The Method of Solving Structural Reliability with Multiparameter Correlation Problem
Correlation among variables must be considered to accurately reflect the level of structure reliability. This problem has referential value to engineering practice and has attracted attention from relevant scholars and industries. In this paper, Copula function was used to build the joint probability density function among all variables. The key is to describe the correlation among variables, solve the correlation parameter of Copula function, and select the type of correlation structure among variables. The correlation parameter of Copula function was solved using Pearson linear correlation coefficient and maximum likelihood estimation. Based on the Akaike information criteria (AIC) and Bayesian information criteria (BIC), the optimal Copula function was selected, and the correlation structure among variables was determined. Monte Carlo method, which is based on Nataf inverse transformation, was introduced and used to evaluate the reliability of the correlated variable. Finally, this paper proposed the reliability calculation method based on dual neural network and direct integration by establishing the dual neural network of original and integrand functions. Compared with the Monte Carlo method, the proposed method can be utilized to efficiently and precisely calculate the structure reliability of multiple correlated random variables.
In structural reliability analysis, variables are often correlated, for instance, the positive correlation between the seismic peak displacement and permanent displacement  and among the stresses of the weld fatigue damage  and the significant negative correlation between the shear strength parameters of rock and soil and the parameters of pile-load-displacement curve [3, 4]. Therefore, the correlation among variables must be considered to rationally analyze the structural reliability. The traditional calculation method for structural reliability includes first-order  and second-order moments , which are only limited to the linear correlation of variables. A 2D or multidimensional distribution model is often used to characterize the correlation among variables, such as 2D lognormal distribution  and multidimensional normal distribution . However, one of the drawbacks of these models is the requirement of same-edge distribution variables, which greatly limit their application in reliability analysis.
The emergence and development of Copula theory solve the above problems and provide a new way to construct joint probability density function of correlated variables . Copula theory was first proposed by Sklar in 1959 . Under the framework of Copula function, variables could follow arbitrary edge distributions, and the linear and nonlinear correlations among variables could be defined . Copula functions were introduced in computational structural reliability because of its capacity to handle the arbitrary correlations of variables. Li et al. used the 2D Frank Copula function to construct the joint probability distribution function of two seismic attenuation models . Based on the Copula function, Tang et al. presented the load-displacement hyperbolic probabilistic analysis of pile foundation and used the Copula function to establish the joint probability distribution function of hyperbolic parameters . Huang et al. established a Copula function model of rock mass shear strength parameters based on measured data . Liu et al. used Copulas to develop a reliability model for systems with s-dependent degradation processes. The proposed model accommodates the assumptions of s-dependence among the degradation processes and allows for different marginal distributions . For two-component and multiple-component systems with multiple failure modes, Liu and Fan established the mixed Copula models for time-independent reliability analysis of series systems, parallel systems, series-parallel systems, and parallel-series systems . Based on the Copula function, Xu et al. constructed a joint probability density function, which was integrated on the failure field to calculate the failure probability of the structure . In reliability-based design optimization (RBDO) of the structure, the exact joint probability density function of the input relevant variables is necessary to obtain the optimal design . Lee et al. used the Copula function to consider the correlation among variables and solve the problem of RBDO [19, 20] and found that the correlation significantly affected RBDO results [21–25]. They also analyzed the influence of the confidence level for Copula function on RBDO results  and used the Bayesian and Markov chain Monte Carlo methods to select the Copula function . For RBDO with varying standard deviations (STDs), Cho et al. used the Copula function for the design of sensitivity and then improved the efficiency of the calculation .
The calculation methods for structural reliability are classified in three categories. The first method solves the structural response or the probability feature quantity of performance function [29–31]; however, the computational accuracy in this method is heavily dependent on the form of performance functions. The second method was the Monte Carlo method, which directly applies sampling statistics [32–34]. The disadvantage of this method is the large computational complexity. The third method was direct integration , which has encountered mathematical difficulties in the multiple integral and quantitative description of computational accuracy.
