Intelligent prediction method for lead ion effect on soil crack fracture dimension under dry-wet cycle

Abstract

The application relates to an intelligent prediction method for lead ion to soil crack fractal dimension under dry-wet cycles, and relates to the technical field of soil crack fractal dimension prediction. The technical scheme is as follows: sample data is collected and standardized; the sample data is divided into prediction samples and training samples; the particle swarm optimization algorithm and the sparrow optimization algorithm are used to optimize the BiLSTM, GRU, Xgboost and RVM model parameters of the training samples; the optimal solution of the particle swarm optimization algorithm and the sparrow optimization algorithm is used to establish a prediction model of the soil body crack fractal dimension which meets the requirements; the prediction samples are brought into the BiLSTM, GRU, Xgboost and RVM model parameters, the obtained prediction results are compared with the actual values, fractal comparison is carried out, and the model feasibility is verified. The application effectively improves the prediction accuracy and efficiency, reduces the research cost and time, promotes soil scientific research and technological innovation, is not only helpful for promoting the development and innovation of soil scientific research, but also provides strong technical support for the realization of the environmental protection and sustainable development goals.

Description

Technical Field

[0001] This invention relates to the technical field of the intersection of civil engineering and artificial intelligence, specifically to the field of soil fissure fractal dimension prediction technology, and particularly to an intelligent prediction method for the effect of lead ions on the fractal dimension of soil fissures under wet-dry cycles. Background Technology

[0002] In recent years, heavy metal pollution in mining soils has become an increasingly serious problem worldwide, urgently requiring global solutions. The impact of heavy metals on soil is multifaceted, affecting its physical, chemical, biological, and ecosystem properties. Therefore, research on the effects of heavy metals on soil properties has become a hot topic in the environmental field. For example, Za studied the impact of long-term Zn and Cu pollution in Spolic Technosols on water-soluble organic matter and soil biological activity; Huang conducted in-depth research on how soil particle size affects the environmental behavior of heavy metals in soil; and Song studied the effects of wet-dry cycles and cadmium pollution on the mechanical properties and structure of red clay. Furthermore, heavy metal pollution exacerbates the formation and distribution of soil fissures, leading to problems such as damage to soil integrity, reduced soil strength, and increased soil permeability, posing a serious threat to local engineering construction and long-term sustainable development. Therefore, studying the influence of heavy metals on soil fissures is of great significance.

[0003] Soil fissures are influenced by climate through repeated wet-dry cycles, such as rainfall, groundwater level fluctuations, and surface water evaporation. Rayhani et al. reported that the size and permeability coefficient of soil samples increased after wet-dry cycles with increasing plasticity index and clay content. Lu explored the microstructure under wet-dry cycle conditions and found that fissures increased significantly after wet-dry cycles. Xu demonstrated that internal cracks in loess were more developed when the salinity was higher after wet-dry cycles. Previous studies have mostly focused on the impact of wet-dry cycles on fissure development. However, in areas with heavy metal pollution, as wet-dry cycles continue, soil integrity is disrupted, leading to increased diffusion range and concentration of metal ions, which further affects fissure development. Therefore, studying the development patterns of soil fissures under the coupling effects of wet-dry cycles and different heavy metal ions can provide a more reliable scientific basis for the management of heavily polluted soils. How to accurately and quickly predict the changing parameters of soil fissures under wet-dry cycle conditions is an important foundation for further research on soil fissure development. Early studies mainly used measurement tools such as soil slicing and sand filling to obtain fissure-related parameters, but these methods are inefficient and have poor accuracy. The development of digital imaging has provided new ways to obtain soil fracture parameters, such as radar and fiber optic methods. However, these methods are expensive, and there is a need to find more efficient and economical methods to obtain and predict soil fracture parameters under wet-dry cycle conditions.

[0004] With the rapid development of computer technology, artificial intelligence algorithms, with their self-learning and efficiency-enhancing capabilities, as well as their cost-reduction advantages, have become a key driving force for technological innovation in soil science research. For example, Prasad et al. used a hybrid intelligent extreme learning machine to predict soil moisture; Araghi used a wavelet neural network (WNN) to predict soil temperature relative to surface air temperature; and Zhang et al. proposed an intelligent method for quickly identifying soil particle size distribution using a convolutional neural network (CNN) within a deep learning framework. Therefore, using artificial intelligence algorithms to predict the evolution of soil fracture parameters under wet-dry cycles has become an inevitable trend. BiLSTM (Bi-directional Long Short-Term Memory), as an improved version of LSTM (Long Short-Term Memory), not only inherits the core advantages of LSTM in processing sequential data but also effectively overcomes its limitations in contextual prediction. GRU (Gate Recurrent Unit), with its fewer parameters and higher efficiency, occupies an important position in prediction models. XGBoost (eXtremeGradient Boosting) excels in numerous machine learning tasks due to its high flexibility and fast computation speed. Relevance Vector Machine (RVM) is renowned for its simple control parameters and efficient sample fitting capabilities. These BiLSTM, GRU, XGBoost, and RVM algorithms have been widely applied and validated in multiple fields. Fractal dimension is an important parameter for studying soil fracture evolution, providing an effective means for quantitatively describing fracture morphology, assessing development degree, quantifying roughness, and analyzing fracture network connectivity. Therefore, constructing intelligent prediction models for fractal dimension using BiLSTM, GRU, XGBoost, and RVM provides a new approach to studying the evolution of soil fracture parameters. However, parameter settings for BiLSTM, GRU, XGBoost, and RVM algorithms often rely on manual trial and error, which not only increases subjectivity but also affects the accuracy and predictive ability of the models. Summary of the Invention

[0005] To address the aforementioned technical problems in existing technologies, this invention provides an intelligent prediction method for the fractal dimension of soil fractures caused by lead ions under wet-dry cycles. By applying PSO and SSA algorithms to the parameter optimization of BiLSTM, GRU, Xgboost, and RVM models, the accuracy and prediction performance of these models are effectively improved. This provides strong support for achieving efficient and intelligent fractal dimension prediction.

[0006] To achieve the above objectives, the present invention provides the following technical solution:

[0007] A smart prediction method for the fractal dimension of soil fissures by lead ions under wet-dry cycles, comprising the following steps:

[0008] S1. Collect sample data and perform standardization processing;

[0009] S2. The standardized sample data is divided into prediction samples and training samples;

[0010] S3. The training samples are optimized using the particle swarm optimization algorithm and the sparrow optimization algorithm to find the parameters of BiLSTM, GRU, Xgboost and RVM models. Based on the optimal solutions of the particle swarm optimization algorithm and the sparrow optimization algorithm, a prediction model of the fractal dimension of soil fractures that meets the requirements is established.

[0011] S4. Input the predicted samples into the parameters of BiLSTM, GRU, Xgboost and RVM models, compare the obtained prediction results with the actual values, and verify the feasibility of the model.

[0012] Furthermore, the steps of the particle swarm optimization algorithm are as follows:

[0013] T1. Initialize the position, fitness, and velocity of the particle swarm;

[0014] T2. Determine the optimal position and fitness of the particle swarm;

[0015] T3. When the number of iterations is less than the maximum number of iterations, update the particle's velocity and position using formula (1):

[0016]

[0017] Where w is a non-negative inertia factor used for algorithm convergence, c1 and c2 are learning factors used to adjust the optimal weights, and r1 and r2 are random numbers between [0,1].

[0018] T4. Calculate the fitness of the next generation population;

[0019] T5. Update the optimal position and fitness of the particle swarm.

