# An Omnidirectional Near-Field Comprehensive Damage Detection Method for Composite Structures
## 2 Ultrasonic Phased-Array Damage Imaging Method
### Near-Field Sampling Ultrasonic Phased-Array Damage Imaging Method  
The near-field sampling principle of the proposed ultrasonic phased-array imaging method is as follows: every position in the near-field region is considered as a virtual point source, and the pixel value of every point source is considered as an emission transducer array synthesis amplitude of the received signal component; the pixel values of all points are displayed in image form, and then the structural damage imaging process is conducted, where the highest-energy pixels are judged to be a damage position \(P(r,\theta)\). The principle of sampling near-field phased-array ultrasound imaging is presented in Figure 2.  
Then, the sampling phased-array data acquisition method is adopted, where the data collection process and sensing time determination of the transducer array elements are performed without distinction between the angle acquisition signal and the damage scattering signal, which is determined using the time delay given by Equation (3). The scanning cycle of this method is short, which can improve the efficiency of data acquisition and processing analysis, so a real-time data assessment is possible. Where the center of a linear array is at the coordinate origin, and M piezoelectric elements represent the transducer array with the array number M; the \(i\)th piezoelectric element excitation signal received at the \(j\)th piezoelectric element is labeled as \(A_{ij}(t)\), \((i,j=0,\,1,\,2,\,\ldots\,,\,M-1)\). When all piezoelectric elements are excited one by one, and when one element is excited, the other piezoelectric elements act as receivers, so that a complete data set \(A_{ij}(t)\), \((i,j=0,\,1,\,2,\,\ldots\,,\,M-1)\) can be obtained. Figure 1: The near-field wavefront diagram of a one-dimensional array. Any point source in the monitoring area can be synthesized by using the transmitting and receiving signals from various piezoelectric elements in the transducer array, which provides a theoretical basis for the generation of damage imaging monitoring. As shown in Figure 2, for the transmitting piezoelectric element \(i\) and the receiving piezoelectric element \(j\) at the respective distance from the point \(P\)\(r_{i}\) and \(r_{j}\), the time \(t_{pij}\) the signal needs to reach \(P\) with the ultrasonic wave velocity \(c\) can be expressed as \((r_{i}+r_{j})\), then the focused energy component \(A_{pij}(t_{pij})\), \((i,j=0,1,2,\ldots,M-1)\) is delayed according to the energy superposition principle of a phased-array, which is given by:  
\[\Delta t_{Pij}=\frac{r_{i}+r_{j}-2r}{c}=\frac{\sqrt{r^{2}+{x_{i}}^{2}-2 rx_{i} \cos\theta}+\sqrt{r^{2}+{x_{j}}^{2}-2 rx_{j}\cos\theta}-2r}{c} \tag{3}\]  
\[x_{i}=(i-\frac{M-1}{2})d \tag{4}\]  
\[x_{j}=(j-\frac{M-1}{2})d \tag{5}\]  
where \(x_{i}\) and \(x_{j}\) are the distances between the \(i\)th and \(j\)th piezoelectric elements from the coordinate origin, respectively. According to Equation (3), the time delay in the near-field can be used to determine the polar angle \(\theta\) (in a one-dimension linear array \(\theta\in[0^{\circ},180^{\circ}]\)) and the pole size \(r\), where the step of the polar angle is \(1^{\circ}\), and the step of the polar size is 1 mm. The delayed near-field focusing energy component \(A_{pij}\) (\(\Delta t_{pij}\)), \((i,j=0,1,2,\ldots,M-1)\) is then superimposed, and finally, the pixel size at point \(P\) in the direction of angle \(\theta\) is given by:  
\[S_{P}=\alpha\sum_{i,j=0}^{M-1}A_{ij}(\Delta t_{pij}) \tag{6}\]  
where \(\alpha\) is the signal conversion coefficient. In this work, \(\alpha\) is taken to be 1, regardless of the influence of a distance on the waveform. By analogy, in a detection area from \(0^{\circ}\) to \(180^{\circ}\) direction, the virtual point source with the biggest amplitude in the angle \(\theta\) direction given by Equation (7) represents a damage point \(P\).