Evaluation of Time-domain Damping IdentificationMethods for Flutter-constrained Optimization

Kevin E. Jacobson, Jan F. Kiviaho, Graeme J. Kennedy, and Marilyn J. SmithDaniel Guggenheim School of Aerospace Engineering, Georgia Institute of Technology, Atlanta,

Georgia, USA

Abstract

As engineers increasingly pursue air vehicle designs with slender, lightweightaerostructures, it has become necessary to model nonlinear aeroelastic effects ear-lier in the design process. As a result, high-fidelity analysis tools are required thatcan efficiently identify flutter within the flight envelope, and find design modifi-cations to alleviate adverse aeroelastic behavior. To address these requirements,we have systematically evaluated automated methods to identify the damping ofan aeroelastic system from a time-domain simulation. The techniques evaluatedinclude the log decrement method, envelope function methods that use the Hilberttransformation, the half-power bandwidth method, and the matrix pencil method.The optimal approach was determined to be the matrix pencil method due to its ro-bustness to noise and its ability to handle multi-component signals over short timesimulations. Identification of the flutter boundary of the AGARD-445.6 wing isdemonstrated with an automated method based on the adjoint sensitivities and thematrix pencil method. While the matrix pencil method is robust, identification ofthe flutter boundary using gradient-based optimization is shown to be sensitive tothe initial dynamic pressure guess.

Keywords: flutter, aeroelasticity, adjoint method, CFD, optimization, matrixpencil method

Nomenclature

c Prony series complex amplitude

L Pencil parameter

M Model order

Preprint submitted to Journal of Fluids and Structures March 20, 2019

s Prony series complex exponent

U, V Left and right singular vectors

w(n) Matrix pencil noise term

Y Hankel Matrix

zv Hilbert transform of z

α Exponential growth coefficient

ζ Damping ratio

λ Eigenvalue vector

Σ Singular value vector

Ψ, Φ Left and right eigenvectors

ω Frequency

1. Introduction

Aeroelasticity plays an important role in aerospace vehicle design. For in-stance, the certification of fixed wing aircraft requires sufficient flutter marginsthroughout the flight envelope, while launch vehicles are susceptible to a broadrange of aeroelastic problems. The prediction of flutter and other aeroelasticphenomena often involves coupled nonlinear physics, such as structural stiffnessor damping nonlinearity, geometric nonlinearity, or aerodynamic nonlinearity in-cluding shock movement or flow separation [1]. The complexity of these problemsis increasing as advanced materials permit engineers to improve performance withthinner and more flexible designs where nonlinearities can exist simultaneouslyin both structures and aerodynamics. Aeroelastic fixed wing aircraft design ispredominantly conducted with linear aerodynamic tools followed by wind tunneland flight tests [2]. However, lower-order or linear tools are insufficient to predictcomplicated aeroelastic behaviors such as the transonic flutter dip. Use of theselower-fidelity methods can lead to heavy, overly conservative structural designsor the need to modify configurations when problems arise late in the design pro-cess. Advancements in solution algorithms and more powerful computers permithigh fidelity multidisciplinary modeling, based on finite element analysis (FEM),

2

and computational fluid dynamics (CFD) to be more tractable and will push thesetools to be accessible earlier in the design cycle. If more accurate models for thecomplex aeroelastic phenomena can be included earlier in the design process, air-craft designers will have more confidence in their designs before flight tests, andthe need for overly conservative designs or costly late-stage modifications can bereduced.

To address this need for high-fidelity aeroelastic design tools, we have system-atically evaluated automated methods to identify the damping of an aeroelasticsystem from time-domain simulations that are also amenable for gradient-baseddesign optimization. Time-domain flutter analysis is typically performed by excit-ing the aeroelastic system about a static equilibrium point. This excitation can becreated by imposing an instantaneous change in structural state or applying impul-sive external forces to the structure. A system identification method is then usedto extract signal damping from the response of the system to the excitation. Theanalysis conditions are then adjusted with a root finding method such as bisectionor the secant method [3, 4, 5] to find the flutter point where the minimum dampingis zero. However, secant or bisection-type methods cannot easily be incorporatedwithin an optimization problem formulation, since their solution process wouldneed to be differentiated. Instead, the damping estimate can be included as a con-straint within an optimization problem formulation. Gradient-based optimizationwith the adjoint method allows the computational cost of CFD-based optimizationproblems to be tractable, even with problems that have hundreds or thousands ofdesign variables.

There are only a few examples of time-domain optimizations considering flut-ter in the literature. Mani and Mavriplis [6] and Palaniappan et al. [7] minimizedthe structural state vector of the final time steps of an initially fluttering airfoil.This essentially maximizes a combination of the aerodynamic damping and aeroe-lastic stiffness without directly calculating them. In a more complex design prob-lem, it would not be sufficient to maximize the damping or stiffness since it couldlead to overly conservative design. The other set of examples of CFD-based flut-ter constraints come from Zhang et al. [8, 9, 10], who demonstrated two differentmethods. Their first method is a minimization of the square of the lift coefficient,which is similar to the optimizations of Mani and Mavriplis and Palaniappan etal., since it minimizes the response of the system, which acts as a maximization ofthe aerodynamic damping. Their second method is a flutter constraint computedfrom a Hilbert transformation based envelope function that will be included inthe comparisons in this work. The envelope function bounds the oscillations inthe system’s response, and the growth or decay of this bounding function can be

3

used as a representation of the behavior of the oscillations and overall stability.Zhang et al. have applied this method to optimization problems where the objec-tive is flutter point identification, where the Hilbert-transform estimated dampingis driven to zero [9]. This constraint was then applied in a weight minimizationproblem [10], where the constraint drove the damping to zero in the optimizeddesign.

This paper studies damping identification methods for application to gradient-based optimization with flutter constraints. In the context of optimization, can-didate methods must be automated and robust. It is also desirable that the meth-ods be accurate, even with relatively short time intervals, since shorter simula-tions can lead to more analyses for a given computational budget. Within thispaper, the methods are compared on signals with characteristics expected in CFD-based analysis such as noise and multi-component signals. Noise in CFD signalscan arise from various sources, including high-frequency initial transients causedby impulsive excitations or resolved turbulence. Automated identification of theflutter boundary of the AGARD-445.6 wing is demonstrated with the selectedmethod, the matrix pencil method.

