Identifying structural variability using Bayesian inference R.P. Dwight 1 , H. Haddad-Khodaparast 2 , J.E. Mottershead 2 1 TU Delft, Aerospace Faculty, Aerodynamics Group, P.O. Box 5058, 2600GB Delft. The Netherlands. e-mail:
[email protected] 2
University of Liverpool, School of Engineering, Brownlow Hill, Liverpool, L69 3GH, United Kingdom e-mail:
[email protected]
Abstract A stochastic approach is proposed for estimating the variability in structural parameters present in a large set of metal-frame structures, given only measurements of modal frequency performed on a subset of the structures. The key step is a new statistical model relating simulation and experiment, including terms representing not only the measurement noise, but also the unknown structural variability. This latter is modelled by random variables whose hyper-parameters are themselves stochastic, and these hyper-parameters are estimated by Bayes’ theorem. The evaluation of the posterior distribution is efficiently performed by combining a number of modern numerical tools: kriging surrogates for the finite-element analysis, probabilistic collocation uncertainty quantification, and Markov chain Monte-Carlo. The method is demonstrated for a metal-frame model with two uncertain parameters, using data from specially designed experiments with controlled variability. The output probability densities on the structural parameters are useful for input to subsequent uncertainty quantification.
1
Introduction
The problem of estimating the variability present in a collection of nominally identical engineered structures is examined. Consider the situation in which a large number of structures are produced by some manufacturing process, and subsequently a sample of these structures is tested by measuring their modal frequencies. Small differences between the structures will result in variability in their modal frequencies. From this information we estimate the variability in physical parameters of the structure such as beam stiffnesses, and in model parameters such as damping coefficients. Thanks to cancellation of errors it is often easier to accurately predict these differences (a.k.a. deltas) than absolute values. Armed with such structural variability information we are in a position to accurately predict variability in the performance of structures in their application setting, using standard techniques of uncertainty quantification (UQ) [11, 22, 29]. An potential application domain is the maintenance of airframes. The structure of an aircraft degrades slowly over time. Over a fleet of aircraft there will be substancial variation in the extent of this degradation, depending on initial manufacturing variability and service history of individual aircraft. It is desirable to estimate this variability, to determine whether the fleet as a whole can be considered safe, without measuring every plane. An additional difficulty is that the airframe is generally inaccessible to direct measurement. Indirect measurements, such as ground vibration testing of the airframe are feasible, but do not deliver the properties of the structure which are the quantities of interest (QoIs). By first using the method described in this paper to assess variability in the airframes from ground vibration data, subsequent aeroelastic simulations can be per-
formed with the identified uncertain structure to evaluate the risk of flutter. Flutter is of particular concern as recent work has shown the potential for high sensitivity of flutter boundaries to structural variability [26, 7]. Our method for the estimation of structural variability is based on Bayesian inference with a novel statistical model. A Bayesian analysis can be separated into two major components: creation of a statistical modelling and numerical evaluation of that model. The statistical model is needed to relate the output of the simulation code (e.g. a finite-element (FE) code) to the observed measurement data, and should account for differences between the two due to various errors. A simple and commonly used statistical model is (see e.g. [10, 27]): d = m(θ) + ,
(1)
where d is the measurement value, m(θ) is the simulation’s prediction of the measurement value for a particular parameter value θ, and is a random variable representing measurement noise. The assumption is that there exists some “true” value of θ for which the simulation corresponds best to the measurement. In the case of a variable structure however, θ cannot be assumed to have one true value, and therefore we introduce here the model d = m (Θ(φ)) + (2) where Θ(φ) is a random variable parametrized by a vector of hyper-parameters φ, and represents the aleatory variability amongst the structures. Here m(Θ) denotes the random variable obtained by propagating Θ through the simulation code m(·), for whose pdf there is in general no closed-form expression - this pdf is evaluated numerically in this work with probabilistic collocation. To complete the statistical model, the parametric family of pdfs from which Θ is selected must be chosen. Finally, to apply Bayesian updating we must choose priors on the hyper-parameters φ. Thus there are two levels of stochastic modelling involved: pdfs on the hyper-parameters represent epistemic uncertainty; the pdfs on the structural parameters represent real variation in these quantities across the population of structures, i.e. aleatory uncertainty. Applying Bayes’ theorem results in a posterior joint-pdf on the hyper-parameters. Sampling the posterior numerically is complicated by the two-level nature of the statistical model - however we show that it is feasible to combine several modern numerical methods in an appropriate way to sample efficiently. In particular Kriging interpolation is used as a cheap surrogate for the FE code, probabilistic collocation is used to evaluate m(Θ), and sampling is performed with Markov-Chain Monte-Carlo (McMC). This particular combination might be novel, but we consider that (given the statistical model) it could have been arrived at by any sufficiently well-informed practitioner – it presented rather as an example of how quite complex statistical models with expensive simulation codes may be evaluated efficiently. We emphasise that Kriging is used here only as a convenient surrogate, and not as a component of the statistical model as in the framework of Kennedy and O’Hagan [18].
