Low Rank Tensor Approximations for the Stochastic Steady-State Structural Dynamic Response

Abstract eng:
The robust design of mechanical systems requires a taking into account of different sources of uncertainties in the model in order to predict their impact on the response. The dynamical response of structures can notably be highly perturbed with low variability on the model. It then reveals necessary to develop reliable and efficient tools for the prediction of the dynamical random response. Functional approaches have emerged in the last two decades for uncertainty quantification in computation engineering and have lead to the so-called spectral stochastic methods for the propagation of uncertainties through physical models (see e.g. reviews [1,2,3]). In the dynamical framework, these methods suffer from the high sensitivity of the dynamical response to the input uncertainties for they require either very large size systems to be solved or a very important number of evaluations of the dynamical response. In order to tackle such problems, in [4] we have proposed a model reduction in order to considerably reduce the computational cost of Galerkin spectral stochastic methods for the computation of the steady-state dynamic response in the low frequency band. A tensor based approximation method, namely the Generalized Spectral Decomposition method [5], is used to obtain a quasi-optimal low rank separated representation of the solution on reduced deterministic and stochastic bases. However, depending on the level and the type of the perturbations, the rank of the approximation of the dynamic stochastic response may be high and the method thus loses its efficiency. We here propose a sample based approach using tensor approximation methods in which a work upstream, adapted classification and/or change of variable, is performed on the data automatically. Tensor approximation methods are then applied on each set of data each having lower rank approximation than the original entire data set.

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