2. Damping Calculation Methods

There are several methods to compute flutter damping that are popular in thenumerical analysis literature. These include the log decrement method, envelopefunctions [8], half-power bandwidth method [11, 12, 13], and Prony series meth-ods [14].

2.1. Log Decrement MethodOne of the most common methods for estimating damping is the log decrement

method. The log decrement is computed as the natural log of the ratio of two peaksin the signal defined by

δ =1n

ln(

z(t)z(t +nT )

),

where n is an integer number of peaks and T is the period of the signal. Thedamping ratio is then computed as

ζ =1√

1+(2π

δ

)2.

4

t

z

0 1 2 3 4 5­6

­4

­2

0

2

4

6

Figure 1: Multi-component signal given by Equation 1.

The log decrement method is commonly selected in analysis for its simplicity.Since it requires only two maxima, the method is applicable to short duration sig-nals, which are typical in CFD-based analysis due to computational cost. Tech-nically, the method is only applicable for a single component, single frequencysignal. Although the log decrement method is not sufficiently robust to consis-tently handle signals with multiple components, it can sometimes be applied tosuch cases. For example, given the signal

z(t) = 5e−t sin(50t)−2.5e−0.75t sin(45t) , (1)

the damping estimate obtained using the two peaks indicated in Figure 1 is 0.965which is within 5% of the damping of the larger amplitude signal. The under-predicted damping is related to the lower damping associated with the smalleramplitude component of the signal. To apply the log decrement method to othersimilar signals, the signal peaks must be intelligently selected. The marked peaksin Figure 1 were chosen by observing the peak pattern and selecting peaks thatwere the highest local peak in the pattern, i.e., the ones where the local maximaimmediately to the left and right are of lower magnitude. While the process leadsto a reasonable damping estimate in this case, it does not work in general as willbe demonstrated in later examples.

2.2. Envelope Functions - Hilbert TransformationsOne set of methods similar to the log decrement method is the use of envelope

functions. In these methods, an envelope function enclosing the signal responseis constructed and the behavior of the envelope is utilized to estimate the overall

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stability of the system. The simplest form of an envelope function is a discreteenvelope formed from connection of the successive local maxima and minima.This is similar to the peak-based analysis of the log decrement method. A moresophisticated approach is the use of a Hilbert transformation, which defines theenvelope function at every time instance of the original signal. The Hilbert trans-formation is the harmonic conjugate of the original signal. Given a signal z(t), theHilbert transform is defined as

zv(t) =1π

∫∞

−∞

z(τ)t− τ

dτ. (2)

Following the description of Zhang et al. [9], the signal can then be representedas a complex value function given by the harmonic conjugate pair

y(t) = z(t)+ izv(t) = A(t)eiθ(t), (3)

where A(t) is the instantaneous envelope amplitude and θ(t) is the instantaneousphase of the signal. The instantaneous frequency of the signal is then the rate ofchange of the phase

ω(t) =dθ(t)

dt. (4)

Assuming that the signal is an exponentially decaying sinusoid, the instantaneousenvelope amplitude can be expressed as

A(t) = e−ζ (t)ω(t)t . (5)

Next, an approximation is made that the damping ratio and instantaneous fre-quency are approximately constant. This is not strictly valid for the Hilbert trans-formation when the signal is not periodic or not infinitely long, since the integral inEquation 2 will not be defined for all time. However, this approximation enablesthe damping ratio to be estimated by taking the natural logarithm of Equation 5:

ζ (t) =− 1ω

d log(A)dt

. (6)

Using this definition of damping ratio, Zhang et al. then form a flutter constraintto ensure non-negative damping.

To demonstrate the Hilbert transformation damping calculation, the method isapplied to the following signal:

z(t) = 5e−t sin(50t) , t ∈ [0,2] . (7)

6

t

z

0 0.5 1 1.5 2

­5

­2.5

0

2.5

5

Signal

Amplitude Envelope

(a) Signal and instantaneous envelope.

t

Fre

qu

en

cy

Da

mp

ing

Ra

tio

0 0.5 1 1.5 20

50

100

150

­6

­5

­4

­3

­2

­1

0

1

Frequency

Damping Ratio

(b) Instantaneous frequency and dampingratio.

Figure 2: Hilbert transformation example.

Figure 2a shows this signal and the instantaneous envelope amplitude. Becausethe signal is over a finite interval, there is some error in the instantaneous enve-lope amplitude compared to the analytic envelope, ±5e−t . Figure 2b illustratesthe instantaneous frequency found from Equation 4 and the damping ratio fromEquation 6. The frequency is not a constant value of 50 due to numerical errorfrom the finite length of the signal. Since the instantaneous damping is calculatedfrom the envelope function and frequency, there is also error in the damping, par-ticularly near the beginning and end of the time interval. The expected dampingratio value is 0.02, but the value reaches as low as -6.3 at the end of the time his-tory. The mean value of the positive instantaneous damping ratio over the timeinterval can be used as a signal damping ratio estimate. For this case, this meanvalue estimate is 0.193, which has an error of 3.5% compared to 0.02.

Envelope function methods require that the behavior of the bounding functionis representative of the stability of the underlying signal. This will always be truefor a sufficiently long time interval. However, for the case of a signal with a smallamplitude component that is growing and an initially large amplitude componentthat is decaying, a relatively long time interval may be required for the grow-ing component to dominate the signal and be detected by the envelope function.This is not ideal for CFD-based aeroelastic optimization where the length of thesimulation should be as short as possible due to computational cost. More sophis-ticated methods have been developed to enhance envelope function techniques,such as improved Hilbert transforms [15]. These more advanced methods can re-duce the envelope function error at the start and end of the time window, but theystill assume the global bounding function behavior is representative of the signalstability.

7

t

Z

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5­4

­2

0

2

4

(a) Time domain signals.

