Cross correlation random field generation method based on fast Fourier transform

Abstract

The invention discloses a cross correlation random field generation method based on fast Fourier transform. The cross correlation random field generation method comprises the following steps: setting region and frequency domain parameters; secondly, setting a mean value, a standard deviation and a distribution form of a random field, and completing conversion from spatial correlation to frequency domain correlation by adopting a square exponential autocorrelation function; then, constructing a cross correlation coefficient matrix, decomposing to obtain an upper triangular matrix, multiplying two groups of random matrixes by the triangular matrix to obtain a cross correlation type random matrix, and obtaining an overall frequency domain point coefficient expression form according to the frequency domain symmetry; and finally, generating a random field through inverse Fourier transform, and correcting the mean value and the standard deviation of the random field by adopting mean value regression and standard deviation regression. The method supports efficient processing of ten millions of units, adapts to normal / logarithmic normal distribution, accurately simulates multi-parameter cross correlation, and improves the accuracy and safety of engineering design.

Description

Technical Field

[0001] This invention belongs to the field of simulation research on spatial variability of soil and rock, specifically involving a method for generating cross-correlation random fields based on fast Fourier transform. Background Technology

[0002] Spatial variability of soil and rock masses refers to the variation in the physical and mechanical properties of soil and rock materials within a spatial range. This variability originates from factors such as the natural distribution characteristics, sedimentary processes, and diagenesis of soil and rock masses. For example, the differences in the mechanical parameters of soil and rock at different spatial locations affect the macroscopic mechanical behavior of soil and rock masses, thereby altering their performance in engineering, such as foundation settlement, bearing capacity of the foundation, and groundwater flow. Furthermore, different mechanical parameters of geotechnical materials, such as cohesion and internal friction angle, often exhibit certain cross-correlation and may show a non-normal distribution trend (log-normal distribution). This characteristic has been verified in several papers, such as Jiang, S., Li, D., Zhang, L., and Zhou, C. 2014. Slope reliability analysis considering spatially variable shearstrength parameters using a non-intrusive stochastic finite element method. Engineering Geology, 168, 120-128 and Qu, C., Liu, Y., Guo, H., Fang, H., Cheng, K., Yuan, H., and Chen, Y. 2025. Probabilistic Assessment of Soil–Rock Mixture Slope Failure Considering Two-Phase Rotated Anisotropy Random Fields. International Journal for Numerical and Analytical Methods in Geomechanics, 49, 1113-1125.

[0003] To address the complexity and uncertainty brought about by spatial variability, researchers often employ random field theory to simulate the spatial variability of soil and rock masses and the cross-correlation of material parameters, thereby improving the accuracy and safety of engineering designs. Common random field generation methods include covariance matrix decomposition (e.g., Chinese invention patent, titled: A Method for Modeling Non-Stationary Random Fields with Cross-Correlation of Soil and Rock Mass Parameters, patent application number: 202211404802.X) and Karhunen-Loève series expansion (e.g., Chinese invention patent, titled: A Method for Simulating Three-Dimensional Spatial Variability of Earth-Rock Dam Foundation Based on Discontinuous Galerkin Method, patent application number: 202110362390.7). However, covariance matrix decomposition is inefficient when dealing with large-scale matrix decomposition problems; while the Karhunen-Loève series expansion, which involves spectral decomposition of the covariance matrix, approximates the random field values, significantly improving efficiency. However, when the number of elements in the calculation is too large (N>10,000), the traditional Karhunen-Loève series expansion method still cannot effectively handle the situation. Although some researchers have improved the Karhunen-Loève series expansion method by discretizing the covariance matrix and then performing spectral decomposition to increase the number of computational units, this approach introduces further errors into the method, as seen in the paper Lin, X., Tang, XS, Li, DQ, and Wang, S. (2025). Bi-coupled discontinuous Galerkin method for realizing large-scale 3-D random fields with non-separable autocorrelation functions. Computers and Geotechnics, 185, 107356.

