Implicit Performance Functions Research Articles

AKSE: A novel adaptive Kriging method combining sampling region scheme and error-based stopping criterion for structural reliability analysis

The reliability analysis of complex structures usually involves implicit performance function and expensive-to-evaluate computational models, which pose a great challenge for the estimation of failure probability. In this paper, an adaptive Kriging-based method is proposed for the efficient estimation of failure probability with high accuracy. Starting from a small set of initial design of experiments (DoE), a Kriging model is constructed and iteratively refined by adding judiciously selected sample points to the DoE. To effectively select more sample points for updating the Kriging model, a new learning function is proposed for the selection of representative samples following the idea of the penalty function method in optimization. Besides, the inclusion of the term that measures the distance between the candidate samples and those in DoE controls the density of samples, and thus enables efficient exploration of the regions of interest. To improve the efficiency of the algorithm, this learning function is integrated with a sampling region scheme to filter out sample points in regions with rather low probability density from the candidate sampling pool. Moreover, a convergence criterion based on the error-based stopping criterion is developed to terminate the learning process at an appropriate stage. Hence, the proposed method is referred to as the Adaptive Kriging method combining Sampling region scheme and Error-based stopping criterion (AKSE). To further explore the possible schemes for the improvement of the proposed method, the combinations of AKSE with a dynamic Kriging method and the bootstrap variance are also investigated. For rare failure events, the Monte Carlo simulation (MCS) in AKSE is replaced by the importance sampling (IS) to establish an improved version of AKSE, namely the AKSE-IS. The performance of the proposed algorithm is evaluated through five numerical examples and the results demonstrate the superior performance of the proposed method both in terms of accuracy and efficiency.

Structural Reliability Assessment Based on Enhanced Conjugate Unscented Transformation and Improved Maximum Entropy Method

Estimation of statistical moments remains one of the main topics of stochastic analysis whose accuracy greatly affects reliability analysis results. In this work, conjugate unscented transformation (CUT) methods, which can balance accuracy and efficiency, are introduced for the statistical moment estimation of responses. Because of the drawbacks of existing CUT methods, a family of enhanced conjugate unscented transformation (ECUT) methods, including ECUT-4, ECUT-6, and ECUT-8 methods, is proposed for statistical moment estimation by combining the original CUT, variable transformation, and exact dimension reduction method. Then the probability density function of a performance function is reconstructed by the improved maximum entropy method (IMEM) with the available statistical moments as constraints. To demonstrate the accuracy and effectiveness of the proposed methods, five numerical examples, including linear and nonlinear, low-dimensional and high-dimensional, and explicit and implicit performance functions, are investigated, in which the results obtained from the proposed methods are compared with other existing moment methods and a Monte Carlo simulation (MCS) method. The results of the examples show that the proposed ECUT-6 and ECUT-8 methods have fairly high accuracy and efficiency for structural reliability analysis.

AN ITERATIVE HDMR APPROACH FOR ENGINEERING RELIABILITY ANALYSIS

Engineering reliability analysis has long been an active research area. Surrogate models, or metamodels, are approximate models that can be created to replace implicit performance functions in the probabilistic analysis of engineering systems. Traditional 1st-order or second-order high dimensional model representation (HDMR) methods are shown to construct accurate surrogate models of response functions in an engineering reliability analysis. Although very efficient and easy to implement, 1st-order HDMR models may not be accurate, since the cross-effects of variables are neglected. Second-order HDMR models are more accurate; however they are more complicated to implement. Moreover, they require much more sample points, i.e., finite element (FE) simulations, if FE analyses are employed to compute values of a performance function. In this work, a new probabilistic analysis approach combining iterative HDMR and a first-order reliability method (FORM) is investigated. Once a performance function is replaced by a 1st-order HDMR model, an alternate FORM is applied. In order to include higher-order contributions, additional sample points are generated and HDMR models are updated, before FORM is reapplied. The analysis iteration continues until the reliability index converges. The novelty of the proposed iterative strategy is that it greatly improves the efficiency of the numerical algorithm. As numerical examples, two engineering problems are studied and reliability analyses are performed. Reliability indices are obtained within a few iterations, and they are found to have a good accuracy. The proposed method using iterative HDMR and FORM provides a useful tool for practical engineering applications.

