# A Nonlinear Method for Characterizing Discrete Defects in Thick Multilayer Composites
## 4 Results and Discussion
### Recurrence Analysis of Defect Areas  
According to the RPs of defect-free areas in Figure 8, chaotic behavior has been found in the ultrasonic signal. Despite the low porosity of the specimen, scattering and data noise may exist and form the feeble differences of cases in Figure 8, together with chaos. The statistical results of the RQA variables in Figure 10 reveal that the difference in RPs of different defect-free areas are in a relatively constant range, and the range can be described with the median and IQR of RQA variables. In other words, each value in the range of RQA variables represents a state of the system and the defect-free ultrasonic pulse echo system can be described by a set of RQA statistics: RR\({}_{\text{median}}\) = 0.1, RR\({}_{\text{IQR}}\)\(\approx\) 0.4; DET\({}_{\text{median}}\) = 0.79, DET\({}_{\text{IQR}}\)\(\approx\) 0.45; LAM\({}_{\text{median}}\) = 0.85, and LAM\({}_{\text{IQR}}\)\(\approx\) 0.4. In order to discover the effect of defects on the states of the ultrasonic pulse echo system, the areas where the defects of 0.5 mm, 0.7 mm, and 1 mm were located were tested and the time series are shown in Figure 11. The signals of 0 mm were from experiment set Defect-free-1, while those of 0.5 mm from Defective-1, 0.7 mm from Defective-2, and 1 mm from Defective-3. No recognizable defect echo could be found in the waveform either. The RPs of the second half of each signal were calculated and the  
Figure 10: Statistical results of recurrence quantification analysis (RQA) variables of defect-free areas; (a) RR; (b) percent determinism (DET); and, (c) laminarity (LAM). Calculate the RQA variables of the second half of the defect areas with the same embedding dimension and delay in 4.1, and the threshold was chosen so that the mean value of RRs of all results equal to 0.1. The statistics of RR, DET, and LAM are shown in Figure 13. In Figure 13a, the RR\({}_{\text{median}}\) of different defect diameter are approximately equal, while the RR\({}_{\text{IQR}}\) decreases as the defect size increases and tend to be constant to 0.01, which is much smaller than the defect-free RR\({}_{\text{IQR}}\) of 0.04. In Figure 13b, DET\({}_{\text{IQR}}\) of different defect diameter are approximately equal while the DET\({}_{\text{median}}\) decreases as the defect size increases. The decrease is rather small and the DET\({}_{\text{median}}\) tend to be constant around 0.75 which is sufficiently different from the value of defect-free areas. The defect size here is 0.7 mm, which is about twice the wavelength of the ultrasonic pulse. The same trend can be seen in the statistics of LAM in Figure 13c. When comparing with results of defect-free areas in Figure 10, the statistical values of the RQA variables of defective areas are different and related to the size of the defect. For defect-free areas, the second half of backscattered signals is rather instable and shows chaotic characteristics. As the defect size increases, no significant change in chaotic characteristics is found, while the instability of the signal due to scattering and data noise is weakening. That is, the effect of defects on the signal modality is like a stabilizer, which makes the chaotic structure of the signal clearer. The larger the defect size, the more obvious the stabilization effect until the defect size is about twice the wavelength of the ultrasonic pulse, and in this paper, defects of 0.7 mm and 1 mm can be distinguished in the results of RQA, while 0.5 mm cannot. In summary, using statistics of RQA variables, it is able to characterize the instability of the signal and reveal the effect of discrete geometry variation in the form of blind holes. The chaotic structure of the signal is more stable due to the presence of simulated defects and it is related to the size of the defect. Figure 12: RPs of the second half of backscattered signals of defect areas with different diameter: (**a**) 0 mm; (**b**) 0.5 mm; (**c**) 0.7 mm; and, (**d**) 1 mm.