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Stochastic modeling of uncertainties in computationalstructural dynamics - Recent theoretical advances

Christian Soize

To cite this version:Christian Soize. Stochastic modeling of uncertainties in computational structural dynamics - Re-cent theoretical advances. Journal of Sound and Vibration, Elsevier, 2013, 332 (10), pp.2379-2395.�10.1016/j.jsv.2011.10.010�. �hal-00743699�

Stochastic modeling of uncertainties in computationalstructural dynamics - Recent theoretical advances

C. Soize∗

Universite Paris-Est, Laboratoire Modelisation et Simulation Multi-Echelle, MSME UMR 8208 CNRS, 5 bdDescartes, 77454 Marne-la-Vallee Cedex 2, France

Abstract

This paper deals with a short overview on stochastic modeling of uncertainties. Weintroduce the types of uncertainties, the variability of real systems, the types of prob-abilistic approaches, the representations for the stochastic models of uncertainties, theconstruction of the stochastic models using the maximum entropy principle, the prop-agation of uncertainties, the methods to solve the stochastic dynamical equations, theidentification of the prior and the posterior stochastic models, the robust updating ofthe computational models and the robust design with uncertain computational models.We present recent theoretical advances in this field concerning the parametric and thenonparametric probabilistic approaches of uncertainties in computational structural dy-namics for the construction of the prior stochastic models of both the uncertainties onthe computational model parameters and on the modeling uncertainties, and for theiridentification with experimental data. We also present the construction of the posteriorstochastic model of uncertainties using the Bayesian method when experimental dataare available.

Keywords: Uncertainties, stochastic modeling, structural dynamics, computationalmodel.

1. Introduction

This paper is devoted to a short overview on stochastic modeling of uncertainties andon related topics, including recent theoretical advances. Many references are givenin order to give an idea of the large number of works published in this field. How-ever, this area is too vast to being exhaustive. We have then limited the references tothose adapted to the guideline of the synthesis proposed in this paper. In Section 3,an overview on stochastic modeling of uncertainties is presented, introducing the typesof uncertainties, the variability of real systems, the types of probabilistic approachesand the representations for the stochastic models of uncertainties. The constructionof the prior stochastic models using the maximum entropy principle is recalled. The

∗Corresponding authorEmail address:christian.soize@univ-paris-est.fr (C. Soize)

Preprint submitted to Elsevier October 19, 2012

different problems related to the propagation of uncertainties and the methods to solvethe stochastic dynamical equations are briefly presented. The fundamental problemrelative to the identification of the prior and the posterior stochastic models is devel-oped. All these tools allow the robust updating of the computational models and therobust design with uncertain computational models to be performed. Sections 4 and 5deal with recent theoretical advances concerning the parametric and the nonparamet-ric probabilistic approaches of uncertainties in computational structural dynamics forthe construction of the prior stochastic models of both the uncertainties on the com-putational model parameters and on the modeling uncertainties. Finally, Section 6 isdevoted to the construction of the posterior stochastic model of uncertainties using theBayesian method when experimental data are available.

2. Comments concerning notation used

In this paper, the following notations are used:(1) A lower case letter is a real deterministic variable (e.g.x).(2) A boldface lower case letter is a real deterministic vector (e.g.x = (x1, . . . , xN ).(3) An upper case letter is a real random variable (e.g.X).(4) A boldface upper case letter is a real random vector (e.g.X = (X 1, . . . , XN)).(5) An upper case letter between brackets is a real deterministic matrix (e.g.[A ]).(6) A boldface upper case letter between brackets is a real random matrix (e.g.[A]).(7) Any deterministic quantities above (e.g.x, x, [A ]) with an underline (e.g.x, x, [A ])means that these deterministic quantities are related to the mean model (or to the nom-inal model).

3. Short overview on stochastic modeling of uncertainties and on related topics

3.1. Uncertainties and variability

Thedesigned systemis used to manufacture thereal systemand to construct the nomi-nal computational model (also called themean computational modelor sometimes, themean model) using a mathematical-mechanical modeling process for which the mainobjective is the prediction of the responses of the real system in its environment. Thereal system, submitted to a given environment, can exhibit a variability in its responsesdue to fluctuations in the manufacturing process and due to small variations of the con-figuration around a nominal configuration associated with the designed system. Themean computational model which results from a mathematical-mechanical modelingprocess of the design system, has parameters which can be uncertain. In this case,there areuncertainties on the computational model parameters. In an other hand, themodeling process induces some modeling errors defined as themodeling uncertainties.It is important to take into account both the uncertainties on the computational modelparameters and the modeling uncertainties to improve the predictions of computationalmodels in order to use such a computational model to carry out robust optimization,robust design and robust updating with respect to uncertainties. Today, it is well under-stood that, as soon as the probability theory can be used, then the probabilistic approachof uncertainties is certainly the most powerful, efficient and effective tool for modeling

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and for solving direct and inverse problem. The developments presented in this paperare limited to the probabilistic approaches.

3.2. Types of approach for stochastic modeling of uncertaintiesTheparametric probabilistic approachconsists in modeling theuncertain parametersof the computational modelby random variables and then in constructing the stochas-tic model of these random variables using the available information. Such an approachis very well adapted and very efficient to take into account the uncertainties on thecomputational model parameters as soon as the probability theory can be used. Manyworks have been published in this field and a state-of-the-art can be found, for instance,in [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Nevertheless, the parametric probabilistic approachdoes not allow the modeling uncertainties to be taken into account (see for instance[12, 13]).

Concerningmodeling uncertaintiesinduced bymodeling errors, it is today well un-derstood that the prior and posterior stochastic models of the uncertain parameters ofthe computational model are not sufficient and do not have the capability to take intoaccount modeling uncertainties in the context of computational mechanics (see for in-stance [12, 13, 14, 15]). Two main methods can be used to take into account modelinguncertainties (modeling errors).(i) The first one consists in introducing a stochastic model of theoutput-prediction-error, considered as a noise, which is the difference between the real system outputand the computational model output [12]. In this approach, the modeling errors and themeasurements errors are simultaneously taken into account and cannot really be sepa-rately identified. Note that if no experimental data are available, such a method cannotbe used because, generally, there are no information for constructing the stochasticmodel of such a noise. Such an approach is simple and efficient but requires exper-imental data. It should be noted that a lot of experimental data are required in highdimension. However, with such an approach, the posterior stochastic model of theuncertain parameters of the computational model strongly depends on the stochasticmodel of the noise which is added to the model output and which is often unknown.In addition, for many problems, it can be necessary to take into account the modelingerrors at the operators level of the mean computational model (for instance, to take intoaccount the modeling errors on the mass and stiffness operators in order to analyze thegeneralized eigenvalue problem related to a dynamical system, or, if the design param-eters of the computational model are not fixed but run through an admissible set ofvalues in the context of the robust design optimization).(ii) The second one is based on thenonparametric probabilistic approachof model-ing uncertainties (modeling errors) which has been proposed in [13] as an alternativemethod to the output-prediction-error method in order to take into account modelingerrors at the operators level and not at the model output level by the introduction of anadditive noise. It should be noted that such an approach allows a prior stochastic modelof modeling uncertainties to be constructed even if no experimental data are available.The nonparametric probabilistic approach is based on the use of a reduced-order modeland the random matrix theory. It consists in directly constructing the stochastic mod-eling of the operators of the mean computational model. The random matrix theory

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[16] (see also [17, 18, 19] for such a theory in the context of linear acoustics) is used toconstruct the prior probability distribution of the random matrices modeling the uncer-tain operators of the mean computational model. This prior probability distribution isconstructed by using the Maximum Entropy Principle [20] (from Information Theory[21]) for which the constraints are defined by the available information [13, 14, 22, 15].Since the paper [13], many works have been published in order to validate the nonpara-metric probabilistic approach of model uncertainties with experimental results (see forinstance [23, 24, 25, 26, 27, 28, 15, 29]), to extend the applicability of the theoryto other areas [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], to extend the theoryto new ensembles of positive-definite random matrices yielding a more flexible de-scription of the dispersion levels [42], to apply the theory for the analysis of complexdynamical systems in the medium-frequency range, including vibroacoustic systems,[43, 44, 23, 45, 25, 26, 27, 28, 46, 47, 48, 39], to analyze nonlinear dynamical systems(i) for local nonlinear elements [49, 50, 37, 51, 52, 53, 54] and (ii) for distributed non-linear elements or nonlinear geometrical effects [55].

