PSI - Issue 78

Shahin Sayyad et al. / Procedia Structural Integrity 78 (2026) 277–284

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function of Eq. 2, which includes both frequency and mode shape terms with corresponding weighting factors of 0.7 and 0.3, respectively. Table 2 shows the calibrated mechanical parameters derived from the optimization procedure.

Table 2. Values of updated parameters using PSO and GA under different objective functions and number of modes

PSO Result (10 Modes) F 1 Swarm Size: 50 Max Iteration: 200

GA Result (10 Modes) F 1 Population Size: 200 Generations: 50

PSO Result (5 Modes) F 2 Swarm Size: 50 Max Iteration: 200

GA Result (5 Modes) F 2 Population Size: 200 Generations: 50

Parameter Bounds [L b , U b ]

Mechanical Parameters

Input value

N

Unit

1 E 1 (Bell Chamber)

[1000, 3000] [1000, 3000] [1000, 3000] [1000, 3000] [1000, 3000]

1500 2600 2600 2600 1500 0.14 0.14 0.14 0.14 0.14

1603.24 2721.96 2874.85 2450.76 1661.3

1554.85 2780.59 2688.46 2959.72 1641.48

1476.27 2486.73 2569.87 2411.47 1266.03

1499.25 2591.77 2620.99 2796.79 1381.77

MPa MPa MPa MPa MPa

2

E 2 (Shaft)

3 E 3 (Base/Battered Wall)

4 5 6 7 8 9

E 4 (Vault/Arch) E 5 (Stair Ramps) ν 1 (Bell Chamber)

[0.1, 0.2] [0.1, 0.2] [0.1, 0.2] [0.1, 0.2] [0.1, 0.2]

0.13 0.11 0.16 0.18 0.10

0.12 0.15 0.13 0.14 0.15

0.15 0.13 0.11 0.18 0.16

0.14 0.15 0.13 0.10 0.12

- - - - -

ν 2 (Shaft)

ν 3 (Base/Battered Wall)

ν 4 (Vault/Arch) ν 5 (Stair Ramps) γ 1 (Bell Chamber)

10 11 12 13 14 15

[10000, 20000] 17000 18169.98 [10000, 20000] 18000 19611.37 [10000, 20000] 18000 17453.95 [10000, 20000] 11000 10982.98 [10000, 20000] 17000 10153.84

17714.21 19099.19 14662.03 16038.17 14378.42

16524.36 17593.55 14078.13 10089.80 14776.80

16985.43 N/m3 17889.52 N/m 3 19578.01 N/m 3 11851.26 N/m 3 18289.86 N/m 3

γ 2 (Shaft)

γ 3 (Base/Battered Wall)

γ 4 (Vault/Arch) γ 5 (Stair Ramps)

The calibration outcomes for the mechanical parameters of the San Giuseppe bell tower, obtained through GA and PSO, are illustrated in Fig. 3 to enhance comprehension. The graph shows the percentage errors produced by each technique across two distinct objective functions.

Estimation Error of Mechanical Parameters Using PSO and GA

0.00

0.00

0.00

20.00

Error

10.00

0.00

E1

E2

E

E

E

1

2

1

2

PSO esult (10 Modes) 1 Swarm Size 0 Ma teration 200

GA esult (10 Modes) 1 Population Size 200 Generations 0

PSO esult ( Modes) 2 Swarm Size 0 Ma teration 200

GA esult ( Modes) 2 Population Size 200 Generations 0

Fig. 3. Calibration error of mechanical parameters using GA and PSO.

Overall, the performance of both algorithms improved when the F2 formulation was employed, especially in the determination of elastic moduli. This indicates that integrating mode shapes into the objective function led to a more accurate calibration of the mechanical properties, even when a limited subset of modes was retained. Moreover, it is evident that for the majority of parameters, the GA yielded smaller errors than the PSO, confirming its greater precision and robustness for the present application. For the elastic moduli (E 1 to E 5 ), the results show that all four configurations (PSO-F 1 , GA-F 1 , PSO-F 2 , and GA-F 2 ) provide low error values, mostly below 10%. Among these, GA under F 2 yielded the best accuracy, especially for parameters E 1 to E 3 , making it the more effective approach in calibrating the mechanical properties of the finite element model of the bell tower. The Poisson’s ratios (ν 1 to ν 5 ), however, demonstrate relatively higher estimation errors, particularly ν 3 and ν 4, where PSO errors exceed 20%. In contrast, GA maintains a more stable performance with generally lower errors in both F 1 and F 2 settings. This further