PSI - Issue 37

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

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If we follow the graph, we can point out four configurations that provide the maximum value for our objective function. Layout of these configurations are shown in Fig. 2 on the two upper rows of images, with bright blue being the hard materials and violet being the soft material and a value for the objective function of 3953.61 MPa. Using the results from this chart we could analyze the results obtained when we ran the Greedy algorithm. On 4 different runs we obtained the configurations shown on the 3 rd and 4 th rows of images in Fig. 2, with the associated objective function values. As it can be seen, none of these distributions of materials points to one of the 4 global maxima we obtained from the brute force run. One iteration comes close to it, but the other 3 converge to solutions that are not nearly close to an optimal solution. We can conclude that the Greedy algorithm does not guarantee convergence to a global maximum, as it stops if the next iteration does not find a higher objective function.

Fig. 2. Variation of the objective function for all possible distributions of material obtained by brute force.

4. Simulated annealing To avoid this drawback, we needed to modify the algorithm in such a way that it would not block in a local maximum. Thus, we implemented a simulated annealing algorithm, which is similar to the Greedy one, as it uses the same method of consecutive iterations, the difference being that we made it capable of accepting worse solutions. Let ’ s consider a part of our previous chart, represented now in Fig. 3, with every green dot representing values of the objective function for different distributions of materials. If the Greedy algorithm starts at the initial random configuration at the lower left, it will make 3 iterations where it finds better configurations but the next one provides a lower value, so this proves to be a barrier to the program. To be able to reach the global maximum we defined another two parameters, one called acceptance probability, and the other one temperature. The acceptance probability can have various expressions, but it generally depends on the difference between the new solution and the old one, and the temperature value. Due to the way the probability is calculated, when the temperature is higher, is it more likely that the algorithm accepts a worse solution.(Nikolaev and Jacobson, 2010)