An active learning Kriging model combined with directional importance sampling method for efficient reliability analysis

In this paper, the active learning Kriging model (ALK), which has been studied extensively in recent years, has been expanded by combining with the directional importance sampling (DIS) method. The directional sampling method can reduce the dimensionality of the variable space by random sampling or interpolation in the direction of vector diameter, which can improve the efficiency of reliability analysis. It is especially suitable for the surfaces whose limit state is spherical or near-spherical. By introducing the control coefficient and constructing the directional importance sampling density function, the sampling efficiency can be further improved in the design point domain. A novel reliability analysis method called ALK-DIS method is proposed. The greatest advantage of the proposed method is its ability on great computational efficiency and dealing with small failure probability problem In addition, due to the excellent performance of directional sampling method in dealing with multi-failure model reliability problems, the ALK-DIS method has the advantage of being applied to system reliability analysis in this paper successfully. The applicability, feasibility and efficiency of the proposed method are proved on examples which contain linearity equation, non-linear numerical example, non-linear oscillator and system reliability engineering problems.     

Bounds approximation of limit-state surface based on active learning Kriging model with truncated candidate region for random-interval hybrid reliability analysis

This article reports a brand-new methodology based on active learning Kriging model for hybrid reliability analysis (HRA) with both random and interval variables. Unlike probabilistic reliability analysis, the limit state surface (LSS) of HRA is projected into a banded region in the domain of random variables. Only approximating the bounds of the banded region is able to meet the accuracy requirement of HRA. In the proposed methodology, the HRA problem is innovatively transformed into a traditional system reliability analysis (SRA) problem with numerous failure modes. And then a basic idea from the field of SRA is borrowed into HRA, and the so-called truncated candidate region (TCR) for HRA is proposed. In each iteration, the negligible region which probably does not influence the bounds estimation of failure probability is truncated from the original candidate region, and the optimal training point is chosen from the TCR. After several iterations, the TCR will converge to the true ideal candidate region, that is, the candidate region without the inner part of LSS, and the added training points will be driven to the region around the bounds of LSS. The performance of the proposed method is compared with relevant methods by five case studies.     

Structural reliability analysis under evidence theory using the active learning kriging model

Structural reliability analysis under evidence theory is investigated. It is rigorously proved that a surrogate model providing only correct sign prediction of the performance function can meet the accuracy requirement of evidence-theory-based reliability analysis. Accordingly, a method based on the active learning kriging model which only correctly predicts the sign of the performance function is proposed. Interval Monte Carlo simulation and a modified optimization method based on Karush–Kuhn–Tucker conditions are introduced to make the method more efficient in estimating the bounds of failure probability based on the kriging model. Four examples are investigated to demonstrate the efficiency and accuracy of the proposed method.     

Time-variant reliability global sensitivity analysis with single-loop Kriging model combined with importance sampling

This paper develops a novel failure probability-based global sensitivity index by introducing the Bayes formula into the moment-independent global sensitivity index to approximate the effect of input random variables or stochastic processes on the time-variant reliability. The proposed global sensitivity index can estimate the effect of uncertain inputs on the time-variant reliability by comparing the difference between the unconditional probability density function of input variables and the conditional probability density function in failure state of input variables. Furthermore, a single-loop active learning Kriging method combined with metamodel-based importance sampling is employed to improve the computational efficiency. The accuracy of the results obtained by Kriging model is verified by the reference results provided by the Monte Carlo simulation. Four examples are investigated to demonstrate the significance of the proposed failure probability-based global sensitivity index and the effectiveness of the computational method.     

