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The reliability analysis of engineering structural systems with limit state functions defined implicitly by time-consuming numerical models (e.g. finite element analysis structural models) requires the use of efficient solution strategies in order to keep the required computational costs at acceptable levels. In this paper, an adaptive Kriging surrogate model with active refinement is proposed to solve component reliability assessment problems (i.e. involving one single design point) with nonlinear and time-consuming implicit limit state functions with a moderate number of input basic random variables. The proposed model, in the first stage, uses an adaptive Kriging-based trust region method to search for the design point in the standard Gaussian space and predict an initial failure probability based on the first-order reliability method as well as sensitivity factors for the input basic random variables. This initial prediction is then verified or improved efficiently in a second stage using Monte Carlo simulation with importance sampling based on a Kriging surrogate model defined iteratively around the design point using an active refinement algorithm. A convergence criterion that detects the stabilization of the failure probability prediction during the active refinement process is also proposed and implemented. The usefulness of the proposed adaptive Kriging surrogate model in terms of accuracy and efficiency for reliability assessment of engineering structural systems is shown in the paper with two relevant numerical examples, involving a highly nonlinear analytical limit state function in two-dimensions and an advanced nonlinear finite element analysis structural model in a larger dimensional space. [ABSTRACT FROM AUTHOR]

Estimating rare event probability with accuracy is of great interest for safety and reliability applications. In this paper, we focus on rare events which can be modeled by a threshold exceedance of a deterministic input–output function with random inputs. Some parameters of this function or density parameters of input random variables may be fixed by an experimenter for simplicity reasons. From a risk analysis point of view, it is not only interesting to evaluate the probability of a critical event but it is also important to determine the impact of such tuning of parameters on the realization of a critical event, because a bad estimation of these parameters can strongly modify rare event probability estimations. In the present paper, we present an example of island particle algorithm referred to as sequential Monte Carlo square (SMC 2 ). This algorithm gives an estimate of the law of random phenomena that leads to critical events. The principles of this statistical technique are described throughout this article and its results are analysed on different realistic aerospace test cases. [ABSTRACT FROM AUTHOR]

We consider a wide class of models that includes the highly reliable Markovian systems (HRMS) often used to represent the evolution of multi-component systems in reliability settings. Repair times and component lifetimes are random variables that follow a general distribution, and the repair service adopts a priority repair rule based on system failure risk. Since crude simulation has proved to be inefficient for highly-dependable systems, the RESTART method is used for the estimation of steady-state unavailability and other reliability measures. In this method, a number of simulation retrials are performed when the process enters regions of the state space where the chance of occurrence of a rare event (e.g., a system failure) is higher. The main difficulty involved in applying this method is finding a suitable function, called the importance function, to define the regions. In this paper we introduce an importance function which, for unbalanced systems, represents a great improvement over the importance function used in previous papers. We also demonstrate the asymptotic optimality of RESTART estimators in these models. Several examples are presented to show the effectiveness of the new approach, and probabilities up to the order of 10-42 are accurately estimated with little computational effort. [ABSTRACT FROM AUTHOR]

• A new hybrid condition-/age-based maintenance policy is proposed for deteriorating units whose observed degradation paths are simultaneously affected by temporal variability, unit to unit variability, and measurement error. • The proposed maintenance policy uses in a combined manner a perfect and an imperfect measurement procedure. • The actual (hidden) degradation phenomenon process is described by using a gamma process with random effect. • The measurement error is modeled as a non-Gaussian random variable that depends in stochastic sense on the measured degradation level. • A comparative study with other maintenance policies is developed to demonstrate the utility of the proposed approach in terms of long-run average maintenance cost rate. In this paper, a hybrid condition-/age-based maintenance policy is proposed for deteriorating units whose observed degradation paths are affected by temporal variability, unit to unit variability, and measurement error. The hidden degradation process is described by using a gamma model with random effect. The measurement error is modeled as a non-Gaussian random variable. Units are assumed to fail when their degradation level exceeds a given threshold. The suggested maintenance policy consists in using the information obtained at a predetermined inspection time to decide whether to immediately replace the unit or to postpone its replacement to a successive predetermined time. The inspection involves measuring the degradation level of the unit. It is supposed that an imperfect procedure is firstly adopted, which provides a perturbed measurement of the true degradation level. However, it is also supposed that a more expensive perfect measurement procedure can be optionally used, depending on the result provided by the imperfect one. The proposed maintenance policy is applied to a real-world inspired case of a pipeline. The performances of the policy are measured in terms of the long-run average maintenance cost rate. Obtained results demonstrate the affordability and utility of the proposed approach. [ABSTRACT FROM AUTHOR]

• k -out-of- n systems with discretely distributed component lifetimes are studied. • The number of failed components upon system failure is described. • The distribution of this number is found. • Its behavior over time is investigated. • Illustrative examples are presented. In this paper, we examine properties of k -out-of- n systems when the component lifetimes are discretely distributed. The primary focus is the random variable which represents the number of failed components upon system failure. For this variate we derive the probability mass function under the most general setting of possibly dependent and heterogeneous components. We also present conditional probabilities for this variable given some information about the time of system failure. In addition to these results, we show that, in the special case of series systems, this variable exhibits some aging properties, and we establish several characterizations of probability distributions based on such properties. For illustration, we provide three numerical examples. [ABSTRACT FROM AUTHOR]

• A Bayesian inference model is proposed to predict time-variant response. • An active learning is proposed to estimate failure probability of time-variant problem. • The proposed method is efficient and accurate for engineering applications. Time-variant reliability analysis aims at estimating the probability that an engineering system successfully performs intended missions over a certain period of time under various sources of uncertainty. In order to perform the time-variant reliability analysis efficiently, this paper presents a high dimensional model representation (HDMR) model combined with an active learning strategy to estimate the failure probability of dynamic problem. Firstly, the HDMR meta-model is established in the augmented input space (random variables and time) based on Gaussian process regression technique. Then, a reliability analysis approach incorporating epistemic uncertainty is proposed, and a learning function is introduced to update the experimental design sequentially. Finally, the Monte Carlo simulation method is applied for time-variant failure probability assessment based on the well-developed HDMR meta-model. Two engineering applications are used to demonstrate the effectiveness of the proposed method for time-variant reliability analysis. [ABSTRACT FROM AUTHOR]

• Global sensitivity analysis is extended to the context of imprecise probability theory (parametric p-boxes). • Imprecise Sobol' indices are defined and computed from a sparse polynomial chaos expansion (PCE) of the computational model. • The sparse PCE is built in an augmented space of input parameters and associated hyperparameters. • A smart experimental design using phantom points is devised to reduce the number of runs of the computational model. Global sensitivity analysis aims at determining which uncertain input parameters of a computational model primarily drives the output quantities of interest. Sobol' indices are now routinely applied in this context when the input parameters are modelled by classical probability theory using random variables. In many practical applications however, input parameters are affected by both aleatory and epistemic uncertainty. In this respect, imprecise probability theory have become popular in the last decade. In this paper, we consider that the uncertain input parameters are modelled by parametric probability boxes (p-boxes). We propose interval-valued (so-called imprecise) Sobol' indices as an extension of their classical definition. An original algorithm based on the concepts of augmented space, isoprobabilistic transforms and sparse polynomial chaos expansions is devised to allow for the computation of these imprecise Sobol' indices at extremely low cost. In particular, phantoms points are introduced to build an experimental design in the augmented space (necessary for the calibration of the sparse PCE) which leads to a smart reuse of runs of the original computational model. The approach is applied to three analytical and engineering examples which illustrate the efficiency of the algorithms against brute-force double-loop Monte Carlo simulation. Image, graphical abstract [ABSTRACT FROM AUTHOR]

