## 5 Illustrative Examples
### Vibration Reduction Optimization Design of S-Shaped Curved Stiffened Shell
At the same time, when the maximum energy relative error is 7.3%, the relative error of the objective frequency band energy sum is 0.09%. Considering the calculation time and accuracy, when the maximum RMSE of the energy-frequency curves of the FOM and the ROM is less than 0.10, the ROM has a relatively high precision. The convergence condition is satisfied after the continuous increase of 18 sample configurations. A global POD ROM is constructed based on 23 sample configurations. The 20 sample configurations mentioned above are used to check the accuracy of the updated ROM. The RMSE is shown in Figure 8a. The RMSE corresponding to all verified configurations is less than 0.10, and the maximum value is 0.066, which corresponds to the second sample configuration. The results based on the ROM and the FOM are shown in Figure 8b, 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 115.2 s, which is 96.9% less than the average single computation time of the FOM (3752.8 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 in adaptively updating the global POD ROM, LHS is used to directly sample 23 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 9. The RMSE with CV-Voronoi sequence sampling method is generally low. The RMSE with LHS method is significantly lower at the 1, 11, and 18 validation configurations. It should be noted that the above comparison is based on the premise that the number of LHS is known. But in engineering practice, it is difficult to directly determine the appropriate sampling number without prior experience. This further reflects the advantages of CV-Voronoi sequence sampling method in constructing the global POD ROM. Figure 6: Accuracy of the verified configurations computed by initial global POD ROM: **(a)** RMSE values; and **(b)** energy–frequency curves of the No.1 verified configuration. Figure 8: Accuracy of the verified configurations computed by updated global POD ROM: (**a**) RMSE values; and (**b**) energy–frequency curves of the No.2 verified configuration. Figure 7: Maximum relative errors of the frequency response energy and the RMSE values of the energy–frequency curves of the FOM and the test ROM. Figure 9: RMSE of verification configuration calculated by ROM constructed with CV–Voronoi sequence sampling and LHS direct sampling. Then, the modal superposition method and the modal acceleration method are used, respectively, for frequency response analyses of the initial configuration in comparison to the global POD ROM. The first 100 mode shapes are selected. The results in Table 1 show that the global POD ROM has higher efficiency and accuracy. It should be noted that when the structure has multiple components or different materials, the damping matrix does not satisfy the mode shape orthogonalization, and the frequency response analysis cannot reduce the order through the mode shape space. The global POD ROM has more potential in the order reduction calculation of complex structures because it does not consider the structure of the actual physical model. Furthermore, the global POD ROM is used instead of the FOM for the vibration reduction optimization design of the S-shaped curved stiffened shell. 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.