Reliability Analysis with Adaptive Response Surfaces

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 time-variant reliability analysis method for non-linear limit-state functions with the mixture of random and interval variables

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

As engineering applications become increasingly complex, they are often characterized by implicit response functions that are both expensive to evaluate and nonlinear in their behavior. Reliability assessment given this type of response is dicult with available methods. Current reliability methods focus on the discovery of a single most probable point of failure, and then build a low-order approximation to the limit state at this point. This creates inaccuracies when applied to engineering applications for which the limit state has a higher degree of nonlinearity or is multimodal. Sampling methods, on the other hand, do not rely on an approximation to the shape of the limit state and are therefore generally more accurate when applied to problems with nonlinear limit states. However, sampling methods typically require a large number of response function evaluations, which can make their application infeasible for computationally expensive problems. This paper describes the application of ecient global optimization to reliability assessment to provide a method that eciently characterizes the limit state throughout the uncertain space. The method begins with a Gaussian process model built from a very small number of samples, and then intelligently chooses where to generate subsequent samples to ensure the model is accurate in the vicinity of the limit state. The resulting Gaussian process model is then sampled using multimodal adaptive importance sampling to calculate the probability of exceeding (or failing to exceed) the response level of interest. By locating multiple points on or near the limit state, more complex limit states can be modeled, leading to more accurate probability integration. By concentrating the samples in the area where accuracy is important (i.e. in the vicinity of the limit state), only a small number of true function evaluations are required to build a quality surrogate model. The resulting method is both accurate for any arbitrarily shaped limit state and computationally ecient even for expensive response functions. This new method is applied to a collection of example problems that currently available methods have diculty

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.

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

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.

Fatigue cracks are an ongoing problem for aluminum high-speed vessels, and preventing fatigue cracks caused by wave loading is expected to be a significant challenge for future aluminum high-speed ferries and military vessels. To aid in this effort, a hot-spot fatigue design approach using first-order reliability methods (FORM) is constructed. Two different limit state functions are investigated, and the accuracy and consistency of the FORM method for the highly nonlinear fatigue limit state equations are evaluated through a comparison with Monte Carlo simulation results. The sensitivity of the resulting safety index to changes in the input variables, and their uncertainties, are presented graphically. The method is compared to existing design standards for four simple structural details.

The Support Vector Machine (SVM) response surface method (RSM) is proposed on fast Markov chain simulation for the problem with implicit limit state function usually encountered in engineering reliability analysis and design. In the proposed method, Markov chain is used to generate the samples in the important region of the limit state function, and the SVM is employed to construct the response surface by use of these samples. Since Markov chain can adaptively simulate the samples in the important region, and the candidate state but not Markov state is used as the training samples for SVM, the proposed method can well approximate the limit state equation in the zone surrounding the design point, and can make full use of information provided by Markov chain simulation. In addition, the iterative strategy is adopted to improve the convergence speed of the failure probability. Moreover, the proposed method uses the SVM regression method to construct the response surface, which can automatically apply the Structural Risk Minimization (SRM) inductive principle in approximating the limit state equation, thus it can approximate the failure probability with high accuracy. Finally applications in a numerical example and an engineering example indicate that the proposed method owns good performance in calculating efficiency and accuracy.

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.

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.

Aircraft conceptual design using metaheuristic-based reliability optimisation

This study describes an efficient application of Rackwitz Fiessler algorithm to search out all of design points, utilized in the multimodal importance sampling simulation to estimate structural failure probabilities. The initial points to execute Rackwitz Fiessler algorithm are proposed to set in the respective quadrants of the basic random variable space and all of possible design points are determined. Numerical examples to estimate the failure probability of structural system with multiple and nonlinear limit state function by using multimodal importance sampling using all of design points determined are presented to illustrate that the proposed method gives accurate estimations effectively.