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This work deals with an extension of the classical construction of reduced-order models (ROMs) that are obtained through modal analysis in computational linear structural dynamics. It is based on a multilevel projection strategy and devoted to complex structures with uncertainties. Nowadays, it is well recognized that the predictions in structural dynamics over a broad frequency band by using a finite element model must be improved in taking into account the model uncertainties induced by the modeling errors, for which the role increases with the frequency. In such a framework, the nonparametric probabilistic approach of uncertainties is used, which requires the introduction of a ROM. Consequently, these two aspects, frequency-evolution of the uncertainties and reduced-order modeling, lead us to consider the development of a multilevel ROM in computational structural dynamics, which has the capability to adapt the level of uncertainties to each part of the frequency band. In this thesis, we are interested in the dynamical analysis of complex structures in a broad frequency band. By complex structure is intended a structure with complex geometry, constituted of heterogeneous materials and more specifically, characterized by the presence of several structural levels, for instance, a structure that is made up of a stiff main part embedding various flexible sub-parts. For such structures, it is possible having, in addition to the usual global-displacements elastic modes associated with the stiff skeleton, the apparition of numerous local elastic modes, which correspond to predominant vibrations of the flexible sub-parts. For such complex structures, the modal density may substantially increase as soon as low frequencies, leading to high-dimension ROMs with the modal analysis method (with potentially thousands of elastic modes in low frequencies). In addition, such ROMs may suffer from a lack of robustness with respect to uncertainty, because of the presence of the numerous local displacements, which are known to be very sensitive to uncertainties. It should be noted that in contrast to the usual long-wavelength global displacements of the low-frequency (LF) band, the local displacements associated with the structural sub-levels, which can then also appear in the LF band, are characterized by short wavelengths, similarly to high-frequency (HF) displacements. As a result, for the complex structures considered, there is an overlap of the three vibration regimes, LF, MF, and HF, and numerous local elastic modes are intertwined with the usual global elastic modes. This implies two major difficulties, pertaining to uncertainty quantification and to computational efficiency. The objective of this thesis is thus double. First, to provide a multilevel stochastic ROM that is able to take into account the heterogeneous variability introduced by the overlap of the three vibration regimes. Second, to provide a predictive ROM whose dimension is decreased with respect to the classical ROM of the modal analysis method. A general method is presented for the construction of a multilevel ROM, based on three orthogonal reduced-order bases (ROBs) whose displacements are either LF-, MF-, or HF-type displacements (associated with the overlapping LF, MF, and HF vibration regimes). The construction of these ROBs relies on a filtering strategy that is based on the introduction of global shape functions for the kinetic energy (in contrast to the local shape functions of the finite elements). Implementing the nonparametric probabilistic approach in the multilevel ROM allows each type of displacements to be affected by a particular level of uncertainties. The method is applied to a car, for which the multilevel stochastic ROM is identified with respect to experiments, solving a statistical inverse problem. The proposed ROM allows for obtaining a decreased dimension as well as an improved prediction with respect to a classical stochastic ROM, Ce travail de recherche présente une extension de la construction classique des modèles réduits (ROMs) obtenus par analyse modale, en dynamique numérique des structures linéaires. Cette extension est basée sur une stratégie de projection multi-niveau, pour l'analyse dynamique des structures complexes en présence d'incertitudes. De nos jours, il est admis qu'en dynamique des structures, la prévision sur une large bande de fréquence obtenue à l'aide d'un modèle éléments finis doit être améliorée en tenant compte des incertitudes de modèle induites par les erreurs de modélisation, dont le rôle croît avec la fréquence. Dans un tel contexte, l'approche probabiliste non-paramétrique des incertitudes est utilisée, laquelle requiert l'introduction d'un ROM. Par conséquent, ces deux aspects, évolution fréquentielle des niveaux d'incertitudes et réduction de modèle, nous conduisent à considérer le développement d'un ROM multi-niveau, pour lequel les niveaux d'incertitudes dans chaque partie de la bande de fréquence peuvent être adaptés. Dans cette thèse, on s'intéresse à l'analyse dynamique de structures complexes caractérisées par la présence de plusieurs niveaux structuraux, par exemple avec un squelette rigide qui supporte diverses sous-parties flexibles. Pour de telles structures, il est possible d'avoir, en plus des modes élastiques habituels dont les déplacements associés au squelette sont globaux, l'apparition de nombreux modes élastiques locaux, qui correspondent à des vibrations prédominantes des sous-parties flexibles. Pour ces structures complexes, la densité modale est susceptible d'augmenter fortement dès les basses fréquences (BF), conduisant, via la méthode d'analyse modale, à des ROMs de grande dimension (avec potentiellement des milliers de modes élastiques en BF). De plus, de tels ROMs peuvent manquer de robustesse vis-à-vis des incertitudes, en raison des nombreux déplacements locaux qui sont très sensibles aux incertitudes. Il convient de noter qu'au contraire des déplacements globaux de grande longueur d'onde caractérisant la bande BF, les déplacements