# A Novel Shape-Adjustable Surface and Its Applications in Car Design
## 4 Modeling Optimization
The drag coefficient \(C_{d}\) could be calculated according to:  
\[C_{d}=\frac{D}{\frac{1}{2}\cdot\rho\cdot v^{2}\cdot A} \tag{13}\]  
where \(\rho\) is air density, \(v\) is vehicle speed (100 km/h), and \(A\) is the vehicle projection area in the windward direction. Since the adjustment of shape parameters only modifies the surface details, \(A\), \(\rho\) and \(v\) remain unchanged, and the value of \(C_{d}\) is only related to \(D\). In this paper, the application of shape parameters in the optimization of the vehicle drag coefficient \(C_{d}\) will be explained in the order of one, two and three variables, respectively. In order to simplify the calculation, the variation range of shape parameters would be accurate to 0.5. As a contrast, the \(C_{d}\) of  
Figure 12: The key isocurves of hood surfaces in the model M0. Figure 13: The pretreatment of computational fluid dynamics (CFD) on the the model M0. the original M0 model (the \(\alpha_{u}\), \(\gamma_{u}\), \(\alpha_{v}\), and \(\gamma_{v}\) of the hood surface are all valued as 1) is 28.40% from the CFD simulation. When only \(\alpha_{u}\) is used as a variable, according to CFD simulation, calculation and statistics, under the condition of keeping \(\gamma_{u}\), \(\alpha_{v}\) and \(\gamma_{v}\) unchanged and valued as 1, the value curve of \(C_{d}\) caused by \(\alpha_{u}\in 1\pm 5\) is shown in Figure 14. According to the algorithm for determining the range of shape parameters in Section 3.3, the curvature meets the requirement only when \(\alpha_{u}\in[-1.5,\,2]\), which is marked as the red area. It could be concluded that \(C_{d}\) takes the optimal value 28.14% when \(\alpha_{u}\) is \(-1.5\). When \(\alpha_{u}\) and \(\gamma_{u}\) are taken as variables, the value surface of \(C_{d}\) with \(\alpha_{u}\), \(\gamma_{u}\in 1\pm 5\) is displayed in Figure 15. The ranges of \(\alpha_{u}\) and \(\gamma_{u}\) that meet the curvature requirements are \([-1.5,\,2]\) and \([-1.5,\,2.5]\), respectively. There are 64 parameter sets suitable for CFD simulation in these ranges, since some parameter sets at the limit (e.g., \(\alpha_{u}=-1.5\) and \(\gamma_{u}=-1.5\)) do not meet the requirements. It could be concluded that \(C_{d}\) takes the optimal value 27.85% when \(\alpha_{u}=-1\) and \(\gamma_{u}=-1.5\). As the hood surface is left-right symmetrical, the values of \(\alpha_{v}\) and \(\gamma_{v}\) are the same as they control the deformation of the surface in the left and right directions, respectively. Therefore, only three shape parameters at most could be optimized as variables on the hood surface. The range of \(\alpha_{v}\) (\(\gamma_{v}\)) that meets the curvature requirements is [\(-0.5\), \(5\)]. Since some parameter sets at the limit (e.g., \(\alpha_{u}=-1.5\), \(\gamma_{u}=-1.5\) and \(\alpha_{v}=\gamma_{v}=-0.5\)) do not meet the requirements, finally, there are 754 parameter sets suitable for CFD simulation in these ranges. The value of \(C_{d}\) in this range is illustrated in Figure 16. It could be concluded that \(C_{d}\) takes the optimal value 27.72% at \(\alpha_{u}=-1.5\), \(\gamma_{u}=-0.5\), \(\alpha_{v}=\gamma_{v}=0\). For those surfaces which are not on the symmetrical plane, all four shape parameters should be used as variables to optimize the \(C_{d}\). Its optimization method is similar to that mentioned above. Figure 17 depicts the mean curvatures of the original hood surface and optimized surface. For this particular surface, with the four unchanged boundary curves and the four decreasing shape parameters, the mean curvature of the surface tends to be the same as that of the whole surface, but there is also a significant sudden change at the front end. CFD simulation confirms that such a change is beneficial to its aerodynamic performance. For the other models M1-M50, the topological relationship between surfaces is the same as M0. Therefore, this method can also be applied to it, even though they have different boundary curve  
Figure 17: The change in mean curvature on the hood surface before and after optimization. shapes.