GPU-Accelerated Anisotropic Random Field and Its Application in the Modeling of a Diversion Tunnel

Abstract

In this paper, a GPU-accelerated Cholesky decomposition technique and a coupled anisotropic random field are suggested for use in the modeling of diversion tunnels. Combining the advantages of GPU and CPU processing with MATLAB programming control yields the most efficient method for creating large numerical model random fields. Based on the geological structural characteristics of red-bedded soft rocks in central Yunnan, anisotropic rock random fields and tunnel excavation with various rotation degrees are simulated. In the comparison of anisotropic random fields specifically, the relationship between the anisotropic rotation angle and the plastic zone, as well as the multiple measurements for the overall safety factor, are analyzed. The distribution of the plastic zone after excavation has a significant relationship with the random parameters of the anisotropic random field. When the stronger or weaker random parameters are located in the surrounding rock of the cavern, they will cause a change in the radius of the plastic zone. The overall safety factor of the anisotropic random field is relatively stable, with an average value of about 2, which mainly depends on the strength of the random parameter of the rock mass. Based on the random fluctuation of the suggested value in the engineering report, the simulation result is safe. This study can provide theoretical and technical support for the design and construction of relevant rock engineering in the red-bedded soft rock region of central Yunnan.

1. Introduction

With the rapid improvement of engineering construction in China, along with the rapid development of ground space resources and the swift completion of infrastructure and transport, the ground space resources in flat areas suitable for construction are gradually being used up. In order to ensure the high quality of life of the people and to continue the rapid development of society and the economy, seeking new space resources and utilizing underground spaces while developing water resources, such as sea, lakes and rivers, especially the massive development of deep underground space resources, is an inevitable trend [1,2]. Geological material, with typical non-homogeneous properties, makes up the surrounding rock body of a tunnel or subterranean chamber. However, in the conventional study of geotechnical engineering, the rock is typically taken to be a continuous medium with homogenous qualities. The randomness and spatial variability of the mechanical characteristics of the real engineering rock cannot be taken into account by this abridged method. Engineers sometimes find it difficult to adequately foresee possible risks in subterranean engineering during the design stage because of a lack of a thorough grasp of the spatial heterogeneity of geological rock materials. The stability analysis of subterranean cavern engineering therefore places tremendous importance on how to properly study and evaluate the unpredictability of rock characteristics [3,4,5,6,7].

Reliability theory has been used for a long period of time in structural engineering and mechanical engineering, but its application in geotechnical engineering is relatively recent. In 1965, Casagrande pointed out in his lecture that risk calculation in geotechnical engineering cannot be fully quantitative, and that research on uncertainty should be broken through. Since then, reliability theory has been gradually applied in ground geotechnical engineering, such as in the reliability analysis of surrounding rock, foundation, and slope stability, etc. Beacher and Ingra proposed the use of the stochastic finite element method to analyze the uncertainty of predicting the total and differential settlement of large flexible foundations, suggesting that the prediction of differential settlement in this method, with obvious differences between one and two dimensions, is mainly due to the two-dimensional random field in insufficient randomness [8]. Phoon and Quek investigated the effect of spatially varying soil media on the settlement of monopile foundations using a stochastic finite element method, assuming the soil media to be linearly elastic and modeling what is defined as a uniform random field. The randomness of the soil modulus was investigated by means of the random field coefficient of variation and fluctuation law, and it was found that the uncertainty of pile head settlement depends on both deterministic and stochastic parameters [9,10]. Brzzkala and Pula discussed three fundamental sources of randomness by means of the Monte Carlo stochastic finite element method: the random shape of the subsoil, random material parameters, and random loads [11]. Then, Nishimura proposed an inverse analysis of the Monte Carlo stochastic finite element method by means of consolidation tests [12]. Vanmarcke made a pioneering contribution to the reliability analysis method. He proposed that random variation is a fact of life. Different disciplines are increasingly required to use random fields to model complex variation and interdependence patterns because deterministic processing and routine statistics are often not sufficient. Therefore, based on this, a random field model that can describe the properties of the soil is established, and the concept of correlation distance and computationally generated random field was proposed [13]. Simultaneously, more scholars have conducted studies on the randomness of soil parameters. Lumb, by studying the probability distribution of the shear strength intensity cohesion (c) and friction angle (φ) of soils, concluded that most soil property parameters obey normal distributions [14]. Suchomel and Masin compared the finite element method based on the Taylor series expansion with the finite element method of the spatial random field of the relevant parameters c and φ, and discussed the problem of the spatial variability of the parameters [15]. Fenton and Gordon systematically presented the way in which random fields are generated and how the generated random fields are analyzed for risk scenarios using different parameters, also comparing the differences in the evaluation of three different random field generators in a cross-sectional manner [16,17]. The method can effectively consider various damage mechanisms caused by the spatial variability of soil properties in probabilistic slope stability assessment [18]. Kavvadas and Karlaftis et al. proposed a probabilistic assessment approach to address spatial variability due to site uncertainty in slope stability analysis [19].

