PSI - Issue 37

Iulian Constantin Coropețchi et al. / Procedia Structural Integrity 37 (2022) 755 – 762 Coropetchi et al./ Structural Integrity Procedia 00 (2019) 000 – 000

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and even for a small difference between the two values of the effective stiffness the penalization becomes important. Because the two values used for the optimization are in the same range, it is indicated in the specialized literature (Kramer, 2017) that a simple way to fulfil multi-objective optimization is just to add the two values, as done in (4). The third formulation in (5) is an updated version of the second one and reduces the sensitivity when penalizing the solutions that do not ensure equal values for and correcting the two effective stiffnesses. 3. Methods 3.1. Brute force algorithm To verify if our algorithm can reach an optimal solution, we used a brute force approach in which we evaluated every possible combination with the specified conditions. The brute force approach computes every possible configuration for the two materials and evaluates their objective and with the results we can sort the solutions and determine the best ones. Without material restrictions, we have 2 16 solutions meaning 65536. If we consider that we have a 50% material 1 restriction we can use the general relation presented in (6). = − = ! !( − )! In equation (6) the used notations are: n – number of cells (finite elements); p – number of cells filled with material 1. Using 16 for n and 8 for p, our parameters, we obtain a total of 12870 possible solutions. Based on the results obtained for all the possible solutions we build the Pareto frontier, presented in Fig. 3, which contains the optimal solutions for the multi objective optimization for maximizing both the effective stiffness on X and Y direction. The Pareto frontier is a set of nondominated solutions, being chosen as optimal, if no objective can be improved without sacrificing at least one other objective. To build this we must evaluate the results obtained with the brute force approach and determine which solutions dominate in terms of maximizing the effective stiffness on the two orthogonal directions. In Fig. 3, on the dashed line we obtain the solutions which have equal values for E eff_x and E eff_y . (6)

Fig. 3. Pareto frontier of all solutions.

If we look closer to the area where the dashed line intersects the Pareto frontier, presented in Fig. 4, we can see that the best solution that ensures that E eff_x and E eff_y have equal values are the ones presented in the left with equal values for the effective stiffness on X and Y direction. At the same time, there are 2 solutions on the Pareto frontier that do not have equal values for the effective stiffness on X and Y directions, as the values are very close with under 0.5% difference. For the top left configuration, the moduli are E x = 3989.024 MPa and E y = 3989.320 MPa. For the bottom right configuration, very close to the previous one, the moduli are switched between them, that is E x = 3989.320 MPa