Multilevel reduced-order model for uncertainty quantification in computational structural dynamics
Résumé
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.