## 4 Finite Element Method (FEM) Simulation and Analysis of Impact Echo Test
The ABAQUS/Explicit method used a central difference rule to integrate the equations of motion explicitly through time, using the kinematic conditions at one increment to calculate the kinematic conditions at the next increment. The intact pile (reference pile) was modeled using an elastic modulus \(E_{1}=38\),400 MPa, density \(\rho\) = 2400 kg/m\({}^{3}\), and Poisson ratio \(\mu=0.20\), while pile sections with material defects were modeled by a lower elastic modulus property \(E_{2}=20\),000 MPa. The material properties of the bearing soil layer are \(E_{b}=5000\) MPa, density \(\rho_{\rm b}=2000\) kg/m\({}^{3}\), and Poisson ratio \(\mu_{\rm b}=0.30\). The impact was modeled by a suddenly applied maximum pressure of \(P=10\),000 Pa at a portion of the top quarter section area of the pile with a half-sine amplitude and 0.00008 s duration that was consistent with the maximum force  
Figure 8: Various cases of intact and defective piles simulated by FEM: (**a**) Intact and uniform pile; (**b**) Pile with weak concrete throughout the length; (**c**) Weak concrete at a section; (**d**) Neck defect; (**e**) Bulge imperfection; (**f**) Notch, crack, or break; (**g**) Void; (**h**–**j**) Multiple defects. generated with typical handheld hammers. Velocity responses were recorded at the top center of the pile at a sampling frequency of 50 kHz. The properties of the intact pile (reference pile) and bearing layer are listed in Table 2, while the properties of the defective piles in Figure 8 are summarized in Table 3.  
Figure 9 shows the original signals of the velocity record from the wave reflection responses taken at the top center of the pile for the different simulation cases. As could be seen in Figure 9, various types of wave dispersion existed due to the very complicated forms of defects present in the piles. The defects included not only a sudden change in cross-section area at fracture, crack, or void but also gradual variation in cross-section at necking or bulging and change in material properties, which occurred in parts of the concrete. Wavelet packet decomposition was applied to the velocity signals after nonlinear amplification with the one-dimensional wavelet packet analysis function 'wpdec' in Matlab and using the \"db2\" wavelet packet filter \(\Psi\) and \"Shannon entropy\" [29]. By applying wavelet packet transform (WPT), low-frequency and high-frequency components, as noted by the left and right branches, respectively, of the WPT decomposition tree shown in Figures 10a and 2, could be obtained from the original waveforms of the signals according to each defect previously considered. Considering the case in Figure 8e, whose original signal generated from ABAQUS simulation is shown in Figure 9e, if WPT was applied up to the third level of decomposition, the following low-frequency approximations shown in Figure 10b were obtained. As could be seen in Figure 10b, the noise was effectively removed, and the feature parameters could then be easily identified. Furthermore, after \begin{table}
\begin{tabular}{c c c} \hline \hline
**Property** & **Unit** & **Description or Quantity** \\ \hline \hline \multicolumn{3}{c}{Intact pile:} \\ \multicolumn{3}{c}{Pile type} \\ \multicolumn{3}{c}{Pile diameter, \(D\)} \\ \multicolumn{3}{c}{Pile length, \(L\)} \\ \multicolumn{3}{c}{Elastic modulus, \(E_{1}\)} \\ \multicolumn{3}{c}{GPa} \\ \multicolumn{3}{c}{Density, \(\rho\)} \\ \multicolumn{3}{c}{kg/m\({}^{3}\)} \\ \multicolumn{3}{c}{Poisson ratio, \(\mu\)} \\ \multicolumn{3}{c}{Dimensionless} \\ \hline \multicolumn{3}{c}{Bearing layer:} \\ \multicolumn{3}{c}{Elastic modulus, \(E_{\text{b}}\)} \\ \multicolumn{3}{c}{Density, \(\rho_{\text{b}}\)} \\ \multicolumn{3}{c}{Rg/m\({}^{3}\)} \\ \multicolumn{3}{c}{Poisson ratio, \(\mu_{\text{b}}\)} \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Properties of the intact pile (reference pile) and bearing layer.