Due to the geological body uncertainty, the identification of the surrounding rock parameters in the tunnel construction process is of great significance to the calculation of tunnel stability. The ubiquitous-joint model and three-dimensional numerical simulation have advantages in the parameter identification of surrounding rock with weak planes, but conventional methods have certain problems, such as a large number of parameters and large time consumption. To solve the problems, this study combines the orthogonal design, Gaussian process (GP) regression, and difference evolution (DE) optimization, and it constructs the parameters identification method of the jointed surrounding rock. The calculation process of parameters identification of a tunnel jointed surrounding rock based on the GP optimized by the DE includes the following steps. First, a three-dimensional numerical simulation based on the ubiquitous-joint model is conducted according to the orthogonal and uniform design parameters combing schemes, where the model input consists of jointed rock parameters and model output is the information on the surrounding rock displacement and stress. Then, the GP regress model optimized by DE is trained by the data samples. Finally, the GP model is integrated into the DE algorithm, and the absolute differences in the displacement and stress between calculated and monitored values are used as the objective function, while the parameters of the jointed surrounding rock are used as variables and identified. The proposed method is verified by the experiments with a joint rock surface in the Dadongshan tunnel, which is located in Dalian, China. The obtained calculation and analysis results are as follows:
The ability to identify the mechanical parameters of surrounding rock of a tunnel based on field monitoring information is crucial for the construction mechanics in the geotechnical engineering field. The identification of surrounding rock parameters is also called the back analysis. Using different monitoring information, the stress back analysis, displacement back analysis, and mixed back analysis can be concocted. The combined displacement–stress back analysis method has been widely applied to geotechnical engineering [
Different constitutive models can describe a different mechanical behavior of surrounding rock. In the 1970s, a surrounding rock of a tunnel was simplified into a uniform elastic constitutive model, and the parameter inversion was mainly to identify the elastic modulus. Kavanagh et al. [
Rich research results have been achieved in the field of tunnel surrounding rock parameter identification. However, limited studies have been conducted using the ubiquitous-joint model and three-dimensional (3D) model. The main reason is that the complex constitutive model and 3D numerical calculation are time-consuming, which makes the parameters identification more difficult [
(1) Optimization algorithms have certain shortcomings; for instance, the GA includes encoding and decoding, and its operation is complex; the PSO algorithm is relatively simple, but its mathematical theory is not strict enough. The optimization algorithms can also be limited to the local optimal problem and become precocious easily [
(2) The ANNs are based on the empirical risk minimization and have the disadvantages of requiring a large amount of training data, over-learning, and poor generalization ability.
(3) Although the SVM can be used for the sequential minimum optimization and other fast learning algorithms, its operating speed is slow when the number of learning samples is too large, and the estimated output is not probabilistic [
The Gaussian process (GP) regression combines nuclear machine learning and Bayesian reasoning, which provides many outstanding advantages, such as flexible non-parametric inference, adaptive parameter acquisition, and simple implementation process [
The remainder of this paper is organized as follows. In Section 2, the background of back analysis of tunnel surrounding rock is introduced. In Section 3, the parameter identification method based on the GP-DE is introduced. In Section 4, the application of the proposed method in engineering is presented. In Section 5, the parameters of GP and DE are analyzed. Finally, conclusions are drawn in Section 6.
With the rapid development of computational mechanics and computer technology, numerical analysis methods have begun to be applied in the field of rock mechanics research, with strong universal applicability. However, the numerical analysis method is under a strong dependence on the selection of model parameters. The accuracy of the parameters will directly affect the final calculation results. How to obtain surrounding rock parameters effectively and reasonably has become a key issue in tunnel engineering calculations [
According to different kinds of monitored information, back analysis is divided into stress back analysis [
The selection of constitutive model is the basis for determining the accuracy of numerical calculation of the tunnel surrounding rock. The constitutive models used in the back analysis are generally divided into elastic models [
The trend of back analysis is to adopt more complex constitutive models which can describe the actual rock mass, however, there no report about back analysis of ubiquitous joint model parameters can be seen now. The numerical calculation based on ubiquitous joint model concerns more parameters, the relation between parameters and monitoring data is more complex. Therefore the general back analysis methods are limited. The intelligent optimization algorithms with better robust and the RSM with better nonlinear fitting capabilities should be adopted. Therefore, this study proposes the new GP-DE algorithm for ubiquitous-joint model parameters inversion, and comprehensively utilizes displacement and stress information, finally provides an effective back analysis method for the surrounding jointed rock of tunnel engineering.