1.1
Context of the present work
Deterministic techniques for identifying parameters of individual structures are well established [32, 9], and applied in industrial settings. They typically do not take into account measurement noise however. Statistical methods for the treatment of measurement noise in model updating were established in 1974 by Collins et al. [4] and later by Friswell [8]. In these approaches, randomness arises only from the measurement noise, and the updated parameters still take unique values. Quantifying the effect of variability in structures using probability has a long history, notable is the development of stochastic finite-element formulations [21, 30], but the corresponding inverse problem of estimating structural variability has been tackled only more recently, and in a deterministic context. In 2006 Mares et al. [25] first derived a gradient-regression formulation to treat test-structure variability, obtaining means and standard deviations of structural parameters. Haddad et al. [20] developed an interval-fitting method, where the variability in the structural parameters is modelled as an interval, whose end-points are deduced from the experimental variability. Bayesian inference is a well established statistical technique, see for example the introductory text by Gelman et al. [10]. An important development was application of Bayesian statistical methodology to the analysis
of complex computer models, notable is the contribution of Kennedy and O’Hagan [18], which attempts to represent all sources of error within the statistical model. Since then Bayesian techniques have become widespread in many fields. In structural modelling Bayesian approaches were considered first in 1998 by Beck and Katafygiotis [3, 17], and subsequently Kershchen et al. [19] and Mares et al. [24]. It was seen in these articles that issues of ill-conditionedness and non-uniqueness common to all parameter estimation problems are naturally resolved in the probabilistic framework. On the other hand it was concluded that a large amount of experimental data was necessary to identify parameter values narrowly - and the conclusion was made that Bayesian approaches required more data than comparable deterministic techniques [3]. This has since become accepted wisdom in the structural model updating community. We disagree with this conclusion, and part of the motivation for this article is to demonstrate that Bayesian techniques are entirely appropriate for structural model updating. Other recent work confirms this stance [39]. With respect to multiple levels of probabilistic modelling, Ghanem and Doostan [12] previously modelled two levels. Epistemic uncertainty in polynomial chaos coefficients due to limited calibration data was estimated; the polynomial chaos itself modelling an epistemic or aleatory uncertainty of primary interest. The formalism is therefore able to represent imprecise uncertainties quite generally. The numerical aspect of the Bayesian inference problem is vital for computational tractability, particularly for large numbers of calibration parameters, and significant recent progress has been made in algorithms for approximating the posterior. For example Marzouk et al. [28, 27] and Soize [35] both use polynomial chaos as surrogates for expensive terms in the statistical model, and this approach has been generalised to sparse grids by Zabaras et al. [38]. Improvements to Markov-chain Monte-Carlo (McMC) techniques such as adaptivity, delayed acceptance Metropolis-Hastings, and multi-fidelity sampling, with judicious use can reduce the number of simulations needed substantially [5]. In a particularly creative approach El Moselhy et al. [31] describe a parameterized map from the prior to posterior measure, and then setup an optimization problem to identify this map; though the high-dimensionality of the map and the stochastic nature of the objective function result in an procedure of comparable expense to McMC. Ultimately posterior sampling might not be feasible at all, then reduced representations of the posterior are necessary. For example Bashir et al. [2] exploit Hessian information to approximate the posterior by a Gaussian. An alternative avenue is to substantially modify the statistical methodology in order to make the numerics more tractable, see Goldstein’s Bayes-linear approach [14]. In the current work the FE simulation code has low cost and we consider a limited number of calibration parameters (4). However the two-level nature of the statistical model demands an evaluation of statistical moments (due to Θ in (2)) for each evaluation of the posterior – i.e. posterior sampling involves a UQ operation at each McMC proposal. Therefore we outline a numerical framework that we expect to be suitable for substantially larger problems than the one we consider here. Application to industrially relevant cases is a goal reserved for a future paper.
