# Optimizing Vehicle Body Cross-Sections Using a Parametric Mathematical Model
## 5 Cases Study
### Optimization of Body Beam Cross-Section Based on Cross-Section Parameters  
The geometric dimensions of the 17 beam unit cross-sections that make up the side enclosure are selected as the parameter variables, with mass minimization as the objective function and loading point displacement, i.e., stiffness, constrained, and the specific mathematical expressions are as follows:  
\[\begin{split} min&=\rho\sum\limits_{i=1}^{17}\big{[} h_{i}b_{i}-(h_{i}-2t_{i})(b_{i}-2t_{i})\big{]}l_{i}\\ s.t.&\Delta Z_{4}\leq 0.5\;\text{mm}\\ & 30\;\text{mm}\leq h_{i}\leq 80\;\text{mm}\\ & 30\;\text{mm}\leq b_{i}\leq 80\;\text{mm}\\ & 2\;\text{mm}\leq t_{i}\leq 3\;\text{mm}\end{split} \tag{28}\]  
where \(l_{i}\) is the length of the ith beam structure\(l_{i}\) is a known quantity, which can be obtained from the simplified geometric model of the body, and the density of the aluminum alloy material (\(\rho\)) is known.  
For the single-objective optimization problem, the stochastic gradient genetic algorithm is chosen to converge to the global optimal solution quickly and efficiently. The convergence of the objective function is shown in Figure 7. Under the working condition of satisfying the bending stiffness, the lightest mass of the body is 0.0502 tons.  
Figure 7: Objective function.  
The results of the above optimized geometrical dimensional parameters of the beam unit cross-section are substituted into the Equation (26) for the calculation of the deformation \(\Delta Z_{4}\) at the loading point in the loading direction:  
\[\Delta Z_{4}=f(\{X\}_{1},\cdots,\{X\}_{i},\cdots\{X\}_{17})=0.38\;\mathrm{mm} \tag{29}\]  
The left-hand sidewall 1 and the right-hand sidewall 2 are symmetrical parts, so the deformation of the right-hand sidewall 2 is the same as that of the left-hand sidewall 1, and thus the average deformation of the whole car can be characterized by the deformation of the left-hand sidewall 1 under the action of loading point loading point loading, so the deformation of the whole car is also 0.38 mm. The body bending stiffness is calculated as:  
\[K_{bend}=\frac{2F}{\Delta Z_{4}}=\frac{2\times 1000}{0.38}=5128\;\mathrm{N/mm} \tag{30}\]  
In view of the effectiveness of the above design method, this paper carries out finite element analysis of the simplified bending stiffness of the marker car. The simplification is carried out under the premise of ensuring that the actual structural mechanical properties are fully reflected, and the established finite element model is shown in Figure 8a:  
After calculation, the deformation of the body under bending condition is shown in Figure 8b, and the left and right loading point displacements are 0.456 mm and 0.467 mm,  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Unit Number** & **H (mm)** & **B (mm)** & **T (mm)** \\ \hline  
\hline \hline \end{tabular}
\end{table}
Table 1: Cross-sectional parameters of optimized sidewall beams.  
Figure 8: The finite element model of the benchmark vehicle. (**a**) Finite element modeling of body bending stiffness; and (**b**) Body bending stiffness simulation analysis results.  
respectively. Substituting into the bending stiffness calculation formula Equation (29), the result is obtained as:  
\[K_{\text{bend}}^{\prime}=\frac{2\times 1000}{\left(0.456+0.457\right)/2}=4381\ \text{N}/\text{mm} \tag{31}\]  
From the analysis results, it can be seen that after the optimization of the established meshless model by stochastic approximation genetic algorithm, the mass of the whole vehicle is reduced by 30%, and the stiffness of the whole vehicle (5128 N/mm) is larger than that of the benchmark vehicle (4381 N/mm). This ensures that the vehicle stiffness performance is optimized, and the body mass is the lightest. Meanwhile, compared with the conceptual design method of CAE technology, the modeling time is greatly reduced, and the computational efficiency is greatly improved by the analytical method compared with the finite element method. Figure 9 shows the prototype of the optimized body skeleton structure.