An Improved Monte Carlo Reliability Analysis Method Based on BP Neural Network

Abstract

In order to reduce the number of calculations of the finite element model and the number of invocations of the implicit surrogate model in structural reliability analysis, an improved Monte Carlo reliability analysis method based on the Back Propagation (BP) neural network algorithm is proposed in this paper. The method mainly includes the improvement of the sampling method and Monte Carlo Simulation (MCS): (1) The key points generated by the random integer method and random sampling are used to uniformly cover the global design space. (2) The BP model is used to predict the sample points near the failure surface, and the sample points that are closer to the failure surface are screened out as validation sets and added to the training set for repeated iterative training to obtain the BP neural network prediction model with high prediction accuracy for the sample points near the failure surface. (3) The product of the probability density function of random variables in each sample point is defined as the weight, and then the concept of the weight critical value is proposed. When calculating the reliability of the MCS method, the sample points whose weights are greater than the critical value are considered as reliable; otherwise, the BP model is called to judge whether they are failed, thus greatly reducing the number of invocations of the BP model. Finally, the accuracy and efficiency of the Improved BP-MCS are verified by five examples, which shows that it has high practical value in engineering.

1. Introduction

In engineering structures such as bridges, tunnels, and slopes, there are several uncertain parameters, including external loads, mechanical properties of materials, and structural dimensions. These factors introduce significant uncertainty in the structural performance, which may lead to premature failure and make it difficult to achieve the expected service life. Structural reliability refers to the probability that an engineering structure will fulfill its intended function under specified time and service conditions, generally measured by the failure probability of the structure. There are various approximate methods for calculating structural failure probability, such as the first-order reliability method (FORM) [1], the second-order reliability method (SORM) [2], the response surface method (RSM) [3], importance sampling (IS) [4], and Monte Carlo simulation (MCS) [5]. MCS is an analysis method based on mathematical statistics, which defines the frequency of failure sample points in all sample points as failure probability. When the number of sample points is sufficiently large, the results exhibit high confidence [6]. However, when the calculation time of a single sample point is long (such as finite element calculation) or the failure probability is low, resulting in a large number of samples required, the huge number of sample points will greatly increase the time cost, reducing the applicability of this method.

In order to solve the problem of high time cost of the Monte Carlo reliability analysis method when performing a large number of sample point calculations, many scholars have proposed different methods for calculating surrogate models. The general approach is to replace the time-consuming finite element model (FEM) with an approximate model. Commonly used approximate models include polynomial response surface [7], kriging model [8], support vector machine (SVM) [9], and artificial neural network (ANN) [10,11], among others. The polynomial response surface method is simple and uses an explicit structural performance function, leading to short computation times, but it is only suitable for weakly nonlinear cases. The kriging model offers high accuracy, but its large initial sample size often requires acceleration methods for optimization [12,13,14]. Echard et al. proposed AK-MCS to reduce the calculation times of the finite element model while ensuring accuracy, and proposed AK-MCS-U, AK-MCS-EFF, and other active learning methods to accelerate [15]. With the increasing application of machine learning algorithms in civil engineering, implicit structural performance functions constructed by intelligent algorithms (such as SVM [16,17], ANN [18,19], BPNN [20], etc.) have also been applied to structural reliability analysis. These intelligent algorithms possess strong nonlinear fitting and generalization capabilities [21], which result in higher accuracy for the prediction models they generate. Consequently, the MCS failure probability results are more precise, though the computation time is longer than that of methods using explicit structural performance functions. Comprehensively comparing the scope of application and limitations of different surrogate models, this paper adopts the machine learning algorithm with higher prediction accuracy for the construction of the surrogate model in the MCS method, but the calculation time of the implicit structural performance function formed by the surrogate model is longer, and how to cut down the number of calculations of the surrogate model is to be explored.

While ensuring the accuracy of the failure probability calculation, how to efficiently select sample points that can represent the global design space is the key to reduce the calculation time. Rashki et al. took the probability density value of sample points as the weight, proposed to distribute sample points uniformly in the design space, and calculate the failure probability by approaching the most likely failure point (MPP) [22]. Compared with MCS, this method reduces the calculation times of the finite element model while retaining the accuracy, but still adopts the traditional approximate calculation method to approximate MPP. Okasha sorted the weights of sample points and avoided the calculation of sample points with very low weights by setting convergence criteria for failure probability [23]. This method further reduces the calculation cost but still evaluates a large number of reliable points. Chen et al. proposed a Monte Carlo reliability analysis method (BP-MCS) based on the BP neural network [24]. The method takes weight and distance to the failure surface as screening criteria, selects points near the failure surface from MCS sample points and adds them to the training set, and trains the BP model repeatedly until the accuracy meets the requirements. The BP model obtained by training has high prediction accuracy, but the training set still selects many sample points with very low weight. When calculating MCS failure probability, the number of calls of BP model is still consistent with the number of MCS sample points, resulting in high calculation cost. Therefore, based on the BP-MCS method proposed in the literature [24], this paper considers updating the sampling strategy, the method of constructing the BP model, and the MCS method of the method to a larger extent in order to enhance its applicability.

In summary, the high computational cost of the Monte Carlo reliability analysis method based on the surrogate model is mainly reflected in the following two aspects: first, the high number of finite element model calculations required for constructing the surrogate model; and second, the high number of calls of the surrogate model during the MCS reliability calculation. Therefore, this paper proposes an improved Monte Carlo reliability analysis method based on the BP neural network. The subsequent research content of this paper is mainly based on the following improvement ideas: firstly, the sampling strategy of the sample points required for constructing the surrogate model is explored, then an iterative construction method of high-precision surrogate model is proposed, and finally, the MCS method is improved in order to reduce the number of calls to the surrogate model. In order to reduce the calculation times of FEM, the key sample points generated by the random integer method and random sampling are used to uniformly cover the global design space, with additional points concentrated near the failure surface to improve the prediction accuracy of the BP model. In order to reduce the call times of the implicit surrogate model, the concept of weight critical value is proposed in combination with the definition of weight. When calculating the failure probability of MCS, if the weight of sample points is greater than the critical value, it is considered reliable. At this time, there is no need to call the BP model to calculate, and a large number of sample points located in reliable areas are greatly screened out. It is realized that the BP neural network prediction model with higher accuracy is constructed with fewer sample points to judge the failure points and calculate the failure probability of engineering structures quickly.

2. Calculation Theory of the BP-MCS Method

2.1. BP Neural Network Algorithm

The finite element model (FEM) is usually used to calculate the structural response value of engineering structures, and the mapping relationship between the actual engineering response value and the influencing variables can be represented by the mapping relationship between the finite element model response value and the influencing variables. With the continuous application of the machine learning algorithm in the field of civil engineering, the BP neural network, as an intelligent prediction algorithm with strong nonlinear mapping ability and high prediction accuracy, has been widely used in the prediction of actual engineering structure response [25]. In the structural reliability analysis, the trained BP neural network model can be used to replace the finite element model with long calculation time, thus improving the calculation efficiency and reducing the calculation cost.