[0020] Furthermore, the steps of the sparrow optimization algorithm are as follows:

[0021] U1. Initialize the location and fitness of the sparrow population;

[0022] U2. Sort the fitness values ​​and determine the first generation of discoverers, followers, and scouts;

[0023] U3. When the number of iterations is less than the maximum number of iterations, for each discoverer, i from 1 to DN, a random number R2 is generated, and the discoverer's position is updated using formula (2):

[0024]

[0025] Where: DN is the number of discoverers, t is the current iteration number, and T is... max is the maximum number of iterations, α is a random number and α∈(0,1], R2∈[0,1] and ST∈[0.5,1] are the warning and safety values ​​respectively, Q is a normally distributed random number, and L is a 1×d matrix with all elements equal to 1;

[0026] U4. For each follower, i ranges from 1 to FN, update the follower position using formula (3):

[0027]

[0028] Where: FN is the number of followers, X B The discoverer obtains the optimal position, X. w Represents the global worst value; A is a one-dimensional vector of length d, A + =A T (AA T ) -1 When i > N / 2, it means that sparrows with low fitness are very hungry and need to fly to other places to forage for food.

[0029] U5. For each scout, i ranges from 1 to VN, update the scout position using formula (4):

[0030]

[0031] Where: VN is the number of scouts, X gB The current global optimum is given by f, where β is the step size control parameter following a normal distribution (0,1), K is a random number between [-1,1], ε is a constant to avoid a denominator of 0, and f is the value of f. i It is the fitness value of an individual sparrow, f g and f w These are the current best and worst fitness values, respectively.

[0032] U6. Update the optimal fitness and location of the sparrow population.

[0033] Furthermore, in step S1, standardization is performed using formula (5):

[0034]

[0035] In the formula: x i For the i-th influencing factor, s is the mean of the sample data and s is the standard deviation of the sample data.

[0036] Furthermore, the following steps are also included: analyzing the mean square error of the prediction results using formula (6):

[0037]

[0038] In the formula, MSE represents the mean square error, y 0,i Let y represent the actual value of the i-th sample. i Let represent the predicted value of the i-th sample, and n represent the total number of samples.

[0039] Furthermore, the following steps are also included: analyzing the root mean square error of the prediction results using formula (7):

[0040]

[0041] In the formula, RMSE represents the root mean square error, and y 0，i Let y represent the actual value of the i-th sample. i Let represent the predicted value of the i-th sample, and n represent the total number of samples.

[0042] Furthermore, the following steps are also included: analyzing the mean absolute error of the prediction results using formula (8):

[0043]

[0044] In the formula, MAE represents the mean absolute error, y 0，i Let y represent the actual value of the i-th sample. i Let represent the predicted value of the i-th sample, and n represent the total number of samples.

[0045] Furthermore, the following steps are also included: analyzing the mean absolute percentage error of the prediction results using formula (9):

[0046]

[0047] In the formula, MAPE represents the mean absolute percentage error, y 0，i Let y represent the actual value of the i-th sample. i Let represent the predicted value of the i-th sample, and n represent the total number of samples.

[0048] Furthermore, in step S1, sample data is collected through the following steps:

[0049] S1.1 Select silty clay, red clay, and expansive soil from 5-7m underground at the construction site;

[0050] S1.2 Sample Preparation: After air-drying and crushing the obtained soil sample, it was sieved through a 2mm sieve. The sieved soil sample was then dried in a 105℃ oven until constant weight. A lead ion solution was then prepared and evenly sprayed into the soil sample for thorough stirring. The initial moisture content of the lead-contaminated soil sample was 20%, and the lead ion content was 0%, 0.03%, 0.06%, 0.09%, and 0.12% of the dry soil mass, respectively. A contaminated soil sample with a dry density of 1.45 g / cm3, a diameter of 61.8 mm, and a height of 20 mm was prepared according to the static method. The sample was humidified by vacuum saturation for 12 h, and then dehumidified in a 45℃ constant temperature oven for 24 h, which constitutes one wet-dry cycle.

[0051] S1.3 Digital Image Processing: First, the acquired sample image is precisely cropped to ensure that the cropped image edge is perfectly aligned with the inner wall of the ring cutter. Then, the image is imported using software and grayscale conversion is performed. Advanced noise reduction algorithms are applied to the grayscale image to improve image quality, and crack identification technology is used to accurately extract crack features. Finally, image analysis technology is used to calculate key parameters such as crack area and fractal dimension to obtain high-precision analysis results.

[0052] Furthermore, in step S1.3, two key indicators are selected: crack area and fractal dimension; crack area is defined as the projected area of ​​the crack on a two-dimensional plane; fractal dimension is used to quantify complex geometric structures with self-similar characteristics; the fractal dimension of the soil crack pattern is determined by the box-counting method, and the calculation method is as shown in equation (10):

[0053] lnN i =A-Dlnε i (10)

[0054] In the formula: A is a constant; D is the fractal dimension; ε i and N i Let lnN be the segmentation size of the i-th level and the corresponding number of self-similar blocks, respectively. i and lnε i The relationship exhibits a linear one, and its slope D is the fractal dimension of the fracture. The larger the fractal dimension of the fracture network, the denser the fracture distribution and the more complex the fracture structure distribution characteristics.

[0055] The beneficial effects of this invention are:

[0056] Compared with existing technologies, the intelligent prediction method for the fractal dimension of soil fissures by lead ions under wet-dry cycles described in this invention has the following technical features or beneficial effects:

[0057] (1) Improve prediction accuracy and efficiency

[0058] Application of the intelligent prediction model: This invention constructs a prediction model for the influence of lead ions on the fractal dimension of soil fissures under wet-dry cycles by introducing advanced intelligent algorithms such as BiLSTM, GRU, Xgboost, and RVM. These algorithms, with their powerful self-learning and data processing capabilities, can significantly improve prediction accuracy. Further optimization of model parameters using Particle Swarm Optimization (PSO) and Sparrow Optimization (SSA) algorithms further enhances the model's predictive performance, making the prediction results more accurate and reliable.

[0059] Standardization and Error Analysis: The collected sample data were standardized to eliminate dimensional differences between different data points, improving the model's generalization ability. Error analysis metrics such as mean squared error (MSE), root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) were used to comprehensively evaluate the prediction results, ensuring their accuracy and reliability.

[0060] (2) Reduce research costs and time

[0061] Highly efficient data processing: This invention utilizes intelligent algorithms to rapidly process and analyze large amounts of data, significantly improving data processing efficiency and reducing research costs compared to traditional research methods (such as soil slicing and sand filling measurement tools). By automatically optimizing model parameters through optimization algorithms, the tedious process of manual trial and error is avoided, further shortening the research cycle.

[0062] Reduced experimental resource consumption: Employing intelligent prediction methods can reduce reliance on actual soil samples, thereby lowering the consumption and waste of experimental resources. Furthermore, intelligent prediction methods can provide guidance for experimental design, optimize experimental protocols, and improve experimental efficiency.

[0063] (3) Promote soil science research and technological innovation

[0064] Innovative Research Methodology: This invention applies advanced intelligent algorithms to the field of soil science research, providing a new approach and method for studying the evolution of soil fracture parameters. This innovative research method not only improves the accuracy and efficiency of research but also promotes technological innovation and progress in soil science research.

[0065] Promoting Interdisciplinary Integration: The implementation of this invention involves the intersection and integration of multiple disciplines such as computer science, artificial intelligence, and environmental science, promoting communication and cooperation among these disciplines. This interdisciplinary research model helps broaden research perspectives and promotes the development and innovation of related disciplines.

[0066] In summary, the beneficial effects of this invention are mainly reflected in improving prediction accuracy and efficiency, reducing research costs and time, and promoting soil science research and technological innovation. These beneficial effects not only help promote the development and innovation of soil science research, but also provide strong technical support for the achievement of environmental protection and sustainable development goals. Attached Figure Description

[0067] To more clearly illustrate the technical solutions of the embodiments of the present invention, the present invention will be described in detail below with reference to the accompanying drawings and detailed embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort. Wherein:

[0068] Figure 1 These are images from the first test after saturation;

[0069] Figure 2 This is a diagram of a fracture observation and photography device;

[0070] Figure 3 This is a flowchart of the crack image processing;

[0071] Figure 4 It is a graph showing the change of crack area in silty clay, red clay and expansive soil with different lead ion contents as a function of wet-dry cycle.