Highlights • The concept of inertia in physics is introduced into MSSs reliability analysis. • A critical threshold is set for each deteriorating output performance. • Both constant and random thresholds are considered. • Analytical deductions are presented to compare the availability variations. Abstract For the repairable multi-state component (MSC), we introduce a continuous-time Markov process to analyze the availability of the element in this paper. When output performance of the MSC is lower than demand but higher than zero, the component will not fail to work immediately under some circumstances. Based on the concept of inertia in physics, the inertial effect of deteriorating output performances is considered. The influence of inertial effect makes the component operate longer than ever before, resulting in failure delay. To measure the probability influence of inertial effect on availability quantitatively, a critical threshold is set for each deteriorating output state. Both non-negative constants and random variables of the critical thresholds are modeled respectively. Comparing with the original system described with a Markov process, the new situation with inertial effect is recorded as the new system and described with a generic stochastic process. Availability variations of the original system and the new system show the influence of inertial effect on the reliability. A numerical example of a power supply system, which is a typical multi-state system (MSS), is presented to illustrate the results obtained. Relevant studies may be extended in many fields such as queuing theory and quality management. Graphical abstract Image, graphical abstract [ABSTRACT FROM AUTHOR]

Highlights • A moment-based reliability analysis method is proposed. • It provides a novel perspective for evaluating structural reliability. • The method is efficient in solving multi-dimensional reliability problems. Abstract Reliability analysis is a widely-used tool to measure a structure's ability of fulfilling safety and serviceability requirements. Existing methods for reliability assessment have, for the most part, been developed based on using the probability distribution functions of input random variables accounting for the uncertainties arising from both structural properties (e.g., material strength, geometry) and external loads. This paper develops a moment-based method for reliability assessment, which relies on the moment information of input variables rather than the probability distribution functions. It is shown that the proposed method is useful in solving multi-dimensional reliability assessment problems with improved efficiency compared with Monte Carlo simulation. The implementation, validity and efficiency of the proposed method are demonstrated through illustrative examples. [ABSTRACT FROM AUTHOR]

Highlights • The approach takes into account the complexity of failure mechanisms as a system and integrates both expert knowledge and diagnostic information. • The diagnostic algorithm enables to detect active failure mechanisms and track their progression based on diagnostic information from different sources. • The prognostic algorithm enables to predict the occurrence of failure modes dynamically as new information becomes available. • The simplicity of the algorithms and graphical representation of the results helps to build decision-makers' trust. • A case study on hydroelectric generator stator is proposed. Abstract The aim of this paper is to propose a holistic multi-failure mode prognosis approach that takes into account the complexity of failure mechanisms as a system. Model assumptions are first proposed by experts and then formalized using graph theory and stochastic models. The prognosis approach relies on a diagnostic algorithm that combines diagnostic information from different sources (e.g., measurements and inspections) to detect active failure mechanisms and track their progression, and a prognostic algorithm that predicts failure mode occurrences dynamically as new information becomes available. Furthermore, the approach identifies undetectable failure mechanisms where no symptoms have yet been measured. The relative simplicity of the algorithms and graphical representation of the results helps to build decision-makers' trust. In addition, the approach is a means of capturing acquired knowledge and available data. A case study of a hydroelectric generator stator is proposed. The resulting multi-state degradation model identified more than 150 failure mechanisms discretized in 70 physical states and leading to three failure modes. Three historical failure and one online case studies are presented, based on diagnostic data from Hydro-Québec's generating fleet. In two of the case studies, the failure mode occurrence could have been predicted more than eight years in advance. [ABSTRACT FROM AUTHOR]

Highlights • Quantify the effect of partial monitoring of multistate components in a system. • The quantification is done by calculating preposterior losses. • Efficient simulations by using discrete event simulations. • Framework for optimal monitoring planning balancing costs with information loss. Abstract For large multicomponent systems it is typically too costly to monitor the entire system constantly. In the present paper we consider a case where a component is unobserved in a time interval [0, T ]. The time T is a stochastic variable with a distribution which depends on the structure of the system and the lifetime distribution of the other components. Different systems will result in different distributions of T. The main focus is on how the unobserved period of time affects what we learn about the unobserved component during this period. We analyse this by considering one single component in three different cases. In the first case we consider both T as well as the state of the unobserved component at time T as given. In the second case we allow the state of the unobserved component at time T to be stochastic, while in the third case both T and the state are treated as stochastic variables. In all cases we study the problem using preposterior analysis. That is, we investigate how much information we can expect to get by the end of the time interval [0, T ]. The methodology is also illustrated on a more complex example. [ABSTRACT FROM AUTHOR]

• The error model of a multi-DOF redundant parallel mechanism with hybrid uncertainties is established and verified. • A hybrid motion reliability method is developed for the mechanism with hybrid uncertainties. • The different effects and adaptability of motion reliability method of mechanism is shown. Accuracy is one of the most important properties of mechanisms. However, the inherent uncertainties will cause an error between the actual motion and the desired motion, leading to a motion reliability problem. The error model and motion reliability of a type of multi-DOF (degrees of freedom) redundant parallel mechanism (MDRPM) with hybrid uncertainties are studied. Firstly, the error model of the mechanism is established and verified. Then, the motion inside the joints is analyzed, and the joint clearance is regarded as an interval variable, while the other variables are treated as random variables. On this basis, a hybrid motion reliability method based on the quasi-Monte Carlo simulation method (QMCSM) is developed for a mechanism with hybrid uncertainties. Compared with the traditional simulation method, the amount of computation required by the QMCSM is significantly reduced, and it is more suitable for solving the large-scale motion reliability problem of the MDRPM. A numerical simulation and experiments are performed. The simulation results show that the feasibility and adaptability of the motion reliability method in this paper are different. The experimental results demonstrate that the error model is effective, so the motion reliability analysis based on the error model is reliable. [ABSTRACT FROM AUTHOR]

• An efficient variable selection-based Kriging model method is proposed for the reliability analysis of slopes. • Two single-layered slopes and a real multiple-layered slope are explored to demonstrate the proposed approach. • The variable selection method can reduce the dimension of the input variables with little impact on the safety factor of slopes. • The proposed approach is suitable and efficient for reliability analysis of slopes with spatially variable soils. The random finite element method (RFEM) is an efficient tool to demonstrate the spatial variability of soil properties during reliability analysis of slopes, but it requires remarkable model evaluations and computational efforts. In this paper, an efficient variable selection-based Kriging model method is proposed to approximate the finite element analysis model in reliability analysis of slopes. The variable selection technique successfully remedies the "curse of dimensionality" within Kriging model induced by the numerous random variables in random field discretization. The implementation procedure of this method for the reliability analysis of slopes is introduced in detail. Two typical examples of soil slopes, as well as a real complex slope are subsequently analyzed to illustrate the validity of the proposed method. The results show that the local loss of variability of random field due to the variable selection method has little impact on the safety factor of slopes. The proposed method can significantly reduce the number of finite element analysis and obtain accurate results in reliability analysis of slopes considering the spatial variability of soil properties. [ABSTRACT FROM AUTHOR]