Uncertainty analysis and reliability improvement of planetary roller screw mechanism using active learning Kriging model

The uncertainties in the geometry, material and operation conditions may cause structural failure of the planetary roller screw mechanism (PRSM). The uncertainty analysis model is the key to the reliability assessment of the PRSM, however, the relevant studies have been rarely reported in the past. This paper focuses on establishing a preliminary mathematical model of the PRSM considering uncertain factors. The quasi-Monte Carlo (QMC) method is introduced to improve the solving efficiency of the multidimensional and nonlinear implicit limit state function (LSF). Then, the parameter sensitivities of the uncertain factors to the load distribution and contact characteristics are comprehensively ranked by the design of experiment (DoE). The computational cost for constructing the active learning Kriging (ALK) model of PRSM is decreased by only selecting the most sensitive variables. Moreover, the ALK model and QMC method (ALK-QMC) are combined to explore how the main factors affect the structural reliability of PRSM, which further guides the implementation of multi-objective optimization to improve the reliability by the developed NSGA-II-Downhill algorithm. Finally, the theoretical model and optimization results are verified by the finite element method.     

Active learning method combining Kriging model and multimodal-optimization-based importance sampling for the estimation of small failure probability

A novel method which combines the active learning Kriging (ALK) model with important sampling is proposed in this paper. The main aim of the proposed method is to solve problems with very small failure probability and multiple failure regions. A surrogate limit state surface (LSS) which strikes a balance between the Kriging mean and variance is proposed. In each iteration, important samples of the surrogate LSS are generated, optimal training points are chosen, the Kriging model is updated and the surrogate LSS is refined. After several iterations, the surrogate LSS will converge to the true LSS. To obtain all the local and global most probable points (MPPs) on the surrogate LSS in each iteration, a recently proposed evolutionary algorithm from the field of multimodal optimization is introduced. In this way, none of the potential failure regions is missed and the unbiasedness of the proposed method is guaranteed. The contribution factor of each MPP is defined and a weighted multimodal instrumental sampling density is formulated. In this way, more attention is paid to the more important failure regions and training points are further saved. The performance of the proposed method is verified by six case studies.     

基于Kriging模型的充液管道共振非概率可靠性分析

为解决传统概率可靠性在解决流固耦合问题方面的不足,研究了结构不确定参量用超椭球凸集模型和区间变量共同描述下的非概率共振可靠性问题。针对隐式极限状态函数难以求解的问题,引入Kriging模型和超立方抽样技术应用于非概率可靠性分析。该方法用Kriging模型作为近似模型描述原结构,并在计算过程中不断更新近似模型。考虑管道与液体之间的耦合作用,利用有限元软件对所建立的简单管道系统进行模态计算并且结合防共振理论进行充液管路系统的流固耦合振动非概率可靠性分析,用优化的方法计算可靠性指标。工程算例分析表明该方法的合理性,能完善流固耦合管道系统的防共振可靠性分析方法与理论。     

Reliability analysis in fracture mechanics using the first-order reliability method and Monte Carlo simulation

The development of a Windows‐based framework to undertake probabilistic fracture mechanics studies is reported. For a selective library of standard case problems, the reliability index of critical and sub‐critical (fatigue) problems with stochastic definition is evaluated. Both first‐order reliability method (FORM) as well as and Monte Carlo simulation method (MCS) techniques are used in critical crack growth, and only MCS is adopted for fatigue problems. Numerical predictions for the stress intensity factors (SIF) were validated with NASA/FLAGRO and reliability predictions were validated with both RELTRAN and VaP. With the advent of powerful and inexpensive personal computer, and the user‐friendliness of graphical user interface, programs such as the one developed will indeed make it possible for engineer to correctly account for the stochastic nature of most fracture problems they are confronted with.     

Error-based stopping criterion for the combined adaptive Kriging and importance sampling method for reliability analysis

Metamodel-based method is a wise reliability analysis technique because it uses the metamodel to substitute the actual limit state function under the predefined accuracy. Adaptive Kriging (AK) is a famous metamodel in reliability analysis for its flexibility and efficiency. AK combined with the importance sampling (IS) method abbreviate as AK–IS can extremely reduce the size of candidate sampling pool in the updating process of Kriging model, which makes the AK-based reliability method more suitable for estimating the small failure probability. In this paper, an error-based stopping criterion of updating the Kriging model in the AK–IS method is constructed and two considerable maximum relative error estimation methods between the failure probability estimated by the current Kriging model and the limit state function are derived. By controlling the maximum relative error, the accuracy of the estimate can be adjusted flexibly. Results in three case studies show that the error-based stopping criterion based AK–IS method can achieve the predefined accuracy level and simultaneously enhance the efficiency of updating the Kriging model.     