Because of the excavation unloading effect, the surrounding rock deformation toward the empty surface increases continuously, which causes that part of the surrounding rock to change from the elastic stage to the plastic stage, and the loose zone occurs. In this process, the joints combined with the rock matrix of the surrounding rock all play important roles, which can be expressed by a ubiquitous-joint model. The rock matrix and joints considered in the ubiquitous-joint model satisfy the Mohr–Coulomb yielding criteria. The direction of the joint surface and the stress state can be represented by the three directions (
Therefore, the relationship between the stress tensors
where [
Further, the shear yield criterion and tensile yield criterion of joints can be respectively expressed as follows:
where
where
The potential function g
Based on
The identification of the jointed surrounding rock parameters is essentially an optimization problem. The upper and lower limits are defined based on the specific physical meaning of model parameters. Assuming that there are
where
As shown in
The GP is a fast developing machine learning method that has good nonlinear performance, but suffers from the problems of high dimension, and small sample regression. According to [
Assume
where
The prior distribution of the observed target value
where
For the test sample (
where
When the training set
where
The covariance function is used to measure the similarity degree between the training sample and prediction samples. According to
where
The GP-based surface should be trained by representative data samples before it can map the complex nonlinear relation between the jointed parameters and displacements. The data samples can be obtained by model tests, field tests, numerical simulation, and other methods. In this study, the data samples are collected using the orthogonal design, uniform design, and numerical simulation. In the GP training process, hyper-parameters
where
It is difficult to select the above parameters artificially, so in this study, an intelligent optimization algorithm DE is used to optimize
The original population that includes the NP individuals is generated randomly and uniformly distributed in the solution space as follows:
where
(2) Mutation operation
For each target vector of the
where
(3) Crossover operation
In order to increase the diversity of the population, a new test vector
where
(4) Selection operation
The new generation of the population obtained by the selection operation is expressed as:
where
Based on
In
The specific algorithm steps are as follows:
By using the orthogonal and uniform design methods, the six parameters of the joint model, including the joint surface cohesion, joint surface dilatation angle, joint surface friction angle, joint surface tensile strength, joint surface dip, and joint surface dip angle, are designed in different parameter combination schemes. For each analysis task, the training datasets that correspond to the jointed surrounding rock parameters and displacement and stress values at key points are constructed. Numerical analysis is used to calculate the dataset for each set of the orthogonal and uniformly experimental schemes. To improve the generation performance of the GP, the test dataset is selected from the training dataset and used to assess the applicability of the GP. The GP model is constructed such that to describe the nonlinear relationship between the joint parameters and key displacement and stress values. The orthogonal design samples are used to train the GP model, and the uniform design samples are used to test the GP model. The parameters of the DE-GP model, including the population size, evolutionary generation number, and hyper-parameter ranges of the GP kernel function, are initialized. Each hyper-parameter group is considered as an individual in the GP. A dataset is randomly generated in the solution space as the initial population according to The operations of mutation and crossover are performed according to If the preset termination conditions for the iteration number or the minimal error are satisfied, and the identified parameters are given, the GP training procedure terminates, and the algorithm turns to the next step; otherwise, the algorithm returns to Step 5. The parameters of the DE-GP model, including the population size, evolutionary generation number, and the parameter range for the jointed rock, are initialized. A dataset is randomly generated in the solution space as the initial population according to The operations of mutation and crossover are performed according to If predefined termination conditions for the iteration number or the minimal error are satisfied, the optimal parameters are obtained; otherwise, the algorithm returns to Step 8.