[0004] Given the efficiency issues of existing random field implementation methods, further research and improvement are needed. The Fast Fourier Transform (FFT) method can generate random fields with extremely high efficiency, but due to the relatively complex principle behind its generation, there are currently few papers and patents related to this method. Furthermore, existing FFT methods do not fully consider the issues of cross-correlation random fields, distribution types, and errors in mean and standard deviation. Summary of the Invention

[0005] The purpose of this invention is to provide a method for generating cross-correlation random fields based on fast Fourier transform, which solves the efficiency problem in existing random field technology, can easily handle tens of millions of computing units, and takes into account the cross-correlation and distribution type of random fields, while also solving the error problems that exist in random fields.

[0006] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0007] A method for generating cross-correlation random fields based on Fast Fourier Transform includes the following steps:

[0008] S1 defines the region's range, sets the horizontal and vertical lengths of the region, selects the number of horizontal and vertical spatial points, and calculates the horizontal bandwidth and frequency domain points, as well as the vertical bandwidth and frequency domain points.

[0009] S2: Set the mean and standard deviation of the random field, select the distribution form of the random field, set the horizontal and vertical correlation lengths, and use the squared exponential autocorrelation function to convert spatial correlation to frequency domain correlation.

[0010] S3: Set the cross-correlation coefficient and construct the cross-correlation matrix. Decompose the cross-correlation matrix to obtain the upper triangular matrix of the cross-correlation coefficient. Multiply the upper triangular matrix with two sets of random matrices to obtain the cross-correlation type random matrix. Based on the frequency domain symmetry, obtain the expression form of the overall frequency domain point coefficients.

[0011] S4: Based on the distribution pattern, a cross-correlated random field is generated through inverse Fourier transform, and mean regression and standard deviation regression are used to correct the mean and standard deviation of the random field.

[0012] Further optimization, step S1 specifically includes: constructing the horizontal length D of the spatial region D. x Vertical length D y And set the number of grid points N in the horizontal direction. x And the number of grid points N in the vertical direction y The number of computing units is N x *N y =N; where N x and N y The size must satisfy a power of 2. The distance Δx between two nodes in the horizontal direction and the distance Δy between two nodes in the vertical direction are given by equations (1.1) and (1.2):

[0013]

[0014] S12: Calculate the horizontal bandwidth Δω x With vertical bandwidth Δω y Specifically, see formulas (1.3) and (1.4):

[0015]

[0016] S13: For any spatial point (x) j y k ) and frequency domain point (ω) l ,ωm The calculations are as follows: (1.5)-(1.8)

[0017]

[0018] Further optimization, step S2 specifically includes:

[0019] S2.1: Set the mean μ of the random field X Standard deviation σ X And select the distribution form of the random field value, including two types: normal distribution and log-normal distribution.

[0020] When the distribution is normal, the mean μ does not need to be processed. X and σ X When the distribution is log-normal, it is necessary to handle the μ... X and σ X After completing the logarithmic transformation, both cases are uniformly represented as μ and σ, as shown in formulas (2.1) and (2.2):

[0021]

[0022] It is also important to note here that for the log-normal distribution, μ = ln(μ) X )-0.5*(σ X ) 2 and These are the mean and standard deviation after logarithmic transformation. This approach is only to facilitate the generation of the log-normal random field in step S4.1, while the mean and standard deviation of the log-distributed random field remain μ. X and σ X .

[0023] S2.2: Set the horizontal correlation length L x Vertical related length l y The quadratic exponential autocorrelation function ρ(x,y) is used, as shown in formula (2.3):

[0024]

[0025] Here, x and y are the spatial horizontal and vertical distance variables, respectively; the role of the correlation length can be described as follows: when the distance between two points is less than the correlation length in a certain direction, the random field values ​​at the two points exhibit a strong correlation. For a detailed explanation of this concept, please refer to Chinese Patent, Title: A Method for Modeling Cross-Correlation Non-Stationary Random Fields of Soil and Rock Mass Parameters, Application No.: CN202211404802.X.

