Surrogate models are popular tool to approximate the functional relationship of expensive simulation models in multiple scientific and engineering disciplines. Successful use of surrogate models can ...provide significant savings of computational cost. However, with a variety of surrogate model approaches available in literature, it is a difficult task to select an appropriate one at hand. In this paper, we present an overview of surrogate model approaches with an emphasis of their application for variance-based global sensitivity analysis, including polynomial regression model, high-dimensional model representation, state-dependent parameter, polynomial chaos expansion, Kriging/Gaussian Process, support vector regression, radial basis function, and low rank tensor approximation. The accuracy and efficiency of these approaches are compared with several benchmark examples. The strengths and weaknesses of these surrogate models are discussed, and the recommendations are provided for different types of applications. For ease of implementations, the packages, as well as toolboxes, of surrogate model techniques and their applications for global sensitivity analysis are collected.
The time-dependent failure credibility (TDFC) can reasonably measure the safety level of the time-dependent structure under the fuzzy uncertainty, but the direct optimization algorithm to estimate ...the TDFC requires large computational cost and even results in locally optimal solutions. Therefore, an efficient method is proposed for estimating the TDFC by combining the fuzzy simulation and the single-loop Kriging model. In the proposed method, fuzzy inverse transformation theorem is firstly used to transform the estimation of the TDFC into a sample classification problem, in which the candidate sample pool generated by fuzzy simulation (FS) is classified into the failure group and the safety one. For improving the efficiency of the classification, a Kriging model is adaptively trained by an elaborate U-learning function in the candidate sample pool. After the candidate sample is divided into the failure group and the safety one by the convergent Kriging model, the TDFC can be estimated as a byproduct easily. The innovation of the proposed method includes two aspects: establishing the idea of the fuzzy simulation combined with the single-loop Kriging model to estimate TDFC efficiently and robustly, and designing an elaborate U-learning function to improve the efficiency of training the single-loop Kriging model. The presented examples validate the efficiency of the proposed method under the acceptable precision.
To efficiently analyze the time-dependent reliability is still a challenge today for many applications. This paper aims at modifying the original single-loop Kriging surrogate method to make it more ...efficient especially for assessing the small time-dependent failure probability. The first contribution of the proposed method is that the radial-based importance sampling scheme is nested in the single-loop Kriging surrogate model-based time-dependent reliability analysis method. By the radial-based importance sampling scheme, the optimal hypersphere can be searched and the samples inside the optimal hypersphere can be removed from the candidate sampling pool. Besides, the samples outside the optimal hypersphere are divided into several sub-candidate sampling pools by the in-process hyperspheres. By decreasing the size of candidate sampling pool in each updating process of Kriging model, the training time of updating Kriging model can be reduced so that the efficiency of time-dependent reliability analysis is enhanced. The second contribution of the proposed method is that the Kriging model-based dichotomy is embedded skillfully to efficiently find the hyperspheres layer after layer until the optimal hypersphere is found. The third contribution of the proposed method is that a modified learning function is constructed from selecting the most easily identifiable failure time during the time period of interest to efficiently update the Kriging model in each sub-candidate sampling pool. Finally, the accuracy and efficiency of the proposed method are verified by three examples.
Structural reliability analysis aims at computing failure probability with respect to prescribed performance function. To efficiently estimate the structural failure probability, a novel two-stage ...meta-model importance sampling based on the support vector machine (SVM) is proposed. Firstly, a quasi-optimal importance sampling density function is approximated by SVM. To construct the SVM model, a multi-point enrichment algorithm allowing adding several training points in each iteration is employed. Then, the augmented failure probability and quasi-optimal importance sampling samples can be obtained by the trained SVM model. Secondly, the current SVM model is further polished by selecting informative training points from the quasi-optimal importance sampling samples until it can accurately recognize the states of samples, and the correction factor is estimated by the well-trained SVM model. Finally, the failure probability is obtained by the product of augmented failure probability and correction factor. The proposed method provides an algorithm to efficiently deal with multiple failure regions and rare events. Several examples are performed to illustrate the feasibility of the proposed method.
