First and second order approximate reliability analysis methods using evidence theory

With a great capability of dealing with epistemic uncertainty, evidence theory has been utilised to conduct reliability analysis for engineering systems recently. Unfortunately, the discontinuous nature of uncertainty quantification using evidence theory incurs a huge computational cost. This paper proposes an efficient method to improve the computational efficiency of evidence theory for reliability analysis. In this method, evidence variables are transformed into random variables. Support vector regression is used to construct the approximation model of the limit-state function. The most probable point (MPP) of the approximate reliability problem with only random variables is searched out. Based on the MPP, the most probable focal element (MPFE) of the original problem with evidence variables is identified. According to the MPFE and the monotonicity of the limit-state function, contributions of some focal elements to belief and plausibility in evidence theory can be judged directly. Hence, the number of focal elements involved in the calculation of extreme values of the limit-state function is reduced. Four numerical examples are utilised to test the performance of the proposed method. Results indicate that the proposed method can reduce the computational cost on reliability analysis under epistemic uncertainty while ensuring the high accuracy of reliability analysis results.

In order to improve the computational efficiency of reliability analysis for MDO, a hierarchical hybrid reliability analysis strategy is proposed. This strategy is composed of two layers, the sequential optimization and reliability assessment (SORA) method is adopted in the upper level to decouple the reliability analysis from MDO, which guarantee that the reliability analysis and deterministic MDO can be executed in a serial way. In the lower layer, the collaborative optimization (CO) strategy is integrated both in deterministic MDO and reliability analysis. Furthermore, based on the traditional theory of most probable point (MPP), the reliability analysis combined with the CO strategy is conducted concurrently. And also, the deterministic MDO is implemented in parallel with the CO strategy. Finally, the proposed strategy is validated with an illustrative example and the results are compared with the existing reliability analysis methods in MDO, showing that the proposed strategy is efficient and competent for evaluating design feasibility in MDO environment with uncertainties.

In this paper, an efficient method for the reliability analysis of systems with nonlinear limit states is described. It combines optimization-based and simulation-based approaches and is particularly applicable for problems with highly nonlinear and implicit limit state functions, which are difficult to solve by conventional reliability methods. The proposed method consists of two major parts. In the first part, an optimization-based method is used to search for the most probable point (MPP) on the limit state. This is achieved by using adaptive response surface approximations. In the second part, a multi-modal adaptive importance sampling method is proposed using the MPP information from the first part as the starting point. The proposed method is applied to the reliability estimation of a vehicle body-door subsystem with respect to one of the important quality issues -- the door closing effort. The superiority of the proposed method, in terms of efficiency and accuracy, is demonstrated with a numerical example of highly nonlinear limit state problem, as well as an automotive door closing effort application. A generalized framework for reliability estimation is also proposed for problems with large numbers of random variables and complicated limit states.

Traditional reliability-based design optimization(RBDO)is either computational intensive or not accurate enough.In this work,a new RBDO method based on Support Vector Machines(SVM)is proposed.For reliability analysis,SVM is used to create a surrogate model of the limit-state function at the Most Probable Point(MPP).The uniqueness of the new method is the use of the gradient of the limit-state function at the MPP.This guarantees that the surrogate model not only passes through the MPP but also is tangent to the limit-state function at the MPP.Then Importance Sampling(IS)is used to calculate the probability of failure based on the surrogate model.This treatment significantly improves the accuracy of reliability analysis.For optimization,the Sequential Optimization and Reliability Assessment(SORA)is employed,which decouples deterministic optimization from the SVM reliability analysis.The decoupling makes RBDO more efficient.The two examples show that the new method is more accurate with a moderately increased computational cost.

In this work, the reliability analysis is employed to take into account the uncertainties in a structure. Reliability analysis is a tool to compute the probability of failure corresponding to a given failure mode. In this study, one of the most commonly used reliability analysis method namely first order reliability method is used to calculate the probability of failure. Since finding the most probable point (MPP) or design point is a constrained optimization problem, in contrast to all the previous studies based on the penalty function method or the preference of the feasible solutions technique, in this study one of the latest versions of the differential evolution metaheuristic algorithm named improved (μ+λ)-constrained differential evolution (ICDE) based on the multi-objective constraint-handling technique is utilized. The ICDE is very easy to implement because there is no need to the time-consuming task of fine tuning of the penalty parameters. Several test problems are used to verify the accuracy and efficiency of the ICDE. The statistical comparisons revealed that the performance of ICDE is better than or comparable with the other considered methods. Also, it shows acceptable convergence rate in the process of finding the design point. According to the results and easier implementation of ICDE, it can be expected that the proposed method would become a robust alternative to the penalty function based methods for the reliability assessment problems in the future works.

Uncertainties in multidisciplinary design optimization (MDO) have a significant influence on the whole design process of engineering systems. The most probable point (MPP) based reliability analysis is an approach that utilizes the safety index β to measure the effect of uncertainties. Collaborative optimization (CO) is a two-level optimization method specially created for large-scale distributed-analysis applications. Simulated annealing-based collaborative optimization (SA—CO) is one of the improved forms of CO that overcomes the difficulty of convergence given the existing of highly nonlinear consistency constraints. By combining the MPP-based reliability analysis method with SA—CO, we present a new collaborative reliability analysis method under the environment of MDO to deal with uncertainties existing in MDO, that is, MPP—SA—CO. Demonstrated by two typical examples, the proposed method inherits the advantages of CO. Also, accurate and efficient results are obtained by employing simulated annealing algorithmic as the system level optimizer and it features response surface instead of disciplinary optimization.

