## 1 Introduction
However, in transient analysis, as shown in Figure 1b, for the spherical wave incident from the free boundary, due to the lack of reasonable standardization for the spring component parameters in the approximate expression of the spherical wave [32], the absorption effect of the VSB is unsatisfactory in the numerical simulation of geological exploration and structural damage detection in semi-infinite space. Figure 1: Schematic of the seismic wave field. (**a**) Unbounded input waves. (**b**) Free boundary input waves. With an attenuation term introduced into the wave equation, the infinite element (IEM) uses the shape functions to describe the decay of the displacement amplitude and the propagation of different waves, and the accuracy of the IEM depends on the wave number and decay factor selected [33]. They can be accurately determined only for a single type of wave but not for multiple types of waves. As such, spurious reflections will occur for the other types of waves passing through the FEM-IEM interface [34]. The PML is one of the most popular boundary schemes in recent years, which includes Classical PML [18; 19], M-PML [20; 21], C-PML [35; 36], and joint boundary schemes framed by the PML, such as IEM-PML [37] and VB-PML [38]. The PML defines the attenuation process of the outgoing wave from the near-field boundary to infinity through stretching and attenuation functions. The absorption effect of the PML on outgoing waves depends highly on the value of the stretching function. Although there are many available stretching functions [39; 40; 41], there is a lack of research on the selection of the best parameters for actual use [37]. Furthermore, the PML cannot be directly applied to commercial finite element software and requires programming to implement [42]. Even though some scholars have applied it to ABAQUS [5], the process is not convenient as it needs to be realized through ABAQUS-UEL combined with programming in computer languages, such as Fortran and Python. The IDM scheme currently includes two main forms: one is to derive the mass and stiffness damping coefficients from the frequency and attenuation factor of the seismic source, and the classical method is the Caughey absorbing layer method (CALM) proposed by [PERSON] et al. [22; 23]. However, it is not convenient to cycle through two damping coefficients for a continuous absorber layer. The other is to calculate the maximum damping coefficient and determine the damping coefficients for each layer according to the designed attenuation rules, e.g., the non-reflective boundary (NRB) proposed by [PERSON] et al. [24; 25]. Compared to the CALM, the NRB only requires determining the mass damping coefficient, greatly improving efficiency. However, the NRB requires some numerical experience to determine the maximum damping coefficient and brings about the probability of destabilization for low-frequency pulses. Grasping the propagation characteristics of seismic waves in a geological body is a prerequisite for geological exploration and structural damage detection. In order to effectively deal with the reflection of seismic waves input from a free boundary at an artificially truncated boundary and ensure the accuracy and efficiency of numerical seismic wave field simulation, this study proposes, based on Rayleigh damping and using the incremental damping method, a multi-directional viscous damping absorbing boundary (MVDB) for the finite element time domain analysis. The MVDB sets up a certain absorption area in the periphery of the calculation area with consistent physical properties and does not require solving the control equations. The boundary seismic wave field is absorbed only by changing the damping value of the absorption area. With a simple implementation process, stable calculation, and effective absorption of the boundary seismic wave field, the MVDB can be well applied to the finite element software numerical seismic wave field simulation in geological exploration and structural damage detection.