Active Learning Reliability Method Research Articles

P-AK-MCS: Parallel AK-MCS method for structural reliability analysis

In recent years, the active learning reliability method that combines the Kriging model and Monte Carlo simulation (AK-MCS) has emerged as a promising approach due to its computational efficiency and accuracy. However, the commonly used learning functions, such as the expected feasibility function (EFF), U function, H function, and expected risk function (ERF), can only select one training point at each iteration which is time-wasteful when parallel computing is available. Therefore, this paper proposes a parallel active learning Kriging strategy, namely P-AK-MCS, for structural reliability analysis. By introducing an influence function that reflects the impact of the added point on the original learning function, four parallel learning functions are constructed: pseudo-U (PU) function, pseudo-EFF (PEFF), pseudo-H (PH) function, and pseudo-ERF (PERF). These functions aim to identify multiple training points at each iteration without requiring additional functional evaluations. The effectiveness of the proposed method is validated using four examples. The results demonstrate that compared to the standard AK-MCS, the proposed P-AK-MCS significantly reduces the number of computation loops and greatly decreases computational costs. Moreover, the total number of functional evaluations required is similar to that of the standard AK-MCS and remains insensitive to the number of multiple training points.

CSP-free adaptive Kriging surrogate model method for reliability analysis with small failure probability

In the field of reliability engineering, the active learning reliability method that amalgamates Kriging model and Monte Carlo simulation has been devised and proven to be efficacious in reliability analysis. Nevertheless, the performance of this method is sensitive to the magnitude of candidate sample pool, particularly for systems with small failure probabilities. To surmount these limitations, this paper proposes an active learning method that obviates the need for candidate sample pools. The proposed method comprises two stages: construction of surrogate model and Monte Carlo simulation for failure probability estimation. During the surrogate model construction stage, the surrogate model is iteratively refined based on the representative samples selected by solving the optimization problem facilitated by the particle swarm optimization algorithm. To achieve an optimal balance between solution accuracy and efficiency, the penalty intensity control and the density control for the experimental design points are incorporated to modify the objective function in optimization. The performance of the proposed method is evaluated using numerical examples, and results indicate that by leveraging an optimization algorithm to select representative samples, the proposed method overcomes the limitations of traditional active learning methods based on candidate sample pool and exhibits exceptional performance in addressing small failure probabilities.

Look-ahead active learning reliability analysis based on stepwise margin reduction

According to the concept of limit-state margin probability function, a new look-ahead learning function called stepwise margin reduction (SMR) is proposed for active learning reliability analysis. SMR aims to select the best next point as the one minimizing, in expectation, the integrated margin probability function when adding such a new point. The plain definition of SMR involves a double integral, and three-fold contributions are made to reducing the associated computational burden. First, the closed-form expression of inner integral is well deduced, which avoids burdensome Gaussian–Hermite quadrature or drawing simulations of Gaussian process. Second, thanks to the locality of analytical expression of inner integral, truncated-integral scheme (TIS) is devised for the outer integral to avoid serious computer memory issue. Third, the candidate pool is pruned to accelerate the selection of best next point per iteration. The efficacy of SMR-based active learning reliability method is illustrated on two analytical functions, three numerical examples and one real-world engineering problem. Results demonstrate that in SMR, the TIS performs better than the traditional limited-integral scheme. Then, in comparison with existing pointwise and look-ahead learning functions, SMR gains favorable advantage in term of both computational accuracy and efficiency.

An adaptive Kriging-based structural reliability analysis method combing dichotomy and improved convergence criterion

Traditionally, analysis requires very large number of calculations; some engineering applications in particular require time-consuming numerical efforts. Classical reliability analysis methods with combinations of surrogate models can relieve the computational burden. An active learning reliability method that combines kriging and Monte Carlo simulation has drawn a great deal of attention in recent years. However, it is often challenging to find the most suitable sample point and speed up the convergence. In this paper, a method is proposed to deal with this problem, in which two strategies are employed: an active learning method is proposed to search the most suitable sample point in a specific region using the dichotomy method; an improved convergence criterion is developed based on standard deviation to improve the convergence speed. In addition, taking advantage of the excellent characteristics of Subset Simulation, a method for dealing with small failure probability problems is proposed based on the above strategies. The efficiency and accuracy of the proposed approaches are tested using three numerical examples and one finite element example. Results demonstrate its accuracy.

