Computational Mechanics manuscript No.(will be inserted by the editor)

O. Ezvan · A. Batou · C. Soize · L. Gagliardini

Multilevel model reduction for uncertainty quantificationin computational structural dynamics

Revised manuscriptR1 submitted on October 11, 2016

Abstract This work deals with an extension of the redu-ced-order models (ROMs) that are classically constructedby modal analysis in linear structural dynamics of com-plex structures for which the computational models areassumed to be uncertain. Such an extension is based ona multilevel projection strategy consisting in introducingthree reduced-order bases (ROBs) that are obtained byusing a spatial filtering methodology of local displace-ments. This filtering involves global shape functions forthe kinetic energy. The proposed multilevel stochasticROM is constructed by using the nonparametric prob-abilistic approach of uncertainties. It allows for affect-ing a specific level of uncertainties to each type of dis-placements associated with the corresponding vibrationregime, knowing that the local elastic modes are moresensitive to uncertainties than the global elastic modes.The proposed methodology is applied to the computa-tional model of an automobile structure, for which themultilevel stochastic ROM is identified with respect toexperimental measurements. This identification is per-formed by solving a statistical inverse problem.

Keywords High modal density · Reduced-OrderModel · Uncertainty quantification · Broad frequencyband · Structural dynamics

1 Introduction

An extension of the classical modal analysis for con-structing a reduced-order model (ROM) in computa-tional linear structural dynamics is presented. This ex-

O. Ezvan · A. Batou · C. Soize (�)Universite Paris-Est, Laboratoire Modelisation et SimulationMulti Echelle, MSME UMR 8208 CNRS, 5 bd Descartes,77454 Marne-la-Vallee, FranceE-mail: christian.soize@univ-paris-est.fr

L. GagliardiniPSA Peugeot Citroen, Direction Technique et Industrielle,Centre Technique de Velizy A, Route de Gisy, 78140 VelizyVillacoublay, France

tension is based on a multilevel projection strategy. Thecomputational model is assumed to be uncertain. Thispaper is a continuation of the work published in [1],for which three new ingredients are presented: (i) themethodology for the construction of the multilevel modelreduction is adapted for allowing the probabilistic modelof uncertainties to be implemented; (ii) the effective im-plementation of a multilevel probabilistic model of uncer-tainties and the statistical inverse problem for its exper-imental identification; (iii) the experimental validationof the methodology for a very complex computationalmodel of an automobile for which experimental measure-ments have been performed for 20 manufactured cars ofthe same type. The complete and detailed developmentscan be found in [2].

Nowadays, it is well recognized that the predictions instructural dynamics over a broad frequency band by us-ing a computational model, based on the finite elementmethod [3; 4; 5], must be improved in taking into ac-count the model uncertainties induced by the modelingerrors, for which the role increases with the frequency.This means that any model of uncertainties must accountfor this type of frequency evolution. In addition, it isalso admitted that the parametric probabilistic approachof uncertainties is not sufficiently efficient for reproduc-ing the effects of modeling errors. In such a framework,the nonparametric probabilistic approach of uncertain-ties can be used, but in counter part requires the intro-duction of a reduced-order model for implementing it.Consequently, these two aspects, frequency-evolution ofthe uncertainties and reduced-order modeling, lead us toconsider the development of a multilevel reduced-ordermodel in computational structural dynamics, which hasthe capability to adapt the level of uncertainties to eachpart of the frequency band. This is the purpose of thepaper.

In structural dynamics, the low-frequency (LF) bandis generally characterized by a low modal density andby frequency response functions (FRFs) exhibiting iso-

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lated resonances. These are due to the presence of long-wavelength displacements, which are global (the conceptof global displacement will be clarified later). In contrast,the high-frequency (HF) band is characterized by a highmodal density and by rather smooth FRFs, these beingdue to the presence of numerous short-wavelength dis-placements. The intermediate band, the medium-frequen-cy (MF) band, presents a non-uniform modal density andFRFs with overlapping resonances [6]. For the LF band,modal analysis [7; 8; 9; 10; 11; 12; 13; 14] is a well-knowneffective and efficient method, which usually provides asmall-dimension ROM whose reduced-order basis (ROB)is constituted of the first elastic modes (i.e. the firststructural vibration modes). Energy methods, such asstatistical energy analysis [15; 16; 17; 18; 19; 20; 21; 22],are commonly used for the HF band analysis. Variousmethods have been proposed for analyzing the MF band.A part of these methods are related to deterministicsolvers devoted to the classical deterministic linear dy-namical equations [6; 23; 24; 25; 11; 26; 27; 28; 29; 30;31; 32]. A second part are devoted to stochastic lineardynamical equations that have been developed for tak-ing into account the uncertainties in the computationalmodels in the MF band (which plays an important role inthis band), see for instance [33; 34; 35; 36; 37; 38; 39; 40].

In this paper, we are interested in the dynamical analy-sis of complex structures over a broad frequency band.By complex structure is intended a structure with com-plex geometry, constituted of heterogeneous materialsand more specifically, characterized by the presence ofseveral structural levels, for instance, a structure that ismade up of a stiff main part embedding various flexiblesub-parts. For such structures, it is possible having, inaddition to the usual global-displacements elastic modesassociated with their stiff skeleton, the apparition of nu-merous local elastic modes, which correspond to predom-inant vibrations of the flexible sub-parts. For such com-plex structures, which can be encountered for instancein aeronautics, aerospace, automotive (see for instance[41; 42; 43]), or nuclear industries, two main difficultiesarise from the presence of the local displacements. First,the modal density can be very high as soon the low-frequency band, yielding a high-dimension ROM whenthe modal analysis is used (several thousands of elasticmodes can be required for such a low-frequency band).Second, such ROMs may suffer from a lack of robustnesswith respect to uncertainty, because of the presence ofthe numerous local displacements, which are known to bevery sensitive to uncertainties. It should be noted that,for such a complex structure, the engineering objectivesmay consist in the prediction of the global displacementsonly, that is to say on predicting the FRFs of observationpoints belonging to the stiff parts.

There is not much research devoted to the dynamic anal-ysis of structures characterized by the presence of nu-

merous local elastic modes intertwined with the globalelastic modes. In the framework of experimental modalanalysis, techniques for the spatial filtering of the shortwavelengths have been proposed [44], based on regular-ization schemes [45]. In the framework of computationalmodels, the Guyan condensation technique [46], basedon the introduction of master nodes at which the inertiais concentrated, allows for the filtering of local displace-ments. The selection of the master nodes is not obviousfor complex structures [47]. Filtering schemes based onthe lumped mass matrix approximations have been pro-posed [48; 49; 50], but the filtering depends on the meshand cannot be adjusted. In [16] a basis of global dis-placements is constructed using a coarse mesh of a finiteelement model, which, generally, gives big errors for theelastic energy. In order to extract the long-wavelengthfree elastic modes of a master structure, the free-interfacesubstructuring method has been used [32]. Other compu-tational methods include image processing [51] for iden-tifying the global elastic modes, the global displacementsas eigenvectors of the frequency mobility matrix [52], orthe extrapolation of the dynamical response using a fewelastic modes [53]. In the framework of slender dynamicalstructures, which exhibit a high modal density in the LFband, simplified equivalent models [54; 55] using beamsand plates, or homogenization [56] have been proposed.Using these approaches, the construction of a simplifiedmodel is not automatic, requires an expertise, and a val-idation procedure remains necessary. In addition, theseapproximations are only valid for the LF band.

For a complex structure for which the elastic modes maynot be either purely global elastic modes or purely localelastic modes, the increase of the dimension of the ROM,which is constructed by using the classical modal anal-ysis, can be troublesome. The methodology that wouldconsist in sorting the elastic modes according to whetherthey be global or local cannot be used because the elas-tic modes are combinations of both global displacementsand local displacements.

Another solution would consist in using substructuringtechniques for which reviews can be found in [57; 58; 59]and for which a state of the art has recently been done in[60]. A brief summary is given hereinafter. The concept ofsubstructures was first introduced by Argyris and Kelseyin 1959 [61] and by Przemieniecki in 1963 [62] and wasextended by Guyan and Irons [46; 63]. Hurty [64; 65] con-sidered the case of two substructures coupled through ageometrical interface. Finally, Craig and Bampton [66]adapted the Hurty method. Improvements have beenproposed with many variants [67; 68; 69; 70; 71], in par-ticular for complex dynamical systems with many ap-pendages considered as substructures (such as a diskwith blades) Benfield and Hruda [72]. Another type ofmethods has been introduced in order to use the struc-tural modes with free geometrical interface for two cou-

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pled substructures instead of the structural modes withfixed geometrical interface (elastic modes) as used in theCraig and Bampton method and as proposed by Mac-Neal [73] and Rubin [74]. The Lagrange multipliers havealso been used to write the coupling on the geometricalinterface [75; 76; 77; 78].

The substructuring techniques would require to discardthe component modes associated with flexible sub-parts,hence removing their associated local displacements. Un-fortunately, for the complex structures considered, thereis no clear separation between the skeleton and the sub-structures for which the displacements would be local.For instance, with fixed thickness, the curvatures of ashell induce stiffened zones with respect to the rigidityof the flat zones. Consequently, in addition to the variousembedded equipments within the structure, the complexgeometry of the structure is responsible for the fact thatthere can be no separation of the several structural lev-els, but rather a continuous series of structural levels. Insuch conditions, the notion of local displacement is rel-ative. It should be noted that in contrast to the usuallong-wavelength global displacements of the LF band,the local displacements associated with the structuralsub-levels, which can also appear in the LF band, arecharacterized by short wavelengths, similarly to HF dis-placements. As a result, for the complex structures con-sidered, there is an overlap of the three vibration regimes,LF, MF, and HF.

Concerning the taking into account of uncertainties inthe computational model, the probabilistic framework iswell adapted to the construction of the stochastic mod-els, to the stochastic solvers, and to solve the associatedstatistical inverse problems for the identification of thestochastic models (for the finite dimension and for theinfinite dimension). Hereinafter, we present a brief back-ground that is limited to the probabilistic frameworkfor uncertainty quantification. Several probabilistic ap-proaches can be used depending on the sources of uncer-tainties in the computational model (model-parameteruncertainties, model uncertainties induced by modelingerrors, and variabilities in the real dynamical system).

(i) Output-predictive error method. Several methods arecurrently available for analyzing model uncertainties. Themost popular one is the standard output-predictive er-ror method introduced in [79]. This method has a majordrawback because it does not enable the ROM to learnfrom data.

(ii) Parametric probabilistic methods for model-parameteruncertainties. An alternative family of methods for an-alyzing model uncertainties is the family of paramet-ric probabilistic methods for the uncertainty quantifi-cation. This approach is relatively well developed formodel-parameter uncertainties, at least for a reasonablysmall number of parameters. It consists in construct-ing prior and posterior stochastic models of uncertain

model parameters pertaining, for example, to geome-try, to boundary conditions, to material properties, etc[80; 81; 82; 83; 84; 85; 86; 87; 88; 89; 90; 91; 92; 93; 94].This approach was shown to be computationally efficientfor both the computational model and its associatedROM (for example, see [95; 96]), and for large-scale sta-tistical inverse problems [97; 98; 99; 100; 101; 102]. How-ever, it does not take into account neither the model un-certainties induced by modeling errors introduced duringthe construction of the computational model, nor thosedue to model reduction.

