reducing uncertainty in aeroelastic flutter boundaries using

IFASD-2011-71

REDUCING UNCERTAINTY IN AEROELASTIC FLUTTER BOUNDARIES USING EXPERIMENTAL DATA Richard P. Dwight1 , Hester Bijl1 , Simao Marques2 , and Ken Badcock3 1 Faculty

of Aerospace Engineering Technische Universiteit Delft, 2600GB, The Netherlands [email protected] 2 School

of Aerospace Engineering Queen’s University Belfast, BT9 5AH, Northern Ireland 3 Department

of Engineering University of Liverpool, L69 3GH, United Kingdom

Keywords: aeroelasticity, flutter, uncertainty quantification, data assimilation, inverse problems, model updating, Bayes’ theorem, probabilistic collocation, Markov-Chain Monte-Carlo Abstract: Flutter prediction as currently practiced is usually deterministic, with a single structural model used to represent an aircraft. By using interval analysis to take into account structural variability, recent work has demonstrated that small changes in the structure can lead to very large changes in the altitude at which flutter occurs (Marques, Badcock, et al., J. Aircraft, 2010). In this follow-up work we examine the same phenomenon using probabilistic collocation (PC), an uncertainty quantification technique which can efficiently propagate multivariate stochastic input through a simulation code, in this case an eigenvalue-based fluid-structure stability code. The resulting analysis predicts the consequences of an uncertain structure on incidence of flutter in probabilistic terms – information that could be useful in planning flight-tests and assessing the risk of structural failure. The uncertainty in flutter altitude is confirmed to be substantial. Assuming that the structural uncertainty represents a epistemic uncertainty regarding the structure, it may be reduced with the availability of additional information – for example aeroelastic response data from a flight-test. Such data is used to update the structural uncertainty using Bayes’ theorem. The consequent flutter uncertainty is significantly reduced across the entire Mach number range. 1 INTRODUCTION Flight flutter tests can be dangerous and costly, whereas computational methods may not accurately predict the flutter boundary [1]. As such there is a long history in aeroelasticity of combining limited flight-test data with physical models to estimate the flutter envelope. A successful and widely-used approach was proposed by Zimmerman and Weissenburger [2] based on an analytic model of the aeroelastic phenomena, and has been modified multiple times to account for uncertainty information in order to evaluate robustness [1, 3]. The present work takes one step towards the goal of performing a comparable stochastic analysis of flight-test data on the basis of modern high-fidelity PDE simulations. The increase in modeling accuracy should lead to a reduction in the quantity of flight-test data necessary, but a major challenge is the high computational expense of the basic fluid-structure interaction (FSI) simulation.

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First we consider the sub-problem of accounting for uncertainty due to structural variability in the aeroelastic simulation. This uncertainty arises from airframe material and manufacturing variability, as well as wear – and results in substancial uncertainty in the aeroelastic response of the system [4–6]. Probability is a standard framework for describing uncertainty, allowing use of the tools of mathematical statistics, in particular Bayes’ theorem [7]. In this context a structural parameter of an aircraft wing is represented by a random variable described by a probability density function (pdf), which represents either aleatory or epistemic uncertainty in that parameter [8]. Aleatory uncertainty describes real physical randomness, e.g. variability in airframes across a fleet of aircraft. It may be desirable to model the behavior of the fleet as a whole, in which case the variability in the structures could be represented as a pdf. Since aleatory uncertainty has a physical significance, it may be estimated, but can not be reduced by measurement or analysis [9]. An example of estimating aleatory uncertainty in structures using observations of modal frequencies is given in [10]. In many problems, the dominant uncertainties arise from a lack of knowledge rather than true randomness. This is epistemic uncertainty, and may be reduced if additional information becomes available. In the Bayesian framework, epistemic uncertainties are also represented using pdfs - in which case the probability represents our degree of confidence in some proposition. The numerical tools applied in this work are agnostic to the type of uncertainty, aleatory or epistemic; the distinction is purely one of interpretation. We consider epistemic uncertainty in the structural parameters of an elastic wing. It is assumed that there exists a single “true” value for each structural parameter of that wing, whose precise value is unknown to us, but since the wing has been manufactured to certain tolerances, we can make an initial informed guess formulated as a pdf, the Bayesian prior. Using probabilistic collocation (PC), the prior can be propagated through a flutter simulation code to obtain an initial estimate of the corresponding uncertainty in the flutter altitude. The second sub-problem considered is the introduction of experimental observations to the analysis, which comes naturally in the probabilistic framework via Bayes’ theorem. The result of an experimental observation of the wing will be dictated only by the true structural parameter values, and not by our epistemic uncertainty. This data is therefore potentially informative about the true structure, bearing in mind that the experimental observations are subject to error (observational noise) that can also be represented as a stochastic variable. Combining the prior and data using Bayes’ theorem gives us an updated pdf, the posterior, which represents our updated knowledge regarding the parameter values given the data. We would like this to be more specific than our prior, so that the corresponding uncertainty in the flutter altitude will be reduced. The probabilistic analysis of a complex fluid-structure coupled system presented here, is made feasible due to two advances in the efficiency of numerical techniques. The most important component is a recently developed efficient eigenvalue solver [11, 12], which estimates the stability properties of a fluid-structure coupled system directly - without performing an unsteady simulation, see Section 2. Uncertainty quantification is accomplished with probabilistic collocation (PC) [13,14], which is non-intrusive (i.e. requires no time-consuming modification of the simulation code), and converges spectrally - reaching a high accuracy with only a few samples of the simulation code in each dimension, see 2