The stochastic study of the mechanical characteristics of geological materials now uses three primary approaches. One is to use a calculating approach that takes into account the interval distribution of parameters such as the interval finite element method [20,21] or the fuzzy finite element method [22,23]. These techniques allow for the direct input of interval parameters and calculation-based simulation results within a certain interval range. However, the inherent model of non-linear materials makes the use of these approaches challenging [24]. The second strategy involves modeling heterogeneous media at the mesoscale using the statistical characteristics of the microstructure [25,26,27,28,29,30,31]. Nevertheless, the scope of this solution’s analysis is insufficient for sizable geotechnical projects. The third strategy involves utilizing random field theory to describe the spatial variability of the geological structural characteristics of the cavern envelope [32,33]. To examine the dependability of ground settlement, Beacher and Ingra [8] used the Taylor expansion of the stochastic finite element approach. R. Suchomel and D. Masin [34] compared probabilistic approaches for determining slope stability and analyzed the impact of spatial variability on slope stability, while taking into account the spatial variable cohesion (c) and the friction angle (φ) of the soil mechanical characteristics.

The research mentioned above makes it clear that the random field approach can more accurately describe the geographical variability of geotechnical material properties. The random field approach is simple to integrate into both open-source and for-profit software. However, the random field modeling method mentioned above mainly relies on the CPU of the computer for calculation, which is expensive in terms of both time and economic costs. At present, the random field method for geotechnical materials is mainly applied to the stability analysis of slopes. Less research has been done regarding the analysis of excavation in underground caverns or tunnels. Thus, in this study, analyses of the stability and reliability of a tunnel after excavation in central Yunnan were carried out.

First, in this paper, the principles of random field theory are introduced. Since the types and properties of rocks are constantly changing with the change of spatial position in engineering geology, an effective generation technique for the random field of subterranean caves is created, and the modeling theory of an anisotropic random field is refined. The covariance matrix decomposition method may encounter the issue of sluggish computing as a result of a high number of model meshes. An effective random field generation method with GPU acceleration is suggested. It is based on the Cholesky decomposition theory and GPU acceleration technology. MATLAB was used to create software for an automated control calculation. An anisotropic rock random field model with various rotation angles was developed to examine the stability and safety factors of the tunnel after excavation, while taking into account the geological structural features of red-bedded soft rocks in central Yunnan.

2. Methodology

2.1. Basic Theory of Random Fields

Many processes around us exhibit variations in time and space. For example, if one considers the measurement of wind speed at a specific location, the signal has a random character. At another location close to this measurement, the wind speed will be different from, and related to, this signal. When this variability is mapped into a model, the concept of random fields comes to mind. According to Vanmarcke’s theory [13], a distinction can be made between time-independent uncertainties in the properties of random media and spatiotemporal processes, in which properties at different points in space vary randomly over time. In this study, random fields are divided into continuous and discrete random fields. The simple discrete random field is shown in Figure 1.

There are different methods to generate random fields. Currently, two main types of random field generation methods are commonly used [17,35,36]. Among them, the first type is the discrete random field, and the second type is the continuous random field. In the first category, a generator of spatially correlated random variables is combined with a discretization method, resulting in a discrete random field. For each node in the random field grid, a random variable that is related to the other nodes in the random field is evaluated. These spatially correlated random variables must be assigned to elements or integration points in the finite element model. The second type of random field generator is the so-called series expansion method, such as the KL expansion method (Karhunen–Loeve expansion) and the expansion linear optimal estimation method. A random field is represented by a sum of functions multiplied by random variables, and this method produces a continuous random field. In a finite element model, the continuous random field must be integrated numerically. This boils down to the sum of the evaluated values of the continuous functions at the integration points multiplied by the weights of the corresponding integration points.

