robustness analysis by a probabilistic approach for propagation of uncertainties in a component mode...

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Robustness analysis by a probabilistic approach for propagation of uncertainties in a component mode synthesis context S.-A. Chentouf a , N. Bouhaddi a,n , C. Laitem b a Institut Femto-st, UMR 6174,De´partement de Me´canique Applique ´e R.Chale´at -Universite´de, Franche Comte´- 24 Chemin de l’Epitaphe, 25000 Besanc - on, France b ALSTOM Transport, 7 avenue de Lattre de Tassigny, 25290 Ornans, France article info Article history: Received 15 September 2010 Received in revised form 4 April 2011 Accepted 27 April 2011 Available online 18 May 2011 Keywords: Dynamics Uncertainties Non-parametric Robustness Combined approximations CMS abstract Modelling uncertainties in an industrial application require a thorough knowledge of their sources and types. Uncertainties can be split into aleatory and epistemic types. Using parametric and non-parametric methods successively can be an adapted approach to model these uncertainties types on a given finite elements model (FEM). However, we propose in this paper to proceed more appropriately by introducing a hybrid approach combining the parametric and non-parametric methods. This approach consists of applying, on a given FEM, parametric and non-parametric methods simultaneously with respect to uncertainties types of each model region. Complexity and size of industrial FEMs often impose model reductions. This introduces necessarily the problem of reduction basis robustness. We are interested in the effectiveness of two methods for model reduction in the case of a hybrid model of uncertainties. We consider the case of component mode synthesis (CMS) based on normal modes of clamped interfaces components. Therefore, we analyze robustness of two methods based on improved Craig–Bampton’s basis: the first one is enriched by static residual vectors (ESRV), the second one is a variant of the combined approximations method (VCA) adapted to CMS. Finally, a dynamic application on a railway electric motor stator, allows comparing methods’ performances in terms of robustness and gain in computing time. Conclusion highlights relevance of the combined approximations method when using a hybrid approach for modelling uncertainties. & 2011 Elsevier Ltd. All rights reserved. 1. Introduction Construction of numerical models to predict the dynamical behaviour of complex assemblies is a real challenge for manufacturers in different industrial areas. Usually, in design step, a numerical model is built in order to predict a mean dynamical response of the system. However, presence of different types of uncertainties can have significant effects on the dynamical behaviour of this model. Consequently, a deterministic mean, or nominal, model is not sufficient to analyze structure dynamics. Robust analysis is usually inevitable during design and it is of a great importance for manufacturers, particularly in railway industry. Improving robustness analysis depends on the process of identification, modelling and propagating of uncertainties for a studied structure. According to previous works, different uncertainties appellations can be cited [1,2]. Nevertheless one can distinguish epistemic and aleatory uncertainties as two main categories. Epistemic or reducible uncertainties are due to a lack of knowledge in mechanism modelling of some physical phenomena such as interfaces or material behaviour. Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/jnlabr/ymssp Mechanical Systems and Signal Processing 0888-3270/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2011.04.014 n Corresponding author. Tel.: þ33 3 81 66 60 56; fax: þ33 3 81 66 67 00. E-mail addresses: [email protected] (S.-A. Chentouf), [email protected] (N. Bouhaddi), [email protected] (C. Laitem). Mechanical Systems and Signal Processing 25 (2011) 2426–2443

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Contents lists available at ScienceDirect

Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 25 (2011) 2426–2443

0888-32

doi:10.1

n Corr

E-m

claude.l

journal homepage: www.elsevier.com/locate/jnlabr/ymssp

Robustness analysis by a probabilistic approach for propagation ofuncertainties in a component mode synthesis context

S.-A. Chentouf a, N. Bouhaddi a,n, C. Laitem b

a Institut Femto-st, UMR 6174, Departement de Mecanique Appliquee R.Chaleat -Universite de, Franche Comte- 24 Chemin de l’Epitaphe, 25000 Besanc-on, Franceb ALSTOM Transport, 7 avenue de Lattre de Tassigny, 25290 Ornans, France

a r t i c l e i n f o

Article history:

Received 15 September 2010

Received in revised form

4 April 2011

Accepted 27 April 2011Available online 18 May 2011

Keywords:

Dynamics

Uncertainties

Non-parametric

Robustness

Combined approximations

CMS

70/$ - see front matter & 2011 Elsevier Ltd. A

016/j.ymssp.2011.04.014

esponding author. Tel.: þ33 3 81 66 60 56;

ail addresses: [email protected]

[email protected] (C. Laitem).

a b s t r a c t

Modelling uncertainties in an industrial application require a thorough knowledge of

their sources and types. Uncertainties can be split into aleatory and epistemic types.

Using parametric and non-parametric methods successively can be an adapted

approach to model these uncertainties types on a given finite elements model (FEM).

However, we propose in this paper to proceed more appropriately by introducing a

hybrid approach combining the parametric and non-parametric methods. This

approach consists of applying, on a given FEM, parametric and non-parametric methods

simultaneously with respect to uncertainties types of each model region. Complexity

and size of industrial FEMs often impose model reductions. This introduces necessarily

the problem of reduction basis robustness. We are interested in the effectiveness of two

methods for model reduction in the case of a hybrid model of uncertainties. We

consider the case of component mode synthesis (CMS) based on normal modes of

clamped interfaces components. Therefore, we analyze robustness of two methods

based on improved Craig–Bampton’s basis: the first one is enriched by static residual

vectors (ESRV), the second one is a variant of the combined approximations method

(VCA) adapted to CMS. Finally, a dynamic application on a railway electric motor stator,

allows comparing methods’ performances in terms of robustness and gain in computing

time. Conclusion highlights relevance of the combined approximations method when

using a hybrid approach for modelling uncertainties.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Construction of numerical models to predict the dynamical behaviour of complex assemblies is a real challenge formanufacturers in different industrial areas. Usually, in design step, a numerical model is built in order to predict a meandynamical response of the system. However, presence of different types of uncertainties can have significant effects on thedynamical behaviour of this model. Consequently, a deterministic mean, or nominal, model is not sufficient to analyzestructure dynamics. Robust analysis is usually inevitable during design and it is of a great importance for manufacturers,particularly in railway industry.

Improving robustness analysis depends on the process of identification, modelling and propagating of uncertainties fora studied structure. According to previous works, different uncertainties appellations can be cited [1,2]. Nevertheless onecan distinguish epistemic and aleatory uncertainties as two main categories. Epistemic or reducible uncertainties are dueto a lack of knowledge in mechanism modelling of some physical phenomena such as interfaces or material behaviour.

ll rights reserved.

fax: þ33 3 81 66 67 00.