• An accurate and robust approach is proposed for reliability analysis. • The exact dimension-reduction model is obtained by contribution-degree analysis. • A sample points selection strategy is proposed for evaluating coefficients of PCE. • A convergence criterion of PCE approximation is proposed based on information entropy. • The proposed method is applied to reliability analysis of industrial robots. Polynomial chaos expansion (PCE) is considered an excellent method for accurately and efficiently reliability analysis in various engineering problems. However, it becomes practically infeasible in scenarios characterized by high-dimensional input random variables and thousands of training data. To solve this problem, this paper proposes a new PCE-based surrogate-assisted method. To start with, the interacting variables are screened out from original high-dimensional input random variables by contribution-degree analysis (CDA). The original high-dimensional performance function is decomposed into a lower-dimensional component function composed of the interacting variables and multiple one-dimensional component function containing only one non-interacting variable. Then, PCE combined with a sample points selection strategy is used to fit the lower-dimensional component function, and a full PCE is utilized for fitting each one-dimensional component function. Three methods, namely the Hadamard's inequality of the design matrix, the rank revealing QR factorization of the design matrix and the maximum entropy, are implemented to realize sample point selection. Four examples are investigated to demonstrated the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper presents an efficient approach to compute the bounds on the reliability of a structure subjected to uncertain parameters described by means of imprecise probabilities. These imprecise probabilities arise from epistemic uncertainty in the definition of the hyper-parameters of a set of random variables that describe aleatory uncertainty in some of the structure's properties. Typically, such calculation involves the solution of a so-called double-loop problem, where a crisp reliability problem is repeatedly solved to determine which realization of the epistemic uncertainties yields the worst or best case with respect to structural safety. The approach in this paper aims at decoupling this double loop by virtue of the Augmented Space Integral. The core idea of the method is to infer a functional relationship between the epistemically uncertain hyper-parameters and the probability of failure. Then, this functional relationship can be used to determine the best and worst case behavior with respect to the probability of failure. Three case studies are included to illustrate the effectiveness and efficiency of the developed methods. • Augmented reliability formulation for coping with aleatoric and epistemic uncertainty. • Bayes' theorem allows tracking dependency with respect to epistemic uncertainty. • Implementation considers different simulation methods. [ABSTRACT FROM AUTHOR]

In structural reliability analysis, it is often desirable to evaluate the relative contributions of random variables to the variability of the limit-state function in the failure domain. Based on the relative contributions, one can effectively reduce the dimension of the reliability problem or obtain useful insight and information. However, existing reliability importance measures, which are available as a by-product of reliability analysis by first-order reliability method (FORM), may not capture the contributions of random variables accurately when the limit-state surface shows a large curvature around the design point or multiple critical subdomains exist in the failure domain. To address the issue, this paper proposes a Generalized Reliability Importance Measure (GRIM) that can deal with multiple critical failure regions, large curvatures of limit-state surfaces and the correlation between the input random variables. By introducing Gaussian mixture and the regional participation factor, the failure domain is divided into several subdomains, and the relative contributions of random variables in each critical domain are evaluated. To facilitate the computations of GRIMs, the cross-entropy-based adaptive importance sampling technique (CE-AIS-GM) is employed to identify the locations of critical subdomains, their relative contributions and corresponding importance vectors. Eight numerical examples covering a variety of component and system reliability problems demonstrate the proposed method and its merits. The test results confirm robust performance against the number of important regions or the dimension. The proposed GRIMs and computational procedure are expected to provide more reliable measures for a wide range of component and system reliability problems. The supporting source code and data are available for download at https://github.com/TyongKim/GRIM . [ABSTRACT FROM AUTHOR]

In this paper, a new method for efficient reliability analysis is proposed. The proposed method utilizes the Voronoi cells to partition the random-variate space into several sub-spaces and the kernel density estimation to approximate the failure probability in each sub-space. The optimal bandwidth for the kernel is also suggested. Then, the failure probability can be conveniently evaluated by a weighted summation over each sub-space (sampling point). Since the weight for each sub-space (sampling point) is not identical, this method is referred to as the unequal-weighted sampling method for reliability analysis. Numerical implementation procedure of the proposed method is also outlined. Several numerical examples are investigated to verify the proposed method, where the results are compared with those of Monte Carlo simulation and subset simulation methods. It is demonstrated that the proposed method can achieve the tradeoff of accuracy and efficiency for reliability analysis. Problems to be further studied are also pointed out. [ABSTRACT FROM AUTHOR]

The presence of uncertainties is inevitable in engineering design and analysis, where failure in understanding their effects might lead to the structural or functional failure of the systems. The role of global sensitivity analysis in this aspect is to quantify and rank the effects of input random variables and their combinations to the variance of the random output. In problems where the use of expensive computer simulations is required, metamodels are widely used to speed up the process of global sensitivity analysis. In this paper, a multi-fidelity framework for global sensitivity analysis using polynomial chaos expansion (PCE) is presented. The goal is to accelerate the computation of Sobol sensitivity indices when the deterministic simulation is expensive and simulations with multiple levels of fidelity are available. This is especially useful in cases where a partial differential equation solver computer code is utilized to solve engineering problems. The multi-fidelity PCE is constructed by combining the low-fidelity and correction PCE. Following this step, the Sobol indices are computed using this combined PCE. The PCE coefficients for both low-fidelity and correction PCE are computed with spectral projection technique and sparse grid integration. In order to demonstrate the capability of the proposed method for sensitivity analysis, several simulations are conducted. On the aerodynamic example, the multi-fidelity approach is able to obtain an accurate value of Sobol indices with 36.66% computational cost compared to the standard single-fidelity PCE for a nearly similar accuracy. [ABSTRACT FROM AUTHOR]

Dynamic event tree (DET) analysis, one of the main dynamic Probabilistic Safety Assessment methods, provides a framework to capture the effect of dynamics on the risk estimate. Depending on how continuous stochastic variables (CSVs) are treated, DETs can be classified into discretization- or sampling-based methods. The accuracy of the estimate and required computational resources depend on the method chosen as well as the nature of the problem. CSVs also include variable initial conditions, some of which significantly impact accident evolution. This work compares alternative DET methods in terms of numerical accuracy and computational resources for a case study of a chemical batch reactor problem, a system sensitive to both accident dynamics as well as variable initial conditions. The reference solution is a computationally intensive analog Monte Carlo simulation. The results show that the DET methods fairly match reference results with significantly less computation required. Further, in light of epistemic uncertainties of model parameters, this paper presents a comparison of DETs that includes detailed analyses of contributors of risk and its uncertainty, which unfolds the strengths and weaknesses of discretization and sampling based DETs. [ABSTRACT FROM AUTHOR]

Markov models are widely used for describing the behavior of multi-state systems, whose transition rates can be approximated through Bayesian inference based on data from field collections and experiments. The estimates of transition rates departing from the same state can be dependent. In this paper, we investigate the influence of uncertainties associated with the transition rates due to the finiteness of the available data, upon the transient and steady state probabilities. Variance-based methods are employed to understand how the uncertainty in the model output can be apportioned to the model inputs. To deal with dependencies among the transition rates, the dependent transition rates are represented by a group of independent random variables. Uncertainty importance measures (UIMs) are derived based on the total sensitivity indices to rank the transition rates based on their contributions to the output variance. Therefore, actions to improve the accuracy of their estimation can be appropriately guided to reduce the output variance. The extended Fourier amplitude sensitivity test is used for the quantification of the UIMs. A numerical example is provided to illustrate the approaches. [ABSTRACT FROM AUTHOR]

