Efficient sampling methods for global reliability sensitivity analysis

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.

A novel method for global reliability sensitivity analysis by adaptive bivariate multiplicative dimensional reduction integral

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.

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.

This study presents an innovative method to estimate the reliability sensitivity based on the low-discrepancy sampling which is a new technique for structural reliability analysis. Two advantages are contributed to the method: one is that, by developing a general importance sampling procedure for reliability sensitivity analysis, the partial derivative of the failure probability with respect to the distribution parameter can be directly obtained with typically insignificant additional computations on the basis of structural reliability analysis; and the other is that, by combining various low-discrepancy sequences with the above importance sampling procedure, the proposed method is far more efficient than that based on the classical Monte Carlo method in estimating reliability sensitivity, especially for problems of small failure probability or problems that require a large number of costly finite element analyses. Examples involving both numerical and structural problems illustrate the application and effectiveness of the method developed, which indicate that the proposed method can provide accurate and computationally efficient estimates of reliability sensitivity.

Systems with random variables and random excitations exist widely in various engineering problems. Extending the traditional global reliability sensitivity to this double-stochastic system has important guiding significance for its design optimization. However, because there is a certain coupling between the randomness of variables and the randomness of excitation, this coupling mechanism is difficult to determine in practical projects. Therefore, it is difficult to extend the traditional reliability sensitivity analysis method to this double-stochastic system. In this research, it is assumed that there is no correlation between variables and excitations. Then, combining the first-passage method–based dynamic strength formula and the variance-based sensitivity analysis method, an approximate global reliability sensitivity analysis method for this double-stochastic system is proposed. In order to improve the computational efficiency, a nested loop method based on seven-point estimation is proposed for reliability sensitivity analysis. In order to verify the accuracy and efficiency of the proposed method, a Monte Carlo simulation is given as a reference. Three examples are studied and discussed to illustrate the practicality and feasibility of the proposed method.

Copula-based methods for global sensitivity analysis with correlated random variables and stochastic processes under incomplete probability information

Reliability analysis techniques combining with various surrogate models have attracted increasing attention because of their accuracy and great efficiency. However, they primarily focus on the structures with continuous response, while very rare researches on the reliability analysis for structures with discontinuous response are carried out. Furthermore, existing adaptive reliability analysis methods based on importance sampling (IS) still have some intractable defects when dealing with small failure probability, and there is no related research on reliability analysis for structures involving discontinuous response and small failure probability. Therefore, this paper proposes a novel reliability analysis method called AGPC-IS for such structures, which combines adaptive Gaussian process classification (GPC) and adaptive-kernel-density-estimation-based IS. In AGPC-IS, an efficient adaptive strategy for design of experiments (DoE), taking into consideration the classification uncertainty, the sampling uniformity and the regional classification accuracy improvement, is developed with the purpose of improving the accuracy of Gaussian process classifier. The adaptive kernel density estimation is introduced for constructing the quasi-optimal density function of IS. In addition, a novel and more precise stopping criterion is also developed from the perspective of the stability of failure probability estimation.The efficiency, superiority and practicability of AGPC-IS are verified by three examples

Structural Reliability Analysis (SRA) is a useful tool in structural engineering. Uncertainty in input parameters and model uncertainties in the analysis predictions are explicitly modelled by random variables. With this methodology, the uncertainties involved are handled in a consistent and transparent way. Compared to a deterministic analysis, SRA provides improved insight in how the various uncertainties involved influence the results. The main results from SRA is the calculated probability of structural failure, but other useful results such as uncertainty importance factors and design points being the most likely combination of all variables at failure represent helpful information. The present paper illustrates some the features using SRA for two different types of application. The first application is the use of SRA as a tool for code calibration and the second shows the application of SRA to a problem where common practice is likely to be rather conservative and therefore leading to unacceptable results, but where the degree of conservatism is not known. Two examples are chosen to illustrate code calibration; i.e. hull girder ultimate limit state (ULS) for tankers and ULS for mooring design in the ULS for floating offshore vessels. Code calibration involves both SRA and design analysis following the code. It is shown how the design analysis can be modified in order to better reflect a chosen target reliability level across a selected set of test cases representative for what the code should cover. Fatigue of subsea wellhead systems is selected as an example of a special case when application of existing rules may lead to unsatisfactory results which are likely to be rather conservative. It is shown how results can be presented in terms of the accumulated probability of fatigue failure as a function of time. This may be a more suitable basis for decision making than a calculated fatigue life from a standard analysis. It is also illustrated how importance factors from the SRA can be used as guidance on how to prioritize effort in order to improve prediction of the fatigue damage. The present paper is not intended to be detailed in all input and analysis methodology, but draw the attention towards the possibilities and benefits of applying SRA in structural engineering, where the examples are used to illustrate this potential.

