MS5.3 - Computational methods for stochastic dynamics

Efficient reliability analysis of complex systems is essential in risk engineering and management. The goal is not only to determine the reliability of a system but also to understand system behavior that leads to critical failure and assess possible effects. A thorough reliability analysis additionally conducts a qualitative and quantitative description of the uncertainty accompanying possible damage and therefore leading to different levels of risk. Reliability is necessary for all engineering fields and specific measures are often described in design codes. However, in the context of civil engineering large, unusually designed, or important structures that are built in areas where influences of extreme external hazards can not be neglected are required to undergo an in-depth analysis for possible rare failures. Furthermore, modern structures are more and more embedded in a larger digitalized analysis framework that aims to predict critical behavior in real-time, putting possible effects of single structures in a larger context or controlling so-called morphable structures to withstand external influences. For these challenges, highly efficient and robust algorithms are necessary to perform complex simulations faster and more efficiently. In this work, a novel framework that allows an efficient time-dependent reliability analysis for stochastic dynamic systems with very small failure probabilities based on Markov chain Monte Carlo simulation techniques and the Probability Density Evolution Method is proposed. Mainly an exploration and exploitation of the multi-dimensional random space are conducted based on performance criteria utilized in e.g. Subset simulation, which is well established in the field of reliability analysis. However, the Subset simulation does only provide deterministic output values of the analyzed system and is specifically tailored to determine small probabilities of failure. The Probability Density Evolution method provides the complete time-dependent multivariate probability density function of desired output target quantities and can yield a robust probabilistic description of structural behavior. However, it has been shown that for high stochastic dimensions, strong non-linearities, and very small failure probabilities in the system, the classical PDEM approach does not always perform satisfactorily. The presented work introduces a simple framework to combine the strengths of both methods by identifying specific failure paths of the system contributing higher to the failure region, assessing their influence on the reliability by analyzing the stochastic input space, and providing a full probabilistic output of desired target quantities of complex systems. This enables the distinct description of possible system behaviors in a safe, intermediate, or critical failure domain and highly efficient estimation of failure probabilities. A critical discussion of the proposed method in light of other existing frameworks such as AK-PDEM and other stochastic simulation techniques is provided. The proposed framework is then tested on practical engineering examples common in the structural safety analysis. A finite element model of a structure under the influence of random seismic hazards is analyzed toward maximum displacement and stresses. Finally, an assessment of the results and a conclusion are given.

This paper is concerned with the prediction of the high frequency shock response of a complex system that has uncertain properties. A fundamental aspect of the method is the representation of the response in a combination of the time and frequency domains. This type of description is commonly used in predicting the response of a deterministic system to non-stationary random excitation (for example in earthquake dynamics), where a formalism known as the Priestley spectrum is used. However the present concern is with the deterministic transient loading of a random system and it is not entirely obvious that a description in the form of the Priestley spectrum can be employed – the random ensemble involved is the ensemble of random systems rather than the ensemble of random loading. It is shown here that an approach which is completely analogous to the Priestley spectrum can in fact be applied to the present problem, and this forms the basis for the derivation of a set of transient statistical energy analysis (TSEA) which govern the system response. The equations predict both the mean and variance of the system response, the system randomness being described by a model based on the Gaussian Orthogonal Ensemble (GOE) of random matrices. The governing equations are presented and then applied to a range of example systems.