# Back Analysis of Geotechnical Engineering Based on Data-Driven Model and Grey Wolf Optimization
## 5 Numerical Example and Application  
### Numerical Example  
A circular tunnel is excavated in a continuous, homogeneous, and isotropic rock mass. The hydrostatic far-field stress \(p_{0}\) and uniform support pressure \(p_{i}\) are shown in Figure 5. When support pressure \(p_{i}\) is not enough to meet critical pressure \(p_{cr}\), a plastic zone will exist. The values of \(p_{cr}\) could be computed as follows:  
\[p_{cr}=\frac{2p_{0}-\sigma_{c}}{k+1} \tag{28}\]  
where \(\sigma_{c}\) notes the uniaxial compression strength. It could be obtained using the following equation. \[\sigma_{c}=\frac{c(k-1)}{\tan\varphi} \tag{29}\]  
Figure 4: Flowchart of the proposed back analysis framework. where \(\varphi\) and \(c\) denote the cohesion and the friction angle, respectively. \(k\) could be determined as follows:  
\[k=\frac{1+\sin\varphi}{1-\sin\varphi} \tag{30}\]  
According to the Mohr-Coulomb criterion, [PERSON] (1993) analyzed and inferred the inward displacement of tunnel wall \(u_{ip}\) and the plastic zone radius \(r_{p}\)([PERSON], 1993). \[\frac{r_{p}}{r_{0}}=\bigg{[}\frac{2(p_{0}+s)}{(k+1)(p_{i}+s)}\bigg{]}^{\frac{1 }{1-1}} \tag{31}\]  
\[\frac{u_{ip}}{r_{0}}=\bigg{[}\frac{(1+\mu)}{E}\bigg{]}\bigg{[}2(1-\mu)(p_{0}-p _{cr})\big{(}\frac{r_{p}}{r_{0}}\big{)}^{2}-2(1-2\mu)(p_{0}-p_{i})\bigg{]} \tag{32}\]  
where \(E\) notes the elastic modulus and \(\mu\) notes [PERSON]'s ratio. The values are computed as follows:  
\[s=\frac{\sigma_{c}}{k-1} \tag{33}\]  
In this study, back analysis was utilized to identify the far-field stress \(p_{0}\), cohesion \(c\), and friction angle \(\varphi\) based on the deformation of the surrounding rock mass in the tunnel. Five horizontal direction monitoring points were placed to record the deformation of surrounding rock mass in the circular tunnel. The distance between the center of the tunnel and the five monitored points are 1.0 m, 1.2 m, 1.6 m, 1.8 m, and 2.0 m, respectively. The displacements of monitored points could be calculated using the above formula (Equation (32)). The tunnel radius is 1.0 m. The value of far-field stress \(p_{0}\), cohesion \(c\), and friction angle \(\varphi\) are 32 MPa, 6.8 GPa, 3.2 MPa, and 32\({}^{\circ}\), respectively. The displacements of 5 monitored points were calculated by the analytical solution and adopted as field measurement to back-calculate the unknown parameters of the surrounding rock mass using the proposed method. The snapshots were constructed and generated based on the experimental design and a numerical method. Once the snapshots are obtained, the surrogate model could be established according to the ROM algorithm. Figure 6 shows the calculated displacement comparison between the ROM and the analytical solution (Equation (32)). The predicted displacement using the ROM surrogate model is in good agreement with the analytical solution. It shows that the ROM captured well the nonlinear function mapping between unknown properties and the tunnel deformation. The ROM-based surrogate model provides a feasible way to replace the analytical solution in back analysis. Figure 5: Numerical example—a circular tunnel. ROM-based back analysis was utilized to identify the unknown properties of the surrounding rock mass using the above ROM surrogate model and to predict displacement in the tunnel. The far-field stress \(p_{0}\), cohesion \(c\), and friction angle \(\varphi\) are 32.01 MPa, 6.73 GPa, 3.40 MPa, and 31.03\({}^{\circ}\), respectively (Table 1). The relative error is \(-0.03\%\), \(1.02\%\), \(-6.25\%\) and \(3.03\%\), respectively. The maximum relative error is less than 7%. It shows that the identified parameters agree with the actual parameters using the proposed back analysis framework. The displacements comparison between the predicted by back analysis, ROM surrogate model, and calculated by the analytical by actual parameters are shown in Figure 7. Figure 8 shows the stress and displacement of the surrounding rock mass in the tunnel calculated based on the different methods.