Comput MechDOI 10.1007/s00466-014-1008-2

ORIGINAL PAPER

Entropy propagation analysis in stochastic structural dynamics:application to a beam with uncertain cross sectional area

A. Batou · T. G. Ritto · Rubens Sampaio

Received: 28 November 2013 / Accepted: 18 March 2014© Springer-Verlag Berlin Heidelberg 2014

Abstract This paper investigates the impact of differentprobabilistic models of uncertain parameters on the responseof a dynamical structure. The probabilistic models of theuncertain parameters are constructed using the maximumentropy principle, where different information is considered,such as bounds, mean value, etc. Nested probabilistic modelsare constructed with increasing information; as the informa-tion given increases, the level of entropy of the input modeldecreases. The response of the linear dynamical model isgiven in the frequency domain, and the propagation of theinput uncertainty throughout the computational model is ana-lyzed in terms of Shannon’s entropy. Low and high frequen-cies are analyzed because uncertainties propagate differentlydepending on the frequency band. A beam discretized bymeans of the finite element method with random cross sec-tional area (random field) is the application analyzed.

Keywords Uncertainty propagation · Input/outputentropy · Stochastic structural dynamics · Parameteruncertainties

A. Batou (B)Laboratoire de Modélisation et Simulation Multi Echelle,MSME UMR 8208, Université Paris-Est, 5 bd Descartes,77454 Marne-la-Vallée, Francee-mail: anas.batou@univ-paris-est.fr

T. G. RittoCentro de Tecnologia, Department of Mechanical Engineering,Federal University of Rio de Janeiro, Ilha do Fundão,Rio de Janeiro 21945-970, Brazile-mail: tritto@mecanica.ufrj.br

R. SampaioDepartment of Mechanical Engineering, PUC-Rio,Rua Marquês de São Vicente, 225, Rio de Janeiro 22453-900, Brazile-mail: rsampaio@puc-rio.br

1 Introduction

This research concerns the prediction of the dynamicalresponse of a structure in the presence of parameter uncer-tainties. These uncertainties are related to the variabilityappearing during the manufacturing process or during thecycle life (uncontrolled damages) of the structure. Theseuncertainties propagates into the response of the structureyielding more or less uncertainties in the quantities of inter-est. If these output uncertainties are non-negligible, then theyhave to be taken into account in order to predict the dynamicalresponse and support decision makers.

In the context of the probabilistic methods for uncertaintyquantification, the uncertain parameters are replaced by ran-dom variables. Then, the first step consists in constructing aprobabilistic model for these parameters, i.e., construct theprobability density functions (pdf) for the random variablesmodelling the uncertain parameters. The Shannon entropy[15] measures the relative uncertainty associated to a pdf. Themaximum entropy (MaxEnt) principle [4,5,15] in the contextof Information Theory is a powerful method which allowsthe pdf of a random variable to be constructed from a set ofavailable information. This method consists in choosing thepdf which maximizes the entropy (and thus the uncertainty)under the constraints defined by the available information.Therefore, the entropy (uncertainty) level in the input para-meters depends on the amount of information available forthe uncertain parameters. It is then interesting to analyse howthis level of entropy propagates into the quantities of interest.

Usually, the propagation of uncertainties is analysedthrough the observation of the input/output variance. Indeed,in general, the variance increases if the uncertainty relatedto a probability distribution increases. Nevertheless, in somespecific cases (for instance, a bi-modal distribution for whichthe distance between the two peaks is large), a probability

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distribution with low uncertainty can have a large variance.Then for these specific cases, the variance is not a reliablemeasure of uncertainty. For this reason entropy-based sensi-tivity indexes have been introduced in [10] as an alternativeto the classical Sobol indexes [16] which are variance-based.Entropy measures were also recently used for identificationof structural dynamics and damage detection. In [12] thetransfer entropy measure was used for damage detection in asimple discrete dynamical system and in [1] a relative entropymeasure was used to perform an inverse problem in a beamstructure.

In the present paper, we provide a general methodologyto analyze the input-entropy propagation into the dynami-cal response of a structure. Such an analysis is important inorder to quantify the effects of the probabilistic modellingof the input parameters on the uncertainty related to the out-put quantities of interest. Then such an analysis can provideimportant information related to (1) the sensitivity of the out-puts with respect to the variability of the inputs (this informa-tion can be obtained using a classical variance-based sensi-tivity analysis) and (2) the robustness of the output variabilitywith respect to the choice of the input probability model. Ifthis robustness is good enough then a coarse probabilisticmodelling of the input parameters is sufficient in order topredict the random dynamical response with a good robust-ness. This latter point is important in order to determine thequantity of information that is needed for the construction ofthe input probability model. For instance, if a uniform dis-tribution and a beta distribution for the inputs have the sameeffect on the output then the uniform distribution will be pre-ferred since it only requires the bounds of the inputs (given bythe manufacturing tolerances). On the contrary, if the effectof the input probability model on the output variability is notnegligible, then it is important to analyze if the decrease ofthe input entropy induces a decrease or an increase of the out-put entropy. In the latter case a fine input probability model(constructed using a large amount of information) is requiredin order to avoid an underestimation of the output uncertainty(measured by the output entropy).

