# Shape Optimization of the Streamlined Train Head for Reducing Aerodynamic Resistance and Noise
## 3 Train Aerodynamic Model  
### Parametric Model of Streamlined Head  
Because of the good symmetry of the streamlined head shape, only half of the streamlined head needs to be parameterized. The streamlined head is very complex, which is composed of several sub-surface pieces through certain continuous order splicing. In this paper, the left half of the streamlined head shape is composed of B-spline surfaces. Control points are firstly established on the streamlined head, 12 B-spline curves are established from control points, and 7 B-spline surfaces are finally established from these 12 B-spline curves. Thus, the left half of the streamlined head shape is established [18], which is shown in Figure 2. On the basis of Figure 2, the three-car formation train model will be easily established.  
In Figure 2, five design variables are extracted, which correspond to the longitudinal line C1, the horizontal contour lines C2 and C3, the central control line C4, the control line of nose height C5.  
Lines C1, C2, and C3 have similar deformation form. The deformation of C1 is achieved by modifying values of vertical coordinates of control points, while the deformations of C2 and C3 are achieved by modifying values of horizontal coordinates of control points. Specifically, increase vertical coordinate of the longitudinal midpoint of C1 by \(x_{1}(x_{1}>0\) indicating that this point moves upward, and \(x_{1}<0\) indicating that this point moves downward), and vertical coordinates of both ends of C1 do not change. For points between midpoint and endpoints, their vertical coordinates increase according to a linear law. Similarly, the horizontal coordinates of the longitudinal midpoints of C2 and C3 increase by \(x_{2}\) and \(x_{3}\), respectively, where, \(x_{2}>0\) or \(x_{3}>0\) indicates that the point moves inward. \(x_{2}<0\) or \(x_{3}<0\) indicates that the point moves outward.  
The deformation of C4 is mainly a deformation of curve concavity and convexity. It is necessary to make the two endpoints of the control line fixed. Meanwhile, the maximum change happens at the midpoint of the control line. The following equation is used here for deformation:  
\[y_{4,\text{new}}(i)=y_{4,\text{old}}(i)\times\left(1+\frac{x_{4}\times(i-1) \times(n_{4}-i)}{(i-1)\times(i-1)+(n_{4}-i)\times(n_{4}-i)}\right) \tag{7}\]  
where \(n_{4}\) denotes the number of control points on C4, \(y_{4,\text{old}}(i)\) denotes the value before deformation, \(y_{4,\text{new}}(i)\) denotes the value after deformation, and \(i\) denotes \(i\)-th control point.  
For the change of nose height, it is only necessary to multiply the vertical coordinate of the spline curve C5 by a factor \(x_{5}\), when \(x_{5}>1.0\), nose height becomes larger; when \(x_{5}<1.0\), nose height becomes smaller.  
It should be noted that curves associated with deformed curves should also be deformed accordingly to ensure continuity and smoothness of the surface.  
When each optimization variable is specified, the coordinates of control points are modified by the self-programming process to obtain a new streamlined head shape. Table 1 shows the value range of each optimization variable and the corresponding deformation. The second row in Table 1 shows the streamlined head shapes for each optimization variable at the lower bound, and the third row shows the ones at the upper bound.