Optimization of blasting parameters based on response surface method and NSGA-II

Abstract

Based on the response surface method and NSGA-II blasting parameters optimization method, the response surface method is used to select the borehole diameter, decoupling coefficient and borehole spacing as the influencing factors for test design, and the simulation of the blasting crack propagation under the three influencing factors is obtained by finite element simulation. The central crack length and the radius of the crushing zone are defined to represent the degree of blasting crack propagation, and the central crack length and the radius of the crushing zone are calculated to obtain the parameter data set of the relationship between the borehole diameter, decoupling coefficient and target function. The response surface regression model between the blasting parameters and the blasting crack propagation effect of the ore rock is established by using Design-Expert software, and the fitting model is output after the optimization of the fitting formula by residual and variance analysis significance analysis. The output fitting model is optimized and solved by using NSGA-II multi-objective optimization algorithm, and the Pareto front optimal solution set is obtained. The optimal solution is selected to better search the entire solution space, and to make up for the shortcomings of the response surface method in dealing with complex nonlinear relationships, which can handle multiple interrelated performance indicators, optimize more comprehensively and has other advantages.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of mining blasting, in particular to a blasting parameter optimization method based on response surface method and NSGA-II. BACKGROUND

[0002] With the gradual depletion of surface easy-to-mine mineral resources, the future trend of mineral resource exploitation is gradually shifting to deep and difficult-to-mine deposits. As a low-cost and high-efficiency mining method for underground mineral resources, natural caving method has the advantages of high efficiency, large scale, small amount of rock drilling, low cost, etc. The key to the successful implementation of natural caving method lies in the caving characteristics of the ore rock, which directly affects the problems such as large block rate, caving rate and caving arch in the production process of the mine.

[0003] In order to improve the caving characteristics of the ore rock, one of the ways currently adopted is to create artificial cracks in the ore body through blasting, change the structural characteristics of the ore body to enhance the caving property of the ore body, so that the ore body can continuously and stably caving and achieve the desired broken size. However, this technology usually relies on empirical rules in practical application, and there is a lot of uncertainty. In the mines that have adopted this technology, due to improper selection of blasting parameters, it may still be difficult for the ore rock to caving, and even cause production stagnation, directly affecting the production efficiency and economic cost. Therefore, how to effectively optimize the blasting parameters to promote crack propagation has become the key to ensure the successful implementation of natural caving method.

[0004] Response surface method can be used to study and optimize multiple factors that affect the response of a system. It considers multiple variables and determines the interaction between them, which can be used to optimize the analysis of blasting parameters. However, the traditional response surface method is usually based on a simplified model such as quadratic polynomial, which is easily affected by the selection of initial points and falls into local optimal solution, making it difficult to achieve global optimization. At the same time, when the system has complex nonlinear relationship or the distribution of response variables is non-normal, it is difficult to establish a suitable model for the response surface method.

[0005] To solve the above problems, Chinese patent CN114956749B discloses a method for determining the mixing proportion of mine filling body, which can determine the mixing proportion of mine filling by comprehensive test combined with the optimization method of NSGA-II, but does not consider the influence of the interaction between factors and the shortcomings compared with the response surface method; CN106746946B discloses a method for optimizing the mixing ratio of filling material, which uses the response surface test method to optimize the mixing ratio scheme of filling material, but this optimization method will fall into local optimal solution and has the problem of incomplete optimization.

[0006] Therefore, it is of great significance to develop a blasting parameter optimization method based on response surface method and NSGA-II. SUMMARY

[0007] The objective of this invention is to overcome the shortcomings of the prior art and provide a method for optimizing blasting parameters based on response surface methodology and NSGA-II, which can both [achieve certain goals] and [achieve other goals].