Section 3. Bayesian updating is performed on the surrogate model created by PC, and requires no additional simulation runs, Section 4. The method is applied to the Goland wing [15], which has been used extensively as an aeroelastic test-case. Seven structural parameters are regarded as uncertain – having been identified as the only structural parameters with a substancial influence on flutter by a preliminary sensitivity analysis, their impact on flutter uncertainty is discussed in Section 5. No experimental data currently exists for this case, therefore artificial data is generated using simulation with added noise. The Bayesian updating of the model results in a corresponding reduction of both structural and flutter uncertainty, Section 6. 2 FLUTTER PREDICTION WITH AN EIGENVALUE SOLVER The unsteady fluid-structure system consisting of an elastic wing in high-speed flow, is approximated by coupling the compressible Euler equations to a structural model for the wing - with surface forces and displacements interpolated between the fluid mesh and the structural model. The flow discretization is based on the University of Liverpool’s parallel multiblock solver - a cell-centered finite-volume code, operating on curvilinear body-fitted meshes [16]. The structural discretization uses a finite-element model, analyzed with MSC.Nastran. The coupling approach used is described in [17]. In order to estimate the stability properties of the aeroelastic system, we could integrate the discrete system forward in time for each parameter-set and condition of interest but this is computationally expensive. Instead we perform a linear stability analysis of the coupled discrete system [4] about a stationary solution. The coupled system may be concisely written: dwf + Rf (wf , ws , α) = 0 dt dws + Rs (wf , ws , α) = 0, dt where Rf is the residual of the fluid discretization, including all boundary conditions and coupling terms. It is an Nf -dimensional vector function of the fluid and structure degrees of freedom, wf and ws , as well as a vector α, containing the stochastic parameters of the structural model. The structure discretization is Rs , of size Ns , and we assume that the system is well-posed, wf has dimension Nf and ws has dimension Ns . ¯ = [w ¯ f, w ¯ s ] to the spatially discretized steady equations The linear stability of a solution w is solely determined by the eigenvalues λ, of the coupled Jacobian matrix, which satisfy: " ∂R ∂R # f f vf Af f Af s vf v ∂wf ∂ws = =λ f . ∂Rs ∂Rs v A A v vs s sf ss s ∂wf ∂ws ¯ w

This is a very large sparse eigenvalue problem, so a technique is introduced to solve it cheaply. Firstly, as is conventional in aircraft aeroelasticity, the dimension of the structural problem is reduced by considering only a small number of modes Nm , obtained by solving the structural eigenvalue problem in isolation. Then the large, linear eigenvalue problem can be reduced to a non-linear eigenvalue problem of dimension Nm using a Schur

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decomposition: the eigenvalues λ satisfy: S(λ)vs =λvs , S(λ) :=Ass − Asf (Af f − λI)−1 Af s , which can be solved with Newton’s method. This is still an expensive procedure because of the repeated inversions of (Af f −λI), where Af f is fixed, but λ changes at each Newton iteration. This term is therefore replaced with a series expansion (Af f − λ0 − ∆λ)−1 = (Af f − λ0 I)−1 + ∆λ(Af f − λ0 I)−2 + . . .

