Rock-soil mass large-scale non-Gaussian random field simulation method based on loop nested matrix

Abstract

The invention discloses a rock-soil mass large-scale non-Gaussian random field simulation method based on a loop nested matrix. The method comprises the following steps: determining marginal distribution of material parameters of a selected constitutive model; solving L moments and L moment ratios of the first five orders of the parameters according to marginal distribution; determining an autocorrelation function of a random field, and if correlation exists among parameters, establishing a cross-correlation matrix; obtaining a correlation matrix of the underlying Gaussian random field at the initial grid point; according to the correlation matrix, establishing a non-negatively definite cyclic nested matrix; generating an independent or cross-correlation Gaussian random field; establishing a finite element model of geotechnical engineering; establishing a Gaussian random field sample corresponding to the finite element model; and non-Gaussian random field simulation and reliability analysis are realized. By adopting the above method, a technical route of combining the loop nesting method and the four-order Hermite polynomial model based on the L moment is adopted, the random field conforming to a given related structure and marginal distribution is generated, the calculation complexity is reduced, and the calculation efficiency is improved.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of geotechnical engineering, and particularly to a large-scale non-Gaussian random field simulation method for geotechnical body based on cyclic nested matrix. BACKGROUND

[0002] The spatial variability of soil material is a key factor in the safety evaluation of geotechnical engineering, and is usually simulated by random field method. In recent years, with the increasing attention to the quantification of uncertainty and risk assessment in geotechnical engineering, researchers have gradually realized that the three-dimensional spatial variability of geotechnical material will significantly affect the failure mechanism and reliability of geotechnical structure, and cannot be completely represented by simplified one-dimensional or two-dimensional random field. However, three-dimensional random field simulation has the bottleneck problem of low computational efficiency, so it is of great engineering significance to develop efficient three-dimensional simulation methods.

[0003] Although the existing random field simulation methods (such as Cholesky decomposition method, Karhunen-Loève expansion method, etc.) can theoretically realize the simulation of three-dimensional Gaussian random field, they face significant computational bottlenecks in practical applications. Specifically, the computational complexity of Cholesky decomposition method is cubic with the size of the correlation matrix, resulting in a sharp increase in computational load in three-dimensional cases; Karhunen-Loève expansion method requires eigenvalue decomposition of large-scale matrix, and when the dimension of the correlation matrix is high, the computational efficiency is greatly reduced; spectral representation method, optimal linear estimation method, etc. perform well in one-dimensional or two-dimensional simulation, but it is difficult to effectively extend to the three-dimensional scene commonly encountered in engineering. In addition, these methods not only consume time due to matrix decomposition and other mathematical operations, but also require excessive storage of correlation matrix, making large-scale, high-resolution three-dimensional random field simulation face serious memory limitations. Therefore, developing efficient and low-storage three-dimensional random field simulation methods has become one of the key challenges in current research.

[0004] A large number of field test data show that the parameters of geotechnical materials often exhibit significant non-Gaussian characteristics, and their statistical characteristics are often described by lognormal distribution with large coefficient of variation. In non-Gaussian random field simulation, although the traditional equal-probability transformation method is simple and direct, it will change the original correlation structure of the random field, especially under strong non-Gaussian conditions, which may lead to significant errors. Based on the L-matrix, the Hermite polynomial transformation has become a research hotspot in this field due to its strong ability to capture non-Gaussian statistical characteristics. However, the transformation process needs to be performed for each element of the correlation matrix, which will bring a heavy computational burden. In addition, this method needs to process each element of the correlation matrix individually, which still faces high computational cost in large-scale three-dimensional simulation. This limitation makes it difficult for existing methods to balance computational efficiency and statistical accuracy, and there is an urgent need to develop more efficient non-Gaussian random field simulation techniques. SUMMARY

[0005] The application aims to provide a large-scale non-Gaussian random field simulation method for rock-soil mass based on cyclic nested matrix to solve the technical problems mentioned in the background.

[0006] To achieve the above-mentioned purpose, the application provides a large-scale non-Gaussian random field simulation method for rock-soil mass based on cyclic nested matrix, and the steps include:

[0007] S1, determining the constitutive model of the rock-soil material used in numerical simulation, determining the material parameter with larger variation coefficient, and determining the marginal distribution function of each material parameter of the model according to the indoor test result;

[0008] S2, solving the first five order L moments and L moment ratios of the material parameters according to the marginal distribution of the material parameters;

[0009] S3, determining the expression of the parameter correlation function and the fluctuation range according to the actual situation, determining the initial grid spacing according to the fluctuation range, judging whether there is mutual correlation between the material parameters, if there is, determining the mutual correlation coefficient in advance and establishing the mutual correlation matrix;

[0010] S4, using the fourth order Hermite polynomial based on L moment to establish the relationship between the correlation structure of the underlying Gaussian random field and the correlation structure of the target random field, and obtaining the correlation matrix of the underlying Gaussian random field at the initial grid points;

[0011] S5, establishing a non-negative cyclic nested matrix according to the obtained correlation matrix of the Gaussian random field at the initial grid points;

[0012] S6, if there is no mutual correlation between the parameters, obtaining the Gaussian random field of two independent parameters based on the established cyclic nested matrix; if there is mutual correlation between the parameters, establishing two standard normal distribution random vectors with mutual correlation, and then obtaining the Gaussian random field of two correlated parameters based on the cyclic nested matrix and the mutual correlation matrix;

[0013] S7, establishing a finite element model of the rock-soil engineering to obtain the coordinates of the center points of the units;

[0014] S8, establishing the Gaussian random field sample corresponding to the finite element model according to step S6;

[0015] S9, establishing the model of the transformation from Gaussian variable to non-Gaussian based on the fourth order Hermite polynomial, and realizing the simulation of the non-Gaussian random field according to the Gaussian random field of the finite element model obtained in step S8.

[0016] Preferably, the calculation formula of L moment and L moment ratio in step S2 is:

[0017] L moment is obtained from order statistics, and the expression of L moment integral form is:

[0018]

[0019] where λ X,k (x) is the kth order L-moment of the parameter X(x); X(x) is a random parameter; x represents a position, and the parameter at any position in the random field is a random number; F[·] is the cumulative distribution function of X(x); ω k {F[X(x)]} is the kth order weight function, which is consistent in form with the shifted Legendre polynomial:

[0020]

[0021] where F denotes the cumulative distribution function, and P i represents the Legendre polynomial, represents the shifted Legendre polynomial, i = 0, 1, 2, 3, 4;

[0022] The L-moments represent the kurtosis, skewness, and hyper-skewness of the marginal distribution of the random field, and the calculation formula is:

[0023] τ X,k = λ X,k / λ X,2 (3)

[0024] where τ X,3 , τ X,4 , and τ X,5 are the L-skewness, L-kurtosis, and L-hyper-skewness of the random variable X, respectively.