The proposed method was verified by the experiment with the Dadongshan tunnel, which is part of the Bohai Avenue Engineering of high way in Dalian city, China. It includes two super large section tunnels with a small clear distance. The lengths of the western and eastern tunnels are 1,113.8 and 1,110.3 m, respectively. The height and width of tunnel clearance are 10.1 and 18.2 m, respectively, and the maximum depth of the tunnel from the ground surface is 155 m. The minimum thickness of the rock column between the two holes is 15.9 m. The map location and field scene of the tunnel under study are shown in
The regional strata were mainly thick quartz sandstone of the qiaotou formation (Q
The tunnel interval mileage K7 + 620 m–K7 + 660 m, where the third group of joint faces was located, was selected as a study object. The 3D numerical model established by the geotechnical engineering software FLAC
The surrounding rock medium was mainly apoplexy fossil English sandstone. The parameters of the corresponding rock matrix were as follows: Young’s modulus was 1200 MPa, Poisson’s ratio was 0.2, cohesion was 1.0 MPa, and the angle of internal friction was 30
The data given in
Parameter | Minimum value | Maximum value | Mean value |
---|---|---|---|
Joint cohesion ( |
0.05 | 0.1 | 0.075 |
Joint friction angle ( |
20 | 30 | 25 |
Joint dilatation angle ( |
20 | 25 | 22.5 |
Joint tensile strength ( |
0.02 | 0.06 | 0.04 |
Joint dip angle ( |
16 | 20 | 18 |
Joint dip direction ( |
190 | 198 | 194 |
Sample No. | A |
D |
BC (mm) | P1 (kPa) | P2 (kPa) | P3 (kPa) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 0.05 | 20 | 20 | 0.02 | 190 | 16 | 20.78 | 15.75 | 10.65 | 77.60 | 133.4 | 75.37 |
2 | 0.05 | 21 | 22 | 0.03 | 192 | 17 | 19.14 | 15.49 | 10.31 | 77.68 | 124.9 | 74.25 |
3 | 0.05 | 22 | 24 | 0.04 | 194 | 18 | 17.92 | 15.23 | 10.00 | 77.48 | 122.3 | 73.56 |
4 | 0.05 | 23 | 26 | 0.05 | 196 | 19 | 16.94 | 15.02 | 9.72 | 78.14 | 124.9 | 73.23 |
5 | 0.05 | 24 | 28 | 0.06 | 198 | 20 | 16.20 | 14.85 | 9.50 | 76.83 | 130.9 | 71.66 |
6 | 0.06 | 20 | 22 | 0.04 | 196 | 20 | 17.29 | 15.30 | 10.13 | 80.10 | 123.7 | 72.06 |
7 | 0.06 | 21 | 24 | 0.05 | 198 | 16 | 16.89 | 14.70 | 10.00 | 78.54 | 122.0 | 73.47 |
8 | 0.06 | 22 | 26 | 0.06 | 190 | 17 | 16.14 | 14.37 | 9.70 | 72.24 | 122.9 | 73.02 |
9 | 0.06 | 23 | 28 | 0.02 | 192 | 18 | 15.93 | 14.54 | 9.45 | 71.48 | 128.4 | 69.61 |
10 | 0.06 | 24 | 20 | 0.03 | 194 | 19 | 18.50 | 15.81 | 10.35 | 81.10 | 128.2 | 72.78 |
11 | 0.07 | 20 | 24 | 0.06 | 192 | 19 | 15.88 | 14.42 | 9.80 | 75.20 | 123.7 | 72.73 |
12 | 0.07 | 21 | 26 | 0.02 | 194 | 20 | 15.58 | 14.50 | 9.54 | 74.26 | 128.1 | 71.33 |
13 | 0.07 | 22 | 28 | 0.03 | 196 | 16 | 15.28 | 14.09 | 9.47 | 72.00 | 130.1 | 69.74 |
14 | 0.07 | 23 | 20 | 0.04 | 198 | 17 | 17.45 | 15.22 | 10.36 | 82.45 | 125.3 | 72.14 |
15 | 0.07 | 24 | 22 | 0.05 | 190 | 18 | 16.65 | 14.77 | 10.01 | 75.31 | 124.2 | 72.77 |
16 | 0.08 | 20 | 26 | 0.03 | 198 | 18 | 15.08 | 14.12 | 9.55 | 73.91 | 129.4 | 71.44 |
17 | 0.08 | 21 | 28 | 0.04 | 190 | 19 | 14.64 | 13.84 | 9.29 | 68.89 | 132.6 | 65.28 |
18 | 0.08 | 22 | 20 | 0.05 | 192 | 20 | 16.36 | 14.87 | 10.15 | 78.17 | 125.9 | 71.41 |
19 | 0.08 | 23 | 22 | 0.06 | 194 | 16 | 15.98 | 14.35 | 10.01 | 75.46 | 122.6 | 72.63 |