(1)

where λ = λ0 + ∆λ, and λ0 is chosen as the purely structural eigenvalue of the corresponding mode. The terms on the RHS of (1) are not evaluated explicitly, rather only the products: Asf (Af f − λ0 I)−1 Af s , etc. of which the most costly part is Nm linear solves of Nf × Nf sparse matrices. Details of the construction of the Jacobian are given in [18]. In particular Af s and Asf are evaluated by finite differences. An example of the output of this solver is given in Figure 1. The real-part of the aeroelastic eigenvalue Re(λ), of the first 4 structural modes are shown as functions of the altitude, for the Goland wing case, described in Section 5, AoA 0◦ and Mach 0.5. Two curves a shown representing the same calculation on fine and coarse CFD meshes. As the altitude drops, the 1st mode starts to interact with the 2nd and becomes unstable. Note that the small change in solution between the coarse and fine CFD meshes suggests that the discretization error (of the fluid part), is small.

Figure 1: Left: Coarse CFD mesh. Right: Real component of aeroelastic eigenvalue for the first 4 modes of the Goland wing at AoA 0◦ and Mach 0.5 in clean configuration - computed on coarse and fine CFD meshes.

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3 PROBABILISTIC COLLOCATION FOR UNCERTAINTY PROPAGATION Probabilistic collocation [13, 14, 19], is a means of computing statistics of the output of a computational model, given pdfs on the model’s input parameters. The method uses a polynomial expansion based on Lagrange polynomials to approximate the response of the model in the uncertain parameter space. Gaussian quadrature weighted by the pdf of the uncertain input is applied to compute the mean, variance, and higher moments of the output. By choosing the support points and Gauss rule appropriately, it is possible to achieve decoupling of the equations for different parameter values (collocation), and a higher-order approximation of the mean and variance of the output (spectral convergence). The PC method is derived as follows: consider a PDE, R(α(ω))w(x, t, ω) = S(α(ω)).

(2)

and its solution w(x, t, ω), depending on not only space x and time t, but also a random event ω. In general ω ∈ Ω, a set of possible outcomes to be defined. In the following we consider M independent continuous uncertain parameters, for which it is convenient to specify Ω = [0, 1]M , with ω equally likely to take any value in Ω (i.e. uniformly distributed). Without loss of generality consider a single scalar uncertain parameter of the system α(ω), modeled as a random variable A, with probability density function (pdf) fα (·), and cumulative density function Fα (·). For higher dimensions we use tensor products of the rules derived here. The solution w is approximated in the parameter space by: w(x, t, ω) ≈

Np X

wi (x, t)Li Fα−1 (ω) ,

(3)

i=1

where the dependence on the stochastic event ω has been separated out and discretized by a sum of Lagrange polynomials, Li . The function Fα−1 (·) = α(·) works to maps events ω into the parameter space, taking into account their distribution. As an example: if the single parameter is normally distributed with mean 0 and variance 1, then Fα−1 (ω) = erf −1 (ω), where erf(·) is the Gauss error function. In order to obtain equations for wi we use a Galerkin projection, and require that * + Np X R(α) wi (x, t)Li (α), Lk (α) = hS(α), Lk (α)iA , ∀k = 1, . . . , Np . i=1

(4)

A

where the inner product with respect to A is defined as Z ∞ ha, biA = abfα (α) dα. −∞

Consider a numerical approximation to this inner product. In particular it is reasonable to aim to approximate the integral: Z ∞ EA (w) = w(α)fα (α) dα (5) −∞