(S.-A. Chentouf), [email protected] (N. Bouhaddi),

Nomenclature

A Generic representation for: M, C or K

cðoÞ Random vector of generalized coordinatesCm Field of complex mx1 vectorsC ,M ,K Nominal damping, mass and stiffness matricesC,M,K Random damping, mass and stiffness matricesE{ } Expected valueE Young modulusEI Flexion rigidityf ðoÞ Vector of external forcesf Vector of junction forcesfDðoÞ Vector of efforts relative to model modifica-

tionsFDðoÞ Static loadings basisI Identity matrixI Quadratic momentIN Identity matrix of (N�N) dimensionsL Lower triangular matrix issued from the Cho-

lesky factorization of A

DM,DK Mass and stiffness modifications matricesrnb Vectors issued from binomial series decom-

position of nth eigenvectorr�b Concatenation of rnb vectorsRb Improved reduction basis (variant of com-

bined approximations method)

RD Series of static residual vectorsRD Improved series of static residual vectorsT0 Condensation basis relative to the nominal

modelTs Robust condensation basis (substructure s)yðoÞ Random vector of dofs

Greek letters

d Non-parametric dispersion levelln nth eigenvalueq DensityS Diagonal matrix with positive real numberso Pulsation (rad s�1)

Subscripts

�1 Inversec Index of the condensed modelT Transpose

Operators

:�: Euclidean norm:�:F Frobenius norm

S.-A. Chentouf et al. / Mechanical Systems and Signal Processing 25 (2011) 2426–2443 2427

Aleatory or irreducible uncertainties are due to randomness of the structure parameters such as variability of its physicalproperties and assembling process or effects of its service conditions (temperature, hygrometry, etc.). This family can alsoinclude uncertainties due to variability of measuring process.

Different mathematical formalisms for modelling uncertainties can be detailed and compared (Klir [1]). Nevertheless,one can classify modelling approaches on stochastic (or probabilistic) and non-stochastic approaches. Generally,pertinence of a modelling formalism depends on the processed problem (inverse, direct) and on uncertainties type. Theprobabilistic approach is retained in this paper.

For a nominal FEM, the classical method for propagating uncertainties by a probabilistic model consists of calculatingN deterministic responses corresponding to N system perturbations entries using the stochastic finite elements method(SFEM) or a Monte Carlo simulation (MCS).

Two probabilistic methods for modelling uncertainties are used in this paper, namely parametric and non-parametricapproaches. The first one is appropriate for modelling irreducible uncertainties while the second one takes into account, ina more global manner, all types of uncertainties.

In parametric approach, uncertain parameters of the FEM such as material or geometric parameters are considered asrandom quantities. A deterministic mapping of variation bands of uncertain parameters generates random constitutivematrices of the system. This approach is widely used as an efficient method for modelling data uncertainties [2–8].However, it should be noted that it is unable to manage epistemic uncertainties. Indeed, this category is not attached orcaused by quantifiable sources. It is generally due to a lack of knowledge of local physical phenomena such as interfacesfrictions or material models. Furthermore, modelling system uncertainties, through its main parameters, supposes thatmodel parameterization is sufficient to introduce representative perturbations on constitutive matrices and thisassumption can be questionable. For instance, when modelling a multilayered thick structure by an anisotropichomogenization method, models often fail to capture local heterogeneities.

The non-parametric approach [9–14] to model uncertainties is introduced to take into account epistemic uncertaintiesin a more appropriate and global manner. Theoretical concepts of the method are developed in Ref. [9]. In this paper, thisapproach is used in a linear context. Indeed, vibration amplitudes are supposed to be sufficiently small and localnonlinearities are supposed to be negligible. Unlike parametric approach, where modelling uncertainties are focused onparameters randomness, non-parametric approach focuses on propagating uncertainties, in a global manner, on modelmatrices. Generalized matrices issued from the nominal finite elements model are replaced by random matrices.Randomness of each matrix is controlled by a real positive parameter called the dispersion parameter.

It would be interesting to carry out a hybrid approach, combining parametric and non-parametric methods, to adynamic industrial complex system. Indeed, in an assembled mechanical system, one can easily classify components andsubassemblies according to uncertainties types. This should be appreciable in design engineering where modelling

S.-A. Chentouf et al. / Mechanical Systems and Signal Processing 25 (2011) 2426–24432428

difficulties and misconceptions sources are often suspected and listed with respect to characteristics of each model region.In this paper, uncertainties are taken into account in a stator of a traction motor model through a hybrid approach in lowfrequencies domain.

In a MCS for propagating uncertainties, FEM size is an important issue since it determines numerical resources requiredto solve eigenvalue problems during a robustness analysis. Therefore, reduced models are usually introduced to carry outthis analysis with realistic computation times.

Soize [10] proposed the extension of the non-parametric approach to a dynamic reduction method. Its principleconsists of shifting the nominal non-parametric model from a large finite element model to a reduced one. Thus, modellingnon-parametric uncertainties is no longer based on large size random matrices, issued from nominal FEM, but from thereduced corresponding one.

In this paper, we chose to proceed differently by introducing uncertainties on the global nominal FEM. Consequently,random matrices are firstly constructed according to the non-parametric approach then reduced via Craig–Bampton’smethod [15]. Thus, unlike the approach proposed Ref. [10], uncertainties are introduced in the global physical model, andnot in the reduced one. In addition to the fact that this method is more global for introducing dispersion on the systemby perturbing all its physical dofs, it is essential especially when using non-parametric approach on some selectedcomponents as it will be used in the hybrid approach.

Using reduced models while propagating uncertainties introduces necessarily the concept of robustness of reduction basisespecially when uncertainties levels are very high. Furthermore, and because of the large number of eigenvalue problemstreated while propagating uncertainties by a MCS, it should be uninteresting or impossible to calculate a reduction basis foreach modified system of matrices in design engineering. Approximate reanalysis approach is used in this paper to overcomethis problem. Thus, in order to reduce numerical costs, reduction basis of each modified system is in fact an improvement of theonly nominal, or non-modified, reduction basis. Many approaches exist for improvement of reduction basis [16–21]. In thispaper, a variant of the combined approximations method (VCA) is proposed and adapted to component mode synthesis (CMS).It is used and compared to the enrichment by static residual vectors (ESRV) based on Craig–Bampton’s method.