• A new method for representing uncertainties in a non-simply connected spatial domain. • An effective method for parameterize high dimensional uncertainties. • Propagate input aleatoric uncertainties to the responses of a topological structure. Quantification and propagation of aleatoric uncertainties distributed in complex topological structures remain a challenge. Existing uncertainty quantification and propagation approaches can only handle parametric uncertainties or high dimensional random quantities distributed in a simply connected spatial domain. There lacks a systematic method that captures the topological characteristics of the structural domain in uncertainty analysis. Therefore, this paper presents a new methodology that quantifies and propagates aleatoric uncertainties, such as the spatially varying local material properties and defects, distributed in a topological spatial domain. We propose a new random field-based uncertainty representation approach that captures the topological characteristics using the shortest interior path distance. Parameterization methods like PPCA and β -Variational Autoencoder (β VAE) are employed to convert the random field representation of uncertainty to a small set of independent random variables. Then non-intrusive uncertainties propagation methods such as polynomial chaos expansion and univariate dimension reduction are employed to propagate the parametric uncertainties to the output of the problem. The effectiveness of the proposed methodology is demonstrated by engineering case studies. The accuracy and computational efficiency of the proposed method is confirmed by comparing with the reference values of Monte Carlo simulations with a sufficiently large number of samples. [ABSTRACT FROM AUTHOR]

In order to improve the performance and prolong the life of the equipment, it is usually necessary to maintain the equipment in practice. Remaining useful life prediction with imperfect maintenance activities is of great importance to prognostics and health management. In this paper, a multi-phase Wiener process-based degradation model is constructed to characterize the degradation process subjected to imperfect maintenance activities. The beta distribution is introduced to describe the residual degradation coefficient caused by the imperfect maintenance activity. The hyper-parameters of residual degradation coefficient are estimated through the maximum likelihood estimation and Newton iteration method. To reflect the unit heterogeneity, the drift coefficient and diffusion coefficient of each phase are synchronously defined as random variables. Furthermore, the analytical forms of remaining useful life are obtained by the convolution operator. The approximate expression of probability density function is derived by the Monte Carlo simulation approach. In the end, a numerical study and a practical study of gyroscopes are provided to demonstrate the practicality and effectiveness of the proposed method. • A multi-phase Wiener propose-based degradation model is improved. • Drift and diffusion coefficients are synchronously reviewed as random variables. • The approximate expressions of PDF for RUL are derived. • The method is accurate and superior based on a numerical study and gyro datasets. • The model realizes accurate RUL prediction under imperfect maintenance activities. [ABSTRACT FROM AUTHOR]

In this paper, a single-loop approach for time-variant reliability evaluation is proposed based on a decoupling strategy and probability distribution reconstruction. The most attractive feature of the proposed method is that the reliability at a specified time instant can be captured by performing time-invariant reliability analysis only once. In this method, the expansion optimal linear estimation is first employed to discretize the loading stochastic process. Then, a decoupling strategy that decouples the loading stochastic process and degradation processes is developed to formulate a single-loop method for time-variant reliability analysis, where an equivalent extreme value limit state function (EEV-LSF) is obtained. To improve the accuracy and robustness, the Box–Cox transformation is applied to get a transformed EEV-LSF. The maximum entropy method with fractional exponential moments is employed to robustly derive the probability distribution of transformed EEV-LSF. Once the probability distribution is captured, the time-variant failure probability can be readily computed. To handle a large number of random variables, a weighted sampling method is applied for moment assessment to ensure an efficient solution. Numerical examples including a complex real-world case are studied to validate the proposed method, where pertinent Monte Carlo simulations and PHI2 method are conducted for comparisons. • A decoupling strategy is proposed to formulate a single-loop method. • The PDF of transformed extreme value is reconstructed by FEM-MEM. • Time-invariant reliability analysis needs to be performed only once. • The proposed method shows satisfactory accuracy and efficiency. [ABSTRACT FROM AUTHOR]

This paper proposes an extended sample-based approach for efficient sensitivity analysis of the group of random variables by extending the original sample-based approach proposed by Jia and Taflanidis. The original sample-based approach can efficiently estimate the sensitivity indices (i.e., Sobol' indices) for individual and low-dimensional inputs by using kernel density estimation (KDE). However, due to the curse of dimensionality of KDE, it is challenging to directly apply the original sample-based approach to estimate sensitivity indices for high-dimensional groups of inputs. To address the challenges, the proposed approach converts estimating the sensitivity index for groups of inputs with potentially high dimensionality to estimating the sensitivity indices for individual inputs in the group and their lower-order (e.g., second or third) interactions using the idea of variance decomposition, thus limiting the dimension of the required KDE to be low (e.g., only two or three) if higher-order interactions are negligible. Different estimators for the main and total sensitivity indices are proposed to reduce the error accumulation and improve estimation accuracy. Overall, the proposed approach can be used to efficiently estimate the sensitivity indices (including main and total sensitivity indices) for both individual and groups of inputs when higher-order interactions have negligible contributions. The good efficiency and accuracy of the proposed approach are demonstrated through three benchmark examples. • Proposes extended sample-based approach for efficient sensitivity analysis. • Efficiently calculates sensitivity indices to groups of variables. • Addresses high-dimensional group of variables by variance decomposition. • Proposes accurate estimators for both main and total sensitivity indices. [ABSTRACT FROM AUTHOR]

The understanding of many physical and engineering problems involves running complex computational models. Such models take as input a high number of numerical and physical explanatory variables. The information on these underlying input parameters is often limited or uncertain. It is therefore important, based on the relationships between the input variables and the output, to identify and prioritize the most influential inputs. One may use global sensitivity analysis (GSA) methods which aim at ranking input random variables according to their importance in the output uncertainty, or even quantify the global influence of a particular input on the output. Using sensitivity metrics to ignore less important parameters is a form of dimension reduction in the model's input parameter space. This suggests the use of meta-modeling as a quantitative approach for nonparametric GSA, where the original input/output relation is first approximated using various statistical regression techniques. Subsequently, the main goal of our work is to provide a comprehensive review paper in the domain of sensitivity analysis focusing on some interesting connections between random forests and GSA. The idea is to use a random forests methodology as an efficient non-parametric approach for building meta-models that allow an efficient sensitivity analysis. Apart its easy applicability to regression problems, the random forests approach presents further strong advantages by its ability to implicitly deal with correlation and high dimensional data, to handle interactions between variables and to identify informative inputs using a permutation based RF variable importance index which is easy and fast to compute. We further review an adequate set of tools for quantifying variable importance which are then exploited to reduce the model's dimension enabling otherwise infeasible sensibility analysis studies. Numerical results from several simulations and a data exploration on a real dataset are presented to illustrate the effectiveness of such an approach. • Global Sensitivity Analysis ranks inputs according to their importance on output. • Random Forests is an efficient non-parametric approach for building meta-models. • Random forests variable importance measure is used to define sensitivity measures. • Provide a comprehensive review paper in sensitivity analysis using random forests. • Focus on connections between random forests and Global Sensitivity Analysis. [ABSTRACT FROM AUTHOR]