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as accurately as possible, in order to approximate the mean of w accurately. If we use an npoint Gauss rule weighted with fα , then the result will be exact provided w is a polynomial of order less than 2n in α. If A is distributed uniformly then this is the standard GaussLegendre quadrature rule. If A is normally distributed, a rule with collocation points at the roots of Hermite polynomials is obtained. More generally, we can derive a rule for any weight numerically using the Golub-Welsch algorithm [20], together with the discredited Stieltjes procedure [21] to obtain recurrence relations for polynomials orthogonal with respect to the weight. Given this choice of quadrature rule, we ask what the most appropriate choice for the support points of the Lagrange polynomials in (3) are. In PC they are chosen as the support points of the Gauss rule αi , given which Li (αj ) = δij , and (4) simplifies to R(αi )wi (x, t) = S(αi ), i.e. the equations at the support points completely decouple, and a collocation method results. PC(M ) is defined as the PC rule with order M , i.e. M + 1 support points, but due to the Gauss quadrature, approximations of E will be of order 2M − 1. In the remainder of this text PC is used principally as a surrogate model for the simulation code, to which Monte-Carlo is applied to obtain approximations of output pdfs. The PC support points for rules of various orders, and uniform, normal, and log-normal distributions are plotted in Figure 2. Sub-figure (a) demonstrates that points are confined to regions of non-zero probability - zero probability regions will not contribute to (5). In sub-figures (b) and (c), for low order rules the support is concentrated around high-probability regions, with points increasingly being pushed to the tails as the order increases. This behavior is a result of the assumption that a high-degree polynomial is a good fit of the simulation response. If this assumption is true, adding points at some distance from the region of high probability will improve the fit everywhere. If not, the fit may be poor or oscillatory. Therefore we generally restrict ourselves to PC rules of order ≤ 4 (5 supports in each parameter direction), and use a convergence study to assess the accuracy of the approximation. 4 BAYESIAN UPDATING OF STRUCTURAL UNCERTAINTY In addition to propagating known structural uncertainties through the model to the output, it is possible to do the reverse: given known output (with corresponding uncertainty) determine the structural parameters. The known output might be available from flight tests for a limited set of conditions, and the uncertainty a consequence of measurement tolerances. This is known variously as model parameter estimation, model calibration, and system identification, and is accomplished in a stochastic setting using Bayes’ theorem [7]. The first step is to define a prior: a probability distribution on the structural parameters encoding all information which is known about the parameters prior to observing the data, with pdf ρ0 (α). 6

(a) Uniform - U(0,1) 1.0 ρ ( ξ)

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Figure 2: Support nodes for PC rules of orders 1 to 10, with (a) uniform, (b) normal, and (c) log-normal distributions in the variable ξ. Pdfs are indicated with a solid line.

The second step is to describe the relationship between the model and the data, this is the statistical model. Let the vector of observed quantities be denoted d (the data). Let H(w, α) be an observation operator, which takes a model state w and parameter vector α, and returns the model’s approximation of the observed quantities d. The model state w(α) satisfies the model equation: R(w, α) = 0. Under the assumptions: (1) that the noise in the measurements d is known and described by the random variable , and (2) that there is no modeling error (for the correct choice of α), then the model and data are related as: d = H(w, α) + s.t. R(w, α) = 0. Given which the probability of observing data d given parameters α (the likelihood) is ρ(d|α) := ρ (d − H(w, α)), where ρ (·) is the pdf of .

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For a prior and a likelihood, Bayes’ theorem gives an explicit expression for the posterior, the probability of parameters given the observed data: ρ(α|d) ∝ ρ(d|α)ρ0 (α), where the constant of proportionality (independent of α) is not usually of interest. In the Bayesian framework this posterior is regarded as the answer to the question: What is known about α?, and is therefore the updated estimate of the parameters. If we require a deterministic estimate of α – rather than the probabilistic posterior – then a reasonable choice is the most-likely value of α, the maximum a posteriori estimate (MAP estimate): αMAP := argmax ρ(α|d). α

5 FLUTTER UNCERTAINTY FOR THE GOLAND WING The Goland wing test-case analyzed here is identical to that considered in [4]. It had a chord of 6 feet, and a span of 20 feet, is rectangular in plan, with a symmetric 4%-thick parabolic profile, see Figure 1. The structural model is cantilevered wing, is described in detail in [22], and outlined in Figure 3. Digital versions of the geometry, structural model, and example CFD meshes are available at [23].

(a) FE model

(b) Skins

(c) Spars

(d) Ribs

Figure 3: Main components of the structural model for the Goland wing.

The Goland wing as a test-case is: simple, both structurally and geometrically; flexible, with a variety of complex aeroelastic phenomena arising readily; and sensitive, to the presence or absence of a tip-store. As such it is a good model for the kinds of aeroelastic behavior that arise in transport and military aircraft applications. In this work all results are for the Goland wing in clean configuration (no store), at zero angle-of-attack (AoA), with compressible inviscid flow. 8

