A model reduction method to analyze the dynamic behavior of vibrating structures with uncertain parameters
Duc-Thinh Kieu  1  , Marie-Laure Gobert  1  , Sébastien Berger  1@  , Jean-Mathieu Mencik  1  
1 : Laboratoire de Mécanique Gabriel Lamé
Université de Tours : EA7494, Institut National des Sciences Appliquées - Centre Val de Loire : EA7494, Université d'Orléans : EA7494, Université d'Orléans : EA7494

Assessing the dynamic response of vibrating structures which are described by means of finite element (FE) models with many degrees of freedom (DOFs) is usually computationally cumbersome. The fact that the manufacturing process and the material properties of structures are usually subject to variability means a dispersion of the physical parameters which can be important. The parameters are therefore considered as uncertain DOFs, which makes FE models more complex. The Monte Carlo (MC) method is commonly used to analyze the propagation of uncertainties through FE modeling. However, it requires a large number of simulations which are therefore very cumbersome in terms of CPU times.

This work aims at developing a low cost computational strategy to compute the harmonic response of vibrating structures having uncertain parameters. The strategy works by considering the Craig-Bampton method to reduce the physical DOFs of a FE model [1]. Also, a generalized Polynomial Chaos (gPC) expansion is considered to describe the propagation of uncertainties and estimate the Quantities of Interest (QoIs), e.g., the displacement at some measurement points, or an energy quantity. Intrusive and non-intrusive methods are both considered to apply the gPC expansion [2, 3]. In the first case, the Galerkin projection method is used to express the FE dynamic equilibrium equation of a structure in a gPC subspace; as for the non-intrusive method, it requires a non-negligible number of simulations of the FE model to be performed to estimate the gPC coefficients, and further the statistics (i.e., mean and variance) of the QoIs. The probability law of the QoIs can be obtained by considering the gPC expansions along with the MC method with 10,000 trials.

In this work, both the intrusive and non-intrusive methods are applied to model an academic structure made of two rectangular Kirchhoff-Love plates connected together across one of their edges by means of a lineic density of torsional springs with an uncertain stiffness. Comparisons with the results obtained from a reference MC solution involving 10,000 simulations of the FE model show good agreement and substantial reduction of the computational effort. The advantages and drawbacks of the intrusive and non-intrusive methods are discussed through numerical comparisons, as well as the influence of the Craig-Bampton reduction method on the estimation of the QoIs of the FE model.

[1] R. R. Craig, M. C. C. Bampton, Coupling of substructures for dynamic analyses, AIAA Journal 6 (7) (1968) 1313–1319

[2] R. G. Ghanem, P. D. Spanos, Stochastic Finite Element Method: Response Statistics, Springer New York, New York, NY, 1991, 101–119. doi:10.1007/978-1-4612-3094-6_4.

[3] D. Xiu, G. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM Journal on Scientific Computing 24 (2) (2002) 619_644. doi: 10.1137/S1064827501387826.

Keywords: Intrusive and non-intrusive generalized Polynomial Chaos, uncertainty propagation, Craig-Bampton method

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