## 1 Introduction  
Numerical seismic wave field simulation based on the elastic wave equation is important to study the kinematics and dynamics characteristics of seismic wave propagation in geologic bodies, which is widely used in seismic design [1], geologic survey [2], and non-destructive structure detection [3]. Currently, the numerical seismic wave field simulation methods include the finite element method (FEM) [4; 5], the finite difference method (FDM) [6; 7], and the boundary element method (BDM) [8; 9]. The FEM has become the most commonly used method for numerical seismic wave field simulation in geochemical engineering due to its advantages in discretizing the mesh and free boundary treatment of irregular geological structures [10; 11], and a variety of commercial software programs have been developed based on the FEM, such as ABAQUS ([https://www.3 ds.com/products/simulia/abaqus](https://www.3 ds.com/products/simulia/abaqus)), COMSOL ([[https://www.comsol.com/](https://www.comsol.com/)]([https://www.comsol.com/](https://www.comsol.com/))), and ANSYS ([[https://www.ansys.com/](https://www.ansys.com/)]([https://www.ansys.com/](https://www.ansys.com/))) (all accessed on 1 March 2024). During simulation, artificially truncating the numerical model is often necessary due to the limitation of computational and storage capacities, and the artificially truncated boundaries must be effectively dealt with so that the seismic wave does not reflect at the boundaries, which is key to numerical seismic wave field simulation. Two main ideas have been proposed to deal with artificially truncated boundaries in numerical wave field simulations using FEM software. One is to deal with the seismic wave field reflection at the boundary through free-field stress equilibrium at the nodes of the artificial boundary, such as the viscous boundary (VB) [12], the viscous-spring boundary (VSB) [13], and the equivalent viscous-spring boundary (EVSB) [14]. The other is introducing attenuation terms in the wave equations to absorb the stress and displacement fields at the boundary, such as the infinite element boundary (IEM) [15], the perfectly matched layer (PML) [16; 17; 18; 19; 20; 21], and the incremental damping method (IDM) [22; 23; 24; 25]. In geological surveys and structural damage detection, seismic waves are usually input from the free boundary at the ground surface, and the existing boundary processing methods fail to deal effectively with the boundary reflection problem of the surface wave and body wave. The VSB achieves the purpose of absorbing the reflected wave field at the boundary by setting up damping units to approximately equalize the stresses on the boundary. Compared with the VB, the VSB consisting of damping and spring units exhibits superior stability and accuracy in effectively simulating elastic recovery within an infinite domain [26]. The accuracy of its solution depends on the form of the approximate expression for the incident wave. The approximate expression for the incident wave of the VSB is mainly established by the stress-displacement constitutive relationship of the plane wave and is, therefore, suitable for endogenous seismic wave situations [27]. As shown in Figure 1a, the seismic sources of these structures are set deep underground, and the seismic waves in the plane form are transmitted from the underground to the structure through absorbing boundaries, making this method widely used in the analysis of dynamic soil-structure interaction [28]. For example, it is used to study the kinetic energy effects generated by natural earthquakes on structures such as dams, tunnels, and coasts (see e.g., [26; 29; 30; 31]).