the sensitivity method in stochastic model updating

The Sensitivity Method in StochasticModel Updating

John E. Mottershead, Michael Link, Tiago A.N. Silva, Yves Goversand Hamed Haddad Khodaparast

Abstract Probabilistic and interval model updating methods are described, withparticular attention paid to variability in nominally identical test structures due, forexample, to the effect of accumulated manufacturing tolerances, or degradation ofperformance caused by wear of engineering components. In such cases the updatingparameter distributions are meaningful physically either as PDFs or as intervals.Stochastic model updating is an inverse problem, generally requiring multipleforward solutions, which may be carried out very efficiently by the use of surro-gates, in place of full FE models. The procedure is illustrated by experimentalexamples, including model updating of (i) a frame structure with uncertain locationsof two internal beams and (ii) the DLR AIRMOD structure, which displaysvibration characteristics very similar to those of a real aircraft.

Keywords Model updating � Sensitivity method � Probabilistic � Interval

1 Introduction

Deterministic model updating of finite element models [1–4] has become a maturetechnology. It is a classical inverse problem in the sense that a measurable output isused to correct a set of analytical parameters that are themselves inaccessible to

J.E. Mottershead (&)Centre for Engineering Dynamics, University of Liverpool, Liverpool L69 3GH, UKe-mail: [email protected]

M. LinkInstitute for Statics and Dynamics, University of Kassel, 34109 Kassel, Germany

T.A.N. SilvaLAETA, IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

Y. GoversGerman Aerospace Centre (DLR), Institute of Aeroelasticity, 37073 Göttingen, Germany

H.H. KhodaparastCollege of Engineering, University of Swansea, Swansea SA2 8PP, UK

© Springer International Publishing Switzerland 2015J.K. Sinha (ed.), Vibration Engineering and Technology of Machinery,Mechanisms and Machine Science 23, DOI 10.1007/978-3-319-09918-7_5

65

measurement. As with most inverse problems the resulting system of equations areill-posed and require regularisation, discussed in full by Ahmadian et al. [5].

Variability in performance has become one of the major challenges facingresearchers and engineering scientists concerned with the robust reliability of engi-neering systems. This has resulted in the emergence of powerful probabilistic andnon-probabilistic (interval, fuzzy etc.) methods combined with modern computersystems and codes. An aspect of uncertainty of particular interest is the inherentvariability that arises from the effect of accumulated manufacturing tolerances orfrom wear so that nominally identical engineering systems behave differently.

The problem of variability in the dynamics of nominally identical test structureswas addressed by Mares et al. [6, 7] using a multivariate gradient-regressionapproach combined with a minimum variance estimator. Haddad Khodaparast et al.[8] developed a first-order perturbation approach whereas the method of Hua et al.[9] required the computation of second-order sensitivities. Govers and Link [10]defined an objective function for the identification of updating-parameter covari-ances with forward propagation of parameters at each iteration. This approach wasmodified by Rui et al. [11] who used a polynomial chaos expansion (PCE) as asurrogate for the full finite element model. Fang et al. [12] used the analysis ofvariance (ANOVA) for the selection of updating parameters together with Monte-Carlo simulation (MSC). Adhikari and Friswell [13] used a sensitivity approach andthe Karhunen-Loève expansion to represent distributed parameters.

Non-probabilistic model updating methods were developed by HaddadKhodaparast et al. [14] and Erdogan and Bakir [15] using interval and fuzzy-arithmetic-based procedures. In the former, a Kriging predictor is used as a sur-rogate and, in the latter, the membership functions of each updating parameter isdetermined by minimising an objective function using a genetic algorithm.

This paper gives a brief overview of probabilistic and interval updating methodsbased on first-order output sensitivities. This is followed by experimental exampleproblems designed to illustrate the application of stochastic model updatingmethods, including the updating of geometrical configuration, made possible by theuse of a trained surrogate model. Finally the example of probabilistic and intervalupdating of the DLR AIRMOD structure is presented using modal data obtained bymultiple disassembly and reassembly of bolted joints—resulting in variability dueprincipally to slight differences in joint tightness in each of the separate tests.