2.2. Parameter Definitions for Random Fields

The correlation length or volatility is a measurement of random field variability. In this paper, it will be denoted by $L_c$, and the correlation length is defined by Vanmarcke [13] as $L_c=\underset{D\to \infty }{\lim }D\gamma (D)$. Through Equation (1), the correlation length can be expressed as:

$$L_c={\displaystyle {\int }_{-\infty }^{\infty }\rho }(\Delta x)d\Delta x=2{\displaystyle {\int }_0^{\infty }\rho }(\Delta x)d\Delta x$$

(1)

Therefore, the value of the correlation length is equal to the integration of the correlation function over the full domain. A necessary condition for the existence of $L_c$ is finite and is given by the following equation:

$$\underset{D\to \infty }{\lim }\frac{1}{D}{\displaystyle {\int }_0^D\Delta }x\rho (\Delta x)d\Delta x=0$$

(2)

In practice, it can be difficult to determine the correlation length. The dispersion to be determined from experimental data can be high, and the correlation length depends largely on a variety of factors. The effect of the correlation length in the random field is shown in Figure 2. It shows the random field for a model with a length of 50 m and a width of 50 m. The left panel uses a correlation length of 2.5, and the right panel uses a correlation length of 10.

Exponential (Exp) and squared exponential (SExp) related functions [37] are often used in engineering practice. For concrete structures, SExp-type correlation functions are more often used. The Exp-type correlation function is more suitable for modeling the spatial variability of soil properties. The two main correlation functions can be expressed as follows:

$$\mathrm{Exp}: \rho \left(\Delta x_1,\Delta x_2\right)=c_1+\left(1-c_1\right)\exp \left(-\sqrt{{\left(\frac{\Delta x_1}{L_{c,1}}\right)}^2+{\left(\frac{\Delta x_2}{L_{c,2}}\right)}^2}\right)$$

(3)

$$\mathrm{SExp}: \rho \left(\Delta x_1,\Delta x_2\right)=c_1+\left(1-c_1\right)\exp \left[-\left({\left(\frac{\Delta x_1}{L_{c,1}}\right)}^2+{\left(\frac{\Delta x_2}{L_{c,2}}\right)}^2\right)\right]$$

(4)

where $c_1$ is the threshold value of the correlation function, and $L_{c,1}$ and $L_{c,2}$ are the correlation length.

2.3. Anisotropic Random Field

It is important to keep in mind the notion of correlation length in order to examine the modeling theory of anisotropic random fields in geotechnical bodies. The crucial parameter for measuring the spatial variability of geological materials is the phase correlation length. The random field’s anisotropic properties are determined by the length’s difference in various orientations. The correlation length contour can be used to describe the random field’s structural properties. The correlation length contour is created by joining the locations in space that have the same coefficients as the circle’s center, as seen in Figure 3.

The correlation lengths in the two coordinate axes’ primary directions are shown by $L_{c,1}$ and $L_{c,2}$. $L_c$ is the correlation length in a coordinate space direction. The coordinate axes’ angle of rotation, with respect to the horizontal datum, is given by the symbol θ. As a result, the contour switches from being an isotropic circle to an ellipse when the correlation lengths along the two coordinate axes’ primary directions are different. This defines the features of the random field with horizontal anisotropy. The length of the projection on the axis also varies when the axes are rotated by an angle dependent on the horizontal line. The rotational anisotropy of the random field is characterized by this circumstance.

The notion of random field production must be appropriately changed if the anisotropy of a rock is to be described by a random field. This work suggests a random field generating technique that is suitable to the anisotropy of rock materials, in combination with the random field theory discussed above. The necessary length is adequately enlarged along a particular main axis direction where anisotropic features must be represented. As shown in Figure 3b, by lengthening the correlation in the axial direction for the horizontal anisotropy, the necessary anisotropic random field may be attained. In contrast, the rotational anisotropy seen in Figure 3c is not in the direction of the main axis. The relevant length projections along the X-axis direction and the Y-axis direction can be obtained by changing the angle θ of the reference coordinates, rather than using the standard coordinate axes as a reference.

2.4. GPU-Accelerated Anisotropic Random Fields

In this work, a generator of spatially correlated random variables is coupled with a discretization technique to calculate a random variable that is associated with other nodes in the random field for each node in the random field grid. In the finite element model, each of these spatially associated random variables is given a cell or integration point. The covariance matrix decomposition method and the midpoint approach are used in tandem to create the random field. Using the covariance matrix decomposition method, a group of related random variables can be generated, and the corresponding generation formula is as follows:

$${\mathrm{z}}_{\mathrm{c}}\left(\mathrm{x}\right)=\mathrm{L}\mathsf{\chi }$$

(5)

Among them ${\mathrm{z}}_{\mathrm{c}}\left(\mathrm{x}\right)$ is a vector containing spatially correlated random variables, $\mathsf{\chi }$ is a vector of independent zero-mean, unit variance, and normally distributed random variables, and is a matrix obtained by Cholesky decomposition of the correlation matrix.