1.2
Validation of the calibration procedure
To investigate the effectiveness of the proposed method we apply it to an artificial structural test-case with known variability. A common challenge in inverse problems is the limited opportunity for validation. Validating statistical models against twin problems is unsatisfactory since all sources of error are exactly known and explicitly included by the practitioner. Validation could be performed by comparing calibration results with measurements of calibration parameters, but generally such measurement data is scarce; they are typically unmeasurable, otherwise calibration would not be necessary. In order to validate our method we use a unique structural modal frequency experiment designed to have easily measurable parameters, in which controlled variability (corresponding to uncertainty) can be explicitly introduced [15]. The variability which our method predicts can therefore be compared with the true known variability. Because we are dealing with a real experiment, both model discrepancy and experimental bias
will be present, which are not accounted for in the statistical model (2). The drawback is that the case is quite contrived. The calibration parameters correspond to the locations of structural components, which is not likely to be the dominant source of structural uncertainty in real cases. The experiment, and the criteria by which the calibration result shall be judged, are described in detail in Section 2.
1.3
Outline
The remainder of this paper is laid out as follows: in Section 2 the test-case is introduced, the experimental setup, computational model, and the Kriging surface serving as a surrogate for the computational model. This surrogate is essential for the efficiency of the parameter fitting procedure. In Section 3 a statistical model for the structural variability in the test-case is constructed, and algorithms for evaluating the model are described: probabilistic collocation (PC) for uncertainty quantification, and Markov-Chain Monte-Carlo (McMC) for sampling the Bayesian posterior. The results obtained using these techniques applied to the test-case data are studied in Section 4, and conclusions are briefly presented in Section 5.
2
Test-case for structural variability
In order to evaluate Bayesian model updating, we select a test-case on which it is possible to perform experiments with controlled structural variability. Such a test-case is provided by Haddad-Khodaparast et al. [15]. It consists of the metal frame structure depicted in Figure 1, with two internal vertical beams which may be positioned independently at one of three horizontal locations. The positions of these two beams are our structural uncertainties θ = (θ1 , θ2 ) where θ ∈ (0, 4) × (0, 4).
Figure 1: The metal-frame structure (left), the finite-element model of same (center), and parameterization of the vertical beam locations with θ1 and θ2 (right).
2.1
Experimental setup
In the experimental structure the beams can each be fixed at one of 3 positions θi = {1, 2, 3}, i ∈ {1, 2} using bolts. This gives 9 combinations of beam locations. Modal tests using an instrumented hammer were carried out for the frame in fixed-free conditions, and measurements of response frequencies for 6 basic
mode-shapes are then performed. The resulting frequency data is listed in Table 1. The explicitly introduced variability dominates variability from other sources — including re-assembly and measurement noise. A limitation of the experiment is that the beams can take only one of 3 positions. In order to simulate the situation where the beam locations vary continuously between θ = 1 and θ = 3, the experimental data is interpolated quadratically. What is more we want to simulate the situation where θ is a sample from some Θ, a random variable distributed with a known distribution. We specify the “true” RV Θtrue ∼ [U (1, 3), U (1, 3)] ,
(3)
i.e. uniformly distributed independently in each component. We generate 9 samples from (3), and evaluate the modal frequencies at these parameter values using the quadratic response. This interpolated experimental data will be used in the calibration, and the goal of the calibration is to identify the true distribution (3). This procedure and the resulting frequency data is intended to emulate a situation in which continuous manufacturing variability is present in the structure. The resulting frequency data is considered to represent the data that might be obtained when testing the response frequencies of a sample of nominally identical (but in fact varying) structures sampled at random from a larger population. In that case the values of θ in each test are unknown. In the use of this experimental data in the following analysis, we never use our knowledge about the values of θ at which the data was obtained - nor the distribution of the actual θ in (3). We attempt to deduce the latter using only the frequency information.
