Efficient Structural Reliability Analysis Research Articles

Adaptive Kriging-based method with learning function allocation scheme and hybrid convergence criterion for efficient structural reliability analysis

Adaptive Kriging-based method with learning function allocation scheme and hybrid convergence criterion for efficient structural reliability analysis

Active Kriging-based conjugate first-order reliability method for highly efficient structural reliability analysis using resample strategy

Efficient structural reliability analysis method is crucial to solving reliability analysis of complex structural problems. High-computational cost and low-failure probability problems greatly limit the efficiency in structural reliability analysis problems, causing the safety and reliability of the structure to be questioned. In this work, a highly efficient structural reliability analysis method coupling active Kriging algorithm with conjugate first order reliability method (AK-CFORM) is proposed. Specifically, the resample strategy is considered to reduce the number of samples evaluated in each active learning process; the uniform sampling is used to better balance global and local optimal problems; the conjugate map is used to improve the robustness of analytical first order reliability method; and the approximate numerical differential formula is proposed to solve the problems of non-convergence when solving the gradient of the Kriging surrogate model. Finally, three numerical cases and four engineering cases are used to illustrate the effectiveness and robustness of the proposed method. The results show that the proposed AK-CFORM has greater advantages in the number of calling system response and surrogate model with robust and accurate performance.

A novel adaptive Kriging method combining Hessian matrix and an efficient F-scoreβ-based stopping criterion for structural reliability analysis

Reliability analysis of complex structures or highly nonlinear performance functions generate an enormous computational burden and affects computational efficiency. It is crucial to accurately assess structural reliability. However, current methods do not meet the accuracy and efficiency requirements as the complexity and nonlinearity of the model gradually increase. To address these problems, a novel active learning Kriging method combined with a Hessian matrix was proposed for efficient structural reliability analysis. The predicted mean of the Kriging model was used as an element of the Hessian matrix to describe the degree of nonlinearity of the model. Based on the fact that the iterative process is greatly influenced dramatically by the sample points in proximity to the limit state surface (LSS), a candidate sample pool (CSP) is generated, where the sample points located in the vicinity of the LSS with a larger curvature are chosen to add to the design of experiments (DoE). A stopping criterion about F-scoreβ based on the number of the misclassified points is proposed to judge the model fitting effect, which can be used to effectively identify the LSS. Two mathematical cases and two finite element cases were used to illustrate the high accuracy and efficiency of the method, which a novel manner of the nonlinear reliability calculation is provided.

Efficient estimation of structural probability of failure with adaptive kriging and multi-layered hyperball-based importance sampling

ABSTRACT The need for more resilient infrastructures entails an accurate and efficient structural reliability analysis. In recent years, Kriging-based meta-modeling has constantly been adopted to reduce the computational cost in estimating the probability of failure. Active learning models focus on most informative samples to increase computational efficiency. Nevertheless, the efficiency gained from the models may diminish when dealing with a rare event with a relatively small probability of failure. To resolve the difficulty, this study proposes a new method, Adaptive Kriging with Multi-layered Hyperball-based Importance Sampling (AK-MHIS), to estimate the structural probability of failure. AK-MHIS uses an adaptive multi-layered hyperball as its candidate sample pool with a new sample filtering mechanism and a more robust stopping condition for the active-learning phase. The performance of AK-MHIS method is validated in benchmark cases. The verification results confirm the superior performance of AK-MHIS to previous methods.

An enhanced uniform simulation approach coupled with SVR for efficient structural reliability analysis

For structural reliability analysis with low failure probability, traditional simulation methods are time consuming approaches, which is a great challenge for estimating the failure probability. In this paper, a novel hybrid enhanced sampling method named as enhanced uniform importance sampling coupled with support vector regression (EUIS-SVR) is proposed. In EUIS-SVR, the samples are simulated by inter-cell with a lower assumed failure interval compared to traditional uniform importance sampling method. Then by approximating a general failure scaling formula, a quasi-linear interval is selected for estimating the failure probability using support vector regression model. Six numerical and engineering cases are studied for showing the abilities for accuracy and efficiency of proposed method compared to simulation reliability methods. The proposed EUIS-SVR method was found more efficient than existing simulation methods.

