Computers and Structures 120 (2013) 24–32

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Computers and Structures

journal homepage: www.elsevier .com/locate/compstruc

Stochastic structural dynamic analysis using Bayesian emulators

0045-7949/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compstruc.2013.01.013

⇑ Corresponding author. Tel.: +44 (0)1792 513177; fax: +44 (0)1792 295157.E-mail address: f.a.diazdelao@swansea.ac.uk (F.A. DiazDelaO).

F.A. DiazDelaO a,⇑, S. Adhikari a, E.I. Saavedra Flores a,b, M.I. Friswell a

a Civil and Computational Engineering Centre, College of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdomb Departamento de Ingeniería en Obras Civiles, Universidad de Santiago de Chile, Av. Ecuador 3659, Santiago, Chile

a r t i c l e i n f o

Article history:Received 7 August 2012Accepted 21 January 2013

Keywords:MetamodelGaussian process emulatorStructural dynamicsStochastic finite element methodHomogenisationComposite materials

a b s t r a c t

The probabilistic characterisation of a frequency response function can be very valuable for structuraldesign and control. Unfortunately, obtaining a sufficiently large sample for wide ranges of vibrationcan easily become unaffordable. A variety of modelling assumptions can dramatically add to the compu-tational cost: nonproportional damping, multiscale material properties, and high-resolution finite ele-ment analysis are some examples. This paper explores Bayesian emulators as surrogates for expensivefinite element models in structural dynamics. We demonstrate the effectiveness of the method by per-forming uncertainty analysis of the frequency response of a nonproportionally damped plate made ofa carbon fibre/epoxy composite material.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The consideration of uncertainties in numerical models to obtainprobabilistic descriptions of the vibration response is becomingmore desirable for industrial-scale finite element models. Verylarge finite element models [1,2] are used for complex engineeringdynamical systems such as helicopters, automobiles and aircraft.For mid and high-frequency applications, the calculation of re-sponse is necessary for a wide range of frequency. Larger size mod-els, the consideration of uncertainty and a wide frequency range ofinterest make stochastic structural dynamics particularly challeng-ing for computational methods. Stochastic finite element-basedmethods [3–9] have been proposed for uncertainty propagation instatic or low-frequency vibration problems. For dynamic problems,perturbation-based approaches utilising random eigenvalue prob-lems [10,11] have been used for the forced vibration response of lin-ear dynamical systems. These methods are, however, normallyrestricted to low levels of random variation in the parameters. Thisproblem can be avoided if a simulation-based approach is adopted.The disadvantage is that a direct Monte Carlo approach can be com-putationally expensive. In this paper we develop an efficient simu-lation approach based on Gaussian process emulators for thefrequency response function (FRF) of stochastic dynamical systems.

A computer code or implementation of a mathematical model,also known as a simulator [12], can be understood as a functiong : X! Rd whose domain is an input space X. A simulator is deter-ministic whenever the same input x 2X results in the same

output y = g(x). Deterministic simulators are widely used in sci-ence and engineering to represent and study intricate phenomena.However, complex models can be too computationally intensive, inwhich case it is common to employ less expensive approximations.In the literature, these approximations are known as surrogatemodels, metamodels, response surfaces, or auxiliary models,among others [13–20]. One such surrogate modelling strategy,known as Gaussian process emulation, is based on the analysisand design of computer experiments [21,22] and on the conceptsof Bayesian statistics. Using this approach, it is possible to obtaina statistical approximation to the output of a simulator after eval-uating a small number n of design points fxign

i¼1 in the input do-main X, hence reducing the required computer processing time.Broadly speaking, the emulation works by generating a small setof training runs fxi;gðxiÞgn

i¼1 that are treated as data used to up-date some prior beliefs about the simulator’s output. These beliefsare represented by a Gaussian stochastic process prior distribution.After conditioning on the training runs and updating the prior dis-tribution, the mean of the resulting posterior distribution approx-imates the output of the simulator at any untried input x 2X,whereas it reproduces the known output at each design point.Gaussian process emulators have been implemented in various sci-entific fields in order to alleviate the computational burden ofexpensive simulators with encouraging results. These fields in-clude climate prediction [23,24], environmental science [25], med-icine [26–28], deterministic structural dynamics [29], stochasticstructural dynamics [30], reservoir forecasting [31], hydrogeology[32], quality control [33], heat transfer [34], multi-scale analysis[35], stochastic finite elements [36], and domain decomposition[37] among many others.

