## 5 Numerical Examples
### Example 1: Thin Plate  
As described in Reference [15], the minimum structural weight of the plate was to be found with the local frequency constrained to be not lower than 60 Hz. The FE model of the plate clamped at one end is shown in Figure 2, and it used a total of 20 \(\times\) 20 elements of 50 \(\times\) 50 mm. The left side of the plate was fixed, and the materials in the white area and the black area were \(M_{1}\) and \(M_{2}\), respectively, shown in Table 1. The dimensions of the plate were \(L_{1}=1\) m, \(L_{2}=1\) m and the thickness \(T=0.02\) m.  
Figure 1: Optimization flowchart. The first four orders of natural frequencies and modal shapes of the initial plate are shown in Figure 3. It was obvious that the third order mode was the local mode of the initial plate and the initial local frequency was 21.382 Hz. A reinforced stiffener, whose cross section is shown in Figure 4, was placed where the maximum displacement occurred to improve the local frequency. The initial design was \(H=W=0.01\) m and \(t_{1}=t_{2}=0.002\) m. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Material** & \(E\) & \(\
u\) & \(\rho\) \\ \hline \(M_{1}\) (white) & 200 \(\times\) 10\({}^{9}\) N/m\({}^{2}\) & 0.3 & 7800 kg/m\({}^{3}\) \\ \(M_{2}\) (black) & 200 \(\times\) 10\({}^{3}\) N/m\({}^{2}\) & 0.3 & 78 kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material property. Figure 3: The first four order mode shape of the initial model. Figure 2: FE model of the plate. Performing a modal analysis and the first four orders of the natural frequencies and modal shapes of the plate with the reinforced stiffener is shown in Figure 5. The natural frequencies changed slightly and the third order mode was also the local mode. In this way, modifying the design of the cross section to satisfying the local frequency constraint with the minimum mass increasement was the optimization problem. The nodal equilibrium forces of the local mode was obtained directly from the modal analysis, and what followed was to establish a static load condition according to above-obtained forces. Then, we continued the static analysis based on the above-established load condition. It was apparent that the nodal displacements from the static analysis shown in Figure 6 were consistent with the third order modal shape from the modal analysis shown in Figure 5, conforming to the assumptions made when defining static loads. Figure 4: FE model of the stiffened plate. Figure 5: The first four order mode shape of the model with the reinforced stiffener. The approximate optimization model was then established with the objective of structural mass, the design variables of cross-section dimensions shown in Table 2, and the constraint of nodal displacement. In the initial optimization problem, the local frequency was constrained to be not lower than 60 Hz, and the nodal displacements could also be calculated by Equation (16) accordingly. Therefore, in the approximate problem, the magnification of the constrained local frequency was \(P=60/22.504=2.67\), the nodal displacements on the reinforced stiffener were constrained less than \(u_{max}=4.12\) m/\(2.67^{2}=0.58\) m.  
The optimization results are listed in Table 3. The iteration history is shown in Figure 7. The final structural weight increased by 0.09 kg compared with the initial weight, complying with the above Assumption 1. The maximum displacement from the static analysis was 0.359 m, satisfying displacement constraints condition of the approximate optimization model. \begin{table}
\begin{tabular}{l l l l} \hline
**Design Variables** & **Initial Design** & **Lower Bound** & **Upper Bound** \\ \hline \(H\)/m & 0.01 & 0.001 & 0.1 \\ W/m & 0.01 & 0.001 & 0.1 \\ \(t_{1}\)/m & 0.002 & 0.001 & 0.1 \\ \(t_{2}\)/m & 0.002 & 0.001 & 0.1 \\ \hline \end{tabular}
\end{table} 
Table 2: Design variables.