## 4 Numerical Results
### Comparisons and Discussion  
To confirm the validity of the numerical results, the \(\boldsymbol{x_{1.20}}^{*}\), \(\boldsymbol{x_{5.20}}^{*}\), and \(\boldsymbol{x_{9.12}}^{*}\) designs were evaluated using a frequency domain solver (finite elements method with adaptive meshing) implemented in CST Studio [52]. The response characteristics (shown in Figure 11) obtained using the time and frequency domain methods were similar, which indicates that the EM models used in the optimization process were valid. Furthermore, the plots of Figure 11 show that, at the lower corner frequency of 3.1 GHz, the \(\boldsymbol{x_{1.20}}^{*}\), \(\boldsymbol{x_{5.20}}^{*}\), and \(\boldsymbol{x_{9.12}}^{*}\) designs featured radiation efficiencies of 76%, 89%, and 99%, respectively. Note that for the \(\boldsymbol{x_{1.20}}^{*}\) design, an efficiency of over 90% was achieved above the 3.4 GHz frequency, which is considered acceptable, keeping in mind its small footprint. Overall, the considered antennas were characterized by average in-band efficiencies of 93%, 97%, and 98%, respectively. Clearly, the high values were due to the utilization of a low-loss substrate for implementation of the structures. Figure 11: Comparison of EM simulation results obtained using solvers based on the finite integration technique (FIT) and finite elements method (FEM) for the designs (**a**) \(\boldsymbol{x_{1.20}}^{*}\), (**b**) \(\boldsymbol{x_{5.20}}^{*}\), and (**c**) \(\boldsymbol{x_{9.12}}^{*}\). The dotted lines represent the total efficiency, evaluated using the FIT method. Figure 10: Realized gain patterns obtained at 4 GHz (-), 7 GHz (- -), and 10 GHz (-) for the optimized antenna designs (**a**) \(\boldsymbol{x_{1.20}}^{*}\), (**b**) \(\boldsymbol{x_{5.20}}^{*}\), and (**c**) \(\boldsymbol{x_{9.12}}^{*}\). The top and bottom plots represent the characteristics obtained in the x–y and x–z planes (see Figure 9), respectively. Figure 12a shows a comparison of the optimal solutions obtained at different iterations of the nested loop (represented using a varying number of parameters). As can be seen, for the majority of the designs, increasing the dimensionality resulted in improvement in the structure responses. However, for certain solutions, a larger number of variables led to performance degradation. This was due to insufficient precision of the spline approximation when the number of knots increased, which led to undesirable frequency shifts at the edges of the operational bandwidth. Notwithstanding this, for most of the designs, the imperfections were corrected by the optimization algorithm. The computational cost of bi-objective optimization corresponded to 4389 evaluations of the EM antenna model (\(\sim\)159.7 h of CPU time). The cost breakdown, with emphasis on individual nested loops, is gathered in Table 1. It should be noted that the average numerical expenses associated with the identification of a single Pareto-optimal design corresponded to about 488 \(R\) model evaluations, which is low when having in mind the dimensionality of the problem (up to 46 independent parameters). It is also worth noting that optimization of the designs represented using more than 30 variables amounted to 57% of the overall cost. For the sake of comparison, the antenna at hand was also optimized using a standard population-based multi-objective evolutionary algorithm (MOEA) with fitness sharing, mating restrictions, and Pareto dominance tournament selection [53]. The setup of the benchmark routine was as follows. The population size and the number of generations were set to 200 and 50 (a total of 10,000 EM simulations), respectively, whereas the probability of mutation and crossover were fixed at 0.2 and 0.5, respectively. Since the discussed algorithm did not support adjusting the antenna dimensionality, the number of parameters was set to 30 (i.e., \(d_{4}=12\)). Furthermore, the corner designs obtained using the method in Section 3 were included into the initial population.