# Assessing Voided Reinforced Concrete by Numerical Modelling of Impact-Generated Rayleigh Waves
## 2 Simulation Model
### Model Boundary Conditions  
The absorbing layer with increasing damping (ALID) method is employed in this paper to mitigate boundary reflections in numerical simulations of wave propagation. The presence of boundary reflections is confirmed to decrease the quality of a wavefield through significant wave energy scattering, making detection of a wave front and the subsequent wave components difficult and inaccurate. ALID restores the outgoing waves as they are in contact with the model boundaries through gradual absorption of wave energy. This helps to prevent waves from being reflected, scattered and \"contaminating\" the first arrivals of wave energies, which all lead to complications when extracting components of R-waves from simulated data. Within the context of this study, it is considered necessary to apply the ALID for optimising the modelling conditions so that direct wave propagation behaviour could be observed and evaluated effectively.  
In terms of boundary condition setting, the numerical model was fixed at three edges: bottom face and the two side faces, leaving only the top face free to deform in the vertical direction. The ALID was applied to the three fixed edges of the concrete using 10 successive layers. The main parameter in ALID functions through the advancing mass damping coefficient \(\alpha\) as it moves between the absorbing layers. The stiffness damping coefficient \(\beta\) receives a value of zero during damping layers to prevent efficiency reduction. The use of non-zero stiffness damping in absorption definition techniques leads to better results, as it enhances wave absorption and minimises artificial reflections at the interface between the absorbing layers and the concrete. However, this approach requires extremely short time increments, substantially increasing computational expenses. The damping layers share all the same attributes related to materials and mesh element characteristics and dimensions as the concrete model. This uniformity between the concrete and absorbing  
Figure 5: Rayleigh damping coefficients, (**a**) concrete and (**b**) reinforced steel.  
layers ensures that the material properties, mesh characteristics, and dimensions remain consistent, minimising impedance discrepancies. As a result, unwanted wave reflections at their interface are weakened, strengthening reflection attenuation. The gradual increase in the damping coefficient, \(\alpha\), can be obtained by Equation (6), as proposed by [PERSON] et al. (2012) [24], which was used in this study to define the smooth transition of wave propagation behaviour from the concrete model to the damping layers.  
\[\alpha(x)=\alpha_{max}X(x)^{p} \tag{6}\]  
The value of \(X(x)\) varied from 0 to 1 (innermost layer to outermost layer). The value of \(P\) was set to be 3, as suggested by [PERSON] & [PERSON] (2018) [25]. In the calculation, the value of the maximum mass-damping coefficient \(\alpha_{max}\) was taken as 2.5 \(\times\) 10\({}^{4}\).  
On the other hand, the stiffness damping coefficients of the damping layers were all taken as zero. This was proposed by [PERSON] (2008) [21] to avoid drastic losses in computational efficiency. Because the stiffness damping coefficients are much larger than the average value for general structures, especially towards the outer layers, very fine time increments are required, leading to an increase in computational time, making the method less economical. Except for the damping coefficients, all other properties of the damping layers were defined in the same way as those defined for the concrete model, including the material properties, mesh element type, and sizes. As exemplified by Figure 6, by replacing the infinite element with an absorbing boundary with increasing damping layers, boundary reflections have been effectively reduced to such an extent that they cause no significant contamination to the first-arriving wave groups.