Frequency [rad/s]

Po

we

r

0 25 50 75 100 125 1500

0.2

0.4

0.6

0.8

1

(b) Power spectrums.

Figure 3: Half-power example with a growing and decaying signal.

2.3. Half-power Bandwidth MethodA common technique for estimating damping that is used in the experimental

materials and structures community is the half-power bandwidth method [11, 12,13]. In this method, the signal is first transformed to the frequency domain. Inthe power spectrum, a decaying or growing periodic signal will appear as a peakat the signal frequency ω∗. The width of the peak in the power spectrum at thehalf-power points, ω1 and ω2, can then be used to estimate the damping ratio

ζ =12

ω2−ω1

ω∗, (8)

where the half-power points have an amplitude smaller than the peak by a fac-tor of 1/

√2. While this technique is robust to noise and able to distinguish

multiple signal components, it is difficult to automate for signals that containboth growing and decaying components. Once the signal is transformed into thefrequency domain, it cannot easily be determined if the oscillation amplitude isgrowing or decaying. For example, the two signals, z(t) = 4.5e−t sin(20π t) andz(t) = 0.05et sin(20π t) over the time range [0.0,4.5], shown in Figure 3a, havethe identical power spectra shown Figure 3b. For these simple signals, it is easyto distinguish which one is gowning or decaying. However, when there are mul-tiple components, it is not as straight forward to apply the half-power bandwidthmethod.

2.4. Prony Methods - the Matrix Pencil MethodRather than approximate the signal as constant amplitude sinusoids, Prony

series methods decompose the signal into a sum of damped sinusoids

z(t) =∞

∑k=0

cke(αk+iωk)t , (9)

8

where ck is a complex number related to the amplitude (ak = mod(ck)) and phase(φk = arg(ck)), αk is the growth factor, and ωk is the frequency. To utilize theProny series representation, the values of ck, αk, and ωk must be obtained basedon a set of signal samples. There are several approaches in the literature forobtaining these coefficients. Common methods include root finding techniquesor minimizing point-wise error using unconstrained optimization methods [14].However, these methods are not always noise tolerant, do not always provide arational basis for the selection of the number of terms in the Prony series, and canbe computationally expensive.

In this work, we utilize the matrix pencil method [16, 17] to obtain the valuesof the coefficients ck, αk, and ωk in the Prony series. The matrix pencil method isselected for this work over other Prony series methods because it can provide anautomated way to select the number of Prony series terms using the results froman eigendecomposition. The matrix pencil method utilizes a set of evenly spacedsamples of the input signal at the points zn = z(n∆t). Next, the infinite series (9)is truncated to include only the first M terms, where M is the model order. Finally,in a departure from other Prony series methods, a noise term, w(n), is embeddedin the signal decomposition, resulting in the discrete representation

zn =M−1

∑k=0

ckeskn +w(n), n = 0,1, . . . ,N−1. (10)

Here, the complex exponent sk includes the uniform time step and the growth andfrequency terms, sk = (αk + iωk)∆t. Based on this decomposition of the signal,the exponents, sk, are computed using the technique of Sarkar and Pereira [18]outlined in Algorithm 1. The optional first step in this algorithm subsamples thesignal down to N samples, which reduces the problem size for the eigenvalue andsingular value decomposition steps later in the algorithm. Next, the pencil param-eter L is set. This parameter is an important factor that controls noise filtering andis typically selected to be between N/3 and N/2. After the pencil parameter ischosen, the Hankel matrix Y ∈ RN×L+1 is formed

Y =

z0 z1 · · · zLz1 z2 · · · zL+1...

... . . . ...zN−L−1 zN−L · · · zN−1

.

9

Next, the singular value decomposition of Y is computed. The model order, M,in Equation 10 is selected based on the number of singular values that exceed aspecified relative tolerance. The first M right singular vectors are kept in V, andthe remaining vectors are discarded. These discarded singular vectors are associ-ated with signal noise. From these filtered right singular vectors, two matrices areformed: V1 which is the rows 0 through L−1 and V2 which is rows 1 through L.The matrix A is formed as the product of the left pseudoinverse of the transposeof V1 and the transpose of V2 such that

A =[VT

1]+ VT

2 ,

where (·)+ denotes the left pseudoinverse. Finally, an eigenvalue decompositionof A is computed, where the eigenvalues approximate the exponents such thatsk≈ λk(A). Finally, the growth factor for each component of the signal is obtainedfrom the real part of these eigenvalues

αk =Re[ln(λk(A))]

∆t, k = 0, . . . ,M−1.

For bodies with many modes of vibration, there are potentially many componentsto the signal. Rather than requiring one adjoint solution for each component, aKS function [19] is applied to aggregate the exponents, and the maximum growth(minimum damping) is approximated as

c(ααα,ρ) =1ρ

ln

[M−1

∑k=0

eραk

]

= αmax +1ρ

ln

[M−1

∑k=0

eρ(αk−αmax)

]≈max

kαk,

(11)

where αmax = maxk αk, and ρ is the aggregation parameter. The KS function (11)smoothly approximates the maximum growth factor. The second form of the KSfunction is mathematically equivalent, but avoids numerical issues associated withlarge exponents.

The matrix pencil method has several advantages over other methods of esti-mating the damping. While other methods can include noise filtering as a prepro-cessing step, the noise term is embedded within the matrix pencil method, makingit inherently more robust. Unlike the half-power and log decrement approaches,the matrix pencil method can identify growing or decaying components of a multi-component signal in an automatic and differentiable manner.