基于贝叶斯统计推断的框架结构损伤诊断研究

提出一种基于贝叶斯推理的非线性结构模型修正方法,同时考虑激励的随机性,建立了复合随机振动系统的动力可靠度分析方法。利用实测结构动力响应主分量的瞬时特征参数作为非线性指标构建似然函数,结合拒绝延缓自适应(Delayed Rejection and Adaptive Metropolis, DRAM)算法和高斯过程替代模型实现了非线性结构模型修正及其参数的不确定性量化。根据首次超越破坏准则,利用广义概率密度演化方法,分别对仅考虑激励随机性的确定性模型和同时考虑结构参数与激励不确定性的复合随机振动模型进行动力可靠度分析,并利用蒙特卡洛随机抽样方法验证了计算结果的准确性。研究结果表明：基于振动响应瞬时特征参数的贝叶斯推理方法能够快速、准确地实现结构的非线性模型修正及其参数的不确定性量化。与具有初始设计参数名义值的确定性模型相比,考虑参数不确定性的复合随机模型的动力可靠度总体偏低,因此,在结构安全评估中应考虑非线性模型参数不确定性的影响,使评估结果更加安全、可靠。     

An efficient method for estimating failure probability bounds under random-interval mixed uncertainties by combining line sampling with adaptive Kriging

For addressing the low efficiency of structural reliability analysis under the random-interval mixed uncertainties (RIMU), this paper establishes the line sampling method (LS) under the RIMU. The proposed LS divides the reliability analysis under RIMU into two stages. The Markov chain simulation is used to efficiently search the design point under RIMU in the first stage, then the upper and lower bounds of failure probability are estimated by LS in the second stage. To improve the computational efficiency of the proposed LS under RIMU, the Kriging model is employed to reduce the model evaluation numbers in the two stages. For efficiently searching the design point, the Kriging model is constructed and adaptively updated in the first stage to accurately recognize the Markov chain candidate state, and then it is sequentially updated by the improved U learning function in the second stage to accurately estimate the failure probability bounds. The proposed LS under RIMU with Kriging model can not only reduce the model evaluation numbers but also decrease the candidate sample pool size for constructing the Kriging model in two stages. The presented examples demonstrate the superior computational efficiency and accuracy of the proposed method by comparison with some existing methods.     

Active learning polynomial chaos expansion for reliability analysis by maximizing expected indicator function prediction error

Assessing the failure probability of complex aeronautical structure is a difficult task in presence of uncertainties. In this paper, active learning polynomial chaos expansion (PCE) is developed for reliability analysis. The proposed method firstly assigns a Gaussian Process (GP) prior to the model response, and the covariance function of this GP is defined by the inner product of PCE basis function. Then, we show that a PCE model can be derived by the posterior mean of the GP, and the posterior variance is obtained to measure the local prediction error as Kriging model. Also, the expectation of the prediction variance is derived to measure the overall accuracy of the obtained PCE model. Then, a learning function, named expected indicator function prediction error (EIFPE), is proposed to update the design of experiment of PCE model for reliability analysis. This learning function is developed under the framework of the variance-bias decomposition. It selects new points sequentially by maximizing the EIFPE that considers both the variance and bias information, and it provides a dynamic balance between global exploration and local exploitation. Finally, several test functions and engineering applications are investigated, and the results are compared with the widely used Kriging model combined with U and expected feasibility function learning function. Results show that the proposed method is efficient and accurate for complex engineering applications.     

