# Surrogate Based Optimization of Aerodynamic Noise for Streamlined Shape of High Speed Trains
## 3 Local Shape Function Parametric Approach  
In this paper, a parametric approach called Local Shape Function (LSF) which is based on FFD method [24] and NURBS method [25] has been designed.  
The whole processes are as follows:  
1. For a given geometry, deformation regions should be divided firstly.
2. Mesh the deformation regions, and obtain the coordinate values of every grid point. In order to keep the smooth transition of the surface, the structural grids have been utilized for mesh discretization, as shown in Figure 3.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Aerodynamic Loads** & **Total-Cd** & **Head-Cd** & **Middle-Cd** & **Tail-Cd** \\ \hline
**Experiment** & 0.326 & 0.125 & 0.082 & 0.119 \\
**CFD** & 0.310 & 0.118 & 0.079 & 0.113 \\
**Error** & 4.91\% & 5.60\% & 3.66\% & 5.00\% \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of aerodynamic coefficients between experimental results and numerical results.  
Figure 2: Wind tunnel train model and mesh configuration for numerical simulation: (**a**) wind tunnel model; and (**b**) mesh configuration.  
Choose the deformation function of each region, which can be selected randomly, but smooth transition between adjacent regions should be ensured.
4. Choose a weight factor \(W_{i}\) for each deformation function, which determines the maximum deformation value of each region.
5. Calculate the increments \(\Delta\) of coordinates of all grid points by the deformation functions and \(W_{i}\).
6. Get the coordinates of the deformed shape by summing \(\Delta\) and the coordinates of the original shape.
7. According to the coordinates of the deformed shape, the deformed surface can be fitted exactly, then a deformation process is done.  
In the above process, Step (3) is the most crucial. The deformation surfaces are different from each other due to the different choices of deformation functions. Inappropriate deformation functions will easily lead to irrational deformation surfaces. Trigonometric functions, exponential functions, logarithmic functions, polynomial functions and NURBS functions are all commonly used for deformation functions.  
Due to the symmetrical design along the longitude of the train, only one side of the symmetrical plane of the streamline is parameterized. As a result, the design parameters can be reduced by half. The parametric surfaces are separated into four deformation regions, as shown in Figure 4a. The width of the streamline is controlled by Zone1, and a control point Point1 is set here, extracting its y coordinate as the third design parameter \(w_{1}\). The slope of the cab window is controlled by Zone2, and the second control point Point2 is set here, extracting its z coordinate as the fourth design parameter \(w_{2}\). Nose height is controlled by Zone3, and another control point Point3 is set here, extracting its z coordinate as the second design parameter \(w_{3}\). Nose drainage is controlled by Zone4, and a control point Point4 is set here, extracting its y coordinate as the first design parameter \(w_{4}\). For simplicity, all the deformation functions in this paper are trigonometric functions. Figure 4b shows the deformation in the height of the nose and the cab window. As seen above, the deformation method can ensure the surface smoothness and smooth transition among different deformation regions.  
Figure 4: Deformation zones and local deformations: (**a**) schematic of deformation zones; and (**b**) the deformation of the nose and the cab window.  
Figure 3: Schematic of surface deformation by Local Shape Function (LSF) method.