Concerning the coupling of the parametric probabilistic approach of the uncertaintieson the computational model parameters with the nonparametric probabilistic approachof modeling uncertainties (modeling errors), a methodology has recently been pro-posed in computational mechanics [56]. Thisgeneralized probabilistic approachofuncertainties in computational dynamics uses the random matrix theory. The proposedapproach allows the prior stochastic model of each type of uncertainties (uncertain-ties on the computational model parameters and modeling uncertainties) to be sepa-rately constructed and identified. The modeling errors are not taken into account withthe usual output-prediction-error method but with the nonparametric probabilistic ap-proach.

3.3. Types of representation for the stochastic modeling of uncertaintiesA fundamental question is the construction of the prior stochastic model of uncer-tainties on the computational model parameters and also the prior stochastic model ofmodeling uncertainties. Such a prior stochastic model can then be used to study thepropagation of uncertainties through the mechanical system which is analyzed. If ex-perimental data are available for the mechanical system, then they can be used (1) toidentify the parameters of the prior stochastic model [29] using, for instance, the max-imum likelihood method [57, 58] or (2) to construct a posterior stochastic model [12]using, for instance, the Bayesian method (see for instance [59, 60, 61, 62, 58]).

Two main methods are available to construct the prior stochastic model of a randomvectorθ �→ X(θ) = (X1(θ), . . . , XN(θ)) defined on a probability space(Θ, T ,P),with values inRN and whose probability distribution onRN is denoted byPX(dx) (thisrandom vector can be the approximation in finite dimension of a stochastic process ora random field). It should be noted the two following methods can also be used forrandom matrices, random fields, etc. We introduce the spaceL2

N of all the second-order random variables such that

E{‖X‖2} =

∫

RN

‖x‖2 PX(dx) < +∞ , (1)

4

in which E is the mathematical expectation, wherex = (x1, . . . , xN ) and‖x‖2 =x21 + . . .+ x2

N is the square of the Euclidean norm of vectorx.

(i)- The first method is a direct approach which consists in directly constructing theprobability distributionPX(dx) onRN in using, for instance, the maximum entropyprinciple (see Section 3.4).

(ii) - The second one is an indirect approach which consists in introducing a repre-sentationX = h(Ξ) for which X is the transformation by a deterministic nonlinear(measurable) mappingh (which has to be constructed) of aR np -valued random vari-ableΞ = (Ξ1, . . . ,Ξnp

) whose probability distributionPΞ(dξ) is given and then isknown. ThenPX is the transformation ofPΞ by mappingh. Two main types of meth-ods can be used.

(ii.1) - The first one corresponds to the spectral methods such as the Polynomial Chaosrepresentations (see [63, 64] and also [65, 66, 67, 68, 69, 70, 71, 72, 73]) which can beapplied in infinite dimension for stochastic processes and random fields, which allowthe effective construction of mappingh to be carried out and which allow any randomvariableX in L2

N , to be written as

X = Σ+∞j1=0 . . .Σ

+∞jnp=0 aj1,...,jnp

ψj1(Ξ1)× . . .× ψjnp(Ξnp

) , (2)

in which ψjk are given real polynomials related to the probability distribution ofΞ

and whereaj1,...,jnpare deterministic coefficients inRN . Introducing the multi-index

α = (j1, . . . , jnp) and the multi-dimensional polynomialsψα(Ξ) = ψj1(Ξ1)× . . .×

ψjnp(Ξnp

) of random vectorΞ, the previous polynomial chaos decomposition can berewritten as

X = Σα aα ψα(Ξ) . (3)

The multi-dimensional polynomialsψα(Ξ) are orthogonal and normalized,

E{ψα(Ξ)ψβ(Ξ)} =

∫

Rnp

ψα(ξ)ψβ(ξ)PΞ(dξ) = δαβ , (4)

in which δαα = 1 andδαβ = 0. The coefficientsaα = aj1,...,jnpare vectors inRN

which completely define mappingg and which are given by

aα = E{Xψα(Ξ)} . (5)

The construction ofh then consists in identifying the vector-valued coefficientsa α. IfΞ is a normalized Gaussian random vector, then the polynomials are the normalizedHermite polynomials. Today, many applications of such an approach have been carriedout for direct and inverse problems. We refer the reader to [65, 74, 75, 76, 77, 78,79, 80, 81] and in particular, to Section 3.7 for a short overview concerning the iden-tification and inverse stochastic problems related to the parametric and nonparametricprobabilistic approaches of uncertainties.

(ii.2) - The second one consists in introducing a prior algebraic representationX =

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h(Ξ, s) in which s is a vector parameter with small dimension (which must be identi-fied), whereΞ is a vector-valued random variable with a given probability distributionPΞ and whereh is a given nonlinear mapping. For instance, tensor-valued randomfields representations constructed with such an approach can be found in [82, 83, 84].

In theory, method (ii.1) allows any random vectorX in L 2N to be represented by a

Polynomial Chaos expansion. In practice, the representation can require a very largenumber of coefficients to get convergence yielding very difficult problems in high di-mension for the identification problem and then requires adapted methodologies andadapted methods [85, 86]. In general, method (ii.2) does not allow any random vectorX in L2

N to be represented but allows a family of representations to be constructed in asubspace ofL2

N whens runs through all the admissible space ofs (but, in counter part,the identification ofs is effective and efficient).

3.4. Construction of the stochastic models using the maximum entropy principle underthe constraints defined by the available information

The measure of uncertainties using the entropy has been introduced by Shannon [21]in the framework of Information Theory. The maximum entropy principle (that is tosay the maximization of the level of uncertainties) has been introduced by Jaynes [20]to construct the prior stochastic model of a random variable under the constraints de-fined by the available information. This principle appears as a major tool to constructthe prior stochastic model (1) of the uncertain parameters of the computational modelusing the parametric probabilistic approach, (2) of both the uncertainties on the compu-tational model parameters and modeling uncertainties, using the nonparametric proba-bilistic approach and (3) of the generalized approach of uncertainties corresponding toa full coupling of the parametric and nonparametric probabilistic approaches.Let x = (x1, . . . , xN ) be a real vector and letX = (X1, . . . , XN ) be a second-orderrandom variable with values inRN whose probability distributionPX is defined by aprobability density functionx �→ pX(x) onRN with respect to dx = dx1 . . . dxN andwhich verifies the normalization condition

∫RN pX(x)dx = 1. In fact, it is assumed

that X is with values in any bounded or unbounded partX of RN and consequently,the support ofpX is X . The available information defines a constraint equation onRµ written asE{g(X)} = f in whichE is the mathematical expectation,f is a givenvector inRµ and wherex �→ g(x) is a given function fromRN into Rµ. Let C be theset of all the probability density functionsx �→ pX(x) defined onRN , with values inR+, with supportX , verifying the normalization condition and the constraint equationE{g(X)} = f. The maximum entropy principle consists in findingpX in C whichmaximizes the entropy (that is to say which maximizes the uncertainties),

pX = argmaxp∈C

S(p) , S(p) = −∫

RN

p(x) log(p(x))dx (6)

in which S(p) is the entropy of the probability density functionp. Introducing theLagrange multiplierλ ∈ Lµ ⊂ Rµ (associated with the constraint) whereLµ is thesubset ofRµ of all the admissible values forλ, it can easily be seen that the solutionof the optimization problem can be written as

pX(x) = c0 1X (x) exp(− < λ, g(x) >) , ∀x ∈ RN (7)