[0025] Preferably, the formula for establishing the cross-correlation matrix in step S3 is:

[0026]

[0027] where β and η represent two cross-correlated parameters, and r βη is the correlation coefficient.

[0028] Preferably, the formula for establishing the relationship between the correlation structure of the underlying Gaussian random field and the correlation structure of the target random field in step S4 includes establishing the cross-correlation structure between the autocorrelation structure and the parameters, specifically including:

[0029] Autocorrelation structure: based on the correlation function expression determined in step S3 and the initial grid point spacing, the correlation coefficients between all grid points and the first grid point are calculated to form the first row r of the correlation matrix R, r k = r(|x k -x1|)k = 1, 2, 3,..., n;

[0030] Based on the fourth-order Hermite polynomial of the L-moment, the target non-Gaussian random field XNG (x) and the underlying Gaussian random field U(x) is given by:

[0031]

[0032] where He i [·] denotes the i-th order Hermite polynomial, i = 1, 2, 3, 4; U(x) denotes the value of the Gaussian random field at x; X NG-sta (x) denotes the normalized non-Gaussian random field; a and h j denote the constant coefficients of the polynomial, j = 3, 4, 5; μ X (x) and σ X (x) denote the mean and standard deviation of the target random field, respectively;

[0033] The fourth-order Hermite polynomial formula is:

[0034]

[0035] Since X NG-sta (x) has been normalized, its variance is 1, and the expression of the constant coefficient a is only related to h j , which is given by:

[0036]

[0037] Substituting the above formula into the covariance expression E[X NG-sta (x)X NG-sta (x)] of X NG-sta (x), we obtain the transformation relationship between the autocorrelation function value of the underlying Gaussian random field and the autocorrelation function value of the target non-Gaussian random field, which is given by:

[0038]

[0039] where r is the autocorrelation function value of the normalized non-Gaussian random field, r U is the correlation function value of the Gaussian random field;

[0040] Cross-correlation structure: transform the cross-correlation matrix in step S3 into the cross-correlation matrix p G between the underlying non-Gaussian random fields, and the transformation formula is:

[0041]

[0042] where r U-cro is the element in the cross-correlation matrix of the underlying Gaussian random field, r X-cro is the element in the cross-correlation matrix of the target non-Gaussian random field, and β, η denote the parameters of the two cross-correlations.

[0043] Preferably, the h j is determined by the statistical characteristics of the marginal distribution of the non-Gaussian random field and is solved based on the marginal distribution according to and r U The inversion of the underlying Gaussian random long correlation matrix is performed according to the one-to-one correspondence between the two, and the correlation matrix R G of the underlying Gaussian random field is obtained.

[0044] The h j Solving process includes:

[0045] According to the equal probability transformation theory Φ[U(x)] = F[X(x)], the kth order L-moment of the normalized non-Gaussian random field X NG-sta (x) is determined by the standard normal distribution function, and the formula is:

[0046]

[0047] In the formula, Φ(U) represents the standard normal distribution function, U represents the variable of the standard normal distribution, and U(x) represents the underlying Gaussian random field;

[0048] The standard normal distribution function is represented by the error function erf(·), and the formula is:

[0049]

[0050] Using the error function and the parity of the weight function ω k , the expression of the first five order L-moments of any distribution is obtained, and the formula is:

[0051]

[0052] Solving the integral in the L-moment expression, and bringing it into the L-moment ratio formula to obtain the relationship between the L-moment ratio and the coefficient h j in the Hermite polynomial model, and the value of h j is solved according to the relationship.

[0053] Preferably, the step S5 of establishing a non-negative circulant nested matrix includes:

[0054] First, a circulant matrix of a one-dimensional random field in the x direction is established, and the first row s G of the circulant matrix S is established according to the first row r G of the autocorrelation matrix R G , and the circulant matrix S is constructed according to s G , s G The construction formula is:

[0055]

[0056] In the formula, and denote the kth element of R G and s G , respectively;

[0057] According to perform a three-dimensional fast Fourier transform on s G , where the real part of s G must be all non-negative;

[0058] Then a circulant matrix of the three-dimensional random field is constructed, and according to the method of constructing the circulant matrix S of the one-dimensional random field, the first row r G of the autocorrelation matrix R num is sequentially circulant embedded in the x, y and z directions to form a three-dimensional matrix s num of size (2x num -1)×(2y G -1)×(2z num -1), wherein x num , y num and z num are the number of discrete points in the three directions of the initial grid, respectively.

[0059] Preferably, step S6 comprises:

[0060] Based on the constructed circulant nested matrix, a Gaussian random field of two independent parameters is obtained:

[0061] Generate a random variable ξ = ξ1 + iξ2 of standard normal distribution in complex space, where ξ1, ξ2 ~ N(0, I), I is the unit matrix, and the size of ξ is (2x num -1)(2y num -1)(2z num -1)×1, and combine the obtained Perform a three-dimensional fast Fourier transform again to obtain the realization of the random field in the complex space:

[0062]

[0063] Take the first (x num y num z 11 ) elements of the real part and the imaginary part of e, which correspond to two independent realizations of the Gaussian random field, respectively;

[0064] Based on the circulant nested matrix and the cross-correlation matrix, a Gaussian random field of two cross-correlation parameters is obtained:

[0065] Generate a random variable ξ = ξ1 + iξ2 of standard normal distribution in complex space, where ξ1, ξ2 ~ N(0, I), I is the unit matrix, and the size of ξ is (2x 12 , ξ12 , ξ 21 , and ξ 22 , each of which has a size of (2x num -1)(2y num -1)(2z num -1) x 1, A = [ξ 11 ξ 12 ], B = [ξ 21 ξ 22 ], and the cross-correlation matrix ρ G is decomposed by Cholesky to obtain a lower triangular matrix L, and the random matrices Xran1 and Xran2 are obtained according to the lower triangular matrix L, and the formula is as follows:

[0066]

[0067] wherein, Xran1 and Xran2 each have a size of (2x num -1)(2y num -1)(2z num -1) x 2, and each matrix has cross-correlation between two columns of random variables, and the two random matrices are independent of each other;

[0068] The first column of Xran1 and Xran2 forms a matrix C, and the second column forms a matrix D, the first column of the matrix is a real number part, and the second column is an imaginary number part, C and D are converted into two random vectors in a complex space, and two Gaussian random field samples of the cross-correlation parameters are generated according to the following formula:

[0069]

[0070] The first (x num y num z num ) elements of the real and imaginary parts of e1 and e2 are taken, corresponding to two independent realizations of the two Gaussian random fields;

[0071] According to the required number of random fields RFnum, the above steps are repeated RFnum / 2 times.