20 | 0.08 | 24 | 24 | 0.02 | 196 | 17 | 15.66 | 14.36 | 9.74 | 75.32 | 123.2 | 72.65 |
21 | 0.09 | 20 | 28 | 0.05 | 194 | 17 | 14.22 | 13.52 | 9.30 | 69.04 | 134.3 | 65.72 |
22 | 0.09 | 21 | 20 | 0.06 | 196 | 18 | 15.84 | 14.50 | 10.16 | 78.91 | 124.6 | 71.03 |
23 | 0.09 | 22 | 22 | 0.02 | 198 | 19 | 15.44 | 14.44 | 9.87 | 77.69 | 124.1 | 70.76 |
24 | 0.09 | 23 | 24 | 0.03 | 190 | 20 | 15.00 | 14.07 | 9.56 | 71.57 | 125.2 | 71.72 |
25 | 0.09 | 24 | 26 | 0.04 | 192 | 16 | 14.69 | 13.71 | 9.47 | 70.20 | 126.9 | 70.15 |
Sample No. | A |
D |
BC (mm) | P1 (kPa) | P2 (kPa) | P3 (kPa) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 0.05 | 21 | 22 | 0.04 | 196 | 20 | 17.82 | 16.31 | 9.62 | 78.21 | 121.2 | 17.82 |
2 | 0.06 | 22 | 26 | 0.03 | 198 | 19 | 16.64 | 15.75 | 9.32 | 79.94 | 130.4 | 16.64 |
3 | 0.07 | 20 | 20 | 0.06 | 194 | 18 | 17.72 | 13.79 | 9.68 | 77.21 | 117.3 | 17.72 |
4 | 0.08 | 24 | 24 | 0.02 | 192 | 17 | 14.78 | 13.54 | 9.12 | 71.98 | 122.5 | 14.78 |
5 | 0.09 | 23 | 28 | 0.05 | 190 | 16 | 14.63 | 13.69 | 8.98 | 71.58 | 127.9 | 14.63 |
Using the engineering monitoring data as the control value, the back analysis of the joint parameters was carried out. The CR was set to 0.9,
The back analysis joint surface parameters were used to carry out the numerical simulation calculation under the reinforcing measures of the initial lining and pipe-roof of the tunnel, and the displacement variation values of the monitoring points were calculated. The numerical simulation results were compared with the field monitoring data of the K7 + 650 m section, and the comparison results are shown in
0.081 | 20.98 | 27.98 | 0.036 | 190.65 | 18.84 |
Measured values | The field measured values | Back analysis calculation values | Relative error (%) |
---|---|---|---|
A |
14.62 | 14.34 | 1.92 |
D |
13.73 | 13.31 | 3.06 |
BC | 9.22 | 9.48 | 2.82 |
P1 | 68.76 | 66.43 | 3.39 |
P2 | 131.61 | 130.12 | 1.13 |
P3 | 65.24 | 62.25 | 4.58 |
The fitness values of the GP-DE algorithm population in the first, 3rd, 10th, and 30th generations are presented in
The changes in the six parameters of the GP-DE algorithm with the number of iterations are presented in
In order to analyze the stability of a jointed surrounding rock and select a rational supporting scheme, the inversion parameters of the jointed rock were used to conduct a numerical simulation to compare the plastic zone of the surrounding rock with different supporting schemes. The four possible reinforcing measures were: initial lining, initial lining + pipe roof, initial lining + pipe roof + pre-stressed anchor, and initial lining + pipe roof + advance grouting anchor. The plastic zones of the tunnel under different reinforcing measures are shown in
Under the condition of surrounding rock with joints, the effect of the pre-stressed bolt was small, which was related to the large deformation in the low-grade surrounding rock and the pre-stress loss. When the surrounding rock was grouted, the effect of reinforcement was obviously improved. Therefore, for the reinforcement of a rock wall of a tunnel with a small spacing, it is preferred to grout the surrounding rock to improve the surrounding rock conditions and parameters. Therefore, the scheme of the initial lining + pipe roof + advance grouting anchor is suggested.
