# Seismic Stability of Dual Tunnels in Cohesive-Frictional Soil Subjected to Surcharge Loading
## 3 AFELA Model  
As an advanced method, FELA has been widely used in engineering practices [6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16]. The FELA can assess accurate ultimate loads by a combination of the limit theorems of plasticity and finite elements [35]. Strictly close lower bound (LB) results and upper bound (UB) results can be obtained by a reasonable construction of statically admissible stress field and kinematically admissible velocity field, respectively. Additionally, the true ultimate bearing capacity is located between the LB and UB results. To achieve precise results, a discontinuous finite element formulation is introduced into the present study for discrete UB [36] and LB [37] issues. The use of the discontinuous finite element formulation in the static form of limit analysis leads to two nonlinear optimization models, which can be presented as the same form [38; 39]:  
\[\textbf{Minimize} \lambda \tag{2}\] \[\textbf{Subject to} \sigma=p_{0}+\lambda p\] (3) \[f(\sigma_{j})\leq 0,j=1,\ldots,n_{\sigma} \tag{4}\]  
where \(\lambda\) is the load multiplier, \(\text{B}^{\text{T}}\) is the operator of discrete equilibrium type, \(\sigma=(\sigma_{1},\ldots,\sigma_{n_{\sigma}})^{\text{T}}\) is the vector of \(\sigma_{j}\), \(n_{\sigma}\) is the sum of discrete stresses, \(p_{0}\) and \(p\) are the vectors of multiplier force and prescribed force, and \(f\) is the yield function. The authors developed a nonlinear optimization algorithm to solve Equations (2)-(4). This algorithm is an improved version of the feasible arc interior point algorithm (FAIPA) [40]. To reach a precise result, a self-developed adaptive remeshing technique [41] is employed in the AFELA code. This technique is developed on the basis of bounds gap error estimator, which can decompose the overall bounds gap, discrepancy of the UB and LB calculated with identical meshes into positive contributions from all elements in the mesh [38; 39]. This adaptive meshing strategy is composed of adjusting element sizes in the field to equably distribute local errors (measured by elemental bound gaps) to each element of this mesh. Through the use of the adaptive meshing program, more accurate results can be obtained. More detailed information about the self-developed adaptive meshing technique proposed is presented as follows:  
\[d_{n}(\textbf{x})=\beta(\textbf{x})\cdot d_{o}(\textbf{x}),\textbf{x}\in \Omega \tag{5}\]  
\[\text{s.t.}\underline{\beta}\leq\beta(\textbf{x})=\frac{\sqrt{4\Delta/\left( \sqrt{3N}\textit{el}\cdot p(\textbf{x})\right)}}{d_{o}(\textbf{x})}\leq\overline {\beta} \tag{6}\]  
Figure 1: Problem definition. where \(Nel\) is the element number of the optimized mesh; \(d_{o}\) is the distribution of element sizes of original mesh; \(d_{n}\) is the distributions of element sizes for the optimized mesh; \(\beta(\mathbf{x})\) is the distribution for the decreasing or increasing rate of element sizes in the optimized mesh subject to the upper and lower limits of \(\overline{\beta}\) and \(\beta\); \(\rho(\mathbf{x})\) is the distribution for the intensity of the dissipation gap; and \(\Delta\) is the overall dissipation gap, which can be decomposed into elemental contributions \(\Delta^{\epsilon}\) from all the elements in the entire mesh [35]:  
\[\Delta=\sum_{elements}\Delta^{\epsilon}=\lambda^{\text{UB}}-\lambda^{\text{LB}} \tag{7}\]  
where \(\Delta^{\epsilon}\) is the elemental dissipation gap and \(\lambda^{\text{UB}}\) and \(\lambda^{\text{LB}}\) are the UB and LB results of ultimate load multipliers, respectively. As space is limited, a further description will not be introduced here. Detailed information regarding the code can be seen in [PERSON]'s work [41]. Figure 2 presents a detailed mesh arrangement for dual tunnels in cohesion-frictional soil by the self-developed code mentioned above. This study adopts four refinement steps to reach the high precision of computation. The initial and final number of elements is set as 2000 and 8000 to guarantee accuracy and efficiency.