# Shape Optimum Design by Basis Vector Method Considering Partial Shape Dependence
## 6 Application Example
### Stress Distribution Optimization of the Engine Mount Bracket  
Figure 9 shows a model of an engine mount bracket for fixing a passenger car engine. The load conditions occur when applying sudden braking, and the uniform load in the oblique direction is applied along the lower half inner diameter of the upper hole. The value of the uniform load is 6000 N in the axial direction, 23,000 N in the lateral direction, and 45,000 N in the downward direction.  
The dotted line in Figure 10 represents the range in which the structural shape under design can be changed and, in the optimization calculation, these change ranges represent the basis vectors. Furthermore, by adjusting the ratio and combination of each basis vector, the optimum shape can be obtained.  
The objective function of the optimization calculation minimizes the weight of the structure. Concerning the constraint condition, the maximum stress of each part is set to the allowable stress value of 140 MPa or less. Optimum solutions are obtained by optimization, and the optimal parameters corresponding to the shape change pattern represented by the dotted line in Figure 10 are shown in Table 2. Regarding the table, column 1 is the shape parameter, column 2 is the maximum value of the shape parameter, column 3 is the initial value of the shape parameter, column 4 is the optimum value of the shape parameter, and column 5 is the adoption rate of the shape change pattern.  
Figure 8: Stress distribution along face of blade side and rotor side. (**a**) Stress distribution along face of blade side, (**b**) Stress distribution along face of rotor side.  
Figure 9: An engine mount bracket for fixing a passenger car engine.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Maximum** & **Initial** & **Optimum** & **Adopt. (\%)** \\ \hline X\({}_{1}\) & 5.0 & 10.0 & 6.9 & 62\% \\ \hline X\({}_{2}\) & 8.4 & 0.0 & 8.4 & 100\% \\ \hline X\({}_{3}\) & 18.0 & 0.0 & 18.0 & 100\% \\ \hline X\({}_{4}\) & 54.0 & 45.0 & 52.9 & 88\% \\ \hline X\({}_{5}\) & 25.0 & 15.0 & 20.0 & 50\% \\ \hline X\({}_{6}\) & 60.0 & 30.0 & 57.0 & 90\% \\ \hline X\({}_{7}\) & 6.6 & 0.0 & 4.6 & 70\% \\ \hline X\({}_{8}\) & 7.0 & 0.0 & 7.0 & 100\% \\ \hline X\({}_{9}\) & 5.4 & 0.0 & 5.4 & 100\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Change of design variables (mm).  
Figure 10: The structure and shape change area.  
Considering the stress distribution diagram shown in Figure 11, in the optimization calculation, each of these change ranges is represented by a basis vector. Concerning the original shape, the maximum Mises stress is 98.4 MPa, and there is a significant difference from the allowable value, which shows that useless material is used more than necessary. Regarding the optimized shape, the maximum Mises stress is 136.8 MPa higher than that of the original shape. Still, the difference from the allowable value is small, the stress in each part is dispersed, and the performance of the material can be fully utilized.  
Compared to the original in Figure 11, concentrated stress is reduced and stress is dispersed. Weight decrease, as the objective function, is 49.9% lighter than the original shape. Additionally, it shows that the requirements for the relative relationship between the external boundary shape and the inner hole are all satisfied and the partial shape subordination problem is solved.