In previous work [24], a sensitivity analysis on all 32 structural parameters of the model was performed, using a coupled solver with a vortex panel method for the aerodynamics. This method rapidly returns the derivative of the stability eigenvalue with respect to each structural parameter. Seven parameters were identified as responsible for ∼ 99.5% of the sensitivity of the eigenvalue (depending on flow conditions). These parameters are: the thicknesses of the upper and lower skins, the thicknesses of the leading- and trailing-edge spars, and the areas of the leading-, center- and trailing-edge spar caps. The nominal values of these parameters, as would be used for example in a deterministic simulation are: sdet = (0.0155, 0.0155, 0.0006, 0.0006, 0.0416, 0.1496, 0.0416), respectively, where the units vary between the parameters. In the following we neglect the influence of the remaining 25 parameters - if their influence was deemed to be significant, they might be included in the analysis cheaply using a first-order perturbation method [24]. 5.1 Probabilistic collocation convergence study for 2 structural parameters Before applying PC to the full set of 7 uncertain variables, a convergence study for 2 structural parameters and PC rules of orders 1 to 6 is made to establish the accuracy of the scheme. We examine the error in the approximated mean, sd, skewness and kurtosis of the real component of the eigenvalue for a fixed altitude and Mach number, see Figure 4(a). The error is absolute and computed with respect to the PC(6) result. 0

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Figure 4: Probabilistic collocation for the Goland wing flutter problem at 9600 ft. Input: first 2 structural parameters, output: real component of dominant eigenvalue. (a) Convergence of 4 statistical moments, (b) response surface using PC(5), where the square box is the edge of the uniform distribution (±10% in each s1 , s2 ), the black circles are the PC node locations.

For PC(1) the error in the mean and standard deviation is already not far from numerical rounding error (we do not expect the flutter analysis to be more accurate than the squareroot of machine precision ∼ 1.4 × 10−8 ), so convergence is only apparent for the first two increases in order. Skewness and kurtosis, on the other hand, involve higher powers of the response, and non-linear effects play a greater role. Nevertheless with PC(2) the highest 9

relative error (in kurtosis) is only 2%. Bearing in mind that the uncertain input is in this case subjective and somewhat arbitrary, these results suggest that PC(2) is absolutely sufficient for the analysis, and even PC(1) may give reasonable results. Clearly the accuracy of PC strongly depends on the regularity of the simulation response within the range of parameter variation. The PC(5) approximation of this surface (a 5thdegree polynomial fit) is plotted in Figure 4(b). The cause of low errors in the moments is apparent: the surface is smooth. Clearly PC would be much less effective if this were not the case; and although we can not guarantee smoothness, for small parameter variations, and integrated quantities, response surfaces relevant for engineering practice are often smooth. To put this limitation in context: to approximate an irregular and discontinuous response in an n-dimensional space would require perhaps O(10n ) samples for reasonable resolution, given which Monte-Carlo may be the only recourse. 5.2 Probabilistic collocation for 7 structural parameters Given the convergence analysis of Section 5.1, we apply PC(2) to the full 7-uncertainparameter flutter analysis, requiring 37 = 2187 samples from the simulation code . For comparison, the interval-uncertainty method used in [4] required two optimizations to determine the upper- and lower-bounds of the output interval. The output for PC(2) is plotted in Figure 5. The main subfigure (top left) shows the real part of the eigenvalue of the first aeroelastic mode (responsible for flutter in this case), plotted as a function of altitude, as in Figure 1. The multiple lines of varying thickness represent the uncertainty in this value caused by the specified structural uncertainty. The true line has a probability of 1/2 of lying to the right of the P = 1/2 line, a probability of 1/3 of lying to the right of the P = 1/3 line, etc. The histogram to the right approximates the pdf of the real eigenvalue at the red vertical line in the main subfigure. It can be seen to be far from either Gaussian or uniform due to the strong non-linearities in the eigenvalue as a function of the structural parameters. The histogram below approximate the pdf of the flutter altitude (i.e. the pdf of Re(λ) = 0), from which the probability of flutter can be quickly evaluated. Flutter is certain for an altitude < 6, 000 ft, and guaranteed not to occur for an altitude > 14, 000 ft. This plot can also be used to judge how informative experimental data is likely to be regarding the structural parameters and therefore plan flight-tests. For example, imagine we were able to obtain, via a flight-test, data on the real eigenvalue component at a single altitude. In order to estimate the structural parameters it would be beneficial to choose an altitude for which the spread of uncertainty due to the parameters is large. Knowledge of the true eigenvalue at 20000 ft might tell us about the accuracy of the simulation, but it contains almost no information about parameter values. An experiment at 7000 ft on the other hand would give us quite specific parameter information, but is well inside the flutter regime and therefore not practical. Hence we recover quantitatively the established fact that flight-tests close to the flutter boundary are necessary to obtain flutter predictions. In this case we can say that if structural parameter uncertainty is significant we must test at a maximum of 15000 ft (where the uncertainty first becomes visible in Figure 5). But care is required, as at 13500 ft there is already some chance of flutter. Given the response surface provided by PC it is cheap to evaluate the uncertainty caused by each parameter individually and in any combination, see Figure 6. The left-hand col10