2 Deterministic Model Updating

The deterministic model updating problem is usually cast as,

zm � zj� � ¼ Sj hjþ1 � hj

� � ð1Þ

where typically zm is a vector of measured natural frequencies and mode-shapeterms, zj is the corresponding vector of predictions determined from an analytical

66 J.E. Mottershead et al.

model with parameters hj and j denotes the iteration index. The updated vector ofparameters is then given by,

hjþ1 ¼ hj þ Tj zm � zj

� � ð2Þ

where the transformation matrix Tj is generally the weighted pseudo inverse of thesensitivity matrix Sj.

In the following analysis we consider a ‘cloud’ of data points in multi-dimensionalspace so that each point represents a separate test. There is a corresponding cloud ofpredicted points and the objective is to adjust the statistics of the parameters so that theposition and orientation of the cloud, determined according to the mean values andcovariances, or alternatively the upper and lower bounds, of the predictions, ismade toagree with the data cloud.

3 Probabilistic Model Updating

The probabilistic model updating approach proceeds by iterative correction toparameter means and covariances. An assumption of statistical normality is there-fore implied. However, this is a mathematical constraint rather than a practicaldrawback, because structural measurements are likely to be limited to mean valuesand standard deviations (or upper and lower bounds as addressed by intervalupdating, described in Sect. 4). Equivalently to Eq. (2) the stochastic modelupdating equation may be written as,

�hjþ1 þ Dhjþ1 ¼ �hj þ Dhj þ �Tj þ DTj� �

�zm þ Dzm � �zj � Dzj� � ð3Þ

where the overbar and D denote the mean and variability on the mean respectively.Then by separating the zeroth-order and first-order terms, Khodaparast et al. [8]obtained the following two expressions:

O D0� �

: �hjþ1 ¼ �hj þ �Tj �zm � �zj

� � ð4Þ

O D1� �: Dhjþ1 ¼ Dhj þ �Tj Dz

m � Dzj� �þ DTj �z

m � �zj� � ð5Þ

Equation (4) is the updating equation for the parameter mean values, exactly thesame as Eq. (12) given by Govers and Link [10].

Using assumptions of (i) small parameter variability, (ii) neglecting the thirdright-hand-side term of Eq. (5) and (iii) statistical independence of the updatedparameters Dhj (and hence the predictions Dzj) from the measurements Dzm, Silvaet al. [16] showed three previously developed expressions for the updated covari-ance Cov Dhjþ1; Dhjþ1

� �to be equivalent. The first of the three expressions, due to

Khodaparast et al. [8] is expressed as,

The Sensitivity Method… 67

Cov Dhjþ1; Dhjþ1� � ¼Cov Dhj; Dhj

� �� Cov Dhj; Dzj� �

�TTj

� �TjCov Dzj; Dhj� �þ �TjCov Dzj; Dzj

� ��TTj þ �TjCov Dzm; Dzmð Þ�TT

j

ð6Þ

The second expression, developed by Govers and Link [10] using the Frobeniusnorm for the minimisation of the difference between measured and analytical outputcovariances, is given by,

Cov Dhjþ1; Dhjþ1� � ¼Cov Dhj; Dhj

� �þ �TjCov Dzm; Dzmð Þ�TTj

� �TjCov Dzj; Dzj� �

�TTj

ð7Þ

Forward propagation is required to determine Cov Dzj; Dzj� �

in both Eqs. (6) and(7), and Cov Dhj; Dzj

� �and Cov Dzj;Dhj

� �in Eq. (6). This is achieved by application

of the relationships,

Cov Dzj; Dzj� � ¼ �SjCov Dhj; Dhj

� ��STj ð8Þ

Cov Dhj; Dzj� � ¼ Cov Dhj; Dhj

� ��STj ð9Þ

Cov Dzj;Dhj� � ¼ �SjCov Dhj; Dhj

� � ð10Þ

However, direct substitution of Eqs. (8), (9) and (10) into either Eqs. (6) or (7)leads immediately to the third expression, given for example by Hart [17], page 47(for the general case of linearly related sets of random variables),

Cov Dhjþ1; Dhjþ1� � ¼ �TjCov Dzm; Dzmð Þ�TT

j ð11Þ

where it is seen that the updated parameter covariance matrix is computed onceusing the final transformation matrix (assuming �Tj ¼ T �hj

� �) from the converged

solution of the means. There is no need for forward propagation.