10

Algorithm 1 Matrix pencil-based method for estimating and aggregating damping1: Given: t, z, ∆t, ρ

2: Z←H(t,z) . Interpolate the simulation data z to N ≈ O(100) samples Z3: L← N/2−1 . Set pencil parameter L between N/3 and N/24: for i = 0→ N−L−1 do5: for j = 0→ L do6: Yi, j = Zi+ j . Fill Hankel matrix Y with samples Z7: end for8: end for9: U,ΣΣΣ,VT ← SVD(Y) . Singular value decomposition of Hankel matrix Y

10: M← f (ΣΣΣ) . Choose model order M as a function of the singular values11: V = V(:,0:M−1) . Keep only the first M singular vectors12: V1 = V(0:L−1,:) . Keep first L rows13: V2 = V(1:L,:) . Keep the second through L+1 rows

14: A =[VT

1]+ VT

2 . Compute A matrix from pseudoinverse of VT1 and VT

215: λλλ ,ΨΨΨ,ΦΦΦ← EIG(A) . Compute eigenvalues λλλ , left and right eigenvectors, ΨΨΨ

and ΦΦΦ

16: λλλ = λλλ (0:M−1) . Keep M eigenvalues17: ααα = Re(ln(λλλ ))/∆t . Compute damping from eigenvalues18: α = c(ααα,ρ) . Use KS function c with parameter ρ to aggregate damping

11

3. Evaluation of the Damping Prediction Methods for CFD-based Analysis

The next set of examples study the effectiveness of the different methods de-scribed above. These methods are applied to various signals with characteristicsthat may be encountered during a CFD analysis including noise, multi-componentsignals, and inaccurate dereferencing. Finally, the methods are compared onmodal responses from CFD analysis from the X-57 Maxwell [20].

3.1. Decaying Sinusoidal Signal with NoiseThis study examines the tolerance of the methods to noise. The example signal

is a decaying sinusoid with noise:

z(t) = 5e−t sin(50t)+w(t), t ∈ [0,2] ,

where w(t) is a white noise term based on a normal distribution. Figure 4 showsthe example signal with normal white noise with standard deviations of 5% and10% of the initial signal amplitude, respectively. Table 1 compares the predictionsof the log decrement, half-power, and matrix pencil methods for levels of noisebetween 0% and 10% percent. The log decrement method is automated with apeak identification method that includes height and minimum distance filteringfor noise tolerance, as would be required in an optimization framework. For theHilbert transformation, the mean of the instantaneous damping is taken for themiddle three-fifths of the signal to avoid the regions of large numerical error. Forthe half-power bandwidth method, a fast Fourier transform (FFT) converts thesignal to the frequency domain. To avoid assuming a shape of the power spec-trum, the half-power points are the discrete points from the FFT power spectrumclosest to the half-power amplitude. The two columns of matrix pencil results inTable 1 are the predictions before and after the application of the KS aggregationfunction with ρ = 50. At zero noise, the log decrement, Hilbert transformation,and matrix pencil method are nearly exact with less than 0.3% error, while thehalf-power method is within 2% of the exact minimum damping. The error inthe half-power method arises from the limited resolution of the FFT. Having alonger signal history would lead to a finer frequency resolution. As the noise levelincreases, the accuracy of the log decrement and Hilbert transformation methoddegrades more rapidly than the half-power or matrix pencil methods. At the 10%noise level, the log decrement-predicted exponent coefficient is in error by 44%,and the Hilbert transformation is 91.3% too low. The noise tolerance of the logdecrement and Hilbert transformation methods could be improved by adding a

12

t

y

0 0.5 1 1.5 2­6

­4

­2

0

2

4

6

(a) 5% noise.

t

y

0 0.5 1 1.5 2­6

­4

­2

0

2

4

6

(b) 10% noise.

Figure 4: Decaying sine with noise example problem.

Table 1: Identification of exponent coefficient α with the various methods for the noisy sine signal.

Noise (%) Log Dec. Hilbert Half-power Matrix Pencil Matrix Pencil (KS)

0.0 -0.99977 -0.99769 -1.01905 -1.00000 -0.993070.1 -0.97772 -1.04763 -1.01905 -1.00022 -0.993291.0 -0.93360 -0.58932 -1.01905 -1.00142 -0.994495.0 -0.78456 -0.21975 -1.01905 -0.99631 -0.98938

10.0 -0.56002 -0.08667 -1.03077 -1.02429 -1.01736

filtering step to the process. Meanwhile, without additional steps, the more noise-tolerant half-power and matrix pencil methods stay within 3.1% and 2.5% of theexact exponent, respectively, even with significant noise levels.

3.2. Sensitivity to DereferencingIn many aeroelastic flutter problems, the equilibrium or reference point about

which the oscillations occur is not z = 0 as was the case in the previous example.Here, dereferencing refers to the shifting of the signal to make it oscillate aboutz = 0 as required by some of the damping identification methods. Inaccuracy inidentifying the reference point can lead to errors in the damping prediction. Thesame signal from the first example is used in this case, except the noise term isreplaced by a constant offset to model an error in the estimate of the equilibriumpoint:

z(t) = 5e−t sin(50t)+ z0, t ∈ [0,2] ,

where z0 is the dereferencing error. Table 2 illustrates how the methods handlevarious errors in the reference point. The signal decomposition methods: the ma-trix pencil and the half-power methods, both identify components with zero fre-quency, i.e., the reference point, therefore the predicted damping does not change

13

Table 2: Identification of exponent coefficient α with the various methods for improperly derefer-enced signals.

Offset, z0 Log Dec. Hilbert Half-power Matrix Pencil Matrix Pencil (KS)

0.0 -0.99977 -0.99769 -1.01905 -1.00000 -0.993070.01 -0.99536 -1.00805 -1.01905 -1.00000 -0.99307

-0.01 -1.00427 -0.98689 -1.01905 -1.00000 -0.993070.1 -0.95744 -1.10734 -1.01905 -1.00000 -0.99307

-0.1 -1.04674 -0.89515 -1.01905 -1.00000 -0.993070.5 -0.82181 -1.67749 -1.01905 -1.00000 -0.99307

-0.5 -1.30377 -0.53749 -1.01905 -1.00000 -0.99307

as the reference offset is varied. The log decrement and Hilbert transformationmethods do show sensitivity to dereferencing. When the offset is 10% of the ini-tial amplitude, the log decrement method has up to 30.4% error while the Hilberttransform method has as much as 67.7% error.

3.3. Multi-component SignalThe next example is a multi-component signal where a smaller amplitude term

is growing:

z(t) = 5e−t sin(50t)+2e−0.5t sin(20t)−0.5e0.05t sin(40t)+w(t), t ∈ [0,2] .