[0008] The objective of this invention is achieved through the following technical solution:

[0009] This paper proposes a blasting parameter optimization method based on response surface methodology (RSM) and NSGA-II. Addressing the problem of inappropriate blasting parameter selection and difficulty in caving due to the empirical application of blasting weakening techniques in natural caving methods, this method combines RSM with the NSGA-II multi-objective optimization algorithm. NSGA-II's advantages are utilized to avoid getting trapped in local optima during optimization, thus better searching the entire solution space. Furthermore, this method compensates for the limitations of RSM in handling complex nonlinear relationships. Specifically, RSM is used sequentially to select borehole diameter, decoupling coefficient, and borehole spacing as influencing factors for experimental design. Based on the experimental design, finite element simulation is used to obtain simulated blasting crack propagation diagrams under different borehole diameters, decoupling coefficients, and borehole spacings. The blasting crack propagation is characterized by defining the central crack length and the radius of the crushing zone. The degree of crack propagation was determined by calculating the length of the central crack and the radius of the crushing area using simulation data files. Further processing yielded a parameter dataset relating the borehole diameter, decoupling coefficient, and objective function. A response surface regression model was established using Design-Expert software to establish the relationship between blasting parameters and the crack propagation effect of blasting in the rock. The fitted model was optimized through residual analysis, variance analysis, and significance analysis, and then output as a fitted model. The NSGA-II multi-objective optimization algorithm was used to optimize and solve the output fitted model, obtaining the Pareto front optimal solution set. Finally, the optimal solution was selected based on the actual situation, thus better searching the entire solution space and compensating for the shortcomings of the response surface method in handling complex nonlinear relationships. It can handle problems with multiple interrelated performance indicators, making the optimization more comprehensive.

[0010] Compared with the prior art, the present invention has the following advantages or effects:

[0011] (1) Because the research is conducted through numerical simulation, it makes up for the cumbersome nature and high uncertainty of field experiments.

[0012] (2) It is superior to the response surface methodology used in the experiment, so it considers both the influence of a single factor on the objective function and the influence of the interaction between multiple factors on the objective function.

[0013] (3) In addition, since the NSGA-II optimization method is used instead of the traditional optimization method, the global optimal solution is output instead of the local optimal solution, which makes up for the shortcomings of the traditional optimization method.

[0014] (4) In addition, since the response surface methodology and the NSGA-II optimization method are combined, it also has the advantages of both the response surface methodology and the NSGA-II optimization method.

[0015] In summary, this invention fits a model between each influencing factor and the objective function, enabling predictive analysis of the objective function. It also considers the effects of single factors and the interactions between factors on the objective function, and can determine the order of importance of each factor. Furthermore, this invention employs the NSGA-II optimization method to optimize the experimental results, which can avoid the optimization results from getting trapped in local optima, thereby achieving parameter optimization. Attached Figure Description

[0016] Figure 1 This is a schematic diagram of the response surface optimization process for a blasting parameter optimization method based on the response surface methodology and NSGA-II proposed in this invention.

[0017] Figure 2 This is a schematic diagram of the NSGA-II multi-objective optimization algorithm based on response surface methodology and NSGA-II.

[0018] Figure 3 This is a schematic diagram of the normal distribution of the objective function residuals for the blasting parameter optimization method based on response surface methodology and NSGA-II.

[0019] Figure 4 This is a schematic diagram of refined mesh generation based on the response surface methodology and the NSGA-II blasting parameter optimization method.

[0020] Figure 5 This is a schematic diagram of the normal distribution of the objective function residuals for the blasting parameter optimization method based on response surface methodology and NSGA-II.

[0021] Figure 6 This is a schematic diagram of the optimal solution set of the Pareto front based on the response surface methodology and the NSGA-II method for optimizing blasting parameters.

[0022] The present invention will now be described in further detail with reference to the accompanying drawings. Detailed Implementation

[0023] Reference Figures 1-6This paper proposes a blasting parameter optimization method based on response surface methodology (RSM) and NSGA-II. Addressing the problem of inappropriate blasting parameter selection and difficulty in caving ore due to the empirical blasting weakening technique in natural caving methods, this method combines RSM with the NSGA-II multi-objective optimization algorithm. NSGA-II's advantages are utilized to avoid getting trapped in local optima during optimization, thus better searching the entire solution space. Furthermore, it compensates for the shortcomings of RSM in handling complex nonlinear relationships. Specifically, RSM is used sequentially to select borehole diameter, decoupling coefficient, and borehole spacing as influencing factors for experimental design. Based on the experimental design, finite element simulation is used to obtain simulated blasting crack propagation diagrams under different borehole diameters, decoupling coefficients, and borehole spacings. The blasting crack propagation is characterized by defining the central crack length and the radius of the crushing zone. The degree of crack propagation was determined by calculating the length of the central crack and the radius of the crushing area using simulation data files. Further processing yielded a parameter dataset relating the borehole diameter, decoupling coefficient, and objective function. A response surface regression model was established using Design-Expert software to establish the relationship between blasting parameters and the crack propagation effect of blasting in the rock. The fitted model was optimized through residual analysis, variance analysis, and significance analysis, and then output as a fitted model. The NSGA-II multi-objective optimization algorithm was used to optimize and solve the output fitted model, obtaining the Pareto front optimal solution set. Finally, the optimal solution was selected based on the actual situation, thus better searching the entire solution space and compensating for the shortcomings of the response surface method in handling complex nonlinear relationships. It can handle problems with multiple interrelated performance indicators, making the optimization more comprehensive.

