responses,” Journal of Air-craft , Vol. 30, 1993, pp. 660–668.

16Raveh, D. E., Levy, Y., and Karpel, M., “Aircraft Aeroe-lastic Analysis and Design Using CFD-Based Unsteady Loads,”41st Structures, Structural Dynamics, and Materials Confer-ence, No. 2000-1325, Atlanta, GA, April 2000.

17Roger, K. L., “Airplane Math Modeling Methods for ActiveControl Design,” Agard-cp-228, Aug. 1977.

18Karpel, M., “Time Domain Aeroservoelastic Modeling Us-ing Weighted Unsteady Aerodynamic Forces,” Journal of Guid-ance, Control, and Dynamics, Vol. 13, No. 1, 1990, pp. 30–37.

19Gupta, K. K., Voelker, L. S., Bach, C., Doyle, T., andHahn, E., “CFD-Based Aeroelastic Analysis of the X-43 Hy-personic Flight Vehicle,” Proceedings of the 39th AerospaceSciences Meeting and Exhibit , No. 2001-0712, Reno, CA, Jan.2001.

20Cowan, T. J., Jr., A. S. A., and Gupta, K. K., “Accelerat-ing CFD-Based Aeroelastic Predictions Using System Identifi-cation,” 36th AIAA Atmospheric Flight Mechanics Conferenceand Exhibit , No. 2001-0712, Boston, MA, Aug. 1998, pp. 85–93,AIAA-98-4152.

21Cowan, T. J., Jr., A. S. A., and Gupta, K. K., “Devel-opment of Discrete-Time Aerodynamic Model for CFD-BasedAeroelastic Analysis,” Proceedings of the 37th Aerospace Sci-ences Meeting and Exhibit , No. 1999-0765, Reno, CA, Jan. 1999,AIAA-99-0765.

22Rodrigues, E. A., “Linear/Nonlinear Behavior of the Typ-ical Section Immersed in Subsonic Flow,” Proceedings of the42nd AIAA/ASME/ASCE/AHS/ASC Structures, StructuralDynamics, and Materials Conference and Exhibit , No. 2001-1584, Seattle, WA, April 2001, AIAA-01-1584.

23Juang, J.-N. and Pappa, R. S., “An Eigensystem Realiza-tion Algorithm for Modal Parameter Identification and ModelReduction,” Journal of Guidance, Control, and Dynamics,Vol. 8, 1985, pp. 620–627.

24“Registered Product of the MathWorks, Inc.” .25Kim, T., “Efficient Reduced-Order System Identification

for Linear Systems with Multiple Inputs,” AIAA Journal ,Vol. 43, 2005, pp. 1455–1464.

26Silva, W. A., “Recent Enhancements to the Develop-ment of CFD-Based Aeroelastic Reduced Order Models,”48th AIAA/ASME/ASCE/AHS/ASC Structures, StructuralDynamics, and Materials Conference, No. AIAA Paper No.2007-XXXX, 2007.

27Krist, S. L., Biedron, R. T., and Rumsey, C. L., “CFL3DUser’s Manual Version 5.0,” Tech. rep., NASA Langley ResearchCenter, 1997.

28Bartels, R. E., “Mesh Strategies for Accurate Computa-tions of Unsteady Spoiler and Aeroelastic Problems,” AIAAJournal of Aircraft , Vol. 37, 2000, pp. 521–525.

29Bartels, R. E., Rumsey, C. L., and Biedron, R. T., “CFL3DVersion 6.4: General Usage and Aeroelastic Analysis,” NASATM 2006 214301 , April 2006.

30Roe, P. L., “Approximate Riemann Solvers, ParameterVectors, and Difference Schemes,” Journal of ComputationalPhysics, Vol. 43, 1981, pp. 357–372.

31Juang, J.-N., Phan, M., Horta, L. G., and Longman, R. W.,“Identification of Observer/Kalman Filter Markov Parameters:Theory and Experiments,” Journal of Guidance, Control, andDynamics, Vol. 16, 1993, pp. 320–329.

32Juang, J.-N., Applied System Identification, Prentice-HallPTR, 1994.

33Silva, W. A., “Simultaneous Excitation of Multiple-InputMultiple-Output CFD-Based Unsteady Aerodynamic Systems,”48th AIAA/ASME/ASCE/AHS/ASC Structures, StructuralDynamics, and Materials Conference, No. AIAA Paper No.2007-XXXX, 2007.

34Eykhoff, P., System Identification: Parameter and StateIdentification, Wiley Publishers, 1974.

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