F.A. DiazDelaO et al. / Computers and Structures 120 (2013) 24–32 25

In this paper we study the application of Gaussian process emu-lation to the frequency response of complex engineering dynamicalsystems in the presence of uncertainty. The choice of the finite ele-ment method is based on its wide range of applicability in differentkinds of problems such as static problems, modal analysis and sta-bility analysis [38]. This paper follows previous work on the emu-lation of a deterministic frequency response [29], that is, the FRFdepends only on the frequency level and not on any stochasticparameter. An earlier study of emulators in the context of stochas-tic structural dynamics can be found in [30], where the FRF of anaircraft model was emulated. In their study, the authors focus onan undamped FRF. Additionally, they mention that their choice ofthe uncertain parameters (and their distributions) is somewhatarbitrary. In this paper, we choose nonproportional damping asone of the sources of numerical cost. In addition to this, the prob-ability distribution of the random parameter results from anhomogenisation approach, which takes into account a realisticphysical description of the material under study. This work canalso be considered a refinement of the approach in [37]. In that pa-per, the design points upon which the emulator is built are not as-sumed to follow any particular probabilistic structure. In thispaper, the inputs follow a probability distribution determined bythe physical properties of the structure under study. It is thanksto that probabilistic nature that it is possible to sample from the in-put space in order to estimate statistical summaries of the output,despite the computational cost.

The paper is organised as follows: In Section 2, the motivationof this study is established. The FRF of a simple spring-mass systemis used as an introductory example. In Section 3, the general theorybehind Gaussian process emulators is briefly reviewed. Section 4discusses how to perform the analysis of the uncertainty propa-gated by a simulator using Gaussian process emulators. In Sec-tion 5, an expensive finite element simulator of the frequencyresponse of a plate is studied. Finally, Section 6 offers someconclusions.

2. Motivation for the present study

In the context of stochastic structural dynamics, let the relevantsimulator be represented by g : X�H! R, where X # [0,1) con-tains frequency levels x and H is the set of realisations of a ran-dom parameter h. The output of the simulator is the FRF of thesystem. Some authors [12] note that, even if h is a random param-eter, the distinction between stochastic and deterministic simula-tors is artificial: when implemented numerically, a set ofrealisations of h is a sequence of pseudo-random numbers. Thus,for a fixed seed that initialises the pseudo-random number gener-ator, such a sequence is deterministic. Suppose that a statistic ofthe output g(x,h), such as the mean, the variance, or a specific per-centile, is to be obtained. This task may easily become burden-some, since even obtaining a few instances of the desiredstatistic would require g(x,h) to be evaluated at each frequency le-vel repeatedly for a potentially large number, S, of realisations of h.

Consider the nonproportionally damped [39] spring-mass sys-tem shown in Fig. 1. Suppose for the time being that the simulatorof the corresponding FRF is computer intensive and that there is noparametric uncertainty. To obtain the FRF, we begin with the equa-tion of motion of a damped M-degree-of-freedom linear structuralsystem [40,41], given by

M€uðtÞ þ C _uðtÞ þ KuðtÞ ¼ fðtÞ ð1Þ

where fðtÞ 2 RM is the forcing vector, uðtÞ 2 RM is the response vec-tor and M, C; K 2 RM�M are respectively the mass, damping andstiffness matrices.

Eq. (1) can be expressed in terms of the excitation frequency le-vel x as

DðxÞ�uðxÞ ¼ fðxÞ ð2Þ

where �uðxÞ and fðxÞ are the Fourier transforms of u and f respec-tively. The complex symmetric matrix D(x), known as the dynamicstiffness matrix, is given by

DðxÞ ¼ �x2Mþ ixCþ K ð3Þ

where i is the unit imaginary number. If D(x) is non-singular, thenthe response vector is �uðxÞ ¼ DðxÞ�1�fðxÞ. D 2 CM�M , but often onlythe moduli of the response vector components are relevant in prac-tice. Hence the simulator of the FRF is

gðxÞ ¼ j½�x2Mþ ixCþ K��1�fðxÞj ð4Þ

The domain X of Eq. (4) is the set of possible values of the frequencylevel x. The cardinality of X (denoted by jXj) depends on the sizeand resolution of the frequency domain, e.g. 1 to 100 Hz with incre-ments of 1 Hz. Let the mass of each block be 0.8 kg, the stiffness ofeach spring be 1 N m�1, and the viscous damping constant associ-ated with each block be 0.2 N s m�1. It can be shown that the mass,stiffness and damping matrices of this simple system are

M ¼m 0 00 m 00 0 m

0B@1CA K ¼

2k �k 0�k 2k �k

0 �k 2k

0B@1CA C ¼

c 0 00 0 00 0 c

0B@1CAð5Þ

where m = 0.8 kg, k = 1 N m�1, and c = 0.2 N s m�1. Assume the forc-ing vector is f = [0,1,0]> N.

Suppose that we introduce uncertainty to the governing equa-tion of the dynamical system shown in Fig. 1. This would be justi-fied because uncertainty is practically unavoidable in the realisticdescription of most physical systems [42,43]. Randomness can beintroduced by uncertain material properties, uncertain boundaryconditions or unknown manufacturing tolerances. Suppose thatany of the matrices in Eq. (1) is a random matrix, that is, it includesa random parameter h 2H. Without loss of generality, if the stiff-ness matrix K is random, then Eq. (4) becomes