The objectives and contributions of the present papers are:(1) provide a general methodology to analyze the propaga-tion of uncertainty throughout a linear dynamical structurein terms of input/output entropy, (2) analyze how increasingentropy in the probabilistic model of the parameters affectsthe entropy of the response; for this purpose, nested inputprobabilistic models are constructed using the MaximumEntropy principle, and (3) analyze the uncertainty propaga-tion in low and high frequencies to highlight the stochastichomogenisation that can occur for the random response inthe low frequency band.

This paper is organized as follows. In Sect. 2, the reduced-order nominal computational model is constructed. Then,Sect. 3 is devoted to (1) the construction of the stochastic

computational which is derived from the reduced nominalcomputational model and (2) the estimation of the outputentropy. Finally, in Sect. 4, an application related to the ran-dom response of an Euler beam with random cross sectionalarea (random field) is presented.

2 Reduced nominal computational model

In this section the reduced nominal computational model isconstructed using the finite element method and the modelreduction is performed using a classical modal analysis.

2.1 Nominal computational model

We are interested in the dynamical response of a three-dimensional damped structure having a linear behavior. Thisstructure is made up of a linear dissipative elastic mediumoccupying an open bounded domain Ω , with boundary∂Ω = Γ0 ∪ Γ . Let x be the coordinates of a point in Ω .The external unit normal to ∂Ω is denoted by n. Let u(x, ω)be the displacement field defined in the frequency domainwith values in C3. We assume that u = 0 on part Γ0 and,consequently, there is no rigid body displacements. Let thevector f surf(x, ω) denote the surface force field applied onthe boundary Γ and let the vector f vol(x, ω) denote the vol-ume force field applied in Ω . We are interested in the lin-ear response around a static equilibrium considered as thereference configuration defined by Ω . The boundary valueproblem in the frequency domain is written, for all ω, as

−ω2ρ u − div σ = f vol in Ω ,

u = 0 on Γ0 ,

σ n = f surf on Γ ,

(1)

where ρ(x) is the mass density and σ (x, ω) is the second-order stress tensor, in which {div σ (x, ω)} j =∑3

k=1 ∂σ jk(x, ω)/∂xk . The stress tensor σ (x, ω) will berelated to the strain tensor ε(x, ω) by a constitutive equationwhich is written for a non-homogeneous anisotropic dissi-pative elastic medium as σh�(x, ω) = ah�jk(x) ε jk(x, ω) +iω bh�jk(x) ε jk(x, ω) in which ah�jk(x) and bh�jk(x) are thefourth-order real tensors related to the elastic and dissipa-tive parts and which must satisfy symmetry and positivenessproperties.

In this paper, in order to improve the readability, weare analyzing only one parameter field of interest which isdenoted by x �→ h(x). The generalization for several parame-ter fields is straightforward. The parameters can be materialparameters such as the mass density or the Lame parameters,or other model parameters such as the section for a beammodel, the thickness for a plate model, etc.

The computational model is constructed using the finiteelement (FE) method applied to the weak formulation of the

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dynamical problem defined by Eq. (1). Let x1, . . . , xN bethe set of the N positions where the field h(x) is discretized.These positions can correspond to the integration points ofthe FE model. If the mesh is sufficiently fine the field h(x)can be assumed as constant in each element and the dis-cretizations points are the barycenters of the elements. Leth = (h(x1), . . . ,h(xN )) ∈ RN be the vector of the dis-cretized system parameters, that latter will be replaced by arandom vector (see Sect. 3).

We are interested in the frequency response of the structureon the frequency band of analysis B = [0, ωmax]. Let m bethe total number of degrees-of-freedom in the FE model. Forallω ∈ B, the vector y(ω) ∈ Rm of the m degrees-of-freedomis the solution of the following matrix equation(−ω2[M(h)] + iω[D(h)] + [K(h)]

)y(ω) = f (ω) , (2)

in which [M(h)], [D(h)] and [K(h)] are the (m × m) mass,damping and stiffness matrices and where f(ω) is the vectorof the external forces.

2.2 Reduced-order nominal computational model

The reduced nominal computation model is constructedusing the modal analysis reduction method. Let Ch be theadmissible set for the vector h. Then for all h in Ch , the nfirst eigenvalues 0 < λ1(h) ≤ λ2(h) ≤ · · · ≤ λn(h) asso-ciated with the elastic modes {φ1(h), φ2(h), . . . , φn(h)} aresolutions of the following generalized eigenvalue problem

[K(h)]φ(h) = λ(h)[M(h)]φ(h) . (3)

The reduced-order nominal computation model is obtainedby projecting the nominal computation model on the sub-space spanned by the n first elastic modes calculated usingEq. (3). Let [Φ(h)] be the m×n matrix whose columns are then first elastic modes. We then introduce the approximation

y(ω) = [Φ(h)] q(ω) , (4)

in which the vector q(ω) is the vector of the n general-ized coordinates and is the solution of the following reducedmatrix equation(−ω2[M(h)] + iω[D(h)] + [K (h)]

)q(ω) = f(ω; h) , (5)

in which [M(h)] = [Φ(h)]T [M(h)] [Φ(h)], [D(h)] =[Φ(h)]T [D(h)] [Φ(h)] and [K (h)]=[Φ(h)]T [K(h)] [Φ(h)]are the n × n mass, damping and stiffness generalizedmatrices, and where f(ω; h) = [Φ(h)]T f(ω) ∈ Rn isthe vector of the generalized forces. The random responseis calculated at nfreq frequencies belonging to frequencyband B. We then introduce the observation vector z =O BS(y(ω1), . . . , y(ωnfreq)) in Rnobs , where nobs is the num-ber of outputs observed.

