## 4 Numerical Results
### Case II: Broadband Patch Antenna  
The second example is a broadband patch antenna with a narrow ground plane shown in Figure 11[56]. The structure is described by five parameters \(\mathbf{x}=[W\,L\,dW\,W_{g}\,h_{r}]^{T}\). The computational model is implemented in CST Microwave Studio and evaluated using the transient solver (-400,000 cells, simulation time 94 seconds). The substrate parameters, dielectric permittivity \(\varepsilon_{r}\) and height \(h\) are the operating conditions being a part of the objective space. The model incorporates the SMA connector. The design optimality is understood as minimization of the antenna reflection within at least 10-percent fractional bandwidth symmetric w.r.t. the target center frequency \(f_{0}\). We aim at constructing the surrogate model within the objective space defined by the following ranges of the center frequency and surrogate parameters: 3.0 GHz \(\leq\)\(f_{0}\)\(\leq\)\(6.0\) GHz, permittivity \(2.0\)\(\leq\)\(\varepsilon_{r}\)\(\leq\)\(5.0\) and height \(0.5\) mm \(\leq\)\(h\)\(\leq\)\(1.0\) mm. There are thirteen reference designs assigned (shown in Figure 12), corresponding to \(\{\)\(\ell_{0}\)\(,\)\(\varepsilon_{r}\)\(,\)\(h\)\(\}=\) {3,0.2,0.0.5, 3,0.2,0.1, 0}, {3,0.5,0.0,5}, {3,0.5,0.1,0}, {4,5,3,5,0.75}, {4,5,3,5,0.5}, {4,5,3,5,1.0}, {4,5,2,0,0.75}, {4,5,5,0,0.75}, {6,0,2,0.0,5}, {6,0,2,0,1,0}, {6,0,5,0,0,5}, and {6,0,5,0,1,0}. Figure 10: Antenna of Figure 6 optimized for \(f_{1}=2.45\) GHz, \(f_{2}=3.3\) GHz: (**a**) photographs of the prototype, (**b**) reflection \(|S_{11}|\), (**c**) end-fire realized gain, (**d**) \(yz\)-plane radiation pattern at both operating frequencies (2.45 GHz and 3.3 GHz on the left- and right-hand-side, respectively). Simulated and measured characteristics shown in gray and black, respectively. The parameter space \(X\) is determined by the lower and upper bounds for design variables \(I=[12.5\) 10.0 4.0 8.0 0.02]\({}^{T}\), and \(\mathbf{u}=[40.0\) 34.0 17.0 10.0 0.2]\({}^{T}\). Model validation has been arranged similarly as for the first example. Table 2 shows the numerical data for the models constructed using the training data sets of different sizes. The benchmark methods include conventional models established in the parameter space \(X\) (kriging interpolation and radial basis functions), as well as the nested kriging with fixed-thickness domain. The nested kriging models have been constructed for the two values of \(T_{\max}\), 0.025 and 0.05. The obtained results are consistent with those reported in Section 4.1. In particular, the accuracy of both nested kriging models (with the fixed-thickness and variable-thickness domains) is essentially the same; however, the number of samples required by the variable-thickness domain model is considerably smaller. Figure 11: Broadband patch antenna [56]: (a) geometry (ground plane shown using light gray shade), (b) 3D views. Figure 12: Allocation of the reference designs for the antenna of Figure 11: (a) objective space, (b) selected three-dimensional projections of the parameter space. The computational savings are close to forty percent. The responses for the variable-thickness nested kriging model at the selected testing designs along with the corresponding EM simulation data are shown in Figure 13.  
The proposed surrogate has been employed for antenna optimization to provide an additional validation. The objective was to assess whether variable-thickness domain affects the model capability to reach the optimum design across the objective space. The results obtained for selected objective vectors have been shown in Figure 14. It can be noted that the designs produced by variable- and fixed-thickness domain models are of comparable quality, which corroborates the design utility of the presented technique. One of the optimized designs, corresponding to \(f_{0}\) = 4.8 GHz, \(\varepsilon_{r}\) = 3.38, \(h\) = 0.51 mm, has been fabricated and measured for additional validation. The antenna structure has been implemented on Rogers RO4003C substrate.