gðx; hÞ ¼ j½�x2Mþ ixCþ KðhÞ��1�fðxÞj ð6Þ

where

KðhÞ ¼2kðhÞ �kðhÞ 0�kðhÞ 2kðhÞ �kðhÞ

0 �kðhÞ 2kðhÞ

0B@1CA ð7Þ

The domain of Eq. (6) is X �H, where X is the set of admissible fre-quency levels and H is the set of realisations of the random param-eter h. As noted before, a fixed seed implies that the simulator in Eq.(6) is deterministic, as it produces the same (scalar) output giventhe same stream of pseudo-random numbers. For notational conve-nience, the seed does not appear explicitly as a parameter of thesimulator. Fig. 2(a) shows S = 150 realisations of the spring-massFRF for the third degree of freedom, where x 2X ={0,0.01, . . . ,2.99,3} and h is assumed to be uniformly distributedin H = [0.9,1.1]. Additional to the 150 realisations of g(x,h), themean FRF (the dotted line) for all x 2X is shown. The direct calcu-lation of the mean model output would involve 150 � 301 evalua-tions of g(x,h).

The metamodelling scheme explored in this paper is based onobserving that the input domain X �H offers the possibility ofsampling the design points from H, whilst fixing those in X. Thatis, if the output of the simulator g(x,h) is treated as a function of honly, then it is possible to build an emulator at every point in X

Fig. 1. Three-degree-of-freedom nonproportionally damped spring-mass system; m = 0.8 kg, k = 1 N m�1, c = 0.2 N s m�1.

0 0.5 1 1.5 2 2.5 3−40

−30

−20

−10

0

10

20

30

40

ω [Hz]

FRF

[dB

]

η(ω,θ) envelopeMean η(ω,θ)∀ωω* =1.809

Mean η(ω*,θ)∀θ

0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.110

12

14

16

18

20

22

24

26

28

30

θ ∼ U[0.9,1.1]

FRF

[dB

]

η(ω*,θ) envelope

Mean η(ω*,θ)∀θTraining points

Fig. 2. 150 Realisations of the frequency response function of the third degree of freedom of a spring-mass system, with x 2 X = {0,0.01, . . . ,2.99,3}, h � U[0.9,1.1]. (a)Realisations of g(x,h) and mean output; (b) realisations of g(x⁄,h) and mean output.

26 F.A. DiazDelaO et al. / Computers and Structures 120 (2013) 24–32

and approximate a statistic of interest using design points takenfrom H. If the number of selected design points per emulator is lessthan S, then the total number of evaluations of g(x,h) is reduced.Fig. 2(a) illustrates this idea. For x⁄ fixed, the mean of 150 realisa-tions of g(x⁄,h) is a point on the mean FRF curve. On the otherhand, Fig. 2(b) shows the same 150 realisations and their mean(the dotted straight line), this time with h as the abscissas. Forillustration purposes, consider choosing 6 design points from H(the diamonds). Using these training runs, a Gaussian process emu-lator could approximate the realisations of g(x⁄,h) and the mean ofthe resulting values would in turn be used to approximate themean output at x⁄. If the same procedure is followed for everyx 2X, then g(x,h) would be evaluated 6 � 301 times. This repre-sents only 4% of the original computational effort. Additionally, ifthe time taken to emulate the statistic for every point in X is rea-sonably short, there might be an improvement in the overall com-puter processing time required. In practice, the choice of thenumber of design points greatly depends on the smoothness ofthe function to be emulated and the computational resourcesavailable.

3. Brief overview of Gaussian process emulators

3.1. Basic definitions

Let g(x,h) be an expensive deterministic simulator, such thatthe cost of running it is affordable only at a limited number of in-put points. The uncertainty about the output of g(x,h) can be de-scribed through a probability distribution [44]. It is common toassume the following stochastic representation

gðx; hÞ ¼ hðx; hÞ>bþ Zðx; hÞ ð8Þwhere h(x,h) is a vector of known functions and b is a vector of un-known coefficients. Some authors [27,45] note that h(x,h) shouldbe chosen to reflect the available information about the functionalform of g(x,h). Other authors [13] consider that a constant mean(that is h = 1) is flexible enough to capture complex functional rela-tionships. In this paper we adopt this view. The function Z(x,h) is astochastic process with mean zero and correlation functionC[(x,h), (x,h0)]. Let y = [g(x1,h1), . . . ,g(xn,hn)]> be a vector of obser-vations corresponding to the design points fðxi; hiÞgn

i¼1. Theseobservations are used to update the prior distribution of the simu-lator. An analytically convenient choice for the prior is the followingGaussian process distribution [26]

gðx; hÞjb;r2 � Nðhðx; hÞ>b;r2C½ðx; hÞ; ðx; hÞ�Þ ð9Þwhere h(x,h) and b are defined as above. The correlation functionC[(x,h), (x,h0)] is such that

C½ðx; hÞ; ðx; h0Þ� ¼ exp �½ðx; hÞ � ðx; h0Þ�>B½ðx; hÞ � ðx; h0Þ�� �

ð10Þwhere B is a diagonal positive definite matrix. The diagonal of B is avector b 2 Rn of smoothness parameters. These parameters quantifythe rate at which the output varies as the input varies. In order toestimate them, a density function f(Bjy) can be derived and a max-imum likelihood estimator obtained [46]. The choice of a Gaussianprocess is made for much the same reasons that Gaussian distribu-tions are so frequently used in statistics: they are convenient, flex-ible and quite often realistic [47]. This does not imply thatuncertainty (v.g. geometric, material) in the physical system beingemulated is necessarily assumed to be Gaussian. However, the flex-

F.A. DiazDelaO et al. / Computers and Structures 120 (2013) 24–32 27

ibility of the Gaussian process allows the emulation of complex in-put/output relationships.