Consequently, the key to this approach is the construction of the correlation matrix. The correlation coefficient matrix for a collection of N random variables gathered in the vector y is defined as follows, after loading the grid file to retrieve the node coordinates:

$$R_{ij}=\frac{\mathrm{Cov}\left(y_i,y_j\right)}{\sqrt{\mathrm{Var}\left(y_i\right)\mathrm{Var}\left(y_j\right)}}=\left[\begin{array}{cccc}1& \rho \left(y_1,y_2\right)& \dots & \rho \left(y_1,y_n\right)\\ & 1& & \rho \left(y_2,y_n\right)\\ & & \ddots & ⋮\\ \mathrm{symmetric} & & & 1\end{array}\right]$$

(6)

where $\rho \left(y_i,y_j\right)$ is the correlation coefficient of $y_i$ and $y_j$.

This matrix must be decomposed such that multiplication of $L\chi$ yields a vector ${\chi }_c$ of normally distributed random variables with zero mean, unit variance, and specific correlations. The correlation matrix is symmetric and positive definite, and the correlation coefficient matrix $R$ is decomposed into an upper triangular matrix and lower triangular matrix by Cholesky decomposition:

$$\begin{array}{ll}R& =\mathrm{Cov}\left[{\chi }_c,{\chi }_c\right]=\mathrm{E}\left[{\chi }_c{\chi }_c^T\right]-0\cdot 0=\mathrm{E}\left[{\chi }_c{\chi }_c^T\right]\\ & =\mathrm{E}\left[L\chi L{\chi }^T\right]=L\mathrm{E}\left[\chi {\chi }^T\right]L^T=LI{L}^T=L{L}^T\end{array}$$

(7)

After obtaining the correlation matrix, it can be decomposed using the Cholesky decomposition method. The characteristic decomposition of the square matrix needs to be derived through the following factorization:

$$R=Q\mathsf{\Lambda }Q$$

(8)

The correlation matrix is nonpositive, or all of the eigenvalues of the matrix are no longer positive, when the SExp correlation function is used, and the number of nodes is greater than the correlation length. The matrix must be changed, in this instance. To get reliable results, the parameters in the modified Cholesky decomposition process are adjusted to zero when they fall below the tolerance threshold. An “approximate” Cholesky decomposition is produced by this algorithm. The correlation matrix is decomposed to provide the eigenvalues and eigenvectors. Equation (9) is used to generate the correlation matrix’s decomposition.

$$R=Q\tilde{\mathsf{\Lambda }}\tilde{\mathsf{\Lambda }}Q=L{L}^T\to L=Q\tilde{\mathsf{\Lambda }}$$

(9)

where $Q$ is the matrix containing the eigenvectors and $\mathsf{\Lambda }$ is the diagonal matrix.

Following the aforementioned derivation, Equation (10) can be used to build the final random field CRV (correlated random variables) computation.

$$CRV=\mu +L\times RV\times \sigma$$

(10)

where $\mu$ is the standard normal distribution mean, $\sigma$ is the standard deviation of the standard normal distribution, and $RV$ is the standard normal distribution random quantity.

Although the conventional approach to creating random fields using enhanced Cholesky decomposition is effective, it is computationally intensive and has few real-world applications. To be useful, the effectiveness of random field creation must be increased. The modified Cholesky decomposition’s related eigenvalue decomposition operation may be found in MATLAB. In the random field generation approach, the correlation matrix decomposition is carried out by MATLAB (R2021b) programming. GPU acceleration techniques are used to increase efficiency because when the model is large and complex, the coordinate data is comparatively much larger, and the length of its coordinate data produces a huge array of correlation matrices that take a long time to decompose.

In computing, CPUs are characterized by overall process control and logical processing in algorithms. The GPU, on the other hand, is characterized by its ability to process data centrally and perform calculations. It is easy to compare the two and conclude that although the CPU has a strong computing power for each core processor, it has few cores and cannot perform well in parallel computing; on the contrary, although the computing power of each core is not outstanding, the GPU has more cores and can handle multiple computing tasks in parallel, making it more powerful than the CPU in terms of parallel computing performance.

In the case of the GPU, it is tasked with composing and displaying images of millions of pixels on the screen, meaning that there are millions of tasks to be processed in parallel at the same time, so the GPU is designed to handle many tasks in parallel, rather than single-threaded tasks, as is a CPU. The CPU and GPU architectures are therefore very different, with the CPU having many functional modules and being able to adapt to complex computing environments, while the GPU’s composition is relatively simple, with the processor and controller currently taking up the majority of the hardware area. A simple diagram is shown below (Figure 4), which shows that GPUs have more logical algorithm units (ALUs) and are also multi-threaded and parallel, without too many control and storage functions, and that they are highly efficient and good at processing dense and large amounts of data and complex algorithms. The CPU, on the other hand, needs to control the entire computation process and store the data at the same time. The number of logical units ithemselves is small, and only one line of data can be processed in a time slot, making it inefficient for large and complex arrays.