Ultimately, the guideline of this paper is to perform a robustness analysis of a traction motor stator. A hybrid methodfor modelling parametric and non-parametric uncertainties is used. Because of finite elements models’ large sizes, thisanalysis is carried out in a reduced model space. Problem relates to an approximate reanalysis under parametric and non-parametric uncertainties models, which can be considered as local perturbations. The VCA method is adapted to CMS andapproximate reanalysis approach and is compared to the ESRV performances.

This paper is organized as follows:

In Section 2, dynamic equations issued from the parametric and non-parametric modelling uncertainties methods arebriefly presented in a reduced model context.

Propagating uncertainties scheme using a MCS is exposed and the approximate reanalysis is introduced. – The ESRV principle, based on Craig–Bampton’s normal modes components, is briefly presented. – A variant of the combined approximations method (VCA) is than detailed and adapted to component mode synthesis

(CMS) based on Craig–Bampton’s modes components.

– Section 3 is devoted to numerical applications. ESRV and VCA methods’ performances are firstly compared under parametric

and non-parametric approaches applied to an academic example. Industrial application is then used to validate efficiency ofthe proposed VCA. A hybrid method for modelling uncertainties is applied to a railway stator electric motor model providedby ALSTOM Transport, Ornans (France). This assembled structure was modelled by three dimensional finite elements andcontains about 115,000 dofs. VCA method adapted to CMS is finally compared to classical Craig–Bampton’s method in anapproximate reanalysis scheme. Two methods are confronted to exact reanalysis.

2. Robustness analysis performed in a context of parametric and non-parametric uncertainties approach

2.1. Parametric and non-parametric approaches for modelling uncertainties

We consider vibrations of a free linear structure, weakly damped, without prestresses and oscillating around anequilibrium state. We are interested in the behaviour of this structure in a low-frequency band. Then we can write for eachpulsation o in this band, the dynamic matrix equation of correspondent (m�m) deterministic finite element model:

ð�o2Mþ ioCþK ÞyðoÞ ¼ f ðoÞ ð1Þ

where

yðoÞ and f ðoÞ are, respectively, the Cm vectors of dofs and external forces, – M , C and K are real positive-definite symmetric (m�m) matrices. They correspond, respectively, to mass, damping and

stiffness FEM matrices.

We consider that this equation describes behaviour of the nominal model structure. We use this formulation tointroduce parametric and non-parametric approaches for modelling uncertainties.

S.-A. Chentouf et al. / Mechanical Systems and Signal Processing 25 (2011) 2426–2443 2429

From a practical standpoint, one can consider that simulating uncertainties’ effects on a dynamical system consists instudying its response to local or global perturbations introduced on its FEM matrices.

2.1.1. Parametric model for aleatory or irreducible uncertainties

This approach [6–8] consists firstly of investigating nominal model by identifying all potentially influent parameters onits dynamical response. It consists secondly of considering these parameters as independent random variables {Xi} thatconstitute the random vector X. We consider in this paper that parameters are modelled by uniform distributionscharacterized by their means mi and variation bands [ai�mi, bi�mi]. Finally, input variables, or parameters, randomness issimulated using a random sampling method. In this paper, we chose to use the Latin Hypercube method.

Thus, one can transform Eq. (1) to a random finite element matrix equation, where M , K and C are, respectively,replaced by M(X), K(X) and C(X) representing, respectively, the random mass, stiffness and damping matrices, dependingon random vector of parameters X. yðoÞ is replaced by yðoÞ, which is the Cm random vector of dofs.

2.1.2. Non-parametric model for epistemic uncertainties

The non-parametric approach [9,10,13] allows building a probabilistic model from the nominal one by replacing itsgeneralized matrices by random mass (M), stiffness (K) and damping (C) matrices such that one can write Eq. (1) as

ð�o2Mþ ioCþKÞyðoÞ ¼ f ðoÞ ð2Þ

where f ðoÞ is deterministic.If we consider thatA 2 M,K ,Cf g, so A are (N�N) real (dim(A)¼N), diagonal random matrices such as:A¼ AþDA, where A, A and DA correspond, respectively, to: non-parametric, nominal and variation matrices.According to the non-parametric approach, randomness on A is ensured by introducing the random matrices GA such as

A¼ LTAGALA ð3Þ

where LTALA ¼ A. The probability distribution of random matrices GA is derived from the problem of maximum of entropy

[9]. GA, and consequently A, are independent random matrices with a dispersion level controlled by a positive real dA,independently from N. Furthermore, construction of random matricesA has to verify the following properties:

EfGAg ¼ GA ¼ INA and Ef:G�1A :2

F go1 ð4Þ

where E GAf g is the expected value of GA and :G�1A :2

F ¼ TrðG�1A G�T

A Þ is the square of the Frobenius norm.dA is defined by the following expression:

dA ¼Ef:GA�GA:

2

F g

:GA:2

F

( )1=2

ð5Þ

and verifies 0odAoffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðn0þ1Þ=ðn0þ5Þ

pwith 1rn0rN.

2.2. Model reduction adapted to the propagation of uncertainties: approximate reanalysis approach

FEM size consists of a major problem when using a MCS method for propagating uncertainties. Indeed, model sizedetermines numerical resources required while solving eigenvalue problems during a robustness analysis. Thus, it shouldbe more appropriate to carry out this analysis in a reduced model context: numerical resources gain is appreciablebetween solving (N�N) or (m�m), (m5N), eigenvalue problem. Very large size models of assembled structures arecommonly used in many industrial areas. Therefore, dynamical sub-structuring or component mode synthesis is often anappropriate approach to reduce problem size. Let us recall that the main idea of this method consists firstly ondecomposing global model to sub-structures, secondly on condensing each substructure through a reduction basis Ti andfinally on assembling reduced sub-structures models as superelements. Because of its efficiency and its numericalimplementation simplicity, Craig–Bampton’s reduction method [15] is widely used in industrial applications. However,construction of reduction Craig–Bampton’s basis Ti for each modified problem, while propagating uncertainties with astochastic approach, remains a numerical problem. An economic solution consists of using approximate reanalysistechniques (Fig. 1). Main objective of this solution is to improve robustness of reduction basis in order to ensure both animportant reduction ratio and a reasonable precision on predicting modified model behaviour. Indeed, one candemonstrate, on one hand that recalculating reduction basis, at each iteration, has a dissuasive numerical cost, and onthe other hand, that using nominal reduction basis is insufficient to predict behaviour of modified system. Theapproximate reanalysis proposes a compromise between these two aspects.