After conditioning on the training runs and updating the priordistribution (9), the mean of the resulting posterior distributionapproximates the output of g(x,h) at any untried input (x,h),and reproduces the known output at each design pointfðxi; hiÞgn

i¼1. The variance of the posterior distribution quantifiesthe uncertainty that arises from having only a limited number ofevaluations of g(x,h) [24]. Very conveniently, the posterior is alsoa Gaussian process distribution, obtained as follows: LetH = [h(x1,h1), . . . ,h(xn,hn)]>, and A 2 Rn�n such that Aij = C[(xi,hi),(xj,hj)] "i, j 2 {1, . . . ,n}. Then

yjb;r2 � NðHb;r2AÞ ð11Þ

To incorporate the information y and obtain the distribution ofg(x,h)jy, we use the following theorem, a demonstration of whichcan be found in [48].

Theorem 1. Letz 2 RN be a random vector such that z � Nðl;RÞ.Partition z as (z1,z2)>, where z1 2 RN�n and z2 2 Rn. Consequently,

partition l = (l1,l2)> and R ¼ R11 R12R21 R22

� �, so that E[zj] = lj and

Cov(zj,zk) = Rjk. Then, z1jz2 � Nðel; eRÞ, where el ¼ l1þR12R

�122 ðz2 � l2Þ and eR ¼ R11 � R12R

�122 R21.

It follows that

gðx; hÞjy; b;r2 � Nðm�ðx; hÞ;r2C�½ðx; hÞ; ðx; hÞ�Þ ð12Þ

where

m�ðx; hÞ ¼ hðx; hÞ>bþ tðx; hÞ>A�1ðy �HbÞ ð13Þ

C�½ðx; hÞ; ðx; h0Þ� ¼ C½ðx; hÞ; ðx; h0Þ� � tðx; hÞ>A�1tðx; h0Þ ð14Þ

tðx; hÞ ¼ ½C½ðx; hÞ; ðx1; h1Þ�; . . . ;C½ðx; hÞ; ðxn; hnÞ��> ð15ÞRemoving the conditioning on b using standard integration tech-niques [26], we obtain the posterior distribution

gðx; hÞjy;r2 � Nðm��ðx; hÞ;r2C��½ðx; hÞ; ðx; hÞ�Þ ð16Þ

where

m��ðx; hÞ ¼ hðx; hÞ>bb þ tðx; hÞ>A�1ðy �HbbÞ ð17Þ

C��½ðx; hÞ; ðx; h0Þ� ¼ C�½ðx; hÞ; ðx; h0Þ� þ ðhðx; hÞ>

� tðx; hÞ>A�1HÞðH>A�1HÞ�1ðhðx; h0Þ>

� tðx; h0Þ>A�1HÞ> ð18Þ

bb ¼ ðH>A�1HÞ�1H>A�1y ð19ÞTo estimate r in Eq. (16), let q be the rank of H. Then

br2 ¼ y>ðA�1 � A�1HðH>A�1HÞH>A�1Þyn� q� 2

ð20Þ

Finally, it can be shown that

gðx; hÞ �m��ðx; hÞbr ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðn�q�2ÞC��ðx;hÞ

n�q

q � tn�q ð21Þ

which is a Student’s t-distribution with n � q degrees of freedom(not to be confused with the degrees of freedom in a finite elementmethod context).

From the above discussion, it can be seen that Gaussian processemulation consists in updating the prior distribution (9), whichcontains subjective information, by adding the objective informa-tion y in order to obtain the posterior distribution (16). This en-ables the calculation of the predictive mean m⁄⁄(x,h) given the

data y. This mean is a fast approximation of g(x,h) for any (x,h)in the input domain.

As mentioned before, the simulator can be treated as function ofh only. For a fixed x⁄, an emulator can be built as outlined in Algo-rithm 1.

Algorithm 1. Gaussian process emulation

Input: Design points fðx�; hiÞgni¼1 for a fixed x⁄

Output: Predictive mean m⁄⁄(x⁄,h) � E[g(x⁄,h)jy] andvariance r2C⁄⁄[(x⁄,h), (x⁄,h)] � Var[g(x⁄,h)jy]begin

1. Select n design points fðx�; hiÞgni¼1

2. Obtain the vector of observationsy = [g(x⁄,h1), . . . ,g(x⁄,hn)]>

3. Update the prior distribution (9) using y and obtain theposterior distribution (16)

4. Compute m⁄⁄(x⁄,h) and r2C⁄⁄[(x⁄,h), (x⁄,h)] for anyuntried (x⁄,h)

end

4. Uncertainty propagation

4.1. FRF statistics

Consider a dynamical system whose FRF is simulated by Eq. (6).Suppose that we wish to obtain a statistic of the FRF, such as themean or the variance. Let jXj = N and ‘ 2 {1, . . . ,N}. Also, leth‘s� �S

s¼1 ¼ h‘1; . . . ; h‘S� �

be a set of realisations of a random parameterh associated to a fixed frequency level x‘ in X ¼ fx‘gN

