Reliability Sensitivity Indices Research Articles

An innovative adaptive Kriging-based parallel system reliability method under error stopping criterion for efficiently analyzing the global reliability sensitivity index

An innovative adaptive Kriging-based parallel system reliability method under error stopping criterion for efficiently analyzing the global reliability sensitivity index

Efficient Methods for Reliability Sensitivity Analysis of Distribution Parameters and Their Application

To efficiently evaluate the influence of the distribution parameters of the input variables on the failure probability of engineering structures and improve the reliability and safety of engineering structures in a targeted manner, new methods for the global reliability sensitivity analysis (RSA) of distribution parameters are proposed in this study based on the cubature formula (CF), surrogate sampling probability density function (SSPDF), and quasi-Monte Carlo (QMC) method. By introducing CF, the proposed methods can effectively improve the computational efficiency of the nested expectation and variance operators in the reliability sensitivity indices of the distribution parameters. Based on the concept of SSPDF, a surrogate importance sampling probability density function was developed. This not only overcomes the problem of the computational effort of propagating parameter uncertainty to the failure probability function (FPF), which depends on the dimensionality of the parameters; it also further improves the efficiency of the RSA of the parameters in the case of a small failure probability. Finally, by incorporating the idea of the QMC method, the process of calculating the reliability sensitivity indices of the parameters is reduced from a double-loop to a single-loop one. Three engineering examples are used in this study to demonstrate the efficiency and accuracy of the new algorithms.

Reliability sensitivity analysis for water hammer-induced stress failure of fluid-conveying pipe

The water hammer, resulting from sudden valve closures or similar operations, is a primary cause of leakage and damage in fluid-conveying pipe systems. This work aims to investigate the reliability sensitivity of fluid-conveying pipes under the water hammer. A reliability model for the fluid-conveying straight pipe, involving uncertainty and fluid-structure interaction, is developed based on the stress-strength interference theory. Then the method of characteristics is employed to analyze the nonlinear behaviors of the pipe. To avoid excessive model evaluations, the Kriging surrogate strategy is used to evaluate rapidly the statistics of nonlinear responses. Furthermore, the Kriging model coupled with the single-loop Monte Carlo simulation is presented for estimating reliability sensitivity indices. The accuracy of results is verified by the Monte Carlo simulation. The results reveal that the proposed method is feasible for the reliability sensitivity analysis of fluid-conveying pipe under the water hammer. Moreover, the wall thickness has the most obvious influence on stress failure, followed by the inner radius, while the pipe length, Young's modulus, density of the pipe material, and Poisson ratio show a low effect. This work can not only guide the safety design of pipe but also enrich the theory and application of statistical analysis and reliability sensitivity evaluation for liquid-conveying pipe systems.

Variance-based reliability sensitivity with dependent inputs using failure samples

Reliability sensitivity analysis is concerned with measuring the influence of a system’s uncertain input parameters on its probability of failure. Statistically dependent inputs present a challenge in both computing and interpreting these sensitivity indices; such dependencies require discerning between variable interactions produced by the probabilistic model describing the system inputs and the computational model describing the system itself. To accomplish such a separation of effects in the context of reliability sensitivity analysis we extend on an idea originally proposed by Mara and Tarantola (2012) for model outputs unrelated to rare events. We compute the independent (influence via computational model) and full (influence via both computational and probabilistic model) contributions of all inputs to the variance of the indicator function of the rare event. We compute this full set of variance-based sensitivity indices of the rare event indicator using a single set of failure samples. This is possible by considering d different hierarchically structured isoprobabilistic transformations of this set of failure samples from the original d-dimensional space of dependent inputs to standard-normal space. The approach facilitates computing the full set of variance-based reliability sensitivity indices with a single set of failure samples obtained as the byproduct of a single run of a sample-based rare event estimation method. That is, no additional evaluations of the computational model are required. We demonstrate the approach on a test function and two engineering problems.

