Verification of polynomial chaos surrogates in the framework of structural vibrations with uncertainties

Abstract

Surface response models, such as polynomial chaos Expansion, are commonly used to deal with the case of uncertain input parameters. Such models are only surrogates, so it is necessary to develop tools to assess the level of error between the reference solution (unknown in general), and the value provided by the surrogate. This is called a posteriori model verification. In most works, people usually search for the mean quadratic error between the reference problem and the surrogate. They use statistical approaches such as resampling or cross-fold validation, residual based approaches, or properties of the surrogate such as variance decay. Here, we propose a new approach for the specific framework of structural vibrations. Our proposition consists of a residual-based approach combined with a polynomial chaos expansion to evaluate the error as a full random variable, not only its mean square. We propose different variants for evaluating the error. Simple polynomial interpolation gives good results, but introducing a modal basis makes it possible to obtain the error with good accuracy and very low cost.