# Shape Optimum Design by Basis Vector Method Considering Partial Shape Dependence
## 8 Summary and Conclusions  
While analyzing the three-dimensional shape optimization problem, the following conclusions were obtained in detail during the study of the internal shape control problem, the so-called partial shape subordination problem with shape change of the external structure boundary. 1. The partial shape subordination problem was classified into three basic modes: the corresponding relation of the line segment formed by any two points, the proportional relationship of the distance  
Figure 16: Remeshing problem of analysis model, (**a**) \(w_{i}=0\), (**b**) \(w_{i}=1\), (**c**) \(w_{i}=-1\). between any three points, and the maintaining relationship of the angle formed by any three points. It was theoretically proved that the method of controlling the internal shape correctly with the change of the shape of the external boundary was used in the analysis of complex structural shape optimization problems. Thus, the problem of partial shape subordination which has not been adequately solved. 2. To solve the shape optimization problems of complex structures, including some shape related problems, the basic vector method was proposed, and an actual optimization example was used to verify the proposed optimization method. The results show that the optimization method was effective for the partial shape subordination problem in complex structural shape optimization problems. 3. Regarding the most important reference vector-making problem in the application of a reference vector, the study put forward the method of making a reference vector based on the smoothing method, and made a reference vector correctly according to the requirements of optimization design. Additionally, the combination of CAD shape corresponding to the basic shape change mode was applied to more practical optimization problems. 4. To use the linear combination of multiple reference vectors to represent the shape change of shape optimization, before the implementation of shape optimization, the relationship between the reference vectors was studied. The MAC value used in evaluating the relationship between vibration modes was used, and the reference vectors were linear. The method of quantity was confirmed to be helpful in improving the stability of the analysis and calculation of complex shape optimization problems. 5. During the process of shape optimization analysis, along with the change of the structure shape, the resolution reset changed greatly, and the quality of some analytic grids was significantly reduced. To avoid the problem where the normal structural analysis could not be continued, before the implementation of shape optimization and to avoid that every basic shape change pattern will have an analytic grid, the study proposed a checking method and remeshing method to solve the problem. During the process of shape optimization calculation, the degradation of the analysis grid resulted in the decrease of calculation. 6. Here, two typical cases were verified by the basic vector method. Similar problems with the case are very common in practice. The research results of this paper can be well verified from the two typical cases (steam turbine and engine mount bracket). It was seen from the result that the stress value of the final optimization result (solid line) was low, and the stress value of each point was uniformly distributed, which avoided the occurrence of stress concentration. Taken from this result, the effectiveness of the basic vector method proposed in this paper was well proved. This method can be well solved, which is of great help to actual engineering problems. Based on the results of this research, the partial shape dependency problem was able to be solved and it made a step forward toward a practical application that effectively solves the complicated structure optimization problem frequently found in the design site.