Reliability sensitivity analysis based on a two-stage Markov chain Monte Carlo simulation

The reliability sensitivity index based on the safety/failure classification of model predictions is a valuable tool to measure how uncertain parameters affect the failure of engineering systems. The key to estimate this index is to estimate the failure-conditional probability density function (PDF) of parameters, which can be regarded as a posterior PDF from the perspective of Bayesian inference. Due to the binary property of the failure indicator function, it is difficult to directly sample the failure-conditional PDF. In this work, a new efficient sampling method is proposed to estimate the failure-conditional PDF and the reliability sensitivity index through a two-stage Markov chain Monte Carlo (MCMC) simulation. In the first stage, a different criterion based on the distance to the failure domain is adopted to update the chain and finally get a first sample in the failure domain. Then, starting with this sample, a normal Markov chain is run in the second stage to simulate the failure-conditional PDF and estimate the reliability sensitivity index. The preconditioned Crank-Nicolson algorithm (a special MCMC method) is adopted to deal with high-dimensional problems (with many uncertain parameters), and adaptive parameter tuning is used to enhance the performance of the proposed two-stage MCMC method. Several test examples show the high efficiency of the proposed two-stage MCMC method compared to subset simulation. The method is currently designed for problems with a single failure domain, yet it could be extended.

Global Reliability Sensitivity of the Structure Under Fuzzy Uncertainty and Its Estimations

Global reliability sensitivity (GRS) index under fuzzy uncertainty is defined as the mean of the absolute difference between unconditional failure credibility and conditional failure credibility, which is important to quantify the effect of fuzzy inputs on failure credibility of the structure. The solution of this index requires nested analysis of conditional failure credibility and fuzzy mean, which introduces a large amount of computation. To efficiently estimate the GRS under fuzzy uncertainty, a method combining fuzzy simulation (FS) with fuzzy first-order and second moment (FS–FFOSM) method as well as a fuzzy sequential optimization and reliability assessment (FSORA) method are proposed. Compared with the direct double-loop FS method, in which both the conditional failure credibility and the fuzzy mean are estimated by FS, the proposed FS–FFOSM uses a more efficient FFOSM method to estimate the conditional failure credibility in the inner loop, which improves the computational efficiency of estimating GRS under fuzzy uncertainty. To further improve the computational efficiency, this paper also transforms the estimation of GRS into the solution of the optimization model under the constraint of failure credibility and establishes the FSORA method to solve the optimization model. The FSORA method uses a series of cycles of deterministic optimization and inverse design point analysis corresponding to the required fuzzy reliability index by GRS to reduce the iterations, and then the computational efficiency is further improved. Several examples illustrate the computational efficiency and accuracy of the proposed methods.

A functional global sensitivity measure and efficient reliability sensitivity analysis with respect to statistical parameters

Sensitivity analysis and reliability assessment are two important aspects of structural and system safety. Epistemic uncertainty with respect to probabilistic model of input parameters due to lack of knowledge is present in many scarce-data applications and complicates the characterization of uncertainty in model response. In this article, we present two importance measures to evaluate the impact of distribution parameters on the probability distribution function (PDF) of the output and the failure probability. The epistemic uncertainty associated with the distribution parameters is modeled as random variables. A modified extended polynomial chaos expansion (MEPCE) approach is introduced in which aleatory and epistemic random variables are modeled and propagated simultaneously while allowing the separate assessment for any single epistemic variable. A MEPCE-based kernel density estimation (KDE) construction provides a composite map from each epistemic variable to the response PDF. The functional global sensitivity index of the PDF with respect to the distribution parameters is thus derived, as a function of output, which is both more informative and more efficient than standard scalar sensitivity measures. Reliability sensitivity indices can be readily evaluated by integrating the global sensitivity index function over the failure zone. Three illustrative examples are used to demonstrate the proposed methodology.