A recursive dimension-reduction method for high-dimensional reliability analysis with rare failure event

A new dimension-reduction method (DRM), called ’subset active subspace method (SASM)’, is proposed to compute small failure probabilities encountered in high-dimensional reliability analysis of engineering systems. The basic idea is to introduce a recursive procedure to improve the efficiency, accuracy and applicability of the conventional active subspace method (ASM). For the reliability problems with a rare event, SASM firstly transfers the original high-dimensional reliability problem into a low-dimensional reliability problem in a proper failure domain. Then, a simplified low-dimensional surrogate model is built in order to improve the result of reliability analysis by increasing significantly the samples with a minimum additional computational effort. The proposed method is verified by three nonlinear numerical examples, including theoretical and industrial, explicit and implicit performance functions. Besides, some other existing methods are also investigated and compared to the proposed method. It is found that the proposed method can keep the trade-off between accuracy and efficiency.

An efficient method by nesting adaptive Kriging into Importance Sampling for failure-probability-based global sensitivity analysis

Failure-probability-based global sensitivity (FP-GS) analysis can measure the effect of the input uncertainty on the failure probability. The state-of-the-art for estimating the FP-GS are less efficient for the rare failure event and the implicit performance function case. Thus, an adaptive Kriging nested Importance Sampling (AK-IS) method is proposed in this work to efficiently estimate the FP-GS. For eliminating the dimensionality dependence in the calculation, an equivalent form of the FP-GS transformed by the Bayes’ formula is employed by the proposed method. Then the AK model is nested into IS for recognizing the failure samples. After all the failure samples are correctly identified from the IS sample pool, the failure samples are transformed into those subjected to the original conditional probability density function (PDF) on the failure domain by the Metropolis–Hastings algorithm, on which the conditional PDF of the input on the failure domain can be estimated for the FP-GS finally. The proposed method highly improves the efficiency of estimating the FP-GS comparing with the state-of-the-art, which is illustrated by the results of several examples in this paper.

A Novel Risk Evaluation Procedure Using a Kriging-Based Surrogate Modeling for Offshore Structures

An innovative reliability estimation procedure for Jacket-type offshore platforms is proposed. The information on risk is extracted using multiple deterministic analyses using advanced mathematical theories resulting in compounding beneficial effects. The platforms are modeled by finite elements. Nonlinearity and major sources of uncertainty are incorporated. The wave loading model is realistically developed to satisfy the underlying physics. It is applied in three-dimension in time domain incorporating the uncertainties in the parameters. Implicit performance functions are expressed explicitly using several advanced factorial schemes and significantly improved response surface method. The efficiency and accuracy of the method are improved using a comprehensively improved Kriging-based surrogate modeling technique. The risk is evaluated for the serviceability and strength limit state functions for a jacket-type offshore platform using about two hundred nonlinear finite element analyses. The procedure was verified using simulation techniques. The concept can be considered an alternative to the conventional random vibration techniques and the standard Monte Carlo simulation procedure.

Research on Bridge Structure Reliability Evaluation due to Vessels Collison Based on a Statistical Moment Method

Damage to bridge structures caused by vessel collision is a risk for bridges crossing water traffic routes. Therefore, safety around vessel collision of existing and planned bridges is one of the key technical problems that must be solved by engineering technicians and bridge managers. In the evaluation of the reliability of the bridge structure, the two aspects of vessel‐bridge collision force and structural resistance need to be considered. As there are many influencing parameters, the performance function is difficult to express by explicit function. This paper combines the moment method theory of structural reliability with finite element analysis and proposes a statistical moment method based on finite element analysis for the calculation of vessel‐bridge collision reliability, which solves the structural reliability problem with a nonlinear implicit performance function. According to the probability model based on current velocity, vessel velocity, and vessel collision tonnage, the estimate points in the standard normal space are converted into estimate points in the original state space through the Rosenblatt reverse transform. According to the estimate points in the original state space and the simplified dynamic load model of vessel‐bridge collision, the sample time‐history curve of random vessel‐bridge collision force is generated, the dynamic response of the bridge structure and the structural resistance of the bridge are calculated by establishing a finite element model, and the failure probability and reliability index of the bridge structure is calculated according to the fourth‐moment method. The statistical moment based on the finite element analysis is based on the finite element analysis and the moment method theory of structural reliability. The statistical moment of the limited performance function is calculated through a quite small amount of confirmatory finite element analysis, and the structural reliability index and failure probability are obtained. The method can be widely used in existing finite element analysis programs, greatly reducing the number of finite element analyses needed and improving the efficiency of structural reliability analysis.

