This paper refers to the Lagrangian mathematical model and the RackwitzFiessler transform model in modeling advanced mathematics applications. In this paper, the Lagrangian interpolation method and RackwitzFiessler transform are used to calculate the correlation coefficient efficiently. The purpose of this algorithm is to speed up the process of probabilistic modeling. This paper uses integral probability transformation to characterize the output's uncertainty by the production's joint distribution function. The purpose of this process is to improve the reliability of the calculation. Through the algorithm analysis, it is found that the method is reasonable and efficient.
Keywords
 Lagrangian mathematical model
 RackwitzFiessler transform
 Advanced mathematics
 Mathematical modeling
MSC 2010
 60G70
Reliability theory is developing and improving rapidly. It has been paid more attention to geotechnical engineering and has been widely used [1]. Scholars go through the linear transformation method and the Rosenblatt transformation. Scholars have done a lot of research to consider the correlation between random variables. Some scholars have introduced the correlated variable independence method based on Cholesky decomposition into the stochastic response surface method. But the technique only works in the case of correlated standard variables. The most widely used is the RackwitzFiessler transform. The transformation takes the marginal cumulative distribution function and covariance matrix of random variables as basic information. This paper adopts the RackwitzFiessler probability transformation and merges this transformation with the Lagrange multiplier method. At the same time, this paper proposes a primary reliability method based on RackwitzFiessler probability transformation and the Lagrange multiplier method [2]. In this paper, the rational polynomial method is introduced into the derivation of the inverse function expression of the distribution function. Finally, we demonstrate the effectiveness of the method proposed in this paper through a numerical example analysis.
We use the principle of equal probability transformation to introduce the standard normal random variable
Φ( ) and Φ^{−1} ( ) are the standard normal cumulative distribution function and the inverse of the cumulative distribution function. We find the following rules based on the RackwitzFiessler distribution theory [4]. When
The coefficient
In the formula
We can obtain the following theorem according to the firstorder secondmoment theory of reliability analysis [8–9]. We can prove that this distance is an essential indicator in reliability analysis: reliability indicator
After normalizing the normal variables, we have the following relationship:
If each random variable follows a general distribution, we can perform Gaussian transformation.
Suppose the inverse matrix of the lower triangular decomposition matrix
The formula (6) can be rewritten as:
Therefore, the standard regular space mathematical model of structural reliability index with correlation variables can be obtained from equation (9) as follows:
In the formula,
The value of the derivative of the function
The function of the function concerning the variable
(16) can also be written as:
In:
The coefficient
Rational polynomial coefficients for variable
/  /  /  /  

/  /  /  


/  /  



/  




Then from equations (16) to (19), we can get:
The limit state equation
We can get
We transform
In the formula, the distribution parameter
Ψ_{x1} (
Comparison of calculation results between this method and other methods
Calculation method  Checkpoint  Function function  Reliable indicator  

x1  x2  g(x)  β  
Nataf transform  6.4348  20.7033  −3.6335E05  2.6822 
Linear RackwitzFiessler Transform  6.4366  20.7148  −2.0729E06  2.6822 
Extended Algorithm  
Normal distribution transformation  6.4622  20.8801  −1.3240E06  2.6604 
The method of this paper  6.4314  20.6815  1.2120E07  2.6865 
It can be seen from Table 2 that:
The verification points and reliability indexes obtained by the method in this paper are consistent with the calculation results of other methods.
The absolute value of the function obtained by the method in this paper and the absolute value of the function calculated by other methods are relatively small. This shows that this paper's calculation method has higher accuracy than other methods.
The RackwitzFiessler transform and the Lagrange multiplier method are straightforward to calculate the structural reliability. Its calculation result is reasonable. Engineers may not program after obtaining the correlation coefficient
Comparison of calculation results between this method and other methods
Calculation method  Checkpoint  Function function  Reliable indicator  

x1  x2  g(x)  β  
Nataf transform  6.4348  20.7033  −3.6335E05  2.6822 
Linear RackwitzFiessler Transform  6.4366  20.7148  −2.0729E06  2.6822 
Extended Algorithm  
Normal distribution transformation  6.4622  20.8801  −1.3240E06  2.6604 
The method of this paper  6.4314  20.6815  1.2120E07  2.6865 
Rational polynomial coefficients for variable x
/  /  /  /  

/  /  /  


/  /  



/  




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