## 5 Back-Analysis of Rock Parameters with GOA-GPR-FLAC\({}^{\text{3D}}\)
### Case Study  
Taking a quarter circular tunnel as an example, a back-analysis of rock mechanical parameters is carried out. The tunnel radius is 3.5 m; the thickness of the tunnel surrounding rock in the x-direction and z-direction is 16.5 m; the buried depth of the tunnel is 50 m; the bulk density of the surrounding rock is 2000 kg/m\({}^{3}\); the initial in situ stress field in the rock is mainly composed of a gravity stress field, and the initial in situ stress is \(\sigma_{0}=10\) MPa. In this case, the classic Mohr-Colomb elastoplastic constitutive model is adopted. The ideal elastic-plastic constitutive model based on the Mohr-Colomb failure criterion is one of the most widely used constitutive models in the field of rock engineering [31]. The model has few input parameters and can well describe the mechanics of surrounding rock after underground engineering excavation, and can be used to distinguish whether rock mass is fractured according to whether the grid element of numerical calculation has entered the plastic state. The elastic modulus, cohesion and internal friction angle need to be inversely analysed. In the FLAC\({}^{\text{3D}}\) software, the generated Lagrangian finite volume mesh covers the entire physical region that needs to be analysed. The smallest meshes may contain only one element, but most are defined by meshes of hundreds, thousands or even millions of elements [32]. Users generally control the grid to adapt to various shapes of 3D problem domains by matching and connecting the built-in grid generator. Figure 9 shows the division of the tunnel mesh. The tunnel model used a hexahedron with 8-nodes and 6-quadrilateral faces as the finite difference element body, which is called a zone in FLAC\({}^{\text{3D}}\). Through the mesh prototype library that comes with the software, the embedded cylindrical radial-gradient rectangular mesh required for this case was directly generated. The adopted grid was composed of 1000 zones with 1386 grid points. The size and density settings of the grid elements were based on the adopted methods [33; 34], considering the surrounding rock properties, strain and integration technique (implicit or explicit). The closer it is to the excavation surface, the smaller the mesh scale is. Multiple computational models with a grid density from small to large were compared. When the mesh density was increased, the change in the calculation results was small, and the correspondingly finer mesh was used. In the process of the FLAC\({}^{\text{3D}}\) calculation and simulation, the boundary conditions can be divided into two categories: stress boundary conditions and displacement boundary conditions [35]. The boundary referred to here can be the real boundary that actually exists in the real situation, or it can be the artificial boundary assumed by the user in order to make the model closed. Displacement boundary conditions cannot be set directly in FLAC\({}^{\text{3D}}\). In order to apply a known displacement value to the boundary, it is necessary to define the velocity value at the boundary when the calculation reaches the specified number of steps. In this case, since the tunnel model only simulates a small part of the surrounding rock, the displacement boundary of the model needs to be constrained to prevent the model from large deformation at the artificial boundary. The specific displacement boundary  
Figure 8: Flowchart of the GOA-GPR-FLAC\({}^{\text{3D}}\) method. conditions defined by adopting the penalty method [36, 37] were as follows: the bottom surface of the model was constrained along the z-direction; the left and right surfaces were constrained along the x-direction, and the front and rear surfaces were constrained along the y-direction. Gradient stress was applied to the entire model from the top of the model along the direction of gravity to simulate the in situ stress of surrounding rock as the stress boundary condition.