• A new single-loop strategy for time-variant system reliability analysis is proposed. • The multiple response Gaussian process is adopted to depict the correlation between time-variant limit state functions. • The subset simulation is introduced into the proposed single-loop strategy to estimate the small failure probability. • The updated strategy and stopping criterion for the proposed single-loop time-variant reliability analysis are provided. This paper proposes a time-variant system reliability analysis method by combining multiple response Gaussian process (MRGP) and subset simulation (SS) to solve the small failure probability problem. One common method for time-variant reliability analysis is based on the double-loop procedure where the inner loop is the optimization for extreme values and the outer loop is extreme-value-based reliability analysis. In this paper, a new single-loop strategy is firstly proposed to decouple the double-loop procedure by using the best value in current initial samples to approximate the extreme value, thus the extremal optimization in inner loop can be avoided. Then the MRGP model is used to construct the surrogate model of extreme value response surface for time-variant system reliability analysis based on the approximated extremums. Meanwhile, the Kriging model is also constructed based on the initial samples to assist in searching the new sample point. Furthermore, for selecting the new point that resides as close to the extreme value response surface as possible from the Monte Carlo simulation (MCS) sample pool, three learning functions (U-function, EFF-function and H-function) are respectively used to find the new random variable sample point based on the MRGP model and the expected improvement (EI) function is used to find the new time sample point based on the Kriging model. Finally, for reducing the size of candidate sample pool and the computing burden, the SS method is combined with the MRGP model to deal with the small failure probability problem. The effectiveness of the proposed method is also demonstrated by several examples. [ABSTRACT FROM AUTHOR]

Reliability analysis of balanced system has become a key factor for the safe operation of some important equipment and the perfect completion of some important missions. As some real systems consisting of several subsystems with different levels of performance and under degradation processes when in operations are relatively common, they have received widely concerned in the recent years, but are rarely in the new reliability research field of balanced systems, the researches on the balanced systems by considering multi-state are rare, and even few researches are in combination with degradation processes. In this paper, a new balanced system for the multi-state system under degradation processes by considering new failure criteria is put forth. The system consists of two subsystems and fails when either subsystem enters into absorbing states, a new model is proposed and applied to two special cases by assuming that the thresholds of the sojourn time in upcoming unbalanced subsets are constant and random variables respectively, and this paper is the first to extend balanced systems into competing risks. The related expressions of the reliability indexes are derived by employing the theory of aggregated stochastic processes. Finally, numerical examples are presented and some future research topics are also discussed. [ABSTRACT FROM AUTHOR]

• A new method is proposed for system reliability analysis with hybrid uncertainties. • The significance of composite projection outlines is determined. • Three system learning functions are built to refine composite projection outlines. • A new composite-projection-outline-based active learning method is proposed. • The accuracy and efficiency of the proposed method are validated by five examples. This paper investigates the system reliability analysis under random and interval variables (SRA-RI). When performance functions in SRA-RI are replaced by surrogate models, it is found that the composite projection outlines on the composite limit-state surface should be well approximated so as to accurately assess system failure probability bounds. Then a composite-projection-outline-based active learning Kriging (CPOK) method is proposed in this paper. To refine the approximated composite projection outlines, three system learning functions are defined in CPOK for parallel, series and mixed systems, respectively. Based on these three functions, new points around the composite projection outlines are sequentially selected and used for the update of Kriging models. Meanwhile, prediction uncertainties of Kriging models are quantified and used for terminating the update of Kriging models. Finally, system failure probability bounds are evaluated by Monte Carlo simulation. The advantages of CPOK are validated by four numerical examples and a piezoelectric energy harvester example. [ABSTRACT FROM AUTHOR]

• Hybrid uncertainty is represented by probability-box and dempster-shafer Structure. • A dependency bounds analysis method is proposed for non-deterministic dependency problem. • Hybrid uncertainty and non-deterministic dependency are integrated and propagated in bayesian network. • A comparison of the proposed method with the frechet inequalities method and 2-stage monte carlo method is performed. • High narrowing rates during execution time are obtained by the proposed method. In reliability assessment, a difficulty is to handle a complex system with hybrid uncertainty (aleatory and epistemic uncertainty) and dependency problem. Probability-box is a general model to represent hybrid uncertainty. Arithmetic rules on the structure are mostly used between independent random variables. However, in practice, dependency problems are also common in reliability assessment. In addition, in most real applications, there is some prior information on the dependency of components, but the available information may be not enough to determine dependent parameters. The issue is named non-deterministic dependency problem in the paper. Affine arithmetic is hence used to produce dependent interval estimates. The arithmetic sometimes has a better effect than probability-box arithmetic (interval arithmetic) in dealing with dependency problem. Bayesian network is a commonly used model in reliability assessment. Under Bayesian network framework, this paper proposes a dependency bounds analysis method that combines affine arithmetic and probability-box method to handle hybrid uncertainty and non-deterministic dependency. For the sake of illustration, this method is applied to two real systems. To show the advantages of the proposed method, the proposed method is compared with the Frechet inequalities and 2-stage Monte Carlo method in the second case study. [ABSTRACT FROM AUTHOR]

In this paper we investigate a cybersecurity model: An attacker can launch multiple attacks against a target with a termination strategy that says that the attacker will stop after observing a number of successful attacks or when the attacker is out of attack resources. However, the attacker's observation of the attack outcomes (i.e., random variables indicating whether the target is compromised or not) has an observation error that is specified by both a false-negative and a false-positive probability. The novelty of the model we study is the accommodation of the dependence between the attack outcomes, because the dependence was assumed away in the literature. In this model, we characterize the monotonicity and bounds of the compromise probability (i.e., the probability that the target is compromised). In addition to extensively showing the impact of dependence on quantities such as compromise probability and attack cost, we give methods for finding the optimal strategy that leads to maximum compromise probability or minimum attack cost. This study highlights that the dependence between random variables cannot be assumed away, because the results will be misleading. [ABSTRACT FROM AUTHOR]

This paper investigates the optimal age replacement policy for used systems, such as second-hand products, which start their second life-cycle in a more severe environment with an initial age that is uncertain. This uncertain age is modelled as a random variable following continuous probability distributions. A mathematical model is developed to minimize the total expected cost per unit of time for these systems on an infinite time horizon. Optimality and existence conditions for a unique optimal solution are derived and used in a numerical procedure to solve the problem. Numerical experiments are provided to demonstrate the added value and the impacts of the random initial age on the optimal replacement policy. [ABSTRACT FROM AUTHOR]

Fault tree analysis remains a significant alternative for modeling and analyzing reliability and failure modes. Traditionally, the probabilities of the basic events are assumed as point estimates from which the fault tree is used to estimate the probability of the top failure event. This paper instead proposes using this framework for a different purpose: framing the model parameters describing the basic events as random variables, which collectively contribute to the statistics of the estimate of the top failure event. An approach for uncertainty propagation via statistical moments is developed for both static and dynamic gates with specific consideration of the effects of the dependency conditions for different distribution types. Assuming there is a cost to acquire model parameters with reduced uncertainty, this framing can thus be used to identify the best set of model parameters possible within a constrained budget to minimize uncertainty in the resulting estimate of the top failure event. With these models in place, the statistical moments of the top event can be connected to all lower levels, and the statistics in the estimate of the top event can be optimized with a given cost constraint (e.g. allocating resources to reducing uncertainty in the knowledge of basic events to minimize variance in the estimate of the top failure event). This optimization of the uncertainty in the top event is demonstrated with a case study. • The prior information at the basic event levels can vary notably in fault trees. • An approach for uncertainty propagation via statistical moments is developed. • The approach depends on the distribution types when dependencies exist. • Sensitivity analysis is leveraged for optimizing resource allocations. • Constrained optimization problems are formulated to improve system safety. [ABSTRACT FROM AUTHOR]