Neural networks can approximate any functions and thus could be properly applied to structural reliability. Papadrakakis et al. combined neural network with Monte Carlo simulation to analyze the reliability of elastoplastic structure; the time-consuming calculation of the Monte Carlo simulation was reduced, and the efficiency of the calculation was improved . Lopes et al. analyzed reliability using neural network instead of finite element analysis. The neural network had advantages in computational efficiency when compared with Monte Carlo method . Zuo et al. used neural network to fit the performance function of the structure. The values of performance function and partial derivatives were obtained at the point of mean values. Hence, the moments of performance function were calculated based on the moments of random variables . Cheng and Li used the neural network to simulate the limit state equation of long-span bridge. The genetic algorithm was used to train the network, and the failure probability of the structure was obtained . Meng et al. utilized BP neural network for nonlinear mapping function trains to obtain the explicit expression of stress in response to the random variable. A study analyzed reliability and the sensitivity of metal structure by combining random perturbation theory and first-order second-moment method . Elhewy et al. proposed a response surface method based on artificial neural networks and consequently reduced the computational complexity for reliability analysis . Li et al. proposed a structural reliability method, which is based on neural network and direct integration, to solve the reliability of independent variables . Dai and Cao developed a new neural network model based on wavelet support vector machine for reliability analysis; this model has extended the application of wavelet neural network to a high dimension .
This paper proposed a direct integration method that is based on dual neural network and could be applied for reliability calculation. The proposed method is composed of two neural networks with similar structure, multiple inputs, single output, and single hidden. By designing the function relation between the weights of two neural networks, one neural network is able to approximate integrands, whereas the other approximates original function. Therefore, the above networks were called integrand and original function neural networks. We only need to train the integrand neural network. Thus, the weights of original function neural network were provided directly by the function relation between the weights of the two neural networks. We subsequently used the original function neural network to calculate the multiple integral. In the proposed method, the integrand could easily obtain the sample data that were directly trained, and the integral computational accuracy would be greatly improved. Therefore, the proposed method can efficiently and accurately solve structural reliability problems.
First, we introduced the method by constructing the joint probability density function of correlation variable based on Copula function. This method includes the solution of correlation parameter and the selection of the structural type of the Copula function. Second, we introduced Nataf-Monte Carlo method. Third, we proposed direct integration based on dual neural network, which is in turn based on the integral form of reliability computation and the normalization method of the integral area. Fourth, we compared the proposed technique with Monte Carlo method and verified its effectiveness in simulation. Finally, the full-text conclusion and future prospects were presented.
2. Construction of Joint Probability Density Function Based on Copula Function for Correlated Variables
2.1. Copula Function
Copula theory was first proposed by Sklar in 1959. Sklar pointed out that any multidimensional joint distribution function could be decomposed into a corresponding edge distribution function and a Copula function. The Copula function determines the correlation among variables, including the size of the correlation coefficient and the type of the correlation structure . According to its strict definition stated by Nelsen, the Copula function is a function that associates the joint distribution function of the variable with its edge distribution function. In essence, Copula function is also a joint distribution function. For -dimensional cases, the Copula function was defined as the -dimensional joint distribution function. The edge distribution was in the n space .
According to Sklar theorem , the joint distribution function of variables , could be expressed aswhere is the edge distribution function of the variable : . is the Copula function, and is the correlation parameter of the Copula function.
From (1), the joint probability density function could be obtained aswhere is the edge probability density function of variables and is the density function for .
2.2. Correlation Parameter of Copula Function
The correlation parameter of the Copula function characterizes the correlation among variables. The method of solving the correlation parameter is different for different types of Copula functions.
When the Copula function of multidimensional variable was an Ellipse Copula function, the number of correlation parameters was the same as that of the correlation coefficients among variables and had one-to-one correspondence. The correlation parameter of any two variables and was . The relationship between and Pearson linear correlation coefficient was 
When the Copula function of the multidimensional variable was the Archimedean Copula function, the multidimensional Archimedean Copula function had only one correlation parameter because only one generator existed. This parameter describes the overall correlation between the multidimensional variables . Maximum likelihood estimation is generally used to solve the correlation parameter .