[0026] S2.3: When generating random fields, the spatial correlation of the Fast Fourier Transform method is divided into discrete spectral density functions G(ω) in the first quadrant. x ,ω y Complete the transformation from ρ(x,y) to G(ω). x ,ω y The conversion is shown in formula (2.4):

[0027] Combining the quadratic exponential autocorrelation function expression in formula (2.3), G(ω) x ,ω y The derivation process of ) is as follows:

[0028]

[0029] in,

[0030]

[0031] It can be seen that S(ω) x ) and S(ω y The form is the same, with S(ω) x Let's take an example and perform the calculation:

[0032] First, regarding S(ω) x Differentiate:

[0033]

[0034]

[0035] Here,

[0036]

[0037] therefore:

[0038]

[0039] Formula (2.4-6) can be simplified to:

[0040]

[0041] The general solution to this differential equation is:

[0042]

[0043] Where C is an indeterminate constant; let ω x =0, substitute into formula 2.4-2, and find S(0):

[0044]

[0045] Substitute S(0) into formula (2.4-8):

[0046]

[0047] Find:

[0048]

[0049] Similarly, in formula (2.4-3), S(ω) y It can be written as:

[0050]

[0051] S(ω) x ) and S(ω y Substituting into formula (2.4-1) and simplifying, we get formula (2.5):

[0052]

[0053] For any frequency domain point (ω) l ,ω m Discrete spectral density G(ω) l ,ω m ), just use ω l Replace ω x ω m Replace ω y You can get it.

[0054] Further optimization, step S3 specifically includes the following steps:

[0055] S3.1: Set the cross-correlation coefficient ρ α,β Construct the cross-correlation matrix and decompose it to obtain the upper triangular matrix M of the cross-correlation coefficient, as shown in formula (3.1):

[0056]

[0057] Where, ρ α,β The value range of is [-1, 1], where α and β are the representative variables of the two random fields, such as the cohesion and internal friction angle of soil and rock materials; when ρ α,β When ρ takes negative values, the spatial variability of α is negatively correlated with the spatial variability of β, while when ρ takes negative values... α,β When the values ​​are positive, the spatial variability of α is positively correlated with the spatial variability of β.

[0058] S3.2: Generate two elements of size (N) x / 2+1)×(N y Given matrices ε1 and ε2 ( / 2+1), whose internal elements follow a Gaussian distribution, and construct a cross-correlation random matrix εα and ε β This process only requires performing the operation as shown in formula (3.2) on each element ε1(l,m) and ε2(l,m) of ε1 and ε2 to obtain ε. α and ε β Each element ε a (l,m) and ε β (l,m):

[0059]

[0060] Where l = 1, 2, ..., N x / 2+1, m=1,2,…,N y / 2+1.

[0061] S3.3: Frequency domain point (ω) l ,ω m The coefficients in the expression include real coefficients A. l,m and complex coefficient B l,m Both are about variance. and The random value, where and The value of is related to the values ​​of l and m, specifically including the following four:

[0062] (1) When l=1,1+N x / 2 and m=1,1+N y / 2 o'clock:

[0063]

[0064] (2) When l=1,1+N x / 2 and 2≤m≤N y / 2 o'clock:

[0065]

[0066] (3) When 2≤l≤N x / 2 and m=1,1+N y / 2 o'clock:

[0067]

[0068] (4) When 2≤l≤N x / 2 and 2≤m≤N y / 2 o'clock:

[0069] Sure and Afterwards, the randomness component is determined by ε. t (l,m) provides, therefore Al,m and B l,m The specific form is:

[0070]

[0071] in, and They are respectively represented as and The square root of ε t (l,m) represents the cross-correlation type random matrix ε t Each element; t represents the type, which can be the representative variable α or β of the random field, i.e., ε. t =ε α orε β .

[0072] For N x / 2+1 <l≤N x N y / 2+1 <m≤N y A l,m and B l,m In general, this can be obtained through symmetry. The principle of symmetry is introduced in the book "Fenton, GA, and Griffiths, DV (2008). Risk assessment in engineering technology. New York: John Wiley & Sons". The symmetry relationship derived from this principle is as follows:

[0073]

[0074] Where n equals 1 or 1+N γ / 2,N γ =N x or N y Thus far, any A has been obtained. l,m and B l,m , 1≤l≤N x , 1≤m≤N y .

