Accurate geomechanical parameters are critical in tunneling excavation, design, and supporting. In this paper, a displacements back analysis based on artificial bee colony (ABC) algorithm is proposed to identify geomechanical parameters from monitored displacements. ABC was used as global optimal algorithm to search the unknown geomechanical parameters for the problem with analytical solution. To the problem without analytical solution, optimal back analysis is timeconsuming, and least square support vector machine (LSSVM) was used to build the relationship between unknown geomechanical parameters and displacement and improve the efficiency of back analysis. The proposed method was applied to a tunnel with analytical solution and a tunnel without analytical solution. The results show the proposed method is feasible.
Numerical analysis plays an important role in construction and design of geotechnical engineering [
There are mainly three types of displacement back analysis methods: inverse solving method, atlas method, and direct (i.e., optimal) method [
In this paper, artificial bee colony (ABC) algorithm was chosen for its biological and evolutionary appeal in finding the set of unknown parameters that best matches the modeling prediction with the measured displacement data. Least square support vector machine (LSSVM) was used to replace numerical analysis to present the relationship between unknown geomechanical parameters and displacement of geotechnical structure. Firstly, the idea and algorithm of ABC were presented in Section
The artificial bee colony (ABC) algorithm was originally developed in 2005 by Karaboga [
In the algorithm, the first half of the colony consists of employed artificial bees and the second half constitutes the onlookers. The number of the employed bees or the onlooker bees is equal to the number of solutions in the population. At the first step, the ABC generates a randomly distributed initial population of
Once initialization is completed, the artificial bees are used to conduct the search for the best food resource (solution). Procedures can be described as follows [
Employed bees determine a food source within the neighborhood of the food source through their memory.
Employed bees share their information with onlookers within the hive and then the onlookers select one of the food sources.
Onlookers select a food source within the neighborhood of the food sources chosen by them to produce and exploit the new food resources.
An employed bee of the sources that have been abandoned by onlookers becomes a scout and starts to search for a new food source randomly.
In the ABC algorithm, a candidate food position can be produced from the memory of bees, which is defined as
An artificial onlooker bee chooses a food source based on the probability of food source. The probability of being selected for fitness,
In ABC algorithm, a food source whose position cannot be improved further through a predetermined number of cycles is assumed to be abandoned by onlookers.
Each candidate source position
There are three control parameters in the ABC, the number of food sources which is equal to the number of employed or onlooker bees (
Optimization algorithm is critical to back analysis. In this section, ABCbased back analysis was presented to identify the geomechanical parameters of a circular tunnel with analytical solution.
A circular tunnel is excavated in a continuous, homogeneous, isotropic, initially elastic rock mass and subjected to a hydrostatic far field stress
A circular tunnel subjected to hydrostatic far field stress and uniform support pressure.
According to the MohrCoulomb criterion, the normal stress
The deformation of surrounding rock of tunnel is as follows.
Elastic zone
Plastic zone
An error function, in this work, is defined as the minimum error between the displacements predicted by the analytical model based identified parameters and the actual measured displacements. It can be expressed as
ABCbased back analysis is combined ABC with the analytical solution (see (
Flowchart of ABCbased back analysis.
The displacement of monitored point of tunnel can be computed by the above formula. In this study, six monitored points were used in circular tunnel to monitor the displacements at the horizontal direction for ABC search. The distance between central of tunnel and 6 monitored points is 1.0 m, 1.1 m, 1.3 m, 1.5 m, 1.7 m, and 2.1 m, respectively (see Figure
Parameters of tunnel model.







30.0000  7000.0000  3.4500  30.0000  0  0 
Identified parameters using ABCbased back analysis.





ABCbased back analysis  6893.04951  3.5065  29.99284 
Actual value  7000.0000  3.4500  30.0000 
Relative error (%)  1.5279  −1.6377  0.0239 
Position of monitored point in circular tunnel.
The comparison of displacement between actual and recognized parameters.
The comparison of stress between actual and recognized parameters.
Relationship between fitness value and cycle.
The variation of identified parameter with the cycle.
The performances of ABC are demonstrated with different searching ranges (Table
The ranges of identified parameters.
Range 1  Range 2  Range 3  