BP neural network is mainly composed of input layer, hidden layer, and output layer. The input layer contains various influence variables and is linked to the hidden layer through different mapping functions and link weights. The output layer is generally the response value of the structure, and its main structure is shown in Figure 1. When constructing the BP neural network, it is necessary to have a training sample set consisting of input variables $X_i(i=\mathrm{1,\; 2}, \cdots , n)$ and output variables $Y_j\left(X_i\right)(j=\mathrm{1,\; 2}, \cdots , m)$. In the training process of the BP model, it is necessary to compare the accuracy of the predicted value $Y_j^{\prime }\left(X_i\right)$ with the actual value $Y_j\left(X_i\right)$, and carry out feedback correction on the link weight of the network structure according to the error back calculation, thus completing the whole learning process. When training the BP model, its nonlinear prediction ability and generalization ability can be improved by adjusting the basic parameters (such as weight and bias) and hyperparameters (such as the number of hidden layers, the number of hidden layer nodes, and the learning rate) of the algorithmic structure [26], where the hyperparameters are determined before the algorithm is trained, and the basic parameters of the network are continuously and adaptively adjusted to obtain the optimal network performance in the process of network training.

2.2. BP-MCS Calculation Method

Monte Carlo Simulation (MCS) mainly generates a large number of sample points and determines whether sample points fail by calculating the structural performance function values of each sample point. Finally, the frequency of failure points in all sample points is taken as the failure probability of the structure, namely:

$$\underset{N_{\mathit{MC}}\to +\mathrm{\infty }}{\lim }\left|{\displaystyle \frac{N_f}{N_{MC}}}-P_f\right|<\epsilon$$

(1)

According to Bernoulli’s law of large numbers, when the total number of samples $N_{MC}$ is large enough, the frequency of failure points converges to the failure probability $P_f$, and the difference between the two is less than any small positive number $\epsilon$. According to the coefficient of variation ($COV$) test of failure probability, $N_{MC}$ must be large enough to calculate a convincing and reasonable failure probability when MCS is used. The influence of $N_{MC}$ on calculation accuracy is mainly controlled by the coefficient of variation [15]:

$${COV}_{P_f}=\sqrt{{\displaystyle \frac{1-P_f}{P_f{N}_{MC}}}}$$

(2)

$COV$ reflects the deviation of the calculated results from the actual values. The larger the $N_{MC}$, the smaller the deviation and the more accurate the result. After the calculation of failure probability is completed, the coefficient of variation test should be carried out to ensure that there are enough sample points, and the general coefficient of variation can be taken as $COV=0.05$ [15]. For actual engineering structures, the failure probability is generally very small, usually on the order of 10⁻⁴ or even smaller [27]. According to Equation (2), at least 4 × 10⁶ finite element analyses are needed to obtain a more accurate failure probability, and the calculation cost is very high.

In the traditional BP-MCS method, when judging whether the sample point is failed, it is put into the BP model instead of the finite element model so as to shorten the calculation time of each sample point and improve the calculation efficiency. However, when using MCS to calculate the failure probability, it is still necessary to call the BP model to calculate at least 4 × 10⁶ times. In general, the traditional polynomial response surface has an explicit structural performance function, so its failure probability calculation time is shorter. However, for the implicit structural performance function constructed by intelligent algorithms such as the BP neural network, the calculation time of failure probability is relatively long. With the increase of the MCS sample number, the calculation time also doubles, especially when the failure probability is small. Therefore, how to reduce the number of BP model calls is the key to reducing the calculation time of BP-MCS failure probability, and to build a BP model near the failure surface with high prediction accuracy is the premise to improve the calculation accuracy of failure probability.

3. Calculation Theory of Improved BP-MCS Method

The prediction accuracy of the BP neural network for sample points near the failure surface largely determines the calculation accuracy of failure probability. In order to improve the prediction accuracy of the BP model near the failure surface, this paper proposes a sampling method of sampling points encryption near the failure surface. For low-dimensional problems (the number of variables < 4), the key points generated by setting “*” shape and random integer method cover the whole design space evenly; For high-dimensional problems (the number of variables ≥ 4), the whole design space is evenly covered by setting the key points generated by random integer method and random sampling of variables. Through finite element calculation, the failure of key points is determined, and the region containing the failure surface is identified. In this region, a set of additional sample points is generated by reducing the standard deviation, thereby constructing a BP neural network with high accuracy. Finally, the sample points near the failure surface are predicted by the BP model. Taking the distance from the sample point to the failure surface as the screening criterion, the sample point with a closer distance is substituted into the FEM for testing and added to the training set. Iterative training is repeated to obtain a BP neural network prediction model with high prediction accuracy for the sample points near the failure surface.

To solve the problem that the calculation time of implicit structural performance function constructed by the BP neural network is long in MCS reliability analysis, an improved calculation method of BP-MCS is proposed in this paper. In order to reduce the number of BP model calls when calculating MCS failure probability, the product of random variable probability density function is defined as the weight of sample point ${\omega }_i$. Considering that the probability of failure point of engineering structure is very low, that is, its weight is very small, there must be a weight-critical value ${\omega }_0$. When the weight of the sample point extracted by MCS is ${\omega }_i>{\omega }_0$, the sample point is in the reliable region, and there is no need to call the BP model to calculate the structural response value at this time. However, when the weight of sample points is ${\omega }_i<{\omega }_0$, it needs to be substituted into the BP model to judge whether it is failed. This method filters out most of the sample points located in the reliable region from the probability of occurrence, which greatly reduces the number of calls of the BP model, thus shortening the calculation time.

3.1. Definition of Weight ${\omega }_i$

In the design space, a sample point $X_i$ contains multiple random variables, denoted as $X_i^j(j=\mathrm{1,\; 2}, \cdots , m)$, and m is the number of random variables. Each random variable in a sample point obeys a different probability distribution, and the probability density function value of the sample point in the design space is used as its weight. Because the distribution of each random variable has its own probability density function, when they are independent of one another, the weight of sample point $X_i$ can be defined as the product of probability density functions, that is:

$${\omega }_i=f_1\left(x_i^1\right)f_2\left(x_i^2\right)\cdots f_m\left(x_i^m\right)$$

(3)

When each random variable is not independent of one another, it is replaced by the joint probability density function of each variable for calculation, that is:

$${\omega }_i=f_1\left(x_i^1\right)f_2\left(x_i^2\right)\cdots f_{m-1, m}\left({x_i^{m-1},x}_i^m\right)$$

(4)

where: $f_m$ is the probability density function of the m-th random variable; $f_{m-1, m}$ is the joint probability density function of the m − 1 and m random variables.