(iii) Nonparametric probabilistic approach for modelinguncertainties. For modeling uncertainties due to model-ing errors, a nonparametric probabilistic approach wasintroduced in [103], in the context of linear structuraldynamics. The methodology is made up of two steps.For the first one, a linear ROM of dimension n is con-structed by using the linear computational model withm degrees of freedom (DOFs) and a ROB of dimensionn. For the second step, a linear stochastic ROM is con-structed by substituting the deterministic matrices un-derlying the linear ROM with random matrices for whichthe probability distributions are constructed using theMaximum Entropy (MaxEnt) principle [104; 105]. Theconstruction of the linear stochastic ROM is carried outunder the constraints generated from the available in-formation such as some algebraic properties (positive-ness, integrability of the inverse, etc.) and some statisti-cal information (for example, the equality between meanand nominal values). This nonparametric probabilisticapproach has been extended for different ensembles ofrandom matrices and for linear boundary value problems[106; 107]. It was also experimentally validated and ap-plied for linear problems in composites [108], viscoelastic-ity [109], dynamic substructuring [110; 111], vibroacous-tics [41; 40], robust design and optimization [112], etc.More recently, the nonparametric approach has been ex-tended to take into account some nonlinear geometricaleffects in structural analysis [113; 114], but it does nothold for arbitrary nonlinear systems, while the work re-cently published [115] allows for taking into account anynonlinearity in a ROM.

Recently, a new methodology [116] has been proposedfor constructing a stochastic ROM devoted to dynami-cal structures having numerous local elastic modes in thelow-frequency range. The stochastic ROM is obtained byimplementing the nonparametric probabilistic approachof uncertainties within a novel ROM whose ROB is con-stituted of two families, one of global displacements andanother one of local displacements. These families are ob-tained through the introduction, for the kinetic energy,of a projection operator associated with a subspace ofpiecewise constant functions. The spatial dimension ofthe subdomains, in which the projected displacementsare constant, and which constitute a partition of the do-main of the structure, allows for controlling the sepa-

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ration between the global displacements and the localdisplacements. These subdomains can be seen as macro-elements, within which, using such an approximation, nolocal displacement is permitted. It should be noted thatthe generation of a domain partition for which the gen-erated subdomains have a similar size (that we call uni-form domain partition), necessary for obtaining a spa-tially uniform filtering criterion, is not trivial for com-plex geometries. Based on the Fast Marching Method[117; 118], a general method has been developed in or-der to perform the uniform domain partition for a com-plex finite element mesh, and then implemented for thecase of automobile structures [42]. In later work [1; 2],the filtering methodology has been generalized throughthe introduction of a computational framework for theuse of any arbitrary approximation subspace for the ki-netic energy, in place of the piecewise constant approxi-mation. In particular, polynomial shape functions (withsupport the whole domain of the structure) have beenused for constructing a global-displacements ROM foran automobile. This generalization allows for carryingout an efficient convergence of the global-displacementsROM with respect to the so-defined filtering (in contrast,constructing several uniform domain partitions of differ-ent characteristic sizes can be, in practice, very time-consuming). In addition, a multilevel ROM has been in-troduced. The ROB of this ROM is constituted of sev-eral families of displacements, which correspond to theseveral structural levels of the complex structure. Moreprecisely, a multilevel ROM has been presented for whichthe ROB is constituted of three families, namely the LF-, MF-, and HF-type displacements. Each family is ob-tained by performing a spatial filterings of local displace-ments. It should be noted that multilevel substructuringtechniques can be found in the literature [119; 120; 121],whose purpose is to accelerate the solution of large-scalegeneralized eigenvalue problems.In the present paper, latter filtering methodology (in-volving polynomial shape functions) is reused, but wepresent a novel formulation for the construction of amultilevel ROM, which allows for implementing a prob-abilistic model of uncertainties that is adapted to eachvibration regime. This way, the amount of statistical fluc-tuations for the LF-, MF-, and HF-type displacementscan be controlled using the multilevel stochastic ROMthat is obtained.

The objective of this paper is double. The first one is toprovide a multilevel stochastic ROM that is able to takeinto account the heterogeneous variability introduced bythe overlap of the three vibration regimes. The secondone is to provide a predictive ROM whose dimension issmaller than the dimension of the classical ROM con-structed by using modal analysis. Both these objectivesare to be fulfilled by means of efficient methods that arenon-intrusive with respect to commercial software.

The paper is organized as follows. In Section 2, the ref-erence computational model is introduced and the clas-sical construction of the ROM is performed by usingmodal analysis. Based on this deterministic ROM, theclassical stochastic ROM is then constructed by usingthe nonparametric probabilistic approach of uncertain-ties. In Section 3, the methodology devoted to the spa-tial filtering of the global and of the local displacementsis presented. These developments are used in Section 4for defining the multilevel ROM. In this section, the nu-merical procedure is also detailed and the constructionof the multilevel stochastic ROM is given. Finally, inSection 5, the proposed methodology is applied to anautomobile, for which the multilevel stochastic ROM isidentified by using experimental measurements. The re-sults are compared to those obtained with the classicalstochastic ROM.

Notations

DOF: degree of freedom.FEM: finite element model.FRF: frequency response function.HF: high frequency.LF: low frequency.MF: medium frequency (or mid frequency).ROB: reduced-order vector basis.ROM: reduced-order model.

C-NROM: classical nominal ROM.C-SROM: classical stochastic ROM.ML-NROM: multilevel nominal ROM.ML-SROM: multilevel stochastic ROM.

Cp: Hermitian space of dimension p.Rp: Euclidean space of dimension p.Sc: vector subspace for the classical ROM.Sg: vector subspace for the global-displacements ROM.S`: vector subspace for the local-displacements ROM.St: vector subspace for the multilevel ROM.

SH: vector subspace for the scale-H ROM.SL: vector subspace for the scale-L ROM.SM: vector subspace for the scale-M ROM.SR: vector subspace for the reduced kinematics.SLM: vector subspace for the scale-LM ROM.

d: maximum degree of the polynomial approximation.m: dimension of the FEM (number of DOFs).n: dimension of Sc.r: dimension of SR.ng: dimension of Sg.n`: dimension of S`.nt: dimension of St.nH: dimension of SH.nL: dimension of SL.nM: dimension of SM.

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nLM: dimension of SLM.

[B]: ROB of SR such that [B]T [M][B] = [Ir].[B`]: ROB of SR such that [B`]T [M`][B`] = [Ir].[Ip]: identity matrix of dimension p.[M]: mass matrix of the FEM.[M`]: lumped approximation of [M].

[Φ]: ROB of the classical ROM.[Φg]: global-displacements ROB.[Φ`]: local-displacements ROB.[Φt]: ROB of the scale-t ROM.[ΦH]: HF-type displacements ROB.[ΦL]: LF-type displacements ROB.[ΦM]: MF-type displacements ROB.[ΦLM]: ROB of the scale-LM ROM.[Ψ ]: ROB of the multilevel ROM.

2 Classical reduced-order model

In this section, we introduce the reference computationalmodel and we present the very well known modal analy-sis method as well as the construction of the associatedstochastic ROM that is obtained by using the nonpara-metric probabilistic approach of uncertainties [103]. Thisway, basic notions that will be reused later are intro-duced. In addition, the multilevel stochastic ROM pro-posed in this paper will be compared to classical stochas-tic ROM.

2.1 Reference computational model

The vibration analysis is performed over a broad fre-quency band – denoted as B – by using the finite elementmethod. Let m denote the dimension (number of DOFs)of the finite element model. For all angular frequency ωbelonging to B = [ωmin, ωmax], the m-dimensional com-plex vector U(ω) of displacements is the solution of thematrix equation,

(−ω2[M] + iω[D] + [K] )U(ω) = F(ω) , (1)

in which F(ω) is the m-dimensional complex vector ofthe external forces and where, assuming the structure isfixed on a part of its boundary, [M], [D], and [K] are thepositive-definite symmetric (m×m) real mass, damping,and stiffness matrices.

2.2 Classical nominal reduced-order model

For all α = 1, . . . ,m the elastic modes ϕα with associ-ated eigenvalues λα are the solutions of the generalizedeigenvalue problem,

[K]ϕα = λα[M]ϕα . (2)

The first n eigenvalues verify 0 < λ1 ≤ λ2 ≤ . . . ≤λn < +∞ and the normalization that is chosen for theeigenvectors is such that

[Φ]T

[M][Φ] = [In] , (3)

in which [Φ] = [ϕ1 . . .ϕn], and where [In] is the identitymatrix of dimension n. Such a normalization with theunit generalized mass is always adopted in this paper inwhich several generalized eigenvalue problems are intro-duced. In practice, only the first n elastic modes withn � m (associated with the lowest eigenvalues or thelowest eigenfrequencies fα =

√λα/2π in Hz) are calcu-

lated. The (m × n) real matrix [Φ] is the ROB of theclassical nominal reduced-order model (C-NROM). Thevector subspace spanned by the ROB of the C-NROM isdenoted by Sc. Using the C-NROM, displacements vec-tor U(ω) belong to Sc and we have

U(ω) ' [Φ]q(ω) =

n∑α=1

qα(ω)ϕα , (4)

where the n-dimensional complex vector of generalizedcoordinates q(ω) = (q1(ω) . . . qn(ω)) is the solution ofthe reduced-matrix equation,

(−ω2[M] + iω[D] + [K] )q(ω) = f(ω) , (5)

in which f(ω) = [Φ]TF(ω), [D] = [Φ]

T[D][Φ] is, in general,

a full matrix and where diagonal matrices [K] and [M]are such that

[K] = [Φ]T

[K][Φ] = [Λ] , [M] = [Φ]T

[M][Φ] = [In] , (6)

in which [Λ] is the matrix of the first n eigenvalues.

2.3 Classical stochastic reduced-order model

The classical stochastic reduced-order model (C-SROM)is constructed by using the nonparametric probabilisticapproach of uncertainties [103] within the C-NROM. Inthis nonparametric approach, each nominal reduced ma-trix of dimension n, say [A] (= [M], [D], or [K]), is re-placed by a random matrix, [A] (= [M], [D], or [K]) forwhich the probability distribution has been constructedby using the maximum entropy principle [104; 105] underthe following constraints:

– Matrix [A] is with values in the set of all the positive-definite symmetric (n× n) real matrices.

– E{[A]} = [A] , with E the mathematical expectation,which means that the mean matrix is chosen as thenominal matrix.

– E{||[A]−1||

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F } < +∞ for insuring the existence of asecond-order solution of the stochastic ROM where||.||F is the Frobenius norm.

Introducing the Cholesky factorization [LA]T

[LA] of ma-trix [A] in which [LA] is an upper-triangular matrix, theconstruction of random matrix [A] is given by

[A] = [LA]T

[Gn(δA)][LA] , (7)

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where the random matrix [Gn(δA)] (see [103]) is positive-definite almost surely, for which its mean value is [In]and which is parameterized by a dispersion parameterδA defined by

δ2A =1

nE{||[Gn(δA)]− [In]||2F } . (8)

The hyperparameter δA of random matrix [Gn(δA)] hasto verify 0 < δ < δmax with δmax given by

δmax =

√n+ 1

n+ 5. (9)

Using the Monte-Carlo simulation method [122], the C-SROM allows for computing the random displacementsvector U(ω) associated with U(ω),

U(ω) = [Φ]Q(ω), (10)

in which the random complex vector Q(ω) of the gen-eralized coordinates is obtained by solving the randommatrix equation,

(−ω2[M] + iω[D] + [K] )Q(ω) = f(ω) . (11)

The classical ROM is built upon the use of the elasticmodes that are present in the frequency band of analysis,B (or a frequency band slightly wider). In this manner,the dimension of the computational model is reducedwhile its accuracy is preserved for this band. For thecomplex structures under consideration, numerous localdisplacements are intertwined with the global displace-ments. As a result, among the elastic modes present inB, many have little contribution to the robust dynam-ical response of the stiff skeleton of the structure thatis provided by the C-SROM. Consequently, we presentthe construction of an adapted ROM that is based on aROB in which local displacements have been filtered.

3 Global-displacements reduced-order model

In this section, the construction of a new ROM is pre-sented. It is based on the use of a global-displacementsROB instead of the classical ROB made up of elasticmodes because the classical ROB can include numerouslocal elastic modes. In Section 3.1, we present the con-struction of an unusual mass matrix that is associatedwith a reduced kinematics for the kinetic energy. In Sec-tion 3.2, we use this mass matrix for obtaining unusualeigenvectors that constitute the global-displacements ROB(this unusual mass matrix is not used as the mass matrixfor computing the response of the dynamical system).In section 3.3, an efficient and nonintrusive algorithm isproposed for implementing the ROB. Finally, in Section3.4, we give the construction of a ROB that is consti-tuted of the complementary local displacements that areneglected in the global-displacements ROM.