An efficient method for estimating time-dependent global reliability sensitivity

Time-dependent global reliability sensitivity can quantify the effect of input variables in their whole distribution ranges on the time-dependent failure probability. To efficiently estimate it to help researchers control the time-dependent failure probability, a novel method is proposed. The proposed method transforms the estimation of unconditional-conditional time-dependent failure probabilities into that of the unconditional-conditional probability density functions (PDFs) of the minimum of time-dependent performance function. Firstly, the minimum of time-dependent performance function is evaluated by adaptive Kriging, and its unconditional-conditional fractional moments are estimated by multiplicative dimensional reduction method (M-DRM). Then, the maximum entropy (MaxEnt) constrained by these fractional moments is used to estimate the unconditional-conditional PDFs, on which the unconditional-conditional time-dependent failure probabilities can be obtained. Finally, the one-dimensional Gaussian quadrature is applied to estimate the time-dependent global reliability sensitivity indices. Due to the high efficiency of adaptive Kriging for estimating the minimum of time-dependent performance function, the avoidance of dimensional curse by M-DRM, and the high efficiency of MaxEnt constrained by fractional moments for estimating PDF, the proposed method can reduce the computational cost dramatically.

Reliability and reliability sensitivity analysis of structure by combining adaptive linked importance sampling and Kriging reliability method

The application of reliability analysis and reliability sensitivity analysis methods to complicated structures faces two main challenges: small failure probability (typical less than 10−5) and time-demanding mechanical models. This paper proposes an improved active learning surrogate model method, which combines the advantages of the classical Active Kriging – Monte Carlo Simulation (AK-MCS) procedure and the Adaptive Linked Importance Sampling (ALIS) procedure. The proposed procedure can, on the one hand, adaptively produce a series of intermediate sampling density approaching the quasi-optimal Importance Sampling (IS) density, on the other hand, adaptively generate a set of intermediate surrogate models approaching the true failure surface of the rare failure event. Then, the small failure probability and the corresponding reliability sensitivity indices are efficiently estimated by their IS estimators based on the quasi-optimal IS density and the surrogate models. Compared with the classical AK-MCS and Active Kriging – Importance Sampling (AK-IS) procedure, the proposed method neither need to build very large sample pool even when the failure probability is extremely small, nor need to estimate the Most Probable Points (MPPs), thus it is computationally more efficient and more applicable especially for problems with multiple MPPs. The effectiveness and engineering applicability of the proposed method are demonstrated by one numerical test example and two engineering applications.

Global reliability sensitivity analysis based on state dependent parameter method and efficient sampling techniques

In order to efficiently assess the influence of input variables on the failure of the structural systems, an improved global reliability sensitivity analysis (SA) method is proposed in this paper. The new method is based on the state dependent parameter (SDP) method and the efficient sampling techniques. In the new method, the efficient sampling techniques are first used to generate samples that are more efficient for reliability analysis, and then the SDP method is further employed to estimate the global reliability sensitivity index by using the same set of samples as the reliability analysis. Two efficient sampling methods, e.g., importance sampling (IS) and truncated importance sampling (TIS), are employed in this paper, and the strategies of combining these methods with the SDP method for global reliability SA are discussed. Compared with the existing SDP method, the new method is more efficient for global reliability SA of structural systems. Three examples are used in the paper to demonstrate the efficiency and precision of the new methods.

