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In this paper, an efficient numerical procedure is proposed, which is adapted to the modal analysis of a fully-loaded spent nuclear fuel canister exhibiting distinct structural levels associated with a hierarchy of components. On the one hand, the fully-loaded spent nuclear fuel canister is constituted of a repetition of identical components, resulting in a pseudo-periodicity; and the components of a given level are separated from each other and only connected to their upper level through localized attachments. This gives rise to an advantageous structural connectivity that can be exploited for efficiency. On the other hand, the necessary fine mesh resolution of the small levels leads to a high-dimensional computational model and, in addition, the independent resonant vibrations of each of the components produce a very large number of vibration eigenmodes. The aforementioned opportunities and difficulties are respectively leveraged and tackled by an adapted method that combines domain decomposition, shift–invert Lanczos eigenvalue solver, and Craig–Bampton substructuring technique. A parallel eigensolution via spectrum slicing is facilitated by an efficient block factorization by Schur complement due to the sparsity of the Craig–Bampton matrices. A computational gain of four orders of magnitude is obtained, at the expense of negligible errors that are exactly characterized. The proposed methodology enables a high-fidelity vibration analysis of the sealed fully-loaded spent nuclear fuel canister, useful for non-intrusive inverse identification of the structural integrity of the internal structural levels holding the nuclear material.

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.

Invited Lecture; National audience; Dans ces travaux, nous nous intéressons au comportement vibratoire des structures complexes en basses et moyennes fréquences. La complexité des structures visées ici induit une très forte densité modale dès les basses fréquences. Deux problèmes se posent alors pour l'analyse en basses et moyennes fréquences de ces structures. Le premier problème est relatif à la construction d'un modèle réduit de faible taille. En effet, la forte densité modale ne permet pas d'utiliser les modes élastiques classiques pour construire un tel modèle. Le deuxième problème est relatif à la forte sensibilité de la réponse dynamique aux incertitudes liées aux paramètres du système mais aussi aux incertitudes liées aux choix de modélisation. Il est important de prendre en compte ces incertitudes afin d'estimer la variabilité des quantités d’intérêt pour la famille de systèmes représentée par le modèle numérique. Pour traiter ces deux problèmes, nous présentons une méthode permettant d'extraire de manière séparée une base réduite de l'espace des déplacements globaux et une base réduite de l'espace des déplacements locaux. Étant donnée qu'en basses fréquences l'énergie mécanique est principalement portée par les déplacements globaux, la base globale permet de bien approximer la réponse dans cette bande de fréquence et donc de construire un modèle très réduit. En moyennes fréquences, les contributions locales ne sont plus négligeables et sont fortement sensibles aux incertitudes. Nous introduisons alors un modèle probabiliste d'incertitudes permettant de contrôler les fluctuations globales et les fluctuations locales de manière séparée. Une extension multi-échelle permettant de réduire la taille du modèle réduit local sera aussi présentée.

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.

International audience; The research addressed here concerns the construction of a stochastic reduced-order model for complex structure exhibiting many local and some global eigenmodes in the mid-frequency range but also in the low-frequency range. The methodology proposed here consists in splitting the admissible displacement space into two subspaces: a global displacement space and a local displacement space. The bases of these two subspaces are constructed by solving two separated eigenvalue problems for which the kinetic energy is modified in order to filter the local or global contributions for the displacements. This modification of the kinetic energy is performed by constructing adapted projection operators. Then the reduced-order model can be constructed by truncating and concatenating the global and local bases. The system-parameters uncertainties and the modeling uncertainties are taken into account using a nonparametric probabilistic approach. The global/local separation allows for constructing a probabilistic model for which the fluctuations of the global displacements and the fluctuations of the local displacements can be controlled separately. The methodology is validated through a numerical application related to the complex computational model of an automotive vehicle. Several families of projection operators are constructed and compared with respect to (1) the size the reduced-order model and (2) the effectiveness of the global/local separation.[1]C. Soize, A. Batou, Stochastic reduced-order model in low-frequency dynamics in presence of numerous local elastic modes, Journal of Applied Mechanics - Transactions of the ASME, 78(6) :061003 (2011).

International audience; In structural dynamics, the use of the vibration eigenmodes (elastic modes) allows for obtaining an accurate small-dimension reduced-order model (ROM) for the low-frequency range analysis. For some complex structures with distinct structural levels (presence of flexible parts attached to a stiff master part), numerous local elastic modes are intertwined with the usual global elastic modes, yielding high-dimension ROM. To circumvent this difficulty, a general method is proposed in order to construct a small-dimension ROM whose reduction vector basis is constituted of the global displacements only. The method is applied to an automobile complex structure.

National audience; Dans ce travail, on s'intéresse à la construction de modèles réduits prédictifs adaptés aux structures complexes pour lesquelles on trouve de nombreux modes locaux entrelacés avec des modes globaux dès les basses fréquences (BF), induisant une forte densité modale et conduisant à des modèles réduits de dimension élevée avec la méthode classique d'analyse modale. Nous proposons pour cette situation une nouvelle approche de construction du modèle réduit basée sur l'extraction d'une base des déplacements globaux.

International audience; This paper deals with the construction of a multilevel stochastic reduced-order model (ROM) devoted to the robust dynamical analysis of complex structures in a broad frequency band. In particular, we are interested in complex structures characterized by the presence of several structural scales (such as a stiff main body supporting flexible parts). In such a case, in addition to the usual global elastic modes (long-wavelengthmodes), numerous local elastic modes (associated with the flexible parts) appear. These local elastic modes, which are numerous in the low-frequency band already, exhibit high-frequency behavior, namely high modal density, small wavelength, and high sensitivity to uncertainty. In such a context, the low- , medium-, and highfrequency (LF, MF, HF) vibration regimes overlap. The objectives are to adapt the modeling of uncertainty to each vibration regime and, also, to deal with the unusually high dimension of the classic ROMs, which is due to the numerous local elastic modes. Thanks to a spatial filtering method of local displacements (that is based on the use of global polynomial shape functions for the kinetic energy), three successive filterings allow three families of displacements to be constructed, namely the LF-, MF-, and HF-type displacements. The filtering of the most local displacements allows the final dimension of the proposed ROM to be reduced. The multilevel reduced-order basis that is obtained yields a multilevel ROM. Using the nonparametric probabilistic approach of uncertainties with this ROM allows for obtaining a stochastic ROM for which the levels of uncertainty can be controlled independently for each type of displacements.The methodology is applied to a detailed finite element model of a car for which FRF measurements are available on a broad frequency band. Unlike a classic stochastic ROM constructed with the nonparametric approach, for which the probability law of each random reduced matrix is controlled by a unique dispersion hyperparameter, three hyperparameters are introduced for each random matrix of the multilevel stochasticROM. These stochastic ROMs are identified with respect to the measurements and are compared.