## 2 MVDB Implementation  
[PERSON] [22; 23] proposed the CALM based on Rayleigh damping and second-order Caughey damping. The CALM relates the damping to the generalized Maxwell's equations by determining the quality factor (Q) through a linear viscoelastic rheological model [43] and then calculating the minimum damping frequency (\(\omega_{min}\)) and the mass damping coefficient (\(\alpha\)) and stiffness damping coefficient (\(\beta\)) based on the relationship (see e.g., [23; 44]) between Q\({}^{-1}\) and the damping ratio (\(\zeta\)). With high solution stability and absorption effect, the CALM can serve as a simple alternative to the PML [45; 46] and can be implemented in FEM software. [PERSON] [24; 25] examined the Lamb wave dispersion characteristics in thin plate structures and introduced the NRB suitable for FEM transient dynamic analysis. The NRB determines the maximum mass damping coefficient (\(\alpha_{max}\)) in the absorbing layer through numerical experience and then calculates the damping coefficient (\(\alpha\)) in each absorbing layer according to the damping control equation (see e.g., [24]). The NRB can effectively attenuate the Lamb wave reflection at the boundary and is easily implemented in ABAQUS, like the CALM. As its implementation process is more convenient than that of the CALM, the numerical simulation cost is greatly shortened (i.e., the storage capacity of computers). The CALM and NRB effectively suppress outgoing wave reflection at the boundary in practical engineering, but they are sometimes not ideal. For example, the attenuation mechanism of the CALM is relatively complicated, and the computational accuracy depends on the quality factor (Q). The \(\alpha_{max}\) of the NRB depends on some numerical experience and may fail to absorb low-frequency pulses. This section introduces the basic ideas and implementation routes of the MVDB from the basic dynamics equations and demonstrates the computational results. In the time integration scheme, the central difference method is used in this study to solve the dynamics equations. Under the basic assumptions of elasticity theory (homogeneity and linear elasticity), the force vectors at any node \(b\) in space at time \(t\) can be derived from the basic dynamics equations as follows [47,48]:  
\[[M]\Big{\{}\ddot{X}(t)\Big{\}}+[D]\Big{\{}\dot{X}(t)\Big{\}}+[K]\{X(t)\}=\{F_ {b}(t)\} \tag{1}\]  
where \([M]\) is the mass matrix, \([K]\) is the stiffness matrix, \([D]\) is the damping matrix, and \(\{X(t)\}\), \(\Big{\{}\dot{X}(t)\Big{\}}\), and \(\Big{\{}\ddot{X}(t)\Big{\}}\) are the displacement components and the first-order and second-order derivatives of the displacement components, respectively. \([M]\Big{\{}\ddot{X}(t)\Big{\}}\), \([D]\Big{\{}\dot{X}(t)\Big{\}}\), and \([K]\{X(t)\}\) are the nodal loads due to acceleration-induced mass, nodal loads due to velocity-induced damping, and nodal loads due to displacement-induced stiffness, respectively. In the center difference method, the velocity and acceleration can be approximated as follows using equal time steps \(\Delta t_{i}=\Delta t\):  
\[\Big{\{}\dot{X}(t_{i})\Big{\}}=\frac{\{X(t_{i+1})\}-\{X(t_{i-1})\}}{2\Delta t} \tag{2}\]  
\[\Big{\{}\ddot{X}(t_{i})\Big{\}}=\frac{\{X(t_{i+1})\}-2\{X(t)\}+\{X(t_{i-1})\} }{\left(\Delta t\right)^{2}} \tag{3}\]  
Therefore, the equations of motion for any node on the absorbing layer at the moment \(t_{i}\) can be obtained by bringing Equations (2) and (3) into (1), taking into account the damping as follows:  
\[\{F_{b}(t_{i})\}=[M]\frac{\{X(t_{i+1})\}-2\{X(t)\}+\{X(t_{i-1})\}}{\left( \Delta t\right)^{2}}+[D]\frac{\{X(t_{i+1})\}-\{X(t_{i-1})\}}{2\Delta t}+[K]\{X (t_{i})\} \tag{4}\]  
The load on the \(b\) node can be minimized when the damping force at the \(b\) node in the end absorbing layer is approximately balanced with the acceleration and the residual nodal load generated by the stiffness.