An efficient variance-based global sensitivity index based on C-DRM and Taylor expansion

Sensitivity analysis plays an important role in quantifying the impact of input uncertainty on model response uncertainty. Through sensitivity analysis, we can grasp the crucial parameters and gain a deeper understanding of the model behavior. In this paper, an effective analytical solution for solving the variance-based global sensitivity index is proposed. Firstly, the original performance function is approximated as the sum of a series of univariate functions using the conventional dimensional reduction method (C-DRM). Then, the Taylor series expansion is used to expand the univariate function as unary linear function and unary quadratic functions. Finally, the analytical solutions of the variance-based global sensitivity index based on unary linear function and unary quadratic functions are derived respectively. The computational cost of the proposed method is completely concentrated on the calculation of the partial derivative information of the performance function with respect to each variable. As long as the partial derivative information is obtained, the variance-based global sensitivity index can be obtained directly by the proposed method without any additional computational cost. For simple explicit performance functions, the derivative information can be directly derived analytically. For complex explicit or implicit performance functions, the derivative information can be estimated by some simple numerical difference methods. Five examples are studied to investigate the accuracy and efficiency of the proposed method.

Kernel principal component analysis-based Gaussian process regression modelling for high-dimensional reliability analysis

An efficient reliability method is presented to address the challenge inherent in the high-dimensional reliability analysis. The critical contribution is an elegant implementation of combining the kernel principal component analysis (KPCA)-based nonlinear dimension reduction and the Gaussian process regression (GPR) surrogate model by introducing a nonintrusive, joint-training scheme. This treatment leads to an optimal KPCA-based subspace in which an accurate low-dimensional GPR model, denoted as the KPCA-GPR model, can be readily achieved. Then, the KPCA-GPR model is combined with the active learning (AL) -based sampling strategy and the Monte Carlo simulation (MCS). In this regard, the newly-added sample at each iteration can be employed simultaneously to both update the estimated KPCA-based subspace and refine the GPR model built on that subspace, which alleviates the 'curse of dimensionality' to some extent and improves progressively the failure probability estimation provided by the low-dimensional GPR model. In order to demonstrate the effectiveness of the proposed method, two numerical examples are investigated, involving both the analysis of high-dimensional, nonlinear, explicit performance functions and the static/dynamic reliability analysis of a truss structure with implicit performance functions. Numerical results indicate that the proposed method is of both accuracy and efficiency for high-dimensional reliability problems.

Structural reliability analysis via dimension reduction, adaptive sampling, and Monte Carlo simulation

An improved reliability method, i.e., AL-AS-GPR-MCS, is developed taking advantage of active subspace (AS)-based dimension reduction technique, Gaussian process regression (GPR) surrogate model, active learning (AL)-based sampling strategy, and Monte Carlo simulation (MCS). In this method, the AL sampling strategy and the AS-based dimension reduction are incorporated into the GPR construction, allowing that the newly added sample in each iteration can be employed simultaneously to update the identified AS of original high-dimensional space and to refine the low-dimensional GPR model over the discovered AS. The failure probability is then estimated by the MCS using the constructed GPR model. In order to verify the versatility of the proposed method, four numerical examples are investigated, involving reliability analyses of both explicit performance functions and shear frame structures with implicit performance functions. It is revealed that satisfactory accuracy enhancement and computational cost savings are achieved by the AL-AS-GPR-MCS, since it is capable of discovering the latent AS of the original high-dimensional space which alleviates the curse of dimensionality, and refining the low-dimensional GPR construction within the discovered AS by the AL sampling strategy. Therefore, the proposed method is a powerful and promising tool for dealing with structural reliability problems with strong nonlinearities and high stochastic dimensions.