locaux associés aux sous-parties flexibles de la structure, qui peuvent alors apparaître dès la bande BF, sont caractérisés par de courtes longueurs d'onde, similairement au comportement dans la bande hautes fréquences (HF). Par conséquent, pour les structures complexes considérées, les trois régimes vibratoires BF, MF et HF se recouvrent, et de nombreux modes élastiques locaux sont entremêlés avec les modes élastiques globaux habituels. Cela implique deux difficultés majeures, concernant la quantification des incertitudes d'une part et le coût numérique d'autre part. L'objectif de cette thèse est alors double. Premièrement, fournir un ROM stochastique multi-niveau qui est capable de rendre compte de la variabilité hétérogène introduite par le recouvrement des trois régimes vibratoires. Deuxièmement, fournir un ROM prédictif de dimension réduite par rapport à celui de l'analyse modale. Une méthode générale est présentée pour la construction d'un ROM multi-niveau, basée sur trois bases réduites (ROBs) dont les déplacements correspondent à l'un ou l'autre des régimes vibratoires BF, MF ou HF (associés à des déplacements de type BF, de type MF ou bien de type HF). Ces ROBs sont obtenues via une méthode de filtrage utilisant des fonctions de forme globales pour l'énergie cinétique (par opposition aux fonctions de forme locales des éléments finis). L'implémentation de l'approche probabiliste non-paramétrique dans le ROM multi-niveau permet d'obtenir un ROM stochastique multi-niveau avec lequel il est possible d'attribuer un niveau d'incertitude spécifique à chaque ROB. L'application présentée est relative à une automobile, pour laquelle le ROM stochastique multi-niveau est identifié par rapport à des mesures expérimentales. Le ROM proposé permet d'obtenir une dimension réduite ainsi qu'une prévision améliorée, en comparaison avec un ROM stochastique classique

International audience; Commonly, the low-frequency range [1] is characterized by the presence of a few dozen isolated eigenfrequencies that are associated with global modes, in which case the modal analysis method leads to an effective and efficient small-dimension reduced-order model (ROM). We present a new method for the robust prediction of frequency response functions (FRF) of complex structures exhibiting a high modal density. We consider complex structures for which there are more than hundreds or thousands modes in the low-frequency range. This unusual feature can be due to the presence of several structural scales within the complex geometry of the structure. Small flexible components attached to the stiff skeleton of the structure induce the presence of numerous local modes intertwined with the usual global modes of the stiff skeleton. For such complex structures, besides the absence of separation of scales, the global displacements (or global modes) cannot easily be identified because coupled with the large-amplitude local displacements (or local modes). First, the proposed method [2] allows for constructing a ROM of smaller dimension, which is obtained by introducing a subspace of global displacements. The construction of the latter is based on the introduction of high-degree polynomial shape functions. The basis of the global-displacements subspace is constituted of the eigenmodes calculated using such an approximation for the kinetic energy. The choice of the polynomial degree allows for controlling the filtering between the so-called global and local displacements, as well as the resulting dimension and accuracy of the so-called global ROM. Furthermore, it is well known that local displacements are in general more sensitive to uncertainties than global displacements. The nonparametric probabilistic approach [3] allows all sources of uncertainty to be globally accounted for by randomizing each reduced matrix whose probability density function, constructed applying the maximum entropy principle, is parameterized by a unique dispersion hyperparameter. In order to separately control the uncertainty level of the displacements of each of the scales, we propose a multilevel ROM, based on the introduction of orthogonal subspaces. The basis of each of these subspaces is constructed by using, notably, the aforementioned polynomial approximation for the kinetic energy, with an adapted polynomial degree. Each basis is constituted of displacements associated with a given structural scale. Then, the multilevel stochastic ROM is obtained by setting different dispersion hyperparameters for constructing the random matrix blocks associated with each scale. The method is applied to the complex computational model of a car and the dispersion hyperparameters of a multilevel stochastic ROM composed of three scales are identified with respect to experimental FRF measurements over a wide frequency band.[1] R. Ohayon and C.Soize, 1998, Structural acoustics and vibration, Academic Press.[2] O. Ezvan, A. Batou, and C. Soize, Multilevel reduced-order computational model in structural dynamics for the low- and medium-frequency ranges, Computers and Structures 160 (2015) 111-125.[3] C.Soize, A nonparametric model of random uncertainties for reduced matrix models in structural dynamics, Probabilistic Engineering Mechanics 15(3) (2000) 277-294.

International audience; We present the construction of a multilevel stochastic reduced-order model devoted to the robust prediction of frequency response functions of complex linear dy-namical systems that are characterized by the presence of several structural scales in which there are local displacements in addition to the usual global displacements, and which are associated with the distinct low-, medium-, and high-frequency bands. As the levels of uncertainties are different in the three frequency bands, a multilevel stochastic reduced-order model using several orthogonal subspaces associated with the several types of displacements is developed. The objective of the paper is to demonstrate the capability of the multilevel stochastic reduced-order model to adapt the stochastic modeling of uncertainties to each one of the three frequency bands.