3 Stochastic computational model

In this section, the stochastic computational model is derivedfrom the reduced-order nominal computational model intro-duced in the previous section. The random variables, the ran-dom vectors and the random fields are denoted using upper-case letters. The uncertain parameter field of the dynami-cal system is modeled by a random field, which means thatthe field {h(x), x ∈ Ω} is modeled by the random field{H(x), x ∈ Ω}. Furthermore, it is assumed that the ran-dom field {H(x), x ∈ Ω} is homogeneous. Therefore, in thecontext of the FE discretization introduced in the previoussection, the vector h which corresponds to the spatial dis-cretization of h(x), is modeled by a random vector H. Theprobabilistic model of this random vector is constructed usingthe MaxEnt principle. Finally, the stochastic reduced-ordercomputational model is presented.

3.1 Probabilistic model of uncertainties

In this section, the probability distribution of the random vec-tor H is constructed using the MaxEnt principle. Two generalcases are considered. For the first case, the available informa-tion is introduced independently for each component of therandom vector H yielding independent components for therandom vector H. For the second case, additional constraintsrelated to the correlation between components are introducedyielding a dependence between these components.

3.1.1 Independent information case

For this particular case, the random variable H is denotedHnc. The available information is introduced independentlyfor each component of the random vector Hnc. Since it isassumed that the random field {H(x), x ∈ Ω} is homoge-neous, the available information is the same for all the com-ponents of the random vector Hnc. The support of the pdf ofeach component Hnc

i is denoted by K such that K ⊂ R. Theavailable information for the components Hnc

i are related tophysical properties or available statistics on this components.For each component, Hnc

i , i = 1, . . . , N , the available infor-mation is written as the following μ constraints

E{g(Hnci )} = f , (6)

in which h �→ g(h) is a given function from R into Rμ, E{.}is the mathematical expectation, and f is a given vector inRμ. Equation (6) can be rewritten as

∫

R

g(hi )phnci(hi )dhi = f . (7)

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An additional constraint related to the normalization ofthe joint pdf pHnc(h) is introduced such that∫

RN

pHnc(h)dh = 1 . (8)

It should be noted that the available information in Eq. (6) isrestricted to expectations of given functions. Therefore, theMaxEnt principle does not allow taking into account almostsure properties of given realizations of the random variables.The Shannon entropy of the joint pdf h �→ pHnc(h) is definedby

S(pHnc) = −∫

RN

pHnc(h) log(pHnc(h))dh , (9)

where log is the natural logarithm. This functional measuresthe uncertainty associated with pHnc(h). Let C be the set ofall the pdf defined on RN with values in R+, verifying theconstraints defined by Eqs. (7) and (8). Then the MaxEnt prin-ciple consists in constructing the probability density functionh �→ pHnc(h) as the unique pdf in C which maximizes theentropy S(pHnc). By introducing a Lagrange multiplier λ0

in R+ associated with Eq. (8) and N Lagrange multipliersλi associated with Eq. (7) and belonging to an admissibleopen subset Lμ of Rμ, it can be shown [4,5] that the MaxEntsolution, if it exists, is defined by

pHnc(h) =N∏

i=1

{1K(hi )} csol0 exp

(

−N∑

i=1

〈λsoli , g(hi )〉

)

, (10)

in which the indicator function hi �→ 1K(hi ) is such that it isequal to 1 if hi ∈ K and is zero otherwise. In Eq. (10), csol

0 =exp(−λsol

0 ), 〈x, y〉 = x1 y1+· · ·+xμyμ and λsol0 and λsol

i arerespectively the values of λ0 and λi for which Eqs. (7) and(8) are satisfied. Equation (10) shows that the componentshi of the random vector H are independent random variablesfor which the pdfs are given for i in {1, . . . , N } by

pHnci(hi ) = 1K(hi ) csol

i exp(−〈λsoli , g(hi )〉) . (11)

This result is obvious and could be stated at the beginningby applying the MaxEnt principle on each component inde-pendently. Using the normalization condition, the parametercsol

i can be eliminated and Eq. (11) can be rewritten as

pHnci(hi ) = 1K(hi ) ci (λ

soli ) exp(−〈λsol

i , g(hi )〉) , (12)

in which ci (λi ) is defined by

ci (λi ) =⎧⎨

⎩

∫

Kexp(−〈λi , g(hi )〉) dhi

⎫⎬

⎭

−1

. (13)

The N Lagrange multipliers λi are then calculated usingEqs. (7), (12) and (13). The integrals can be calculated explic-itly for some particular cases of available information. Since,

the dimension of these integrals is one, they can be calculatedusing any numerical integration method.

