PSI - Issue 37

Andrzej Katunin et al. / Procedia Structural Integrity 37 (2022) 292–298 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

296

5

̃ ( ) = ̃ ( ) ̃ ( ), ( ) = (2 ⌊ ⁄2⌋ 2 1 ),

(5)

where ̃ and

(6)

̃ ( ) = √ 2 +1 ( ) − 2 ( ), ≥ 0, ( 1 , 2 ) = (2 − 1 ) (2 − 2 )

(7)

(8)

is the product of the 1D windows, and = { 1, [− 1 2 , 1 2 ] 0, [−2,2] , 0 ≤ ≤ 1 , 2 ≤ 1 ≤ 2 +1 , −2 − ⁄2 ≤ 2 1 ≤ 2 − ⁄2 . The discrete version of (4), called by Candès et al. (2006) the digital CT can be represented by: ( , , ) = ∑ ̂ 1 , 2 [ 1 , 2 − 1 ] ̃ [ 1 , 2 ] ( 2 ( 1 1 1, + 2 2 2, )), where i = √−1 is the imaginary unit. In this study, the implementation of (10) in Matlab ® environment shared by the Curvelet.org team was used for The determined SDIs according to the algorithm presented in section 2.2 for all 3 considered damage scenarios are given in Fig. 3(a). The calculated curvelet coefficients according to (10) are depicted in Fig. 3(b). In general, the presented SDIs plots points out the damage locations for all the considered cases, however some false-positive damage indications caused by the measurement and processing noise are also observed. SDI results clearly show the through the-width crack for the case A, nevertheless many SDI values have passed the pre-assigned damage index threshold at the boundary of the plate resulting in false-positive damage identification. For the cases B and C, SDIs have successfully determined the location and partially the planar shape of the extrusion damage, while other two damage sites for the case C are barely visible in the plot. In comparison, curvelet coefficient maps (Fig.3(b)) show considerably higher damage identification accuracy. Obtained results demonstrate excellent performance of CT not only by clearly indicating the location of every considered damage site but, moreover by giving a valuable information about the damage signature, e.g. planar shape and size. It must be also noted that CT plots are less noisy compared to SDI ones and do not produce the boundary effect which is typical for most of DI approaches (evident in this study too). (9) (10) processing the mode shapes. 3. Results and comparisons