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in which< x , y >= x1y1+. . .+xµyµ and where1X is the indicatrix function of setX .The normalization constantc0 and the Lagrange multiplierλ are calculated in solvinga nonlinear vectorial algebraic equation deduced from the normalization condition andfrom the constraint equation. This algebraic equation can be solved using appropriatedalgorithms. Then, it is necessary to construct a generator of independent realizations ofrandom variableX whose probability density function is that which has been built. Insmall dimension (N is a few units), there is no difficulty. In high dimension (N hundredsor thousands), there are two major difficulties. The first one is related to the calculationof an integral in high dimension of the typec0

∫X

g(x) exp(− < λ, g(x) >)dx. Sucha calculation is necessary to implement the algorithm for computingc 0 andλ. The sec-ond one is the construction of the generator oncec 0 etλ have been calculated. Thesetwo aspects can be solved using the Markov Chain Monte Carlo methods (MCMC) (seefor instance [62, 87, 88, 58]). The transition kernel of the homogeneous (stationary)Markov chain of the MCMC method can be constructed using the Metropolis-Hastingsalgorithm [89, 90, 91, 92] or the Gibbs algorithm [93] which is a slightly different al-gorithm for which the kernel is directly derived from the transition probability densityfunction and for which the Gibbs realizations are always accepted. These two algo-rithms construct the transition kernel for which the invariant measure isPX. In general,these algorithms are effective but can not be when there are regions of attraction thatdo not correspond to the invariant measure. These situations can not be easily detectedand are time consuming. Recently, an approach [41] of the class of the Gibbs methodhas been proposed to avoid these difficulties and is based on the introduction of an Itostochastic differential equation whose unique invariant measure isPX and is the ex-plicit solution of a Fokker-Planck equation [94]. The algorithm is then obtained by thediscretization of the Ito equation.

3.5. Random Matrix Theory

The random matrix theory were introduced and developed in mathematical statistics byWishart and others in the 1930s and was intensively studied by physicists and math-ematicians in the context of nuclear physics. These works began with Wigner in the1950s and received an important effort in the 1960s by Wigner, Dyson, Mehta andothers. In 1965, Poter published a volume of important papers in this field, followed,in 1967 by the first edition of the Mehta book whose second edition [16] published in1991 is an excellent synthesis of the random matrix theory. We refer the reader to [19]and [18] for an introduction to random matrix theory presented in the context of Me-chanics. Concerning multivariate statistical analysis and statistics of random matrices,the reader will find additional developments in [95] and [96].

3.5.1. Why the Gaussian orthogonal ensemble cannot be used if positiveness propertyis required

The Gaussian Orthogonal Ensemble (GOE), for which the mean value is the unitymatrix [In] (see [22]), is the set of random matrices[GGOE]) which can be written as[GGOE] = [In] + [BGOE] in which [BGOE] belongs to the GOE (see [16]), for which themean value is the zero matrix[0], and consequently, would be a second-order centeredrandom matrix with values inM S

n(R) such thatE{[BGOE]} = [ 0 ] andE{‖[BGOE]‖2F } <

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+∞, and for which the probability density function, with respect tod[B] = 2n(n−1)/4

Π1≤i≤j≤n d[B]ij , is written as

p[BGOE]([B]) = Cn × exp

{− (n+1)

4δ2tr{[B]2}

}. (8)

The constantCn of normalization can easily be calculated andδ is the coefficient ofvariation of the random matrix[GGOE] which is such thatδ2 = n−1E{‖ [GGOE] −[In] ‖2F } because‖ [In] ‖2F = n. The real-valued random variables{[BGOE]jk, j ≤ k}are statistically independent, second order, centered and Gaussian. It can be seen that[GGOE] is with values inMS

n(R) but is not positive definite. In addition,

E{‖[GGOE]−1‖2} = +∞ . (9)

Consequently,[GGOE] is not acceptable if positiveness property and integrability of theinverse are required.

3.5.2. Ensemble SG+0 of random matricesThe GOE cannot be used when positiveness property and integrability of the inverseare required. Consequently, we need new ensembles of random matrices which will beused to develop the nonparametric probabilistic approach of uncertainties in compu-tational solid and fluid mechanics and which differ from the GOE and from the otherknown ensembles of the random matrix theory.

The objective of this section is then to summarize the construction given in [13, 14, 22,15] of the ensemble SG+0 of random matrices[G0] defined on the probability space(Θ, T ,P), with values in the setM+

n (R) of all the positive definite symmetric(n× n)real matrices and such that

E{[G0]} = [In] , E{log(det[G0])} = C , |C| < +∞ . (10)

The probability distributionP[G0] = p[G0]([G ]) dG is defined by a probability densityfunction [G ] �→ p[G0]([G ]) fromM+

n (R) intoR+ with respect to the volume element

dG on the setMSn(R) of all the symmetric(n × n) real matrices, which is such that

dG = 2n(n−1)/4 Π1≤j≤k≤n dGjk . This probability density function can then verifythe normalization condition,

∫

M+n (R)

p[G0]([G ]) dG = 1 . (11)

Let‖A‖2F =∑n

j=1

∑nk=1[A ]2jk be the square of the Frobenius norm of symmetric real

matrix [A]. Let δ be the positive real number defined by

δ =

{E{‖ [G0]− E{[G0]} ‖2F}

‖E{[G0]} ‖2F

}1/2

=

{1

nE{‖ [G0]− [In] ‖2F }

}1/2

, (12)

which will allow the dispersion of the stochastic model of random matrix[G 0] to becontrolled. Forδ such that0 < δ < (n+1)1/2(n+5)−1/2, the use of the maximum en-tropy principle under the constraints defined by the above available information yields

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the following algebraic expression of the probability density function of random matrix[G0],

p[G0]([G ]) = 1M+n (R)([G ])× CG0

×(det [G ]

)(n+1) (1−δ2)

2δ2 × e−(n+1)

2δ2tr[G ] , (13)

in which1M+n (R)([G ]) is equal to 1 if[G ] ∈ M+

n (R) and is equal to zero if[G ] /∈M+

n (R), where tr[G ] is the trace of matrix[G ], wheredet [G ] is the determinant ofmatrix [G ] and where the positive constantCG0

is such that

CG0=(2π)−n(n−1)/4

(n+ 1

2δ2

)n(n+1)(2δ2)−1{Πn

j=1Γ(n+1

2δ2+

1−j

2

)}−1

, (14)

and where, for allz > 0,Γ(z) =∫ +∞

0 tz−1 e−t dt. Note that{[G0]jk, 1 ≤ j ≤ k ≤ n}are dependent random variables.

Let ‖G‖ = sup‖x‖≤1 ‖[G ] x‖ be the operator norm of matrix[G ] which is such that‖[G ] x‖ ≤ ‖G‖ ‖x‖ for all x in Rn. Let ‖G‖F be the Frobenius norm of[G ] whichis defined by‖G‖2F = tr{[G ]T [G ]} =

∑nj=1

∑nk=1[G ]2jk and which is such that

‖G‖ ≤ ‖G‖F ≤ √n ‖G‖. It is proven [14] that

E{‖[G0]−1‖2} ≤ E{‖[G0]

−1‖2F } < +∞ . (15)

In general, the above equation does not imply thatn �→ E{‖[G 0]−1‖2} is a bounded

function with respect ton, but, in the present case, we have the following fundamentalproperty,

∀n ≥ 2 , E{‖[G0]−1‖2} ≤ Cδ < +∞ , (16)

in whichCδ is a positive finite constant which is independent ofn but which dependsonδ. The above equation means thatn �→ E{‖[G0]

−1‖2} is a bounded function from{n ≥ 2} intoR+.