3.1 Reduced kinematics for the kinetic energy

In order to filter local displacements, a reduced kine-matics is introduced for the mass matrix. This reducedkinematics is intended to be such that the local displace-ments cannot be represented. Instead of using the clas-sical local shape functions of the finite elements for theconstruction of the mass matrix, we propose the use ofr global shape functions, which span a vector subspacedenoted by SR and which constitute the columns of a(m × r) real matrix [B]. These global shape functionsare polynomials for which the maximum degree is de-noted by d and that are defined on the whole domain ofthe structure. The construction of [B] is given in [1; 2]and the dimension r of the reduced kinematics is writtenas

r = (d+ 1)(d+ 2)(d+ 3)/2 . (12)

The value of degree d will allow for controlling the filter-ing of the global displacements and of the local displace-ments. In order to determine an appropriate value of d,a convergence analysis will be carried out with respectto both the deterministic and the stochastic quantitiesof interest. In the present paper, the strategy proposedconsists in approximating the kinetic energy while keep-ing the elastic energy exact. Such a construction is de-tailed in [1; 2]). The mass matrix corresponding to thisreduced kinematics will be called the reduced-kinematicsmass matrix.

First, the kinetic energy Ek(V(t)) associated with anytime-dependent real velocity vector V(t) of dimension mis given by

Ek(V(t)) =1

2V(t)

T[M]V(t) . (13)

Let Vr(t) be the orthogonal projection of V(t) onto sub-space SR with respect to the inner-product defined bymatrix [M]. Assuming the columns of [B] to be orthonor-malized with respect to [M], the projector [P] that is suchthat Vr(t) = [P]V(t) is written as

[P] = [B][B]T

[M] . (14)

The rank of the (m×m) real matrix [P] is equal to r ≤ m.The reduced kinetic energy Erk(V(t)) = Ek(Vr(t)) is thenwritten as

Erk(V(t)) =1

2V(t)

T[Mr]V(t) , (15)

in which them-dimensional reduced-kinematics mass ma-trix [Mr] = [P]

T[M][P] is positive semidefinite with a

rank equal to r and can be written as

[Mr] = [M][B][B]T

[M] . (16)

3.2 Global-displacements reduced-order basis

In order to span the global-displacements space denotedby Sg (and which is a subspace of Rm), mass matrix [M]

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is replaced by [Mr] and yields the following generalizedeigenvalue problem (that differs from the one used forcomputing the elastic modes and that cannot be usedfor computing them),

[K]ψgα = σgα[Mr]ψgα , (17)

in which the eigenvectors ψgα consist of global displace-ments and where σgα are the associated eigenvalues. Thefirst r eigenvalues are such that 0 < σg1 ≤ σg2 ≤ . . . ≤σgr < +∞ and the eigenvalues of rank greater than r areall infinite. Their associated eigenvectors are orthogonalto vector subspace SR.

It should be noted that (i) the r eigenvectors ψgα arenot orthogonal with respect to mass matrix [M] and (ii)they will be used for the projection of the computationalmodel defined by Eq.(1), which involves mass matrix [M]and not [Mr]. Let us introduce the (m× ν) real matrix,[Ψg] = [ψg1 . . .ψ

gν ], in which ν is a given truncation order

such that

ν ≤ r . (18)

The global-displacements ROB is defined by the firsteigenvectors of the dynamical system, which are con-strained to belong to the vector space spanned by theν columns of the (m × ν) real matrix [Ψg]. The global-displacements ROB is denoted by [Φg] whose columnsϕgα are written as

ϕgα = [Ψg]rα , (19)

in which rα are the eigenvectors of the small-dimensiongeneralized eigenvalue problem,

([Ψg]T

[K][Ψg]) rα = λgα([Ψg]T

[M][Ψg]) rα . (20)

Introducing the matrix [R] = [r1 . . . rng], in which ng is a

given truncation order that will be defined after, matrix[Φg] can be written as

[Φg] = [Ψg][R] . (21)

Using Eq. (20) yields

[Φg]T

[K][Φg] = [Λg] , [Φg]T

[M][Φg] = [Ing] . (22)

The eigenvalues verify 0 < λg1 ≤ λg2 ≤ . . . ≤ λgν <+∞. Only the first ng eigenvectors (associated with the

lowest frequencies fgα =√λgα/2π) are kept for defining

[Φg] = [ϕg1 . . .ϕgng

]. The dimension ng of the global-displacements subspace Sg is deduced from a cutoff fre-quency, f c, for which ng verifies

fgng≤ f c. (23)

In addition, ng satisfies the inequality,

ng ≤ ν . (24)

Cutting frequency f c is a data that must be chosengreater or equal to the upper bound ωmax/2π of fre-quency band B and that must be adjusted through the

analysis of the FRFs. It should be noted that trunca-tion order ν cannot directly be deduced from the value√σgν/2π because the eigenvalues σgα are not the eigenfre-

quencies of the dynamical system. For ν ≤ r, the follow-ing inequality can be proved,

λgν ≤ σgν , (25)

for which the difference between λgν and σgν can be sig-nificant.

For brevity, no notation is introduced for the equationsrelated to the global-displacements ROM, which wouldbe similar to Eqs. (4) and (5) and which, anyway, wouldnot be used.

3.3 Numerical implementation for the construction ofthe global-displacements ROB

As the mass matrix [Mr] is a full (m×m) matrix, this ma-trix is not assembled. In addition, the eigenvalue problemdefined by Eq. (17) requires the knowledge of matrices[M] and [K], which can involve problems for the commer-cial software. The purpose of this section is to presentan efficient method for the construction of the global-displacements ROB.

Let [M`] be the diagonal matrix that is a lumped ap-proximation of mass matrix [M]. For avoiding the use of[M], the following approximation of [Mr] is introduced,

[Mr] ' [M`][B`][B`]T

[M`] , (26)

in which the construction of [B`] is similar to the one of

[B] (see Section 3.1) but is such that [B`]T

[M`][B`] =[Ir]. The projector [P] defined by Eq. (14) is then ap-proximated by

[P] ' [B`][B`]T

[M`] . (27)

Equation (26) consists in applying the reduced kinemat-ics (associated with a given maximum degree for thepolynomial approximation) to a diagonal lumped ap-proximation [M`] of consistent mass matrix [M], insteadof applying it to [M]. Since a diagonal approximationcan be obtained by a kinematic reduction for which thedisplacement field is constant over each finite element,the error induced by Eq. (26) is related to the one ob-tained by approximating the polynomial shape functionsby functions that are a constant in each finite element.For a polynomial with a low degree, the spatial varia-tion is slow and, as it is assumed that the FEM is con-structed upon a fine mesh, such an approximation cantherefore be considered negligible. Nevertheless, in thispaper, high-degree polynomial shape functions have beenused. In order to estimate the maximum effect of suchan approximation given by Eq. (26) (for the case of high-degree polynomial shape functions), a prior analysis hasbeen carried out, consisting in computing deterministic

8

quantities of interest (such as moduli of FRFs in log-scale) by two different global-displacements ROMs, onewith the lumped mass matrix and the other one with theconsistent matrix, and in comparing the results given bythe two ROMs, which have shown that the error is effec-tively negligible.

A double projection method is presented, which allowsthe explicit use of matrix [K]. It consists in projectingEq. (17) onto subspace Sc that is associated with theclassical ROB made up of the elastic modes. This sub-space is supposed to provide, upon the use of a suffi-ciently large value of the number n of elastic modes,an accurate representation on frequency band B. Then,without loss of model fidelity, such a projection can beobtained by writing

Sg ⊆ Sc (28)

that is satisfied if ψgα is written as

ψgα = [Φ]sα , (29)

in which sα is a n-dimensional real vector that has tobe calculated as follows. By using Eq. (29), the projec-tion of Eq. (17) yields the following reduced-dimensiongeneralized eigenvalue problem,

([Φ]T

[K][Φ]) sα = σgα([Φ]T

[Mr][Φ]) sα . (30)

The matrix [Φ]T

[K][Φ] is the diagonal matrix [Λ] definedin Eq. (6), which is available. By using Eq. (26), the full

matrix [Φ]T

[Mr][Φ] can be computed as

[Φ]T

[Mr][Φ] ' [N ][N ]T, (31)

in which the (n× r) real matrix [N ] that is defined by

[N ] = [Φ]T

[M`][B`] (32)

is also available. Introducing [S] = [s1 . . . sν ] and usingEq. (29), matrix [Ψg] can be rewritten as

[Ψg] = [Φ][S] . (33)

Denoting as [Σg] the diagonal matrix of the first ν eigen-values σgα and recalling the choice of a unit generalizedmass normalization, it can then be deduced that the re-duced matrices involved in Eq. (20) are such that

[Ψg]T

[K][Ψg] = [Σg] and [Ψg]T

[M][Ψg] = [S]T

[S] . (34)

Remark 1. Physical interpretation of the filtering strat-egy. Introducing the (m×n) real matrix [Φr] = [P][Φ], we

obtain [Φ]T

[Mr][Φ] = [Φr]T

[M][Φr]. The following inter-pretation of Eq. (30) can be given. While reduced stiff-

ness matrix [Φ]T

[K][Φ] is the projection of stiffness ma-trix [K] onto the basis [Φ] of the elastic modes (includingboth the global and the local displacements), the reduced

mass matrix [Φr]T

[M][Φr] is the projection of mass ma-trix [M] onto displacements represented by matrix [Φr],which belong to subspace SR and in which some local

displacements are filtered.

Remark 2. Computational efficiency. It should be noted

that the numerical rank, R , of [N ][N ]T

is such that R ≤r and R ≤ n. Similarly to Eq. (18), truncation orderν must satisfy the inequality ν ≤ R. For solving thegeneralized eigenvalue problem defined by Eq. (30), threecases are considered.

– For r < n, a thin SVD (see [123]) of the (n× r) realmatrix [Λ]−1/2[N ] is performed for a lower cost.

– For r � n, R = n is verified and matrix [N ][N ]T

ispositive definite. For this case, the usual algorithmsare used.

– For the intermediate case for which R is close to n,the SVD approach is more efficient in order to obtaina good accuracy.

3.4 Local-displacements reduced-order basis

In the rest of the paper, it is assumed that the above nu-merical implementation of the global-displacements ROBis used. In this section, we present the construction of alocal-displacements ROB. The vector subspace associ-ated with the local-displacements ROB, denoted by S` ,is the orthogonal complement of subspace Sg of Sc withrespect to the inner-product defined by matrix [M]. Inparticular, Sc is the orthogonal direct sum of Sg with S`,

Sc = Sg ⊕ S` . (35)

Thanks to this definition of S` , its ROB, denoted by [Φ`],satisfies the orthogonality condition

[Φg]T

[M][Φ`] = [0], (36)

as well as the following equality

[Φ`] = [Φ][Q`], (37)

in which [Q`] is the (n×n`) real matrix of the coordinatesin the basis defined by [Φ], and where the dimension n`is such that

n` = n− ng . (38)

Let [Qg] be the (n× ng) real matrix such that

[Qg] = [S][R] . (39)

Using Eqs. (21), (33), and (39) yield

[Φg] = [Φ][Qg] . (40)

From Eqs. (3), (36), (37), and (40), the following orthog-onality property can be deduced,

[Qg]T

[Q`] = [0] . (41)

Let [Z] be the (n×n`) real matrix whose columns are theright-singular vectors associated with the n` zero singu-

lar values of the SVD of [Qg]T

(and which, consequently,

9

is an algebraic basis of the nullspace of [Qg]T

). By con-struction, matrix [Z] verifies

[Z]T

[Z] = [In`] . (42)

Equation (41) is satisfied for [Q`] expressed as

[Q`] = [Z][U ] , (43)

in which [U ] is a (n`×n`) real matrix of coordinates in thebasis defined by [Z]. The local-displacements ROB rep-resented by matrix [Φ`] is then defined by the first eigen-vectors of the dynamical system, which are constrainedto belong to the vector space spanned by the n` columnsof the (m × n`) real matrix [Φ][Z]. The columns ϕ`α of[Φ`] are thus written as

ϕ`α = [Φ][Z]uα , (44)

in which, thanks to Eqs.(6) and (42), it can be deducedthat the columns uα of [U ] are the eigenvectors of thefollowing standard eigenvalue problem,

([Z]T

[Λ][Z])uα = λ`α uα . (45)

In this Section 3, a general method has been presentedfor obtaining a global-displacements ROM, for whichthe construction of the associated ROB depends on thechoice of parameters d (maximum degree of the polyno-mial approximation), ν (truncation order), and f c (cut-off frequency). The dimension ng of the ROM resultsfrom the values of these parameters. Suitable values ofd and ν must be tuned in order to obtain a smallerdimension ng ≤ n while preserving the fidelity of thecomputational model. As higher frequencies are charac-terized by vibrations of shorter wavelength, the choiceof the values for d and ν strongly depend on the valueof frequency f c. The construction of a ROB made upof local displacements, which is complementary to theglobal-displacements ROB, has also been proposed andwill be useful for the construction of a multilevel ROM.