Many approaches exist for improvement of reduction basis [16–21]. In this paper, the variant of the combinedapproximations method is proposed and adapted to CMS and approximate reanalysis. It is used and compared toEnrichment by Static Residual Vectors based on the Craig–Bampton’s method performances method under parametric andnon-parametric analyses.

Fig. 1. Reanalysis cycle with parametric or non-parametric perturbations in a CMS context. : approximate reanalysis:Ts¼T0, : approximate

reanalysis: Ts¼ improved T0 and : exact reanalysis.

S.-A. Chentouf et al. / Mechanical Systems and Signal Processing 25 (2011) 2426–24432430

2.3. Improvement of reduction basis robustness in a CMS context using Craig–Bampton’s normal modes components

As shown in Fig. 1, improvement in reduction basis robustness depends on modification matrices (DM,DK). Theseperturbations are issued from parametric or/and non-parametric uncertainties modelling approaches. Improvementmethods treated in this paper deals with perturbations independently from the approach chosen for modellinguncertainties.

We note that Craig–Bampton’s method is used in this paper as a reference method of model reduction. It is based on theenrichment of the Guyan static transformation by normal modes of clamped interfaces sub-structures. The basisconstituted from these normal modes associated with static transformation is an excellent Ritz basis, whose validityrange increases considerably compared to a standard Guyan transformation.

When used in a reanalysis approach, one can show that increasing normal Craig–Bampton’s modes number in thenominal basis, which is relative to nominal model (T0), improves basis robustness up to a certain perturbation level.However, improving robustness by this process is often monotonous or inefficient for high level modifications. Principle ofimprovement methods is to complete or to enrich nominal reduction basis in an efficient manner, by taking into account inthe improved reduction basis Ts, global modification matrices (DM,DK).

2.3.1. Enrichment by static residual vectors (ESRV)

The main idea of the method is to complete, in an efficient manner [17,18], Craig–Bampton’s basis with vectors issuedfrom static responses due to system modifications matrices. These vectors are susceptible to improve reduction basiscapacity of predicting behaviour of the modified system.

Let us consider dynamical equilibrium of the modified substructure, which is not subject to any dynamical strain on allof its dofs:

½ðZ ðoÞþDZðoÞ�yðoÞ ¼ 0 ð6Þ

with

ZðoÞ ¼ K�o2M and DZðoÞ ¼DK�o2DM ð7Þ

S.-A. Chentouf et al. / Mechanical Systems and Signal Processing 25 (2011) 2426–2443 2431

One can introduce the concept of a force fDðoÞ associated to system modificationsDZðoÞ. Thus Eq. (6) is equivalent to arelationship expressing the dynamical equilibrium of nominal, or non-modified, system under fDðoÞ solicitations such asfDðoÞ ¼ �ZðoÞyðoÞ.

The dynamical response of modified system can be expressed by a classical reduction basis T0 corresponding to initialsystem, enriched with static residuals RD such as

yðoÞ � T0cðoÞþRDfDðoÞ ð8Þ

Reduction basis Ts, which is common to both nominal and modified system, is a concatenation of Craig–Bampton’sreduction basis, T0 relative to nominal system, and a series of static residual vectors associated to a series of static loads FDrepresenting system modifications DZðoÞsuch as

Ts¼ ½T0^RD� ð9Þ

The effort vector associated to system modifications fDðoÞ depends on modified system response yðoÞ. We consider thatwe can approximate yðoÞ by the response of nominal system such as yðoÞ � T0cðoÞ.

The method adapted in this paper is based on normal modes of the clamped interfaces components (used also inCraig–Bampton’s method). T0,t is a truncated basis of these modes. So we can write:

FDðoÞ ¼ �DZðoÞT0,tcðoÞ ð10Þ

where FDðoÞ is a series of static effort vectors associated to system modifications. Thus, we can calculate a series of staticresidual vectors associated to system modifications by RD ¼ K�1FDðoÞ, where K�1is the inverse of stiffness modified matrix.

We should note that the ESRV method can be used in a parametric modifications context where FDðoÞ is aconcatenation of series of vectors related to successive parameters modifications. It can also be used in a non-parametriccontext where modifications are localized in some model regions characterized by different dispersion levels and whereFDðoÞ is also a concatenation of different series of vectors. Therefore, RD vectors are not necessarily independent. In orderto select more appropriate enrichment vectors and to ensure that selected vectors are independent, singular valuedecomposition is carried out on RDand ‘‘m’’ dominant directions are kept.

After calculating reduction basis and condensing each substructure, assembling phase is carried out. Restitution phasefor each substructure is finally done to express results on global physical dofs.

2.3.2. Proposed variant of combined approximations method (VCA)

Efficiency of the method has been shown in reanalysis applications [19–21]. Principle of the method (Kirsch [19]) is toconstruct, for a given structure, a robust reduction basis in an efficient manner by a binomial series expansion of some ofits initial modes.

In this section, the main steps for establishing reduction basis by the combined approximations method are summarized.

2.3.2.1. Variant of combined approximations method principle. Let us consider dynamical equilibrium of modified sub-structure, which is not subject to any dynamical strain on any of its dofs. Let us consider that M and K are, respectively, themass and stiffness matrices of nominal or non-modified structure. For each eigenmode n, we can write:

K rn ¼ lnMrn ð11Þ

where rn and ln are, respectively, the nth eigenvector and eigenvalue of initial structure.Dynamical equilibrium of modified structure is written as

ðKþDKÞrn ¼ lnMrn ð12Þ

where rn and ln are, respectively, the nth eigenvector and eigenvalue of modified structure with

M¼MþDM ð13Þ

We can write after a left multiplication by K�1(K is regular):

rn ¼ ðIþBÞ�1lnK�1Mrn ð14Þwith

B¼ K�1DK ð15Þ

we introduce an approximation on rnsuch as

rn � lnK�1Mrn ð16Þ

so Eq. (14) can be written as

rn ¼ ðIþBÞ�1rn ð17Þ

Eq. (17) consists of a binomial series, which allows us to express modified eigenvector rn using a reduction basis rnb such as

rnb ¼ ½rn1,rn2,. . .,rns � ð18Þ

S.-A. Chentouf et al. / Mechanical Systems and Signal Processing 25 (2011) 2426–24432432

Basis vectors are calculated by the following recurrence relation:

rn1 ¼ rn

rni ¼�Brni�1i¼ f2,. . .sg:

(ð19Þ

Thus we can express eigenvectors of the modified model using vectors of the basis constructed from eigenvectors of thenominal or, non-modified model, and taking into account modifications matrices (DM,DK). We can reduce the globalproblem dimensions to