2.2
Model problem setup
In order to relate modal frequencies to structural paramaters, a physical model of the structure is needed. Complementarily to the experiments, simulations are performed using the MSC Nastran FE structural analysis software [34], and FE models of the frame - shown in Figure 1. Eight-noded solid elements (CHEXA) are used, and the bolted joint connections are modelled using rigid elements over an area three times greater than the cross-section of the bolts. For preliminary studies of simulation accuracy it is convenient to simulate the 9 θ-configurations investigated in the experiment. In addition another 16 configurations are simulated in order to provide sufficient sampling of the θ-space to build a response surface. For the purposes of assessing the accuracy of the simulation code, the data from both simulations and experiments is presented in Table 1 for the 9 θ-configurations studied experimentally. Results are in the form of frequencies associated with specific mode shapes. Six low-order mode shapes which are seen in all 9 cases have been selected, and other frequencies discarded - therefore the six frequencies listed are not necessarily the six lowest frequencies. For more complete modal data see [15]. Notable is the good agreement of the simulation and the experiment, with average relative errors less than 2%. This is attributable to the simple physics, and simple test-case. In order to perform UQ for uncertain inputs θ, it is necessary to simulate the frequency response for arbitrary values of θ in the feasible range (0, 4) × (0, 4). We interpolate the 25 FE results with a response surface, which also reduces the computational cost of evaluating the frequencies. The response surface methodology chosen is Gaussian process interpolation, also known as Kriging interpolation [36, 6]. Originating in the field of geostatistics, Kriging itself may be derived from Bayesian principles [37]. In this paper use use it only as an interpolation technique. A Kriging surface is constructed independently for each frequency with respect to the variables θ1 and θ2 . Quadratic regression and Gaussian correlation functions are used. The accuracy of the interpolation is assessed by comparing the surrogate with the original code at 3 new locations in the parameter space, see Table 2. The error due to the surrogate is in all cases below 2%, which is regarded as acceptable.
Experimental modal frequencies (Hz): (θ1 , θ2 ) (1,1) (1,2) (1,3) (2,1) (2,2) (2,3) (3,1) (3,2) (3,3)
Mode 1 22.54 23.78 23.33 24.31 24. 24.34 23.3 23.794 22.577
Mode 2 27.84 27.43 26.92 24.38 24.65 24.43 26.59 27.088 27.497
Mode 3 47.63 49.85 48.94 47.17 48.36 47.13 48.83 49.785 47.536
Mode 4 81.19 79.41 74.6 76.68 80.83 76.63 74.38 79.311 81.122
Mode 5 256.4 222.84 219.94 220.52 254.23 220.27 219.48 222.013 255.603
Mode 6 312.39 306.56 299.94 304.76 309.67 304.66 299.72 305.946 311.538
Mode 5 259.05 227.9 224.41 225.85 258.02 225.85 224.41 227.9 259.05
Mode 6 316.49 311.63 304.92 310. 314.89 310. 304.92 311.63 316.49
Simulation modal frequencies (Hz): (θ1 , θ2 ) (1,1) (1,2) (1,3) (2,1) (2,2) (2,3) (3,1) (3,2) (3,3)
Mode 1 22.59 23.97 23.52 24.53 24.25 24.53 23.52 23.97 22.59
Mode 2 27.27 26.97 26.4 24.25 24.49 24.25 26.4 26.97 27.27
Mode 3 48.14 50.47 49.43 47.77 48.93 47.77 49.43 50.47 48.14
Mode 4 80.89 79.65 74.73 76.69 80.93 76.69 74.73 79.65 80.89
Key: Nr. 1 2 3 4 5 6
Mode 1st in-plane bending 1st out-of-plane bending 1st torsion 2nd in-plane bending 2nd out-of-plane bending 2nd torsion
FE RMS error (Hz) 0.18 0.33 0.59 0.23 4.68 5.10
Relative error (%) 0.78 1.26 1.23 0.29 2.02 1.67
Table 1: Experimental and finite-element simulation modal frequency data for the frame structure. Only six modes are considered; higher-order in-plane bending modes observed in both data-sets are not used or tabulated here. The average discrepancy between simulation and experiment is given as the root-meansquared (RMS) error.