Active sparse polynomial chaos expansion for system reliability analysis

Surrogate model techniques have been widely used for structural systems with expensively evaluated simulations. However, their application to system reliability problems meets the challenge since approximating multiple implicit performance functions is required to estimate small failure probabilities. In this paper, we demonstrate the potential of a newly active learning method based on polynomial chaos expansion (PCE) to alleviate the problem. The method employs Bayesian compressed sensing (BCS) technique which incorporates a parameterized prior to establish the PCE with a sparse structure. Then, a new composite learning function is developed to update sparse PCEs adaptively by iteratively selecting new training points near the system failure surface until the preset target accuracy is satisfied. This criterion can identify important failure modes and thus guarantee that computation cost will be spent only on areas with a significant probability content. Three examples including a parallel and two series systems are used to demonstrate the effectiveness of the proposed method.

An efficient algorithm for estimating time-dependent failure credibility by embedding double-loop adaptive Kriging in dichotomy searching

Time-dependent failure credibility (TDFC) in a given service time of interest can reasonably quantify the safety of the structure with fuzzy inputs. However, the estimation of TDFC requires high computational cost because it involves multi-layer optimization, especially for problems with the implicit performance functions in engineering. In order to improve the efficiency of estimating TDFC under satisfied precision, this paper proposes an efficient dichotomy searching algorithm (DSA) combined with double-loop adaptive Kriging (D-AK) model (shorten as D-AK-DSA). In the proposed D-AK-DSA, a double-loop adaptive Kriging model is constructed to provide the directions for the dichotomy searching TDFC. In the inner loop, the AK model is constructed to surrogate the relation of the performance function and time variable at the given fuzzy input realization so as to obtain the minimum performance function with respect to the time for the outer loop, and in the outer loop, the AK model is constructed to identify the sign of the upper/lower boundary of the minimum performance function with the fuzzy inputs at a given membership interval so as to provide the dichotomy searching direction for TDFC. By the proposed dichotomy searching strategy, accurately solving the values of the upper/lower boundaries in evaluating TDFC is innovatively replaced by identifying the signs of them, which greatly reduces the difficulty of constructing the outer AK model. At the same time, the proposed method enhances the convergence of the outer AK surrogate model by establishing a reasonably improved learning function. Three examples verify the accuracy and efficiency of the proposed D-AK-DSA method.

An adaptive trivariate dimension-reduction method for statistical moments assessment and reliability analysis

An adaptive trivariate dimension-reduction method is proposed for statistical moments evaluation and reliability analysis in this paper. First, the raw moments of the performance function can be estimated by means of the trivariate dimension-reduction method, where the trivariate, bivariate and univariate Gaussian-weighted integrals are involved. Since the trivariate and bivariate integrals control the efficiency and accuracy, delineating the existence of bivariate and trivariate cross terms is performed, which could significantly reduce the numbers of trivariate and bivariate integrals to be evaluated. When the cross terms exist, the trivariate and bivariate integrals are numerically evaluated directly by the high-order unscented transformation, where the involved free parameters are provided. When the cross terms don’t exist, the trivariate and bivariate integrals can be further decomposed to be the lower-dimensional integrals, where the high-order unscented transformation is again adopted for numerical integrations. In that regard, the first-four central moments can be computed accordingly and the performance function’s probability density function can be reconstructed by fitting the shifted generalized lognormal distribution model based on the first-four central moments. Then, the failure probability can be computed by a one-dimensional integral over the performance function’s probability density function in the failure domain. Three numerical examples, including both the explicit and implicit performance functions, are investigated, to demonstrate the efficacy of the proposed method for both the statistical moments assessment and reliability analysis.