The normal white noise component has a standard deviation of 0.25. Figure 5 il-lustrates this signal and its power spectrum. The superposition of the componentsof the signal and the short time interval makes it difficult to identify that there isan unstable mode by visual inspection. With no noticeable pattern, the log decre-ment method does not produce a consistent damping estimate between any setsof peaks. The relatively short length of the time interval creates a low frequencyresolution for the FFT which affects the accuracy of the half-power method. Thepower spectrum does contain three distinct peaks corresponding to the three sinu-soidal components of the signal. Some interpolation strategy could be applied, butthe approximation of the half power of the peak at 40 rad/s would still be difficultsince the peak is essentially a single point. Additionally, the power spectrum doesnot discriminate between growing and decaying components. Extra steps could beadded to the half-power algorithm to identify the correct combination of growingand decaying components using a reconstruction of the signal. However, thesemodifications would add complexity and nondifferentiable conditional statements

14

t

y

0 0.5 1 1.5 2­6

­4

­2

0

2

4

6

(a) Signal.

Frequency [rad/s]

Po

we

r

0 20 40 60 80 100

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

(b) Power Spectrum.

Figure 5: Multi-component sinusoid with noise example problem.

Table 3: Matrix pencil breakdown of the multi-component sinusoid example.

Component Frequency [rad/s] Exponent (αk) Amplitude

0 50.016 -0.99780 2.48821 -50.016 -0.99780 2.48822 19.9957 -0.50051 0.999223 -19.9957 -0.50051 0.999224 40.016 0.052669 0.250175 -40.016 0.052669 0.25017

to the algorithm. Table 3 shows how the matrix pencil method decomposes thenoisy multi-component signal. The absolute value of the frequencies match theexact frequencies to three significant digits, and the addition of the amplitudes forthe positive and negative frequency pairs produces amplitudes of 4.976, 1.998,0.500 (compared to the original amplitudes of 5.0, 2.0, and 0.5). The predictedexponents are also very accurate. For the decaying components, the exponent er-ror is less than 0.3%, while for the unstable components it is 5.3%, which is verygood since the standard deviation of the noise is 50% of the initial amplitude ofthe unstable signal. Not only is the matrix pencil method able to capture the smallunstable component of the signal, it also provides accurate exponent predictionswith a relatively short time signal.

Because the Hilbert transformation method was found to be very sensitive tonoise in the first example, it was applied to the same signal without the noise term.The envelope function and instantaneous damping are shown in Figure 6. The en-velope function indicates significantly more oscillatory behavior than observed inFigure 2. Since the single constant frequency assumption is violated, the damp-

15

t

z

0 0.5 1 1.5 2

­6

­4

­2

0

2

4

6

(a) Signal and instantaneous envelope.

t

Da

mp

ing

Ra

tio

0 0.5 1 1.5 2­3

­2

­1

0

1

2

3

(b) Instantaneous damping ratio.

Figure 6: Multi-component signal without noise.

ing ratio prediction in Equation 6 is not applicable. Therefore, the instantaneousdamping ratio oscillates between positive and negative values and does not givea good indication of the behavior of the signal. While this is a simple implemen-tation, it is unlikely that any envelope function-based method would identify theunstable component of the signal, since these methods are designed for overallbehavior and the unstable component is diluted by the decaying components overthe short time interval.

3.3.1. Matrix Pencil Sensitivity to Model NumberThe noise filtering in the matrix pencil method is accomplished by selecting

a model order that is lower than the full order of the Prony series decompositionproblem in line 10 of Algorithm 1. If the model number is too high, the noise maynot be filtered out. If the model number is too low, the matrix pencil method maynot have enough components to accurately reproduce the signal. The effect of themodel number is studied on the following signal

z(t) =5e−t sin(50t)+2e−0.5t sin(20t)

−0.5e0.05t sin(40t)+0.1+w(t), t ∈ [0,2] ,(12)

where the noise level w(t) has a standard deviation of 0.5, which is 10% of thelargest amplitude component. Ideally, the matrix pencil method would selectM = 7 as the model number, which accounts for the constant offset and a positiveand negative frequency term for the three sinusoidal components in the signal. Ta-ble 4 shows how the matrix pencil method’s decomposition of signal changes asthe model number is increased. With a model number of M = 2, the matrix pencilmethod approximates the signal with a damped sinusoid that is close to the largestcomponent of the signal. As the model number increases to M = 7, the zeroth

16

and first components become closer representations of the original signal’s largestcomponent, and the other components of the signal begin to appear. When themodel number is M = 6, the smallest amplitude oscillation appears in the decom-position. For this signal, it is important to capture this new component becauseit is the harmonic that is undergoing exponential growth. At a model number ofM = 7, the decomposition is most accurate, as expected. The least accurate valueat a model number of M = 7 is the exponent coefficient of the growing signal withan absolute and relative error of 0.0028 and 5.6%, respectively. When the modelnumber is greater than M = 7, some of the noise is retained as seen in the highfrequency components for the model numbers M = 8 and M = 9. For a modelnumber of M = 8, the last component is a high-frequency, low-amplitude oscil-lation which has an extremely large damping that does not affect the ability ofthe KS function to determine the maximum exponent coefficient. As the modelnumber is increased to a value of M = 9, the noise contributions appear as unsta-ble high frequency components. This would be an undesirable situation during anoptimization because the optimizer would try to reduce the damping of a signalcomponent that is not physically meaningful. From this analysis, there are onlytwo choices of model number, M = 6 and M = 7, that include enough componentsto capture the important oscillatory terms without being polluted by noise.

17

Table 4: Matrix pencil method’s sensitivity to the model number parameter.