In today’s computer architecture, the GPU alone cannot perform a compute unified device architecture (CUDA) parallel computation, but must be used in conjunction with the CPU to perform a high-performance parallel computation. The CPU is responsible for the overall program flow, while the GPU is responsible for the specific computational tasks, and when the GPU threads have completed their computational tasks, we copy the results of the GPU computation to the CPU. Once the GPU threads have completed their computation, we copy the results from the GPU to the CPU to complete a computation. Therefore, at this step of the correlation matrix decomposition, the efficiency of the GPU parallel computation increases significantly as the matrix gets progressively larger, and the efficiency of the cyclic simulation increases dramatically. When it comes to the modified Cholesky decomposition, after the efficient computational decomposition in the GPU, the required eigenvector matrix and eigenvalue matrix are transferred back to the CPU for the next computation, and the comparison of the time required is shown in the figure below. As can be seen in Figure 5, the advantage of the GPU acceleration gradually emerges as the array of the operational matrix becomes larger, and its operation speed and efficiency far exceed that of the CPU, which saves a great deal of time when performing multiple round-robin operations to compute probability problems.

3. Simulation of the Red Layer in Central Yunnan

3.1. Geological Background

The primary rock type surrounding the tunnel of the Central Yunnan Water Diversion Project is the soft rock of the Central Yunnan red beds. Under the influence of high ground stress (or relatively high ground stress), groundwater, or self-swelling properties, the rock loses or partially loses its self-supporting capacity, resulting in plastic deformation, with progressive and significant time effects. This deformation cannot be effectively restrained, as depicted in Figure 6. The structure of the rock stratum of the weak rock and soil body pertains to the spatial distribution and arrangement between the sedimentary layers in its rock stratum, such as the block-like thick layer to layer-like thin structure, which indicates the thickness of a single layer of the rock stratum, and the sandwich-like and interlayer-like structure that demonstrates the spatial combination and arrangement of the rock stratum units. These units can be categorized based on lithology, layer thickness, and rock, among other factors, at each site. It is apparent that the original rock stratigraphic structure of the weak rock and soil body is closely related to the engineering rock structure type, and the former serves as the foundation of the latter.

The soft rocks in the red bed of Central Yunnan occur in three main forms: the first is mudstone in a laminated structure dominated by soft mudstone, the second is mudstone in an interbedded structure of soft and hard sandy mudstone, and the third is soft and weak mudstone interbedded in a laminated structure dominated by thick bedded hard rock.

In the first type of rock structure in the soft rock, the form of damage in the surrounding rock will mainly be section compression, as shown in Figure 7. Due to the mud-rich and soft nature of the soft rocks in the Central Yunnan red bed, the rocks surrounding the tunnel are characterized by significant plastic deformation or rheology. At the same time, the soft rocks in the Central Yunnan red bed are prone to disintegration, swelling and softening, which will further deteriorate the stability conditions of the surrounding rocks of the tunnel and intensify the plastic deformation of the surrounding rocks. Specifically, the soft rocks that can easily absorb water and swell are prone to strong swelling deformation damage. Within this type of rock structure, soft rocks are dominant, while sandstone, conglomerate, and other hard rocks exist as a sandwich-like structure, with soft rocks playing a pivotal role in the deformation of the surrounding tunnel rocks. Thus, in sections with a significant burial depth, the surrounding rock’s deformation will primarily occur via sectional compression, and softer rock quality coupled with higher ground stress or deeper tunnel burial will result in more pronounced sectional compression deformation.

For the soft rock in the second rock mass structure (Figure 8), the deformation and failure form of the surrounding rock will also be mainly section compression, but some tunnel sections will experience the top arch collapse deformation and failure form. Compared to the first type of soft rock deformation, this type is affected by a certain proportion of hard rock and therefore experiences some degree of restraint in section compression deformation. The deformation magnitude will be reduced, and the time effect of the deformation will be prolonged. Furthermore, in some tunnel sections where the hard rock proportion is high and the rock mass integrity is good, the surrounding rock of the tunnel may exhibit crown collapse deformation failure due to the presence of weak interlayers in the soft rock.

In the soft rock in the third rock mass structure (Figure 9), the stability of the surrounding rock of the tunnel in the soft rock position is poor, and the deformation and failure form of the surrounding rock will mainly occur as the collapse of the crown. Soft rock exists in the form of weak interlayers in this type of rock mass structure, often constituting areas with unstable surrounding rock in the tunnel section. Affected by environmental conditions, the surrounding rock of the tunnel is prone to collapse, deformation, and failure of the roof arch.