An Active Learning Polynomial Chaos Kriging metamodel for reliability assessment of marine structures

Metamodel combined with simulation type reliability method is an effective way to determine the probability of failure (Pf) of complex structural systems and reduce the burden of computational models. However, some existing challenges in structural reliability analysis are minimizing the number of calls to the numerical model and reducing the computational time. Most research work considers adaptive methods based on ordinary Kriging with a single point enrichment of the experimental design (ED). This paper presents an active learning reliability method using a hybrid metamodel with multiple point enrichment of ED for structural reliability analysis. The hybrid method (termed as APCKKm-MCS) takes advantage of the global prediction and local interpolation capability of Polynomial Chaos Expansion (PCE) and Kriging, respectively. The U learning function drives active learning in this approach, while K-means clustering is proposed for multiple point enrichment purposes. Two benchmark functions and two practical marine structural cases validate the performance and efficiency of the method. The results confirm that the APCKKm-MCS approach is efficient and reduces the computational time for reliability analysis of complex structures with nonlinearity, high dimension input random variables, or implicit limit state function.

A fast-convergence algorithm for reliability analysis based on the AK-MCS

In the field of reliability engineering, assessing the probability of failure of an event is usually a computationally demanding task. One way of tackling this issue is by metamodelling, in which the original computational-expensive model is approximated by a simpler metamodel. A method called AK-MCS for Active learning reliability method combining Kriging and Monte Carlo Simulation, was developed for the metamodel construction, and proved effective in reliability analysis. However, the performance of the AK-MCS algorithm is sensitive to the candidate size and the Kriging trend. Moreover, it cannot take advantage of parallel computing, a highly efficient way to speed up the simulation process. Focusing on the identified issues, this study proposes three methods to improve algorithm performance: the candidate size control method, multiple trends method, and weighted clustering method. These three methods are integrated into the AK-MCS structure, with their individual and combined performances being tested using four examples. Results suggest that all three methods contribute to the improvement of algorithm efficiency. When the three methods working together, the computational time is reduced significantly, and in the meantime, higher accuracy can be achieved.

Closed-Form Solution and Reliability Analysis of Deep Tunnel Supported by a Concrete Liner and a Covered Compressible Layer Within the Viscoelastic Burger Rock

Time-dependent rock behavior can significantly affect the design and the construction method of tunnel support lining. As an innovation, a concrete liner covered with a compressible layer is thus of great interest to reduce the underground space project costs. Indeed, the compressible material usually exhibits very high deformability under oedometric loading (30–60% volumetric deformation under a weak variation in axial stress). This helps to absorb the convergence of the excavated wall of the squeezing rock, and thus limit the transmission of stress to the inner concrete lining. This study firstly aims at deriving a closed-form solution for a deep-circular tunnel excavated within the viscoelastic Burger rock and supported by a double-layer concrete/compressible material. The inner concrete layer is assumed to be linear elastic, while the outer compressible layer is described by a tri-linear elastic model. The analytical solution is derived under the integral equation form. Secondly, the effects of uncertainty of time-dependent mechanical properties of host rock on the failure probability of tunnel for a period of 100 years are studied by introducing the closed-form solution into the well-known Kriging-based reliability analysis AK-MCS (Active learning reliability method combining Kriging and Monte Carlo Simulation) method. Notably, the benefit of the compressible layer is highlighted from the reliability analysis.