Model Number 2 3 4 5 6 7 8 9 Exact

Frequency 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000Amplitude 0.2199 0.1443 0.0989 0.1262 0.1011 0.1000

α — — — — — —Frequency 49.7461 49.7848 49.8486 49.8539 49.9990 49.9978 50.0069 50.0050 50.0000Amplitude 2.4144 2.3739 2.4943 2.5102 2.5024 2.5040 2.4809 2.4595 2.5000

α -0.9480 -0.9164 -1.0035 -1.0202 -1.0103 -1.0019 -0.9894 -0.9723 -1.0000Frequency -49.7461 -49.7848 -49.8486 -49.8539 -49.9990 -49.9978 -50.0069 -50.0050 -50.0000Amplitude 2.4144 2.3739 2.4943 2.5102 2.5024 2.5040 2.4809 2.4595 2.5000

α -0.9480 -0.9164 -1.0035 -1.0202 -1.0103 -1.0019 -0.9894 -0.9723 -1.0000Frequency 20.0375 20.0622 19.9872 19.9987 20.0072 20.0229 20.0000Amplitude 0.9899 1.0172 1.0103 1.0001 0.9970 1.0064 1.0000

α -0.4726 -0.5223 -0.5042 -0.5000 -0.4912 -0.5006 -0.5000Frequency -20.0375 -20.0622 -19.9872 -19.9987 -20.0072 -20.0229 -20.0000Amplitude 0.9898 1.0172 1.0103 1.0001 0.9970 1.0064 1.0000

α -0.4726 -0.5223 -0.5042 -0.5000 -0.4912 -0.5006 -0.5000Frequency 39.9903 40.0040 40.0257 40.0735 40.0000Amplitude 0.2429 0.2491 0.2364 0.2543 0.2500

α 0.0729 0.0528 0.0900 0.0400 0.0500Frequency -39.9903 -40.0040 -40.0257 -40.0735 -40.0000Amplitude 0.2429 0.2491 0.2364 0.2543 0.2500

α 0.0729 0.0528 0.0900 0.0400 0.0500Frequency 7852.41 2736.23Amplitude 0.0787 0.2766

α -1788.46 0.0125Frequency -2736.23Amplitude 0.2766

α 0.0125

18

Singular Value Number

No

rma

lize

d S

ing

ula

r V

alu

e

0 500 1000 1500 2000 2500

10­5

10­4

10­3

10­2

10­1

100

1% Noise

10% Noise

Figure 7: Normalized singular values for the decomposition of Equation 12.

In general, the best model number will not be known a priori. One approachto select the model number is based on the magnitude of the singular values. Fig-ure 7 shows the singular values normalized by the largest singular value for noiselevels of 1% and 10%. The first six singular values appear as pairs of approxi-mately equal value. The seventh singular value is one order of magnitude smallerthan the largest. These first seven singular values are essentially equal for thetwo different noise levels, while the magnitude of the remaining singular valuesis related to the noise in the signal. The one order of magnitude difference ofthe noise-related singular values is consistent with the difference in noise level.Because the noise term is random, it is logical that the energy would be equallydistributed among the range of the possible frequencies, leading to approximatelyequal singular values. One distinguishing feature that delineates the true signalsingular values from the noise is the significant drop of magnitude between thelowest true component magnitude and those of the noise. Therefore, the modelnumber in optimizations presented in this work is calculated as the number ofnormalized singular values that are above the largest singular values times a tol-erance. For the optimizations presented in this work, this tolerance is set between0.1 and 0.01 under the assumption that RANS-based simulations will not have thelevel of noise of this example signal.

19

t

z

0 0.2 0.4 0.6 0.8 1 1.2

­0.004

­0.002

0

0.002

0.004

(a) Mode 5.

t

z

0 0.2 0.4 0.6 0.8 1 1.2 1.4

­0.02

­0.01

0

0.01

0.02

(b) Mode 6.

t

z

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8­0.02

­0.015

­0.01

­0.005

0

0.005

0.01

0.015

0.02

(c) Mode 7.

t

z

0 0.5 1 1.5

­0.003

­0.002

­0.001

0

0.001

0.002

0.003

(d) Mode 9.

Figure 8: X-57 modal responses with instantaneous envelope functions.

3.4. X-57 Maxwell System IdentificationThe final comparison between the damping calculation methods uses the modal

responses of a structure coupled to CFD. While the other tests have allowed ver-ifying methods against exact results and testing sensitivity to particular signalfeatures, this test represents more realistic data, but the true signal decomposi-tion is not known. For this type of problem, consistent results across the methodsis the best indication that they are working well. The modal displacements aretaken from the X-57 Maxwell simulations with FUN3D from Heeg et al. [20].Four modes, illustrated in Figure 8, are selected as representative of the types ofbehaviors observed in the full set of modes. As would be done during optimiza-tion, an initial time period has been removed so that the initial transients afterthe modal excitations have disappeared and do not affect the calculated damping.Mode six is a relatively clean signal with very low damping. Mode five is a verylow damping response with more variation in amplitude. Mode seven representsa sinusoid with higher damping, and mode nine is a decaying signal with multiplecomponents.

Table 5 gives the predicted damping ratios for different modes with the vari-ous system identification methods. For the first three modes, the table illustratesthat there is reasonably good agreement between Hilbert transform and matrix

20

Table 5: Predicted damping ratios of X-57 modal responses.

Mode Number Log Decrement Hilbert Half-power (min) Matrix Pencil (KS)

5 0.00033542 0.00014745 ± 0.00170843 0.000150136 0.00017212 0.00008456 ± 0.00185588 0.000070537 0.00264498 0.00258143 ± 0.00089262 0.002557679 0.00203437 0.22148883 ± 0.00092640 0.00085409