[0072] Preferably, the S8 specifically comprises:

[0073] According to the established finite element model, the coordinates of each grid center point are extracted, and three-dimensional linear interpolation calculation is performed according to the random field values at each grid point in the initial grid to obtain Gaussian random field sample values corresponding to the coordinates of each grid center point; RFnum times of interpolation calculation are performed on the generated Gaussian random field samples on the initial grid to obtain Gaussian random field samples corresponding to the finite element model.

[0074] Preferably, the S9 specifically comprises:

[0075] Based on the implementation of the RFNum number of Gaussian random field samples of step S8, combined with the calculated fourth-order Hermite polynomial model coefficient h j , according to the relationship formula between the target non-Gaussian random field X NG (x) and the correlation structure of the Gaussian random field U(x), the implementation of the RFNum number of target non-Gaussian random fields X NG (x) is obtained.

[0076] Therefore, the present application adopts the above-mentioned large-scale non-Gaussian random field simulation method for rock-soil mass based on cyclic nested matrix, which has the following beneficial effects:

[0077] (1) The simulation method of the present application generates a cyclic matrix by three-dimensional non-negative embedding of the correlation matrix, and the first row or the first column can include all data in the entire matrix, overcoming the difficulty of requiring a large amount of memory for storing the correlation matrix, laying a foundation for random field simulation of large-scale projects, and on this basis, the eigenvalues of the cyclic matrix are obtained by means of fast Fourier transform, and then the Gaussian random field is generated, which is efficient and time-saving.

[0078] (2) The fourth-order Hermite polynomial model based on L-matrix can accurately capture the statistical characteristics of the marginal distribution of the non-Gaussian random field, and can realize non-Gaussian transformation under the condition of keeping the correlation structure unchanged.

[0079] (3) With the special properties of the cyclic matrix, the inversion process of the underlying Gaussian random field correlation structure is simplified, which is suitable for non-Gaussian random field simulation of large-scale geotechnical engineering material parameters.

[0080] The technical solutions of the present application will be further described in detail below through the drawings and examples. BRIEF DESCRIPTION OF DRAWINGS

[0081] Figure 1 is a method flowchart;

[0082] Figure 2 is a system analysis flowchart;

[0083] Figure 3 is a regular non-Gaussian random field simulation flowchart;

[0084] Figure 4 is a schematic diagram of the relationship between the autocorrelation matrix (Toeplitz matrix) and the cyclic nested matrix;

[0085] Figure 5 is the distribution of the samples of the Gaussian random field before and after interpolation and the value of the Gaussian random field;

[0086] Figure 6 is a non-Gaussian random field sample schematic diagram;

[0087] Figure 7 Verification results of the random field correlation structure and marginal distribution after 1000 simulations;

[0088] Figure 8 Verification results of the random field for different fluctuation ranges;

[0089] Figure 9 Finite element model diagram for a simple slope;

[0090] Figure 10 Simulation diagram of a three-dimensional non-Gaussian random field simulation system;

[0091] Figure 11 Schematic diagram of a certain sampling result of cohesion and friction angle parameter random field;

[0092] Figure 12 Statistical results of slope stability calculation. DETAILED DESCRIPTION

[0093] To make the purposes, technical solutions and advantages of the embodiments of the present application clearer, the technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are some embodiments of the present application, rather than all the embodiments of the present application, and therefore should not be understood as limitations of the present application.

[0094] Embodiment 1

[0095] The present application provides a large-scale non-Gaussian random field simulation method for rock-soil mass based on cyclic nested matrix, which is illustrated by taking the Mohr-Coulomb constitutive model as an example, as shown in Figure 1 The steps include:

[0096] S1, determine the constitutive model of the rock-soil material used in numerical simulation, determine the material parameters with larger variation coefficients, and determine the marginal distribution function of each material parameter of the model according to the results of indoor tests.

[0097] The soil samples taken from the field are used for indoor tests to obtain the constitutive model parameters of the rock-soil material, including determining the dry density, elastic modulus, Poisson's ratio, cohesion and internal friction angle and other parameters of the soil body. Statistical analysis is performed on the soil sample parameters of all sampling points, and the results show that the cohesion (c) and internal friction angle of the rock-soil material have larger variation coefficients, and therefore can be regarded as random fields. The probability density function of the parameters, i.e., the marginal distribution function F[·] of the parameter random field, is obtained. In addition, there is a significant mutual correlation between the two parameters, which is usually described by a mutual correlation coefficient. The variation coefficients of other parameters are smaller, and therefore they can be assumed to be constants.

[0098] S2, solving the parameter first five order L-moments and L-moment ratio according to the marginal distribution of the material parameters, specifically comprising:

[0099] The L-moment is obtained from the order statistics, and the kth order L-moment can be defined as:

[0100]

[0101] In the formula, λ X,k (x) is the kth order L-moment of the parameter X(x); X(x) is a random parameter; x represents a position, and the parameter at any position in the random field is a random number; F[·] is the cumulative distribution function of X(x), and X k-q:k (x) represents the kth order statistics at x;

[0102] The expression of the integral form of the L-moment is:

[0103]

[0104] In the formula, ω k {F[X(x)]} is the kth order weight function, k=1, 2, 3, 4, 5, which is consistent in form with the shifted Legendre polynomial:

[0105]

[0106] In the formula, F represents the cumulative distribution function, and P i represents the Legendre polynomial, represents the shifted Legendre polynomial, i=0, 1, 2, 3, 4;

[0107] The L-moment ratio is used to represent the kurtosis, skewness and hyper-skewness of the marginal distribution of the random field, and the calculation formula is:

[0108] τ X,k =λ X,k / λ X,2 (3)

[0109] In the formula, τ X,3 , τ X,4 and τ X,5 are the L-skewness, L-kurtosis and L-hyper-skewness of the random variable X, that is, the third order, fourth order and fifth order L-moments used in the application.

