## 3 Results and Discussions  
The simulation is repeated by keeping the crack at any of the predefined locations, as depicted in Figure 4, and some additional scenarios are simulated for comparison purposes. Figure 7 displays a comparison of system responses with different crack sizes at an offset of 50% (A1050S). The blue waveform is the system response with the original sized crack, whereas the red waveform is the response of the system when simulated with crack width doubled (i.e., width set as 4 mm). There is a clear decrease in the amplitude of the received response in case of the bigger crack scenario. Figure 8 represents a comparison of system responses for a single crack and a pair of cracks at an offset of 50%. The waveform represented in blue is the system response with a single crack at 50% offset location (A1050S), whereas the red waveform represents the system response when a pair of cracks is introduced around 50% offset (A1050M). The exact location of 1 crack is at 45% offset, and its dimensions are 10 mm in length and 1 mm in width. The second crack is at the 50% offset location with 15 mm length and 2 mm width. The comparison clearly shows a decrease in the magnitude of the received wave at sensing points for the case of a two-crack system as compared to the single crack setting. Figure 6: Graphical representation of support vector machine showing the maximum margin hyperplane. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \begin{tabular}{c} **Crack** \\ **(Axis 2)** \\ \end{tabular} & \multirow{2}{*}{**Amplitude 1**} & \
\begin{tabular}{c} **Frequency 1** \\ **KHz** \\ \end{tabular} & \multirow{2}{*}{**Phase 1**} & \multirow{2}{*}{**Amplitude 2**} &
\begin{tabular}{c} **Frequency 2** \\ **KHz** \\ \end{tabular} & \multirow{2}{*}{**Phase 2**} & \multirow{2}{*}{**RMSE**} \\ \hline
12\% & 37.39 & 295.8 & \(-\)17.3 & 36.42 & 300.8 & 2.785 & 1.47 \\
25\% & 4.47 & 262.00 & \(-\)1.92 & 1.41 & 193.70 & \(-\)1.11 & 1.52 \\
37\% & 5.02 & 273.90 & \(-\)7.60 & 3.06 & 321.30 & \(-\)5.45 & 1.37 \\
50\% & 4.29 & 303.40 & 7.49 & 1.54 & 221.50 & 13.11 & 1.17 \\
63\% & 4.21 & 314.00 & 2.70 & 1.31 & 146.70 & \(-\)8.49 & 1.19 \\
75\% & 2301.00 & 295.60 & \(-\)16.93 & 2301.00 & 295.60 & 5.04 & 1.53 \\
87\% & 5.47 & 293.20 & 7.05 & 2.91 & 357.80 & 7.27 & 1.80 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Optimized parameters and error terms for the numerical model for the cracks along axis 2. Figure 9: Comparison of the actual response and optimized mathematical model for the pipe with multiple damage locations indicated via percentage of offset between exciter and sensor locations along axis 1. Figure 10: Comparison of the actual response and optimized mathematical model for the pipe with damage in axis 2. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Crack** & **Amplitude 1** & \
\begin{tabular}{c} **Frequency 1** \\ **KHz** \\ \end{tabular} & **Phase 1** & **Amplitude 2** &
\begin{tabular}{c} **Frequency 2** \\ **KHz** \\ \end{tabular} & **Phase 2** & **RMSE** \\ \hline
12\% & 1.94 & 323.90 & \(-4.05\) & 2.33 & 275.80 & \(-8.29\) & 0.47 \\
25\% & 1.73 & 329.50 & \(-7.25\) & 3.41 & 284.60 & \(-11.90\) & 0.49 \\
37\% & 2.02 & 282.90 & 13.71 & 2.08 & 350.10 & 15.17 & 0.49 \\
50\% & 3.22 & 259.70 & \(-4.42\) & 3.20 & 335.10 & \(-11.60\) & 1.47 \\
63\% & 5.31 & 262.80 & \(-5.89\) & 1.90 & 207.30 & \(-10.00\) & 2.03 \\
75\% & 2.49 & 361.60 & \(-20.10\) & 7.42 & 279.50 & \(-9.80\) & 1.41 \\
87\% & 2.76 & 305.00 & 3.53 & 0.40 & 192.30 & 0.28 & 0.47 \\ \hline \end{tabular}
\end{table} 
Table 5: Optimized parameters and error terms for the numerical model for the cracks along axis 3. SVM was used to classify the pipe models in two possible categories, i.e., pipes with crack and pipes without a crack. The test dataset was applied to the designed SVM model without presenting the output data label corresponding to the particular response. The predicted output label of SVM was compared to the actual output label of each response and the prediction accuracy of the developed model resulted as 87%.