## 2 Numerical Approach
### The Numerical Model  
This paper utilizes FLAC\({}^{\text{3D}}\), which is based on the finite difference method, to perform calculations. Geotechnical engineering problems are typically expressed as differential equations, including basic equations such as the balance equation, geometric equation, and constitutive equation, as well as boundary conditions. By replacing each derivative with a finite difference approximation formula, the finite difference method transforms the problem of solving partial differential equations into the problem of solving algebraic equations. FLAC\({}^{\text{3D}}\) takes nodes as the calculation objects and concentrates both forces and  
Figure 2: Composition and dimensions of double-layered shield tunnel. masses at each node, then solves them in the time domain using the equation of motion. The equation of node motion can be expressed as follows:  
\[\frac{\partial v_{i}^{l}}{\partial t}=\frac{F_{i}^{l}(t)}{m^{l}} \tag{1}\]  
where \(F_{i}^{l}(t)\) is the unbalance force component of node \(l\) at time \(t\) in direction \(i\), which can be derived from the principle of virtual work. \(m^{l}\) is the concentrated mass of node \(i\). FLAC\({}^{\text{3D}}\) uses the velocity to calculate the strain increment of the element in a time step, as follows:  
\[\Delta e_{ij}=\frac{1}{2}(v_{i,j}+v_{j,i})\Delta t \tag{2}\]  
where \(v_{i,j}\) is the partial derivative of \(v\) with respect to the vector component \(x_{i}\) of the node position. By using the constitutive equation, the stress increment can be obtained from the strain increment. Then, by accumulating the stress increment of each time step, the total stress can be obtained. The aforementioned shield tunnel project was utilized as an exemplar for the formulation of a numerical model, depicted in Figure 4. The model's dimensions were 36.1 m \(\times\) 100 m \(\times\) 100 m (X \(\times\) Y \(\times\) Z). The lateral side of the model represented the boundary with a normal displacement constraint, while the bottom surface served as the fully constrained boundary. As this paper primarily focused on the mechanical properties of shield tunnel structures rather than the effect of strata deformation or strata properties, the Mohr-Coulomb constitutive model, known for its simplicity and wide applicability [41; 42], was employed for the strata. Table 1 presents the specific physical-mechanical parameters for the strata. The shield tunnel was located at the center of the model. Tunnel excavation was simplified to the excavation of the entire section at once. In the shield tunnel project upon which this paper relies, the soil cover near the riverbed was thicker, and the water head was higher. Consequently, the tunnel at the riverbed experienced a higher water and soil load compared to the adjacent sides. To streamline modeling and expedite model calculations, a trapezoidal stress boundary was implemented at the upper part of the model to represent the actual loading conditions. Within the model, \(q_{1}\) and \(q_{2}\) represented the soil and water load at the typical sections 1 and 3, respectively. Figure 3: Schematic diagram of geological cross-section along the tunnel’s longitudinal axis. Combined with the real engineering project, a three-dimensional longitudinal refined numerical calculation model of a double-layered shield tunnel was established in the present research, and the longitudinal mechanical properties of the shield tunnel were studied. The established model was a double-layered shield tunnel with 100 rings and a total length of 100 m. A single segment ring consists of six standard blocks, two adjacent blocks, and one capping block, and the segments were assembled in a staggered manner. The secondary lining was placed inside the segment lining and composed of several homogeneous secondary lining rings (Figure 5).