The generator of independent realizations (which is required to solve the random equa-tions with the Monte Carlo method) can easily be constructed using the following al-gebraic representation. Random matrix[G0] can be written (Cholesky decomposition)as[G0] = [L]T [L] in which [L] is an upper triangular(n×n) random matrix such that:

(1) random variables{[L]jj′ , j ≤ j′} are independent;

(2) for j < j ′, the real-valued random variable[L]jj′ is written as[L]jj′ = σnUjj′ inwhichσn = δ(n + 1)−1/2 and whereUjj′ is a real-valued Gaussian random variablewith zero mean and variance equal to1;

(3) for j = j ′, the positive-valued random variable[L]jj is written as[L]jj = σn

√2Vj

in whichσn is defined above and whereVj is a positive-valued gamma random variablewhose probability density function ispVj

(v) = 1R+(v) 1Γ(aj)

vaj−1 × e−v, in which

aj = n+12δ2 + 1−j

2 . It should be noted that the probability density function of eachdiagonal element[L]jj of the random matrix[L] depends on the rankj of the element.

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3.5.3. Ensemble SG+ε of random matricesLet L2

n be the set of all the second-order random variables, defined on probabilityspace(Θ, T , P ), with values inRn, equipped with the inner product≪ X,Y ≫=E{< X ,Y >} and with the associated norm|||X||| =≪ X,Y ≫1/2.

Let 0 ≤ ε ≪ 1 be a positive number as small as one wants. The ensemble SG+ε is

defined as the ensemble of all the random matrices which are written as

[G] =1

1 + ε{[G0] + ε [In]} , (17)

in which [G0] is a random matrix which belongs to ensemble SG+0 .

Let [G] be in SG+ε with ε ≥ 0 fixed as small as one wants (possibly,ε can be equal tozero and in such a case, SG+

ε = SG+0 and then,[G] = [G0]). It can easily be seen

thatE{[G]} = [In] . (18)

Letb(X,Y) be the bilinear form onL2n×L2

n such thatb(X,Y) =≪ [G]X ,Y ≫. Forall X in L2

n, we haveb(X,X) ≥ cε|||X|||2 (19)

in which cε = ε/(1 + ε). The proof can easily be obtained. We haveb(X,X) =1/(1 + ε) ≪ [G0]X ,X ≫ +ε/(1 + ε)|||X|||2 ≥ cε|||X|||2, because, for all[G0] inSG+

0 , and for allX in L2n, we have≪ [G0]X ,X ≫ ≥ 0.

Finally, for all ε ≥ 0, it can be proven that

E{‖[G]−1‖2F

}< +∞ . (20)

3.6. Propagation of uncertainties or what are the methods to solve the stochastic dy-namical equations

Concerning the methods and formulations to analyze the propagation of uncertain-ties in the computational model, the choice of a specific method depends on the de-sired accuracy on the model output and on the nature of the expected probabilisticinformation. These last two decades, a growing interest has been devoted to spec-tral stochastic methods, which provide an explicit representation of the random modeloutput as a function of the basic random parameters modeling the input uncertain-ties [63, 64, 97, 98, 99]. An approximation of the random model output is soughton suitable functional approximation bases. There are two distinct classes of tech-niques for the definition and computation of this approximation. The first class iscomposed by direct simulation methods [100, 74, 101, 102, 103, 104]. These meth-ods are often called non-intrusive since they offer the advantage of only requiring theavailability of classical deterministic codes. The second class of techniques rely onGalerkin-type projections of weak solutions of models involving differential or partialdifferential equations [? 106, 107, 68, 108, 109, 103, 110, 111, 112]. However, forapplications with high complexity, alternative solution techniques must be provided

10

in order to drastically reduce computational requirements. Recently, some alterna-tive methods, based on the construction of optimal separated representations of thesolution, have been proposed. They consist in representing the solution on optimal re-duced bases of deterministic functions and stochastic functions (scalar-valued randomvariables). Several strategies have been proposed for the extraction of reduced basesfrom approximate Karhunen-Loeve expansions (or classical spectral decompositions)of the solution [113, 103]. Another method, called Generalized Spectral Decomposi-tion method, has been recently proposed for the construction of such representationswithout knowinga priori the solution nor an approximation of it [69, 114, 70]. A majoradvantage of these algorithms is that they allow a separation of deterministic problemsfor the computation of deterministic functions and stochastic algebraic equations forthe computation of stochastic functions. In that sense, they can be considered as partlynon-intrusive techniques for computing Galerkin-type projections. However, it doesnot circumvent the curse of dimensionality associated with the dramatic increase inthe dimension of stochastic approximation spaces when dealing with high stochasticdimension and high approximation resolution along each stochastic dimension. Re-cently, multidimensional extensions of separated representations techniques have beenproposed [115, 116]. These methods exploit the tensor product structure of the solutionfunction space, resulting from the product structure of the probability space defined byinput random parameters.Finally, the direct Monte Carlo numerical simulation method (see for instance [117, 87,88] is a very effective and efficient method because this method (1) is non-intrusive,(2) is adapted to massively parallel computation without any software developmentsand (3) is such that its convergence can be controlled during the computation and (4)the speed of convergence is independent of the dimension. The speed of convergenceof the Monte Carlo method can be improved using advanced Monte Carlo simulationprocedures [118, 119, 120, 121], subset simulation technics [122], important samplingfor high dimension problems [123], local domain Monte Carlo Simulation [124].

3.7. Identification of the prior and posterior stochastic models of uncertainties

The identification of the parameters of the stochastic model of uncertainties (paramet-ric and nonparametric probabilistic approach) is of the class of the statistical inverseproblems (see for instance [62]).

3.7.1. Identification of the stochastic model of random variables in the context of theparametric probabilistic approach of uncertainties on the computational modelparameters

Let us assume that experimental data are available for observations related to the ran-dom computational model output. The experimental identification of the parameters ofthe prior probability distribution of the random variable modeling uncertainties of thecomputational model parameters can be performed, either in minimizing a distance be-tween the experimental observations and the random model observations (least-squaremethod for instance), or in using the maximum likelihood method (see for instance[57],[58]). Such an approach is described in Sections 4.5 and 5.3, and many applica-tions can be found in the literature. In the domain of structural dynamics, vibrations

11

and vibroacoustics, we refer, for instance, the reader to [125] for the stochastic systemidentification for operational modal analysis, [77] for the stochastic inversion in acous-tic scattering, [126] for the stochastic modeling of a nonlinear dynamical system usedfor producing voice and [127] for the computational model for long-range non-linearpropagation over urban cities. With such an identification, we then obtained an optimalprior probability distribution. The Bayesian method then allows a posterior probabilitydistribution to be constructed from the optimal prior probability distribution and theexperimental data. Many works have been published in the literature (see for instancetextbooks on the Bayesian method such as [59, 60, 61, 62] and papers devoted to theuse of the Bayesian method in the context of uncertain mechanical and dynamical sys-tems such as [12, 128, 129, 130, 131, 132, 133]. We will use such a Bayesian approachin Sections 4.6 and 6.