In this paper, two new sampling strategies are proposed to estimate the Morris’ screening sensitivity measure and its improved version. The two new sampling strategies, which employ random sampling and quasi-random sampling respectively, compute the elementary effects of each factor at the same initial point and with a same step size in the input space. The new quasi-random sampling strategy performs better than the radial based sampling strategy and the new random sampling strategy performs almost the same with the radial based sampling strategy. Then, the improved version of the Morris’ screening sensitivity measure is applied to estimate the effects of the epistemic uncertainty of random variables’ distribution parameters on the failure probability using the new quasi-random sampling strategy. During this process, the principle of maximum entropy, fractional moments and dimension reduction method are used to estimate the failure probability with a good accuracy and a low computational demand. Several examples are employed to demonstrate the reasonability and the efficiency of the proposed strategy. [ABSTRACT FROM AUTHOR]

• A hybrid reliability analysis model involving interval processes is considered. • An equivalent uncertainty transformation is developed for time-dependent analysis. • An active learning method based on deep neural network is proposed. • Three cases are used to demonstrate the efficacy of the proposed method. Performing time-dependent reliability analysis is an effective way to estimate the failure probability of structural system throughout its lifetime. In the engineering practices, uncertain parameters with sufficient sample and limited sample may exist simultaneously. The uncertain parameters with limited sample data are difficult to construct its precise probabilistic characteristics during estimating the accurate time-dependent reliability. To address this issue, this paper first develops a new hybrid time-dependent reliability model involving interval processes. Then, to reduce the high dimensionality, an extension method based on equivalent stochastic process transformation approach is proposed to transform the stochastic processes and the interval processes into corresponding equivalent random variables respectively. In particular, an instantaneous reliability model is constructed to envelope all potential system failures that may occur during the time interval. In order to identify the instantaneous failure surface accurately, an active learning method is proposed based on the deep neural network model and the weighted sampling method. With the constructed deep neural network model, the new hybrid time-dependent reliability can be evaluated by performing the Monte Carlo Sampling. Three numerical examples are used to verify the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This paper presents a hierarchical partitioning strategy (HP) for probability space to improve the accuracy of metamodel and avoid the memory problems of computer in dealing with small failure probability events. By the continuous partition of probability space, the construction of adaptive Kriging metamodel is intelligently divided into two steps. Thereafter, the well-trained Kriging via two steps is used to approximate the relationship between the extreme value of response and basic random variables of the system. By combining the probability density evolution method with well-trained Kriging metamodel, the reliability of the investigated stochastic system can be readily obtained with a one-dimensional integration operation. Subsequently, a new reliability analysis method is proposed, called HP-AK-PDEM (H ierarchical P artitioning strategy-based A daptive K riging combining P robability D ensity E volution M ethod). To demonstrate the accuracy and efficiency of the proposed method, three analytical performance functions with nonlinear features and a ten-story shear-frame structure are addressed. The comparative study of different reliability methods is carried out as well. Numerical results demonstrate that the presented method has better accuracy and higher efficiency in dealing with the problems of small and rare failure probability issues encountered in engineering structures. • A hierarchical partition strategy is proposed to train the more accuracy metamodel. • The two-stage method for training metamodels is presented. • A high accuracy reliability (namely HP-AK-PDEM) analysis frame is presented. [ABSTRACT FROM AUTHOR]

• Moving improved importance-sampling least-squares (MIIL) is put forward. • Probabilistic substructure (PS)-based MIIL comes up by incorporating PS with MIIL. • DC-PS-MIIL is presented by adding distributed collaborative strategy into PS-MIIL. • Confidence level-based strain-life model is proposed and combined with DC-PS-MIIL. • This paper provides theoretical guidance for uncertainty quantification problems. The objective of this paper is to develop an effective reliability approach to improve the numerical accuracy and computational efficiency of the low-cycle fatigue (LCF) damage prediction for turbine disks. The approach is called as distributed collaborative (DC) probabilistic substructure (PS)-based moving improved importance-sampling least-squares (MIIL), DC-PS-MIIL. In this approach, the MIIL model is firstly developed by combining an improved importance sampling (IIS) with moving least squares (MLS), to reduce the computation burden and improve the precision of surrogate model; then, the PS-MIIL comes up by integrating probabilistic substructure (PS) method into MIIL, to address the interface forces of turbine disks; finally, DC-PS-MIIL is proposed by incorporating distributed collaborative strategy with PS-MIIL. The comprehensive analysis procedure with DC-PS-MIIL is shown to simplify the probabilistic assessment of complex structures for improving numerical accuracy and simulation efficiency. Taking the LCF damage prediction of turbine disks as an example, DC-PS-MIIL is demonstrated to be an effective reliability approach. Also, numerical results show that the confidence levels, applied cycles and coefficients of variation (CV s) of random input variables have important impact on the LCF damage reliability for turbine disks. [ABSTRACT FROM AUTHOR]

• Probability-space surrogate models with analytical solutions. • Surrogate training using single iterations of multidisciplinary analysis. • Simultaneous uncertainty and multidisciplinary analyses using Bayesian strategy. • Computationally efficient reliability-based and robustness-based optimization. • Illustration using a three-discipline aircraft design example. This paper proposes a probability-space surrogate modeling approach for computationally efficient multidisciplinary design optimization under uncertainty. This paper uses a probability-space surrogate as opposed to an algebraic surrogate so that the probability distributions of the required outputs at a given design input can naturally be obtained without repeated Monte Carlo runs of an algebraic surrogate at different realizations of the uncertain variables. We consider three probability-space surrogates with analytical solutions for prediction and inference - Multivariate Gaussian, Gaussian Copula, and Gaussian Mixture Model, and investigate their applicability to perform multidisciplinary design optimization under uncertainty. All the input design and random variables, coupling variables, objective and constraint functions are incorporated within the probability-space surrogate, which helps analytically obtain the distributions of coupling variables, objective and constraint functions at desired design inputs while enforcing multidisciplinary compatibility. The training points for the probability-space surrogates are obtained by performing one-pass analysis through the disciplinary models at different realizations of the input variables. The proposed methodology is demonstrated for reliability-based design optimization (RBDO) and reliability-based robust design optimization (RBRDO) in an aircraft design example. The performance of the probability-space surrogates is compared against a Kriging algebraic surrogate and fully coupled Monte Carlo analysis. [ABSTRACT FROM AUTHOR]

• A general network model is proposed to model a power grid system with the performance sharing mechanism. • Both the performance of each generator and the demand of each node in the network are assumed to be random variables. • The optimal performance sharing policy is formulated for any given system structure. • The optimal allocation of generators, which minimizes the expected system unsupplied demand is studied. • Case studies are presented to illustrate the applications. Performance sharing is an effective policy for a power grid system to satisfy the power demand of different districts to greatest extent. Through transmission lines, the districts with sufficient power can share the redundant power with the districts with power deficit. The existing research has incorporated the performance sharing mechanism into systems with simple structures such as parallel systems and series-parallel systems. However, little concentration has been spent on more complex structures. This necessitates the need of this paper that models a power distribution with a more complex reliability structure. We assume that the system is composed of generators and nodes. Both the performance of each generator and the demand of each node in the network are assumed to be random variables. This paper first proposes a dynamic performance sharing policy to minimize the unsupplied demand for a given system with fixed capacity and demand. The optimal allocation of generators, which minimizes the expected system unsupplied demand, is then studied. Numerical examples are proposed to illustrate the applications of the proposed procedures. [ABSTRACT FROM AUTHOR]