Reliability sensitivity analysis with subset simulation: application to a carbon dioxide storage problem

Reliability sensitivity analysis (RSA) is a sensitivity analysis to measure the effect of modelling parameters on the predicted reliability of a system. It can be used for reliability-based design, safety management, etc. The output-classification-based version of RSA compares the failure-conditional probability density function (PDF) of model parameters with their unconditional PDF to measure sensitivity. The main challenge is to estimate failure-conditional PDFs. Usually, these PDFs can be estimated through the failure samples obtained by Monte Carlo simulation. However, practical systems usually have a small failure probability. For such cases, the brute-force Monte Carlo simulation requires a larger number of samples to obtain enough failure samples. Therefore, the computational cost is very high. In this paper, we propose to use subset simulation to estimate the output-classification-based reliability sensitivity index. Subset simulation introduces a series of intermediate failure events which are easier to sample from, and then iteratively samples in each constrained failure region until the target failure event is reached. Compared to brute-force Monte Carlo simulation, subset simulation samples in a direction towards the target failure domain. Therefore, the failure samples can be obtained more efficiently. We apply subset simulation to perform RSA for a carbon dioxide storage benchmark problem. We show that subset simulation can estimate the output-classification-based reliability sensitivity index more efficiently compared to brute-force Monte Carlo simulation.

Dynamic Reliability and Global Sensitivity Analysis for Hydraulic Pipe Based on Sparse Grid Integral Method

Abstract Due to the extremely rough working environment, aero-hydraulic pipes face serious dynamic failure problems in applications for practical engineering. This paper proposes a dynamic reliability and moment-independent global sensitivity analysis (GSA) method to evaluate the dynamic reliability and the effects of random input variables on the dynamic reliability of aero-hydraulic pipes. Based on the Miner criterion for the cumulative damage of structural fatigue, this paper establishes the dynamic reliability analysis method under the condition of double random vibration. In order to further analyze the influence of the uncertainty of each random variable of pipe on its dynamic reliability, a moment-independent global sensitivity index for dynamic reliability based on cumulative distribution function is proposed in this paper. The index can reflect the effects of random variables on dynamic reliability quantitatively. Based on the proposed GSA method of dynamic reliability, a sparse grid integral (SGI) method is introduced to solve the dynamic reliability and moment-independent global sensitivity index, with high computational efficiency. Finally, the effects of clamp supports, diameters, and curvature of curved pipe on the dynamic reliability and GSA are analyzed through a hydraulic piping example.

An efficient method for estimating the parameter global reliability sensitivity analysis by innovative single-loop process and embedded Kriging model

Abstract For evaluating the effect of uncertain distribution parameter on the uncertainty of the failure probability function, the parameter global reliability sensitivity index (PGRSI) is researched and the high efficient estimation algorithm is proposed to approximately estimate the PGRSI. The used definition form of the PGRSI is the expectation of absolute differences between the unconditional expectation of failure probability function and the conditional expectation of failure probability function, which involves a triple-loop evaluation. For efficiently estimating the PGRSI, this paper innovatively inducts the Bayes theorem and the law of total expectation in the successive intervals without overlapping in order to convert the triple-loop evaluation to a single-loop one. The number of actual model evaluations of the single-loop Monte Carlo simulation (MCS) is independent of the dimensionality of uncertain distribution parameters, and the single-loop MCS only needs one set of input-output samples. Besides, the single-loop MCS can be regarded as a classification problem which needs to identify the failure or safety states of samples in the one set. To further enhance the computational efficiency of the proposed single-loop process, the adaptive Kriging (AK) model is embedded in the single-loop MCS, where the Kriging model is used to approximate the actual limit state function to classify the samples. The number of actual model evaluations of the Kriging model nested single-loop process only generates in the process of constructing the AK model. The results of case studies illustrate the accuracy and efficiency of the proposed method.

Structural reliability and reliability sensitivity analysis of extremely rare failure events by combining sampling and surrogate model methods

