Step-by-step decomposition-space mapping method for generating 3d material parameter random field of material points

Abstract

The application provides a step-by-step decomposition-space mapping three-dimensional material point material parameter random field generation method, and belongs to the technical field of water conservancy and hydropower. First, a three-dimensional geometric model of a structure is established according to engineering geological data, high-precision mesh division is carried out, each mesh is converted into a material point carrying mass and volume, and a material point model is obtained. Second, a background mesh element is determined, and a background mesh covering the space occupied by the entire structure is generated. Third, a correlation function reflecting the material parameter properties is selected, a step-by-step decomposition method is used, and a material parameter random field on the background mesh element is generated. Finally, the value of the material parameter of the material point model is determined through a space mapping technology, and a three-dimensional material point material parameter random field is generated. The application is based on the material point method, and realizes the generation of a three-dimensional random field of a geotechnical engineering material parameter through the joint application of a step-by-step decomposition-space mapping technology, thereby providing technical support for the simulation of large deformation of a geotechnical engineering considering spatial variability.

Description

Technical Field

[0001] This invention belongs to the field of water conservancy and hydropower technology, and relates to a method for generating random fields of three-dimensional material point material parameters by stepwise decomposition-spatial mapping. Background Technology

[0002] Large deformation and failure of soil seriously affect the safe operation of structures, and grid-based numerical simulation methods (such as the finite element method) are insufficient to meet the needs of large deformation analysis. The material point method (MPM) breaks free from the constraints of element meshes and records variable information (mass, displacement, velocity, acceleration, strain, stress, etc.) on material points, effectively avoiding problems such as mesh distortion. It also has high computational stability and efficiency, showing unique advantages in simulating large deformation and failure of soil and rock masses.

[0003] Previous approaches to large deformation problems in soil and rock masses have largely employed deterministic analysis methods. These methods rely on single, deterministic mechanical parameters for mechanical calculations and stability analysis, neglecting the spatial variability of the physical and mechanical properties of soil and rock masses. Consequently, they fail to adequately reflect the actual structural response. In reality, influenced by multiple factors such as sedimentary environment and stress history, the physical and mechanical properties of soil and rock masses often differ significantly at different spatial locations, exhibiting substantial spatial variability. This inherent spatial variability is the primary source of uncertainty in soil and rock material parameters, directly impacting soil deformation characteristics and failure modes. Therefore, uncertainty analysis considering the spatial variability of soil is gradually replacing traditional deterministic analysis. Random field theory, by introducing concepts such as correlation functions, correlation distances, and coefficients of variation, can effectively characterize the randomness and spatial correlation of soil material parameters, clearly reflecting the spatial distribution differences of material parameters. It has become an important theoretical framework for studying the spatial variability of soil.

[0004] However, applying random field theory to the simulation of large deformations in large-scale geotechnical engineering still faces many challenges: on the one hand, the computational memory requirements are too high; on the other hand, generating random fields corresponding to complex geometric models remains difficult. For example, Chinese invention patent CN202411706982.6 discloses a method for realizing random fields in fractured rock masses that considers the spatial variability of three-dimensional geological structures. This method uses shape function interpolation to map from the background grid random field to the random field of rock mass units, thereby generating random fields for three-dimensional rock mass units with complex geometric structures. However, because it directly decomposes and operates on the overall covariance matrix, the computational efficiency is low and the program implementation is complex. Chinese invention patent CN201910594391.7 discloses a random field modeling method for geotechnical parameters. This method uses an improved covariance matrix decomposition method for random field modeling, which reduces the amount of computation while ensuring accuracy. However, its applicability is mainly limited to standard cuboids, making it difficult to reflect the complex geometric shapes of structures.

