Bayesian inference with Subset Simulation: Strategies and improvements

A new methodology for incorporating stochastic models in integrity maintenance management for high-pH stress corrosion cracking (SCC) in gas pipelines has been presented. The generation of new crack features along with the growth of existing ones are included in the stochastic modeling, by means of nonhomogeneous Poisson process (NHPP) and nonhomogeneous gamma process (NHGP), respectively. The dependence (correlation) among the growths of individual cracks is considered by employing the Gaussian copula method. Data from multiple in-line inspections (ILI) are used to evaluate the parameters of the stochastic models by means of Bayesian updating. Hence, a hierarchical Bayesian framework is developed that can efficiently account for the ILI associated measurement errors and the probability of detection (PoD) of the crack features. The Bayesian updating is performed through subset simulation, a structural reliability method (SRM) conjointly with the data augmentation (DA) technique. Multiple simulated crack features from different ILI inspections are employed for the implementation and validation of the methodology. At the end, the time-dependent system reliability is evaluated along with a parametric study that examines the impact of correlations among stochastic growths of crack features on the posterior growth model and system reliability.

Highly efficient Bayesian updating using metamodels: An adaptive Kriging-based approach

Structural reliability is an engineering method that focuses on the calculation and prediction of the probability of failure for a structural system. Structural reliability is used by many pipeline operators to assess the probability of failure, and these probability of failure estimates are then used to manage safety, environmental, and operational risks for threats such as corrosion, cracking, and third-party damage. In the structural reliability approach, the probability of failure is obtained from a multi-dimensional integral, for which the solution is commonly estimated numerically using Direct Monte Carlo (DMC) simulation. DMC is straight-forward and robust but requires a significant amount of computational effort to estimate small probabilities, which are typical of threats to pipeline integrity. Subset Simulation is an approach that improves on the efficiency issues of DMC by representing the rare event probability as the product of a number of more frequent events, which are each estimated separately. Previous work published at IPC in 2020 by Bandstra and Fraser [1] showed very close agreement between DMC and Subset Simulation for a single defect reference case with a single limit state. This study extends that initial work by applying Subset Simulation to various failure models for a variety of pipeline defects. Subset Simulation is applied to the CSA corrosion model, PRCI MAT-8 crack model, and the EPRG dent failure model, and the performance is evaluated by comparing the results to DMC. For each of these comparisons, simulations are performed across a large-scale grid of validation cases that consider a range of pipeline and defect sizes. An approach that utilizes Subset Simulation to handle multiple limit states is also presented and applied for relevant failure models and the results are evaluated against DMC.

Heliostats represent the most important maintenance cost in the solar power tower plant. The aim of this work is to provide a failure probability analysis for heliostat design in order to minimize this maintenance cost. Based on mechanics of material study and wind aerodynamic analysis, a performance function, with five random variables, has been developed wherein the random variables are: wind speed, inside and outside pedestal diameters, pedestal yield stress and mirror mass. Four main methods have been proposed: first order reliability method, second order reliability method, Monte Carlo (MC) method and subset simulation (SS) method. The variation of failure probability with the variation of pedestal wall thickness and wind speed, for different outside diameters and heliostat azimuth and elevation angles, has been simulated. The results show that SS is more efficient and accurate for small failure probabilities; however, MC is more accurate for high failure probabilities.

• An active learning Kriging methodology for Bayesian updating is presented. • The BUS method recasts the updating problem into an equivalent reliability problem, solved with a Kriging metamodel. • A multi-fidelity approach is employed in the active learning Kriging model, where models of different accuracy are adopted. • Multi-fidelity Bayesian optimization is used for calculating the constant c of the BUS method. • An improved version of the Quantified Active learning Subset Simulation method is proposed to define the posterior samples. The paper presents a multi-fidelity framework for efficient Bayesian updating using active learning Kriging models. The updating problem is solved as an equivalent probabilistic problem using the Bayesian updating with Structural reliability (BUS) method. The proposed approach leverages an ensemble surrogate modeling strategy that combines two Kriging models of multiple fidelities, thereby enhancing predictive capacity while reducing computational cost by exploiting information from models of different fidelity levels. A key contribution is the integration of multi-fidelity Bayesian optimization in order to determine the optimal BUS constant, ensuring accurate transformation of the posterior into a reliability problem. Furthermore, an improved version of the Quantified Active Learning Subset Simulation (qAK-SuS) method, previously proposed by the authors, is proposed in order to efficiently estimate small failure probabilities and obtain the posterior distribution. The proposed framework, called qAK-BUS, offers a robust solution for Bayesian inference in engineering systems by smartly balancing model fidelity, prediction variance, and learning efficiency. Three numerical examples and an engineering application that involves the updating of a reinforced concrete bridge are examined for demonstrating and validating the proposed framework.