For an m-dimensional independent normal distribution, the weight of each sample point is:

$${\omega }_i={\displaystyle \frac{1}{{\left(\sqrt{2\pi }\right)}^m{\sigma }_1{\sigma }_2\cdots {\sigma }_m}}\mathrm{e}\mathrm{x}\mathrm{p}(-\left[{\displaystyle \frac{{\left(x_1-{\mu }_1\right)}^2}{2{\sigma }_1^2}}+{\displaystyle \frac{{\left(x_2-{\mu }_2\right)}^2}{2{\sigma }_2^2}}+\cdots +{\displaystyle \frac{{\left(x_m-{\mu }_m\right)}^2}{2{\sigma }_m^2}}\right])$$

(5)

where: ${\mu }_m$ is the mean of the m-th random variable; ${\sigma }_m$ is the standard deviation of the m-th random variable.

When it is an m-dimensional standard normal distribution, the weight of each sample point is:

$${\omega }_i={\displaystyle \frac{1}{{\left(\sqrt{2\pi }\right)}^m}}\mathrm{e}\mathrm{x}\mathrm{p}(-{\displaystyle \frac{1}{2}}\left(x_1^2+x_2^2+\cdots +x_m^2\right))$$

(6)

It can be seen from Equations (5) and (6) that the weight of the sample point is the largest at the center point $\left({\mu }_1,{\mu }_2,\cdots ,{\mu }_m\right)$. By observing the exponential term, ${\omega }_i$ is essentially equivalent to the distance between the sample point and the center point. The closer the sample point is to the center point, the greater the weight. The farther away from the center point, the smaller the weight. According to the normal distribution law, a large number of sample points randomly selected by MCS will have an aggregation effect to the central point $\left({\mu }_1,{\mu }_2,\cdots ,{\mu }_m\right)$, and the distribution of sample points is not uniform enough at this time. In order to solve this problem, this paper uses random integer method to design the sample points evenly.

3.2. Construction of Sample Points

For one-dimensional normal distribution, $\left[\mu -5\sigma , \mu +5\sigma \right]$ contains almost all sample points in the whole design space. Therefore, when sampling low-dimensional problems (the number of variables m < 4) by the random integer method, the sampling range can be defined as all integer points within [−5, 5], and then the sample points can be constructed according to the mean $\mu$ and standard deviation of variables $\sigma$, that is:

$$x_i={\mu }_i+a{\sigma }_i$$

(7)

where: $a$ is the integer value extracted by random integer method.

Since $a$ is an integer point in [−5, 5], the sample points constructed are very evenly distributed in the global design space. However, with the increase of variable dimension, the agglomeration effect of a large number of sample points to the center point becomes more and more obvious. At this time, it is necessary to explore the sampling frequency and extraction range of the random integer method.

Considering that the random integer method used to construct sample points is similar to the orthogonal experiment method with multi-factor and multi-level, when the sampling range of random integer method is [−3, 3], the orthogonal experiment design with multi-factor 7 level can be carried out. Based on the number of test groups used in the orthogonal test method, when the variable number m is 2~8, the orthogonal test table can be L₄₉(7⁸), and the number of test groups is 49. Taking the three-dimensional standard normal distribution as the research object, 49 groups of sample points were sampled by the random integer method and orthogonal test method, respectively, and the distribution of sample points is shown in Figure 2. Figure 2 shows that the spatial distribution of the sample points adopted by the orthogonal test method has the characteristics of stratification, and each factor is orthogonal in pairs and each level has the same number of occurrences. Therefore, it is found that the weights are discontinuous and have the characteristics of hierarchical distribution when calculated by Equation (6). However, the sample points adopted by the random integer method effectively avoid this phenomenon, and the sample points are evenly distributed in the design space and the weights are continuously distributed, which shows that the random integer method is essentially an optimization of the orthogonal test method. The orthogonal test method mainly selects representative sample points to reflect the global design space, so the number of experimental groups can be used as the sampling times of the random integer method. The orthogonal tests of different variable dimensions are shown in

Table 1. Orthogonal test table taken for different variable dimensions.

Dimensions of Standard Normal Distribution	Orthogonal Test Table
2~8	L49(78)
9	L64(89)
10~16	L98(716)

. When the number of variables m is 9, the orthogonal test table can be L₆₄(8⁹), and then the pseudo-horizontal method can be used to replace the excess level with the existing level. Considering that the sampling times of the random integer method are related to the dimensionality of random variables, and the number of experimental groups of the orthogonal test method should be included, the sampling times of random integer method can be 10 × m groups.

Since normal distribution can be normalized into standard normal distribution, this paper takes m-dimensional standard normal distribution as the research object and uses MCS to carry out random sampling of different orders of magnitude. The obtained sample point weight orders are shown in

Table 2. MCS sample points weight order of magnitude.

Number of Sampling	Order of Magnitude	Dimensions of Standard Normal Distribution
		2	3	4	5	6	7
107	max	1.59 × 10−1	6.35 × 10−2	2.53 × 10−2	1.01 × 10−2	4.00 × 10−3	1.59 × 10−3
	min	1.62 × 10−8	2.08 × 10−9	5.09 × 10−10	1.97 × 10−11	3.92 × 10−12	5.97 × 10−14
106	max	1.59 × 10−1	6.35 × 10−2	2.53 × 10−2	1.01 × 10−2	3.95 × 10−3	1.56 × 10−3
	min	4.74 × 10−7	1.44 × 10−8	1.01 × 10−9	2.23 × 10−10	1.23 × 10−11	6.49 × 10−13
105	max	1.59 × 10−1	6.35 × 10−2	2.52 × 10−2	9.97 × 10−3	3.92 × 10−3	1.47 × 10−3
	min	1.13 × 10−6	4.62 × 10−7	1.25 × 10−8	3.25 × 10−9	3.80 × 10−10	1.44 × 10−11
104	max	1.59 × 10−1	6.33 × 10−2	2.50 × 10−2	9.92 × 10−3	3.90 × 10−3	1.46 × 10−3
	min	3.61 × 10−5	3.01 × 10−6	4.32 × 10−7	4.72 × 10−8	3.03 × 10−9	5.02 × 10−10

. At the same time, the random integer method is used to carry out the statistics of weight orders in different extraction ranges, and the specific results are shown in

Table 3. Random integer method sample points weight order of magnitude (Number of samples N = 10 × m).

