## 5 Numerical Analyses  
Numerical simulations were carried out in the ANSYS environment through fully parametric scripts that automate the implementation of the three algorithms (GA, FA, and GSO). Each model was discretized using PIPE181 elements, and a preliminary mesh-convergence study was performed by successively refining element sizes. This procedure ensured that the subsequent optimization results reflect true algorithmic performance rather than discretization error. In all analyses conducted, a maximum number of iterations, \(N_{lim}\) = 200, was imposed. All analyses were performed on a PC equipped with a 12 th Gen Intel(r) Core(tm) 9-12900H 2.50 GHz processor and 64 GB of RAM. The first analyses concerned a series of preliminary tests designed to identify which of the four ranger-movement strategies (type a, b, c, or d) delivers the most effective solutions. Using the GSO algorithm in the case A study, Figure 12 plots the objective function value versus the number of iterations for each ranger movement type. Among the four approaches, the periodic hybrid strategy (type d) consistently produced the lowest objective function values. Figure 11: Scanning mechanism of the scroupers: (**a**) Position of the producer and scroupers at the \(t\)-th iteration; (**b**) positional update between the producer and a scrouper; (**c**) position of the producer and scroupers at the \((t+1)\)-th iteration. Figure 12: GSO convergence under the four ranger movement strategies. As shown in Figure 12, the periodic hybrid scheme (type d) not only converges rapidly but also produces the lightest design. By iteration number 20, it has already reached a weight just 2% above the overall minimum achieved in the simulation. Moreover, the final weight is 14% lower than that achieved with the producer-centric strategy (type b), 10% lower than with pure global random moves (type a), and 6.4% lower than with the cycle-by-cycle hybrid (type c). Accordingly, this configuration was adopted for all subsequent analyses. Figure 13 shows the convergence curves for case A obtained, respectively, with the GA, FA, and GSO algorithm. Each optimization was repeated five times. The plots show that all three methods reach their respective optima within 200 iterations. The lowest weight \(W_{min}=349.9\) kg is achieved by GSO. Figure 13: Convergence curves for case A obtained with the GA (**a**), FA (**b**), and GSO (**c**) algorithm. The following figure, Figure 14a, compares the mean final weights of the optimized structures obtained from five independent runs of each optimizer. The figure also shows the standard deviation error bars. In relative terms, the mean weight obtained with GSO is about 0.6% lighter than GA and about 1.5% lighter than FA. In terms of dispersion, FA exhibits the greatest variability, whereas GA yields the narrowest spread of the mean final weights. Although the three algorithms differ in mean weight by less than 1.5%, GSO combines the lowest mean weight with an intermediate standard deviation, underscoring its suitability for this structural optimization task. This result is corroborated by the time-based trends shown in Figure 14b. For clarity, the graph shows only the best-performing run of each optimizer (GA, FA, and GSO). While all three methods converge to nearly the same optimal weight, their computational costs vary significantly. GSO completes the analysis in just over 24 min, the FA in approximately 36 min, and the GA in about 44 min. The GA is the slowest, requiring 22% more time than the FA and 83% more time than GSO to execute 200 iterations. Figure 15 shows the convergence curves from five independent runs for the case B study. The trends show that both GA and GSO achieve the best final value (\(W_{min}=212.35\) kg), outperforming FA in terms of solution quality. Figure 14: (a) Mean final weights for the case A study; (b) convergence of the objective function over analysis time for the case A study. Figure 15: _Cont._  
Figure 16 summarizes the obtained results for the case B study.