‘¼1. Then, atevery x‘ the mean FRF is estimated by

bg1 x‘; h‘1; . . . ; h‘S

� �¼ 1

S

XS

s¼1

g x‘; h‘s

� �ð22Þ

whereas the variance of the FRF is estimated by

bg2 x‘; h‘1; . . . ; h‘S

� �¼ 1

S� 1

XS

s¼1

g2 x‘; h‘s

� �� 1

S

XS

s¼1

g x‘; h‘s

� �" #224 35

ð23ÞThe most straightforward approach to estimate bg1 and bg2 is

Monte Carlo simulation (MCS): For ‘ = 1, . . . ,N, simulate

gðx‘; h‘sÞ

� �S

s¼1 and compute Eq. (22) or Eq. (23). The number ofevaluations of g(x,h) would be S � N. Nevertheless, if g(x,h) is com-putationally expensive, MCS can quickly become impracticable.Observe however that the input domain X �H allows the sam-pling of design points from H for every frequency level in X. Thatway, the number of evaluations can be reduced: For ‘ = 1, . . . ,N,

generate L‘ design points h‘j

n oL‘

j¼1. Then, compute the training runs

h‘j ;gðx‘; h‘j Þ

n oL‘

j¼1and emulate p runs gðx‘; h

‘kÞ

� �p

k¼1 at each of the

untried inputs h‘1; . . . ; h‘pn o

. The number of evaluations of g(x,h)

would be L = L1 + � � � + LN. If L < S, the computational burden of eval-uating g(x,h) would is reduced compared to Monte Carlo simula-tion, since L < S) L < S � N.

In the procedure described above, for every frequency level x‘,

the design points h‘j

n oL‘

j¼1are realisations of a random parameter

and therefore are drawn from a probability distribution FHðhÞ.Such distribution can be specified or elicited through expert opin-ion. The uncertainty in this input distribution is propagatedthrough the simulator g(x,h). Our aim is to characterise the result-ing output distribution Y = g(x,h). If g(x,h) were not an expensivesimulator, the most straightforward uncertainty analysis would be

28 F.A. DiazDelaO et al. / Computers and Structures 120 (2013) 24–32

to draw a large sample or realisations of h from the input distribu-tion FHðhÞ and then run the simulator in each of them. This wouldresult in a sample of outputs from which any statistic SðYÞ couldbe estimated as in Eq. (22) or Eq. (23). In principle, the predictivemean m⁄⁄(x,h) of a Gaussian process emulator could be used asa direct replacement of g(x,h) in order to carry out the simpleuncertainty analysis just described. However, m⁄⁄(x,h) is only anapproximation to g(x,h), and therefore this approach does notincorporate the corresponding uncertainty. The solution to thisproblem was proposed by Oakley and O’Hagan [49] with a modi-fied Monte Carlo procedure summarised in Algorithm 2. In ourcase, the algorithm is applied for every x‘ 2X.

Algorithm 2. GPE uncertainty analysis

Input: Sample hif gL‘i¼1 from the input distribution FHðhÞ

Output: Summary SðYÞ of the output distribution Ybegin

1. Draw a sample fhigL‘i¼1 from the distribution FHðhÞ

2. Draw a random function g(j) from the emulator posteriordistribution (16)

3. Evaluate gðjÞðh1Þ; . . . ;gðjÞðhL‘ Þ4. Obtain SjðYÞ, the Monte Carlo estimate of SðYÞ5. Repeat steps 2–4 and obtain the samplefS1ðYÞ; . . . ;SNg ðYÞg

6. Use the sample obtained in step 5 to estimate any sum-mary of the distribution of SðYÞ

end

4.2. Nonparametric regression

The strategy described above can be used to generate statisticalsummaries of the FRF for a finite set of frequencies x1, . . . ,xN in X.Suppose that, additionally, we wanted to estimate statistics of theFRF for any frequency level in the interval [x1,xN]. Clearly, a point-wise approach like the one described so far would be unfeasible.We propose to solve this by running a Gaussian process regressionbased on the data set fðx‘;Sðx‘ÞÞj‘ ¼ 1; . . . ;Ng, where each Sðx‘Þis the statistic computed using uncertainty analysis previously dis-cussed. This is a type of non-parametric regression, in the sensethat we are not constraining the unknown function to belong toa prescribed functional family (linear, for example). It can beshown [50] that, analogously to Eq. (10), the covariance functionwill be of the form

CovðWðx; hÞ;Wðx0; hÞÞÞ ¼ r2wCðx;x0Þ þ R ð24Þ

where W(x,h) is a Gaussian process and R is a diagonal matrix with½R�ii ¼ r2

i . We can readily estimate the variances r21; . . . ;r2

N

� �with

the uncertainty analysis provided by Algorithm 2.Summarizing, for every x in X (which is a finite set), we draw

design points from H, evaluate the simulator to obtain trainingruns, and estimate a summary of the FRF. We then use those esti-mates as new training runs and provide an estimate of the sum-mary for any x between x1 and xN.