A semi-analytical Monte Carlo simulation method for system's reliability with load sharing and damage accumulation

A novel approach for assessing a systems' reliability with dependency structures, load sharing, and damage accumulation and reversal is proposed in this paper. It is a blend of analytical reliability analysis performed at the component level, and is based on understanding the failure mechanism of the components, and a Monte Carlo simulation for the entire system to assess the reliability at the system level incorporating the dynamics of the system behavior as the components interact, share loads, and age over time. Model reduction is deployed to reduce the complexity and accelerate the simulation and convergence of the analytical methods such as FORM and SORM performed at the component level. Numerical examples are provided to illustrate the usability and performance of the method.     

An efficient method for estimating time-dependent failure possibility by combining adaptive Kriging with adaptive truncated fuzzy simulation

Time-dependent failure possibility (TDFP) can measure the structural safety level for a time interval of interest under fuzzy uncertainty, but its calculational cost is unaffordable by using fuzzy simulation (FS) due to a required large size of FS candidate sampling pool (CSP). Although time-dependent adaptive Kriging model (T-AK) combined with FS (T-AK-FS) was presented to reduce the number of calling performance function, a large FS CSP still makes training T-AK time-consuming. To improve its efficiency, an adaptive truncated FS (ATFS) with T-AK (T-AK-ATFS) is proposed by CSP size reduction approach. By T-AK-ATFS, the largest safety hypercube in fuzzy standard space is adaptively searched, in which the samples are in safety states and can be removed from the FS CSP. Moreover, T-AK is adaptively trained to search the largest safety hypercube and estimate TDFP simultaneously. In adaptively searching process, the FS CSP is divided into several sub-CSPs, on which training T-AK is more time-saving. Overall, strategies of T-AK-ATFS include proposing ATFS to reduce the FS CSP, adaptively searching the largest safety hypercube, estimating the TDFP with the same T-AK and training T-AK in the sub-CSPs sequentially. Verified by examples, these strategies make T-AK-ATFS more efficient than existing FS and T-AK-FS.     

压杆稳定的非概率可靠性凸集合模型分析方法

 针对概率可靠性模型对原始数据要求高的局限性,用凸集合模型来描述影响压杆稳定可靠性分析中的不确定参数,利用一阶Taylor展开式基于凸集合模型讨论了压杆稳定分析中不确定性参数对压杆稳定响应的影响,提出了压杆稳定的非概率可靠性度量的指标.此方法对数据的要求低,不用求概率密度函数而且计算简便.通过对工程实例的计算,其结果表明所提出的方法是一种简便而实用的分析方法.     

System‐level reliability sensitivity analysis by using weighted average simulation method

In addition to reliability analysis, investigation of the uncertainties' effect on the safety of structures can be regarded as one of the great concerns in engineering fields. The present study provides an efficient perturbation‐based reliability sensitivity analysis approach based on weighted average simulation method (WASM) to attain uncertainties effects on the structures safety. Without asking additional samplings and/or requiring to function derivation (that is necessary in score function method), the proposed approach simultaneously uses the finite difference and weight flexibility feature of WASM to estimate the parameter sensitivities of a reliability problem. The proposed method has also been successfully applied to the improved version of WASM to obtain reliability based sensitivity results with very few samples. The accuracy and efficiency of the proposed method is examined by solving five analytical and engineering examples with highly nonlinear performance functions and system‐level reliability problems. For each example, results are compared with those obtained by Monte Carlo simulation and common reliability methods. It is shown that the method is capable of solving real world system‐level engineering problems efficiently and accurately.     

A moving Kriging interpolation‐based meshless method for numerical simulation of Kirchhoff plate problems

This paper mainly proposes an alternative way for numerical implementation of thin plates bending based on a new improvement of meshless method, which is combined between the standard element‐free Galerkin method and one different shape functions building technique. The moving Kriging (MK) interpolation is applied instead of the traditional moving least‐square approximation in order to overcome Kronecker's delta property where the standard method does not satisfy. Obviously, the deflection of the thin plates is approximated via the MK interpolation. To illustrate this approach, numerical analysis is examined in both regular and irregular systems. Three examples with different geometric shapes of thin plates undergoing a simply supported boundary are performed. In addition, two important parameters of the present method are also analyzed. A good agreement can be found among the proposed, analytical and finite element methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Adaboosting‐based dynamic weighted combination of software reliability growth models