[0024] The specific process steps and conditions of this invention can be further described as follows:

[0025] (1) The Box-Behnken Design experimental design method of response surface was used to conduct a three-factor, three-level experimental design for borehole diameter, decoupling coefficient and borehole spacing. The design level values ​​and coding values ​​are shown in Table 1.

[0026] Table 1 Design Level Values ​​and Coding Values

[0027]

[0028] (2) Numerical models for each level were constructed in Hypermesh modeling software according to Table 1. The models were constructed in a pseudo-two-dimensional form with a thickness of 1 cm. Z-axis constraints were added to all nodes, meaning that the detonation effect of the explosives only propagated in the x and y directions. Non-reflective boundaries were defined around the model to simulate actual boundary effects. The models were uniformly set to detonate the explosives simultaneously. The model calculation size was 20m × 20m (large square area), and the air domain size was 15m × 15m (small square area). The schematic diagram of the model is shown below. Figure 3 As shown in the diagram, the refined mesh model of the borehole is divided into the following figures: Figure 4 As shown.

[0029] (3) In the finite element simulation LS-DYNA numerical simulation software, appropriate rock material keywords, explosive model keywords, air material keywords and rock material failure keywords are selected to simulate the dynamic behavior of rock under explosive impact load. In this case, the rock material keywords selected are (*MAT_PLASTIC-KINEMATIC), the explosive model keywords are (*MAT HIGH_EXPLOSIVE_BURN), the air material keywords are (*MAT_NULL), and the rock material failure keywords are (*MAT_ADD_EROSION).

[0030] (4) Input the parameters of each material model into the keywords. The rock material parameters used in the case are shown in Table 2, the explosive material reference is shown in Table 3, and the air material parameters are shown in Table 4.

[0031] Table 2. Plastic-Kinematic parameters of ore and rock materials of a certain ore body.

[0032]

[0033] Table 3 Material parameters of the explosive model

[0034]

[0035] Table 4 Material Parameters for the Air Model

[0036]

[0037] In Table 3, A, B, R1, R2, and ω are the material constants of the explosive model, and in Table 4, C0 to C6 are the material constants of the air model.

[0038] (5) Output all the k files with processed keywords in hypermesh, and import the k files into LS-DYNAManager for solving.

[0039] (6) The solution results were processed in the LS-Prepost post-processing software. The length of the central crack and the radius of the crushing area under each model parameter were statistically analyzed. The result data were organized and the dataset output by the numerical simulation is shown in Table 5.

[0040] Table 5 Datasets from Numerical Simulation Output

[0041]

[0042] (7) Using Design-Expert software, multivariate nonlinear fitting was performed on the simulation dataset output in Table 5 to establish a response model relating the central crack length, the radius of the crushing zone, the borehole diameter, the decoupling coefficient, and the borehole spacing. The response model consists of two sets:

[0043]

[0044] R1 2 =0.9669

[0045]

[0046] R3 2 =0.9894

[0047] In the formula: y1 is the length of the central crack, m; y2 is the radius of the crushing zone, m; x1 is the borehole diameter, mm; x2 is the decoupling coefficient; x3 is the borehole spacing, m. y1 represents the squared correlation coefficient R1 of the nonlinear model fitting. 2 The correlation coefficient R3 of the y2 nonlinear model fitting is 0.9669. 2 The value is 0.9894, indicating that the two models have a high degree of reliability in fitting each other.

[0048] (8) Use Design-Expert software to perform residual analysis on the fitted model. For example... Figure 5 As shown, the data points of the central crack length response fitting model and the crushing area radius response fitting model are all distributed around the fitting curve, indicating that the two sets of response fitting models have a good fit and can accurately reflect the relationship between influencing factors and objective function.

