## 5 The Application of ETRM for Damage Imaging of Composite Laminates  
The specimen, PZT and the experiment setup are the same as in Section 4. Eight circular PZTs, with a distance of 140 mm between elements, are bounded on the specimen, and the layout is shown in Figure 11. The dimension of the inspected area is 280 mm \(\times\) 280 mm. An added mass, with dimension of 30 mm \(\times\) 20 mm \(\times\) 5 mm and centering at \((46,-46)\) mm, is placed on the specimen to simulate delamination damage. A coordinate system is established with the origin set at the center of the sensor array. The coordinates of the eight PZTs are shown in Table 2. \begin{table}
\begin{tabular}{c c} \hline
**Name** & **Coordinates** \\ \hline Damage center & \((46,-46)\) \\ P1 & \((-140,140)\) \\ P2 & \((0,-140)\) \\ P3 & \((140,-140)\) \\ P4 & \((140,0)\) \\ P5 & \((140,140)\) \\ P6 & \((0,140)\) \\ P7 & \((-140,140)\) \\ P8 & \((-140,0)\) \\ \hline \end{tabular}
\end{table} 
Table 2: Coordinates of the damage center and piezoelectrics (PZTs). Figure 11: Photo of the composite laminates and sensing path of the sensor array. Figure 10: The DI value changes with _weight_. In the active sensor array, the eight PZTs take turns as actuator while the rest of them are listening. As shown in Figure 11, there are 16 paths available for damage imaging. A typical chirp signal with frequency sweep from 10 kHz to 600 kHz over a 1-ms window is excited in turn. The mode extraction method proposed in Section 4 is used to extract the direct wave components of A0 mode. Then the ETRM is used to calculate the narrow-band reconstructed signal corresponding to a five-cycle 40-kHz tone burst excitation. For instance, the comparison between the reconstructed signals and the original input signal of Path 1 and Path 2 is shown in Figure 12a. With damage on the Path 1, the associated reconstructed signal significantly deviate from the original input signal. While, they are almost the same for the healthy Path 2. The DI values of all the 16 paths are shown in Figure 12b. It can be seen that the DI values of Path 1, Path 5 and Path 10 are significantly higher than those of other paths, indicating the presence of damage on these paths. Subsequently, a probabilistic imaging algorithm is introduced to estimate the location of damage [37; 38]. In this algorithm, DI value is used to represent the severity of damage along a sensing path. The inspection area that enclosed in the sensor array is meshed into uniform grids. The probability of damage present at a certain node \(N_{i}\) (\(i\)=1,..., \(n\times n\)) in the inspection area, is defined as,  
\[P(i)=\sum_{j=1}^{m}DI_{j}f\big{(}z_{ij}\big{)}, \tag{16}\]  
where, \(DI_{j}\) is the \(DI\) value of the \(j\)-th sensing path, and \(f(z_{ij})\) is the Gaussian distribution function defined as follows [39; 40],  
\[f\big{(}z_{ij}\big{)}=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{z_{ij}^{2}}{ 2\sigma^{2}}\right), \tag{17}\]  
where, \(\sigma\) is the standard variance, \(z_{ij}\) indicates the distance from node \(N_{i}\) to the \(j\)-th sensing path, it is defined as,  
\[z_{ij}=\frac{\big{|}\big{(}y_{jA}-y_{jS}\big{)}x_{Ni}+\big{(}x_{jS}-x_{jA}\big{)} y_{Ni}+x_{jA}y_{jS}-x_{jS}y_{jA}\big{|}}{\sqrt{\big{(}x_{jS}-x_{jA}\big{)}^{2}+ \big{(}y_{jA}-y_{jS}\big{)}^{2}}}, \tag{18}\]  
where, \((x_{jA}\), \(y_{jA})\) and \((x_{jS}\), \(y_{jS})\) are the coordinates of actuator and sensor of the \(j\)-th sensing path, respectively, \((x_{Ni}\), \(y_{Ni})\) is the coordinate of node \(N_{i}\). Figure 13a-c show the imaging results (i.e., \(\sigma\) = 0.01) as the _weight_ takes 0.1, 0.2 and 0.3, respectively. Each individual image is normalized by its maximum values. The actual center of the simulated damage is indicated by a red 'o', and the estimated center is denoted by a blue '+'. As Figure 13 shows,  
Figure 12: (**a**) The comparison between the reconstructed signals and the original input signal of Path 1 and Path 2 and (**b**) the DI values of the 16 paths.