Number of Sampling (10 × m)		20	30	40	50	60	70
Dimensions of Standard Normal Distribution		2	3	4	5	6	7
[−3, 3]	max	9.65 × 10−2	2.34 × 10−2	5.65 × 10−3	5.03 × 10−4	2.01 × 10−4	4.86 × 10−5
	min	2.39 × 10−4	1.06 × 10−6	3.86 × 10−10	9.33 × 10−12	1.12 × 10−12	2.23 × 10−14
[−4, 4]	max	5.85 × 10−2	2.34 × 10−2	5.65 × 10−3	1.37 × 10−3	5.46 × 10−4	2.42 × 10−6
	min	1.79 × 10−8	9.67 × 10−10	4.29 × 10−12	1.56 × 10−15	2.09 × 10−19	2.79 × 10−23
[−5, 5]	max	9.65 × 10−2	2.34 × 10−2	9.32 × 10−3	9.21 × 10−6	3.68 × 10−6	4.43 × 10−8
	min	2.21 × 10−12	2.96 × 10−16	3.96 × 10−20	5.89 × 10−26	3.18 × 10−27	4.25 × 10−31

.

As can be seen from Table 2 and Table 3, when using the random integer method to construct sample points, for low-dimensional problems (m < 4), the weight order of sample points constructed in [−5, 5] can reach the minimum order of magnitude required when MCS is used to extract 10⁷ times. For high-dimensional problems (m ≥ 4), the weight order of sample points constructed in [−3, 3] can reach the minimum order of magnitude required for 10⁷ extraction with MCS. When the value range is [−4, 4] or [−5, 5], the minimum weight order of sample points is too small compared with MCS; that is, many sample points with extremely low probability are added. With the increasing number of random variables, the maximum weight order of sample points extracted by random integer method is also decreasing, and it cannot reach the maximum order of magnitude required by MCS for 10⁷ extraction times. Therefore, the missing area can be covered by random normal distribution of sample points extracted at the center point. Thanks to the characteristics of uniform design, the sample points constructed by random integer method are evenly distributed in the whole design space.

In view of the above statistical phenomena, this paper proposes a sampling method that can cover the global design space and distribute evenly. For low-dimensional problems, the range of the random integer method is [−5, 5]. At this time, there are 11 integer points, and the 10 × m key points generated by it are not enough to reflect the whole design space. Therefore, combined with the key point method proposed in reference [28], it is proposed to set the “*”-shaped key points combined with 10 × m groups of key points generated by the random integer method to evenly cover the global design space. The distribution of “*”-shaped key points is shown in Figure 3, and there are 25 groups of two-dimensional problems and 43 groups of three-dimensional problems. As can be seen from Figure 3, the “*”-shaped key points are evenly distributed in the design space. However, as the dimensionality of the variables increases, the number of required “*”-shaped key points increases dramatically and the applicability needs to be improved.

For high-dimensional problems, the value range of the random integer method is [−3, 3], then the global design space is uniformly covered by setting the key points generated by the random integer method and random sampling of variables. In this case, the number of sample points generated by random sampling of variables needs to be discussed. Similar to the method used to determine the sampling times of the random integer method, considering that the weights of the sample points need to maintain continuity and the number of groups of sample points is related to the dimensions of random variables, random sampling of 3~7 × m groups of m-dimensional standard normal distribution is carried out, respectively. and the weights of sample points are calculated. The weights of sample points obtained are shown in

Table 4. Random sampling method sample points weight order of magnitude (Number of samples N = 3~7 × m).

Dimensions of Standard Normal Distribution		4	5	6	7	8
3 × m	max	1.39 × 10−2	5.1 × 10−3	2.02 × 10−3	6.77 × 10−4	2.7 × 10−4
	min	3.91 × 10−4	1.15 × 10−4	3.59 × 10−5	5.23 × 10−7	1.02 × 10−7
4 × m	max	1.88 × 10−2	5.1 × 10−3	2.02 × 10−3	6.77 × 10−4	2.7 × 10−4
	min	3.91 × 10−4	1.09 × 10−5	3.43 × 10−6	5.23 × 10−7	1.02 × 10−7
5 × m	max	1.88 × 10−2	5.1 × 10−3	2.02 × 10−3	6.77 × 10−4	2.7 × 10−4
	min	1.46 × 10−4	1.09 × 10−5	3.43 × 10−6	5.23 × 10−7	1.02 × 10−7
6 × m	max	2.04 × 10−2	5.93 × 10−3	2.02 × 10−3	6.77 × 10−4	2.7 × 10−4
	min	1.46 × 10−4	1.09 × 10−5	3.43 × 10−6	5.23 × 10−7	1.02 × 10−7
7 × m	max	2.04 × 10−2	5.93 × 10−3	2.02 × 10−3	6.77 × 10−4	2.7 × 10−4
	min	1.46 × 10−4	1.09 × 10−5	3.43 × 10−6	5.23 × 10−7	1.02 × 10−7

. According to Table 3 and Table 4, the minimum weight orders of sample points randomly selected are all smaller than the maximum weight orders of 10 × m sample points extracted by the random integer method, which indicates that the combination of the two can effectively cover the missing area generated by the random integer method at the center point. At the same time, it is observed that when the number of random variables is 4~5 and the sampling times is 6 × m group, the weight order tends to be stable. When the number of random variables is 6~8 and the sampling times is 4 × m group, the weight order tends to be stable. At this time, the number of randomly selected sample points is about 30 groups, so the number of randomly selected sample points groups can be set as 30 groups.

3.3. High-Precision BP Neural Network Prediction Model

On the limit state surface, the structure performance function value is 0. In the reliable region, the structure performance function value is positive. In the failure region, the structure performance function value is negative. When using the BP model to calculate MCS reliability, we only need to pay attention to whether the response value of the sample points is reliable, but not to its value. Therefore, it is only necessary to ensure and improve the prediction accuracy of the BP model near the limit state surface on the basis of uniform coverage of the global region, then the BP model can be used to judge whether the sample points extracted by MCS are failed, so as to complete the failure-probability calculation of the structure.

The training steps of the high-precision BP model proposed in this paper are as follows:

(1) Extracting key points. For low-dimensional problems (m < 4), firstly, 10 × m groups of key points are extracted from [−5, 5] by the random integer method, and then the “*”-shaped key points are selected. The combination of the two forms the initial sample point set $X_i$. For high-dimensional problems (m ≥ 4), firstly, 10 × m groups of key points are extracted from [−3, 3] by the random integer method, and then the sample points of m-dimensional random normal distribution are extracted according to the mean ${\mu }_i$ and standard deviation ${\sigma }_i$ of each random variable (the number can be 30 groups). The combination of them constitutes the initial sample point set $X_i$. $X_i$ is substituted into the finite element model for calculation, and the response set $G\left(X_i\right)$ of structural performance function is obtained. The initial sample point set and the response set together constitute the initial set.

(2) Constructing the training set. For low-dimensional problems, the region where the failure surface is located can be determined according to the failure situation of the “*”-shaped key points, and then $0.5{\sigma }_i$ is taken from some key points close to the failure surface to generate 10 groups of sample points, respectively, which are substituted into FEM to calculate the response values. The newly added sample points and the initial set are used together as the training set. For high-dimensional problems, the initial set is directly used as the training set.