[0072] Figure 5 It is a graph showing the change of the fractal dimension of cracks in silty clay, red clay, and expansive soil under different lead ion contents as a function of wet-dry cycles.

[0073] Figure 6 This is a curve showing the change in crack area of ​​silty clay, red clay, and expansive soil under the same lead ion content, under dry-wet cycle conditions.

[0074] Figure 7 The curves showing the change of the fractal dimension of cracks in silty clay, red clay, and expansive soil under the same lead ion content during the wet-dry cycle.

[0075] Figure 8 This is a schematic diagram of a BiLSTM.

[0076] Figure 9 This is a GRU schematic diagram;

[0077] Figure 10This is the schematic diagram of XGBoost;

[0078] Figure 11 This is the schematic diagram of RVM;

[0079] Figure 12 It is a model flowchart;

[0080] Figure 13 The predicted values ​​for three types of soil are shown in the diagram: (a) silty clay, (b) red clay, and (c) expansive soil.

[0081] Figure 14 The following is a chart of evaluation indicators for silty clay: (a) MSE, (b) RMSE, (c) MAE, and (d) MAPE.

[0082] Figure 15 The evaluation index diagram for red clay is shown in the figure: (a) MSE, (b) RMSE, (c) MAE, and (d) MAPE.

[0083] Figure 16 The following is a diagram of expansive soil evaluation indicators: (a) MSE, (b) RMSE, (c) MAE, and (d) MAPE.

[0084] Figure 17 This is a comparison chart showing the effects of different lead ion contents on the development of fissures in silty clay during wet-dry cycles.

[0085] Figure 18 This is a comparison chart showing the effects of different lead ion contents on the development of fissures in red clay during wet-dry cycles.

[0086] Figure 19 This is a comparison chart showing the effects of different lead ion contents on the development of cracks in expansive soil during wet-dry cycles. Detailed Implementation

[0087] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. The following description, in conjunction with the accompanying drawings... Figure 1-19 This paper further explains the intelligent prediction method for the fractal dimension of soil fissures by lead ions under wet-dry cycles, and provides a clear and complete description of the technical solutions in the embodiments of this application. Obviously, the described embodiments are only some embodiments of this application, not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of this application without creative effort are within the scope of protection of this application.

[0088] Understanding soil fissure changes under wet-dry cycles and heavy metal pollution is crucial for developing and implementing soil management strategies. Therefore, this invention investigates the fissure development patterns in soils contaminated with different levels of lead ions under wet-dry cycles. Silty clay, red clay, and expansive soil samples were subjected to different lead ion concentrations and wet-dry cycle experiments. Image processing techniques were used to perform precise quantitative analysis of parameters such as fissure area and fractal dimension. The results show that the presence of lead ions significantly promotes fissure area development in silty clay, red clay, and expansive soil. However, this promoting effect is not unlimited and has a specific concentration threshold. Specifically, the lead ion content threshold for silty clay is at a moderate level, i.e., the lead ion to soil particle mass ratio is 0.09%, while the lead ion content thresholds for red clay and expansive soil are at lower levels, at 0.06% and 0.03%, respectively. Simultaneously, a fracture fractal dimension prediction model was constructed using Bidirectional Long Short-Term Memory (BiLSTM), Gated Recurrent Unit (GRU), Extreme Gradient Boosting (Xgboost), and Relevant Vector Machine Regression (RVM). The RVM model demonstrated superior performance in predicting fracture fractal dimension. For example, in the prediction of silty clay samples, the RVM model significantly reduced key evaluation metrics such as mean squared error (MSE), root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) compared to the BiLSTM, GRU, and Xgboost models. The most significant reduction was in RMSE, decreasing by 56.36%, 37.83%, and 61.61%, respectively. To further improve prediction accuracy, Sparrow Search Algorithm (SSA) and Particle Swarm Optimization (PSO) were introduced to fine-tune the parameters of the four models. Among the optimized algorithms, the SSA algorithm-optimized RVM model performed best, showing reductions of 3.55%, 6.98%, and 16.79% in MSE, RMSE, and MAPE compared to the unoptimized RVM model, respectively. This further validates the effectiveness of the optimization algorithm in improving the predictive performance of the RVM model. Therefore, this invention provides an efficient tool for monitoring and assessing soil pollution, which is of great significance for promoting the sustainable use of land.

[0089] Particle Swarm Optimization (PSO) and Sparrow Search Algorithm (SSA) demonstrate significant advantages in solving complex optimization problems due to their excellent global optimization capabilities, fast convergence, and stability. By applying PSO and SSA algorithms to parameter optimization of BiLSTM, GRU, Xgboost, and RVM models, the accuracy and prediction performance of these models are effectively improved. This provides strong support for achieving efficient and intelligent fractal dimension prediction.

[0090] This invention prepared silty clay, red clay, and expansive soil samples with different lead ion contents (0%, 0.03%, 0.06%, 0.09%, and 0.12%) and conducted wet-dry cycle tests. Soil fracture parameter data for each cycle were obtained using digital image analysis and processing technology. Based on the wet-dry cycle test results, the effects of soil type and different lead ion contents on fracture area and fractal dimension were systematically analyzed. To efficiently and economically obtain the fractal dimension of soil, an intelligent prediction model for fracture fractal dimension based on BiLSTM, GRU, Xgboost, and RVM was proposed, achieving high-precision prediction of the fractal dimension. Furthermore, PSO and SSA algorithms were introduced to optimize the model parameters, further enhancing the model's generalization and computational efficiency, providing an innovative method for obtaining soil fracture parameters.

[0091] Example 1

[0092] This embodiment provides a specific experimental method and process.

[0093] (1) Test materials

[0094] The silty clay, red clay, and expansive soil used in this experiment were sourced from 5–7 meters underground at a construction site and were free from heavy metal contamination. The mineral composition of the silty clay included quartz (94.97%), zeolite (2.46%), and montmorillonite (2.57%); the red clay contained quartz (80.54%), zeolite (15.05%), montmorillonite (2.99%), and kaolinite (1.43%); and the expansive soil contained quartz (67.13%), montmorillonite (2.6%), and calcite (30.27%). The basic physical properties of the three types of soil are shown in Table 1. The pollution source used in this experiment was lead nitrate (Pb(NO3)2) (analytical purity), a white crystalline solid, readily soluble in water, with a melting point of 470℃ and a density of 4.53 g / cm³. 3 Its technical components are shown in Table 2.

[0095] Table 1. Basic physical properties of three types of soil

[0096]

[0097] Table 2 Lead nitrate (Pb(NO3)2) (analytical concentration)

[0098]

[0099] (2) Sample preparation

[0100] After air-drying and crushing the obtained soil samples, they were sieved through a 2mm sieve. The sieved soil samples were then dried in a 105℃ oven until constant weight. A lead ion solution was then prepared and evenly sprayed into the soil samples, followed by thorough mixing. Lead-contaminated soil samples were prepared with an initial moisture content of 20% and lead ion contents of 0%, 0.03%, 0.06%, 0.09%, and 0.12% of the dry soil mass. A dry density of 1.45 g / cm³ was prepared using the static method. 3 A contaminated soil sample with a diameter of 61.8 mm and a height of 20 mm was used. The sample was humidified using a vacuum saturation method for 12 hours, and then dehumidified in a constant temperature oven (45℃) for 24 hours, constituting one wet-dry cycle. The sample after the first saturation is shown below. Figure 1 As shown.

[0101] During the wet-dry cycle, the mass of the sample was weighed and recorded for each desiccation and saturation process. The sample was then placed in a photographic chamber, the distance between the camera and the sample was adjusted, and a suitable focal length was selected to obtain photographs of the sample during each cycle, as shown below. Figure 2 As shown, each soil type underwent 21 cycles at different lead ion concentrations.