pencil methods. Even for the very low damping responses of modes five and six,the Hilbert transformation and matrix pencil damping ratios differ by less than2.0× 10−5. For mode seven, they agree to within 1% of each other. The logdecrement method also agrees fairly well with the Hilbert transformation and ma-trix pencil methods for the first modes, but predicts slightly higher damping forthe modes near zero damping. For mode nine, the log decrement method usingthe peaks indicated by the red symbols in Figure 8d results in a damping estimatethat is 2.38 times larger than the matrix pencil method. The Hilbert transfor-mation method’s damping ratio is about two orders of magnitude larger than thematrix pencil and log decrement results. Since Equation 6 in the Hilbert methodassumes a single mode, while the log decrement and matrix pencil methods arerelatively close, it is likely the Hilbert method is less accurate than the other twoapproaches. The damping ratios predicted by the half-power bandwidth methodare listed as plus or minus because the method does not indicate whether a com-ponent of the signal is growing or decaying. The half-power bandwidth methoddoes not perform well in determining the damping of these signals. The lengthsof the signals create FFT frequency resolutions of approximately 5 rad/s. Linearinterpolation of the FFT-based power spectrum is employed to approximate thehalf-power frequencies. The limited resolution causes the calculated damping ofmode seven to be lower than that of modes five and six, which is inconsistent withthe other methods. While the minimum damping is close to the matrix pencil KSaggregated value for mode nine, this appears to be coincidental as the half-powerbandwidth and matrix pencil methods predict that the minimum damping ratio oc-curs on different modes. Table 6 lists the predicted frequencies computed from thematrix pencil and half-power bandwidth methods. While these methods indicatethe minimum damping occurs on different modes, the frequencies are within theresolution of the FFT frequencies (4.02 rad/s). That is a good indication that thesignal decompositions for the half-power bandwidth and matrix pencil methodswould be similar if the FFT resolution was refined.

21

Table 6: Frequencies (rad/s) predicted for X-57 mode nine response. The starred frequenciesindicate the mode with the smallest predicted damping

Component Half-power Matrix Pencil

1 257.4* 257.62 289.5 288.23 313.7 312.0*4 317.8 319.4

3.5. Summary of Damping Prediction Method ComparisonsDespite its sensitivity to the model number, the matrix pencil method appears

to be the best suited for identifying damping coefficients in CFD-based flutteranalysis and optimization. The log decrement method is not robust enough to con-sistently handle multi-component signals. The Hilbert transformation-based enve-lope function of Zhang [8] does not handle multi-component signals or noise in aneffective manner. While able to decompose the signal into different components,the half-power bandwidth method’s resolution is insufficient for time windowsthat would be typical of CFD calculations. Curve fits other than linear interpola-tion could be applied to the low resolution power spectrum, but any assumed shapeof the power spectrum will heavily influence the damping prediction. The matrixpencil method is noise tolerant and capable of handling multi-component signalswith growing and decaying components. Therefore, the matrix pencil method hasbeen selected as the basis of flutter-based optimization in this work. The followingsections demonstrate its application in flutter-based optimizations.

4. Demonstration of the Matrix Pencil Method for Flutter Identification

In this section, the matrix pencil method is applied to three-dimensional flut-ter prediction of the AGARD 445.6 wing. Experimental flutter data from theTransonic Dynamics Tunnel at NASA Langley was published by Yates [21]. Thewall-mounted wing has a span of 2.5 feet, root chord of 1.83 feet, a quarter chordsweep angle of 45 degrees, and a taper ratio of 0.66. The airfoil is a symmetricNACA 65A004 airfoil, and all flutter data for the test case studied in this work isat zero degrees angle of attack.

The structural model described by Yates is a modal structure; therefore, anadjoint-enabled modal solver was applied to this problem. A second-order back-ward difference formulation was utilized for temporal discretization. The first

22

four modes of the structure were retained for the analysis. The aerodynamics weremodeled on a 446,584 node mesh using an Euler simulation in FUN3D [22, 23],which is an unstructured, node-centered finite volume solver that has been appliedto time-domain aeroelastic problems [24, 25, 26, 27]. The time-domain ArbitraryLagrangian-Euler (ALE) formulation of the Navier–Stokes equations are solvedwith an optimized backwards difference scheme with dual time-stepping [24]. Toaccount for motion of the body in the flow, a linear elasticity analogy is appliedto deform the CFD volume mesh [28] at each time step. FUN3D includes a time-domain discrete adjoint capability that has been applied to moving grid [29, 30]and aeroelastic problems [31, 32].

The coupling of FUN3D and the modal structural solver is performed withFUNtoFEM [33, 32, 34], a Python-based framework developed for both high-fidelity aeroelastic analysis and adjoint-based aeroelastic optimization. The trans-fer of loads and displacements is performed with a method called Matching-basedExtrapolation of Loads and Displacements (MELD) [35].

Each flutter simulation consisted of 6,000 time steps with 10 CFD subiter-ations per time step. The time step size was 5.02× 10−5 or about 220 stepsper cycle of the highest frequency mode. At the 125th step, all four structuralmodes were excited with an impulsive force of magnitude 0.1 times the modalstiffness. The matrix pencil method was applied to the final 3,000 time steps ofeach simulation. The automated identification of the flutter point was formulatedas an optimization problem, which was solved using the sequential least-squaresquadratic programming (SLSQP) optimizer [36] in PyOpt [37]. The only designvariable was the dynamic pressure, and the objective was to minimize the dynamicpressure, subject to a constraint that the KS-aggregated damping predicted by thematrix pencil method is equal to zero. This process was performed for Mach num-bers in the range of 0.499 to 1.141. The sensitivity of the constraint with respect tothe design variable was calculated with the adjoint method driven by FUNtoFEM.Complex step verification could not be performed to verify the accuracy of thegradient since the matrix pencil method itself uses complex arithmetic. Instead,Table 7 provides a comparison of the adjoint-based sensitivity of the damping con-straint with respect to the dynamic pressure for step sizes ∆x = 10−7, 10−8, and10−9. The adjoint and finite-difference gradient values are close to one anotherwith between four to seven digits of agreement. The FUNtoFEM adjoint imple-mentation has previously been verified for coupled time-domain simulations usingthe complex step method for real-valued functions [32]. Kiviaho et al. [34] alsoverified the matrix pencil function derivative on a problem with a known exactsolution.

23

Table 7: Derivative of the damping with respect to dynamic pressure.