[0075] Further optimization involves the following two steps in step S4:

[0076] S4.1: Based on the distribution form of the random field values, a cross-correlation random field is generated through inverse Fourier transform; where, when the distribution form is normal, the generated random field is shown in formula (4.1), and when the distribution form is log-normal, the generated random field is shown in formula (4.2):

[0077]

[0078] in, RF(x j ,y k ) t Indicates at point (x) in space j ,y k Random field values ​​generated at position (j=1,2,…,N) for the representative variable t. x k = 1, 2, ..., N y .

[0079] S4.2: Current random field generation methods all suffer from discrepancies between the random field mean and standard deviation and the setpoint mean μ. X and standard deviation σ X There is an issue of error. A Chinese invention patent, titled "A Random Field Modeling Method for Spatial Variability of Rock Mass Parameters under Complex Geological Structures," application number 202411201166X, proposes a method using mean regression and standard deviation regression, described in detail in step S6. Here, this invention employs this method to eliminate the fluctuation problem of the mean and standard deviation of the random field. The mean regression method requires first calculating the actual mean μ of the random field. real :

[0080] Then, according to formula (4.4), adjust each random value to ensure that the mean is close to the set value μ. X Consistency:

[0081]

[0082] in, For RF(x) j ,y k ) t After mean regression, at spatial point (x j ,y k Update value at );

[0083] In standard deviation regression, the first step is to calculate... Actual standard deviation σ real :

[0084] Recalculate With mean μ X Deviation Dis_RF(x) j ,y k ) t As shown in formula (4.6):

[0085]

[0086] Finally, adjust each random value to match the standard deviation σ. X :

[0087]

[0088] in, For RF(x) j ,y k ) t At the spatial point (x j ,y k The final generated random field values ​​for the representative variable t, j = 1, 2, ..., N x k = 1, 2, ..., N y , t = α or β.

[0089] Compared with the prior art, the present invention has the following beneficial effects:

[0090] 1. This invention effectively solves the problems of low efficiency and inability to handle large-scale computing units in existing technologies by introducing the Fast Fourier Transform method. This method remains highly efficient when dealing with tens of millions of computing units, meeting the needs of the vast majority of engineering projects.

[0091] 2. This invention considers the cross-correlation between random fields of soil and rock material parameters, as well as common distribution types of random fields, such as normal distribution and log-normal distribution.

[0092] 3. This invention utilizes mean regression and standard deviation regression methods, effectively solving the mean and standard deviation error problems of random fields, and providing a reliable generation technology for simulation research on spatial variability of soil and rock. Attached Figure Description

[0093] Appendix Figure 1 The flowchart shows a method for generating cross-correlated random fields based on fast Fourier transform.

[0094] Appendix Figure 2 The diagram shows the implementation effect of Example 1;

[0095] Appendix Figure 3 The diagram shows the implementation effect of Example 5;

[0096] Appendix Figure 4 The diagram shows the implementation effect of Example 3;

[0097] Appendix Figure 5 The diagram shows the implementation effect of Example 4; Detailed Implementation

[0098] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0099] Example 1:

[0100] In this embodiment, the cohesion c and internal friction angle of the soil and rock material are... Constructing a cross-correlated random field involves the following basis generation methods: Figure 1 As shown:

[0101] S1: Determine the area range, set the horizontal and vertical lengths of the area, and select the number of horizontal and vertical spatial points to obtain the horizontal bandwidth and frequency domain points, and the vertical bandwidth and frequency domain points;

[0102] Construct the horizontal length D of spatial region D x 50m, vertical length D y Also set to 50m, and the number of horizontal grid points N is set. x The number of grid points is 128 and the number of grid points in the vertical direction is N. y The number is 128, and the number of computational units is N. x *N y =N=16384; N x and N y The values ​​of both nodes satisfy the power of 2. Therefore, the distances Δx and Δy between two nodes in the horizontal direction and the vertical direction are as follows:

[0103]

[0104] Horizontal bandwidth Δω x With vertical bandwidth Δω y The calculation formula is:

[0105]

[0106] Therefore, for any spatial point (x) j y k ) and frequency domain point (ω) l ,ω m The formula for calculating ) is:

[0107]

[0108] S2: Set the mean and standard deviation of the random field, select the distribution form of the random field, set the horizontal and vertical correlation lengths, and select the quadratic exponential autocorrelation function to realize the conversion from spatial correlation to frequency domain correlation. The specific steps include the following:

[0109] S2.1: Set the mean μ of the random field X =1, standard deviation σ X The value is 0.2, and the random field is set to a normal distribution, so no additional processing is needed for μ. X and σ X As shown in formulas (2.1) and (2.2):

[0110] μ = μ X =1 (2.1)

[0111] σ=σ X =1 (2.2)

[0112] S2.2: Set the horizontal correlation length N x =2m, related vertical length L y =2m, and the quadratic exponential autocorrelation function ρ(x,y) is selected, as shown in formula (2.3):

[0113]

[0114] Where x and y are the horizontal and vertical distance variables, respectively.

[0115] S2.3: Complete the transformation from ρ(x,y) to G(ω) x ,ω y The conversion is shown in formula (2.4):

[0116] The derivation process is shown in formulas (2.4-1) to (2.4-11) of the instruction manual, and finally G(ω) x ,ω y The expression for ) is as follows:

[0117]

[0118] S3: Set the cross-correlation coefficient, construct the matrix, decompose to obtain the upper triangular matrix of the cross-correlation coefficient, generate two sets of random matrices, multiply by the upper triangular matrix of the cross-correlation coefficient to obtain the cross-correlation type random matrix, and obtain the expression form of the overall frequency domain point coefficients based on frequency domain symmetry. This includes the following three steps:

[0119] S3.1: Set the cross-correlation coefficient Construct the cross-correlation matrix and decompose it to obtain the upper triangular matrix M of the cross-correlation coefficient, as shown in formula (3.1):

[0120] Here, ρ α,β =-1 represents the cohesion c and internal friction angle of the soil and rock material. The correlation is completely negative, and the spatial distribution of c within the region is... Completely opposite.

[0121] S3.2: Generate two matrices ε1 and ε2 of size (128 / 2+1)×(128 / 2+1), with their internal elements following a Gaussian distribution, and construct a cross-correlation random matrix ε c and This process only requires performing the operation as shown in Equation 3.2 on each element ε1(l,m) and ε2(l,m) of ε1 and ε2 to obtain ε. c and Each element ε a (l,m) and ε β (l,m):

[0122]

[0123] Where l = 1, 2, ..., 128 / 2 + 1, m = 1, 2, ..., 128 / 2 + 1.

[0124] S3.3: Frequency domain point (ω) l ,ω m The expression for ) is given by the real coefficients A. l,m and complex coefficient B l,m Both are about variance. and The random value, where and The value of depends on the values ​​of l and m, and includes the following four:

[0125] (1) When l = 1, 1 + 128 / 2 and m = 1, 1 + 128 / 2:

[0126]

[0127] (2) When l = 1, 1 + 128 / 2 and 2 ≤ m ≤ 128 / 2:

[0128]

[0129] (3) When 2≤l≤N x / 2 and m=1,1+N y / 2 o'clock:

[0130]

[0131] (4) When 2≤l≤N x / 2 and 2≤m≤N yAt t = 128 / 2:

[0132] Determine and After that, the random part is provided by ε t (l, m), so the specific forms of A l,m and B l,m are:

[0133]

[0134] Where, and are respectively expressed as and the square roots of, ε t (l, m) represents each element of the cross - correlation type random matrix ε t ; t represents the type, which can take the representative variable c of the random field or That is

[0135] For A l,m and B l,m when 128 / 2 + 1 < l ≤ 128, 128 / 2 + 1 < m ≤ 128, they can be obtained by symmetry as follows:

[0136] A l,m = A 128-l+2,128-m+2 , B l,m = - B 128-l+2,128-m+2 (3.9)

[0137] A 128-l+2,m = A l,128-m+2 , B 128-l+2,m = - B l,128-m+2 (3.10)

[0138] A n,128-m+2 = A n,m , B n,128-m+2 = - B n,m (3.11)

[0139] A 128-l+2,n = A l,n , B 128-l+2,n = - B l,n (3.12)

[0140] Where, n is equal to 1 or 1 + N γ / 2, N γ = N x = 128, N γ = N y = 128; So far, any A l,m and B l,m, 1≤l≤128, 1≤m≤128.

[0141] S4: Based on the distribution pattern, a cross-correlated random field is generated through inverse Fourier transform, and mean regression and standard deviation regression are used to correct the mean and standard deviation of the random field. This includes the following two steps:

[0142] S4.1: Based on the distribution form of the random field values, a cross-correlation random field is generated through inverse Fourier transform; where, when the distribution form is a normal distribution, the generated random field is shown in formula (4.1):

[0143]

[0144] in, RF(x j ,y k ) t Indicates at point (x) in space j ,y k The random field values ​​generated at position j = 1, 2, ..., 128 and k = 1, 2, ..., 128 are for the representative variable t.

[0145] S4.2: This method is used to eliminate the fluctuations in the mean and standard deviation of random fields. Mean regression requires first calculating the actual mean μ of the random field. real :

[0146] Then, according to formula (4.4), adjust each random value to ensure that the mean is close to the set value μ. X Consistency:

[0147]

[0148] in, For RF(x) j ,y k ) t After mean regression, at spatial point (x j ,y k Update value at );

[0149] In standard deviation regression, the first step is to calculate... Actual standard deviation σ real :

[0150] Recalculate With mean μ X Deviation Dis_RF(x) j ,y k ) t As shown in formula 4.6:

[0151]

[0152] Finally, adjust each random value to match the standard deviation σ. X :

[0153]

[0154] in, For RF(x) j ,y k ) t At the spatial point (x j ,y k The final generated random field values ​​for the representative variable t, j = 1, 2, ..., 128, k = 1, 2, ..., 128, t = c or

[0155] The above steps were completed in just 0.015 seconds using an Intel CPU with an i7-11800H processor. Figure 2 Figures (a) and (b) show the cohesion c and the internal friction angle, respectively. The random field distribution of c. It is clear from the figure that the spatial distribution of c is similar to... The spatial distributions are completely opposite, proving that the cross-correlation coefficient... It is effective. Figure 2 In the text, (c) and (d) are c and d respectively. The numerical distribution plots of the random fields show that both exhibit normal distribution characteristics, which confirms the correctness of Formula 4.1. As for the mean regression and standard deviation regression methods, they have been described in detail in the Chinese invention patent, entitled "A Random Field Modeling Method for Spatial Variability of Rock Mass Parameters under Complex Geological Structures," application number: 202411201166X, and will not be verified separately.

[0156] Example 2:

[0157] In Example 1, the total number of computing units N is only N x *N y =128×128=16384, therefore Figure 2 The changes in random field values ​​between computational units in (a) and (b) exhibit obvious coarse characteristics, which can be addressed by increasing N. x and N y The value can be used to change this situation.

[0158] Therefore, in this embodiment, N x *N y The values ​​are set to 256×256 and 4096×4096 respectively. The total number of computing units in the second case is 16,777,216, which is nearly 17 million computing units. Figure 3 (a) and Figure 3(b) represents c in the first case. Spatial distribution, Figure 3 (c) and Figure 3 In the second case, (d) represents c and The spatial distribution is shown in the figure. It is clear that as the number of computational units increases, the continuity of the spatial distribution becomes smoother. The computation times for the two cases are 0.024 seconds and 3.93 seconds, respectively. Obviously, even for computational units in the tens of millions that meet the needs of most engineering simulations, the generation time is less than 4 seconds, fully demonstrating the efficiency of this method.