θ = (2.3, 2.0) FE model Kriging Error (%) θ = (2.3, 1.7) FE model Kriging Error (%) θ = (2.3, 0.4) FE model Kriging Error (%)
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6
24.20 24.36 -0.6
24.58 24.58 -0.0
49.21 49.36 -0.3
80.48 80.62 -0.2
248.29 249.06 -0.0
314.53 314.58 -0.0
24.47 24.65 -0.7
24.49 24.48 0.0
49.20 49.36 -0.3
79.68 79.70 -0.0
233.96 237.17 -1.4
313.89 313.84 0.0
23.37 23.66 -1.2
24.18 23.84 1.4
45.80 46.52 -1.6
70.74 71.08 -0.5
227.03 224.78 1.0
303.91 303.00 0.3
Table 2: Comparison of the Kriging response and simulation code output at 3 unsampled points.
3
Bayesian inference for parameter variability
Given the computational model and experimental data described in the previous section, the objective is to determine the true variability present in the structural parameters θ, responsible for the variability observed in the experimental frequency response (3). We construct a statistical model to relate the simulation and data, apply Bayes’ theorem and solve for the posterior numerically. The choice of the statistical model and the accuracy of the simulation code will determine the quality of the calibration result. The numerical method used for evaluating the posterior (assuming it’s converged) will influence only the efficiency of the method. In order to highlight the differences between our statistical model and standard models, we first discuss the model of Kennedy and O’Hagen [18].
3.1
Standard statistical model: Kennedy and O’Hagen
A statistical model that has become standard for calibration of computer models is that of Kennedy and O’Hagan [18]: d = m(θ) + δ + ,
(4)
δ ∼ N (m2 (·), c2 (·, ·)) ,
(5)
∼ N (0, λI),
(6)
δ⊥ ⊥ ,
(7)
where θ are the structural parameters, and
where δ is intended to capture discrepancy in the simulation m(·) (whose output is a prediction of d), and describes some unbiased error in the observation of d. This model implies that all of the information about d that is contained in the value of θ and the function m(·) may be summarized by the single function evaluation m(θ). This is a questionable assumption for several reasons, in particular: given that our simulation is imperfect, why should we contend that there exists a single θ which is sufficient for predicting d? See Goldstein et al. [13] for a more complete discussion of the consequences of this assumption. In any case, in our structural-uncertainty case the assumption is certainly not valid: if our multiple measurements of the 6 model frequencies are denoted d(i) , i ∈ {1, . . . , Nd }, where in this case Nd = 9, then each results from physical processes described by m(θ (i) ) for multiple distinct values θ (i) . Hence (4) is not appropriate. One straightforward path by which we might still employ (4) is to apply it individually to each d(i) , i ∈ {1, . . . , Nd }, i.e. perform Nd calibrations to obtain Nd posterior distributions for each θ (i) corresponding to d(i) . These could be used to obtain sample statistics for the distribution of θ. However this approach has several drawbacks: the number of parameters to estimate is large; we are unable to exploit the fact that in all calibrations the model discrepancy will be the same; we are unable to exploit the fact that due to cancellation of errors simulations can typically predict differences (so-called deltas) much more accurately than absolute values; and finally we cannot take advantage of any prior knowledge of the form of the aleatory uncertainty in θ.
3.2
Generic two-level statistical model
Motivated by the discussion above, we propose a single statistical model predicting all the observed data in one-shot. We incorporating the knowledge that the data is randomly distributed due to random structural parameters, and we attempt to identify directly hyperparameters φ of the distributions of these parameters.
