## 5 Numerical Examples
### Example 2: Satellite Structure  
In this example, the proposed method was applied in a practical satellite composed of a service cabin and a payload cabin. Based on the original design, an FE model consisting of shell and beam elements was established. The boundary condition was to fix the bottom of the joint ring in view of the connecting interface between the launch vehicle and the satellite.  
The modal analysis results indicated that there was a high vibration of densities on the horizontal plate of the satellite. Figure 10 shows the local modal shape of the horizontal plate. From this basis, the design domain was specified as the horizontal plate. Figure 11 shows where the stiffener was placed, influenced by the surrounding structures. Here, the minimum structural weight was to be found with the local frequency to be extended beyond 60 Hz to insulate it from the global frequency.  
With the same techniques used in the previous example to apply the static load, the nodal displacements in the design domain were obtained. Displacements results of the satellite were shown in Figure 11.  
Figure 10: The local modal shape of the satellite.  
Figure 9: Static nodal displacement of the optimized model.  
The concordance between the displacements and the local modal shape could be seen directly from Figures 10 and 12, thus, the approximate scalar finite element model could be developed.  
A rectangular stiffener was placed where the maximum displacement occurred to increase the local frequency. In the approximate problem, the design variables were the cross-section dimensions of the stiffeners. The minimum structural weight was to be found. The initial design was \(H=W=0.01\) m; lower bounds of 0.001 m and upper bounds of 0.1 m were imposed on both design variables.  
In the optimization problem, the local frequency was constrained to be at least 60 Hz, which is 1.43 times the initial. Therefore, in approximate optimization problem, nodal displacements on the reinforced stiffeners were constrained to be at most 0.029.  
The whole optimization took four iterations to reach the optimal solution. In the last two iterations, there was not much change in the objective function. The optimization results are listed in Table 4.  
Figure 11: The arrangement of the stiffener.  
Figure 12: Displacement results of the satellite.  
The weight of stiffener accounted for 1.351% of the local plate weight and 0.043% of the initial total weight. The small percentage also reflected that the results were consistent with Assumption 1. The difference of the frequency order between the initial and optimized design indicated that the order change of local frequency was remarkable. Considering verifying the accuracy of our approximate model, an analysis for the local mode using the plate with the optimized parameters resulted in the modal shape shown in Figure 13. The displacements constraint and the frequency were both satisfied, and there was also a small relative error of 0.54%. Upon further observation of the optimized mode, approximate similarity of the optimized and initial modal shape was found, which corresponded to Assumption 2. This example verified the effectiveness of the proposed method in practical problem.