An augmented weighted simulation method for high-dimensional reliability analysis

doi:10.1016/j.strusafe.2021.102117

Structural Safety

Volume 93, November 2021, 102117

https://doi.org/10.1016/j.strusafe.2021.102117 Get rights and content

Highlights

•
A novel augmented weighted simulation method is proposed.
•
A new optimization method is proposed to compute the threshold values.
•
A space reduction strategy is developed.
•
The coefficient of variation of the augmented weighted simulation method is derived.
•
The accuracy of the proposed method is significantly improved.

Abstract

In the reliability analysis of mechanical systems, sampling method is widely used due to the universality and practicability. However, the computation of high-dimensional problems encounters tremendous numerical difficulties, especially when the performance function is highly nonlinear. In this study, an augmented weighted simulation method (AWSM) is proposed in order to tackle this difficulty. The basic idea of AWSM is introducing a series of intermediate events into weighted simulation method (WSM), in which a new optimization method is constructed to reasonably determine each intermediate event. In this way, the failure event is divided to a sequence of conditional events, and the failure probability is accordingly converted to the product of conditional probabilities. Furthermore, a space reduction strategy is proposed to increase the probability of the samples generated in each conditional event, which greatly improves the sampling efficiency. Also, the coefficient of variation of AWSM is derived. Two mathematical examples and four engineering examples are tested, and the results demonstrate the efficiency and accuracy of the proposed method for high-dimensional problems.

Introduction

Structural reliability refers to the capacity of the structure to meet the expected safety, applicability and durability under the specified conditions [1], [2]. It uses the failure probability to represent the safety level of the structural system [3], [4], which is defined as $P_{f} = P [g (x) ⩽ 0] = \int_{g (x) ⩽ 0} f (x) d x$ where $x = {[x_{1}, x_{2}, . . ., x_{n}]}^{T}$ is the vector of random variables, $f (x)$ is the joint probability density function (PDF) of $x$ , $g (x)$ is the performance function, and $F = \{x : g (x) ⩽ 0\}$ represents the failure event located in the failure domain.

The above failure probability can be solved by three different ways: approximate reliability calculation methods [5], [6], [7], moment methods [8], [9], [10] and sampling methods [11], [12], [13]. Approximate reliability calculation methods require evaluating the gradients of the performance function at the most probable point, in which the first order reliability method and second order reliability method [14], [15], [16] are two distinguished representatives. Both of them are commonly applied in mechanical systems because of the simplicity and efficiency. However, when the performance function is highly nonlinear with high-dimensional random variables, their results often encounter the inaccurate problems [17], [18].

Different from the approximate reliability calculation methods, the moment methods calculate the failure probability from the mathematical definition of the reliability [19], [20]. The basic idea of the moment methods is to obtain the statistical moments of the structural response according to the random information of input variables, and then the moment information are converted to the failure probability of the output response [21], [22]. Since the moment methods use the random information from limited samples, the computational results are susceptible to the position of the samples [23], [24].

Comparing with approximate reliability calculation methods and moment methods, sampling methods are insensitive to the types of random variables and the nonlinear degree of limit state function, which provide a universal tool for reliability assessment [25], [26]. Among them, Monte Carlo simulation (MCS) is the most well-known method for approximating the failure probability, and it is often used as a reference solution to test the accuracy of other methods [27]. However, it requires generating huge number of samples to ensure the computational accuracy, which leads to the unaffordable computational cost in practical application. To improve the computational efficiency of MCS, an active learning reliability method combining Kriging and Monte Carlo Simulation (AK-MCS) was proposed by Echard et al. [28] to construct a learning function to update the design of experiment, which improves the efficiency to a large extent. However, the Kriging model may lead to the inaccurate results for high-dimensional problems, especially involving nonlinear performance function [29], [30].

In addition, a series of advanced methods have been also developed, such as important sampling (IS) [31], [32], line sampling (LS) [33], [34] and subset simulation (SS) [35], [36]. IS moves the sampling center to the limit state surface, in this case, the sample has a high probability to fall into the vicinity of the limit state surface, which can promote the sampling efficiency to a certain extent [37], [38]. LS transforms a high-dimensional sampling problem into multiple one-dimensional conditional probability problems in the standard normal space, and the results indicate that the computational efficiency can be significantly enhanced comparing with MCS [39], [40]. SS reduces the computational cost by converting the small failure probability into the multiplication of large conditional probabilities, and thus it is effective to solve the small failure probability problem in time-consuming engineering applications [41], [42].