[0102] (3) Digital Image Processing

[0103] In recent years, with the development of computer science, digital image processing technology has been widely used in the quantitative analysis of fracture network morphology. First, the acquired sample image is precisely cropped to ensure that the cropped image edges are perfectly aligned with the inner wall of the ring cutter. Then, the image is imported using software and grayscale conversion is performed. Based on this, advanced noise reduction algorithms are applied to the grayscale image to improve image quality, and fracture identification technology is used to accurately extract fracture features. Finally, image analysis techniques are used to calculate key parameters such as fracture area and fractal dimension to obtain high-precision analysis results. The process is as follows: Figure 3 As shown.

[0104] To accurately quantify the crack development characteristics under different lead ion concentrations and wet-dry cycles, two key indicators were selected: crack area and fractal dimension. Crack area is defined as the projected area of ​​the crack on a two-dimensional plane and is an important parameter for assessing the degree of crack development; its increase directly reflects the intensification of crack development. Fractal dimension, as a fundamental concept in fractal geometry, provides a powerful tool for quantifying complex geometric structures with self-similar characteristics. Lu et al. used fractal theory to conduct an in-depth analysis of the morphological characteristics of soil cracks and used fractal dimension as an indicator to quantify the complexity of crack morphology. An increase in fractal dimension indicates an increase in the complexity of the crack structure. Currently, the box dimension method has been widely recognized as the standard method for determining the fractal dimension of soil crack patterns. The calculation of this method is as shown in equation (1):

[0105] lnNi =A-Dlnε i (1)

[0106] In the formula: A is a constant; D is the fractal dimension; ε i and N i Let lnN be the segmentation size of the i-th level and the corresponding number of self-similar blocks, respectively. i and lnε i The relationship exhibits a linear one, and its slope D is the fractal dimension of the fracture. The larger the fractal dimension of the fracture network, the denser the fracture distribution and the more complex the fracture structure distribution characteristics.

[0107] Example 2

[0108] This embodiment provides an analysis of the test results.

[0109] (1) Description of crack development in the specimen

[0110] Figure 17 / 18 / 19 show the crack development of silty clay, red clay, and expansive soil samples under drying conditions of 0, 5, 10, 15, and 20 wet-dry cycles with different lead ion contents. It can be observed that without wet-dry cycles (N=0), the soil surfaces of different samples are smooth and the soil particles are interconnected at different lead ion concentrations. After 5 wet-dry cycles, obvious cracks are formed in all three types of soil samples. At 10, 15, and 20 cycles, the silty clay, red clay, and expansive soil continue to deepen and widen along the initial cracks, and the number of cracks increases continuously. Furthermore, when the expansive soil has severe particle loss on one side, cracks are also observed on the other side, indicating that the cracks have penetrated the soil sample. During the wet-dry cycle, white particles appeared on the surface of the silty clay and red clay samples containing lead ions during the drying stage. These white particles are Pb(NO3)2 crystals. The number of precipitated white particles increases with increasing lead ion content. This is mainly because the soil sample undergoes expansion and contraction during the wet-dry cycle, leading to changes in internal stress, which in turn creates cracks and increases the permeability of the soil sample. This makes it easier for lead ions that were originally adsorbed by soil particles to migrate and precipitate with the moisture.

[0111] Different lead ion contents have varying effects on crack development in silty clay, red clay, and expansive soil. (I) In silty clay, no significant crack formation was observed in the uncontaminated samples during the 5th cycle; however, significant crack formation was observed after the addition of lead ions, indicating that lead ions promote crack development in silty clay. This is because the double-layer structure of silty clay is stable when uncontaminated with lead ions, thus no significant crack formation was observed. With the addition of lead ions, due to their large hydration radius and charge density, they exert a stronger attraction on the negative charge on the surface of clay particles, thereby compressing the diffusion layer of the double-layer, reducing the repulsive force between clay particles, and making it easier for clay particles to approach each other, thus promoting crack formation. When the lead ion content was low (0.03% and 0.06%), cracks mainly developed at the edges; when the lead ion content increased by 0.09%, cracks in the sample began to extend from the center to the edges, causing the sample to split into four independent blocky structures. When the lead ion content continued to increase to a higher level (0.12%), the area of ​​crack formation decreased compared to the medium lead ion content (0.09%). This indicates that a lead ion content of 0.09% is the critical point for promoting crack area development; above this concentration, the promoting effect of lead ions on crack area growth begins to decrease. (II) In the fifth cycle, no cracks were formed in the middle of the red clay sample that was not contaminated with lead ions; the cracks mainly developed at the edges. When lead ions were added, the cracks developed from the middle to the edges. When the lead ion content was low (0.03% and 0.06%), small cracks were formed in the middle; at a medium content (0.09%), deep cracks appeared in the middle of the sample; at a high content (0.12%), the crack development was consistent with that of silty clay, that is, the area of ​​crack formation decreased compared to the medium lead ion content (0.09%). The crack development pattern in expansive soil differs from that in silty clay and red clay. Cracks form in the center of expansive soil samples that are not contaminated with lead ions. This is because calcite in expansive soil readily dissolves and precipitates during wet-dry cycles. When water evaporates from the soil, calcite dissolves, reducing the soil volume; conversely, when the soil absorbs water again, calcite reprecipitates, increasing the soil volume. This repeated volume change alters the stress distribution within the soil, leading to crack development from the center. Furthermore, regardless of the presence or absence of lead ions, cracks still form from the center, indicating that lead ion concentration has a relatively small impact on crack development in expansive soil.

[0112] (2) The influence of different factors on fracture index

[0113] 2.1 Lead ion concentration

[0114] Figure 4 The curves showing the change of crack area with wet-dry cycle for silty clay, red clay, and expansive soil under different lead ion contents are presented. Figure 5The curves showing the change of fracture fractal dimension with wet-dry cycles for silty clay, red clay, and expansive soil under different lead ion contents are presented. To fully illustrate the change of fractal dimension with the number of cycles, the fractal dimension curves are smoothed using the adjacent averaging method. Figure 5 ).

[0115] (Ⅰ) In Figure 4 In (a), when the number of wet-dry cycles ranged from 1 to 18, the fracture area growth curve of the uncontaminated silty clay sample (0%) was consistently lower than that of the lead-contaminated sample. This phenomenon further confirms that lead ions have a significant promoting effect on the fracture area growth of silty clay. In particular, during the first four cycles, the effect of moderate lead ion content (0.09%) on fracture area growth was the most significant. Specifically, at the end of the fourth cycle, the fracture area of ​​the sample with a lead ion content of 0.09% was 2.81 times, 2.23 times, 1.64 times, and 1.72 times that of samples with lead ion contents of 0%, 0.03%, 0.06%, and 0.12%, respectively. These results indicate that a moderate level (0.09%) of lead ions is the critical content that promotes the fracture area growth of silty clay; when the lead ion content exceeds this threshold, its promoting effect on fracture area growth begins to slow down. Figure 5 In (a), for silty clay samples uncontaminated with lead ions, the fractal dimension generally showed an increasing trend with the increase of the number of wet-dry cycles. In contrast, when the lead ion content in the sample increased to 0.03%, the fractal dimension exhibited a pattern of first decreasing, then increasing, and finally stabilizing. Further increasing the lead ion content to 0.06% and 0.09%, the fractal dimension showed an initial increase, followed by a decrease, and finally gradually stabilized. However, for silty clay samples with a lead ion content as high as 0.12%, the fractal dimension showed a trend of first increasing and then decreasing.