The local correlation coefficient
The GP-DE model was compared with the general GP and ANN models, and comparison results are given in
In the GP-DE algorithm, the control of the DE algorithm is relatively complex, involving many influencing factors, and it plays a decisive role in the GP-DE algorithm. The back analysis of the joint parameters in the construction site was conducted to discuss the DE algorithm from the two aspects, the control parameters, and different difference strategies.
In the DE, the variation factor
For the DE/Best/1 difference strategy and
When the DE/Best/1 difference strategy was used, and
Sample No. | DE-GP (%) | GP (%) | ANN (%) | ||||||
A |
D |
BC | A |
D |
BC | A |
D |
BC | |
1 | 0.81 | 2.52 | 5.07 | 2.00 | 3.89 | 6.10 | 1.61 | 6.06 | 5.77 |
2 | 1.16 | 3.75 | 2.14 | 2.69 | 6.02 | 3.60 | 2.61 | 5.01 | 5.81 |
3 | 5.19 | 4.80 | 1.77 | 3.14 | 8.49 | 6.61 | 3.23 | 9.89 | 6.71 |
4 | 3.92 | 3.61 | 3.07 | 8.25 | 6.00 | 6.74 | 5.96 | 3.95 | 5.44 |
5 | 1.03 | 0.68 | 2.63 | 2.84 | 1.42 | 3.54 | 3.35 | 0.86 | 5.78 |
P1 | P2 | P3 | P1 | P2 | P3 | P1 | P2 | P3 | |
1 | 2.01 | 1.55 | 3.82 | 4.21 | 2.18 | 4.99 | 2.09 | 2.85 | 5.96 |
2 | 2.84 | 1.94 | 2.39 | 3.34 | 3.25 | 3.86 | 3.37 | 2.53 | 5.02 |
3 | 2.37 | 2.78 | 4.40 | 3.07 | 3.65 | 5.34 | 4.59 | 4.30 | 5.08 |
4 | 0.63 | 0.91 | 0.51 | 1.38 | 1.04 | 1.29 | 1.50 | 2.49 | 3.09 |
5 | 2.32 | 1.37 | 1.18 | 6.71 | 2.99 | 1.84 | 3.12 | 1.45 | 1.62 |
In the DE algorithm, different difference strategies can be used to achieve mutation operation, as given in
Aimed at solving the problems of jointed rock parameters identification based on the ubiquitous-joint model and 3D numerical model, the paper proposed a hybrid GP-DE algorithm.
The main contributions of this work can be summarized as follows:
1. In this paper, an orthogonal design, GP, DE, and ubiquitous-joint model are combined to develop a hybrid method for the joint parameters identification of surrounding rock of a tunnel. The proposed method makes full use of the advantages of GP and DE, improves the computing speed, and avoids the problem of result limitation to the local optimal solution. Since hyper-parameters of the GP affect its forecast precision, the DE is used to optimize the GP parameters. Compared with the general GP and ANN models, the proposed GP-DE model has a smaller forecast error.
2. The proposed method is verified by the experiment with a real tunnel, namely, the proposed method is used to invert and analyze the parameters of a joint rock surface in the Dadongshan tunnel project. The results of the back analysis are compared with the field monitored values. The relative error of 4.58% is obtained, which can be considered as a good result. The calculation results show that the proposed method has fast calculation speed, good convergence, and high precision, which can meet the engineering requirements. The variations in the DE parameters have an effect on the convergence speed. Through the calculation and analysis, it has been found that the recommended parameters’ values are
3. The identified jointed parameters are considered in the tunnel construction scheme selection and surrounding rock stability analysis. The results show that under the condition of surrounding rock with joints, it is preferred to grout the surrounding rock and rock wall to improve the surrounding rock conditions and parameters. The scheme of the initial lining + pipe roof + advance grouting anchor is suggested.
The results and conclusions presented in this work have guiding significance for tunnel engineering. The paper provides an effective means for the tunnel information construction and feedback analysis with the jointed surrounding rock. However, in the calculation process, different covariance functions need to be selected according to different learning samples.
According to the above discussion, and on the basis of fully grasping the characteristics of various covariance functions, a combined covariance function can be used to improve the accuracy and practicability of the model.