5. Numerical investigation: nonproportionally damped platemade of a composite material

In the introductory example in Section 2, the uncertainty in thestiffness matrix K(h) in Eq. (6) was assumed to depend in a uni-formly distributed parameter h. This choice, although illustrative,was made arbitrarily. In this section, we present a model in whichthe uncertainty in the stiffness is determined by the microstruc-ture of a composite material. We assume this material to bedefined by carbon fibres embedded in an epoxy matrix, such that

there is uncertainty in the diameter of each fibre. This uncertaintyis explicitly modelled and then propagated through a finite ele-ment code. This propagation results in an uncertain FRF, the meanand variance of which we aim to estimate. We employ a homoge-nisation-based multiscale model to compute the mechanical re-sponse of the material and we assume the presence ofnonproportional damping.

5.1. Finite element modelling

In order to implement the metamodelling strategy outlined inSection 4, consider a square plate. The plate is excited by an unit har-monic force and the frequency response is measured at one of thenodes. The geometric and physical properties are: length 1 m, width1 m and thickness 1 mm. A finite element model of the plate isshown in Fig. 3(b). We assume that the plate has a damping patchattached. The resulting damping is nonproportional. The calculationof the FRF for this kind of nonproportionally damped systems can bevery expensive [51,52]. In general, it is not possible to represent theresponse in terms of undamped modes. In such cases the responseneeds to be expressed in terms of the complex modes of the system[53]. Even for a relatively small case, solving the linear system (2) foreach frequency level can be very resource-consuming.

5.2. Material modelling

The material is assumed to be a carbon fibre/epoxy compositewhose overall mechanical response is obtained by the computa-tional homogenisation of the representative volume element(RVE) shown in Fig. 3(a). To capture the actual mechanical re-sponse of the composite material, we adopt a framework basedon the periodic boundary displacement fluctuations model [54],widely adopted in the modelling of periodic media. This type ofmodel offers the possibility to describe more accurately the stressresponse under complex strain paths, although it suffers from thedrawback of excessive computing costs. These costs are usually ac-ceptable when finite element analyses of single RVEs are carriedout in the context of deterministic problems (refer, for instance,to Refs. [55–57]). However, when parametric uncertainty is consid-ered in the context of multiscale problems, the computational costof solving the computer model may rise by several orders of mag-nitude when compared to deterministic single-scale analyses. Formore details about homogenisation-based multiscale constitutivetheory, refer to Appendix A.

We consider a periodic linear elastic medium with circular rein-forcing fibres distributed uniformly. Following [58], for the carbonfibres we assume an isotropic material with Young’s modulusE = 235 GPa and Poisson’s ratio m = 0.2. The longitudinal directionof the fibres coincides with the X axis shown in Fig. 3. The trans-verse directions of the material are defined by the Y and Z axes.The epoxy material is assumed to be isotropic with Young’s mod-ulus E = 4.6 GPa and Poisson’s ratio m = 0.36 [58]. A typical finiteelement mesh consists of 23,471 nodes and 93,776 four-noded lin-ear tetrahedral elements. Since the density of the carbon fibres andthe epoxy matrix is similar, we adopt the same value for bothmaterials, namely 1.5 kg/m3.

5.3. Uncertainty modelling

In order to introduce uncertainty in the material definition, wepropose a modelling strategy which perturbs the diameter of eachfibre shown in Fig. 3(a). Since we are interested in calculating theoverall mechanical properties of the composite, all the diametersare perturbed simultaneously about a baseline value, resulting ina first set of homogenised elastic properties. In this study, thebaseline value for the corresponding diameters of each fibre is ta-

(a)

00.2

0.40.6

0.81

00.2

0.40.6

0.81

−0.5

0

0.5

1

X direction (length)

Output

Input

Y direction (width)

Fixed edge

(b)Fig. 3. Finite element model of a composite plate with a damping patch. The material and geometric properties are: length 1 m, width 1 m, thickness 1 mm. A microscopicfinite element mesh was adopted for the computation of the homogenised mechanical properties of the composite material. (a) Microscopic FE model; (b) macroscopic FEmodel.

0 500 1000 1500−140

−135

−130

−125

−120

−115

−110

Sample Size

Mea

n Fr

eq. R

espo

nse

[dB

]

(a)

0 500 1000 1500300

400

500

600

700

800

Sample Size

Var

ianc

e Fr

eq. R

espo

nse

(b)Fig. 4. Mean (a) and variance (b) of the frequency response function (FRF) at x = 71 Hz for different sample sizes. Both statistics seem to stabilise for sample sizes greater thanor equal to 1000.