Knc yn ¼ lnMn

c yn ð20Þ

with

Knc ¼ ðr

nbÞ

T KrbMc ¼ ðrnbÞ

T Mrb ð21Þ

Then we can solve the eigenvalue problem of the reduced system and obtain

yn ¼ yn1 yn2 . . . yns� �

ð22Þ

We can finally expand results on global physical dofs by

rn ¼ rnbyn ¼ rn1yn1þrn2yn2þ . . .þrns yns ð23Þ

Let us recall that the previous steps are applied for each eigenmode n such that we can obtain finally

r�b ¼ r1b r2

b . . . rsb

h i, which is a global series taking into account contribution of all eigenvectors. Noting that r�b can

contain redundant information relative to a unique modification, we introduce an improvement to the combinedapproximations method to overcome this problem. In order to reduce redundant information, to ensure that vectors keptare independent and to select the more appropriate direction from this series of vectors, a singular value decomposition is

carried out on r�bsuch as r�b ¼USVt . We select ‘‘m’’ dominant directions and write

r�b ¼U1S1Vt1þU2S2Vt

2 ð24Þ

where S1 is a (m�m) diagonal matrix containing highest singular values in a decreasing order. Highest singular values are

selected according to a chosen criterion and it is common to take U1,S1,Vt1 such as S1ð1,1Þ=S1ðm,mÞr10�6.

Finally, we calculate an efficient sub-basis relative to system modifications by

Rb ¼U1S1 ð25Þ

According to results presented by Kirsch and Weisser [19–22], the CA method ensures robustness to reductionmethodology in condensation applications. It would be appreciable, especially for industrial applications, to propose anextension of this method in a CMS context.

2.3.2.2. Variant of combined approximations method extended to CMS approach. In this paragraph, we proposed an extensionof combined approximations method to CMS. Indeed, the method can be adapted to a system decomposed into sub-structures being applicable to each of them. The assembly of reduced sub-structures has finally to be carried out.

Let us consider the nominal model of a sub-structure (k). In order to use, as announced above, normal modes of blockedjunction, which are used also in Craig–Bampton’s method as well as in the ESRV, we consider that the sub-structure isnaturally clamped at its interfaces dofs. Mass and stiffness matrices used in this development correspond to a blocked-junctions sub-structure.

We calculate normal modes of this sub-structure by solving the following eigenvalue problem: ðK�lMÞr ¼ 0.Equilibrium of modified sub-structure, which is subject only to junction efforts f , can be written as

½ðKþDKÞ�lM�r¼ f ð26Þ

After a left multiplication of both sides of Eq. (26) by K�1, and by introducing the approximation on r such asr ¼ lK�1Mr (16), we obtain

r� ðIþBÞ�1rþðIþBÞ�1K�1f ð27Þ

where B¼ K�1DK

We can develop Eq. (27) such as

r� ðIþBÞ�1rþðIþK�1DKÞ�1K�1f ð28Þ

S.-A. Chentouf et al. / Mechanical Systems and Signal Processing 25 (2011) 2426–2443 2433

By developing ðIþK�1DKÞ�1at the first order, we can write Eq. (28) as

r� ðIþBÞ�1rþðI�K�1DKÞK�1f ð29Þ

or

r� ðIþBÞ�1rþK�1f�K�1DKK�1f ð30Þ

K�1DKK�1f is considered as a second order term and will be neglected such as rcan be approximated by

r� ðIþBÞ�1rþK�1f ð31Þ

which can be written as

r� rbyþRf ð32Þ

with R¼ K�1(K is regular) and y is the vector of dofs issued from the reduced problem using the combined approximationsmethod presented above.

By splitting the previous relationship between junction dofs, relative to ‘‘j’’ index, and intern dofs, relative to ‘‘i’’ index,we can rewrite Eq. (32) as

rj

ri

" #¼

rbj

rbi

" #yþ

Rjj Rji

Rij Rii

" #fj

0

� �ð33Þ

Solutions on the internal dofs are expressed in terms of those related to junction dofs by

ri ¼ RijR�1jj rjþðrbi�RijR

�1jj rbjÞy ð34Þ

which can be also written as

rj

ri

" #¼

Ijj 0

j1 j2

" #rj

y

" #ð35Þ

where

j1 ¼ RijR�1jj , j2 ¼ ðrbi�RijR

�1jj rbjÞ ð36Þ

In order to ensure a high value of rank for the reduction basis, singular value decomposition is carried out on j2 such asdirections corresponding to highest singular values are selected and kept. In this step, we use the same approach than thatdeveloped in Eqs. (24) and (25).

After assembling and solving reduced eigenvalue problem, restitution phase of each substructure on the global physicaldofs, is carried out.

An appreciable gain of the method consists on the fact that a reduction basis, corresponding to the modified system, isnot systematically re-constructed for each system modification. A binomial series expansion is used to calculate andupdate, for each modification matrices, an efficient and robust reduction basis. Updated basis vectors are based on nominalnormal modes of Craig–Bampton’s components.

3. Numerical simulations

3.1. Academic model study

3.1.1. Model description

In this section, we compare, through an academic example, robustness performances, of reduction methods exposedabove, while carrying out an approximate reanalysis in a CMS context. Modifications introduced in sub-structure modelssimulate uncertainties taken into account in this example successively by parametric and non-parametric approaches.

Process adopted consists firstly of building a finite elements model that can be decomposed to sub-structures. Themodel is secondly parameterized. Then, modifications are introduced in mass and stiffness matrices. Two series of N¼2000reanalysis cycles are done. Modifications are due, in first and second N series, respectively, to parametric and non-parametric modelling uncertainties approaches.

At each reanalysis cycle, modified mass and stiffness matrices are calculated for each model sub-structure. Processesof reduction models are carried out by 1—Craig–Bampton (CB) method based on the nominal sub-structure basis,2—Craig–Bampton method enriched by ESRV approach and 3—Combined approximations method based on Craig–Bampton’smodes components, adapted to CMS, (VCA).

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Results obtained with each method are treated according to the following criteria:

a.

TabPar

P

EEEEEhqh

The frequency criterion: it consists on relative differences between n¼20 first eigenfrequencies of the approximatereanalysis solution and those of exact or non-reduced ones.

b.

The vector criterion: it consists on correlation between the truncated basis of n¼20 first eigenvectors of theapproximate reanalysis solution and those of the exact solution. Correlation level between the two bases is quantifiedby the MAC criterion.