[0116] (II) In Figure 4(b) The data presented show that the fracture area of ​​the red clay samples increased significantly in the first four cycles, especially in the lead-contaminated samples, where the rate of increase was significantly higher than that of the uncontaminated samples. Specifically, at the end of the fourth cycle, the fracture areas of samples with lead ion contents of 0.03%, 0.06%, 0.09%, and 0.12% were 2.39 times, 6.75 times, 5.19 times, and 2.20 times that of the uncontaminated samples, respectively. These data indicate that the increase in fracture area is most significant when the lead ion content reaches 0.06%. However, as the lead ion content further increases, its promoting effect on fracture area growth shows a decreasing trend. This phenomenon suggests that a lead ion content of 0.06% constitutes a critical threshold, below which lead ions play a significant catalytic role in the expansion of fracture area in red clay. When the lead ion concentration exceeds this threshold, its promoting effect on fracture area development begins to weaken. During cycles 1-21, the fracture area growth curves of red clay samples with lead ion contents of 0.03% and 0.06% were consistently above those of the samples without lead ion contamination, indicating that lead ions continuously promoted the fracture area growth of red clay during the wet-dry cycle. Figure 5 In (b), during the 2nd to 18th cycles, the fractal dimensions of the red clay samples with lead ion content of 0.06% and 0.09% were higher than those of the samples without lead ion contamination, indicating that at this lead ion content, the fracture morphology of the red clay tends to have a more complex evolution trajectory.

[0117] (III) Figure 4 In (c), it was observed that the fracture area of ​​the expansive soil samples showed an increasing trend in the first four cycles. Notably, the fracture area curve of the contaminated sample with a lead ion content of 0.03% consistently remained above that of the uncontaminated sample. This phenomenon reveals that at this specific content, the presence of lead ions significantly promoted the development of fracture area. However, when the lead ion content increased to 0.06% and 0.09%, the fracture area curves were below those of the uncontaminated sample, indicating that the growth of fracture area was inhibited at these levels. In particular, the fracture area was minimal when the lead ion content reached 0.09%. This is because in expansive soil, at low contents (0.03%), the free expansion rate of the sample is increased, thereby promoting the development of fracture area. As the lead ion content increases, the free expansion rate of the expansive soil decreases, thus inhibiting the development of fracture area. In the first 12 cycles, the fracture area of ​​the expansive soil developed slowly. After 12 cycles, the fracture area first increased rapidly and then fluctuated. Figure 5(c) The fractal dimension of expansive soil fissures generally increased with the number of cycles. In the first three cycles, the fractal dimension of the uncontaminated expansive soil samples showed an increasing trend, while the fractal dimension curve of the lead-contaminated samples showed a decreasing trend, indicating that the presence of lead ions led to a decrease in the fractal dimension in the first three cycles. This is because the presence of lead ions affects the surface charge of the sample particles in the first three cycles, thereby changing the interaction between particles and causing a rearrangement and distribution of particles. This rearrangement affects the pore structure of the sample, leading to a decrease in the fractal dimension. However, after 17 cycles, the fractal dimension curve of the lead-contaminated samples was found to be below that of the uncontaminated samples, revealing that after 17 cycles, the lead ion concentration had an inhibitory effect on the fractal dimension.

[0118] 2.2 Types of Soil

[0119] Figure 6 The curves showing the change in fracture area of ​​silty clay, red clay, and expansive soil with wet-dry cycles under the same lead ion content are presented. (I) In Figure 6In the first three cycles, the curve of the fracture area of ​​expansive soil was always above that of silty clay and red clay, indicating that the fracture area of ​​expansive soil expanded faster than that of silty clay and red clay. For example, in the first three cycles, when the lead ion content was 0%, 0.03%, and 0.12%, the fracture area of ​​expansive soil was 2.63 times, 1.21 times, and 1.75 times that of silty clay, respectively, and 5.31 times, 3.33 times, and 3.69 times that of red clay, respectively. This is because expansive soil has a large proportion of calcite (30.27%) and kaolinite (2.6%). Montmorillonite has high expansibility and ion exchange capacity, while the presence of calcite can provide additional cation exchange sites, which allows for faster adsorption of lead ions, leading to double-layer compression, thereby reducing the repulsive force between particles and thus rapidly increasing the fracture area. Furthermore, after 14 wet-dry cycles, the fracture area of ​​expansive soil significantly increased, exceeding that of silty clay and red clay. Silty clay and red clay contain a high proportion of quartz, whose high chemical stability helps maintain a stable microstructure during wet-dry cycles, making significant changes in fracture area less likely in the later stages of the cycles. In contrast, expansive soil contains a larger proportion of montmorillonite and calcite. The water absorption and expansion and water loss shrinkage characteristics of montmorillonite directly promote fracture formation, while the dissolution-precipitation behavior of calcite during wet-dry cycles indirectly promotes fracture propagation by altering the pore structure. The synergistic effect of these two factors leads to a significant increase in the fracture area of ​​expansive soil. Additionally, it can be observed that in the early stages, the fracture area of ​​silty clay is generally higher than that of red clay, but in the later stages, the fracture area of ​​red clay generally exceeds that of silty clay. The mineral composition of silty clay is mainly quartz, accounting for 94.97%, with small amounts of zeolite (2.46%) and montmorillonite (2.57%). Quartz is a mineral with high hardness and good stability. In the early stages of cycling, the silty clay sample maintains structural stability when dry, reducing its water retention capacity, thus making it easier to form cracks in the early stages of cycling. Zeolite (15.05%), montmorillonite (2.99%), and kaolinite (1.42%) in red clay will absorb water and expand, and lose water and shrink. With the accumulation of cycling, this expansion and contraction will weaken the cohesion between soil particles, thereby destroying the soil structure and increasing the crack area.

[0120] Figure 7The fractal dimension of silty clay, red clay, and expansive soil under the same lead ion content is shown as a function of wet-dry cycles. (I) Under conditions without lead ion contamination (0%), the fractal dimension of silty clay, red clay, and expansive soil all showed an increasing trend with the increase of wet-dry cycles, reflecting the increasing complexity of soil fracture structure with the progress of cycles. However, with the accumulation of cycles, the changes in fractal dimension of different samples began to differ. Especially after the completion of the 4th cycle, the fractal dimension of expansive soil was significantly higher than that of silty clay and red clay. By the 21st cycle, the fractal dimension of expansive soil increased by 5% and 4% compared to silty clay and red clay, respectively. (II) Under the condition of lead ion content of 0.03%, silty clay exhibited a higher fractal dimension than red clay and expansive soil in the first 10 wet-dry cycles. However, this trend reversed after more than 10 cycles, at which point the fractal dimensions of expansive soil and red clay surpassed those of silty clay. (III) Under the condition of a lead ion content of 0.06%, the fractal dimensions of red clay and expansive soil showed a continuous increasing trend, while the fractal dimension of silty clay remained relatively stable. After 4 wet-dry cycles, the fractal dimension of red clay significantly exceeded that of silty clay and expansive soil. With the accumulation of cycles, by the 21st cycle, the fractal dimension of expansive soil finally exceeded that of red clay, showing that the complexity of its fracture development increased during the cycle. (IV) When the lead ion content increased to 0.09%, the fractal dimension of red clay surpassed that of silty clay and expansive soil after the 4th wet-dry cycle. Furthermore, at the end of 21 cycles, the fractal dimension of red clay increased by 8% and 5% compared to silty clay and expansive soil, respectively. This phenomenon indicates that under the influence of lead ions, the complexity of the fracture structure of red clay significantly increases after multiple wet-dry cycles compared to silty clay and expansive soil. (V) When the lead ion content is 0.12%, after 8 wet-dry cycles, the fractal dimension of expansive soil is higher than that of red clay and silty clay. This phenomenon indicates that at this lead ion level, the complexity of the fracture network in expansive soil increases significantly in the later stages of the wet-dry cycle, thus indicating that the fracture development of expansive soil tends to become more complex.

[0121] Example 3

[0122] This embodiment provides a detailed explanation of the construction of an intelligent prediction model for the fractal dimension of soil fissures.

