## 1 Introduction
[25] proposed the 2.5D FEM-singular boundary method (FEM-SBM) to deal with soil-tunnel interaction problems under the moving train load. The other category is the study under external source loads such as seismic motions [26; 27; 28; 29]. [PERSON] et al. [26] used the 2.5D finite element/infinite element method to analyze the seismic response of longitudinal invariant tunnels. [PERSON] et al. [27] used a 2.5D finite element indirect boundary element (FE-IBE) model to analyze the difference in seismic response of tunnels embedded in single-phase and two-phase media. Meanwhile, by using 2.5D FE-BE method, the dynamic response of segmented tunnels [28] and long lined tunnel in a water-saturated poroviscoelastic half-space [29] under earthquake were analyzed. In the domain-based method, it is necessary to introduce artificial boundary conditions to consider the radiation damping effect of infinite domain [30; 31; 32; 33; 34; 35; 36]; this also applies in the 2.5D finite element method. In this case, the artificial boundary needs to be established by 2.5D forms. A 2.5D form of infinite element boundary is first developed by [PERSON] and [PERSON] [17], in which the choice of wave number is different from that of the previous infinite element method. The 2.5D boundary element condition was deduced by [PERSON] et al. [19], which was used to simulate the ground vibration caused by train moving loads. However, for some complex infinite domain problems, the fundamental solution is complicated, which limits the application of the boundary element method to a certain extent. [PERSON] et al. [27] established 2.5D indirect boundary element method, which is easier to simulate the layered site than boundary element method in solving tunnel seismic response. In addition to the boundary element method, other artificial boundaries require finite element spatial domain and mesh discretization. At the beginning of this century, a consistent absorbing boundary for horizontally layered soil was derived by [PERSON] [37]. The thin layer method was popularized to obtain the zigzag boundary of multi-layer soil [38; 39]. The zigzag boundary can transform the boundary into a nonvertical line. Since the zigzag boundary can adapt to complex geometric shapes, it is easy to attach to the structural surface. However, the zigzag boundary is limited to a rigid base. At the same time, in order to further improve the computational efficiency, [PERSON] et al. [40]established a substructure technique to simulate the wave scattering problem. Combined with the site response analysis method and the seismic input method, [PERSON] et al. [41] has established a substructure analysis method for seismic response of the soil-structure system, which is suitable for multi-medium coupling conditions. [PERSON] and [PERSON] [42] derived an accurate dynamic stiffness matrix of a homogeneous half space, which constitutes an accurate boundary condition of an infinite underlying half space. However, since this matrix is not a polynomial of wave numbers, it cannot be coupled with the zigzag boundary. Later, [PERSON] [43] deduced the stiffness matrix of the homogeneous half space, which can be assembled in the overall stiffness matrix to deal with the wave problem in layered half space. Compared with previous numerical analyses of 3D tunnel models, the 2.5D method has a greater advantage in terms of saving computational costs. In addition, the novelty of this article mainly includes the following two aspects. (1) The paper first derives the 2.5D zigzag boundary condition for 2.5D finite element method. Compared with other boundary conditions, the formula is simpler and can be attached to the structure surface, effectively reducing the calculation domain. (2) The paper proposes a 2.5D substructure method by combining the 2.5D finite element method, 2.5D zigzag boundary condition and seismic motion input method, where the 2.5D calculation model only includes structure domain, resulting in fast and efficient computation.