4 Multilevel reduced-order model

4.1 Formulation of the multilevel ROM

In this section, the previous developments are used in theconstruction of a multilevel ROM, for which three ROBsassociated with the low-, medium-, and high-frequencybands (LF, MF, HF) are introduced. In contrast to theHF band, the LF band is associated with long-wavelengthglobal displacements, while the MF band is a combina-tion of global and local displacements with more or lessshort wavelength. For the complex structures considered,there is an overlap of the three vibration regimes. For in-stance, numerous local elastic modes can be found in thelow-frequency band. The previously introduced filteringstrategy is therefore used in order to separate the LF-,

MF-, and HF-type displacements. The filtering method-ology presented in Sections 3.3 and 3.4 can be repre-sented by the following mapping,

F1 : (Sc ; d, ν, f c) 7−→ (Sg ,S`) , (46)

which will be used for defining the multilevel ROM. Asthe classical ROM, the multilevel ROM is devoted tothe vibration analysis over whole band B, which can bedecomposed into the three following bands,

B = BL ∪ BM ∪ BH . (47)

We now present the basic ideas concerning the construc-tion of the multilevel ROM based on the introductionof three successive filterings, which are defined throughmapping F1 . We introduce the cutoff frequency f cH forwhich the value has to be chosen by considering the valueof the upper bound of BH . The (global-displacements)vector subspace St , which includes the totality of theremaining considered displacements and which is associ-ated with the multilevel ROM, is given by

(St ,S⊥t ) = F1(Sc ; dH, νH, fcH) , (48)

in which the values of the parameters dH and νH aretuned in order to obtain a decreased dimension whilepreserving the fidelity of the ROM up to frequency f cH.It should be recalled that subspace Sc is assumed to beassociated with an accurate classical ROM over band B.The (local-displacements) vector subspace S⊥t , verifyingSc = St⊕S⊥t , is not used (the local displacements span-ning this subspace are discarded in order to decrease thedimension of the proposed ROM). For carrying out thenext filterings, similarly to the double projection methodassociated with Eq. (28), the computational model is nowprojected onto subspace St, which is supposed to be as-sociated with a sufficiently accurate representation.

Similarly, we introduce the cutoff frequency f cM whichis be chosen by considering the value of the upper boundof BM . The (local-displacements) vector subspace, SH ,associated with the HF vibration regime is given by

(SLM ,SH) = F1(St ; dM, νM, fcM) , (49)

where SLM is the complementary (global-displacements)subspace belonging to St . The values of the parametersdM and νM are tuned such that the ROB associatedwith SLM yields an adequate representation up to fre-quency f cM . We have the decomposition St = SLM⊕SH.

Finally, the cutoff frequency f cL is introduced, which ischosen by considering the value of the upper bound ofBL . The (global-displacements) vector subspace, SL , as-sociated with the LF vibration regime, is given by

(SL ,SM) = F1(SLM ; dL, νL, fcL) , (50)

where SM is the complementary (local-displacements)subspace belonging to SLM. The values of the param-eters dL and νL are tuned such that the ROB associ-ated with SL and mainly made up of global displace-ments, yields a sufficiently accurate representation up

10

to frequency f cLIt should be noted that, with a similarmanner as the previous one, the proposed constructionmakes the assumption that subspace SLM constructedwith a higher polynomial degree (that has been tunedfor covering a broader) frequency band (LF ∪ MF)),is associated with a sufficiently accurate representation(for the LF band). Finally, we have the decompositionSLM = SL ⊕ SM and consequently,

St = SL ⊕ SM ⊕ SH . (51)

4.2 Implementation of the multilevel nominal ROM

4.2.1 Numerical procedure

In this section, the numerical implementation is detailed.We summarize the steps introduced in Sections 3.3 and3.4 that are devoted to the construction of the global-displacements ROB and of the local-displacements ROB.Similarly to Eq. (46), the procedure is then compactedin a mapping, F2 , which allows for defining the alge-braic quantities associated with the multilevel ROM for-mulated in Section 4.1. In the following, the steps forcalculating the outputs of mapping F2 are given.

For this, it is assumed that maximum degree d of thepolynomial approximation, truncation order ν, and cut-off frequency f c are given and some new notations areintroduced in order to generalize the numerical proce-dure. Let [Λ0] be a diagonal matrix of dimension n0whose diagonal elements are strictly positive. Let [Q0]be a (n1 × n0) real matrix for which n1 ≥ n0 and such

that [Q0]T

[Q0] = [In0]. Let [N0] be a (n1×rmax) real ma-

trix for which rmax ≥ r , with r = (d+1)(d+2)(d+3)/2.These three matrices are the input parameters of map-ping F2 in addition to the filtering parameters that ared, ν, and f c. The outputs of mapping F2 will allow usto define the construction of all the matrices involved inthe multilevel ROM. It should be noted that, aside fromthe multilevel ROM, mapping F2 allows for constructingthe global- and the local-displacements ROBs defined inSection 3 by using, as inputs the quantities [Λ0] = [Λ],[Q0] = [In] with n1 = n0 = n, and [N0] = [N ].

The matrix [N0] is associated with a reduced kinemat-ics for which the maximum degree of the polynomials isgreater or equal to d, and which can already have beenused during a previous filtering. Let [N0

r ] be the (n1× r)matrix constituted of the first r columns of [N0]. In ad-dition, the previous filtering is associated with a changeof basis defined by matrix [Q0]. Let [Nr] be the (n0 × r)real matrix defined by

[Nr] = [Q0]T

[N0r ] . (52)

The eigenvectors sα and the associated eigenvalues σgαare calculated (similarly to Eq. (30)) as

[Λ0] sα = σgα([Nr][Nr]T

) sα . (53)

Introducing the matrix [S] = [s1 . . . sν ] and the ma-trix [Σg] of the eigenvalues σgα, the eigenvectors rα andthe associated eigenvalues λgα are calculated (similarly toEq. (20)) as

[Σg] rα = λgα([S]T

[S])rα . (54)

Dimension ng of the global-displacements ROB is themaximum integer α verifying the inequality fgα ≤ f c, in

which fgα =√λgα/2π. Then, matrix [Λg] is made up of

the ng eigenvalues λgα. Let [R] be the matrix such that[R] = [r1 . . . rng ] and let [Qg] be the matrix defined by

[Qg] = [S][R]. Introducing [C] = [Qg]T

, the followingSVD is performed,

[C] = [UC ][ΣC ][VC ]T. (55)

The columns of [VC ] associated with the n` zero singularvalues in [ΣC ] allow for obtaining the (n0×n`) real ma-trix [Z] with n` = n0 − ng. Finally, the eigenvectors uαand the associated eigenvalues λ`α are calculated (simi-larly to Eq. (45)) as

([Z]T

[Λ0][Z])uα = λ`α uα , (56)

which allows for obtaining the matrix [U ] = [u1 . . .un`]

and the diagonal matrix [Λ`] of the n` eigenvalues λ`α,followed by the construction of matrix [Q`] = [Z][U ].

The procedure summarized hereinbefore allows the fol-lowing mapping to be constructed,

([Λg], [Qg], [Nr], [Λ`], [Q`]) =

F2([Λ0], [Q0], [N0]; d, ν, f c) , (57)

in which the outputs [Qg] and [Q`] verify the followingorthonormality properties,

[Qg]T

[Qg] = [Ing], [Q`]

T[Q`] = [In`

],

[Qg]T

[Q`] = [0] . (58)

These properties can be deduced from Eqs. (20), (34),(37), (41), (42), (43), (45), and from the fact that theunit generalized mass normalization is used.

4.2.2 Construction of the reduced-order bases

First filtering. In Eq. (48), vector subspace St is obtainedby the projection onto Sc in which dH , νH , and f cH arethe filtering parameters. Let nt be the dimension of themultilevel ROM and let rH = (dH+1)(dH+2)(dH+3)/2.The real matrices [Λt], [Qt], and [N t], respectively ofdimensions (nt×nt), (n×nt), and (n× rH) are definedas the first three outputs of mapping F2 such that

([Λt], [Qt], [N t],∼,∼) =

F2([Λ], [In], [N ]; dH, νH, fcH) , (59)

11

in which [N ] = [Φ]T

[M`][B`]. The symbol ∼ indicatesthat the corresponding output variables are not calcu-lated. The ROB [Φt] associated with St is given by

[Φt] = [Φ][Qt] , (60)

in which, thanks to Eq. (58), [Qt] verifies

[Qt]T

[Qt] = [Int] . (61)

Second filtering. In Eq. (49), vector subspaces SLM andSH have their ROB defined through the following out-puts of mapping F2 ,

([ΛLM], [QLM], [NLM], [ΛH], [QH]) =

F2([Λt], [Qt], [N t]; dM, νM, fcM) , (62)

with dM ≤ dH, νM ≤ nt, and where (similarly to Eq. (58))the following properties are verified,

[QLM]T

[QLM] = [InLM ], [QH]T

[QH] = [InH ],

[QLM]T

[QH] = [0] , (63)

with nLM = dim(SLM) and nH = dim(SH). The ROBs[ΦLM] and [ΦH] of subspaces SLM and SH are given by

[ΦLM] = [Φt][QLM] , [ΦH] = [Φt][QH] . (64)

Third filtering. In Eq. (50), vector subspaces SL and SMhave their ROB defined through the following outputs ofmapping F2,

([ΛL], [QL],∼, [ΛM], [QM]) =

F2([ΛLM], [QLM], [NLM]; dL, νL, fcL) , (65)

with dL ≤ dM, νL ≤ nLM, and where (similarly toEq. (58)) the following properties are verified,

[QL]T

[QL] = [InL ], [QM]T

[QM] = [InM ],

[QL]T

[QM] = [0] , (66)

with nL = dim(SL) and nM = dim(SM) . The ROBs[ΦL] and [ΦM] of subspaces SL and SM are given by[ΦL] = [ΦLM][QL] and [ΦM] = [ΦLM][QM] , which yields

[ΦL] = [Φt][QLM][QL] , [ΦM] = [Φt][QLM][QM] .(67)

4.2.3 Construction of the reduced-order models

Scale-S reduced-order model. For S representing any oneof the symbols t , LM , H , L , and M , when using thescale-S ROM, displacements vector U(ω) belong to thesubspace SS that is defined as the space spanned by thecolumns of matrix [ΦS ], and are written as

U(ω) ' [ΦS ]qS(ω) . (68)

The nS -dimensional complex vector of generalized co-ordinates qS(ω) is the solution of the reduced-matrixequation,

(−ω2[MS ] + iω[DS ] + [KS ] )qS(ω) = fS(ω) , (69)

in which it can easily be proved that

fS(ω) = [ΦS ]TF(ω) , [MS ] = [ΦS ]

