the multibody model is built parametrically and they are related to the same file, being then possible to define the parameters just once. Since the modal parameters are not a direct output of Virtual.Lab Motion, the state space matrices need to be extracted from the results file, and then the procedure shown in section 2 can be applied by defining a Matlab script. To automate the process and run the optimization, the commercial software Noesis Optimus is used. The workflow created with the different tasks that need to be performed for each design is shown in figure 4. Fig. 4 process automation in Noesis Optimus The algorithm selected to optimize the model is the Self Adaptive Evolution algorithm available in the software, which is based on genetic evolution techniques. The total number of design evaluated during the first optimization is 4200, coming from 30 generations of 140 individuals each; the number of parents, that is the number of optimal points from the previous generations used to evaluate the new individuals is set to 28. For the second optimization run, 20 generations of 200 individuals are evaluated (with 40 parents per new population), resulting in a total of 4000 design evaluations. The results after each of the two runs are shown in table 3. Table 3: results for the single objective optimization MODE # EXP/NUM 1ST RUN 2ND RUN - MAC ε MAC ε 1/1 0.8696 0.0034 0.8698 0.0026 2/2 0.9537 0.0028 0.9535 7.34E-5 3/3 0.8479 0.0259 0.8536 0.0317 4/4 0.8543 0.1282 0.8582 0.121 5/6 0.9325 0.0106 0.9328 0.0215 6/5 0.7649 0.0780 0.7667 0.0718 SUM 5.223 0.249 5.2345 0.2487 By comparing the results in table 3 with table 2, it can be seen how the MAC values are not changing too much. On the other hand, there are significant changes in the errors between the natural frequencies. In particular, the errors between the frequencies of the first three correlated modes are significantly reduced already after the first optimization run. A different trend can be observed for the forth modes, where the error between the corresponding modes is drastically increased after the first run and then reduced after the second, but still showing a bigger value if compared with the original. For the last two modes, the errors remain more or less constant. The reason for this behavior is probably related to the choice of the objective function. It is a trade off between the MAC values and error between natural frequencies for all the modal data considered. Moreover, by assigning no weights to the correlation indices used, they are given the same importance. The result is that the optimum point found by applying this methodology shows a better overall behavior, not guarantying an increased correlation for all the indices used. By analyzing the results of the optimization, it is observed that only few of the parameters considered in the two run show a significant correlation with the objective function, partially explaining the limited improvement obtained. 5. MULTI-OBJECTIVE OPTIMIZATION RESULTS To verify the influence of the weights on the results, a multi-objective optimization is also performed. In this case, all the parameters are considered at once, resulting in a relatively big design space. Between the different algorithms available, the Non-dominated Sorting Differential Evolution (NSDE) is selected [9]. The advantage of this algorithm is that it creates a Pareto front with uniformly distributed points and assigns weights to the objectives with respect to the location of the point in the front. For this problem, all the variables are considered together and with the same ranges defined for the single-objective problem. Starting from the original model, an initial random population of 256 individuals is considered; after this, 35 new 354
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