[0159] Example 3:

[0160] In this embodiment, by changing the horizontal correlation length L x Vertical related length L y This allows for changes in the length of spatial correlation. In N x *N y When L = 4096 × 4096, let L = 4096 × 4096 respectively. x =10m, L y =2m, and L x =2m, L y =10m, and L x =10m, L y =10m. Figure 4 (a) and Figure 4 (b) represents c in the first case. Spatial distribution, Figure 4 (c) and Figure 4 In the second case, (d) represents c and Spatial distribution, Figure 4 (e) and Figure 4 In the third case, (f) represents c and Spatial distribution. From Figure 4 As can be seen, the spatial variation length increases continuously with the change of the relevant length, which verifies the correctness of formula (2.5).

[0161] Example 4:

[0162] Considering that random fields also have a log-normal distribution type, this type is tested in this embodiment. In N x *N y When L = 4096 × 4096, let L x =10m, L y =2m, the mean μ X =1, standard deviation σ X =0.2, substitute into formulas (2.1) and (2.2) to calculate:

[0163] μ = ln(μ) X )-0.5*(σ X ) 2 =ln(1)-0.5*(0.2) 2 =-0.02 (2.1)

[0164]

[0165] Next, substitute the above calculation results into formula (4.2):

[0166] Figure 5 (a) and Figure 5 (b) shows the cohesion c and the internal friction angle. The random field spatial distribution; Figure 5 (c) and Figure 5 (d) in the text represents c and The random field numerical distribution plot. As can be seen, Figure 5 (c) and Figure 5 (d) indicates that both exhibit the characteristics of a log-normal distribution, proving the correctness of formula (4.2).

[0167] Based on the above four embodiments, it can be concluded that the present invention, by introducing the Fast Fourier Transform method, not only achieves high efficiency but also fully considers distribution types and cross-correlation, successfully realizing the construction of random fields, and providing a reliable and efficient generation technology for the simulation study of spatial variability of soil and rock.

[0168] The above description is merely an embodiment of the present invention and does not limit the patent scope of the present invention. Any equivalent structural or procedural transformations made based on the content of the present invention's specification and drawings, or direct or indirect applications in other related technical fields, are similarly included within the patent protection scope of the present invention.

Claims

1. A method for generating cross-correlation random fields based on Fast Fourier Transform, characterized in that, Includes the following steps: S1: Determine the area range, set the horizontal and vertical lengths of the area, select the number of horizontal and vertical spatial points, and calculate the horizontal bandwidth and frequency domain points, as well as the vertical bandwidth and frequency domain points; S2: Set the mean and standard deviation of the random field, select the distribution form of the random field, set the horizontal and vertical correlation lengths, and use the squared exponential autocorrelation function to convert spatial correlation to frequency domain correlation. S3: Set the cross-correlation coefficient and construct the cross-correlation matrix. Decompose the cross-correlation matrix to obtain the upper triangular matrix of the cross-correlation coefficient. Multiply the upper triangular matrix with two sets of random matrices to obtain the cross-correlation type random matrix. Based on the frequency domain symmetry, obtain the expression form of the overall frequency domain point coefficients. S4: Based on the distribution pattern, a cross-correlated random field is generated through inverse Fourier transform, and mean regression and standard deviation regression are used to correct the mean and standard deviation of the random field.

2. The method for generating cross-correlation random fields based on Fast Fourier Transform according to claim 1, characterized in that, Step S1 specifically includes: S11: Construct the horizontal length D of spatial region D x Vertical length D y Set the number of grid points N in the horizontal direction. x and the number of grid points N in the vertical direction y The number of computing units is N x *N y =N; where N x and N y All satisfy powers of 2; The distance Δx between two nodes in the horizontal direction and the distance Δy between two nodes in the vertical direction are given by equations (1.1) and (1.2): S12: Calculate the horizontal bandwidth Δω x With vertical bandwidth Δω y Specifically, see formulas (1.3) and (1.4): S13: For any spatial point (x) j y k ) and frequency domain point (ω) l ,ω m The calculations are as follows: (1.5)-(1.8) 3. The method for generating cross-correlation random fields based on Fast Fourier Transform according to claim 2, characterized in that, Step S2 specifically includes: S2.1: Set the mean μ of the random field X Standard deviation σ X The distribution form of the random field values ​​is selected, including two types: normal distribution and log-normal distribution, as shown in formulas (2.1) and (2.2): S2.2: Set the horizontal correlation length L x Vertical related length L y The quadratic exponential autocorrelation function ρ(x,y) is used, as shown in formula (2.3): Where x and y are the horizontal and vertical distance variables, respectively; S2.3: Transform spatial correlation into a discrete spectral density function G(ω) in the frequency domain. x ,ω γ Its expression is formula (2.5):