The reliability of high-reliable and long-lifespan products in the presence of competing risks are commonly derived based on a known acceleration model and a latent independent competing risks model. For the cases with unknown acceleration models and dependent competing risks, this paper proposes a copula based partially accelerated competing risks model using a tampered random variable transformation. The reliability function and dependence structure of the partially accelerated competing risks data are derived between the real lifetime and the accelerated lifetime of each failure cause. In consideration of the parametric constraints and complicated joint posterior distribution, Hamiltonian Monte Carlo method within MCMC procedures is utilized to obtain Bayesian estimation of model parameters and reliability characteristics by defining unconstrained parameters using a specific transformation function. A simulation study is conducted to investigate the estimation performance, showing that the transformation forms have little influence on the point estimation of model parameters, and the constrained parametric Bayesian estimation method is efficient to evaluate the reliability for the proposed model. Finally, a real data set from a light-emitting diode partially accelerated life test is presented for further illustration of Bayesian reliability analysis. • Copula based step-stress partially accelerated competing risks model is developed. • Properties of reliability function using real and accelerated copulas are studied. • Constrained Bayesian method is proposed to estimate parameters and reliability. • Posterior density function and its partial derivatives in HMC method are derived. [ABSTRACT FROM AUTHOR]

Many system reliability problems involve performance functions with a discontinuous distribution. Such situations occur in both connectivity- and flow-based network reliability problems, due to binary or multi-state random variables entering the definition of the system performance or due to the discontinuous nature of the system model. When solving this kind of problems, the standard subset simulation algorithm with fixed intermediate conditional probability and fixed number of samples per level can lead to substantial errors, since the discontinuity of the output can result in an ambiguous definition of the sought percentile of the samples and, hence, of the intermediate domains. In this paper, we propose an adaptive subset simulation algorithm to determine the reliability of systems whose performance function is a discontinuous random variable. The proposed algorithm chooses the number of samples and the intermediate conditional probabilities adaptively. We discuss two MCMC algorithms for generation of the samples in the intermediate domains, the adaptive conditional sampling method and a novel independent Metropolis–Hastings algorithm that efficiently samples in discrete input spaces. The accuracy and efficiency of the proposed algorithm are demonstrated by a set of numerical examples. • We introduce adaptive effort subset simulation for network reliability assessment. • The proposed algorithm modifies the adaptive selection of the intermediate domains. • We propose a novel independent M–H sampler to sample in discrete input spaces. • Numerical results demonstrate the efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

Reliability sensitivity analysis aims at identifying the source of structure/mechanism failure, and quantifying the effects of each random source or their distribution parameters on failure probability or reliability. In this paper, the time-dependent parametric reliability sensitivity (PRS) analysis as well as the global reliability sensitivity (GRS) analysis is introduced for the motion mechanisms. The PRS indices are defined as the partial derivatives of the time-dependent reliability w.r.t. the distribution parameters of each random input variable, and they quantify the effect of the small change of each distribution parameter on the time-dependent reliability. The GRS indices are defined for quantifying the individual, interaction and total contributions of the uncertainty in each random input variable to the time-dependent reliability. The envelope function method combined with the first order approximation of the motion error function is introduced for efficiently estimating the time-dependent PRS and GRS indices. Both the time-dependent PRS and GRS analysis techniques can be especially useful for reliability-based design. This significance of the proposed methods as well as the effectiveness of the envelope function method for estimating the time-dependent PRS and GRS indices are demonstrated with a four-bar mechanism and a car rack-and-pinion steering linkage. [ABSTRACT FROM AUTHOR]

The failure processes of heterogeneous repairable systems are often modeled by non-homogeneous Poisson processes. The common way to describe an unobserved heterogeneity between systems is to multiply the basic rate of occurrence of failures by a random variable (a so-called frailty) having a specified parametric distribution. Since the frailty is unobservable, the choice of its distribution is a problematic part of using these models, as are often the numerical computations needed in the estimation of these models. The main purpose of this paper is to develop a method for estimation of the parameters of a nonhomogeneous Poisson process with unobserved heterogeneity which does not require parametric assumptions about the heterogeneity and which avoids the frequently encountered numerical problems associated with the standard models for unobserved heterogeneity. The introduced method is illustrated on an example involving the power law process, and is compared to the standard gamma frailty model and to the classical model without unobserved heterogeneity. The derived results are confirmed in a simulation study which also reveals several not commonly known properties of the gamma frailty model and the classical model, and on a real life example. [ABSTRACT FROM AUTHOR]

In this paper, a discrete complex reliability system subject to internal failures and external shocks, is modelled algorithmically. Two types of internal failure are considered: repairable and non-repairable. When a repairable failure occurs, the unit goes to corrective repair. In addition, the unit is subject to external shocks that may produce an aggravation of the internal degradation level, cumulative damage or extreme failure. When a damage threshold is reached, the unit must be removed. When a non-repairable failure occurs, the device is replaced by a new, identical one. The internal performance and the external damage are partitioned in performance levels. Random inspections are carried out. When an inspection takes place, the internal performance of the system and the damage caused by external shocks are observed and if necessary the unit is sent to preventive maintenance. If the inspection observes minor state for the internal performance and/or external damage, then these states remain in memory when the unit goes to corrective or preventive maintenance. Transient and stationary analyses are performed. Markov counting and reward processes are developed in computational form to analyse the performance and profitability of the system with and without preventive maintenance. These aspects are implemented computationally with Matlab . [ABSTRACT FROM AUTHOR]

In this paper, we address the issue of conducting a sensitivity analysis of complex models with both static and dynamic uncertain inputs. While several approaches have been proposed to compute the sensitivity indices of the static inputs (i.e. parameters), the one of the dynamic inputs (i.e. stochastic fields) have been rarely addressed. For this purpose, we first treat each dynamic as a Gaussian process. Then, the truncated Karhunen–Loève expansion of each dynamic input is performed. Such an expansion allows to generate independent Gaussian processes from a finite number of independent random variables. Given that a dynamic input is represented by a finite number of random variables, its variance-based sensitivity index is defined by the sensitivity index of this group of variables. Besides, an efficient sampling-based strategy is described to estimate the first-order indices of all the input factors by only using two input samples. The approach is applied to a building energy model, in order to assess the impact of the uncertainties of the material properties (static inputs) and the weather data (dynamic inputs) on the energy performance of a real low energy consumption house. [ABSTRACT FROM AUTHOR]

This paper studies reliability for multi-component systems subject to dependent competing risks of degradation wear and random shocks, with distinct shock sets. In practice, many systems are exposed to distinct and different types of shocks that can be categorized according to their sizes, function, affected components, etc. Previous research primarily focuses on simple systems with independent failure processes, systems with independent component time-to-failure, or components that share the same shock set or type of shocks. In our new model, we classify random shocks into different sets based on their sizes or function. Shocks with specific sizes or function can selectively affect one or more components in the system but not necessarily all components. Additionally the shocks from the different shock sets can arrive at different rates and have different relative magnitudes. Preventive maintenance (PM) optimization is conducted for the system with different component shock sets. Decision variables for two different maintenance scheduling problems, the PM replacement time interval, and the PM inspection time interval, are determined by minimizing a defined system cost rate. Sensitivity analysis is performed to provide insight into the behavior of the proposed maintenance policies. These models can be applied directly or customized for many complex systems that experience dependent competing failure processes with different component shock sets. A MEMS (Micro-electro mechanical systems) oscillator is a typical system subject to dependent and competing failure processes, and it is used as a numerical example to illustrate our new reliability and maintenance models. [ABSTRACT FROM AUTHOR]