2.3. Selection of the Optimal Copula Function
Different Copula functions could describe different correlations among variables. Table 1 lists various Copula function types . Therefore, selecting the optimal Copula function, which could best fit the correlation among variables, is necessary when constructing the joint probability density function among variables. In this paper, we used Akaike information criterion (AIC) and Bayesian information criterion (BIC) to select the optimal Copula function. The best Copula function has the smallest AIC or BIC values and is well fitted for the correlated structure among variables. For multidimensional distribution models, the AIC and BIC values were expressed as follows [45, 46]: where is the number of correlation parameters . In the Ellipse Copula function, , whereas in the Archimedean Copula function, .
3. Monte Carlo Method Based on Nataf Inverse Transformation
Monte Carlo is a numerical simulation method that is widely used in structural reliability engineering [47, 48]. Based on statistical sampling theory, Monte Carlo method solves structural reliability by statistical sampling or random simulation of random variables. This method could obtain the true failure probability if the distribution type of the variable is known, and the number of sampling times is sufficient. In reliability calculation, Monte Carlo method is considered a precise calculation method . In this section, the Nataf-Monte Carlo method was introduced to calculate the structural reliability of correlated variables. This method was defined as Nataf-MCS.
The Nataf distribution model was proposed by Nataf in 1962 . The famous Nataf transform is a representation form of the Nataf distribution model, which includes positive and inverse transforms. Nataf transforms could convert correlated variables into independent variables. Nataf inverse transforms convert the independent variables into correlated variables.
If the cumulative distribution function of the variables were , then the correlation coefficient matrix was . The concrete steps were given by the Nataf-Monte Carlo method to calculate the structural reliability of the relevant variables as follows [49–51].
(1) The correlation coefficient matrix of the standard normal distribution variables was solved.Ang et al. established the following empirical equation  for various distribution types under normal circumstances :where the coefficient is a function of the correlation coefficient and the edge distribution . The equation for the coefficient was provided in literature .
(2) The standard normal distribution variables were correlated, and their correlation coefficient matrices were . was decomposed by Gorensky to obtain the lower triangular matrix .
(3) The traditional random number generation method was applied to produce independent variables , which all followed the standard normal distribution.
(4) Based on the lower triangular matrix , the independent normal random variables were transformed into standard normal distribution variables .
(5) The relevant standard normal distribution variables were mapped to the original spatial variables using the principle of equal probability transformation.
(6) The original spatial variables were substituted into the functional functions of the reliability analysis, and the structure response was calculated: . The number of times was the number of failure sample points . The unbiased estimate of the structural failure probability was  as follows:
4. Structural Reliability Calculations Based on Dual Neural Network and Direct Integration for Correlated Variables
Structural reliability could be expressed as the probability of function . could be expressed as a function of basic variables ; that is, . The joint probability density function of the correlated variables was , introducing a weight function: Structural reliability could be expressed as multiple integrals as follows:In (16), and were the mean and standard deviation, respectively, of random variable . and were positive integers, which were determined by accuracy requirements. In general, and .
The construction of dual neural network for multiple integral calculation was discussed in the following section.
We constructed a single hidden layer BP neural network to establish the mapping relationship between the input variable and the original function . The network structure is shown in Figure 1.
The relationships between the output variables and input variables could be expressed asThe derivation of (17) could be given bywhere . Equation (16) could be rewritten as a function of the relationship between the output and input variables as follows:
The neural network completed the mapping relationship between the input variable and the integrand . The network structure is shown in Figure 2.
Let the original function of the integrand be . According to multiple integral theory, function is the weighted algebraic sum of each vertex in a hypercube and could be expressed as
Thus, the original function and integrand neural networks were combined as a dual neural network. The two networks, namely, integrand network and original function network, had three layers, inputs, single output, and unit in hidden layer. In the integrand network, the connection weights and threshold of input layer to hidden layer unit were and , respectively; the connection weights and threshold of hidden layer to output layer were and 0, respectively; and the activation function of the hidden layer units was . In the original function network, the connection weights and threshold of input layer to hidden layer unit were and , respectively; the connection weights and threshold of hidden layer to output layer were and , respectively; and the activation function of the hidden layer units was . When the integrand network approximated the integrand in the integral, the original function network simultaneously approximated the original function.
