## 5 Illustrative Examples
### Vibration Reduction Optimization Design of S-Shaped Curved Stiffened Shell
Figure 4: Geometric parameters of the S-shaped curved stiffened shell structure: (**a**) reference line, and (**b**) cross-section along the reference line. The design variables for vibration reduction optimization of the S-shaped curved stiffened shell include a total of 11 parameters, including the thickness of five sections of skin (S1, S2, S3, S4, S5), the thickness of five sections of axial stiffeners (\(T1\), \(T2\), \(T3\), \(T4\), \(T5\)), and the thickness of ring stiffeners (\(Tu\)). The optimization objective is to minimize the energy in the interest frequency range. The constraints are that the optimized configuration mass is not higher than the initial configuration mass, and the optimized first resonance frequency is not less than the initial value. The optimization formulation is:  
\[ \begin{array}{rl}\min:&\mathit{Te}=&\sum_{\omega_{a}}^{\omega_{b}}\overline{ E}(\omega)\\ \text{s.t.}:&\mathit{x}_{i}^{lb}\leq&\mathit{x}\leq\mathit{x}_{i}^{up},i=1, \ldots,I\\ &\mathit{Mass}(\mathit{x})\leq\mathit{Mass}_{0}\\ &\mathit{freq}(\mathit{x})\geq\mathit{freq}_{0}\end{array}  \tag{26}\]  
where \(\mathit{x}\) is the vector of design variables, and \(\mathit{x}_{i}^{lb}\) and \(\mathit{x}_{i}^{ub}\) are the lower and upper bound of the design variables, respectively. \(\mathit{Mass}(\mathit{x})\) represents the mass of the structure, and \(\mathit{Mass}_{0}\) is the mass of the initial configuration (657.92 kg). The term \(\mathit{freq}(\mathit{x})\) is the first resonance frequency; \(\mathit{freq}_{0}\) is the first resonance frequency of the initial configuration (60.1 Hz). \(\mathit{Te}\) is the energy sum of the interest frequency range [\(\omega_{a}\),\(\omega_{b}\)]. Five sample configurations are sampled in [1.5 mm, 4.5 mm] as the initial sample configurations, and the initial reduced basis matrix is obtained to reduce the order of the stiffness matrix, damping matrix and mass matrix to construct the initial ROM. In order to verify the prediction accuracy of the initial reduced basis matrix in the entire design space, the LHS is used to obtain 20 sample configurations, and the energy-frequency curves are obtained based on the FOM and the ROM respectively, and then RMSEs are calculated. As shown in Figure 6a, the maximum RMSE is 1.082, and the corresponding results of the ROM and the FOM of the No.1 sample configuration are shown in Figure 6b. The curve deviation of the two is obvious, indicating that the accuracy based on the initial reduced basis 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 adaptively global POD ROM update. An inappropriate truncation threshold increases offline time under the premise of limited improvement in the global POD ROM accuracy. To confirm the truncation threshold for the global POD ROM update, a simple test is done. For a certain sample configuration, the first 5, 10, 15, 20, 25, 30, and 35 mode shapes are used, respectively, to construct the ROM by the POD method. The ROM has a higher accuracy only for the confirmed sample configuration, and its global accuracy is low. For the FOM and each of the ROMs distinguished by the number of mode shapes,  
Figure 5: Energy–frequency response of initial configuration calculated by the FOM of the S-shaped curved stiffened shell. the maximum relative error in the frequency response energy at the frequency point and the RMSE value of the energy-frequency curves are calculated, as shown in Figure 7. When the number of mode shapes for ROM is 25, the RMSE value is 0.09, and the maximum energy relative error is 7.3%. With the increase in the number of mode shapes, the accuracy gains of the ROM decrease. The global POD ROM has the same rule. If the truncation threshold is less than 0.1, more time will be spent, and the accuracy improvement will be limited.