## 5 Illustrative Examples
### Vibration Reduction Optimization Design of PAF
Figure 18: RMSE of verification configuration calculated by reduced-order model constructed with CV–Voronoi sequence sampling and LHS direct sampling. The optimized results are shown in Figure 12:0 The optimized configuration can ensure that the first-order resonant frequency and mass satisfy the constraints, while the energy sum of the interest frequency range decreases significantly. The optimized energy sum of the interest frequency range is 1450.0 db, which is 31.5% lower than the initial value (2117.8 db). The optimized configuration has no resonance frequency in the [95 Hz, 105 Hz], and the energy-frequency curve has no peak value in this frequency range, which is also the main reason why the energy sum of interest decreases significantly. Figure 20 shows the energy-frequency curves before and after the vibration reduction optimization of the PAF structure. The energy-frequency curve of the optimized configuration is calculated based on the global POD ROM and the FOM, which almost completely coincide with each other. The _Te_ is 1439.72 db and 1450.0 db, respectively, and the relative error is only 0.7%. The accuracy and credibility of the optimized analysis results based on the global POD ROM are verified. Next, the optimization ability of the global POD ROM is discussed. In the offline stage, it takes 28.4 h to construct the global POD ROM. In the online stage, the vibration reduction optimization based on the global POD ROM uses 50 initial sample configurations to construct the initial surrogate model. The optimization iteration is 150 steps, and the time is 7.7 h, so the total time is 36.1 h. However, with the same time-consuming condition, the FOM can only finish 23 frequency response analyses, and it is difficult to complete the initial sampling if the same 50 initial samples are used. For comparison, 15 sample configurations are used to construct the initial surrogate model, and eight steps of optimization iteration  
Figure 12:1 Iterative convergence curve for vibration reduction optimization of the PAF (ROM). \begin{table}
\begin{tabular}{c c c c c c} \hline  & **Lower Bound** & **Upper Bound** & **Initial Value** & \
\begin{tabular}{c} **Optimized Value** \\ **(ROM)** \\ \end{tabular} &
\begin{tabular}{c} **Optimized Value** \\ **(FOM)** \\ \end{tabular} \\ \hline \(T\)1 [mm] & 1.50 & 4.50 & 3.00 & 4.50 & 4.39 \\ \(T\)2 [mm] & 1.50 & 4.50 & 3.00 & 4.27 & 3.39 \\ \(T\)3 [mm] & 1.50 & 4.50 & 3.00 & 3.40 & 3.72 \\ \(T\)4 [mm] & 1.50 & 4.50 & 3.00 & 2.56 & 2.15 \\ \(T\)5 [mm] & 1.50 & 4.50 & 3.00 & 4.47 & 3.74 \\ \(T\)6 [mm] & 1.50 & 4.50 & 3.00 & 3.11 & 1.50 \\ \(T\)7 [mm] & 1.50 & 4.50 & 3.00 & 1.50 & 2.97 \\ \(T\)8 [mm] & 1.50 & 4.50 & 3.00 & 1.52 & 2.94 \\ \(T\)9 [mm] & 1.50 & 4.50 & 3.00 & 1.97 & 3.27 \\ _Te_ [db] & - & - & 2117.8 & 1450.0 & 1821.4 \\ mass [kg] & - & - & 5.64 & 5.59 & 5.59 \\ resonant frequency [Hz] & - & - & 37.6 & 40.6 & 39.2 \\ \hline \end{tabular}
\end{table} 
Table 3: Vibration reduction optimization results of PAF. are performed. The convergence curve is shown in Figure 21, and the optimization results are shown in Table 3. By comparison, the vibration reduction optimization result based on the global POD ROM are reduced by 20.0% compared with that directly using the FOM, which verifies that the proposed approach has better optimization ability.