3.7.2. Identification of the stochastic model of random matrices in the context of thenonparametric probabilistic approach of both the uncertainties on the compu-tational model parameters and modeling uncertainties

General developments concerning the experimental identification of the parameters ofthe prior probability distribution of random matrices used for modeling uncertaintiesin computational mechanics can be found in [15]. Many works have been publishedin this field, such as [79] in micromechanics, [78] for the identification and predictionof stochastic dynamical systems in a polynomial chaos basis, [23] for the experimentalidentification of the nonparametric stochastic model of uncertainties in structural dy-namics [134, 24, 29], in structural acoustics for the low- and medium-frequency ranges[26, 28], in nonlinear structural dynamics [49, 50]. The identification of the gener-alized probabilistic approach of uncertainties can be found in [56] and also below inSections 5.3 and 6.

4. Parametric probabilistic approach of uncertainties in computational structuraldynamics

4.1. Introduction of the mean computational model in computational structural dy-namics

The developments are presented for computational models in structural dynamics. Thedynamical system is a damped fixed structure around a static equilibrium configurationconsidered as a natural state without prestresses and subjected to an external load.For given nominal values of the parameters of the dynamical system, the basic finiteelement model is called themean computational model. In addition, it is assumed thata set of model parameters has been identified as uncertain model parameters. Thesemodel parameters are the components of a vectorx = (x1, . . . , xnp

) belonging to anadmissible setCpar which is a subset ofRnp . The dynamical equation of the meancomputational model is then written as

[M (x)] y(t) + [D(x)] y(t) + [K(x)]y(t) + fNL(y(t), y(t); x) = f(t; x) , (21)

in whichy(t) is the unknown time response vector of them degrees of freedom (DOF)(displacements and/or rotations);y(t) andy(t) are the velocity and acceleration vectors

12

respectively;f(t; x) is the known external load vector of them inputs (forces and/ormoments);[M (x)], [D(x)] and [K(x)] are the mass, damping and stiffness matricesof the mean computational model, which belong to the setM +

m(R) of all the positive-definite symmetric(m×m) real matrices;(y, z) �→ fNL(y, z; x) is the nonlinear map-ping modeling local nonlinear forces.

4.2. Introduction of the reduced mean computational model

In the context of the parametric probabilistic approach of model-parameter uncertain-ties, the parameterx is modeled by a random variableX. The mean value ofX willbe the nominal valuex = (x1, . . . , xnp

) of the uncertain model parameterx and thesupport of its probability distribution onRnp is Cpar.

For allx fixed inCpar ∈ Rnp , let{φ1(x), . . . ,φm(x)} be an algebraic basis ofRm con-structed, for instance, either with the elastic modes of the linearized system or of theunderlying linear system, or with the POD (Proper Orthogonal Decomposition) modesof the nonlinear system). Below, it is assumed that the elastic modes of the underlyinglinear system are used. In such a framework, there are two main possibilities to con-struct the reduced mean computational model.

(1) The first one consists in solving the generalized eigenvalue problem associated withthe mean mass and stiffness matrices forx fixed to its nominal valuex. We then obtainthe elastic modes of the nominal mean computational model which are independent ofx and which depend only onx which is fixed. In this case, whenx runs throughC par,matrices[M (x)] and[K(x)] have to be projected on the subspace spanned by the elasticmodes of the nominal mean computational model. For very large computational model(m can be several tens of millions) such an operation is not easy to perform with usualcommercial softwares which often are black boxes.(2) The second one consists in solving the generalized eigenvalue problem associatedwith the mean mass and stiffness matrices for each requiredx belonging toC par. Inthis case, the elastic modes of the mean computational model depend onx. In the con-text of the parametric probabilistic approach of model-parameter uncertainties, we thenhave to solve a random generalized eigenvalue problem and such an approach is betteradapted to usual commercial softwares and allows a fast convergence to be obtainedwith respect to the reduced order dimension. In addition, some algorithms have beendeveloped in this context for random eigenvalue problems of large systems [135, 136].In order to limit the developments, we will focus the presentation using this second ap-proach. The extension to the first approach is straightforward from a theoretical pointof view (see for instance [13]). Finally, it should be noted that the random generalizedeigenvalue problem can also be considered in a polynomial chaos decomposition forwhich an efficient approach has been proposed [137]. Such an ingredient can be addedwithout difficulty in the developments presented below but would induce additionaldifficulties in the understanding which could mask the ideas of the method proposed.

For each value ofx given inCpar, the generalized eigenvalue problem associated with

13

the mean mass and stiffness matrices is written as

[K(x)]ϕ(x) = λ(x) [M (x)]ϕ(x) , (22)

for which the eigenvalues0 < λ1(x) ≤ λ2(x) ≤ . . . ≤ λm(x) and the associatedelastic modes{ϕ1(x),ϕ2(x), . . .} are such that

< [M (x)]ϕα(x) ,ϕβ(x)>= µα(x) δαβ , (23)

< [K(x)]ϕα(x) ,ϕβ(x)>= µα(x)ωα(x)2 δαβ , (24)

in whichωα(x) =√λα(x) is the eigenfrequency of elastic modeϕα(x) whose nor-

malization is defined by the generalized massµα(x) and where<u , v>=∑

j ujvj isthe Euclidean inner product of the vectorsu andv. For each value ofx given inC par,the reduced mean computational model of the dynamical system is obtained in con-structing the projection of the mean computational model on the subspaceH n of Rm

spanned by{ϕ1(x), . . . ,ϕn(x)} with n ≪ m. Let [φ(x)] be the(m × n) real matrixwhose columns are vectors{ϕ1(x), . . . ,ϕn(x)}. The generalized forcef(t; x) is anRn-vector such thatf(t; x) = [φ(x)]T f(t; x). For all x in Cpar, the generalized mass,damping and stiffness matrices[M(x)], [D(x)] and[K(x)] belong to the setM +

n (R) ofall the positive-definite symmetric(n× n) real matrices, and are defined by

[M(x)]αβ = µα(x) δαβ , [D(x)]αβ =< [D(x)]ϕβ(x) ,ϕα(x)> , (25)

[K(x)]αβ = µα(x)ωα(x)2 δαβ , (26)

in which, generally,[D(x)] is a full matrix. Consequently, for allx in C par and for allfixed t, the reduced mean computational model of the dynamical system is written asthe projectionyn(t) of y(t) onHn which yields

yn(t) = [φ(x)] q(t) ,

(27)

[M(x)] q(t) + [D(x)] q(t) + [K(x)] q(t) + FNL(q(t), q(t); x) = f(t; x) ,

in whichFNL(q(t), q(t); x) = [φ(x)]T fNL([φ(x)] q(t), [φ(x)] q(t); x) and whereq(t) ∈Rn is the vector of the generalized coordinates. In the particular case for whichfNL = 0, the corresponding equation, in the frequency domain, is written as

−ω2[M(x)] q(ω) + iω[D(x)] q(ω) + [K(x)] q(ω) = f(ω; x) . (28)

in whichq(ω) =∫R e−iωtq(t) dt andf(ω; x) =

∫R e−iωtf(t; x)dt

Below, we will denote byn0 the value ofn for which the responseyn is converged toy, with a given accuracy, for all values ofx in Cpar.