If we consider that the support K is compact then, if noavailable information is introduced, the MaxEnt distributionis the uniform distribution. Hence, if an available informationis added, a constraint is added and the research manifold forthe maximum of the entropy is reduced yielding a smallermaximum. More specifically, if we introduce the two randomvariables H1 and H2, with the same support, for which theavailable information are respectively defined by

E{g1(H1)} = f 1 , (14)

E{g1(H2)} = f 1 , E{g2(H2)} = f 2 , (15)

in which the functions g1 and g2 are independent. Thenthe available information related to H1 is included in theavailable information related to H2. Let S1 be the maximumentropy related to H1 with the constraint defined by Eq. (14)and S2 be the maximum entropy related to H2 with the con-straint defined by Eq. (15). We then have

S1 ≥ S2 . (16)

By proceeding this manner, it is possible to create nestedprobabilistic model with increasing information (decreasingentropy).

3.1.2 Dependent information case

For this case, a correlation structure is introduced for the ran-dom field {H(x), x ∈ Ω} and, therefore, for the random vec-tor H. This correlation structure can be introduced throughthe Pearson correlation coefficient [6] defined by

rH(x, x′) = CH(x, x′)√CH(x, x)CH(x′, x′)

, (17)

in which CH(x, x′) is the covariance between H(x) andH(x′) defined by CH(x, x′) = E{(H(x)− m H (x)) (H(x′)−m H (x′))} in which m H (x) = E{H(x)}. We then haverH(x, x) = 1 for all x in Ω . Since it is assumed that therandom field {H(x), x ∈ Ω} is homogeneous, the correlationcoefficient is rewritten as

rH(x − x′) = CH(0, x − x′)σ 2

H

, (18)

in which σ 2H = CH(0, 0). In most of the correlation model

that can be found in the literature, the correlation coeffi-cient is isotropic and, therefore, only depends on the distancebetween x and x′. For instance, the exponential correlationmodel is given by

rH(x − x′) = exp

(

−‖ x − x′ ‖l2

)

, (19)

in which l is a positive real variable. Now, let [TH ] be thematrix defining the correlation coefficients for the N points

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where the stochastic process {H(x), x ∈ Ω} is discretized.We then have

[TH]i j = rH(

xi − x j). (20)

The matrix [TH ] completely defines the correlation struc-ture of the random vector H. In this paper, we will consideronly cases where this matrix is positive-definite, althoughit can be positive semi-definite. The possible zero eigen-values can be eliminated by introducing the regularization[T reg

H ] = [TH ] + α[IN ] in which [IN ] is the identity matrixand where 0 < α � 1. The diagonal elements of matrix[TH ] are equal to one and the other entries are lower thanone. Therefore, using Hadamard inequality, it can be shownthat

0 < det ([TH ]) ≤ 1 , (21)

the equality arising for the uncorrelated case. In this paper,the homogeneous correlation length following the directionek is defined by

Lkc =

+∞∫

0

| rH(η ek)|dη . (22)

For instance, using the correlation model in Eq. (19) yieldsL1

c = L2c = L3

c = √π l/2. Other definitions of the correla-

tion length can be found in the literature [8].The objective of this section is to take into account a

correlation structure in the construction of the probabilitymodel of the random vector H, which means that H hasto verify N (N − 1)/2 additional constraints defined for1 ≤ i ≤ (N − 1), i < j ≤ N by

E{(Hi − m Hi ) (Hj − m Hj )} = σ 2H [T g

H]i j , (23)

in which the matrix [T gH] is the target value for [TH] and where

m H = E{H}. There are several ways to take into accountthese constraints. Let us analyze three different ways.

(1) The first one consists in directly imposing the constraintsdefined by Eq. (23) in the construction of the MaxEntdistribution. This way is numerically prohibitive in high-stochastic dimension because it yields N (N−1)/2 addi-tional Lagrange multipliers to be identified. Furthermorethe components of the random vector H would be depen-dent and then the integrals involved in Eqs. (7), (12) and(13) could not be calculated using a numerical integra-tion method. It should be noted that a new algorithm (see[2,18]) has recently been developed in order to addressthis problem in a relatively high stochastic dimension(N < 10, 000).

(2) The second one consists in constructing the random vec-tor H as a transformation of the independent (thus uncor-related) random vector Hnc, obtained in Sect. 3.1, in the

following way. Let the autocorrelation matrix of ran-dom vector Hnc be E{(Hnc − mHnc) (Hnc − mHnc)T } =σ 2

H [IN ]. The transformation is defined by

H = [V ](Hnc − mHnc)+ mHnc , (24)

in which mHnc = E{Hnc} and the matrix [V ] is relatedto the Cholesky factorization of matrix [TH ], i.e, [TH ] =[V ][V ]T . Then it can be verified that E{H} = mH =mHnc and E{(H − mH) (H − mH)

T } = σ 2H [TH ] and,

therefore, Eq. (23) is verified. Nevertheless, the randomvector H constructed using Eq. (24) does not verify theconstraints defined by Eq. (6) which is verified only bythe random vector Hnc. Furthermore, the transformationalso modify the support of the probability distribution.Therefore, if the correlation between the componentsis strong, the construction defined by Eq. (24) is notadapted.