[0005] To address this challenge, this invention innovatively proposes a stepwise decomposition-spatial mapping theory, aiming to generate random fields of material point parameters for large-scale geotechnical engineering projects stepwise with relatively low computational cost. This method leverages the separability of correlation function matrices to significantly reduce the dimensionality and computational complexity of matrix operations. Combined with spatial mapping methods, it effectively reflects the complex geometry of geotechnical structures, thereby achieving efficient generation of three-dimensional random fields of material point parameters for complex structures. This provides technical support for the simulation and subsequent numerical calculations of large deformation problems in soils considering spatial variability. Summary of the Invention

[0006] This invention addresses the problem of low efficiency in generating random fields of material parameters for large and complex structures in geotechnical engineering large deformation simulation, which fails to reflect complex geometries. It proposes a stepwise decomposition-spatial mapping method for generating three-dimensional random fields of material parameters. This invention reduces the amount of matrix computation required in the random field generation process through stepwise decomposition and utilizes spatial mapping technology to characterize the complex geometries of geotechnical structures, thereby generating a three-dimensional random field of material parameters. This provides technical support for geotechnical engineering large deformation simulation considering spatial variability.

[0007] To achieve the above objectives, the technical solution adopted by the present invention is as follows: A method for generating random fields of material parameters for three-dimensional material points through stepwise decomposition and spatial mapping, comprising the following steps: S1. Based on engineering geological data, a three-dimensional geometric model of the structure is established, and the three-dimensional geometric model is meshed with high precision. Each mesh is then converted into a material point carrying mass and volume, thus obtaining a material point model. Specifically: S1.1, Based on the actual engineering characteristics of the structure, a three-dimensional geometric model is established and subjected to high-precision meshing to obtain a mesh model, so as to ensure the accurate representation of complex geometric shapes; S1.2, Based on geological exploration data, assign corresponding density parameters to each part of the grid model; S1.3 calculates the coordinates, mass, and volume information of the material point corresponding to each grid cell, realizing the transformation from the grid model to the material point model.

[0008] S2, based on the 3D geometric model and material point model established in S1, generates a background mesh covering the entire space occupied by the structure. Specifically: S2.1, Based on the three-dimensional geometric model of the structure, determine the range of the background mesh to ensure that it can cover the entire space occupied by the structure; S2.2, the background grid cell spacing is determined based on the average spacing of the material points. To balance computational accuracy and efficiency, the background grid spacing is calculated according to formula (1): (1) In the formula, Background grid spacing; The average spacing between the material points; The number of material points; , These represent two distinct material points; 、 、 They represent material point direction, direction and Direction coordinates; 、 、 They represent material point direction, direction and Direction coordinates.

[0009] Furthermore, the background grid cells are cubes, and their side length is the spacing between the background grid cells.

[0010] S2.3, Based on the background grid range and background grid cell spacing specified in S2.1 and S2.2, determine the background grid cells in... x, y, z Number of positions in three directions 、 、 This generates a background mesh that covers the entire space occupied by the structure.

[0011] S3, on the background mesh elements established in S2, select a correlation function that reflects the material parameter properties, and use a stepwise decomposition method to efficiently generate a random field of material parameters on the background mesh elements. Specifically: S3.1 Select a correlation function that reflects the properties of material parameters and establish the correlation matrix of the background mesh element in three directions; Furthermore, a separable single exponential correlation function is used, and background grid elements are established according to formula (2) in... x, y, z Correlation matrix in direction , , Their dimensions are respectively , , ; (2) In the formula: , , These represent the center points of the two background grid cells at... , , The separation distance in the direction; , , They represent in , , The directional separation distance is , , The two background mesh cells in , , Relevant function values ​​in the direction; , , Random fields are respectively , , The range of fluctuations in direction.

[0012] S3.2, based on the three directional correlation matrices established in S3.1, uses a stepwise decomposition method to efficiently generate the random field of material parameters for the background mesh element; Furthermore, the background grid totals n Units, of which n = A by n A standard normal random field composed of spatial locations X The covariance matrix is ​​generated by the covariance matrix decomposition method, as shown in equation (3). Theoretically, equation (3) can be used for large-scale three-dimensional random field simulation. However, if the Cholesky decomposition is directly applied to the total correlation matrix to generate the random field, the matrix decomposition... The implementation of random fields requires, respectively n 3 and n 2 The computation time and memory usage of sub-floating-point operations increase with... n The number of simulations increases dramatically with the increase in the number of elements, thus limiting them to small-scale three-dimensional random field simulations.