In summary, the surrounding rock of the water diversion tunnel in central Yunnan is a typical heterogeneous rock-soil medium with obvious spatial variability. Traditional geotechnical engineering analysis mostly assumes that the properties of the surrounding rock materials are uniform, and the randomness of rock and soil parameters cannot be considered. A large number of research results show that the randomness of geotechnical parameters has an important impact on underground cavern engineering. As a typical heterogeneous material, the surrounding rock of the Central Yunnan Water Diversion Project must be evaluated in regards to the spatial anisotropy of the rock and soil parameters. Focusing on the typical spatial distribution form of the surrounding rock of the water diversion tunnel in central Yunnan, this paper constructs the corresponding anisotropic random field distribution form.

3.2. Modeling and Analysis Process for Red-Bedded Soft Rock Tunnels in Central Yunnan

A section of a hydraulic tunnel with a horseshoe-shaped design was selected for analysis in the Central Yunnan Water Diversion Project. This particular section of the hydraulic tunnel is situated at a depth of roughly 600 m, with a gently sloping ground surface. The numerical model used in the study is depicted in Figure 10, featuring dimensions of 80 m in length, 80 m in height, a bottom wall of 8.72 m, and a height of 9.43 m. The model was divided into separate rock and tunnel zones, and a hexahedral structural grid was utilized to divide the model mesh. The total number of meshes in the numerical model was 7225, with a total of 14,594 nodes.

The depth of burial and the mass of the rock in this portion of the hydraulic tunnel may be used to determine the initial ground tension. The fluctuation value of the internal friction angle $\phi$ is proposed to be between 25 and 35 degrees, and the fluctuation value of the cohesion $c$ of the rock strength parameter is suggested to be between 0.1 and 0.4 MPa. Other rock mechanics parameters show values indicated from the site engineering geological survey report, as illustrated in

Table 1. Suggested values of random field parameters for red layer excavation tunnels in central Yunnan.

Threshold of Correlation Function	Cohesion $\mathit{c}$ (MPa)	Friction Angle $\mathit{\phi }$ (°)	Young’s Modulus (GPa)	Poisson’s Ratio	Density (kg/m3)	Geostress (MPa)
0.5	0.1–0.4	25–35	20	0.25	2500	14.5

.

To analyze the excavation process of caverns using anisotropic random field theory, the first step is to construct a model and divide it into a mesh using meshing software. Then, the pre-processing of the subdivided grid is controlled by code programming, the coordinates of the grid center point are derived, and the coordinate matrix is generated in MATLAB. Finally, the correlation matrix is generated from the coordinate matrix, and the random field matrix is generated using the covariance matrix decomposition method. Based on the Mohr–Coulomb constitutive model, the attributes that need to be assigned are selected, and the anisotropic random field matrix is substituted into the matrix to generate an initial random assignment model of the tunnel based on the random field theory, and its numerical simulation is performed. The maximum displacement and the point coordinates of the maximum displacement during the simulated excavation process are recorded, and conclusions are drawn after multiple cycles. The whole process can be accomplished by executing operating system commands with MATLAB. The specific flow chart is shown in Figure 11.

4. Analyses on the Influence of the Law of Anisotropic Random Fields on the Safety Factor

4.1. Simulation Conditions and Method

Based on the method of anisotropic random field modeling and generation, this paper selects an exponential correlation function that is more suitable for describing the spatial variability of rock and soil mass. On the basis of the isotropic random field, the correlation length in the y-axis direction is kept unchanged at 2.5, and the correlation length in the x-axis direction is changed to 10. The other parameters are shown in Table 1. The safety factor is calculated 100 times for each angle, and according to the influence of parameter randomness on the mechanical characteristics of the surrounding rock, the plastic zone of the anisotropic random field with different rotation angles is compared and analyzed to obtain the safety factor. Using the method above, the numerical simulation of the excavation of an anisotropic random field cavern chamber with different rotation angles is conducted. The simulation results are analyzed as follows.