We note that for this example, frequency range of interest is [0,500] Hz.The validation model used is described in Fig. 2. It consists on a gantry structure naturally fixed, which can be

decomposed on 2 sub-structures and a residual one that will not be reduced. Global finite elements model contains 426dofs. Sub-structures 1 and 2 contain, respectively, 214 and 104 dofs. Armature section used has a rectangular geometry,whose dimensions are ‘‘b’’ and ‘‘h’’ as indicated in Fig. 2. Parameterization chosen for the problem contains 6 solidproperties zones: P1yP6 such as parameters included in each zone are summarized in Table 1. The following conventionsare adopted: Ei: Young modulus of solid zone i. hi: h dimension of the section corresponding to solid zone i. Ii: quadraticmoment of solid zone i. qi: density of solid zone i.

For parametric modelling application, selected parameters are considered as random variables with uniformdistributions which characteristics are summarized in Table 1.

For non-parametric modelling application, dispersion levels applied aredKs1¼0.25, dMs1¼0.05, dKs2¼0.2, where dKs1, dMs1 and dKs2 are non-parametric dispersion levels, respectively,

relative to: stiffness matrix of the sub-structure 1, mass matrix of the sub-structure 1 and stiffness matrix of the sub-structure 2.

We should note that dispersion levels introduced in this model, both in parametric and non-parametric approaches, arehigh levels. This can be physically admitted for parametric dispersion levels that are about 80% for E1 and E2 or 40% for E4.Concerning non-parametric levels, by referring to orders of magnitude used in industrial applications [12], we can find that

Fig. 2. Academic model parameters. : sub-structure-1 (214 dofs) and : sub-structure-2 (104 dofs).

le 1ameterization of academic model.

arameter Mean Dispersion (%)

1¼E6 2.1�1011 N/m2 80

2 2.1�1011 N/m2 80

3 1�1011 N/m2 15

4 2.1�1011 N/m2 40

3I3 1181.25 N m2 30

1 15�10�3 m 50

2 7800 kg/m3 15

5 15�10�3 m 20

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dK¼0.13 corresponds at least to a parametric dispersion of 8% or 12%. Thus, we consider that dK¼0.20 or dK¼0.25 are highlevels. Use of such high dispersion levels aims to reveal efficiency of improvement reduction basis methods exposed above.

3.1.2. Comparison between reduction methods performances: results synthesis

Let us recall that a series of N¼2000 reanalysis cycles were carried out for both parametric and non-parametricuncertainties modelling methods. In order to compare robustness performances of Craig–Bampton (CB), CB enriched bythe ESRV and combined approximations (VCA) using normal modes of clamped interfaces components, we have to notethat the following settings were adopted:

For each nominal or unmodified sub-structure ‘‘i’’, a truncated basis of ‘‘ni’’ normal modes with blocked junction wascalculated. n1¼45 and n2¼20 normal modes were calculated, respectively, for sub-structures 1 and 2.

At each iteration and for each sub-structures ‘‘i’’, a set of ‘‘mi’’ first normal modes (mioni) was used to performimprovement approaches exposed above (ESRV and combined approximations methods). m1¼18 and m2¼8 normalmodes were used for updating improved reduction basis at each approximate reanalysis cycle.

A binomial series expansion on 3 terms is used for combined approximations method. – For the three model reduction methods, n1¼45 and n2¼20 are the common numbers of blocked junction modes. In

other words:J CB’s reduction basis contains n1 and n2 normal modes of blocked junction relative to nominal or unmodified sub-

structures,J Improved reduction basis by ESRV method contains n1 and n2 modes based on nominal and updated normal modes.J Improved reduction basis by combined approximations methods contains n1 and n2 modes based on updated

normal modes.J Thus, three methods ensure the same reduction model ratio. We note that global and reduced model contains,

respectively, 426 and 173 dofs, this means that the reduction ratio equals to 40%.

In order to ensure a reference solution, a non-reduced (exact) analysis is carried out at each cycle. We note that, thethree reduction model methods and the exact solution are applied simultaneously for each modification.

For both sets of results, we calculate upper and lower values of the confidence region corresponding to a probabilitylevel equal to 0.96.

For the frequency criterion results, we compare the three reduction methods performances by confronting definedlower and upper values.

For vector criterion results, we compare the three reduction methods’ performances by confronting defined lowervalues, which correspond to the most severe configurations.

In order to verify the precision of nominal CB’s reduction basis to predict nominal behaviour of the model, we plot, inFig. 3, the frequency response function (FRF) of condensed nominal model confronted to the exact one. Fig. 3 illustrates ahigh precision of the CB’s method to predict the nominal model.

Fig. 3. Precision of CB’s reduction basis to predict the nominal model behaviour. Exact model and Condensed model.

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Concerning results relative to parametric approach for modelling uncertainties, it can be observed (Figs. 4–6) that bothenrichment methods have significantly better robustness performances than the non-updated CB’s method. Reductionbasis vectors of the two improvement methods are therefore more efficient for describing modified model behaviour thaneigenvectors relative to non-updated basis. Indeed, on one hand, frequency results relative to non-updated basis oscillatebetween [�30, 16]%, which is larger than [�5, 12]% that corresponds to enriched methods results. On the other hand,eigenvector criterion confirms the same trend. It should be noted that for a common size reduction basis, combinedapproximations improvement method delivers slightly better results than the ESRV one.

Concerning non-parametric approach for modelling uncertainties we note (Figs. 7–9) that trend revealed withparametric modifications is more pronounced especially for the VCA method. Indeed, we can clearly appreciate theadvantage provided by VCA improvement method versus ESRV enrichment one, especially when observing eigenvectorcriterion. We can note that ESRV method did not improve at all nominal CB’s basis. It is therefore inappropriate forpredicting non-parametric perturbations. Robustness of VCA method can be justified by efficiency of its improvementprocess based on a binomial series expansion of each nominal normal mode.

In addition to robustness performances (Figs. 4–9), we can quantify in Fig. 10 advantages, in terms of numericalcomputation times, provided by model reduction methods.

At each reanalysis cycle, ESRV time process equals to 25% of non-reduced problem time resources. This allows a 75%gain ratio in time computation. For the VCA method, this gain ratio equals to 73%. Globally, Fig. 10 illustrates an importantgain for reduction methods. Though, they always have to be correlated to reduction model ratio which equals to 40% forthis academic example.

With regard to the quality prediction in approximate reanalysis provided by proposed VCA method, with regard to thefact that it is more appropriate than ESRV improvement method especially for non-parametric perturbations, with regardalso to quasi-equivalent CPU times requested by the two improvement processes, and finally to the fact that it can beadapted to CMS, we select this improvement method for the following industrial application. Uncertainties are taken intoaccount in this industrial case by a hybrid approach combining parametric and non-parametric uncertainties models withrespect to structure zones of the studied model.