The aim of this article is to study the reliability analysis, parametric reliability sensitivity analysis and global reliability sensitivity analysis of structures with extremely rare failure events. First, the global reliability sensitivity indices are restudied, and we show that the total effect index can also be interpreted as the effect of randomly copying each individual input variable on the failure surface. Second, a new method, denoted as Active learning Kriging Markov Chain Monte Carlo (AK-MCMC), is developed for adaptively approximating the failure surface with active learning Kriging surrogate model as well as dynamically updated Monte Carlo or Markov chain Monte Carlo populations. Third, the AK-MCMC procedure combined with the quasi-optimal importance sampling procedure is extended for estimating the failure probability and the parametric reliability sensitivity and global reliability sensitivity indices. For estimating the global reliability sensitivity indices, two new importance sampling estimators are derived. The AK-MCMC procedure can be regarded as a combination of the classical Monte Carlo Simulation (AK-MCS) and subset simulation procedures, but it is much more effective when applied to extremely rare failure events. Results of test examples show that the proposed method can accurately and robustly estimate the extremely small failure probability (e.g. 1e–9) as well as the related parametric reliability sensitivity and global reliability sensitivity indices with several dozens of function calls.

An efficient method for estimating global reliability sensitivity indices

Global reliability sensitivity index, defined as the average absolute difference between the original probability density function (PDF) of input and the conditional one on the failure event, can measure the effect of input on the failure probability and provide information for the safety design. However, efficiently estimating the global reliability sensitivity index is still a challenge for engineering structures with rare failure events and implicit complex limit state functions. Thus, a novel method by combining the dynamic Kriging with Monte Carlo simulation (abbreviated as DK-MCS) is proposed for addressing this issue. The novelty of DK-MCS is twofold. One advantage is that the DK model is inserted in MCS process for accurately recognizing failure samples from all MCS samples by the iteratively updated DK model instead of the actual implicit limit state function, thus the computational cost of DK-MCS is significantly less than that of MCS while the precision of DK-MCS is the same as that of MCS. The other is that a new criteria is presented for selecting informative training points to effectively refine the DK model, where the informative training points with the largest contribution to the failure probability are selected by the cross validation method. Therefore, the iteration process for building the DK model can converge more rapidly. Several examples demonstrate the proposed DK-MCS method can greatly save computational cost of estimating the global reliability sensitivity indices and keep the same precision as MCS.

Two Efficient AK-Based Global Reliability Sensitivity Methods by Elaborative Combination of Bayes’ Theorem and the Law of Total Expectation in the Successive Intervals Without Overlapping

The global reliability sensitivity index (GRSI) can measure the effect of model input variable on the failure probability of the structure and provide guidance for the reliability-based design optimization. In this paper, to efficiently estimate the GRSI, an equivalent form of the GRSI is derived by elaborative combination of Bayes’ theorem and the law of total expectation in the successive intervals without overlapping, and it not only makes the total computational cost independent of the dimensionality of model inputs, but also avoids approximating the probability density function approximation used in the existing Bayes’ theorem based global reliability sensitivity analysis. For further improving the efficiency of estimating the GRSI by the equivalent form, two algorithms are presented by nesting the adaptive Kriging (AK) into Monte Carlo simulation (MCS) and importance sampling (IS), respectively, which are abbreviated as AK-MCS and AK-IS. Results of one numerical example and four engineering applications show that the number of model evaluations by the AK-IS is less than $\text{2}\% $ of that by direct IS, and the model evaluation number by AK-MCS is less than $\text{4}\% $ of that by direct MCS under the convergent condition. The results illustrate that the proposed methods for estimating the GRSI are practical for engineering applications.

Reliability sensitivity analysis method for time-dependent problem based on first-passage method

The sensitivity index plays a critical role in the design of product and is used to quantify the impact degree of the uncertainty of the input variable to the uncertainty of the interest output. This paper presents a new local reliability sensitivity method and a global reliability sensitivity analysis method of time-dependent reliability problems. Firstly, according to the Poisson's assumption-based first-passage method, the local reliability sensitivity index is directly obtained by calculating the partial derivative of the failure probability to the distribution parameters of input random variable. Then, the moment-independent global reliability sensitivity index of the time-dependent problems is derived based on the concept of moment-independent. Finally, the efficiency and accuracy of the proposed method are verified with the reference results of Monte Carlo simulation.

