# Improved Acoustic Emission Tomography Algorithm Based on Lasso Regression
## 4 Numerical Simulation
### Different Types of Damage  
Numerical models were created to demonstrate the efficacy and confirm the functionality of the proposed damage detection approach in the LASSO algorithm. Usually, the concrete structure loses structural integrity due to earthquake, fire, impact, or malicious damage and structural overloads (such as static overloading), and defects appear. As a method of inspection, the types of concrete failure suitable for this non-destructive testing technique by AE tomography have hole destruction, concentrated holes, and spalling or edge peeling [22].  
The following numerical simulation was based on actual concrete damage sorts, and three different damage types were studied to simulate the situation in the fundamental structure. The velocity of elastic wave propagation in concrete was assumed as 4000 m/s, and the standard slowness was determined from the velocity. In the simulation, the blue part in each model is the healthy area with a velocity of 4000 m/s, and the red part is the damaged area with a velocity of 3000 m/s; the source locations that were randomly generated in the model are expressed as the white spheres. Case 1 was a simulation of the concrete slab that appeared with damaged holes, shown in Figure 8a. The simulation model in Figure 8b represented the size of the model and the location of the receivers. The low-velocity cell denoted the damaged area. The model information is displayed in Table 2. The damaged area of case 2 was a concentrated hole, shown in Figure 9. The damage was located at the center of the model, and it consisted of four cells. The precise information is described in Table 3. Case 3 was a simulation of the damage at the edge of the concrete frame, and the simulation model is shown in Figure 10; the damage location was in the upper right corner with a lower velocity, and it consisted of six cells for analysis. The exact information is shown in Table 4. Every simulation model owned a uniform initial elastic wave velocity distribution of 4000 m/s. The calculation steps remain the same in Section 4.1.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Damage Percent (\%)** & **LASSO** & **SIRT** \\ \hline
10 & 0.32010 & 0.27366 \\
20 & 0.38114 & 0.18681 \\
30 & 0.52836 & 0.13941 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Normalized errors of the L2-norm.  
Figure 8: Case 1 of the damage target: (**a**) real damage; (**b**) original distribution.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Case** & **Model Size** & **Damage Area Size** & \begin{tabular}{c} **Initial** \\ **Velocity** \\ \end{tabular} & **Sensor No.** & **Mesh No.** & **Events No.** & \
\begin{tabular}{c} **Relay Points for** \\ **Each Cell No.** \\ \end{tabular} &
\begin{tabular}{c} **Measurement** \\ **Paths No.** \\ \end{tabular} \\ \hline
1 & 1 m \(\times\) 0.6 m & 0.05 m \(\times\) 0.05 m & 4000 m/s & 16 & 240 & 100 & 12 & 1600 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Model information for case 1.  
Figure 10: Original distribution for the damage target in case 3.  
Figure 9: Original distribution for the damage target in case 2.  
The results of three different simulation models with different damage types, different damage ranges, and different numbers of elastic wave events are illustrated in Figures 11-13. According to the results, it is demonstrated that the defected area was correctly identified. In comparison with the reconstructed velocity distribution of the SIRT algorithm, the damaged classification of hole damage, concentrated hole, and edge damage, the LASSO algorithm could accurately identify the lower velocity region in the model. Notably, the LASSO algorithm described accurate identification in the high-velocity region of the healthy region in all of the numerical simulation models and few shadows exist in identified velocity distribution. The LASSO algorithm is suitable for AE tomography techniques to identify typical concrete damage types with relatively accurate reconstructed velocity distribution results.