# A Nonlinear Method for Characterizing Discrete Defects in Thick Multilayer Composites
## 4 Results and Discussion  
### Recurrence Analysis of Defect-Free Areas  
The purpose of this paper is to evaluate the discrete defects in thick multilayer CFRP by recurrence analysis of the second half of backscattered signals. The signal modality is affected by not only defects, but also material structures. Therefore, an area without artificial defects in the specimen has been analyzed first. Four signals have been randomly selected, and the second half of backscattered signals was magnified, as shown in Figure 6. The first half of each signal has the same resonance noise structure and large amplitude, while the second half has smaller amplitude and seem to be different from each other. \begin{table}
\begin{tabular}{c c c} \hline
**Experiment Set** & **Defect Diameter (mm)** & **Test Times** \\ \hline Defect-free-1 & 0 & 100 \\ Defect-free-2 & 0 & 100 \\ Defect-free-3 & 0 & 100 \\ Defective-1 & 0.5 & 100 \\ Defective-2 & 0.7 & 100 \\ Defective-3 & 1 & 100 \\ \hline \end{tabular}
\end{table} 
Table 1: Arrangement of experiments. Figure 6: Ultrasonic signals randomly selected in results of experiment set Defect-free-1. (**a**) signal 1 (**b**) signal 2 (**c**) signal 3 (**d**) signal 4. For further recurrence analysis of the signal, the mutual information method has been used for determination of the delay time (Lag). The relationship between the mutual information value and Lag has been obtained, as shown in the Figure 7a. The first local minimum of the mutual information value was the optimal time delay value. As a result, the optimal delay time \(\tau=3\). Next, the false nearest neighbor algorithm has been used to obtain the embedding dimension. Figure 7b shows the relationship between the ratio of false nearest neighbors and the value of the embedded dimension. The embedding dimension is considered to be the best when the ratio of false nearest neighbors is close to zero. Thus, the optimal embedding dimension \(m=5\). It has been verified by calculation that the embedding dimension and the delay time of all signals are the same. Nonoptimal embedding parameters may cause many small blocks or even diagonal lines perpendicular to the main diagonal line [24], which should be carefully checked in the RP. A lot of methods could be used to select the threshold, which need to be determined according to specific problems. The most commonly used threshold is that with a fixed value. In order to find the statistical behavior of the RQA parameters, while both the DET and LAM are variables that are based on the total amount of recurrence points according to Equations (4) and (5), the recurrence point rate RR needs to be determined first. Usually, RR takes 0.1; the threshold should be adjusted so that the statistical mean of RRs of all signals is 0.1. The results show that the requirement can be met when the threshold is 0.28 for defect-free signals. The RPs of the second half of the four signals were calculated using the parameters above, as shown in Figure 8. It can be seen that the structures of the figures are similar and mainly composed of some special structures: short diagonal lines, small black blocks, and vertical and horizontal lines. According to the meaning of structures in RP, short diagonal lines and small blocks can be regarded as the suggestion of chaos. Short diagonal lines indicate that the periodic behavior of the ultrasonic pulse only lasts for a short time, and small black blocks represent different states of the ultrasonic pulse. Ultrasonic pulses travel the material thickness from the front surface echo to the back wall echo twice and may reflect multiple times in the material. Thus, it is reasonable to the existence of chaotic components in the ultrasonic signal, especially those with large number of plies and complex lay-up. Besides, vertical or horizontal lines represent time segments that remain unchanged or change very slowly, and they are typical behaviors of the state of the laminate.