Recently, Rashki et al. [43] proposed a new weighed simulation method (WSM) by assigning the weight index for each sample, and the failure probability was defined as the ratio of the weight of samples falling into the failure domain to the weight of the total samples. The uniformly distributed samples can sufficiently cover the random space [44], and thus more samples can be generated in the failure domain for solving the high-dimensional problems. Okasha [45] extended the WSM into the field of reliability-based design optimization, in which the firefly algorithm was adopted to search the global optimum. Besides, Xu et al. [46] utilized the Voronoi cells to divide the random space into several sub-spaces, and the failure probability was evaluated by the weighted summation over each sub-space. However, the construction of high order Voronoi cells is difficult for high-dimensional problems [47].

In this paper, an augmented weighted simulation method (AWSM) is proposed to promote the computational accuracy and efficiency of WSM for high-dimensional reliability analysis. Firstly, the random space is divided into several subset failure domains by introducing intermediate events into WSM, which converts the failure probability into the product of a series of conditional probabilities. Secondly, the threshold values of the intermediate events are determined by constructing a new optimization formulation to ensure the rationality of the determination of the intermediate events. Thirdly, a space reduction strategy (SRS) is proposed by sufficient use of samples.

The remainder of this paper is organized as follows: Section 2 reviews the classical sampling methods, including MCS, SS and WSM. Section 3 describes the proposed AWSM in detail. Section 4 provides two mathematical examples and four engineering examples with high-dimensional input variables for demonstrating the validity of AWSM. The conclusions are drawn in Section 5.

Section snippets

Monte Carlo simulation

MCS generates a large number of samples in random space based on the joint PDF of random variables. The failure probability of MCS is defined as the ratio of the number of samples in the failure region $F$ to the total number of samples, which is formulated as $P_{f}^{MCS} = \frac{1}{N} \sum_{j = 1}^{N} I_{F} (x_{j})$ where $N$ is the total number of samples, $I_{F} (x)$ is the index function with respect to the performance function. If $x_{j} \in F$ , $I_{F} (x_{j}) = 1$ ; otherwise, $I_{F} (x_{j}) = 0$ . Although MCS is very simple and universal, it always needs generating a

The proposed method

In this section, an augmented weighted simulation method (AWSM) is proposed to further improve the computational efficiency and accuracy of WSM, which contains five parts: the definition of AWSM, the determination of threshold values, SRS, the computation of coefficient of variation (COV) and the computational procedure of AWSM.

Demonstrative examples and comparative study

In this section, six numerical examples are given to test the efficiency and accuracy of AWSM. The results obtained by AWSM are verified by comparing with those of MCS, polynomial chaos expansion (PCE), dimension-reduction method (DRM), AK-MCS, SS, IS and WSM. The results of SS and IS are computed by UQLAB [59]. The number of samples in each layer of SS is 1000, and the conditional probability is 0.1. The maximum number of iterations is 10. The error is evaluated by the relative error $ε = |P_{f} - P_{f}^{MCS}|$

Conclusions

In this paper, an augmented weighted simulation method (AWSM) is proposed by introducing a series of reasonable intermediate events, in which a new optimization method is constructed by using the variance to obtain the threshold values reasonably and efficiently. Then, a space reduction strategy is developed to narrow each random sub-space, and thus the intermediate event can reuse the samples from the previous conditional events. In this case, the small failure probability is expressed as the

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The supports of the National Natural Science Foundation of China (Grant No. 11972143), the Fundamental Research Funds for the Central Universities of China (Grant Nos. JZ2020HGPA0112, JZ2020HGTA0080) and State Key Laboratory of Reliability and Intelligence of Electrical Equipment (Grant No. EERI_KF2020002) are much appreciated.