3.2. Industrial application

This application consists of taking into account both aleatory and epistemic uncertainties in a concrete finite elementsmodel. We are interested in a stator of railway electric motor, which is a complex structure assembly composed bydifferent heterogeneous zones that are: the stator laminated core, windings parts inserted in internal core teeth and steelframe which maintains the external assembly cohesion of the stator (Fig. 11). Stator core is made of hundreds of stackedlaminated thin sheets treated with polymer resin. However, its structure is simplified and is modelled by a one-block

Fig. 4. Parametric approach – eigenfrequency criterion for comparison between reduction methods performances – lower values of the confidence region

corresponding to a probability level equal to 0.96. CB, VCA and ESRV.

Fig. 5. Parametric approach – eigenfrequency criterion for comparison between reduction methods performances – upper values of the confidence

region corresponding to a probability level equal to 0.96. CB, VCA and ESRV.

Fig. 6. Parametric approach – eigenvector criterion for comparison between reduction methods performances – lower values of the confidence region

corresponding to a probability level equal to 0.96. CB, VCA and ESRV.

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homogeneous and orthotropic solid (1). Concerning windings, they consist on assembled coils made of thin plates ofcopper coated with insulation material. Because of great difficulty to realize a detailed model, windings model is alsosimplified and is modelled by solid homogeneous and orthotropic zones: internal composite parts inserted in core teethare modelled by homogeneous 3D beams (2-a), and external parts that have a braided complex structure according to realstructure are modelled by cylindrical solids (2-b). Concerning steel frame, it consists on an assembly of different steel parts

Fig. 7. Non-parametric approach – eigenfrequency criterion for comparison between reduction methods performances – lower values of the confidence

region corresponding to a probability level equal to 0.96. CB, VCA and ESRV.

Fig. 8. Non-parametric approach – eigenfrequency criterion for comparison between reduction methods performances – upper values of the confidence

region corresponding to a probability level equal to 0.96. CB, VCA and ESRV.

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welded on stator core. The nominal model used contains about 115,000 dofs. It is based exclusively on 3D solid elements.All parts are rigidly interconnected and any contact phenomenon is taken into account.

In order to deal with uncertainties problem in a CMS context, the FEM is decomposed axially into 4 sub-structures(Fig. 11) with about 30,000 dofs for each one. Each sub-structure contains parts from principal components (1), (2) and (3),and is affected by their relative uncertainties. Reduction model ratio is about 90% for each sub-structure.

Fig. 9. Non-parametric approach – eigenvectors criterion for comparison between reduction methods performances – lower values of the confidence

region corresponding to a probability level equal to 0.96. CB, VCA and ESRV.

Fig. 10. CPU time comparison between ESRV and VCA methods adapted to CMS. 1—exact problem reanalysis, 2—ESRV and 3—VCA. reduction basis

construction, condensation step, assembling step, eigenvalue problem solution and expansion step.

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3.2.1. Hybrid approach for modelling uncertainties

We note that stator core and winding parts are important sources of uncertainties, in one hand, because of strongassumptions introduced in the FEM and in the other hand because of lack of knowledge concerning local phenomena thatsit in these zones (interfaces phenomena, localized movements in external winding parts, etc.). For this reason, the non-parametric approach is particularly adapted for modelling these types of uncertainties. Concerning steel frame, we notethat its behaviour is better known by manufacturer and consists essentially on random uncertainties relative to materialparameters. In this case, parametric approach for modelling uncertainties is appropriate and sufficient.

Therefore, we introduce a unique hybrid approach to take into account aleatory and epistemic uncertainties withrespect to model regions uncertainties.

Concerning parametric approach, 3 parameters are selected. They consist of 3 Young’s modulus relative to steel frame.They are considered as random variables with a uniform distribution, whose characteristics are: mean¼2.1�1011 N/m2,dispersion¼7%.

Fig. 11. Stator components: (1)—stator core, (2-a)—windings (internal parts), (2-b)—windings (external parts), (3)—steel frame, (4)—axial decomposi-

tion into 4 sub-structures.

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For non-parametric approach, the following dispersion levels are applied:

dKs1¼0.25: stiffness dispersion level associated to external windings region 2-b – dMs1¼0.15: mass dispersion level associated to external windings region 2-b – dKs2¼0.20: stiffness dispersion level associated to stator core 1 and internal windings 2-a.

3.2.2. Robustness performances of VCA method extended to CMS

By analogy with academic example paragraph, the same criteria are used to discuss results sets obtained in this session.We are interested in 10 eigenfrequencies and eigenvectors evolutions calculated in [0,600]Hz band. N¼500 reanalysiscycles are carried out to obtain these sets of results. ni¼30 eigenmodes are calculated for each nominal sub-structure.

In addition to the proposed VCA method, we also tested performances of the CB’s one. Indeed, this standard method isthe most common and popular one used in industrial applications because of its numerical implementation simplicity.

With important dispersion levels introduced, especially concerning non-parametric dispersion levels, we can clearlyobserve (Figs. 12–14) that robustness performances of the VCA method are higher than those relative to a non-updatedCB’s basis. Indeed, frequency relative errors are maintained in [1;7]% while non-updated the CB’s basis does not ensurebetter performances than errors included in [9;45]% for 10 selected eigenmodes. Concerning eigenvector criterion, lowerMAC values, corresponding to worst and most severe correlation values, are about 0.88 for the VCA method versus 0.40 forresults provided by non-updated CB’s method. Furthermore, the VCA adapted to CMS ensures 60% of gain ratio in terms ofCPU resources. Let us recall that model reduction ratio is about 90% for each sub-structure. We can note a difference in CPUtime performance comparing to academic example where time gain ratio was 73% for a reduction ratio of 40%. This ismainly due to the number of sub-structures (4) and their size (about 30,000 dofs for each sub-structure) which increaseconsiderably reduction basis construction and assembling step computing times. However, we note that 60% of gain ratioin terms of CPU time is an appreciable value.