An efficient structural reliability analysis method with active learning Kriging-assisted robust adaptive importance sampling

Due to deterioration effects and a variety of manmade or natural hazards, structures and infrastructure systems can face serious challenges for the functionality and safety during the service life. Thus, it is crucial to accurately assess the structural reliability and perform reasonable management actions. Simulation methods can yield more accurate reliability analysis results compared with approximation methods. However, as structures and infrastructure systems become complex, evaluating the engineering models can be very time-consuming. To address the gap, this paper proposes an active learning Kriging-assisted method for efficient structural reliability analysis. The proposed method consists of two stages. In the first stage, Kriging-based random walk, i.e., MCMC, is performed to explore the failure domain. Then, in the second stage, a Gaussian mixture distribution is established as the quasi-optimal sampling distribution for importance sampling. With the active learning framework that enlarges the training database in the MCMC and importance sampling process, the proposed method can achieve robust exploration of the whole failure domain, and also accurate failure probability estimates. Two classical numerical examples first validate the performance and advantages of the proposed method for handling problems with multiple failure domains and rare events. Then, a real-world application demonstrates the feasibility of applying the proposed method for quantitatively assessing the failure risk for prestressed-concrete continuous rigid-frame bridges during the service life.

Efficient structural reliability analysis via a weak-intrusive stochastic finite element method

This paper presents a novel methodology for structural reliability analysis by means of the stochastic finite element method (SFEM). The key issue of structural reliability analysis is to determine the limit state function and corresponding multidimensional integral that are usually related to the structural stochastic displacement and/or its derivative, e.g., the stress and strain. In this paper, a novel weak-intrusive SFEM is first used to calculate structural stochastic displacements of all spatial positions. In this method, the stochastic displacement is decoupled into a combination of a series of deterministic displacements with random variable coefficients. An iterative algorithm is then given to solve the deterministic displacements and the corresponding random variables. Based on the stochastic displacement obtained by the SFEM, the limit state function described by the stochastic displacement (and/or its derivative) and the corresponding multidimensional integral encountered in reliability analysis can be calculated in a straightforward way. Failure probabilities of all spatial positions can be obtained at once since the stochastic displacements of all spatial points have been known by using the proposed SFEM. Furthermore, the proposed method can be applied to high-dimensional stochastic problems without any modification. One of the most challenging problems encountered in high-dimensional reliability analysis, known as the curse of dimensionality, can be circumvented with great success. Three numerical examples, including low- and high-dimensional reliability analysis, are given to demonstrate the good accuracy and the high efficiency of the proposed method.

Stochastic configuration network for structural reliability analysis

Structural reliability analysis is of critical importance for ensuring the safety of many engineering systems. This paper proposes an efficient structural reliability analysis method based on stochastic configuration networks (SCNs), which is constructed by learning the real performance function of the structure. In this article, two kinds of innovative surrogate models of performance function are constructed based on the trained SCNs. To improve the approximation accuracy and efficiency of SCNs for complex high-dimensional performance functions, the improved high-dimensional model representation is applied to decompose the multivariate function into a combination of several low-dimensional functions. An innovative SCN-based learning strategy is proposed to simultaneously approximate all the low-dimensional component functions. By performing Monte Carlo Simulation on the established SCN-based surrogate models, we can successfully evaluate the failure probability of many complex multi-dimensional engineering structures. The proposed SCN-based reliability analysis method provides a powerful tool for estimating the failure probability of complex multi-dimensional engineering structures. Effectiveness of the proposed method in terms of accuracy and efficiency as well as stability is demonstrated through several numerical experiments.