[0110] After the marginal distribution function F[·] of the random field is solved by S1, the third to fifth order L-moment ratios can be solved by means of formula (1) to formula (3), which are used for the calculation of Hermite polynomial coefficients in the subsequent steps.

[0111] S3, determine the parameter correlation function expression and fluctuation range, and determine the initial grid spacing according to the fluctuation range, establish the autocorrelation structure of the random field at the initial grid point and the cross-correlation matrix between the material parameters.

[0112] The correlation function expressions commonly used in geotechnical engineering include but are not limited to exponential type, Gaussian type, triangular type, exponential cosine type and second-order autoregressive type, which need to be selected according to actual conditions. The specific correlation function expressions are:

[0113] Exponential type:

[0114]

[0115] Gaussian type:

[0116]

[0117] Triangular type:

[0118]

[0119] Exponential cosine type:

[0120]

[0121] Second-order autoregressive type:

[0122]

[0123] Where (x i ,y i ,z i ) and (x j ,y j ,z j ) are the coordinates of any two points, l x ,l y and l z are the fluctuation ranges in three orthogonal coordinate directions. It has been verified that the ratio of the fluctuation range to the initial grid spacing of the random field will affect the accuracy of the random field simulation and also affect the non-negative embedding of the correlation matrix in the second step. Generally speaking, the fluctuation range needs to be more than twice the initial grid spacing, that is, 2min(|x i -x j |)<l x , 2min(|y i -y j |)<l y , 2min(|z i -z j |)<l z .

[0124] The parameter correlation function is exponential, and the initial grid point correlation structure of the random field and the mutual correlation matrix between the material parameters are established based on the statistical results of the mutual correlation between the parameters in step S1 as follows:

[0125]

[0126] In the formula, c represents the cohesion, represents the internal friction angle.

[0127] S4, the relationship between the correlation structure of the underlying Gaussian random field and the target random field is established based on the fourth-order Hermite polynomial of L-moment, and the correlation matrix of the underlying Gaussian random field at the initial grid point is obtained.

[0128] The relationship between the correlation structure of the underlying Gaussian random field and the target random field includes the establishment of the autocorrelation structure and the mutual correlation structure between the parameters, which specifically includes:

[0129] First, the autocorrelation structure is established: according to the correlation function expression determined in step S3 and the initial grid point spacing, the correlation coefficients between all grid points and the first grid point are calculated to form the first row r of the correlation matrix R, r k = r(|x k -x1|)k = 1, 2, 3,..., n. Note that the initial grid requires that the spacing in each direction remains unchanged, so r contains the correlation information between any two points of the target non-Gaussian random field.

[0130] The loop nesting method for the discretization of the random field needs to use the Gaussian property of the random field, and the relationship between the target non-Gaussian random field X NG (x) and the correlation structure of the Gaussian random field U(x) must be established in advance, which is specifically established based on the fourth-order Hermite polynomial of L-moment, and the expression is:

[0131]

[0132] In the formula, He i [·] represents the i-th order statistical Hermite polynomial, i = 1, 2, 3, 4; U(x) represents the value of the Gaussian random field at x; X NG-sta (x) represents the normalized non-Gaussian random field; a and h j represent the constant coefficients of the polynomial, j = 3, 4, 5; μ X (x) and σ X (x) represent the mean and standard deviation of the target random field, respectively.

[0133] The first four-order Hermite polynomial formula is:

[0134]

[0135] Since X NG-sta (x) has been standardized, its variance is 1, i.e. The expression of constant a can be obtained only in terms of h j , which is:

[0136]

[0137] It can be found that after the parameter a is expressed in terms of h j , only h j to be solved is left in equation (7). Equation (5) - equation (7) is brought into the covariance expression of X NG-sta (x), E[X NG-sta (x)X NG-sta (x)], to obtain the conversion relationship between the autocorrelation function value of the underlying Gaussian random field and the autocorrelation function value of the target non-Gaussian random field, which is:

[0138]

[0139] In the formula, is the autocorrelation function value of the normalized non-Gaussian random field, r U is the correlation function value of the Gaussian random field.

[0140] Then, the cross-correlation structure is established based on the autocorrelation structure: the cross-correlation matrix in equation (4) is converted into the cross-correlation matrix p G between the underlying non-Gaussian random fields, and the conversion formula is:

[0141]

[0142] In the formula, r U-cro is an element in the cross-correlation matrix of the underlying Gaussian random field, r X-cro is an element in the cross-correlation matrix of the target non-Gaussian random field, c represents cohesion, and p G represents the internal friction angle. Compared with the autocorrelation matrix, the cross-correlation matrix is generally small, for example, the cross-correlation between two parameters in equation (4) is represented by a two-dimensional square matrix, and after the conversion of equation (9), the cross-correlation matrix p G of the underlying Gaussian random field corresponding to the two parameters can be obtained.

[0143] The coefficients h j in equation (8) and equation (9) can be completely determined by the statistical characteristics of the marginal distribution of the non-Gaussian random field, so and r UThe one-to-one correspondence between them can guarantee that the correlation coefficient value between any two points of the standard non-Gaussian random field can be solved by interpolation. This step is called inversion of the correlation matrix of the underlying Gaussian random field in the application. Through the processing of formula (8), the correlation matrix R G .

[0144] where h j Based on the L matrix ratio solving of the marginal distribution, the solving process includes:

[0145] According to the equal probability transformation theory Φ[U(x)] = F[X(x)], the kth order L moment of the standardized non-Gaussian random field X NG-sta (x) is determined by the standard normal distribution function, and the formula is:

[0146]

[0147] In the formula, Φ(U) represents the standard normal distribution function, and U represents;

[0148] The standard normal distribution function is represented by the error function erf(·), and the formula is:

[0149]

[0150] Using the error function and the parity of the weight function ω k , the expression of the first five order L moments of any distribution is obtained, and the formula is:

[0151]

[0152] Solving the integral in the L moment expression, and bringing it into the L moment ratio formula, i.e. formula (3), the relationship between the L moment ratio τ j and the coefficient h j in the Hermite polynomial model is obtained, as shown in Table 1. It can be found that there is a linear relationship between the L moment ratio τ j and the coefficient h j . Only the 3rd to 5th order L moment ratios of the marginal distribution function of the random field are needed to solve h3, h4 and h5 required by formula (5), formula (8) and formula (9).