4.3. Methodology of the parametric probabilistic approach of model-parameter uncer-tainties

The methodology of the parametric probabilistic approach of model-parameter uncer-tainties consists in modeling the uncertain model parameterx (whose nominal value is

14

x) by a random variableX defined on a probability space(Θ, T ,P), with values inR np .Consequently, the generalized matrices in Eq. (27) become random matrices[M(X)],[D(X)] and[K(X)] and, for all fixedt, the generalized external forcef(t; x) becomes arandom vectorf(t;X). The mean values of these random matrices are denoted by[M ],[D] and[K]. We then have

E{[M(X)]} = [M ] , E{[D(X)]} = [D] , E{[K(X)]} = [K] , (29)

in whichE is the mathematical expectation. It should be noted that the mean matrices[M ], [D] and [K] are different from the matrices[M(x)], [D(x)] and [K(x)] of themean (nominal) computational model. The parametric probabilistic approach of un-certainties then consists in replacing the mean computational model by the followingstochastic reduced computational model,

Y(t) = [φ(X)]Q(t) , (30)

[M(X)] Q(t) + [D(X)] Q(t)+ [K(X)]Q(t) +FNL(Q(t), Q(t);X) = f(t;X) , (31)

in which for all fixedt, Y(t) is anRm-valued random vector andQ(t) is anRn-valuedrandom vector. As soon as the stochastic model of random vectorX is constructed,the stochastic computational model defined by Eqs. (30) and (31) can be solved usingthe methods presented in Section 3.6, in particular the direct Monte Carlo method,taking into account thatnp can be very large and that we have to solve a reduced-orderstochastic computational model.

4.4. Construction of the prior stochastic model of model-parameter uncertainties

The unknown probability distribution ofX is assumed to be defined by a probabilitydensity functionx �→ pX(x) fromRnp intoR+ = [0 ,+∞[. Under the assumption thatno experimental data are available to constructpX, the prior model can be constructedusing the maximum entropy principle (see Section 3.4). For such a construction, theavailable information has to be defined. Sincex belongs toCpar, the support ofpX mustbeCpar and the normalization condition must be verified. We then have,

supppX = Cpar ⊂ Rnp ,

∫

Rnp

pX(x)dx =

∫

CparpX(x)dx = 1 . (32)

Since the nominal value ofx is x ∈ Cpar, an additional available information consistsin writing that the mean valueE{X} of X is equal tox which yields the followingconstraint equation, ∫

Rnp

x pX(x)dx = x . (33)

In general, an additional available information can be deduced from the analysis of themathematical properties of the solution of the stochastic computational model underconstruction. The random solutionQ of the stochastic computational model defined byEq. (31) must be a second-order vector-valued stochastic process (because the dynami-cal system is stable) which means that, for allt, we must haveE{‖Q(t)‖2} < +∞. Inorder that such a property be verified, it is necessary to introduce a constraint which can

15

always be written as the equationE{g(X)} = γ onRµX , in whichγ = (γ1, . . . , γµX)

is a given vector inRµX with µX ≥ 1 and wherex �→ g(x) = (g1(x), . . . , gµX(x)) is a

given mapping fromRnp intoRµX . Consequently, the additional available informationdefines the following constraint equation,

∫

Rnp

g(x) pX(x)dx = γ . (34)

Let C be the set of all the probability density functionspX defined onRnp with valuesinR+ such that Eqs. (32) to (34) hold. The prior modelpX ∈ C can then be constructedusing the maximum entropy principle (see Section 3.4). Since Eq. (34) is a vectorialequation of dimensionµX , the solutionpX of the maximum entropy principle dependson the freeRµX -valued parameterγ. However, parameterγ has generally no physicalmeaning and it is better to expressγ in terms of anRµX -valued parameterδX whichcorresponds to a well defined statistical quantity for random variableX. In general,δ X

does not run throughRµX but must belong to an admissible setCX which is a subsetof RµX . Consequently,pX depends onx andδX and is rewritten as

x �→ pX(x; x, δX) with (x, δX) ∈ Cpar× CX ⊂ Rnp × RµX . (35)

4.5. Estimation of the parameters of the prior stochastic model of the uncertain modelparameter

The value ofn is fixed to the valuen0 for which, for all values ofx in Cpar, the responseyn is converged toywith a given accuracy. The prior stochastic modelpX(x; x, δX) ofrandom variableX relative to the model-parameter uncertainties, depends on parame-tersx andδX belonging to admissible setsCpar andCX . If no experimental data areavailable, thenx is fixed to the nominal value andδX must be considered as a param-eter to perform a sensitivity analysis of the stochastic solution. Such a prior stochasticmodel of model-parameter uncertainties then allows the robustness of the solution tobe analyzed in function of the level of model-parameter uncertainties controlled byδ X .

For the particular case for which a few experimental data exist,x can be updated andδX can be estimated with the experimental data. The updating ofx and the estimationof δX must then be performed with observations of the systems for which experi-mental data are available. There are several possibilities in the choice of such observa-tions satisfying these criteria (for instance, the first eigenfrequencies and the associatedstructural elastic modes if an experimental modal analysis is available (see for instance[138]), the frequency response functions on a given frequency band, etc). LetW be therandom vector which is observed. For all(x, δX) ∈ Cpar× CX ⊂ Rnp ×RµX , and forall w fixed, the probability density function ofW is denoted bypW(w; x, δX). On theother hand, it is assumed thatνexp independent experimental valueswexp

1 , . . . ,wexpνexp are

measured. The optimal value(x opt, δoptX ) of (x, δX) can be estimated by maximizing

the logarithm of the likelihood function (maximum likelihood method [57],[58]),

(x opt, δoptX ) = arg max

(x,δX )∈Cpar×CX

{νexp∑

r=1

log pW(wexpr ; x, δX)} . (36)

16

The quantitiespW(wexpr ; x, δX) are estimated using the independent realizations ofW

calculated with the stochastic computational model (using the Monte Carlo method)and the multivariate Gaussian kernel density estimation method (see for instance [139,140]).

4.6. Posterior probability model of uncertainties using output-prediction-error methodand the bayesian method

Let ppriorX (x) = pX(x; xopt, δ

optX ) be the optimal prior probability density function ofX,

constructed with the optimal value(xopt, δoptX ) of (x, δX) which has been determined

in Section 4.5, solving the optimization problem defined by Eq. (36). As we haveexplained, the parametric probabilistic approach does not have the capability to takeinto account the modeling errors (modeling uncertainties). A possible method to takeinto account the modeling errors is to use the output-prediction-error method (with thedifficulties that we have indicated in Section 3.2-(i)). With such an approach the poste-rior probability density functionppost

X (x) of X can be estimated using the experimentaldata associated with observationW introduced in Section 4.5 and the Bayesian method.

In order to apply the Bayesian method [59, 58, 61, 60] to estimate the posterior proba-bility density functionppost

X (x) of random vectorX, the output-prediction-error method[141, 12, 62] is used and then, an additive noiseB is added to the observation in orderto simultaneously take into account the modeling errors and the measurements errorsas explained in Section 3.2. The random observed outputW out with values inRmobs,for which experimental datawexp

1 , . . . ,wexpνexp are available (see Section 4.5), is then

written asWout = W + B , (37)

in which W is the computational model output with values inRmobs. The noiseBis aRmobs-valued random vector, defined on probability space(Θ, T ,P), which isassumed to be independent ofX, and consequently, which is independent ofW. Itshould be noted that this hypothesis concerning the independence ofW andB couldeasily be removed. It is also assumed that the probability density functionp B(b) ofrandom vectorB, with respect todb, is known. For instance, it is often assumed thatBis a centered Gaussian random vector for which the covariance matrix[C B] = E{BBT }is assumed to be invertible. In such a case, we would have

pB(b) = (2π)−mobs/2 (det[CB])−1/2 exp(−1

2< [CB]

−1b , b >) . (38)

The posterior probability density functionppostX (x) is then estimated by

ppostX (x) = L(x) pprior

x (x) , (39)

in whichx �→ L(x) is the likelihood function defined onRnp , with values inR+, suchthat

L(x) =Π

νexpr=1 pWout|X(w

expr |x)

E{Πνexpr=1 pWout|X(w

expr |Xprior)}

. (40)

17

In Eq. (40),pWout|X(wexpℓ |x) is the values, for the experimental data, of the conditional

probability density functionw �→ pWout|X(w|x) of the random observed outputWout

givenX = x in Cpar. Taking into account Eq. (37) and asB andW are assumed to beindependent, it can easily be deduced that

pWout|X(wexpr |x) = pB(wexp

r − h(x)) , (41)

in which w = h(x) is the model observation depending on{yn(t), t ∈ J } whichis computed using Eq. (27). It should be noted that the posterior probability densityfunction strongly depends on the choice of the stochastic model of the output additivenoiseB.