The performance of ABC using different searching ranges.
Population size is key parameters of ABC. To study the effect of the colony size on the convergence rate of the ABC algorithm, five different colonies that consisted of 20, 50, 100, 200, and 400 bees were used. The fitness versus cycle numbers is shown in Figure
The convergence of different population size.
In the above section, ABCbased back analysis was used to the circular tunnel with analytical solution. To the practical engineering, it is difficult to get the analytical solution. The procedure with numerical solution is timeconsuming. Regression analysis is a good approach to build the relation between geomechanical parameters and field monitored information. In this study, least square support vector machine (LSSVM) was adopted to present the relationship between geomechanical parameters and displacement based on numerical analysis.
The least square support vector machine (LSSVM) was originally developed by Suykens and Vandewalle [
LSSVM is used in this study to map the nonlinear relationship between geomechanical parameters such as Young’s modulus, cohesion, geostress coefficients, and monitored displacements. The mathematical model of least square support vector machine is defined as
In order to obtain
After the LSSVM model, representing the nonlinear relation between the displacement and a parameter, is obtained, it can be used to predict displacements at monitored points instead of numerical analysis. ABC is used to search the optimal parameter to be identified based on the error function (see (
To verify the model, we suppose there is a tunnel (see Figure
Identified in situ stress and angle in different stages.


Angle  

Actual  20.0000  10.0000  30.0000 
Stage 1  19.9583  10.0614  30.0104 
Stage 2  20.6493  10.8171  33.3676 
Stage 3  20.0252  10.0376  30.623 
Training samples and model parameters of LSSVM.
Number of samples 



Displacement 
 

MP1  MP2  MP3  MP1 
MP1 
MP2 
MP2 
MP3 
MP3  





 
1  10.0000  5.0000  20.0000  −0.8380  −1.3600  1.5500  −0.0231  −2.0200  −1.5100  1.4473  2.0149  −0.8992  −0.3815  1.5989  2.2484 
2  10.0000  7.5000  25.0000  −0.4990  −2.3300  1.3900  −0.0687  −1.6700  −1.5800  1.6424  0.8880  −0.9801  −0.3294  1.6348  1.9749 
3  10.0000  10.0000  30.0000  0.0000  −3.1300  1.4000  −1.4400  1.4000  −1.4400  2.1479  0.2439  −0.9786  −1.6870  4.9088  2.1843 
4  12.5000  12.5000  35.0000  0.0000  −3.9100  1.7500  −1.8000  −1.7500  −1.8000  2.0307  −0.3980  −0.5684  −1.8560  1.4959  1.7655 
5  15.0000  15.0000  40.0000  −0.0001  −4.7000  2.0900  −2.1600  −2.1000  −2.1700  2.0040  −1.0849  −0.2202  −2.1514  1.2404  1.4127 
6  15.0000  5.0000  25.0000  −2.0000  −1.4700  2.0800  0.8610  −3.1900  −2.7200  0.2187  1.8194  −0.3108  0.5215  0.3286  0.9391 
7  15.0000  7.5000  30.0000  −1.6800  −2.5600  1.8300  0.1890  −2.7700  −2.8200  0.5089  0.6915  −0.5137  −0.1191  0.5530  0.7972 
8  15.0000  10.0000  35.0000  −1.2300  −3.4700  1.7400  −0.5740  −2.4200  −2.7500  0.6722  0.0683  −0.5353  −0.5142  0.7871  0.7058 
9  15.0000  12.5000  40.0000  −0.6420  −4.1900  1.8300  −1.3800  −2.1800  −2.5200  1.0483  −0.3389  −0.5100  −1.0033  1.0326  0.8752 
10  15.0000  15.0000  20.0000  −0.0001  −4.7000  2.0900  −2.1600  −2.1000  −2.1700  2.2964  −1.2063  −0.4593  −2.4334  1.6207  1.6580 
11  20.0000  5.0000  30.0000  −3.4100  −1.9500  2.2700  1.8500  −4.2500  −4.3300  −0.9584  1.4147  −0.1741  1.3821  −0.5279  −0.4169 
12  20.0000  7.5000  35.0000  −3.0700  −3.2100  1.9200  1.1000  −3.6000  −4.3700  −0.4940  0.2093  −0.4605  0.5409  0.1538  −0.3109 
13  20.0000  10.0000  40.0000  −2.5800  −4.2600  1.7400  0.2750  −3.1500  −4.3100  −0.1430  −0.6938  −0.6499  −0.1060  0.5071  −0.3365 
14  20.0000  12.5000  20.0000  −1.2600  −3.6100  3.0300  −0.7560  −3.7300  −2.9900  0.9442  −0.0545  0.4200  −0.9120  0.0125  0.7845 
15  20.0000  15.0000  25.0000  −0.9990  −4.6500  2.7900  −1.3700  −3.3400  −3.1500  1.2917  −1.1019  0.1791  −1.6037  0.4438  0.6994 
16  25.0000  5.0000  35.0000  −5.0300  −2.8100  2.2000  2.9600  −5.3200  −6.2900  −2.3159  0.7126  −0.2344  2.3232  −1.4578  −2.0741 
17  25.0000  7.5000  40.0000  −4.5700  −4.3400  1.7200  2.0100  −4.2700  −6.2200  −1.7211  −0.8042  −0.7054  1.2559  −0.3011  −1.8612 
18  25.0000  10.0000  20.0000  −2.5600  −2.5500  4.0000  0.6760  −5.3900  −3.8400  −0.0681  0.8422  1.2115  0.2427  −1.3147  0.1347 
19  25.0000  12.5000  25.0000  −2.5800  −3.8300  3.5200  0.2050  −4.8900  −4.2900  −0.1085  −0.2739  0.7151  −0.1537  −0.8165  −0.2612 
20  25.0000  15.0000  30.0000  −2.3100  −5.0100  3.2000  −0.3910  −4.4100  −4.5200  0.2387  −1.3780  0.5252  −0.8035  −0.4328  −0.4061 
21  30.0000  5.0000  40.0000  −7.0100  −4.2700  1.9200  4.2500  −6.3900  −8.5200  −4.4142  −0.7485  −0.5555  3.7206  −2.5680  −4.4067 
22  30.0000  7.5000  20.0000  −4.1800  −1.5000  5.1000  2.1700  −7.4200  −4.8300  −1.6564  1.9159  2.3915  1.6943  −3.4243  −0.8180 
23  30.0000  10.0000  25.0000  −4.3200  −3.0600  4.3800  1.8900  −6.5200  −5.5400  −1.6996  0.3741  1.5582  1.3646  −2.3453  −1.4311 
24  30.0000  12.5000  30.0000  −4.1800  −4.5200  3.7600  1.3100  −5.8600  −5.9500  −1.5641  −0.8404  1.0366  0.8632  −1.8027  −1.7517 
25  30.0000  15.0000  35.0000  −3.8900  −5.8500  3.3200  0.5890  −5.2300  −6.1900  −1.3480  −2.2716  0.7182  0.1455  −1.3269  −2.1053 

