## 5 Illustrative Examples
### Vibration Reduction Optimization Design of PAF
The initial mass of the PAF is 5.64 kg (without satellite), and the initial first resonance frequency is 37.6 Hz. Figure 14: Geometry parameters of PAF. Figure 13: Schematic diagram of PAF [38; 39]. Five sample configurations are sampled in the design space [1.5 mm, 4.5 mm] as the initial sample configuration set to form the initial ROM. In order to verify the prediction accuracy of the initial ROM in the whole design space, 20 sample configurations are sampled by LHS, and the energy-frequency curves are obtained based on the FOM and the ROM respectively, and the RMSE of the two are computed. As shown in Figure 16a, the maximum RMSE is 0.533, and the corresponding results of the ROM and the FOM of the No.13 sample configuration are shown in Figure 16b. The curve deviation of the two is obvious, indicating that the accuracy of the initial ROM is insufficient, and the reduced basis matrix needs to be updated to improve the global accuracy of the ROM. The CV-Voronoi sequence sampling method is used to increase the sample configurations to realize the adaptive global POD ROM update. According to the same test in Section 5.1, for this problem, when the maximum RMSE of energy-frequency curves of the FOM and the ROM is less than 0.15, the ROM has a relatively high precision. Therefore, the RMSE truncation threshold is set as 0.15 in this example. The convergence condition is satisfied after the continuous increase of 10 sample configurations. A global POD ROM is constructed based on 15 sample configurations. The 20 sample configurations mentioned above are used to check the accuracy of the updated ROM. The RMSE is shown in Figure 17a. The RMSE corresponding to all verified configurations is less than 0.15, and the maximum value is 0.146, which corresponds to the No. 3 sample configuration. The results based on the ROM and the FOM are shown in Figure 17b, and the curves of the two are almost completely coincidental. Compared with the accuracy of the initial ROM, the accuracy based on the updated global POD ROM is greatly improved. This shows that the updated ROM has higher prediction accuracy in the design space. The average single computation time based on the global POD ROM is 138.6 s, which is 97.6%  
Figure 16: Accuracy of the verified configurations computed by initial global POD ROM: (**a**) RMSE values; and (**b**) energy–frequency curves of No.13 verified configuration. Figure 15: Energy frequency response of initial configuration calculated by FOM of PAF. less than the average single computation time of the FOM (5849.6 s), indicating that the proposed approach can significantly improve the computational efficiency of frequency response analysis. To verify the advantages of the CV-Voronoi sequence sampling method to adaptively update the global POD ROM, LHS is used to directly sample 15 configurations to construct a global POD ROM. Then, the same 20 sample configurations are used to verify the accuracy of the ROM constructed by LHS. The obtained RMSE is compared with that of the ROM constructed adaptively by CV-Voronoi sequence sampling, as shown in Figure 18. The RMSE with the CV-Voronoi sequence sampling method is generally low. The RMSE with LHS method is significantly lower at the 3, 6, 11, 16 and 17 validation configurations. This reflects the advantages of the CV-Voronoi sequence sampling method in constructing the global POD ROM. Furthermore, the global POD ROM is used instead of the FOM for the vibration reduction optimization design of the PAF structure. In the optimization design stage, 50 sample configurations are selected to calculate the corresponding responses to construct the initial Kriging surrogate model. The upper limit of iteration is 300. The iteration stops when the maximum number of iteration steps is reached or Formula (27) is satisfied. The iterative convergence curve is shown in Figure 19.  
Figure 17: Accuracy of the verified configurations computed by updated global POD ROM: (**a**) RMSE values; and (**b**) energy–frequency curves of No.3 verified configuration.