(3) Calculating the training set’s weight and constructing the prediction set. Calculating the weights ${\omega }_i$ of all sample points in the training set and arranging them in descending order. After the first failure point with $G<0$ is found, the weight of this point is taken as the boundary, respectively; m sample points close to 0 with greater weight are found upward, and (m − 1) sample points close to 0 with greater weight are found downward. Taking $0.5{\sigma }_i$ at these sample points to generate 10 groups of sample points to form a prediction set, the number of sample points in the prediction set is 2m × 10. At this time, these sample points do not need to be substituted into FEM for calculation.

(4) Training the BP model. Taking the sample point $X_i$ of the training set as input and the response value $G\left(X_i\right)$ as output, the initial BP model is trained by the MATLAB R2021a Neural Network toolbox, and the response $G^{\prime }\left(X_i\right)$ of the prediction set is calculated by the BP model.

(5) Verifying the model accuracy. The sample points of the prediction set are arranged in ascending order according to the size of $G^{\prime }\left(X_i\right)$, and then the sample points whose $G^{\prime }\left(X_i\right)$ is close to 0 are screened out. For the first iteration, all the sample points with $G^{\prime }\left(X_i\right)$ close to 0 can be taken as the validation set and put into FEM to calculate the response, and the error between the predicted value and the real value can be compared. For subsequent iterations, the number of sample points in the validation set can be 15% of the number of training sets each time [24], and at this time, the validation set takes the sample points with large weight and $G^{\prime }\left(X_i\right)$ close to 0. If the BP model meets the required prediction accuracy, output the model. Otherwise, add the sample points from the validation set to the training set, return to step (3) to generate a new prediction set, and retrain the BP model. Repeat the above steps until the prediction accuracy meets the requirements.

When training the BP model, its fitting and prediction accuracy can be evaluated using the determination coefficient ($R^2$), mean absolute error ($MAE$), and mean bias error ($MBE$). Higher model accuracy is indicated by $R^2$ values approaching 1, with $MAE$ and $MBE$ values close to 0 [29]. Prior to using the BP model for prediction, it must demonstrate robust fitting capability on the training set to enhance prediction accuracy. Therefore, the BP model should satisfy the following criteria on the training set before proceeding to further accuracy testing: $R^2>99\%$, $MAE<0.1$ and $MBE<0.01$. When verifying the model’s accuracy on the validation set, the BP model must meet the following standards to ensure reliable failure probability calculations: $R^2>98\%$, $MAE<0.1$ and $MBE<0.05$.

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^N{\left(G\left(X_i\right)-G^{\prime }\left(X_i\right)\right)}^2}{{\sum }_{i=1}^N{\left(G\left(X_i\right)-\overline{G\left(X_i\right)}\right)}^2}}$$

(8)

$$MAE={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N\left|G^{\prime }\left(X_i\right)-G\left(X_i\right)\right|$$

(9)

$$MBE={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N(G^{\prime }\left(X_i\right)-G\left(X_i\right))$$

(10)

3.4. The Steps of Reliability Calculation for Improved BP-MCS Method

Because the BP model constructs the implicit structure-performance function, it takes a longer time to predict the structure response than the explicit structure performance function. When the failure probability is small, the number of MCS calculations reaches more than 10⁶, which makes the calculation time increase dramatically. Considering that the failure probability of the actual engineering structure is very low, meaning the weight of its failure point is very small, then there must be a weight critical value ${\omega }_0$. When the weight of sample points extracted by MCS is ${\omega }_i>{\omega }_0$, these sample points are considered reliable, and there is no need to call the BP model to calculate the structural response value at this time. However, when the weight of sample points is ${\omega }_i<{\omega }_0$, the area contains all failure points, so it is necessary to call the BP model to determine whether failure occurs. Therefore, most of the sample points located in the reliable domain are filtered out in the probability of occurrence, which significantly reduces the number of calls of the BP model and shortens the calculation time.

The detailed implementation steps of the improved BP-MCS reliability analysis method proposed in this paper are as follows:

(1) The BP model with high prediction accuracy for the sample points near the failure surface is obtained by training. See 3.3 for details.

(2) Selecting the weight-critical value ${\omega }_0$. For the weight-critical value ${\omega }_0$, it can be selected through the sample points of the training set and the validation set after each BP model accuracy test: Arrange the sample points of the training set and the validation set according to their descending weights, finding the first failure point with $G<0$, and return the weight ${\omega }_1$ of the previous sample point with $G>0$. If there is a sample point very close to 0⁺ before the first failure point, the weight ${\omega }_1$ of this sample point is returned. To avoid not taking all failure points into account, the amplification factor $k$ can be added. According to the follow-up research, $k$ can be 1.1~1.3, then the critical weight value is:

$${\omega }_0=k{\omega }_1$$

(11)

(3) Using the improved MCS method to calculate the failure probability. Firstly, $N_{MC}$ sample points are extracted by MCS, and the weight of each sample point is calculated. When the weight of the sample point is ${\omega }_i>{\omega }_0$, the sample point is reliable, and there is no need to call the BP model. When ${\omega }_i<{\omega }_0$, the BP model should be called to judge whether it is failed. If the predicted value $G^{\prime }\left(X_i\right)<0$, the sample point will be failed. If the total number of failure points is $N_f$, the structural failure probability is:

$$P_f={\displaystyle \frac{N_f}{N_{MC}}}$$

(12)

(4) Testing the variation coefficient of failure probability according to Equation (2) to ensure that the number of sample points meets the requirements. If $COV<0.05$, the failure probability is output. Otherwise, go back to step (3) to generate more MCS sample points for calculation, and the initial sample number can be 10⁶. The above iterative process is repeated until the convergence requirement is met, and the final failure probability $P_f$ is output.

Combined with the training steps of the high-precision BP model, the specific calculation flow of the improved BP-MCS method is shown in Figure 4.

4. Examples Verification

In this paper, classical examples in relevant literature are cited for algorithm testing, and the calculation times $N_{FEM}$ of the mathematical formula or finite element model, the call times $N_{call}$ of the surrogate model, the coefficient of variation ${COV}_{P_f}$, the calculation accuracy $\epsilon$ of failure probability, and the calculation time $t$ are taken as evaluation indexes to verify the efficiency, accuracy, and applicability of the improved BP-MCS algorithm to engineering structures.

4.1. Reliability Problem of Series System

Example 1 considers a widely used benchmark problem with weak nonlinearity, which is a series system with four branches. The structural performance function is defined as:

$$G\left(x_1,x_2\right)=min\left\{\begin{array}{c}{g}_1=3+0.1{\left(x_1-x_2\right)}^2-\left(x_1+x_2\right)/\sqrt{2}\\ g_2=3+0.1{\left(x_1-x_2\right)}^2+\left(x_1+x_2\right)/\sqrt{2}\\ g_3=x_1-x_2+7/\sqrt{2}\\ g_4=x_2-x_1+7/\sqrt{2}\end{array}\right\}$$

(13)

where $x_1$ and $x_2$ are two independent random variables, both of which obey the standard normal distribution.