References (59)

S. Marelli et al.
An active-learning algorithm that combines sparse polynomial chaos expansions and bootstrap for structural reliability analysis
Struct Saf
(2018)
M.A. Valdebenito et al.
Fuzzy failure probability estimation applying intervening variables
Struct Saf
(2020)
J. Wu et al.
On reliability analysis method through rotational sparse grid nodes
Mech Syst Signal Pr
(2021)
B.A. Bradley et al.
Accuracy of approximate methods of uncertainty propagation in seismic loss estimation
Struct Saf
(2010)
Z. Meng et al.
A novel experimental data-driven exponential convex model for reliability assessment with uncertain-but-bounded parameters
Appl Math Model
(2020)
Y.M. Low
A new distribution for fitting four moments and its applications to reliability analysis
Struct Saf
(2013)
X. Zhang et al.
Structural reliability analysis based on the concepts of entropy, fractional moment and dimensional reduction method
Struct Saf
(2013)
C.D. Eamon et al.
Evaluation of accuracy and efficiency of some simulation and sampling methods in structural reliability analysis
Struct Saf
(2005)
W. Yun et al.
An efficient sampling method for variance-based sensitivity analysis
Struct Saf
(2017)
N.-C. Xiao et al.
A new adaptive sequential sampling method to construct surrogate models for efficient reliability analysis
Reliab Eng Syst Safe
(2018)

Z. Zhang et al.

First and second order approximate reliability analysis methods using evidence theory

Reliab Eng Syst Safe

(2015)

X. Zhang et al.

Maximum entropy distribution with fractional moments for reliability analysis

Struct Saf

(2020)

W. He et al.

A novel structural reliability analysis method via improved maximum entropy method based on nonlinear mapping and sparse grid numerical integration

Mech Syst Signal Pr

(2019)

D.V. Rosowsky

Evolution of probabilistic analysis of timber structures from second-moment reliability methods to fragility analysis

Struct Saf

(2013)

Q. Pan et al.

An efficient reliability method combining adaptive support vector machine and Monte Carlo simulation

Struct Saf

(2017)

R. Vaisman et al.

Splitting sequential Monte Carlo for efficient unreliability estimation of highly reliable networks

Struct Saf

(2016)

H. Zhang et al.

Interval Monte Carlo methods for structural reliability

Struct Saf

(2010)

B. Echard et al.

AK-MCS: An active learning reliability method combining Kriging and Monte Carlo Simulation

Struct Saf

(2011)

J. Zhang et al.

An active learning reliability method combining Kriging constructed with exploration and exploitation of failure region and subset simulation

Reliab Eng Syst Safe

(2019)

T. Zhou et al.

Active learning and active subspace enhancement for PDEM-based high-dimensional reliability analysis

Struct Saf

(2021)

S. Geyer et al.

Cross entropy-based importance sampling using Gaussian densities revisited

Struct Saf

(2019)

I. Papaioannou et al.

Reliability sensitivity estimation with sequential importance sampling

Struct Saf

(2018)

I. Depina et al.

Reliability analysis with Meta model line sampling

Struct Saf

(2016)

M. de Angelis et al.

Advanced line sampling for efficient robust reliability analysis

Struct Saf

(2015)

F. Wang et al.

Subset simulation for non-Gaussian dependent random variables given incomplete probability information

Struct Saf

(2017)

H. Dai et al.

Wavelet density-based adaptive importance sampling method

Struct Saf

(2015)

Z. Lu et al.

Reliability sensitivity method by line sampling

Struct Saf

(2008)

H.J. Pradlwarter et al.

Application of line sampling simulation method to reliability benchmark problems

Struct Saf

(2007)

Z. Wang et al.

Hamiltonian Monte Carlo methods for subset simulation in reliability analysis

Struct Saf

(2019)

Cited by (16)

