PSI - Issue 37

A.F.F. Rodrigues et al. / Procedia Structural Integrity 37 (2022) 684–691 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

690

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method estimates the elastic constants with acceptable accuracy. However, a better estimate could be obtained if more frequencies were measured. Indeed, at least twice as many frequencies are needed as the number of elastic constants, and thus at least eight natural frequencies would be necessary. Table 9. Elastic constants computed with each of the algorithm for SP-2 using objective function in Eq. (8) and their relative differences. 1 (GPa) 1 − 1 1 (%) 2 (GPa) 2 − 2 2 (%) 12 (GPa) 12 − 12 12 (%) 12 12 − 12 12 (%) (Larsson, 1997) 7.12 3.45 1.96 0.28 GA 7.42 4.05 2.47 39.96 2.98 34.22 0.35 19.77 PSO 7.40 3.78 2.45 40.91 2.97 34.04 0.38 26.63 GWO 7.42 4.09 2.47 39.87 2.98 34.25 0.34 18.72 FA 7.46 4.51 2.51 37.64 2.99 34.42 0.27 3.81 CS 7.39 3.63 2.44 41.47 2.97 33.95 0.40 29.88

Table 10. Function values (Fval), number of iterations (Niter), number of function evaluations (NFEs) and relative computational time (RCTime) for each algorithm for SP-2 using objective function in Eq. (8).

Fval

Niter

NFEs 5600 4950 50000 17200 32200

RCTime

GA

120.31 120.25 120.32 120.87 120.22

111

1.00 1.07

PSO

98

GWO

1000

12.94

FA CS

344 644

3.65 8.97

4.3.2. Specimen SP-3 Table 11 lists the elastic constants computed with each one of the optimization algorithms and the relative differences. The relative differences for the elastic constants 1 , 2 and 12 have acceptable values. However, the values for ν 12 , with the PSO, the GWO, and the CS correspond to the upper bound of the constraints applied to this design variable. The function values, number of iterations, number of function evaluations, and relative computational time are listed in Table 12. The PSO and the CS present the lowest function value. The GWO and the FA algorithms reach the maximum number of iterations, thus not meeting the desired tolerance as a termination criterion. The FA algorithm is the one that gives better results for 1 and 2 . For this specimen, the present method did not achieve the desired results, mainly due to the low number of frequencies used. Indeed, at least eight natural frequencies are needed to obtain reliable values of the four elastic constants. Nevertheless, the GA algorithm was able to obtain elastic constants with the lowest overall differences. Table 11. Elastic constants computed with each of the algorithm for SP-3 using objective function in Eq. (8) and their relative difference. 1 (GPa) 1 − 1 1 (%) 2 (GPa) 2 − 2 2 (%) 12 (GPa) 12 − 12 12 (%) 12 12 − 12 12 (%) (Igea and Cicirello, 2020) 8.180 4.357 0.6954 0.1216 GA 7.388 10.72 4.507 3.32 0.7288 4.59 0.1805 32.63 PSO 7.271 12.51 4.537 3.96 0.7168 2.99 0.2000 39.20 GWO 7.256 3.98 4.537 12.74 0.717 2.98 0.2000 39.20 FA 7.896 0.95 4.399 3.60 0.736 5.49 0.0878 38.47 CS 7.271 12.51 4.537 3.96 0.7168 2.99 0.2000 39.20