A novel active learning reliability method combining adaptive Kriging and spherical decomposition-MCS (AK-SDMCS) for small failure probabilities

Structural reliability analysis for small failure probabilities remains a challenging task, despite the significant progress made by the active learning reliability methods (ALRMs) represented by AK-MCS (ALRMs combining adaptive Kriging and Monte Carlo simulations). In order to address this issue, advanced ALRMs with improved computational efficiency than AK-MCS have been proposed, however at the expense of sacrificing the generality and simplicity of AK-MCS. Therefore, a novel ALRM combining adaptive Kriging and spherical decomposition-MCS (AK-SDMCS) was proposed to alleviate the computational cost and preserve the advantages of AK-MCS simultaneously. The key of AK-SDMCS lies in the spherical decomposition-MCS (SDMCS), which not only decomposes the computationally intensive active learning of AK-MCS into a series of easy-to-compute active learning in the divided subsets but also reduces the total number of candidate samples by focusing the sampling region on the most probable failure regions (MPFRs) and adaptively allocating samples in the divided subregions. As a result, AK-SDMCS significantly alleviates the computational cost of active learning of Kriging model. Since AK-SDMCS is built in the framework of AK-MCS, the advantages of AK-MCS can be preserved. Besides, the active learning strategy is modified to further increase the computational efficiency. Thus, AK-SDMCS provides an efficient, easy-to-implement and generalized option for reliability analysis of small failure probabilities. The application and efficiency of AK-SDMCS are demonstrated by three academic examples.

AK-MSS: An adaptation of the AK-MCS method for small failure probabilities

Structural reliability analysis aims to estimate the failure probability of a structure with respect to the performance function. This estimation may be a difficult task when the computation of a structural response requires large computational efforts. Although simulation-based methods can be used directly, they may require a large number of calls to the performance function for small failure probabilities. Metamodels (such as Kriging) that can replace the original performance function can be applied to address the computational costs. Among these methods, the active learning reliability method combining Kriging and Monte Carlo simulation (AK-MCS) is efficient, except for small failure probabilities and system reliability analysis. In this paper, a modified algorithm that combines the AK-MCS and the modified subset simulation (MSS) is proposed to estimate small failure probabilities. The strategy replaces the initial population with a large population that is generated by the MSS. No prior knowledge about the probability level is needed, and the sample size will adaptively change according to the estimation that is obtained in the last iteration and the target coefficient of variation. Therefore, the limit state can be covered by the new population, which is important for refining the Kriging model. The efficiency and accuracy of the proposed algorithm are illustrated using several examples.

An Improved Structural Reliability Analysis Method Based on Local Approximation and Parallelization

The Kriging-based reliability method with a sequential design of experiments (DoE) has been developed in recent years for implicit limit state functions. Such methods include the efficient global reliability analysis, the active learning reliability method combining Kriging and MCS Simulations. In this research, a novel local approximation method based on the most probable failure point (MPFP) is proposed to improve such methods. In this method, the MPFP calculated in the last iteration is the center of the next sampling region. The size of the local region depends on the reliability index obtained by the First Order Reliability Method (FORM) and the deviation distance of the standard deviation. The proposed algorithm, which approximates the limit state function accurately near MPFP rather than in the whole design space, can avoid selecting samples in regions that have negligible effects on the reliability analysis results. In addition, a multi-point enrichment technique is also introduced to select multiple sample points in each iteration. After the high-quality approximation of limit state function is obtained, the failure probability is calculated by the Monte Carlo method. Four numerical examples are used to validate the accuracy and efficiency of the proposed method. Results show that the proposed method is very effective for an accurate evaluation of the failure probability.

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.

Probabilistic analysis of strip footings resting on spatially varying soils using kriging metamodeling and importance sampling

This paper aims at computing the probability of failure of strip footings resting on a spatially varying soil and subjected to a vertical load. The active learning reliability method (called AK-IS) which is a combination of kriging metamodeling and importance sampling (IS) is used. The AK-IS technique significantly reduces the computation time with respect to the classical active learning reliability technique (called AK-MCS) combining kriging with Monte Carlo Simulations (MCS) by sampling around the design point. It was shown that the critical realization corresponding to the design point exhibits a perfect symmetry about the central vertical axis of the foundation.

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

Subset simulation (SS) is a powerful reliability analysis method by transforming a rare failure event into a sequence of multiple intermediate failure events with larger probabilities. Recently, the metamodel-assisted SS method has attracted great attention to improve the efficiency of reliability analysis for time-consuming performance functions. However, in cases with highly nonlinear performance functions and small failure probabilities, it is still difficult to balance the estimation accuracy of failure probability and the computational cost on the construction of metamodels. To address this challenge, an active learning reliability method combining Kriging constructed with exploration and exploitation of failure region and subset simulation (AKEE-SS) is proposed in this paper. The exploration and exploitation of failure region benefit from the samples in the first and last levels of SS, respectively. To control the influence of metamodel error on the estimation of failure probability, two error measure functions are developed to quantify the influence and be considered in the termination conditions of metamodel update. Five numerical examples are used to test the performance of AKEE-SS. Results indicate that AKEE-SS is an accurate and efficient reliability analysis method for problems with highly nonlinear performance functions and small failure probabilities.

