# Assessing Voided Reinforced Concrete by Numerical Modelling of Impact-Generated Rayleigh Waves
## 2 Simulation Model
### Element Type, Size and Wave Velocity  
Both the concrete and steel were modelled as a 2D planar deformable shell structure. The selected element type was a 4-node, bi-linear plain strain element with reduced integration. There is a lesser number of integration points in an element with reduced integration. Thus, the element is more computationally economical to apply. However, a common problem with reducing integrated elements is the susceptibility to zero-energy mode, which produces unrealistic deformation (i.e., hourglassing) ([PERSON], 2008) [15]. The 'hourglass' effect can be eliminated by either using very fine meshes or ABAQUS's built-in hourglass control. In this study, ABAQUS's built-in control was used. The size of the mesh elements is directly linked to the accuracy of the model ([PERSON], 1990) [16]. The size of the elements \(L_{e}\) is limited by the shortest wavelength \(\lambda_{min}\) of the waves applied to the model. The model is susceptible to spurious oscillations if the meshes are not fine enough ([PERSON], 1996) [17]. Also, studies have shown that the velocity obtained from the numerical model will differ significantly from the actual velocity due to period elongation and amplitude decay ([PERSON] & [PERSON], 1972) [18]. Therefore, very fine meshes are required to accommodate rapid spatial variation with high-frequency waves. The number of elements per wavelength N (i.e., mesh density) is suggested to be 20 to achieve good spatial resolution (Swanson Analysis Systems, 1992) [19], as shown in Equation (1). \[L_{e}=\frac{\lambda_{min}}{\text{N}}=\frac{\lambda_{min}}{20} \tag{1}\]  
After several trials, the element size of the model was limited to 0.002 m to achieve optimised computing performance. The highest frequency \(f_{max}\) of the impact source adopted in this study was 50 kHz. The velocity of the R-wave, \(V_{R}\), and P-wave components, VP, was taken as 2400 m/s and 4000 m/s. The values chosen fall within the range of typical  
Figure 1: Model configuration schematic. Figure 2: Distributed voids located below the sensor array at (**a**) left edge, (**b**) middle and (**c**) right edge of the latter. velocities, i.e., between 1800 and 2500 m/s, and 3500 and 4500 m/s for R-wave and P-wave, respectively ([PERSON], 2007) [20]. Table 1 presents the calculated wavelengths (\(\lambda_{R}\) and \(\lambda_{P}\)) and mesh densities (\(N_{R}\) and \(N_{P}\)) for R-wave and P-wave, respectively. The velocity error \(V_{Err}\) is related to the mesh density \(N\) by Equation (2) ([PERSON], 2008) [21], which is a function of the element size \(L_{e}\), wave frequency f and wave velocity \(v\). Figure 3 shows that adopting a wave frequency of 50 kHz, the velocity errors for the R-wave and P-wave components could reach a maximum of 0.313%, which is considered insignificant. \[V_{Err}=\frac{180}{N^{2}}=\frac{180}{(\frac{\lambda}{L_{e}})^{2}}=\frac{180L_ {e}{}^{2}\text{f}^{2}}{\text{v}^{2}} \tag{2}\]  
The element's aspect ratio is also critical. Quadrilateral elements were chosen in the modelling. The quadrilateral elements have better accuracies with the same node point arrangement as a triangular mesh ([PERSON], 1990) [16]. The irregular meshing was avoided whenever possible in the modelling since any element distortion could possibly deteriorate the accuracy ([PERSON], 1990) [16]. In this study, areas that required meshing were partitioned into regions with regular shapes to produce the best consistency in element sizing and distribution. On the other hand, voids were modelled as \"acoustic media\" using a four-node linear 2D acoustic quadrilateral element with reduced integration. Distributed voids took the shape of circles and were randomly packed within the block, which created much distortion in the mesh. The elements around the voids were extremely irregular, which would drastically deteriorate the accuracy of the model if left unmodified. This study first tried partitioning the faces around the voids and applied a medial algorithm with the mesh control.