4.2. Simulation Results and Analysis

At a rotation angle of 0 degrees, the correlation coefficient matrix was obtained through covariance matrix decomposition and subsequently inserted into the cohesion and friction angle parameters. One of the 100 simulations was chosen to derive the standard anisotropy random field, as illustrated in Figure 12a,b. On the bottom wall of the excavation, weaker horizontal strips of rock exist that possess lower cohesion and friction angle magnitudes compared to those of the surrounding rock. This phenomenon is akin to the intercalated mudstone type stratigraphic formation in the red layer of central Yunnan. The reduced parameters in this area result in a larger radius of the plastic zone on both sides of the bottom wall. Conversely, the strip of rock situated to the right of the vault exhibits stronger parameter magnitudes than the surrounding rock, causing a smaller radius of the plastic zone in this direction, as illustrated in Figure 12c. The average safety factor value in 100 calculation cycles is 2.05, and the safety factor ranging from 2 to 2.05 occurs most frequently, with a frequency of 22 times. Most of the safety factors fall within the interval of 1.95 to 2.15, indicating that the overall excavation simulation process is relatively safe. The frequency of safety factors that are less than 1.9 is minimal.

By altering the projection coordinates’ angle between two points in space, the correlation function can be generated at different angles. Consequently, when the projection angle is set at 15°, an anisotropic random field with a rotation angle of 15° is obtained, and one of the 100 simulation cycles’ outcomes is selected, as illustrated in Figure 13a,b. At the excavation hole’s vault, a 15° strip-shaped rock mass with a higher cohesion and friction angle than the surrounding rock mass exists, akin to the siltstone-type stratum structure in the red bed in central Yunnan. The stronger parameter in the region results in a smaller plastic zone’s radius near the vault. Conversely, the banded rock mass located on the right wall possesses weaker parameters than the surrounding rock mass, and the plastic zone’s radius in this direction is larger than that of the left wall area, as shown in Figure 13c. After 100 calculation cycles, the average safety factor value is 2.06, and the safety factor occurs most frequently in the [2, 2.05] interval, with 25 occurrences. Most safety factors are located in the [2, 2.15] interval. The overall excavation simulation process is relatively safe, and the frequency of a safety factors less than 1.9 is low.

When the projection angle is 30 degrees, an anisotropic random field with a rotation angle of 30° is obtained, and the simulation results of one of the 100 cycles are selected, as shown in Figure 14a,b. At 30° of the vault of the excavation hole, there is a band of rock mass across the excavation hole, and its cohesion and friction angle are higher than those of the surrounding rock mass. It can be clearly seen that the radius of the plastic zone at the line connecting the bottom of the left wall and the right side of the vault is small, and the spatial variability of the rock mass along this direction is weak. The distribution of the plastic zone presents a certain angle, perpendicular to the anisotropy direction of the random field, as shown in Figure 14c. In 100 cycle calculations, the average value of the safety factor is 2.09, and the safety factor occurs most frequently in the [2.1, 2.15] interval, with 27 occurrences. Most of the safety factors are located in the [2, 2.15] interval. Overall, the excavation simulation process in a relatively safe state, and the frequency of a safety factor less than 1.9 is low.

An anisotropic random field with a rotation angle of 45° was created when the projection angle was 45°, and the outcomes of one of the 100 simulation cycles were chosen, as shown in Figure 15. The parameters in this area will weaken, resulting in a larger radius of the plastic zone in this direction, with the plastic zone distribution angle being similar to the random field anisotropy angle, as shown in Figure 15a,b. At the top of the excavation vault, there is a 45° strip of rock with a lower cohesion and friction angle magnitude than the surrounding rock, passing through the core of the cavern through the excavation tunnel, as displayed in Figure 15c. The overall excavation simulation process is in a relatively safe state, and the frequency of safety factors less than 1.9 is low. The average value of the safety factor after 100 calculation cycles is 2.05, and the safety factor in the interval of [2, 2.05] occurs most frequently with 26 occurrences.

Figure 16a,b illustrates the simulation outcomes of one of the 100 cycles when the projection angle is 60° and an anisotropic random field with a rotation angle of 60° is created. A 60° narrow strip-shaped rock mass with stronger cohesion and friction angles than those in the surrounding rock mass is present on the left side of the excavation hole. The radius of the plastic zone will decrease because the rock mass characteristics in this region are stronger than those of other rock masses close to the cavern. While the anisotropy on the right is not immediately apparent, the distribution angle of the plastic zone on the left is comparable to the anisotropy angle of the random field. According to Figure 16c, the shift is not significant. The safety factor has an average value of 2.07 in computations involving 100 cycles, and it appears 26 times, most commonly in the interval [2.1, 2.15]. The [2, 2.15] interval contains the majority of the safety considerations. The overall excavation simulation process is in a relatively safe state, and the frequency of safety factors less than 1.9 is minimal.