[0153] Table 1 Relationship between L moment ratio and Hermite polynomial coefficient

[0154]

[0155] S5, according to the correlation matrix of the obtained Gaussian random field at the initial grid points, a non-negative circulant nested matrix is established.

[0156] Specifically, it includes:

[0157] Step S4 obtains the autocorrelation matrix RG Since the initial grid keeps the same interval in all directions, R G is a Toeplitz matrix whose elements on the diagonals parallel to the main diagonal are equal, the first row r G contains all the data of the whole matrix For a one-dimensional random field in x direction, the first row of the circulant matrix S can be obtained by cyclic embedding using equation (13) Each element of the next row vector of S can be obtained by moving the elements of the previous row vector to the right by one position, which belongs to a special Toeplitz matrix, so the first row s G The circulant matrix S can be easily constructed as follows:

[0158]

[0159] where, and denote the kth element of R G and s G respectively;

[0160] Then construct the circulant matrix of the three-dimensional random field For a three-dimensional random field, according to the one-dimensional method, the first row of the autocorrelation matrix R G is cyclically embedded in x, y and z directions in turn to form a three-dimensional matrix s num of size (2x num -1) x (2y num -1) x (2z G -1), where x num , y num and z num are the number of discrete points of the initial grid in the three directions The above steps realize the three-dimensional cyclic embedding of the autocorrelation matrix of the underlying Gaussian random field.

[0161] The three-dimensional cyclic embedding method used in the present application requires that the circulant matrix be non-negative definite, that is, the eigenvalues of S must be non-negative. The real part of sG must be non-negative. The autocorrelation matrix that meets this requirement can realize non-negative embedding, otherwise the autocorrelation matrix needs to be reconstructed. It has been verified that different correlation functions have different non-negative embedding conditions. For the exponential correlation function in Table 1, the fluctuation range cannot exceed 100 times the initial grid interval, and for the Gaussian correlation function, the ratio of the fluctuation range to the initial grid interval in the corresponding direction cannot exceed 5. The larger the fluctuation range relative to the initial grid, the more likely it is to appear negative embedding. Therefore, when using the present application, the non-negative embedding of the autocorrelation matrix needs to be verified, and the ratio of the initial grid interval to the fluctuation range is adjusted to meet the non-negative embedding condition.

[0162] S6, two interrelated standard normal distribution random vectors are established, and then based on the cyclic nested matrix and the interrelated matrix, the Gaussian random field of two interrelated parameters is obtained.

[0163] Specifically includes:

[0164] In the complex space, a random variable ξ = ξ1 + iξ2 of standard normal distribution is generated, where ξ1, ξ2 ~ N(0, I), I is a unit matrix, and ξ 11 , ξ 12 , ξ 21 and ξ 22 have the size of (2x num -1)(2y num -1)(2z num -1)×1, A = [ξ 11 ξ 12 ], B = [ξ 21 ξ 22 ], the interrelated matrix ρ G is decomposed by cholesky to obtain a lower triangular matrix L, and the random matrices Xran1 and Xran2 are obtained according to the lower triangular matrix L, and the formula is:

[0165]

[0166] Wherein, Xran1 and Xran2 have the size of (2x num -1)(2y num -1)(2z num -1)×2, and each matrix has interrelated random variables between two columns, and the two random matrices are independent of each other;

[0167] The first column of Xran1 and Xran2 is composed of matrix C, and the second column is composed of matrix D, the first column of the matrix is the real part, and the second column is the imaginary part, C and D are converted into two random vectors in the complex space, and two interrelated parameter Gaussian random field samples are generated according to formula (16):

[0168]

[0169] Take the first (x num y num z num ) elements of the real and imaginary parts of e1 and e2, corresponding to two independent realizations of two Gaussian random fields;

[0170] According to the required number of random fields RFnum, the steps of obtaining interrelated or independent Gaussian random field samples are repeated RFnum / 2 times.

[0171] S7, a finite element model of geotechnical engineering is established, and the coordinates of the center points of the units are obtained.

[0172] S8, Gaussian random field samples corresponding to the finite element model are established according to step S6. Specifically, it comprises:

[0173] According to the finite element model established in step S7, the coordinates of each grid center point are extracted, and the Gaussian random field sample values corresponding to each grid center point coordinate are obtained by performing three-dimensional linear interpolation calculation according to the random field values at each grid point in the initial grid. RFnum times of interpolation calculation are performed using the generated Gaussian random field samples on the initial grid of RFnum to obtain Gaussian random field samples corresponding to the finite element model. It has been verified that the error caused by interpolation has little to do with the interval of the interpolation points, and is mainly affected by the ratio of the initial grid interval and the fluctuation range. The smaller the fluctuation range is relative to the initial grid interval, the larger the interpolation error is. This is contrary to the condition of non-negative embedding in step S5. A suitable initial grid interval can be determined according to the calculation and verification of the two steps.

[0174] S9, a model for converting Gaussian variables to non-Gaussian is established based on the fourth-order Hermite polynomial, and non-Gaussian random field simulation is realized according to the non-Gaussian random field corresponding to the finite element model obtained in step S8. Specifically, it comprises:

[0175] Based on the RFNum Gaussian random field samples realized in step S8, combined with the calculated fourth-order Hermite polynomial model coefficients h j , RFNum target non-Gaussian random fields X NG (x) are obtained according to the relationship formula between the target non-Gaussian random field X NG (x) and the Gaussian random field U(x) related structure, i.e. formula (5).

[0176] S10: The non-Gaussian random field corresponding to the finite element model is brought into the finite element calculation, and the influence of the spatial variability of the material parameters on the engineering reliability is analyzed from the probability angle by Monte Carlo simulation.