F.A. DiazDelaO et al. / Computers and Structures 120 (2013) 24–32 29

ken as 17.84% of the unit side of the RVE. This results in a baselinevolume fraction of 0.40 (refer to [59]). The assumed input distribu-tion for each diameter is chosen to be Gaussian with a mean equalto the baseline value and with a variance such that 8.89% of thebaseline value equals 1 standard deviation [59]. Consequently,for each fibre, the diameter di is distributed as di � Nð0:1784;½0:0889� 0:1784�2Þ and thus each radius ri is distributed asri � Nð0:0892;0:00792Þ. The procedure is repeated again, for a sec-ond set of homogenised mechanical properties, and so on. We thentake the volume fraction as the random parameter h. The uncer-tainty induced by this random volume fraction is propagatedthrough the finite element model such that, for every realisationof h, we obtain an FRF for a predetermined range of vibration.While the probability distribution of the radii were taken fromthe literature [59], the distribution of the volume fraction h forour material model was determined as follows: The volume frac-tion for the RVE with the above characteristics is naturally definedas

h ¼ pX16

i¼1

r2i ð25Þ

Since frig16i¼1 is a sample of independent normal random variables, it

can be shown [60] that the sum in Eq. (25) is distributed as p times

a noncentral chi-squared distribution with 16 degrees of freedom.The noncentrality parameter k is given by

k ¼X16

i¼1

li

ri

� �2

ð26Þ

5.4. Numerical study

A preliminary study was carried out by running a series of tenexperiments, each with S = 100 Monte Carlo simulations of theFRF. The frequency domain X ranges from 1 to 200 Hz. The resolu-tion is 1 Hz, therefore jXj = N = 200. The frequency 71 Hz was iden-tified as consistently exhibiting the widest spread between itsmaximum and its minimum frequency response. This was the casefor every one of the 10 repetitions of the experiment. Conse-quently, for 71 Hz fixed, the mean and variance of the FRF wascomputed for sample sizes S ranging from 1 to 2000. Fig. 4 showsthat both the mean and the variance seem to stabilise around asample size of 1000. Therefore, this value was chosen for S. Thecorresponding 1000 realisations are shown in Fig. 5.

The Monte Carlo mean and variance of the FRF at every x 2Xwere taken as the benchmark against which the accuracy of theemulation approach was compared. Note that it was the output

0 20 40 60 80 100 120 140 160 180 200−400

−350

−300

−250

−200

−150

−100

−50

0

50

100

Frequency [Hz]

FRF

[dB

]

Fig. 5. The envelope resulting from S = 1000 Monte Carlo simulations of thefrequency response simulator g(x,h).

0 20 40 60 80 100 120 140 160 180 200−400

−350

−300

−250

−200

−150

−100

−50

0

50

100

Frequency [Hz]

Mea

n FR

F [d

B]

MCS benchmarkEmulators’ predictive means

(a)Fig. 6. Emulation and simulation of the mean frequency response (a) and variance of thethe input domain. The shaded areas represent 95% probability bounds.

Fig. 7. Histograms of the simulated output mean (a) and variance (b) and of the emulareflects the accuracy of the emulation strategy.

30 F.A. DiazDelaO et al. / Computers and Structures 120 (2013) 24–32

of the simulator, not the value of the quantity the simulator isattempting to predict, that was taken into account to determinethe accuracy of the emulator approximation. The sizes of the setsof design points for each of the 1000 emulators corresponding toevery frequency level had to be determined. Ideally, one shouldaim at having as many design points as possible to minimise theuncertainty in the output of the simulator. Intuitively, the more de-sign points available, the maximum distance between any two de-sign points either remains the same or decreases. If the amount ofinformation increases, then the approximation would be bound toimprove. It is therefore a choice of the modeller to decide betweena better approximation and the use of computational resources.Since for each fx‘g200

‘¼1 the functional form of g(x,h) is unknown,an exploration of H to determine every individual number of de-sign points can be very costly. Instead, it was decided to take 100design points for each emulator. This choice was made to showthat the mean and the variance of the response can be estimatedby generating only 10% of the original 1000 realisations of theFRF. Each set of design points was generated from the distributionof h, a noncentral chi squared with 16 degrees of freedom and

0 20 40 60 80 100 120 140 160 180 200−200

0

200

400

600

800

1000

1200

1400

Frequency [Hz]

FRF

Var

ianc

e

MCS benchmarkEmulators’ predictive means

(b)frequency response (b). The emulation was performed for every frequency level in

ted output mean (a) and variance (b). The degree of similarity of the distributions

F.A. DiazDelaO et al. / Computers and Structures 120 (2013) 24–32 31

noncentrality parameter k = 64.43. Due to the absence of priorknowledge, h(x,h) = 1 was assumed for every emulator. Thesmoothness parameters were obtained by the maximum likelihoodmethod outlined in [46,61].

Fig. 6(a) and (b) show, respectively, a comparison between theemulated mean and variance of the FRF against the MCS bench-mark. The emulated curves were built with the nonparametricregression discussed in Section 4.2. They provide estimates of themean and the variance for every frequency in 0–200 Hz. 95% prob-ability bounds are also provided. There was an improvement inthe computation time, due to the fact that the simulator wasevaluated only 200 � 100 = 20,000 times, as opposed to200 � 1000 = 200,000 times. Each emulator was run at virtuallyno computational cost. The saving in CPU time therefore implieda speed-up factor of around 10 when adopting the proposedapproach.