Method Value

Adjoint -16.24943742Finite difference, ∆x = 1×10−7 -16.24943337Finite difference, ∆x = 1×10−8 -16.24253229Finite difference, ∆x = 1×10−9 -16.74305849

Figure 9 compares the flutter dynamic pressure determined by the FUNtoFEMoptimization, experimental values, and results obtained with the FUN3D modalsolver from Silva et al. [38]. Each optimization based on the matrix pencil methodsuccessfully determined the flutter dynamic pressure to within 0.1 psf across therange of Mach numbers in four to seven design cycles. Like the inviscid FUN3Dmodal solution from Silva et al., the FUNtoFEM solution predicts a transonic flut-ter dip at M∞ = 0.96 but overpredicts the flutter dynamic pressure above Mach 1.The discrepancy between the FUNtoFEM and inviscid FUN3D results are likelydue to the difference in mesh resolution. The mesh in the inviscid FUN3D resultshas two million nodes, while the FUNtoFEM mesh has less than five hundredthousand nodes. However, in this regime above Mach 1, the flutter dynamic pres-sure predicted by inviscid methods and experimental results diverge. Since thepurpose of this study is to explore the effectiveness of the matrix pencil basedflutter constraint rather than achieve spatially converged solutions, the discrep-ancy between the FUN3D and FUNtoFEM Euler solutions was deemed accept-able. Since the present model captures the correct trends, the fidelity is sufficientto model the important physics in the problem.

When flutter is present, there are at least two points in the design space wherethe aeroelastic damping is zero: both at q = 0 and q = qflutter. Since gradient-based optimization methods search for a local optimum from an initial guess, itis important to characterize the robustness of the proposed optimization methodas the initial dynamic pressure is varied. Figure 10 shows the convergence of theflutter identification optimization for M∞ = 1.072 from various initial dynamicpressures. The time histories for the first mode (the flutter mode) at some pointsare also given in the plot to illustrate the convergence from these initial dynamicpressures. The highest initial dynamic pressure is well above the flutter dynamicpressure, so the rapidly increasing magnitude of the oscillations causes the meshdeformation to produce negative cell volumes, causing the simulation and opti-mization to fail. From the initial dynamic pressures of q = 144.0 psf and 100.8

24

Mach Number

Flu

tte

r D

yn

am

ic P

res

su

re [

lb/f

t 2]

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20

50

100

150

200

250

300

350

ExperimentFUN3D ­ RANS

FUN3D ­ EulerFUNtoFEM ­ Euler

Figure 9: AGARD 445.6 flutter boundary. FUN3D results are from Silva et al. [38].

psf, the optimizations converge to the same dynamic pressure. At the final designcycles, the damping ratio for these cases is less than 10−5, indicating that it is theflutter dynamic pressure. Although the initial dynamic pressures are close, theoptimizations that start from values of q = 43.2 psf and 57.6 psf diverge becausethey start on either side of the maximum damping peak. At a dynamic pressure ofq = 43.2 psf both the damping and the derivative of the damping with respect tothe dynamic pressure are positive, which forces the optimization toward the localminimum at q = 0. Because the damping coefficient at a value of q = 57.6 psf is0.72 and near to the maximum damping, the derivative of the damping coefficientwith respect to the dynamic pressure is -0.005972, which has a negative but veryshallow slope. Therefore, the optimizer takes a large step to a dynamic pressureof 178.15 psf, which is where the tangent line intercepts zero damping. At a valueof q = 178.15, the simulation fails due to negative volumes generated in the CFDvolume mesh caused by unbounded oscillations.

Although this analysis demonstrates that the behavior the method can dependon the initial design conditions, there are other considerations that could improvethe robustness in a full multidisciplinary design problem. For a real design prob-lem, the flutter identification process would include an inequality constraint wherethe flutter dynamic pressure, qflutter, must be above the flight condition by somesafety factor. This would prevent the optimization from converging to a value ofq = 0, because the optimizer would have the ability to adjust other design vari-ables, such as structural or geometric variables, to increase the flutter dynamic

25

Design Cycle

Dy

na

mic

Pre

ss

ure

[lb

/ft^

2]

0 1 2 3 40

50

100

150

200

q0=180.0

q0=144.0

q0=100.8

q0=57.6

q0=43.2

Figure 10: AGARD 445.6 flutter optimization history at M∞=1.072 from various initial conditions.

pressure until this new flutter clearance constraint is satisfied. However, in thisAGARD 445.6 problem, the design variable only controls the flow conditions andthe flutter behavior at a given condition cannot change. For the optimizationsthat fail due to negative volumes at high dynamic pressures, the unconstrainedSLSQP optimizer cannot recover from an evaluation failure. More robust opti-mization algorithms [39] are fault-tolerant and are able to reduce the changes indesign variables and continue the optimization in the feasible design space. Atthe highest initial dynamic pressure it would not be possible to backtrack becausethe failure occurs at the first design cycle. In this case, the user could restart theoptimization with a reduced dynamic pressure or excitation magnitude.

5. Conclusions

In this paper, methods to compute the characteristics of flutter have been eval-uated for a time-domain CFD-based methodology. The log decrement, envelopefunctions using the Hilbert transformation, half-power bandwidth, and matrix pen-cil methods were evaluated. The evaluation criteria considered the aeroelasticdamping estimation capabilities for cases with multi-component signals, shortsignal intervals, and robustness of the estimate to noisy input signals. The ma-trix pencil method proved to be the most robust method to estimate the dampingacross a range of characteristic input signals. Aeroelastic simulation data from theX-57 Maxwell were also used to validate the prediction capability of the matrix

26

pencil method. While the matrix pencil method provided good damping estimatesfor time domain signals, gradient-based optimization with this method must behandled with care. Since multiple conditions give zero damping, automated flut-ter identification methods based on optimization techniques may fail to give thedesired flutter point. Nevertheless, the matrix pencil method provides a robusttechnique to estimate aeroelastic damping and can be used to automatically findthe flutter point.

6. Acknowledgments

Funding for this work was provided through the NASA Transformative Toolsand Technologies program with grant number NNX15AU22A with Technical Mon-itor Steve Massey. Computational resources were provided by the NASA High-End Computing Program through the NASA Advanced Supercomputing Divi-sion at Ames Research Center. The authors would like to thank Jennifer Heegfrom NASA Langley Research Center for providing the X-57 Maxwell modal re-sponses.

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