The outcomes of one of the 100 cycles of simulation are chosen when the projection angle is 75°, and an anisotropic random field is created with a rotation angle of 75°, as illustrated in Figure 17a,b. The rock characteristics deteriorate in this area, increasing the radius of the plastic zone, and there is a small strip of rock to the left of the excavation entrance that has lower cohesiveness and friction angle magnitudes than those of the surrounding rock. On the right side of the vault of the excavation opening there are stronger random field parameters and weaker random field parameters at the base of the right wall, which are reflected in the comparative plastic zone. The radius of the plastic zone on the upper side is significantly smaller than that on the lower side, as shown in Figure 17c. The safety factor has an average value of 2.04 during 100 cycles of computations, and it appears 31 times, most commonly in the [2, 2.05] interval. The [2, 2.15] interval is where the majority of safety factors are found. The overall excavation simulation process is in a relatively safe state, and the frequency of safety factors less than 1.85 is low.

When the projection angle is 90°, the anisotropic random field with a rotation angle of 90° is obtained, and the simulation results of one of the 100 cycles are selected, as shown in Figure 18a,b. On the right side of the excavation hole, there is a longer distribution of strip-like rock masses, whose cohesion and friction angle size are lower than those of the surrounding rock masses. Comparing the mechanical properties of the rock masses in the left area, the radius of the plastic zone caused by the weakening of rock parameters in this area is significantly larger than that in the left side, as shown in Figure 18c. In 100 cycles of calculation, the average value of the safety coefficient is 2.06, and the safety coefficient in the interval of [2.1, 2.15] occurs most frequently, with 23 occurrences. Most of the safety coefficients are located in the interval of [1.95, 2.15], and the overall excavation simulation process is in a relatively safe state. The frequency of safety coefficients less than 1.9 is low.

Based on the above simulations of anisotropy with tunnel excavation at seven different angles, seven sets of statistical values of safety coefficients were obtained. To study the variability of anisotropy with at different projection angles, box plots were drawn, as shown in Figure 19. Deformation damage analysis was carried out by comparing the mean and variance to the tunnel excavation simulation for the Central Yunnan Diversion Project. A small number of outliers exist for the anisotropic random field safety factor at 15°, 30°, and 75°, with the smallest median safety factor of 2.04 at 0°, 45°, and 75°. The variance of the safety factor of the anisotropic random field of 30° is the largest, and the variance of the safety factor of the anisotropic random field of 75° is the smallest. The maximum factor of safety for removing outliers in all simulations was 2.34, and the minimum factor of safety was 1.84. The safety factors of the seven anisotropic random fields with different angles are relatively safe, and the median of 15° is safe, but the variance is large, and some dangerous outliers appear. The safety factor of 30° has a large fluctuation range and poor calculation stability. Although the variance of 75° is small, there are many outliers. Compared with the overall safety factor statistics, there is a lower safety factor, and there is a risk of instability.

5. Conclusions

In this paper, the random field model is used to describe the spatial variability of the surrounding rock parameters of an underground tunnel. Based on this model, the stability analysis of the excavation of the underground tunnel is carried out. This study is based on the excavation of deep underground caverns in the Central Yunnan Water Diversion project. The effects of the anisotropy of the random field and the randomness of the parameters on the state of the surrounding rock are studied. The stability of the anisotropic random field with different rotation angles is also analyzed. The main conclusions and results are as follows:

(1)

The focus of this paper is on the covariance matrix decomposition method, which can be used to obtain stable independent random fields through Cholesky decomposition. However, due to the large number of grids and correlation matrix arrays, the method can result in slow calculations. To address this issue, the paper proposes a GPU acceleration technology, which enables the parallel operation of local matrix decomposition and overall serial calculations. By combining the advantages of the GPU, the proposed method efficiently generates random fields.

(2)

In this study, numerical simulation of the excavation stability of hydraulic tunnels was carried out for the anisotropic random field of red-bedded soft rocks in central Yunnan. The Mohr–Coulomb model was chosen for the rock material, and the internal friction angle and cohesion were set as random parameters, while other parameters were defined as constants. The safety factors of the anisotropic random field with different rotation angles are compared. The distribution of the plastic zone after excavation has a significant relationship with the random parameters of the anisotropic random field. When the stronger or weaker random parameters are located in the surrounding rock of the cavern, it will cause the change in the radius of the plastic zone. The overall safety factor of the anisotropic random field is relatively stable, with an average value of about 2, which mainly depends on the strength of the random parameters of the rock mass. Based on the random fluctuation of the value suggested in the engineering report, the simulation result is safe.

In summary, the GPU-accelerated anisotropic random field modeling method proposed in this paper can better consider the anisotropy of red-bedded soft rock and more realistically simulate the deformation and failure of tunnel excavation using software calculations. At the same time, based on the parallel computing and cycle efficiency of the GPU, this method can greatly improve modeling efficiency, saving time and cost.