Software reliability growth models (SRGMs) are very important for software reliability estimation and prediction and have been successfully applied in the critical airborne software. However, there is no general model which can perform well for different cases. Thus, some researchers proposed to obtain more accurate estimation and prediction than one single model by combining various individual SRGMs together. AdaBoosting is a commonly used machine learning algorithm for combining several weak predictors into a single strong predictor to significantly improve the estimating and forecasting accuracy, which may be very suitable for the combination of SRGMs. Hence, two novel AdaBoosting‐based combination approaches for improving the parametric SRGMs are presented in this paper. The first one selects several variations of one original SRGM for obtaining the self‐combination model (ASCM). The second selects several various candidate SRGMs for obtaining the multi‐combinational model (AMCM). Finally, two case studies are presented and the results show that: (1) the ASCM is fairly effective and applicable for improving the estimation and prediction performance of its corresponding original SRGM without adding any other factors and assumptions; (2) the AMCM is notably effective and applicable for combining SRGMs because it has well applicability and provides a significantly better reliability estimation and prediction power than the traditional SRGMs and also yields a better estimation and prediction power than the neural‐network‐based combinational model. Copyright © 2011 John Wiley & Sons, Ltd.     

A non‐stationary covariance‐based Kriging method for metamodelling in engineering design

Metamodels are widely used to facilitate the analysis and optimization of engineering systems that involve computationally expensive simulations. Kriging is a metamodelling technique that is well known for its ability to build surrogate models of responses with non‐linear behaviour. However, the assumption of a stationary covariance structure underlying Kriging does not hold in situations where the level of smoothness of a response varies significantly. Although non‐stationary Gaussian process models have been studied for years in statistics and geostatistics communities, this has largely been for physical experimental data in relatively low dimensions. In this paper, the non‐stationary covariance structure is incorporated into Kriging modelling for computer simulations. To represent the non‐stationary covariance structure, we adopt a non‐linear mapping approach based on parameterized density functions. To avoid over‐parameterizing for the high dimension problems typical of engineering design, we propose a modified version of the non‐linear map approach, with a sparser, yet flexible, parameterization. The effectiveness of the proposed method is demonstrated through both mathematical and engineering examples. The robustness of the method is verified by testing multiple functions under various sampling settings. We also demonstrate that our method is effective in quantifying prediction uncertainty associated with the use of metamodels. Copyright © 2006 John Wiley & Sons, Ltd.     

Second‐order reliability method‐based inverse reliability analysis using Hessian update for accurate and efficient reliability‐based design optimization

First‐order reliability method (FORM) has been mostly utilized for solving reliability‐based design optimization (RBDO) problems efficiently. However, second‐order reliability method (SORM) is required in order to estimate a probability of failure accurately in highly nonlinear performance functions. Despite accuracy of SORM, its application to RBDO is quite challenging due to unaffordable numerical burden incurred by a Hessian calculation. For reducing the numerical efforts, a quasi‐Newton approach to approximate the Hessian is introduced in this study instead of calculating the true Hessian. The proposed SORM with the approximated Hessian requires computations only used in FORM, leading to very efficient and accurate reliability analysis. The proposed SORM also utilizes a generalized chi‐squared distribution in order to achieve better accuracy. Furthermore, SORM‐based inverse reliability method is proposed in this study. An accurate reliability index corresponding to a target probability of failure is updated using the proposed SORM. Two approaches in terms of finding an accurate most probable point using the updated reliability index are proposed. The proposed SORM‐based inverse analysis is then extended to RBDO in order to obtain a reliability‐based optimum design satisfying probabilistic constraints more accurately even for a highly nonlinear system. The numerical study results show that the proposed reliability analysis and RBDO achieve efficiency of FORM and accuracy of SORM at the same time. Copyright © 2014 John Wiley & Sons, Ltd.