Hybrid enhanced Monte Carlo simulation coupled with advanced machine learning approach for accurate and efficient structural reliability analysis

The accurate estimations of the failure probability with low-computational burden play a vital role in structural reliability analyses. Due to high-calculation cost and time-consuming Monte Carlo simulation (MCS), this paper focused on developing a novel enhanced MCS approach with an advanced machine learning method, namely hybrid enhanced MCS (HEMCS), for achieving accurate approximation of failure probability with high-efficiency computations. The failure probability is approximated by a probabilistic model using a scalar factor less than one which is multiplied on the capacity term of performance function. An adaptive input for scalar factor is proposed by the coefficient of variation of failure probability for active region in the training process of artificial neural network (ANN) with multilayer back-propagation algorithm. Four analytical optimization training approaches, active region and number of hidden nodes are discussed for accurate approximation of ANN models. The results of the HEMCS are compared with several analytical reliability methods for numerous engineering problems. Laminated composite plate and turbine bladed disk are selected to illustrate the capability of HEMCS for approximation of failure probability. The proposed method provides higher flexibility for the prediction of failure probability when compared to the enhanced MCS which also offers significantly more accurate and computationally efficient results for high-nonlinear problems.

Efficient structural reliability analysis based on adaptive Bayesian support vector regression

To reduce the computational burden for structural reliability analysis involving complex numerical models, many adaptive algorithms based on surrogate models have been developed. Among the various surrogate models, the support vector machine for regression (SVR) which is derived from statistical learning theory has demonstrated superior performance to handle nonlinear problems and to avoid overfitting with excellent generalization. Therefore, to take the advantage of the desirable features of SVR, an Adaptive algorithm based on the Bayesian SVR model (ABSVR) is proposed in this study. In ABSVR, a new learning function is devised for the effective selection of informative sample points following the concept of the penalty function method in optimization. To improve the uniformity of sample points in the design of experiments (DoE), a distance constraint term is added to the learning function. Besides, an adaptive sampling region scheme is employed to filter out samples with weak probability density to further enhance the efficiency of the proposed algorithm. Moreover, a hybrid stopping criterion based on the error-based stopping criterion using the bootstrap confidence estimation is developed to terminate the active learning process to ensure that the learning algorithm stops at an appropriate stage. The proposed ABSVR is easy to implement since no embedded optimization algorithm nor iso-probabilistic transformation is required. The performance of ABSVR is evaluated using six numerical examples featuring different complexity, and the results demonstrate the superior performance of ABSVR for structural reliability analysis in terms of accuracy and efficiency.

An improved active learning method combing with the weight information entropy and Monte Carlo simulation of efficient structural reliability analysis

A significant challenge of surrogate model-based structural reliability analysis (SRA) is to construct an accurate approximated model of the nonlinear limit state function (LSF) with high order and high dimension effectively. As one of the sequential update-strategies of design of experiment (DoE), the active learning method is more attractive in recent years due to greatly reduces the burden of reliability analysis. Although the active learning method based on information entropy learning function H and the line simulation (AK-LS) is a powerful tool of SRA, the computational burden from the iterative algorithm is still large during the learning process. In this research, an improved learning criterion, named the weight information entropy function (WH), is developed to update the DoE of Kriging-based reliability analysis. The WH learning function consists of the information entropy function and an adaptive weight function (W). Locations in the variable space and probability densities of the samples are taken accounted into the WH learning function, which is the most important difference from the H learning function. The samples that are closer to the LSF and has a greater probability density can be preferentially selected into the DoE comparing to others by changing the weight of information entropy during the learning process. The WH learning function can efficiently match the limit state function in an important domain rather than the entire variable space. Consequently, the approximated model of LSF via Kriging interpolation can be constructed more effectively. The new active learning method is developed based on Kriging model, in which WH learning function and Monte Carlo simulation (MCS) are employed. Finally, several engineering examples with high non-linearity are analyzed. Results shown that the new method are very efficient when dealing with intractable problems of SRA.

