# Seismic Isolation Effect of Non-Water Reacted Two-Component Polymeric Material Coating on Tunnels
## 3 Numerical Model  
### Infinite Element Model  
The concept of infinite elements was first presented by [PERSON] (1973) and [PERSON] (1992) [32; 33]. Further constitutive work has been done by some other researchers to apply the infinite elements to the solution of static and dynamic problems in engineering practice.  
The main idea in forminga static infinite element is to derive it from an element displacement shape function, and the element displacement shape function is the product of the Lagrange interpolation function (decay function). The infinite element is mapped onto a finite one by using some mapping techniques, and the mapping techniques are also employedodynamic infinite elements. But considering the difficult mechanism of wave propagation in infinite elements, the wave propagation function in its dynamic infinite elements is replaced by the decay function in the static infinite elements [32].  
Although [PERSON] and [PERSON]. (1993), [PERSON] et al. (2015), and [PERSON] et al. (2019) [34; 35; 36] have successfully used the finite and infinite elements coupled method and the related wave input procedure in infinite media to solve the dynamic response, the application of this method to study the dynamic response of polymeric seismic isolation around tunnels has not been done before.  
2D finite element models with infinite element boundaries are made for the nonlinear numerical analysis by ABAQUS. According to the Code for the seismic design of urban rail transit structures [37], the distance between the numerical model boundary and underground structure should be three times larger than the structure height.  
The finite element (FE) model in Figure 5 is 50 m wide and 50 m deep. The diameter of the tunnel is 5.5 m, and the thickness of the liner is 0.35 m, as shown in Figure 5.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Specimen** & & **Dynamic Elastic** & **Average** & **Dynamic Elastic** & **Average** \\
**Density** & **No** & **Modulus of the** & **Dynamic Elastic** & **Modulus of the** & **Dynamic Elastic** \\
**(g/cm\({}^{3}\))** & & **First Mode/MPa** & **Modulus/MPa** & **Second Mode/MPa** & **Modulus/MPa** \\ \hline \multirow{3}{*}{0.22} & 1 & 69.4 & & 73.0 & \\  & 2 & 71.0 & 72.26 & 81.3 & 81.87 \\ \cline{1-1}  & 3 & 76.4 & & 91.3 & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Results of the dynamic elastic modulus.  
Figure 5: Geometry and mesh discretization of the numerical model (**left**), lining, and coating (**right**).  
Incompatible modes of 4-node bilinear elements are employed in the surrounding rock, lining, and polymeric material. There are 17,283 nodes and 16,941 elements with an average dimension of 0.3 m in the FE model. Nodal points from number 1 to number 16 on the lining and polymeric layer are shown in Figure 5. These points' stress will be analyzed in the results.  
To deal with contact, a tie constraint is adopted between the surrounding soil and NRTCPM coating, and is adopted between the NRTCPM coating and the lining in order to improve the calculation of the stability and efficiency.  
In the mesh center, the characteristic dimension of the elements should satisfy the condition that the spacing of the finite element nodes must be smaller than around one-eighth to one-sixth of the wavelength related to the maximum frequency component of the input motion [38].  
A detail of the mesh before excavation and after equipping the NRTCPM layer and lining is shown in Figures 6 and 7.