[0123] Fractal dimension, as a key indicator for quantifying the complexity of fracture systems, can accurately depict the self-similarity and scale-invariance characteristics of fracture networks. In-depth analysis of the dynamic changes in fractal dimension is crucial for revealing the evolutionary patterns of fractures under diverse environmental conditions, thus providing a scientific basis for understanding the intrinsic mechanisms and evolutionary laws of fracture development. However, traditional fractal dimension measurement techniques often introduce significant random errors due to the subjectivity of human operation. In contrast, artificial intelligence algorithms, with their ability to identify complex patterns from massive datasets, significantly enhance the accuracy and stability of prediction results. Therefore, using artificial intelligence algorithms to predict the fractal dimension of soil fractures can not only effectively improve the accuracy and efficiency of prediction work but also deepen the understanding of fracture complexity. This has significant application value for deepening the understanding of fracture development mechanisms, optimizing engineering design schemes, and improving the safety of engineering structures.

[0124] (1) Basic Principles

[0125] 1.1BiLSM

[0126] BiLSTM (Bi-directional Long Short-Term Memory) consists of two LSTM units, one processing the forward sequence and the other the reverse sequence of data. Data first enters the network through the input layer, then is processed by the forward and reverse LSTM units; subsequently, the outputs of these two LSTM units are combined to generate the final output. In BiLSTM, x... t This represents the input at time point t, while h t This represents the network output at time point t. This structure allows BiLSTM to consider contextual information simultaneously, thereby improving the model's ability to understand sequence data. For example... Figure 8 As shown, the architecture of BiLSTM allows it to effectively integrate information from both directions to generate a more comprehensive output.

[0127] 1.2GRU

[0128] GRU (Gated Recurrent Unit) is an advanced recurrent neural network that introduces a gating mechanism into the traditional RNN, enabling it to effectively process long sequences of data while mitigating the vanishing gradient problem. The core of GRU consists of two key gating units: the update gate and the reset gate. The update gate determines which information to retain and which to discard, while the reset gate decides whether to ignore previous memories. Figure 9 This demonstrates the basic structure of the GRU, where x t Represents input at different points in time; h t-1 and ht Represents the hidden state at the corresponding time point; Gr represents resetting the gate, σ represents the sigmoid activation function, and Gu represents updating the gate. This represents the output of the hidden layer of the network.

[0129] 1.3Xgboost

[0130] XGBoost (eXtreme Gradient Boosting) is a high-efficiency machine learning algorithm based on gradient boosting. It optimizes GBDT by introducing a second-order Taylor expansion to handle the loss function, enhancing model stability and expressive power. XGBoost accelerates tree construction through greedy algorithms and quantile approximation algorithms, and uses hyperparameters to strictly control model complexity, effectively preventing overfitting. Furthermore, it features automatic handling of missing values, support for sparse data, and parallel computation, improving the algorithm's efficiency and scalability. Its principle is as follows: Figure 10 As shown.

[0131] 1.4RVM

[0132] Relevance Vector Machine (RVM) is an innovative machine learning algorithm that combines Bayesian statistics and sparsity optimization to automatically select "relevance vectors" from the data that are crucial for prediction. These vectors play a core role in the model, while the weights of other data points approach zero, achieving model sparsity. During training, RVM optimizes the weights and precision parameters by maximizing the marginal likelihood function, selecting key vectors, reducing model complexity, and improving computational efficiency. Furthermore, RVM can quantify prediction uncertainty and provide confidence assessments, making it suitable for scenarios that require handling uncertainty. Its principle is as follows: Figure 11 As shown.

[0133] (2) Optimization Algorithm

[0134] 2.1 Particle Swarm

[0135] Particle Swarm Optimization (PSO) is an evolutionary computation technique based on swarm intelligence. It randomly generates a group of particles in the solution space, each representing a solution to a problem. These particles use their own optimal position and the optimal position information of the entire swarm to iteratively adjust their position and velocity using Equation (2), thereby gradually exploring and approaching the optimal solution. Due to its easy-to-set parameters, efficient search capabilities, and fast convergence, PSO is widely adopted in many fields such as pattern recognition and parameter optimization. Its pseudocode is shown in Table 3.

[0136]

[0137] Where w is a non-negative inertia factor used for algorithm convergence, c1 and c2 are learning factors, often used to adjust the optimal weights, and r1 and r2 are random numbers between [0,1]. The code execution steps are shown in Table 3.

[0138] Table 3. Pseudocode for Particle Swarm Optimization Algorithm

[0139]

[0140] 2.2 Sparrow Optimization Algorithm

[0141] The Sparrow Search Algorithm (SSA) is an optimization algorithm that simulates the foraging and predator-avoiding behavior of sparrows. It has attracted attention due to its simplicity, efficiency, and fast convergence. In this algorithm, the sparrow population is divided into three roles according to a specific ratio: discoverers, followers, and scouts, each undertaking different tasks to achieve the optimization objective. Discoverers are responsible for finding food and guiding the population, followers forage based on information from discoverers, and scouts monitor the surrounding environment and warn of potential dangers. Discoverers, followers, and scouts update their positions using Equations 3, 4, and 5, respectively. Its pseudocode is shown in Table 4.

[0142]

[0143] Where: t is the current iteration number, T max X is the maximum number of iterations, α is a random number ∈ (0,1], R2∈[0,1] and ST∈[0.5,1] are the warning and safety values ​​respectively, Q is a normally distributed random number, and L is a 1×d matrix with all elements equal to 1. B The discoverer obtains the optimal position, and Xw represents the global worst value. A is a one-dimensional vector of length d. + =A T (AA T ) -1 When i > N / 2, it means that sparrows with low fitness are very hungry and need to fly to other places to forage. gB f is the current global optimum, β is the step size control parameter following a (0,1) normal distribution, K is a random number between [-1,1], and ε is a constant to avoid a denominator of 0. i It is the fitness value of an individual sparrow, f g and f w These are the current best and worst fitness values, respectively.

[0144] Table 4. Pseudocode of the Sparrow Optimization Algorithm

[0145]

[0146]

[0147] (3) Constructing a soil fissure fractal dimension prediction model

[0148] The geometric parameters of soil fissures provide a comprehensive description of the fissures. For example, the fissure area describes the projected area of ​​the fissure in two-dimensional space, the perimeter and length describe the boundary and one-dimensional size of the fissure, and the dip angle and shape factor provide information on the complexity of the fissure's direction and shape. These characteristics are indirectly related to the fractal dimension, and the number of cycles and lead ion content also affect the fractal dimension. Therefore, this invention selects the number of cycles, lead ion concentration, area, dip angle, shape factor, perimeter, length, and width of the soil fissure as input parameters, and the fractal dimension as the output parameter. In the wet-dry cycle test on silty clay, red clay, and expansive soil, each sample underwent 21 wet-dry cycles, with each cycle including one wet-dry alternation stage, and each alternation stage producing 10 sets of data. Based on this, a total of 210 sets of sample data were obtained for each soil type. This invention selects the first 190 sets of fissure data from the wet-dry cycle as training samples and the last 20 sets as test samples. Predictive models for the fractal dimension of silty clay, red clay, and expansive soil are established respectively.

[0149] This invention combines PSO and SSA optimization algorithms with BiLSTM, GRU, Xgboost, and RVM models respectively to adaptively obtain optimal parameter values. Using the optimized models, a nonlinear mapping relationship is established between the number of cycles, lead ion concentration, soil fissure area, dip angle, shape factor, perimeter, length, width, and fractal dimension, thus establishing a predictive model for the fractal dimension of soil fissures. The prediction steps of the model are as follows:

[0150] Step 1: Divide the collected sample dataset into two parts: a training set and a test set. To prevent the influence of differences in the dimensions of influencing factors on the results, normalize the samples using the following formula:

[0151]

[0152] In the formula: x i For the i-th influencing factor, s is the mean of the sample data and s is the standard deviation of the sample data.

[0153] Step 2: Initialization, setting parameters for each model, optimizing the parameters of BiLSTM, GRU, Xgboost and RVM models using PSO and SSA, and establishing a prediction model for the fractal dimension of soil fractures that meets the requirements based on the optimal solutions of SSA and PSO algorithms.

[0154] Step 3: Input the test data into the prediction model of soil fracture fractal dimension, compare the predicted results with the actual values ​​to verify the feasibility of the model.

