PSI - Issue 83

Amal Lahrizi et al. / Procedia Structural Integrity 83 (2026) 162–170

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The Particle Swarm Optimization (PSO) algorithm is employed to pinpoint, from multiple potential solutions, the specific pair of crack location and depth that induces corresponding variations in natural frequencies. Given the inherent randomness in this approach, the average results from 10 attempts are presented as the final outcomes. After 9 iterations, the most optimal results for all deterioration scenarios are obtained and summarized in Table 2. The reported errors underscore the predictive efficacy of the PSO for crack detection, with the maximum error occurring in case 1 (i.e., 3.05% for crack location and 0.56% for depth estimation). For the other scenarios, the predictions for crack location and depth prove successful, with errors of less than 1%. Graphs 4(a) and 4(b) visually represent the trajectory of the optimal value of the objective function during the optimization process with the PSO algorithm. These visualizations also indicate that convergence is achieved after around 60 iterations and that predictions of crack location and depth are rapidly obtained with a reduced number of iterations.

Table 2. . It presents the exact and predicted results of clamped-clamped Functionally Graded Material (FGM) beam cracks obtained through the application of the Particle Swarm Optimization (PSO) algorithm.

Exact crack (mm)

Predicted crack (mm)

Error%

Case No.

Location

Depth

Location

Depth 2,0113

Location

Depth 0,56% 1,73% 1,48% 1,35% 0,77% 1,57% 0,70% 0,47% 1,53%

1 2 3 4 5 6 7 8 9

40 40 40

2 3 4 2 3 4 2 3 4

41,22

3,05% 0,24% 1,11% 1,63% 0,08% 0,56% 0,21% 0,37% 0,30%

40,098 40,442 101,634 100,0751 100,5607 275,565 276,023 275,831

3,052 4,059 2,027 3,023 4,063 1,986 3,014 4,061

100 100 100 275 275 275

(a)

(b)

Fig. 4. Figure depicts the value of the cost function with respect to the number of generations for two distinct cases: (a) case number 9 of the fixed beam and (b) case number 1 of the clamped-clamped beam. 5. Conclusions In conclusion, this paper highlights the crucial importance of structural integrity monitoring and early damage detection in modern industry, given the potential catastrophic consequences of structural defects such as cracks. The