Despite being a common procedure, abdominal hernia treatments still require improvement due to the number of relapses and other postoperative issues. In silico testing can be employed to predict the behavior of the complex abdominal wall and implant systems. Here, uncertainty quantification and sensitivity analysis are required in order to optimize the parameters of hernia repair models. This paper concerns the modeling of an abdominal wall and implant using the finite element method. A Gasser-Ogden-Holzapfel (GOH) material model is used for the abdominal wall and an orthotropic material model for the implant. The parameters of the GOH model and the orientation of the implant are assumed to be uncertain. Regression-based polynomial chaos expansion is used as a meta-modeling method for uncertainty propagation and global sensitivity analysis. The maximum force in the connection between the implant and native tissue is considered as the quantity of interest. A failure risk criterion is also defined and presented. It has been found that the significance of the material parameters depends on the type of implant that is analyzed. Likewise, the risk of connection failure varies considerably depending on the implant used. Models with different types of implant produce very diverse results. Moreover, these differences also appear in the global sensitivity index and the risk of connection failure. This would indicate that specific implant designs and material properties are crucial to the success of hernia repair surgery.

An enhanced method for improving the accuracy of small failure probability of structures

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

Structural reliability analysis (SRA) has been used to calculate the probability of mooring line failure in an intact mooring system as a function of the magnitude of the safety factor applied in design. A range of different units have been considered, comprising ship shaped units and semisubmersibles at different water depths from 100 m to 2200 m. Environmental conditions representative for the Norwegian continental shelf and the Gulf of Mexico are used in the analyses, and the characteristics of the results in the different environments are compared and discussed. Analyses for Brazilian environment are currently ongoing, but not included here. Time domain analysis is applied to obtain the short-term, extreme value distribution of line tension, conditional on stationary metocean conditions. A large number of different conditions are considered. A response surface is used to interpolate on the distribution parameters in order to describe the tension response in varying conditions. Joint probabilistic models of the metocean environment corresponding to the different geographical locations have been applied, taking account of wind, wave and current and their respective heading angles. A continuous model is used for the metocean conditions at the Norwegian continental shelf, whereas a hurricane model is used in the Gulf of Mexico. The effects of uncertainties in the response calculation are included. The mooring line component strength is based on strength data from break load tests. Conventional catenary chain-wire chain systems as well as polyester moorings are considered. With the probability of failure as a function of the safety factor, it is shown that present regulations result in a significant scatter in reliability level between the cases. Safety factors have been calibrated considering all cases. Alternative design formats are proposed and tested including a format with 2 safety factors. Calibration results are provided as a function of the target reliability level. The final recommendation on target reliability level is given in an accompanying paper at OMAE 2017, comprising both the ULS and the ALS. It is demonstrated that alternative design formats can provide a more consistent safety level across the cases. A different design philosophy is needed for the Gulf of Mexico in order to achieve acceptable risk. Options for design are discussed. The present work provides a unique and comprehensive set of results, where advanced reliability methods are used in combination with detailed response calculations in the time domain. The results provide a basis for calibration of mooring design for ULS and subsequently for regulators to update their rules. The work has been carried out as part of the NorMoor Joint Industry Project, with participants from oil companies, engineering companies, rig-owners, manufacturers and marine authorities. This paper is the first one in a series of three at OMAE 2017, where the second deals with structural reliability analysis of the ALS and the third one provides summary and recommendations for safe mooring design in ULS and ALS.

Global sensitivity analysis (GSA) is useful to recognize important inputs for assigning priority and unimportant inputs for simplifying models by exploring whole distribution ranges. Meanwhile, regional sensitivity analysis (RSA) is also studied for finding the contribution of the critical region of an input, which can be viewed as the complementary of GSA. However, there is a lack of GSA and RSA research in presence of fuzzy uncertainty. Thus, a new global sensitivity index (GSI) under the fuzzy uncertainty is devoted on the credibility distribution function (CrDF), a comprehensive distribution description under the fuzzy uncertainty. The CrDF-based GSI is defined by the fuzzy expectation of the difference between the CrDF and the conditional CrDF of the output on fixing the fuzzy input over its whole distribution range, which can quantify the contribution of the fuzzy input to the output CrDF. Then, a new fuzzy RSA technique, the contribution to this CrDF based index (shortened by CCI) plot, is also proposed, and it can assess the effect of given regions of important inputs on output CrDF. Besides, mathematical properties of the CrDF based GSI and the CCI plot are discussed, and their solution are established by use of the fuzzy simulation with the same set of samples. After the accuracy of established fuzzy simulation solution for the CrDF based GSI and the CCI plot are verified by an analytical example, other examples are used to demonstrate the reasonability and applicability of proposed CrDF based GSI and CCI plot under fuzzy uncertainty.

An adaptive parallel learning dependent Kriging model for small failure probability problems