# Optimizing Vehicle Body Cross-Sections Using a Parametric Mathematical Model
## 5 Cases Study  
### Establishment of Fully Parametric Mathematical Model of Body Structure  
The body skeleton shown in Figure 1a is used as a benchmark vehicle for optimization analysis. The simplified geometric model of the body shown in Figure 1b can be decomposed into three parts, namely, the left-hand sidewall 1, the right-hand sidewall 2, and the 11 cross beams 3, according to the symmetry, as shown in Figure 6a. The method of establishing the mathematical model of the body structure is discussed below using the body shown in Figure 6a as an example. In this example, the left side enclosure 1 is selected as the object of study, and it is decomposed into J skeleton beams, and each skeleton beam is numbered. As shown in Figure 6b, the left side enclosure 1 is decomposed into three skeleton beams, each of which is numbered as j, j = 1, 2, 3. And each skeleton beam is marked with coupling points, node numbers, and beam unit numbers, with a total of 17 nodes and 17 beam units. Skeleton beam 1 (Figure 6b), skeleton beam 2 (Figure 6c), and skeleton beam 3 (Figure 6d) are chain beam structures. Figure 5: Optimization process of cross-section of body mathematical model. Figure 6: Geometric model decomposition of body structure. (**a**) Simplified Vehicle Body Geometry Model: (**b**) Skeleton Chain 1; (**c**) Skeleton Chain 2; and (**d**) Skeleton Chain 3. For skeleton beam 1, construct the transfer equation for skeleton beam 1 with the expression:  
\[S_{7}^{R}=\prod_{i=1}^{7}\lambda_{i}H_{i}R_{i}S_{1}^{L} \tag{13}\]  
For the skeleton beams, the load column vectors at the coupling points--\(F_{1},F_{2},F_{6}\) for nodes 1, 2, and 6, are unknown. The boundary conditions are as follows:  
For beam 1:  
\[S_{1}^{L}=\left[O_{2}\;F_{0}\;1\right]^{T} \tag{14}\]  
\[S_{7}^{R}=\left[O_{2}\;F_{7}\;1\right]^{T} \tag{15}\]  
\(S_{1}^{L}\) and \(S_{7}^{R}\) are the leftmost and rightmost state vectors of the first and seventh units, respectively. \(F_{0}\) and \(F_{7}\) are the applied forces at nodes 0 and 7, and O\({}_{2}\) is the vector of other known parameters. For beam 2:  
\[S_{15}^{R}=\prod_{i=8}^{15}\lambda_{i}H_{i}R_{i}S_{8}^{L} \tag{16}\]  
\[S_{8}^{L}=\left[O_{2}\;F_{1}^{2}\;1\right]^{T} \tag{17}\]  
\[S_{15}^{R}=\left[O_{2}\;F_{6}^{2}\;1\right]^{T} \tag{18}\]  
Here, \(S_{8}^{L}\) and \(S_{15}^{R}\) are the state vectors at nodes 8 and 15 of beam 2. For beam 3:  
\[S_{17}^{R}=\prod_{i=16}^{17}\lambda_{i}H_{i}R_{i}S_{16}^{L} \tag{19}\]  
\[S_{16}^{L}=\left[O_{2}\;F_{1}^{2}\;1\right]^{T} \tag{20}\]  
Figure 6:0  
In Equations (21)-(23), \(S_{16}^{L}\) is the leftmost state vector of the 16 th girder unit in the skeleton girder 3, and \(S_{17}^{R}\) is the rightmost state vector of the 17 th girder unit in the skeleton girder 3; \(F_{2}\) and \(F_{9}\) are the applied forces at nodes 2 and 9, respectively, and \(O_{2}\) is the vector of other known parameters. Notate the transfer equation and boundary conditions for the above skeleton beam 3 as:  
Figure 6:1  
For the overall system, the coupling relationships between the nodes are given by:  
\[ \begin{cases}F_{1}^{2}=\lambda_{1}F_{1}^{1}\\ F_{2}^{3}=\lambda_{2}F_{2}^{2}\\ F_{6}^{2}=\lambda_{6}F_{6}^{1}\\ F_{9}^{3}=\lambda_{9}F_{9}^{2}\end{cases}  \tag{23}\]  
where F1, F2 are the forces at node 1 between beams 1 and 2, F6, F7 at node 6 between beams 2 and 1, and so on. The functional equation for the coupling points is:  
\[g_{1}=0 \tag{24}\]  
The transfer equations, boundary conditions, and functional relationships of the coupling points for each of the coupled skeleton beams are given by:  
\[ \begin{cases}f_{1}=0\\ f_{2}=0\\ f_{3}=0\\ g_{1}=0\end{cases}  \tag{25}\]Solving Equation (27) yields the state vector of any node. Similarly, the transfer function of the skeleton beams of other parts of the body can be obtained and based on the coupling relationship and topological relationship of each part, the whole vehicle skeleton chain body bending stiffness is obtained.