T[M][ΦS ] = [InS ] ,

[DS ] = [ΦS ]T

[D][ΦS ] , [KS ] = [ΦS ]T

[K][ΦS ] = [ΛS ] .(70)

Multilevel nominal reduced-order model. The (m × nt)ROB of the multilevel nominal ROM (ML-NROM) isrepresented by matrix [Ψ ] that is written as

[Ψ ] = [ [ΦL][ΦM][ΦH] ] . (71)

Using the ML-NROM, displacement vector U(ω) is ap-proximated by

U(ω) ' [Ψ ]q(ω)

= [ΦL]qL(ω) + [ΦM]qM(ω) + [ΦH]qH(ω) , (72)

in which q(ω) = (qL(ω),qM(ω),qH(ω)) with qL(ω) inCnL , qM(ω) in CnM , and qH(ω) in CnH . The complexvector q(ω) is the solution of the reduced-matrix equa-tion,

(−ω2[M ] + iω[D] + [K] )q(ω) = f(ω) , (73)

in which

f(ω) = [Ψ ]TF(ω) , [M ] = [Ψ ]

T[M][Ψ ] ,

[D] = [Ψ ]T

[D][Ψ ] , [K] = [Ψ ]T

[K][Ψ ] . (74)

Let us define [WL] = [QLM][QL] , [WM] = [QLM][QM] ,and [WH] = [QH] . Let ([A], [A]) representing any one ofthe symbols ([M ], [M]), ([D], [D]), and ([K], [K]). FromEqs. (64), (67), and (71), it can be deduced that a block-

writing of [A] = [Ψ ]T

[A][Ψ ] can be written as

[A] =

[ALL ] [ALM ] [ALH ][AML] [AMM] [AMH][AHL ] [AHM ] [AHH ]

, (75)

for which the matrix blocks are defined as follows. For Iand J in {L,M,H}, the (nI ×nJ ) real matrix [AIJ ] isgiven by

[AIJ ] = [W I ]T

[At][WJ ] , (76)

in which, thanks to Eq. (60), [At] = [Qt]T

[A][Qt] with

[A] = [Φ]T

[A][Φ]. Moreover, Eqs. (61), (63), and (66)yield

[W I ]T

[W I ] = [InI ] , [W I ]T

[WJ ] = [0] if I 6= J , (77)

from which it can be deduced that [M ] = [Int ] .

12

4.3 Multilevel stochastic reduced-order model

Similarly to the C-SROM, the multilevel stochastic ROM(ML-SROM) is based on the nonparametric probabilisticapproach of uncertainties. This approach allows for tak-ing into account both the model-parameter uncertaintiesand the model uncertainties induced by the modeling er-rors. The ML-NROM previously presented is based onthe use of three orthogonal ROBs represented by thematrices [ΦL], [ΦM], and [ΦH], which are constitutedof LF-, MF-, and HF-type displacements, respectively.For instance, as explained in Section 4.1, the LF-typedisplacements consist of long-wavelength global displace-ments, in contrast to the short-wavelength local displace-ments of the HF band. When a small design change isperformed in the structure, the local displacements thatexist in the modified part of the structure are likely tovary a lot, whereas the shape of the global displacementsis not really modified. Subsequently and as it is wellknown, the local displacements are more sensitive to un-certainties than the global displacements.

For each given random matrix [A] representing [M], [D]or [K] of the ML-SROM, which is associated with thecorresponding deterministic matrix [A] representing [M ],[D] or [K] of the ML-NROM, three dispersion hyperpa-rameters, δLA , δMA , and δHA are introduced. These hyper-parameters are intended to allow each type of displace-ments to be affected by a particular level of uncertainties.For S equal to L, M or H, the dispersion hyperparam-eter δSA is such (see Eq. (8)) that

(δSA)2

=1

nSE{||[GnS (δSA)]− [InS ]||2F } . (78)

We define the random matrix [GA] with values in theset of all the positive-definite symmetric (nt × nt) realmatrices, such that

[GA] =

[ GnL(δLA) ] [ 0 ] [ 0 ][ 0 ] [GnM(δMA )] [ 0 ][ 0 ] [ 0 ] [ GnH(δHA ) ]

, (79)

in which the random matrices [GnL(δLA)], [GnM(δMA )],and [GnH(δHA )], with dimensions (nL×nL), (nM×nM),and (nH × nH), are statistically independent and areconstructed similarly to the (n× n) random matrix [A]defined by Eq. (7). Performing the Cholesky factorization

[A] = [LA]T

[LA], in which [LA] is an upper-triangularmatrix, the random matrix [A] is constructed as

[A] = [LA]T

[GA][LA] . (80)

The ML-SROM allows the random displacements vectorU(ω) associated with U(ω) to be obtained as

U(ω) = [Ψ ]Q(ω), (81)

in which the Cnt-valued random variable Q(ω) is thesolution of the random matrix equation,

(−ω2[M] + iω[D] + [K] )Q(ω) = f(ω) . (82)

For all ω in B, the random equation defined by Eq. (82)is solved with the Monte-Carlo simulation method.

4.4 Comments about the construction and theidentification of the ML-SROM

A multilevel ROM has been presented for which the con-struction depends on filtering parameters. Three filter-ings are successively performed. For each filtering, thefiltering parameters have to be chosen such that the ap-proximation is satisfactory for the considered frequencyband. The satisfaction criterion can be based either ondeterministic or on stochastic quantities of interest. InSection 4.3, a multilevel stochastic ROM has been intro-duced, for which the uncertainty is controlled by threedispersion hyperparameters associated with each one ofthe three families, namely, LF-, MF-, and HF-type dis-placements, which constitute the ROB of the ROM. Inthe framework of the nonparametric probabilistic ap-proach of uncertainties (which uses a random matrix the-ory), the dispersion hyperparameters allow for control-ling the level of statistical fluctuations for each reducedmatrix of the ROM (around its mean matrix) . Thus,the hyperparameters take into account uncertainty in aglobal manner for each type of forces (the inertial forces,the damping forces, and the elastic forces). The statis-tical inverse identification (or calibration) of the hyper-parameters of the ML-SROM depends on the filteringparameters, which has been made for constructing theML-NROM. For example, one set of filtering parameterscan lead us to a low-dimension ROM with low eigen-value density while another set of filtering parameterscan lead us to a high-dimension ROM with high eigen-value density. With fixed dispersion hyperparameters,the stochastic ROM with a high dimension is likely toexhibit a higher level of uncertainty than the ROM witha low dimension.

5 Application to an automobile structure

5.1 Problem definition

5.1.1 Experimental measurements: excitation force,observation points, and frequency band of analysis

Experimental measurements of some FRFs have beencarried out for ne = 20 nominally identical cars over abroad frequency band, B = 2π × [10 , 900] Hz. For eachcar, the same excitation force is applied to one of theengine fasteners and the acceleration (following a givendirection) is measured at two locations referenced as ob-servation 1 that is far away from the excitation force andas observation 2 that is close to the excitation force.

5.1.2 Computational model

The finite element model is very dense and complex, in-cluding several kinds of elements (springs, bars, beams,

13

plates, shells, volume elements), rigid bodies, and con-straint equations. The total number of DOFs is m =7, 872, 583. A view of the finite element model is dis-played in Fig. 1.

Fig. 1 View of the finite element model of the automobile.

5.1.3 Modal density characterizing the dynamics anddefinition of the LF, MF, and HF bands

An intensive computational effort has been carried outfor calculating the 24, 578 elastic modes that are in thefrequency band [0 , 2200] Hz. The graph of the modaldensity corresponding to these 24, 578 elastic modes isdisplayed in Fig. 2. From this calculation, it can be seen

Frequency (Hz)

0 500 1000 1500 2000

Modal density (

Hz

-1)

0

5

10

15

Fig. 2 Modal density calculated with the computationalmodel.

that there are 7, 258 elastic modes in frequency bandB. There are several possible definitions of the frequencybands BL , BM , and BH for a complex dynamical system.The definitions greatly depend on the use that is madeof the bands defined. In the present framework devotedto the construction of a multilevel model reduction instructural vibrations, we choose the approach proposedin [40], which is based on the analysis of the graph of theunwrapped phase as a function of the frequency in loga-rithmic scale for observation 1 that is far away from the

excitation force and consequently, for which the propaga-tion follows a long path. It is recalled that the unwrappedphase corrects the radian phase angles by adding multi-ples of ± 2π when absolute jumps between two consecu-tive sampled frequencies are greater than or equal to thejump tolerance of π radians. It is known that in the LFrange, the phase rotates of π around an eigenfrequencywhile, in the HF band, the unwrapped phase decreasesquasilinearly. Figure 3 displays the graph of the un-wrapped phase obtained with the computational model,which is compared to the 20 graphs that correspondto the experimental measurements. The analysis of thisfigure allows for defining the following frequency bandsBL = [10, 70] Hz, BM =]70, 300] Hz, and BH =]300, 900]Hz. There are 149 elastic modes in low-frequency bandBL , 1, 202 elastic modes in medium-frequency band BM ,and 5, 897 elastic modes in high-frequency band BH , or,in average, about 2.5 modes per Hz in BL , 5 modes perHz in BM , and 10 modes per Hz in BH . A modal densityof 2.5 modes per Hz is quite high for the LF band of sucha structure. This unusual feature is due to the presenceof numerous local displacements in addition to the usualglobal displacements. For higher frequencies, the densityof local elastic modes keeps increasing, which yields alarge number of elastic modes for the modal analysis.

10 70 300 900

−150

−100

−50

0

Frequency (Hz) in log scale

Un

wra

pp

ed

ph

ase

Fig. 3 Graph of the unwrapped phase as a function of thefrequency in logarithmic scale for observation 1. Computa-tional model (thick line) and 20 experimental measurements(thin lines).

5.1.4 Damping model for the automobile

The damping matrix [D] does not correspond to the fi-nite element discretization of physical damping and dis-sipation phenomena. The damping model is introducedat the ROM level. For each ROM, the damping matrixis constructed by using a modal damping model. In theLF band and in the MF band, a multi-parameter modaldamping model is fitted by using the experimental FRFs.In the HF band, a one-parameter modal damping modelis identified by using the experimental FRFs, for whichthe parameter is denoted by cH (for the details of such aconstruction, see [2]). For the deterministic ROMs, three

14

cases have to be considered in order to properly definethe damping model, depending on which ROM is used.

– The reduced damping matrix of the C-NROM in-volved in Eq. (5) is defined by [D] = 2[Ξ(cH)][Λ]1/2.The diagonal matrix [Λ] is defined by Eq. (6) and[Ξ(cH)] is a (n×n) diagonal matrix of modal damp-ing rates that depend on parameter cH, which has tobe identified with respect to the experimental mea-surements, in a deterministic framework. In such acase, Eq. (5) consists of a diagonal matrix equation.

– The reduced damping matrix of the scale-S ROM in-volved in Eq. (69) is defined by the equation [DS ] =2[ΞS(cH)][ΛS ]1/2. The diagonal matrix [ΛS ] is de-fined by Eq. (70) and [ΞS(cH)] is a (nS×nS) diagonalmatrix of damping rates, which depend on parametercH that has to be identified with respect to the exper-imental measurements, in a deterministic framework.In such a case, Eq. (69) consists of a diagonal matrixequation.

– Concerning the ML-NROM, reduced stiffness matrix[K] involved in Eq. (73) is a full matrix. In orderto solve latter matrix equation, one possibility is toperform a change of basis in order to diagonalize thereduced matrices [K] and [M ]. Doing so leads us backto scale-t ROM, for which the definition of the damp-ing matrix [Dt] is given in the item just before.

For the stochastic ROMs, two cases have to be consideredin order to properly define the random matrix of thedamping model, depending on which stochastic ROM isused.