Reliability analysis of complex geotechnical engineering is time-consuming since its performance function is highly nonlinear and implicit. In this paper, an adaptive sequential sampling metamodeling-based method is proposed to deal with such problems. Gaussian process regression (GPR), utilized to approximate the real performance function, is constructed by the initial design of experiments (DOEs). Based on the geometric meaning of the most probable point (MPP) in the first-order reliability method (FORM), the potential MPP, which is a point infinitely close to the limit-state surface and with the minimum distance to the origin while considering a distance constraint, is searched and added to the DOE to refine the GPR. Then, the Monte Carlo simulation (MCS) is adopted to evaluate the failure probability by the refined GPR. The above two procedures are repeated until the stopping criterion is reached. Three examples, including one mathematical example and two geotechnical engineering problems, are analyzed. The results show the proposed method requires fewer performance function calls and is an efficient, accurate, and robust reliability analysis method.

This paper proposes a novel second-order reliability method (SORM) using non-central or general chi-squared distribution to improve the accuracy of reliability analysis in existing SORM. Conventional SORM contains three types of errors: (1) error due to approximating a general nonlinear limit state function by a quadratic function at most probable point (MPP) in the standard normal U-space, (2) error due to approximating the quadratic function in U-space by a hyperbolic surface, and (3) error due to calculation of the probability of failure after making the previous two approximations. The proposed method contains the first type of error only which is essential to SORM and thus cannot be improved. However, the proposed method avoids the other two errors by describing the quadratic failure surface with the linear combination of non-central chi-square variables and using the linear combination for the probability of failure estimation. Two approaches for the proposed SORM are suggested in the paper. The first approach directly calculates the probability of failure using numerical integration of the joint probability density function (PDF) over the linear failure surface and the second approach uses the cumulative distribution function (CDF) of the linear failure surface for the calculation of the probability of failure. The proposed method is compared with first-order reliability method (FORM), conventional SORM, and Monte Carlo simulation (MCS) results in terms of accuracy. Since it contains fewer approximations, the proposed method shows more accurate reliability analysis results than existing SORM without sacrificing efficiency.

A hybrid self-adaptive conjugate first order reliability method for robust structural reliability analysis

Dimension reduction for constructing high-dimensional response distributions by accounting for unimportant and important variables

In this study, an effective method for reliability-based design optimization (RBDO) is proposed enhancing sequential optimization and reliability assessment (SORA) method by convex approximations. In SORA, reliability estimation and deterministic optimization are performed sequentially. The sensitivity and function value of probabilistic constraint at the most probable point (MPP) are obtained in the reliability analysis loop. In this study, the convex approximations for probabilistic constraint are constructed by utilizing the sensitivity and function value of the probabilistic constraint at the MPP. Hence, the proposed method requires much less function evaluations of probabilistic constraints in the deterministic optimization than the original SORA method. The efficiency and accuracy of the proposed method were verified through numerical examples.

Profust reliability analysis is investigated in this article. The main aim is to address problems with very small failure probabilities. First, a single-layer multimodal importance sampling (MMIS) method based on the most probable points (MPPs) in the fuzzy failure region is proposed. Secondly, to reduce function calls, an adaptive method fusing kriging model and MMIS is developed. In this method, a surrogate fuzzy failure region (SFFR) based on the kriging prediction information is defined. The MPPs in the SFFR are explored, the important samples are generated, the optimal training point is chosen, and the kriging model and the SFFR are updated. The learning process is executed iteratively until the SFFR converges to the true one. To find all MPPs and maintain unbiasedness, an intelligent multimodal optimization method based on multi-objective optimization is introduced. Four case studies are used to verify the advantages of the proposed method.

Reducing the surrogate model-based method computation without loss of prediction accuracy remains a significant challenge in structural reliability analysis. The unbalanced probability density, important information in critical region and information redundancy of added sample points are ignored in most of traditional surrogate-based methods, resulting in heavy computational burden. In this work, an active learning combining adaptive Kriging method and weighted penalty (AK-WP) is proposed to analyze the reliability of engineering structures. Firstly, an active learning and weighted penalty function (WPLF) is the result of integrating active learning method, weighted function and penalty function, which is proposed to find the most probable point (MPP). Meanwhile, to avoid redundant information, the best suitable MPP is determined by a proposed distance law established between the found MPP and the existing design of experiment (DoE). Secondly, the Kriging model is refined according to best suitable MPP in each iteration. Thirdly, the failure probability is estimated by the Monte Carlo sample points from the n-ball domain until the convergence condition is satisfied. The accuracy and efficiency of the proposed method are demonstrated by some numerical examples including the highly nonlinear, the small probability problems and implicit function, as well as a real engineering application.

This paper proposes an adaptive probability analysis method that can effectively generate the probability distribution of the output performance function by identifying the propagation of input uncertainty to output uncertainty. The method is based on an enhanced hybrid mean value (HMV+) analysis in the performance measure approach (PMA) for numerical stability and efficiency in search of the most probable point (MPP). The HMV+ method improves numerical stability and efficiency especially for highly nonlinear output performance functions by providing steady convergent behavior in the MPP search. The proposed adaptive probability analysis method approximates the MPP locus, and then adaptively refines this locus using an a posteriori error estimator. Using the fact that probability levels can be easily set a priori in PMA, the MPP locus is approximated using the interpolated moving least-squares method. For refinement of the approximated MPP locus, additional probability levels are adaptively determined through an a posteriori error estimator. The adaptive probability analysis method will determine the minimum number of necessary probability levels, while ensuring accuracy of the approximated MPP locus. Several examples are used to show the effectiveness of the proposed adaptive probability analysis method using the enhanced HMV+ method.

Aircraft conceptual design using metaheuristic-based reliability optimisation