[0013] (3) In the formula: 、 They are respectively Random field vectors, independent standard normal random vectors, The mean and standard deviation are respectively and According to the two , , The order corresponds to the individual cells in the background grid. Specifically, in , , In the direction, the position number is , , The unit, corresponding to and The first in There are elements, and their indices are: ; for The total correlation matrix of the background grid cells, utilizing its symmetric positive definiteness, is used to... Perform Cholesky decomposition. This is the obtained decomposition matrix.

[0014] Furthermore, the total correlation matrix of the background grid cells Size is The scale is enormous. By utilizing the property of separable single exponential correlation functions, the total correlation matrix can be expressed as correlation matrices in each direction. , , The Kronecker product, and on , , Perform Cholesky decomposition, see formula (4): (4) In the formula: , , The background grid cells are respectively in x, y, z The correlation matrix in the direction has the following dimensions: , , ; For Kronecker product; , , Correlation matrix , , The Cholesky decomposition matrix; Applying the properties of the Kronecker product and combining the decomposition formulas of the correlation matrices in the three directions mentioned above, the decomposition matrix of the total correlation matrix is ​​obtained according to formula (5). Based on the number of positions of the background grid in each direction, the independent standard normal random vectors are decomposed. ( )according to , , The order is rearranged in three dimensions to obtain a three-dimensional matrix. ( Corresponding to in , , The position number on the axis is ( , , Substituting formula (5) into formula (3), the three-dimensional random field on the background grid unit is realized in a stepwise but equivalent form, as shown in formula (6): (5) (6) In the formula: for The three-dimensional random field matrix, and its correspondence with the background grid cells are as follows: same; Defined as a matrix sum matrix No. Multiplication of dimensions =1, 2, 3.

[0015] S4, based on the random field of material parameters on the background mesh elements established in S3, uses spatial mapping technology to determine the values ​​of material parameters for the material point model, thus completing the generation of the random field of material parameters for the three-dimensional material points of the complex structure. Specifically: S4.1, Determine the background grid cell in which each material point is located based on its spatial coordinates; The background grid cell is a cube with a side length of S2.2. The background grid spacing is determined by substituting the spatial coordinates of the material point into formula (7) to obtain the background grid cell where the material point is located. x, y, z The position number on the axis determines the background grid cell where the material point is located; (7) In the formula: The background mesh cells where the material points are located are respectively x, y, z Position number on the axis; They are the material points at , , Coordinates on the axis The center points of the initial background grid cells are respectively at , , Coordinates on the axis; This refers to the size of the background grid cells, i.e., the background grid spacing.

[0016] S4.2 maps the material parameter values ​​corresponding to the background mesh elements where the material points are located to the material points, thereby completing the generation of the random field of material parameters for the three-dimensional material points of the complex structure.

[0017] Compared with the prior art, the beneficial effects of the present invention are as follows: (1) This invention simulates the spatial variability of material parameters of complex structures based on the stepwise decomposition-spatial mapping idea, providing technical support for large deformation analysis that considers spatial variability.

[0018] (2) By introducing a stepwise decomposition mechanism of the relevant matrix, the present invention significantly reduces the dimension and computational complexity of matrix operations.

[0019] (3) The present invention uses spatial mapping technology to enable the random field of three-dimensional material points to reflect the complex geometry of actual engineering structures.