Structural reliability assessment by salp swarm algorithm–based FORM

AbstractThe first‐order reliability method (FORM) is well recognized as an efficient approach for reliability analysis. Rooted in considering the reliability problem as a constrained optimization of a function, the traditional FORM makes use of gradient‐based optimization techniques to solve it. However, the gradient‐based optimization techniques may result in local convergence or even divergence for the highly nonlinear or high‐dimensional performance function. In this paper, a hybrid method combining the Salp Swarm Algorithm (SSA) and FORM is presented. In the proposed method, a Lagrangian objective function is constructed by the exterior penalty function method to facilitate meta‐heuristic optimization strategies. Then, SSA with strong global optimization ability for highly nonlinear and high‐dimensional problems is utilized to solve the Lagrangian objective function. In this regard, the proposed SSA‐FORM is able to overcome the limitations of FORM including local convergence and divergence. Finally, the accuracy and efficiency of the proposed SSA‐FORM are compared with two gradient‐based FORMs and several heuristic‐based FORMs through eight numerical examples. The results show that the proposed SSA‐FORM can be generally applied for reliability analysis involving low‐dimensional, high‐dimensional, and implicit performance functions.

AK-PDF: An active learning method combining kriging and probability density function for efficient reliability analysis

An important challenge in structural reliability is to reduce the number of calls to evaluate the performance function, especially the complex implicit performance functions. To reduce the computational burden and improve the reliability analysis efficiency, a new active learning method is developed to consider the probability density function of samples based on the learning function U in an active learning reliability method that combines the kriging and Monte Carlo simulation. In the proposed method, the proposed active learning function contains two parts: part A is based on function U, and part B is based on the probability density function and function U. By changing the weights of parts A and B, the sample points close the limit-state function, and those in the region with a higher probability density function have more weight to be selected compared to the others. Subsequently, the kriging model can be constructed more effectively. The proposed method avoids a large number of time-consuming function evaluations, and the recommended weight is also reported. The performance of the proposed method is evaluated through three numerical examples and one engineering example. The results demonstrate the efficiency and accuracy of the proposed method.

An adaptive multiple-Kriging-surrogate method for time-dependent reliability analysis

• Multiple-Kriging-surrogate is simultaneously constructed for time-dependent structure. • Adaptive strategy is established to identify surrogates and new training samples. • Man-made subjectivity for regression trend is avoided with adaptive selecting procedure. • Time-dependent failure probability can be accurately estimated by the most accurate local model. For accurately and efficiently estimating the time-dependent failure probability (TDFP) of the structure, a novel adaptive multiple-Kriging-surrogate method is proposed. In the proposed method, the multiple Kriging models with different regression trends (i.e., constant, linear and quadratic) are simultaneously constructed with the highest accuracy, on which the TDFP can be obtained. The multiple regression trends are adaptively selected based on the size of sample base, the maximum differences of multiple models and the global accuracy of multiple models. After that, the most suitable multiple regression trends are identified. The proposed method can avoid man-made subjectivity for regression trend in general Kriging surrogate method. Furthermore, better accuracy and efficiency will be obtained by the proposed multiple surrogates than just using a fixed regression model for some engineering applications. Five examples involving four applications with explicit performance function and one tone arch bridge under hurricane load example with implicit performance function are introduced to illustrate the effectiveness of the proposed method for estimating TDFP.

Reliability of Jack-up against Punch-through using Failure State Intelligent Recognition Technique

The preload operation of jack-up in complex multi-layered foundation requires enhanced understanding of its behaviour in punchthrough accident and suitable safety analysis tools for the assessment of their reliability for a particular site. In this study, reliability analysis model of jack-up against punch-through is established considering structural uncertainty. In order to identify the failure state, an improved reliability solution method has been developed based on Sparse Auto-Encoder (SAE) deep learning network model. Sparse self-coding algorithm is used to the training of the deep network, and Softmax regression model is established to solve the identification and classification problem of the output layer. The first application of the technique was the study of HYSY 941 jack-up platform. More specifically, numerical calculations of structure ultimate bearing capacity have been undertaken, and the influence of model parameters on the prediction accuracy of the failure state is discussed. The results show that implicit performance function can be constructed accurately using SAE-MC method by reflecting the relationship between different critical safety state and structural vulnerability. Compared with traditional BP neural network, deep learning network has higher prediction accuracy to failure probability. The dynamic risk grade in the process of preload operation can be determined quantitatively using the reliability analysis method mentioned in this paper.