•An improved AK-MCS method is proposed to estimate the small failure probability.•An optimal β-sphere is searched without any extra model evaluations.•Kriging model is ceaselessly updated layer by ...layer outside the current β-sphere.•The adaptive Kriging model is finished until an optimal β-sphere is founded.•The candidate sampling pool in the proposed algorithm is remarkably reduced.
The pivotal problem in reliability analysis is how to use a smaller number of model evaluations to get more accurate failure probabilities. To achieve this aim, an iterative method based on the Monte Carlo simulation and the adaptive Kriging (AK) model (abbreviated as AK-MCS) has been proposed in 2011 by Echard et al. But for small failure probability, the number of the candidate points is extremely large for convergent solution. These points need to be evaluated by the current Kriging model to select the best next sample for updating the Kriging model in AK-MCS method, and the large candidate points will make the adaptive updating process of Kriging model much more time-consuming. Therefore, to improve the applicability of the AK-MCS method for small failure probability, the adaptive radial-based importance sampling (ARBIS) is employed to reduce the number of candidate points in the AK-MCS method, and an ARBIS combined with AK model method (abbreviated as AK-ARBIS) is proposed. The idea of the ARBIS is adaptively to find the optimal β-sphere, i.e., the largest sphere of the safe domain, and then samples inside the optimal β-sphere is directly recognized as safety and do not need to call the true limit state function to judge their states (safe or failed). During the adaptive process of finding the optimal β-sphere, the Kriging model is ceaselessly updated layer after layer based on the U learning scheme in each sampling pool which only contains the samples between the current spherical rings. The updating process of Kriging model stops until the optimal β-sphere is adaptively found and the convergent condition is satisfied. By finding the optimal β-sphere, the total number of candidate samples is reduced which only includes the samples outside the optimal β-sphere. Besides, the whole candidate sampling pool is partitioned into several sub-candidate sampling pool sequentially. The proposed method not only inherits the advantage of the AK-MCS but also reduces the reliability analysis time of the AK-MCS from two aspects. One is the size reduction of the candidate sampling pool, the other is the reduction of the actual limit state function evaluations because the sampling points locating inside the adaptively searched optimal β-sphere do not need to participate in the training process. By analyzing a highly nonlinear numerical case, a non-linear oscillator system, a simplified wing box structural model, an aero-engine turbine disk and a planar ten-bar structure, the effectiveness and the accuracy of the proposed AK-ARBIS method for estimating the small failure probability are verified.
•Use metamodel to estimate the time-dependent failure probability (TDFP).•The proposed methods reduce computational cost drastically.•AK-co-IS is based on the optimal time-dependent design ...point.•AK-co-SS transforms small TDFP into a series of larger conditional ones.
For efficiently estimating the time-dependent failure probability, two new methods named as the active learning Kriging (AK) coupled with importance sampling (AK-co-IS) and AK coupled with subset simulation (AK-co-SS) are proposed. The proposed methods are based on the fact that the AK coupled with Monte Carlo simulation (AK-MCS) method has been proved to be a very efficient method. However, for problem with small time-dependent failure probability or long service time, the size of candidate sample pool generated by MCS would be so large that the efficiency of AK-MCS is reduced. Therefore, the AK-co-IS and AK-co-SS are proposed to highly enhance the computational efficiency by greatly reducing the candidate sample pool size. And these two methods reduce the candidate sample pool size respectively by searching the optimal time-dependent design point to increase the ratio of failure samples and converting a rare event simulation problem into sequence of more frequent event ones. Through iteratively constructing the AK model to be convergent by the U-learning function in the IS and SS sample pools, respectively, the computational cost of estimating the time-dependent failure probability would reduce drastically compared with AK-MCS. Several examples are used to illustrate the efficiency and accuracy of the proposed methods.
