# Optimization Design of Lattice Structures in Internal Cooling Channel of Turbine Blade
## 4 Results and Analysis
### Sensitivity Analysis
#### 4.1.1 Overall and Single Sensitivity Analysis for the Results of Op-I  
Figure 7a shows that both for PLS or XLS, the Pareto front was an exponential distribution, approximately. \(Nu\) of XLS was always bigger than that of PLS, under the same \(\overline{\rho}\), as XLS was regarded as two PLS composed together. Thus, the heat conduction of XLS was better than PLS. Tables 3 and 4 show the distribution for the Pareto solutions of \(Nu\) and \(\overline{\rho}\) for Op-I, respectively. It can be found that the Pareto front was mainly concentrated in the region of higher \(Nu\) both for PLS and XLS. For PLS, the optimal Pareto solutions of \(Nu\) between 130 and 148 were 73.45% of the total set of solutions of \(Nu\). For XLS, the optimal Pareto solutions of \(Nu\) between 150 and 172 were 79.1% of the total set of solutions of \(Nu\). It can also be found that the set of Pareto solutions of \(\overline{\rho}\) were relatively uniform in both PLS and XLS. Therefore, Op-I is more sensitive to \(Nu\) than to \(\overline{\rho}\).  
Figure 7: Influence of main parameters of genetic algorithm on the stability: (**a**) Sizes of population; and (**b**) algebras of heredity.  
As discussed above, under the same \(\overline{p}\), \(Nu\) of XLS was greater than \(Nu\) of PLS, so only XLS was selected for the analysis of single sensitivity.  
As shown in Figure 7b, the optimal solutions of \(D\) concentrated between 2.3 mm and 2.9 mm, and the total distribution was relatively uniform. Figure 8c shows that the optimal solution of \(\omega\) was between \(52^{\circ}\) and \(66^{\circ}\), which was mainly concentrated between \(58^{\circ}\) and \(66^{\circ}\). The diffusion level of \(\omega\) was bigger than \(D\). Therefore, \(D\) had a greater impact on Op-I than \(\omega\). In addition, the correlation coefficient \(r\) was used to represent the single sensitivity of each parameter, and its calculation formula is as follows:  
\[r_{XY}=\frac{\sum\big{(}X-\overline{X}\big{)}\big{(}Y-\overline{Y}\big{)}}{ \sqrt{\sum\big{(}X-\overline{X}\big{)}^{2}}\sqrt{\sum\big{(}Y-\overline{Y} \big{)}^{2}}}=\frac{\sum xy}{\sqrt{\sum x^{2}\sqrt{\sum y^{2}}}}=\frac{S_{XY}} {S_{X}S_{Y}} \tag{32}\]  
where \(x=X-\overline{X}\), \(y=Y-\overline{Y}\), \(S_{XY}\) is the total variation of samples, \(S_{XY}=\sum\big{(}X-\overline{X}\big{)}\big{(}Y-\overline{Y}\big{)}/(n-1)\); \(S_{X}\) is the standard deviation of X sample, \(S_{X}=\sqrt{\sum\big{(}X-\overline{X}\big{)}^{2}/(n-1)}\); \(S_{Y}\) is the standard deviation of Y sample, and \(S_{Y}=\sqrt{\sum\big{(}Y-\overline{Y}\big{)}^{2}/(n-1)}\). Among them, the positive sign indicates a positive correlation, and the negative sign indicates a negative correlation.  
At the same time, the larger these values, the stronger is the correlation between two variables, and the stronger is the single sensitivity, and vice versa.  
Figure 8d shows that the set of Pareto optimization solutions of \(\omega\) was the most sensitive for \(Nu\), while the set of Pareto optimization solutions of \(D\) was the most sensitive for \(\overline{p}\).  
In conclusion, the overall performance of Op-I was that the objective function of \(Nu\) had the highest sensitivity for Op-I, \(\omega\) had the highest correlation or sensitivity for \(Nu\), and \(D\) had the highest correlation or sensitivity for \(\overline{p}\).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\overline{p}\) (\%) & [0.5, 3) & [3, 6) & [6, 9) & [9, 12) & [12, 15] \\ \hline Pareto (PLS) & 65 & 51 & 33 & 38 & 39 \\ \hline Pareto (XLS) & - & 82 & 56 & 39 & 43 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The distribution for Pareto solutions of \(\overline{p}\) for Op-I.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(Nu\) & [116, 120) & [120, 130) & [130, 139) & [139, 148] \\ \hline Pareto (PLS) & 23 & 37 & 81 & 85 \\ \hline \(Nu\) & [141, 150) & [150, 160) & [160, 172] & - \\ \hline Pareto (XLS) & 46 & 76 & 98 & - \\ \hline \hline \end{tabular}
\end{table}
Table 3: The distribution for Pareto solutions of \(Nu\) for Op-I.