[0049] (9) When performing ANOVA on the fitted model using Design Expert software, the P-value and F-value are obtained. These parameters help assess the significance and influence of factors in the model. In ANOVA, the P-value represents the significance of a factor or the difference between factors. Generally, a P-value less than 0.05 indicates a significant difference, and a P-value less than 0.01 indicates a highly significant difference. The P-value can be used to determine the influence of a factor on the experimental results, including the significance of a single factor and the significance of interactions. The F-value represents the influence of each factor on the response value, i.e., the influence of a single factor. The larger the F-value, the greater the influence of that single factor on the response value. The results of the ANOVA are shown in Table 6. As shown in Table 7, the minimum F-test value of the two models is F = 19.46 > F. 0.05The p-value (3,13) = 3.41 indicates that both models are significant and statistically meaningful, and can effectively reflect the relationship between the response value and various influencing factors. The significance test for the two established models showed p-values ​​≤ 0.001, indicating high reliability for both models. Analysis of the models reveals that in the model for the radius of the crushing region y2, the p-value for the borehole spacing x3 is larger than that for the model fitting the central crack length y1, indicating that the influence of borehole spacing on the radius of the crushing region is smaller than that on the central crack length. In the model for the radius of the crushing region y2, the interaction term x1x2 (p = 0.002) between the borehole diameter and the decoupling coefficient is smaller than that in the model for the central crack length y1 (p = 0.707), indicating that the interaction between the borehole diameter and the decoupling coefficient has a more significant impact on the radius of the crushing region. The variance analysis of the fitted models is shown in Table 6.

[0050] Table 6. Analysis of Variance of Fitting Model

[0051]

[0052] (10) The NSGA-II multi-objective optimization algorithm is used to simultaneously optimize the two fitted models to obtain the optimal solution set of the Pareto front.

[0053] (11) Determine the objective function and constraints for multi-objective optimization. In this case, the objective functions are the central crack length model function and the crushing zone radius model function, with no constraints.

[0054] (12) Set the upper and lower limits of the influencing factors of each function. According to the BBD response surface experimental design, the borehole diameter range is 120mm to 165mm, the decoupling coefficient range is 1.2 to 1.8, and the borehole spacing range is 6m to 12m.

[0055] (13) Initialize the population size and generation within the range of the above influencing factors. In this case, the population size is set to 200 and the generation to 50.

[0056] (14) Classify each individual in the population according to the non-dominated order to form multiple levels or "non-dominated fronts" and determine the dominant solutions of each individual in the solution space. For each individual i in the population, there are two parameters n(i) and S(i), where n(i) is the number of solution individuals that dominate individual i in the population, and S(i) is the set of solution individuals dominated by individual i. Find all individuals in the population that satisfy n(i) = 0, and then store these individuals in set A. For each individual j in the current set A, iterate through the set of individuals n(j) it dominates, and subtract 1 from n(k) of each individual k in set S(j). If n(k) equals 0 after subtracting 1, then store individual k in another set B. Take set A as the first-level non-dominated individual set and assign the same non-dominated order to the individuals in this set. Perform a similar classification operation on set B and assign the corresponding non-dominated order to the individuals in it. Iterate this process until all individuals are classified. In this case, the number of iterations is set to 100.

[0057] (15) Calculate the crowding degree of individuals in each level to maintain population diversity. After calculating the sum of the sub-objective function differences of individuals, sort them in ascending order. Assign infinite distance values ​​to boundary solutions, which are not included in the sorting. The clustering distance of individual i for r sub-objectives is:

[0058]

[0059] Where F distance (i) represents the crowding distance. For individual i in objective function f k The value on.

[0060] (16) Selection operations are performed using non-dominated ranking and crowding to build the next generation of population. Selection operations typically involve selecting individuals with the best ranking and those with high crowding to preserve diversity and promote population evolution.

[0061] (17) Use the Simulated Binary Crossover (SBX) genetic operators (crossover and mutation) to perform evolutionary operations on the selected individuals to generate new individuals.

[0062] (18) Repeat the above steps until the predetermined number of iterations is reached or the termination condition is met. In this case, the number of iterations is set to 100.

[0063] (19) Generate the optimal solution set of the Pareto front. The optimal solution set of the Pareto front generated in this case is as follows: Figure 6 As shown.

