## 4 Numerical Results
### Case I: Dual-Band Microstrip Dipole Antenna
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Number of** \\ **Training Samples** \\ \end{tabular} } & \multicolumn{3}{c}{**Relative RMS Error**} & \multicolumn{3}{c}{**Relative RMS Error**} \\ \cline{2-7}  & \begin{tabular}{c} **Conventional** \\ **Models (Domain** \\ **X)** \\ \end{tabular} & \begin{tabular}{c} **(Original) Nested Kriging** \\ **with Fixed Domain** \\ **Thickness** \\ \end{tabular} & \begin{tabular}{c} **Number** \\ **of** \\ **Training** \\ **Samples** * \\ \end{tabular} & \begin{tabular}{c} **Nested Kriging with** \\ **Variable-Thickness** \\ **Domain [this Work]** \\ \end{tabular} \\ \cline{2-7}  & \begin{tabular}{c} **Kriging** \\ **50** \\ \end{tabular} & \begin{tabular}{c} **RBF** \\ **55.6\%** \\ \end{tabular} & \begin{tabular}{c} **58.1\%** \\ \end{tabular} & \begin{tabular}{c} 14.5\%** \\ \end{tabular} & \begin{tabular}{c} 18.9\%** \\ \end{tabular} & \begin{tabular}{c} 15 \\ \end{tabular} & \
\begin{tabular}{c} 25.7\%** \\ \end{tabular} &
\begin{tabular}{c} 31.2\%** \\ \end{tabular} \\ \hline
50 & 46.7\% & 46.9\% & 11.2\% & 13.5\% & 19 & 18.3\% & 22.5\% \\
200 & 41.9\% & 44.1\% & 6.9\% & 7.8\% & 55 & 8.9\% & 12.1\% \\
400 & 36.9\% & 41.2\% & 6.1\% & 6.6\% & 108 & 5.9\% & 6.9\% \\
800 & 35.5\% & 37.5\% & 4.1\% & 4.9\% & 208 & 5.0\% & 6.7\% \\
1600 & 33.2\% & 35.8\% & 3.9\% & 4.2\% & 405 & 4.3\% & 3.8\% \\ \hline \hline \end{tabular}  
* The number of training samples is determined by the relative volume of the variable-thickness and fixed-thickness domains (see the design of experiments procedure description at the end of Section 3).  
\end{table} 
Table 1: Modeling Results for Antenna of Figure 6.  
Figure 7: Allocation of the reference designs for the dual-band antenna of Figure 6: (**a**) objective space, (**b**) selected three-dimensional projections of the parameter space. Supplementary validation has been conducted to determine whether variable-thickness domain affects the design utility of the surrogates. Figure 9 shows the initial and the optimized designs obtained for the selected objective vectors. It can be observed that the designs obtained by optimizing the fixed- and variable-thickness domain surrogates are comparable. Figure 8: Dipole antenna: reflection responses at the selected test designs: EM model (—), nested kriging with variable-thickness domain obtained using 400 training samples and \(T_{\text{max}}\) = 0.02 (o). Figure 9: Optimization of the antenna of Figure 6 using the variable-thickness nested kriging surrogate for the four pairs of operating frequencies: (**a**)\(f_{1}\) = 2.45 GHz,\(f_{2}\) = 3.3 GHz, (**b**)\(f_{1}\) = 3.2 GHz,\(f_{2}\) = 5.0 GHz, (**c**)\(f_{1}\) = 3.4 GHz,\(f_{2}\) = 4.8 GHz, (**d**)\(f_{1}\) = 2.2 GHz,\(f_{2}\) = 2.7 GHz. Shown are: the initial design obtained from the first-level kriging model (‘...’), response of the optimized variable-thickness nested kriging surrogate (‘-’), response of the optimized fixed-thickness surrogate (o), and EM-simulated antenna response at the variable-thickness surrogate model optimum (—). This indicates that restricting the domain does not compromise the model ability to yield high-quality designs. This was to be expected because the very definition of the variable-thickness domain is founded on reducing the thickness in the vicinity of the reference designs (where the image of the first-level model agrees perfectly with the optimum design manifold) and increasing it between the reference points (where the two manifolds deviate from each other). The design optimized for \(f_{1}=2.45\) GHz and \(f_{2}=3.3\) GHz has been fabricated and measured for additional validation. Figure 10 shows the photographs of the antenna prototype, the reflection and realized gain characteristics, as well as the H- and E-plane radiation patterns. The agreement between the simulation and measurement data is satisfactory.