An active learning reliability method with multiple kernel functions based on radial basis function

Surrogate models combined with Monte Carlo simulation (MCS) are effective ways to address failure probability problems, which involve time-consuming computer codes, in structural reliability analysis. Recently, many active learning functions based on the Kriging model have been developed to conduct adaptive sequential sampling due to its predicted value and variance. However, effective methods for learning functions based on other surrogate models to address failure probability problems remain sparse. Hence, this paper presents a new adaptive sampling function that combines the radial basis function (RBF) approximate model with MCS to calculate the failure probability of structures. The new learning function is named the active RBF method with multiple kernel functions and Monte Carlo simulation (ARBFM-MCS). It uses several RBF kernel functions to estimate the local uncertainty of the predicted values based on interquartile range (IQR) to formulate the active learning function. This method can establish several surrogate models simultaneously for application to different problems. The stopping criterion of this method is that if one of the errors calculated by the various RBF models meets the demand, the learning process stops automatically. Furthermore, the method is also compared with the k-fold cross-validation approach based on a single RBF kernel function and some other approaches presented in the literature. Six numerical examples are considered to verify the accuracy and the efficiency of the proposed method. The results reveal that the multiple kernel function method is effective and has the same accuracy level as other methods. Moreover, for these examples, the method often calls the limit state function in fewer times to obtain accurate failure probabilities.

AK-MCSi: A Kriging-based method to deal with small failure probabilities and time-consuming models

Reliability analyses still remain challenging today for many applications. First, assessing small failure probabilities is tedious because of the very large number of calculations required. Secondly, mechanical system models can require considerable numerical efforts. To deal with these problems, classical reliability analysis methods may be combined with those of meta-modeling, to enable the construction of a model like the former numerical model but with fewer time-consuming evaluations. Among these approaches, the Active Learning Reliability Method, combining Kriging and Monte Carlo Simulations, has received attention over the last few years. This method has several drawbacks, such as the difficulty to assess small failure probabilities and its inability to parallelize computations. The proposed paper focuses on the improvement of such a method to solve both these issues. It introduces a sequential Monte Carlo Simulation technique to deal with small failure probabilities. A multipoint enrichment technique is also proposed to allow parallelization and thus to reduce numerical efforts. Both these new techniques give rise to the proposal of a new, more conservative stopping condition for learning. The efficiency of this new method, called AK-MCSi, is then demonstrated using three examples for which the results show a significant reduction in the time required and/or the number of iterations needed for an accurate evaluation of the failure probability.

A Sequential Kriging reliability analysis method with characteristics of adaptive sampling regions and parallelizability

The sequential Kriging reliability analysis (SKRA) method has been developed in recent years for nonlinear implicit response functions which are expensive to evaluate. This type of method includes EGRA: the efficient reliability analysis method, and AK-MCS: the active learning reliability method combining Kriging model and Monte Carlo simulation. The purpose of this paper is to improve SKRA by adaptive sampling regions and parallelizability. The adaptive sampling regions strategy is proposed to avoid selecting samples in regions where the probability density is so low that the accuracy of these regions has negligible effects on the results. The size of the sampling regions is adapted according to the failure probability calculated by last iteration. Two parallel strategies are introduced and compared, aimed at selecting multiple sample points at a time. The improvement is verified through several troublesome examples.

