# Shape Optimum Design by Basis Vector Method Considering Partial Shape Dependence
## 7 Discussion
### Mutual Independence Problem of Basis Vectors  
When changing the structure shape using the shape transformation Equation (3), for the combination of arbitrary weighting coefficients \(w_{i}\), the shape vector obtained by calculation using Equation (7) is considered to be represented by a linear combination of the basis vectors. Here, the mutual relationship between each basis vector is considered.  
Figure 15 shows a structure composed of two elements, and its nodes 1, 2, and 3 are defined as shape change nodes. The basis vectors corresponding to the original shape and the basic shape change pattern shown in Figure 15a-d are expressed by Equations (36)-(39).  
\[\begin{split}\alpha_{0}&=\{10,40,10,20,10,0\}^{T} \\ \alpha_{1}&=\{30,40,20,20,10,0\}^{T}\\ \alpha_{2}&=\{10,40,20,20,30,0\}^{T}\\ \alpha_{3}&=\{20,40,20,20,20,0\}^{T}\\ \end{split} \tag{36}\]  
Further, the shape change vector represented by the difference between the basic shape change pattern and the basis vector of the original shape is expressed by Equation (37).  
\[\begin{split}\beta_{1}&=\alpha_{1}-\alpha_{0}=\{20, 0,10,0,0,0\}^{T}\\ \beta_{2}&=\alpha_{2}-\alpha_{0}=\{0,0,10,0,20,0\}^{T} \\ \beta_{3}&=\alpha_{3}-\alpha_{0}=\{10,0,10,0,10,0\}^{T} \\ \end{split} \tag{37}\]  
Here, to evaluate the correlation between the two shape change vectors \(\beta_{1}\) and \(\beta_{2}\), it is necessary to evaluate using the MAC (Modal Assurance Criterion) value calculated using the \(MAC(\beta_{1},\beta_{2})\) value calculation, Equation (38), used for evaluating the mutual relationship of the vibration modes.  
\[MAC(\beta_{1},\beta_{2})=\frac{\left(\beta_{1}^{T}\beta_{2}\right)^{2}}{\left( \beta_{1}^{T}\beta_{1}\right)\left(\beta_{2}^{T}\beta_{2}\right)} \tag{38}\]  
Figure 14: An original shape having a thick walled cylindrical shape.  
Using Equation (38), it is understood that \(0\leq MAC(\beta_{1},\beta_{2})\leq 1\). When \(MAC(\beta_{1},\beta_{2})=1\) shape change vectors \(\beta_{1}\) and \(\beta_{2}\) have a linearly independent relationship, it indicates that there is no mutual influence between shape change patterns represented by \(\beta_{1}\) and \(\beta_{2}\). When \(MAC(\beta_{1},\beta_{2})=0\) it indicates that there is a proportional relation \(\beta_{1}=K\beta_{2}\) between shape change vectors \(\beta_{1}\) and \(\beta_{2}\), and eliminating one in \(\beta_{1}\) and \(\beta_{2}\) does not affect the shape optimization result.  
Therefore, with respect to the shape change vector of Equation (37), the MAC value between each shape change vector is calculated using Equation (38), and the result becomes Equation (39).  
\[MAC(\beta_{1},\beta_{2})=0.04\;\;MAC(\beta_{2},\beta_{3})=0.6\;\;MAC(\beta_{1},\beta_{3})=0.6 \tag{39}\]  
Considering the case of shape optimization by the basis vector method, it is ideal if there is a mutual linear independent relationship between each shape change vector; however, as shown in Figure 7, when a plurality of basic shape change patterns are set along the same direction with respect to one boundary shape, as shown in Equation (39), a certain degree of change--even though there is a linear relationship since it is possible to express shape change accurately by a combination of shape change vectors--is verified by the results of numerical analysis that the result of shape optimization is hardly affected.  
However, if the linear combination between the shape change vectors is potent and the MAC value is too close to 1, then the design sensitivity components corresponding to the design variable may interact with each other. When the corresponding basic shape change modes are adjusted, for example,  
Figure 15: An example for independence of basis vectors. (**a**) Original shape, (**b**) Basis shape 1, (**c**) Basis shape 2, (**d**) Basis shape 3.  
two basic shape change modes are changed to one. It can greatly improve the analytical efficiency and quality of shape optimization.