(3) The third way, which is the one adopted in the presentwork, consists in introducing a Nataf transformation[9,11,17] of the random vector H. This transformationconsists in introducing the following change of variablefor each component of random vector H

Hi = F−1Hnc

i(Φ(Ui )) , (25)

in which U1, . . . ,UN are the components of the Gaus-sian centered random vector U for which the covari-ance matrix is denoted by [CU ]. In Eq. (25), the functionu �→ Φ(u) is the cumulative distribution function of thenormalized Gaussian random variable and the functionv �→ F−1

Hnci(v) is the reciprocal function of the cumu-

lative distribution of the random variable Hnci which is

the i th component of the random vector Hnc constructedfollowing Sect. 3.1. By construction, the random vectorH verifies the constraints defined by Eq. (6). It can beshown that the joint probability density function of therandom vector H is such that

pH(h)= 1√det([CU ]) exp

(

−u(h)T ([CU ]−1−[IN ])u(h)2

)

pHnc (h),

(26)

in which u(h) = (Φ−1(FHnc1(h1), . . . ,

Φ−1(FHncN(hN )) and where the function h �→ pHnc(h)

is the probability density function of random vector Hnc

constructed following the Sect. 3.1, which can be rewrit-ten as

pHnc(h) =N∏

i=1

pHnci(hi ) , (27)

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in which the pdf h �→ pHnci(h) is defined by Eq. (12).

The correlation matrix [CU ] has to be calculated suchthat the constraints defined by Eq. (23) is verified, yield-ing an inverse problem which is difficult to solve [9].The direct choice [CU ] = [TH ] generally gives verygood results. For the classical pdfs, a correction of [CU ]in order to improve the verification of Eq. (12) can beconstructed [9] if necessary. It can be noted that thePearson correlation coefficients for the random vectorH and U are different but the Spearman rank correlationcoefficients [6] for the random vectors H and U, whichare defined as the Pearson correlation coefficients forthe random vectors (FHnc

1(Hnc

1 ), . . . , FHncN(Hnc

N )) and(Φ(U1), . . . , Φ(UN )), respectively, are equal by con-struction. In the rest of this paper, the approximation[CU ] = [TH ] is chosen. Then, it can be proven that theentropy S(pH) of the pdf pH is such that

S(pH) = S(pHnc)+ 1

2log(det([TH ])) . (28)

Thus, using Eq. (22) it can be verified that

S(pH) ≤ S(pHnc) , (29)

the equality arising for the uncorrelated case. This means thateven if the correlation has not been introduced directly dur-ing the construction of the MaxEnt distribution, it induces adecrease of the entropy anyway, as it corresponds to an addi-tional information. In general, the value of log(det([TH ]))depends on the correlation coefficient model and on the cor-relation length. For the 1-D version of the exponential cor-relation model introduced in Eq. (19) for a domain [0, 1]discritized into 100 regularly spaced points, the functionLc �→ log(det([TH (Lc])) is plotted on Fig. 1. It can be seenthat this function decreases very rapidly, which means thatas the correlation increases, the entropy decreases. This cor-responds well to the intuition that the information of theprobabilistic model increases when the correlation lengthincreases.

Finally, generators of independent realization of randomvector H can easily be constructed using Eq. (25) and clas-sical generators of Gaussian centered random vectors.

3.2 Random response of the stochastic computationalmodel

The stochastic computational is derived from the reducednominal computational model introduced in Sect. 2.2, forwhich the deterministic vector h of the discretization of theuncertain fields is replaced by the random vector H. Then, then first random eigenvalues 0 < Λ1(H) ≤ · · · ≤ Λn(H) asso-ciated with the random elastic modes {ψ1(H), . . . , ψn(H)}

0 0.2 0.4 0.6 0.8−700

−600

−500

−400

−300

−200

−100

0

100

Lc

log(

det(

TH

))

Fig. 1 Graph of the function Lc �→ log(det([TH (Lc]))

are solutions of the following random eigenvalue problem

[K(H)]ψ(H) = Λ(H)[M(H)]ψ(H) . (30)

Then for all ω in B, the random response Y(ω) of thestochastic reduced-order computational model, is written as

Y(ω) = [Ψ (H)] Q(ω) , (31)

in which the random vector Q(ω) of the random generalizedcoordinates, is the solution of the following random reduced-order matrix equation,(−ω2[M(H)] + iω[D(H)]+[K (H)]

)Q(ω)= f(H). (32)