[0177] In this example, S8 specifically comprises:

[0178] In order to enhance the practicability of the method in the probabilistic reliability analysis of geotechnical engineering, a large-scale non-Gaussian random field simulation system of rock and soil based on cyclic nested matrix is also provided, which comprises a simulation platform and an analysis module. The simulation platform encapsulates steps S2-S6 and steps S8-S9, and the analysis module comprises a three-dimensional random field simulation platform and a finite element calculation software, which realizes steps S7 and S10. For example, Figure 2As shown, the discrete point coordinates of the real model are completed by the modeling and meshing module in the finite element calculation software, and are input to the three-dimensional random field simulation platform in the form of a file. The marginal distribution parameters of the random field, the mutual correlation coefficients between the parameters, and the autocorrelation function are also input to the random field simulation platform by the user. After determining the input parameters, the random field is simulated in batches and statistically verified. The simulation results are output in the form of a file and the relative errors of the average PDF curve and the theoretical curve, the relative errors of the correlation coefficients between any discrete point and other points, the relative errors of the L-moment ratio, and the relative errors of the first two order moments are shown to evaluate the rationality and accuracy of the random field simulation. Finally, after parameterized modeling and several times of Monte Carlo simulation, the random finite element calculation is performed, and the reliability evaluation index is output for probability reliability analysis.

[0179] Example 2

[0180] Unlike Example 1, there is no mutual correlation between the constitutive model parameters of the adopted geotechnical material. In step S3, only the parameter correlation function expression needs to be determined. In step S4, the correlation matrix of all parameters of the underlying Gaussian random field at the initial grid points is obtained according to the determined autocorrelation function, i.e., the autocorrelation structure correlation matrix. Since there is no mutual correlation between the parameters, two independent Gaussian random fields are obtained based on the established loop-nested matrix in step S6, and the remaining steps are the same as in Example 1. The specific process is shown in Figure 1 .

[0181] Generating two independent Gaussian random fields includes:

[0182] In the complex space, a random variable ξ = ξ1 + iξ2 of standard normal distribution is generated, where ξ1, ξ2 ~ N(0, I), I is the unit matrix, and the size of ξ is (2x num -1)(2y num -1)(2z num -1) × 1. The obtained Again, a three-dimensional fast Fourier transform is performed to obtain the random field realization in the complex space:

[0183]

[0184] where e takes the first (x num y num z num ) elements of the real and imaginary parts, respectively, corresponding to two independent realizations of the Gaussian random field. According to the required number of random fields RFnum, the steps of obtaining mutually correlated or independent Gaussian random field samples are repeated RFnum / 2 times.

[0185] Example 3

[0186] To further verify the effectiveness of the present application, a regular non-Gaussian random field simulation is used as an example for further illustration.

[0187] As shown in Figure 3 , first, the initial grid range is determined according to the model range Ω = {100m, 100m, 100m}, and the fluctuation range l x = l y = l z = 40m. The initial grid spacing in each direction is taken as 2m, and the mean value of the parameters is assumed to be 5, the coefficient of variation COV = 0.4, and the parameters obey a lognormal distribution, i.e. the marginal distribution of the target non-Gaussian random field. The first five order L-moments and L-moment ratios of the marginal distribution of the random field are shown in Table 2, which are solved by equations (1) to (3). According to the relationship between the L-moment ratios in Table 1 and the Hermite polynomial coefficients, the linear equation set is solved to obtain the values of the coefficients h i : h3 = 0.1926, h4 = 0.0241, and h5 = 0.0023.

[0188] Table 2 Marginal distribution statistical characteristics

[0189]

[0190] According to step S3, the exponential function in Table 1 is selected as the autocorrelation function of the random field, and the cross-correlation is temporarily ignored. According to the selected autocorrelation function, the first row r of the parameter autocorrelation matrix is established by step S4, i.e. the correlation coefficient between the first discrete point in the initial grid and all discrete points. Then, by means of the linear interpolation method, r is converted into RG by using the relationship between the autocorrelation of the underlying Gaussian random field and the autocorrelation of the target non-Gaussian random field.

[0191] According to step S5, the RG is sequentially implemented in nested loops in the x, y, and z directions to obtain the loop-nested matrix S, and all elements in the first row of S are contained in sG. Taking a one-dimensional correlation matrix containing eight discrete points as an example, Figure 4 the relationship between the autocorrelation matrix (Toeplitz matrix) and the loop-nested matrix is shown. To determine whether S is a non-negative definite matrix, only the fast Fourier transform of sG is required, and when the real part of the transformed result is all positive, S is definitely non-negative, and if there is a negative number, the initial grid spacing needs to be adjusted and recalculated.

[0192] According to step S6, the Gaussian random field sample is obtained, for this example, the simulation of 125,000 discrete points at a time only takes 0.15s, the efficiency is very high, and the memory occupied during running is only 15200KB, which has very large potential to realize simulation of large-scale and high-resolution random field. Compared with existing matrix decomposition method, KL expansion method and other methods, it is found that the method adopted in the application consumes less memory and has high calculation efficiency. On a computer with 13th Gen Core TM i5-13600KF, 32GB RAM, a three-dimensional random field with a maximum of 5065.4375 million discrete points can be generated.

[0193] According to step S8, assuming that 100 points are uniformly inserted in each direction of the region Ω, the Gaussian random field sample value on the initial three-dimensional grid obtained by the above steps is used to generate a random field with 1 million discrete points by linear interpolation. The three-dimensional random field before and after interpolation of a certain sample is shown in Figure 5 (a) and (b). The probability distribution of all discrete points of the sample is shown in Figure 5 (c), which is a composite standard normal distribution and the distribution of the data before and after interpolation is consistent.

[0194] According to step S9, the Gaussian random field is converted into a non-Gaussian random field that meets the target correlation structure and marginal distribution. A certain sampling of a certain sample of the non-Gaussian random field is shown in Figure 6 (a); the distribution of all data in the random field sample is shown in Figure 6 (b). The histogram distribution result is consistent with the lognormal distribution with a mean of 5 and a coefficient of variation of 0.4. Finally, the above process is repeated 500 times to generate 1000 non-Gaussian random fields, and the marginal distribution is counted, and the related results are shown in Figure 7 (a), the average probability density curve is in good agreement with the theoretical probability density curve. In addition, correlation verification is also a key step to prove the rationality of the random field model. Taking the x direction as an example, taking all discrete points in the yz plane as the reference, the difference between the correlation decay curve of all discrete points along the x axis direction and the reference point and the theoretical curve is discussed. The verification result is shown in Figure 7 (b), the mean of all correlation decay curves is consistent with the theoretical curve. Select different fluctuation ranges for simulation, and the mean curve of the correlation function and the marginal distribution function is compared with the theoretical curve as shown in Figure 8 , where (a) is the comparison of the mean of the probability density curve with the theory, (b)-(d) are the comparison of the mean of the correlation coefficient decay curve in the x, y and z directions with the theoretical curve. The average curve is obtained from 500 calculations with the same parameters, and the results show that the mean is in good agreement with the theoretical value.