4 Interval Model Updating

Interval model updating is carried out by updating a large number of deterministicmodels. This would be prohibitively expensive were the full FE model to be usedand it is for this reason that a meta-model is used to provide a highly efficientmapping between the input parameters and the output measurements. Noassumption is placed on the probability distribution which is described only in


terms of upper and lower bounds on the updating parameters. The interval modelupdating problem is formally described as,

~zm � ~zj ¼ ~Sj ~hjþ1 � ~hj� � ð12Þ

where ~�ð Þ represents an interval vector or matrix. An advantage of the meta-modelapproach, developed by Haddad Khodaparast et al. [14] is that it avoids the use ofinterval arithmetic, which is known to be unduly conservative. The accuracy of thetechnique depends on the type of meta-model, the sampling used and the behaviourof the outputs within the range of parameter variation. The Kriging meta-model wasused due to its excellent performance in the prediction of FE model behaviour(particularly with non-smooth behaviour) and also the provision of a probabilisticinterpretation of the residual between the outputs of the FE model and the surrogateitself. The interval model updating procedure is illustrated in Fig. 1. Firstly, themean values of updating parameters are found by deterministic model updatingusing the mean values of measured data. If the solution is unique, the vector ofupdated parameter mean values can be represented by a point in the parameterspace. In the next step, an initial hypercube around this point is constructed and themeta-model (the Kriging predictor in the present case) is then used to map the spaceof the initial hypercube of updating parameters to the space of outputs. Care shouldbe taken to make sure that the mapping is good enough to represent the relationshipbetween inputs and outputs with sufficient accuracy. This can be achieved byincreasing the number of training samples until the Mean Squared Error (MSE) atan unsampled point falls below a threshold, when the Kriging model is deemed tobe accurate enough. Equation 6 may then be solved by taking all the measured data,represented by circles in Fig. 1, and mapping them in the reverse direction (usingthe Kriging predictor) to find the vertices of the updated parameter hypercube. The

Fig. 1 Interval model updating using the Kriging predictor (Reprinted from [14] with permissionfrom Elsevier)


need for a highly computationally efficient meta-model is illustrated by the fact thata large number of inversions are required for interval model updating. Furtherdetails on the mathematical development of the method are provided by HaddadKhodaparast et al. [14].

5 Experimental Examples

Stochastic model updating procedures and their effectiveness are illustrated by twoexperimental examples, both of which are designed to reproduce the conditions thatwould be experienced in the case of multiple nominally-identical structures each withvariability arising from, for example, manufacturing tolerances or degradation due towear. The objective is to estimate the spread of uncertainty in parameters deemed tobe responsible for observed variability in dynamic behaviour from modal tests.

In the first example, a frame structure is tested nine times to determine thenatural frequencies. In each of the nine tests, two internal beams were placed indifferent configurations and intervals were determined on the extreme positions ofthe two beams. The use of a meta-model enabled the parameterization of beamlocations, which could not have been done by conventional model updating withoutre-meshing of the FE model at each iteration.

The second example is the DLR AIRMOD structure, a replica of the GARTEURSM-AG19 benchmark described by Balmes [18]. Probabilistic and interval modelupdating was carried out using data selected from 130 disassemblies and reas-semblies of the structure with a modal test carried out at each stage. Details of thetest procedure, carried out at DLR Göttingen, can be found in [19, 20].

5.1 Frame Structure with Internal Beams

The two internal beams of the test structure shown in Fig. 2 can each be placedindependently in 3 locations (1, 2, 3 in Fig. 2b), making a total of nine differentconfigurations. In each case natural frequencies were determined in modal testsdescribed by Haddad Khodaparast [14]. The FE model, shown in Fig. 2c, wasconstructed using 8-noded CHEXA elements and a preliminary deterministic modelupdating exercise was completed in order correct the FE boundary conditions.A Kriging model was then constructed based on a CCD design using the 9 samplescorresponding to the 9 different beam positions. This model was deemed suffi-ciently accurate according to a tolerance placed on the mean square error (MSE) atbeam locations (θ1 and θ2 shown in Fig. 2b) away from the 9 training points.Interval model updating was then carried out using the continuous mapping fromupdating parameters, θ1 and θ2, to natural frequencies provided by the mean pre-diction of the trained Kriging model.