[0064] (20) Select the optimal solution based on specific needs. In this case, increasing the length of the central crack is more conducive to the propagation of rock mass cracks. Therefore, the optimal solution is selected as a central crack length of 4.73m and a crushing area radius of 1.12m.

[0065] As described above, the present invention can be well implemented. The above embodiments are only the best implementations of the present invention, but the implementation of the present invention is not limited to the above embodiments. Other changes, modifications, substitutions, combinations, and simplifications made without departing from the spirit and principle of the present invention should be considered equivalent substitutions and are all included within the protection scope of the present invention.

Claims

1. A method for optimizing blasting parameters based on response surface methodology and NSGA-II, characterized in that... A combined approach using response surface methodology (RSM) and the NSGA-II multi-objective optimization algorithm was employed. Specifically, RSM was used sequentially to select borehole diameter, decoupling coefficient, and borehole spacing as influencing factors for experimental design. Based on the experimental design, finite element simulations were used to obtain simulated blasting crack propagation diagrams under different borehole diameters, decoupling coefficients, and borehole spacings. The degree of blasting crack propagation was characterized by defining the central crack length and the radius of the crushing zone. The central crack length and the radius of the crushing zone were calculated from the simulation data file, and a parameter dataset relating borehole diameter, decoupling coefficient, and objective function was compiled. A response surface regression model between blasting parameters and the blasting crack propagation effect in the rock was established using Design-Expert software. After optimizing the fitted equation through residual analysis, variance analysis, and significance analysis, the fitted model was output. The NSGA-II multi-objective optimization algorithm was used to optimize and solve the output fitted model, obtaining the Pareto front optimal solution set. Finally, the optimal solution was selected based on the actual situation.

2. The method according to claim 1, characterized in that: The specific process steps and conditions are as follows: (1) The Box-Behnken Design experimental design method of response surface methodology was used to conduct a three-factor, three-level experimental design for borehole diameter, decoupling coefficient and borehole spacing; (2) Numerical models at each level were constructed in the Hypermesh modeling software according to the experimental design. The numerical models were constructed in a pseudo-two-dimensional form with a thickness of 1 cm. Z-axis constraints were added to all nodes, meaning that the detonation effect of the explosive only propagated in the x and y directions. Non-reflective boundaries were defined around the numerical models to simulate the actual boundary effects. The numerical models were uniformly set to detonate the explosives simultaneously. (3) Select rock material keywords, explosive model keywords, air material keywords and rock material failure keywords in the finite element simulation LS-DYNA numerical simulation software to simulate the dynamic behavior of rock under explosive impact load; (4) Input the parameters of each material model into the keywords; (5) Output all the k files with processed keywords in Hypermesh, and import the k files into LS-DYNA Manager for solving; (6) Process the solution results in LS-Prepost post-processing software, and statistically analyze the central crack length and crushing area radius under each model parameter, and organize the result data; (7) The output data simulation dataset was fitted with multivariate nonlinear fitting using Design-Expert software to establish a response model relating the central crack length, the radius of the crushing area to the borehole diameter, the decoupling coefficient and the borehole spacing. (8) The residual analysis of the fitted model was performed using Design-Expert software. The data points of the fitting model of the central crack length response and the fitting model of the crushing area radius response were distributed around the fitting curve. The two sets of response fitting models had a good fit and could accurately reflect the relationship between the influencing factors and the objective function. (9) Use Design Expert software to perform analysis of variance on the fitted model to obtain P-values ​​and F-values. This helps to assess the significance and influence of factors in the model. The P-value represents the significance of a factor or between factors. Generally, a P-value less than 0.05 means a significant difference, and a P-value less than 0.01 means a highly significant difference. The P-value is used to judge the influence of factors on the experimental results, including the significance of individual factors and the significance of interactions. The F-value represents the influence of each factor on the response value, that is, the influence of a single factor. The larger the F-value, the greater the influence of a single factor on the response value. The minimum F-test value between the two models is F = 19.46 > F. 0.05 (3,13)=3.41, indicating that the model is significant and statistically meaningful, and can reflect the relationship between the response value and each influencing factor; (10) The NSGA-II multi-objective optimization algorithm is used to simultaneously optimize the two fitted models to obtain the optimal solution set of the Pareto front; (11) Determine the objective function and constraints for multi-objective optimization. The objective function is the model function of the central crack length and the model function of the crushing zone radius. There are no constraints. (12) Set the upper and lower limits of the influencing factors of each function. According to the BBD response surface experimental design, the borehole diameter range is 120 mm to 165 mm, the decoupling coefficient range is 1.2 to 1.8, and the borehole spacing range is 6 m to 12 m. (13) Initialize the population size and generation within the range of the above influencing factors, setting the population size to 200 and the generation to 50; (14) Classify each population individual according to the non-dominated order to form multiple levels or "non-dominated fronts" and determine the dominant solution of the individual in the solution space. For each population individual i, there are two parameters n(i) and S(i), where n(i) is the number of solution individuals that dominate individual i in the population and S(i) is the set of solution individuals dominated by individual i. Find all individuals in the population that satisfy n(i)=0 and store these individuals in set A. For each individual j in the current set A, iterate through the set of individuals n(j) it dominates and subtract 1 from n(k) of each individual k in set S(j). If n(k) equals 0 after subtracting 1, store individual k in another set B. Use set A as the first level non-dominated individual set and assign the same non-dominated order to the individuals in the set. Perform similar classification operations on set B and assign the corresponding non-dominated order to the individuals in it. Iterate this process until all individuals are classified. The number of iterations is set to 100. (15) Calculate the crowding degree of individuals in each level to maintain population diversity. After calculating the sum of the differences of the sub-objective functions of individuals, sort them in ascending order. Assign infinite distance values ​​to boundary solutions and do not participate in the sorting. The clustering distance of individuals i in r sub-objectives is: Where F distance (i) represents the crowding distance. For individual i in objective function f k The value on; (16) Using non-dominated ordering and crowding for selection operations to build the next generation of population. The selection operations involve selecting individuals with the best order and individuals with high crowding to preserve diversity and promote population evolution. (17) Perform evolutionary operations on selected individuals using the genetic operators crossover and mutation to generate new individuals; (18) Repeat the above steps until the predetermined number of iterations is reached or the termination condition is met. In this case, the number of iterations is set to 100. (19) Generate the optimal solution set of the Pareto front; (20) Select the optimal solution based on specific needs.

