# Shape Optimum Design by Basis Vector Method Considering Partial Shape Dependence
## 6 Application Example
### Shape Optimization of Steam Turbine Implantation Section
Considering Figure 7, a relatively low distribution is seen in the upper part on the rotor side and the lower part on the blade side in the stress distribution of the initial shape. As a result of averaging the load sharing rate by optimization, the tooth tip inclination angle becomes larger with respect to the vertical plane, and the lateral width of the moving blade increases. Since the shape and centrifugal force of the blade implant are limited, the average stress value is determined by the ratio of centrifugal force and cross-section, so it is difficult to reduce the comprehensive stress level only by adjusting the internal shape. Therefore, the purpose of the optimization problem is to balance the stress distribution in the implant. Accompanying the increase of the width and the movement of the material, the stress on the blade side is lower. Since the stress on the rotor side tends to be higher, the angle of the contact surface becomes acute relative to the horizontal plane, and the angle of the tooth surface clamp becomes larger. Additionally, to reduce the stress concentration of the first hook, the stress balance should be maintained to increase the radius of curvature. Table 1 shows the change in the basis vector coefficient corresponding to each shape change pattern. The change range of each coefficient is from \(-1\) to \(1\), and the initial value is \(0\). Regarding the optimum shape, it coincides with the shape shown in Figure 6 where the coefficient becomes \(1\), is adopted \(100\%\), and the coefficient becomes minus changes in the direction opposite to the shape shown in Figure 6.  
Figure 7: Shape and stress distribution of fixed part of blade. Figure 8a is a distribution showing the Mises stress value for the initial shape and the optimum shape along with the surface layer of the blade. The vertical axis is the stress value, and the hook corresponding to the horizontal axis is shown in Figure 3. Similarly, Figure 8b shows the stress distribution along with the surface layer on the rotor side. Going from there, the concentrated stress of the first hook on the blade side and the fourth hook on the rotor side is reduced, in which stress is high in the initial shape. Additionally, it is found that the maximum stress at each hook is almost the same value, the stress difference of each hook is decreased, and the overall stress balance is improved. Further, as shown in Figure 7, it is understood that the assembling dimension of the blade implant portion (the distance between the hook and the hook) is not changed as compared with the original. After optimization, it is easy to confirm that the size of the position of the hook coincides with the original. That is, the position of the vertex of each hook becomes a straight line, and all three requirements for shape change are satisfied. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variable** & **Low Limit** & **Up Limit** & **Initial** & **Optimal** \\ \hline \(\mathrm{X_{1}}\) & \(-1.00\) & \(1.00\) & \(0.00\) & \(-0.401\) \\ \hline \(\mathrm{X_{2}}\) & \(-1.00\) & \(1.00\) & \(0.00\) & \(1.000\) \\ \hline \(\mathrm{X_{3}}\) & \(-1.00\) & \(1.00\) & \(0.00\) & \(0.745\) \\ \hline \(\mathrm{X_{4}}\) & \(-1.00\) & \(1.00\) & \(0.00\) & \(-0.573\) \\ \hline \(\mathrm{X_{5}}\) & \(-1.00\) & \(1.00\) & \(0.00\) & \(0.214\) \\ \hline \(\mathrm{X_{6}}\) & \(-1.00\) & \(1.00\) & \(0.00\) & \(1.000\) \\ \hline \(\mathrm{X_{7}}\) & \(-1.00\) & \(1.00\) & \(0.00\) & \(0.956\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Change of Design Variable Coefficients. Figure 8: _Cont._