# Shape Optimum Design by Basis Vector Method Considering Partial Shape Dependence
## 6 Application Example  
### Shape Optimization of Steam Turbine Implantation Section  
The movable wing of the steam turbine for thermal power generation equipment is composed of a blade and a rotor. To replace the blade easily, the root of the tree-type blade is directly inserted into the rotor for assembly, as shown in Figure 5.  
When the steam turbine operates, centrifugal force is created due to the rotation of the rotor action and the concentrated stress is generated in the mutually contacting parts of the tree-type implanted part, which connects the blades and the rotating rotor. When there is improper internal stress in parts, it will seriously reduce the fatigue strength of parts; on the contrary, it will increase the life or fatigue strength of parts. Since most defects where cracks occur are seen partially, it is necessary to reduce the concentrated stress of the steam turbine as much as possible. How to obtain the optimum shape of the tree-type e implantation part is an important issue. Here, to simplify the analysis problem, as shown in Figure 5, one blade is cut out from a large number of blades installed along the circumferential direction of the rotating rotor. Further considering  
Figure 5: FEM model for fixed part of the blade. the periodic symmetry in the circumferential direction, as shown in the figure, half of the blade implanting portion is made into the FEM model, the boundary constraint condition is cyclically symmetrical in the circumferential direction, the centrifugal force is converted from the mass of the blade effective portion, and the rotational speed of the rotor is applied to the root portion of the blade. The structural design of the implant portion, upon considering the assembled condition of the parts, etc., as the basic pattern of the shape changes, is shown in Figure 6.  
\(x_{1}\): Changing the slope of the tooth tip  
\(x_{2}\): Length of implanted part  
\(x_{3}\): Width of implanted part  
\(x_{4}\): Width of implanted part  
\(x_{5}\): Change in the inclination of the tooth contact surface  
\(x_{6}\): The radius of curvature of the first hook  
\(x_{7}\): Change in the slope of the first hook  
Considering the influence of fatigue stress, the constraint condition is the stress difference between the hooks being less than or equal to the allowable value of the stress in each hook, the stress difference between the hooks is added below the allowable value, and the number of constraint conditions is 27. However, the seven shape variables shown in Figure 6 cannot be changed freely, there is a partial shape dependency relationship among them, and some shapes should be the same as far as possible. To obtain simple shapes that are easy to process, the shape changes are shown in Figure 6.  
(1): Tooth tip points \(a\), \(b\), \(c\) are always on the same straight line  
(2): The distance between \(\overline{ab}\) and \(\overline{bc}\) of a tooth tip point are always equal  
(3): The curved contour line of each hook is always the same  
An optimum shape will be determined on the premise that the requirements relating to these three shape changes are forcibly satisfied. To realize this problem, following the shape indicated by the basic shape change pattern shown in Figure 7 and using the smoothing method discussed in the previous section to generate a basis vector, shape optimization can be performed according to the algorithm shown in Figure 4.  
Figure 6: Basis vectors to change shape of fixed part. Aiming for the optimization constraint condition, the maximum mises stress value on the blade side and the rotor side in each hook is limited to the allowable amount or less. Furthermore, constraints are added to the stress difference between the hooks so stress is not concentrated on one hook, but dispersed over the whole. The number of constraint conditions is 27 in total. Figure 7 shows the stress distribution of the optimum shape and initial shape obtained by optimization calculation.