Based on its weak nonlinearity, many scholars had adopted different improved methods to analyze this explicit problem. The calculation results of the proposed method and related literature results are shown in

Table 5. Reliability calculation results of series system.

Method	${\mathit{N}}_{\mathit{F}\mathit{E}\mathit{M}}$	${\mathit{N}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{l}}$	${\mathit{P}}_{\mathit{f}}$	${\mathit{C}\mathit{O}\mathit{V}}_{{\mathit{P}}_{\mathit{f}}}$	$\mathit{\epsilon }$
MCS	106	106	2.233 × 10−3	0.0211	—
Kriging [30]	600	106	2.450 × 10−3	0.0202	9.72%
AK-MCS-U [15]	96	106	2.233 × 10−3	0.0211	0.00%
AK-MCS-EFF [15]	101	106	2.232 × 10−3	0.0211	0.04%
AK-MCS-IRS [28]	21	106	2.23 × 10−3	0.0212	0.13%
DRL [27]	2597	5 × 105	2.314 × 10−3	0.0208	3.63%
Direct BP-MCS	88	106	2.262 × 10−3	0.021	1.30%
Improved BP-MCS	88	15,714	2.243 × 10−3	0.0211	0.45%

. As can be seen from Table 5, some commonly used methods are mainly acceleration methods based on the Kriging model and the deep learning theory (DRL). These methods focus on how to reduce $N_{FEM}$, but do not take into account the computational burden brought by the calculation of a large number of MCS sample points. The sampling times of MCS adopted by the proposed method in this paper is 10⁶, and the BP model adopted by Direct BP-MCS in the table is consistent with the Improved BP-MCS.

According to Table 5, AK-MCS-IRS greatly reduces the calculation times of mathematical formulas compared with other methods, but this method is only applicable to low-dimensional problems [28]. The Improved BP-MCS algorithm proposed in this paper is only 88 times, which is lower than some commonly used AK-MCS improvement algorithms such as AK-MCS-U and AK-MCS-EFF. The DRL method based on deep learning theory, which is as high as 2597 times, is relatively poor in practicality. By comparing the call times of the surrogate model, it is found that the Improved BP-MCS algorithm is reduced from 10⁶ to 15,714, while other methods are consistent with MCS, so the computation workload can be greatly reduced. At the same time, the relative error of the failure probability obtained by the Improved BP-MCS is only 0.45%, indicating that the calculation accuracy is higher.

Based on the MATLAB platform, the failure probability calculation time of example 1 explicit polynomial, Direct BP-MCS, and Improved BP-MCS are calculated, and the calculation results are shown in

Table 6. Example 1 calculation time of failure probability by different methods.

Method	${\mathit{N}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{l}}$	$\mathit{t}$ (s)
Polynomial	106	19.6
Direct BP-MCS	106	7325.1
Improved BP-MCS	15,714	144.5

. By comparing the calculation time of explicit polynomial and Direct BP-MCS, it can be seen that the time cost of calculating the failure probability by using the implicit structural performance function is increased by about 373 times compared with the explicit structural performance function. However, when the improved method is adopted, the calculation time is only increased by about seven times.

Figure 5 shows the sample set finally used by Improved BP-MCS. It can be found that the key points used are evenly distributed globally, and the sample points added by iteration are mainly concentrated near the failure surface. For the amplification factor $k$, the failure probability is calculated for $k$ from 1.0 to 1.5, and the calculation results are shown in

Table 7. Calculation results of failure probability under different amplification coefficients.

$\mathit{k}$	${\mathit{P}}_{\mathit{f}}$	$\mathit{\epsilon }$	${\mathit{N}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{l}}$	$\mathit{t}$ (s)
1.0	2.197 × 10−3	1.61%	14,438	126.7
1.1	2.243 × 10−3	0.45%	15,714	144.5
1.2	2.239 × 10−3	0.27%	17,502	151.6
1.3	2.245 × 10−3	0.54%	18,595	155.6
1.4	2.248 × 10−3	0.67%	20,059	164.1
1.5	2.227 × 10−3	0.27%	21,163	173.3

. As can be seen from Table 7, with the increase of the amplification factor, the number of calls of the BP model also gradually increases, resulting in the increase of the calculation time of the failure probability, but its relative error gradually becomes stable. When $k$ is 1.1, the method proposed in this paper has high precision, and the relative error is only 0.45%, so the amplification coefficient $k$ of subsequent examples is 1.1. The weight adopted by the algorithm is ${\omega }_1=0.00228$, then the critical value of the weight is ${\omega }_0=0.00251$, and the critical region is shown as the circle in Figure 5. When the weight of the sample point ${\omega }_i>{\omega }_0$, the sample point is in the circular region, and the sample point is reliable. When ${\omega }_i<{\omega }_0$, meaning the sample point is outside the circular region, it needs to be substituted into the BP model to determine whether it is failed.

4.2. An Explicit Polynomial with a Low Probability of Failure

Example 2 is an explicit polynomial with a low probability of failure. The random variables $x_1$ and $x_2$ are independent of each other and obey the standard normal distribution. The structural performance function is:

$$G\left(x_1,x_2\right)=0.5{(x_1-2)}^2-1.5{\left(x_2-5\right)}^3-3$$

(14)

For this two-dimensional problem, Echard et al. [31] used MCS, Importance sampling (IS) method, and AK-IS method to calculate it, Huang et al. [32] used the classical AK-MCS to analyze its reliability, and Chen et al. [24] directly solved the failure probability of this problem by combining the BP neural network. This paper adopts the improved BP-MCS method to solve the problem, in which the total number of MCS sample points is 5 × 10⁷, and the calculation results are shown in

Table 8. Reliability calculation results of two-dimensional explicit polynomials.

Method	${\mathit{N}}_{\mathit{F}\mathit{E}\mathit{M}}$	${\mathit{N}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{l}}$	${\mathit{P}}_{\mathit{f}}$	${\mathit{C}\mathit{O}\mathit{V}}_{{\mathit{P}}_{\mathit{f}}}$	$\mathit{\epsilon }$
MCS	5 × 107	5 × 107	2.85 × 10−5	0.0265
IS	19 + 104	5 × 107	2.86 × 10−5	0.0239	0.35%
AK-IS	19 + 7	5 × 107	2.86 × 10−5	0.0239	0.35%
AK-MCS	27	4 × 106	2.88 × 10−5	0.0932	1.04%
BP-MCS	124	2 × 107	3.22 × 10−5	0.0387	11.49%
Improved BP-MCS	77	24,155	2.86 × 10−5	0.0265	0.35%

.

