## 4 Numerical Results
### Case I: Dual-Band Microstrip Dipole Antenna  
Our first example is a dual-band dipole antenna shown in Figure 6[55]. The structure is implemented on Rogers RO4003 substrate (\(\varepsilon_{r}=3.38\), \(h=0.76\) mm) and described by eight independent parameters: \(\mathbf{x}=[L_{rr}\ d\ W_{s}\ W_{d}\ S\ L_{d}\ L_{gr}\ W_{gr}]^{T}\) (all dimensions in mm except those with \(r\)-subscript which are relative). The parameters \(W_{r}=5\), \(L_{s}=5\), and \(L_{0}=25\) are fixed. The feed line width \(W_{0}=4.5\) mm is calculated to ensure 50 ohm impedance. Other parameters are \(L_{r}=L_{rr}((W_{s}-W_{0})/2-W_{d}-d)\), \(W_{g}\) = \(W_{gr}W_{s}\), \(L_{g}=L_{gr}(L_{0}-W_{g}/2+W_{0}/2)\), and \(g=W_{d}\). The computational model is implemented in CST Microwave Studio (\(\sim\)900,000 mesh cells, simulation time 250 seconds). The surrogate model is to be constructed for the objective space parameterized by the operating frequencies \(f_{1}\) and \(f_{2}=Kf_{1}\) for 2.0 GHz \(\leq\)\(f_{1}\)\(\leq\) 3.5 GHz, and 1.2 \(\leq\)\(K\)\(\leq\) 1.6. Nine reference designs (for their allocation see Figure 7) are assigned, optimized for all combinations of \(f_{1}\)\(\in\) [2.0, 2.75, 3.5] GHz and \(K\)\(\in\) {1.2, 1.4, 1.6}. The parameter space \(X\) is determined by the lower and upper bounds for design variables, \(I\) = [0.55 1.0 47.0 2.5 3.8 3.8 0.5 0.24]\({}^{T}\), and \(\mathbf{u}\) = [1.0 2.5 77.0 6.5 6.0 6.0 1.0 0.5]\({}^{T}\), set up using the reference points. For the sake of computational efficiency, the reference designs are obtained using the feature-based optimization framework [31]. Numerical verification of the proposed approach has been realized by constructing the surrogate model for several training data sets of various sizes as reported in Table 1. The nested kriging with variable-thickness domain has been compared to conventional surrogates (kriging and radial basis functions) as well as the (original) nested kriging with fixed domain thickness [51]. The nested kriging  
Figure 6: Dual-band dipole antenna geometry [55]. Ground plane shown using the light-gray shade. models have been constructed for two values of \(T_{\text{max}}\), 0.02 and 0.05. It can be observed that both nested kriging models (with the fixed-thickness and variable-thickness domains) exhibit comparable predictive power for the training data set sizes from 200 samples up (for original nested kriging model) and 50 samples for the proposed model, which is much better than that of the conventional surrogates. For smaller data sets, the results for the proposed model are not representative because of very limited numbers of samples (15 and 19, respectively), which makes the estimation of the predictive power unreliable. It should be emphasized that the number of training samples required by the variable-thickness models is significantly smaller. The computational savings are as high as 75 percent with the average of 70 percent over the considered training data sets. Figure 8 shows the surrogate responses for the variable-thickness nested kriging model at the selected testing designs along with the corresponding EM simulation data. The visual agreement between the characteristics is satisfactory.