## 6 Conclusions  
This study introduces a new boundary-reflected wave processing scheme (MVDB) for structural damage detection and geological exploration. Compared with the traditional absorbing boundary, the MVDB does not need to define the wave equation of the absorbing region or derive the expression of the unilateral outgoing wave, and a better absorbing effect can be achieved only by setting up a certain thickness of the damping incremental absorbing layer at the periphery of the calculation region. Compared with the CALM, the MVDB can set boundary damping without increasing the grid cells of the absorbing region and only needs to solve the mass damping coefficients. Therefore, it is more advantageous for treating reflected waves on artificially truncated boundaries. With the MVDB, the geometrically centered explosion source and the input spherical wave on the free boundary are simulated, respectively, both of which verify the effectiveness of the proposed method. The main conclusions are as follows:  
1. The MVDB implementation process is simple, and the physical meaning is clear. Independent of the space-time discrete step, its calculation process is stable. According to the numerical results of a two-dimensional uniform soil, the MVDB effectively attenuated the wave energy of the body wave and R wave in the endogenous and exogenous seismic wave field under the vertical incidence of the P wave, showing high accuracy and stability. The MVDB is also applicable to the oblique incidence problem on the free boundary. 2. The thickness of the absorbing region \(L\) determines the total damping value of the absorbing region and the smoothness of the damping growth, making \(L\) the key factor affecting the wave energy attenuation, and the energy attenuates to about \(1/4\), \(1/140\), and \(1/230\) of the original when \(L\) is \(\lambda_{min}\), \(2\lambda_{min}\), and \(3\lambda_{min}\). Based on the numerical results, the \(2\)-\(3\lambda_{min}\) thickness of the absorption region can meet the general engineering requirements for accuracy. 3. When the thickness of the absorption region \(L\) is constant, and the thickness of a single absorption layer is set as a discrete grid size \(\Delta l\), the attenuation trend of the boundary wave field energy is relatively stable without interlayer reflection, thus achieving a better absorption effect. 4. The damping amplitude control coefficient \(k\) in the MVDB is key in determining the initial damping value and smoothing the damping growth. When the thickness of the absorbing region is thinner, an excessively large \(k\) triggers interlayer reflections due to the uneven damping growth in the absorbing layer. A better absorption effect is ensured for incident waves of different frequencies when \(k\) is between \(1.02\) and \(1.12\). As a new boundary scheme, the MVDB provides a promising way to study seismic wave fields in semi-infinite space. In future work, specific research tasks will be carried out based on the MVDB, such as advanced prediction of tunneling and non-destructive testing of anchor rods. Additionally, the efficiency of the MVDB in 3D mode also needs to be evaluated. Conceptualization, J.Z. and Y.Y. ; writing--original draft, J.Z., Y.M. and J.L. ; writing, J.Z. and J.L. ; data curation, J.Z., H.X. and J.L. ; methodology, J.Z. and H.X. ; numerical simulation, J.Z. and R.Z. ; project administration and supervision Y.Y. and H.X. ; resources, H.X. and R.Z. All authors have read and agreed to the published version of the manuscript. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline _f (Hz)_ & **0–100** & **100–500** & **500–2 k** & **2–5 k** & **5–10 k** & **10–30 k** & **30–50 k** & **50 k–0.1 M** & **0.1–0.5 M** & **0.5–1 M** \\ \hline \(k\) (\(L=2\lambda\)) & 1.10 & 1.09 & 1.05 & 1.046 & 1.043 & 1.04 & 1.035 & 1.033 & 1.03 & 1.025 \\ \(k\) (\(L=3\lambda\)) & 1.12 & 1.10 & 1.07 & 1.05 & 1.05 & 1.045 & 1.04 & 1.04 & 1.035 & 1.035 \\ \hline \hline \end{tabular}
\end{table}