Distribution-free polynomial chaos expansion surrogate models for efficient structural reliability analysis

In complex stochastic high-dimensional reliability studies, polynomial chaos expansion (PCE) has been widely used to build surrogate models in lieu of prohibitively expensive Monte Carlo simulation (MCS). PCE relies on parametric distributions for associated variables and appropriate basis functions. However, incomplete or imperfect information on the stochastic variables can limit its use; accepted parametric forms for variable distributions, for instance, may not be justified when variables display multimodal character or mixed discrete-continuous support. Also, the dependency structure among the random variables may be complex, which can make probabilistic mapping or transformation to independent variables needed for PCE cumbersome. Nonlinearities in such transformations can affect the accuracy of PCE surrogate models and lead to slower convergence relative to “truth” system computations of desired QoIs (quantities of interest). To address these challenges, a distribution-free PCE approach is proposed. We compute joint raw moments of underlying random input variables for Gram-Schmidt orthogonalization in developing surrogate models. Using illustrative examples, we demonstrate the proposed approach as an efficient and accurate surrogate model-building alternative to traditional PCE.

Efficient structural reliability analysis by using a PGD model in an adaptive importance sampling schema

One of the most important goals in civil engineering is to guarantee the safety of the construction. Standards prescribe a required failure probability in the order of 10^{-4} to 10^{-6}. Generally, it is not possible to compute the failure probability analytically. Therefore, many approximation methods have been developed to estimate the failure probability. Nevertheless, these methods still require a large number of evaluations of the investigated structure, usually finite element (FE) simulations, making full probabilistic design studies not feasible for relevant applications. The aim of this paper is to increase the efficiency of structural reliability analysis by means of reduced order models. The developed method paves the way for using full probabilistic approaches in industrial applications. In the proposed PGD reliability analysis, the solution of the structural computation is directly obtained from evaluating the PGD solution for a specific parameter set without computing a full FE simulation. Additionally, an adaptive importance sampling scheme is used to minimize the total number of required samples. The accuracy of the failure probability depends on the accuracy of the PGD model (mainly influenced on mesh discretization and mode truncation) as well as the number of samples in the sampling algorithm. Therefore, a general iterative PGD reliability procedure is developed to automatically verify the accuracy of the computed failure probability. It is based on a goal-oriented refinement of the PGD model around the adaptively approximated design point. The methodology is applied and evaluated for 1D and 2D examples. The computational savings compared to the method based on a FE model is shown and the influence of the accuracy of the PGD model on the failure probability is studied.

Adaptive scaled unscented transformation for highly efficient structural reliability analysis by maximum entropy method

The approximation of the probability density function (PDF) of the performance function, especially for the tail distribution, is of paramount importance in structural reliability analysis. In this paper, a new method is proposed to derive the PDF of the performance function with accuracy and high efficiency. The derivation is based on the maximum entropy method(MEM), where the fractional moments are adopted as constraints. Since the MEM dose not involve deterministic structural analysis, the efficiency and accuracy of the proposed method is dependent on the evaluation of fractional moments. In this regard, an adaptive scaled unscented transformation (ASUT) is developed to obtain the fractional moments with only a few of sample evaluations. The proposed ASUT is applicable to problems with correlated/uncorrelated random variables. Besides, it can circumvent the so-called “curse of dimensionality” to some extent. Thus, the proposed method could be taken as a general tool for highly efficient structural reliability analysis. Numerical examples involving explicit and implicit performance functions are used to illustrate the implementation of the proposed method, which shows the great efficacy of the proposed method, particularly for a high-dimensional problem. The problems to be further investigated are also pointed out.

A new bivariate dimension reduction method for efficient structural reliability analysis

This paper presents a new bivariate dimension reduction method (BDRM) for statistical moments evaluation and structural reliability analysis with accuracy and efficiency. A high-order unscented transformation (HUT) is introduced to evaluate the two-dimensional integrals involved in BDRM, and the free parameter involved in HUT is suggested. In this regard, the proposed BDRM can be formulated accordingly for statistical moments assessment. Then, the performance function’s probability density function (PDF) is reconstructed by the shifted generalized lognormal distribution (SGLD) with the available statistical moments as constraints. Thus, the failure probability can be straightforwardly obtained by a simple integral over the PDF. The proposed method is verified by five numerical examples, including linear and non-linear, explicit and implicit performance functions. Besides, some other existing methods are also employed to demonstrate the advantages of the proposed method. It is found that the proposed method can keep the trade-off of accuracy and efficiency for structural reliability analysis.