Fast integration algorithms for time-dependent structural reliability analysis considering correlated random variables

During the service life of civil infrastructures, their performance, functionality, and serviceability are essentially time-dependent reliability problems as both structural resistance and load effect are related with time. With respect to the practical engineering, the performance function is usually complicated, multi-dimensional and implicit and involves both correlated and non-correlated random variables; thus the computational process of time-dependent reliability analysis is relatively challenging. To address this issue, the time-dependent failure probability of structures with complicated, multi-dimensional and implicit performance functions involving correlated random variables is mathematically formulated in this paper. A novel and efficient approach is then proposed to compute the time-dependent failure probability. In the proposed method, the integral of time-dependent failure probability with respect to time domain and random-variate space are estimated by Gauss-Legendre quadrature and point-estimate method based on bivariate dimension-reduction integration, respectively. Accordingly, the time-dependent probability of failure within a given period is decomposed into a series of probabilities of failure with time-dependent random variables at specified time points conditioned on the estimating points of the time-independent random variables, which can be evaluated from state-of-the-art techniques for reliability assessment. The efficiency and accuracy of the proposed method are demonstrated by three numerical examples with either explicit or implicit performance functions involving correlated random variables, in which Monte Carlo simulations are employed for comparison.

A New Sequential Surrogate Method for Reliability Analysis and its Applications in Engineering

In reliability analysis for the practical engineering problems with the time-consuming model, it has become an important challenge that how to obtain accurate reliability assessment with a minimum number of calls. In order to reduce the computational cost, this paper develops a new sequential surrogate method combining adaptive kriging and Markov chain Monte Carlo simulation with a novel learning strategy for reliability analysis. The proposed method is named AK-MCMC, which takes full advantage of the classification feature of reliability analysis based on the surrogate models, and it can efficiently approximate the classification boundary of the performance function. First, the learning strategy is developed to sequentially pick out the informative samples for updating the experimental design samples. Then, a new stopping criterion is adopted to guarantee the classification accuracy of the constructed kriging model. In this way, the proposed method skillfully makes reliability evaluation independent of an adaptive iterative process, which greatly improves the efficiency of model refinement. Finally, the proposed method is applied to several examples, which contain small failure probability problem, non-linearity problem, and engineering problem with an implicit performance function. In particular, the efficiency of the proposed AK-MCMC method is proved for the problems with small failure probability.

A Concise Transformation Combined With Adaptive Kriging Model for Efficiently Estimating Regional Sensitivity on Failure Probability

Regional sensitivity on failure probability (RS-FP) can quantify the effect of the input region of interest (IRoI) on the failure probability and provide useful information for reliability based design optimization. Traditional methods for estimating RS-FP require huge computational cost, especially for implicit performance function and rare failure event in engineering applications. In order to alleviate these issues, Adaptive Kriging (AK) model based methods involving AK model inserted Monte Carlo Simulation (AK-in-MCS) and AK model inserted Importance Sampling (AK-in-IS) are carried out for efficiently estimating the RS-FP. Furthermore, the RS-FP can be estimated by the conditional probability of IRoI on failure domain by employing the probability product principle. Based on this transformation, AK-in-MCS and AK-in-IS can be much conveniently organized for identifying the conditional probability just as a byproduct of estimating the failure probability. Since the original complicated estimation process has been alleviated by the concise transformation, and the AK model is efficiently trained to identify the failure samples from the sample pool generated by MCS or IS, the computational cost of estimating RS-FP is greatly reduced. At the same time, the detailed geometric interpretation for the “contribution to failure probability (CFP) plot” is discussed. Several examples containing numerical and engineering examples are introduced to demonstrate the accuracy and efficiency of the proposed methods.