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P(flutter) < 1/25000 P(flutter) = 1/6 P(flutter) = 1/3 P(flutter) = 1/2 P(flutter) = 2/3 P(flutter) = 5/6 P(flutter) = 1

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Figure 5: Eigenvalue against altitude plot for the dominant eigenmode at Mach 0.5. Calculated with PC(2). The uncertainty in the eigenvalue at a given altitude is outlined by probability intervals. PDFs at the conditions Re(λ) = 0 and altitude = 9600ft are plotted below, and to the right respectively.

umn displays the variability in flutter altitude due to one parameter at a time ±10%. Evident is the similar magnitude of the influence of parameters 1,2,3,4 and 6, and the relatively lower influence of 5 and 7. The uniform input distributions have lead to relatively square output pdfs, and an interval analysis techniques would capture all relevant information in these cases. In the right-hand column the cumulative effect of adding uncertainty in one variable after another is observed, and the output pdf rapidly approaches a normal distribution. The histogram on the far bottom-right shows the cumulative effect of ±10% uncertainty in all structural variables, and the range in which flutter may or may not occur is extremely large. Clearly the usefulness of the simulation is limited if the uncertainty bounds are so large, therefore we examine methods for reducing the epistemic uncertainty by identifying more accurately the structural parameters. 6 REDUCING STRUCTURAL UNCERTAINTY WITH OBSERVATIONS Assuming experimental observations of the response of the system are available at a limited set of flow conditions, it is often possible to reduce the epistemic uncertainty in inputs, and thereby obtain an improved predictive capability at other flow conditions. For 11

the Goland wind we assume that a single piece of data is available at each of the following Mach numbers: M∞ = (0.5, 0.6, 0.7, 0.8). 6.1 Generating “experimental” data For the Goland case no experimental data is currently available. In order to verify the data assimilation methods studied here, we use a simulation code to generate surrogate data in the following manner. Fixed (non-stochastic) values are chosen to represent the “true” parameter values of the system, strue , distinct from the nominal deterministic parameter values sdet . “True” in this sense means that, under assumptions of zero modeling discrepancy and zero measurement noise, given these values of the parameters the data will be reproduced exactly by the model (i.e. a twin experiment in the terminology of inverse problems). There may be multiple values of s for which this is the case, but it will in general not be the case for sdet . The corresponding artificial data is therefore generated as dtrue := H(w, strue ), R(w, strue ) = 0 where R and H represent elements of the Schur eigenvalue code, and H evaluates the flutter altitude. Hereafter the values of strue are considered unknown, and one output of the Bayesian analysis will be a stochastic approximation of strue . It is unrealistic to expect error-free experimental data. Therefore we add normally distributed noise, with standard deviation σd = 500 ft to dtrue : noise ∼ N (0, σd2 ). The artificial data used in the follow is therefore: d := dtrue + noise . There is no requirement that noise be uniformly distributed. A distribution with heavy tails (e.g. a Cauchy distribution) may be more appropriate in the case of potentially unreliable experimental data. 6.2 Applying Bayesian updating to Goland flutter Using the data manufactured in Section 6.1, the Bayesian updating procedure described in Section 4 is applied to the Goland wing flutter case. The prior is taken to be uniform ±10% distributions in each parameter independently, corresponding to the previous UQ analysis. The likelihood term is modeled as a Gaussian using the known noise level introduced into the data, and the posterior is sampled with Markov-Chain Monte-Carlo applied to the PC(2) response surface constructed in the previous section . As a consequence of the use of the response surface, no additional runs of the Schur code are necessary for the fitting procedure. The effect of the fitting procedure on the joint pdf for s1 and s2 , and pdf for flutter speed are show in Figure 7. The joint pdf is displayed on the left by plotting samples from 12