[0020] In summary, this invention, based on the material point method, achieves the generation of a three-dimensional random field for geotechnical engineering material parameters through the combined application of stepwise decomposition and spatial mapping techniques. This provides technical support for the simulation of large deformations in geotechnical engineering considering spatial variability and improves the reliability of numerical simulations of large deformations in geotechnical engineering. Attached Figure Description

[0021] Figure 1 This is a schematic diagram of the main process of the method of the present invention; Figure 2 This is a schematic diagram of the geometric model of an earth-rock dam; Figure 3 A schematic diagram of the grid division for an earth-rock dam; Figure 4 A schematic diagram of the material point model for an earth-rock dam; Figure 5 A schematic diagram of the cohesive random field of a three-dimensional background mesh element; Figure 6 A schematic diagram of the cohesive random field profile of a three-dimensional background mesh element; Figure 7 This is a schematic diagram of the cohesive random field profile of a three-dimensional background mesh element. Figure 8 A schematic diagram of the random field of cohesion in an earth-rock dam; Figure 9 A schematic diagram of the random field profile of cohesion in an earth-rock dam; Figure 10 This is a schematic diagram of the random field profile of cohesion in an earth-rock dam. Detailed Implementation

[0022] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, but the scope of protection of the present invention is not limited thereto.

[0023] Example 1: Three-dimensional random field simulation of material parameters (cohesion) of an earth-rock dam, see [link to example]. Figure 1 This includes the following steps: S1. Based on the engineering geological data of the earth-rock dam, a three-dimensional geometric model of the earth-rock dam is established. The three-dimensional geometric model is then meshed with high precision, and each mesh is converted into a material point carrying mass and volume, thus obtaining a material point model. Specifically: S1.1, Based on the actual engineering characteristics of earth-rock dams, a three-dimensional geometric model is established, such as... Figure 2 As shown, this model illustrates the geometric dimensions of an earth-rock dam: a height of 50.0m, a length along the river of 180m, and an axial length of 275m. Based on this, the model is meshed with high precision to obtain a mesh model, ensuring accurate representation of the complex geometry. The mesh model of the earth-rock dam is based on... Figure 3 ; S1.2, Based on geological exploration data, assign corresponding density parameters to each part of the grid model. In this embodiment, the soil density is 1.8 g / cm³. 3 ; S1.3 calculates the coordinates, mass, and volume information of the material point corresponding to each grid cell, realizing the transformation from the grid model to the material point model, referring to... Figure 4 It showcases a material point model of an earth-rock dam.

[0024] S2, based on the 3D geometric model and material point model established in S1, generates a background mesh covering the entire space occupied by the structure. Specifically: S2.1. Based on the three-dimensional geometric model of the earth-rock dam, determine the range of the background grid to ensure that it can cover the entire space occupied by the structure. In this embodiment, the background grid size is 280m×180m×50m, which is a standard cuboid shape. S2.2, the background grid cell spacing is determined based on the average spacing of the material points. In order to balance the calculation accuracy and efficiency, the background grid spacing is calculated according to formula (1). In this embodiment, the background grid spacing is 0.5m. (1) In the formula, Background grid spacing; The average spacing between the material points; The number of material points; , These represent two distinct material points; 、 、 They represent material point direction, direction and Direction coordinates; 、 、 They represent material point direction, direction and Direction coordinates.

[0025] Furthermore, in this embodiment, the background grid unit is a cube, and its side length is the same as the background grid unit spacing of 0.5m.

[0026] S2.3, Based on the background grid range and background grid cell spacing specified in S2.1 and S2.2, determine the background grid cells in... x, y, z Number of positions in three directions 、 、 This generates a background mesh that covers the entire space occupied by the structure. In this embodiment, x The direction is the dam axis. y The direction is along the river. z If the direction is vertical, then =560 、 =360 、 =100.

[0027] S3, on the background mesh elements established in S2, select a correlation function that reflects the cohesive properties of the material, and use a stepwise decomposition method to efficiently generate a cohesive random field on the background mesh elements. Specifically: S3.1 Select a correlation function that reflects the cohesive properties of the material and establish the correlation matrix of the background mesh element in three directions; This embodiment uses a separable single exponential correlation function, and establishes background grid units according to formula (2) in... x, y, z Correlation matrix in direction , , Their dimensions are respectively , , ; (2) In the formula: , , These represent the center points of the two background grid cells at... , , The separation distance in the direction; , , They represent in , , The directional separation distance is , , The two background mesh cells in , , Relevant function values ​​in the direction; , , Random fields are respectively , , In this embodiment, the fluctuation range in direction is taken as 50m, 40m, and 25m respectively.