A stress-strength interference model in its interval form, with uncertain quantities of structures described by interval numbers, is used for reliability analysis. Such structural reliability analysis problem is called structural interval reliability analysis. A discrete possibility degree method for structural interval reliability is proposed. By comparing equally-spaced-points of structural interval stress value and structural interval strength value, a partition for comparison relations (big, equal or small) between interval stress and interval strength is built. Based on this partition, new concepts for interval reliability analysis are defined, such as structural safety possibility degree and structural failure possibility degree, etc. By one simple example, the feasibility and effectiveness of discrete possibility degree method for structural interval reliability are demonstrated.

Maintenance, rehabilitation and replacement of existing bridges by using cost effective management are increasingly important topics in efforts to deal with aging and deteriorating infrastructure. Recently, engineers have shifted the emphasis from design and construction of new bridges to maintenance and rehabilitation of existing bridges. This paper uses the structural reliability method with a predefined performance function to assess the component performance of the main load carrying members of an inner span in a multi-span bridge. The performance of the components is a function of various random variables with certain probabilistic distributions. The analysis is extended using the same performance function to evaluate the reliability of the bay as subsystem and bridge as a system. System reliability in this case is defined as the probability that a system (bridge) performs its intended function under operating conditions for a specified period of time using second order reliability method (SORM) and Monte Carlo simulations. In addition, deterioration in the main carrying elements (components) due to environmental impacts is introduced and the sensitivity parameters for certain random variables are re-evaluated to determine the inspection and maintenance period of the bridge. The deterioration is imposed as a percentage reduction in certain random variables. The paper involves use of multi reliability analysis at component, sub-system and system levels. Results show that probability of failure (Pf) increases with decrease in distance of the bridge from source of deterioration. Reliability index (β) decreased with increase in the age of the bridge because of corrosion of reinforcing steel. Results also show that the rate of probability of failure for bridges exposed to deteriorating effects increased dramatically after 45-50 years.

Recent brittle two-way shear failures have made assessing strength models a topic of interest. In addition, the economic alternatives have made lightweight reinforced concrete slabs (LWRCS) a popular choice of construction in many areas around the world. However, researchers have rarely tackled the two-way shear strength models of LWRCS using accurate structural reliability analysis methods. Most of the research investigations evaluated the performance of the two-way shear strength models for the LWRCS with respect to an experimental database using statistical measures. While it did not implement structural reliability methods, which could include the variability due to the following: (1) the governing variable’s random distribution; (2) the size of the experimental database; and (3) errors in measurements, construction, and design. Thus, this current study aims to apply structural reliability methods for the two-way shear of LWRCS. Two internationally renowned design codes were selected using an extensive experimental database. A brief outline of existing structural reliability methods is presented and compared. A selected reliability method was used to assess the reliability of the design-chosen codes. The reliability and sensitivity indices were calculated for the American and European design code provisions of two-way shear of lightweight concrete and compared with each other as well as international standard reliability. The American design code were found to be less reliable compared with the European, which might be due to not considering the flexure reinforcement ratio in the model. In addition, model accuracy is the main factor affecting reliability, however, the variability of basic parameters including concrete density, concrete strength and cross section depth have larger effect compared to flexure reinforcement ratio and loading area dimensions. Moreover, in order to achieve target reliability indices value of 3.8, 3.7, or 3.6 a reduction factor value of 0.8, 0.75, or 0.70 could be implemented. These findings could be helpful in design code developments. However, the results are limited to the selected variability of the random variables.

