## 1 Introduction  
In seismic engineering problems, the study of tunnel dynamic characteristics has become increasingly important. The safety of tunnels under earthquakes has attracted extensive attention [1; 2; 3; 4; 5]. Some observation results show that when seismic wave propagates along the longitudinal direction, it will cause tension, compression and bending deformation of the tunnel [6]. As an underground structure with a large span in the longitudinal direction, tunnel deformation in the longitudinal direction is more remarkable than that in the transverse direction, which should be highly considered in seismic design. The common seismic analysis methods of underground tunnels include analytical solutions [7; 8], model experiments [9] and numerical methods [10; 11; 12; 13; 14]. Compared with the assumption limitation of the analytical method and the long period of experimental testing, numerical methods can efficiently and accurately analyze complex soil-structure dynamic interaction. For a 3D simplified soil-tunnel model, soil around the tunnel is denoted by using soil springs to consider the soil-tunnel interaction [15]. Based on the concurrent discretization of the entire computation domain with coarse and fine-scale finite element meshes, [PERSON] et al. [10] used a multi-scale method to simulate dynamic responses of a long tunnel. [PERSON] and [PERSON] [11] established a 3D soil-tunnel structure interaction model to simulate the longitudinal response of the tunnel when subjected to an asynchronous earthquake. Based on the time-domain finite element method with the viscous-spring artificial boundary condition, [PERSON] et al. [13] conducted a study of the seismic responses of the long lined tunnels under the obliquely incident earthquake. [PERSON] et al. [12] addressed the problem of underground tunnels when subjected to asynchronous seismic shaking along the axis by a comparison with the synchronous case that is usually considered in the tunnel design. Both sides of the tunnel model are truncated, resulting in some numerical errors in the seismic response of the tunnel. Therefore, the 3D finite element model needs to take a long distance in the longitudinal direction to ensure that the tunnel seismic response results are accurate. In recent years, some researchers have become increasingly interested in 2.5D modeling methods for tunnel-soil system dynamic interaction analysis, because in long lined tunnels, 2.5D finite element models are more efficient and accurate than 3D models. [PERSON] and [PERSON] [16] first used a special type of wave transmission element to simplify the 3D system into a finite element plane, where each node of the plane element has three degrees of freedom, while retaining the propagation effects of waves. [PERSON] and [PERSON] [17] introduced the 2.5D method, which applies Fourier transform of the longitudinal coordinate to reduce the problem's dimensionality and named it the 2.5D finite element/infinite element method. According to the different positions of the excitation source, the dynamic interaction analysis of the tunnel-soil system is mainly divided into two categories. One is the internal source problem, such as the vibration analysis of tunnel-soil system under moving train load [17; 18; 19; 20; 21; 22; 23; 24; 25]. [PERSON] et al. [19] coupled the finite element boundary element (FE-BE) method to analyze the vibration problem of the soil-tunnel system caused by moving trains load in the ground or tunnels. [PERSON] et al. [20] studied the dynamic interaction between tunnels and layered soil under train load by using the general 2.5D FE-BE method. [PERSON] et al. [22] used measurement data of dynamic response of tunnel to verify 2.5D coupled FE-BE model under the passage of a train. [PERSON] et al. [23] proposed a coupled 2.5D FEM-perfectly matched layer method to study the effect of soil stiffness on soil-tunnel interaction in the frequency domain. Recently, [PERSON] et al.