[0028] S3.2, based on the three directional correlation matrices established in S3.1, uses a stepwise decomposition method to efficiently generate the cohesive random field of the background grid cells; Furthermore, in this embodiment, the background grid comprises a total of n ( n = ) units, one of which n A standard normal random field composed of spatial locations X The covariance matrix is ​​generated by the covariance matrix decomposition method, as shown in equation (3). Theoretically, equation (3) can be used for large-scale three-dimensional random field simulation. However, if the Cholesky decomposition is directly applied to the total correlation matrix to generate the random field, the matrix decomposition... The implementation of random fields requires, respectively n 3 and n 2 The computation time and memory usage of sub-floating-point operations increase with... n The number of simulations increases dramatically with the increase in the number of elements, thus limiting them to small-scale three-dimensional random field simulations.

[0029] (3) In the formula: 、 They are respectively The random field vector and the independent standard normal random vector are used in this embodiment. U The mean and standard deviation are 15 and 1, respectively. 、 According to both , , The order corresponds to the individual cells in the background grid. Specifically, in , , In the direction, the position number is , , The unit, corresponding to and The first in There are elements, and their indices are: ; for The total correlation matrix of the background grid cells, utilizing its symmetric positive definiteness, is used to... Perform Cholesky decomposition. This is the obtained decomposition matrix.

[0030] Furthermore, the total correlation matrix of the background grid cells Size is The scale is enormous. By utilizing the property of separable single exponential correlation functions, the total correlation matrix can be expressed as correlation matrices in each direction. , , The Kronecker product, and on , , Perform Cholesky decomposition, see formula (4): (4) In the formula: , , The background grid cells are respectively in , , The correlation matrix in the direction has the following dimensions: , , In this embodiment, they are respectively , , ; For Kronecker product; , , Correlation matrix , , The Cholesky decomposition matrix; Applying the properties of the Kronecker product and combining the decomposition formulas of the correlation matrices in the three directions mentioned above, the decomposition matrix of the total correlation matrix is ​​obtained according to formula (5). Based on the number of positions of the background grid in each direction, the independent standard normal random vectors are decomposed. ( )according to , , The order is rearranged in three dimensions to obtain a three-dimensional matrix. ( Corresponding to in x, y, z The position number on the axis is ( , , Substituting formula (5) into formula (3), the three-dimensional random field on the background grid unit is realized in a stepwise but equivalent form, as shown in formula (6): (5) (6) In the formula: for The three-dimensional random field matrix, and its correspondence with the background grid cells are as follows: same; Defined as a matrix sum matrix No. Multiplication of dimensions =1, 2, 3.

[0031] S4, based on the background mesh element cohesion random field established in S3, determines the cohesion values ​​of the material point model through spatial mapping technology, thereby completing the generation of the three-dimensional material point cohesion random field for complex structures. Specifically: S4.1, Determine the background grid cell in which each material point is located based on its spatial coordinates; The background grid cell is a cube with a side length of S2.2. The background grid spacing is determined by substituting the spatial coordinates of the material point into formula (7) to obtain the background grid cell where the material point is located. x, y, z The position number on the axis determines the background grid cell where the material point is located; (7) In the formula: The background mesh cells where the material points are located are respectively x, y, z Position number on the axis; They are the material points at , , Coordinates on the axis The center points of the initial background grid cells are respectively at , , In this embodiment, the coordinates on the axes are taken as 0.0, 0.0, and 0.0, respectively; a This represents the size of the background grid cells, i.e., the background grid spacing is 0.5m.

[0032] S4.2 maps the cohesion parameters corresponding to the background mesh elements where the material points are located to the material points, thereby completing the generation of the three-dimensional material point cohesion random field for complex structures.