Use of Adaptive Kriging Metamodeling in Reliability Analysis of Radiated Susceptibility in Coaxial Shielded Cables

In this paper, reliability analysis is applied to assess electromagnetic risk due to external electromagnetic waves in a coaxial shielded cable. Generally, the classical sampling method-Monte Carlo (MC)-is used to conduct thousands or millions of runs of the numerical model. As this latter is time-demanding, the computational cost becomes prohibitive. As an alternative, an advanced metamodeling method based on Kriging is proposed for efficiently assessing small failure probabilities. This method is called AK-MCS. It consists in an active learning reliability method combining Kriging and MC simulation. Metamodeling reduces the computational run time by replacing the true numerical model by an inexpensive surrogate model. The interpolated surrogate model presents the advantage to be rapidly handled by MC simulations to calculate the failure probability. AK-MCS method is based on an adaptive strategy to enrich the design of experiments (DOE) with significant sampling points near the failure region, and hence, Kriging model is ameliorated and so is the assessment of failure probability. Three numerical examples of radiated susceptibility in a shielded cable are conducted to demonstrate the efficiency of the underlined Kriging approach. Reliable metamodels can be obtained through a small number of runs of the true numerical model, and it is found that the computed failure probabilities are very accurate compared to crude MC simulation results.

A hybrid algorithm for reliability analysis combining Kriging and subset simulation importance sampling

To solve the problem of large computation when failure probability with time-consuming numerical model is calculated, we propose an improved active learning reliability method called AK-SSIS based on AK-IS algorithm [1]. First, an improved iterative stopping criterion in active learning is presented so that iterations decrease dramatically. Second, the proposed method introduces Subset simulation importance sampling (SSIS) into the active learning reliability calculation, and then a learning function suitable for SSIS is proposed. Finally, the efficiency of AK-SSIS is proved by two academic examples from the literature. The results show that AK-SSIS requires fewer calls to the performance function than AK-IS, and the failure probability obtained from AK-SSIS is very robust and accurate. Then this method is applied on a spur gear pair for tooth contact fatigue reliability analysis.

A combined Importance Sampling and Kriging reliability method for small failure probabilities with time-demanding numerical models

Applying reliability methods to a complex structure is often delicate for two main reasons. First, such a structure is fortunately designed with codified rules leading to a large safety margin which means that failure is a small probability event. Such a probability level is difficult to assess efficiently. Second, the structure mechanical behaviour is modelled numerically in an attempt to reproduce the real response and numerical model tends to be more and more time-demanding as its complexity is increased to improve accuracy and to consider particular mechanical behaviour. As a consequence, performing a large number of model computations cannot be considered in order to assess the failure probability. To overcome these issues, this paper proposes an original and easily implementable method called AK-IS for active learning and Kriging-based Importance Sampling. This new method is based on the AK-MCS algorithm previously published by Echard et al. [AK-MCS: an active learning reliability method combining Kriging and Monte Carlo simulation. Structural Safety 2011;33(2):145–54]. It associates the Kriging metamodel and its advantageous stochastic property with the Importance Sampling method to assess small failure probabilities. It enables the correction or validation of the FORM approximation with only a very few mechanical model computations. The efficiency of the method is, first, proved on two academic applications. It is then conducted for assessing the reliability of a challenging aerospace case study submitted to fatigue.

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

An important challenge in structural reliability is to keep to a minimum the number of calls to the numerical models. Engineering problems involve more and more complex computer codes and the evaluation of the probability of failure may require very time-consuming computations. Metamodels are used to reduce these computation times. To assess reliability, the most popular approach remains the numerous variants of response surfaces. Polynomial Chaos [1] and Support Vector Machine [2] are also possibilities and have gained considerations among researchers in the last decades. However, recently, Kriging, originated from geostatistics, have emerged in reliability analysis. Widespread in optimisation, Kriging has just started to appear in uncertainty propagation [3] and reliability [4,5] studies. It presents interesting characteristics such as exact interpolation and a local index of uncertainty on the prediction which can be used in active learning methods. The aim of this paper is to propose an iterative approach based on Monte Carlo Simulation and Kriging metamodel to assess the reliability of structures in a more efficient way. The method is called AK-MCS for Active learning reliability method combining Kriging and Monte Carlo Simulation. It is shown to be very efficient as the probability of failure obtained with AK-MCS is very accurate and this, for only a small number of calls to the performance function. Several examples from literature are performed to illustrate the methodology and to prove its efficiency particularly for problems dealing with high non-linearity, non-differentiability, non-convex and non-connex domains of failure and high dimensionality.