Statistical inverse identication of the multilevel stochastic reduced- order model: application to an automobile 

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Reduced kinematics for the kinetic energy

In order to lter local displacements, a reduced kinematics is introduced for the mass matrix. This reduced kinematics is intended to be such that the local displacements cannot be represented.Instead of using local shape functions within the usual nite elements, we propose the use of r global shape functions, which span a vector subspace, SR, and which constitute the columns of a (m r) real matrix, [B]. It should be noted that the support of these shape functions is the whole domain of the structure and that they are approximated within the nite element basis. In this work, the reduced kinematics used consists of polynomial shape functions.

Construction of the polynomial basis

The objective of this section is the construction of basis matrix [B] of subspace SR . For all m-dimensional vector v belonging to SR , there exists a r-dimensional real vector, c , such that v = [B] c : (3.1).
In the work initialized in [120], the construction of the reduced kinematics is based on a uniform domain partition of the struture, , into Ns subdomains 1; : : : ; Ns .
For complex nite element models, such domain partitioning is not a straightforward task. In [45, 46], uniform domain partitions of the nite element mesh of automobiles were performed using an algorithm [45] based on the Fast Marching Method [121, 122]. In this thesis, the use, for the kinematics reduction, of more accurate approximations (compared to the piecewise constant approximation), allows for avoiding such a domain partitioning.

Polynomial reduced kinematics

In this work, we propose a polynomial approximation of maximum degree d over the entire domain of the structure. To do so, N multivariate orthogonal polynomials p() are used, where for = 1; : : : ;N multi-index () belongs to some set Kd that is dened as follows. Let Kd be the set of vectors = (1; 2; 3) for which integers 1 , 2 , and 3 verify 3 2 1 d : (3.2). It can be deduced (number of possible combinations) that, N = (d + 1)(d + 2)(d + 3)=6 : (3.3). The orthogonality for the polynomials is dened with respect to mass matrix [M] . Denoting as Nf the number of free nodes of the nite element model, the approximate displacement vj of node 2 f1; : : : ;Nfg following direction j is written as vj = XN =1 p()(x ) cj .

Global-displacements reduced-order basis

In order to span the global-displacements space, denoted by Sg (and which is a subspace of Rm), mass matrix [M] is replaced by [Mr] for obtaining the generalized eigenvalue problem (that diers from the one used for computing the elastic modes and that cannot be used for computing them),[K] g = g [Mr] g ; (3.24) in which the eigenvectors g consist of global displacements and where g are the associated eigenvalues. The rst r eigenvalues are such that 0 < g 1 g 2 : : : g r < +1 and the eigenvalues of rank greater than r are all innite, their associated eigenvectors being orthogonal to vector subspace SR . It should be noted that (i) the r eigenvectors gare not orthogonal with respect to mass matrix [M] and (ii) they will be used for the projection of the computational model dened by Eq.(2.1), which involves mass matrix [M] and not [Mr]. Let us introduce the (m ) real matrix, [ g] = [ g 1 : : : g ], in which is a given truncation order such that r : (3.25).
The global-displacements ROB is then dened by the rst eigenvectors of the dynamical system, which are constrained to belong to the vector space spanned by the columns of the (m) real matrix [ g]. The global-displacements ROB is denoted by [g] whose columns ‘g are thus written as ‘g = [ g]r ; (3.26).

Formulation of the multilevel reduced-order model

In this section, the previous developments are used in the construction of a multilevel ROM, for which three ROBs associated with the low-, medium-, and highfrequency bands (LF, MF, HF) are introduced. In contrast to the HF band, the LF band is associated with long-wavelength global displacements, while the MF band is a combination of global and local displacements with more or less short wavelength. For the complex structures considered, there is an overlap of the three vibration regimes. For instance, numerous local elastic modes can be found in low frequencies. The previously introduced ltering strategy is therefore used in order to separate the LF-, MF-, and HF-type displacements. The ltering methodology presented in Sections 3.3 and 3.4 can be condensed into the following mapping, F1 : (Sc ; d; ; fc) 7􀀀! (Sg ; S`) ; (4.1).
which will be used for dening the multilevel ROM. Like the classical ROM, the multilevel ROM is devoted to the vibration analysis over whole band B, which can be decomposed into three bands, such that B = BL [ BM [ BH : (4.2).
We now present the basic ideas concerning the construction of the multilevel ROM, based on the introduction of three successive lterings, which are dened through mapping F1 .