3. The method according to claim 2, characterized in that: The numerical model calculation size in step (2) is 20 m × 20 m, and the air domain size is 15 m × 15 m.

4. The method according to claim 2, characterized in that: The rock mass material keywords selected in step (3) are *MAT_PLASTIC-KINEMATIC, *MAT_PLASTIC-KINEMATIC, the explosive model keyword is *MAT HIGH_EXPLOSIVE_BURN, the air material keyword is *MAT_NULL, and the rock mass material failure keyword is *MAT_ADD_EROSION.

5. The method according to claim 2, characterized in that: The response model for step (7) is as follows: y1=-200.1+0.713x1+194.5x2+2.09x3-0.00283x1 2 -56.95x2 2 -0.0846x3 2 +0.0375x1x2+0.0112x1x3-2.114x2x3; y2=-2.47-0.0173x1+7.784x2-0.2112x3+0.000026x1 2 -3.217x2 2 +0.01689x3 2 +0.01466x1x2-0.001263x1x3+0.0528x2x3; In the formula: y1 is the length of the central crack, m; y2 is the radius of the crushing zone, m; x1 is the borehole diameter, mm; x2 is the decoupling coefficient; x3 is the borehole spacing, m; y1 is the squared correlation coefficient R1 of the nonlinear model fitting. 2 The correlation coefficient R3 of the y2 nonlinear model fitting is 0.9669. 2 The value of 0.9894 indicates that the two sets of models have high reliability in fitting.

6. The method according to claim 2, characterized in that... The significance test of the two models established in step (9) is P≦0.001, which is highly significant, indicating that both models have high reliability. In the model with the radius of the crushed area y2, the P value of the borehole spacing x3 is larger than that of the borehole spacing x3 in the model fitting the central crack length y1, indicating that the influence of the borehole spacing on the radius of the crushed area is smaller than that on the central crack length. The interaction term x1x between the borehole diameter and the decoupling coefficient is relatively stable. 2， P=0.002, in the model relative to the central crack length y1, the interaction term x1x between the borehole diameter and the decoupling coefficient. 2， The small value of P=0.707 indicates that the interaction between the borehole diameter and the decoupling coefficient has a more significant impact on the radius of the crushing zone.

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