4. Concluding remarks

In order to perform a robustness analysis of a stator of a railway electric motor, a hybrid method was presented formodelling parametric and non-parametric uncertainties. Problem relates to an approximate reanalysis under parametricand non-parametric uncertainties models and was treated in a reduced model context because of finite elements largesizes. The combined approximations method was adapted to CMS in order to be applied to approximate reanalysis process.In addition to its advantage in terms of computation time comparing to a non-reduced analysis (Fig. 15), it has been shownthat it is more efficient in terms of robustness performances compared to the ESRV enrichment method especially whileusing non-parametric approach. Let us recall that each of Craig–Bampton, the ESRV enrichment method and the VCAmethod, proposed in this paper, use normal modes of clamped interfaces. Therefore comparison between these methodswas allowed and explored. A robustness analysis using hybrid modelling uncertainties method was carried out for anindustrial stator of a railway electric motor. For the same reduction basis sizes, combined approximations method ensuredhigher robustness performances than Craig–Bampton’s approach using non-updated calculated basis.

Fig. 12. Hybrid approach – eigenfrequency criterion for comparison between reduction methods performances – lower values of the confidence region

corresponding to a probability level equal to 0.96. CB and VCA.

Fig. 13. Hybrid approach – eigenfrequency criterion for comparison between reduction methods performances – upper values of the confidence region

corresponding to a probability level equal to 0.96. CB and VCA.

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Nevertheless, we can highlight difficulties and remarks when using this robustness analysis. Indeed, hybrid method formodelling uncertainties reveals a difficulty for estimating non-parametric dispersion level and linking it to physical orstructural skills. A method proposed by Capiez-Lernout et al. [12], and which establish a probabilistic link betweenparametric and non-parametric dispersion levels can be cited as an interesting solution. However, it should be moreappreciable to establish tables of correspondences or statistical relationships between typical physical phenomena andnon-parametric dispersion levels.

Fig. 14. Hybrid approach – eigenvector criterion for comparison between reduction methods performances – lower values of the confidence region

corresponding to a probability level equal to 0.96. CB and VCA.

Fig. 15. Industrial application: CPU time performances of VCA method adapted to CMS: 1—Exact problem reanalysis, 2—VCA. reduction basis construction,

condensation step, assembling step, eigenvalue problem solution and expansion step.

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We can also note a possibility for increasing the VCA prediction by introducing a convergence criterion relative to thenumber of normal modes and binomial development order for each of them. It should be also interesting to analyzeprediction quality of the VCA method adapted to CMS with respect to normal modes nature (Craig–Bampton’s componentsmodes, Martinez free interfaces modes, etc.).

Acknowledgements

The authors thank ALSTOM-Transport Company for the financial support (SEME Project 2007) and for providing statorsof electric motors used to establish and validate the nominal FEM.

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References

[1] G. Klir, Generalized information theory: aims, results, and open problems, Reliability Engineering and Systems Safety 85 (1-3) (2004) 21–38.[2] F. Massa, Gestion des Imperfections en Phase de Conception des Structures Mecaniques. (Ph.D. thesis), Universite de Valenciennes et du

Hainaut-Cambresis, Valenciennes, 2005.[3] R. Ibrahim, Structural dynamics with parameter uncertainties, Applied Mechanical Reviews 40 (3) (1987) 309–328.[4] R. Singh, C. Lee, Frequency response of linear systems with parameter uncertainties, Journal of Sound and Vibration 168 (1) (1993) 507–516.[5] S. Adhikari, M.I. Friswell, K. Lonkar, A. Sarkar, Experimental case studies for uncertainty quantification in structural, Probabilistic Engineering

Mechanics 24 (4) (2009) 473–492.[6] G.I. Schueller, A state-of-the-art report on computational stochastic mechanics, Probabilistic Engineering Mechanics 12 (4) (1997) 197–321.[7] G.I. Schueller, Computational stochastic mechanics: recent advances, Computers and structures 79 (22–25) (2001) 2225–2234.[8] M. Guedri, N. Bouhaddi, R. Majed, Reduction of the stochastic finite element models using a robust dynamic condensation method, Journal of Sound

and Vibration 297 (2006) 123–145.[9] C. Soize, Maximum entropy approach for modelling random uncertainties in transient elastodynamics, Journal of the Acoustical Society of America

109 (5) (2001) 1979–1996.[10] C. Soize, A Non-parametric model of random uncertainties for reduced matrix models in structural dynamics, Probabilistic Engineering Mechanics

15 (3) (2000) 277–294.[11] C. Chen, D. Duhamel, C. Soize, Probabilistic approach for model and mata uncertainties and its experimental identification in structural dynamics:

case of composite sandwich panels, Journal of Sound and Vibration 294 (1–2) (2006) 64–81.[12] E. Capiez-Lernout, M. Pellissetti, H. Pralwarter, G.I. Schueller, C. Soize, Data and model uncertainties in complex aerospace engineering systems,

Journal of Sound and Vibration 295 (3–5) (2006) 923–938.[13] S. Adhikari, A Non-parametric Approach for Uncertainty Quantification in Elastodynamics, in: Proceedings of the 47th AIAA/ASME/ASCE/AHS/ASC

Structures, Structural Dynamics & Materials Conference, New-port, Rhode Island, USA, 2006.[14] S. Adhikari, Wishart random matrices in probabilistic structural mechanics, Journal of Engineering Mechanics 134 (12) (2008) 1029–1044.[15] R.R. Craig Jr., M.C.C. Bampton, Coupling of substructures for dynamic analyses, AIAA Journal 6 (7) (1968) 1313–1319.[16] E. Balmes, Parametric families of reduced finite element models. Theory and applications, Mechanical Systems and Signal Processing 10 (4) (1996)

381–394.[17] G. Masson, Synth�ese Modale Robuste Adaptee �a l’Optimisation de Mod�eles de GrandeTaille. (Ph.D. thesis), Universite de Franche-Comte, Besanc-on,

2003.[18] G. Masson, B. Ait Brik, S. Cogan, N. Bouhaddi, Component mode synthesis (CMS) based on an enriched ritz approach for efficient structural

optimization, Journal of Sound and Vibrations 296 (4–5) (2006) 845–860.[19] U. Kirsch, Design-oriented analysis of structures—unified approach, Journal of Engineering Mechanics 129 (3) (2003) 264–272.[20] U. Kirsch, A unified reanalysis approach for structural analysis, design, and optimization, Structural and Multidisciplinary Optimisation 25 (2003)

67–85.[21] U. Kirsch, Combined approximations—a general reanalysis approach for structural optimization, Structural and Multidisciplinary Optimisation 20

(2000) 97–106.[22] T. Weisser, N. Bouhaddi, Reanalyse Dynamique de Structures Par une Variante de la Methode des Approximations Combinees. 9e colloque National

en Calcul des Structures, Giens, France, 1 (2009) 695–700.