[0195] Embodiment 4

[0196] The application can also be used in engineering with irregular geometry, using a simulation system to evaluate the probabilistic reliability of the structure based on considering the spatial variability of material parameters. Figure 9 Taking a simple three-dimensional slope shown in FIG. 8 as an example, the model contains 36,000 hexahedral elements and 40,986 nodes, and the interpolation points in step S8 are replaced by the center point coordinates of each element in the slope finite element model. The soil density is 1800 kg / m 3 , the Poisson's ratio is 0.3, and the Young's modulus is 20 MPa. The cohesion c and the internal friction angle are random fields with a correlation coefficient of-0.5, a mean value of 30 kPa and 15° respectively, and a marginal distribution of lognormal distribution, and a variation coefficient from 0.2 to 0.7. The autocorrelation function takes the exponential function in Table 1, and the fluctuation range is 20 m in the horizontal direction lx and lz, and 2 m in the vertical direction ly. The above parameters are input into the three-dimensional non-Gaussian random field simulation system as random field simulation parameters, Figure 10 and FIG. 9 is a screenshot of the simulation system, showing various relative errors. The various relative errors show that the simulation accuracy of the random field is high, which can support the subsequent reliability analysis. Figure 11 (a) and (b) are a sample of two parameters respectively, according to step S10, the probabilistic reliability analysis is carried out by using the Monte Carlo method with the finite element software, and the safety factor of the simple slope obtained by 1000 times of static finite element calculation is as shown in FIG. 10. Figure 12 As can be seen from FIG. 10, as the variation coefficient of the random field variable increases, the distribution range of the safety factor becomes wider, and the mean value becomes smaller, so when the simulation parameters take the mean value instead of the random field, the safety of the slope may be overestimated.

[0197] Therefore, the application adopts the above-mentioned large-scale non-Gaussian random field simulation method for rock-soil mass based on cyclic nested matrix, adopts the technical route of combining the cyclic nested method with the fourth-order Hermite polynomial model based on L matrix, generates a random field conforming to the given correlation structure and marginal distribution, reduces the calculation complexity, and improves the calculation efficiency.

[0198] Finally, it should be noted that: the above embodiments are only used to illustrate the technical solutions of the application but not to limit them, although the application has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solutions of the application can still be modified or replaced by equivalents, and these modifications or equivalent replacements cannot make the modified technical solutions deviate from the spirit and scope of the technical solutions of the application.

Claims

1. A method for simulating large-scale non-Gaussian random fields of rock and soil based on cyclic nested matrices, characterized in that the steps include: S1. Determine the constitutive model of the geotechnical materials used in the numerical simulation, determine the material parameters with large coefficient of variation, and determine the marginal distribution function of each material parameter of the model based on the indoor test results; S2. Calculate the first five L moments and L moment ratios of the parameters based on the marginal distribution of the material parameters; S3. Determine the parameter correlation function expression and fluctuation range according to the actual situation, and determine the initial grid spacing according to the fluctuation range to determine whether there is mutual correlation between the material parameters. If so, determine the mutual correlation coefficient in advance and establish a mutual correlation matrix; S4. Using the fourth-order Hermite polynomial based on L moment, a relationship between the correlation structure of the underlying Gaussian random field and the correlation structure of the target random field is established to obtain the correlation matrix of the underlying Gaussian random field at the initial grid points; S5. Establishing a non-negative definite cyclic nested matrix based on the obtained correlation matrix of the Gaussian random field at the initial grid points; S6. If there is no mutual correlation between the parameters, obtain the Gaussian random fields of the two independent parameters based on the established cyclic nested matrix; if there is mutual correlation between the parameters, establish two mutually correlated standard normal distribution random vectors, and then obtain the Gaussian random fields of the two mutually correlated parameters based on the cyclic nested matrix and the cross-correlation matrix; S7. Establish a finite element model for geotechnical engineering and obtain the coordinates of the unit center point; S8. Establishing a Gaussian random field sample corresponding to the finite element model according to step S6; S9. A model for converting Gaussian variables to non-Gaussian variables is established based on fourth-order Hermite polynomials, and non-Gaussian random field simulation is achieved according to the Gaussian random field of the finite element model obtained in step S8.

2. The method for simulating large-scale non-Gaussian random fields of rock and soil based on cyclic nested matrices according to claim 1 is characterized in that: The calculation formulas for L moment and L moment ratio in step S2 are: The L moment is obtained from the order statistics, and the integral form of the L moment is expressed as: Where λ X,k (x) is the kth L-moment of the parameter X(x); X(x) is a random parameter; x represents the position, and the parameter at any position in the random field is a random number; F[·] is the cumulative distribution function of X(x); ω k {F[X(x)]} is the k-th order weight function, which is consistent with the shifted Legendre polynomial in form: Where F refers to the cumulative distribution function, P i represents the Legendre polynomial, represents the shifted Legendre polynomial, i = 0, 1, 2, 3, 4; The L moment ratio is used to express the kurtosis, skewness and excess skewness of the marginal distribution of the random field. The calculation formula is: t X,k =λ X,k / l X,2 ; Where, τ X,3 ,τ X,4 and τ X,5 are the L-skewness, L-kurtosis and L-excess skewness of the random variable X respectively.

3. The method for simulating large-scale non-Gaussian random fields of rock and soil based on cyclic nested matrices according to claim 2 is characterized by: The formula for establishing the cross-correlation matrix in step S3 is: In the formula, β and η represent two mutually correlated parameters, r βη is the correlation coefficient.