[0155] Based on the above modeling steps, soil fracture fractal dimension prediction models were developed using BiLSTM, PSO-BiLSTM, SSA-BiLSTM, GRU, PSO-GRU, SSA-GRU, Xgboost, PSO-Xgboost, SSA-Xgboost, RVM, PSO-RVM, and SSA-RVM. The model flow is as follows: Figure 12 As shown.

[0156] Example 4

[0157] This embodiment provides a detailed explanation of the prediction result analysis.

[0158] To better reflect the overall prediction performance of each model, a comparative analysis was conducted on the mean absolute error, mean squared error (MSE), root mean squared error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) of the prediction results of these 12 models. The closer the values ​​of MSE, RMSE, MAE, and MAPE are to 0, the better the prediction performance. The mathematical formulas are shown in equations (7), (8), (9), and (10). Figure 13 These are the predicted values ​​for the fractal dimensions of three types of soil.

[0159]

[0160] In the formula y 0，i Let y represent the actual value of the i-th sample. i This represents the predicted value of the i-th sample. n represents the total number of samples.

[0161] Figure 14The evaluation indices for (silty clay) provide a more intuitive and clear comparison of the MSE, RMSE, MAE, and MAPE of the BiLSM, PSO-BiLSTM, SSA-BiLSTM, GRU, PSO-GRU, SSA-GRU, Xgboost, PSO-Xgboost, SSA-Xgboost, RVM, PSO-RVM, and SSA-RVM models. Compared to the BiLSTM, GRU, and Xgboost models, the RVM model shows significant improvements in all indices: MSE decreased by 33.94%, 21.15%, and 38.04%, respectively; RMSE decreased by 56.36%, 37.83%, and 61.61%, respectively; MAE decreased by 17.05%, 9.74%, and 21.88%, respectively; and MAPE decreased by 21.97%, 15.50%, and 26.54%, respectively. Further comparisons showed that the SSA-RVM and PSO-RVM models reduced MAE by 23.58% and 18.30% respectively compared to the RVM model; and reduced MAPE by 16.79% and 11.77% respectively. Overall, the SSA-RVM model achieved the best MSE, RMSE, and MAE scores among all models. Although the PSO-BiLSTM model achieved the best MAPE score, it is noteworthy that its MAPE value was only 0.14% higher than the SSA-RVM model. This slight difference indicates that the SSA-RVM model not only demonstrates strong stability but also high reliability in predicting silty clay samples.

[0162] Figure 15 In the evaluation of fractal dimension prediction for red clay, the RVM model achieved significant performance improvements across multiple evaluation metrics compared to the BiLSTM, GRU, and XGBoost models. Specifically, the RVM model reduced mean squared error (MSE) by 48.46%, 51.82%, and 52.70%, respectively; root mean square error (RMSE) by 28.21%, 30.58%, and 31.22%, respectively; mean absolute error (MAE) by 8.16%, 17.55%, and 12.78%, respectively; and mean absolute percentage error (MAPE) by 10.69%, 17.83%, and 14.56%, respectively. Furthermore, the SSA-RVM and PSO-RVM models showed significant reductions in MAE and MAPE compared to the RVM model, at 55.49% and 58.28%, and 54.45% and 58.39%, respectively. Considering all indicators, the PSO-RVM model demonstrated the best performance among all models in terms of MSE, RMSE, MAE, and MAPE, thus verifying the superior performance of the PSO-RVM model in predicting the fractal dimension of red clay.

[0163] Figure 16 In the evaluation of expansive soil fractal dimension prediction, the predictive performance of BiLSTM, GRU, XGBoost, and RVM was analyzed in depth. The XGBoost algorithm excelled in the two key evaluation metrics, MAE and MAPE, standing out with the smallest error values. The RVM algorithm achieved the best results in MSE and RMSE, demonstrating its significant advantage in prediction accuracy. However, when PSO and SSA algorithms were applied to optimize the parameters of these models, the SSA-RVM model not only comprehensively outperformed BiLSTM, GRU, XGBoost, and RVM in all evaluation metrics (MSE, RMSE, MAE, and MAPE), but also significantly outperformed the BiLSTM, GRU, and XGBoost models optimized by PSO and SSA. This significant result highlights the efficiency and reliability of the SSA-RVM model in predicting the fractal dimension of expansive soil, further confirming its superior performance in this field.

[0164] In a comprehensive analysis of fractal dimension predictions for silty clay, red clay, and expansive soil samples, the results show that the RVM model outperforms the BiLSTM, GRU, and XGBoost models on several key evaluation metrics. This finding validates the efficiency and reliability of the RVM model in predicting different soil samples. Furthermore, the RVM model optimized with SSA and PSO exhibits a significant improvement in predictive performance. Particularly in the prediction of silty clay and expansive soil samples, the SSA-RVM model achieved optimal performance on evaluation metrics such as MSE, RMSE, and MAE. In the prediction of red clay samples, the PSO-RVM model demonstrated superior performance, surpassing all other models. These results not only indirectly confirm the potential of combining optimization algorithms with the RVM model in predicting soil fracture parameters but also highlight its significant value in practical applications.

[0165] This invention conducted systematic wet-dry cycle tests on silty clay, red clay, and expansive soil samples under different lead ion contents, and applied advanced digital image analysis technology to accurately measure the fracture parameters of the samples during each wet-dry cycle. The results revealed significant differences in fracture development among the three soil samples under the same lead ion content. Specifically, in the first three cycles and after the fourteenth cycle, the fracture development of expansive soil significantly exceeded that of silty clay and red clay. Under conditions of no lead ion addition or a lead ion content of 0.12%, expansive soil exhibited a more complex fracture structure than silty clay and red clay in the later stages of the cycle. However, under a lead ion content of 0.09%, the fracture structure of red clay was more complex in the later stages of the cycle. Furthermore, the presence of lead ions significantly promoted the expansion of fracture area in silty clay, red clay, and expansive soil. It is noteworthy that this promoting effect does not increase indefinitely, but tends to stabilize after reaching a specific content threshold. Specifically, the lead ion content threshold for silty clay is at a moderate level (0.09%). In contrast, the lead ion content thresholds for red clay and expansive soil are relatively low, at 0.03% and 0.06%, respectively.

[0166] To obtain fractal dimension efficiently and economically, intelligent fractal dimension prediction models of BiLSTM, GRU, Xgboost, and RVM are proposed. Furthermore, to improve the accuracy of these models, SSA and PSO algorithms are introduced to optimize the parameters of these four models. Results show that compared with BiLSTM, GRU, and Xgboost models, the RVM model performs superiorly in predicting fracture fractal dimension, and its prediction accuracy is significantly enhanced after parameter optimization. In silty clay, compared with the RVM model, the SSA-RVM model reduced MSE by 6.88%, RMSE by 3.54%, and MAE by 23.59%. In red clay, compared with the RVM model, the PSO-RVM model reduced MSE by 76.84%, RMSE by 51.87%, and MAE by 58.28%. In expansive soil, compared with the RVM model, the SSA-RVM model reduced MSE by 20.96%, RMSE by 11.05%, and MAE by 32.19%. Comparative analysis validates the potential of combining optimization algorithms with RVM models to predict soil crack parameters.

[0167] Given the impact of lead ions on soil fissure development, differentiated management strategies need to be formulated and implemented based on specific soil characteristics and varying lead content. Effective control of fissure development can mitigate risks such as damage to soil structural integrity, decreased soil bearing capacity, and increased soil permeability, ensuring the stability of local infrastructure construction and promoting the achievement of long-term sustainable development goals. Future research should focus on expanding the scope of soil types and pollutants to verify the generalizability of the model developed in this invention. Furthermore, long-term field studies can further understand the applicability of research results and models in the real world, improving their predictive accuracy under different environmental conditions. This will provide crucial decision-making support for policymakers and environmental management experts, ensuring the sustainable use of soil resources.

