# Shape Optimum Design by Basis Vector Method Considering Partial Shape Dependence
## 7 Discussion
### Creation of Basis Vectors  
Prior to shape optimization using the basis vector method, it is necessary to create a basic shape change pattern as a reference for the shape change and a basis vector represented by its node coordinates. Additionally, compared to the original shape, the basic shape is required to be the same as that of the original model, and only the coordinate values of the basis vector and the original vector having the same node number are different from each other.  
Here, the paper proposed a basis vector creation method making full use of the finite element method analysis mesh smoothing technique. Specifically, as explained using the simple example shown in Figure 12, a basis vector can be created according to the following procedure:  
1. According to the design scheme of the original shape, the analytical model used to generate the basic vector is divided into three parts: the fixed part (\(a_{1}\),\(a_{2}\),\(a_{5}\)), the specified part (\(b_{1}\),\(b_{2}\),\(b_{5}\)), and other shape-related changes as shown in Figure 12a.
2. A new node coordinate is given to the designated part (\(b_{1}\),\(b_{2}\),\(b_{5}\)) of the shape change, and the node position of the designated part is updated as shown in Figure 12b.
3. To enable the shape-related change part, excluding the fixed part (\(a_{1}\),\(a_{2}\),\(a_{5}\)) whose shape is not changed and the designation part (\(b_{1}\),\(b_{2}\),\(w_{5}\)) to forcibly change, a smoothing calculation of the analysis mesh is performed using Equation (35). \[x_{i}=\frac{1}{N}{\sum_{j=1}^{l}{x_{j}}}\ \ y_{i}=\frac{1}{N}{\sum_{j=1}^{l}{y_{j}}}\ \ z_{i}=\frac{1}{N}{\sum_{j=1}^{l}{z_{j}}}\] (35) Considering Equation (35), \(x_{i}\), \(y_{i}\), \(z_{i}\) are the new coordinate values of the nodes \(i\) of the shape-related change part, \(x_{i}\), \(y_{i}\), \(z_{i}\) are the neighboring node \(i\) coordinate values adjacent to the node, and \(j\) is a surrounding node-set adjacent to nodes \(i\).
4. Smoothing calculation of the mesh, with respect to the repetitive shape-related change part, is performed using Expression (35) and, as a result, a smooth basis shape can be obtained. Here, the basis shape obtained by the number of repetitive smoothing calculations is also different. The basic shape can be finely adjusted according to the actual shape optimization request so the number of repetitive calculations is small in Figure 12c, and the shape with many repetitive calculations is shown in Figure 12d.
5. Basis vectors can be created directly by sequentially arranging the node coordinate values of the basic shape obtained last.  
When actually generating a basis vector there are two methods; one method is moving a shape designation portion directly, as shown in Figure 13, and the other is a projection method to a CAD shape in which a shape designation portion is accurately created, as shown in Figure 14.  
Figure 13a shows an original shape in which a circular hole is drilled in the center of a thick square plate, and Figure 13b is a basic shape change pattern that moves only the shape designation portion around the inner hole in the upward direction.  
Figure 14a shows an original shape having a thick-walled cylindrical shape, the inner surface of which is a fixed portion, and the outer surface is a shape designating portion. Shown in Figure 14b, a CAD shape to be the destination of the shape designation portion is prepared.  
Figure 14c shows the outer surface of the cylinder is projected to the CAD shape, at the same time, the analysis mesh of the thick portion of the cylinder is subjected to smoothing processing and, finally, a basic shape change pattern is shown in Figure 14d.  
Figure 12: Creation of basis vectors.  
Figure 13: A shape designation portion.