In the practical engineering, to solve the small failure probabilities with correlated high-dimensional variables, the subset simulation (SS) is combined together with the Monte Carlo and importance sampling (IS) method. The samples from the probability density functions (PDF) of the importance sampling are used to construct the intermediate failure events, by which the small failure probabilities are turned into a Markov chain (MC), which is a continuous product made of a series large failure probability or conditional failure probability (CFP) which is easily answered, on which the structural reliability can be efficiently simulated by directly obtaining the samples with correlated ones. Finally, the 3 planet carriers of 3 grade planetary reducers in shield tunneling machine(STM) are as examples to check the algorithm above, the results show that the SS of the IS with correlated variables can highly simulate failure probability.

These are the proceedings of the twenty-first working conference of the International Federation of Information Processing (IFIP) Working Group 7.5 on Reliability and Optimization of Structural Systems, which took place at the University of California, Berkeley, USA, on August 19–21, 2024. This volume contains 15 selected papers from the 20 presentations delivered at the conference. The conference was supported by Pacific Earthquake Engineering Research (PEER) Center, and by the University of California, Berkeley, which provided outstanding facilities in the conference venue and remarkable logistic support throughout the event. The purpose of the WG7.5 Working Group is to promote modern theories and methods of structural and system reliability and optimization, to stimulate research, development, and applications in these areas, to foster the dissemination and exchange of information, and to encourage education on those subjects. The main themes of the conference included structural reliability methods, engineering risk and resilience analysis, Bayesian methods, reliability-based design optimization, and sensitivity analysis, along with their applications in civil engineering, dynamics, and natural hazards. The conference facilitated engaging discussions among participants, both during and after the technical sessions. Notably, a significant number of presentations were delivered by promising young researchers, who contributed fresh perspectives and innovative ideas to the field.

The reliability of building structures subjected to ground motion is a time-consuming, complex, and iterative computation problem involving multidiscipline theories such as probability theory, reliability theory, and dynamic structural analysis. In order to address this challenge, this study proposes a highly efficient approach based on the subset simulation and a deep learning-based surrogate model. The subset simulation is an efficient sampling strategy that significantly diminishes the number of samples to compute when determining the failure probability, especially for very small ones. On the other hand, the surrogate model based on a deep learning algorithm can deliver equivalently accurate structures’ responses compared to the well-known finite element method with markedly smaller time complexity, given appropriate available training data. The efficiency and effectiveness of the proposed approach are demonstrated in detail through three case studies with increasing complexity: a 1-DoF problem, a 2D frame, and a 3D structure based on experimental data showing a reduction up to two orders of magnitude in time complexity compared to the Monte Carlo simulation using finite element method.

Time-dependent reliability is the probability that a system will perform its intended function successfully for a specified time. Unless many and often unrealistic assumptions are made, the accuracy and efficiency of time-dependent reliability estimation are major issues which may limit its practicality. Monte Carlo simulation (MCS) is accurate and easy to use but it is computationally prohibitive for high dimensional, long duration, time-dependent (dynamic) systems with a low failure probability. This work addresses systems with random parameters excited by stochastic processes. Their response is calculated by time integrating a set of differential equations at discrete times. The limit state functions are therefore, explicit in time and depend on time-invariant random variables and time-dependent stochastic processes. We present an improved subset simulation with splitting approach by partitioning the original high dimensional random process into a series of correlated, short duration, low dimensional random processes. Subset simulation reduces the computational cost by introducing appropriate intermediate failure sub-domains to express the low failure probability as a product of larger conditional failure probabilities. Splitting is an efficient sampling method to estimate the conditional probabilities. The proposed subset simulation with splitting not only estimates the time-dependent probability of failure at a given time but also estimates the cumulative distribution function up to that time with approximately the same cost. A vibration example involving a vehicle on a stochastic road demonstrates the advantages of the proposed approach.

Structural reliability analysis based on improved wolf pack algorithm AK-SS

Assessing small failure probabilities by AK–SS: An active learning method combining Kriging and Subset Simulation

Corrosion failure probability analysis of buried gas pipelines based on subset simulation