This equation can be solved using the Monte Carlo sim-ulation method [13]. The pdf of the random observationvector Z = O BS(Y(ω1), . . . ,Y(ωnfreq)) is denoted bypZ, and the observation entropy is defined by S(pZ) =− ∫Rnobs pZ(z) log(pZ(z))dz. Let Z1, . . . ,Zns be ns inde-

pendent realizations of the random observation vector Z.Then, the observation entropy S(pZ) can be estimated usingthe plug-in estimator [3]

S(pZ) = − 1

ns

ns∑

i=1

log(

pZ(Zi )), (33)

in which pZ is a multi-dimensional kernel density estimatorfor the random variable Z. If nobs = 1, then the entropy canalso be estimated using a numerical integration of the integral− ∫

A⊂R log( pZ(Z)) pZ(Z)dZ, in which the compact supportA can be delimited by the extremal values of the sampling(Z1, . . . ,Zns ) for instance. For large values of nobs the esti-mation of pZ using a multi-dimensional kernel density esti-mator becomes difficult. In this case, other estimators suchas the nearest neighbour estimator [7] can be used. Anotherapproach consists in introducing the variable Znc = [V ]−1Z,

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in which the matrix [V ] is related to the Cholesky factor-ization of the covariance matrix of random vector Z, i.e,E{ZZT } = [CZ] = [V ][V ]T . Then, the entropy of randomvector Z is such that

S(pZ) = S(pZnc)+ 1

2log(det([CZ])) . (34)

It can be verified that the covariance matrix of Znc is theidentity matrix and then the components of Znc are uncor-related but dependent. By neglecting the dependence of thecomponents of Znc, the entropy S(pZ) of random vector Zcan be approximated by a pseudo-entropy S(pZ) defined by

S(pZ) =nobs∑

j=1

S(pZncj)+ 1

2log(det([CZ])) , (35)

which can be estimated using the plug-in estimator

S(pZ) = − 1

ns

nobs∑

j=1

ns∑

i=1

log(

pZncj(Znc

j,i ))

+1

2log

(

det

(1

ns

ns∑

i=1

Zi ZTi

))

(36)

in which pZncj

is a one-dimensional kernel density estima-tor for the random variable Znc

j . It should be noted that forGaussian random vectors, the non-correlation implies theindependence and, therefore, the entropy and the pseudo-entropy are equals, i.e., S(pZ) = S(pZ).

4 Application

A clamped-clamped Euler–Bernoulli beam is considered forthe application of entropy propagation. The beam has 1 meterlength, diameter of 1 × 10−2 m, and material with elasticmodulus of 210 GPa and density of 7,850 kg/m3.

First, the nominal model and its deterministic responseare presented, then the stochastic model is constructed fromthe nominal model and entropy propagation is analyzed.

4.1 Nominal model

The nominal computational is constructed as described inSect. 2. The beam is discretized into 500 finite elements andHermitian shape functions have been used. The frequencyband of analysis is B = [0, 314] kHz. Then the reduced-ordermodel is constructed using 75 normal modes so that a goodconvergence of the response in frequency band B is reached.A proportional damping model with damping rate 0.01 hasbeen used to construct the generalized damping matrix.

The first natural frequency of the structure is about 45 Hz.A vertical point force equal to one in the frequency band Bis applied at x = 0.70 m. The frequency response in velocity

0 0.5 1 1.5 2 2.5 3

x 105

10−4

10−3

Frequency (Hz)

Spe

ed (

mH

z−1 /s

)

0 1000 2000 3000

10−4

10−3

10−2

10−1

Frequency (Hz)

Spe

ed (

mH

z−1 /s

)

(a)

(b)

Fig. 2 Velocity spectrum of the deterministic response at x = 0.40. aB = [0, 314] kHz and b B = [0, 3] kHz

{ω �→ iω×y(ω), ω ∈ B}, with i2 = −1, is calculated usingEqs.( 4) and (5). The observation vector is the modulus ofthe transversal frequency response in velocity observed atx = 0.40 m and calculated for nfreq = 20, 000 frequenciesregularly spaced in the frequency band B. Figure 2a showsthe values of this observation vector, and Fig. 2b shows azoom in the low frequency band [0, 3] kHz.

4.2 Stochastic model

The stochastic model is derived from the nominal modelintroduced in the previous section and for which the diameteris replaced by a random field yielding randomness in both thekinetic energy and elastic energy. Several random field mod-els are successively introduced corresponding to successiveaddition of information. Four classes, which will be depictedin the next section, are considered for the input probabilis-

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tic model: uniform independent, beta independent, uniformcorrelated and beta correlated.