It can be seen from Table 8 that the number of sample point calculations of Improved BP-MCS is 77, which is obviously lower than that of MCS, IS, and BP-MCS, but there is still a certain gap with AK-IS and AK-MCS. In terms of surrogate model calls, the algorithm reduces the calculation times of 5 × 10⁷ to 24,155 times, which indicates that the calculation workload reduced by this method is more obvious when the failure probability is small. At the same time, the relative error of failure probability calculated by this method is 0.35%, indicating that the accuracy is high.

Compared with the BP-MCS algorithm adopted by Chen et al. [24], the Improved BP-MCS algorithm proposed in this paper has a large degree of improvement in accuracy and calculation cost. The specific explicit polynomial and the calculation time of the failure probability of Improved BP-MCS are shown in

Table 9. Example 2 calculation time of failure probability by different methods.

Method	${\mathit{N}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{l}}$	$\mathit{t}$ (s)
Polynomial	5 × 107	945.7
Direct BP-MCS	5 × 107	—
Improved BP-MCS	24,155	1335.0

. According to Table 9, the calculation time of the Improved BP-MCS is very close to the explicit polynomial, while the calculation time of the Direct BP-MCS may take 2,763,403 s according to the conversion of the calculation time of the Improved BP-MCS. This shows that compared with Direct BP-MCS, the time cost of the Improved BP-MCS is reduced by thousands of times, which proves the high efficiency of the method when the failure probability is low.

Figure 6 shows the final sample set used in Improved BP-MCS. It can be found that the key points adopted are distributed uniformly globally, and the sample points added by iteration mainly focus on the failure surface. Due to the low failure probability, the weight critical value adopted by the Improved BP-MCS is lower, that is, ${\omega }_0=7.7\times {10}^{-5}$, and the critical region is shown as a circle in Figure 6. A large number of sample points extracted by MCS will be concentrated in the circular region, at which time the sample points are reliable and there is no need to call the BP model for calculation, so the efficiency of this method is explained in space when the failure probability is low.

4.3. Dynamic Response of Nonlinear Oscillator

Example 3 involves a nonlinear undamped single-degree-of-freedom system, as shown in Figure 7, and its structural performance function is defined as:

$$G\left(X\right)=3R-\left|{\displaystyle \frac{2F_1}{M{\omega }_0^2}}\sin \left({\displaystyle \frac{{\omega }_0{T}_1}{2}}\right)\right|,{\omega }_0=\sqrt{{\displaystyle \frac{C_1+C_2}{M}}}$$

(15)

where $X=\left[R, F_1, M, C_1, C_2, T_1\right]$ is a random variable with normal distribution, and the distribution parameters are shown in

Table 10. Random variable distribution of nonlinear oscillator.

Random Variables	Distribution	Mean	Standard Deviation
R	Normal	0.5	0.05
F1	Normal	1	0.2
M	Normal	1	0.05
C1	Normal	1	0.1
C2	Normal	0.1	0.01
T1	Normal	1	0.2

. Aiming at the reliability problem of six-dimensional random variables, this paper uses Direct BP-MCS and Improved BP-MCS to calculate the failure probability based on the same BP model and compares the results with those of different methods. The sampling times of MCS are all 7 × 10⁴, and the calculation results are shown in

Table 11. Reliability calculation results of nonlinear oscillator.

Method	${\mathit{N}}_{\mathit{F}\mathit{E}\mathit{M}}$	${\mathit{N}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{l}}$	${\mathit{P}}_{\mathit{f}}$	${\mathit{C}\mathit{O}\mathit{V}}_{{\mathit{P}}_{\mathit{f}}}$	$\mathit{\epsilon }$
MCS	7 × 104	7 × 104	2.834 × 10−2	0.0221
Kriging [15]	600	7 × 104	3.086 × 10−2	0.0212	8.89%
AK-MCS [31]	530	7 × 104	2.852 × 10−2	0.0221	0.64%
AK-MCS-U [28]	212	7 × 104	2.841 × 10−2	0.0221	0.25%
Direct BP-MCS	93	7 × 104	2.809 × 10−2	0.0222	0.88%
Improved BP-MCS	93	37,957	2.79 × 10−2	0.0223	1.55%

.

Table 11 shows that compared with the classical methods such as Kriging, AK-MCS, and AK-MCS-U, the Improved BP-MCS algorithm has greatly reduced the number of sample sets, only 93 groups. The number of calls of the BP model is reduced from 7 × 10⁴ to 37,957, which shows that the improvement effect of the algorithm is reduced when the failure probability is large, and the relative error of the failure probability is 1.55%.

The calculation time of failure probability of explicit structure performance function, Direct BP-MCS, and Improved BP-MCS are shown in

Table 12. Example 3 calculation time of failure probability by different methods.

Method	${\mathit{N}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{l}}$	$\mathit{t}$ (s)
Explicit structural performance function	7 × 104	3.8
Direct BP-MCS	7 × 104	519.5
Improved BP-MCS	37,957	290.5

. When the failure probability is large, the calculation time of the explicit structure performance function is only 3.8 s, while the calculation time of MCS for the implicit structure performance function constructed by BP model is greatly increased, but the Improved BP-MCS still has a large degree of reduction compared with the Direct BP-MCS.

4.4. Multivariate Response Surface Polynomial

Example 4 comes from a vehicle side collision problem [33], which has random variables with seven-dimensional normal distribution. The specific distribution parameters are shown in

Table 13. Random variable distribution of multivariable response surface polynomials.

Random Variables	Distribution	Mean	Standard Deviation
x1	Normal	1.38	0.3
x2	Normal	1.38	0.3
x3	Normal	1.38	0.3
x4	Normal	1.38	0.3
x5	Normal	0.3	0.06
x6	Normal	0	10
x7	Normal	0	10

, and its explicit structural performance function is defined as:

$$G\left(X\right)=0.489x_1{x}_4+0.843x_2{x}_3-0.0432x_5{x}_6+0.0556x_5{x}_7+0.000786x_7^2-0.75$$

(16)

For this polynomial response surface, the results calculated by different methods are shown in

Table 14. Reliability calculation results of multivariable response surface polynomials.

Method	${\mathit{N}}_{\mathit{F}\mathit{E}\mathit{M}}$	${\mathit{N}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{l}}$	${\mathit{P}}_{\mathit{f}}$	${\mathit{C}\mathit{O}\mathit{V}}_{{\mathit{P}}_{\mathit{f}}}$	$\mathit{\epsilon }$
MCS [34]	5 × 106	5 × 106	1.57 × 10−4	0.0357
AK-IS [34]	108	5 × 106	1.5 × 10−4	0.0365	4.46%
SS [35]	3.7 × 105	5 × 106	1.6 × 10−4	0.0354	1.91%
AK-SS [35]	12 + 55.5	5 × 106	1.533 × 10−4	0.0361	2.36%
Direct BP-MCS	127	5 × 106	1.592 × 10−4	0.0354	1.40%
Improved BP-MCS	127	132,466	1.55 × 10−4	0.0359	1.27%

. Among them, the Improved BP-MCS algorithm needs 127 groups of sample points, which greatly reduces the calculation workload compared with the subset simulation (SS). Compared with the accelerated methods (AK-IS, AK-SS), the number is slightly larger, but it is still in a lower sample size range. The number of calls of surrogate model is reduced from 5 × 10⁶ to 132,466, which greatly reduces the calculation costs, and the relative error of failure probability is only 1.27%.