Our statistical model is d = m(Θ(φ)) + ,
(8)
Θ ∼ A(φ)
(9)
∼ N (0, λI),
(10)
where θ in (4) has been replaced by the random variable Θ(φ), which is assumed to be distributed according to a family of distributions A parameterized by φ. For example A could be a multivariate normal distribution parameterized by mean and covariance matrix; or be defined over a polynomial chaos expansion parameterized by its coefficients, following [12]. The goal is to estimate φ. To evaluate the likelihood ρ(d|φ) under model (8) it is necessary to compute the distribution of m(Θ), i.e. an uncertainty quantification operation that is not necessary under model (4). This substantially increases the computational cost of the calibration. As in the KOH model it is in principle desirable to account for the model discrepancy by adding a δ-like term in (8). However in practice – and in our case in particular – little is known about the form of the model discrepancy. In the absence of specific prior information, modelling δ requires a large number of parameters, and a large amount of data to identify. Therefore we neglect modelling discrepancy in the following, and require that the simulation is sufficiently accurate. In order to reduce computational cost, and reduce the sensitivity of the calibration to model discrepancy, we can optionally approximate m(Θ) by a random variable M distributed according to a multivariate normal distribution with the same mean and covariance as m(Θ): M ∼ N (µm , Σm ),
(11)
µm := E [m(Θ)] , Σm
(12)
:= Cov [m(Θ)] = E (m(Θ) − µm )(m(Θ) − µm )T .
(13)
and the statistical model becomes: d = M + .
(14)
The primary motivation for this approximation is that we do not want to require the simulation to be able to reproduce all fine details of the true response. This is unlikely to be possible for simulations of complex structures. It is sufficient if the model is able to reproduce the overall variability in the response due to parameter variation, something that models tend to be better at than predicting absolute values. This choice of a Gaussian approximation also has the effect of making the method much more robust to modelling discrepancy and measurement noise, than if e.g. a kernel density estimate were used.
3.3
Statistical model for the artificial test-case
In this section we specialise the statistical model described in Section 3.2 to the structural test-case. We use the knowledge that individual θ are sampled from a uniform distribution, to choose the family of A: d = m(Θ1 (φ), Θ2 (φ)) + , Θ1 ∼ Θ2 ∼
+ U (φ− 1 , φ1 ) + U (φ− 2 , φ2 )
∼ N (0, λI).
(15) (16) (17) (18)
where the vector of hyperparameters is defined as + − + φ := [φ− 1 , φ1 , φ2 , φ2 ],
(19)
so the true distribution of Θ in (3) corresponds to φtrue := [1, 3, 1, 3].
(20)
In a more realistic test-case one might suppose that the wear in the structure might be subject to random processes resulting in a normal distribution by the central limit theorem. Then the hyperparameters are the mean and covariance of that normal distribution. As in the generic model we approximate m(Θ) by a multivariate normal random variable.
3.4
Evaluation of the posterior
Given the statistical model, the aim is to identify the posterior ρφ (φ|d), which we consider the definitive solution of the inverse problem. It is identified by application of Bayes’ theorem ρA,B (a ∩ b) = ρA (a|B = b)ρB (b) = ρB (b|A = a)ρA (a),
(21)
to the statistical model (14). In this notation ρA (a) is the probability that the continuous random variable A takes the value a. Further ρA,B (a ∩ b) is the probability that A takes the value a and B takes the value b. Finally ρA (a|B = b) is the probability that A takes the value a given that B takes the value b. If A and B are independent then ρA (a|B = b) = ρA (a), and ρA,B (a ∩ b) = ρA (a)ρA (b). By applying Bayes’ theorem recursively we obtain: ρΦ (φ|d) ∝ ρ(d|Φ = φ)ρΦ (φ), ∝ ρ(d|Φ = ∝ ρ(d|Φ =
(22)
− − + − φ)ρΦ (φ+ 1 ∩ φ2 |Φ1 = φ1 ∩ Φ2 + − − − φ)ρΦ (φ+ 1 |Φ1 = φ1 )ρΦ (φ2 |Φ2
= =
− − φ− 2 )ρΦ (φ1 ∩ φ2 ) − − φ− 2 )ρΦ (φ1 )ρΦ (φ2 )
(23) (24)
where constants of proportionality are neglected. The likelihood ρ(d|Φ = φ) has the form: ρ(d|Φ = φ) ∝
∝
Nd Y i=1 Nd Y i=1
ρ(d(i) |Φ = φ)
(25)
1 (i) T −1 (i) exp − (d − µm (φ)) Σ(φ) (d − µm (φ)) , 2
(26)
where d(i) refers to the observed values of all modal frequencies of the ith observed structure, and Σ(φ) := Σm (φ) + λI,
(27)
has a contribution from the experimental noise. For the remaining terms in (23) it is equally easy to write down explicit expressions given their definitions in the previous sections. An explicit expression for the pdf of the posterior is thereby obtained. In order to evaluate the likelihood (25) we must evaluate µm (φ) and Σm (φ), from the deterministic model m(θ), and a distribution Θ(φ). This is an uncertainty quantification (UQ) problem on the Kriging surface for the modal frequencies. Given that the UQ must be performed a large number of times when fitting the model, it must be an efficient and reliable procedure. Experience shows the probabilistic collocation (PC) method [23, 1] well suited to this role. We extract information from ρφ (φ|d) in two ways: firstly an optimization algorithm is used to identify the MAP estimator: φMAP := argmax ρφ (φ|d). (28) φ
The Nelder-Mead Simplex method [33] is used, and typically attains a local optimum within ∼ 100 evaluations of the posterior. Secondly samples are generated from the posterior with Markov-Chain Monte-Carlo (McMC). In the following we use Metropolis-Hastings McMC [16], with 105 samples, plus a burn-in of 104 samples.