4. The method for generating cross-correlation random fields based on Fast Fourier Transform according to claim 3, characterized in that, Step S3 specifically includes the following steps: S3.1: Set the cross-correlation coefficient ρ α,β Construct the cross-correlation matrix and decompose it to obtain the upper triangular matrix M of the cross-correlation coefficient, as shown in formula (3.1): Where, ρ α,β The value range is [-1, 1], and α and β are the representative variables of the two random fields, respectively. S3.2: Generate two elements of size (N) x / 2+1)×(N y The random matrices ε1 and ε2 ( / 2+1) have elements that follow a Gaussian distribution. These are obtained by constructing a cross-correlation random matrix ε using formula (3.2). α and ε β : Where l = 1, 2, ..., N x / 2+1, m=1,2,…,N y / 2+1; S3.3: Frequency domain point (ω) l ,ω m The coefficients in the expression include real coefficients A. l,m and complex coefficient B l,m Both are about variance. and random value, and Based on the values ​​of l and m, there are four possible cases: (1) When l=1,1+N x / 2 and m=1,1+N y / 2 o'clock: (2) When l=1,1+N x / 2 and 2≤m≤N y / 2 o'clock: (3) When 2≤l≤N x / 2 and m=1,1+N y / 2 o'clock: (4) When 2≤l≤N x / 2 and 2≤m≤N y / 2 o'clock: Sure and Afterwards, the randomness component is determined by ε. t (l,m) provides, therefore ε l,m and B l,m The specific form is as follows: in, and They are respectively represented as and The square root of ε t (l,m) represents the cross-correlation type random matrix ε t Each element; t represents the type, which can be the representative variable α or β of the random field, i.e., ε. t =ε α orε β ; For N x / 2+1 <l≤N x N y / 2+1 <m≤N y A l,m and B l,m In other words, it can be obtained through the following symmetry relationship: Where n equals 1 or 1+N γ / 2,N γ =N x or N y .

5. The method for generating cross-correlation random fields based on Fast Fourier Transform according to claim 4, characterized in that, Step S4 includes the following steps: S4.1: Based on the distribution form of the random field values, a cross-correlation random field is generated through inverse Fourier transform; where, when the distribution form is normal, the generated random field is shown in formula (4.1), and when the distribution form is log-normal, the generated random field is shown in formula (4.2): in, RF(x j ,y k ) t Indicates at point (x) in space j ,y k Random field values ​​generated at position (j=1,2,…,N) for the representative variable t. x k = 1, 2, ..., N y ; S4.2: Eliminate the fluctuation problem of mean and standard deviation in random fields through mean regression and standard deviation regression. Mean regression requires first calculating the actual mean of the random field. Then, according to formula (4.4), adjust each random value to ensure that the mean is close to the set value μ. X Consistency: in, For RF(x) j ,y k ) t After mean regression, at spatial point (x j ,y k Update value at ); In standard deviation regression, the first step is to calculate... Actual standard deviation σ real As shown in formula (4.5): Recalculate With mean μ X Deviation Dis_RF(x) j ,y k ) t As shown in formula (4.6): Finally, adjust each random value to match the standard deviation σ. X As shown in formula (4.7): in, For RF(x) j ,y k ) t At the spatial point (x j ,y k The final generated random field values ​​for the representative variable t, j = 1, 2, ..., N x k = 1, 2, ..., N y , t = α or β.

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