Fig. 7 shows a comparison between the distributions of the sim-ulated and emulated mean and variance for both strategies. Notehow the histogram for the emulation strategy resembles that ofMCS very closely for both statistics.

6. Conclusions

In this paper, Gaussian process emulators were employed as aneffective approach to alleviate the cost of characterising the ran-dom response of a structure subject to vibration. This cost doesnot only arises from the need to obtain a large sample of responsesfor every frequency level in the selected domain, but also dependsand is increased by the modelling of the material properties or bythe presence of nonproportional damping. Hence, the selected casestudy was a nonproportionally damped plate made of a carbon fi-bre/epoxy composite material. We were able to model the uncer-tainty induced by the random diameter of each reinforcingcarbon fiber as a non-central chi-squared distribution. We thenpropagated this uncertainty through a multiscale finite elementmodel coupled with a Gaussian process emulator for each fre-quency level in order to estimate the mean and the variance ofthe frequency response of the plate.

It was shown that if the output of the simulator is treated as afunction of the random parameter only, then it is possible to buildan emulator at every point in the frequency domain and approxi-mate the statistic of interest. The mean and the variance of the fre-quency response function estimated with Gaussian processemulators was benchmarked against a full Monte Carlo run. Themetamodelling approach closely approximated the desired statis-tics, and the number of runs of the original expensive finite ele-ment simulator was reduced by one order of magnitude.

Acknowledgments

F.A. DiazDelaO, E.I. Saavedra Flores, and M.I. Friswell acknowl-edge funding from the European Research Council through GrantNo. 247045 entitled ‘‘Optimisation of Multiscale Structures withApplications to Morphing Aircraft’’. E.I. Saavedra Flores alsoacknowledges the support of the Department of Civil Engineering,University of Santiago, Chile. S. Adhikari gratefully acknowledgesthe support of the Royal Society through the award of Wolfson Re-search Merit award.

Appendix A. Homogenisation-based multiscale constitutivetheory

The main assumption in the homogenisation-based multiscaleconstitutive theories of solids is that the macroscopic or homoge-nised strain tensor e at any arbitrary point x of the macroscopic

continuum is the volume average of the microscopic strain tensorfield el over the domain Xl of a representative volume element(RVE) of material. Similarly, the macroscopic or homogenisedstress tensor field r is assumed to be the volume average of themicroscopic stress tensor rl, over Xl. Models of the present typecan be found, for instance, in [62,54,63].

For any instant t, the above conditions can be expressed math-ematically as

eðx; tÞ ¼ 1Vl

ZXl

elðy; tÞ dV and rðx; tÞ ¼ 1Vl

ZXl

rlðy; tÞ dV

ðA1Þ

in which Vl is the volume of the RVE associated to point x, and y isthe local RVE coordinates. Furthermore, the microscopic strain ten-sor el can be related to the local displacement field ul by means ofthe standard expression el �rsul, in which rs is the symmetricgradient operator. In addition, it is possible to decompose the dis-placement field ul as a sum of a linear displacement e(x, t)y, whichrepresents a homogeneous strain, and a displacement fluctuationfield ~ul. The displacement fluctuations field represents local varia-tions about the linear displacement e(x, t)y and does not contributeto the macroscopic scale strain.

By taking into account the Hill–Mandel Principle of Macro-homogeneity [64,65], which establishes that the macroscopicstress power must equal the volume average of the microscopicstress power over Xl, the virtual work equation for the RVE canbe reduced toZ

Xl

rlðy; tÞ : rsgdV ¼ 0 ðA2Þ

with g representing the virtual kinematically admissible displace-ments field of the RVE.

In order to make problem (A2) well-posed, a set of kinematicalconstraints upon the selected RVE is required. In what follows, thechoice of this set will coincide with the widely used periodicboundary displacement fluctuations model, which is typicallyassociated with the modelling of periodic media. Here, the funda-mental kinematical assumption consists of prescribing identicaldisplacement fluctuation vectors for each pair of opposite pointsy+ and y� on the RVE boundary @Xl, such that:

~ulðyþ; tÞ ¼ ~ulðy�; tÞ ðA3Þ

By describing the RVE response by means of a generic local dissipa-tive constitutive theory, the microscopic stress tensor rl is repre-sented by a functional of the history of el, which is expressedsymbolically as rlðy; tÞ ¼ Fyðet

lðyÞÞ. The functional Fy associatedwith point y maps the strain history, et

l, up to time t, into the stressrl of time t. In view of the above constitutive assumption and theadditive decomposition of the microscopic displacement field, theequilibrium equation (A2) leads to

Gðe; ~ul;gÞ �Z

Xl

Fyf½eðx; tÞ þ rs ~ulðy; tÞ�tg : rsg dV ¼ 0 ðA4Þ

where we have defined G as virtual work functional.

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