– For constructing the random reduced damping ma-trix [D] of the C-SROM involved in Eq. (11), Eq. (7)is not used. In order to solve Eq. (11), the randomgeneralized eigenvalue problem associated with ran-dom reduced matrices [K] and [M] is solved, whichyields the diagonal matrix, [Λ], of the random eigen-values and the matrix, [Φ], of the associated randomeigenvectors. Equation (11) is then projected onto thestochastic ROB [Φ]. In order to obtain a diagonal ma-trix equation, random damping matrix [D] of the C-SROM is then constructed similarly to deterministicreduced damping matrix [D] of the C-NROM givenhereinbefore, except that diagonal matrix [Λ] is re-placed by [Λ]. The parameter cH has to be identifiedwith respect to the experimental measurements, in astochastic framework.

– In Eq. (80), the definition given to the random damp-ing matrix [D] of the ML-SROM involved in Eq. (82)is not used. In order to solve Eq. (82), the randomgeneralized eigenvalue problem associated with ran-dom reduced matrices [K] and [M] is solved, whichyields the diagonal matrix, [Σ], of the random eigen-values and the matrix, [Ψ ], of the associated randomeigenvectors. Equation (82) is then projected onto thestochastic ROB [Ψ ]. In order to obtain a diagonal ma-trix equation, the random reduced damping matrix

[D] of the ML-SROM is then constructed similarlyto the deterministic reduced damping matrix [D] ofthe ML-NROM given just before, except that diago-nal matrix [Λt] is replaced by [Σ]. The parameter cHhas to be identified with respect to the experimentalmeasurements, in a stochastic framework.

5.1.5 Definition of the observations

Let U(1)(ω), . . . ,U(nj)(ω) be the nj scalar observationsthat are made up of DOFs or combinations of DOFsof the displacements vector U(ω). In this application,we have nj = 2 (corresponding to the two experimentalobservations). For each j, the computed observation isdefined as the modulus ω 7→ ucj(ω) in dB scale,

ucj(ω) = 20 log10|U(j)(ω)| , (83)

in which |.| is the modulus of a complex number. Thecounterpart for the experimental measurements of thek = 1, . . . , ne cars (with ne = 20) is denoted by ω 7→uej,k(ω) for which the definition is the same as the defi-

nition of ucj(ω).

5.1.6 Defining the objective function used for theconvergence analyses of the deterministic computationalROMs

In order to define a distance between the computed FRFsand the experimental FRFs, an objective function, Jd, isdefined by

J2d =

1

nj

nj∑j=1

1

ne

ne∑k=1

1

|B|

∫B

(ucj(ω)− uej,k(ω))2dω , (84)

with |B| = ωmax−ωmin . It should be noted that the con-struction of Jd would remain unchanged if, in Eq. (84),the displacements were replaced by their correspondingaccelerations.

5.1.7 Defining the objective function used for theidentification of the stochastic computational ROMs

The filtering parameters and the hyperparameters of thestochastic computational ROMs have to be identifiedwith respect to the experimental measurements (solv-ing a statistical inverse problem). Let Uej (ω) be the real-valued random variable for which uej,1(ω), . . . , uej,ne

(ω)are ne independent realizations. Let U cj (ω) be the real-valued random variable corresponding to the determinis-tic quantity ucj(ω). The following objective function, Js ,is introduced

Js =1

nj

nj∑j=1

Js,j , (85)

15

in which the objective function Js,j associated with ob-servation j = 1, . . . , nj is written as

Js,j =1

|B|

∫B

OVL (U cj (ω), Uej (ω)) dω . (86)

In Eq. (86), the function (X,Y ) 7→ OVL (X,Y ) is definedby

OVL (X,Y ) = 1− 1

2

∫R|pX(x)− pY (x)| dx , (87)

in which X and Y are real-valued random variables, forwhich pX and pY are the probability density functions.Function OVL is known as the overlapping coefficient[124]. For all j = 1, . . . , nj and all ω in B, nsim real-izations of random variable U cj (ω) are computed by us-ing the Monte-Carlo simulation method. The probabilitydensity functions are estimated by using the kernel den-sity estimation method. It can easily be proved that thevalues of Js, of Js,j for j = 1, . . . , nj , and of the OVLfunction, are between 0 and 1 (with 1 meaning a perfectmatch).

5.2 Classical nominal ROM and classical stochasticROM

In a first step, the dimension n of the C-NROM (de-terministic) is calculated by performing a convergenceanalysis of Jd as a function of n. For this value of n, theC-NROM is used for computing the deterministic FRFsthat are compared to their experimental counterparts. Ina second step, for the value of n determined in the firststep, the hyperparameters of the C-SROM (stochastic)are identified by maximizing the objective function Js.Using the identified values of the hyperparameters, theC-SROM is used for estimating the confidence regionsof the random FRFs, which are compared to the exper-imental measurements.

5.2.1 First step: C-NROM

Convergence analysis of Jd as a function of n. The C-NROM is obtained by using Eqs. (4) and (5). A conver-gence analysis of the C-NROM is performed with respectto its dimension n and to parameter cH that controlsthe damping model. For each value of n, the optimalvalue of cH is identified, in minimizing Jd. For each pair(n , cH(n)), Fig. 4 displays the graph of Jd as a functionof n and shows that convergence is reached starting fromn = 7,000. Nevertheless, we choose n = 8,450 (for whichfn = 1,000 Hz), in order (1) to have subspace Sc suffi-ciently rich for the construction of subspace St ⊆ Sc ofthe multilevel ROM and (2) to have a classical ROM as-sociated with eigenfrequencies covering more than wholeband B for obtaining a satisfying stochastic ROM.

Deterministic FRFs and experimental comparisons. The

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

50

100

150

200

250

300

350

400

n

Jd

Fig. 4 Convergence of the C-NROM with respect to itsdimension: value of Jd for several values of n (black crosses).Horizontal light-gray line: value of Jd for n = 8, 450.

100 200 300 400 500 600 700 800 900

−90

−80

−70

−60

−50

−40

−30

−20

Frequency (Hz)

Accele

ration (

dB

,m/s

2)

Fig. 5 Observation 1: experimental FRF measurements(black lines), deterministic FRF using the C-NROM (grayline)

100 200 300 400 500 600 700 800 900−80

−70

−60

−50

−40

−30

−20

−10

Frequency (Hz)

Acce

lera

tio

n (

dB

,m/s

2)

Fig. 6 Observation 2: experimental FRF measurements(black lines), deterministic FRF using the C-NROM (grayline)

FRFs provided by the C-NROM, associated with obser-vation 1 (acceleration related to the displacement ω 7→uc1(ω)) and with observation 2 (acceleration related tothe displacement ω 7→ uc2(ω)), are plotted in Figs. 5 and6 and are compared to the ne = 20 experimental FRFs(accelerations related to displacements ω 7→ uej,k(ω) with

k = 1, . . . , ne). These figures clearly show that

– the experimental variabilities strongly increase withthe frequency (it should be noted that the relativeimportant experimental variabilities in the LF bandfor observation 1 that can be seen in Fig. 5 is due tomeasurement noise).

16

– significant differences between the prediction of thecomputational model and the experimental measure-ments can be observed, in particular in the MF band.

– the big experimental variabilities cannot be repre-sented by a deterministic computational model, butrequire the prediction of confidence regions with astochastic computational model.

5.2.2 Second step: C-SROM

Experimental identification of the C-SROM. The C-SROM

Fig. 7 Plot of function (δM, δK) 7→ Js(δM, δK) for the iden-tification of hyperparameters δM and δK of the C-SROM.Black dots: sampling points, black cross: optimal point.

0 5 10 15 20 25 30 35 400.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

nsim

Js

Fig. 8 Convergence of objective function Js (black line), ofJs,1 (dark-gray line), and of Js,2 (light-gray line) with respectto number of simulations nsim.

is obtained by using Eqs. (10) and (11) for which thedispersion hyperparameters and damping parameter cHmust be identified with respect to the experimental mea-surements, by solving a statistical inverse problem. Therandom damping matrix [D] of the C-SROM is the onedefined in Section 5.1.4 and consequently, there is no dis-persion hyperparameter δD. The statistical inverse prob-lem consists in computing the optimal values δoptM , δoptK ,

and coptH of the optimization problem defined as the max-imization of the objective function Js with respect tothe three hyperparameters δM, δK, and cH in the setof their admissible values. The optimization problem is

not convex and a trial method is used by introducing afine grid of the admissible set with a nonhomogeneousdistribution of the sampling points (that can be viewedin Fig. 7). For each point (δM, δK) in the grid, cH 7→Js(δM , δK , cH) is maximized yielding the optimal valuec?H(δM, δK). Figure 7 displays the graph of function (δM, δK) 7→Js(δM, δK , c

?H(δM, δK)) calculated at the sampling points

of the grid. It can be seen that there is a quasi-symmetrywith respect to δM = δK axis. It can then be deduced theoptimal solution, for which δoptM = 0.13 and δoptK = 0.11.

Concerning the calculation of the objective function, foreach sampling point of the grid, the value of the objectivefunction Js is estimated using nsim Monte-Carlo simula-tions. Figure 8 presents the convergence of the objectivefunction Js as a function of nsim evaluated at the op-timal point (δoptM , δoptK , coptH ). It can be seen that, for areasonable precision of Js (of about 0.01), convergenceis reached quite fast for nsim = 40, which can be consid-ered as a good compromise between the numerical costand the accuracy (it would make little sense to carryout a very fine statistical estimation if not exploring theparameter space with a sufficiently fine grid – and viceversa).

Concerning the computational cost, for each independentrealization of random matrices [M] and [K] in Eq. (11),the matrix equation is diagonalized (solving the general-ized eigenvalue problem of reduced dimension associatedwith the conservative dynamical system) for obtainingan efficient resolution over (1) the frequency samplingand (2) the sampling of parameter cH.

Confidence regions of the random FRFs and experimen-tal comparison. The confidence region (corresponding

Fig. 9 Observation 1: experimental FRF measurements(black lines), random FRF using the identified C-SROM(gray region), and overlap function OVL (black line under-neath)

to a probability level of 95%) of each FRF is computedby using the identified C-SROM. A convergence anal-

17

Fig. 10 Observation 2: experimental FRF measurements(black lines), random FRF using the identified C-SROM(gray region), and overlap function OVL (black line under-neath)

Fig. 11 Observation 1, zoom into band BL ∪ BM: experi-mental FRF measurements (black lines), random FRF usingthe identified C-SROM (gray region), and overlap functionOVL (black line underneath)

ysis of the confidence region with respect to the num-ber nsim of realizations in the Monte-Carlo simulationmethod has been performed. A good convergence of theconfidence region is reached for nsim = 10, 000. The cor-responding values of objective functions Js,1 and Js,2 forthe identified C-SROM are Js,1 = 0.62 and Js,2 = 0.56,which yields Js = 0.59 (the statistical estimation is donewith nsim = 10,000). The results for observation 1 andobservation 2 are displayed in Figs. 9 and 10. On eachfigure, it can be seen the confidence region, the 20 ex-perimental measurements, and the OVL function ω 7→OVL(U cj (ω), Uej (ω)) defined by Eq. (87). This OVL func-tion is plotted between two horizontal lines. The lowerhorizontal line corresponds to the value 0 and the up-per one to the value 1. In Figs. 9 and 10, it can be seenthat the C-SROM does not perfectly represent most ofthe experimental FRFs in the LF and MF bands. Thiscan be explained by the discrepancies of the C-NROMwith respect to the experiments and by the too narrowconfidence regions provided by the C-SROM in the LF

Fig. 12 Observation 2, zoom into band BL ∪ BM: experi-mental FRF measurements (black lines), random FRF usingthe identified C-SROM (gray region), and overlap functionOVL (black line underneath)

Fig. 13 Observation 1, zoom into band BL: experimentalFRF measurements (black lines), random FRF using theidentified C-SROM (gray region), and overlap function OVL(black line underneath)

Fig. 14 Observation 2, zoom into band BL: experimentalFRF measurements (black lines), random FRF using theidentified C-SROM (gray region), and overlap function OVL(black line underneath)

band (and to a lesser extent in the MF band). This canbe better seen in Figs. 11 and 12 that present a zoom of

18

Figs. 9 and 10 into band BL ∪ BM, and in Figs. 13 and14 that present a zoom of Figs. 9 and 10 into band BL. Itcan also be seen that OVL function confirms the not per-fectly correct prediction of the C-SROM in the LF andMF bands. The reason why such a one-level stochasticmodel of uncertainties is not sufficient for predicting theconfidence regions in all the frequency band is that theeffects of uncertainties on the FRFs are not homogeneousin the frequency band. It should be noted that, with thisone-level stochastic ROM, if one would want to obtainbroader confidence regions in the LF band, one could usegreater values for the dispersion hyperparameters but insuch a case, one would consequently obtain too broadconfidence regions in the HF band. The introduction ofa multilevel stochastic ROM allows for improving theprediction as demonstrated in the next section.

