## 3 Results
### Optimization Summary  
Optimization results obtained from the metamodel were verified using the FEM method and are compared in Figure 21. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name** & **Baseline** & **Candidate 1** & **Candidate 2** & **Candidate 3** \\ \hline Shell\_1\_THCK (in) & & & 0.100 & \\ Shell\_2\_THCK (in) & & & 0.100 & \\ Shell\_3\_THCK (in) & & & 0.100 & \\ FWD Hook THCK (in) & & & 0.150 & \\ AFT Hook THCK (in) & & & 0.090 & \\ FWD Hook Lean (degree) & 90 & 80.577 & 80.013 & 80.122 \\ AFT Hook Lean (degree) & 90 & 80.329 & 80.343 & 80.446 \\ Nozzle Shell THCK (in) & & & 0.060 & \\ FWD Hook Pos (in) & 0 & 0.037 & 0.038 & 0.036 \\ AFT Hook Pos (in) & 0 & \(-\)0.050 & \(-\)0.050 & \(-\)0.050 \\
1 st System mode Frequency (Hz) & 118.0 & 124.4 & 124.4 & 124.4 \\ Geometry Surface Area (in\({}^{2}\)) & 18.965 & 18.978 & 18.978 & 18.979 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Optimization results—Objective function \(J_{\text{2B}}\). Figure 21: _Cont._  
Figure 20: Convergence history for the first mode’s natural frequency. The maximum error was found for the metamodel with 10 parameters set and is equal to 2%. The results have been summarized Figure 22 and Table 8, showing candidates obtained from optimization processes together with a baseline for reference using validated FEM data. The highest frequency shift is obtained for the scenario with objective function \(J_{1}\) (10 geometrical variables set) and is equal to 21%. At the same time, this solution shows the highest impact on model area so in consequence it also affects the component's weight. The scenario reported as objective function \(J_{\text{2A}}\) (geometrical parameters associated with nozzle only) is not able to reach the same frequency shift as for the optimization process  
Figure 21: Metamodel validation with FEM model results. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Objective Function**} \\ \cline{2-5}  & \multicolumn{1}{c}{\(J_{\text{1}}\)} & \(J_{\text{2A}}\) & \(J_{\text{2B}}\) \\ \hline Frequency, Hz & 118.03 & 143.2 & 128.2 & 123.6 \\ Model area, in\({}^{2}\) & 18.965 & 19.111 & 19.047 & 18.972 \\ Frequency shift, \% & & 21\% & 9\% & 5\% \\ Effectiveness & & & & \\ Area/Freq shift & & 0.006 & 0.008 & 0.001 \\ (Area cost per 1 Hz) & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Results of the optimization process. Figure 22: Optimization summary results with baseline for reference. with all parameters included. Moreover, effectiveness was calculated (Equation (28)), so the cost in terms of area per one Herz is similar between \(J_{1}\) and \(J_{2\text{A}}\). \[Effectiveness=\frac{A_{\text{{\tiny{CANDIDATE}}}}-A_{\text{{\tiny{BASELINE}}}}}{f_{ \text{{\tiny{CANDIDATE}}}}-f_{\text{{\tiny{BASELINE}}}}} \tag{28}\]  
The only valuable solution from a weight perspective is \(J_{2\text{B}}\) (parameters related only to forward and rear hook position and leaning). The obtained frequency shift is 5% and is limited by the geometrical parameter's lower and upper bounds. A geometry comparison between baseline configuration (blue lines) and proposed design (red lines) has been reported in Figure 23 for scenario \(J_{1}\) and Figure 24 for scenario \(J_{2\text{B}}\). The main difference is related to the casing shell thickness between the rails and the hook leaning. The number of iterations varies from 13 for scenario \(J_{1}\) to 16 for scenario \(J_{2\text{B}}\). The main reason why the iteration number increases is related to the available parameters for the optimization task and how the convergence stability percentage criterion is defined. The denominator in Equations (25) and (26) will be greater for scenario \(J_{1}\) compared to scenario \(J_{2\text{B}}\), leading to a tighter convergence criterion for scenarios with parameters associated with the hook position and leaning. To keep the same absolute criterion on output percentage should be adjusted.