An efficient global reliability sensitivity analysis algorithm based on classification of model output and subset simulation

The global reliability sensitivity analysis measures the effect of each model input variable on the failure probability, which is very useful for reliability-based optimization design. The aim of this paper is to propose an alternative method to estimate the global reliability sensitivity indices by one group of model input–output samples. Firstly, Bayes formula is used to convert the original expression of global reliability sensitivity index into an equivalent form where only the unconditional failure probability and the failure-conditional probability density function (PDF) of each model input variable are required. All global reliability sensitivity indices can be simultaneously estimated by this new equivalent form, and the computational cost of the process is independent of the dimensionality of model input variables. Secondly, to improve the efficiency of sampling which aims at calculating the unconditional failure probability and estimating the failure-conditional PDF of every model input simultaneously, subset simulation method is extended to achieve these two aims. In the proposed procedure, subset simulation is used to estimate the unconditional failure probability, and Metropolis-Hastings algorithm is employed to convert the samples in failure domain from the current PDF in subset simulation to the PDF corresponding to the original PDF of model inputs for estimating the failure-conditional PDF of each model input variable. Thirdly, Edgeworth expansion is employed to approximate the failure-conditional PDF of each model input variable. Finally, the global reliability sensitivity index can be easily computed as byproducts using the unconditional failure probability and the failure-conditional PDF of each model input in failure probability analysis, and this process does not need any extra model evaluations after the unconditional failure probability analysis is completed by subset simulation. A headless rivet model, a roof truss structure and a composite cantilever beam structure are analyzed, and the results demonstrate the effectiveness of the proposed method in global reliability sensitivity analysis.

Global Reliability Sensitivity Analysis Based on Maximum Entropy and 2-Layer Polynomial Chaos Expansion.

To optimize contributions of uncertain input variables on the statistical parameter of given model, e.g., reliability, global reliability sensitivity analysis (GRSA) provides an appropriate tool to quantify the effects. However, it may be difficult to calculate global reliability sensitivity indices compared with the traditional global sensitivity indices of model output, because statistical parameters are more difficult to obtain, Monte Carlo simulation (MCS)-related methods seem to be the only ways for GRSA but they are usually computationally demanding. This paper presents a new non-MCS calculation to evaluate global reliability sensitivity indices. This method proposes: (i) a 2-layer polynomial chaos expansion (PCE) framework to solve the global reliability sensitivity indices; and (ii) an efficient method to build a surrogate model of the statistical parameter using the maximum entropy (ME) method with the moments provided by PCE. This method has a dramatically reduced computational cost compared with traditional approaches. Two examples are introduced to demonstrate the efficiency and accuracy of the proposed method. It also suggests that the important ranking of model output and associated failure probability may be different, which could help improve the understanding of the given model in further optimization design.

A modified importance sampling method for structural reliability and its global reliability sensitivity analysis

The importance sampling method is an extensively used numerical simulation method in reliability analysis. In this paper, a modification to the importance sampling method (ISM) is proposed, and the modified ISM divides the sample set of input variables into different subsets based on the contributive weight of the importance sample defined in this paper and the maximum super-sphere denoted by β-sphere in the safe domain defined by the truncated ISM. By this proposed modification, only samples with large contributive weight and locating outside of the β-sphere need to call the limit state function. This amelioration remarkably reduces the number of limit state function evaluations required in the simulation procedure, and it doesn’t sacrifice the precision of the results by controlling the level of relative error. Based on this modified ISM and the space-partition idea in variance-based sensitivity analysis, the global reliability sensitivity indices can be estimated as byproducts, which is especially useful for reliability-based design optimization. This process of estimating the global reliability sensitivity indices only needs the sample points used in reliability analysis and is independent of the dimensionality of input variables. A roof truss structure and a composite cantilever beam structure are analyzed by the modified ISM. The results demonstrate the efficiency, accuracy, and robustness of the proposed method.