populations of 150 individuals each are considered. The Pareto front resulting from the optimization is shown in figure 5, where goal 1 on the x axis is ω J and goal 2 is ψ J (see equation 10). Fig. 5 Pareto front for the multi-objective optimization problem By observing the front, it can be seen how the non-dominated points present a small variation of the MAC values; on the other end, the values of the objective function representing the errors between the natural frequencies are distributed in a wider range. Table 4 shows the results obtained by selecting 4 points along the Pareto front; the location on the front gives the relative weights for the two objective function considered. Table 4: results of the multi-objective optimization for a set of optimal points with different weights MODE # EXP/NUM w( ψ J )=0 w( ω J )=1 w( ψ J )=0.09 w( ω J )=0.91 w( ψ J )=0.28 w( ω J )=0.72 w( ψ J )=0.55 w( ω J )=0.45 - MAC ε MAC ε MAC ε MAC ε 1/1 0.8676 0.0024 0.8644 0.0438 0.8425 0.0461 0.828 0.05 2/2 0.9527 0.0036 0.9472 0.0005 0.9483 0.1992 0.9364 0.37 3/3 0.8533 0.0047 0.8756 0.0654 0.8950 0.3852 0.9279 0.754 4/4 0.8571 0.0979 0.8695 0.0714 0.9007 0.1917 0.9437 0.512 5/6 0.9282 0.0337 0.9276 0.1399 0.9351 0.1519 0.9339 0.21 6/5 0.7867 0.0712 0.8112 0.0041 0.8463 0.0071 0.889 0.0614 SUM 5.2456 0.2558 5.2955 0.3251 5.368 0.982 5.459 1.958 By comparing the results in table 4 with that on table 3 for the single objective optimization and the original design in table 2, it is immediately clear how bigger improvements can be obtained for the error between the natural frequencies than for the MAC values. By considering a weight of 1 for the error between the natural frequencies and 0 for the MAC, a relatively good result can be obtained, that is relatively similar to the one of the single objective optimization. By using a higher weight associated to the MAC values, some improvement can be obtained, but on the other hand the error between natural frequencies is significantly increased, leading to non-acceptable solutions. The better solution can probably be considered the one with w( ψ J )=0.09 and w( ω J )=0.91, where both objective function are improved with the respect to the original design and the single objective optimization results. By processing the results, both in the single and in the multi-objective optimization run, a very low correlation between the design variables and the objective functions is observed. Besides the modeling approximation discussed in section 3, the relatively high number of design parameters to be optimized at once and the formulation of the objective functions can both be reasons for this behavior. By the way, the optimal solutions selected in the two cases show some improvements with 355
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