the distribution (from MCMC) as black crosses. On these plots sdet is plotted as a blue square, and strue as a red circle. The top row of the figure shows the original pdfs, without addition of data: the s-sampling is uniform, and the pdf on the flutter speed is the same as that in Figure 5. Each row then adds one of the 4 scalar pieces of data (at Mach 0.5 to 0.8), and a Bayesian updating is performed each time. The bottom row in Figure 7 is then the fitted model using all 4 data points. The corresponding distributions on flutter speed for 3 other Mach numbers are shown in Figure 8. Apparent is that the addition of this data increases the specificity of the flutter altitude estimates. Furthermore, the addition of data at one Mach number increases the specificity of the model across all Mach numbers, as expected given that structural parameters are better identified. From the scatter plots in Figure 7 it is apparent that the true values of s themselves have not been narrowed down much, indicating the different informativeness of the observed data with respect to different variables. The data from Figures 7 and 8 is presented in tabular form in Table 1, where the reduction in standard-deviation σflutter , the movement of the mean µflutter , and the altitude at which flutter is certain with respect to the amount of data are readily identifiable. 7 CONCLUSIONS Identification and reduction in uncertainty in flutter behavior due to uncertainty in structural parameters has been performed using efficient numerical techniques. Very large flutter altitude variabilities are seen to result from moderate variability in many structural parameters, and this relationship is captured well by a probabilistic representation of uncertainty. Using a very small amount of experimental data it has been shown how these uncertainties can be reduced substantially, improving the predictive capability of the simulation. Further work will concentrate on the use of real flight-test data, e.g. from systematic excitation across a frequency range, rather than the artificially generated data used here. 8 REFERENCES [1] Lind, R. and Brenner, M. (1997). Robust flutter margins of an F/A-18 aircraft from aeroelastic flight data. Journal of Guidance, Control and Dynamics, 20(3), 597–604. [2] Zimmerman, N. and Weissenburger, J. (1964). Prediction of flutter onset speed based on flight testing at subcritical speeds. Journal of Aircraft, 1(4), 190–202. [3] Khalil, M., Sarkar, A., and Poirel, D. (2010). Application of Bayesian inference to the flutter margin method: New developments. ASME Conference Proceedings, 2010(54518), 1143–1151. doi:10.1115/FEDSM-ICNMM2010-30041. [4] Marques, S., Badcock, K., Khodaparast, H., et al. (2010). Transonic aeroelastic stability predictions under the influence of structural variability. Journal of Aircraft, 47(4), 1229–1239. [5] Lindsley, N., Pettit, C., and Beran, P. (2006). Nonlinear plate aeroelastic response with uncertain stiffness and boundary conditions. Struc. Infrastruct. Eng., 2(3–4), 201–220. 13

Mach = # data 0 1 2 3 4

0.5: µflutter 9683 10644 10957 10850 10961

σflutter 1160 459 348 293 258

Pflutter = 1 5757 8827 9527 9484 9880

5/6 8530 10203 10619 10566 10712

2/3 9161 10445 10807 10723 10850

1 1/2 1/3 1/6 < 25000 9680 10199 10836 13795 10644 10842 11090 12596 10959 11108 11292 12412 10849 10976 11133 12066 10962 11073 11210 12025

Mach = # data 0 1 2 3 4

0.6: µflutter 20291 21201 21498 21396 21504

σflutter 1098 434 329 276 241

Pflutter = 1 16578 19466 20151 20116 20505

5/6 19200 20783 21178 21129 21271

2/3 19798 21013 21357 21277 21399

1/2 20289 21202 21500 21395 21505

1/3 20781 21388 21640 21514 21609

1 1/6 < 25000 21384 24248 21623 23013 21813 22894 21662 22516 21737 22474

Mach = # data 0 1 2 3 4

0.7: µflutter 29868 30751 31040 30940 31049

σflutter 1067 423 321 270 233

Pflutter = 1 26283 29056 29710 29703 30058

5/6 28808 30342 30728 30680 30824

2/3 29390 30567 30902 30824 30949

1/2 29865 30752 31042 30938 31050

1/3 30344 30934 31179 31055 31150

1 1/6 < 25000 30932 33801 31161 32513 31350 32425 31200 32062 31275 31986

Mach = # data 0 1 2 3 4

0.8: µflutter 40187 41125 41437 41327 41454

σflutter 1140 465 358 303 260

Pflutter = 1 36413 39244 39938 40003 40372

5/6 39053 40674 41090 41036 41203

2/3 39671 40924 41284 41197 41342

1/2 40177 41126 41441 41326 41455

1/3 40688 41327 41593 41456 41566

1 1/6 < 25000 41325 44552 41577 43017 41783 42993 41619 42694 41706 42506

Table 1: Flutter altitude statistics in feet at 4 Mach numbers for varying amounts of experimental data. Row # data = 3 signifies flutter data at Machs 0.5, 0.6 and 0.7 was assimilated.

