Hybrid FORM-Sampling simulation method for finding design point and importance vector in structural reliability
Introduction
There are naturally various types of uncertainties in structural analyses, namely material characteristics, geometric properties, and imposed loads. Generally, in structural reliability analysis, the effect of uncertainties can be expressed in the LSF using resistance parameter (R) and load effect (L), as given in Eq. (1) [1], [2], [3], [4]. This function differentiates between the safe region and the failure region. The main objective in reliability problems is to calculate the failure probability based on the multiple integral of Eq. (2) [5], [6], [7]. where is the failure probability and is the LSF in the main space (X-space). specifies the failure region while indicate the safe region. stands for the joint probability density function of random variable and denoted the cumulative distribution function of the standard normal distribution based on which the reliability index is calculated. When the LSF has an extremely high nonlinearity degree and several random variables, the multiple integral of Eq. (2) would be dramatically difficult or even impossible. To tackle this problem, reliability methods including simulation methods [8], [9], [10], [11], [12], [13] and approximation methods [14], [15], [16], [17], [18] are being developed.
To approach the real structural behavior, the uncertainty of variables must be incorporated in the calculations. Therefore, there is a failure probability for each structure. Mayer et al. [19] proposed the first mathematical formulation for structural reliability problems in which the load effect and resistance parameters are found the best random variables for expressing the structural probabilistic behavior. Their ideas were further developed by Freudenthal [20], [21].
Among the most central approximation methods, one can refer to the first- and second-order reliability methods (FORM and SORM). These two methods are founded on the first and second order Taylor expansions, respectively, which are approximated from the LSF at the design point. An optimization model is employed in these methods to find the design point or the most probable point (MPP). Based on the definition by Hasoer-Lind [22], the design point has the shortest distance from origin in the standard normal coordinate system or U-space (see Fig. 1). To transform independent random variables from X-space to U-space, the transformation is applied. The distance of design point from the origin in the standard normal space is called reliability index . Despite the simplicity of approximation methods to calculate the design point, these methods suffer from some drawbacks, namely dependence of response on the start point of search, necessity for linearization of the LSF, necessity for differentiability of the LSF about the design point, and also problem of convergence to the local response.
Simulation methods like Monte Carlo (MC) have been founded on the production of random samples based on the sampling probability function and evaluation of each random sample regarding the LSF. One of the major drawbacks of simulation methods is the lack of design point calculation. Furthermore, more accurate calculation of the probability of failure requires the generation of a large number of random samples and evaluation of the LSF which extremely lengthen the computational time, particularly where the LSF is implicit. Many simulation methods have been proposed to tackle this problem, known as variance reduction methods [23], [24], [25], [26]. The importance sampling method is one of the most central variance reduction methods in which the number of generated random samples is raised in the failure zone. This is carried out by the transformation of the sampling density function to the point with the highest probability density on the LSF, i.e. the point with the shortest distance from the origin in the standard normal space, as presented in Eq. (3). Although the number of random samples is reduced, there is still a large number of random samples remained.
Above-mentioned methods are not being broadly used for real large-scale structures yet, since they incur high computation costs so that it is even impossible to calculate the reliability index in many problems. In other words, in the structural reliability analysis the performance function of structural systems is required to evaluate LSF, and generally, it is required to use finite element (FE) model, which is computationally time consuming [27]. In the recent years, the meta-modeling approaches including neural network [28], Kriging [29], [30], M5 model tree [31], [32] were implemented as effective tools to evaluate the LSF in structural reliability analysis.
One of the most effective methods to decrease computation costs in reliability problems is to apply the ANN. Using ANN, one can accurately calculate the LSF in a drastically short time which is significantly beneficial to the LSFs obtained by the output of a finite element model.
In the present paper, a new algorithm is proposed to calculate reliability index most efficiently by combining the modified importance sampling method and ANN. In each iteration of the proposed algorithm by limiting the SD of random variables of sampling density functions, the location of predicted design point is improved via generating random samples and, consequently, the more accurate design point is reached step-by-step. Furthermore, ANN is used to define the LSF.
Actually, combining FORM with sampling simulation method enables the proposed algorithm to find the design point, while in conventional simulation methods it is not possible to calculate the design point. On the other hand, the proposed method can successfully eliminate the problems of non-convergence which is very troublesome in the approximation method. The proposed hybrid FORM-Sampling simulation method has significantly reduced the number of samples with desirable accuracy. While simultaneously using the ANN model has improved the computational speed of the proposed algorithm and reduced the elapsed run time.
In fact, the proposed algorithm offers some outstanding advantages, namely accurate calculation of the design point and importance vector using the simulation method, reduction in the number of random samples compared to other available methods, and benefiting from the generated points in the safe region to calculate the design point on the LSF.
Section snippets
Importance vector
One of the most important information obtained from the secondary generation of FORM is the importance vector , calculated using Eq. (4). This vector provides specific information on the importance of random variables and their effect on the structural failure probability that can be beneficial for a diverse range of objectives. Or instance, it can be applied to reduce the dimensions of the problem so that random variables with lower importance can be removed from the random status and
Approximation of LSF using artificial neural network
ANNs are broadly used in the engineering applications to solve the complex problems that cannot be solved through conventional methods. The proposed methodologies can be applied to modeling, classification, pattern recognition, estimation, forecasting and more [39], [40], [41], [42], [43]. In this respect, ANNs can be very useful tools for the structural reliability problems. Since the reliability analysis is not practically feasible in many large and real problems.
For the calculation of the
Proposed algorithm
If no information on the design point and failure region is accessible at the beginning of the sampling operations, approximation methods cannot be conducted or any reason, and the LSF is the output of a finite element model, then the proposed algorithm is capable of accurately calculating the design point and failure probability in short computation time based on modified the SD by approximating the LSF with ANN.
In this method, the information on the failure region and design point increases
Numerical examples
To show the robust performance of the proposed algorithm, some examples have been used in this part. Some examples have been selected from conventional standard examples to evaluate the performance of the proposed algorithm in comparison to other conventional methods. The results obtained by the proposed algorithm have been compared to the results of the FORM approximation method and also the simulation Monte Carlo and shift importance sampling (SIS) method. The FORM method has been applied to
Conclusion
The present study proposed an efficient algorithm using the simulation method and FORM method to solve reliability problems. This algorithm can accurately calculate reliability index, design point, and importance vector with substantially fewer samples than that needed for available methods.
The proposed algorithm is able to approach the design point step-by-step by combining the importance sampling and limiting the standard deviation of the sampling density function, by generating fewer random
CRediT authorship contribution statement
Kiyanoosh Malakzadeh: Conceptualization, Methodology, Software, Validation, Formal analysis, Data curation, Writing - review & editing. Maryam Daei: Conceptualization, Methodology, Investigation, Writing - original draft, Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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