4.2.1 Construction of the diameter random fields

The random fields are discretized at the middle of each ele-ment of the mesh, thus N = 500. The first diameter randomfield model corresponds to the case developed in Sect. 3.1.1for which the components of random vector Hnc are inde-pendent. For this first model the available information is onlyrelated to the support K which is constrained to be [blow, bup],in which blow = 0.75 × 10−2 m is the lower bound andbup = 1.25 × 10−2 m is the upper bound. Then applyingMaxEnt principle yields a uniform distribution for the inde-pendent components of random vector Hnc denoted Hnc,unif

for this first model, i.e., for i in {1, . . . , N }

pHnc,unifi

(hi ) = 1

|K|1K(hi ) , (37)

in which |K| = 5 × 10−3 m is the size of the support K.The entropy of the marginal distribution pHnc,unif

i(hi ), called

marginal entropy in this paper, is then

S(pHnc,unifi

(hi )) = log(|K|) , (38)

which shows that the entropy (and then the uncertainty)increases as the length of the support increases. For the appli-cation, we then have S(pHnc,unif

i(hi )) � −5.298. Since the

components of random vector Hnc,unif are independent, thetotal entropy of the distribution pHnc,unif of random vectorHnc,unif is S(pHnc,unif ) = N × S(pHnc,unif

i(hi )) � −2.65 ×

103.The second diameter random field model also corresponds

to independent components. For this second model the avail-able information is related to (1) the support K which isconstrained to be [blow, bup], (2) the behaviour of the pdf inthe neighbourhood of the lower bound through the constraintE{log(Hnc

i − blow)} = clow < +∞ and (3) the behaviour ofthe pdf in the neighbourhood of the upper bound through theconstraint E{log(bup − Hnc

i )} = cup < +∞. Then applyingMaxEnt principle yield a beta distribution for the independentcomponents of Hnc denoted Hnc,beta for this second model,i.e., for i in {1, . . . , N }

pHnc,betai

(hi )= |K|1−α−β

B(α, β)1K(hi ) (H

nci − blow)

α−1 (bup−Hnci )

β−1(39)

in which α ≥ 1 and β ≥ 1 depend on clow and cup andwhere B(α, β) is the classical Beta function. The case α = 1and β = 1 is a special case which correspond to the uni-form distribution and then corresponds to the first diameterrandom fields model. Instead of setting clow and cup, α andβ are directly set to given values . The marginal entropy for

pHnc,betai

(hi ) is then

S(pHnc,betai

(hi )) = log(|K|)+log(B(α, β))−(α − 1) Ψ (α)

−(β−1) Ψ (β)+(α+β − 2) Ψ (α+β) ,(40)

in which Ψ is the Digamma function defined by Ψ (x) =d(log(Γ (x)))/dx where Γ (x) is the classical Gamma func-tion. Then since α ≥ 1 and β ≥ 1, we have asexpected S(pHnc,beta

i(hi )) ≤ S(pHnc,unif

i(hi )). For the appli-

cation and for α = 2 and β = 2, we then haveS(pHnc,beta

i(hi )) � −5.423. The total entropy of the distribu-

tion pHnc,beta of random vector Hnc,beta is then S(pHnc,beta ) =N ×S(pHnc,beta

i(hi )) � −2.71×103. 5,000 independent real-

izations of random vectors Hnc,unif and Hnc,beta have beengenerated. It can be seen a good convergence of the plug-inestimators to the theoretical values.

The third diameter random field model corresponds tothe case developed in Sect. 3.1.2 for which the componentsof random vector H are dependent. The available informa-tion for the components is the same as for the first diam-eter random field model. The correlation coefficient modelis the Gaussian exponential one introduced in Eq. (19) withl = 2 Lc/

√π in which Lc is the correlation length. Then

the joint pdf of random vector H denoted Hunif for this thirdmodel is given by Eqs. (26) and (27) with pHnc

i(hi ) defined

by Eq. (37). Then the entropy of the marginal distributionare pHnc

i(hi ) = pHnc,unif

i(hi ). The total entropy of the dis-

tribution pHunif of random vector Hunif is then S(pHunif ) =N × S(pHnc,unif

i(hi ))+ 0.5 × log(det([TH ])) ≤ S(pHnc,unif ).

The fourth diameter random field model is constructed asthe third diameter random field model but using marginalavailable information of the second diameter random fieldmodel. For this fourth model, the joint pdf of random vectorH denoted Hbeta. We then have

S(pHnc,unif ) ≥ S(pHnc,beta ) ≥ S(pHbeta ) . (41)

The same way, if the correlation length increases, the entropydecreases:

S(pHnc,unif ) ≥ S(pHunif,L1 ) ≥ S(pHunif,L2 ) , (42)

S(pHnc,beta ) ≥ S(pHbeta,L1 ) ≥ S(pHbeta,L2 ) , (43)

where L1 < L2. The objective of the next section is to ana-lyze whether this ordering is propagated to the stochasticdynamical responses of the beam. We expect the entropy ofthe response to increase if the entropy of the input randomfield increases.

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0.5 1 1.5 2 2.5 3

x 105

10−5

10−4

10−3

Frequency (Hz)

Spe

ed (

mH

z−1 /s

)MeanUpper limit (95%)Lower limit (95%)

0 500 1000 1500 2000 2500 3000

10−4

10−3

10−2

10−1

Frequency (Hz)

Spe

ed (

mH

z−1 /s

)

Nominal

Mean

Upper limit (95%)

Lower limit (95%)

(a)

(b)

Fig. 3 Modulus of the velocity stochastic response Znc,unif (a) andzoom in the low frequency band [0, 3,000] Hz (b)

4.2.2 Analysis of the random frequency responses

Let Znc,unif , Zunif , Znc,beta and Zbeta be the random obser-vation vectors corresponding to the four random vectorsHnc,unif , Hunif , Hnc,beta and Hbeta, respectively.