• A unified non-Gaussian stochastic model from limited observations is developed. • The developed model is based on polynomial chaos (PC) and fractional moments. • The developed model can be embedded into PC framework for stochastic propagation. • Two numerical examples demonstrate the developed model. The reasonable representation of input random fields is the key element in the reliability analysis of practical engineering systems. In most engineering applications, the characterization of a random field often relies on limited measurements. Although the simulation of random fields with complete probabilistic information has been quite well-established, reconstructing a random field from limited observations is still a challenging task. In this paper, we develop a methodology for constructing non-Gaussian random model from limited observations based on polynomial chaos (PC) and fractional moments for real-life problems. Our method begins with the reduce-order representation of measurements by Karhunen-Loève (KL) expansion, followed by the PC representation of KL coefficients. The PC coefficients are further modeled as random variables, whose distributions are determined by a modified maximum entropy principle with fractional moments (ME-FM) procedure and a ME-FM-based bootstrapping. In this way, the developed non-Gaussian model enables to quantify the inherent randomness and the statistical uncertainty of the observed non-Gaussian field simultaneously. Since the developed non-Gaussian model is embedded into the well-established PC framework, our method facilitates the implementation of PC-based stochastic analysis in practical engineering applications, in which only limited probabilistic measures are available. Two numerical examples demonstrate the application of the developed method. [ABSTRACT FROM AUTHOR]

• K-L expansion method was adopted to generate random field with different spatial correlation lengths • A critical spatial correlation length existed for predicting time-dependent settlement of soft soils • A single random variable analysis is not always the most conservative analysis in predicting settlement of soft soils • It is essential to perform random field analysis to reduce the risk of the design to be applied in construction. Low embankment strategy is one of the effective methods to control time-dependent settlement of soft soils in infrastructure construction projects. Spatial variability of soil characteristics is a crucial factor, affecting the reliability of predictions of the long-term settlement in soft soils. In this paper, the time-dependent behaviour of soft soils is analysed incorporating spatial variability of elastic visco-plastic model parameters. Standard Gaussian random fields for correlated elastic-plastic model parameter (λ / V) and the initial creep coefficient (ψ 0 / V) are generated adopting Karhunen-Loeve expansion method based on the spectral decomposition of correlation function into eigenvalues and eigenfunctions. Then the generated random fields are incorporated in the proposed non-linear elastic visco-plastic (EVP) creep model. The impacts of spatially variable elastic visco-plastic model parameters (i.e. ψ 0 / V and λ / V) on long-term settlement predictions are evaluated through random field analysis (RF) with different spatial correlation lengths, and results are then compared to a single random variable (SRV) analysis. The probability of failure (PF) is calculated adopting RF and SRV analysis to determine the critical spatial correlation length, resulted in a maximum probability of failure. This study can be employed by design engineers to determine the critical spatial correlation length for safe design in the absence of adequate data to determine the exact spatial correlation length. The results also confirm that SRV analysis is not always the most conservative analysis in predicting time-dependent settlement of soft soils; and it is essential to perform RF analysis considering the spatial correlation length to reduce the risk and increase the reliability of the design to be applied in construction. [ABSTRACT FROM AUTHOR]

• Direct probability integral method (DPIM) for multimodal distributions is proposed. • DPIM presents high efficiency and accuracy for static/dynamic reliability analysis. • GF-discrepancy method is superior in solving probability density integral equation. • Uncertainty propagation in structures with multimodal distributions is revealed. For practical structures, some input random variables follow multimodal distributions, and conventional reliability analysis methods may result in large computational errors. In this paper, a novel direct probability integral method (DPIM), which decouples governing equation of structure and the probability density integral equation (PDIE), is proposed to address the static and dynamic reliability assessment of structures involving random variables with multimodal distributions. Firstly, the multimodal probability density functions (PDFs) of input random variables are established by the Gaussian mixture model. Then, using numerical integration and smoothing technique of Dirac delta function, the PDFs of structural responses with multimodal random variables are achieved by solving the PDIE, in which three numerical integration algorithms, namely, the Quasi-Monte Carlo approach, the sparse grid approach, and Generalized F-discrepancy-based point selection approach are employed. Further, the reliability of static structure can be readily obtained by integrating the PDF of response function, while the dynamic reliability can be evaluated by DPIM combining with extreme value distribution of stochastic process. Finally, several examples demonstrate the superiority of DPIM, and the Generalized F-discrepancy-based point selection approach has the highest accuracy and efficiency for solving PDIE. The characteristics of uncertainty propagation in structures involving multimodal distributions of input random variables are revealed. [ABSTRACT FROM AUTHOR]

• A comparison between the S-N curve and FM, fatigue assessment methods, was performed. • This comparison was achieved by developing a non-intrusive stochastic expansion. • The developed framework calculates reliability index values in manageable computation time. • The initial crack size is the most critical random variable affecting fatigue reliability. This paper investigates from a structural reliability assessment (SRA) perspective the fatigue reliability using the S-N curve approach compared with the fracture mechanics (FM) approach for a typical welded offshore wind turbine (OWT) jacket support structure. A non-intrusive formulation was developed for an OWT jacket support structure in 50 m deep water, consisting of a sequence of steps. First, stochastic parametric 3D (three-dimensional) Finite Element Analysis (FEA) simulations are performed, taking into account stochastic variables such as wind loads, wave loads and soil properties using facilities within the software package ANSYS. Secondly, the FEA results are post-processed using an Artificial Neural Network (ANN) response surface modelling technique deriving the performance functions expressed in terms of stochastic variables. Finally, the First Order Reliability Method (FORM) is applied in calculating the reliability index values of components. The developed framework was applied to elucidate the fatigue damage process, including the small to long crack transition amongst other stages, for structural steels used for OWT jacket applications. The FM formulation investigated includes a crack growth formulation based on the bilinear crack growth law, considering both segments of the crack growth law as non-correlated and correlated in calculating the reliability index (RI). Sensitivity analysis results showed a strong dependence of the structure's reliability levels on the uncertainties of the crack growth law constants measured in terms of coefficient of variation (COV). Also investigated, was the reliability of the structure reassessed and updated in the presence of assumed structural health monitoring/ condition monitoring (SHM/CM) data. The results from the case study revealed that fracture reliability is highly sensitive to the initial crack size. It is recommended to apply the S-N curve method at the design stage while the FM approach applied towards the end of the design life as the structure approaches failure. [ABSTRACT FROM AUTHOR]

• Quantifies the uncertainty of GNATS simulation in en-route trajectory as randomness in waypoints coordinates. • Information fusion of observation data and physical constraints to predict aircraft trajectory • Encode physical constraints in trajectory simulation as a Bayesian-Entropy model selection framework • Predicts runway overrun based on historical data from Sherlock Data Warehouse Eliminating accidents while maintaining the integrity of the National Airspace System is one of the central objectives of the Next Generation Air Transportation System. This paper presents a Bayesian framework for accurate trajectory and accident prediction in National Airspace System using a high-fidelity trajectory simulation platform. Various uncertainties in aircraft trajectory prediction due to pilot behavior and weather effects are included as random variables in the simulations. Bayesian-Entropy method fuses available observation data (e.g., positioning system) with existing physical constraints (e.g. runway location) to update these simulation parameters. The posterior distributions of parameters are used to predict the probability of an adverse incident and time-remaining to incident. The proposed Bayesian updating scheme offers a flexible and rigorous way for adverse incident diagnostics and prognostics in current and future Air Traffic Management. Two realistic examples are given to show that it is possible to derive advance warning using the proposed methodology. The approach integrates data from a simulation model, with real-time traffic data streams and available physical constraints, using the Bayesian-Entropy information fusion methodology. This advance warning will allow the pilots/controllers to take actions to mitigate adverse incident in the National Airspace System. [ABSTRACT FROM AUTHOR]