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[6] Pettit, C. L. and Beran, P. S. (2006). Spectral and multiresolution wiener expansions of oscillatory stochastic processes. Journal of Sound and Vibration, 294(4–5), 752– 779. [7] Tarantola, A. (2004). Inverse Problem Theory and Methods for Model Parameter Estimation. Society for Industrial and Applied Mathematics. [8] Oden, T., Moser, R., and Ghattas, O. (2010). Computer predictions with quantified uncertainty. ICES REPORT 10-39, The Institute for Computational Engineering and Sciences, University Texas at Austin. [9] Oberkampf, W. and Roy, C. (2010). Verification and Validation in Scientific Computing. Cambridge University Press, Cambridge. [10] Dwight, R., Haddad-Khodaparast, H., and Mottershead, J. (2011). Identifying structural variability using bayesian inference. Mechanical Systems and Signal Processing. [11] Badcock, K. and Woodgate, M. (2007). On the fast prediction of transonic aeroelastic stability and limit cycles. AIAA Journal, 45(6), 1370–1381. [12] Timme, S., Marques, S., and Badcock, K. (2011). Transonic aeroelastic stability analysis using a Kriging-based Schur complement formulation. AIAA Journal, 49(6), 1202–1213. [13] Babuˇska, I., Nobile, F., and Tempone, R. (2007). A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Journal of Numerical Analysis, 45(3), 1005–1034. [14] Loeven, G. (2010). Efficient Uncertainty Qunatification in Computational Fluid Dynamics. Ph.D. thesis, TU Delft, Department of Aerodynamics. [15] Goland, M. (1945). The flutter of a uniform cantilever wing. Journal of Applied Mechanics, 12, 197–208. [16] Badcock, K., Richards, B., and Woodgate, M. (2000). Elements of computational fluid dynamics of block structured grids using implicit solvers. Progress in Aerospace Sciences, 36(5-6), 351–392. [17] Woodgate, M., Badcock, K., Rampurawala, A., et al. (2005). Aeroelastic calculations for the Hawk aircraft using the Euler equations. Journal of Aircraft, 42(4), 1005– 1012. [18] Badcock, K., Woodgate, M., and Richards, B. (2004). The application of sparse matrix techniques of the CFD based aeroelastic bifurcation analysis of a symmetric aerofoil. AIAA Journal, 42(5), 883–892. [19] Loeven, G. and Bijl, H. (2008). Probabilistic collocation used in a two-step approach for efficient uncertainty quantification in computational fluid dynamics. CMES Computer Modeling in Engineering & Sciences, 36(3), 193–212. [20] Golub, G. H. and Welsch, J. H. (1969). Calculation of gauss quadrature rules. Mathematics of Computation, 23(106), 221–230.

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[21] Gander, M. J. and Karp, A. H. (2001). Stable computation of high order gauss quadrature rules using discretization for measures in radiation transfer. Journal of Quantitative Spectroscopy Radiative Transfer, 68, 213–223. [22] Beran, P., Knot, N., Eastep, F., et al. (2004). Numerical analysis of store-induced limit-cycle oscillations. Journal of Aircraft, 41(6), 1315–1326. [23] Badcock, K. and J.E.Mottershead (2010–). Ecerta project - aeroelastic test-cases. [24] Khodaparast, H., Mottershead, J., and Badcock, K. (2010). Propagation of structural uncertainty to linear aeroelastic stability. Computers and Structures. (to be published).

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Figure 6: Histograms of flutter altitude at Mach 0.5 under ±10% variability of structural parameters. Calculated with PC(2). Left column: Flutter variability due to individual parameters. Right column: Cumulative effect of variability in parameters s0 through si , for i ∈ {0, · · · , 6}. Bottom right is therefore the total effect of all parameter variability.

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Figure 7: Left column: MCMC sampling of the PC(2) 7d response surface. Samples locations (black crosses), nominal parameter values (blue square), true parameter values (red circle). Right column: Histograms of flutter speed based on MCMC sampling. The rows show the effect of increasing the amount of experimental data employed.

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Figure 8: Columns: Histograms of flutter speed at various Mach numbers. Rows: Influence of increasing the amount of experimental data used in the analysis.

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