4. The method for simulating large-scale non-Gaussian random fields of rock and soil based on cyclic nested matrices according to claim 3 is characterized by: In step S4, establishing a relationship between the underlying Gaussian random field correlation structure and the target random field correlation structure includes establishing an autocorrelation structure or a cross-correlation structure between parameters, specifically including: Autocorrelation structure: Based on the correlation function expression determined in step S3 and the initial grid point spacing, the correlation coefficients between all grid points and the first grid point are calculated to form the first row r, r of the correlation matrix R. k =r(|x k -x1|) k=1,2,3,...,n; Establish the target non-Gaussian random field X based on the fourth-order Hermite polynomial of L moment NG The relationship between (x) and the correlation structure of the Gaussian random field U(x) is expressed as: In the formula, He i [·] represents the i-th order statistical Hermite polynomial, i = 1, 2, 3, 4; U(x) represents the Gaussian random field value at x; X NG-sta (x) represents the normalized non-Gaussian random field; a and h j represents the constant coefficient of the polynomial, j = 3, 4, 5; μ X (x) and σ X (x) represents the mean and standard deviation of the target random field; The fourth-order Hermite polynomial formula is: Due to X NG-sta (x) has been standardized, its variance is 1, and the expression of the constant coefficient a is only related to h j The formula is: Substitute the above formula into X NG-sta The covariance expression E[X NG-sta (x)X NG-sta (x)], and the conversion relationship between the autocorrelation function value of the underlying Gaussian random field and the autocorrelation function value of the target non-Gaussian random field is obtained. The formula is: Where, is the autocorrelation function value of the normalized non-Gaussian random field, r U is the correlation function value of the Gaussian random field; Cross-correlation structure: Convert the cross-correlation matrix in step S3 into the cross-correlation matrix ρ between the underlying non-Gaussian random fields G , the conversion formula is: Where r U-cro is the element in the underlying Gaussian random field cross-correlation matrix, r X-cro is the element in the cross-correlation matrix of the target non-Gaussian random field, β and η represent the two cross-correlation parameters.

5. The method for simulating large-scale non-Gaussian random fields of rock and soil based on cyclic nested matrices according to claim 4 is characterized in that: In step S4, h j Determined by the statistical characteristics of the marginal distribution of non-Gaussian random fields, and solved based on the marginal distribution, according to and r U The one-to-one correspondence between them is used to invert the underlying Gaussian random long correlation matrix to obtain the underlying Gaussian random field correlation matrix R G ; The h j The solution process includes: According to the equal probability transformation theory Φ[U(x)]=F[X(x)], the standardized non-Gaussian random field X NG-sta The kth L moment of (x) is determined by the standard normal distribution function, which is: Where Φ(U) represents the standard normal distribution function, U represents the standard normal distribution variable, and U(x) represents the underlying Gaussian random field; The standard normal distribution function is represented by the error function erf(·), which is expressed as follows: Using error function and weight function ω k The parity of the first five L moments of any distribution is obtained, and the formula is: Solve the integral in the L moment expression and substitute it into the L moment ratio formula to obtain the L moment ratio and the coefficient h in the Hermite polynomial model j According to the relationship, we can solve h j The value of .

6. The method for simulating large-scale non-Gaussian random fields of rock and soil based on cyclic nested matrices according to claim 5, characterized in that: The step S5 of establishing a non-negative definite loop nested matrix includes: First, establish the circulant matrix of the one-dimensional random field in the x direction, according to the autocorrelation matrix R G The first row of r G Create the first row s of the circulant matrix S G , according to s G Construct circulant matrix S, s G The construction formula is: Where, and Respectively represent r G and s G The kth element of ; according to to s G Perform a three-dimensional fast Fourier transform, where The real parts of must all be non-negative; Then, the circulant matrix of the three-dimensional random field is constructed. According to the method of establishing the circulant matrix S of the one-dimensional random field, the first row r of the autocorrelation matrix is G , sequentially expand the loop embedding in the x, y and z directions to form a (2x num -1)×(2y num -1)×(2z num -1) a three-dimensional matrix of size s G , where x num 、y num and z num are the number of discrete points of the initial grid in three directions.

7. The method for simulating large-scale non-Gaussian random fields of rock and soil based on cyclic nested matrices according to claim 6, characterized in that: Step S6 includes: Based on the established loop nested matrix, we get the Gaussian random field of two independent parameters: Generate a standard normally distributed random variable ξ=ξ1+iξ2 in complex space, where ξ1,ξ2~Ν(0,I), I is the identity matrix, and the size of ξ is (2x num -1)(2y num -1)(2z num -1)×1, combined with Perform the three-dimensional fast Fourier transform again to obtain the random field realization in the complex space: Take the first of the real and imaginary parts of e (x num y num z num ) elements, corresponding to two independent realizations of the Gaussian random field; Based on the cyclic nested matrix and the cross-correlation matrix, the Gaussian random fields of the two cross-correlation parameters are obtained: Generate a standard normally distributed random variable ξ=ξ1+iξ2 in complex space, where ξ1,ξ2~Ν(0,I), I is the identity matrix, ξ 11 ,ξ 12 ,ξ 21 and ξ 22 The size of both is (2x num -1)(2y num -1)(2z num -1)×1, let A=[ξ 11 ξ 12 ],B=[ξ 21 ξ 22 ], for the cross-correlation matrix ρ G Perform Cholesky decomposition to obtain a lower triangular matrix L. According to the lower triangular matrix L, the random matrices Xran1 and Xran2 are obtained. The formula is: Among them, the size of Xran1 and Xran2 are both (2x num -1)(2y num -1)(2z num -1)×2, there is mutual correlation between the two columns of random variables in each matrix, and the two random matrices are independent of each other; The first columns of Xran1 and Xran2 form the matrix C, and the second columns form the matrix D. The first column of the matrix is ​​the real part, and the second column is the imaginary part. Convert C and D into two random vectors in the complex space, and generate Gaussian random field samples of two cross-correlation parameters according to the following formula: Take the first (x) of the real and imaginary parts of e1 and e2 num y num z num ) elements, corresponding to two independent realizations of two Gaussian random fields; Repeat the above steps RFnum / 2 times according to the required number of random fields RFnum.

8. The method for simulating large-scale non-Gaussian random fields of rock and soil based on cyclic nested matrices according to claim 7, characterized in that: The S8 specifically includes: According to the established finite element model, the coordinates of each grid center point are extracted, and three-dimensional linear interpolation calculations are performed based on the random field values ​​at each grid point in the initial grid to obtain the Gaussian random field sample values ​​corresponding to the coordinates of each grid center point; the Gaussian random field samples on the generated RFnum initial grids are used to perform RFnum interpolation calculations to obtain the Gaussian random field samples corresponding to the finite element model.

9. The method for simulating large-scale non-Gaussian random fields of rock and soil based on cyclic nested matrices according to claim 8, characterized in that: The S9 specifically includes: Based on the RFNum Gaussian random field samples implemented in step S8, combined with the calculated coefficients h of the fourth-order Hermite polynomial model j , according to the target non-Gaussian random field X NG The relationship formula between (x) and the correlation structure of Gaussian random field U(x) is used to obtain RFNum target non-Gaussian random fields X NG Implementation of (x).

Images