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Multilevel stochastic reduced-order model

Similarly to the C-SROM, the multilevel stochastic ROM (ML-SROM) is based on the nonparametric probabilistic approach of uncertainties. This approach allows for taking into account both the model-parameter uncertainties and the model uncertainties induced by the modeling errors. The ML-NROM previously presented is based on the use of three orthogonal ROBs represented by the matrices [L], [M], and [H], which are constituted of LF-, MF-, and HF-type displacements, respectively. For instance, as explained in Section 4.1, the LF-type displacements consist of long-wavelength global displacements, in contrast to the short-wavelength local displacements associated with the HF regime. When a small design change is performed in the structure, the local displacements that exist in the modied part of the structure are likely to vary a lot, whereas the shape of the global displacements is not really modied. Subsequently and as it is well known, the local displacements are more sensitive to uncertainties than the global displacements.

Damping model for the automobile

The provided nite element model does not include a damping matrix, [D], and consequently, the damping model is introduced at the ROM level. For each ROM constructed in this application, the physical damping is introduced through the use of a modal damping model. In the LF band and in the MF band, a multiparameter modal damping model is tted by using the experimental FRFs. In the HF band, a one-parameter modal damping model is identied by using the experimental FRFs, for which the parameter is denoted by cH. For the deterministic ROMs, three cases have to be considered in order to properly dene the damping model, depending on which ROM is used.
The damping matrix of the C-NROM involved in Eq. (2.5) is dened by [D] = 2[(cH)][]1=2, in which the diagonal matrix [] is dened by Eq. (2.6) and where [(cH)] is a (n n) diagonal matrix of modal damping rates, which depend on parameter cH that has to be identied with respect to the experimental measurements, in a deterministic framework. In such a case, Eq. (2.5) consists of a diagonal matrix-equation.
The damping matrix of the scale-S ROM involved in Eq. (4.27) is dened by [DS] = 2[S(cH)][S]1=2, with [S] the diagonal matrix involved in Eq. (4.28) and where [S(cH)] is a (nS nS) diagonal matrix of damping rates, which depend on parameter cH that has to be identied with respect to the experimental measurements, in a deterministic framework. In such a case, Eq. (4.27) consists of a diagonal matrix-equation.
Concerning the ML-NROM, reduced stiness matrix [K] involved in Eq. (4.31) is a full matrix. In order to solve latter matrix equation, one possibility is to perform a change of basis in order to diagonalize reduced matrices [K] and [M]. Doing so leads us back to scale-t ROM, for which the denition of the damping matrix [Dt] is given in the item before.

Table of contents :

1 Introduction 
1.1 Context of the research
1.2 Position of the research
1.3 Objectives of the research
1.4 Strategy of the research
1.5 Manuscript layout
2 Classical reduced-order model 
2.1 Reference computational model
2.2 Classical nominal reduced-order model
2.3 Classical stochastic reduced-order model
3 Global-displacements reduced-order model 
3.1 Reduced kinematics for the kinetic energy
3.1.1 Construction of the polynomial basis
3.1.2 Reduced-kinematics mass matrix
3.2 Global-displacements reduced-order basis
3.3 Numerical implementation
3.4 Local-displacements reduced-order basis
4 Multilevel reduced-order model 
4.1 Formulation of the multilevel reduced-order model
4.2 Implementation of the multilevel nominal reduced-order model
4.2.1 Numerical procedure
4.2.2 Construction of the reduced-order bases
4.2.3 Construction of the reduced-order models
4.3 Multilevel stochastic reduced-order model
5 Statistical inverse identication of the multilevel stochastic reduced- order model: application to an automobile 
5.1 Problem denition
5.1.1 Experimental measurements (excitation force and observation points) and frequency band of analysis
5.1.2 Computational model
5.1.3 Modal density characterizing the dynamics and denition of the LF, MF, and HF bands
5.1.4 Damping model for the automobile
5.1.5 Denition of the observations
5.1.6 Dening the objective function used for the convergence analyses of the deterministic computational ROMs
5.1.7 Dening the objective function used for the identication of the stochastic computational ROMs
5.2 Classical nominal ROM and classical stochastic ROM
5.2.1 First step: C-NROM
5.2.2 Second step: C-SROM
5.3 Multilevel nominal ROM and multilevel stochastic ROM
5.3.1 First step: ML-NROM
5.3.2 Second step: ML-SROM
5.4 Complementary results
5.4.1 Deterministic analysis of the contribution of each of the ROBs
5.4.2 Stochastic sensitivity analysis
6 Conclusions and future prospects 


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