5. Numerical Examples
5.1. Example 1
The performance function of a system was assumed as , where is the lognormal distribution random variables with mean value , standard deviation , and coefficient of variation . was an extreme value distribution of type I with mean value , standard deviation , and coefficient of variation . The correlation coefficient of and was . The structural reliability was calculated.
Based on the Copula function, the 2D joint probability density function of the random variables and was established and could be written aswherewhere , , , and .
Calculated by AIC and BIC criteria, in example 1, Plackett Copula, Frank Copula, number 16 Copula, and Gaussian Copula were selected to construct the 2D joint probability density function of random variables and . The values of the correlation parameter of each Copula function were obtained from (3) and are shown in Table 2.
and were substituted for (13) to obtain the integrand.
The integrand neural network had 2 inputs, single output, and 20 units in hidden layer. The transfer functions of hidden layer and output layer were and Purelin function.
The ranges of and and were divided into 50 parts. Two variables crossed each other to form the network input samples. The network output value of the corresponding sample points was calculated by the integrand. The training sample set is shown in Table 3.
This example used Levenberg-Marquardt training algorithm. Error convergence curve of training integrand network after 1000 steps is shown in Figure 3.
The relationship of the dual neural network was used to construct the original function network. The original function network structure was the same as the integrand network. The integrand network training sample set of each vertex in a hypercube was calculated and simulated using the original function network. The sample set is shown in Table 4. The results were substituted into (18) to obtain the result of reliability.
For comparison, we provide the results from Nataf-Monte Carlo method and the proposed method. The results calculated using Nataf-Monte Carlo method with sampling 1,000,000 were viewed as accurate solutions The relative errors are shown in Table 5.
The ratio of the calculation workloads of the Nataf-MCS and the proposed method can be obtained from the following equation:where is a single workload for structural analysis. is the calculation times of the Nataf-MCS. is the calculation times required to construct the network training sample. is a single workload for neural network training. is the neural network training times.
We can compare the computational efficiency of the two methods using the ratio of calculation workload. In this paper, . From the calculation in this example, .
From Table 5, the accuracy of the proposed method was close to that of the Nataf-MCS. The maximum relative error was only 2.00%. Based on ratio , the calculation workload of Nataf-MCS was 80 times higher than that of the proposed method. With regard to computational efficiency, the proposed method saves calculation time and greatly improves the computational efficiency of reliability.
5.2. Example 2
In the cantilever beam shown in Figure 4, Y is the horizontal load on the beam, is the lateral load on the beam, and represents the beam width. The performance function of structure . Considering the correlation between variables, the reliability of the cantilever beam was analyzed .
Calculated by AIC and BIC criteria, in example 2, Frank Copula, Gumbel Copula, and Clayton Copula were selected to construct the 3D joint probability density function of random variables Y, Z, and .
The 3D joint probability density function and the performance function in example 2 were substituted into (13) to calculate the integrand.
The structure of the integrand neural network could also be constructed. The integrand neural network had three inputs and single output. After adjustment, the number of hidden layer units is . The transfer functions of hidden layer and output layer were and Purelin function, respectively.
Variables , , and were separately divided into 30 parts in the range of . The three variables intersect each other to form the integrand network input sample point. The network output value of the corresponding sample point was calculated by the integrand function, and the training sample point set is shown in Table 8.
We used the Levenberg-Marquardt training algorithm. When the training reached 100 steps, the error of the integrand network reached the convergence state. The error convergence curve is shown in Figure 5.
The integrand network training sample set of each vertex in a hypercube was calculated and simulated with the original function network. The sample set is shown in Table 9. The results were substituted into (18) to obtain the result of reliability.
Nataf-MCS was introduced in Section 3. The sampling was 1,000,000 times, and the reliability of example 2 was 0.9998. In this example, . Table 10 shows the reliability values calculated by the two methods.