[0168] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A method for intelligently predicting the fractal dimension of soil fissures caused by lead ions under wet-dry cycles, characterized in that, The steps are as follows: S1. Collect sample data and perform standardization processing; S2. The standardized sample data is divided into prediction samples and training samples; S3. Particle swarm optimization and sparrow optimization algorithms are combined with BiLSTM, GRU, Xgboost, and RVM models respectively to adaptively obtain optimal parameter values. Using the optimized models, a nonlinear mapping relationship is established between the number of cycles, lead ion concentration, soil fissure area, dip angle, shape factor, perimeter, length, width, and fractal dimension. A prediction model for the fractal dimension of soil fissures is then established. The prediction steps of the model are as follows: Step 1: Divide the collected sample dataset into two parts: a training set and a test set. To prevent the influence of differences in the dimensions of influencing factors on the results, normalize the samples using the following formula: In the formula: For the i-th influencing factor, s is the mean of the sample data, and s is the sample standard deviation; Step 2: Initialization, setting parameters of each model, and optimizing the parameters of BiLSTM, GRU, Xgboost and RVM models using particle swarm optimization and sparrow optimization algorithms. Based on the optimal solutions of particle swarm optimization and sparrow optimization algorithms, a prediction model for the fractal dimension of soil fractures that meets the requirements is established. S4. Input the predicted samples into the parameters of BiLSTM, GRU, Xgboost and RVM models, compare the predicted results with the actual values ​​using fractal analysis to verify the feasibility of the models; based on the above modeling steps, write soil fracture fractal dimension prediction models for BiLSTM, PSO-BiLSTM, SSA-BiLSTM, GRU, PSO-GRU, SSA-GRU, Xgboost, PSO-Xgboost, SSA-Xgboost, RVM, PSO-RVM and SSA-RVM.

2. The intelligent prediction method for the fractal dimension of soil fissures by lead ions under wet-dry cycles as described in claim 1, characterized in that, The steps of the particle swarm optimization algorithm are as follows: T1. Initialize the position, fitness, and velocity of the particle swarm; T2. Determine the optimal position and fitness of the particle swarm; T3. When the number of iterations is less than the maximum number of iterations, update the particle's velocity and position using formula (1): Where w is a non-negative inertia factor used for algorithm convergence, c1 and c2 are learning factors used to adjust the optimal weights, and r1 and r2 are random numbers between [0, 1]. T4. Calculate the fitness of the next generation population; T5. Update the optimal position and fitness of the particle swarm.

3. The intelligent prediction method for the fractal dimension of soil fissures by lead ions under wet-dry cycles as described in claim 1, characterized in that, The steps of the sparrow optimization algorithm are as follows: U1. Initialize the location and fitness of the sparrow population; U2. Sort the fitness values ​​and determine the first generation of discoverers, followers, and scouts; U3. When the number of iterations is less than the maximum number of iterations, for each discoverer, i from 1 to DN, a random number R2 is generated, and the discoverer position is updated using formula (2): Where: DN is the number of discoverers, t is the current iteration number, and T is... max is the maximum number of iterations, α is a random number and α∈(0,1], R2∈[0,1] and ST∈[0.5,1] are the warning and safety values ​​respectively, Q is a normally distributed random number, and L is a 1×d matrix with all elements equal to 1; U4. For each follower, i ranges from 1 to FN, update the follower position using formula (3): Where: FN is the number of followers, X B The discoverer obtains the optimal position, X. w Represents the global worst value; A is a one-dimensional vector of length d, A + =A T (AA T ) -1 When i > N / 2, it means that sparrows with low fitness are very hungry and need to fly to other places to forage; U5. For each scout, i ranges from 1 to VN, update the scout position using formula (4): Where: VN is the number of scouts, X gB The current global optimum is given by β, which is a step size control parameter following a normal distribution (0, 1), K is a random number between [-1, 1], ε is a constant to avoid a denominator of 0, and f is the result of f. i It is the fitness value of an individual sparrow, f g and f w These are the current best and worst fitness values, respectively. U6. Update the optimal fitness and location of the sparrow population.

4. The intelligent prediction method for the fractal dimension of soil fissures by lead ions under wet-dry cycles as described in claim 1, characterized in that, It also includes the following steps: The mean square error of the prediction results is analyzed using formula (6): In the formula, MSE represents the mean square error, y 0,i Let y represent the actual value of the i-th sample. i Let represent the predicted value of the i-th sample, and n represent the total number of samples.

5. The intelligent prediction method for the fractal dimension of soil fissures by lead ions under wet-dry cycles as described in claim 1, characterized in that, It also includes the following steps: The root mean square error of the prediction results is analyzed using formula (7): In the formula, RMSE represents the root mean square error, and y 0，i Let y represent the actual value of the i-th sample. i Let represent the predicted value of the i-th sample, and n represent the total number of samples.

6. The intelligent prediction method for the fractal dimension of soil fissures by lead ions under wet-dry cycles as described in claim 1, characterized in that, It also includes the following steps: The mean absolute error of the prediction results is analyzed using formula (8): In the formula, MAE represents the mean absolute error, y 0,i Let y represent the actual value of the i-th sample. i Let represent the predicted value of the i-th sample, and n represent the total number of samples.

7. The intelligent prediction method for the fractal dimension of soil fissures by lead ions under wet-dry cycles as described in claim 1, characterized in that, It also includes the following steps: The mean absolute percentage error of the prediction results is analyzed using formula (9): In the formula, MAPE represents the mean absolute percentage error, y 0，i Let y represent the actual value of the i-th sample. i Let represent the predicted value of the i-th sample, and n represent the total number of samples.

8. The intelligent prediction method for the fractal dimension of soil fissures by lead ions under wet-dry cycles as described in claim 1, characterized in that, In step S1, sample data is collected through the following steps: S1.1 Select silty clay, red clay, and expansive soil from 5-7m underground at the construction site; S1.2 Sample Preparation: After air-drying and crushing the obtained soil sample, it was sieved through a 2mm sieve. The sieved soil sample was then dried in a 105℃ oven until constant weight. A lead ion solution was then prepared and evenly sprayed into the soil sample for thorough stirring. The initial moisture content of the lead-contaminated soil sample was 20%, and the lead ion content was 0%, 0.03%, 0.06%, 0.09%, and 0.12% of the dry soil mass, respectively. A contaminated soil sample with a dry density of 1.45 g / cm3, a diameter of 61.8 mm, and a height of 20 mm was prepared according to the static method. The sample was humidified by vacuum saturation for 12 h, and then dehumidified in a 45℃ constant temperature oven for 24 h, which constitutes one wet-dry cycle. S1.3 Digital Image Processing: First, the acquired sample image is precisely cropped to ensure that the cropped image edge is perfectly aligned with the inner wall of the ring cutter. Then, the image is imported using software and grayscale conversion is performed. Advanced noise reduction algorithms are applied to the grayscale image to improve image quality, and crack recognition technology is used to accurately extract crack features. Finally, image analysis technology is used to calculate the crack area and fractal dimension to obtain high-precision analysis results.

9. The intelligent prediction method for the fractal dimension of soil fissures by lead ions under wet-dry cycles as described in claim 8, characterized in that, In step S1.3, two key indicators are selected: crack area and fractal dimension; crack area is defined as the projected area of ​​the crack on a two-dimensional plane; fractal dimension is used to quantify complex geometric structures with self-similar characteristics; the fractal dimension of the soil crack pattern is determined by the box-counting method, and the calculation method is as shown in equation (10): lnN i =A-Dlnε i (10) In the formula: A is a constant; D is the fractal dimension; ε i and N i Let ln N be the segmentation size of the i-th level and the corresponding number of self-similar blocks, respectively; if ln N i and ln ε i They exhibit a linear relationship, and the slope D is the fractal dimension of the crack. The larger the fractal dimension of the crack network, the denser the crack distribution and the more complex the crack structure distribution characteristics.

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