5.3 Multilevel nominal ROM and multilevel stochasticROM

In a first step, a deterministic analysis is carried out inorder to find suitable filtering parameters, which affectthe reduction of the dimension of the model. In fact,this deterministic analysis is not sufficient in itself, asthe final objective is the experimental identification ofthe ML-SROM. Consequently, in a second step, all thefiltering parameters defining the multilevel ROB and allthe dispersion hyperparameters are simultaneously iden-tified. To this end, first, a temporary choice of filteringparameters defining the ML-NROM is done, based onthe deterministic analysis. Then, a sensitivity analysiswith respect to the dispersion hyperparameters allowsfor decreasing the number of dispersion hyperparame-ters to be identified. Based on this assumption, a 3Dcoarse grid allows for finding initial values for the dis-persion hyperparameters. All the other parameters beingfixed, the filtering parameters of the HF band and thedispersion hyperparameter of the HF band are simulta-neously identified in a precise way. Finally, defining thevalues of the other filtering parameters independently ofthe global optimization problem, the remaining disper-sion hyperparameters are identified at a low cost. Usingthe identified ML-SROM, the confidence regions of therandom FRFs are statistically estimated and are thencompared to the experimental measurements.

5.3.1 First step: ML-NROM

Convergence analysis of Jd as a function of dimension ntdeduced from a first filtering. For constructing the multi-level ROM, the first step consists in defining the filteringof local displacements, which is devoted to the reductionof the final dimension nt of the proposed ROM. This stepcorresponds to either Eq. (49) or Eq. (59). It dependson the maximum degree dH of the polynomials of thereduced kinematics of mass matrix [Mr], on truncation

order νH that is indirectly related to the upper boundof the frequency band (through the value

√σtνH/2π in

which σtα is the eigenvalue of rank α in Eq. (54), in-volved in the mapping F2 in Eq. (59)) and to the cutofffrequency f cH. It should be noted that since by construc-

tion one has St ⊆ Sc , the frequency f tnt=

√λtnt

/2π ver-

ifies the inequality f tnt≤ fn. Therefore, choosing f cH =

fn = 1, 000 Hz would automatically yield nt = νH. Thevalue of f cH is taken as 925 Hz. It should be noted thatthe scale-t ROM (given by Eqs. (68) and (69)) yieldsthe same response as the ML-NROM (by construction).Exploring the possible values of parameters dH and νHallows the construction of the ML-NROM to be adjusted.It is recalled that λtνH ≤ σtνH . For fixed dH , some sam-pling points are explored in a segment in which param-eter νH verifies

f cH ≤√σtνH/2π ≤ 10× f cH . (88)

Figure 15 presents a plot of function Jd with respect toparameters dH and νH , with dH ranging from 2 to 40and with the corresponding values of νH deduced fromEq. (88). For νH ≥ 4,000, it can be seen that the value ofJd is only subjected to small fluctuations. It should alsobe noted that, in general, for a fixed νH between 2,500and 4,000, the value of Jd is greater (i.e. is less good) fora greater value of dH. This is due to the fact that for anincreasing value of dH, the value of νH has to increase inorder that the quantity

√σtνH/2π reach the upper bound

of the frequency band. Indeed, a larger maximum degreedH means an increasing presence of local displacements,which means more basis vectors kept in the constructionof the ROM.

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

1000

2000

3000

4000

5000

6000

7000

DH

Hν

Fig. 15 Plot of function Jd with respect to parameters dHand νH (lighter gray level meaning higher Jd)

Figure. 16 shows the convergence of the ML-NROM withrespect to its dimension nt. For this, the value of Jd isplotted as a function of nt for several ROMs (all the sam-pling points). Likewise, the convergence of the C-NROMis given on the same figure, in order to put into evidencethe faster convergence of the ML-NROM towards a valuecorresponding to a reasonable accuracy, compared to the

19

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

50

100

150

200

250

300

350

400

nt

Jd

Fig. 16 For the C-NROM, graph of n 7→ Jd(n) (blackcrosses) and graph of n 7→ Jd(8,450) (horizontal light-grayline). For the ML-NROM, graph of nt 7→ Jd(nt) (gray dots).

C-NROM. This is explained by the fact that the ML-NROM lacks local displacements (and thus has a lowerdimension), which are not really important for repre-senting the dynamical responses. For a low maximumdegree dH of the polynomials (and for a low truncationorder νH), the lower dimension nt that is obtained for theROM is associated with more important modeling errors.For constructing the ML-NROM, a good compromise be-tween the value of nt and the accuracy associated withthe value of Jd is sought. In fact, the purpose of the mul-tilevel ROM is to better represent the variabilities of theexperimental measurements in the frequency band (i.e.to obtain random FRFs that are able to represent theexperimental measurements) and consequently, the iden-tification of parameters dH and νH depends on the val-ues taken by the dispersion hyperparameters of the ML-SROM (coupled problem). The previous exploration ofthe possible values of parameters dH and νH is thus notsufficient. Nevertheless, it allows for defining a subregionin which the global identification (taking into accountthe coupling of the dispersion hyperparameters with thefiltering parameters) of the ML-SROM will be restrained.Concerning the computational cost, one sampling pointinvolves solving an eigenvalue problem of dimension νH(associated with Eq. (20) or Eq. (54)). In addition, it im-plies solving, for each value of the maximum degree dHof the polynomials, an eigenvalue problem of dimensionn (associated with Eq. (30) or Eq. (53)), which can besolved at a lower cost if dH (and thus column dimensionrH of matrix [N t] in Eq. (59)) is low, through a SVD (asexplained in Remark 2 of Section 3.3). Furthermore, foreach value of maximum degree dH of the polynomials,the matrix [N t], which is constructed in Eq. (52) withinthe use of mapping F2 in Eq. (59), can be obtained byextracting the first rH columns of a matrix [N t] asso-ciated with a reduced kinematics of dimension rH,max

that satisfies rH,max ≥ rH for all considered values ofrH. This way, the construction of matrix [B`] and thematrix product with (m × n) matrix [Φ] are done onceand for all.

Deterministic FRFs and experimental comparisons. Fig-

100 200 300 400 500 600 700 800 900

−90

−80

−70

−60

−50

−40

−30

−20

Frequency (Hz)

Acce

lera

tio

n (

dB

,m/s

2)

Fig. 17 Observation 1: experimental FRF measurements(black lines), deterministic FRF using the identified ML-NROM (gray line)

100 200 300 400 500 600 700 800 900−80

−70

−60

−50

−40

−30

−20

−10

Frequency (Hz)

Acce

lera

tio

n (

dB

,m/s

2)

Fig. 18 Observation 2: experimental FRF measurements(black lines), deterministic FRF using the identified ML-NROM (gray line)

ures 17 and 18 show the deterministic FRFs obtained us-ing the ML-NROM with dH = 34 and νH = 4,250, whichare the estimated optimal values obtained by solving theglobal stochastic optimization problem presented in thenext section. It can be seen that the ML-NROM, withreduced dimension nt = 4,232, yields a satisfactory pre-diction with respect to that of the C-NROM.

5.3.2 Second step: ML-SROM

Global experimental identification of all the parametersdefining the ML-SROM. For the inverse identification ofthe ML-SROM, the global stochastic optimization prob-lem requires identifying the successive input parameters(filtering parameters) of function F1 in Eqs. (48), (49),and (50) used for defining the ML-NROM. Based on theML-NROM, the ML-SROM is constructed upon disper-sion hyperparameters δLK , δLM , δMK , δMM , δHK , and δHMthat also have to be identified. The random reduceddamping matrix [D] of the ML-SROM is defined in Sec-tion 5.1.4 and consequently, this random matrix is notcontrolled by dispersion hyperparameters δLD, δMD , andδHD . The cutoff frequencies f cL and f cM are chosen as theupper bounds of bands BL and BM, that is to say f cL =70 Hz and f cM = 300 Hz. It is recalled that f cH = 925 Hz.To sum up, in addition to cH and to the 6 dispersionhyperparameters, the ML-SROM is defined by the 6 fil-

20

tering parameters, (dH , νH), (dM , νM), and (dL , νL).

Construction of a first version of the ML-NROM. First,from the numerical exploration of parameters (dH , νH)in a deterministic framework as described in latter sec-tion (see also Fig. 15), a first version of the scale-t ROM,given by dH = 20 and νH = 3,900, is chosen. Then, suc-cessive numerical explorations of the parameters (dM , νM)and (dL , νL) are carried out in a similar manner by us-ing the scale-LM and scale-L ROMs (these ROMs wereintroduced in Section 4.2.3). These explorations yield(dM = 12, νM = 800) and (dL = 6, νL = 275). It shouldbe noted that the coupling with the dispersion hyperpa-rameters to be identified is, at this step, not taken intoaccount.

Sensitivity analysis of the ML-SROM with respect to thedispersion hyperparameters. A sensitivity analysis of theML-SROM associated with latter definition of the ML-NROM is carried out. It shows that for a given scale Sequal to L,M or H, the influence of parameters δSM andδSK (associated with the statistical dispersion of reducedmass and stiffness matrices) is similar. Therefore, only3 dispersion hyperparameters δL , δM , and δH have tobe identified: δL = δLK = δLM , δM = δMK = δMM , andδH = δHK = δHM .

First identification of the hyperparameters of the ML-SROM using a coarse 3D grid. A first identification iscarried out using a coarse δL × δM × δH grid, for whichthe boundaries are deduced from the previous sensitiv-ity analysis. The 3D grid is constituted of 540 samplingpoints, defined by the cartesian product of the followingsets δL × δM × δH:

δL = ( 0.15 , 0.20 , 0.25 , 0.30 , 0.35 , 0.40 , 0.45 , 0.50 , . . .

0.55 , 0.60 ) ,

δM = ( 0.10 , 0.15 , 0.20 , 0.25 , 0.30 , 0.35 , 0.40 , . . .

0.45 , 0.50 ) ,

δH = ( 0.05 , 0.07 , 0.09 , 0.11 , 0.13 , 0.15 ) . (89)

As for the C-SROM, only nsim = 40 Monte-Carlo simu-lations are used for estimating the objective function Jsassociated with each sampling point. The optimal pointfound is δL = 0.25 , δM = 0.25 , δH = 0.11 , which issufficiently far from the grid boundary. Concerning thecomputational cost, similarly to the C-SROM, for eachindependent realization of random matrices [M] and [K]in Eq. (82), the matrix equation is diagonalized, by solv-ing an eigenvalue problem of dimension nt. In addition,damping parameter cH is identified for each samplingpoint. It should be noted that, compared to dimensionn = 8,450 of the classical ROM, the final dimensionnt = 4,232 of the identified multilevel ROM allows forobtaining a nonnegligible gain of about a factor of ten(the complexity of a full eigenvalue problem being ap-

proximatively cubic).