—  —  —  —  —  —  —  —  —  −2.4124  −3.4816  2.5241  0.3809  −3.7541  −3.9253 
The cross section of tunnel and parameters.
Comparison between monitored displacement and predicted displacement using identified parameters.
Stage 1
Stage 2
Stage 3
Calculated stress comparison between using theory value and identified value at stage 3.
σ_{x} using theory parameters
σ_{x} using identified parameters
σ_{y} using theory parameters
σ_{y} using identified parameters
The performance of LSSVM is very important to back analysis. The predicted displacement using LSSVM is in well agreement with the calculated displacement using theory and identified parameters (shown in Figure
Predicted displacement using LSSVM with calculated displacement using theory and identified parameters.
Stage 1
Stage 2
Stage 3
In this study, the RBF kernel function was adopted. The relationship between fitness and cycle is listed in Figure
Fitness with different parameters of kernel function.
The performance of LSSVM with different parameters of kernel function.
The paper presents a new methodology called back analysis based on ABC. ABC is used to identify the geomechanical parameters based on monitored displacements. Results of circular tunnel with the analytical solution illustrate clearly that ABC is effectively able to search parameters of geomaterial and has proved ABC has powerful global optimal performance. To improve the efficiency of back analysis, LSSVM was used to present the relationship between geomechanical parameters and displacement instead of numerical analysis. Results of horseshoe tunnel without the analytical solution demonstrate that LSSVM presents well the nonlinear relationship between geomechanical parameters and monitored displacements. The proposed approach improves the efficiency and precision of back analysis and makes it possible to be applied to more complex engineering problem.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research was also supported by the National Fund of Science in China (no. 41072224, 51104057).