Six natural frequencies were used for interval updating, these being the first andsecond in-plane bending, first and second out-of-plane bending and the first and


Fig. 2 Test structure and FE model (Reprinted from [14] with permission from Elsevier). a Teststructure, b Beam locations, c FE Model

Table 1 True, initial and updated beam locations

True parameters Initial parameter error%

Updated parameter error%

θ1 θ2 θ1 θ2 θ1 θ21.0 1.0 60 60 3.7 2.0

1.0 2.0 60 20 −0.2 7.6

1.0 3.0 60 −20 0.2 2.8

2.0 1.0 −20 60 1.8 −9.8

2.0 2.0 20 20 6.5 −0.1

2.0 3.0 20 −20 −2.4 3.0

3.0 1.0 −20 60 −0.6 −11.0

3.0 2.0 −20 −20 −0.3 −8.4

3.0 3.0 −20 −20 −2.2 −0.6

Table 2 Measured, initial and updated frequency intervals (Hz)

Measured Initial FE Updated FE Initial FE error % Updated FE error %

1 22.5–24.3 21.6–24.6 22.6–24.6 [−4.1, 1.1] [0.1, 1.1]

2 24.4–27.8 23.7–35.5 23.9–27.5 [−3.0, 27.6] [−2.1, −1.3]

3 47.1–49.9 43.7–67.6 45.1–50.6 [−7.2, 35.6] [−4.2, 1.4]

4 74.4–81.1 71.1–82.5 74.0–81.4 [−4.4, 1.6] [−0.5, 0.2]

5 219.5–256.4 224.1–267.3 224.1–259.5 [2.1, 4.3] [2.1, 1.2]

6 299.7–312.4 300.3–339.7 303.6–317.2 [0.2, 8.7] [1.3, 1.5]


second twisting modes. Interval updating is carried out in practice by a series ofdeterministic updates using the measured natural frequencies at each 9 differentbeam configurations. The true, initial and updated parameters are listed in Table 1whereas Table 2 shows the measured, initial and updated natural frequencies. It isseen that the maximum error of 35.6 % in the natural frequencies is reduced to4.2 %. A sample showing the initial and updated spaces in the plane of the first andsecond natural frequency is shown in Fig. 3 (black dots denote measurements).

Fig. 4 AIRMOD structure and FE model. a Structure, b FE Model (Reprinted from [21] withpermission from Elsevier)

Fig. 3 Space of first and second natural frequencies (Reprinted from [14] with permission fromElsevier). a Initial space, b Updated space

VTP/HTP

(a) (b) (c) (d)

VTP/Fuselage Wing/Fuselage Winglet

Fig. 5 Bolted joints. a VTP/HTP, b VTP/Fuselage, c Wing/Fuselage, d Winglet (Reprinted from[21] with permission from Elsevier)


5.2 DLR AIRMOD Structure

The AIRMOD structure, shown in Fig. 4 is made of aluminium and consists of sixbeam-like components connected by five bolted joints, illustrated in Fig. 5. The wingspan of AIRMOD is 2.0 m, the length of the fuselage is 1.5 m and the height is

Table 3 Mean value updating results (Italics rows active modes, Non-italics rows passive modes)

