# Shape Optimum Design by Basis Vector Method Considering Partial Shape Dependence
## 3 Basis Vector Method Used for Shape Optimization  
To deal with the partial shape dependency problem mentioned in the previous section, the basis vector method is applied. The basis vector method is a method in which a movement of a certain group of nodes is regarded as a pattern of one shape change, corresponding with one shape design variable for some shape change patterns and adds each weighting factor to obtain a new shape. Here, the principle of the basis vector method will be explained using the example shown in Figure 2. Figure 2a shows the initial shape, Figure 2b shows the basic shape change pattern 1 in which the entire shape falls forward. Figure 2c shows the basic shape change pattern 2, in which the upper-end face of the shape is reduced by half in the front-rear direction. Figure 2d shows the basic shape change pattern 3 in which the upper-end face of the shape is reduced by half in the lateral direction. However, the mesh structure (element connectivity) of the four analysis models shown in Figure 2 are the same, and only the node coordinate values are different. Here, the node coordinate values of the four analysis models are arranged in the same order, represented by vectors \(\vec{\alpha_{0}}\), \(\vec{\alpha_{1}}\), \(\vec{\alpha_{2}}\), \(\vec{\alpha_{3}}\) respectively, \(\vec{\alpha_{0}}\) is referred to as an original vector while \(\vec{\alpha_{1}}\), \(\vec{\alpha_{2}}\) and \(\vec{\alpha_{3}}\) are called basis vectors. Figure 2: An example of original and basis shapes. (**a**) Original shape, (**b**) Basis Shape 1, (**c**) Basis Shape 2, (**d**) Basis Shape 3. Further, each coordinate vector \(\overrightarrow{\alpha_{0}}\), \(\overrightarrow{\alpha_{1}}\), \(\overrightarrow{\alpha_{2}}\), \(\overrightarrow{\alpha_{3}}\) is multiplied by a weighting coefficient \(x_{0}\), \(x_{1}\), \(x_{2}\), \(x_{3}\) respectively. Adding the results, the linear combination creates a new node coordinate vector as follows:  
\[\overrightarrow{\alpha}=x_{0}\overrightarrow{\alpha_{0}}+x_{1}\overrightarrow {\alpha_{1}}+x_{2}\overrightarrow{\alpha_{2}}+x_{3}\overrightarrow{\alpha_{3}} \tag{5}\]  
When applying Equation (5) to the topological optimization problem in practice, the original vector \(\overrightarrow{\alpha_{0}}\) and the basis vectors \(\overrightarrow{\alpha_{1}}\), \(\overrightarrow{\alpha_{2}}\), \(\overrightarrow{\alpha_{3}}\) do not always change while the reference data of shapes change at all times. By changing the design variable \(x_{0}\), \(x_{1}\), \(x_{2}\), \(x_{3}\), the corresponding node coordinates of the new shape can be calculated directly using Equation (5). However, when giving the same value to the original vector \(\overrightarrow{\alpha_{0}}\) and the basis vectors \(\overrightarrow{\alpha_{1}}\), \(\overrightarrow{\alpha_{2}}\), \(\overrightarrow{\alpha_{3}}\), this paper considers the geometric requirement that the new node coordinate vector obtained by Equation (5) becomes the same. The node coordinates of the new shape corresponding to that can be calculated directly using Equation (5):  
\[x_{0}+x_{1}+x_{2}+x_{3}=1 \tag{6}\]  
Substituting Equation (6) into Equation (5), the coordinate calculation formula of the basis vector method used for shape optimization is obtained as follows:  
\[\overrightarrow{\alpha}=\overrightarrow{\alpha_{0}}+x_{1}(\overrightarrow{ \alpha_{1}}-\overrightarrow{\alpha_{0}})+x_{2}(\overrightarrow{\alpha_{2}}- \overrightarrow{\alpha_{0}})+x_{3}(\overrightarrow{\alpha_{3}}- \overrightarrow{\alpha_{0}}) \tag{7}\]  
Substituting \(x_{1}=0\), \(x_{2}=0.5\), \(x_{3}=0.5\) into Equation (7), the analysis model shown in Figure 3 can be formed using the obtained coordinate vector. Comparing Figures 1 and 3, the shape obtained by combining the base shape 1 with 0%, the basis shape 2 at 50% and the basis shape 3 at 50%, according to the value of the design variable, is shown in Figure 3. Therefore, using the design variables \(x_{1}\), \(x_{2}\), \(x_{3}\), it may be said to be the adoption rate of each basis shape.