Dynamic pruning-based Bayesian support vector regression for reliability analysis
2024, Reliability Engineering and System Safety
Adaptive surrogate-based reliability analysis methods have garnered significant attention due to their potential to enhance computational efficiency in accurately estimating failure probability. However, the candidate sample pool remains constant for most surrogate-based reliability methods, and traversing the candidate sample pool one by one will reduce the efficiency of surrogate modeling. More importantly, maintaining a static sample pool may lead to the inclusion of samples that contribute minimally to the construction of the surrogate modeling, thereby impacting the accuracy and efficiency of the reliability analysis. Accordingly, this paper leverages the robust performance of Bayesian support vector regression to propose a dynamic pruning strategy for the candidate sample pool to estimate failure probability efficiently. A dynamic pruning strategy is presented to streamline the process further, iteratively reducing the candidate sample pool. An adaptive learning algorithm is then introduced, integrating the U function and the sparsity of training samples. This is complemented by a formulated convergence condition, contributing to an ideal surrogate model. The proposed approaches showcase superior efficiency and accuracy through illustrations using well-known benchmark problems and complex reliability analysis problems involving small failure probability and high-dimensional limit state function.
Monte Carlo and variance reduction methods for structural reliability analysis: A comprehensive review
2023, Probabilistic Engineering Mechanics
Monte Carlo methods have attracted constant and even increasing attention in structural reliability analysis with a wide variety of developments seamlessly presented over decades. Along the way, a number of specialized reviews and benchmark studies have been provided from time to time, aiming at summarizing and comparing selected few approaches in detail, mainly from an implementation point of view. In contrast, the aim of the present survey is to play a comprehensive role as a methodological guidebook on Monte Carlo simulation and its related, especially variance reduction, techniques through a covering of 444 references in the relevant literature. To achieve this goal, we present an extensive review of formulations and techniques along with insightful summaries of developments of existing numerical methods, ranging from the general formulation, sub-categories and variants, to their combined uses with other simulation techniques and surrogate models, as well as the key advantages and assumptions.
An eccentric radial-based importance sampling method for reliability analysis
2023, Expert Systems with Applications
By introducing a Beta-hypersphere centered at the origin of a standard normal space coordinate system and with reliability index Beta as the radius in the safety domain, traditional Radial-Based Importance Sampling (RBIS) method can improve the efficiency of the reliability analysis by avoiding the evaluation of the Limit State Function (LSF) of the safety sample located in the Beta-hypersphere. However, the evaluation of LSF in most safety samples outside the Beta-hypersphere cannot be avoided in the RBIS. In order to further improve the efficiency of the RBIS, an Eccentric RBIS (ERBIS) method is proposed by introducing an eccentric hypersphere. Compared to the Beta-hypersphere, the eccentric hypersphere is centered along the straight line connecting the origin of the coordinate system and the most probable failure point, and its radius is made larger than that of the Beta-hypersphere, so that it can envelope more safety samples than the Beta-hypersphere and avoid the excess evaluation of LSF. Meanwhile, an optimization model is established for obtaining the eccentric hypersphere locating the safety region and with the largest radius, and two strategies, including the sampling method and sequential decoupling method, are proposed for searching the optimal eccentric hypersphere. Fourteen numerical examples, a ten-bar truss structure example and an aero-engine turbine disk example are presented to demonstrate the advantages of the proposed ERBIS method. The presented examples show that almost 80%-100% LSF evaluations required in the RBIS method could be avoided in the proposed ERBIS method.
An efficient method for accessing structural reliability indexes via power transformation family
2023, Reliability Engineering and System Safety
Evaluating the large reliability index of an implicit and nonlinear performance function (PF) is still a challenging problem in practical engineering. To overcome this problem, an efficient method by using multiple techniques is proposed. A strategy is proposed to select the optimal transformation parameter (OTP) that makes the transformed PF approximately normally distributed. This strategy is established based on the power transformation family including Box-Cox and dual power transformation, in which three formulations, i.e., maximum likelihood estimation, Jarque–Bera test, and absolute skewness minimization criterion are first introduced to estimate the transformation parameters. Further, the selective factors corresponding to the three different formulations can be estimated based on the minimum distance between the pair of the skewness and kurtosis of the transformed PF and that of a normal random variable, where the first four moments of the transformed PF are evaluated from the high-order unscented transformation. Then, the OTP can be chosen as the one with minimum selective factor. Eventually, the reliability index can be obtained based on the first four moments of the transformed PF and the cubic normal distribution. The accuracy and efficiency of the proposed method are demonstrated through four numerical examples with nonnormal, nonlinear and implicit PFs.