5. Nonparametric probabilistic approach of uncertainties in computational struc-tural dynamics. The generalized probabilistic approach

The nonparametric probabilistic approach of uncertainties has been introduced in [13,14, 15] to take into account both the model-parameter uncertainties and the modelingerrors inducing model uncertainties without separating the contribution of each oneof these two types of uncertainties. Recently, in [56], an improvement of the non-parametric approach of uncertainties, called the generalized probabilistic approach ofuncertainties, has been proposed and allows the prior stochastic model of each typeof uncertainties (model-parameter uncertainties and modeling errors) to be separatelyconstructed. Below, we summarized this approach which allows modeling errors to betaken into account.

5.1. Methodology to take into account model uncertainties (modeling errors)

Let (Θ′, T ′,P ′) be another probability space. The probability space(Θ, T ,P) is de-voted to the stochastic model of model-parameter uncertainties using the parametricprobabilistic approach (see Section 4), while(Θ ′, T ′,P ′) is devoted to the stochas-tic model of model uncertainties (modeling errors) using the nonparametric proba-bilistic approach. Therefore, for allx fixed in Cpar, the matrices[M(x)], [D(x)] and[K(x)] are replaced by independent random matrices[M(x)] = {θ ′ �→ [M(θ′; x)]},[D(x)] = {θ′ �→ [D(θ′; x)]} and [K(x)] = {θ′ �→ [K(θ′; x)]} on probability space(Θ′, T ′,P ′) and defined, below, in Section 5.2. The nonparametric probabilistic ap-proach of uncertainties then consists in replacing in Eq. (31) the dependent randommatrices[M(X)], [D(X)] and [K(X)] by the dependent random matrices[M(X)] ={(θ, θ′) �→ [M(θ′;X(θ))]}, [D(X)] = {(θ, θ′) �→ [D(θ′;X(θ))]} and[K(X)] = {(θ, θ′) �→[K(θ′;X(θ))]}, defined onΘ×Θ′. It can easily be deduced that

E{[M(X)]} = [M ] , E{[D(X)]} = [D] , E{[K(X)]} = [K] . (42)

Consequently, the stochastic reduced computational model, defined by Eqs. (30) and(31), is replaced by the following stochastic reduced computational model,

Y(t) = [φ(X)]Q(t) , (43)

[M(X)] Q(t) + [D(X)] Q(t) + [K(X)]Q(t) + FNL(Q(t), Q(t);X) = f(t;X) , (44)

18

in which for all fixed t, Y(t) = {(θ, θ ′) �→ Y(θ, θ′; t)} is anRm-valued randomvector andQ(t) = {(θ, θ′) �→ Q(θ, θ′; t)} is anRn-valued random vector definedon Θ × Θ′. Let X(θ) be any realization of random variableX for θ in Θ. For all xin Cpar, let [M(θ′; x)], [D(θ′; x)] and[K(θ′; x)] be any realizations of random matrices[M(x)], [D(x)], [K(x)] for θ′ in Θ′. The realizationY(θ, θ′; t) of the random variableY(t) and the realizationQ(θ, θ ′; t) of the random variableQ(t) verify the deterministicequations

Y(θ, θ′; t) = [φ(X(θ))]Q(θ, θ′; t) , (45)

[M(θ′;X(θ))] Q(θ, θ′; t) + [D(θ′;X(θ))] Q(θ, θ′; t)

+[K(θ′;X(θ))]Q(θ, θ′; t) + FNL(Q(θ, θ′; t), Q(θ, θ′; t);X(θ))

= f(t;X(θ)) . (46)

5.2. Construction of the prior stochastic model of the random matricesAs explained in [56], the dependent random matrices[M(X)], [D(X)] and[K(X)], in-troduced in Eq. (44), are written as

[M(X)]=[LM (X)]T [GM ] [LM (X)] ,

[D(X)]=[LD(X)]T [GD] [LD(X)] , (47)

[K(X)]=[LK(X)]T [GK ] [LK(X)] ,

in which, for all x in Cpar, [LM (x)], [LD(x)] and [LK(x)] are the upper triangu-lar matrices such that[M(x)] = [LM (x)]T [LM (x)], [D(x)] = [LD(x)]T [LD(x)] and[K(x)] = [LK(x)]T [LK(x)]. In Eq. (47), the random matrices[GM ], [GD] and[GK ]are random matrices which are defined on probability space(Θ ′, T ′,P ′) with values inM+

n (R). The joint probability density function of these random matrices[G M ], [GD]and [GK ] is constructed using the maximum entropy principle under the constraintsdefined by the available information. Taking into account the available information(see [13, 14, 15, 56]), it is proven that these three random matrices are statisticallyindependent, belong to ensemble SG+

ε defined in Section 3.5.3, and then depend onfree positive real dispersion parametersδM , δD andδK which allow the level of thestatistical fluctuations, that is to say the level of model uncertainties, to be controlled.Let δG = (δM , δD, δK) be the vector of the dispersion parameters, which belongs toan admissible setCG ⊂ R3. Consequently, the joint probability density function of therandom matrices[GM ], [GD] and[GK ] is written as

([GM ], [GD], [GK ]) �→ pG([GM ], [GD], [GK ]; δG) = p[GM ]]([GM ]; δM )

× p[GD ]]([GD]; δD) × p[GK ]]([GK ]; δK) , δG ∈ CG ⊂ R3 . (48)

The probability distributions of random matrices[GM ], [GD] and[GK ] and their alge-braic representations useful for generating independent realizations[G M (θ′)], [GD(θ′)]and[GK(θ′)] are explicitly defined in Section 3.5.3 using Section 3.5.2. From Eq. (47)and for(θ, θ′) in Θ×Θ′, it can be deduced that

[M(θ′;X(θ))] = [LM (X(θ))]T [GM (θ′)] [LM (X(θ))] , (49)

[D(θ′;X(θ))] = [LD(X(θ))]T [GD(θ′)] [LD(X(θ))] , (50)

[K(θ′;X(θ))] = [LK(X(θ))]T [GK(θ′)] [LK(X(θ))] . (51)

19

5.3. Estimation of the parameters of the prior stochastic model of uncertainties

The prior stochastic model of uncertainties then depends on parametersx, δX andδG

belonging to the admissible setsCpar, CX andCG. If no experimental data are available,thenx is fixed to the nominal value andδX andδG must be considered as parametersto perform a sensitivity analysis of the stochastic solution. Such a prior stochasticmodel of both model-parameter uncertainties and model uncertainties then allows therobustness of the solution to be analyzed in function of the level of model-parameteruncertainties controlled byδX and in function of model uncertainties controlled byδG.

If a few experimental data are available, the methodology presented in Section 4.5 canthen be used to updatex and to estimateδX andδG. As previously, letW be the randomvector which is observed and which is deduced from the random solutionY given byEq. (43) with Eq. (44). For all(x, δX , δG) ∈ Cpar× CX × CG, the probability density

function ofW is denoted bypW(w; x, δX , δG). The optimal value(x opt, δoptX , δ

optG ) of

(x, δX , δG) can be estimated by maximizing the logarithm of the likelihood function,

(x opt, δoptX , , δ

optG ) = arg max

(x,δX ,,δG)∈Cpar×CX×CG

{νexp∑

r=1

log pW(wexpr ; x, δX , δG} .

