PSI - Issue 37

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Alexandru Vasile et al. / Procedia Structural Integrity 37 (2022) 857–864 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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Fig. 3. Schematic of the simulated annealing mechanism compared to the Greedy algorithm.

The acceptance probability is defined by the relation _ = exp[( _ ( ) − _ (∗))/ ] acc_prob – acceptance probability; obj_funct(curr) – value of the objective function for the current iteration; obj_funct(*) – highest value of the objective function for all the iterations analysed; T – temperature.

(5)

where:

The temperature value has different functions that have been implemented. Generally, it starts with the value of 1 and decreases every time a new iteration is considered, thus leading to lower acceptance probabilities as the algorithm keeps searching for more distributions. We used a function that takes into the account the number of iterations our program has run up a point. So, the relation for the temperature is given by the relation = ( 1; 1 − _ _ / _ _ ) (6) where: no_curr_iter – number of the current iteration; no_max_iter – predefined parameter that decides how long the algorithm should be able to look for a better solution. Fig. 4 explains how the algorithm works. The first stages are similar to the Greedy algorithm, the difference being this last part of the cycle, where the acceptance probability is calculated, and compared to a random generated number between 0 and 1, or to a predefined probability under which we will not accept any solutions. If the acceptance probability is higher, the program proceeds to decrease the value of T and start over with another random population storing the old ones. Else the program ends, (Gu et al., 2016) . We ran a couple of tests to see which algorithm works better and Fig. 5 represents a visual comparison. We can see that the Greedy stops after 3 iterations with a final objective function value of 3169.86 MPa. When we compiled the simulated annealing algorithm, we noticed that it continues after it reaches that local maximum with another distribution, reaches 4 new local maximums at iteration 5, 8,11 and 14 and then it stops with a final value for the objective function of 3951.07 MPa. We can see that it still has not reached the desired global maximum, but the results are closer to it. We completed 10 more tests to get an overview of the results with slight modifications to the simulated annealing algorithm. The results are presented in the graph in Fig. 5 and Table 3. The green dots show the 4 maximum values