Junho Song

Multi-scale reliability analysis and updating of complex systems by use of linear programming

Complex systems are characterized by large numbers of components, cut sets or link sets, or by statistical dependence between the component states. These measures of complexity render the computation of system reliability a challenging task. In this paper, a decomposition approach is described, which, together with a linear programming formulation, allows determination of bounds on the reliability of complex systems with manageable computational effort. The approach also facilitates multi-scale modeling and analysis of a system, whereby varying degrees of detail can be considered in the decomposed system. The paper also describes a method for computing bounds on conditional probabilities by use of linear programming, which can be used to update the system reliability for any given event. Applications to a power network demonstrate the methodology.

Post-hazard flow capacity of bridge transportation network considering structural deterioration of bridges

The flow capacity of a transportation network can be reduced significantly if its constituent bridges are damaged by natural or man-made hazards. For rapid risk-informed decision making on hazard mitigation and response, it is therefore essential to have a capability to predict the post-hazard flow capacity of the network efficiently and accurately. However, this is a challenging task due to the uncertainty in hazards and structural damage, and the complex nature of the network flow analysis. Moreover, the bridge structures may experience significant deterioration over their life cycle, which requires time-varying network reliability analysis. This paper proposes a new non-sampling-based approach to estimate the time-varying post-hazard flow capacity of a bridge transportation network considering structural deterioration of bridges. The proposed approach evaluates the probabilities of structural damage scenarios efficiently using the matrix-based system reliability method and rapidly computes the corresponding flow capacities using a maximum flow capacity analysis algorithm. The matrix-based framework facilitates the integration of these results to obtain the probabilistic distributions and statistical moments of the network flow capacity. It also enables computing various measures useful for risk-informed decision making, such as the conditional mean and standard deviation of flow capacity given observed structural damage, and component importance measures. In the proposed approach, probability calculation and network flow analysis are performed separately, which renders time-varying post-hazard flow capacity analysis efficient. The proposed approach is demonstrated by a numerical example based on the Sioux Falls network under multiple bridge-deterioration scenarios simulating the progress of deterioration.

Single-Loop System Reliability-Based Design Optimization Using Matrix-Based System Reliability Method: Theory and Applications

This paper proposes a single-loop system reliability-based design optimization (SRBDO) approach using the recently developed matrix-based system reliability (MSR) method. A single-loop method was employed to eliminate the inner-loop of SRBDO that evaluates probabilistic constraints. The MSR method enables us to compute the system failure probability and its parameter sensitivities efficiently and accurately through convenient matrix calculations. The SRBDO/MSR approach proposed in this paper is applicable to general systems including series, parallel, cut-set, and link-set system events. After a brief overview on SRBDO algorithms and the MSR method, the SRBDO/MSR approach is introduced and demonstrated by three numerical examples. The first example deals with the optimal design of a combustion engine, in which the failure is described as a series system event. In the second example, the cross-sectional areas of the members of a statically indeterminate truss structure are determined for minimum total weight with a constraint on the probability of collapse. In the third example, the redistribution of the loads caused by member failures is considered for the truss system in the second example. The results based on different optimization approaches are compared for further investigation. Monte Carlo simulation is performed in each example to confirm the accuracy of the system failure probability computed by the MSR method.

Availability, reliability and downtime of systems with repairable components

Closed-form expressions are derived for the steady-state availability, mean rate of failure, mean duration of downtime and lower bound reliability of a general system with randomly and independently failing repairable components. Component failures are assumed to be homogeneous Poisson events in time and repair durations are assumed to be exponentially distributed. The results are expressed in terms of the mean rates of failure and mean durations of repair of the individual components. Closed-form expressions are also derived for the rates of change of the various probabilistic system performance measures with respect to the mean rate of failure and the mean duration of repair of each component. These expressions provide a convenient framework for identifying important components within the system and for decision-making aimed at upgrading the system availability or reliability, or reducing the mean duration of system downtime. Example applications to an electrical substation system demonstrate the use of the formulas developed in the paper.

Further development of matrix-based system reliability method and applications to structural systems

In efforts to estimate the risk and reliability of a complex structure or infrastructure network, it is often required to evaluate the probability of a ‘system’ event, i.e. a logical function of multiple component events. Its sensitivities with respect to design parameters are also useful in decision-making processes for more reliable systems and in reliability-based design optimisation. The recently developed, matrix-based system reliability (MSR) method can compute the probabilities of general system events including series, parallel, cut-set and link-set systems, and their parameter sensitivities, by use of efficient matrix-based procedures. When the component events are statistically dependent, the method transforms the problem into an integral in the space of random variables which cause the statistical dependence, termed as the common source random variables (CSRVs). One can identify CSRVs by fitting a generalised Dunnett-Sobel (DS) model to a given correlation coefficient matrix. This article introduces two further developments of the MSR method: First, for efficient evaluation, it is proposed that the integral in the CSRV space can be performed using the first- or second-order reliability methods. Second, a new matrix-based procedure is developed to compute the sensitivity of the system failure probability with respect to the parameters that affect the correlation coefficients between the components. In addition, an extensive parametric study is performed to investigate the effect of the error in fitted generalised DS model on the accuracy of the estimates by the MSR method. The further developed MSR method is demonstrated by two examples: system reliability analysis of a three-storey Daniels system structure, and finite element reliability analysis of a bridge pylon system.