[0033] The simulation results of this invention on the cohesion random field of earth-rock dams are summarized, and the simulation results of the background grid cells and the cohesion random field of the earth-rock dam are obtained, as shown in the figure. Figures 5-10 As shown. Specifically, Figure 5 A schematic diagram of the cohesive random field of the cuboid background mesh element is given. Figure 6 , 7 The images show the renderings of two cross-sections of the cuboid background mesh, in which... Figure 6 A schematic diagram of a cross-section of a cohesive random field on a three-dimensional background mesh element; Figure 7 This is a schematic diagram of the cross-section of the cohesive random field on a three-dimensional background mesh unit; Figure 8 A schematic diagram of the cohesion random field of an earth-rock dam is given, where L1=70m and L2=80m. Figure 9 , 10 The images show renderings of two cross-sections of the earth-rock dam. Figure 9 This is a schematic diagram of the random field profile of cohesion in an earth-rock dam. Figure 10 This is a schematic diagram of the second random field profile of cohesion in an earth-rock dam.

[0034] Specifically, for this embodiment, the computational complexity comparison between the stepwise decomposition method of the present invention and the traditional decomposition method for generating random fields is shown in Table 1 below. It can be seen that the stepwise decomposition method inherits the simplicity of the traditional decomposition method and significantly reduces computational complexity. Especially in matrix decomposition, for a given... For three-dimensional random fields, the number of floating-point operations is reduced by 7 orders of magnitude by the stepwise decomposition method, and the memory consumption of the correlation matrix is ​​reduced by 7 orders of magnitude.

[0035] Table 1: Comparison of Decomposition Methods in 3D Random Field Simulation

[0036] from Figures 5-10 The cohesion random field effect diagram and the comparison results in Table 1 show that the present invention has a good effect on the random field simulation of material parameters of large and complex geotechnical structures, and the computational complexity is small. This indicates that the present invention can effectively promote the application of three-dimensional spatial variability simulation in the large-scale deformation analysis of geotechnical structures.

[0037] The above description describes a more feasible specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any modifications, equivalent substitutions, improvements, or adaptive adjustments made within the technical concept and core principles disclosed in the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for generating random fields of material parameters for three-dimensional material points through stepwise decomposition-spatial mapping, characterized in that, The method for generating random field parameters of three-dimensional material points includes the following steps: S1. Based on engineering geological data, a three-dimensional geometric model of the structure is established, and the three-dimensional geometric model is divided into high-precision meshes. Each mesh is then converted into a material point carrying mass and volume, thus obtaining a material point model. S2, based on the three-dimensional geometric model and material point model established by S1, determines the background mesh unit and generates a background mesh covering the entire space occupied by the structure. S3, on the background mesh element established in S2, select a correlation function that reflects the material parameter properties, and use a stepwise decomposition method to generate a random field of material parameters on the background mesh element; specifically: S3.1 Select a correlation function that reflects the properties of material parameters and establish the correlation matrix of the background mesh element in three directions; S3.2, based on the three directional correlation matrices established in S3.1, uses a stepwise decomposition method to generate a random field of material parameters for the background mesh elements; S4, based on the random field of material parameters on the background mesh unit established in S3, uses spatial mapping technology to determine the values ​​of material parameters of the material point model, generating a three-dimensional random field of material point material parameters; specifically: S4.1, Determine the background grid cell in which each material point is located based on its spatial coordinates; S4.2 maps the material parameter values ​​corresponding to the background mesh elements where the material point is located to the material point, generating a three-dimensional random field of material parameters for the material point.

2. The method for generating random fields of material parameters for three-dimensional material points through stepwise decomposition-spatial mapping according to claim 1, characterized in that, Specifically, S1 is: S1.1, Based on the actual engineering characteristics, a three-dimensional geometric model is established, and a high-precision mesh is generated to obtain a mesh model; S1.2, Based on geological exploration data, assign corresponding density parameters to each part of the grid model; S1.3 Calculate the coordinates, mass, and volume information of the material point corresponding to each grid cell, realizing the transformation from the grid model to the material point model, and obtaining the material point model.