(52)Similarly to Section 4.5, the quantitiespW(wexp

r ; x, δX , δG) are estimated using theindependent realizations ofW calculated with the stochastic computational model de-fined by Eqs. (43) and (44) (using the Monte Carlo method) and the multivariate Gaus-sian kernel density estimation method.

6. Posterior stochastic model of uncertainties using the Bayesian method

Let ppriorX (x) = pX(x; xopt, δ

optX ) andpG([GM ], [GD], [GK ]; δ

optG ) be the optimal prior

probability density functions of random vectorX and random matrices[G M ], [GD],[GK ], constructed with the optimal value(xopt, δ

optX , δ

optG ) of (x, δX , δG) which has

been calculated in Section 5.3 solving the optimization problem defined by Eq. (52).The objective of this section is the following. For this given optimal prior probabilitymodelpG([GM ], [GD], [GK ]; δ

optG ) of model uncertainties induced by modeling errors,

the posterior probability density functionppostX (x) of model-parameter uncertainties

is estimated using the experimental data associated with observationW and using thestochastic computational model, the optimal prior probability density functionp

priorX (x)

and the Bayesian method.

6.1. Posterior probability density function of the model-parameter uncertainties

Let wexp1 , . . . ,wexp

νexp be theνexp independent experimental data corresponding to obser-vationW, introduced in Section 4.5 and used in Section 5.3. The Bayesian method (seefor instance [12, 59, 60, 61, 57, 58, 142, 141]) allows the posterior probability densityfunctionppost

X (x) to be calculated by

ppostX (x) = L(x) pprior

x (x) , (53)

20

in whichx �→ L(x) is the likelihood function defined onRnp , with values inR+, suchthat

L(x) = Πνexpr=1 pW|X(w

expr |x)

E{Πνexpr=1 pW|X(w

expr |Xprior)}

. (54)

In Eq. (54),pW|X(wexpℓ |x) is the value, for the experimental data, of the conditional

probability density functionw �→ pW|X(w|x) of the random observationW givenX =x in Cpar. Equation (54) shows that the likelihood functionL must verify the followingequation,

E{L(Xprior)} =

∫

Rnp

L(x) ppriorX (x)dx = 1 . (55)

For a high dimension uncertain model parameter, the usual Bayesian method presentedabove can be not efficient for partial experimental data. In such a case a novel approachderived from Baye’s approach has been proposed in [143].

6.2. Posterior probability density functions of the responses

Let U = (V,W) be the random response vector in whichW is experimentally observedand whereV is not experimentally observed. Random response vectorU is constructedfrom the random solutionY of Eqs. (43) and (44). The probability density functionu �→ pUpost(u) of the posterior random response vectorUpost is then given by

pUpost(u) =∫

Rnp

pU|X(u|x) ppostX (x)dx , (56)

in which the conditional probability density functionpU|X(u|x) of U, givenX = x,is constructed using the stochastic computational model defined in Sections 5.1 and5.2. From Eqs. (53) and (56), it can be deduced that the posterior probability densityfunctionpUpost can be written aspUpost(u) = E{L(Xprior) pU|X(u|Xprior)}. Let Upost

k

be any component of random vectorUpost. The probability density functionuk �→pU

postk

(uk) onR of the posterior random variableU postk is then given by

pU

postk

(uk) = E{L(Xprior) pUk|X(uk|Xprior)} , (57)

in whichuk �→ pUk|X(uk|x) is the conditional probability density function of the realvalued random variableUk givenX = x and which is constructed using the computa-tional model defined in Sections 5.1 and 5.2.

6.3. Computational aspects

We use the notation introduced in Sections 5.1 and 5.2 concerning the realizations.Let Xprior(θ1), . . . ,Xprior(θν) beν independent realizations ofXprior whose probabil-

ity density function isx �→ ppriorX (x). For ν sufficiently large, the right-hand side of

Eq. (57) can be estimated by

pU

postk

(uk) ≃1

ν

ν∑

ℓ=1

L(Xprior(θℓ)) pUk|X(uk|Xprior(θℓ)) . (58)

21

Let ([GM (θ′1)], [GD(θ′1)], [GK(θ′1)]), . . . , [GM (θ′ν ′)], [GD(θ′ν ′)], [GK(θ′ν ′)]) beν ′ in-dependent realizations of([GM ], [GD], [GK ]) whose probability density function is([GM ], [GD], [GK ]) �→ pG([GM ], [GD], [GK ]; δ

optG ). For fixedθℓ, the computational

model defined in Sections 5.1 and 5.2 is used to calculate theν ′ independent realiza-tionsU(θ′1|x), . . . ,U(θ′ν ′ |x) for x = Xprior(θℓ). We can then deduceW(θ ′1|Xprior(θℓ)),

. . . , W(θ′ν ′ |Xprior(θℓ)) and, for all fixedk, we can deduceUk(θ′1|Xprior(θℓ)), . . . ,

Uk(θ′ν ′ |Xprior(θℓ)).

(1) Using the independent realizationsW(θ ′1|Xprior(θℓ)), . . . ,W(θ′ν ′ |Xprior(θℓ)) and

using the multivariate Gaussian kernel density estimation method, forr = 1, . . . , ν exp,we can estimatepW|X(w

expr |Xprior(θℓ)) and then, forℓ = 1, . . . , ν, we can deduce

L(Xprior(θℓ)) using Eq. (54).

(2) For all fixedk, pUk|X(uk|Xprior(θℓ)) is estimated using the independent realizations

Uk(θ′1|Xprior(θℓ)), . . . , Uk(θ

′ν ′ |Xprior(θℓ)) and using the kernel estimation method. From

Eq. (58), it can then be deducedpU

postk

(uk).

7. Conclusion

We have presented a brief survey concerning the state-of-the-art and the recent ad-vances in the field of stochastic modeling of uncertainties in computational structuraldynamics, their propagation and their quantification when experimental data are avail-able or not. Many interesting and important works have been published and are inprogress concerning the methodologies which allow the propagation of uncertainties tobe analyzed. Nevertheless, an excellent stochastic solver will compute a bad stochasticsolution if the stochastic model of the input (the uncertainties) is bad. In particular, if noexperimental data are available, the quality of the random responses directly dependson the quality of the prior stochastic model of uncertainties. For the complex dynami-cal systems, the major source of uncertainties is due to modeling errors and is not dueto model-parameter uncertainties. This is the reason why the overview presented inthis paper is mainly focussed on the methodologies and tools, such as the maximumentropy principle, useful to construct the prior stochastic models of uncertainties andin particular, the prior stochastic model of modeling errors using the nonparametricprobabilistic approach. This latter approach requires the use of adapted ensemblesof random matrices and has been proposed as an alternative method to the output-prediction-error method in order to take into account modeling errors at the operatorslevel and not at the model output level by the introduction of an additive noise. Wehave then detailed the methodologies for the construction of the prior stochastic modelof model-parameter uncertainties (using the parametric probabilistic approach) and theprior stochastic model of modeling errors (using the nonparametric probabilistic ap-proach) in computational structural dynamics. In addition, when experimental data areavailable for some observations of the dynamical system, we have presented method-ologies to perform the identification of the stochastic model of uncertainties, that isto say, the uncertainties quantification. When a few experimental data are available

22

for some observations of the dynamical system, these experimental data can be usedto identify, with the maximum likelihood method, an optimal prior stochastic modelof uncertainties. If a large experimental data basis is available the posterior stochasticmodel of system-parameter uncertainties, in presence of modeling errors in the com-putational model, can be estimated with the bayesian method.

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