## 3 Crack Identification Based on COSMO
### Numerical Simulation  
A two-dimensional model of a steel board embedded with a single crack was simulated by using a commercial software (Comsol Multiphysics V4.3a. COMSOL, Inc., Palo Alto, CA, USA). Figure 1 shows the schematic illustration of an ultrasonic measurement system in the simulation. Two longitudinal transducers with \(60^{\circ}\) wedge as the transmitter were typically used to carry out the ultrasonic inspection on the top surface of the steel board with the length of 220 mm and the height of 30 mm. A fixed distance between the two transducers was kept at a constant interval to make sure the first back wall echo was fully collected by the receiver. The single crack with a depth of \(d\) was located at the middle of the steel board along the x direction, i.e., x = 110 mm. A 0.5 MHz continuous sinusoidal signal with signal-noise ratio (SNR) of 15 dB was applied as the exciting signal. Both the transmitter and the receiver were moved simultaneously to scan the steel board along the x direction. The receiving waveforms, at eight different positions, were spaced by 20 mm on the top of the simulated steel board, recorded, and the corresponding spectrums were then analyzed. Figure 2a depicts spectrums at eight positions of the simulated steel board with crack length \(d\) = 2 mm. According to the COSMO algorithm model, the corresponding spectrums, at eight different observation positions, were saved then a group of \(z\)-scores were calculated by Equation (2) after every scanning process, finally 30 groups of \(z\)-score were obtained by scanning repeatedly 30 times. It is clearly shown in Figure 2b that \(z\)-score of observation points are distributed almost evenly between 0.3 and 1 except \(x\) = 110 mm, while most of \(z\)-score at \(x\) = 110 mm are mainly distributed below 0.4, just right at the crack's position. It is shown that the distribution of \(z\)-scores could be used to locate and identify cracks or defects in materials, i.e., \(z\)-score of damaged regions might be below 0.4. However, the conclusions need to be subjected to hypothesis tests to reach statistical significance, which determines whether a null hypothesis can be rejected or retained. Figure 3a shows the calculated level of significance testing for crack depth of 2, 4, and 5 mm by Equation (4). It could be seen from Figure 3a that the \(p\)-value has a much smaller value than 0.1 around the crack region from 90 to 150 mm, which suggests that the imperfect structure of this region is significant. To make the comparisons and analysis clearly, an indicator called deviation level is defined as,  
\[T=-\lg(p),\]  
Figure 1: Schematic illustration of scanning on a simulated specimen with a single crack. Figure 2: (**a**) Spectrums and (**b**) \(z\)-score distribution of echoes at eight monitoring points (depth of crack d = 2 mm) along x direction. i.e., logarithm transformed \(p\)-value, obviously the small T-value indicates little significant probability of crack. The T-value curve of significance testing is shown in Figure 3b. Obviously, the maximum T-value occurs around the position of crack (\(x\) = 110 mm) for crack depth of \(d\) = 2, 4 and 5 mm, respectively. Furthermore, the maximum T-value increases simultaneously as crack growth, e.g., the maximum T-value is close to 4 when crack depth is 2 mm, and the peak of T-value up to 10 when crack depth equal to 5. This result indicates that the higher level T-value is strongly correlated with the crack depth, which could become an index to exhibit the evolution of crack growth inside materials. When the crack depth changed from 0 mm to 5 mm, the peaks of T-value around the cracks were obtained and thus the relationship curve between the maximum T-value and crack depth d was plotted, as shown Figure 4. It can be seen that as the crack depth increased from 0 to 1, the slope of the curve sharply increased.