3. The method for generating random fields of material parameters for three-dimensional material points through stepwise decomposition-spatial mapping according to claim 2, characterized in that, Specifically, S2 is: S2.1, Based on the three-dimensional geometric model, determine the range of the background mesh to ensure that it can cover the entire space occupied by the structure; S2.2, the background grid cell spacing is determined based on the average spacing of the material points, and the background grid spacing is calculated according to formula (1): （1） In the formula, Background grid spacing; The average spacing between the material points; The number of material points; , These represent two distinct material points; 、 、 They represent material point direction, direction and Direction coordinates; 、 、 They represent material point direction, direction and Direction coordinates; S2.3, Based on the background grid range and background grid cell spacing determined in S2.1 and S2.2, determine the background grid cells in... x, y, z Number of positions in three directions 、 、 This generates a background mesh that covers the entire space occupied by the structure.

4. The method for generating random fields of material parameters for three-dimensional material points by stepwise decomposition-spatial mapping according to claim 3, characterized in that, In S2.2, the background grid unit is a cube, and its side length is the spacing between the background grid units.

5. The method for generating random fields of material parameters for three-dimensional material points by stepwise decomposition-spatial mapping according to claim 4, characterized in that, Specifically, S3.1 is as follows: Using a separable single exponential correlation function, the background grid cells are established according to formula (2) in... x, y, z Correlation matrix in direction , , Their dimensions are respectively , , ; （2） In the formula: , , These represent the center points of the two background grid cells at... , , The separation distance in the direction; , , They represent in , , The directional separation distance is , , The two background mesh cells in , , Relevant function values ​​in the direction; , , Random fields are respectively , , The range of fluctuations in direction.

6. The method for generating random fields of material parameters for three-dimensional material points through stepwise decomposition-spatial mapping according to claim 5, characterized in that, Specifically, S3.2 is as follows: The background grid totals n Units, of which n = A by n A standard normal random field composed of spatial locations X , is generated by the covariance matrix decomposition method, as shown in equation (3) below; （3） In the formula: 、 They are respectively Random field vectors, independent standard normal random vectors, The mean and standard deviation are respectively and According to the two , , The order corresponds to the individual cells in the background grid. Specifically, in , , In the direction, the position number is , , The unit, corresponding to and The first in One element; for The total correlation matrix of the background grid cells, utilizing its symmetric positive definiteness, is used to... Perform Cholesky decomposition; The resulting decomposition matrix; Total correlation matrix of background grid cells Size is The total correlation matrix is ​​expressed as correlation matrices in each direction. , , The Kronecker product, and on , , Perform Cholesky decomposition, see formula (4): （4） In the formula: , , The background grid cells are respectively in x, y, z The correlation matrix in the direction has the following dimensions: , , ; For Kronecker product; , , Correlation matrix , , The Cholesky decomposition matrix; Applying the properties of the Kronecker product and combining the decomposition formulas of the correlation matrices in the three directions, the decomposition matrix of the total correlation matrix is ​​obtained according to formula (5). Based on the number of positions of the background grid in each direction, the independent standard normal random vectors are decomposed. according to , , The order is rearranged in three dimensions to obtain a three-dimensional matrix. Substituting formula (5) into formula (3), the three-dimensional random field on the background grid cell is realized in a stepwise but equivalent form, as shown in formula (6): （5） （6） In the formula: for The three-dimensional random field matrix, and its correspondence with the background grid cells are as follows: same; Defined as a matrix sum matrix No. Multiplication of dimensions =1, 2, 3.

7. The method for generating random fields of material parameters for three-dimensional material points by stepwise decomposition-spatial mapping according to claim 6, characterized in that, Specifically, S4.1 is as follows: The side length of the background grid cell is determined by the background grid spacing of S2.

2. By substituting the spatial coordinates of the material point into formula (7), the background grid cell where the material point is located is obtained. x, y, z The position number on the axis determines the background grid cell where the material point is located; （7） In the formula: The background mesh cells where the material points are located are respectively x, y, z Position number on the axis; The matter points are respectively at , , Coordinates on the axis The center points of the initial background grid cells are respectively at , , Coordinates on the axis; This refers to the size of the background grid cells, i.e., the background grid spacing.

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