System reliability analysis using dominant failure modes identified by selective searching technique

The failure of a redundant structural system is often described by innumerable system failure modes such as combinations or sequences of local failures. An efficient approach is proposed to identify dominant failure modes in the space of random variables, and then perform system reliability analysis to compute the system failure probability. To identify dominant failure modes in the decreasing order of their contributions to the system failure probability, a new simulation-based selective searching technique is developed using a genetic algorithm. The system failure probability is computed by a multi-scale matrix-based system reliability (MSR) method. Lower-scale MSR analyses evaluate the probabilities of the identified failure modes and their statistical dependence. A higher-scale MSR analysis evaluates the system failure probability based on the results of the lower-scale analyses. Three illustrative examples demonstrate the efficiency and accuracy of the approach through comparison with existing methods and Monte Carlo simulations. The results show that the proposed method skillfully identifies the dominant failure modes, including those neglected by existing approaches. The multi-scale MSR method accurately evaluates the system failure probability with statistical dependence fully considered. The decoupling between the failure mode identification and the system reliability evaluation allows for effective applications to larger structural systems.

Finite-element-based system reliability analysis of fatigue-induced sequential failures

When a structural system is subjected to repeated loadings, local fatigue-induced failures may initiate sequential failures and disproportionally large damage in the system. In order to quantify the likelihood of fatigue-induced sequential failures and identify critical failure sequences, a branch-and-bound method employing system reliability bounds (termed the B3 method) was recently developed and successfully demonstrated by a three-dimensional truss example. The B3 method identifies critical sequences of fatigue-induced failures in the decreasing order of their likelihood. Since the identified sequences are disjoint to each other, both lower and upper bounds on system failure probability are easily updated without performing additional system reliability analysis. The updated bounds provide reasonable criteria for terminating the branch-and-bound search without missing critical sequences or estimating the system-level risk inaccurately. Since the B3 method was originally developed for reliability analysis of discrete structures such as truss, however, the method is not applicable to continuum structures, which are often represented by finite element (FE) models. In particular, the method has limitations in describing general stress distributions in limit-state formulations, evaluating stress intensity range based on crack length, and in dealing with slow convergence of the upper and lower bounds for structures with high redundancy. In this paper, the B3 method is further developed for FE-based system reliability analysis of continuum structures by modifying the limit-state formulations, incorporating crack-growth analysis by external software, and introducing an additional search termination criterion. The proposed method is demonstrated by numerical examples including a continuum multi-layer Daniels system and an aircraft longeron structure.

Risk Analysis of Fatigue-Induced Sequential Failures by Branch-and-Bound Method Employing System Reliability Bounds

Various types of structural systems are often subjected to the risk of fatigue-induced failures. If a structure does not have an adequate level of structural redundancy, local failures may initiate sequential failures and cause exceedingly large damage. For the risk-informed design and maintenance of such structural systems, it is thus essential to quantify the risk of fatigue-induced sequential failure. However, such risk analysis is often computationally intractable because one needs to explore innumerable failure sequences, each of which demands component and system reliability analyses in conjunction with structural analyses to account for various uncertainties and the effect of load redistributions. To overcome this computational challenge, many research efforts have been made to identify critical failure sequences with the highest likelihood and to quantify the overall risk by system reliability analysis based on the identified sequences. One of the most widely used approaches is the so-called “bran...

Accelerated Monte Carlo system reliability analysis through machine-learning-based surrogate models of network connectivity

Abstract The two-terminal reliability problem in system reliability analysis is known to be computationally intractable for large infrastructure graphs. Monte Carlo techniques can estimate the probability of a disconnection between two points in a network by selecting a representative sample of network component failure realizations and determining the source-terminal connectivity of each realization. To reduce the runtime required for the Monte Carlo approximation, this article proposes an approximate framework in which the connectivity check of each sample is estimated using a machine-learning-based classifier. The framework is implemented using both a support vector machine (SVM) and a logistic regression based surrogate model. Numerical experiments are performed on the California gas distribution network using the epicenter and magnitude of the 1989 Loma Prieta earthquake as well as randomly-generated earthquakes. It is shown that the SVM and logistic regression surrogate models are able to predict network connectivity with accuracies of 99% for both methods, and are 1–2 orders of magnitude faster than using a Monte Carlo method with an exact connectivity check.

Evaluation of Multinormal Integral and Sensitivity by Matrix-based System Reliability Method

In efforts to estimate the system-level risk and reliability of an engineering system, it is often required to evaluate multinormal integrals efficiently and accurately. Their sensitivities with respect to design parameters are also important in decision-making processes for more reliable systems. This paper proposes to use the recently developed matrix-based system reliability (MSR) method for evaluating multinormal integrals and their sensitivities with respect to design parameters. While most of the existing multinormal calculation methods are applicable to parallel or series system events only, the proposed approach can compute the probabilities of any general system events. The correlation coefficient matrix of the normal random variables is fitted by a generalized Dunnett-Sobel correlation model. This transforms the multinormal problem into an integral in the space of statistically independent standard normal random variables, termed as common source random variables (CSRVs). If many CSRVs are needed for accurate representation of the given correlation coefficient matrix, first- and second-order reliability methods (FORM/SORM) can be used for efficient numerical integration. The paper demonstrates the proposed approach through numerical examples.