In this example, three kinds of Archimedean Copula functions were selected to construct the joint probability density function of variables , , and . The reliability of the proposed method was 0.85%, which was higher than that of Nataf-MCS. The ratio of calculation workload was 35.71, indicating that the proposed method has superior computational efficiency and can reduce the calculation time.
5.3. Example 3
The soil parameters of rock and soil include liquidity index (LI), undrained shear strength , remolded undrained shear strength , preconsolidation stress , and vertical effective stress . The experimental data in literature  were used as an example to illustrate the calculation steps of the proposed method. The performance function of the soil parameters was considered, and the reliability of rock and soil was analyzed under normal use limit state. Test data for soil parameters are shown in Table 11.
The mean , standard deviation , and edge distribution functions type of each variable were obtained by analyzing the experimental data as shown in Table 12.
From the calculation, we obtained the correlation coefficient of any two variables to form the matrix .
Given the positive and negative correlations among variables , , , , and , we need to select the appropriate Copula function from Elliptical Copula function to construct the joint probability density function . Using AIC and BIC criteria, Gaussian Copula was selected for example 2. Afterwards, (3) was used to obtain any two variables related parameters and to form matrix .
The 5D joint probability density function of variables , , , , and can be expressed as the following equation:
The 5D joint probability density function and the performance function were substituted into (13) to calculate the integrand.
The structure of the integrand neural network could also be constructed. The integrand neural network had five inputs and single output. After adjustment, the number of hidden layer units was . The transfer functions of hidden layer and output layer were and Purelin function, respectively.
Variables , , , , and are divided into 10 parts in the range of . The five variables intersect each other to form the input sample points of the integrand network. The network output value of the corresponding sample points was calculated by the integrand function. Given that each variable follows the lognormal distribution, we get when the left interval of the variable . The training sample set is shown in Table 13.
We used the Levenberg-Marquardt training algorithm. When the training reached 100 steps, the error of the integrand network reached the convergence state. The error convergence curve is shown in Figure 6.
We used the same method for the previous example. The sample points of the original function network were obtained and are shown in Table 14. The simulation values were substituted into (18) to obtain the reliability of the structure as shown in Table 15.
The calculation workload of the two methods was substituted into (21), thus obtaining .
The reliability of 5D correlated variables is analyzed using example 3. The reliability of the proposed method was extremely close to that of Nataf-MCS, and the relative error was only 1.16%. From , for the same problem, the workload of the two methods varied greatly. The neural network method has advantages in computational efficiency and reduces the calculation time for structural reliability analysis. This example shows that the proposed method can efficiently solve the reliability problem of multidimensional correlated variables.
Identifying the joint probability density function among variables is necessary to accurately calculate the structural reliability. This paper used the Copula function to construct the joint probability density function among correlated variables. According to Copula theory, the estimation of the edge probability density function among variables is independent of the selection of Copula function. The joint distribution function with arbitrary edge distribution and correlation structure could be constructed. This method was suitable for constructing joint probability density functions of 2D and multidimensional correlation problems. In addition, this paper proposed a Monte Carlo method based on Nataf inverse transformation. Nataf-MCS and the proposed method were used to solve the structural reliability among correlated variables. The proposed method could overcome the difficulty of solving multiple integral problems and achieve accurate calculation of reliability. However, we did not establish the relationship of the original function and every random variable because the training sample set originated from the integrand. We conveniently selected the sample set to improve the accuracy of calculating the reliability. Numerical examples showed that the proposed method is highly precise and efficient in calculating the reliability of multidimensional correlated variables.
Copula function is often used to construct the static variable correlation model. In many applications, the correlation coefficient among variables has the characteristics of being dynamic and time-varying. Thus, a Copula model must be constructed to describe the effect of these factors on structural reliability. This issue must be further explored.
In future work, the proposed method can also be extended for the reliability analysis of large and complex structural problems. In these cases, the workload of computing for dual neural network was negligible as compared with that of constructing network training samples. In addition, (21) can be simplified to . Hence, the proposed method shows efficiency and advantages in analyzing large and complex structural problems.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China under Grant no. 11262014 and Graduate Student Scientific Research Innovation Project of Autonomous Region under Grant no. B20171012805.
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