# Phase Reversal Method for Damage Imaging in Composite Laminates Based on Data Fusion
## 2 Materials and Methods  
### Phase-Reversal Method  
Due to the dispersion and multimodal characteristics of guided waves and anisotropy of composite materials, the propagation of Lamb waves in composite materials is very complex, which restricts the application of damage-detection methods based on Lamb waves in composite materials and limits their accuracy of damage imaging. To overcome these limitations, this paper proposes a phase-reversal imaging method. Compared to the existing damage-detection methods, the proposed method can realize the damage detection of composite materials without prior information on material parameters, dispersion relationship, and influence of anisotropic characteristics of composite materials. The schematic diagram of the phase-reversal method process is presented in Figure 1. In the process of damage imaging based on an ultrasonic guided wave, the guided wave propagates through the structure and interacts with the damage, where the damage can be regarded as a secondary wave source. Therefore, damage localization can be realized by analyzing the scattering wave from the defect. As shown in Figure 1, in phase-reversal imaging theory, sensor arrays with narrow spacing are used for detecting the scattering waves. For each pair of the sensors, the wave velocity of the damage-scattering wave propagating from the damage source to each receiving array element is considered the same within the margin of error, so the anisotropy of composite materials does not affect the phase-reversal method. By using the Fourier transform, the phase spectrum of the damage scattering signal at a pair of sensors can be obtained as follows:  
\[F_{m}(\omega)=\int_{-\infty}^{+\infty}u(x_{m},t)e^{-i\omega t}dt\phi_{m}^{ \omega}=arctan\left[\frac{Im(F_{m}(\omega))}{Re(F_{m}(\omega))}\right](m=1,2) \tag{1}\]where \(u(x_{m},t)\) are the wave signals detected by two sensors; \(x_{m}\) are the positions of the sensors; and \(Im(\bullet)\) and \(Re(\bullet)\) denote the imaginary and real parts of a complex value after the Fourier transform, respectively. After the Fourier transform, the harmonic component corresponding to the damaged scattered wave is given by the following:  
\[u^{\omega}(x_{m},t)=A_{m}^{\omega}e^{i(k_{r}^{\omega}\cdot\vec{x_{m}}^{\omega} -\omega t)}(m=1,2) \tag{2}\]  
where \(A_{m}^{\omega}\) is the amplitude corresponding to the angular frequency, \(\omega\); \(\vec{x_{m}}^{\rightarrow}\) represents the direction vector from the damage point to sensors; and \(k_{r}^{\omega}\) is the wave number of \(\omega\) along the direction of the scattering wave propagation path. The relationship between the phase difference corresponding to each frequency of the damage scattering signal and the difference in propagation path distance, \(\Delta l\), is given by the following:  
\[\Delta l=\Delta\phi(\omega)/|k_{r}^{\omega}| \tag{3}\]  
where \(|k_{r}^{\omega}|\) is the norm of the wave number, as shown in Figure 1; \(\Delta l\) is the difference in the propagation distance of the damage scattering wave from the source to a pair of sensors, and its value is calculated by \(\Delta l=l_{2}-l_{1}\); and the phase difference is calculated by \(\Delta\phi(\omega)=\phi_{2}(\omega)-\phi_{1}(\omega)\). By reversing the phase corresponding to each frequency along the reverse path from the sensors to the damage scattering source, the phase spectrum, \(\Phi^{\omega}\), of the damage scattering signal at the source can be obtained by the following:  
\[\Phi^{\omega}=\overline{\phi^{\omega}}-\Delta\phi^{\omega}\cdot\bar{l}/\Delta l \tag{4}\]  
where \(\overline{\phi^{\omega}}\) denotes the mean value of the phase angle, and its value is calculated by \(\overline{\phi^{\omega}}=\frac{\phi_{2}^{\omega}+\phi_{m}^{\omega}}{2}\); and \(\bar{l}\) is the average propagation distance, and its value is obtained by \(\bar{l}=\frac{l_{1}+l_{2}}{2}\). It is important to note that this phase-reversal process can only be achieved when the distance between the sensors is small enough.