ftest/Hz fFE_ini/Hz ΔfFE_ini/% ΔfFE_upd/%

RBM Yaw 0.23 0.17 – 0.01

RBM Roll 0.65 0.56 – –0.01

RBM pitch 0.83 0.82 –1.00 –0.00

RBM Heave 2.17 2.14 –1.62 –0.02

2nWingBending 5.50 5.65 2.78 0.40

3nWingBending 14.91 15.11 1.35 –0.01

WingTorsionAnti 31.96 33.31 4.25 0.25

WingTorsionSym 32.33 33.62 3.98 –0.42

VtpBending 34.38 35.39 2.94 1.14

4nWingBending 43.89 44.66 1.77 –0.08

1nWingForeAft 46.71 47.21 1.08 0.02

2nWingForeAft 51.88 52.91 1.99 0.05

5nWingBending 58.59 60.59 3.43 1.29

VtpTorsion 65.93 67.69 2.67 –0.05

2nFuseLat 100.05 102.59 2.55 2.07

2nVtpBending 124.56 128.62 3.26 1.48

6nWingBending 129.38 132.08 2.08 –0.14

7nWingBending 141.47 145.91 3.14 0.76

2nHtpBending 205.59 206.73 0.56 –0.04

HtpForeAft 219.07 225.73 3.04 0.11

WingBendingRight 254.73 261.53 2.67 –0.02

WingBendingLeft 255.02 262.64 2.99 0.32

3nWingForeAft 272.08 278.71 2.44 1.44

WingletBendingLeft 303.96 320.15 5.33 2.24

WingletBendingRight 304.32 321.64 5.69 2.35

3nFuseLat 313.68 324.12 3.33 2.39

WingTorsionSym2 328.55 336.31 2.36 0.60

WingTorsionAnti2 331.18 341.15 3.01 0.71

4nWingForeAft 336.21 343.55 2.18 –0.15

2nFuseVert 348.68 359.54 3.12 1.77


0.46 m. The total weight of the structure is 44 kg. Two additional masses of 167grams each are installed at the forward tips of the winglets to ensure better excitationof the wing torsion modes. The complete FE model shown in Fig. 4b, including themain structure, cables, sensors and bungee cords consists of 1440 CHEXA, 6CPENTA and 561 CELAS1, 55 CMASS1, 18 CONM2 and 3 CROD elements. Thestructural behaviour of AIRMOD shows similarities to the dynamical characteristicsof a real aircraft, including mode shapes that show either symmetric or antisymmetricdeformations and closely space modes are found at 32, 255 and 330 Hz.

Modal testing of the AIRMOD structure and assessment of sources of uncer-tainty is discussed in detail in a series of papers by Govers and Link [10, 19, 20].Model updating using both probabilistic and interval methods is described in full byGovers et al. [21]. The means of the parameters were estimated by deterministicmodel updating of 18 parameters using both the eigenvalue and mode-shaperesiduals of the 14 active modes shown in Table 3. It is seen from in the table thatthe mean frequency values of the updated model match perfectly with the test datafor the active mode set. Also, the passive modes (not included in the residual) aresignificantly improved, thereby providing validation of the updated FE model.

The parameter covariances and intervals were estimated using the natural fre-quencies of the same 14 modes (the mode shapes were not included) and results areshown in Fig. 6. Three standard deviations are shown in the case of covarianceupdating.

Updated covariance and interval results for the 14 natural frequencies are pre-sented in Fig. 7. The size and orientation of the 105 frequency clouds show a verygood agreement between test (red) and updated analysis clouds (green) for the activerange. The interval-updated regions (black) are generally greater than those calcu-lated by the covariance method. This is to be expected because the number of termsthat define the multi-variate probability distributions on the updating parameters (themeans and fully populated covariance matrix) is greater than the set of upper- andlower-bounds that define the interval model. The probabilistic method starts with thediagonal covariance matrix at the first iteration (assuming uncorrelated updatedparameters), but becomes a fully populated matrix after convergence. The interval

Fig. 6 Updated parameter intervals: Changes with respect to the initial finite element model(Reprinted from [21] with permission from Elsevier)


Fig. 7 Frequency distributions from updated models (Reprinted from [21] with permission fromElsevier)


methodmakes no assumption about the correlation of the updating parameters, whichare therefore confined to a hypercube, whereas the updated fully-populated covari-ance matrix defines a space of hyper-elliptic parameters. The hypercube creates agreater sample space than that of the hyper-ellipse. Another observation from Fig. 6 isthat the degree of overestimation of the interval results is not of the same order for allthe frequencies, as can be seen by comparing plane f8-f10 to plane f1-f2. One mayrelate this to the sensitivity of the natural frequencies with respect to the parameterswhich are located outside the hyper-ellipse but inside the hypercube. In other words,the greater the sensitivities of the frequencies to parameter variation, the higher willbe the degree of over-estimation seen in the results.

6 Conclusions

An overview of probabilistic and interval methods in finite element model updatingis presented. The study is focused on the problem of multiple nominally-identicaltest structures where the problem is to quantify the certainty in parameters deemedresponsible for observed variability in dynamic behavior. The overview isaccompanied by two experimental example problems designed to demonstrate theworking of the techniques involved.

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http://dx.doi.org/10.1016/j.ymssp.2014.06.003

the sensitivity method in stochastic model updating

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