Monte Carlo Simulations With Novel Sampling Variance Reduction Schemes For Structural Reliability Estimation

Abstract

The work reported in this thesis is in the area of computational structural reliability modelling. The primary focus of the study is on the estimation of structural reliability using Monte Carlo simulations with sampling variance control. We consider both time-invariant and time-variant problems of reliability estimation. For dealing with time-invariant reliability estimation, we employ the frameworks of Markov chain particle splitting and importance sampling based strategies. The problems of time-variant reliability estimation are tackled using two alternative schemes based on Girsanov’s transformation approach: the first scheme employs an adaptive strategy to arrive at the suboptimal Girsanov’s controls, while in the second approach, we develop the controls based on the application of reinforcement learning tools. This thesis is organized into seven chapters. A background of the existing approaches for structural reliability modelling, including analytical and computational methods, is presented in Chapter 1. Chapter 2 contains a critical review of literature pertaining to Monte Carlo simulations based methods equipped with variance reduction schemes for time-invariant and time-variant reliability estimation. The review also covers the more recently developed machine learning based tools for reliability estimation. The Chapter concludes by identifying open questions addressing the formulation of Markov Chain Monte Carlo (MCMC) samplers in the context of particle splitting methods for time-invariant reliability estimation and in developing effective Girsanov’s controls for problems of time-variant reliability estimation. Chapters 3 and 4 address the challenges of devising variance reduction strategies for time-invariant reliability estimation in the presence of geometric complexities in the performance functions, such as (a) discontinuity/rapid changes in performance function values in the parameter space, (b) existence of multiple, possibly unimportant, regions of failure with rapid changes in performance function near one or more of the important regions, and (c) multiple regions of failure with substantial contributions to failure probability. In Chapter 3, we tackle this problem using the particle splitting framework and introduce an improved MCMC sampler based on the replica exchange strategy. This is shown to enhance the capacity of samples to detect and explore important regions of failure when the aforementioned difficulties are present. The application of the bootstrap technique to deduce the sampling variance of the estimator of the probability of failure is also developed. Chapter 4 approaches this problem within an alternative framework of variance reduction technique, namely, the importance sampling technique. In this approach, we utilize the replica exchange based MCMC sampler to generate failure samples, which more effectively account for significant failure regions in the presence of the above-mentioned geometric complexities. The importance sampling density function is then arrived at using the kernel density estimate obtained from the generated samples in the failure region. The illustrative examples presented in these two Chapters include dependent and non-Gaussian random variable models for the parameter uncertainties, nonlinear system behaviour, and nonlinear performance functions where the above-mentioned geometric difficulties arise. We also consider problems rooted in elastic buckling and structural vibration and show that these difficulties arise in such settings. Finally, we consider the reliability estimation problem of a circular arch susceptible to losing stability by snap-through buckling, wherein the performance function is implicitly defined through a finite element model residing in a professional finite element package (Abaqus). In Chapters 5 and 6, we focus on developing variance control strategies for time-variant reliability analysis of randomly excited dynamical systems based on importance sampling achieved via Girsanov’s transformation. In Chapter 5, we propose an adaptive method of formulating state-independent Girsanov’s control force where the iterates of the control forces are taken as the mean of a set of trajectories of random excitations that progressively lead the system towards failure. We subsequently extend this strategy to tackle the problem of reliability estimation of randomly excited dynamical systems with uncertain parameters. In Chapter 6, we express the problem of choosing state-dependent Girsanov’s control force as a constrained stochastic optimal control problem. We then recast this further as a problem in Markov decision process (MDP) which is then solved by employing a deep reinforcement learning technique called the deep deterministic policy gradient (DDPG) algorithm. The illustrative examples contained in these two Chapters include reliability estimation of linear/nonlinear dynamical systems with single/multiple degrees of freedom under non-white, non-stationary, and/or non-Gaussian excitations, including a previously studied benchmark problem. A summary of research contributions made in this study are provided in Chapter 7 along with a set of suggestions for future research.