Opposition-based learning equilibrium optimizer with Levy flight and evolutionary population dynamics for high-dimensional global optimization problems
2023, Expert Systems with Applications
Citation Excerpt :
Finally, the conclusions and the future work are given in Section 6. High-dimensional global optimization problems involving a large number of design variables are widely existed in the science and engineering applications, such as Burro Creek arch bridge (Meng et al., 2021b), large-scale truss optimization design (Panagant et al., 2021). Because the searching space of design variables is exponentially enlarged with the increasing of dimension, most of metaheuristic algorithms may encounter performance degradation or low convergence in high-dimensional optimization problems.
The equilibrium optimizer (EO) is a recently proposed physics-based metaheuristic algorithm inspired by the dynamic mass balance on a control volume. However, EO may encounter local convergence and low accuracy when solving high-dimensional optimization problems. In this work, we propose the opposition-based learning EO algorithm with the Levy flight and the evolutionary population dynamics for high-dimensional global optimization problems, called EOOBLE. Firstly, the opposition-based learning strategy is embedded into the initialization and updating process of EO. Then, the Levy flight strategy is used to improve the exploration ability of EO in the updating mechanism. Moreover, the strategy of evolutionary population dynamics is used in the proposed algorithm to avoid falling into the local optimum. The performance of the proposed algorithm is tested by 25 benchmark functions with dimensions from 100 to 5000, and is compared with 8 state-of-the-art metaheuristic algorithms. The statistical results indicate that the proposed algorithm has better convergence capacity than the compared algorithms. Besides, the proposed algorithm is also compared with different variants of EO, which outperforms the original EO and variants of EO with one or two operators. Therefore, the EOOBLE is a competitive algorithm in solving high-dimensional global optimization problems. Finally, a high-dimensional engineering design problem demonstrates the effectiveness of the EOOBLE.
An adaptive polynomial skewed-normal transformation model for distribution reconstruction and reliability evaluation with rare events
2022, Mechanical Systems and Signal Processing
Evaluating the failure probabilities with rare events remains a challenging task, especially when dealing with complex limit state functions with high-dimensional random inputs. To handle this problem, an adaptive polynomial skewed-normal transformation (A-PSNT) model is proposed in this paper for evaluating the probability distribution of the limit state function and failure probability. First, the polynomial transformation models on the basis of normal distribution are introduced, where the limitations are also pointed out. Then, the skewed-normal distribution is presented, which serves as the basis of the proposed polynomial transformation model. The PSNT with a given order can be initiated, whereby the unknown coefficients can be determined by using the probability weighted moments (PWMs)-matching technique. Further, a tail error criterion is proposed to select the most appropriate order for the PSNT model, which makes the proposed method an adaptive one. To handle the low- to high-dimensional random inputs, two low-discrepancy sampling methods are employed to generate samples for unbiasedly estimating the PWMs of the limit state function. Once the probability distribution of the limit state function is reconstructed by the proposed A-PSNT model, the failure probability, especially for the rare one, can be determined accordingly. The proposed method is validated through numerical examples involving both static and dynamic, low- to high-dimensional problems, where some classical reliability analysis methods are also adopted for comparisons. The results show that the proposed A-PSNT model not only can effectively reconstruct the probability distribution, but also is able to accurately predict the failure probability with rare events.

View all citing articles on Scopus

View full text

An augmented weighted simulation method for high-dimensional reliability analysis

Highlights

Abstract

Introduction

Section snippets

Monte Carlo simulation

The proposed method

Demonstrative examples and comparative study

Conclusions

Declaration of Competing Interest

Acknowledgments

Struct Saf

Struct Saf

Mech Syst Signal Pr

Struct Saf

Appl Math Model

Struct Saf

Struct Saf

Struct Saf

Struct Saf

Reliab Eng Syst Safe

Reliab Eng Syst Safe

Struct Saf

Mech Syst Signal Pr

Struct Saf

Struct Saf

Struct Saf

Struct Saf

Struct Saf

Reliab Eng Syst Safe

Struct Saf

Struct Saf

Struct Saf

Struct Saf

Struct Saf

Struct Saf

Struct Saf

Struct Saf

Struct Saf

Struct Saf