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

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solved with PSO to obtain thirty sets of elastic constants of specimen SP-1. Each optimization problem has initial search agents distributed differently across the search space. The parameters for the optimization problems analyzed in this section are summarized in Table 4 and the constraints applied to the design variables listed in Table 3. The computer used has an Intel® Core™ i7 -9750H CPU @2.60 GHz processor with 16 GB of RAM.

Table 4. Parameters for the optimization problems of SP-1.

Objective function No of frequencies, Number of search agents Tolerance Max. number of iterations Equation (3) 14 100 10 -9 100

The elastic constants computed in each run, as well as the objective function value and computational time, are gathered from the performed optimization. Table 5 lists the averages and standard deviations of these parameters. The average values of the elastic constants are very close to the values reported in (Lopes et al., 2019) with small standard deviations. The shear modulus, 12 , and the Young's modulus 1 show standard deviations with the same order of magnitude. The Young's modulus 2 and the Poisson's ratio, ν 12 , present standard deviations with the same order of magnitude, but smaller than those for the other two elastic constants. The function value presents a small variation proving that despite the different initial search agents, the global function minimum is obtained. Table 5. Average and standard deviation of thirty optimization runs for the elastic constants, function value and computational time for SP-1. 1 (GPa) 2 (GPa) 12 (GPa) 12 Fval Computational Time (h) Average 30.51 27.15 6.38 0.1673 45.66 11.15 Standard deviation 0.0020 1.534E-05 0.0067 6.18E-05 0.0112 3.702 The effect of the different initial search agents is more noticeable in the computation time each optimization took. The computational time presents an average of around eleven hours and a standard deviation of around three point seven hours, which is significant corresponding to almost one-third of the average computational time. The high standard deviation of the computation time is noticeable with the computational time varying from around six hours to as much as twenty hours. In the best-case scenario, the optimization process took a little bit more than six hours using, corresponding to one hundred and thirty-four iterations. In the worst case-scenario, more than twenty hours and three hundred and seventy-four iterations were needed to solve the optimization problem. These specific optimizations, best-case scenario and worst-case scenario together with the data gathered from the thirty other optimizations demonstrate that the time each optimization takes vary with the initial set of search agents. Despite that, the validity of this method for properties identification is verified regardless of the initial set of search agents. Thus, it is possible to demonstrate the independence of the initial population of search agents on the correct elastic constants identification. 4.2. Comparison of results with different number of search agents In this section is studied the influence of the number of search agents for the PSO algorithm. For this analysis the specimen used is SP-1 and the number of search agents varies from ten to one hundred with increments of ten. Table 6 summarizes this optimization problem and the constraints for this specimen design variables are listed in Table 3. Table 7 lists the elastic constants computed using the different number of search agents, the function values and the relative time for the PSO algorithm. It is possible to observe that as the number of search agents increases, the function values decrease, converging to a value of around forty-five. From Table 7, it is also possible to observe that as the number of search agents increases, the differences between the computed values and the results from (Lopes et al., 2019) decrease, staying approximately constant for more than fifty agents. The best results are obtained with thirty or more than fifty search agents, presenting the lowest function values and the lowest relative differences. In view of the above results, it can be concluded that the PSO algorithm should be used with fifty search agents.