A cubature collocation based sparse polynomial chaos expansion for efficient structural reliability analysis

Polynomial chaos expansion (PCE) is widely used to build a surrogate meta-model of the performance function for structural reliability analysis. The number of terms to be determined in PCE grows exponentially with the number of input random variables, which makes the computational effort intractable in practices. Although several sparse PCEs have been developed, a large number of deterministic model evaluations may be still required to achieve a satisfactory accuracy since equal-weighted collocation samples are used. To address such problems, this paper proposes a cubature collocation based sparse PCE for efficient structural reliability analysis. An iterative scheme is actually involved in the proposed method, which automatically selects the significant terms in PCE contributing to the variance of the performance function. The cubature formula not only generates unequal-weighted collocation samples, which has much faster convergent rate, but also provides the target variance of the performance function to terminate the iterative process. In this regard, a weighted regression method is employed in each step to determine the coefficients of PCE. As a consequence, a rather small number of terms in PCE are retained. Since the number of cubature collocation points is relatively small, the construction of a sparse PCE is quite efficient. Several numerical examples are investigated to validate the proposed method for structural reliability analysis. The results show the effectiveness of the proposed method for different reliability problems.

A new Kriging-based DoE strategy and its application to structural reliability analysis

As the numerical model of engineering structure becomes more and more complicated and time consuming, efficient structural reliability analysis is badly in need. To reduce the number of calls to th...

Enriched FR conjugate search directions for robust and efficient structural reliability analysis

This paper proposes three modified conjugate Fletcher- Reeves (FR) algorithms including improved FR (IFR), the spectral-variant FR (SVFR) and modified FR (MFR) for first-order reliability method (FORM) for nonlinear reliability problems. An adaptive finite-step length is proposed to improve the efficiency and robustness of FORM formula using both steepest descent and conjugate search directions. These conjugate nonlinear maps for computing the search direction of FORM are adjusted based on sufficient decent condition to control instabilities of FORM formula. The efficiency and robustness of three proposed conjugate methods are compared with conjugate Hasofer and Lind-Rackwitz and Fiessler (CHL-RF), limited FR (LFR), HL-RF and finite-step length (FSL) through five nonlinear limit state functions. Numerical experiments illustrated that the modified FR versions of conjugate search direction are robust methods for highly nonlinear performance functions and the IFR is more efficient than the other reliability algorithms.

Efficient reliability analysis of structures with the rotational quasi-symmetric point- and the maximum entropy methods

This paper presents a new method for efficient structural reliability analysis. In this method, a rotational quasi-symmetric point method (RQ-SPM) is proposed for evaluating the fractional moments of the performance function. Then, the derivation of the performance function’s probability density function (PDF) is carried out based on the maximum entropy method in which constraints are specified in terms of fractional moments. In this regard, the probability of failure can be obtained by a simple integral over the performance function’s PDF. Six examples, including a finite element-based reliability analysis and a dynamic system with strong nonlinearity, are used to illustrate the efficacy of the proposed method. All the computed results are compared with those by Monte Carlo simulation (MCS). It is found that the proposed method can provide very accurate results with low computational effort.

A hybrid relaxed first-order reliability method for efficient structural reliability analysis

The Hasofer-Lind and Rakwitz-Fiessler (HL-RF) algorithm is widely used for structural reliability analysis in first-order moment method (FORM). However, it meets non-convergence problem including generating periodic and chaotic solutions for highly nonlinear limit state function. In this paper, relaxed HL-RF (RHL-RF) is developed based on a relaxed factor, which is dynamically computed by the second-order interpolation between zero and one. A hybrid relaxed HL-RF (HRHL-RF) method is proposed, in which the HL-RF and RHL-RF are adaptively implemented by using an angle criterion to improve the robustness and efficiency of FORM formula. The angle condition is simply calculated based on the results from the new and previous points. Finally, the performances in terms of robustness and efficiency of the HRHL-RF are compared with several existing FORM methods through five mathematical and structural examples. The results indicate that HRHL-RF method is more robust than the HL-RF and more efficient than other existing methods.