4
Model updating for the metal-frame: Results
We apply the statistical model (14) to the metal frame structure with uncertain beam locations. The objective is to deduce the distribution of Θtrue in (3), from which the observed data was generated. First we construct priors for the hyperparameters, then in Section 4.2 use UQ to determine the variability in numerical results for a given Θ. In Section 4.3 we estimate Θ using Bayesian updating and compare with Θtrue .
4.1
Priors on the hyperparameters
Under the Bayesian framework we must specify a prior distribution for φ, which captures our a priori knowledge regarding the structural variability. The available knowledge will depend strongly on the situation. Here we assume to have no prior knowledge, and therefore attempt to specify an uninformative prior. For this somewhat unrealistic test-case we select a prior stating that all possible uniform distributions with end-points in the interval [0, 4] are equally likely. This is achieved with a joint prior: Φ− 1 ∼ T, Φ+ 1 Φ− 2 Φ+ 2
∼
(29)
U (φ− 1 , 4),
∼ T, ∼
(30) (31)
U (φ− 2 , 4).
(32)
where T has pdf ( −
ρT (φ ) =
4−φ− 8
0
if 0 ≤ φ− ≤ 4 otherwise.
+ These two univariate distributions for Φ− 1 and Φ1 , form a bivariate distribution with the joint pdf ( + − 1 if 0 ≤ φ− + 1 ≤ 4 and φ1 ≤ φ1 ≤ 4 8 ) = ρφ (φ− , φ 1 1 0 otherwise. + which is taken as the prior. The same holds for Φ− 2 and Φ2 .
4.2
Uncertainty quantification with specified input pdfs
Before applying the Bayesian hyper-parameter estimation procedure, we examine the variability present in the frequency data from the experiment and the simulation - given specified parameter uncertainty. In particular we let φ = φdefault := [0.5, 3.5, 0.5, 3.5]. The response variation is visualised in the three plots in the left-hand column of Figure 2. Each plot displays two of the six modal frequencies (in Hertz) on the two axes. The observed experimental data [d1 , . . . , d6 ] is plotted as a scatter of cyan squares in each graph, and corresponds to the data given in Table 1. The gray + − + crosses display the variability in the simulation frequency response for θ ∈ I(φ) = [φ− 1 , φ1 ] × [φ2 , φ2 ] - and are obtained by sampling the region I(φ) on a uniform grid, and obtaining the frequency for each sample from the Kriging surrogate for the simulation. Finally the red dot and three ellipses are contours of the multivariable normal pdf, with mean and covariance given by probabilistic collocation uncertainty quantification on the simulation. The dot is the mean, and the contours represent 1σ, 2σ and 3σ deviations from the mean. This is therefore an image for our model of M from (14). In this artificial test-case the match between the Gaussian and the predictions looks quite poor because the θ are uniformly distributed, and therefore the distributions of fi are not close to Gaussians. For models with Gaussian θ and more parameters (so that the mean-value theorem applies) the match is better.
In the left-hand column Figure 2 the apparent good agreement of experiment and simulation for fi , i = {1, . . . , 4}, and disagreement for f5 and f6 is visible. Since our statistical model currently has no provision for model discrepancy these latter data may distort the calibration, and therefore are neglected in the following.