Table 15. Example 4 calculation time of failure probability by different methods.

Method	${\mathit{N}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{l}}$	$\mathit{t}$ (s)
Polynomial	5 × 106	371.9
Direct BP-MCS	5 × 106	36,521.2
Improved BP-MCS	132,466	1364.1

statistics the failure probability calculation time of the polynomial response surface, Direct BP-MCS, and Improved BP-MCS. When using Direct BP-MCS, its calculation time is as high as 36,521.2 s (about 10 h). When Improved BP-MCS is used, the calculation time is only 1364.1 s, which is about 1/27 of that of Direct BP-MCS. This shows that this method has a great reduction in time cost, but there is still a certain gap in calculation time compared with the explicit response surface polynomial.

4.5. Unequal Span Three-Span Continuous Beam

Example 5 is a simple finite element model, that is, a three-span continuous beam with unequal spans, which is common in engineering structures. The specific layout is shown in Figure 8. The side span is $L=20 \mathrm{m}$, and the middle span is $30 \mathrm{m}$. The rectangular beam section is adopted, with a height of $1.4 \mathrm{m}$ and a width of $0.8 \mathrm{m}$. The middle span acts as a uniform load $Q$, and the middle span bears a concentrated load $P$. See

Table 16. Random variable distribution of Three-unequal-span continuous girder.

Random Variables	Distribution	Mean	Standard Deviation
E (kN/m2)	Normal	3 × 107	0.6 × 107
I (m4)	Normal	0.1829	0.03
q (kN/m)	Normal	10.5	2.1
P (kN)	Normal	320	64

for the random variable distribution law of the unequal span three-span continuous beam.

The maximum allowable deflection of the main beam is $1.5\mathrm{L}/360=1/12 \mathrm{m}$, so the structural performance function of the unequal span three-span continuous beam is defined as:

$$G\left(X\right)={\displaystyle \frac{1.5L}{360}}-w_{max}$$

(17)

where: $w_{max}$ is the maximum deflection of the three-span continuous beam with unequal span, which is calculated by the ANSYS finite element model (version 2022R1). Beam188 element is used to simulate the continuous beam in ANSYS finite element model, with a total of 30 elements. The beam section is rectangular, and the constraints at the support are consistent with those in Figure 8.

For the implicit structural performance function with four random variables, the calculation results of different methods are shown in

Table 17. Reliability calculation results of three-unequal-span continuous girder.

Method	${\mathit{N}}_{\mathit{F}\mathit{E}\mathit{M}}$	${\mathit{N}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{l}}$	${\mathit{P}}_{\mathit{f}}$	${\mathit{C}\mathit{O}\mathit{V}}_{{\mathit{P}}_{\mathit{f}}}$	$\mathit{\epsilon }$
MCS [28]	5 × 106	5 × 106	7.402 × 10−4	0.0164
AK-MCS-U [36]	91	5 × 106	7.417 × 10−4	0.0164	0.2%
AK-MCS-IRS [28]	29	5 × 106	7.300 × 10−4	0.0165	1.38%
Direct BP-MCS	93	5 × 106	7.334 × 10−4	0.0165	0.92%
Improved BP-MCS	93	122,767	7.350 × 10−4	0.0165	0.7%

. The Improved BP-MCS only needs 93 sets of finite element sample points to get 0.7% relative error failure probability, and its $N_{FEM}$ is roughly the same as AK-MCS-U. AK-MCS-IRS has fewer $N_{FEM}$, but its relative error is 1.38% and only applies to low-dimensional problems [28]. Thus, the Improved BP-MCS calculation accuracy is higher, the calculation times of finite element model is less, and the call times of BP model is reduced from 5 × 10⁶ to 122,767 times.

In the calculation process, the calculation time of failure probability of Improved BP-MCS and Direct BP-MCS is counted. When using Direct BP-MCS, the calculation time is as high as 35,939.7 s (about 10 h). When Improved BP-MCS is used, the calculation time is shortened to 1116.2 s, which is about 18.6 min. Compared with Direct BP-MCS, the calculation time of Improved BP-MCS is obviously shortened, which verifies the efficiency of the algorithm.

5. Conclusions

Aiming at engineering structures with complex models, long time-consuming single-FEM calculations and low failure probabilities, this paper proposes an improved Monte Carlo reliability analysis method (Improved BP-MCS) based on the BP neural network algorithm. By calculating the failure probabilities of four explicit mathematical models and one implicit finite element model, the accuracy, efficiency, and engineering applicability of the Improved BP-MCS are verified. The specific conclusions are as follows:

(1) Taking the m-dimensional standard normal distribution as the research object, 10 × m groups of key points were extracted by means of the random integer method. It was found that the weight distribution was relatively uniform, and its minimum order of magnitude could reach that when 10⁷ sample points were extracted using Monte Carlo Simulation (MCS). Moreover, when combined with the key points generated from the random sampling of variables, the global design space could be uniformly covered.

(2) Five numerical examples are used to compare the Improved BP-MCS with some accelerated algorithms. The results indicated that the relative error of the failure probability calculated by the Improved BP-MCS was within 2%. There was little difference in the calculation times of the finite element model, yet the number of invocations of the surrogate model was significantly reduced. The Improved BP-MCS algorithm can be applied to reliability calculate problems of different dimensions.

(3) Comparing the failure probability calculation time of Direct BP-MCS and Improved BP-MCS, the results demonstrated that when the failure probability was low, the Improved BP-MCS required fewer invocations of the BP model, which could substantially reduce the calculation time.

The sampling strategy of random integer method combined with random sampling of variables proposed in this paper can effectively improve the computational accuracy of the surrogate model under fewer finite element model calculations, while the improved Monte Carlo reliability analysis method can effectively reduce the number of surrogate model calls in the calculation of failure probability. It should be noted that the method in this paper has high accuracy and computational efficiency in the reliability problem in high dimensions, but the sampling strategy is different in low dimensions, and the number of finite element model calculations required for constructing the surrogate model is at a medium level, so the unity of the computational method from different dimensions can be considered for subsequent research. Meanwhile, the performance of the Improved BP-MCS algorithm proposed in this paper is based on the nonlinear prediction ability and generalization ability of the BP neural network algorithm, because the BP neural network algorithm has the characteristics that may fall into the local optimal solution and has the initial parameter dependence. Therefore, the subsequent research can be adjusted to find the optimization of the initial parameters of the BP model through some intelligent optimization algorithms such as Genetic Algorithm (GA) and Particle Swarm Optimization (PSO).