Advanced surrogate modeling and machine learning methods in computational stochastic mechanics

Abstract

Parameter uncertainty quantification and methods to predict uncertainty propagation on the response of structural systems have become an essential part of the analysis and design of engineering applications. Over the past decades, the stochastic finite element method (SFEM) has been developed for the study of systems with inherent uncertainties in their parameters (i.e. material parameters), boundary conditions, geometry and loading conditions. The most eminent and widely applied method in the field of SFEM is the Monte Carlo simulation. In practice, this is the only method capable of handling stochastic problems with nonlinearities, dynamic loads, stability problems etc. However, in order to achieve high levels of accuracy, it requires a large number of repeated model evaluations for various realizations of the parameters. As a consequence, the computational cost associated with this method becomes unaffordable in detailed large-scale finite element models or nonlinear dynamic problems, where each model run takes minutes or hours to complete. Based on this conclusion, this thesis presents a series of methodologies based on surrogate modeling and machine learning techniques that have been implemented in order to bypass the computational demands of the direct Monte Carlo simulation. The first part of the dissertation focuses on the probability density evolution method and proposes appropriate formulations for its application in static and nonlinear problems. Also, a more accurate and efficient finite element scheme is proposed based on the Streamline Upwind/Petrov Galerkin method for the solution of the partial differential equations arising in the application of the method. In the continuation of the thesis, the mathematical and computational framework is developed for the application of the Spectral Stochastic Finite Element method in stochastic problems of framed structures that exhibit geometric nonlinearity. Lastly, a novel surrogate modeling methodology is proposed based on the Diffusion Maps manifold learning algorithm. With the proposed methodology, the detailed finite element model is substituted with simpler mathematical functions that are cheap to evaluate, thus, leading to a significant cost reduction for the Monte Carlo simulation without compromising the accuracy.