Concerning random vector Znc,unif , the random responseis plotted on Fig. 3, with the 95 % confidence envelope. Fig-ure 3b shows a zoom in the frequency band [0, 3,000] Hz. Itshould be noted that the peaks of the mean random responseare translated to the left, comparing with the response of thenominal model. For example, instead of a peak in about 830Hz, the mean peak is about 760 Hz.

Figure 5 compares the marginal entropies for random vec-tors Zunif and Zbeta, for different correlation lengths. It canbe seen as expected that S(pZnc,unif

i) ≥ S(pZnc,beta

i) for almost

all frequencyωi . It should be noted that since the randomness

0 200 400 600 800 1000 1200

450

500

550

600

650

700

750

800

850

Nb elements

L1−

Nor

m

0 200 400 600 800 1000 1200

800

850

900

950

1000

1050

Nb elements

L1−

Nor

m

(a)

(b)

Fig. 4 For the uniform case, convergence analysis of the L1-norm ofthe entropy response in the frequency band [0, 314] kHz with respectto the number of beam elements a L = 0.3 m and b L = 0.005 m

1 2 3 4

x 105

−0.5

0

0.5

1

Frequency (Hz)

Mar

gina

l Ent

ropi

es R

espo

nse

Unif, L=0,300Unif, L=0,040Unif, L=0,010Unif, L=0,005Beta,L=0,300Beta, L=0,040Beta, L=0,010Beta, L=0,005

Fig. 5 Marginal entropies marginal entropies for Zunif (dark color)and Zbeta (light color) for different correlation lengths

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0 1000 2000

−1.5

−1

−0.5

0

0.5

1

1.5

Frequency (Hz)

Mar

gina

l Ent

ropi

es R

espo

nse

L=0.300

L=0.040

L=0.010

L=0.005

1.5 2 2.5 3

x 105

0

0.5

1

1.5

Frequency (Hz)

Mar

gina

l Ent

ropi

es R

espo

nse L=0.300

L=0.040

L=0.010

L=0.005

(a)

(b)

Fig. 6 Marginal entropies of the response for different correlationlengths of input uniform probabilistic model. a low frequencies [0,3,000] Hz and b high frequencies [100, 300] kHz

introduced in the model translates the frequency response tolower or higher frequencies, the comparison is not strictlycarried out frequency by frequency but peak by peak. For theuniform case, Fig. 4 shows the convergence of the L1-normof the entropy response in the frequency band [0, 314] kHzwith respect the spatial discretization. It can be seen onthis figure that, as expected, the convergence is faster forlarge correlation lengths. For the smallest correlation length(L = 0.005 m), a good convergence is reached for 500 beamelements (corresponding to 0.002 m for each beam element).Figure 6 details this response for different frequency bands.Since the upcoming conclusion is similar for Zunif and Zbeta,only Zunif is analyzed. Figure 6a shows that, for low frequen-cies, the entropy of the response increases, as the correlationlength increases, which is surprising with regard to the rela-tion S(pHnc,unif ) ≥ S(pHunif ). This result can be explainedusing stochastic homogenization theory (see [14,17]): the

size of the representative volume element (RVE) increaseswith respect to the spatial correlation length. Therefore, forlarge wavelengths, the variability of the response increaseswith respect to the spatial correlation length and then theentropy increases. For higher frequencies, Fig. 6b, the spa-tial wavelengths of the mode shapes are smaller, therefore,no homogenization takes place any more and the expectedresult S(pZnc,unf

i) ≥ S(pZunif

i) is obtained for each frequency

ωi . These results mean that in the low frequency band, themissing of the correlation structure information related tothe input random fields is not conservative and can lead toan underestimation of the uncertainty related to the outputresponse of the structure.

5 Conclusions

A methodology for the entropy propagation analysis in adynamical structure has been presented. This methodologyis based on the MaxEnt construction of the probability distri-bution of the input parameters with more or less informationand in the estimation of the resulting entropy for quantitiesof interest. Concerning the application: (1) For the case ofuncorrelated input random fields, the sensitivity of the outputentropy with respect to the input entropy is low in the lowfrequency range but this sensitivity increases with respectto the frequency due to the decrease of the involved spatialwavelengths for the random displacement. (2) For the case ofcorrelated input random fields, the output entropy increasesas the correlation length increases (and then input entropydecreases) in the low frequency range and decreases as thecorrelation length increases for higher frequencies. Theseresults which can be explained by the stochastic homogeni-sation which takes place in the low frequency range showthat the correlation structure of the input random fields hasto be taken into account in order to avoid an underestima-tion of the output entropy. This result, which is not reachableusing a classical variance-based sensitivity analysis is of pri-mary importance for an engineering point of view and showsthat a missing of information in the construction of the inputprobability model can underestimate the output uncertaintyand then, in the case of a reliability analysis of a dynamicalstructure, the probability of failure can be underestimated.

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