Probabilistic model identification of uncertainties in computational models for dynamical systems and experimental validation

Abstract

We present a methodology to perform the identification and validation of complex uncertain dynamical systems using experimental data, for which uncertainties are taken into account by using the nonparametric probabilistic approach. Such a probabilistic model of uncertainties allows both model uncertainties and parameter uncertainties to be addressed by using only a small number of unknown identification parameters. Consequently, the optimization problem which has to be solved in order to identify the unknown identification parameters from experiments is feasible. Two formulations are proposed. The first one is the mean-square method for which a usual differentiable objective function and an unusual non-differentiable objective function are proposed. The second one is the maximum likelihood method coupling with a statistical reduction which leads us to a considerable improvement of the method. Three applications with experimental validations are presented in the area of structural vibrations and vibroacoustics.

Stochastic modeling of uncertainties in computational structural dynamics—Recent theoretical advances

“…This prior probability distribution is constructed by using the Maximum Entropy Principle [20] (from Information Theory [21]) for which the constraints are defined by the available information [13,14,22,15]. Since the paper [13], many works have been published in order to validate the nonparametric probabilistic approach of model uncertainties with experimental results (see for instance [23,24,25,26,27,28,15,29]), to extend the applicability of the theory to other areas [30,31,32,33,34,35,36,37,38,39,40,41], to extend the theory to new ensembles of positive-definite random matrices yielding a more flexible description of the dispersion levels [42], to apply the theory for the analysis of complex dynamical systems in the medium-frequency range, including vibroacoustic systems, [43,44,23,45,25,26,27,28,46,47,48,39], to analyze nonlinear dynamical systems (i) for local nonlinear elements [49,50,37,…”

“…Such a prior stochastic model can then be used to study the propagation of uncertainties through the mechanical system which is analyzed. If experimental data are available for the mechanical system, then they can be used (1) to identify the parameters of the prior stochastic model [29] using, for instance, the maximum likelihood method [57,58] or (2) to construct a posterior stochastic model [12] using, for instance, the Bayesian method (see for instance [59,60,61,62,58]). …”

“…Many works have been published in this field, such as [79] in micromechanics, [78] for the identification and prediction of stochastic dynamical systems in a polynomial chaos basis, [23] for the experimental identification of the nonparametric stochastic model of uncertainties in structural dynamics [134,24,29], in structural acoustics for the low-and medium-frequency ranges [26,28], in nonlinear structural dynamics [49,50]. The identification of the generalized probabilistic approach of uncertainties can be found in [56] and also below in Sections 5.3 and 6.…”

Post-buckling nonlinear static and dynamical analyses of uncertain cylindrical shells and experimental validation

“…There exists several methodologies adapted to the identification of stochastic computational models, see for instance [46,47,48]. In the present research, the identification procedure is achieved using the maximum likelihood method associated with a statistical reduction of the information [49]. Note that it is particularly adapted to the present case for which the random variables U obs (s i , δ) are not only dependent but also strongly correlated.…”

“…Note that the likelihood function is replaced by an approximation which is constructed as the product of marginal probability density functions of each uncorrelated random variable. Although this assumption modifies the Likelihood function, it has been shown that its combination with the statistical reduction yield accurate estimations [49]. Seeking the maximum of L red (δ) for the experimental identification yields the optimal value δ opt = 0.45.…”