An adaptive method fusing the kriging model and multimodal importance sampling for profust reliability analysis

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.

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 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.

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

Neural Networks Combined with Importance Sampling Techniques for Reliability Evaluation of Explosive Initiating Device

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

Assessing small failure probabilities by AK–SS: An active learning method combining Kriging and Subset Simulation

Aircraft conceptual design using metaheuristic-based reliability optimisation

The surrogate model-based sampling method has been an active area of research due to its evident efficiency over direct sampling method. The fusion of the adaptive kriging (AK) model with the adaptive radial-based importance sampling (ARBIS) method is developed as an improvement of the AK model combined with Monte Carlo simulation (MCS) method especially for estimating the small failure probabilities. In the AK–ARBIS method, the number of candidate samples in each updating process of the kriging model is reduced through partitioning the whole MCS sample domain into different subdomains and removing the samples inside the last subdomain from the participating candidate sampling pool, which can avoid the time-consuming matrix operation involved in kriging predictions in order to find the best training sample in each updating step. In the existing AK–ARBIS method, kriging model stops updating according to the learning function-based stopping criterion, which cannot directly reflect the error of the estimated failure probability. In this regard, this paper proposes the error-based stopping criterion (ESC) enhanced AK–ARBIS method, where the processes of finding the optimal hypersphere and accurately identifying the sign of candidate samples are decoupled in order to successfully embed the ESC. In addition to embedding the ESC into the AK–ARBIS method, two target-oriented candidate sampling reduction strategies are investigated, respectively, to improve the efficiency of finding the optimal hypersphere and accurately identifying the sign of the participating candidate samples. Based on the candidate sampling pool reduction strategies and ESC, both the computational time and the required number of calls to the real limit state function are reduced over the existing AK–ARBIS method. Three case studies are analyzed to demonstrate the accuracy and efficiency of the proposed enhanced AK–ARBIS method, especially for estimating the small failure probabilities.

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.

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.

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 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.

Reliability-Based Design Optimization (RBDO) is a powerful tool in engineering structural design, it tries to find a balance between cost and reliability for structural designs under uncertainty. Several RBDO formulations are developed to solve the RBDO problem, such as double loop methods, single loop methods, and decoupled methods. Despite, these new formulations of RBDO, they are unable to deal for engineering complex problems, due to the computational cost. The Kriging surrogate has been widely used to replace the time-consuming mechanical constraints. In this paper, a single loop RBDO approach (SLA) is coupled with the Kriging surrogate, the most probable points (MPP) of each loop are used as new sample points to update the Kriging model. The Kriging-SLA is running iteratively until it reaches the converge criteria. Compared with other sampling methods, this method can be started with very few training points and converges to the right minimum very efficiently. 2 benchmark examples are used to demonstrate this method.

The lead-zinc sintering process has the characters of complex mechanism, multi-parameter and strong interconnected coupling. This paper presents an intelligent modeling and optimization method based on comprehensive production Indices. First, the neural network prediction model of the comprehensive production indices is proposed, which is synthesizing some techniques, including gray correlation analysis, principal components analysis, and neural network and so on. And the target function is deduced and the model based on comprehensive production Indices is given by using the multi-objective optimization technique. At last, this paper proposes an improved multi-objective particle swarm optimization algorithm based on SPEA2 to calculate the optimization executing parameters. The applied results of actual runs show that the intelligent modeling and optimization method proposed in this paper attains a better industrial effect, and provides an effective and new way to implement the global optimization.