4.3
Bayesian determination of input pdfs
The spread of simulation results in the previous section is much larger than that of the experiment, suggesting that φdefault is an over-estimate of the variability in θ. In order to obtain a better estimate the Bayesian parameter fitting procedure described in the previous section is applied. Metropolis-Hastings is used to sample the posterior, which is visualising by plotting the set of all 1d and 2d marginals, see Figure 3, from which the locations and extent of regions of high probability are immediately evident. Visible is a single major peak in each 2d marginal, corresponding to a sharply identified pair of hyperparameters. This peak is close to φ = [1, 3, 1, 3], thus the procedure retrieves substancial information about the true distribution + − + of Θ. Much smaller secondary peaks produce an inverted “L”-like shape in both the [φ− 1 , φ1 ] and [φ2 , φ2 ] axes, suggesting that the experimental data can be approximately reproduced with a smaller variation in e.g. φ1 . However their probability is substantially less than that of the main peak - leaving no ambiguity regarding the best fit. The corresponding MAP estimator is φMAP = [0.9230, 3.0770, 0.9360, 3.0639], where the starting point for the Nelder-Mead Simplex optimization method was chosen with the help of the histogram. The result is reasonable with respect to the reference point [1, 3, 1, 3], and slightly more conservative - the intervals for θ are slightly over-estimated on both ends. In order to see the effect of this calibrated value of φ, the resulting variability in frequency can be plotted against the experimental data, as in Section 4.2. In the right-hand column of Figure 2 the effect of variability (b) given by φMAP is displayed. The extent of the model domain (area of grey crosses) is substantially reduced compared to Section 4, but all experimental data remains inside that domain (at least for the first 4 frequencies). Also the data remains reasonably probable according to the multi-variate normal distribution model for the data. The estimated variability in θ, given by φMAP , seems to capture roughly the variability seen in the data - as expected.
5
Conclusions and further work
A two-level statistical model for predicting variability in populations of structures has been developed and applied to an experimental test-case with controlled structural variability. In doing so Bayesian model updating for structural dynamics problems has been shown to be an effective tool, not suffering from requirements of large amounts of data or high computational cost, as has been previously suggested. The first flaw has been overcome with an appropriate statistical model, the second with the use of modern response surface and uncertainty quantification techniques. We were able to deduce the explicitly added variability in structure correctly from modal frequency data, validating the statistical and numerical approach applied here. This approach may now be applied to structures with truly unknown variability with some confidence, and this is the subject of future work. Once identified the structural variability information can be used to predict structural reliability and other quantities of interest. However, two issues must be tackled to make this technique viable for realistic airframes. Such structures are much more geometrically and physically complex than the case considered here, with many hundreds of important uncertain parameters. The physical complexity will lead to greater model discrepancy, and this
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Figure 2: Initial and updated spaces of predicted frequency response. Cyan squares: experimental data, gray crosses: range of output of model under variation of θ, red dot: model mean, red ellipses: model 1, 2, and 3 standard-deviations from the mean. Left column: φ = φdefault , corresponding to Θ1 ∼ U (0.5, 3.5), (b) Θ2 ∼ U (0.5, 3.5). Right column: φ = φMAP , corresponding to Θ1 ∼ U (0.9230, 3.0770), Θ2 ∼ U (0.9360, 3.0639).
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Figure 3: Marginal distributions of the posterior ρ(φ|d). Plots on the diagonal are 1d-marginals of the indicated variable, off-diagonal plots are 2d-marginals of the corresponding variables. The point φ = (1, 3, 1, 3) is marked as a red circle. may require including discrepancy terms in the statistical model [18]. It remains to be seen how effective such error modelling can be. Furthermore, the procedure as described does not scale well to large numbers of parameters. The limiting factor is the Kriging response surface and probabilistic collocation UQ, which work well only for dimension . 10; the fundamental issue is the curse of dimensionality. A possible remedy is the use of response surfaces incorporating adjoint gradients, and gradient-enhanced Kriging is being examined for this purpose. In general the ability to determine input distributions on unknown parameters, as demonstrated here, is essential to the success of parametric uncertainty quantification, which itself is essential to the analysis and control of errors in numerical models. At present there is a surfeit of UQ methods, but a dearth of input data. It is hope that this paper will contribute to rebalancing the situation.
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