The output of structure under fuzzy uncertainty can be classified into three cases, i.e., safety-failure case corresponding to that failure and safety both can occur in different membership levels, ...absolute failure case corresponding to that only failure can occur, and absolute safety case corresponding to that only safety can occur in any membership level. The existing fuzzy possibilistic safety degree measure models can only distinguish the structural safety degree in the safety-failure case but play no role in the absolute failure case and absolute safety case. Aiming at addressing this issue, a novel fuzzy possibilistic safety degree measure model is proposed. Before establishing the new fuzzy possibilistic safety degree measure model, a new value-interval ranking technique is first constructed. Then, the new safety possibility and failure possibility are estimated by synthesizing the information in the entire uncertain space based on the proposed value-interval ranking technique. The new fuzzy possibilistic safety degree measure model can distinguish the structural safety degree in all the three cases, and the results coincide with the human’s intuitive cognition. Several examples involving an engineering application with the finite element model are introduced to show the effectiveness of the established fuzzy possibilistic safety degree measure model.
Time-dependent reliability-based design optimization (RBDO) can provide the optimal design parameter solutions for the time-dependent structure, and thus plays a significant role in engineering ...application. Directly solving the time-dependent RBDO needs a nested double-loop optimization procedure, which undoubtedly leads to large computational costs. A novel decoupling method called two-step method (TSM) is proposed to efficiently solve the time-dependent RBDO. In the two-step method, the first step makes the minimum instantaneous reliability index satisfy the reliability target index by solving a transformed time-independent RBDO, and the second step performs time-dependent reliability analysis and deterministic optimization to obtain the optimal design parameters which meet the reliability target. Only a few time-dependent reliability analyses and several deterministic optimizations are involved in the proposed procedure; thus, the time-dependent RBDO can be efficiently solved. Several examples containing one numerical example and two engineering examples are introduced to show the effectiveness of the proposed TSM.
Failure credibility is popular in measuring safety degree of structure under fuzzy uncertainty, but the heavy computational cost is still a challenge in estimating the failure credibility. To ...alleviate this issue, an iterative method combining adaptive Kriging and fuzzy simulation (AK-FS) has been developed by Ling et al. However, for the problem with complex performance function, a large candidate sampling pool is needed in the AK-FS, which makes the training process of the Kriging model fairly time consuming. In order to improve the estimation efficiency of failure credibility through reducing the size of candidate sampling pool in AK-FS, an efficient sample reduction strategy based on adaptive Kriging (SR-AK) is proposed in this paper. In the SR-AK, the estimation of failure credibility is transformed into searching two active points in candidate sampling pool. After updating the Kriging model in each circle, current active points can be easily identified. Then, according to the properties of the active points and the prediction characteristics of Kriging model, the samples in current candidate sampling pool can be divided into two sets, i.e., the samples affect the estimation of active points and the samples have no effect on it. Obviously, the samples in the latter set can be deleted from current candidate sampling pool to reduce its size. By using this sample reduction strategy, the process for training Kriging model is accelerated circle by circle, which is very helpful to save the analysis time and improve the computational efficiency in estimating failure credibility. Four examples are employed to demonstrate the performance of the proposed SR-AK in fuzzy safety degree analysis.
In the presence of random and interval hybrid uncertainty (RI-HU), the safety degree of the structure system can be quantified by the upper and lower bounds of failure probability. However, there is ...a lack of efficient methods for estimating failure probability under RI-HU in present. Therefore, a novel method is proposed in this paper. In the proposed method, the interval variables are extended to the random variables by assigning a priori probability density function, in which the conditional density estimation (CDE)–based method and conditional probability estimation (CPE)–based method are proposed, and the failure probability varying with the interval variables can be obtained by only one group Monte Carlo simulation (MCS). Since the computational complexity of CPE is much lower than that of CDE, the CPE-based method is mainly concerned. In the CPE-based method, the conditional failure probability on a realization of the extended interval vector is approximated by that on a differential region adjacent to the corresponding realization; then, the density function estimation required in the CDE can be avoided. In order to ensure the accuracy of the CPE, a strategy is proposed to adaptively select the differential region, in which the MCS can be combined with the CPE (CPE + MCS) and the adaptive Kriging can be nested into the CPE + MCS for improving the efficiency. To improve the efficiency further, the meta-model importance sampling nested Kriging is combined with the CPE-based method. The presented examples illustrate the superiority of the proposed method over the existing methods.
