## 2 MVDB Implementation
The Rayleigh damping model in dynamic calculations is a classical approach for structural dynamic response analysis, which is easy to implement and has been implemented in most FEM software because the decoupled governing equations are controlled by only two constants (\(\alpha\), \(\beta\)). This study uses Rayleigh damping to describe the outgoing wave energy attenuation in the absorbing layer, and the Rayleigh damping model in the explicit direct integration method can be expressed as follows [49,50]:  
\[[D]=[D_{M}]+[D_{K}] \tag{5}\]  
where  
\[\Big{\{} \begin{array}{l}[D_{M}]=\alpha[M]\\ \left[D_{K}\right]=\beta[K]\end{array}  \tag{6}\]In the above equations, \([D_{M}]\) is the mass damping matrix, \([D_{K}]\) is the stiffness damping matrix, \([M]\) and \([K]\) are the mass and stiffness matrices of the whole model, respectively, and \(\alpha\) and \(\beta\) are the mass damping coefficient and stiffness damping coefficient, respectively. Although second-order Rayleigh damping is generally more concise in mathematical form, the physical meaning is not clear. Thus, reverse calculations are often adopted by the damping ratio (\(\xi\)). Therefore, Equation (5) can be expressed in the form of \(\xi\)[51; 52]. \[\xi_{Rayleigh}=\xi_{M}+\xi_{K}=\frac{\alpha}{2\omega}+\frac{\beta\omega}{2} \tag{7}\]  
where \(\xi_{M}\) is the mass damping ratio, \(\xi_{K}\) is the stiffness damping ratio, \(\xi_{Rayleigh}\) is the Rayleigh damping ratio, and \(\omega\) is the circular frequency corresponding to \(\xi_{M}\) and \(\xi_{K}\), and the variation curve of \(\xi\) with \(\omega\) is shown in Figure 2.  
As shown in Figure 2, the damping ratios corresponding to different frequency bands vary greatly, which can cause serious distortions in the relative responses of different vibration modes. Generally, the interval \((\omega_{m},\omega_{n})\) is chosen for the value of \(\omega\) in the seismic calculation of structures, as shown in the dashed frame. Since the Rayleigh damping ratio in this band is smaller than the actual damping ratio \((\xi_{n})\), the calculated damping in this band becomes smaller than the actual damping. Therefore, the response of the structure becomes greater than the actual response to ensure a safe design (Equation (8)). However, the damping should be maximized so that the outgoing wave is not reflected at the boundary in the wave field simulation. Therefore, the choice of \(\omega\) should be set in the frequency band of interest outside the \((\omega_{m},\omega_{n})\) interval. According to the variation curve of \(\xi\) with \(\omega\) (Figure 2), the value of \(\omega\) is linearly proportional to \(\xi_{K}\) and inversely proportional to \(\xi_{M}\). The circular frequency is in the \((\omega_{m},\omega_{n})\) interval, and \(\xi\) is less than or equal to the actual damping ratio. Therefore, there exists a minimum Rayleigh damping ratio \((\xi_{min})\) as follows:  
\[\xi_{min}=\frac{D}{D_{\rm C}}=\frac{1}{2}\bigg{(}\frac{\alpha}{\omega_{q}}+ \beta\omega_{q}\bigg{)},\ \omega_{m}\leq\omega_{q}\leq\omega_{n} \tag{8}\]  
where \(D\) is the damping coefficient, \(D_{\rm C}\) is the critical damping coefficient, and \(\omega_{q}\) is the circular frequency corresponding to \(\xi_{min}\). When \(\omega\) is outside this frequency band, its damping ratio is all greater than the actual damping ratio, so the damping is also all greater  
Figure 2: Damping ratio versus frequency curve. than the actual damping. In order to maximize the damping effect treatment, the limit is taken for \(\xi\) at this time as follows:  
\[\underset{\omega\to 0}{\text{lim}}(\xi_{M}+\xi_{K})=\underset{\omega\to 0}{\text{lim}} \left(\frac{\alpha}{2\omega}+\frac{\beta\omega}{2}\right) \tag{9}\]  
Since \(\omega\) is a number tending to 0, Equation (7) can be approximated as follows:  
\[\xi_{max}\approx\frac{\alpha}{2\omega},\;\omega\to 0 \tag{10}\]  
To ensure the accuracy and convergence of the analysis area, we only apply damping in the absorption layer.