Polynomial chaos expansions for uncertain dynamical systems – Applications in earthquake engineering

Abstract

In modern science and engineering, the consideration of uncertainties has become indispensable. The framework of uncertainty quantification, which aims at identifying, quantifying and hierarchizing the sources of uncertainties and studying their effects on the outputs of computational models, has been developed in the last decades. To this end, polynomial chaos expansions (PCEs) represent a powerful and versatile tool which is commonly used in various research fields. The objective of this PhD work is to develop PCE methods that can be applied to dynamical systems with uncertain parameters and/or uncertain excitation.

Chapter 2 presents the general mathematical foundation of generalized PCEs and all the aspects associated with their practical computation. An original analytical formulation of derivatives of PCEs, which allows a straightforward computation of sensitivity measures, is introduced.

In Chapter 3, a literature review on PCE methods for uncertain dynamical systems is thoroughly presented. It opens discussions on why pure vanilla PCEs fail to represent the uncertain behaviour of dynamical systems and how to overcome this issue. Successful existing methods are examined, which reveals their common strategy. Nonetheless, most of those methods are intrusive by construction, meaning that they are developed to solve specific uncertain evolution equations. The findings constitute the guidance upon which two non-intrusive, general-purpose methods are proposed in the remaining of the manuscript.

Chapter 4 introduces a PC-based \emph{stochastic time-warping} method to solve problems of random oscillations. The idea is to capture the dynamics characterized by the vibration frequency with the stochastic time-warping process before applying PCEs to represent the effects of uncertainties on the random amplitudes.

In Chapter 5, a more general method is investigated to solve problems of mechanical systems subject to stochastic excitations. The dynamics is handled with a stochastic nonlinear autoregressive with exogenous input (NARX) model, whose stochastic parameters are modelled with PCEs. The use of a sparsity-promoting regression technique is considered for selecting appropriate NARX terms and polynomial chaos functions.

Finally, Chapter 6 features applications of PC-based surrogate models in the context of earthquake engineering.~Predictions of the transient structural responses obtained with the proposed surrogates are used to compute seismic fragility curves. Original non-parametric methods for computing these curves are introduced, which allows one to assess the accuracy of the commonly used parametric methods based on the lognormal format.

The manuscript focuses on applications of PCEs in structural dynamics. However, the developed methods can be easily extended and used in various contexts as some numerical case studies from chemical engineering will illustrate. More importantly, the strategy utilized in the manuscript appears to be a promising research path which differs significantly from existing approaches and shall attract more attention from the uncertainty quantification community.

Keywords

Polynomial chaos expansions, uncertain dynamical systems, earthquake engineering, surrogate modelling, time-warping, nonlinear autoregressive with exogenous input models, fragility curves.

BibTeX cite

@PHDTHESIS{MaiThesis,
 author = {Mai, C. V.},
 title = {Polynomial chaos expansions for uncertain dynamical systems – Applications in earthquake engineering},
 school = {ETH Z\"urich, Z\"urich, Switzerland},
 year = {2016}
}