Precise and simultaneous identification of the filteringparameters of the HF band and of the dispersion hy-perparameter of the HF band. We are now interested inadjusting the values of parameters (dH , νH) , responsi-ble for the dimension nt of the ML-NROM. Supposingthese filtering parameters are sufficiently big, the choiceof their values does not influence the random FRFs inthe LF and MF bands (which have already convergedwith respect to them). The influence of parameter δM

(and especially of parameter δL) is not preponderant forthe random response in the HF band. Consequently, fix-ing the values of δL and δM that have been identified inthe coarse 3D grid, the ML-SROM is identified with re-spect to filtering parameters (dH , νH) and to dispersionhyperparameter δH, simultaneously. It should be notedthat the identification does not consist, at this step, inpicking the parameters that maximize the objective func-tion Js. Rather, it consists in finding a set of parametersfor which Js is close to its maximum but under the con-straint of a reasonably small dimension nt for the ROM.Anyway, it should be recalled that Js is only estimatedusing nsim = 40 realizations. After this step, the param-eters that are chosen are dH = 34, νH = 4,250, andδH = 0.078.

Final stage for the global identification of the ML-SROM.Parameters (dL , νL) , (dM , νM) , δL , and δM remain tobe identified. In order to avoid a 6-dimensional costlyoptimization problem, the 4 filtering parameters are leftunchanged. It should be noted that these parameterscontrol the overlap of subspaces SL , SM , and SH ofthe multilevel ROM, associated with the LF-, MF-, andHF-type displacements. Also, if parameters (dL , νL) and(dM , νM) tended towards infinity, then there would beno overlap of subspaces SL , SM , and SH. Qualitatively,subspace SL is supposed to be composed of long-wavelengthglobal displacements, without numerous local displace-ments. Therefore, based on these physical considerations,it is more suitable to force the values of filtering parame-ters (dL , νL), and (dM , νM) outside the global stochas-tic identification problem of the ML-SROM. Then, pa-rameters δL and δM of the ML-SROM are identified byestimating objective function Js at the sampling pointsof a given 2D grid, the other parameters being fixed.After this final stage, the identified values are δL = 0.4and δM = 0.22. For the identified ML-SROM, the valuesof objective functions Js,1 and Js,2 are Js,1 = 0.65 andJs,2 = 0.64, hence Js = 0.65. Such a value correspondsto a nonnegligible improvement with respect to the C-SROM.

Confidence regions of the random FRFs and experimen-tal comparisons. The random FRFs associated withthe identified ML-SROM are plotted in Figs. 19 and 20.It can be seen that, despite the discrepancies of the ML-

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Fig. 19 Observation 1: experimental FRF measurements(black lines), random FRF using the identified ML-SROM(gray region), and OVL function (black line underneath)

Fig. 20 Observation 2: experimental FRF measurements(black lines), random FRF using the identified ML-SROM(gray region), and OVL function (black line underneath)

NROM (which are similar to those of the C-NROM), theML-SROM is able to represent most of the experimentalFRFs (unlike the C-SROM).

This improvement is due to the increased flexibility ofthe ML-SROM with respect to the C-SROM, particu-larly concerning the capability to adapt the level of un-certainties to each frequency band. It should be notedthat, in general, on one hand the variability of the realsystem is low in the LF band and that, on the otherhand, the robustness of the computational models is bet-ter in this band. In the present case, in the LF and MFbands, the modeling errors are more important than thelevel of variabilities of the real system, hence the largeconfidence intervals provided by the ML-SROM in thesebands. Figures 21 and 22 display a zoom of Figs. 19 and20 into band BL ∪ BM and, Figs. 23 and 24 display azoom of Figs. 19 and 20 into band BL. It can be seenthat the OVL function confirms the improved predictionof the ML-SROM in the LF and MF bands.

Fig. 21 Observation 1, zoom into band BL ∪ BM: experi-mental FRF measurements (black lines), random FRF usingthe identified ML-SROM (gray region), and OVL function(black line underneath)

Fig. 22 Observation 2, zoom into band BL ∪ BM: experi-mental FRF measurements (black lines), random FRF usingthe identified ML-SROM (gray region), and OVL function(black line underneath)

5.4 Sensitivity analysis

5.4.1 Proposed ML-SROM

A sensitivity analysis of the ML-SROM with respect tothe dispersion hyperparameters is presented. The objec-tive of such a sensitivity analysis is to quantify the roleplayed by each dispersion hyperparameter that is relatedto a given type of displacements. Using the identified pa-rameters of the ML-SROM and successively setting tozero 2 dispersion hyperparameters out of the 3 hyperpa-rameters δL , δM , δH allow for quantifying the individ-ual contribution of each scale L ,M , H of displacementsin the random responses. The random responses are ob-tained by using the identified ML-SROM and by setting,successively, δM = δH = 0 (see Fig. 25 for observation 1and Fig. 28 for observation 2), δL = δH = 0 (see Fig. 26for observation 1 and Fig. 29 for observation 2), andδL = δM = 0 (see Fig. 27 for observation 1 and Fig. 30for observation 2). In each figure, the vertical lines indi-

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Fig. 23 Observation 1, zoom into band BL: experimen-tal FRF measurements (black lines), random FRF using theidentified ML-SROM (gray region), and OVL function (blackline underneath)

Fig. 24 Observation 2, zoom into band BL: experimen-tal FRF measurements (black lines), random FRF using theidentified ML-SROM (gray region), and OVL function (blackline underneath)

Fig. 25 Observation 1: random FRF using the identifiedML-SROM but for which δM = δH = 0 is imposed.

cate the boundaries between the LF, MF, and HF bands.Figures 27 and 30 show, for instance, that adding uncer-tainties to the HF-type displacements yields the presenceof uncertainties in the LF and MF bands. It is explainedby the fact that, since dH is greater than dM, some HF-

Fig. 26 Observation 1: random FRF using the identifiedML-SROM but for which δL = δH = 0 is imposed.

Fig. 27 Observation 1: random FRF using the identifiedML-SROM but for which δL = δM = 0 is imposed.

Fig. 28 Observation 2: random FRF using the identifiedML-SROM but for which δM = δH = 0 is imposed.

type displacements are likely to be present in the LF andMF bands.However, despite the absence of MF-type displacementsin the HF band, Figs. 26 and 29 show, for instance,that adding uncertainties to the MF-type displacementsyields the presence of uncertainties in the HF band. Thisis due to the fact that the LF-, MF-, and HF-type dis-placements are not orthogonal with respect to the stiff-ness matrix.

5.4.2 Naive ML-SROM

A sensitivity analysis is carried out by using a naiveML-SROM that is defined as follows. The vector basisassociated with subspace SL is given by the 149 elastic

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Fig. 29 Observation 2: random FRF using the identifiedML-SROM but for which δL = δH = 0 is imposed.

Fig. 30 Observation 2: random FRF using the identifiedML-SROM but for which δL = δM = 0 is imposed.

modes that belong to frequency band BL, the vector ba-sis associated with subspace SM is given by the 1,202elastic modes that belong to frequency band BM, andthe vector basis associated with subspace SH is given bythe next 7,099 elastic modes. Note that this ML-SROMcorresponds to choosing filtering parameters (dL , νL),(dM , νM), and (dH , νH) as going to infinity (no filter-ing). The sensitivity analysis of this stochastic ROM iscarried out using the same set of combinations of thedispersion hyperparameters as in the previous section.Figures 31, 32, and 33 depict the random FRFs obtainedfor observation 1 while Figs. 34, 35, and 36 depict therandom FRFs obtained for observation 2. It can be seenthat, for this naive ML-SROM, the introduction of uncer-tainties for one given vector basis induces the presenceof uncertainties for the corresponding frequency band,while practically none elsewhere. This stochastic ROMis thus not well adapted for modeling uncertainties of acomplex dynamical system for which the LF, MF, andHF vibration regimes overlap.

5.5 Remark concerning the use of the proposedmultilevel ROM

The filtering parameters allow for separating the differ-ent regimes of vibration (LF, MF, and HF), which canexist in the frequency band of analysis. These regimescan directly be viewed by using the deterministic com-

Fig. 31 Observation 1: random FRF using the naive ML-SROM for which δM = δH = 0 is imposed.

Fig. 32 Observation 1: random FRF using the naive ML-SROM for which δL = δH = 0 is imposed.

Fig. 33 Observation 1: random FRF using the naive ML-SROM for which δL = δM = 0 is imposed.

putational model for which the classical ROM is con-structed as the first step of the multilevel ROM and con-sequently, allows for analyzing the dynamical behaviors.The filtering parameters are strongly connected to allthe geometrical and mechanical properties, but do notinduce some problems with respect to the prediction ob-jectives, taking into account the proposed methodologythat allows for fixing the values of the filtering param-eters without using experiments. The hyperparametersthat are introduced in the probabilistic models of uncer-tainties can either be used for performing a sensitivityanalysis with respect to the level of uncertainties if ex-periments are not available or be identified if experimentsare available. In this last case, for a class of dynamicalsystems (such as a class of popular automobiles corre-

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Fig. 34 Observation 2: random FRF using the naive ML-SROM for which δM = δH = 0 is imposed.

Fig. 35 Observation 2: random FRF using the naive ML-SROM for which δL = δH = 0 is imposed.

Fig. 36 Observation 2: random FRF using the naive ML-SROM for which δL = δM = 0 is imposed.

sponding to a given level of technologies), the identifiedhyperparameters can generally be reused (without addi-tional identifications) for similar dynamical systems.

Concerning the numerical cost of the proposed multi-level ROM, for a big computational structural dynamicsmodel (for instance with 10 million of DOFs), which isanalyzed over a very broad frequency band (LF, MF,and HF bands), the stochastic analysis (including theidentification procedure) requires the introduction of aROM in order the calculations to be tractable, in par-ticular if such a model is used for robust design. For ob-taining the C-SROM, one needs to solve Eq. (2) for thefirst eigenvalues/eigenvectors (offline stage), which canbe costly (large-scale generalized eigenvalue problem).

When the Monte-Carlo method is used a a stochasticsolver (that is the case in the paper), each realizationof the C-SROM implies solving a full eigenvalue prob-lem of dimension 8,450. For obtaining the ML-SROM,one reuses the eigenvalues/eigenvectors calculated for theC-SROM. All the following calculations devoted to theconstruction of the multilevel ROM (offline stage) havea negligible cost in comparison to the large-scale gener-alized eigenvalue problem of Eq. (2). Then, each realiza-tion of the ML-SROM implies solving a full eigenvalueproblem of dimension 4,232. Due to the roughly cubiccomplexity of a full eigenvalue problem, the ML-SROMallows for obtaining a speed factor of about 10 comparedto the C-SROM. Nevertheless, the ML-SROM requiresmore parameters to be identified. Still, for this case, theidentification of the ML-SROM remains cheaper.

6 Conclusions

A general method has been presented for the construc-tion of a multilevel stochastic ROM devoted to the ro-bust dynamical analysis of complex systems over a broadfrequency band. The complex systems are constitutedof several structural scales. In the low-frequency range,these scales induce the presence of numerous short-wavelengthlocal displacements that are intertwined with the usuallong-wavelength global displacements. The proposed mul-tilevel ROM is based on the construction of three orthog-onal ROBs whose displacements are either LF-type, ei-ther MF-type, or HF-type displacements, and which areassociated with the overlap of the LF, the MF, and theHF vibration regimes. The construction of these ROBsrelies on a filtering strategy that is based on the introduc-tion of global shape functions for the kinetic energy. It isadmitted that the local displacements are more sensitiveto uncertainties than the global displacements. Combin-ing the use of the nonparametric probabilistic approachof uncertainties with the constructed multilevel ROB al-lows for constructing a multilevel stochastic ROM, forwhich each type of displacements can be assigned to aspecific level of uncertainties. As an application, the mul-tilevel stochastic ROM of an automobile is constructedand is identified by using experimental measurements.Such an approach allows for obtaining a smaller dimen-sion for the stochastic ROM as well as an improved pre-diction, with respect to a classical stochastic ROM.

Acknowledgements This research was supported by AgenceNationale de la Recherche, Contract HiMoDe, ANR-2011-BLAN-00378. The authors thank PSA Peugeot Citroen forproviding the experimental measurements and the computa-tional model.

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